NONLINEAR LIGHT-MATTER INTERACTIONS WITH ENTANGLED
PHOTONS AND BRIGHT SQUEEZED VACUUM
by
TIEMO S. LANDES
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 2022
DISSERATION APPROVAL PAGE
Student: Tiemo S. Landes
Title: Nonlinear Light-Matter Interactions with Entangled Photons and Bright Squeezed
Vacuum
This dissertation has been accepted and approved in partial fulfillment of the require-
ments for the Doctor of Philosophy degree in the Department of Physics by:
Dr. Michael G. Raymer Advisor
Dr. Hailin Wang Chair
Dr. Ben Farr Core Member
Dr. Andrew H. Marcus 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 2022.
ii
© 2022 Tiemo S. Landes
This work is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike
iii
DISSERTATION ABSTRACT
Tiemo S. Landes
Doctor of Philosophy
Department of Physics
December 2022
Title: Nonlinear Light-Matter Interactions with Entangled Photons and Bright Squeezed
Vacuum
We investigate the role of time-frequency entanglement in nonlinear interactions
of both low- and high-gain broadband squeezed vacuum. Our work is motivated
by the large body of research proposing the use of time-frequency entanglement in
nonlinear spectroscopy, as well as reports of enhancements in two-photon absorption
e ciencies 10 orders of magnitude larger than expected.
We theoretically investigate two-photon absorption, deriving a generalized form
capable of rigorously predicting e ciencies for time-frequency entangled states. We
find good agreement between our theory and previously expected e ciencies and
find no explanation for large enhancements reported elsewhere. Experimentally, we
replicate experiments that reported large enhancements, finding no evidence of en-
hancement beyond what is expected by theory.
We further develop an analytically tractable model for broadband squeezed vac-
uum valid at both high- and low-gain, which we apply to sum-frequency genera-
tion and two-photon absorption. Our model demonstrates the persistence of time-
frequency correlations at high gain and predicts the cross-over between scaling regimes
and absolute e ciencies in agreement with previous calculations.
We verify key components of our theory experimentally using both low- and high-
gain squeezed vacuum. We demonstrate the persistence of time-frequency correlations
iv
at high-gain via dispersion sensitivity as well as direct measurement of coherence time
via time-delayed sum frequency generation. We confirm the cross-over between low-
and high-gain regimes via sum frequency generation. Finally, we confirm predictions
about two-photon absorption e ciencies of high-gain squeezed vacuum, by direct
comparison to classical two-photon absorption.
This dissertation contains previously published and unpublished material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Tiemo S. Landes
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene
Lewis and Clark College, Portland, Oregon
DEGREES AWARDED:
Doctor of Philosophy, Physics, 2022, University of Oregon
Masters of Science, Physics, 2019, University of Oregon
Bachelor of Arts, Physics, 2014, Lewis and Clark College
AREAS OF SPECIAL INTEREST:
Nonlinear Quantum Optics
Time-Frequency Entanglement and Bright Squeezed Vacuum
vi
PUBLICATIONS:
M. G. Raymer, T. Landes, “Theory of two-photon absorption with broadband
squeezed vacuum,” Phys. Rev. A, 106, 013717 (2022)
T. Landes, M. Allgaier, S. Merkouche, B. J. Smith, A. H. Marcus, M. G.
Raymer, “Experimental feasibility of molecular two-photon absorption with isolated
time-frequency-entangled photon pairs,” Phys. Rev. Research, 3, 033154 (2021)
M. G. Raymer, T. Landes, A. H. Marcus, “Entangled two-photon absorption
by atoms and molecules: A quantum optics tutorial,” J. Phys. Chem., 155, 081501
(2021) – Cover Issue
T. Landes, M. G. Raymer, M. Allgaier, S. Merkouche, B. J. Smith, A. H. Marcus,
“Quantifying the enhancement of two-photon absorption due to spectral-temporal en-
tanglement,” Opt. Express, 29, 20022-20033 (2021)
M. G. Raymer, T. Landes, M. Allgaier, S. Merkouche, B. J. Smith, A. H. Marcus,
“How large is the quantum enhancement of two-photon absorption by time-frequency
entanglement of photon pairs?”, Optica, 8, 757-758 (2021)
A. Tamimi, T. Landes, J. Lavoie, M. G. Raymer, A. H. Marcus, “Fluorescence-
detected Fourier transform electronic spectroscopy by phase-tagged photon counting,”
Opt. Express, 28, 25194-25214 (2020)
J. Lavoie, T. Landes, A. Tamimi, B. J. Smith, A. H. Marcus, M. G. Raymer,
“Phase-Modulated Interferometry, Spectroscopy, and Refractometry using Entangled
Photon Pairs,” Adv. Quantum Tech., 3, 1900114 (2020)
vii
B. D. Mangum, T. Landes, B. R. Theobald, and J. N. Kurtin, “Exploring
the bounds of narrow-band quantum dot downconverted LEDs,” Photon. Res., 5,
A13-A22 (2017)
viii
ACKNOWLEDGEMENTS
I’d like to express my deepest gratitude to first and foremost my family, whose
love and support enabled me to be in the position I am today. To my parents Gretl,
Rüdiger for instilling in me drive and curiosity. To my sister Franziska for being the
role-model and friend I needed so many times in my life. To Emma for the love,
support, and patience. For always being up for an adventure. I truly cannot imagine
the last few years without you.
To my friends in Eugene–too many to list–for the good times, good food, and
many adventures.
To my teachers at Norman High, David Askey, Thomas Richardson, Leonard Gib-
son, Sandrah Bahan, and Betsy Ballard, I wouldn’t be here today without you. To my
professors at Lewis & Clark, Iva Stavrov, Paul Allen, Shannon O’Leary, and Michael
Broide, for the exceptional instruction and conversations about math, physics, and
more. To Ben Mangum for giving me my first real opportunity and putting me in a
position to learn, grow, and eventually pursue a my doctorate.
To my advisors Michael Raymer, for being continually invested in my project and
giving me the freedom to work at my own pace and in my own way. To Brian Smith
and Andy Marcus for the many insightful discussions and everything you taught me.
To the Post-docs and group members from whom I learned so much.
Thank you.
ix
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Previous Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Where does the enhancement from ETPA come from? . . . . . . . . . . . . 6
Note on the Entangled Two-Photon Absorption Cross-Section,  e . . 10
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Original Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Recent work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
ETPA of molecular solutions . . . . . . . . . . . . . . . . . . . . . . . 15
Work Rebutting ETPA claims . . . . . . . . . . . . . . . . . . . . . . 19
Literature Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Quantum Optics Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Spontaneous Parametric Down Conversion . . . . . . . . . . . . . . . 25
Note on Spectral and Spatial Behavior with Initial Phase-Mismatch . 28
II. THEORY OF ETPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Quantum TPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
One-photon absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Two-photon absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 34
O↵-Resonance Assumption . . . . . . . . . . . . . . . . . . . . . . . . 37
Gaussian Pulse TPA . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Quantifying the Enhancement of Entangled Two-Photon Absorption(Opex) 40
ETPA Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Comparison of EPP and coherent-state TPA . . . . . . . . . . . . . . 48
x
Chapter Page
TPA Gaussian Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Classical Gaussian-Beam TPA . . . . . . . . . . . . . . . . . . . . . . 49
Entangled Two-Photon Absorption . . . . . . . . . . . . . . . . . . . 51
Generalization to Many Molecules . . . . . . . . . . . . . . . . . . . . . . . 52
E↵ects of Broadening on Coherent Contribution . . . . . . . . . . . . 55
Empirical Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 57
Homogeneously Broadened . . . . . . . . . . . . . . . . . . . . . . . . 58
Inhomogeneously Broadened . . . . . . . . . . . . . . . . . . . . . . . 59
Summary of Inhomogeneous Broadening . . . . . . . . . . . . . . . . 59
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
III. EXPERIMENTAL BOUNDS ON ETPA IN R6G . . . . . . . . . . . . . . . 62
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Experimental setup and overview . . . . . . . . . . . . . . . . . . . . 64
Dispersion compensation . . . . . . . . . . . . . . . . . . . . . . . . . 75
Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Note on Chopped Measurements . . . . . . . . . . . . . . . . . . . . . 80
Adverse E↵ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
IV. TPA OF BSV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
TPA of Chopped CW Squeezed Light (Phys Rev. A) . . . . . . . . . . . . 86
Temporal Gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
TPA of BSV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Joint Spectral Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 96
xi
Distinguishable PDC Joint Spectra . . . . . . . . . . . . . . . . . . . 98
Same-Label Joint Spectra . . . . . . . . . . . . . . . . . . . . . . . . 99
Joint Spectrally Resolved Hong-Ou-Mandel Interference . . . . . . . . 100
Sum Frequency Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 102
The E↵ect of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 105
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
V. EXPERIMENTAL TPA OF BRIGHT SQUEEZED VACUUM . . . . . . . . . 108
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
BSV ETPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Experimental Configurations . . . . . . . . . . . . . . . . . . . . . . . 110
JSI Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Collinear (Indistinguishable) Joint Spectra . . . . . . . . . . . . . . . 114
Non-collinear (Distinguishable) Joint Spectra . . . . . . . . . . . . . . 116
HOM Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Comparison of BSV and Classical TPA . . . . . . . . . . . . . . . . . . . . 129
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
VI. SUMMARY AND CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . 136
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Final thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
xii
LIST OF FIGURES
Figure Page
1.2 TPA e ciency given bandwidth relative to TPA transition lineshape. 8
2.1 The anti-diagonal projectionK (x) of the two-photon amplitude (!, !̃). 42
2.2 Two-photon amplitude in frequency domain (!, !̃) and time domain
 (t, t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Inhomogeneous Broadening as a function of  fg/ fg . . . . . . . . . . 56
2.4 E↵ect of TPA probability for a given homogeneous linewidth. . . . . 56
2.5 Cross-section scaled by homogeneous linewidth. . . . . . . . . . . . . 57
3.1 Diagram of experimental figure for Bounding measurements. . . . . . 65
3.2 SFG counts at 532 nm as a function of IR power for three cases: . . . 66
3.3 Fluorescence count rates from laser-driven TPA in Rhodamine-6G . . 68
3.4 Photon pair coincidence rate as a function of pump power. . . . . . . 73
3.5 Marginal spectrum of the entangled photon pairs at various crystal
temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6 Schematic of the prism compressor . . . . . . . . . . . . . . . . . . . 75
3.7 Schematic of (a) the 90  collection geometry and (b) back-reflection
geometry. PMT: photomultiplier tube, SP: short-pass filter, APD
avalanche photodiode. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8 Plots of (a) maximum collected TPA fluorescence as a function of con-
centration, (b) normalized TPA fluorescence intensity as a function of
focal position relative to the front face of the cuvette, and (c) decay of
fluorescence intensity with focal position . . . . . . . . . . . . . . . . 77
4.1 PDC spectra in the low- and high-gain regimes, with characteristic
widths w and b, respectively. . . . . . . . . . . . . . . . . . . . . . . . 88
xiii
4.2 Predicted mean number of molecules excited by TPA per pulse for a
final-state TPA linewidth that is much narrower than the BSV band-
width but broader than the e↵ective PDC pump bandwidth, from Eq.
