HAMILTONIAN FORMULATION FOR SINGLE/FEW PHOTON DETECTION
by
SAUMYA BISWAS
A DISSERTATION
Presented to the Department of Physics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
December 2021
DISSERTATION APPROVAL PAGE
Student: Saumya Biswas
Title: Hamiltonian Formulation for Single/Few Photon Detection
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Jens Noeckel Chairperson
Steven J van Enk Advisor
Jayson Paulose Core Member
Brittany Erickson Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded December 2021
ii
©c 2021 Saumya Biswas
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
iii
DISSERTATION ABSTRACT
Saumya Biswas
Doctor of Philosophy
Department of Physics
December 2021
Title: Hamiltonian Formulation for Single/Few Photon Detection
Fully quantum mechanical models for device models of single photon 
detectors have recently been developed. Detection of single/few photons in both 
inanimate devices and biological eyes have the universal structure of absorption, 
amplification and measurement s tages. Previous models succeeded in developing 
definite models for all stages but the amplification st age. We  write out explicit 
Hamiltonians that can describe such irreversible changes and also measurement 
induced decoherence. The time evolutions created by these Hamiltonians
are solved in the discrete part of the Hilbert space and the desired dynamics
are verified. Previous proposals of minimum noise amplification schemes are 
completed with specific Hamiltonians presented in this d issertation. A  new kind 
of problem where a molecule absorbs two photons sequentially is investigated. The 
proposed Hamiltonian method is matched with a classic method in the field called
iv
“generalized density matrix operator method” to calculate the probabilities for such
sequential absorption of photons.
This dissertation contains previously published and unpublished material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Saumya Biswas
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, Oregon
University of California, Riverside, California
Bangladesh University of Engineering and Technology, Dhaka, Bangladesh
DEGREES AWARDED:
Doctor of Philosophy, 2021, University of Oregon
Master of Science, 2019, University of Oregon
Master of Science, 2016, University of California Riverside
Bachelor of Science, 2012, Bangladesh University of Engineering and
Technology, Dhaka, Bangladesh
AREAS OF SPECIAL INTEREST:
Theoretical Atomic, Molecular and Optical Physics, Condensed Matter
Physics, and Quantum Information Theory
PROFESSIONAL EXPERIENCE:
Graduate Teaching/Research Assistant, University of California Riverside,
2013-2017
Graduate Employee, University of Oregon, 2017-2021
Summer Intern, NumFocus/QuTiP (Google Summer of Code’19),
Summer’2019
GRANTS, AWARDS AND HONORS:
Emanuel Optical, Molecular, and Quantum Sciences Scholarship,
Summer’2021, University of Oregon
Jhamandas Watumull Scholarship, University of Oregon, 2018-19
Dean’s Distinguished Fellowship, University of California Riverside, 2013-14
Dean’s List, Level-2,3,4; Bangladesh University of Engineering and
Technology
vi
PUBLICATIONS:
Biswas, Saumya, and S. J. van Enk. “Heisenberg picture of photodetection.”
Physical Review A 102.3 (2020): 033705.
Biswas, Saumya, and S. J. van Enk. “Detecting two photons with one
molecule.” Physical Review A 104 (2021): 043703.
vii
ACKNOWLEDGEMENTS
I would like to thank my advisor Dr. Steven van Enk for supervising this
work. His input and insight to the project was invaluable. I also thank my
committee chair Dr. Jens Noeckel for conducting the proceedings seamlessly.
Special thanks are owed to Dr. Jayson Paulose and Dr. Brittany Erickson for their
contributions to completing this dissertation. My dissertation committee deserves
my heartfelt gratitude for the meticulous reading and rereading of the manuscript.
I also thank all the professors in the department of physics for their teaching and
mentor-ship. The researchers of DARPA DETECT program were very helpful in
stimulating conversations about the theory developed. I am grateful to my family,
especially my parents for their inspiration. My labmate Tzula Propp and close
friend Wenqian Sun were extremely helpful. Overall, all my friends have been a
constant source of support. This research was supported by DARPA contract No.
W911NF-17-1-0267. The writing of the dissertation was supported by Emanuel
Optical, Molecular and Quantum Sciences Scholarship.
viii
I dedicate this dissertation to my parents.
This dissertation is the product of unionized labor as part of the Graduate
Teaching Fellows Federation, AFT Local 3544.
ix
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Overview and Glossary . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. What happens in a quantum measurement? . . . . . . . . . . . . 2
1.3. Are our eyes special? Like our minds? . . . . . . . . . . . . . . . 9
1.4. How are quantum observables (classically) measured? . . . . . . 13
1.5. Single Photon Detection (SPD) in a device . . . . . . . . . . . . 28
1.6. Genealogy of theories of photodetection and single photon detectors 31
1.7. Can we write a general and useful model? . . . . . . . . . . . . . 37
II. SUMMARY OF RELEVANT THEORY . . . . . . . . . . . . . . . . . 43
2.1. A Photon Wavepacket . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2. Atoms in a single mode Electromagnetic Field . . . . . . . . . . . 49
2.3. Open System dynamics . . . . . . . . . . . . . . . . . . . . . . . 52
2.4. Systems and meters . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5. Linear and nonlinear quantum amplification . . . . . . . . . . . . 63
2.6. Access to some example observables: Techniques of quantum optics 67
2.7. Measurement Methods . . . . . . . . . . . . . . . . . . . . . . . . 75
2.8. Excitation of a Two-level System . . . . . . . . . . . . . . . . . . 78
x
Chapter Page
III. THE HEISENBERG PICTURE OF PHOTODETECTION . . . . . . . 81
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2. A class of model Hamiltonians . . . . . . . . . . . . . . . . . . . 85
3.3. Heisenberg equations of motion . . . . . . . . . . . . . . . . . . . 92
3.4. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . 105
IV. DETECTING TWO PHOTONS WITH ONE MOLECULE . . . . . . 109
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2. Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3. The two photon absorber and its Hamiltonian . . . . . . . . . . . 122
4.4. Two theories for photon absorption . . . . . . . . . . . . . . . . . 124
4.5. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.7. Emulating Photon wavepackets with Auxiliary Cavities . . . . . 144
V. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
APPENDICES
A. HEISENBERG PICTURE EVOLUTION OF OPERATORS . . . . . 152
A.1. Open system dynamics as a homomorphism for operators . . . . 154
A.2. A Complete Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.3. Expansion of evolving operators in orders of time . . . . . . . . . 158
xi
Chapter Page
A.4. Binomial Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.5. Example time independent Hamiltonian: A Cavity mode Driving
a 2LS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B. AMPLIFICATION MECHANISMS AND HAMILTONIANS . . . . . 168
B.1. From master equation to rate equation . . . . . . . . . . . . . . . 171
B.2. New class of Phase Preserving Linear Amplification . . . . . . . . 178
B.3. Quantum Darwinism . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.4. Code Repository . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
xii
LIST OF FIGURES
Figure Page
1.1. Phototransduction cascade. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2. Schematic for the double slit experiment for light. . . . . . . . . . . . . . 15
1.3. Bohr’s original drawing for the thought experiment behind
Bohr-Einstein debate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4. The three parts of the single photon detection process: transmission,
amplification, and measurement. . . . . . . . . . . . . . . . . . . . . . . 17
3.1. From a single photon to a macroscopic signal: . . . . . . . . . . . . . . 86
3.2. The five types of nonzero terms in |F2〉〈F2|(t) as functions of time in
units of γ−11 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3. The term in c(t)(in units of µ/Γ) prop. to |1〉〈1| ⊗ |F2〉〈F2| ⊗ |0〉〈0| . . . 103
3.4. The term in ND(T ), given by Eq. (3.43), proportional to
|1〉〈1| ⊗ |F0〉〈F0| ⊗ |0〉〈0| as a function of the integration time T (in units
of γ−11 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.1. Model of a two-photon detector. . . . . . . . . . . . . . . . . . . . . . . 113
4.2. Top: Populations in the ground state and the two metastable states as
functions of time, when two photons arrive sequentially. Bottom: the
(Gaussian) amplitudes of the “blue” photon (uα) and the “green”
photon (uβ) as functions of time. We chose here γk = γ1 for k = 2, 3, 4
and the time delay between the two input photons is 3/γ1 . . . . . . . . 116
4.3. Top: Populations in the ground state and the two metastable states as
functions of time, when the “green” photon arrives just before the
“blue” photon (the time delay is −1/(4γ1). Bottom: the absolute values
of the amplitudes |uα| of the “blue” photon and |uβ| of the “green”
photon as functions of time. . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4. The cavity modes a1 and a2 each have one excitation to start with. . . 132
xiii
Figure Page
4.5. The steady state occupations of F2 and F4 level for the initial state of
both cavity having one photon. Here we chose γ1 = γ2 = γ3 = γ4 and
κ1 = κ2 = γ1/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6. The occupation of the F2 level as a function of time for an initial state
of both cavities having one photon. . . . . . . . . . . . . . . . . . . . . 140
4.7. Same as the preceding Figure, but plotting the occupation of level F4.
Here the steady-state population of |F4〉 increases with decreasing value
of κ2 since the second photon is more effective at moving population
from |F2〉 to |F4〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.8. ρ24,24(∞) plotted on the vertical z-axis against the standard deviations
of the α and β photons plotted on the two axes on the horizontal plane
for (a) no time delay, (b) 1/γ1,(c) 3/γ1, and (d) 5/γ1 delays (of the
means/centres of the waveshape) of the second photon, β. . . . . . . . . 141
4.9. ρ24,24(∞) plotted on the vertical z-axis against the standard deviations
of the α photon Gaussian waveshape on the x axis and the inverse of
the rate constant (κ) of the β photons plotted on the y axis for (a) no
time delay, (b) 0.5/γ1 delay (delay between the mean of the Gaussian
and the onset of the β photon waveshape). . . . . . . . . . . . . . . . . 142
A.1. The Digraph (brown edges) for the Hamiltonian in eq. (A.32) along
with the edge |6〉〈1| (orange). The edge 1-6 of the Hamiltonian is not
shown (or is hidden under the edge |6〉〈1| ). The edges have arrows on
both ends (bidirectional edges or two edges) because of the hermiticity
of the matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.2. The composite graph (ref. [186]) for H4|6〉〈1|H2 (The arrows of the
graph-edges not shown). H is from eq. (A.32) . . . . . . . . . . . . . . 166
xiv
LIST OF TABLES
Table Page
1.1. Glossary of terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
xv
CHAPTER I
INTRODUCTION
Everything we call real is made of
things that cannot be regarded as
real.
If quantum mechanics hasn’t
profoundly shocked you, you
haven’t understood it yet.
Niels Bohr
Are not rays of light very small
bodies emitted from shining
substances?
Sir Isaac Newton
1.1. Overview and Glossary
Chapter 1 explains the motivation behind writing the entire photodetection
process with the help of a Hamiltonian. The discussion has been kept mostly
nontechnical.
Chapter 2 reviews some of the theoretical formalisms used in the dissertation.
The discussion is technical and is helpful for a smooth read of chapter 3 and 4.
Chapter 3 presents a simple Hamiltonian that embodies the physics of a
typical single photon detector in its entirety. The evolution problem is solved in
the Heisenberg picture for elucidation of the magnitude of gain and noise.
1
Chapter 4 presents the problem of two photon detection with one absorber
molecule.
Chapter 5 presents some novel schemes for amplifying weak signals,
computations of operator evolution in Heisenberg picture with time dependent
Hamiltonians.
Appendix A discusses the general structure of an operator evolution in the
Heisenberg picture. Appendix B discusses further Hamiltonian models that can
facilitate amplification and connect with available measurement theories.
A glossary of terms is presented in the table 1.1 that are not necessarily
technical, but have special meanings in the context of single photon detection.
1.2. What happens in a quantum measurement?
The trademark of nature’s departure from classical behaviour is undoubtedly
quantum superposition. Our perception or knowledge of the physical world is
subject to the measurement induced wavefunction collapse. Knowledge is limited
to the instant of measurement. And the quantum world meets the classical one
through the measurement. Erwin Schrödinger’s gedanken experiment of a cat
simultaneously alive and dead inside a box is a cliché, far too well known to be
repeated here. It has invited interesting accounts from physicists and authors
in the field. Maximilian Schlosshauer ([2]) in his book “Decoherence: and the
quantum-to-classical transition”, writes “no other example has illustrated this
problem of the quantum-to-classical transition more poignantly and drastically
than Schrödinger’s infamous cat, which appears, by the verdict of quantum
theory, to be doomed into a netherworldy superposition of being alive and dead”.
It is widely accepted that measurement yields classical readout results from a
2
Term Meaning
Amplification (non-unitary) The irreversible growth of a microscopic signal
into a classical macroscopic signal.
Amplification (unitary
quantum amplification) The unitary evolution (between final
and initial time) equation for a quantum
mechanical “signal” operator where it is
magnified by some gain and corrupted by
the addition of noise operators during the
evolution.
Continuous measurement Monitoring of the “meter” through a coupling
to a bath of large degrees of freedom.
Information flows from the “meter” to the
bath.
Continuum Solutions of the Schrödinger equation (without
any imposed boundaries) that carry a nonzero
probability current are continuum states.
Decay Rate The uniform (Markovian) coupling of discrete
states to the continuum. The rate at which a
discrete state decays into a continuum.
Discrete State Solutions of the Schrödinger equation confined
by a boundary (higher potential than energy
eigenvalue) that carry zero probability current.
Electron Shelving A method where macroscopic fluorescence is
created with laser from a molecule when it is
in a particular state (shelving state).
Filtering (frequency) Sifting through the optical scatterers.
Reflection and transmission coefficient
dictates whether a particular frequency of
electromagnetic wave passes through or not.
Minimum noise amplification Quantum mechanical amplification of an
operator with addition of minimal noise [1].
POVM Positive Operator Valued Measure: A set of
operators that completely specify a quantum
measurement process.
Reflection coefficient Ratio of the reflected probability current and
incident probability current by a scatterer.
Shelving state The particular state of the absorber molecule
that drives the amplification mechanism.
TABLE 1.1. Glossary of terms.
3
quantum observable. The dephasing that follows through measurement eats
away the quantum attributes of a system and can render the state completely
classical. Since any measurement device we use such as a photodetector can itself
be considered a quantum object, it is in turn performed a measurement on by
the circuity it has. The circuitry, in turn, is measured by the dial or display of
the device and our eyes perform a measurement of the dial/display. This series
or chain of measurement devices is referred to as the “von Neumann chain”
and we can cut it off (“Heisenberg’s cut”) at the final link (of our choice) to the
measurement device whose measurement readout we consider to be classical.
The chain can extend at best to the human mind which records the information
gained from neurological impulses from our eyes. A subsequent observer cannot be
imagined which can find our brain to be in a quantum superposition state. Human
consciousness is not subject to quantum superposition ambiguity, our minds have
an unambiguous idea about what we saw, and no subsequent observer can be
defined that measures our brains. Human consciousness is unique in a world of
quantum objects since it can possess classical knowledge and make up its “mind”
about some observed quantity. Physicist Sean Carroll finds it quite remarkable,
“We are part of the universe that has developed a remarkable ability: We can
hold an image of the world in our minds. We are matter contemplating itself.”
Since measurement process always ends with a human mind learning some new
numbers or images, consciousness based quantum measurement theories have
been propounded. Eugene Wigner, a pioneer of consciousness based quantum
measurement theories adds a twist to the Schrödinger’s cat experiment. Dubbed
as “Wigner’s friend” experiment, the cat is replaced by a (lady)friend of Wigner
who is asked upon opening the box if at a given point in time, the vial was intact
4
or broken. In her mind, she has no doubts about one certain state of the vial at
that point in time (and she will relay that) and we cannot agree with the idea of
the vial being in a superposition state prior to opening the box anymore.
The separateness of soul/mind/consciousness from the physical world
gives rare common ground to major religions and scientific minds. In fact, the
validation or confirmation of some observational fact (such as the Moon orbits
our planet Earth) is not made until some human mind observed and recorded
the information. It is the inspiration behind Einstein’s famous quip, “ Is the
moon there when nobody looks?”. In fact, some thinkers have invoked “God” or
an omnipresent observer with eyes set on all of the universe for the continuation
of matter left unobserved by all humans at any given point in time. In certain
quantum information theories, laws of physics are valid for all of the universe
but the assumed non-physical human consciousness. The other extreme theory
uses many worlds interpretation to preserve the universality of physical laws at
the expense of branching out of universes at every single measurement event that
happens. See ref. [3] for a comprehensive overview.
If we are dead set on preserving the universality of quantum laws and
are economical about the number of universes (many world theoy is a little
extravagant in terms of necessary number of universes), we need a closer look
on why the measuring device can produce classical results measuring a quantum
observable/system and are themselves somewhat distinct from the quantum
system under study. Firstly, a measurement device like galvanometer is made
up of a massive number of atoms (and in turn subatomic particles) whereas
quantum systems are typically a few atoms (or photons or some other particles)
at most. Secondly, the working principle of a measurement device can be explained
5
with classical laws exclusively (laws of electromagnetism for a galvanometer for
example). A proposition called “spontaneous collapse” has been used in some
theories to describe the collapse of the superposition when a macroscopic object is
used in the measurement chain or von Neumann chain. Despite being explainable
by macroscopic classical rule, the origin of the behaviour of a macroscopic object
can come from microscopic components. The bulk (“classical”) resistances of
metals and insulators are to be calculated from microscopic quantum mechanical
description of band theories. The heat conductivity or capacity is also calculated
from microscopic quantum mechanical models before the values can be used in
classical laws. Therefore in a sense, macroscopic objects are quantum mechanical
as well. How could they be different from the few atom quantum system we
investigate?
One possible answer is the presence of cooperative/collective motion of
a large number of particles that indicate the measurement result. The pointer
that moves across the dial in a galvanometer is made up of an enormous number
of atoms which move in unison that human eye can observe. The “avalanche”
of particles in a avalanche photodiode is also a collective motion of a massive
number of particles. Does such cooperative/collective motion result in a collape of
quantum superposition spontaneously? Although macroscopic quantum states have
been realized in superposition, no confirmation of spontaneous collapse of collective
states like superconducting current (in Superconducting Quantum Interference
Devices (SQUIDS) ) has ever been observed. However, it is yet not possible to rule
out the spontaneous collapse theory [3].
To distinguish between a quantum process (reversible) and measurement
process (seems to be irreversible), a third category of theory distinguishes
6
between the quantum systems isolated from external disturbance and quantum
systems affected by thermal noise. It also requires the mechanism for a collapse,
where some of the measurement outcome would be actualised. If a detector
or measurement device is coupled to external degrees of freedom, it can make
a tangible measurement record like with a flash of light, pulse or current or
movement of a pointer. This will be a “measurement process”. If it does couple
to numerous degrees of freedom, there is a minuscule chance of avoiding making
a mark, some telling sign of some change is almost guaranteed. In an effort to
not leave a mark, in an experiment, horizontal/vertical polarized photon may
pass through a H/V polariser and the record and everything associated with it
may be erased and the photon may be reconstructed in the original state; only we
know doing this is prohibitively difficult if not impossible, so postulating it to be
impossible is not illogical. So, a quantum process is reversible, but measurement
process is absolutely irreversible.
And we need not look hard to find theories of irreversibility, since
thermodynamics pre-dates quantum mechanics by years. Irreversibility is predicted
to occur with high likelihood by the second law of thermodynamics in the
relevant timescale. There also exists the ergodic principle which predicts that
a thermodynamic system touches upon every possible (allowed by conservation
laws) configurations if left to evolve on itself. Therefore there will be recurrences
of any given configuration after waiting long enough times– a phenomena known
as Poincaré recurrence. The postulate that works for quantum measurement
problem now becomes one that assumes ergodic principle is incorrect and Poincaré
recurrence does not occur. Important distinctions between reversible quantum
processes and irreversible quantum measurements were discussed by Ilya Prigogine.
7
It is well known that future dynamics of a system can be very sensitive to small
changes in the initial conditions, which is referred to as “strong mixing”. Prigogine
solves the measurement problem with the basic idea that a measurement chain is
broken whenever strong mixing is involved and the second law takes precedence
over quantum mechanical rules. It necessarily implies pure quantum behaviour
is observed only when strong mixing is negligible. This perspective makes
irreversible changes fundamental entities and reversible dynamics are considered
an approximation.
There exists recent theories for explaining quantum measurement without
necessarily falling back on irreversibility, such as “consistent histories" where
the most faithful representation or map of a physical process is constructed [3].
The modern update of the theory, however, is based on quantum decoherece. As
such, most theories, do work with a premise of irreversibility in the measurement
process. A modern contribution, “Quantum Darwinism" by Wojciech Zurek is
gaining ground, even passing initial experimental tests [4]. Quantum Darwinism
explains emergence of classical reality with an environment “watching” over. Since,
we are interested in irreversibility based approaches and so we simply sum them
up with two simple conclusions. First, quantum measurement devices will have
intrinsic irreversibility as a fundamental attribute. Second, our eyes are the final
conceivable measurement device (the latest link in the chain where the cut can
be taken) before the signal is imparted to our consciousness. Naturally, we should
take a closer look at what happens in a human eye in converting optical rays into
neurological pulses.
8
1.3. Are our eyes special? Like our minds?
The process responsible for human perception through vision is of
fundamental importance. Seeing with own eyes has been considered the fail-
safe method for acquiring reliable information – in all adages across all societies.
Efforts to understand the mechanism of light transmission and human vision also
originated in ancient times, notably by Euclid and Ptolemy. Ibn al-Haytham
aka Alhazen, Sir Isaac Newton and Johannes Kepler made crucial contributions
fractions of millennia apart. The 20th century, however, witnessed a revolution
where our knowledge of light and knowledge of quantum theory complemented
each other’s growth. The biggest debates and celebrated problems in the history
of quantum mechanics have been starred by photon, the fundamental quanta
of light. The EPR paradox and the photo-electric effect have revolved around
photons, which originally marked the coming of age of quantum mechanics.
Feynman resolved the enigma of interference pattern created by light in Young’s
double slit experiment, showing that they simultaneously take every possible
path from point A to point B. Photon has enjoyed a unique attention in major
conceptual breakthroughs in physics. All in all, we have come to peace with the
wave and particle dual nature of light/photon and quantum theory has become
the most successful and meaningful theory of our universe. Naturally, scientists
are getting more interested in the response of our eyes to photons, and not just
electromagnetic waves. It is still impressive that, such efforts originated as early
as 1942, with the seminal work by Hecht et al. [5]. Previously, physicist Hendrik
Lorentz had given the problem a fair shake in 1911 freshly equipped with the
knowledge of photons or particles of light [6].
9
Quantum measurement problem and theories ultimately separate mind from
matter. We have measurement devices that can produce numbers as measures of a
physical quantum mechanical observable. If nature is indeed quantum mechanical,
so would be the measurement device itself. So, in effect, our minds perform a
measurement on the measurement device itself, be it a photodetector display or
the interference pattern out of an interferometer. Since the human observer uses
their eyes to read that display, the eye counts as a “quantum” measurement device
as well. In the end, a neurological impulse from the eyes take the information to
the brain or human mind. A mathematical description of the process would hold
clues for the mathematical description of a general single photon detection that
can be applied to photon detection devices as well.
Ref. [7] reports a recent convincing experiment establishing the ability of
human eye for detecting single photons. The ideas originally are credited to Hecht
et al.[5] (see also [8] for a review). Experiments characterizing single photon
responses in mammalian eyes have been reported for a while now [9, 10, 11].
With the advent of modern quantum optics techniques, modern experiments are
employing state of the art quantum optics capabilities (such as heralded single
photons) for measuring attributes of human eye response to single photon signals
[12, 13, 14, 15]. To be fair, the basic mechanism of single photon detection by the
human eye had been “disentangled” at least two decades earlier [6].
Human eyes have two distinct (in shape and functionality) kind of
photoreceptors, rods (approximately 100 million in a retina) and cones
(approximately 3 million in a retina). Cones respond to bright light and facilitates
colour vision. Rods are functional in dim lights and can detect single photons.
Seeing a flash involves less than 10 of these rods. The chain of events in seeing
10
FIGURE 1.1. Phototransduction cascade.
(a) Outer-segment discs (a stack of about 1000) are specialized for photoreception.
In the absence of light, Ca2+ and Na+ ions flow into the outer segment through a
channel gated by cGMP. The photosensitive molecule (called rhodopsin) in the
rods have a prosthetic group 11-cis retinal (light absorbing chromophore) and a
protein component opsin linked together. (b) Schematic of transduction cascade.
Absorption of light causes an isomerization of the 11-cis-retinal group to a all-trans
form. In dim light, an incident photon activates rhodopsin (Rh) as a catalyst
which activates transducin (T), in turn T activates a cGMP phosphodiesterase
(PDE), next activated PDE causes cGMP concentration to fall. (c) The rod
current, I(t) calculated from eq.1.1. Rhodopsin activity, R(t) is taken to be a
decaying exponential. The parameter values and other equations can be found in
[6] (reproduced with permmission from [6])
a dim light, commonly referred to as “phototransduction” in vertebrate rods
can be summarised from the review [16]. The photopigment molecule rhodopsin
changes its structure upon absorption of a photon and becomes active as a catalyst
(fig. (1.1) ). In response to a photon, the isomerization from cis to trans of the
11
prosthetic group in the rod is characterized by a 5Ådisplacement of Schiff-base
nitrogen atom [17]. This is a characteristc of bio-based photodetectors. Photon
transduction brings about structural-chemical changes in their bond structure. The
current in the rods, was modelled with a linear filter system, where the rhodopsin
response function, R(t) acts an an input. The filter function (impulse response),
F(t) can be estimated [6]. ID is the current in the dark, k is a constant, GD is the
cGMP concentration in the dark.
∫ t
I(t) = ID − 3kG2D dτF (τ)R(t− τ) (1.1)
0
The electrical signal change manifests as a closure of some channels in the outer
segment and a dip in the circulating current. The names of the compounds or
types of compounds (in Fig. 1.1(b), see also [17]) are less important for the
purpose here, so they are mentioned briefly in the caption of Fig. 1.1 without
much details. What is more important is the structure of the equation 1.1. The
filtering function acts a temporal and frequency filtering operation. In effect,
the input photon undergoes filtering in passage through the eye lens as well as
in the internal mechanism of eye’s response into the output macroscopic current
signal. The production of a macroscopic signal through the closing and opening of
a gate is equivalent of an amplification process. Modern idea of a general quantum
mechanical model for photodetector resorts to same “filtering+amplification”
strategy as we shall shortly see. A structural change in the photoreceptive
molecule is responsible for initiating the amplification mechanism. We later
discuss how this irreversible change in the structure is a feature of device based
photodetectors as well. Single photon detection in a device has the same working
12
principle. A single photon wavepacket is filtered in, creates a structural change and
incurs a macroscopic signal.
1.3.1. Two-photon detection in a human eye
An interesting question to contemplate would be if human or animal eyes can
do Photon Number Resoling (PNR) detection. Can our eyes tell a single photon
excitation from two photons? We have known for awhile, the toad rods produce
distinct signals in response to 0,1,2 photons and can thus differentiate between
the three possibilities for number of absorbed photons [6]. And very recently Two
Photon Absorption (TPA) has been reported in human vision as well, with the
added intrigue that we have learnt that human eye may be sensitive outside the
visible range. Previously, Rods were believed to have a sensitivity range from
300 to 700nm of wavelength. Recent works have found rods to be sensitive at
wavelengths longer than 800nm and some authors have purportedly perceived
1060nm of infrared light as pale green. [18] The physical mechanism behind it
were suspected to be either Second Harmonic Generation (SHG) or Two Photon
Absorption (2PO). We can be safe in assuming that human eyes have capabilities
far exceeding our previous expectations, also physics that confounds current
theory.
1.4. How are quantum observables (classically) measured?
Einstein and Bohr faced off (“officiated” by mutual friend Ehrenfrest) a
number of rounds (Solvay conferences) in, perhaps, the most consequential mano
a mano in the history of science. A particular bone of contention was Bohr’s
complementarity principle. It refers to the dual nature of the behaviour of the
13
photon (particle and wave) observed, depending on the particular setup of the
experiment. In fig. 1.2, the first or left-most screen has a slit small enough for the
wavelength of the light so as to create single slit diffraction. If the size of the slit
were large, the diffraction pattern would disappear. The second screen (middle)
has two slits that the photon deflected through the first screen’s slit can take if
unobserved. And on the third screen the interference pattern of the two states (one
each for the photon passing through each slit) shows up as alternate bright and
dark spots. If the photon is not observed (through which slit it passes) the wave
nature wins and an interference fringe is observed. If it is observed the particle
nature takes over and interference fringe is lost (Heisenberg’s statement, “the
particle trajectory is created by our act of observing it.”). Bohr’s complementarity
principle proclaims mutual exclusivity of the two natures. Fig. 1.3 shows a
modification of the first screen that Einstein proposed. The slit is suspended
by a spring and can be deflected vertically up or down while the photon passes
through. Einstein argued that from the deflection observed of the first slit and
conservation of momentum principle we can deduce the direction the photon was
deflected when it passed the first screen which in turn gives away whichever slit
the photon went through on the second screen (a piece of information referred
to as “which-path” information). Since the interference is the result of the two
wavefronts created from the two slits, it will not be disturbed by whatever happens
before the second screen. Thus we can obtain both which-path information
and the interference pattern in direct contradiction of Bohr’s complementarity
principle. Bohr successfully countered pointing out that the precise measurement
of the momentum of the first slit would smudge its position (due to Heisenberg’s
indeterminacy principle) and effectively increase the size of the slit making the
14
diffraction from it disappear and destroy the interference pattern on the third
screen.
FIGURE 1.2. Schematic for the double slit experiment for light.
The first slit can deflect the photon towards the top or bottom slit in the middle
screen. An interference pattern emerges on the third screen.
FIGURE 1.3. Bohr’s original drawing for the thought experiment behind Bohr-
Einstein debate.
If the screen has a recoil upward or downward, the “which-path” information
namely the path taken by the photon can be learned.
The round went to Bohr (in fact the match eventually as well), needless to
say. It also serves to warn us against treating macroscopic objects as completely
classical, because if we did then atomic scale measurements that violate
the uncertainty principle may become a possibility. However, later theorists
(Wootters, Zurek, Scully, and Drühl) have shown in certain cases if the which-path
information is partial (and not complete), we can have interference pattern that
15
is partially smudged ([2]). This is the mechanism through which environment can
monitor a system and gain which-path information without completely destroying
the quantum attributes of the system. The system is affected in a way that
quantum phase relationship between certain states is partially lost, a process
called decoherence. The solutions derived in this thesis utilizes the method of
employing an environment to watch the quantum system and absorb “which-path”
information. Before discussing decoherence and strategies of learning which-path
information from the environment, a segment of Bohr’s account of the events is
worth remembering.