4.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 Diagram of experimental setup for collinear joint spectral intensity
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Collinear Joint Spectral Intensities at varying gain with simulations. . 115
5.3 Diagram of experimental setup for non-collinear joint spectral intensity
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4 Non-collinear joint spectral intensities at varying gain with simulations. 118
5.5 Non-collinear joint spectral intensities on a single channel. . . . . . . 118
5.6 Diagram of experimental setup for non-collinear joint spectrally re-
solved HOM interference. . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.7 Spectrally Resolved HOM interference . . . . . . . . . . . . . . . . . 120
5.8 Joint spectrally resolved HOM interference. Data was generated by
taking a JSI measurement at each delay vaule. . . . . . . . . . . . . 121
5.9 HOM interference in terms of the frequency di↵erence. . . . . . . . . 122
5.10 Diagram of experimental setup for SFG characterization. . . . . . . . 123
5.11 Scaling crossover in intermediate gain PDC . . . . . . . . . . . . . . . 125
5.12 Spectrally resolved SFG of collinear BSV, with varying degrees of dis-
persion compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.13 Diagram of experimental setup for direct measurement of spectral-
temporal correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.14 Spectrally resolved SFG of BSV generated in the distinguishable con-
figuration, with temporal delay scanned between signal and idler beams 128
5.15 Schematic of experimental apparatus for comparison of classical TPA
and TPA driven by BSV. . . . . . . . . . . . . . . . . . . . . . . . . . 129
xiv
5.16 Scaling Comparison between classical excitation and BSV . . . . . . . 132
xv
CHAPTER I
INTRODUCTION
Background and Motivation
The central question of this dissertation is, simply put, “How large is the en-
hancement of the two-photon absorption probability in molecules due to time-frequency
entanglement?” This question is of particular interest, because in recent years there
have been a number of experiments indicating the presence of enormous so-called
entangled two-photon absorption (ETPA) e ciencies, many orders of magnitude
stronger than classical two-photon absorption (TPA). [1–8] The claimed e ciencies
are much larger than can be expected from simple heuristic models, and attempts
to replicate some of these experiments have cast doubt on some of these claims [9–
12]. In an attempt to answer this question, we develop quantum mechanical models
for two-photon absorption and sum-frequency generation (SFG) driven by time-
frequency entangled photon pairs generated via spontaneous parametric down con-
version (SPDC), which are discussed in Chapter 2. We next attempt to observe
ETPA experimentally under carefully controlled and very favorable conditions, dis-
cussed in Chapter 3. In Chapter 4 we adapt our theory to the high-gain regime of
SPDC, sometimes referred to as bright squeezed vacuum (BSV). This allows us to
gain insight into the problem from the perspective of experimentally tractable con-
ditions. Chapter 5 details experimental verification of key predictions for the char-
acteristics of BSV as they’re relevant to the question of ETPA.
Taking a step back, it is worth giving a brief overview of the context in which
this question is being asked. In recent years, quantum optics has shifted from a
field in which novel phenomena are explained and observed to a field in which these
phenomena are increasingly used to create technologies which are only possible by
1
utilizing the unexpected, counter-intuitive predictions of quantum mechanics. A
great demonstration of the prominence these technologies have gained is this year’s
Nobel Prize in physics, awarded to Alain Aspect[13–15], John F. Clauser[16], and
Anton Zeilinger[17–20] for pioneering work demonstrating violations of Bell’s in-
equalities. Their work violating Bell’s inequalities is fundamental to our under-
standing of quantum mechanics and the nature of entanglement and locality in
quantum systems. These properties are central to the fields of quantum comput-
ing, quantum information, quantum communication, and quantum metrology.
The work in this dissertation is motivated by quantum spectroscopy, which
falls under the umbrella of quantum metrology or quantum sensing. Quantum spec-
troscopy is an active field of research, with numerous proposals to utilize the time-
frequency correlations present in entangled photon pairs to enhance nonlinear spec-
troscopic methods[21–32]. In this dissertation we focus on two-photon absorption,
which is a process in which two photons from the optical field interact simulta-
neously with the same molecule, atom, or semiconductor bandgap, resulting in an
electronic excitation. The motivation to study TPA comes from the fact that both
photons need to interact simultaneously with the two-photon absorber within the
spectral linewidth of the transition, for which time-frequency entangled photons
seem well suited due to their ability to satisfy both conditions beyond what is pos-
sible classically.
Time-frequency entangled photons are generated via spontaneous parametric
down-conversion, in which one photon from the pump field is split into two photons
referred to as signal and idler, each with roughly half of the energy of the pump
photon. Because this process must conserve energy and momentum, certain degrees
of freedom of the down-converted photons are entangled. While we focus on time-
frequency entanglement, SPDC is also capable of generating entanglement between
polarization and momentum, and it is often an experimental necessity to carefully
2
control the design of the SPDC source to ensure entanglement only in the desired
degree(s) of freedom.
Before delving fully into the search for entangled two-photon absorption, it is
worth taking a quick aside to review other work I’ve done. Beside providing the
foundation for much of my understanding of the physics and experimental consid-
erations, these experiments provide a background for the evolution of the set of ex-
periments in our work.
Previous Projects
The motivation to explore entangled nonlinear interactions came from a pro-
posal to utilize time-frequency entangled photon pairs in order to increase the spec-
tral resolution of a particular type of 2-dimensional fluorescence spectroscopy[24].
In this proposal, entangled photon pairs in a Franson-type interferometer[33] were
recombined on a molecular sample, with both two-photon fluorescence and photon
coincidence transmission being monitored.
My work began doing phase-modulated interferometry with entangled photons
alongside Dr. Jonathan Lavoie, which has been published in Advanced Quantum
Technologies [34]. This work was intended as a precursor to the two-dimensional
spectroscopy proposed in [24]. This work demonstrated the ability to identify spe-
cific quantum pathways via phase-sensitive detection of phase-modulated photon-
counting experiments using an entangled photon source. It also demonstrated the
ability to find information encoded in those channels. The phase modulation tech-
nique and phase-sensitive measurement utilizing a lock-in amplifier was adapted
from methods employed in the Marcus lab [35–38]. In order to track the phase
inside the interferometer, we back-propagated a reference laser of a di↵erent fre-
quency through our interferometer. This allowed us to reduce the sampling require-
ments, enable phase-sensitive measurements, and passively stabilizing the measure-
3
ment, similar to the original configuration.
Subsequent work with Dr. Amr Tamimi, published in Optics Express [39], built
on these experiments to build a generalized photon-counting phase-tagging mea-
surement to replace the action of the lock-in amplifier in previous generations of
the experiment. This work was implemented using a Field-Programmable Gate Ar-
ray to generate ’phase-tags’ of the photon-detection events, based on a synchro-
nization signal derived from a reference waveform generated in the interferometer.
The ’phase-tags’ refer to the relative phase of the interferometer at the time of a
photon-detection event, which is used rather than an analog multiplication of the
signal in traditional phase-sensitive lock-in amplification schemes. One major ben-
efit of this work is the ability to synthesize sum and di↵erence frequencies from an
arbitrary number of reference frequencies, alongside arbitrary harmonics of the fun-
damental and synthesized frequencies.
The phase-tagging work has been built upon by Matthew Brown in Dr. Smith’s
lab to help implement proof-of-principle experiments demonstrating quantum ad-
vantage in the Gottesman protocol for optical very long baseline interferometry us-
ing distributed heralded single photons as the (non)-local oscillator [40]. Various
stabilization techniques based on the phase-modulation scheme and phase-tagging
have been investigated to overcome di culties introduced via phase drift in the ex-
periment.
After these projects wrapped up, the next challenge involved detecting nonlin-
ear signatures driven by time-frequency entangled photon pairs in molecular sam-
ples in order to implement nonlinear spectroscopic methods. It became clear early
on that such experiments would prove di cult, with the low interaction strength of
nonlinear interactions being the central issue. While we were able to observe fluo-
rescence from single-photon absorption of entangled-photons in Cyanine dyes, two-
photon absorption proved elusive both in resonant and non-resonant systems.
4
While the initial experiments were designed to produce entangled photons pairs
centered at 532 nm, this created complications detecting fluorescence from TPA
emitted in the ultraviolet (UV) end of the spectrum. Beyond this, the e ciency
of generating entangled pairs was limited by the stability and strength of the UV
pump laser. The choice of nonlinear medium was also restricted due to the strong
absorption of UV light in many of the nonlinear crystals with the largest nonlinear
coe cients.
These di culties alongside nice experiments demonstrating sum-frequency gen-
eration (SFG) driven by entangled photons by Barak Dayan et al.[41], and reported
observation of entangled two-photon absorption in a similar system by Dmitry Tabakaev
[7], motivated a transition to a system in which the entangled photon pairs were
generated in the near infra-red(NIR) at 1064 nm. At the same time, a realization
that many of the interaction strengths reported elsewhere were orders of magnitude
stronger than heuristic arguments would indicate forced us to carefully evaluate
the theory behind two-photon absorption of arbitrary quantum states. This would
enable a full calculation of TPA rates of entangled photon pairs. Due to its theoret-
ical similarity to TPA and vastly increased interaction strength, we also used SFG
as a test system for understanding non-linear interactions with time-frequency en-
tangled photon pairs.
TPA is central to many of the technologies hoping to exploit time-frequency
entanglement for metrology, and scales with photon flux the same as other non-
linear two-photon interactions. Because of the controversy surrounding the magni-
tude of the e↵ect, our inability to observe the e↵ect, and its implications for other
quantum-enhanced metrology, observing and explaining this e↵ect became the fo-
cus of my research. The di culties faced in observing such a two-photon signal, the
theoretical background, the insights gained, and the directions taken to overcome
these di culties compose the heart of this dissertation.
5
Where does the enhancement from ETPA come from?
Before delving deeper into the dissertation, it is worth taking some time to get
an intuitive understanding of the e↵ects that are claimed to result in the large en-
hancement of two-photon absorption probabilities.
While enhancement from ETPA can come from other degrees of freedom, the
primary considerations are the following: photon number correlations and time-
frequency entanglement.