From Bohr’s reminiscence, “Discussions with Einstein on Epistemological
Problems in Atomic Physics” (1949), “all unambiguous use of space-time concepts
in the description of atomic phenomena is confined to the recording of observations
which refer to marks on a photographic plate or to similar practically irreversible
amplification effects like the building of a water drop around an ion in a cloud-
chamber.” Niels Bohr might have inadvertently outlined the methodology for
amplification of a quantum observable long before quantum theory received wide
acceptance. The amplification mechanism he alluded to is still the method to
obtain classical measurement record from quantum observables. A mechanism
that can create a sizeable effect (perceptible by ordinary human sense organs)
from a microscpic attribute or signal is required. This amplification mechanism
discards any quantumness as it grows in size and can be treated as a classical
signal produced from the microscopic values. It is very similar to the amplification
mechanism discussed in the eye that a photon hitting the rhodopsin molecule
causes. Modern developments in the theory of single photon detection retains
this fundamental structure. The single photon detection process is subdivided
16
into a transduction+amplification part. As reasons that would be discussed
later, transduction for the case of optical photons is explained in terms of a
quantum mechanical transmission probability. Amplification drives the process of
measurement. The DARPA DETECT ([19]) consortium (this dissertation is result
of the collaboration) in the 2010s have summarized this fundamental structure.
FIGURE 1.4. The three parts of the single photon detection process:
transmission, amplification, and measurement.
The input photons with frequency components ωs are transmitted irreversibly via
network of discrete states and absorbed into the atom or molecule that changes in
shape and irreversibly initiates an amplification mechanism at its higher state. In
the figure, a laser with frequency ωL drives a single mode cavity which populates
massively at a optical frequency, ω′ ≈ ωL. This method of creating fluorescence
excitations when a molecule is at a higher state is also called “electron shelving”.
The amplified signal can be considered a classical macroscopic signal and human
eyes can perform a measurement of it trivially (Not unlike Bohr’s idea mentioned
before or the process in the rod cells of human eyes). (Reproduced with permission
from [20] )
Historically, writing down a quantum mechanical description for a
measurement problem had been ripe with challenges and pitfalls and caveats and
unsolved enigmas. At initial stages, Copenhagen interpretation have guided us in
the interpretation of the physical world and their perception in our minds. Over
17
FIGURE 1.5. Photon with frequency ω initiates amplification into a macroscopic
signal via electron-shelving [21, 22, 23]:
A resonant photon lifts an atom (modelled here as a three-level system) enters the
first excited state. A laser beam tuned to the second transition frequency ωL ≈ ω′,
induces fluorescence. (Reprinted with permission from [1])
time, from the shorthand narrative of Copenhagen interpretation for describing
measurements, we have evolved to the more austere von Neumann measurement
which ultimately has fallen short of eliminating the need for an interpretative
process for outcomes. Von Neumann school of thought makes a methodical
attempt to construct an evolution that supersedes the collapse of wavefunction
of the Copenhagen interpretation. The missing link of emergence of classicality
from the quantum domain is nailed down more narrowly.
The work of Wojciech Zurek has been a revelation in Quantum Information
Science (QIS) and quantum measurement problem. The theory of decoherence,
a strictly quantum effect with no classical analogue has become the language
of choice in quantum information experiments. Although we cannot yet discard
the need for an interpretive process in the emergence of classical outcomes in a
measurement process, we can certainly capitalize on the redundant information
18
accrued in the environment over the course of measurement evolution through
decoherence. Zeh and then Zurek have demonstrated that to resolve a problem
called “preferred basis” in quantum measurement theory it is necessary to
treat system and meter to be open systems (coupled to their environment) [2].
Performing a measurement on a closed system energy eigenstate would definitely
perturb it. A closed system maybe preferable for precluding the undesired coupling
effects (decoherence) from environment. But is it not a little too optimistic
hoping to have closed systems absorbing the information from the photon
wavepacket? Since realistically, we would expect optical photons, stray back-
ground radioactivity, air molecules, cosmic muons, solar neutrinos, and even the
ubiquitous 3 K cosmic background radiation to couple to the quantum system of
interest. Therefore, a faithful or realistic model would include an environment that
model the decoherence that any and all quantum systems are subject to.
Our best understanding of the emergence of the classical objective reality
from quantum mechanical observables is the structure of system-environment
interaction– as pointed out by Zurek. Niels Bohr’s intuition in the early days
was correct– an irreversible amplification process will render a quantum system
“classical”. However, the quantum to classical transitions were better understood
only post 2000 with the theories of Quantum Darwinism and decoherence. All
single photon detectors have the physical mechanism where a classical macroscopic
signal is produced in response to the quantum mechanical photon transduction.
Fundamentally there exists no reason for microscopic and macroscopic objects
to be governed by different rules of nature. Physicists have referred to this
appearance of classical behaviour from large objects with the umbrella term
“quantum to classical transition”. In our everyday experience, we observe certain
19
(few in number) features of classical objects like position and speed to be
measurable, underlying quantum mechanical models have a long list of quantum
numbers and observables that could possibly be observed. Yet “position eigenstate”
or “momentum eigenstate” is the quantum observable (of a bouncing basketball, for
example) is what we typically observe. Also, quite puzzlingly, we never observe
classical objects to be in superposition. How do we make sense of it without
incurring the Copenhagen interpretation? The idea of decoherence can resolve
these two puzzles [2].
A fundamental quantum mechanical model of a single photon detector can
utilize an “environment simulator” or “meter” that would become correlated with
the system observables and reveal the system observable through a measurement
of itself. This system-meter(environment) interaction would lead to decoherence
of the system and certain pointer states emerge as the “preferred basis” for the
system, resolving the first puzzle. In the spirit of von Neumann theory, the system
state |ψ〉 and meter (“environment simulator”) is initialized as a product state (the
environment’s initial state is called “ready” state).
The desired evolution will produce the following,
|ψ1〉|“ready”〉 → |ψ1〉|1〉
|ψ2〉|“ready”〉 → |ψ2〉|2〉 (1.2)
The |1〉 and |2〉 states of the environment would indicate the system states at
the end of the evolution. So if we are able to measure the meter (“environment
simulator”) projectively, we are able to measure the system, indirectly. |ψ1〉 and
|ψ2〉 are the states in the double slit experiment when the photon passes through
20
the first or second slit respectively. If unobserved, the superposition of the two
states will evolve accordingly,
√1 | 1( ψ1〉+ |ψ2〉) |“ready”〉 → √ (|ψ1〉|1〉+ |ψ2〉|2〉) (1.3)
2 2
There are two things to note from eq. B.18. Firstly, an entanglement has
been created between the system and meter dynamically. Secondly, the
superposition originally in the photon states have delocalized into a larger system
of system+meter.
The density matrix can now be written out as well as the particle density,
%(x) at the detector screen,
1
ρ̂particle = [|ψ1〉〈ψ1|+ |ψ2〉〈ψ2|+ |ψ1〉〈ψ2|〈2|1〉+ |ψ1〉〈ψ1|〈1|2〉]
2
%(x) ≡ ρ̂particle(x, x) ≡ 〈x|ρ̂particle|x〉
1
= |ψ1(x)|2
1
+ |ψ (x)|22 +Re {ψ1(x)ψ∗2(x)〈2|1〉} (1.4)2 2
The two extremes (Einstein and Bohr’s original stances) of perfect
distinguishability (complete knowledge of “which-path” information) and null
“which-path” information are due to the detector states 〈2|1〉 = 0,
1 1
%(x) = |ψ (x)|21 + |ψ2(x)|2 (1.5)
2 2
and 〈2|1〉 = 1 respectively,
1 1
%(x) = |ψ (x)|21 + |ψ2(x)|2 +Re {ψ1(x)ψ∗2(x)} (1.6)2 2
21
The extraction of the “which-path” information destroys the interference pattern
(in eq. 1.5). This is the process called decoherence, where a quantum system loses
its “quantumness” through interaction with an environment. The second puzzle
of non-observability of interference is understood in this way. Environmental
monitoring causes superposition properties to disappear. Also the intermediate
values of 〈2|1〉 between 0 and 1 is when we have partial “which-path” information
and the interference pattern has a lesser contrast (but not completely washed out).
Eq. 1.2’s evolution would be achieved with an explicit Hamiltonian in a von-
Neumann formulation.
Ĥ = ĤS + ĤE + Ĥint (1.7)
where ĤS and ĤE are the self-Hamiltonians of the system and environment
respectively. Ĥint is the system-environment interaction Hamiltonian.
The choice of Ĥint tries to make different relative environmental states to be
distinguishable or orthogonal (τd being a characeristic time).
〈Ei(t)|Ej(t)〉 ∝ e−t/τd for i 6= j (1.8)
If the relative environmental states are correlated with the center-of-mass of the
system at x, we would want to design Ĥint to produce
〈Ex(t)|Ex′(t)〉 ∝ e−Γtott (1.9)
, where τd is a characeristic rate.
22
Clearly, the proper design and selection of Ĥint enables the measurement of
the system observable to be possible. Ĥint is also responsible for the decoherence of
the system states. For a particular choice of Ĥint, certain basis states of the system
will be more prone to decoherence (their phase relations will decay fast) and some
other basis states will be more robust against phase information losses. System
states can be written in multiple bases of course,
∑ ∑
|ψ〉 = ci|si〉 = ci′|si′〉 (1.10)
i i′
The system states more robust against environmental decoherence are the so-
called pointer states or preferred states or quantities. Therefore classically
certain eigenstates are more accessible, like the position or speed of a basketball.
Alternatively, the preferred pointer states can be thought of as the states that get
least entangled with the environment.
In a typical von-Neumann measurement interaction, A microscopic system, S,
with a Hilbert space HS having basis vecors {|si〉}, and a measuring apparatus A ,
with a Hilbert space HA having basis vectors {|ai〉} (|ai〉 are the pointer states),
(∑ ) ∑
|ψ〉|ar〉 = ci|si〉 |ar〉 → |ψ〉 = ci|si〉|ai〉 (1.11)
i i
Now we assume an environment (for both system and measuring apparatus) which
is initialized in the state |E0〉.
|si〉|ai〉|E0〉 → |si〉|ai〉|Ei〉, for all i (1.12)
23
In most cases it is assumed that mostly the apparatus interacts with the
environment and therefore |ai〉 are the environment superselected robust preferred
states. For the socalled quantum measurement limit, Ĥ = ĤS + ĤE + Ĥint ≈ Ĥint,
and the eigenstates of Ĥint becomes stationary under the evolution. Hence,
measurement superselected pointer states are none other than the eigenstates of
Ĥint. For example, for the following Ĥint,
Ĥint = x̂⊗ Ê (1.13)
environment continuously monitors the position of the system.
e−iĤintt|s〉|E 〉 = |x〉e−ixÊt0 |E0〉 ≡ |x〉|Ex(t)〉 (1.14)
The more general form of the interaction Hamiltonian,
∑
Ĥint = Ŝα ⊗ Êα, (1.15)
α
Ŝα| (α)si〉 = λi |si〉, for all α and i. (1.16)
Eq. 1.18 can produce,
e−iĤintt|si〉|E0〉 = |si〉|Ei(t)〉, a product state (1.17)
Because |si〉 does not get entangled with the environment, |si〉 is an environment-
superselected preferred state, since it is immune to decoherence and does not get
entangled with the environment. For reasons to be mentioned soon, there can
be subspaces of pointer states that are immune to decoherence. If {|si〉} is an
24
orthonormal basis and all |si〉 are simultaneous degenerate eigenstate of each Ŝα,
Ŝ |s 〉 = λ(α)α i |si〉, for all α and i. (1.18)
we get
e−iĤintt|ψ〉|E0〉 = |ψ〉|Eψ(t)〉, a product state (1.19)
Existence of a Decoherence Free Subspace (DFS) is a consequence of a
symmetry in Ĥint. Such dynamical symmetries can be exploited in preserving
quantum information more robustly. System-environment interaction is all we
need to look at for the availability of a DFS. It derives from a symmetry in the
structure of the system-environment interaction, a dynamical symmetry.
The theory development at Los Alamos National Lab led by Zurek has been
a tour de force. The procession of ideas can be time ordered into some interesting
descriptions or names: environment as a witness.
In essence, we should capitalize on Zurek’s seminal ideas of decoherence,
the larger family of problems of quantum-to-classical transitions and quantum
Darwinism. Coupling to the environment would involve decoherence, be it for
measurement purposes or unwarranted coupling to stray baths of particles.
Judicious choices of pointer states (states more robust against decoherence) and
strategies of battling decoherence with Decherence Free Subspaces (DFS) can help
us design photodetectors with superior performance.
1.4.1. Positive Operator Valued Measure (POVM)
POVM are the most general description for quantum measurement. This
elucidates the observables that we can define that we would be interested in for
25
learning about the photon signal. Measurement of an observable in the linear
vector space of the operators acting on HA can be done indirectly through a
projective measurement on another system, B once it is correlated/entangled with
the former. This is formulated as a generalized measurement on A. A generalized
measurementM on A is defined to be a set of operators (Hermitian or otherwise),
Mi ofM in (L(HA), the linear space of operators acting on HA) which yields the
result mi with the probability πi, given by
[ ]
πi = Tr MiρAM
†
i (1.20)
which projects A conditionally into the final mixed state,
M ρ M †i A
ρ iA|i = (1.21)
πi
They have the normalization condition:
∑
M †iMi = 1 (1.22)
i
If Mi are orthogonal projectors, Pi with the conditions, P †i = Pi, P
†
i P
2
i = Pi = Pi,
we have our projective measurement postulates back.
Without the orthogonality postulate, the generalized procedure,M
performed on A can be viewed as resulting from a projective measurement on an
environment simulator B, after it has been entangled to A by a proper unitary
operation, UM. We need the dimension of HB to be equal or larger than the
number of Mi operators. We then define an orthonormal basis set of | (B)ui 〉 for B
and associate one to each Mi.
26
The operation of UM on a A in a statistical mixture is,
( ( )U ρ(A) ⊗ |0(B)〉〈0(B))| U †∑ M M
= U p | (A)φ 〉〈 (A)φ | ⊗ |0(B)M ∑ i i i 〉〈0
(B)| U †M
i
= M ρ(A)
(B) (B)
i M
†
j ⊗ |ui 〉〈uj | (1.23)
i,j
For, an eigenbasis | (B)uj 〉 of B with result mj, a projective measurement of
observable OB of B projects A into ρA|i with probabilities, πi.
POVM is the most general method for the interpretation of measurent
results in Quantum Mechanics. It is a set of positive operators (Π̂k) that add up
to the identity operator and each operator represents a measurement outcome
and projects onto orthonormal quantum states with a calculated weight, (k)wi .
(k)
wi is the conditional probability of obtaining outcome k given the input i, i.e.
(k)
wi = Pr(k|i).
∑
(k) (k)
Π̂k = wi |φi 〉〈
(k)
φi |. (1.24)
i
The experimentalist can at best have a probability distribution for possible inputs
calculated from the measurement outcomes at hand. Knowing (k)wi = Pr(k|i)
values, an experimentalist can calculate P (i|k) through Bayes’ theorem and
update their idea for the inputs i.e. the probability distribution over the inputs.
Knowledge of Π̂k is therefore useful and is calculated theoretically.
27
1.5. Single Photon Detection (SPD) in a device
Many ingenious ideas have been implemented to detect (as well as
create) single photons with high efficiency in a device. The major ideas include
PhotoMultiplier Tube (PMT), Single Photon Avalanche Photodiode, quantum
dot field-effect transistor based detector, Superconducting Nanowire Single
Photon Detector (SNSPD), and up-conversion single photon detector. Few
photon detection have been developed in a Photon Number Resolving (PNR)
fashion. Examples include superconducting tunnel junction (STJ) based detector,
superconducting nanowire-based single photon detector, quantum dot field-effect
transistor-based detector. superconducting transition edge sensor, frequency
up conversion and visible light photon counter [24, 25, 26, 27, 28]. For a short
and concise introduction to the figures of merit (spectral range, dead time, dark
count rate, detection efficiency, timing jitter, photon number resolution, Noise
Equivalent Power (NEP) ) characterizing single photon detectors, see [25, 26].
The state of the art capabilities in terms of the figures of merit are also presented
in [25]. Modern capabilities record The timing jitters in the tens of picoseconds
in superconducting nanowire SPDs [29, 30], dark count rates on the order of a
single dark count per day [31, 32]. Quantum dot field transistor detectors and
superconducting transition-edge sensors have a longer dead time (∼ 10µs) [25].
SNSPDs [31, 32, 33, 34] with low dark count rates (∼ 10Hz) and dead times
(∼ 1ns) are becoming a popular choice these days. The prospects of robust
performance for SPDs in room temperature have also improved [35, 36].
Classic works dating as far back as 1939 [37] deserve a mention. The first
PMT was invented on 4 August 1930 in Soviet Union by L.A. Kubetsky [38],
and avalanche photodiode was invented by Jun-ichi Nishizawa in 1952 [39].
28
These two platforms function based on the photoelectric effect. The myriad of
investigations done on PMT ([40, 41]), superconducting tunnel junction ([42, 43]),
superconducting transition edge sensors ([44, 45]), superconducting nanowire SPD
([31, 32]), single Photon Avalanche photodiodes (SPAD) ([46, 47]), quantum dot
field effect transistor photon detector ([48, 49, 50]) can be found.
Photoelectric effect is at the heart of the longest serving (since 1949) single
photon detector, PMT. Electrons from a photocathode with low work function
(in a vacuum tube) is liberated by incident photon due to photoelectric effect.
A cascade of secondary electrodes biased progressively higher multiplies the
initial single electron current. The Single Photon Avalanche Diode (SAPD)
has a p-n or p-i-n junction reverse biased above its breakdown voltage, the
carriers generated by photon absorption trigger a macroscopic breakdown of
the junction and create an avalanche. In a superconducting Transition Edge
Sensor (TES), a superconducting material is biased at the tipping point of a
superconducting transition; the incident photon heats up the material and it
undergoes a phase transition to lose its superconductivity. Both the energy of
the incident photon and number can be measured through the response in the
current. Ultrathin NbN films of superconducting is biased just below its critical
current for Superconducting Nanowire Single Photon Detectors (SNSPD). An
impinging photon creates a resistive hotspot and blocks the entire channel and a
fast voltage pulse is observed in response to the change in current. Quantum Dot
Field Effect Transistors use the conventional field effect to modulate the channel
conductivity in response to an absorbed photon. More details about individual
platforms can bee found in [25, 27, 51]. Some of the platforms mentioned can have
PNR functionality [45, 52].
29
Photon number resolution has become a sought-after functionality in order
to reduce errors in QIS applications.[53, 54] There exists two classes of methods
for PNR detection process. Certain platforms of single photon detectors produce
output electrical pulses that are commensurate with the number of absorbed
photons [55]. The second class uses spatial or temporal multiplexing [56]. Spatial
multiplexing combines the outputs from an array of detectors [57, 58], temporal
multiplexing uses the same detector but a cascade of beam splitters separates
and delays the input pulse so as to feed them to the detector sequentially [59].
The coherent dynamics of an array of SPDs performing PNR with multiplexing
was investigated theoretically recently [60]. Inspired by biological abilities of
phootodetection, some experiments have been done investigating the abilities of
toad rods with fixed number of photons [61].
DARPA DETECT program has been instrumental in developing a unified
and general natural theory that can provide fundamental description of a single
photon detector valid across all platforms and which facilitates determination
of fundamental constraints between figures of merit. [1, 20, 35, 60, 62, 63, 64,
65, 66, 67, 68, 69, 70] The general model separates the whole process into three
subprocesses transmission, amplification and measurement (fig 1.4). A review
of biological photodetectors and device photodetectors all have this general
underlying structure. Participants of DARPA detect have also formulated new
ideas for weak signal amplification [71, 72, 73]. New device architecture and
concepts were realized by the Sandia Livermore group [74, 75, 76, 77]. Another
particular avenue explored by the group is bio-inspired photodetectors ([36]). They
would be more vulnerable to elevated thermal noise at room temperature and
larger dark count rates, but with a high enough signal to noise ratio interesting
30
new devices may be found. A host of work in the traditional single photon
detectors’ phenomenological models were also conducted [30, 35, 64, 78, 79].
1.6. Genealogy of theories of photodetection and single photon
detectors
Although DARPA detect program developed the general theory for a
single photon detection “device”, it is built on years of progress of our knowledge
of the fundamental natural laws. For the physics of single photon, classical
Maxwellian theory proves inadequate and the need for quantization has driven
the advancement of our understanding of quantum light. The theories of
photodetection can be separated into few genres. Classic works by Glauber,
Kelley and Scully helped reveal fundamental features of the theory of quantum
light [80, 81, 82]. Einstein’s work on the photoelectric effect ([83]) and A-B
coefficients also belong to this fundamental physics genre. Dirac’s formulation of
Quantum Electrodynamics finally enabled the complete quantum treatment of
field and matter and subsequent contributions from others (Wigner, Oppenheimer,
Fermi, Bloch, Weisskopf, Tomonoga, Schwinger, and Feynman) gave us the full
understanding of light-matter interactions [84, 85, 86, 87, 88, 89, 90, 91, 92].
The basic results in the theory of photodetection is reviewd here. For a single
polarization µ of the quantized electromagnetic field, [93]
ʵ(~r, t) = Ê
+
µ (~r, t) + Ê
−∑(~r,√t) = Ê+µ µ (~r, t) +H.c. (1.25)
Ê+
~ωj ~
µ (~r, t) = − ei(kj ·~r−ωjt)̂µâj (1.26)2
j 0
31
where Ê+µ (~r, t) and Ê−µ (~r, t) are positive and negative frequency components
respectively. ̂µ is the polarization vector, ~kj = p~/~ the wave number, p~, 0 the
√
permittivity of free space, ~ Planck’s reduced constant, i = −1, and âj the
annihilation operator corresponding to the mode j.
Glauber’s work in connecting quantized electromagnetic field correlations to
experimentally measurable physical quantities may be the most useful contribution
to quantum mechanical photodetection theory ever [80]. The theory facilitates the
calculation of transition probabilities (that experimental detection is correlated
with) as well as quantum coherence between field components in the framework of
QED. Glauber defines first and second order correlation function of the field (due
to pointlike interaction),
G(1)(~r1, ~r
− + −
2; t1, t2) = 〈ʵ (~r1, t1)ʵ (~r2, t2)〉 = Tr[ρʵ (~r1, t +1)ʵ (~r2, t2)]
G(2)(~r1, ~r2, ~r2, ~r2; t1, t2, t3, t4) = 〈Ê−µ (~r1, t1)Ê− +µ (~r2, t2)ʵ (~r3, t3)Ê+µ (~r4, t4)〉
= Tr[ρÊ−µ (~r1, t1)Ê
−
µ (~r2, t2)Ê
+
µ (~r3, t )Ê
+
3 µ (~r4, t4)] (1.27)
∑
where ρ = i Pi|i〉〈i| is the classical mixture of the (initial) field. Photoelectric
effect can enable an absorber atom to make local field measurements. The
electrons produced in a photoionization process are observed. A detector atom
placed at position ~r in the radiation field has a transition probability (by absorbing
a photon between time t and t+dt) proportional to, [51, 94, 95]
∑ ∑
w1(~r, t) = |〈f |Ê+µ (~r, t)|i〉|2 = 〈i|Ê−µ (~r, t)|f〉〈f |Ê+µ (~r, t)|i〉
f f
= 〈i|Ê−µ (~r, t)Ê+µ (~r, t)|i〉 (1.28)
32
Eq. 1.28 dictate the probability of a photodetection event with a transition in the
(pure) state of the field |i〉 → |f〉. For a classical mixture of the initial field, we
resort to the density operator for the field,
∑
w1(~r, t) = P
− + −
i〈i|ʵ (~r, t)ʵ (~r, t)|i〉 = Tr[ρʵ (~r, t)Ê+µ (~r, t)] = G(1)(~r, ~r; t, t) (1.29)
i
so we can appreciate the physical significance of Glauber’s first correlation
function.
Likewise, the joint probability of observing one photoionization at point ~r2
between t2 and t2 + dt2 and another one at point ~r1 between t1 and t1 + dt1 with
t1 < t2 is proportional to w2(~ri, ~r2; t1, t2)dt1dt2, [94]
w2(~ri, ~r2; t1, t ) = G
(2)
2 (~r1, ~r2, ~r2, ~r1; t1, t2, t2, t1) (1.30)
The other enduring contribution by Glauber is the theory of coherent
states of electromagnetic field [96]. Coherent states were originally formulated
by Schrödinger as he was trying to find solutions to his namesake equation that
follows Neils Bohr’s correspondence principle. Simply put, the principle states
that in the limit of large quantum numbers quantum mechanics follows classical
mechanics. Coherent states are the most classical a quantum mechanical state
can be. The Heisenberg’s uncertainty principle saturates for them. They are the
eigenstates of the annihilation operator of a quantum harmonic oscillator. In terms
of the Bosonic Fock (number) basis, a coherent state is written as
∑∞ ( √ )
|α〉 = exp{−|α|2/2} αn/ n! |n〉 (1.31)
n=0
33
Laser states are coherent states and they are characterized by a single complex
number (or magnitude). They play a crucial role in quantum-to-classical transition
theories, since a “classical” state can be written out in a quantum mechanical basis.
The time-correlation measurements of Hanbury Brown and Twiss ([97]) and
Glauber’s thories inspired a trail of photoelectron counting theories produced
by photoionization by electromagnetic fields (especially Glauber states/coherent
states). The most notable ones are due to Mandel and Kelly-Kleiner and Scully-
Lamb[81, 82, 98]. A particular theme of interest would be the measurement
induced back-action on the field and back-action evading measurement schemes
[99, 100, 101]. The ideas of back-action evading measurement and quantum non
demolition measurements will be popular in modern quantum computing platforms
[102].
The underlying physics of photoionization is quantum mechanical
photoelectric effect. However, the photodetection events recorded in a
photoionization experiment is a classical quantity. On the receiving end, statistical
rules can be used to produce counting formulas such as Mandell’s counting formula
[98],
∫ ∞ 1
P (n, t, T ) = W ne−WP(W )dW (1.32)
0 n!
with
∫ t+T
W = η I(t′)dt′ (1.33)
t
34
where η is the photodetection efficiency, I(t) is the intensity of the electromagnetic
field, and P(W ) is the quasi-probability distribution of the integrated intensity.
Mandell’s formula has had success due to its simplicity and elegance. One
particular example is that, for a thermal distribution of light eq. 1.32 yields a
Bose-Einstein distribution for photodetection events [103]. Photoionization is
the nexus between quantum mechanical interaction of light-matter and classical
measurement results of photodetector click counts. The Carmichael school and
chain of thought from [104] in using the Kelley-Kleiner (KK) ([81]) formula to
derive the photocurrent statistics from quantum-mechanical operator expressions
is a way of defining a Heisenberg cut at the detector. The photocurrent (units
of counts/time), a time series of numbers, can be treated as a classical random
variable. The signal has been demoted into a classical standing, but amplification
of a current carried by bosons/fermions have to abide by quantum laws all the
same ([105]).
The KK formula degenerates into the semiclassical Mandel photon counting
formula if the incident field is a coherent state, a result from the Optical
Equivalence Theorem.
T 〈 Ω̂(T )
k
−Ω̂(T ) 〉 〈 (Qn̄∆T )
k
P (k, ) = : e : = : e−Qn̄∆T :〉
k! k!
where, Ω̂(T ) is the integrated photon flux and enclosure with ‘:’s denote normal
and time ordering. It quantifies the probability of k detection events in a time
window T . This enables us to find, 〈i(t)〉 = eQn̄, indicating the quantum
efficiency, Q to be the number of electrons produced in the photocurrent for each
electron on average. Classical i(t)/e/Q and quantum mechanical, n̂(t) can be
shown to have same statistical features (mean, variance, correlation functions etc.).
35
Another commonly used quantum model for light-matter interaction is the
Jaynes-Cummings model. A two-level atom interacting with a quantized mode of
an optical cavity (or a bosonic field), with or without the presence of light (in the
form of a bath of electromagnetic radiation that can cause spontaneous emission
and absorption)[106].
∑d i ∑d ( )
H + − †JC = ωjAjj − µjk σ a− σ a (1.34)
2
j=1 j=6 k=1
This Hamiltonian commutes with the total excitation number, a†a + σ+σ−.
From ref. [106], if the two possible energy level of the molecule is, ψm,m = 1, 2 and
the number of quanta in the field oscillator is n, Φn, n = 0, 1, 2, 3, ....
Another avenue of theory for relating classical measurement results from an
underlying quantum theory is quantum stochastic processes [107]. They are the
natural language for the description of continuous quantum measurements [108].
Multi-photon Fock space driven absorber atom models have been formulated as a
quantum noise driven process [109]. One interesting contribution to motivating the
photoelectron counting formulae is [110]. Quantum trajectory theory or unravelling
of the quantum master equation is the theoretical method for simulation of
measurement records [111]. Ref. [64] used the following abstract amplification
model, formulated as a continuous measurement.
Aô(t) = D[χ|X〉〈X|]ô(t) (1.35)
χ is the amplification strength, and the designated final internal state being
monitored is X. X is also called the “shelving state”, or the state that drives the
amplification. D is the Liouvillian superoperator representing the measurement.
36
∫
(for example in [64], the stochastic photocurrent, tIt = 2 ′− χ ρXX(t )dt
′ + χdW (t′)
t tm
was used) and a conditioned (upon It) density matrix can be simulated from it.
The quantum optics laboratory methods offer ways of extracting information
from a quantum system under study through continuous measurements. The
quantum observables are mathematically connected to the number or rate of
photoelectron sweeping. For example, The x-quadrature of the system dipole can
be measured from the rate of photoelectrons, (γ a real number) ([112]).
[ ] x̂ =[ĉ+ ĉ
†, ŷ = −i(ĉ− ĉ†)
d ]
E N(t) = Tr (γ2 + γx̂+ ĉ†ĉ)ρI(t) (1.36)
dt
For γ >> 〈c†c〉, eq. 2.81 has the inside the trace a large constant altered
by a small term and a term proportional to x̂. This gives an idea about how
information is extracted in a continuous measurement. It is related to the number
of particles swept per unit time in the output.
1.7. Can we write a general and useful model?
We have learnt that single photon detector models should have the
functionality embodying photon transducton, amplification and measurement.
Biological photon detectors or eyes have photoreceptive molecules that make a
structural change to itself when a photon is absorbed. The changed molecule is in
a “shelving state” and can drive an amplification mechanism. The environmental
states should be discernible to a classical observer. The photon wavepacket
should be stored in preferred basis state of the meter or states robust against
environmental decoherence. We can envisage a system+meter+environment
37
structure for the photodetector. Though the system will be exposed to
environmental decoherence, the dynamics should mostly be dominated by meter-
environment interaction. The meter-environment interaction Hamiltonian,
Ĥmeter−environment dictates the nature of the decoherence that meter states are
subject to. Naturally, the meter states should be chosen so as to enable be robust
against environmental decoherence. The symmetries of Ĥmeter−environment can
facilitate Decoherece Free Subspaces (DFS) and Noise Subsystems (NS). The
“quantum information” of temporal rise in the occupation levels of the absorber
atom is better preserved in the meter states that are robust against as much
decoherence as possible. States that are immune to environmental effects are also
called “dark” states as their evolution as completely decoupled from the evolution
of the environment.
• Unitarity Glauber’s fundamental theory of photoionization calculates
transition probabilities between field states. The transition amplitude for
i→ f is [51, 93],
〈f | Û(i→ f) |i〉 ∝ 〈f | Ê+µ (~r, t) |i〉 (1.37)
The transduction of photon is fundamentally a unitary process. However,
a photodetecto atoms have a special property. The structural change
that happens in a photodetector absorber innitiates a chain of events
going forward and the probability of the photon leaking back out into the
continuum of electromagnetic modes is statistically small (this spontaneous
emission does happen in typically a small fraction of times). Naturally,
practical photodetector physics is modelled with quantum stochastic
38
calculus often that models irreversible quantum processes. Liouville master
equations that model the dynamics of a discrete system’s evolution coupled
to a continuum of modes depict an irreversible dynamics for the discrete
system for the timescale concerned. Excellent works have been done on
modelling irreversible photon transduction into absorber molecules with
a cascaded driven process for modelling photodetection [113, 114, 115].
Since the complete Hamiltonian (system+environment) in these models are
understood, the unitarity of evolution (preservation of commutation rules) is
explicitly verifiable. A photodetector model can capitalize on these theories
[116, 117].
• Amplification of weak signals Bohr’s initial intuition still holds.
Our measurement of quatum observables are only confirmable through
an amplified classical readout. The problem of amplifying weak signals,
therefore, is not new in quantum physics research. Famous examples
of techniques in use are SU(1,1) interferometry [118, 119] and Ramsey
interferometry [120]. A host of new proposals and theories exist on new
techniques for amplifying quantum observables [1, 121, 122]. The theories
offer quantum mechanically allowed evolution of operators that carry the
signal. So these input-output relationships of the operators need to be
commutator relationship preserving. The most precise description of such
unitarity preserving evolution is through a Hamiltonian that describes all the
interactions in play.
Our ability to project onto input photon wavepackets is contingent upon
experimentally obtaining the temporal shape of the rising occupation level
of the “dark state” of the absorber molecule [70]. This is the quantum
39
information we need to protect and amplify and read out. By modern
standards, we do not rely upon Copenhagen interpretation, rather formulate
suitable system+meter+environment who have entwined dynamics in the
spirit of the von Neuman measurement theory. Effectively we measure meter
through the environment. The quantum information in the system and meter
should be protected against measurement backaction as much as possible.
“Which-path” information leaks into the environment from the meter
through decoherence. This redundantly encoded classical information in the
environment is available to all classical observers. Relative environmental
states are preferably clearly distinguishable [2]. A judicious design of
Ĥmeter−environment would be required. Naturally, it would be easier to measure
100spins or atoms in a collective state than an absorber single molecule
state. So 100spins could be the meter that serves the purpose of amplifying
the weak signal from the absorber molecule [71, 72]. Modern quantum
optics techniques allow classical readout of temporal evolution of quantum
observables [123].
• Preserving quantum information The dynamical symmetry of the
evolution, the symmetry of Ĥmeter−environment can help protect against
certain errors in the quantum information. A model for a single photon
detector should consider errors attacking the quantum information. Beyond
a mathematical equivalence, physical formulations should be augmented with
error models that they can possibly protect against.
The occupation in shelving state, X’s occupation can be coupled to a bound
optical cavity mode with a Hamiltonian term, i|X〉〈X|(c − c†). The cavity
mode becomes our “meter”, it needs to interact with the environment or we
40
do not find out any information about its dynamics. If the cavity is coupled
to a bath of electromagnetic (continuum) modes, it would leak out and that
reveals information about it. The bath or “environment” monitors the state
of the oscillator performing continuous weak measurement. If unobserved
by an environment, the driven oscillator (initiated in the ground state, |0〉,
a coherent state with amplitude 0) continues to stay a coherent state with
amplitude being the integral of the coherent drive amplitude [124] (it’s the
time integration of the occupation we need to measure experimentally).
ρ̇ = −(i/~)[Hsys, ρ]
( γ(t) = |X〉〈X)|(t)
Ĥdrive = i~ γ(t)a† − γ∫∗(t)at
aout = ain + α(t) α(t) = dt
′γ(t′) (1.38)
0
With the cavity leaking out at a rate κ to the environment, a cavity
oscillator of resonance frequency ωc (no driving present), initiated in coherent
state |α〉, evolves as a coherent state of decaying magnitude αe−iωcte−κt/2
[125]. With driving proportional to the waveshape ( γ(t) = |X〉〈X|(t) )
we wish to measure, the quadratures of the oscillator bears information
about the waveshape we are interested in, and they can be measured with
a homodyne setup [123]. But it is quite obvious the measurement damping is
going to affect the amplitude of the cavity mode.
Another clarifying comment is in order here. The measurement result of
an atom undergoing spontaneous decay would be a discontinuous one [125].
Such trajectories can be simulated with quantum trajectory methods. So if
41
our desired function γ(t) was driving an atom, and the fluorescence signal
leaking from it would be broken up ones as well. However if it was driving an
ensemble of atoms and their output was monitored the observed signal would
be almost continuous similar to the solutions of the master equation, that
calculates ensemble average dynamics.
The ideas of quantum non demolition observables and deoceherence free
subspaces are useful here. Whether we are monitoring the oscillator
amplitude or the collective atomic operators, they can be made immune
to certain errors that preserve the symmetry of the Ĥmeter−environment
interactions. These ideas are borrowed from quantum information theory
and they can preserve quantum information better. Quantum non demolition
measurements, on the other hand can carry out measurements without
backaction and such methods can help purer information about the photon
wavepackets initiating the detection event.
In summary, the interactions in the model that define the dynamics from a
weak signal to the amplified classical signal need to abide by natural or quantum
mechanical rules. The unitarity requirement is most precisely enforced by a
Hamiltonian formulation. A complete Hamiltonian can elucidate the interrelations
between all constituents of the theory, i.e. particles and modes and shed light onto
fundamental constraints of photodetector figures of merit. With a host of cavity-
QED system apparatus and Hamiltonians representing them at our disposal, we
can design very precise and fundamental descriptions of single photon detectors in
terms of the Hamiltonians.
42
CHAPTER II
SUMMARY OF RELEVANT THEORY
You know only insofar as you can
measure.
Lord Kelvin
2.1. A Photon Wavepacket
Fundamentally, photon is the excitation of the electromagnetic fields.
However, from a practical point of view of an experimentalist working on an
apparatus of photo-electric effect or an atom’s spontaneous decay that definition is
of no particular practical use. Rather definitions of photon tailored to the purpose
at hand may be more useful. Rather more useful definitions can be procured
from a wavepacket point of view. For a photon generating experiment like a
spontaneous emission of an excitation from an atom into a bath of electromagnetic
field or for a photon detection experiment where an excitation is absorbed into an
atom from the bath we can define coherent superpositions of frequency (or time)
components normalized with a particle number of 1.
Classical electrodynamics has a simple form in the reciprocal space and the
electromagnetic field is decomposed into longitudinal and transverse components.
[51, 126]
43
ik · E 1= ρ
0
ik · B = 0
ik× E = −Ḃ
ik× B 1 1= Ė + j (2.1)
c2  c20
A classical electric field can be represented by the complex representation,
or analytical signal E(+)(r, t). [127] All solutions of the Maxwell’s equations,
E(+)(r, t) can be considered a vector in the Hilbert space of modes, the modal
space. See ref. [127] for a review of the definitions and structures of the space. The
mode is defined to be a vector field that is a normalized solution of the Maxwell
equations in vacuum.
For a field propagating in the +z direction,
∑ ∑
E(+)(r, t) = Fi,p,rfi,p,r(r, t) = Fmfm(r, t) (2.2)
i,p,r m
f (T ) (L)i,p,r(r, t) = ifp (x, y, z)fr (t, z) (2.3)
where (T)i are the two orthogonal polarization vectors in the x-y plane, fp (x,y, z)
is the transverse or spatial part, (L)fr (t, z) the longitudinal or temporal. This
separation of mode functions help identify the possible excitations (photons) in the
quantized version of the electromagnetic fields. In a beamlike geometry or a one
dimensional waveguide (oriented along the z-axis), the translationaly symmetric
direction is the z-axis alone and the reciprocal vector representations in eq. has
44
only kz points. The transverse component of the mode function,
(T)
fp (x,y, z) is
determined by the transverse boundary conditions.
Under the paraxial approximation (wave-vectors close to a particular ~k0) and
narrow-band approximation (frequencies close to a central frequency, ω0 = c|k0|);
the E(+)(r, t) can be resolved into a product of envelope functions and the plane
wave, ei(k0z−ωt).
∑
E(+)(r, t) = ei(k0z−ωt)  f (T)i p (x,y, z)f
(L)
r (t, z) (2.4)
i,p,r
The quantum extension of the classical complex field E(+)(r, t), is written in
the Heisenberg picture so as to get the time dependence for the operators. [127]
Ê(r, t) = Ê(+∑)(r, t) + Ê(−)(r, t)
Ê(+) E (1)(r, t) = l âlul(r, t) (2.5)
l
ul(r, t) =  e√ki.r−ωltl
~ωl
E (1)l = 20V
The expansion is in the basis of monochromatic plane wave modes ul(r, t). The
expansion eq. (2.5) is not unique, Ê(+)(r, t) can be expanded into other modal
bases (solutions of the wave equations). [128] Unitary transformations can be
found between modal bases and also between the the corresponding creation and
annihilation operators. [127, 128]
In a waveguide oriented along the z-axis, we once again have kz the only
continuous mode (plane wave mode) index. The transverse functions have mode
45
labels determined by the boundary conditions.
u (r, t) =  f (T)(x,y, z)eki.z−ωltl l p (2.6)
We can identify the degrees of freedoms (DOF) that a photon can have from the
translational symmetry of the problem and polarization. In free space, light has
four DOFs e.g. helicity and the three components of the momentum vector. [129]
In a beamlike geometry or an one dimensional waveguide we have polarization,
two transverse mode profile and the longitudinal (or time) profile ([129]). A
photon can reside in any of the DOFs that light has and quantized electromagnetic
field theories have been developed for all possible photon wavepackets. The
excitations can be separated into two components. The Transverse Electric (TE)
and transverse Magnetic (TM) modes. TMs are defined with the condition Hz = 0
and only electric field has a component parallel to the z-axis (Ez). Knowledge of
Ez is sufficient for complete determination of the relevant modes,
− kz ∂Ez − kz ∂Ez ωµ ∂Ez ωµ ∂EzEx = i , Ey = i , Hx = i , H2 2 2 y = −i (2.7)kρ ∂x kρ ∂y k 2ρ ∂y kρ ∂x
The field-orthogonal temporal modes (connected to the longitudinal DOF or
temporal DOF only) enable us to define a photon wavepacket as a coherent
superposition of monochromatic wave modes [129, 130].
∫
|A † †j〉 = dωfj(ω)â (ω)|0〉 ≡ Âj|0〉 (2.8)
46
In the time domain we have corresponding creation operators,
∫
|Aj〉 = dtf̃j(t)†(t)|0〉 ≡ †j|0〉 (2.9)
The definitions of the Fourier transform are
∫ ∫
â†
1 1
(ω) = √ ∫ dteiωt†(t), †(t) = √ ∫dωe−iωt†(ω) (2.10)2π 2π
1 1
f̃j(t) = √ dωeiωtfj(ω), fj(ω) = √ dte−iωtf̃j(t) (2.11)
2π 2π
The broadband mode operator Âj follows the Bosonic commutation relations [129],
∫ ∫
†j = dωfj(ω)â
†(ω) = [ dtf̃ (t])†j (t) (2.12)
 , †i j = δij (2.13)
Temporal modes are orthogonal in both time and frequency domain and are
important for Quantum Information Science (QIS) applications.
∫ ∫
dωf ∗j (ω)fk(ω) = dtf̃
∗
j (t)f̃k(t) = δjk (2.14)
Similar single photon wave-packet creation operator can be defined for
twisted photon pulse, [131], (where the photon resides in the helicity degree of
freedom)
∫
â†ξλ = d
3kξλ(k)â
†
kλ (2.15)
47
The Spectral Amplitude Function (SAF), ξλ(k) of a twisted pulse with
deterministic Orbital Angular Momentum (OAM) can be written
1
ξλ(k) = √ ηλ(kz, ρ imφkk)e (2.16)
2π
In this dissertation, we are interested in the photon wavepackets in the
temporal or longitudinal DOF. The photon absorber is placed at the end of a
one dimensional waveguide and interacts with any photon or photons travelling
through the waveguide. Two (frequency) un-entangled single-photon wavepackets
in two polarizations (or spatial) modes
|ψ∫2〉 = |ψα〉|ψβ〉 ∫ (2.17)
= dω g (ω )b†1 α 1 1(ω1) dω2gβ(ω2)b
†
2(ω2)|vac〉 (2.18)
We do not consider entangled photons. Parametric Down Conversion (PDC)
can generate spectrally (time) entangled photon pairs. [129, 130]
∫
|Ψ〉 = dωsdωif(ω †s, ωi)â (ω )b̂†s (ωi)|0, 0〉 (2.19)
where the Joint Spectral Amplitude (f(ωs, ωi) ) cannot be factorised as in eq.
(4.22). But the important point is the transverse spatial mode profile is decoupled
which is determined solely by the waveguide’s features.
Note that the two photons in eq. (4.22) need not have the same polarization.
In fact it is extremely unlikely for the two to have the same polarization (as well as
transverse profile) i.e. reside in the same continuum.
48
2.2. Atoms in a single mode Electromagnetic Field
Optical transition and resonance fluorescence problems exploit the dipole
coupling of the field and atom. For a d-level atom, we consider the single mode
field nearly resonant with two of the levels of the atom. So, |e〉 and |g〉 are two of
the d-levels, HI mentioned would fit inside the d-dimensional Hamiltonian for the
d-level [132].
HI = (|g〉〈e|+ |e〉〈g|) 〈g|HI |e〉 (2.20)
Eq. 2.20 only clarifies the structure of the Hamiltonian matrix elements in the
linear space of operators acting on the atomic part of the Hilbert space. The
explicit HI act on the product space of the atom and field Hilbert space and will
be presented shortly (in eq. 2.23).
The single mode electromagnetic field is a classical plane-wave field of the
form,
[ ]
E(x, t) = E ei(kx−ωt)0 + c.c. (2.21)
where, the field amplitude, E0 is taken to be real. Quantization of the
electromagnetic field is not always necessary [132]. We take, ~ω ≈ Ee − Eg, hence
all electronic states except |g〉 and |e〉 will be less important in the rotating frame
of the field..
49
2.2.1. The Dipole Approximation
The d-level atomic Hamiltonian (a d dimensional operator) can be written
as,
∑d ∑d
Ha = (Ha)kj |k〉〈j| = Ej|j〉〈j| (2.22)
k,j=1 j=1
The single mode electromagnetic field with frequency, ω would have the energy of
a single harmonic oscillator with Hf = ~ωa†a and interact with the atom with
the Hamiltonian part, Haf = −d~.E~ . Here d~ = e~r denotes the dipole moment
operator and E~ the external quantum electric field ∑at the position of the atom.†
In the rotating wave defined by, H = eiωt/~a a+i/~
d
j=1 ωjAjjt
0 , and in the long
wavelength approximation (ei~k.~x ≈ 1), the interaction is, (ref. [133, 134])
d
i ∑
HI = H
†
af = − µjk (|j〉〈k|+ |k〉〈j|) (a− a ) (2.23)
2
j=6 k=1
√ ∑ ( ) √
with µ = 2π~ω E~jk β 0.~eβ(xβ) , E =
~ω
jk 0 . Ω = dgeE0/~, where dge is theV 20V
matrix element of the electric dipole operator. The complete Hamiltonian would
be,
∑d i ∑d
H = ωjAjj − µjk (|j〉〈k|+ |k〉〈j|) (a− a†) (2.24)
2
j=1 j 6=k=1
50
2.2.2. The Jaynes Cummings Interaction
The interaction Hamiltonian, (ref. [132, 134])
∑d i ∑d ( −)H = ω A − µ +j jj jk σ + σ (a− a†)
2
∑ j=1 j=6 k=1d di ∑ ( )
= ωjAjj − µ +jk σ a− σ−a† − σ+a† + σ−a (2.25)
2
j=1 j 6=k=1
In the original frame of the laboratory or “lab-frame”, σ+a − σ−a† is accompanied
by no time dependence and −σ+a† + σ−a is accompanied by the time dependence
of e±i2ωt. The Jaynes Cummings Hamiltonian ignores the high frequency
components. Also, σ+ = |j〉〈k| if Ej − Ek = Ee − Eg > 0.
∑d i ∑d ( )
H +JC = ωjAjj − µjk σ a− σ−a† (2.26)
2
j=1 j=6 k=1
This Hamiltonian commutes with the total excitation number, a†a + 1σ
2 z
. We
can call the two possible energy level of the atom or molecule ψm,m = 1, 2 and
the number of quanta in the field oscillator n, Φn, n = 0, 1, 2, 3, ... [106]. Then,
the state vectors, Φn ⊗ ψm form a basis of the Hamiltonian. We can write out the
matrix elements of the Hamiltonian in the said basis,
〈mn|H|m′n′〉 = (E ′ ′m + n~ω)δmm′δnn′ + 〈mn|Hint|m n 〉
Hint = −µ~.E~
〈mn|µ |m′n′z 〉 = µ(1− δmm′)δnn′
√ √
〈mn|H |m′n′int 〉 = ~α(1− δm,m′)[ nδn,n′+1 + n+ 1δn+1,n′ ]. (2.27)
51
2.3. Open System dynamics
The Liouville representation of quantum map, S can be done with the aid
of S expressed in a particular basis. In a d-dimensional quantum system with the
Hilbert space, Hd, operator A is a map, A : Hd → Hd, and S is a mapping for the
operator A,S : L(Hd) → L(Hd). L is the vector space of linear operators acting
on Hd [135]. We define a Liouvillian dissipator map for density matrix, D : ρ →
XρX† − 1X†Xρ − 1ρX†X and a Liouvillian-type dissipator map for operator A,
2 2
D′ : A → X†AX − 1X†XA − 1AX†X, commonly known as superoperators. X is
2 2
called a quantum jump operator. The Schrödinger picture evolution of the density
matrix and the Heisenberg picture evolution of the system operator are determined
by the Liouvillian and the Liouvillian type equation respectively.
i
ρ̇ = Lρ = − [Hsys, ρ] +DXρ (2.28)~
D 1ρ = XρX† − [X†X Xρ−
1
] ρX†X2 2
˙̂
A = L′ i = H ,  +D′sys X (2.29)~
D′ Â = X†X ÂX −
1 1
X†XÂ− ÂX†X
2 2
Choosing {P }d2n n=1 for a Hermitian operator orthonormal basis (w.r.t.
Hilbert-Schmidt inner product), we can represent any operator at any time
with its components in the basis. The inner product of A and B is defined as
52
(A|B) := Tr[A†B].
1 ∑
A = Tr[P
∑ k
A]Pk
d
k
1
S(A) = SnmTr[PmA]Pn
d
n,m
1
Snm = Tr[PnS(Pm)]
d
(Pn|Pm) := Tr[P †nPm]
The operator A and map S both can be specified with the orthonormal
basis {Pn}. This is the most crucial tool for the calculations. The basis {Pn}
can be taken to be time stationary. There exists an under-appreciated method
for solving equations like (2.28) and (2.29) from [136]. For a total system that is
a cascade of three or more constituent quantum mechanical Hilbert spaces, the
density matrix can be expanded in one, two or all three constituent Hilbert space
bases to solve eq. (2.28). The coefficients of expansion are complex numbers if
expanded in all three constituent Hilbert space bases. And they are operators if
expended in less than three constituent Hilbert spaces. The same structure applies
in expanding into vector spaces of operators in solving eq. (2.29) for the evolution
of an operator.
For a cascaded system of a discrete bosonic cavity a (with a maximum
photon of Na), a N level atom F and another discrete bosonic cavity mode, c (with
a maximum photon of Nc) under the action of a system Hamiltonian and some
53
collapse operators, an arbitrary operator, K̂ can be expanded as
∑Na ∑N
K̂ = K̂ij⊗kl(t)|a†a = i, Fk〉〈a†a = j, Fl| (2.30)
∑ i,j=1 k,l=1Na ∑N ∑Nc
= K †ij⊗kl⊗ab(t)|a a = i, F , c†k c = m〉〈a†a = j, F †l, c c = n| (2.31)
i,j=1 k,l=1 m,n=1
The expansion coefficients in eq. (2.30) are operators and in eq. (2.31) are complex
numbers. The basis in either case is time stationary and the time dependence is
entirely in the expansion coefficients. If the Schrödinger picture is opted for, eq.
(2.31) is solved with the density matrix, ρ expanded (gradually) in complete bases.
The two picture solutions are to be consistent with the condition,
Tr[ρ(t)K(t0)] = Tr[ρ(t0)K(t)] (2.32)
An example of such expansion in operator coefficients to solve a Liouvillian
equation (2.28) can be found in [136]. In a problem that is solved in chapter 3,
a cascaded system of a discrete bosonic cavity (Na = 1), a N=3 level atom F,
and a discrete cavity c, with the atom F driving mode c in the state |F2〉 with the
Hamiltonian term HF−c = i|F2〉〈F |(c − c†2 ), we get the following equation set
54
(decoupled from all the other) of equations. This set has an interesting feature.
˙̂
K ′√00⊗00 = D ΓcK̂00⊗00
˙̂
K00⊗22 = i[iF (c− c†), K̂00⊗22] +D′√ΓcK̂00⊗22
˙̂
K00⊗11 = −(γ1 + γ2)K̂00⊗11
+D′√ΓcK̂00⊗11 + γ1K̂00⊗00 + γ2K̂00⊗22
˙̂ ˙̂ −(γ1 + γ2 + κ)K10⊗01 = K01⊗10 = K̂10⊗01
2
√
+ κγ1(K̂
′√
00⊗00 − K̂00⊗11) +D ΓcK̂10⊗01
In the composite operator, 1̂ ⊗ 1̂ ⊗ ĉ(t) operator, all the diagonal operators,
K̂ii⊗jj are ĉ itself. So the first two equations look like the following,
c˙̂ = D′√Γcĉ
c˙̂ = i[iF (c− c†), ĉ] +D′√Γcĉ (2.33)
The two equations are for a cavity mode under dissipative dynamics and a driven
cavity mode under dissipative dynamics.
In essence, every single equation in the set above behaves as a stand-alone
(except for the coupling to other ones) system initialized in a certain density
matrix (in the sense of eq. (2.32) ). The have the appearance of copies of the
same system running in parallel initialized in all the different possibilities. The
off-diagonal elements contribute to the noise in the output signal. Schrödinger
equation evolution and quantum mechanics is determinsitic. Our Heisenberg
evolution is a counting problem of how many initial state matrices can end up in
the observable eigenvalue observed. The probability of an observable expectation
55
value is proportional to the number of initial states that evolved deterministically
would end up in states consistent with it. As our model shows initial unphysical
density matrices contribute to the measurement fluctuations the same way the
physical density matrices do. Mathematically invalid solutions are supported by
nature sometimes. ρ01 is a physically invalid state, but it has a bearing on physical
observable statistics!
We can emulate the multi-photon Fock state equations of ref. [113] with
expanding the density matrix, ρ(t) in a complete basis. In chapter 4, we have
an explicit example of this. The operator evolution calculation is also facilitated
by expanding them in the complete orthonormal basis, where the sparsity of the
couplings between the equations make analytical computation possible.
The variance of a quantum mechanical observable (represented by a
Hermitian matrix M) is
var(M) = 〈M2〉 − (〈M〉)2
var(M) = (Tr[M ×M × ρ(t )])20 − (Tr[M × ρ(t0)])2 (2.34)
∑
For the case of initial density matrix being a pure state, ρ(t0) = i,j δi,i0δj,i0 , we
get an elegant expression for var(M(t)),
∑
var(M(t)) = |M 2i0k(t)| (2.35)
k 6=i0
So off-diagonal couplings in the equation (2.29) for any diagonal element, add to
the variance and hence noise. If the initial density matrix is a mixed state, the
expression for noise would account for the added classical uncertainty as well.
56
2.4. Systems and meters
A brief review from [112]. The initial system state vector is |ψ(t)〉 and the
meter system has initial state vector |θ(t)〉. Hence, the initial combined state
|Ψ(t)〉 = |θ(t)〉|ψ(t)〉 (2.36)
The product state can be written for only at time t. The two systems evolve
together for a time T1 by a unitary evolution operator Û(t + T1, t) ≡ Û(T1). The
combined system-meter at time T1 is
|Ψ(t+ T1)〉 = Û(T1)|θ(t)〉|ψ(t)〉 (2.37)
Let the meter be measured projectively over a time interval T2, and T = T1 +
T2. We can assume the system+meter does not evolve in this interval (T2 may be
negligible or the coupling time-dependent)
The projection operator for the meter is Π̂r = π̂r ⊗ 1̂, where |r〉 forms an
orthonormal basis for the meter Hilbert space. The final combined state is
| 〉 |r〉〈r|Û(T1)|θ(t)〉|ψ(t)〉Ψr(t+ T ) = √ (2.38)
ρr
ρr = Pr[R = r] = 〈ψ(t)|〈θ(t)|Û †(T1)[|r〉〈r| ⊗ 1̂]Û(T1)|θ(t)〉|ψ(t)〉 (2.39)
ρr is the probability of obtaining the value r for the result R.
Post measurement system and meter are disentangled once again.
| 〉 | 〉M̂r|ψ(t)〉Ψr(t+ T ) = r √ (2.40)
ρr
M̂r = 〈r|Û(T1)|θ(t)〉, a measurement operator (2.41)
57
where M̂r is an operator that acts only in the system Hilbert space. The
probability distribution for R can be written as
Φr = 〈ψ(t)|M̂ †rM̂r|ψ(t)〉 (2.42)
2.4.1. Measurement operators and effects
Measurement can be completely specified in terms of the measurement
operators, M̂r. Given that after the measurement of duration T, the result R has
the value r, the conditional state of the system is
|ψr(t+ T )〉
M̂√r|ψ(t)〉= (2.43)
Φr
The probability operators, or effects are defined to be
Êr = M̂
†
rM̂r (2.44)
Their expectation values give the probabilities. Probabilities need to add up to
unity and a completeness condition on the measurement operators is imposed:
∑
Êr = 1̂S (2.45)
r
There are two restrictions on the set of measurement operators: 1) they have to be
positive, 2) Êr : r is a resolution of the identity for the system Hilbert space.
Generalized measurement does not require physical observables.
Measurement results r are simply labels, they can be integers, fractions, real or
complex (unlike physical observables that are real and eigenvalues of a Hermitian
58
operator). The set of Êr constitute a Positive Operator Valued Measure (POVM)
on the space of results r. We do not need a probability distribution over space of r,
but rather only the POVM. For the general case of mixed states, the conditioned
state for the measurement result r is,
J [M̂r]ρ(t)
ρr(t+ T ) =
ρr
ρr = Tr[ρ(t)Êr]
J [Â]B̂ ≡ ÂB̂†
Or = J [M̂r] the operation for r : a superoperator
Superoperators map operators to operators, J is a superoperator. The class
of superoperators (called the completely positive maps) can map physical states to
states.
In summary:
1. It is good enough to assume that we projectively measure the ‘meter’
i.e. the electromagnetic continua, although subsequent measurements are also
done. They involve electronic signal, number plate, retina, brain signal etc. But
macroscopic objects decohere fast. So, a projective measurement on the meter is a
good enough description, we have a classical signal.
2. Measurement disentangles the system and meter.
3. M̂r are the measurement operators, act only on the system and in general,
are not unitary.
4. What state we prepare the meter state at time, t=0, (called the fiducial
state) can give rise to quantum noise. If the measurement basis states of the meter
(that we can take projective measurements on) are different from the fiducial state
59
(even if it is a pure state), we shall have a distribution of outputs, a quantum
noise. If the fiducial state is a mixed state, we get a noise of classical nature on
top of it.
5. There is a way of representing the evolution during T1 + T2 using the
unitary operator,G.
6. The most general formulation of quantum measurement theory: The
operator Or for the result r is a Completely Positive (CP) superoperator. An
operation can always be written as,
∑
Or = J [Ω̂r,j] (2.46)
j
for some set of operators, Ω̂r,j : j which is not unique. Ω̂r,j : j is the basic element
of the theory, which takes the a-priori system state to the conditioned a-posteriori
state:
ρ̃r(t+ T ) = Orρ(t) (2.47)
And now we have completely replaced the measurement operators, M̂r.
It is worth mentioning here that Quantum Darwinism theory takes a different
approach and we discuss that later.
A Back-Action Evading (BAE) measurement is one where the statistics of
an observable does not change. A projective measurement before or after the
BAE would give the same statistics. A Quantum Non Demolition measurement
is a stronger condition, demanding that the observable, X̂ does not evolve in the
Heisenberg picture (all but ensures preserving the statistics).
60
∑ BAE: The total operation for the measurement in question being O =
rOr,
Tr[Π̂xρ] = Tr[Π̂xOρ] (2.48)
QND: X is a QND observable (effects/probabilities are functions of X̂) if
X̂ (= X̂S ⊗ 1̂A) is a constant of motion in the Heisenberg picture.