Perhaps the most quantum feature is the e↵ect of number correlations. Since
entangled photons generated via SPDC are always emitted in pairs, there is an en-
hancement in pair probability over classical light of the same average flux. This is
most pronounced at low-flux. For a classical coherent light source with an average
flux of N̄ = " photons per pulse with " ⌧ 1, the probability of observing a pair
of photons in any given pulse is approximately "2/2, neglecting the possibility of
triplets etc. On the other hand, for an entangled pair source of the same average
flux, the probability of observing a pair in any given pulse is "/2 (again neglect-
ing higher order terms). This is because entangled photons are always emitted in
pairs. This leads to an enhancement in the probability of observing a pair of pho-
tons, PNE of:
"/2 1
PNE ⇡ = . (1.1)
"2/2 "
This can be large for very weak sources, where " is much less than one. The
increased probability of seeing two photons in the same time-window can be read-
ily verified via coincidence counting experiments. Since two-photon absorption can
only occur when two photons are present, this also represents an enhancement of
the TPA probability.
For enhancement due to temporal correlation and frequency anti-correlation,
we can think of this in terms of two separate timescales present in the source of
6
A B C
!"
!!
SFG TPA
Figure 1.1. Intuitive model for ETPA enhancement. (A) shows the joint spec-
tral intensity of time-frequency entangled pairs, alongside the marginal and sum-
frequency distributions. The marginal bandwidth is broad, but simultaneously the
bandwidth of the sum of the two frequencies is narrow, due to the frequency anti-
correlations present in the entangled pairs. (B) Shows the case of sum-frequency
generation, in which signal and idler frequencies are converted to a narrowband
up-converted frequency. (C) shows the case of two-photon absorption of broad-band
time-frequency entangled photon pairs. The narrow spectral anti-correlations yield
e cient overlap with a narrow final-state linewidth.
entangled photons. The first timescale is the duration of the pump generating the
pairs, and the second is the correlation time between the two photons, which is re-
lated to the spectral width by a modified Fourier relationship. An intuitive descrip-
tion in the frequency domain is given in Fig. 1.1. This picture can be quantified in
terms of the ratio of the correlation time to the duration of the pump. This leads
to a temporal enhancement of TPA e ciency, TE due to the increased temporal
overlap of the pairs relative to the pump pulse:
T
TE p⇡ , (1.2)
Te
where Tp is the duration of the pump, and Te is the entanglement or correlation
time of the two photons, which quantifies how closely correlated in time the two
photons are, and is constrained by the marginal bandwidth of the down-converted
pairs, as discussed in more detail in the section on SPDC.
7
classical
102 EPP  n  = 10
-3
fg
EPP  n  = 10
1
fg
EPP  n  = 10
3
fg
100
10-2
10-2 100 102
Bandwidth / fg
Figure 1.2. Comparison of spectral overlap with a two-photon transition lineshape
(proportional to TPA e ciency neglecting number enhancement) between classical
excitation and entangled photon pairs. This comparison is plotted as a ratio of
the spectral bandwidth of the excitation light  , to the spectral linewidth of the
two-photon absorber,  fg. Here, we assume Tc ⇡ 1/ , Te ⇡ 1/ B, Tp ⇡ 1/ N , and
⌧fg ⇡ 1/ fg. The e ciency of ETPA for various values of  N are displayed. It is as-
sumed that  B    N . For classical excitation, the e ciency reaches an asymptotic
value as the bandwidth begins to exceed the TPA linewidth. For entangled pho-
tons, the e ciency can increase arbitrarily, so long as the correlation width ( N)
is narrower than the TPA linewidth. QSE can be visualized as the ratio between
entangled photon pairs (EPP) and classical excitation with   =  B. It follows that
QSE = 1 for  B <  fg.
8
TPE probability(a.u)
While this ratio can be large for long-pulse and CW pumping schemes, in many
cases much of the e ciency increase could be achieved classically by utilizing pulses
of shorter duration. For instance, we can compare the TPA e ciency to a classi-
cal pulse with the same temporal envelope as the correlation time, (or equivalently
with a spectral bandwidth that matches the spectral bandwidth of the PDC). In
this case, the relevant comparison is to a classical pulse with Tc = Te. There is
still an opportunity for spectral enhancement in this case if the bandwidth of the
classical pulse is greater than the TPA linewidth. The scenario in which there is
significant spectral enhancement over a classical pulse with matching spectral band-
width is summarized in Fig. 1.2. In this scenario, the e ciency of spectrally broad
classical excitation is reduced due to a reduction in resonant overlap with the TPA
lineshape. However, due to the tight spectral anti-correlation of the entangled pairs,
there is no loss in resonant overlap in the entangled case, despite being localized
tightly in time.
The calculation leading to this plot is discussed in detail in Chapter 2. When
the classical excitation light is broader than the TPA linewidth, there is decreased
resonant overlap with the TPA linewidth, which limits the attainable e ciency of
the classical pulse. However, broad bandwidth entangled photons don’t su↵er from
this limitation, since their sum can be spectrally narrow. In this case, the quantum
spectral enhancement, QSE is:
⌧
QSE fg⇡ for ⌧fg > Te (1.3)
Te
where ⌧fg is the inverse of the TPA linewidth. In this sense, the QSE = 1 for ⌧fg <
Te, since in this scenario classical excitation could have produced the same resulting
e ciency.
While there can in theory be enhancement from other degrees of freedom (i.e.,
9
polarization & space, which we briefly discuss in the section SPDC), these are in
general not controlled or implicated in these experiments. Spatial entanglement
in particular may play a significant role, if it leads to an interaction volume larger
than possible with a Gaussian beam. We discuss this briefly in Ch. 1, but a full
discussion of this is beyond the scope of this dissertation, and could be a future di-
rection to take this work. In our work, we assume tight focusing, in which the aper-
ture of the optical system ensures that further spatial correlations can be neglected.
Note on the Entangled Two-Photon Absorption Cross-Section,  e
Before moving on, it is worth taking some time to discuss the way interaction
strengths for electronic excitation is usually quantified. For absorption of a sin-
gle photon, the interaction strength for a single molecule and single photon can
be quantified via an absorption cross-section,  (1) which has units of [cm2]. For
strongly absorbing dyes, the absorption cross-section can be on the order of 10 16 cm2
[42]. Similarly, the interaction strength for two-photon absorption can be writ-
ten in terms of a two-photon absorption cross-section  (2) with units of GM =
10 50 [cm4][s]/[photon], named after Maria Göppert-Mayer who first predicted the
e↵ect in 1931[43]. This can be thought of as a cross-section divided by a photon-
flux density. Classically, the total absorption rate at a given frequency is propor-
tional to  (1) +  (2) 2 where   is the photon flux in units of [photons]/[cm2][s].
For entangled two-photon absorption, the TPA rate demonstrates both lin-
ear and a quadratic flux scaling. The scaling is predominantly quadratic at high
gain, while at low gain it is linear with incident photon flux. While these aren’t
inherently separate, they are often cited as such according to the heuristic equa-
tion introduced by [44], R =  e  +  r 2, where R is the rate of ETPA,   is the
entangled-pair flux,  r is the ‘random TPA cross-section’, and  e is the ‘entangled
two-photon absorption cross-section’. While this is a greatly simplified descrip-
10
tion, it is nonetheless one that has gained significant traction in the community
interested in ETPA. It is worth noting, that unlike typical cross-sections, which are
specified by the absorbing sample itself,  e is dependent also on the entangled pair
source and the particular optical system in the experiment.
Despite this dependence on the experimental apparatus, the precedent has
been set within the community interested in ETPA that the e ciency of ETPA is
reported in terms of  e[1, 2, 6, 7, 44]. This is somewhat unfortunate, because  e is
a function of several experimental parameters that a↵ect its magnitude, such as the
correlation time, Te, correlation area, Ae, dispersion, and losses in the system. Val-
ues for  e have been reported with disregard for these parameters, resulting in am-
biguity and leaving room for di↵erences in reported values due solely to controllable
experimental parameters. We have advocated for reporting quantum enhancements
to the TPA e ciency rather than simply reporting  e. A recent proposal to report
the product,  e ⇤ Te ⇤ Ae [8], would have similar benefits while also facilitating com-
parisons to previous work in which the product can be inferred.
In contrast to   , the classical two-photon absorption cross-section,  (2)e , is only
a function of the two-photon absorbing sample itself. It also requires knowledge of
the temporal and spatial properties of the excitation field to predict rates. Cru-
cially, the magnitude of the cross-section itself is entirely determined by the partic-
ular absorber and its preparation, but not on the optical system or source. While
the classical cross-section is implicitly still a function of excitation wavelength, the
wavelength dependence is itself a function of the particular absorber rather than
solely of the experimental apparatus, so its implicit inclusion in the quantity is well
justified.
Despite these considerations, I will refer to  e extensively throughout the dis-
sertation, both to facilitate discussion and comparison to other work, and because
it is decidedly less cumbersome than the full description. I will do my best to in-
11
clude assumptions about experimental parameters when referencing  e.
Literature Review
Now that we understand the big picture, it is worth reviewing work done
previously and the state of research around the question.
Literature surrounding entangled two-photon absorption can be broadly
divided into experimental and theoretical work. Pioneering theoretical work on the
topic described the e↵ect in the 1990s. More recently, entangled photons have been
investigated in various theoretical spectroscopic configurations.
Notably, however, prior to our work there are few examples which critically
evaluate whether the enhancement from such sources is su cient to yield feasible
molecular experiments, though the potential di culty is sometimes mentioned. In
that sense, a major contribution of the work in this dissertation is a feasibility
study for such techniques, with due consideration given to low and high-gain
regimes of spontaneous parametric down-conversion.
Original Work
In 1985, Friberg, Hong, and Mandel [45] describe the first coincidence counting
experiment demonstrating the linear nature of coincidences from SPDC photons.
And in 1987 Janszky and Yushin describe the statistics of coherent, thermal, and
high gain squeezed light[46].
In 1989 J. Gea-Banacloche predicted linear intensity scaling of the squeezed
vacuum component of quadrature squeezed light [47]. And in 1990 nearly
simultaneous results from Javanainen and Gould [48], describe the same e↵ect in an
aptly named article, ‘Linear intensity dependence of a two-photon transition rate.’
The e↵ect is described in terms of correlations between down-converted photons in
an intuitive picture, which is in line with the way we tend to conceptualize the
12
problem currently. In 1991 a similar e↵ect was proposed in interactions with a
three-level atom by Ficek and Drummond [49].
In 1997, Fei, Saleh, and Teich describe ‘Entanglement-Induced Two-Photon
Transparency’ in which destructive interference between signal and idler results in
decreased TPA probability. The heuristic equation,
R =  e +    
2
r , (1.4)
is introduced [44]. This is also the first use of the term ‘entangled two-photon
absorption cross-section.’ In the equation,   is the incident photon flux density,  e
is the aforementioned entangled two-photon absorption cross-section, and  r is the
classical two-photon absorption cross-section, which we will refer to as  (2)
throughout the dissertation. The estimate,
 
  = re , (1.5)
2AeTe
also stems from this work. This work also can be seen as one of the first proposals
to use entangled photons as a spectroscopic tool, which is currently a topic in which
there is significant active research. It is worth noting that the ‘induced two-photon
transparency’ can be understood as interference between signal and idler photons,
rather than a particular molecular e↵ect.
The first experimental demonstration of the TPA of squeezed light is from
Georgiades and Kimble in 1995 [50]. In the paper ‘Nonclassical Excitation for
Atoms in a Squeezed Vacuum,’ they demonstrate TPA of squeezed light using a
Cesium atom in a magnetic optical trap, pumped by a non-degenerate optical
parametric oscillator.
13
Recent work
In the mid 2000s, Avi Pe‘er, Barak Dayan, Asher Friesem, and Yaron
Silberberg published a set of experiments which I consider the most thorough and
well-designed experiments relating to nonlinear interactions driven by parametric
down-conversion.
In their first experimental work on the topic, they demonstrate TPA in a
Rubidium vapor cell with non-degenerate non-collinear parametric down-conversion
generated by a 3 ns pump pulse centered at 516 nm in a low-finesse resonator on
the signal beam [51]. Signal and idler beams were centered at 870 nm and 1270 nm,
respectively. Each had a bandwidth of approximately 100 nm. In this work, they
demonstrate the sharp temporal correlation by adding a delay to the signal beam
via spatial light modulator (SLM). Additionally, by detuning the pump frequency,
they demonstrate the sharp temporal correlations.
This is one of the cleanest examples of TPA by source of parametric
down-conversion, exhibiting tight correlations in time and frequency. However,
several questions of interest were not answered. First, they did not quantify the
degree to which they remained in the low-gain regime, and they did observe
low-e ciency incoherent TPA from their source when detuned from resonance.
However, no scaling information was included, and the presence of the oscillator
cavity complicates the analysis. Finally, no attempt was made to characterize the
e ciency of the process in relation to classical TPA. This latter point is
complicated by the question of whether any resonant intermediate states were
driven due to the large bandwidth of the signal and idler beams.
In subsequent work utilizing a 532 nm CW pump laser driving Type-0,
degenerate, collinear SPDC in a PPKTP crystal, they demonstrate sum-frequency
generation (SFG) driven by the entangled pairs [41]. This SFG displays linear
dependence on entangled pair flux, as well as quadratic dependence on losses
14
introduced between pair generation and up-conversion. The observation of both of
these e↵ects is crucial to verify that the observed e↵ect is the result of a non-linear
process driven by entangled photons. We have strongly advocated for this
demonstration in ETPA experiments.
In additional work on a similar experimental system, they demonstrate
two-photon interference by adding frequency-dependent delays using a spatial light
modulator[52].
Finally, independent theoretical work by Barak Dayan was published in 2007
describing a theoretical framework for nonlinear interactions driven by SPDC[53].
We verify many of the predictions from this work in Chapters 2 and 4. While this
work is thorough, it is left in terms that are not easily translated to simulations or
predictions for practical experiments.
ETPA of molecular solutions
Also in the mid 2000s, ETPA in molecular solutions was investigated. In
contrast to the work done by the Silberberg group, the magnitude of the observed
e↵ects did not have strong theoretical backing. Many of these results are also
lacking some experimental verification steps ruling out other linear-optical e↵ect.
For instance, with only a single exception, all of the experiments referenced in this
section lack verification of quadratic scaling.
The original work on molecular samples was conducted in 2006 by Lee and
Goodson at the University of Michigan [1]. In this work, they investigate TPA in a
prophyrin dendrimer H2TPP using Type-II down-converted photon pairs,
measuring   10 17e ⇡ cm2. This measurement was conducted via di↵erential
transmission measurement, and it is assumed that no other linear source of loss is
present. They claim an entanglement area of 10 2 cm2, but don’t cite an
entanglement time which we can bound at 100 fs, which is the duration of their
15
pump laser. While they claim a 31-order of magnitude enhancement over classical
TPA excitation, this claim is a comparison of an entangled-two-photon cross-section
(units cm2) to a standard two-photon cross-section (units cm4s), which lacks a clear
interpretation.
It is worth noting that, besides the possibility of linear e↵ects causing the
signal interpreted as ETPA, this experiment su↵ered from additional experimental
inconsistencies. For instance, the measured photon-counting rates: 1.3 107⇥ cps in
a single channel, and 3.5 105⇥ cps in coincidence between two channels, are a large
fraction of the repetition rate of their source, which is 8.2⇥ 107 Hz. At these rates
saturation and dead-time e↵ects at their detectors become significant. This can be
seen by the fact that a coherent state would be expected to produce a higher rate
of coincidence events than they observe between their entangled photons.
After this initial work in porphyrin dendrimers, subsequent work in thiophene
dendrimers, which have a naturally large classical TPA cross-sections increasing
from 20 GM to 2000 GM as the dendrimer number increases from 6T to 90T, was
conducted via transmission experiments[3]. The corresponding ETPA cross-sections
ranged from 1.3 10 17 cm2 to 5.9 10 18 cm2⇥ ⇥ In this work they measured up to
3 7⇥ 10 cps with 2.2 5⇥ 10 cps coincidences. This indicates a lower relative
coincidence rate than the previous experiment. The dead time for the APD module
they used (Perkin Elmer SPCM AQR), is 50 ns and they are indicated for use at
rates below 5⇥ 106 cps[54]. Notably, this dead time indicates that after each
detection event, the detectors are blind to the next 4 pulses of their experiment.
Experimental inconsistencies such as these are unfortunately a prominent
feature in the early work in this field. While recent experiments are more carefully
controlled, the inconsistencies in basic characterization of photon pair sources in
these early experiments suggests that caution should be used in interpreting their
results.
16
Similar work is reported in [4], where spatial e↵ects are considered in
bisannulene, triannulene, and tetraannulene. In [55] separate molecules with large
classical two-photon cross-section, but no significant ETPA cross-section are
considered. In 2017, they report a fluorescence based experiment in which up to
2cps of fluorescence were measured in bisannulene[56].
Some of the di culty in this field is due to the fact that the pioneering work
was done in complex molecular systems that were synthesized specifically for ETPA
experiments. This makes replication of these experiments extremely di cult, and
would require expertise in both quantum optics and synthetic chemistry. Because of
this, it is worth noting that this work from 2017 also presents data on Zinc
tetraphenylporphyrin(ZnTPP), which is the first compound that we found
commercially readily available to be included in ETPA studies.
In 2017 Villabona-Monsalve et al. published the first work in the field outside
of the group from Michigan. They considered ETPA in ZnTPP and Rhodamine
B[6]. As with many of the transmission-based experiments, only linear intensity
scaling was verified. In this case, the linear intensity scaling was of the transmitted
coincidence-counts, however this su↵ers from the same issue as raw transmitted
events in which linear losses can result in a false signal. The experiment observed
strong inverse dependence of the cross-section on concentration, which was varied
over several orders of magnitude. The observed absorption rates varied little slowly
with these changes. The cross-sections ranged between 4 10 17 cm2⇥ and
2 10 19 cm2⇥ for ZnTPP and 4⇥ 10 18 cm2 and 2⇥ 10 20 cm2 for Rhodamine B.
For this experiment, pairs were generated by 404 nm CW diode laser in a 1 mm
BBO crystal focused and collimated by two 50 mm lenses, and the beam waist at
the sample was 61 µm. They estimate their entanglement time to be 17 fs.
In 2020, Varnavski and Goodson published work demonstrating a microscope
using a bis(styryl)benzene derivative, in which they successfully image a drop-cast
17
film with fluorescence from entangled-photon pairs[5]. This was later applied to
biological samples [57]. These are some of the most convincing demonstrations of
the ETPA to date, and they rule out many e↵ects that could cause a false signal in
transmission-based experiments. The fluorescence based experiments still see
relative enhancements on the order of around 106 compared to classical excitation.
The rates of fluorescence in this experiment are large enough to easily validate
quadratic scaling, but this test was not included in the publications, which leaves
room for ambiguity about the nonlinear nature of the e↵ect.
A 2020 discussion of modeling ETPA for complex molecular systems, assumes
that ETPA is the inverse process of two-photon fluorescence[58]. The decay times
for the two-photon fluorescence transition are calculated to be exceedingly long,
and coupling to these transitions is predicted to be correspondingly large. While
this perspective is used to justify the many orders of magnitude enhancement seen
in their work, no concrete way to calculate expected rates from arbitrary quantum
states is described. A thorough discussion of the properties necessary to access such
interactions is also absent. In particular, no explanation for why classical states do
not couple to this transition is given.
Work in Rhodamine 6G was conducted by Tabakaev in 2020[7] and later
followed up in [8]. The first spatially filters the entangled photon pairs with 2 m of
single-mode fiber before focusing on the sample, which is located in an integrating
sphere. Counts were collected on a camera at an output port of the integrating
sphere. They report cross-sections of 1.9 10 21 cm2, 9.9 10 22⇥ ⇥ cm2 and
6.4⇥ 10 23 cm2 at 38 µM , 4.5 mM , and 110 mM concentrations respectively. For
their highest-concentration sample, they observed 4⇥ 106 counts from ETPA in
300 s, which corresponds to a rate 1.3⇥ 104 cps of detected fluorescence.
In this experiment, spatial correlations are erased due to spatial filtering in the
single-mode fiber and the beam is focused to a 60 µm waist in the sample cell.
18
They estimate their ‘e↵ective’ flux by estimating the rate of pairs that arrive still
arriving within 140 fs original coherence time of the SPDC after fiber dispersion.
Estimating  e from parameters reported in their paper yields a value of
 est = 2.8 10 33⇥ cm2, taking T 2e = 140 fs, and Ae = A0 = ⇡(60µm) . This leavese
12 orders of magnitude unaccounted for.
In the subsequent work, they utilize a simplified setup without the fiber
filtering [8]. At 160 nW of power they measure 16⇥ 104 counts over 2⇥ 104 s, or
about 8 cps from fluorescence. This rate is approximately 1600 times lower than in
their previous experiment. Despite having significantly higher entangled pair rates
which are not filtered by a single-mode fiber, they estimate their ETPA
cross-section to be 5 10 22cm2⇥ at 5mM. This is roughly half the cross-section
from their previous experiment, despite the large discrepancy in rates and
entanglement areas.
Crucially, this experiment is the only one to date to demonstrate quadratic
scaling in an ETPA experiment. It also observed similar spatial focusing e↵ects to
classical TPA measured via Z-scan. This fluorescence observed in this experiment
had very low rates, around 8 cps, which precluded further study of the detected
light. The experiment also reported an unexplained background signal of around 10
cps, measured in pure ethanol, which was subtracted from the ETPA rates in the
experiment. Regardless of the discrepancy in rates and cross-section measurement,
this experiment demonstrates quadratic scaling of their measured signal, and as
such constitutes the most carefully controlled experiment measuring ETPA to date.