X̂ = Û †(T1)X̂Û(T1) (2.49)
Û(T1) is the system-meter coupling.
2.4.2. Classical and Quantum uncertainties
For two quantum operators, x̂ and ŷ, we take up the definition
δx̂ = x̂− 〈x̂〉, δŷ = ŷ − 〈ŷ〉
where, 〈..〉 refers to complete expectation values arrived at from a combination of
quantum mechanical and classical uncertainties. We have the constraint [137],
〈δx2〉〈 2〉 ≥ 1 |〈 1δy [δx, δy]〉|2 + |〈{δx, δy}〉|2 (2.50)
4 4
In classical noise theory, x̂ and ŷ commutes and, we have,
〈δx2〉〈δy2〉 ≥ σ2xy (2.51)
61
But in quantum mechanical open systems (once we integrate out the bath modes),
we have a mixed state with classical and quantum statistical properties.
√
∆x∆p ≥ 1~2 + σ2
4 xp
(2.52)
Therefore, in our open system model of photodetection, the uncertainty
relationships of canonically conjugate varibles have the extra contribution from the
classical statistics as well. I refer to a discussion in chapter 1 of ref. [137], to point
out that certain operator (or observable) may be measured with more certainty
than others. One quadrature of a harmonic oscillator mode (say p̂) may be a more
attractive choice for measurement observable than the other quadrature (x̂). For
the motion of a free particle without any thermal noise, the classical Hamilton
equations and the quantum Heisenberg equations of motion are of the same form,
mẋ = p ṗ = 0
We now consider the repeated measurement of the position of the free particle at
intervals of τ .
x(t+ τ) = p(t)τ/m+ x(t)
〈δx(t+ τ)2〉 = 〈δx(t)2〉+ (τ/m)2〈δp(t)2〉+ τ/m〈δx, δp〉
A precise measurement of x(t) would cause 〈δp(t)2〉 → ∞ and 〈δx(t + τ)2〉 → ∞.
It becomes impossible to measure x precisely at any subsequent moment. However,
p can be measured repeatedly without introduction of such measurement back
action. p(t+ τ) = p(t).
62
For a specific choice of the Hamiltonian, H̃c(F ) = 2iF cos(ωF t)(c − c†),
p̂ ∝ i(c−c†) (commutes with the Hamiltonian for zero detuning) would be preferred
as a Quantum Non Demolition Observable over the other quadrature variable,
x̂ ∝ (c+ c†) for repeated measurements.
2.5. Linear and nonlinear quantum amplification
In principle, the “input-output relationship” for some operator are always
obtainable with a brute-force calculation.
b̂(t) = eiĤtott/~b̂(0)e−iĤtott/~
for, Ĥtot = Ĥ0 + V̂ (2.53)
b̂(t) = eiĤ0t/~b̂(0)e−iĤ0t/~ (2.54)
A lot of the times, the important part of the transformation can be achieved by
splitting the Hamiltonian, Ĥtot into a tractable part, Ĥ0 and a complex part, V̂ (in
eq. (2.53) ) as is done in the famous Caves’ model. The primary mode, â interacts
with the ancillary mode, b̂ (can be called amplifier’s internal mode)
Ĥ = Hˆ̃I LPA = i~κ(âb̂− â†b̂†)
Û (t) = e−iĤI t/~ = er(âb̂−â
†b̂†)
I ≡ Ŝ(r), r = κt
Ŝ(r) is the two mode squeezing operator. The a mode evolves in the Heisenberg
picture, √
â = Ŝ†âŜ = âcosh(r)− b̂†sinh(r) = gâ− b̂†out g2 − 1 (2.55)
63
The transformation belongs to SU(1,1) algebra. Quantum Mechanics demands
unitarity of evolution (for the system+environment or the universe) and hence
requires commutation relationships to be preserved.
√ √ √
â † †out = Gâin + G− 1b̂i[n = ]Gâin + L̂ (2.56)
L̂, L̂† = G− 1 (2.57)
~
∆L̂∆L̂† ≥ ~[L̂, L̂†] = (G2 − 1) (2.58)
2
The b̂in operator is the amplifier’s internal noise mode which must be added to
preserve canonical commutation relationships. The relationship was generalized
to show that amplifying currents carried by either Fermions or Bosons require
addition of noise currents holding a relationship similar to eq. (2.57) [105]. It
was also noted that, a phase sensitive amplifier can amplify one of the x̂ or p̂
quadratures noiselessly while attenuating the other.
√ √
x̂out = Gx̂in, p̂out = p̂in/ G (2.59)
Eq.s (2.56) and (2.59) are both linear in the annihilation operators. The problem
with linear amplification is that the noise mode is necessarily amplified Eq. (2.56).
More generally, for linear amplifiers,
√ √
âout = e
iĤtot/~â e−iĤtot/~in = Gâin + G− 1b̂†in
↠√â = eiĤtot/~â
†
out out inâine
−iĤtot/~
= G↠⠆ † †in in + (G− 1)b̂inb̂in + G(G− 1)(b̂inâin + âinb̂in)
64
Diagonalizing equations like Eq. (2.60) are the reason the transformation in eq.
(2.55) is given the name Bogoliubov transformation after the famous example of
BCS superconductivity. â†outâout is the diagonalized form. The output excitation
expectation value, 〈n̂out〉 is calculated from either evolved states (Schrödinger
picture) or evolved operators (Heisenberg picture) in accordance with eq. (2.62).
〈n̂out〉 = 〈ψ iĤtot/~in|e ↠â e−iĤtot/~in in |ψin〉 (2.60)
= 〈ψout|â†inâin|ψout〉 The Schrdinger P icture (2.61)
= 〈ψ †in|âoutâout|ψin〉 The Heisenberg P icture (2.62)
Eq. (2.60) enables calculation of the quantity in eq. (2.62) through initial
state relationships alone. If we assume that G excitations are to be considered a
classical macroscopic signal, we can see what initial state can cause such a click
simply from the number of excitations in the initial state. For an initial state,
|ψin〉 = |n̂a = 1, n̂b = 0〉, the quantity n̂out comes out to be G. So a single photon
signal would be amplified to a “click”. However, the initial state, |ψin〉 = |n̂a =
0, n̂b = 1〉 would cause a click just as well (with G-1 excitations), with no input
photons. This would count as a dark count, where a thermal excitation causes a
click in the detector. The linear transformation in eq. (2.60) amplifies noise almost
as much as it does the signal. Linear amplification transformations are plagued
with this noise amplification property that render them non-ideal for the purpose
of single photon detection ([1]).
The particular form of the transformation is mandated by the requirement
of the commutation preservation. It was noted in [1] that, nonlinear amplification
65
transformations can be conceived of, that offer ways of amplifying an input signal
without amplifying the noise excitations.
b̂†outb̂out = Gâ
†
inâ + b̂
†
in inb̂in (2.63)
With such a nonlinear amplification scheme, |ψin〉 = |n̂a = 1, n̂b = 0〉 would
cause a click, |ψin〉 = |n̂a = 0, n̂b = 1〉 would not. So nonlinear amplification (eq.
(2.65) ) mechanism would far outperform linear amplification schemes in terms of
the Signal to Noise (SNR) ratios. Nonlinear amplification can be achieved with the
addition of a single noise mode operator which is not amplified. That one mode of
noise operator is, however, essential. Amplification with minimum noise addition is
called “minimum noise amplification”.
√ √
âout = Gâin + G− 1b̂†in The Caves’ relation(linear)
b̂†outb̂out = b̂
† b̂ +Gâ†in in inâin √Number amplification (nonlinear) (2.64)
b̂ †out = Ŝ (b b)in +G((b†b)in, (2.65)
Ŝ is a unitary operator which is designed so as to preserve the commutation
relationship of the operator, b̂out in eq. (2.65). Number amplification like in eq.
(2.65) has improved SNR performance.
The Heisenberg evolution of an operator yields an input-output relationship
for the operator, which enables easy calculation of the time evolved expectation
values (such as 〈n̂out〉), only eigenvalue equations of the initial state suffice for the
computation. This is always a good way for interpreting results in the Heisenberg
picture. The matrix elements of the operators in some basis help calculate the time
evolution of the expectation values of an operator (for a initial condition) which
66
coincide with the Schrödinger picture since matrix elements are the same across all
pictures.
For the more general case of mixed/entangled states,
Tr[ρ(t0)
†g(t)] = Tr[ρ†(t)g(t0)] (2.66)
Tr[ρ(t0)g(t)] = Tr[ρ(t)g(t0)] (2.67)
Each individual element of g(t), e.g. 〈i|g(t)|j〉 in a chosen basis can be interpreted
as the expectation value of g(t) for a system initialized in ρ(t0) = δr,iδc,j
2.6. Access to some example observables: Techniques of quantum optics
The number operator holds a special place in quantum mechanics, since the
methods of measurement are reliant on photocurrent clicks that are proportional
to intensity of a mode which is proportional to the number operator. Also, we
have talked about why a quadrature of the field may be better suited for low noise
detection in repeated weak measurements over the other. Therefore, we can do
better by finding more suitable observables for low noise detection and better
readout of the steadystate population of the atom. I now talk about a few schemes
of finding more observables inside our absorber atom.
The Balanced Homodyne Detection (BHD) gives us photocurrent
proportional to the difference between two intensities, Ic − Id = 〈n̂cd〉 = 〈ĉ†ĉ− d̂†d̂〉.
ĉ and d̂ are mixed combinations of signal, â and reference b̂, ĉ = √1 (â + ib̂), d̂ =
2
√1 (b̂+ iâ). In a homodyne detection scheme, a strong coherent field local oscillator
2
(lo) is added to the field to be measured (the source field) and continuous
67
photoelectric detection is performed on the sum. For the continuous photoelectric
detection of the field, Ê(t)
[ √ √ ]
Ê(t) = e−iωCt Ê + ξ 2κ∆aˆ̃(t′lo )
Ê(t),Êlo are in units of photon flux. 〈Êlo = Elo, a c-number, is the local oscillator
√
coherent amplitude, t′ is a retarded time, ξ is the collection efficiency, 2κ scales
the source field for 2κ∆aˆ̃†aˆ̃(t) to have units of photon flux.
2.6.1. Quantum Stochastic Calculus
By coupling a microscopic system to a macroscopic system, the quantum
features should be transferrable to the classical objects– a measurement readout.
Decoherence is a consequence of environmental monitoring of a system via system-
entanglement. Environmental monitoring has three main consequences, a) the
suppression of interference effects at the level of the system, b) the selection of
quasi-classical preferred states, and c) the robust and redundant encoding of
information about the preferred states in the environment.
2.6.1.1. Stochastic Difference Equations
At each time, the solution to a stochastic difference equation is a random
variable, which changes variable at each time step (not different random variables
at each time points). y0, y1, y2, ... (noise increments) are random variables (x(0)
may or may not be a random variable), and so x(∆t), x(2∆t), x(3∆t), x(4∆t) are
random variables.
68
Solution of a stochastic Difference equation: calculating the probability
density functions of x(tn) (from the probability densities for all noise increments
(yns) and x(0) ).
2.6.1.2. Stochastic Differential Equation (SDE)
Stochastic Differential Equations are obtained by taking the ∆t → 0 limit of
stochastic difference equations. The solution of a stochastic differential equation is
a probability density for the value of x at all future ts. A process that fluctuates
randomly in time is called a stochastic process. SDEs are thus driven by stochastic
processes. The random increments that drive an SDE are referred to as noise (“the
noise driven”). A set of values of the random increments (values sampled from the
probability density) is called a “realization of the noise”. A particular evolution for
x, given a specific noise realization is called “a sample path for x”.
The full solution to SDE: complete set of all possible sample paths and their
probabilities. Simpler solution: Probability density for x at each time, correlation
of x at one time with another.
Stochastic integral are defined as,
∫T
W (T ) = dW, dW are called “Wiener process” (2.68)
0
The continuum limit of the Gaussian increments, dW, are referred to as
being infinitesimal. A general SDE for a single variable x(t), is,
dx = f(x, t)dt+ g(x, t)dW (2.69)
69
2.6.2. Quantum Trajectories
We take the ideology of ref. [112], in defining a quantum trajectory as
the path taken by the conditional state of a quantum system for which the
unconditioned system evolves continuously. The unconditioned state is the one
obtained by averaging over the random measurement results which condition the
system. Quantum jumps are discontinuous conditioned evolution. Quantum jumps
can be related to photon-counting measurements on the bath.
The evolution of an isolated quantum system in the absence of measurement
is
|ψ(t+ T )〉 = U(T )|ψ(t)〉 = exp(−iHT )|ψ(t)〉 (2.70)
We find the finite differential,
|ψ(t+ τ)〉 − |ψ(t)〉
lim = −iH(t)|ψ(t)〉 = finite (2.71)
τ→0 τ
which governs the continuous evolution. The unconditioned state is obtained
by averaging over all the possible measurement results. For a system we are
monitoring, the differential of the density matrix,
ρ(t+ τ)− ρ(t)
lim = ρ̇(t) (2.72)
τ→0 τ
With T taken to be infinitesimal, the state matrix at time t+dt, averaging
over all possible results,
∑
ρ(t+ dt) = J [M̂r(dt)]ρ(t)
r
70
where, ρ(t + τ) = Lτ [ρ(t)] = ρ(t) + O(τ). O(τ) is a first-order contribution in
τ . One of the operators, Mµ should be of order of unity. We can write it,
M0 = 1− iKτ + O(τ2)
K can also be put in terms of its Hermitian and antiHermitian parts, H and J
respectively.
K +K† K −K†
H = ~ , J = i
2 2
H
K = − iJ
~
C∑learly, if M0 = 1 − Jdt − iH/~dt was the only measurement operator, enforcing
M †r r (dt)Mr(dt) = 1 would require, M
†
0(dt)M0(dt) = 1 − 2Jdt 6= 1 or J = 0 and
the measurement operator would only be an unitary operator (the irreversibility
would not be facilitated). The minimum number of measurement operators are 2
and we must define,
√
M1(dt) = dtc
c†1c1 = 2J
M †M +M †0 0 1M1 = 1
71
In fact, we can add NK − 1 measurement operators, Mµ to the set instead of
just 1 defined as,
√
∑ Mµ∑(dt) = dtcµNK−1
M †µMµ = 1− 2Jdt+ c†µcµ = 1
µ=0 µ=6 0
⇒ 1
∑
= J = c†µcµ2
µ6=0
To the first order in dt,
† − iM0(dt)ρAM0(dt) = ρA [H, ρA]dt− (Jρ~ A
+ ρAJ)dt
i
= ρA − [H −(i~J, ρ~ A† † )
]dt
∑ for µ∑> (0, Mµ(dt)ρAMµ(dt) = cµρAcµNK NK )dt
† − i 1 1MµρAMµ = [HA, ρA]dt+ c † †µρAcµ − cµcµρA − ρ c†A µcµ dt (2.73)~ 2 2
µ=0 µ=1
2.6.3. Stochastic evolution
We now confine the number of our measurement operators to 2. The
probability for the result r=1 is,
[ ]
P1(dt) = Tr [J [M1(dt)]ρ] = dtTr c†cρ (2.74)
Provided that c†c is bounded, P0(dt) = 1 − P1(dt) = 1 − O(dt) is much larger
than P0(dt). Clearly, r=0 is the most likely measurement result each infinitesimal
time, dt. In these majority of dt segments with outcome r=0, the system state
changes infinitesimally (and not unitarily) with the operator, M0(dt) (not a
unitary operator); we call these segments null results. And at random times,
72
happening at a rate of P0(dt)/dt, we get the r=1 result, that we call a detection.
The evolution for these dt segments happen with M1(dt). We call these changes “a
quantum jump”. These discontinuous changes take place in the knowledge of the
observer and not in the physical state of the system.
In a simulation of an experiment where emitted photons from a cavity are
registered with photodetectors, the conditioned state can be thought of as the
state the system was in at the time of emission, since there may be delays from the
emission to detection.
Let us assume, the system state at time t is a pure state, |ψ(t)〉, and the
number of photodetections up to time t is N(t). Then,
dN(t)2 = dN(t) It is either 0 or 1
E[dN(t)] = 〈M †1(dt)M1(dt)〉 = dt〈ψ(t)|c†c|ψ(t)〉
For the dt segment, dN(t)=1, the state ket changes to,
| 〉 √ M1(dt)|ψ(t)〉 √c|ψ(t)〉ψ1(t+ dt) = = (2.75)
〈M † †1(dt)M 〈c c〉(dt)1(dt)〉(t)
For the dt segment, dN(t)=0, the state ket changes to,
|ψ0(t+ dt)〉 = √ { [ ]}M0(dt)|ψ(t)〉 − 1 1= 1̂ dt iH + c†c− 〈c†c〉(t) |ψ(t)〉 (2.76)
〈 † 2 2M0(dt)M0(dt)〉(t)
73
Combining them, we get the Stochastic Nonlinear Schrödinger Equation (SSE),
[ {
d|ψ(t)(〉 = dN(t) √
}
c
〈c†c〉(d
† )
− 1
t)]
− 〈c c〉(dt) − c
†c
+(1 dN(t))dt − iH |ψ(t)〉 (2.77)
2 2
For the general case of mixed/entangled state matrix,
π̂(t) = |ψ(t)〉〈ψ(t)|
{ dπ̂(t) = |dψ(t)[〉〈ψ(t)|+ |ψ](}t)〉〈dψ(t)|+ |dψ(t)〉〈dψ(t)|
1 r̂ρr̂†
dπ̂(t) = dN(t)G[ĉ]− dtH iĤ + c†c π̂(t)G[r̂]ρ = − ρ
2 Tr[r̂ρr̂†]
H[r̂]ρ = r̂ρ+ ρr̂† − Tr[r̂ρ+ ρr̂†]ρ (2.78)
Then, we define,
[ ρ(t) = E[π̂(t)] ]
E [dN(t)g(π̂(t))] = dtE Tr[π(t)c†c]g (π̂(t)) (2.79)
The identity eq. (2.79) can be used to find,
[ ]
dρ = −idt Ĥ, ρ + dtD[ĉ]ρ (2.80)
74
2.7. Measurement Methods
2.7.1. Direct Photodetection
The unitary operator representing the interaction between the system and
the bath in the infinitesimal interval [t, t+ dt) is
Û(t+ dt, t) = exp[ĉdB̂† − ĉ†dB̂ − iĤdt]
dB̂ = dB̂z:=−t
The noise operators are all defined in the interaction frame, as is the Hamiltonian.
The entangled system-bath state after the dt segment [t, tdt) is Û(t+ dt, t)|0〉|ψ(t)〉.
Since, dB̂|0〉 = 0. we need only keep the non-normally ordered second-order terms,
dB̂dB̂† = dt,
[ ]
Û(t+ dt, t)|0〉|ψ(t)〉 = 1̂− dtc†c/2− iĤ |0〉|ψ(t)〉+ dB̂†|0〉c|ψ(t)〉
dB̂†|0〉 is a bath state containing one photon. However, it is non-normalized one-
photon state with a norm of 〈0|dB̂dB̂†|0〉 = dt. The probability of finding one
photon in the bath is
〈0|dB̂dB̂†|0〉〈ψ(t)|c†c|ψ(t)〉 = P1(dt)
P1 was defined in (2.74). Eq. (2.75) gives the system state conditioned on the
bath containing a photon. The probability of finding no photons in the bath is
P0(dt) = 1− P1(dt), and the conditioned system state is given in eq. (2.76).
75
The field state at the beginning of the following interval [t + dt, t + 2dt) is
a vacuum state; it is a vacuum state for a new field operator, which comes from
the part of the field which has been initiated to interact with the system while the
previous part has moved on to be detected.
2.7.2. Homodyne Detection
Eq. (2.80) is invariant under the transformations,
( c→ c+ γ)
Ĥ → Ĥ − i γ∗c− γc†
2
Under the transformation, the measurement operators transform to,
[ √M1(dt) = dt(ĉ+ γ)]
M0(dt) = 1̂−
1 1
dt iĤ + (ĉγ∗ − c†γ) + (c† + γ∗)(ĉ+ γ)
2 2
Therefore, a deterministic master-equation can be unravelled into multiple
stochastic quantum trajectories. Physically, the above transformation can be
realized by homodyne detection. A Beam-Splitter (BS) of transmittance η
performs the transformations below.
√
b̂→ √ηb̂+
√ √1− ηô
ô→ ηb̂− 1− ηô
The strong local oscillator of a coherent field (with the same frequency as the
system dipole), ô is reflected into the path of the transmitted beam itself. The
76
√ √
mode ô can be modelled as, ô = γ/ 1− η + ν̂, γ/ 1− η; the coherent amplitude,
√
γ/ 1− η (|γ|2/(1 − η) being its photon flux) comes in accompanied with vacuum
fluctuations, ν̂.
[ν̂(t), ν̂†(t′)] = δ(t− t′)
The transformation and measurement operators talked about before can be
achieved with η → 1 with the transforation,
b̂→ b̂+ γ
The x-quadrature of the system dipole can be measured from the rate of
photodetections if we take γ to be real.
[ ] x̂ =[ĉ+ ĉ
†, ŷ = −i(ĉ− ĉ†
d ] )
E N(t) = Tr (γ2 + γx̂+ ĉ†ĉ)ρI(t) (2.81)
dt
For γ >> 〈c†c〉, eq. (2.81) has the inside the trace a large constant altered by a
small term and a term proportional to x̂.
The differential in conditioned state and state matrix equations are found as
(with the new measurement operators):
{ [ ]}
1
dρI(t) = dN(t)G[[ĉ+ γ] +{dtH −Ĥ − γĉ− c†c ρI(}t) (2.82)2
ĉ+ γ
( d|ψI(t)〉 = dN(t) √ − 1〈(c† + γ)(c+ γ)〉I(dt)
〈ĉ†ĉ〉 (dt)− ĉ†
)]
I ĉ
†
− 〈ĉ γ + γĉ〉I(t)+dt − γĉ− iĤ |ψ(t)〉 (2.83)
2 2
77
2.8. Excitation of a Two-level System
If we only considered the evolution of a pure state, we could expand it
|ψ(t)〉 = ab(t)|ψR, 0〉+ a1(t)||vac〉R, 1〉 (2.84)
[ ]
and evolve it with the Hamiltonian, √H = i γ1 ĉb̂† †in(t)− ĉ b̂in(t) (which
conserves the number of excitations and the expansion (eq. (2.84) ) is sufficient)
if |ψ(0)〉 = |ψR, 0〉. If we restrict the problem to a single excitation, the Bosonic
annihilation operator of a quantum harmonic oscillator, ĉ can be used in place
of |F0〉〈F1|. One of the basis vector has the excitation in the b-field (amplitude
ab(t) ) and the other has it in the absorbe∫r atom (amplitude a1(t)). Remembering
[ĉ, b̂in] = 0, but [ĉ, b̂†in] 6= 0, and |ψR〉 = dωuα(ω)b†(ω)|vac〉; we can calculate
the Heisenberg Langevin equation of motion for ĉ. If we opted for the Schrödinger
picture, we would find a differential equation for the quantum amplitude, a1(t).
d γ1 √
ĉ(t) = − ĉ(t)− γ1b̂in(t) (2.85)
dt 2
d γ1 √
a1(t) = − a1(t)− γ1uα(t) (2.86)
dt 2
γ
So, the quantity √ 1T (t) = γ e− t1 2 Θ(t) for the two level system are Green’s
functions for the first order differential equation (2.88), (2.85) or (2.86). Another
interpretation is that they are the decaying wavepacket shape leaked into a bath
mode if the atom is initiated in the final state and is allowed to spontaneously
relax [1] (the process of absorption in reverse). These Green’s functions, T(t)
play an important role in the mathematical structure of the problem and also
help identify the physical criteria for efficient excitations. For example, ref. [138]
78
concludes with a Wigner-Weisskopf theory that the input photon needs the time
reversed properties of a spontaneously emitted photon for efficient excitation of a
two level atom. It is quite easy to see from the solution of eq. (2.86)
{√ }− γ− 1a t1(t) = γ1e 2 Θ(t) ∗ uα(t). (2.87)
ρ11,11(t) (same as |a 21(t)| calculated from eq. (2.87) ) and a1(t) in eq. (2.87) are
both maximized if uα(t) has the time reversed shape of the Green’s function T (t)
(spontaneous emission waveshape) because of the convolution between them. The
reciprocity of optimal storage and retrieval holds for a three level Λ atom like ours
as well according to [139]. In the two level absorber the development of the excited
state population is hindered by amplification backaction and for optimal photon
absorber the three level Λ atom is required [64].
For the absortption in the quantum harmonic oscillator’s lowest two states
with one excitation, we have from the scattering theory,
∫
ˆ̃ 1bin/out(ω) = √ dtb̂in/out(t)eiωt
2π
ˆ̃bout(ω) = R(ω)
ˆ̃bin(ω)
ĉ(ω) = −T (ω)ˆ̃bin(ω).
√
γ1
T (ω) =
−iω + γ1
2
−iω − γ1
R(ω) = 2
−iω + γ1
2
And we deduce that in frequency domain, the functions, T (ω) would serve as a
transmission coefficient. In the absence of cubic and higher order terms of the
annihilation operator in the Hamiltonian, the Heisenberg-Langevin operator
79
equation of motion is that of a damped harmonic oscillator. Therefore, it is
the Green’s function for a damped quantum Harmonic oscillator amplitude. It
is an equation of motion shared by the excited state quantum amplitude, the
annihilation operator and the coherence of the generalized density matrix, ρα,0.
We are also able to relate the quantum amplitude of the excited state with
the transmission coefficient and input wavepacket.
a1(t) = −uα(t) ∗ T (t)
1
a1(ω) = −√ uα(ω)T (ω)
2π
The damped quantum harmonic oscillator equation also arises if the density
matrix formalism was used [64],
d γ1 √
ρ01(t) = − ρ01(t)− γ1uα(t) (2.88)
dt 2
80
CHAPTER III
THE HEISENBERG PICTURE OF PHOTODETECTION
I have no satisfaction in formulas
unless I feel their numerical
magnitude.
Lord Kelvin
We construct a class of Hamiltonians that describe the photodetection
process from beginning to end. Our Hamiltonians describe the creation of a
photon, how the photon travels to an absorber (such as a molecule), how the
molecule absorbs the photon, and how the molecule after irreversibly changing
its configuration triggers an amplification process—at a wavelength that may be
very different from the photon’s wavelength—thus producing a macroscopic signal.
We use a simple prototype Hamiltonian to describe the single-photon detection
process analytically in the Heisenberg picture, which neatly separates desirable
from undesirable effects. Extensions to more complicated Hamiltonians are pointed
out.
3.1. Introduction
We may distinguish two traditional types of photodetection theory. The first
tries to determine what quantum field observable is measured when photoelectrons
are produced by photoabsorption and the photoelectrons are subsequently
detected and/or counted. This sort of theory is exemplified by the classic papers
Refs. [80, 81], which take as starting point the Hamiltonian describing the
interaction between photons and an electric dipole, but which do not describe
81
the remainder of the detection process quantum mechanically. The second type
of theory models actual photodetectors phenomenologically, taking great care to
model the many mechanisms involved in converting the initial photon energy to
the final macroscopic current. This type of theory is exemplified by recent work on
the superconducting nanowire detector [140, 141, 142].
Neither of the above types of photodetection theory establishes fundamental
limits of photodetection, that is, platform-independent limitations arising from the
laws of physics on, e.g., single-photon detection efficiency, dark count rates, time
and wavelength resolution and tradeoffs between these figures-of-merit.
For just this purpose—finding fundamental limits on photodetectors—a third
type of theory has been developed in recent years. Here the aim is to develop fully
quantum-mechanical and sufficiently realistic models that include all stages of the
photodetection process, including the crucial amplification process [1, 62, 64, 65,
66, 68, 71, 72]. The point of this paper is to continue this recent work and present
a quantum description of the processes involved in the detection of a single photon,
especially the connection between the photoabsorption and amplification processes.
Moreover, we perform our calculations in the Heisenberg picture. That picture may
not be the most intuitive—it may be easier to follow the trajectory of an excitation
through the system in the Schrödinger picture—but it does have its merits. We
mention two reasons here to use this picture.
First, lower limits on noise accompanying quantum amplification are most
easily derived in the Heisenberg picture. Caves [143] studied linear amplification
of electricmagnetic (EM) field amplitudes and formulated the problem in terms of
the Heisenberg evolution of the annihilation operator of a single discrete EM field
82
mode of the schematic form
√
aout = Gain +Nin, (3.1)
with G > 1 the gain factor and Nin a noise term. The left-hand side here
represents the annihilation operator for the mode to be amplified at the end of
the amplification process, the operators on the right-hand side represent input
operators, i.e., initial values just before the amplification process starts. Ideally,
the number of excitations in the output equals the number of input excitations
multiplied by G, and this would be the case if it were not for the noise term Nin.
The commutator [a, a†] = 1 has to be preserved, i.e., at any time t we must
have [a(t), a†(t)] = 1 for the Heisenberg-picture operators a(t) and a†(t). This
puts a restriction on the noise operator Nin. In particular, it cannot be zero. For
example, phase-insensitive amplification is obtained by setting
√
N †in = G− 1bin, (3.2)
in terms of the creation operator of an additional discrete bosonic mode b. As is
easily verified, the addition of that noise term preserves the commutator. Thermal
excitations in the additional mode b are amplified, too, by a factor of G − 1; and
even if mode b starts in the vacuum state (at zero temperature) the fact that it
is the mode’s creation operator appearing here still causes noise. We refer to the
lower limit on noise reached here as the Caves limit on linear amplification.
Second, in a recent paper [66] we showed that the first part of the photo
detection process (the part preceding amplification, including absorption of
the incident photon) can be described compactly in the Heisenberg picture as
83
well. The input-output relations for the annihilation operators (which now are
continuous-mode operators indexed by frequencies ω) consist of two clearly
distinct terms, one desired, the other undesired but inevitable so as to satisfy the
commutator [aout(ω), a† ′ ′out(ω )] = δ(ω − ω ). We can write
aout(ω) = T (ω)ain(ω) + Ñin(ω), (3.3)
where T (ω) is a complex transmission amplitude, with the physical meaning that
a photon with frequency ω will survive the pre-amplification stage with probability
|T (ω)|2 (we will encounter this interpretation in Eqs. (4.28–3.38)). Here the noise
term is of the form
Ñin(ω) = R(ω)cin(ω), (3.4)
with cin(ω) the annihilation operators for external bosonic modes at frequency ω,
and
|R(ω)|2 + |T (ω)|2 = 1 (3.5)
so as to preserve the commutator [a † ′out(ω), aout(ω )]. Once again, thermal
excitations at arbitrary frequencies ω in the mode cin(ω) contribute noise as soon
as R(ω) 6= 0.