Work Rebutting ETPA claims
As will be argued throughout this dissertation, there is good reason to question
the validity of the many of the results claiming extremely large values of  e. The
first publication calling into question the large enhancements was published by
19
Parzuchowski et al. at NIST Boulder in 2020, [9], shortly thereafter our own
theoretical and experimental work was published [10, 59, 60]. Since then, several
other groups have published work calling the e↵ect into question.
In agreement with our theoretical work, theoretical work by Drago et al.
indicates smaller ETPA cross-sections than reported in non-resonant ETPA. It also
investigates ETPA with a resonant intermediate state, finding that in these
systems, one-photon e↵ects with the resonant state dominate any two-photon
interactions [61].
Work by Mikhaylov et al. (also at NIST Boulder) published in 2022
demonstrates that hot-band absorption can mimic some qualities of ETPA, such as
anti-stokes shifted fluorescence[62]. This suggests another linear e↵ect capable of
explaining apparent ETPA e↵ects, and demonstrate linear scaling of observed
fluorescence with classical excitation at low flux.
In 2022 Acquino et al. replicated the finding of reduced transmission of
entangled photon-pairs at 808 nm through solutions of ZnTPP and Rhodamine B.
However, they did not observe any dependence on delay between the entangled
photons, implicating a linear process in the transmission e↵ect [12].
Work Hickaman, Cushing et al. demonstrate a linear scattering e↵ect in
Rhodamine 6G much larger than classical TPA, which could be interpreted as a
cross-section on the order of 10 21 cm2 in linear transmission experiments [11].
Literature Summary
The above sections have described the current state of research in the field, and
the central ideas motivating research into them. While only a handful of groups
claim to have successfully measured ETPA, there is enough consistency in the rates
of ETPA reported in samples such as ZnTPP and Rhodamine B, to conclude that
there is some consistent e↵ect being measured [6, 12]. Anti-stokes shifted
20
fluorescence microscopy utilizing entangled photons at ultra-low flux [5, 57] has also
been demonstrated. These indicate solid proof of a repeatable optical e↵ect
between the entangled photons and molecular sample. However, quadratic scaling
tests have not been reported on these demonstrations and linear e↵ects cannot be
ruled out.
Some plausible explanations such as hot-band absorption[62], and linear
scattering[11] have been proposed to explain the e↵ects typically attributed to
ETPA. Further, careful fluorescence experiments have not been able to replicate the
e↵ect in a large number of samples [9, 10]. Other groups have been able to replicate
the e↵ect in transmission-based experiments, but conclude that a linear e↵ect is
being observed due to insensitivity to delay between entangled photons[12].
Experimental tests an ultra-high flux of entangled photons have observed
quadratic scaling via fluorescence in Rhodamine 6G [8] and indicate similar spatial
focusing e↵ects to classical TPA measured via Z-scan.
Ultimately, there are many experiments with various experimental strengths
and weakness and no single explanation is likely to explain the e↵ects observed in
each one, especially considering the large discrepancy in entangled pair sources and
samples used. Because of this, significant ambiguity exists around the e↵ects being
measured. Certain experimental tests such as quadratic scaling verification have
been neglected in most work, further complicating matters.
Quantum Optics Basics
In order to discuss the theory behind entangled two-photon absorption, it is
first necessary to lay some groundwork in classical and quantum optics. The first
step of which is to write down the electromagnetic field[63]:
Z sX ~!
E(x, t) = dV j uj(x, y)Aj(t)e
i(!jt kjz) + c.c. (1.6)
2c"0nj
j=H,V
21
Here, we have summed over polarization states, H and V . The transverse
spatial modes are described by uj(x, y), and Aj is the temporal envelope of the
field. While the electromagentic field requires specification of the magnetic field as
well, for optical interactions, it is almost always su cient to consider the electric
field alone. It is often convenient to write this in terms of frequency:
Z Z sX d! ~!
E(x, t) = dV j u (x, y)Ã (!)ei(!j⌧ kjz)j j + c.c. (1.7)
2⇡ 2c"0nj
j=H,V
In order to proceed, we need to quantize the electromagnetic field. In doing so,
quantum mechanical operators are related to the classical quantities, with quantum
operators denoted by hats: O ! Ô. Quantization of the field is typically
accomplished in terms of position and momentum quadratures as can be reviewed
in any quantum optics text. We skip this description and write the electric field
directly in terms of bosonic harmonic oscillator creation and annihilation operators,
â† and â†, which describe quantized excitations of the electric field in a given mode,
sometimes referred to as ‘photons’.
Z Z sX d! ~!
Ê(x, t)(+) = dV j u (x, y)ei(!j⌧ kjz)j âj(!) (1.8)
2⇡ 2c"
= 0
nj
j H,V
The raising and lowering operators are convenient due to their simple
correspondence to the quantized electric field, and their relatively intuitive nature
in the number basis:
p
âj |n > 0i = n |n  1i , â |0i = |0ij
â†
p
|n+1i = n |n+ 1i
j j
h i
â , â†i =  j ij
Any state can be represented in this form, and here we consider the coherent
22
state |↵i, which is an eigenstate of the number operator, â |↵i = ↵ |↵i, where the
average number of photons is 2hni = |↵| . It can be written as:
1
  |
n
↵|2 X ↵
|↵i = e 2 p |ni (1.9)
n!
n=0
The probability of detecting n photons for a coherent state corresponds to a
n nh i
Poisson distribution, peaked around hni: P (n) = 2  hni|hn|↵i| = e .
n!
A pure state can be written as, | i. It is often convenient to consider mixed
states, for which the density matrix ⇢̂ is convenient. In some basis, it can be
written as:
X
⇢̂ = pi | ii h i| (1.10)
i
where the individual terms pi are real and sum to 1. A pure state in this form
can be written as | i h |. The expectation value of an operator, Ô, for a given
state, ⇢̂, can be written as Tr(Ô⇢̂).
The expectation value of an operator hÔi, for a pure state, | i can be written
as h | Ô | i. For a mixed state, ⇢̂, this is written as hOi = Tr[⇢̂Ô] for a mixed
state, ⇢̂.
In order to do something useful with this, we need to consider the dynamics of
the system. This is often accomplished in the Heisenberg picture, where the state,
| i of the field is constant, while the operators Ô(t) are time-varying.
dÔ 1
= ~ [Ô, Ĥ] (1.11)dt i
Where the Hamiltonian for the electric field in vacuum can be expressed as
X
Ĥ = ~!â†(t)â (t) (1.12)
j j
j
where we have left out the factor of 1/2 representing the zero point energy. This
23
yields operator equations of motion in vacuum:
dâj(t) =  i!âj(t)
dt
(1.13)
dâ†(t)
j = i!â†(t)
dt j
Which immediately yield the temporal propagation of the state, âj(t) = âj(0)e i!t
In the Schrödinger picture, operators are invariant and state evolve in time.
Where a state | (t)i evolves in time according to:
i
| (t)i = e  ~ Ĥt | (0)i (1.14)
It is often su cient to approximate this in the perturbative limit as:
i
| (t)i = | (0)i   ~Ĥt | (0)i+ ... (1.15)
Which is su cient for weak interactions, but for higher order terms care must
be taken with considerations to time-ordering e↵ects.
Entanglement
An entangled state, generally, is one that cannot be expressed as the product
of separable states. For bipartite systems, a state can be defined in terms of a
Schmidt decomposition[64–66]:
Xp
 (!, !̃) =  i n(!) n(!̃), (1.16)
n
where an unentangled state is separable and can be written as
 (!, !̃) =  n(!) n(!̃). Some useful ways to quantify the entanglement of a pure
state are the entropy of entanglement, which can be defined as
P ⇣P ⌘ 1
S = 2 
n=1  n log2  n, the Schmidt number or cooperativity, K = n=0  n
24
and the purity, p = Tr⇢̂s. It can be shown that p = 1/K.
While entanglement is much more di cult to treat for mixed states[67], we’re
generally not concerned with quantum information theoretic quantification of
entanglement, as much as we’re interested in the specific benefits of time-frequency
entangled state generated via SPDC for nonlinear interactions, though these are
related via the number of temporal modes in the SPDC state.
Spontaneous Parametric Down Conversion
Spontaneous parametric down-conversion (SPDC) is the process by which
time-frequency entangled photons are generated. SPDC is a three-wave mixing
process, in which the pump field interacts with the signal and idler fields, via
nonlinear coupling in a (non-centrosymmetric) crystal with a  (2) nonlinearity. The
two down-converted fields are referred to as signal and idler for historical reasons.
The Hamiltonian of this interaction reads1:
Z Z Z
dtĤ(t) =  (2) dt dV Ê(+)(x, t)Ê( )(x, t)Ê( )(x, t) + h.c. (1.17)
p s i
For a generalized squeezing we use this Hamiltonian for dynamics. For the low gain
case, we can use Eq. 1.15, and write the state generated via SPDC as:
Z Z
i
| i = |vac  (2)i   ~ dt dV Ê
(+)(x, t)Ê( )(x, t)Ê( )(x, t) |vaci (1.18)
PDC p s i
Writing our field operators in terms of creation and annihilation operators:
1While strictly speaking we should quantize the D-field, the standard form used most often
uses the E-field and is correct up to an integer factor[68].
25
R q
Ê(+)(x, t) = d! ~!2 2 ( ) exp[i(k(!) · x)  !t)]â(!)⇡ "0cn !
Z Z Z Z
d!p d!s d!i
| i = " dte i(!p !s !i)t⇥
PDC
Z 2⇡ 2⇡ 2⇡ (1.19)
dV exp[i k(! ,! ,! ) x]â (! )â†(! )â†p s s · p p s (!s i i) |vaci
The time integral enforces energy conservation:
R
dtei(!p !s !i)t =  (!p   !s   !i), and  k(!p,!s,!s) = k(!p)  k(!s)  k(!i)
determines phasematching. In the paraxial approximation, where the pump is
assumed to be a plane wave:
Z n o Z Z Z L
dV exp i ~k · ~r = dxei kxx dyei kyy dzei kzz (1.20)
V X Y 0
R
Which leads to dxei kxx  ( k ), and  k = 0 k(s)! x x ! x =  k
(i)
x , where we have
X
suppressed the frequency dependence. The same holds for y. For now, we focus on
the collinear configuration in which the further approximation k(i) = k(i)x y = 0 is
made. In this case, the phase-matching is determined by the phase-matching in the
z-component of the fields alone.
Z
L
 (!p,!s,!i) = dz e
i kz(!p,!s,!i)z
0
L ⇣ ⌘
(1.21)
L
sinc  kz(!p,! ,! )L/2 e
i kz(!p,!s,!i) 2
s i
2
We can expand k(s)(!s) as k(!s) = k(!0) + k0(!s   !0) + k00/2(! 2s   !0) , where
!0 = !p/2. We can then write  k = k(!p)  2k(!0)  2k00/2(! 2s   !0) . Where the
k0 terms cancel. The k00 terms have been combined due to the fact that
!s + !p = 2!0. For perfect phase-matching, k(!p) = 2k(!0), and we can write:
L ⇣ ⌘ 00 2 L
 (!p,!s,!i) = sinc k
00(! 2 ik (!s !0) 2s   !0) L/2 e (1.22)
2
26
Assuming the pump has a classical envelope function ↵(!), our state can then be
written as:
Z Z
d! d!
| = " s ii ↵(!s + !i) (! ,!
†
p s,!i)â (!
†
s)â (!i) |vaci (1.23)PDC 2⇡ 2⇡ s i
This state is in general not separable in time-frequency basis, and the temporal
and spectral correlation between signal and idler fields is one of the characteristics
entangled two-photon spectroscopy and ETPA are hoping to utilize. As noted in
[24, 27], a separable (unentangled) field obeys the condition that
 (! + !̃) (⌧   ⌧̃)   1. However, this is not the case for non-separable states, such
as those generated via SPDC.
Spatial Considerations
If we don’t constrain ourselves to the collinear geometry, but keep our previous
approximations, then we have k(s) = (i)x  kx and k
(s) = k(i)y   y . This yields a spatial
distribution symmetric around the pump beam.
Without thinking too hard about directionally varying refractive indices, we
note that ~k(s) 2 = (k(s))2 + (k(s))2 + (k(s))2| | x y z and the same for ~k(i). This yields:
q  !
(k(s))2 + (k(s) 2
k(
)
s) = |~k(s)|2
x y
  ((k(s)x )2 + (k
(s))2) ⇡ |~y k(s)| 1  (1.24)z
2|~k(s)|2
Where the approximation holds for small angles. If we assume that the
phasematching is perfect ( kz = 0) for perfectly on-axis signal, idler and pump,
then for ~k(s) = (0, 0, k(s)),  kz = 0 k(p)z ! z = k
(s)
z + k
(i)
z , and k
(p)
z /2 = |~k(s)| = (i)|~k |.
Evaluating  kz under this condition yields:
 !  !
k(p) (k(s))2 + (k(s))2 k(p) (k(i))2 + (k(i))2
 k = k(p)
z x y z x y
z   1    1  (1.25)z 2 (k(p))2/2 2 (k(p) 2z z ) /2
27
Noting again that k(s) = k(i) (s)x   x and ky = k
(i)
  y yields:
2    
 kz = (k
(s))2 + (k(s))2 (1.26)
k(p)
x y
z
With these geometrical considerations in mind, we can re-evaluate Eq. 1.20 to yield
the k-vector distribution for perfectly degenerate PDC.
Z Z ⇢
L L 2    
 
dzei kzz = dz exp i (k(s))2 + (k(s))2 z
(p) x y
⇢0 0 kz     ⇢      (1.27)2L
= exp i (k(s))2 + (k(
2L
s))2 sinc (k(s))2 + (k(s))2
2k(p)
x y x y
z 2k
(p)
z
This yields a circularly symmetric sinc-shaped k-vector distribution. The
corresponding spatial correlation area is given by the 2-dimensional Fourier
transform of the momentum distribution, a discussion of the spatial correlations
can be found in [69]. This is similar to the time-frequency variables, in which the
correlation time is given by the Fourier transform of the spectral distribution.
Note on Spectral and Spatial Behavior with Initial Phase-Mismatch
We can modify Eq. 1.25 to accommodate imperfect collinear phasematching by
adding a phase o↵set term, k(p) 2 ~k(s)z   | | =  kz0 :
 !  !
(k(s))2 + (k(s) 2( ) ~ ( ) x y ) (k
(s)
x )2 + (k
(s) 2
p s y
)
 kz = k   2|k |+ =  kz0 + (1.28)z
|~k(s)| |~k(s)|
The k-vector distribution then looks like:
(  !)
L (k(s) 2 (s) 22 x ) + (ky )sinc  kz0 + (1.29)2 |~k(s)|
We can compare this to the dependence on the initial phase-mismatch for the
28
frequency spectrum:
⇢ ✓ 00 ◆ L k (! )
sinc2  k 2 0 (! ! )20     0 (1.30)
2 2
Notably, the sign between the two terms in Eqs. 1.29 and 1.30 are opposite of
one another, for a crystal with normal dispersion. This explains why raising the
temperature of a PPLN crystal makes the frequency spectrum more degenerate,
and why lowering the temperature of the PPLN crystal makes the angular spread
more ring-like. We use this ring-like SPDC mode for experiments with
distinguishable signal and idler.
Finally, we can get an intuition for the combined frequency-spatial behaviors
by making a crude approximation in which we combine the two and ignore the
frequency dependence of the angular k-vectors.
(  !)
L (k(s))22 x + (k
(s) 2 00
y ) k (! )
sinc 2 0  (! ! 2  0) (1.31)
2 |~k(s)| 2
This provides the intuition that more extreme frequency mismatches are
present on the outside of the PDC ring. As the ring shrinks into a spot and
disappears, the outside of the ring becomes the center, and so the center point gets
more extreme frequency mismatches. This agrees with expectations and
observations.
29
CHAPTER II
THEORY OF ETPA
Introduction
One major result of this thesis is the application of a general calculation for
non-resonant TPA of an arbitrary quantum state to the problem of ETPA. In this
chapter, we first review in brief the calculation of the TPA probability, following
closely the derivation in [70]. This TPA probability is applied to an entangled
two-photon state, in order to set rigorous bounds on the non-resonant ETPA
probability, as argued in [59, 60]. We also apply this calculation to a coherent state
to identify the regimes in which ETPA has the greatest possible benefit relative to
classical light. We then consider spatial e↵ects for dye molecules in solution, under
the assumption that the pairs propagate in a Gaussian beam. Finally, we consider
the e↵ects of inhomogeneous broadening of the TPA lineshape on the measurement.
The traditional approach, first developed by Maria Göppert-Mayer, uses
perturbation theory for state amplitudes and posits a density of molecular or
photonic states to arrive at the Fermi Golden Rule for the conventional
cross-section for TPA [43]. Accessible textbook treatments are given in the
quantum optics text by [71] and in the nonlinear optics text by [63]. When dealing
with short light pulses or light having time–frequency correlations, a more detailed
treatment is needed, and several such treatments have appeared, a few being Refs.
[28, 47, 48]
Quantum TPA
The calculation is carried out in the interaction picture where
⇢̂I(t) = Û
†
0(t, t0)⇢̂(t)Û
†
0(t, t0), and satisfies @⇢̂I(t) = [ĤI(t), ⇢̂I(t)]. The density
30
P
operator can be described perturbatively as ⇢̂I(t) = ⇢̂ +
1 (n)
0 n=1 ⇢̂0 (t). The
interaction Hamiltonian for the interaction between the optical field and the
molecule is:
ĤI(t) =  d̂IÊI(t). (2.1)
The form of the interaction Hamiltonian, after application of the rotating wave
approximation (RWA) is: Ĥ (t) = d̂( )(t)Ê(+) d̂(+)(t)Ê( )I     with,I I I I
X
d̂( )I (t) = d i j e
i(!i ! )t
ij | i h |
j ,
j,i>j
X (2.2)
d̂(+)I (t) = d i j e
i(!i !j)t
ij | i h | .
j,i<j
where dij = hi| d~ · e  |ji are the electric dipole matrix elements. The
interaction-picture operators commute for di↵erent degrees of freedom commute at
all times.
The calculation assumes the molecule begins in the ground state. The density
N
operator prior to the interaction is the tensor product, ⇢0 = ⇢M ⇢F , of the
molecule and field density operators respectively. The first-order perturbative
solution to the state of the molecule at time t is:
Z
(1) 1
t h i
⇢̂ (t) = ~ dt1 ĤI(t1), ⇢̂I 0 (2.3)i  1
Higher orders can be calculated by iterated application of the perturbation.
The second order solution is the lowest order that can lead to a population in the
excited state and corresponds to single photon absorption. For this, the interaction
Hamiltonian is applied twice. The density operator can be written as:
✓ ◆
1 2
Z Z
t t2 h h ii
⇢̂(2)(t) = ~ dt dt ĤI 2 1 I(t2), ĤI(t1), ⇢̂0 (2.4)i  1  1
31
The same goes for higher-order interactions such as two-photon absorption. Of
interest for two-photon absorption is the fourth order solution:
✓ ◆4 Z Z Z Zt t4 t3 t2
⇢̂(4)
1
(t) = ~ dtI 4 dt3 dt2 dt1⇥i  1  1  1  1
h h h h iiii (2.5)
ĤI(t4), ĤI(t3), ĤI(t2), ĤI(t1), ⇢̂0
Given these building blocks, we can proceed to outline the case of single-photon
and two-photon absorption, respectively. For one-photon absorption, two
interactions lead to a population in the e state of the molecule. For two-photon
absorption, four interactions lead to a population in the f state of the molecule,
where virtual states or real intermediate states e, e0 serve as stepwise interactions in
the perturbative picture.
One-photon absorption
We first start by calculating one photon absorption. The population Pe in a
given excited state |ei resulting from one-photon absorption is equal to the
expectation value: P (2)e = Tr(⇢̂
(2)(t) |ei he|). Where,
I
⇣ ⌘ Z Z1 2 t t2
⇢̂(2)(t) = ~ dt2 dt1[ĤI(t2), [ĤI(t1), ⇢0]] (2.6)I i  1  1
Since the molecule starts in the ground state, ⇢0 = |gi hg| ⇢̂F , the integrand can
be simplified by applying the trace operator within the integral. Noting that
⇢0 |ei he| = |ei he| ⇢0 = 0, all terms where ⇢0 is leading or trailing are zero. This
yields: ⇣ ⌘
Tr [ĤI(t2), [ĤI(t1), ⇢0]] |ei he| =
⇣ ⌘ (2.7)
Tr  ĤI(t2)⇢̂0ĤI(t1) |ei he|  ĤI(t1)⇢̂0ĤI(t2) |ei he|
After application of the rotating wave approximation, the interaction
Hamiltonian is Ĥ (t) d̂( )(t)Ê(+) d̂(+)I ⇡       (t)Ê
( )(t). The neglected terms will
I I I I
32
contribute small corrections to the final result, which are significant only when the
excitation is far from resonance. Using Eq. 2.2 for the dipole operators, we find for
the first term in Eq. 2.7,
Ĥ (̂t )⇢ Ĥ (t ) = d̂( )(t )Ê(+)I 2 M I 1 (t ) |g
(+) ( )
i hg| d̂ (t )Ê (t
I 2 I 2 I 1 I 1)
X (2.8)
d d i j ei!igt2ei!gjt1 Ê(+) ( )ig gj | i h | ⇥ (t2)Ê (t1),I I
i,j
where !ij = !i   !j. The second term in Eq. 2.7 is the same with t1 and t2
swapped. Hence, using permutation inside the trace, the identity, Tr(Â†) = Tr(Â)⇤,
and the initial state, ⇢̂0 = |gi hg|⌦ ⇢̂F , the trace over the molecule and field can be
written separably as:
⇣X ⌘ ⇣ ⌘
Tr d d i e ei!| i h | igt2ei!get1 Tr Ê(+) ( )M ig ge ⇥ F ⇢̂F Ê + c.c. (2.9)I I
i
D E ⇣ ⌘
We can further write Ê( )(t )Ê(+) (+) ( )1 (t2) = TrF Ê ⇢̂F Ê , which is theI I I I
second-order field correlation function, related to g(1)(t1, t2), without normalization.