There are several reasons for wishing to describe the whole photo-detection
process with one Hamiltonian. First, although in a recent paper [1] it was
shown that one can write down commutator-preserving (nonlinear) amplification
relations that beat the above-mentioned Caves limit, no explicit Hamiltonians
84
were considered there that may reach that improved limit. Here, we reach the
same improved limit, but in a new way and with a (fairly simple) Hamiltonian.
Second, that same paper also noticed how one can formally express the idea that
one can amplify at a frequency that differs substantially from the incoming photon
frequency. We show here explicitly how that idea can be implemented, quite
straightforwardly, by a Hamiltonian.
3.2. A class of model Hamiltonians
3.2.1. Description
We wish to represent the whole photo-detection process (including absorption
of the photon and amplification) plus the generation of the photon to be detected,
by a Hamiltonian. We start with what seems to be a minimal model (various
possible extensions of the model are discussed in the concluding Section). There
are 6 quantum systems in total; we have 3 discrete quantum systems a, F , and
c (with small Hilbert-space dimensions, which generate the photon, absorb the
photon, and amplify the signal, respectively) and 3 continuous-mode quantum
systems that connect the discrete systems and that are used to model irreversible
processes (see Fig. 1).
The continuous modes are modeled by bosonic mode operators
b(ωb), d(ωd), g(ωg) with ωi = cki proportional to the wave number ki (using just
1 spatial dimension, the x-axis) of the bosonic excitations of type i = b, d, g.
When there is no confusion possible, e.g., when we integrate over all frequencies ωi,
we will use the symbol ω without subscript to denote those. Positive frequencies
ω > 0 describe waves traveling from left to right (towards positive x), ω < 0 waves
traveling from right to left.
85
1
0
a F
c
FIGURE 3.1. From a single photon to a macroscopic signal:
a cavity a which contains one excitation generates one single-photon wavepacket
(in a continuum b(ω)). That photon is (resonantly) absorbed by a molecule F .
The molecule may decay back to its initial state |F0〉 or it may decay to a different
state |F2〉 by emitting a different photon (in a continuum g(ω)) that escapes. In
the state |F2〉 the molecule’s shape and/or dipole moment have changed. That
physical change triggers an amplification process in another system c, which
eventually reaches a steady state in which spontaneous decay is balanced by a
“classical” drive, thus producing a stream of fluorescence photons (in a continuum
d(ω)) that a classical (human) observer can observe. The final macroscopic signal
may be at a (very) different wavelength than that of the single photon.
System a is a cavity that contains a single excitation that leaks out into the
continuum described by b(ω), and this is the single photon wave packet that we
intend to detect. That single excitation couples to to first leg of a three-level Λ
system—a “molecule” which we denote by F because it is the driving force behind
86
amplification. The photon may excite the molecule from its initial state, the lower
level |F0〉, to an excited state |F1〉. That level could decay back to |F0〉 or it could
decay to the state |F2〉 through coupling to the continuum g(ω) [the excitation in
g thus produced is assumed to escape; this is an irreversible transition]. When the
molecule F is in the state |F2〉 the accompanying physical change in the molecule
triggers an amplification process in system c.
This aspect of the model mimics the mechanism used in the human eye: a
retinal molecule changes its configuration from cis to trans, and that change of
shape in turn induces a conformational change in the protein the retinal binds to.
Further changes in shapes of proteins then finally lead to a change in the charge
distribution, that then can generate an electric signal (see [144]). This idea can be
exploited in a bioinspired photodetector [36, 145] where a chromophore molecule
changes its shape upon absorbing a single photon and thereby changes its dipole
moment, which then affects a (macroscopic) current.
Finally, the many excitations generated by c leak out to the continuum
mode d(ω). The macroscopic signal present in continuum d(ω) is what we then
(classically) observe (see Fig. 1).
3.2.2. Hamiltonian
We follow here the ideas of Gardiner [116, 117] (see also [114]) for describing
how the output of one quantum system may serve as the input for the next
quantum system, without the latter acting back on the first system. This is done
simply by setting the coupling to the right-to-left traveling waves equal to zero.
That is, we only need the positive frequencies here. The (electric) field operators
describing fields that travel from left to right corresponding to the modes b, d,
87
and g are denoted by corresponding capital letters B(x), D(x) and G(x), and are
expanded as
∫
1 ∞
B(x, t) = √ ∫ dω b(ω, t) exp(iωx/c),2π 0
1 ∞
D(x, t) = √ ∫ dω d(ω, t) exp(iωx/c),2π 0
1 ∞
G(x, t) = √ dω g(ω, t) exp(iωx/c). (3.6)
2π 0
It is through these field operators together with their hermitian conjugates
B†(x), D†(x) and G†(x) that the discrete systems interact (at their respective
locations on the x axis) with the continuous modes, where we will make both
Markov and rotating-wave approximations (RWA), as detailed below.
The Hamiltonian is of the following form
H = Ha +Ha−b +Hb
+Hb−F +HF +HF−g +Hg
+HF−c +Hc
+Hc−d +Hd. (3.7)
System a is a single cavity mode with resonance frequency ωa (which, in all
generality, would be time-dependent), whose Hamiltonian we write as (setting
~ = 1 everywhere)
Ha = ω a
†
a a. (3.8)
The cavity mode is located at x = −cτ with τ the time delay between a signal
(a photon) leaving system a and interacting with system F (which we take to be
88
located at x = 0). The cavity mode is coupled to the field B(x = −cτ, t) like so:
√
Ha−b = i κ[aB
†(−cτ, t)−B(−cτ, t)a†], (3.9)
with the field B described by the Hamiltonian (recall we leave out negative
frequencies since they do not couple to the systems of interest)
∫ ∞
Hb = dω ωb
†(ω)b(ω). (3.10)
0
We can anticipate that the main terms contributing to the interaction are those
at ω ≈ ωa. That is, in the Heisenberg picture a(t) ∼ exp(−iωat) and b†(ω, t) ∼
exp(+iωt) so that the main terms not averaging to zero over time come from ω ≈
ωa. In numerical simulations, we always transform to a rotating frame, i.e., we
solve equations for the slowly-varying operators exp(iωat)a(t) rather than a(t),
etcetera.
The next line of (4.37) contains four Hamiltonians that describe how the Λ
system F interacts with two different continua (namely, b and g) at position x = 0.
The four terms are written as
√
Hb−F = i∑γ1[|F0〉 〈F1|B†(x = 0, t)−B(x = 0, t) |F1〉 〈F0|]
HF = ωk|Fk〉〈Fk|
k=0,1,2
√
HF−g = i∫ γ2[|F1〉 〈F2|G†(x = 0, t)−G(x = 0, t) |F2〉 〈F1|]∞
Hg = dω ωg
†(ω)g(ω) (3.11)
0
and we again can anticipate that the most important terms are those with ω ≈
ω1−ω0 := ω10 for the interaction between F and b(ω) and for ω ≈ (ω1−ω2) for the
89
interaction between g(ω) and F . An important parameter is the detuning of the
photon from resonance with the molecular transition from |F0〉 to |F1〉,
δ = ωa − ω10. (3.12)
The third line of (4.37) is going to be crucial as it models the amplification
process and how the F system triggers it. We construct the two Hamiltonians
HF−c and Hc in several steps. First, assume we have a system c that is driven
by an external “force” F . That is, we assume the Hamiltonian for system c
contains a driving term proportional to a parameter F . For example, we may use
a Hamiltonian that describes electron shelving [23, 146], which is used as a method
to perform quantum state measurements on ions: in one state the ion, driven by
a laser, produces large amounts of fluorescence whereas in another state (from
which there is no transition resonant with the laser) it remains dark. The simple
Hamiltonian is [in Section 4.6 below we suggest several more involved examples of
suitable Hamiltonians]:
H̃c(F ) = 2iF cos(ω
†
F t)(c− c ), (3.13)
which in the RWA becomes
H̃c(F ) ≈ iF (exp(iωF t)c− exp(−iωF t)c†). (3.14)
Alternatively, we could introduce yet another pair of bosonic mode operators α
and α† to replace F exp(−iωF t) and F exp(iωF t), respectively, and add another
Hamiltonian Hα = ωFα†α; the initial state of that mode would then be a coherent
90
state with amplitude F , i.e., an eigenstate of α with eigenvalue F (which indeed
can model a strong laser field). This would have the formal advantage of making
our Hamiltonian completely independent of time.
Now in order to couple system c to our quantum system F we replace the
parameter F by the quantum operator
∑
F̃ = Fk|Fk〉〈Fk|, (3.15)
k=0,1,2
so that we replace
H̃c(F ) 7→ H̃c(F̃ ). (3.16)
Note we may apply this substitution trick to any Hamiltonian H̃c(F ) that models
amplification. For the simple example (3.14) we have
H̃F−c = iF̃ (exp(iωF t)c− exp(iωF t)c†). (3.17)
For H̃c we assume the simple form
H̃ †c = ωcc c, (3.18)
appropriate for a bosonic mode c (but we could also use a two-level atom). An
interaction proportional to a projector |F2〉〈F2|, implements the idea (mentioned
above) that it is a physical property of the state |F2〉 that triggers amplification.
For the RWA to apply all we need is that the driving frequency ωF be close to
the frequency ωc, independent of the frequencies ωk for the three states of system
F and independent of ωa, the frequency of the photon to be detected. And so
amplification happens at the frequency ωF , not at ωa. It may be important
91
to be able to amplify at a different frequency so as to suppress thermal noise
(which could lead to dark counts) by amplifying at a frequency ωF such that
(reinserting ~) ~ωF  kT . Pre-amplification thermal excitations at a frequency
ωa [either thermal photons in the input mode a or thermal fluctuations exciting
the transition to level |F1〉] are amplified, and so, too, should be suppressed by
operating at a temperature T such that ~ωa  kT . (For example, this is how our
eyes can detect optical photons at room temperature.)
We may move to a frame rotating at frequency ωF and replace our first
guesses for Hamiltonians H̃F−c and H̃c by the final results
H †F−c = iF̃ (c− c ),
H †c = ∆c c, (3.19)
with ∆ = ωc − ωF the detuning from resonance.
Finally, for the fourth line in (4.37) we stay in the same rotating frame and
write
√
Hc−d = ∫i Γ[cD†(x = 0, t)−D(x = 0, t)c†],∞
Hd = dω (ω − ω †F )d (ω)d(ω) (3.20)
0
3.3. Heisenberg equations of motion
3.3.1. Eliminating the continua
For any operator O that does not explicitly depend on time, we have the
equation of motion
d
O = i[H,O] (3.21)
dt
92
with H the total Hamiltonian (4.37). We choose a time t0 in the past at which we
start the calculation (i.e., we solve the equations for t > t0), and at which time the
Heisenberg and Schrödinger picture operators are taken as equal. Those operators
at that special time are our input operators and thus are also indicated by the
subscript “in.”
We first formally solve the equations for the continuum operators b(ω), g(ω),
and d(ω) and substitute those results into the equations of motion for arbitrary
operators acting on the discrete quantum systems a, F and/or c, thus eliminating
the continua from the description. For example, starting at the end, with modes
d(ω) and the field operator D(x), we obtain [117]
√
D(x = 0, t) = din(t) + Γc(t) (3.22)
with the “free field” given by
∫
1 ∞
din(t) = √ dω d0(ω) exp(−i(ω − ωF )(t− t0)). (3.23)
2π 0
The operator d0(ω) := d(ω, t0) is an initial value for d(ω, t) at time t = t0.
For the field operator G(x) we similarly obtain
√
G(x = 0, t) = gin(t) + γ2 |F1〉 〈F2| (t) (3.24)
with the free field
∫
1 ∞
gin(t) = √ dω g0(ω) exp(−iω(t− t0)). (3.25)
2π 0
93
For the field B(x) which couples both to a and to F we find that at x = 0 it
contains two driving terms
√ √
B(x = 0, t) = bin(t) + κa(t− τ) + γ1 |F0〉 〈F1| (t) (3.26)
with the free field
∫
1 ∞
bin(t) = √ dω b0(ω) exp(−iω(t− t0)). (3.27)
2π 0
At the location x = −cτ of the cavity we get just one driving term
√
B(x = −cτ, t) = bin(t) + κa(t). (3.28)
We can now write down the equations of motion for the operators corresponding to
the three discrete quantum systems. For example, for the cavity mode annihilation
operator a(t) we get
d √ √
a = − 1iωaa− κ[bin + κa]. (3.29)
dt 2 ∫
(The “extra” factor of 1/2 on the r.h.s. comes from the use of t dt′ δ(t − t′)a(t′) =
t0
∫1a(t), where the delta function inside the integral comes from the approximation2
dω exp(−iω(t− t′)) = 2πδ(t− t′) [117]. Instances of the same factor of 1/2 appear
in several equations below.)
We can solve Eq. (3.29) to obtain
a(t) = a(t0) e∫xp[(−iωa − κ/2)(t− t0)]√ t
− κ dt′ exp[(iωa + κ/2)(t′ − t)]bin(t′). (3.30)
t0
94
For the evolution of c we find
d √ 1√
c = −i∆c− Γ(din + Γc)− F̃ . (3.31)
dt 2
The equation for system F and hence for F̃ is more complicated and in general has
to be solved numerically. We can formally solve (3.31)
c(t) = c(t∫0) exp[(−i∆− Γ/2)(t− t0)]t √
− dt′ exp[(i∆ + Γ/2)(t′ − t)][ Γdin(t′) + F̃ (t′)].
t0
(3.32)
and this is an explicit solution provided we can ignore the backaction of system
c on system F , i.e., when the operator F̃ (t) does not depend on “downstream”
system c operators, but only on “upstream” system a operators.
3.3.2. Steady-state solutions
Our Hamiltonian is such that the F system will reach a steady state, The
reason is that the force driving system F is the single photon emitted by the
cavity, and that photon will have disappeared after a few cavity life times κ−1.
The F system then decays to either the |F0〉 state or to the |F2〉 state, and stays
there. The operator F̃ will eventually become constant, apart from fluctuating
noise (Langevin) terms. Eq. (3.32) then shows that the operator c will reach a
steady state, too (all transient effects decay away at a rate Γ), up to noise terms.
We now focus on terms in F̃ proportional to |F0〉〈F0| only. (These terms
describe the response of molecule F when it starts from state |F0〉, where it is
supposed to start. Other nonzero terms are discussed in Section 4.5.) We also
95
assume δ = 0 for the moment (this is the optimum case, of course, for detecting
the photon). Moreover, we set F0 = F1 = 0 and F2 =: µ > 0, so that the
molecule triggers amplification only in the state |F2〉. In that case we simply have
F̃ = µ|F2〉〈F2|, and we find its steady-state value to be
F̃ †ss = µPabsa a⊗ |F0〉〈F0|, (3.33)
where
4γ1γ2
Pabs = (3.34)
(γ1 + γ2)(γ1 + γ2 + κ)
is the probability that the photon transfers the population from the initial state
|F0〉 to |F2〉. This probability is maximized for γ2 21 = γ2 + κγ2, and for κ  γ1,2
this maximum approaches unity arbitrarily closely. This result confirms the “ideal
detection” result of Ref. [64].
We can in fact generalize this result to arbitrary detuning δ. Apart from
obtaining the answer by replacing κ by κ− 2iδ and taking the real part,
[ ]
4γ1γ2
Pabs = Re (3.35)
(γ1 + γ2)(γ1 + γ2 + κ− 2iδ)
we can write the result more insightfully in the form
∫
Pabs = dω |φ(ω)|2|T (ω)|2 (3.36)
where
√
γ1γ2
T (ω) = (3.37)
(γ1 + γ2)/2− i(ω − ω10)
96
is the transmission coefficient describing the transmission of a single excitation
through the Λ system (which for a single excitation is equivalent to a Fabry-Perot
filter cavity) [63, 66] and where
√
1 κ
φ(ω) = √ (3.38)
2π κ/2− i(ω − ωa)
is the (properly normalized) spectral shape of the photon produced by the cavity.
This way of writing the probability can be generalized to other systems than
a three-level molecule by substituting other transmission functions T (ω) that
describe the initial (absorption) stage of the photodetection process, as discussed
in great detail in Ref. [66].
When the system reaches its steady state, the expression for c(t) becomes
css(t) = d̃(t)−
F̃ss
, (3.39)
Γ/2 + i∆
where d̃(t) is a single-mode “noise” annihilation operator given by (for large t, i.e.,
t− t0  1/Γ)
∫ t √
d̃(t) = − dt′ exp[(i∆ + Γ/2)(t′ − t)] Γd ′in(t ) (3.40)
t0
One can verify that
[css(t), c
†
ss(t)] = 1 (3.41)
thanks purely to the noise term.
What we observe in the end is a macroscopic amount of excitations in the
continuum mode d(ω), or, equivalently, the field D(x = 0). The excitations are
97
collected over some finite time interval of duration T from T0 to T0 + T [much later
than t0] with some low efficiency η. We could assume that our signal is determined
by ∫ T0+T 〈 〉
SD(T ) = η dt D
†(x = 0, t)D(x = 0, t) =: ηND(T ), (3.42)
T0
which corresponds to collecting a fixed fraction η of all excitations in the field
D. We could also assume we collect data continuously as a function of T , by
continuously monitoring the field D. Leaving out the noise terms we can write
∫ T ∫ ∫0+T t t 〈 〉
N (T ) = Γ dt dτ dτ ′D F̃ (τ)F̃ (τ
′) ×
T0 t0 t0
exp ((−i∆ + Γ/2)(τ − t) + (i∆ + Γ/2)(τ ′ − t′)) .
(3.43)
〈 〉
Here the expectation value F̃ (τ)F̃ (τ ′) must be calculated using the Quantum
Regression Formula (or Theorem) [147].
Alternatively we could assume we collect a fraction of excitations in a
particular single discrete time-integrated mode
〈 〉
N ′D(T ) = η d
†
TdT (3.44)
where, for example, when ∆ = 0, we choose
∫
√1
T0+T
dT = dt D(x = 0, t), (3.45)
T T
〈 0 〉
which similarly would contain F̃ (τ)F̃ (τ ′) . For either choice, the signal grows
linearly with T once c reaches a steady state. We will focus on the former choice
98
in the numerical calculations, i.e., we assume that ND(T ) contains our macroscopic
signal.
Consider now the noise in our amplification process. We can write our
discrete mode operator dT (in the steady state) as
√
dT,out = Pabs G(a
†a)in ⊗ (|F0〉〈F0|)in + ẽin, (3.46)
where we explicitly added back in the subscripts “out” and “in” to indicate the
operators on the right-hand side are all input operators and the left-hand side is an
output operator. (Recall that we did leave out here other terms to be discussed in
4.5, given that the initial state of our molecule is |F0〉.) The gain factor G here—
which is the gain one gets if the molecule ends up in the desired state |F2〉—is
linear in T
4µ2
G = T, (3.47)
Γ
and ẽin is a single-mode discrete annihilation noise operator fully determined by
din:
∫
1 T0+T √
ẽin = √ dt ( Γd̃(t) + din(t)), (3.48)
T T0
since the operator d̃ is determined by din according to Eq. (3.40).
Note that (i) the noise is not amplified, (ii) the first (gain) term is hermitian
and, therefore, commutes with its hermitian conjugate, so that the presence
of ẽin is sufficient to preserve the commutator. Of course, our operator a†a is
restricted here to the very narrow range of 0 or 1 excitations [so that we can use
99
that (a†a)2 = a†a when we calculate either ND(T ) or d†TdT ]. The main point
is, Eq. (3.46) is not of the Caves form for linear amplification, but is, rather, a
nonlinear minimum-noise form that is akin to but different from the input-output
relation found in Ref. [1].
3.3.3. Numerical integration
Without noise terms, we can find the gain term and the steady-state values
numerically as well. If the Schrödinger-picture evolution equation for the density
operator can be formally solved as ρ(t) = exp(L(t − t0))ρ(t0) with L the time-
independent Liouvillian superoperator, then in the Heisenberg picture observable
O evolves as O(t) = exp(L†(t− t0))O(t0).
For the Heisenberg operator |F2〉〈F2|(t) we plot all nonzero terms (there are
five for δ = 0) as functions of time in Fig. 2. Here are the five types of terms with
their interpretations (where we ignore operators acting on the Hilbert space for the
“downstream” system c)
1. K1 = |1〉〈1| ⊗ |F0〉〈F0|: This term describes how an initial state with 1 cavity
excitation and the molecule starting in |F0〉 transfers the molecule to state
|F2〉 (blue curve).
2. K2 = |0〉〈0| ⊗ |F1〉〈F1|: This term describes how the molecule reaches state
|F2〉 even without a photon present provided it starts in the upper state |F1〉.
It decays to |F2〉 with probability 1/2, given that γ1 = γ2 here (green curve).
3. K3 = |1〉〈1| ⊗ |F1〉〈F1|: This term again corresponds to the molecule starting
in the upper state |F1〉, from which it decays to |F2〉 with probability 1/2.
Initially it behaves like the previous case. However, because of the presence
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of the photon, the molecule can also be transferred to the desired final state
|F2〉 by first decaying to |F0〉 and then absorbing the photon (orange curve).
4. K4 = 1 ⊗ |F2〉〈F2|: This term describes the trivial case where the molecule
starts in |F2〉 and just stays there, independent of the presence or absence of
a photon (dashed purple curve).
5. K5 = a ⊗ |F1〉 〈F |+a†0 ⊗ |F0〉 〈F1|: This term describes the influence of
coherence: if we start with a coherent superposition of no photon and 1
photon, and the molecule is in a superposition of |F0〉 and |F1〉, then the
contributions from |0〉 ⊗ |F1〉 and |1〉 ⊗ |F0〉 to the probabillity of ending up
in |F2〉 interfere destructively (dashed black/red curve). Moreover, this term
does contribute to the signal though the combination K†5K5, see main text.
(There is the similar sixth term, K6 = ia ⊗ |F1〉 〈F0|-ia† ⊗ |F0〉 〈F1| which is
nonzero only for nonzero detuning δ.)
We also plot the amplitude of system c and the total number of excitations
in the field D(x = 0) in Figs. 3 and 4. More precisely, we plot the time evolutions
of the terms in c(t) and ND(T ) = D†(x = 0)D(x = 0) proportional to |1〉〈1| ⊗
|F0〉〈F0| ⊗ |0〉〈0|, which correspond to the system starting in the initial state with
1 excitation in the cavity, the molecule in state |F0〉 and no excitations in mode
c. We choose T0 = t0 in the definition of ND(T ). Note that it is not just the first
term of the five terms we just discussed that contributes to ND(T ): the fifth term
contributes as well and so do the noise terms; together they ensure that ND(T )
scales linearly with Pabs (as it should), rather than quadratically. (Numerically, we
used the Quantum Regression Theorem to calculate ND(T ).)
101
FIGURE 3.2. The five types of nonzero terms in |F2〉〈F2|(t) as functions of time in
units of γ−11 ,
where δ = 0, γ2 = γ1 and κ = γ1/5. The probability Pabs for the photon to trigger
amplification is Pabs = 10/11 in this case. For details, see main text.
102
FIGURE 3.3. The term in c(t)(in units of µ/Γ) prop. to |1〉〈1| ⊗ |F2〉〈F2| ⊗ |0〉〈0|
which describes how the expectation value of the amplitude of our final quantum
system c grows with time (in units of γ−11 ), if the cavity starts with 1 excitation,
the molecule starts in |F0〉 and the mode c itself starts in the vacuum. Parameter
values are µ = Γ = γ2 = γ1, ∆ = δ = 0, and κ = γ1/5. The steady-state value of c
for these values is css = −20µ/11Γ.
103
200
180
160
140
120
100
80
60
40
20
0
0 10 20 30 40 50 60
T in units of 1/ 1
FIGURE 3.4. The term in ND(T ), given by Eq. (3.43), proportional to
|1〉〈1| ⊗ |F0〉〈F0| ⊗ |0〉〈0| as a function of the integration time T (in units of γ−11 ).
Parameter values are as in the previous Fig. This represents the macroscopic signal
produced by detection of a single photon, i.e., the expectation value of the number
of excitations in the field D(x = 0). This expectation value increases linearly with
T once the system has reached a steady state, after a few κ−1, with a slope given
by Pabs4µ2/Γ.
104
expansion coefficient
The behavior of our signal plotted here and the efficiency η with which that
macroscopic signal is measured determine when the photodetector can be reset.
Namely, as soon as a few excitations (in principle, even a single one, if we can
ignore dark counts) have been detected (which requires the number of excitations
to have been of order 1/η), a photon has been detected, and our detector may be
reset in order to be able to detect a next photon. Resetting involves emptying the
system c (which takes several Γ−1) and resetting the molecule to the state |F0〉
(which takes several γ−11 ).
If we include the Langevin terms we could solve the stochastic differential
equations by standard methods (using Ito calculus, for example [117]). Much
more simply, we could determine what the generic form of the noise terms must
be, by making use of the fact that commutators for our output operators like dT
should be preserved. For example, if the steady-state value of dT is dT,ss (which is
expressed in terms of the input operators of our discrete systems) and the noise
term in dT is ein (which is expressed in terms of continuum input operators and
which, therefore, commutes with d †T,ss and dT,ss), then we must have
[ein, e
†
in] = 1− [dT,ss, d
†
T,ss], (3.49)
which limits the operator form for ein severely.
3.4. Conclusions and outlook
We used a fairly simple Hamiltonian model to describe quantum
mechanically the photodetection process from beginning (generation and
absorption of a single photon) to end (amplification). We solved the equations
105
in the Heisenberg picture, because the end result compactly describes the essence
of the whole process including the noise therein (see Eq. (3.46), where we consider
the optimum case Pabs = 1):
√
dout = G(a
†a)in ⊗ |F0〉〈F0|+ ein. (3.50)
Here dout is the annihilation mode operator for a time-integrated mode that
contains the macroscopic output signal (a large number of excitations) that we
ultimately observe classically. We have to place the Heisenberg cut somewhere,
and we place it as far along the whole photodetection process as we can, after
amplification. Unlike for linear amplification [143] where the gain term would be
√
Gain, here the gain term indicates the amplification process is nonlinear [1].
The noise term (needed to preserve the commutator [d, d†] = 1) is just an
√
input mode operator, rather than G− 1e†, which is the noise term accompanying
phase-insensitive linear amplification. That is, the noise in our case is not
amplified, and a vacuum input gives zero noise, unlike for linear amplification.
The projector |F0〉〈F0| projects onto the initial state of a molecule that
triggers the amplification process once it has changed its configuration by
absorbing the photon. Confirming results of Refs. [64, 65] the probability of
absorption (and detection) can indeed be nearly 100%.
Several aspects of our minimal prototype Hamiltonian can be generalized
and/or extended:
(i) In order to detect more than a single excitation we could include more
levels in the F molecule (and more excitations either in the same cavity or in
multiple cavities, all coupled to the same system F ). For example, to be able
to detect a second photon we could introduce two more F levels, such that a
106
transition from |F2〉 to another upper level could occur (triggered by the second
photon), which then could decay to a level |F4〉 where the value of the parameter
F4 would be substantially larger than F2 = µ. This then would allow us
to distinguish the signal produced by the second photon from that produced
(triggered) by just a single photon.
(ii) In [71] a quantum phase transition was proposed and analyzed as a
means of amplifying a weak signal (such as a single photon). Here we could
use a dissipative phase transition [148, 149, 150, 151] to achieve minimum-noise
amplification. The Hamiltonian Hc(F ) we used is for either a driven atom or a
driven cavity. A dissipative phase transition arises even for the simple system
of an atom inside (and coupled to) a cavity, with either the atom or the cavity
driven. The presence of a phase transition may make the amplification process
more robust against deviations from the ideal Hamiltonian.
(iii) The single-photon wavepacket to be detected is fixed here by the
resonance frequency and the decay rate of the cavity that generates the photon.
We could make these two parameters arbitrary functions of time so that an
arbitrary single-photon wavepacket can be created [113, 114, 115]. That should
allow us to formulate the POVM (projecting onto a specific temporal state of the
photon) that describes our detector, as in Ref. [68].
(iv) We assumed a bosonic mode to contain the amplified signal.
Alternatively we may use many spin-1/2 particles, as in the model discussed in
Refs. [71, 72]. This extension would increase the scope of our description to include
fermionic systems.
In conclusion, the main point here was to present a class of Hamiltonians
that describe the photodetection process fully quantum-mechanically from
107
beginning to end, including nonlinear, minimum-noise amplification [1] and near-
perfect photoabsorption [64].
108
CHAPTER IV
DETECTING TWO PHOTONS WITH ONE MOLECULE
We apply input-output theory with quantum pulses [AH Kiilerich, K
Mølmer, Phys. Rev. Lett. 123, 123604 (2019)] to a model of a new type of two-
photon detector consisting of one molecule that can detect two photons arriving
sequentially in time. The underlying process is distinct from the usual two-photon
absorption process where two photons arriving simultaneously and with frequencies
adding up to the resonance frequency are absorbed by a single molecule in one
quantum jump. Our detector model includes a Hamiltonian description of the
amplification process necessary to convert the microscopic change in the single
molecule to a macroscopic signal.
4.1. Introduction
There are two standard ways of detecting two photons in a photon-number
resolved (PNR) manner: (i) an inherent PNR detector produces a different signal
depending on whether one or two photons were absorbed by the detector, (ii)
multiplexed PNR detection [152] exploits multiple single-photon detectors, and
the signal consists of either one or two such detectors “clicking.” An inherent
PNR detector may, for example, be sensitive to the total energy deposited by
the photons [153]. A second type of detector sensitive to two photons makes use
of a process called “two-photon absorption” (TPA) in which one molecule can
absorb two photons that arrive simultaneously and whose frequencies add up to
the resonance frequency. This effect was discovered by Göppert-Mayer in 1931
[154, 155], goes through a virtual intermediate state, and has become an item of
109
modern interest since the realization that this TPA process is sensitive to time-
frequency entanglement between the two incoming photons [156, 157, 158].
For a biological example of multiplexing, we may consider the human eye.
There are about 108 rods, each of which is sensitive to single photons in that they
can absorb one photon at a time [159]. Interestingly, the TPA process occurs in
the human eye, too, where two infrared photons may give rise to the sensation
corresponding to that of light in the visible range [18, 160]. In this case the
detection is not strictly PNR, as the signals from two infrared photons or from
one visible photon are the same.