This leaves the population as:
Z Z
(2) dged
1 1 D E
P = eg dt dt ei!eg(t2 t1) Ê( )(t )Ê(+)(t ) + c.c. (2.10)
e ~2 2 1 I 1 I 2 1  1
where we have evaluated the trace over the molecule. We can introduce dephasing
phenomenologically by taking ei!eg(t2 t1) e ( eg i!eg)(t2 t! 1) [70, 72–75]. In the
frequency domain, this simplifies to:
Z ⌦ ↵
1 d! â†(!)â(!)
P (2) = ~ L
2
0dgedeg + c.c. (2.11)e 2 2⇡  eg   i(!eg   !)
This is proportional to the overlap of the spectrum of the photons with the
absorption line profile, and agrees with the standard calculation for coherent and
33
thermal states.
Two-photon absorption
Moving on to the calculation for two-photon absorption, the fourth order
perturbative solution is needed, where the RWA has been applied preemptively to
the interaction Hamiltonian, keeping only d̂(±)Ê(⌥) terms, such that
V̂ (p)(t) = d̂( p)Ê(p):
X Z Z Z Z1 t t4 t3 t2
⇢̂(4)(t) = ~ dt4 dt4 3 dt2 dt1⇥ 1  1  1  1
p,q,r,s
h h h h iiii (2.12)
V̂ (s)(t ), V̂ (r)(t ), V̂ (q)4 3 (t
(p)
2), V̂ (t1), ⇢̂0 ,
The various polarization states permutations are included in the sum over
p, q, r, s = ± for simplicity. V̂ (+) raises the molecule and lowers the field. V̂ ( ) does
the opposite. The solution has 16 terms contributing to the sum, many of which
can be neglected in most cases. Similar to before, we focus on interactions resulting
in a population, but now in the f state. We write this expectation as
Pf = Tr⇢̂(4) |fi hf |.
Given that the transition of interest is g ! f , the only terms resulting from the
nested commutators that are nonzero are those of the form, V̂ (+)V̂ (+)⇢̂ V̂ ( )V̂ ( )0 ,
identified as:
⇣ ⌘
QDQC = Tr Tr V̂ (+)M F (t4)V̂
(+)(t )⇢̂ ( ) ( )3 0V̂ (t2)V̂ (t1)
⇣ ⌘
QRP = Tr Tr V̂ (+)M F (t4)V̂
(+)(t )⇢̂ V̂ ( )2 0 (t1)V̂
( )(t3) (2.13)
⇣ ⌘
QNRP = TrMTr V̂
(+)(t )V̂ (+)(t )⇢̂ V̂ ( )F 3 2 0 (t1)V̂
( )(t4)
With
X Z Z Z Z1 t t4 t3 t2 X
P = dt dt dt dt QPATHf ~ 4 3 2 1 (2.14)4  1  1  1  1
p,q,r,s
With PATH = {DQC,RP,NRP}, referring to the specific time-ordering of
34
the interaction. Because we are using the interaction picture, operators for the field
commute with those of the molecule. Thus, we can separate the correlation
functions into field and molecule portions, keeping in mind that ⇢̂M = |gi hg|. The
molecular correlation functions each take the form:
C = Tr(d̂( )d̂( )⇢̂ d̂(+)d̂(+)M M |fi hf |) with the appropriate time orderings for the
DQC, RP, and NRP terms.
Accounting for dephasing, the molecular correlation functions can be written
as[72–75]:
X
CDQC = M e ( fe0 i!fe0 )re ( fg i!fg)sfee0g e
 ( eg i!eg)⌧
M
e,e0
X
CNRP = M e ( fe0 i!fe0 )re ( ee0 i!ee0 )se ( eg i!eg)⌧
M fe
0eg (2.15)
e,e0
X
CRP = M e ( fe+i!fe)re ( ee0 i!ee0 )s  ( eg i!eg)⌧
M fe
0eg e
e,e0
as detailed in[70]. Here r = t4   t3, s = t3   t2, and ⌧ = t2   t1, and
Mfee0g = dfe0de0gdgedef . The time variables track the time increases during disjoint
time intervals during which dephasing takes place, allowing the dephasing during
each interval to be considered separately. For each transition at frequency !ij, the
corresponding dephasing rate is  ij.
The field correlation functions take the form,
C = Tr(Ê(+)(t )Ê(+)(t )⇢̂ Ê( )F a b F (tc)Ê( )(td)), again with the appropriate time
ordering. Via the cyclic permutation symmetry of the trace operation, this is
simply the expectation, Ê( )h (tc)Ê( )(t (+) (+)d)Ê (ta)Ê (tb)i, for the state ⇢̂f , where
care must be taken to keep track of the proper time ordering.
It is worth noting that the four-field correlation function is the degree of
second-order temporal coherence, g(2) when normalized by
hÊ( )(t (+)1)Ê (t ) ( ) (+)1 ihÊ (t2)Ê (t2)i[71].
The population in the f state Pf = Tr(⇢̂(4) |fi hf |), can then be written in the
35
form:
1 Xpath
P = M Rpathf ~4 fee
0g e,e0 + c.c., (2.16)
e,e0
where,
Z 1 Z 1 Z 1
RDQC0 = dr ds d⌧e ( fe0 i!fe0 )re ( fg i!fg)se ( eg i!eg)⌧⇥e,e
0 D0 0 E (2.17)
Ê( )(t ( ) (+) (+)1)Ê (t2)Ê (t4)Ê (t3)
with analogous expressions for RP and NRP. It is worth noting that these
expressions are equivalent to those in [72–74], generalized to arbitrary quantum
fields.
These results are most easily applied in the frequency domain. Applying this
transformation yields:
Z 1 0 Z Zd! 1 dd 1DQC d!̃R = !
e,e0
D  1 2⇡  1 2⇡  1 2⇡ E
Ê( )(!0)Ê( )(! + !̃ !0)Ê(+)
(2.18)
  (!)Ê(+)(!̃)
( fe0   i(!fe0   !̃))( fg   i(!fg   (! + !̃)))( eg   i(!eg   !0))
For completeness, we also include RP and NRP terms:
Z 1 0 Z Z
DQC d!
1 d! 1 d!̃
R
e,e0 =
D  1 2⇡  1 2⇡  1 2⇡ E
Ê( )(!0)Ê( )(! + !̃ !0)Ê(+)  (!)Ê(+)(!̃)
( fe0   i(!fe0   !̃))( fgZ  i(!
0
fg  Z (!   !Z)))( eg   i(!
0
eg   ! ))
1 0 1 1 (2.19)
RDQC
d! d! d!̃
e,e0 =
D  1 2⇡  1 2⇡  1 2⇡ E
Ê( )(!0)Ê( )(! + !̃ !0)Ê(+)  (!)Ê(+)(!̃)
( fe   i(!fe   (! + !̃   !0)))( ee0   i(!ee0   (!0   !)))( eg   i(! 0eg   ! ))
These expressions are still quite general, allowing for resonant intermediate
states. While this is some sense concludes the calculation, it is useful to make a few
further approximations, and write things in terms that are more amenable to future
36
calculations. We can substitute the field operator Ê(+)(!) = L0a(!), where
q
L = ~!00 2 . Where A0 is the e↵ective beam area at the molecule’s location."0ncA0
This factor can be expressed in terms of a beam’s propagation for ensemble
calculations, which we describe later in this chapter. This substitutes the four field
correlation function with:
D E
Ê( 
⌦ ↵
)(!0)Ê( )(! + !̃ !0)Ê(+)(!)Ê(+)  (!̃) 4! L0 â
†(!0)â†(! + !̃   !0)â(!)â(!̃)
(2.20)
O↵-Resonance Assumption
The case we are most interested in is the case in is the DQC pathway with no
resonant intermediate states, in which case we can simplify the above expressions
as:
4
RDQC
L
0 = 0 ⇥e,e ( !fe0 + !0)(!eg   !0)
Z 1 0 Z 1 Z 1 ⌦ ↵ (2.21)d! d! d!̃ â†(!0)â†(! + !̃ !0  )â(!)â(!̃)
 1 2⇡  1 2⇡  1 2⇡  fg   i(!fg   (! + !̃))
It is worth taking a moment to relate this quantity to the conventional
two-photon absorption cross-section,  (2). In this case, the integral expression is
independent of the intermediate state pathways taken(e, e0), and the sum from
Eq.2.16 can be evaluated as
✓ ◆
~ 2 DXQC1 !0 dfe0de0gdgedef
~ ⇥ ... (2.22)4 2"0ncA0 0 ( !fe0 + !0)(!eg   !0)e,e
multiplied by the integral expression in Eq. 2.21. Where we have substituted L40
and Mfee0g. Noting that the traditional form of the two-photon absorption
cross-section is written as [63]:
✓ ◆    2 2
(2) ~!0 1 1  
 X d  
  =   fe
deg    (2.23)
"0nc 2  4fg ~   0 (!eg   !0)  e,e
37
(2)
We can write the prefactor in Eq. 2.22 as    fg2 , where it is worth noting that  (2)2A0
is implicitly a function of  fg. Then Eq. 2.16 can be written as:
Z Z Z ⌦ ↵
 (2) fg d!0 d! d!̃ â
†(!0)â†(! + !̃   !0)â(!)â(!̃)
Pf =
2A2
+ c.c. (2.24)
0 2⇡ 2⇡ 2⇡  fg   i(!fg   (! + !̃))
This is the central result from this calculation, which relates the classical
two-photon absorption cross-section to an overlap between the four-frequency
correlation function of an arbitrary input state with the lineshape of the final state,
under the assumption of non-resonant intermediate states, with dephasing
described via Kubo theory.