A process related to TPA is called stepwise two-photon absorption where the
first photon takes the molecule to an actual (rather than a virtual) excited state
and a subsequent photon takes the molecule to an even higher lying excited state,
see e.g., [161]. Taking a molecule to an excited state, however, is not yet sufficient
for implementing a measurement. We also need an amplification process that
produces a macroscopic signal. In the human eye a light-absorbing molecule decays
from the excited state irreversibly to a metastable state, in which the shape of the
molecule has changed. That change in shape triggers a chain reaction of shape
changes in surrounding proteins, eventually producing (or changing) a permanent
dipole moment that in turn triggers a change in a mesoscopic electric current [162],
which then permanently registers the detection of the photon.
Following the example of Refs [36, 60, 64, 65] of taking inspiration from
biological systems to design photo detectors (see also [163]), based on this robust
photo-detection mechanism we propose and model a PNR two-photon detector
consisting of a five-level molecule, as follows (see Fig. 1 and Section III for more
details and reasons for choosing this particular configuration): a ground state |F0〉
110
from which a photon with a frequency ωα ≈ ω1 − ω0 =: ω01 can induce a transition
to an excited state |F1〉, which can then irreversibly decay to a metastable state
|F2〉. In this state the molecule triggers a first amplification process that indicates
and permanently registers the detection of that first photon. Subsequently, a
second photon of a different frequency ωβ ≈ ω3 − ω2 =: ω23 can excite the
molecule to another state |F3〉, from which it can decay to a different metastable
state |F4〉, triggering a second (different) amplification process that indicates the
detection of the second photon. The two photons must arrive sequentially rather
than simultaneously for TPA.
One motivation for this work comes from recent theory efforts to find
fundamental (i.e., device-independent) limits to photo detection [1, 62, 64, 65,
66, 68, 69, 71, 72]. For reasons fully explained in [69] we construct a Hamiltonian
here for the full detection process, including the crucial amplification step. A
second motivation is of a more technical nature. The theoretical description
of two (or more) photons interacting with a quantum system is known to be
considerably more complicated than that of just a single photon interacting with
the same system [109, 113, 114, 115, 164, 165, 166, 167, 168, 169, 170]. Two
types of methods have been developed to tackle this problem. One is based on a
hierarchy of coupled differential equations for generalized density matrix elements
[109, 113, 171] for a quantum system interacting with prescribed multi-photon
pulses. The other method [114, 115] includes virtual cavities that generate the
photons and is thus based on a Hamiltonian description of the quantum system
and the photons. We refer to these two methods as the “generalized density
matrix” and the “Hamiltonian” formulations, respectively. We will use both
methods here, since they each have their own advantages, and we also give the
111
explicit equations (which seem not to have been given before) that link the two
methods. Moreover, we can explain why the methods above yield expressions for
scattered light and for the dynamics of the quantum system in terms of (Hilbert
space) inner products that involve the temporal amplitudes of the incoming
photons [172] on the one hand, and the appropriate response functions of the
system on the other.
This paper is organized as follows. In Section II we first give a synopsis of
some of the results, which can be understood without going into the details of
the derivations. Such details are provided in the remaining Sections. In Section
III we give the Hamiltonian for our 5-level molecule. Section IV describes the
two different methods we used to obtain results: the generalized density matrix
methods is used to obtain analytical results, while the Hamiltonian method is used
to obtain numerical results. We explain why the latter method is so much easier
to use for numerical calculations. Section V ends with conclusions and discusses
possible extensions of our work. In the Appendix we present the transformation
that unifies the two formalisms (generalized density matrix and the Hamiltonian
formulation) used in the paper.
4.2. Synopsis
Since the detailed description of our system is rather involved we first give
here a synopsis of the basic results without any derivations. The results presented
here are quite straightforward to understand. The light-absorbing molecule at the
heart of our detector is described in detail in Fig. 1.
112
|F3>
|F1>
|F4>
|F2>
|F0>
FIGURE 4.1. Model of a two-photon detector.
A light-absorbing molecule starts in the ground state |F0〉. There are two excited
states |F1〉 and |F3〉, indicated in red, and two metastable states |F2〉 and |F4〉. We
assume level |F2〉 corresponds to a shape change [while still being an electronic
ground state], which down the line corresponds to a change in electric dipole
moment, which in turn induces a change in a mesoscopic current or voltage (thus
mimicking the process taking place in the human eye). That mesoscopic change
permanently registers the detection of the first photon. The metastable state |F4〉
corresponds to yet another change of shape, which eventually leads to a change in
a dipole moment, which then in turn can change a mesoscopic current in a way
that is distinct from what the molecule in state |F2〉 accomplished. This distinct
mesoscopic change then registers the second photon. The molecule can detect two
photons, one “blue” photon resonant with the transition from the ground state to
the first excited state, and a “green” photon resonant with the transition from |F2〉
to the second excited state. From each of the two excited states the molecule can
spontaneously decay back to state it came from or to the desired metastable state.
Thus there are four decay rates, indicated from left to right by γ1 . . . γ4, which are
assumed to be more or less of the same order of magnitude. (The spontaneous
transitions are indicated with dashed black lines.) On a time scale much longer
than 1/γ1 the molecule resets by the metastable states decaying back to the
ground state |F0〉 (this resetting is not indicated in the figure).
4.2.1. Detection probabilities
For an incoming single-photon wave packet, the different frequency
components are not all absorbed with 10101%3 efficiency. The probabilty Pα for the
first photon, labeled α, to be detected can be written in the form
∫
Pα = dω |T1(ω)|2|uα(ω)|2. (4.1)
Here uα(ω) is the Fourier component of the incoming wave packet at frequency
ω and may also be referred to as its spectral amplitude. T1(ω) is a complex
transmission amplitude for the molecule to go from the initial state |F0〉 to the
desired state |F2〉 through the intermediate excited state |F1〉 (see Eq. (4.28)
below):
√
γ1γ2
T1(ω) = . (4.2)
(γ1 + γ2)/2− i{ω − ω01}
If γ1 = γ2, the transmission probability |T1(ω)|2 reaches a maximum of 1 at the
resonance frequency ω01 and has a width of about γ1. Thus, a resonant photon
with a narrow width in frequency space (much less than γ1) and whose duration is,
therefore, much longer than γ−11 , can be absorbed with near-unit efficiency, exactly
as was found before in Refs. [64, 65, 66].
A similar result holds for the second photon, labeled β. The only
(important!) difference is that the second photon can be absorbed only when the
molecule is in the state |F2〉. Hence ideally it should arrive after photon α has
been fully absorbed. In that ideal case, the conditional probability of detecting
photon β (with a spectral amplitude uβ(ω)), given that photon α was detected, is
∫
Pβ = dω|T (ω)|22 |uβ(ω)|2, (4.3)
with
√
γ3γ4
T2(ω) = (4.4)
(γ3 + γ4)/2− i{ω − ω23}
114
a second complex transmission amplitude, describing how the molecule can
transition from level |F2〉 to level |F4〉 through the intermediate |F3〉 excited state.
The probability to detect both photons in the more general case when the
two photons do overlap in time can be written in the form
Pα&β = PαPβ − Poverlap, (4.5)
where the (non-negative) “overlap term” will be derived and discussed in Section
4.4.1. We merely note here that the overlap term can be found analytically and is
then written as a convolution involving the two spectral amplitudes uα,β(ω) and
the two transmission amplitudes T1,2(ω). If photon β is delayed by a time much
longer than 1/γ1, then Poverlap → 0, but if photon β entirely precedes photon α,
then Poverlap → PαPβ.
In Figure 2 we plot a numerical result for a case that is not optimal for two
reasons. First, the widths in time of the two incoming single-photon pulses are
equal to 1/(2γ1), which is too short to be close to optimal. Second, the pulses
partially overlap in time. The probability to detect both photons is then about
42%.
In Figure 3 we plot a case that shows how important the delay between the
two photons is. Here photon β arrives just before photon α: while this does not
affect at all the absorption (and detection) of photon α, photon β is now detected
only with a very small probability of about 2%.
115
1.2
F0
F2
1 F4
0.8
0.6
0.4
0.2
0
0 2 4 6 8 10 12
1
|u |
0.8 |u |
0.6
0.4
0.2
0
0 2 4 6 8 10 12
time [units of 1/ 1]
FIGURE 4.2. Top: Populations in the ground state and the two metastable
states as functions of time, when two photons arrive sequentially. Bottom: the
(Gaussian) amplitudes of the “blue” photon (uα) and the “green” photon (uβ) as
functions of time. We chose here γk = γ1 for k = 2, 3, 4 and the time delay between
the two input photons is 3/γ1
Eventually a steady state is reached, with the total population in the three lowest
states adding up to 1. The steady-state population in the ground state (dot-dashed
curve) is 0.346, which equals the probability to not detect any of the photons. The
sum of the steady-state populations in the metastable states is .654 and equals the
probability to detect the “blue” photon. The steady-state population in |F4〉 is
0.418 and equals the probability to detect both photons.
4.2.2. Detector clicks
The generalized density matrix formalism can be used to get analytical
expressions describing “clicks” of our detector in simple cases.
116
amplitude population
1
F0
F2
0.8 F4
0.6
0.4
0.2
0
0 2 4 6 8 10 12
1
|u |
0.8 |u |
0.6
0.4
0.2
0
0 2 4 6 8 10 12
time [units of 1/ 1]
FIGURE 4.3. Top: Populations in the ground state and the two metastable states
as functions of time, when the “green” photon arrives just before the “blue” photon
(the time delay is −1/(4γ1). Bottom: the absolute values of the amplitudes |uα| of
the “blue” photon and |uβ| of the “green” photon as functions of time.
The probability to detect both photons is very small in this case, 0.022. The first
photon is detected with the same probability (0.346) as in the previous figure.
4.2.2.1. One photon
For example, suppose for simplicity that we could measure in what state our
molecule is at a specific time T > t0, given that it started in the state |F0〉 at time
t0, and suppose that we find our molecule in the state |F2〉. This clearly would
implement a measurement of the incoming photon. Thus, ignoring the second
photon for now, given an expression for the population in that level as a function
of time, we can write that probability at time T in the form of the Born rule as
P2(T ) = Tr(|uα〉〈uα|Π1), (4.6)
117
amplitude population
where |uα〉〈uα| is the projector onto the input single-photon wave packet of photon
α, and Π1 is a positive hermitian operator (guaranteeing that P2(T ) is a real non-
negative number). We can always write Π1 in a diagonal form
∑
Π1 = λn|φn〉〈φn| (4.7)
n
with λn real and non-negative, and with {|φn〉} forming an orthonormal basis of
single-photon states. That means the probability P2(T ) can be rewritten as
∑
P 22(T ) = λn| 〈uα|φn〉 | . (4.8)
n
The fact that the Born rule is linear in the input state (represented as a density
operator or matrix) thus explains why this probability can be expressed in terms
of overlaps involving the incoming single-photon wave packet [172, 173]. It also
follows that λn ≤ 1, since λn has the meaning of the probability that an input
photon in the state |φn〉 will be detected.
In our specific case we find that Π1 is of the form
∫ T
Π1 = dtWt|φt〉〈φt| (4.9)
t0
where Wt is a weight per unit of time
γ1γ2
Wt = [1− exp(−(γ1 + γ2)(t− t0))] , (4.10)
γ1 + γ2
118
and the projector projects onto a normalized single-photon state of the form
∫ t
dt′ exp[(γ1√+∫γ2)t′/2] exp(iω ′01(t − t))b† ′1(t )|vac〉|φ 〉 = t0t (4.11)t
dt′ exp[(γ1 + γ2)t′]t0
where b†1(t) is the Fourier transform of b
†
1(ω). (Note that we could equivalently
write t′ − t instead of t′ in the arguments of the γ1,2-dependent exponentials
in both numerator and denominator.) These states |φt〉 are not orthogonal for
different values of t and this type of nonorthogonal states also appears in the
context of spectral filtering [63]. We also note that the transmission function T1(ω)
given above in Eq. (4.2) is the (properly normalized) Fourier transform of the
time-dependent function —which is a Green’s function—appearing in |φt〉. That
transmission function also determines the spectral shape of the photon emitted
spontaneously by the molecule [173].
It is important to note that in Eq. (4.6) Π1 refers only to the detector, and
|uα〉 refers only to the incoming photon. Π1 is called a POVM (Positive-Operator
Valued Measure) element and fully describes the outcome of the measurement
corresponding to finding the molecule in level |F2〉 at time T . It allows us to
calculate for any incoming photon the detection probability (4.6). In particular,
it allows us in principle to infer the type of photon that is detected with the
largest possible probability, by making use of the diagonal form (4.7). The largest
eigenvalue λmax = maxn λn gives the highest possible efficiency ηmax = λmax of
detecting a single photon, and the corresponding eigenstates [there may be more
than one] give the optimal single-photon states that achieve that limit.
119
The interpretation of
∑ ∫ T
Tr(Π1) = λn =
n [ dtWtt0 ]
≈ γ1γ2 − − 1T t0 , (4.12)
γ1 + γ2 γ1 + γ2
(where we ignored an exponentially small term in the second line) is that of a
bandwidth: the effective size of the single-photon Hilbert space covered by this
particular measurement outcome [62]. This bandwidth may be (much) larger than
unity. For a fixed value of γ1 + γ2 the bandwidth is maximized by γ1 = γ2, an
optimal “impedance-matching” condition found before in the same context of
designing an optimal single-photon detector [64, 65, 66]. The bandwidth is then
approximately equal to the total time the detector has been on in units of 2/γ1.
If we would be able to measure if the molecule were in state |F1〉 at time t,
then the corresponding POVM element would be proportional to a pure projector.
But, since we do not know when the upper state spontaneously decayed to
state |F2〉, we do not know t, and hence we get a mixed POVM element. That
is, for fixed T (when we detect the molecule to be in the state |F2〉) there are
different possibilities for time t, each with their own probability Wtdt. That is the
interpretation of (4.9).
The idea that a quantum system absorbs a single-photon wave packet with in
principle 100% efficiency if and only if it is the time-reversed version of a photon
that the system would emit if it started in the final state [174, 175] does not apply
so simply here, because of the presence of irreversible spontaneous decay. If we
imagine we would apply a laser pulse to the |F1〉 → |F2〉 transition to induce
120
stimulated emission, then, as is well known [139, 176], that idea indeed would
apply straightforwardly .
4.2.2.2. Two photons
The more interesting case of detecting the molecule in level |F4〉 at time T
signals the detection of both photons and is described by the POVM element
∫ T ∫ t
Π2 = dt dt
′Wt′Wt,t′ |φt′〉〈φt′ | ⊗ |ψt,t′〉〈ψt,t′ |, (4.13)
t0 t0
with
γ3γ4
W ′t,t′ = [1− exp(−(γ3 + γ4)(t− t ))] , (4.14)
γ3 + γ4
and the single-photon state corresponding to the second photon is
∫ t
′ dτ exp[(γ3√+∫γ4)τ/2] exp(iω23(τ − t))b†(τ)|vac〉|ψ 〉 = t 2t,t′ . (4.15)t′
dτ exp[(γ3 + γ4)τ ]t
The prefactor Wt and the single-photon state |φt〉 appearing here are exactly as
defined before in (4.10) and (4.11). The time-dependent function appearing in
|ψt,t′〉 is once again a Green’s function, and T2(ωb) is its (normalized) Fourier
transform.
There is a double integral over time in (4.13), each integral corresponding to
an irreversible step in the detection process, which makes it uncertain at what time
t we could have found the molecule in state |F3〉 and at what earlier time t′ < t we
could have found the molecule in |F1〉.
121
We may again write down an eigenvalue equation for Π2 [which would have
to be solved numerically] and then write that POVM element in the diagonal form
∑
Π2 = µ
(α,β)
n|φn 〉〈φ(α,β)n |, (4.16)
n
where the projectors | (α,β) (α,β)φn 〉〈φn | project onto specific pure two-photon
(eigen)states, and the eigenvalues 0 ≤ µn ≤ 1 give the corresponding efficiencies
with which those specific two-photon wave packets are detected at time T .
Like we saw for the single-photon case treated above, the bandwidth
∑ ∫ T ∫ t
Tr(Π2) = µn = dt dt
′Wt′Wt,t′ (4.17)
n t0 t0
is the size (dimension) of the two-photon Hilbert space covered by our detector.
4.3. The two photon absorber and its Hamiltonian
To construct the minimal absorber atom or molecule or multi-level system
that can absorb two photons sequentially and produce classical outputs signaling
the final state of the absorber, we consider the five level system of Figure 1 for
efficient photon transduction. Some recent efforts for physically based fundamental
models for photo detection assemble all parts of the process into a single fully
coupled evolution problem [1, 62, 64, 65, 66, 68, 68, 69, 71, 72]. Minimal noise
amplification of the absorbed photon signal has been shown to be optimally done
with continuous quantum measurement [64, 65, 69]. In this scheme, the “shelving
state” or the state in which the absorber produces the amplified classical readout
is continuously measured. To get around the quantum Zeno effect problem with
having the same state to be the photo-excited and shelving state, a three level
122
system is determined to be optimal for single photon detection [64]. Hence we use
the three levels |F0,1,2〉 to detect one photon.
For the two-photon detection scheme, we supplement the molecule with
two more levels. The second photon can lift the molecule from state |F2〉 into
the excited state |F3〉 which can spontaneously relax into the second shelving
state |F4〉. In the latter state the molecule triggers an amplification process which
produces a noticeably different signal than that produced by the shelving state
|F2〉. The absence of a signal and the two different signals from the two levels |F2〉
and |F4〉 help the observer distinguish the number of photons (0, 1, or 2) absorbed
by the molecule. Since the frequency of the amplified signal is independent of
the input photon frequency [1], we can have different shelving states (classically)
driving different oscillators of different frequencies [69]; and hence we can have
distinguishable classical output signals for one or two detected photons.
We wish to calculate the dynamics of the 5-level discrete quantum system F
coupled to the two continua b1 and b2 which contain our two input photons (with
different frequencies). With ~ = 1, the parts of the Hamiltonian in the Markov
approximation for these coupled systems are
∑4
Hsys = ωk|Fk〉〈Fk|, (4.18)
k=0
∫
1(2)
H †bath = dω ωb1(2)(ω)b1(2)(ω), (4.19)
123
∫ [√
− γ1Hint =√ i dω |F1〉〈F0|b1(ω)2π ]
γ3
+ |F3〉〈F2|b2(ω) +H.c. . (4.20)
2π
This part of the Hamiltonian includes spontaneous decay back to |F0〉 and
back to |F2〉. (The radiation field modes are fully described by four degrees of
freedom. Here we fixed the quantum numbers for three of them (polarization
and two transverse spatial degrees of freedom) and explicitly retain only the
spectral/temporal degree of freedom.)
The next and last part of our Hamiltonian is necessary for the purpose of
enabling the additional spontaneous decays of the absorber from |F1〉 to |F2〉 and
from |F3〉 to |F4〉. These two transitions need to be dipole allowed and γ2 and γ4
determine the rates (probability per unit time) of those two processes:
∫ [√
γ2
H1int =√−i dω |F]1〉〈F2|g(ω)2π
γ4
+ |F3〉〈F4|h(ω) +H.c. . (4.21)
2π
in terms of two additional independent (commuting) bosonic modes, described by
annihilation operators g(ω) and h(ω) and their hermitian conjugates.
4.4. Two theories for photon absorption
Restriction of the number of excitations to one or two offers a workaround
for the complications of the multimode nature of the interaction of propagating
light with a nonlinear medium such as a two- or three-level atom. The Fock state
124
master equation formalism by Baragiola et. al. [109], and the set of generalized
density matrices by Gheri et. al. [113] offer suitable theoretical frameworks for
calculating few photon Fock state interactions with a multi-level discrete quantum
system.
An alternate route for having a computationally manageable effective
master equation has recently been developed by Kiilerich et. al. [114, 115] by
restriction of the input pulse to a single time dependent mode. This approach is
appealing to the problem of single photon absorption as the same physical effects
of the incoming wave packet of the multimode bosonic input field is emulated.
As previously formulated by Gheri et. al. [113], an upstream virtual cavity is
introduced whose output serves as the incident field for a system under study.
The incident field generated by the cavity is in a state residing in a specific wave-
packet mode and all other orthogonal modes are designated the vacuum state.
Since we are only interested in the input quantum state and the absorption of
the photon, we only acquire the technique of driving with a quantum pulse from
Ref. [114, 115]. The reflected quantum state is of no interest to us, and only the
transmitted state ([66]) which quantifies the probability of absorption is required
for our purpose.
The generalized density matrices framework developed by Gheri et. al.
[113] suffices for calculating the absorption probabilities and corresponding
POVMs. However, we introduce the virtual upstream cavities and formulate
a Hamiltonian formulation for the entire evolution problem of photo detection
(including amplification to a mesoscopic signal) that we introduced in the previous
publication [69]. The Hamiltonian formulation is versatile and facilitates the
calculations to be done in either the Schrödinger or the Heisenberg picture.
125
The explicit transformation between the generalized density matrices and the
components of the density matrix obtained by the Hamiltonian method is
presented in Appendix A.
4.4.1. Generalized Density Matrix Operators
We assume we have two unentangled single-photon wavepackets in two
orthogonal modes
|Ψ2〉 = |Ψα〉|Ψβ〉, (4.22)
where the individual photon states are defined as
∫∞
|Ψα(β)〉 = dω †a(b)uα(β)(ω)b1(2)(ω)|vac〉. (4.23)
−∞
uα(β) is the properly normalized wave function for photon a (b). The two photons
reside in the two distinct continua b1 and b2. (We will also use the Fourier
transforms of the single-photon amplitudes, which for simplicity we denote by
uα,β(t).)
Following Ref. [113], we can define generalized density matrix operators
for i, j = 0, 2, α, β and derive a set of coupled differential equations for them
that describes the absorption of the two photons. In the following, R denotes
the reservoir or bath, which includes continua other than b1 and b2, such as the
continua g and h introduced above:
[ ]
ρi,j(t) = TrR U(t, t0)ρS(t0)⊗ |Ψi〉〈Ψj|U †(t, t0) . (4.24)
126
Here |Ψ0〉 denotes the vacuum state |vac〉, and |Ψ2〉, |Ψα〉 and |Ψβ〉 are the two-
photon input state and the individual single-photon states introduced above.
Furthermore, ρS(t0) is the initial state of all remaining quantum systems, including
our 5-level molecule and the reservoir R. In our case, each of these generalized
density matrices for fixed values of i and j is a 5x5 matrix, describing the 5 levels
of our molecule.
The generalized density matrices can be expanded in a time independent
complete 5x5 basis, and substitution in the evolution equations yields a set of
coupled differential equations for the coefficients ρik,jl(t) of the expansion,
∑
ρi,j(t) = ρik,jl(t)|Fk〉〈Fl|. (4.25)
k,l
These equations are given in Appendix A. The diagonal generalized density
matrices (for i = j) have a preserved trace of 1, and off-diagonal ones have a
preserved trace of 0 over the evolution [113]. (In the alternative Hamiltonian
formulation shown below a single Hamiltonian (with auxiliary cavities appended)
can embody the complete evolution, and a single density matrix (with preserved
trace of 1) of size 20x20 can embody the complete dynamics [69].)
In order to simplify intermediate equations, we will absorb a time-dependent
phase factor exp(iω01)t) in the definition of the single-photon amplitude uα(t) for
photon a and similarly a factor exp(iω23t) in the amplitude uβ(t) for photon b,
such that both amplitudes can be considered slowly-varying if the photons are
more or less on resonance with their respective transition in the molecule. End
results are quoted in terms of the original amplitudes.
127
The evolution problem is initiated with ρ00,00 = 1 at time t0, i.e., the
molecule is in the |F0〉 state, with any photon yet to come in. The coefficient ρα2,α2
embodies the evolution of the molecule occupation elevated to |F2〉 state driven by
just the first photon α with temporal amplitude uα(t). The solution found is
∫t [
ρ (t) = γ γ dt e−(γ1+γ2)t1α2,α2 1 2 1
∫ t0t1 ∫ t2
γ1+γ2 ∗ γ1+γ2dt e t2 22 uα(t2) dt3e
t
2 3u α(t3) + c.c., (4.26)
t0 t0
This result becomes especially simple when considering the steady-state, obtained
by taking the limit t → ∞. The result further simplifies when we take the limit
t0 → −∞ such that in principle any single-photon wave packet could be absorbed,
irrespective of when it arrives. The equations in those limits are most easily solved
in Fourier space, and we obtain then the same result we had obtained before in
Ref. [69], ∫
ρα2,α2(∞) = Pα = dω |u (ω)|2α |T1(ω)|2, (4.27)
where
√
γ1γ2
T1(ω) = (4.28)
(γ1 + γ2)/2− i{ω − (ω1 − ω0)}
is the transmission coefficient describing the propagation of a single excitation
through the Λ system [63, 66]. (This is Eq. (4.2)) of the Synopsis Section.)
ρα2,α2(t) in Eq. 4.26 can be recast into the more informative form
∫t ∣∣∣∫ ∣t′ √ ∣∣2′ γ1+γ2 (t −t′ρ 2 )α2,α2(t) = dt ∣∣ dt2 γ1γ2e 2 uα(t2)∣∣ .t0
t0
(4.29)
128
This is the form that can be used straightforwardly to obtain the expressions
(4.9)–(4.11) for the POVM element Π1. The quantity inside the integral over t′
is actually γ2 times the population in level |F1〉 as a function of time [65].
The probability of the molecule reaching |F4〉 state driven by the second
photon β has a “nested” structure containing the expression, ρα2,α2(t),
∫t [
ρ −(γ3+γ4)t124,24(t) = γ3γ4 dt1 e
∫ ∫t0 t1 t2
γ3+γ4 γ3+γ4
dt e t2 22 u
∗
β(t2) dt3e
t
2 3u β(t3)ρα2,α2(t3) + c.c.
t0 t0
(4.30)
We may rewrite this expression by changing variables in the complex conjugate
term and by substituting (4.29) to obtain our two-photon POVM (4.13).
As we noted in Section 4.2 the time-dependent functions appearing in the
expression for ρ24,24(t) and other populations of quantum levels can be interpreted
as Green’s functions. Their (normalized) Fourier transforms act as transmission
and reflection coefficients when treating this problem as a scattering problem.
In our case, transmission coefficients T1(ω) (defined above) and T2(ω) (defined
below) play the new role of determining the detection probability of photons with
frequency ω in the limit of t → ∞, as we saw in Eq. (4.27) and as we will show in
the next subsection.
4.4.2. Overlap term
The absorption of the two photons can be completely calculated in the
frequency domain. To that end, we define the Fourier transform of the population
129
in level |F1〉, since we can express all quantities of interest in terms of that
function. We find
∫
F1(ω) = √
1
dx T1(ω + x)uα(ω + x)T
∗
1 (x)u
∗
α(x). (4.31)
2π
√
The 0-frequency componen∫t of 2πF1(ω), equals the detection probability for the√
first photon 2πF 2 21(0) = dx|uα| |T1(x)| =: Pα. In the frequency domain, we
then obtain the Fourier transform of ρα2,α2(t) as
[ ( ) ]
1
ρα2,α2(ω) = γ2 iF1(ω)P + πF1(0)δ(ω) . (4.32)
ω
where P denotes the principal value. In eq. (4.30), if we replace ρα2,α2(t) with its
steady state value Pα, we get an expression identical in form to eq. (4.26) with
different decay rates, and we thus can simply evaluate the result for t → ∞ as the
product PαPβ with Pβ given by (4.3). So, if the second photon arrives long after
the first photon has been completely absorbed (and the absorber raised to the level
|F2〉), the probability of both photons being absorbed becomes the product of their
individual absorption probabilities.
Therefore, we can rewrite the probability of two-photon absorption, ρ24,24(∞)
as a sum of two parts, one being the product of the two absorption probabilities.
We name the other term Poverlap, since we expect the term to vanish if the second
photon comes in after a delay and the two wave functions of the two photons
overlap negligibly. We thus write
ρ24,24(∞) = PαPβ − Poverlap . (4.33)
130
After some algebra, we obtain
1
Poverlap = PαPβ + P
αβ (4.34)
2
where
√ ∫
αβ γ2 γ3γ4P = ∫√ dω u∗β(ω)T2(ω)2π(γ3 + γ4)
× F1(x)dx uβ(ω − x)P + c.c.,
ix
(4.35)
where P denotes the principal value. The following results are borne out in
numerical simulations for different arbitrary wave shapes of the two photons that
are delayed by a long time td >> 1/γ1:
Pαβ −→ −1Pα P β
2 abs abs
Poverlap −→ 0
ρ24,24(∞) −→ PαPβ. (4.36)
4.4.3. Hamiltonian Formulation
In a recent paper, Ref. [69], we developed a “Hamiltonian formulation” that
can describe a single photon detection process in its entirety. We now adapt that
formulation for the detection of two unentangled photons absorbed sequentially.
The most convenient method for solving the dynamical equation set is numerical
integration of the Liouvillian equation in the Hamiltonian formulation [69]. In
the Hamiltonian formulation, we get a single density matrix for the entire system
131
that can be solved easily with well-known vectorization and Trotter decomposition
techniques [177]. From the solution of the single density matrix, the generalized
density matrices can be found easily with the transformation (4.50).
We introduce two auxiliary cavities with damped harmonic motion leaking
one excitation each into the continuous bath modes b1(ω) and b2(ω). These two
excitations mimic the photon wave packets in the two baths that we are trying to
detect. There are two other continuous modes g and h, which are introduced only
to enable the spontaneous relaxation of the molecule F. The Hamiltonian is of the
FIGURE 4.4. The cavity modes a1 and a2 each have one excitation to start with.
These two excitations leak out into their adjacent baths (continuous modes b1 and
b2 respectively) by designing the coupling to the baths in time, thus creating two
single-photon wavecpackets. They respectively drive the |F0〉 to |F1〉 and |F2〉 to
|F3〉 transitions. From the excited levels, the molecule can relax with certain
probabilities either back to the state it came from or to another shelving state.
The two shelving states drive two distinct amplification processes and thus produce
two macroscopically distinct “classical” signals (unrelated in frequency to the
incoming photons) in an output bath d(ω).
following form
H = Ha1 +Ha1−b1 +Hb1 +Ha2 +Ha2−b2 +Hb2
+Hb1−F +Hb2−F +
HF +HF−g +Hg +HF−h +Hh
+HF−c +Hc +Hc−d +Hd. (4.37)
132
The diagonal terms in the Hamiltonian give all the eigen energies of the systems.
For example, for the cavities it features their resonance frequencies,
Ha = ωa a
†
1(2)a1(2), (4.38)1(2) 1(2)
and for the continuous modes, such as g, we have
∫ ∞
H †g = dω ωg (ω)g(ω), (4.39)
0
and similar terms for Hb1 , Hb2 , Hc, Hd and Hh. HF is simply Hsys as defined before
in Eq. (4.18).
The interaction between the cavities and the field modes, as well as the
interaction of the photons with the molecule are mediated by the electric fields
corresponding to modes b1 and b2. Each of the electric field operators of the modes
can be expanded into plane wave basis (also their Hermitian conjugate operators).
For the input fields B1,2(x, t), we expand
∫
1 ∞
B1,2(x, t) = √ dω b1,2(ω, t) exp(iωx/c). (4.40)
2π 0
The molecule is located at x = 0 and the cavities a1 and a2 are located “upstream”
at x = −cτ1 and x = −cτ2 where c is the speed of light and τ1, τ2 are the times
it takes for a photon to travel from the respective cavities to the absorber F. The
cavities are coupled to the fields B1(x = −cτ1, t) and B2(x = −cτ2, t)in the manner:
Ha1(2)−b =1(2)
i[g∗ † †1(2)(t)a1(2)B1(2)(−cτ1(2), t)− g1(2)(t)B1(2)(−cτ1(2), t)a1(2)]
133
As pointed out in previous work [69, 113, 115], the coupling of the virtual cavities
to the fields can be made time dependent for the purpose of creating arbitrary
photon wavepackets, and therefore the Hamiltonian formulation is completely
general for the photo detection process. In this way, we can calculate the evolution
of the complete system with the elements of a single density matrix, instead of the
multiple generalized density matrices in eq. (4.24). All other discrete-continuum
couplings are at position x = 0.
√ ∣∣ 〉 〈 ∣H = i γ F F ∣B†b −F 1 0(2) 1(3) 1(2)(x = 0, t) +H.c.1(2)
√
H †F−g = i γ2 |F2〉 〈F1|G (x = 0, t) +H.c.
√
HF−h = i γ4 |F4〉 〈F |H†3 (x = 0, t) +H.c.
√
Hc−d = i ΓcD
†(x = 0, t) +H.c., (4.41)
where the field operators G(x, t) and H(x, t) are defined in terms of g(ω) and h(ω)
just as the field operator B(x, t) is defined in Eq. (4.40) in terms of b(ω).
The amplification mechanism is embodied in the parts,
∑
F̃ = Fk|Fk〉〈Fk|, (4.42)
k=0,1,..,4
HF−c = iF̃ (c− c†).
The different eigenvalues of the operator F̃ drive a discrete quantum harmonic
oscillator (another cavity, for example) with annihilation operator c by different
classical driving strengths. That driven cavity mode will contain an increasing
number of excitations. We assume here F0 = F1 = F3 = 0 so that no amplification
(no driving) takes place when the molecule is in the corresponding states. The
134
values for F2 and F4(6= F2) are nonzero and drive the amplification process.
Excitations from the driven cavity c leak into the continuum mode d(ω), which can
be observed “classically” when populated massively. Thus d(ω) contains our final
“classical” signal. We will not analyze the macroscopic signal here and refer instead
for further details to Ref. [69], where it is shown that this type of amplification
process yields minimal noise; see also [122].
4.4.4. Invariants of motion
The Hamiltonian formalism preserves the basic idea of the photodetection
process that is meant to be simulated. We can find some operators that commute
with the Hamiltonian and are therefore conserved in time.
∫
I = a†a + ∫ dωb†20 1 1 1(ω)b1(ω)− |F0〉〈F0|, ∫
I = a†21 1a1 + ∫ dωb†1(ω)b1(ω) + |F1〉〈F1|+ ∫ dωg†(ω)g(ω),
I = a†22 2a + ∫ dωb†2 2(ω)b2(ω)− |F2〉〈F2|+ ∫ dωg†(ω)g(ω),
I23 = a†2a2 + d∫ωb†2(ω)b2(ω) + |F3〉〈F3|+ dωh†(ω)h(ω),
I24 = |F4〉〈F †4| − dωh (ω)h(ω),
A conserved quantity of particular interest is I = 1N I + 1I + 1I + 1I −2 20 2 21 2 22 2 23
I24 + 1 . The 1 is added here to give the invariant IN the meaning of the number2 2
of excitations (photons). The invariant takes the values 0,1,2 for the three cases of
0,1,2 input photons, respectively.
The values of these quantities keep track of where the excitations are and
whether the photons will be detected or not. For example, an initial excitation in
135
the a1 cavity means I20 maintains a value of 0 in the entire evolution∫. So as the
eigenvalue of a†1a1 decays from 1 to 0, either the eigenvalues of both dωb
†
1(ω)b1(ω)
and |F0〉〈F0| for t → ∞ are 1 (the photon was not detected) or they are both 0
(the photon was detected).
∫ Similarly, I24 always equals 0, with an eigenvalue of 1 for both |F4〉〈F4| and
dωh†(ω)h(ω) indicating the second photon was detected, and an eigenvalue 0
indicating it was not detected (yet).
4.4.5. The Liouvillian Representation
Due to the continua in our model, the Hilbert space is infinite dimensional.
However, we follow the well established practice of eliminating the continua and
focus our attention on the “system Hilbert space, Hd” (d=2×2×5=20) and are able
to calculate all quantities of interest in the vector space of the linear operators,
L(Hd) acting on the Hilbert space, Hd. We eliminate the continua b1, b2, g, h
and obtain our Liouvillian master equation for the system density operator, ρs
comprised of discrete quantum systems a1, a2, F . Details of the exact method and
validation of quantum mechanical commutation relationships can be found in the
preceding paper Ref. [69]. For the absorption problem, we need not include the
discrete cavity mode c. The Liouvillian master equation for the chosen discrete
quantum parts of the Hamiltonian is
∂
ρs = −i[Hsys, ρs] +D [ρs] . (4.43)
∂t
Eq. 4.43 facilitates numerical calculation in the Schrödinger picture. For a collapse
operator, X, the Lindblad dissipator super-operator (a map, S: L(Hd) → L(Hd))
136
acting on the system density operator, ρs has the form, DX [ρs] = XρsX† −
1ρ X†s X − 1X†Xρs. For time dependent coupling of system and environment,2 2
the collapse operator take time dependent f∣orms〉[114, 115]. The coll∣apse 〉operator
embodying the decay from the upper state ∣F ∣1(3) back to the state F0(2) takes
the form √X = g∗1(2)(t)a1(2) + γ1|F0(2)〉〈F1(3)|. A quantum jump effected by this
operator indicates the corresponding photon was not detected.
The density operator in Eq. 4.43 can be expanded in the partial basis of
the two virtual cavity populations, i.e. four basis states |n,m〉 with n,m = 0, 1
indicating the number of photons inside the cavity. This gives rise to a coupled
equation set of sixteen coefficients. The complete expansion can be found in eq.
4.47. With the transformation in eq. 4.50, we get back the equation set in eq. 4.46
for the generalized density matrix operators, ρab(t) with the substitution.
∫t
− 1 dt′|g ′1(2)(t )|22
u ∗α(β)(t) = g1(2)(t)e
t0 , (4.44)
which is the same result as found in [114, 115]. Inversion of the relationship in
eq. 4.44 gives away the method of varying the couplings g1(2)(t) in time so as to
generate a desired photon wavepacket uα(β)(t) [114, 115]
√ u∗∫ α(β)(t)g1(2)(t) = . (4.45)t
1− dt′|u ′ 2α(β)(t )|
t0
137
FIGURE 4.5. The steady state occupations of F2 and F4 level for the initial
state of both cavity having one photon. Here we chose γ1 = γ2 = γ3 = γ4 and
κ1 = κ2 = γ1/5.
4.5. Results
4.5.1. Exponentially decaying uα(t) and uβ(t)
√
With u (t) = κ e−κ1t/2
√
α 1 Θ(t) and uβ(t) = κ e−κ2t/22 Θ(t), (Θ(t) being the
Heaviside unit step function) either dynamical equation sets 4.46 or 4.43 as well
as quantities like absorption probabilities (eq. 4.33) can be solved analytically.
Through eq. 4.45, we find both couplings √g1(2)(t) = κ1(2) to be constant in
time, This is the only example which can be calculated with a time-independent
system Hamiltonian and collapse operators. As discussed previously, if the two
photons have significant temporal overlap, the second photon may get reflected
before the first is absorbed, and the molecule may end up in |F2〉 instead of being
raised all the way to |F4〉. If we gradually delay the second photon in time with
increasing delay periods (td) and calculate the steady state populations in |F2〉
and |F4〉 from the Liouvillian equation each time, we find that with a longer delay
the second photon is absorbed with higher probability (Fig. 4.5). The sum of
the two populations is always Pαabs(= 10/11), the probability of the first photon
138
being absorbed. For a delay td  1/γ1, the steady state occupation of the F4 level
reaches Pα βabsPabs(= 100/121), as expected.
However, the delay is not the only critical determinant of the absorption
probability of the second photon. (The absorption of the first photon is completely
independent of the second.) The closer to Pα the occupation in |F2〉 has risen when
the second photon arrives, the more efficient the absorption of the second photon
is. So, a longer width of the second photon wavefunction would also increase the
efficiency of the second photon absorption. The overlap between the two photon
wave functions determines the efficiency of the second photon absorption. By
making κ2 smaller we can make the second photon wave function longer in time.
In Figs. 4.6 and 4.7 respectively we plot the average occupation levels of |F2〉 and
of |F4〉 as functions of time for different values for κ2 that make the wavefunction
of the second (β) photon longer. The same colored curves from Figs. 4.6 and 4.7
add up to Pα = 10/11 for t → ∞. With the larger share of the second photon
coming into the detector after the first photon has already populated the F2 level,
the probability of a successful two-photon absorption rises.
4.5.2. Gaussian |u (t)|2α and |u (t)|2β
We numerically calculate the two photon absorption probability, ρ24,24(∞)
for two real Gaussian wavefunctions with varying standard deviations and the
second one delayed by different delays. The results are plotted in Fig. 4.8. A note
on the numerical method is in order here. For repeated calculations with different
values of the parameters, we use eq. 4.33, since it is less demanding than solving
the Liouvillian equation (eq. 4.43) many times. For the numerical calculation of
the principal value, we use
139
FIGURE 4.6. The occupation of the F2 level as a function of time for an initial
state of both cavities having one photon.
The decay rate κ2 of the second cavity (which determines the width in time of the
second photon) is varied. All rates γs are equal, and the rates κ1,2 are given in
units of γ1. The steady-state population of |F2〉 decreases with decreasing value of
κ2, that is, with increasing width of the second photon. The second photon moved
population out of level |F2〉.
FIGURE 4.7. Same as the preceding Figure, but plotting the occupation of level
F4. Here the steady-state population of |F4〉 increases with decreasing value of κ2
since the second photon is more effective at moving population from |F2〉 to |F4〉.
( ) ( )
1 1 1 1
P = + ,
ω 2 ω − i ω + i
and use a sufficiently small .
Curiously, we find that the efficiency plot is symmetric in their standard
deviations for any given delay. The efficiency for a standard deviation σ1 for
140
photon α and standard deviation σ2 for photon β for a given time delay td is the
same as for standard deviations σ2 for photon α and σ1 photon β. The probability
of the second photon absorption improves a lot as the delay is increased. For
some delays there is a peak efficiency for a certain standard deviation and falls
off slightly with even longer standard deviations.
(a) (b) 
(c) (d) 
FIGURE 4.8. ρ24,24(∞) plotted on the vertical z-axis against the standard
deviations of the α and β photons plotted on the two axes on the horizontal plane
for (a) no time delay, (b) 1/γ1,(c) 3/γ1, and (d) 5/γ1 delays (of the means/centres
of the waveshape) of the second photon, β.
4.5.3. Gaussian |uα(t)|2 and exponentially decaying uβ(t)
For completeness we also consider the “mixed” case of one Gaussian wave
packet (for the first photon) and an exponentially decaying wave packet (for the
second photon). The results are plotted in Fig. 4.9 As long as the two photon
wavefunctions overlap, we get a decrease in the ρ24,24(∞) value with increasing the
standard deviation of the α photon. Increase in κ or decrease in the time constant
of the β photon increases P βabs and is responsible for larger ρ24,24(∞).
141
(a) 
(b) 
FIGURE 4.9. ρ24,24(∞) plotted on the vertical z-axis against the standard
deviations of the α photon Gaussian waveshape on the x axis and the inverse of
the rate constant (κ) of the β photons plotted on the y axis for (a) no time delay,
(b) 0.5/γ1 delay (delay between the mean of the Gaussian and the onset of the β
photon waveshape).
4.6. Conclusions
We developed a fully quantum-mechanical model for a photon-number
resolving detector that can detect up to two photons by extending the model of
Ref. [69] to a five-level molecule. Moreover, we used two different methods for
treating the interaction of two photons with a quantum system—the methods
developed by [113] and [109] on the one hand, and by [114, 115] on the other—and
provided the explicit connection between the two. The former method allowed us
to obtain several analytical results in Section 4.2 that characterize our detector,
the latter method is very well suited for numerical calculations, as shown in
Section 4.5.
The model developed in [69] followed the lead by Refs. [60, 64, 65] in taking
inspiration from visual systems appearing in biology. It is an open question
whether our current extension of that model can be found in nature as well: in
142
particular, whether the specific step-wise two-photon absorption process we studied
here occurs in the human eye, just as simultaneous two-photon absorption does
occur [160].
We note two extensions of our work that may be interesting. The first
extension of our model is to another type of five-level molecule that would detect
just one photon, but it would be sensitive to polarization. From the initial state
we could either reach an excited state |F1〉 (as in our actual model) but also an
alternative excited state |F1′〉 (for an orthogonally polarized photon), which would
then decay to a different metastable state |F2′〉. If the signal produced in the latter
state is distinguishable from that produced by |F2〉, then this molecule would
perform a polarization-sensitive single-photon measurement. It is known that some
animals (insects, fish, birds) did develop polarization vision, see, e.g. Ref. [178].
Second, we focused here on the case of two distinguishable input photons,
with different frequencies. The case of two overlapping frequencies (relevant when
the two molecular transitions would have nearly equal transition frequencies)
would reveal two additional features. Both input photons would be able to
drive the two transitions, and the final expression for the two-photon absorption
amplitude would contain two terms, corresponding to two different time orders
in which the “first” and “second” photon could be absorbed. Those two terms
may interfere destructively. That type of effect is certainly interesting but known
[63, 179]. Moreover, the POVM element would involve projections onto entangled
two-photon input states, like it does for standard two-photon absorption [156].
143
Acknowledgments
This work is supported by funding from DARPA under Contract No.
W911NF-17-1-0267.
4.7. Emulating Photon wavepackets with Auxiliary Cavities
We outline the systematic process of deriving the transformations between
the generalized density matrix operators and coefficients (in the expansion in eq.
4.47) in the Hamiltonian formulation. Unlike Gheri et al, we do not introduce
a detuning of the auxiliary cavities for the emulation of the generalized density
matrix equations (Refs. [114, 115] did not either). Gheri et al addressed the
mapping for the problem of photons (one or few) in a single continuum. For the
problem of two photons residing in two continua (or even more complex scenarios),
the procedure outlined here can find the mapping between the two formalisms
(generalized density matrix operator and Hamiltonian formulation) systematically.
The generalized density matrix equations found for the system described in
sections III and IV are
144
ρ̇2,2(t) = L{
√
ρ2,2}(t) + [ γ3uβ(t)[ρα2(t), |F3〉〈F2|(t0)]
√
+ γ1uα(t)[ρβ2(t), |F1〉〈F0|(t0)] +H.c.]
ρ̇α,2(t) = L{ρα,2}
√
(t)− γ3u∗β(t)[ρα,α, |F2〉〈F3|(t0)] +
√ √
γ1uα(t)[ρ0,2, |F1〉〈F0|(t ∗0)]− γ1uα(t)[ρα,β, |F0〉〈F1|(t0)]
ρ̇β,2(t) = L{
√
ρβ,2}(t)− γ1u∗α(t)[ρβ,β, |F0〉〈F1|(t0)] +
√ √
γ3uβ(t)[ρ
∗
0,2, |F3〉〈F2|(t0)]− γ3uβ(t)[ρβ,α, |F2〉〈F3|(t0)]
√
ρ̇α,α(t) = L{ρα,α}(t)− γ1uα(t) [|F1〉〈F0|(t0), ρ0,α(t)]
√
+ γ ∗1uα(t) [|F0〉〈F1|(t0), ρα,0(t)]
√
ρ̇β,β(t) = L{ρβ,β}(t)− γ3uβ(t) [|F3〉〈F2|(t0), ρ0,β(t)]
√
+ γ ∗3uβ(t) [|F2〉〈F3|(t0), ρβ,0(t)]
√
ρ̇α,β(t) = L{ρα,β}(t)− γ1uα(t) [|F1〉〈F0|(t0), ρ0,β(t)]
√
+ γ ∗3uβ(t) [|F2〉〈F3|(t0), ρα,0(t)]
ρ̇0,2(t) = L{ρ0,2}(t)
−√γ u∗ √3 β(t)[ρ0,α, |F2〉〈F3|(t0)]− γ1u∗α(t)[ρ0,β, |F0〉〈F1|(t0)]
√
ρ̇0,α(t) = L{ρ ∗0,α}(t) + γ1uα(t)[ρ0,0, |F0〉〈F1|(t0)]
√
ρ̇0,β(t) = L{ρ ∗0,β}(t) + γ3uβ(t)[ρ0,0, |F2〉〈F3|(t0)]
ρ̇0,0(t) = L{ρ0,0}(t) (4.46)
D is the Lindblad dissipator superoperator and its explicit form depends on
the number and nature of the baths coupled to the system. Due to the coupling
of the total of four continua, we get four collapse operators for the Lindblad
dissipator superoperator, D, namely √γ1|F0〉〈F1|,
√
γ2|F2〉〈F1|,
√
γ3|F2〉〈F3|, and
145
√
γ4|F4〉〈F3|. The density matrix in the Hamiltonian formulation can be expanded
in the complete basis (time dependence restricted to the expansion coefficients):
ρ(t) =
ρ̃2,2(t)|0, 0〉〈0, 0|+ ρ̃α,α(t)|0, 1〉〈0, 1|+ ρ̃β,β(t)|1, 0〉〈1, 0|
+ρ̃β,2(t)|1, 0〉〈0, 0|+ ρ̃0,α(t)|1, 1〉〈0, 1|
+ρ̃2,β(t)|0, 0〉〈1, 0|+ ρ̃α,0(t)|0, 1〉〈1, 1|
+ρ̃α,2(t)|0, 1〉〈0, 0|+ ρ̃0,β(t)|1, 1〉〈1, 0|
+ρ̃2,α(t)|0, 0〉〈0, 1|+ ρ̃β,0(t)|1, 0〉〈1, 1|
+ρ̃0,0(t)|1, 1〉〈1, 1|+ ρ̃0,2(t)|1, 1〉〈0, 0|+ ρ̃2,0(t)|0, 0〉〈1, 1|
+ρ̃β,α(t)|1, 0〉〈0, 1|+ ρ̃α,β(t)|0, 1〉〈1, 0| (4.47)
Here a state |n,m〉 for n,m ∈ {0, 1} indicates the number of photons in the
two cavities, respectively. Since we start the cavities with one photon each, an
input photon in mode α or β will correspond to a cavity state |n,m〉 with n = 0
or m = 0 respectively (the photons have leaked out of their cavities). The F -
operators in the Hamiltonian and Lindbladian of eq. 4.43 act on the expansion
coefficients, ρ̃i,j(t) and the cavity mode annihilation a1, a2 operators act on the
basis elements |n,m〉. Using the expansion, eq. 4.47 in eq. 4.43, we get a set
of coupled differential equations for the 16 coefficients. For example, the two
146
coefficients, ρ̃β,2(t) and ρ̃0,α have the coupled equations,
(
ρ˙̃β,2(t) = )D√γ1|F0〉〈 | +D√ | 〉〈 | +D√F1 γ2 F2 F1 γ3|F2〉〈F3|
+D√ 1 2 2γ4|F4〉〈F3| ρ̃β,2(t)− |g2(t)| [ρ̃β,2] + |g1(t)| ρ̃0,α(t)2
√ √
+g1(t) γ1 [|F0〉〈F1|, ρ̃β,β(t)]− g2(t) γ3 [ρ̃β,α, |F2〉〈F3|]
− √g∗2(t) γ3([|F3〉〈F2|, ρ̃0,2(t)] (4.48)
ρ˙̃ 0,α(t) = D√ √γ1|F0〉〈F1| +D γ3|F)2〉〈F3|
−| 1g1(t)|2 − |g 22(t)| ρ̃0,α
2
√
+g1(t) γ1 [|F0〉〈F1|, ρ̃0,0(t)] (4.49)
The following transformation from the tilde operators gives us back the set
of differential equations in eq. 4.46 with the generalized density matrix operators
defined previously in eq. 4.24. This comes with the substitutions in eq. 4.45.
147
∫
ρ (t) = ρ̃ (t)e dt(|g1(t)|
2+|g2(t)|2)
0,0 0,0 ∫
dt|g (t)|2ρ 1α,α(t) = [ρ̃α,α(t) + ρ̃0,0(t)] e∫
2
ρβ,β(t) = [ρ̃ (t) + ρ̃ (t)] e
dt|g2(t)|
β,β 0,0
ρ2,2 = [ρ̃2,2 + ρ̃α,α(t) + ρ̃∫ β,β
(t)]
2 1 2
ρ (t) = ρ̃ (t)e dt(|g1(t)| + |g2(t)| )0,α 0,α 2∫
1 2
ρβ,2(t) = [ρ̃β,2(t) + ρ̃ (t)] e
dt |g2(t)|
∫ 0,α
2
1
ρ (t) = ρ̃ (t)e dt( |g1(t)|
2+|g 2
2 2
(t)| )
0,β 0,β ∫
1 2
ρα,2(t) = [ρ̃α,2(t) + ρ̃0,β(t)] e
dt |g1(t)|2
∫
1 2
ρ = ρ̃ e dt (|g1(t)| +|g2(t)|
2)
α,β α,β 2∫
1 2
ρ (t) = ρ̃ (t)e dt (|g1(t)| +|g2(t)|
2)
0,2 0,2 2 (4.50)
The coefficients in eq. 4.47 can actually be given the meaning of a density
matrix element with their usual meaning. The Hamiltonian formulation helps us
write the density operator with trace of 1 that embodies all the generalized density
operators in eq. 4.46 (the diagonal(off-diagonal) ones each have a preserved trace
of 1(0) each).
148
CHAPTER V
CONCLUSIONS
We talk about two kinds of amplifications. The first kind is the one Bohr
remarked on, classical measurements of a microscopic quantum observable need
to be the outcome of an irreversible amplification from it. The second kind
are specific unitary (reversible) evolutions of quantum mechanical operators
with explicit gains and noise additions. Quantum measurement theories cannot
completely eliminate the need for an interpretative process in the end of the
quantum mechanical process of the amplification. We do have answers to a number
of puzzles in the measurement process. Quantum mechanical decoherence can be
used to explain and interpret quantum measurement process in most parts. As
progress continues in quantum measurement theories, it behoves us to push the
efforts to write fundamental models and theories for technological platforms like
single photon detectors using our best understanding of nature at the time.
This dissertation accomplishes the task of embodying the entire process of
single/few photon detection with Hamiltonians as well as outlines the procedure
for analysing absorption probabilities. A few well known Hamiltonians are
discussed. The mechanism and physics inherent in these Hamiltonians are well
understood from cavity Quantum Electro-Dynamics. As such the methodology
described can serve as a guideline for designing photon detectors and their
amplification characteristics. The models are capable of depicting the entire
evolution composed of absorption, amplification and measurement. This is
an improvement over the status quo Hamiltonian description of single photon
detectors which discusses definite models for absorption stage only. A number
149
of Hamiltonians are discussed that can implement a wide range of quantum
amplification processes and measurement interpretations in the theoretical
frameworks known today. The measurement section is described with classicality
of large macroscopic signals. But it is compatible with the modern theory of
measurement induced decoherence and readily adaptable to quantum Darwinism
and emergence of classical objective reality theories.
Hamiltonians readily ensure preservation of bosonic canonical commutation
and fermionic canonical anticommutation relationships. Photon absorptions
leading the evolution into a macroscopic number of bosonic excitations in a mode
or macroscopic number of atoms or spins excited or generated can be modelled
with known Hamiltonian models that are explicitly compliant with quantum
mechanical evolution requirements. Bonus features like models with dissipative
phase transitions can make our models further protected against non-ideal
realizations. The macroscopic excitations generation gives us an output quantum
mechanical operator amplified with minimal noise contribution added. The other
desired attribute of near-perfect absorption is built in the absorber molecule in all
cases.
The absorber molecule can be designed with the near-perfect absorption
design into a five level atom that can absorb the two photons sequentially. The
versatility of our Hamiltonian formulation is verified by the model and suitability
of computations. The model can be adapted to implement polarization sensitive
single photon detection capability. The POVM elements show rich structure. If
the two photons have somewhat close central frequencies, the photons become less
distinguishable and the POVM elements involve projectors into entangled photon
pairs and connects with the standard two photon absorber physics.
150
On a more fundamental physics front, we get exciting new prospects
of device level modelling of modern quantum measurement theory and
interpretations. Theories of decoherence and quantum Darwinism can be
incorporated and more profound quantum measurement processes could be
conceptualized. Going beyond a single photon detection problem alone, the more
general problem of amplifying microscopic quantum mechanical observables
into objective reality can be touched upon. Concrete models of generating
macroscopic number of excitations or large signal amplitudes in a quantum
mechanical evolution can benefit the modern evolution of the ideas of the quantum
measurement problem.
151
APPENDIX A
HEISENBERG PICTURE EVOLUTION OF OPERATORS
Knowing the Hamiltonian essentially indicates knowing the input-output
relationship for the amplification evolution, as it comes down to the formal
solution of the Heisenberg equation of motion. The density matrix evolves
accordingly in the Schrödinger equation [180],
ρ(t) = U(t)ρ(0)U †(t), ρ(t) = ρ(0); Schrödinger picture (A.1)
A(t) = U †(t)A(0)U(t), A(t) = A(0); Heisen ∫
berg picture (A.2)
t
U(t) = T exp −i H(t′)dt′ , (A.3)
0
where T is the Dyson time-ordering operator. The decomposition of the output
operator in terms of the input operators tells us the gain of the amplification
as well as noise operators and their contributions. From the point of view of
amplification, computation of eq. (A.2) is of paramount importance. Computation
of eq. (A.2), even with convergent series expansions, can answer the important
question of which operators (signal and noise) do and do not contribute to the
output signal operator. Now we look at methodical ways of counting or enlisting
the operators that should arise in the decomposition of an output operator.
There are methods of solving eq. (A.1) in the Schrödinger picture with series
expansions. For an explicitly time dependent Hamiltonian (time dependent in the
Schrödinger as well as Heisenberg picture), the evolution operator in eq. (A.3) can
152
be expanded in a Dyson series [180],
 ∫ tU(t) = T exp −i H(t′)dt′
∫ ∫ ∫ 0t t t1
− (−1)
2
= 1 i dt1H(t1) + dt1 dt2H(t1)H(t2) + ...
2!
0 0 0
The Hamiltonian does not necessarily commute at different times, i.e.
[H(t1), H(t2)] 6= 0, and the time ordering operator ensures t1 ≥ t2 ≥ t3 ≥ t4....
Average Hamiltonian Theory (AHT) ([180]) and Magnus expansion ([181])
tries to calculate an average solution for a certain time T [182],
 ∫ t
U(t = T ) = T exp−i H(t′)dt′ = e−iHavgT (A.4)
0 ∑
with Havg = Hj
∫ j=0T ∫T ∫t1
(0) 1 1 iH = dt1H(t1), H
(0) = dt1 dt2[H(t1), H(t2)]
T T 2!
0 0 0
Therefore, the computation of the operator at some point in time T, can be
facilitated through
A(t) = U †(t)A(0)U(t) = eiHavgTA(0)e−iHavgT (A.5)
For a Hamiltonian constant in time (in the Schrödinger picture), we trivially have
the same form for the evolution of an operator in the Heisenberg picture, from eq.
(A.2)
A(t) = U †(t)A(0)U(t) = eiHTA(0)e−iHT (A.6)
153
Now, Havg needs to be calculated for each T. And later we see an approximate
evaluation of an operator evolved till T with Havg to a given order in T.
So, numerically the expansion to be mentioned soon is not very useful for
Hamiltonians with explicit time dependence. But the expansion into powers of
T still gives us the structure of evolution and an approximate magnitude for A(t)
at T.
With labels indicating the picture a operator is represented in, we write out
U(t) = e−iHt/~
AH(t) = U
†(t)ASU(t) (A.7)
d i
AH(t) = [H,AH(t)], (A.8)
dt ~
where H is understood to be a Hamiltonian constant in time (in the spirit of eq.
(A.5) and eq. (A.6)).
A.1. Open system dynamics as a homomorphism for operators
Structure preserving maps in algebra and group theory are called
Homomorphism. Fermionic and Bosonic creation and annihilation operators
generate CAR (Canonical Anticommutation Relation) and CCR (Canonical
Anticommutation Relation) algebras respectively. Interactions of two level
and three level atoms can be represented by the two and three dimensional
representation of SU(2) and SU(3) Algebra respectively. The irreversible
transduction of photons can be emulated with the quantum mechanical
rulebook where the system evolution is tracked disregarding the evolution of the
environmental degrees of freedom. Mathematically it amounts to eliminating the
154
bath operators in favour of system operators. However, all quantum mechanical
rules hold in the back of our minds in the description with the all operators and
variables kept track of.
[AB]H (t) = e
iHt/~A e−iHt/~eiHt/~B e−iHt/~S S = AH(t)BH(t) (A.9)
A simple reflection of the Heisenberg evolution of operators eq. (A.9) demands
that the calculated system operator dynamics when appended with the bath
operators we have integrated out must preserve its structure. Mathematically, it
is equivalent of a homomorphism. And therefore CCR, CAR, SU(3) etc. algebra
are expected to be preserved fully in the evolution in the complete space of
‘system+bath’ operators. With the bath coupling for Bosonic operators, the
following general form can be found which with a condition of the white noise
correlations ([bin(t), b† ′ ′in(t )] = δ(t − t )) for the input field operator, bin(t) always
conform to Bosonic commutation relations [183].
∫t
γ √ γ
a(t) = a(t0)exp[−(iωa + )(t− t0)]− γ dt′bin(t′)exp[−(iωa + )(t− t′)], t > (t0A.10)
2 2
t0
155
Eq. (A.9) ensures the preservation of commutation relationships in the Heisenberg
picture at the same time,
[AH(t), BH(t)] = AH(t)BH(t)−BH(t)AH(t)
= eiHt/~A e−iHt/~eiHt/~B e−iHt/~ − eiHt/~S S BSe−iHt/~eiHt/~A e−iHt/~S
= eiHt/~A B e−iHt/~ − eiHt/~( S S BSASe
−iHt/~
)
= eiHt/~ A −iHt/~ −iHt/~SBSe −BSASe
= eiHt/~ (ASBS −BSAS) e−iHt/~
= eiHt/~ ([A ,B ]) e−iHt/~S S
= [AS, BS]H(t)
is the Heisenberg evolution of the commutator in the Schrödinger picture
operators.
A.2. A Complete Basis
The linear space of operators associated with the Hilbert space of {|na〉 ⊗
|Fj〉} is L(S). A complete basis for L(S) is succinctly written as {|q〉〈p|}. The
time evolution of each element,|q〉〈p| of the basis can facilitate the calculation of
correlation function of interest in the system.[184] |q〉〈p| can be written out more
explicitly,
|q〉〈p| = |nq, Fq〉〈np, Fp|
Since any given operator (for example |F1〉〈F1| or â) can be written out
in the basis at t = t0, knowing the time evolution of all |q〉〈p|(t) produces the
156
time evolution of the operator. The evolving operator will have components
along different static basis elements, |k〉〈l|(t0). The time varying component, i.e.
coefficient of |q〉〈p|(t) along |k〉〈l|(t0) is Cqpkl .
∑
|q〉〈p|(t) = Cqpk,l(t)|k〉〈l|(t0) (A.11)
k,l
The coefficients, Cqpk,l(t) can have interesting physical interpretations in problems
like photodetection and amplification [69].
A.2.1. Evolution of basis elements, (|j〉〈i|)(t) in L(S)
Any operator in the linear space of operators acting on Hilbert space, H
can be expanded in terms of basis elements (|j〉〈i|)(t). Their dynamics can be a
powerful computational tool in the calculation of quantities of interest. We recount
some of the equations from ref. [184]. The density matrix, ρ = TrE (ρSE), obeys
the linear master equation,
d ∑
ρij(t) = Mij,i′j′ρi′,j′(t) (A.12)
dt
i′,j′
Now,
TrS {ρ(t)|j〉〈i|(t0)} = TrS {ρ(t0)(|j〉〈i|)(t)}
=⇒ ρij(t) = TrS {ρ(t0)(|j〉〈i|)(t)} = 〈(|j〉〈i|)(t)〉
=⇒ ρij(t) = 〈(|j〉〈i|)(t)〉 (A.13)
157
Now, the identity,
∑ ∑
〈A(t)〉 = ∑ρij(t0)Aji(t) = ρi∑j(t)Aji(t0) = TrρAij ij
〈(|j〉〈i|)(t)〉 = ρij(t)|j〉〈i|ji(t0) = ρij(t0)|j〉〈i|ji(t)
ij ij
(A.14)
For the case, |j〉〈i|ji(t0) = δr,iδc,j ad ρij(t0) = δr,iδc,j,
(ρij(t))S = (|j〉〈i|(t))H
Consequently, if the master equation is,
d ∑
ρij(t) = Mij,i′j′ρi′,j′(t) (A.15)
dt
ij,i′j′
we can use eq. (A.13) to write,
d ∑〈(|j〉〈i|)(t)〉 = Mij,i′j′ρi′,j′(t) (A.16)
dt
ij,i′j′
Eq. (A.16) is written only to be corrected with the noise operators so as to satisfy
the commutation relations,
d ∑〈(|j〉〈i|)(t)〉 = Mij,i′j′ρi′,j′(t) + Fij(t) (A.17)
dt
ij,i′j′
A.3. Expansion of evolving operators in orders of time
Equations (A.7) and (A.8) enable us to write an operator evolving in time
at a particular time with evolution operators having constant Hamiltonians. We
158
define the notation,
[H,AS] = A
H
S [H,AH(t)] = A
H
H(t)
[H, [H,A ]] = AHHS S [H, [H,AH(t)]] = A
HH
H (t)
[H, [H, [H,A ]]] = AHHHS S [H, [H, [H,AH(t)]]] = A
HHH
H (t)
.....................................................
The solution of eq. (A.8) is,
∫t
i
AH(t) = [H,AH(t1)]dt1 (A.18)~
0
Of course, we can substitute the integral for AH(t) in the right hand side, as
many times as we wish,
∫t
i
AH(t) = [H,AH(t1)]dt1 (A.19)~
( )2 ∫t ∫ 0t1
i
AH(t) = [H, [H,AH(t2)]]dt2dt1 (A.20)~
( ) ∫ 0 0t ∫t1 ∫t3 2
i
AH(t) = [H, [H, [H,AH(t3)]]]dt3dt~ 2
dt1 (A.21)
0 0 0
159
In the notation chosen,
∫t
i
AH(t) = A
H
~ H
(t1)dt1
( )2 ∫ 0t ∫t1
i
AH(t) = A
HH
H (t2)dt2dt~ 1
( ) ∫ 0 0t ∫t3 1 ∫t2
i
AH(t) = A
HHH
H (t3)dt3dt2dt~ 1
0 0 0
A.3.1. Expansion in unevolving Schrödinger operators
Continuing from eq. (A.7),
A (t) = eiHt(/~A)e−iHt/~H S ( )2 ( )3
it 1 it 1 it
= 1 + ( ) [H,AS] +( ) [H, [H,(AS])] + [H, [H, [H,AS]]] +(.A...22)~ 2! ~ 3! ~2 3
it H 1 it 1 it= 1 + A + AHH( )S S + A
HHH
S + ...∑ ~ 2! ~ 3! ~∞ n1 it n
= AH n+1
n! ~ S
+ E(t ) (A.23)
n=0
Eq. (A.23) can be verified citing identities in literature [185],
p−1
eτA1Be−τA
τ
1 = B + [A1, B]τ + ...+ [A1, ..., [A1, B]...] +O(τ
p) (A.24)
(p− 1)!
where the error term, explicitly is,
∫τ
τ p−1
O(τ p) = dτ e(τ−τ2)A12 [A1, ..., [A1, B]...]e
−(τ−τ2)A1 2 (A.25)
(p− 1)!
0
160
Therefore, the error term in eq. (A.23) would be
( ) ∫it/~p+1
i pi iE t(tn+1) = dt e ~ (t−t2)H2 [H, ..., [H,A ]...]e− ~ (t−t2)H 2S (A.26)~ p!
0
A.4. Binomial Expansion
The terms in the expansion eq. (A.23) resemble a binomial expansion,
∑n ( )
AH
n n
S = (H − A )
n
S = (−1)rHn−rAr (A.27)
r S
r=0
For example, the term, 4AHS in the expansion of eq. (A.23),
4
AH = AHHHHS S = [H, [H, [H, [H,AS]]]]
= HHHHAS − 4HHHASH + 6HHASHH − 4HASHHH + ASHHHH
The calculation of HnAS terms completely enumerates all the possible time
stationary Schrödinger operators that can show up in the closed system evolution
of a an operator.
A.5. Example time independent Hamiltonian: A Cavity mode Driving
a 2LS
The cavity mode will be assumed to have a maximum occupation of 1. The
driven system is a Two-Level System (TLS) with the two levels, F0 and F1(F1 is
161
higher in energy).
H(t) = H = Hsys(t) +Hint(t),∫ an invariant of time∞
H † †sys(t) = ~ωaa (t)a(t)− ~δ0,1|F0〉〈F0|(t) + ~ dω|ω|b (ω, t)b(ω, t)
∫ ( −∞∞ † )Hint(t) = i~ dωκ1(ω) b (ω, t)a(t)− a†(t)b(ω, t)
∫ −∞∞ ( † − )+i~ dωκ2(ω) b (ω, t)|F0〉〈F1|(t)e iωτ − |F 〉〈F |(t)b(ω, t)eiωτ1 0
−∞
Hsys(t) or Hint(t) are not invariants of time; but their sum is.
Integrating the equation of motion for b(ω, t), the bath mode annihilation
operator can be found to be,
∫t [√ √ ]
b(ω, t) = b (ω)e−iω(t−t ) ′
κ ′ −iω(t−t′) γ0 1
0 + dt a(t )e + |F0〉〈
′
F |(t′)e−iω(t−t +τ)1 (A.28)
2π 2π
t0
Hint(t) is necessarily time dependent, H is the only operator which can be taken to
be conserved in time. Substitution of eq. (A.28) can give the Hint(t).
√ ( )
( )Hint(t) = i(~ κγ1 a†(t− τ)|F0〉〈F1|(t)− |F1〉〈F0|(t)a(t− τ)√ )
+i~ κ b†
√
in(t)a(t)− a†(t)bin(t) + i~ γ b
†
1 in(t− τ)|F0〉〈F1|(t)− |F1〉〈F0|(t)bin(t− τ) (A.29)
Hsys(t) can be evaluated at t = t0, substituting eq. (A.28) for t = t0,
∫∞
Hsys(t0) = ~ω †aa (t0)a(t †0)− ~δ0,1|F0〉〈F0|(t0) + ~ dω|ω|b0(ω)b0(ω)
−∞
162
∫∞
The quantity, ~ dω|ω|b†0(ω)b0(ω) is a constant and do not contribute to any
−∞
dynamical coupling between operators evolving in time in the Heisenberg picture.
Taking that last number to be zero, we have our Hamiltonian, (ignoring the
delay, τ)
H = H(t0) = Hsys(t0) +Hint(t0)
=( ~ω a†
√ ( )
a (t0)a(t0)− ~δ0,1|F0〉)〈F0|(t0) + i(~ κγ1 a†(t0)|F0〉〈F1|(t0)− |F1〉〈F0|(t0))a(t0)√
+i~ κ b†
√
in(t0)a(t
† †
0)− a (t0)bin(t0) + i~ γ1 bin(t0)|F0〉〈F1|(t0)− |F1〉〈F0|(t0)bin(t0) (A.30)
A.5.1. Matrix representation of a system Hamiltonian
Making open system dynamics explicitly unitary (conforming to quantum
mechanical rules) requires tracting the bath operators and a combinatorial
counting problem that may be facilitated by graph theoretical algorithms. Konig
digraphs can be a very useful representation for matrices and matrix multiplication
[186]. For a closed system evolution problem without bath contributions Konig
digraph methods can be formulated most straightforwardly. Our H will be,
( )
H = ~ω †aa (t0)a(t0)− ~δ0,1|F0〉〈 |
√
F0 (t0) + i~ κγ a†1 (t0)|F0〉〈F1|(t0)− |F1〉〈F0|(t0)a(t0)
In the rotating frame defined by,
{ }
U(t) = exp i~ω a†a a− i~δ0,1|F0〉〈F0| ,
the Hamiltonian would be even more compact,
√ ( )
H = i~ κγ a†1 (t0)|F0〉〈F1|(t0)− |F1〉〈F0|(t0)a(t0)
163
In the basis {|na〉 ⊗ |Fj〉},
 0 0 0 0 0 0  0 0 0 −1 0 0
√ 0 0 0 0 0 0

H = i~ κγ  1  (A.31)0 1 0 0 0 0  0 0 0 0 0 0
0 0 0 0 0 0
A.5.2. Weighted Konig Digraph
Hamiltonian matrices are always Hermitian, so they would necessarily have
directed edges going in both directions between two nodes (weighted by complex
conjugate numbers). We choose the following Hamiltonian (chosen to be more
stuffy than the one in eq. (A.31) ) to demonstrate the method. The elements of
the Hamiltonian are taken to be -1 for simplicity.
 ∆1 0 0 −th 0 −th   0 0 0 0 0 0 


0 0 0 0 0 0 
H =  (A.32)


−th 0 0 ∆ 4 0 −th
  0 0 0 0 0 0 
−th 0 0 −th 0 ∆6
Now for the calculation of some Cqpk,l(t) (in eq. (A.11) ), for example, C
61
6,4(t),
we have to calculate all the different contribution from t-powers from the closed
walks from 6 to 4 (k to l). For the coefficient of tn, we find the weight of all walks
164
FIGURE A.1. The Digraph (brown edges) for the Hamiltonian in eq. (A.32) along
with the edge |6〉〈1| (orange). The edge 1-6 of the Hamiltonian is not shown (or is
hidden under the edge |6〉〈1| ). The edges have arrows on both ends (bidirectional
edges or two edges) because of the hermiticity of the matrix.
of length (n+1) from the node k to l (in fig. (A.2) ). If |6〉〈1| appears multiple
times (if only the Hamiltonian has that edge itself), we have multiple permutations
to be summed; each time with a factor of (−1)m; m is the serial (starting from 1,
m ∈ N ) of the edge |6〉〈1| in the composite (cascaded) graph (fig. (A.2) ). The
general relationship is,
∑∞ ( )n
qp 1 iC (t) = δ + tn(−1)mW n+1k,l k,l n! ~ kl
n=1
W n−1kl = Weight of the walks of length (n+1)
from k to l through the edge p → q
m = the serial of the edge p → q in the walk, m ∈ N
If i=j, and there is a self-loop at vertex i, the walk must include that loop (as an
edge) at a serial in the walk.
Fig. (A.2) makes the method more obvious; however the weighted Konig
di-graph (fig. (A.1) ) of the Hamiltonian is all we need for the calculation of any
Cpqk,l(t) coefficient.
165
FIGURE A.2. The composite graph (ref. [186]) for H4|6〉〈1|H2 (The arrows of the
graph-edges not shown). H is from eq. (A.32)
The QuTiP ([187]) function mesolve is used for calculating the expectation
value of any operator in L(S) in the open system evolution. However, it operates
on exponentiating matrices and runs into memory problems for large Hilbert space
sizes. Knowledge of all |q〉〈p|(t) calculated through algorithms of graph theory
might make it possible to calculate these expectation values for very very large
system dimensions with known error terms.
The coefficients Cqpk,l(t) can be calculated recursively. Here is a pseudo code
expressing this idea.
166
Algorithm 1: Recursive function program for A(t)
Input: A(t0), H, N(maxm order of t)
Output: A(t)
1 Initialize n to 1
2 A(t) = recursive function(n).
3 Recursive function(n):
4 Step 0: Set An to 0. m to 1.
5 Step 1: take the product C = HH..A ..HHH of length (n+1) with A at the
serial of m.
6 Step 2. Add (−1)mC to An.
7 if m < N:
8 m → m+1.
9 goto Step 1.
10 else:
11 A(t0) → A(t0) + (it/~)nAn
12 Step 3:
13 if n < N: resursive function(n+1)
14 else: return A(t0)
167
APPENDIX B
AMPLIFICATION MECHANISMS AND HAMILTONIANS
We have discussed a quantum amplification mechanism implemented with a
driven quantum Harmonic oscillator. We now contemplate a host of other cavity-
QED Hamiltonians that can implement such amplification mechanisms.
Ref. [122] discusses a class of nonlinear amplifiers that add a single noise
mode for the amplification evolution.
âout = gf̂in + b̂in (B.1)
f̂in is the quantum mode carrying the signal and b̂in is the added noise mode. It
was facilitated with the interaction Hamiltonian,
( )
ĤI = −iκ f̂ †b̂− f̂ b̂† (B.2)
The input-output transformaton is achieved with a normal operator f̂in
(an operator that commutes with its adjoint). In fact f need not be a modal
operator, it can be a qubit operator such as |0〉〈1| − |1〉〈0| which is normal but
non-Hermitian. Therefore a qubit population operator such as σ̂z can be measured
with minimum noise addition.
Ref. [122] summarizes modern techniques in qubit readout techniques with
the single interaction Hamiltonian in eq. (B.2). Conditional displacement of an
oscillator based on the occupation of a qubit is exploited in some nondemolition
measurements of qubits [188, 189, 190]. Fundamentally, the oscillation frequency
of a cavity mode is dynamically modulated by the occupation of the qubit. The
168
Langevin equation of the cavity mode reads [189],
√
a˙̂ = − 1 1i g̃zσ̂z − κâ− κâin (B.3)
2 2
The input-output boundary condition reads,
√
âout = âin + κâ (B.4)
The mean of the output is,
[ ]
1
αout = 〈
ig̃z
âout〉 = −√ 〈σ̂z〉 1− e− κt2 (B.5)
κ
The homodyne detection of the output signal corresponds to the measurement
operator, M̂(τ), (τ being the integration time and φh the homodyne angle).
∫τ√ [ ]
M̂(τ) = κ dt â†out(t)e
iφh + âout(t)e
−iφh (B.6)
0
For long measurement times, the signal to Noise Ratio (SNR) grows as square root
of time [189, 190]. The interaction Hamiltonian is a special case of the general
relationship in eq. (B.2).
The amplification mechanism formulated in chapters 3 and 4 are different
from the one in [122] in the sense that they include the projector into a third
Hilbert space,
âout = gn̂in ⊗ |ψ0〉〈ψ0|+ b̂in (B.7)
|ψ0〉〈ψ0| = |F2〉〈F2| in the single photon model. The shelving state |F2〉 drives
the amplification process. Since the operator, |ψ0〉〈ψ0| commutes with the
169
measurement Hamiltonian, it counts as a Quantum Non Demolition (QND)
observable. In chapter 4, it was established that to project onto photon
wavepackets, the temporal shape of the occupation of the absorber levels are
necessary. We now consider physical amplification mechanisms where such
quantum signals (carried by a projector operator) can be amplified into large
quantum signals carried by a large number of spins or atoms. Readout techniques
for such large quantum signals into classical signals may enable us to learn the
growing occupation levels in the shelving state within the liberties of quantum
mechanics.
The projector operator of the shelving state |ψ0〉〈ψ0| can be used in other
cavity-QED Hamiltonians to achieve the desired change in some parameter value
that will drive an amplification. A driven version of the Jaynes-Cummings model
can lead to a coherent state steady state ([191]). Therefore, we are able to use
the model for amplification of a change in a parameter, µ caused by an absorbed
photon.
ρ̇ = Lρ = −i[H̃ , ρ] +D√ √dJC κddρ+D κτ τzρz
1
H̃dJC = (∆− ωa)(1 + τz) + λ(τ+d+ τ−d†) + µ(t)(τ+ + τ−) (B.8)
2
µ(t) = µ|ψ0〉〈ψ0|(t) (B.9)
For, µ > λ, as the shelving state grows its occupation, we get to the coherent state
(with amplitude µ/λ, ([191]) ) steady state for the cavity mode. H̃dJC is written
in a rotating frame, ωa is the angular frequency of the ac field driving the qubit τz,
τ are the Pauli spin operators representing a two level atom. The steady state of
the cavity mode, d̂ would be our pointer state. The Heisenberg picture evolution of
170
|ψ0〉〈ψ0|(t) is unchanged if terms are added to the Hamiltonian that it commutes
with. And the evolution of other operators can be calculated from the Hamiltonian
H̃dJC in eq. B.8.
The detuning of different photon signal can lead to different steady state
occupation of the absorber atom. In fact, the switching between distinct steady
states brought about by varying the detuning across certain critical value was
utilized to fashion a quantum transducer in a Kerr resonator.[192] There are a
host of non-linear Kerr models where the steady-states are Schrödinger cat states
of coherent states [193]. These systems involve the physics of driven-dissipative
phase transitions [194]. The heightened susceptibilities of the degrees of freedom
of a system close to a critical point has long been investigated for applications in
metrology, estimation and sensing[195, 196]. First order phase transitions have also
attracted much attention in schemes for amplifying weak signals.[71].
B.1. From master equation to rate equation
Dicke Hamiltonian describes the dynamics of a two level atom/ qubit coupled
to a single mode of electromagnetic cavity mode.
HDicke = ω0Sz + ωa
†a+ λ(a+ a†)(σ+ + σ−)
The Jaynes Cummings Hamiltonian discards terms far from resonance,
H † + † −JC = ω0Sz + ωa a+ λ(aσ + a σ )
The emission from the qubit and the populations are found to be discrete
in time (in a single shot measurement) and can be calculated with the quantum
171
trajectory method. However, the solutions and measurement signals can both
be continuous (solutions from the master equation approach) for an ensemble
of qubits driven and decaying identically. Such a system is described by the
Tavis-Cummings Hamiltonian, where N qubits are coupled identically to a single
electromagnetic cavity mode.
H †
λ † + −
TC = ω0Jz + ωa a+ √ (a+ a )(J + J )
N
The collective spin-operators abide by the SU(2) algebra just like the individual
spin operators [125, 182, 197, 198].
∑N
Ji = σ
j
i , i = x, y, z;
∑ j=1N ∑N ( )
J = σj = σj± ± x ± iσjy = Jx ± iJy
j=1 j=1
[Jx, Jy] = iJz, [Jy, Jz] = iJx, [Jz, Jx] = iJy
[Jz, J±] = ±J±, [J+, J−] = 2Jz (B.10)
The simultaneous eigenstates of J2 and Jz are called Dicke states. With ~ = 1,
J2|j,m〉 = j(j + 1)|j,m〉
√ Jz|j,m〉 = m|j,m〉
J±|j,m〉 = (j ∓m)(j ±m+ 1)|j,m〉
where j ≤ N/2, and |m| ≤ j, j,m multiples of half-integers for spin-1/2 particles.
jmin = 0 for even M (number of TLS) and 1/2 for odd N.
172
The Dicke states are permutationally symmetric quantum superpositions of
spin tensor product states with the same number of excited and unexcited spins.
| N 1
∑
j,m = k − 〉 = √( ) |j1, 1〉...|jk, 1〉|jk+1, 0〉...|jN , 0〉 (B.11)2 N
p
k
= √(1 ) [ ⊗ − ]S |e〉 k ⊗ |g〉(N k) (B.12)
N
k
The Symmetrization operator S enlists all permutations with the same number of
excited (|e〉) and unexcited (|g〉) spins in the sum. The Dicke states with j=N/2
are called the symmetric Dicke states.
Quantum information can transfer from a system to the monitoring
environment. While single-shot measurements yield a discrete in time quantum
jumps, an ensemble of atoms under the same drive and coupling can accumulate
temporal shapes of “click”s that are sufficiently continuous or smooth. The
populations of the master equation’s density operator becomes state populations in
a rate equation. If the environment does not distinguish between individual spins,
the dynamics is split into individual symmetric subspaces evolving separately. The
populations may be labeled by the z-component of the Spin operators m, and we
get a rate equation model for the state populations, pm = ρm,m.
d
pm = −Γm→m±1pm + Γm−1→mpm−1 + Γm+1→mpm+1
dt
if the population transfer with state of m is limited to states with m ± 1. The
symmetries (such as collective spin operator interactions only) in the meter-
environment Hamiltonian also facilitates Decoherence Free Subspaces (DFS) that
can protect against environmental errors [182].
173
B.1.1. The observable |ψ0〉〈ψ0|
The “generalized Dicke model” of ref. [199] shows utility in engineering
arbitrary Dicke states in the steady state.
λ U
H † †gen. Dicke = ω0Jz + ωa a+ √ (a+ a )(J+ + J−) + Jza†a (B.13)
N N
We propose a scheme for amplifying a monotonically increasing amplitude of a
occupation operator through the transformation in eq. (B.7).
λ
H † † + −
U †
gen. Dicke pd = ω0Jz + ωa a+ √ (a+ a )(J + J ) + |ψ0〉〈ψ0| Jza a (B.14)
N N
Since |ψ0〉〈ψ0| commutes with Hgen.Dickepd, it’s equation of motion is not changed
with the addition of the “meter” spin ensemble. Therefore the solution for
|ψ0〉〈ψ0|(t) we obtained before still holds. For any of the collective spin operators,
J∗ ∈ {Jx, Jy, Jz, J+, J−}, we get the Heisenberg equation of motion,
d
J∗ = i [Hgen. Dicke pd, J∗] (B.15)
dt
The derived equations of motion would be exactly the same ones as the ones
derived from the following Hamiltonian,
† λ U(t)H †gen. Dicke spin = ω0Jz + ωa a+ √ (a+ a )(J+ + J−) + Jza†a (B.16)
N N
where, U(t) = U |ψ0〉〈ψ0|(t).
174
− i γ↓ γΦ γ↑ρ̇ = [H, ρ] + LJ− [ρ] + LJz [ρ] + LJ+ [ρ] (B.17)~ 2 2 2
As long as the master equation has system Hamiltonian and collapse operators
only in terms of collective operators Jx, Jy, Jz, J+, J− and the system was
initialized in a symmetric Dicke state the evolution is confined to the (N+1)
symmetric Dicke states. For a closed system evolution (without any bath
coupling), we can solve the Schrödinger equation itself instead of eq. (B.17),
d
i |ψ〉(t) = Hgen. Dicke spin(t)|ψ〉(t) (B.18)
dt
since it is computationally much less demanding. However, decoherence must be
introduced for the purpose of measurement as discussed in chapter 1. The closed
system results can serve as a guiding measure and are easier to compute.
B.1.2. Measurement Models
A macroscopic number of spins/atoms/qubits may cause a discernible
classical signal. But to be more precise about the measurement model, there
must be a transfer of information from the “meter” (quantum system) to the
environment. Our meter here is a large number of spins. The model would
require introduction of decoherence to be a viable model for measurement. If
only collective emission (or other collective operators) are considered, the Hilbert
space size can be restricted to (N+1). It still is a sizeable and time-costing job
computationally.
175
B.1.2.1. Model 1
As discussed in ref. [182], in the semiclassical approach to Nuclear Magnetic
Resonance (NMR) physics, the spin magnetization model is quantum mechanical
and the NMR detection coil is treated semi-classically. So with the variation of the
spin magnetic field calculated from the quantum mechanical Dicke model, we can
calculate the emf induced in the coil,
∫
d
emf = − B̂ ~1.M(t)dΩ (B.19)
dt
B.1.2.2. Model 2
The weak measurement model introduced in ref. [182] can serve as a method
to extract the classical readout 〈Jz〉(t). While strong measurement model demands
the 〈Jz〉(t) value to be repeated in measurements done in quick succession, it
does not necessarily happen in the weak measurement model. In fact, in the weak
measurement model, the measurement device is less precise and the spin ensemble
collapses to a few manifolds of m centred around m0. The conditional probability
distribution p(m|m0) is a distribution with mean m and width w. The POVM
elements, E = M †m M is written as,
l=∑N/2
Em = D(m, l)Πl (B.20)
l=−N/2
In one model,
( ∑ )2 2
D(m, l) = 1/ e−
(k−l) − (m−l)
2w2 e 2w2 (B.21)
k
176
If w is taken to be vanishingly small, D(m, l) becomes a delta function, δ(m − l).
More details of the weak measurement model can be found in ref. [182].
B.1.2.3. Machine Learning/Deep Learning Algorithms
Given that measurement models can be found that enables us to learn
the temporal shape of 〈Jz〉(t), we still have to determine the theoretical shape
of U(t) or ψ0〉〈ψ0|(t) that produced it. Modern Machine learning and Deep
Learning algorithms are one of doing that. A large number of theoretical 〈Jz〉(t)
and the corresponding U(t) can be derived from the theory and used to train.
Experimentally observed 〈Jz〉(t) can lead us to the corresponding U(t).
B.1.3. Fermionic Quantum Amplification
Inverse of the Jordan-Wigner transformation and the transformation
itself are clever ways of adorning a spin with fermionic statistics and vice-versa.
j∑−1 j∑−1
−iπ a†ka
†
+ k
−iπ akak
σ = e a†, σ+ = e a† †k=1 k=1 zj j j j, σj = 2ajaj − 1 (B.22)
Instead of measuring 〈Jz〉(t), the problem can be mapped mapped to a problem of
counting fermions i.e. the number of fermions changing in time would indicate the
source U(t)’s temporal profile.
B.1.4. Single mode amplification
Ref. [1] paper concluded that amplifying in a single mode will outperform
multi-mode amplification. So, a collective mode of a large number of fermions or
177
spins would have superior signal to noise characteristic performance over multi-
mode amplifiers.
B.2. New class of Phase Preserving Linear Amplification
Very recently, from ref. [121], we have learnt of new phase preserving linear
amplifiers (which are not simulable by linear parametric amplifiers). Hamiltonians
can be found for achieving these new kinds of phase preserving linear amplifiers.
d
â = (κ↑ − κ↓) â†â†ââ+ 2κ↑â
dt
We find the Hamiltonian,
i
Hnl = (κ↑ − κ↓) â†â†â†ââ+ 2iκ †↑â â
3
B.3. Quantum Darwinism
Quantum states are understood to be fragile against measurement (they
collapse to eigenstates) or even the smallest leak of information that necessarily
‘reprepare’s it in states consistent with the leaked information [200]. The
measurement and amplification models discussed in this dissertation use it to
advantage. The eigenstates of the bosonic annihilation operator (the coherent
states) in the case of collecting the excitations in the mode d (leaked from mode
c), or the eigenstate of the z-component of the collective spin operator are singled
out, among other states while the system leaks out information about itself. Phase
relations between eigenstates or these “pointer” states are destroyed. Decoherence
178
solves the puzzle of non-observationality of quantum superposition in measurement
theory [200].
The second part of the measurement problem, the emergence of objective
reality, is explainable in the setting of ‘Quantum Darwinism’ as well. In the
coupled evolution of the system, S and environment, E ; information about S is
accumulated in E . Although in the numerical result presented, E is ‘traced out’
and is inaccessible, our observers (plural) are eavesdropping on it, None of whom
need to access the entire E . Only fragments of E suffice to reveal the eigenstates
they need to observe. This is how classical objective reality may emerge. The
eigenstates mentioned are capable of producing multiple informational offspring or
inserting multiple record into E at a robust rate. The proliferation of information
throughout E , enables our observers (plural) to collect it from shards and pieces of
E and agree about the observations.
The models in this dissertation chooses system states robust against
environmental decoherence as both the ‘meter’ states for the absorber molecule
and ‘einselected’ (environment induced superselected) pointer states amenable to
classical readouts. Coherent states of bound cavity modes and Dicke states are
shown to be suitable choices for that purpose.
B.4. Code Repository
Some code examples of amplification and measurement ideas can be found in
[201].
179
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