This equation can be put in a more useful form, by utilizing the change of
variables, z0 = !0   !0, z = !̃   !0, and x = ! + !̃   2!0. This yields:
 (2)
Z
 fg dx 1Pf = 2 ⇥
Z Z 2A0 2⇡  fg   i(!fg   2!0   x)0 ⌦ (2.25)dz dz
â†(z0 + ! )â†(x z0
↵
0     !0)â(x  z + !0)â(z + !0) + c.c.
2⇡ 2⇡
Where the z integrals can be shown to be real. Therefore, only the real part of the
Lorentzian, (  fg2 2 ), contributes. fg+(!fg 2!0 x)
Gaussian Pulse TPA
It is worth applying this theory to a coherent state Gaussian pulse, |↵i of the
qp
2 2
form: ↵(!) = ↵ 2⇡e (! !0) /4 0 . Recalling that â(!) |↵i = ↵(!) |↵i, the average 
R R
number of photons can be written as N = d!2 hâ
†(!)â(!) = d!↵(!)2 = ↵ 2i 2 | | .⇡ ⇡ 0
Utilizing Eq. 2.25, the probability can be written:
 (2)
Z
 
P = fg
dx  fg
f
2A2 2⇡  2
⇥
+ (!   2!   x)2
  0 fg
fg 0
 Z  
(2.26)
  dz
2
↵⇤
 
  (z + !
⇤
0)↵ (x  z   ! ) 0
2⇡  
38
The N2 dependence can be readily observed for this, verifying the expected
quadratic scaling relationship. The overlap integral can be evaluated, by noting the
integral over z serves as a convolution of the Gaussian envelope with itself,
resulting in another Gaussian. For details, see [70]. The resulting probability is:
 (2) 
P = N2 fgf 2 ⇠( fg/2 ) (2.27)2A0
where ⇠(z) = exp(z2)erfc(z), referred to as the scaled complimentary error function,
which is a special case of the Voigt distribution, which results from the convolution
of the Gaussian lineshape of the pulse with the Lorentzian shape of the absorption
line. See Fig.1.2 for the full behavior.
We can identify the two limiting behaviors. In the first, when the pulse is
spectrally narrow, the TPA probability increases linearly as the pulse broadens
spectrally. In the time domain, this can be understood as the interaction strength
scaling with the inverse duration of the pulse. However, once the spectral width of
the pulse surpasses the width of the absorption line, the benefit of shortening the
pulse is o↵set by reduced resonant overlap with the final state, and an asymptotic
e ciency is reached in the short-pulse limit:
8
><> 2 (2)N   p 2 (  ⌧   )
A0 ⇡
fg
Pf = > (2.28)>: (2)N2    fg
2 2
(      )
A fg0
The latter region ( fg ⌧  ) is where the e↵ect of frequency correlations have
the most impact, and where the potential for quantum advantage in the sense of
Eq. 1.3 is greatest.
39
Quantifying the Enhancement of Entangled Two-Photon
Absorption(Opex)
This section utilizes the general form of the probability of a TPA event, Eq.
2.24, to calculate the TPA probability for time-frequency entangled photon pairs.
Additionally, we use the TPA theory to derive bounds strict bounds on the TPA
e ciency of an arbitrary time-frequency entangled state for a given spectral width.
This excludes the possibility of an exotic state leading to higher e ciency than
expected. This section closely follows the work published in [59].
ETPA Calculation
In order to calculate the advantage of utilizing entangled photons, we apply the
two-photon state of time-frequency entangled photon pairs produced via Type-0
collinear SPDC, Eq. 2.33, to the theory developed in the previous section. In the
case where the pump is narrow, we can consider a simplified model of the
joint-spectral amplitude which is analytically tractable, without the need for
numerical simulations. We use this calculation to place bounds on the maximum
e ciency of ETPA, and relate this calculation to the heuristic equation introduced
by [44] (Eq. 1.4): ✓ ◆
N (2)
P = EPP
N
  = EPP
 
f e ⇥ fEPP (2.29)
A0 A0 A0Te
where NEPP < 1 is the mean number of photons in a pulse of entangled photons
with duration Tp. When intermediate-state populations can be neglected and TPA
proceeds only through coherent pathways, the probability for a molecule to
transition from the ground state g to the final state f driven by a pulsed field,
described by any pure state | i, can be rewritten from Eq. 2.24 as:
 (2)
Z Z Z
d! d!̃ d!0
P = (! ! !̃)C(4) 0f 2 L fg     (! ,! + !̃
0
  ! ,!, !̃), (2.30)
A0 2⇡ 2⇡ 2⇡
40
where the frequency-domain field correlation function is:
C(4)(! ,! ,! ,! ) =  â†a b c d h | (!a)â
†(!b)â(!c)â(!d) | i (2.31)
and the real part of the (peak-normalized L(0) = 1) homogeneously broadened TPA
transition is a Lorentzian line-shape, with width  fg:
 2
L(!fg   !   !̃) =
fg
 2
(2.32)
+ (! 2
fg fg
  !   !̃)
Collinear Type-0 or Type-I SPDC pumped by a pulse of finite duration can be
designed to occur in a single spatial-and-polarization mode, as described by:
Z Z
p d! d!̃
| i = 1  "2 |vaci+ "  (!, !̃)â†(!)â†(!̃) |vaci . (2.33)
2⇡ 2⇡
For the current calculation, we neglect higher-order terms representing the
generation of multiple pairs. The case of few to many pairs per pulse is discussed in
later sections. The joint-spectral amplitude  (!, !̃) is determined by the spectrum
of the pump laser and the phase-matching properties of the nonlinear crystal. It is
R R
square-normalized, such that d! d!̃ 22 2 | (!, !̃)| = 1. For Type-0 or Type-I SPDC,⇡ ⇡
the JSA is symmetric under exchange of variables:  (!, !̃) =  (!̃,!). The mean
number of photons in a pulse is N 2EPP = 2" .
The field correlation function is evaluated as:
C(4)(!0,! + !̃ !0  ,!, !̃) = 4"2 ⇤(!0,! + !̃   !0) (!, !̃). (2.34)
The probability is maximized when the pump laser is resonant with the
two-photon transition !p = 2!0 = !fg. Under these assumptions, we can take a few
steps to simplify the equations. We start by inserting Eq.2.34 into Eq. 2.30 and
41
Figure 2.1. The anti-diagonal projection K (x) of the two-photon amplitude
 (!, !̃), and the two-photon absorption profile L(x), assumed to be two-photon
resonant with the center frequency of the EPRP light field. Also shown is the
‘marginal’ projection, M (!) = (1/2⇡)
d!̃
2 | (!, !̃)
2
| , which is the energy spec-
⇡
trum of the EPP and has bandwidth B.
include the same change of variables, used in 2.25, z = !̃ 0 0  !0, z = !   !0 and
x = ! + !̃   2!0: ✓ ◆ Z
N  (2) dx
Pf = 2
EPP
L(x) 2|K (x)| (2.35)
A0 A0 2⇡
with, Z
dz
K =  (!0 + z,! + x  z). (2.36)
2⇡
Here, K represents the integrated amplitude for TPA at a particular two-photon
detuning x. A graphical interpretation is given in Fig. 2.1.
We define the spectral overlap factor, ⌘ as:
Z
dx
⌘ ⌘ 2 L(x) 2|K (x)| (2.37)
2⇡
42
Figure 2.2. Two-photon amplitude in frequency domain (!, !̃) and time do-
main  (t, t). Diagonal and anti-diagonal frequency arguments are ! =
p p D
(! + !̃   2!0)/ 2 and !A = (!   !̃)/ 2. Diagonal and anti-diagonal time ar-
guments are ⌧ = t   t̃ and ⌧̃ = t + t̃/2. The entanglement time is approximated as
the inverse of the marginal bandwidth, B, of the EPP spectrum:Te = 1/B.
Comparison of Eq. 2.35 with 2.29 yields the relationship:
⌘ = fEPPTe (2.38)
Since reported values of  e have been 7  10 orders of magnitude larger than
(2)
estimates would yield given the estimate    e = , it is worth investigating theA0Te
factor fEPP , which would need to be large to account for these discrepancies.
 (2)
 e = ⇥ fEPP (2.39)
A0Te
Given this comparison, we now calculate an upper bound on the
temporal-spectral enhancement possible given a simplified form of the JSA.
The form of the JSA that maximizes the spectral compression onto the
two-photon absorption line L(x) is narrow along the diagonal, and broad along the
anti-diagonal, which is exactly the form generated via Type-0 and Type-I SPDC
with a narrowband pump. In order to find an expression for the upper bound of the
integral, we assume the two-photon amplitude can be written in a factored form as
43
the product of narrow and broad functions,  N(!) and  B(N) centered at !0,
oriented along the diagonals of the !, !̃ plane:
✓ ◆
!   !̃
 (!, !̃) =  N(! + !̃   2!0) B . (2.40)
2
Both functions are square-normalized in d!2 and  B(!) =  B( !) has the⇡
required symmetry. Under these assumptions, the spectral width  N(!) is
determined by the duration of the pump pulse and the  B(!) is determined by the
phase-matching function. The ratio of these two widths is a measure of the amount
of time-frequency entanglement in the EPP field [76]. The more elongated the
two-photon amplitude is, the higher the degree of entanglement. This model can be
a good approximation for Type-0, -I, or -II SPDC, depending on pulse durations
and phase-matching conditions.
The entanglement time can be evaluated under the optimal factorization
assumption. Defining di↵erence and sum times as ⌧ = t  t̃, ⌧̃ = (t  t̃)/2, the
2-dimensional Fourier transform of Eq.2.40 gives the joint temporal amplitude,
 (t, t̃): Z Z
d! d!̃
 (t, t̃) =  (!, !̃)e i!(⌧̃+⌧/2e i!̃(⌧̃ ⌧/2)
2⇡ 2⇡ (2.41)
= 'N(⌧̃)'B(⌧)
where 'N(⌧) and ')B(⌧) are Fourier transforms of  N(!) and  B(!)
respectively. Using a change to diagonal and anti-diagonal frequency variables,
x = ! + !̃   2!0, and y = (!   !̃)/2, we find:
Z
dx
'N(⌧̃) = e
 i!0⌧̃  N(x)e
 ix⌧̃
Z 2⇡ (2.42)
dy
'B(⌧) =  
 iy⌧
B(y)e
2⇡
We quantify the entanglement time Te as the ‘width’ of the di↵erence-time
44
distribution |'N(⌧) 2| , s illustrated in Fig. 2.2.
For this case we can evaluate K (x):
Z
dz
K (x) =  N(x)  B(z   x/2) (2.43)
2⇡
The z-integral in this equation is independent of x since, it represents only a shift
of the center of the distribution, or the bounds of integration. Because of this, the
overlap, ⌘, between L(x) and K (x) becomes factorable into two separate terms:
⌘ = 2⌘N⌘B. With: Z
dx
⌘N = L(x)| N(x)
2
|
 2 Z
⇡    (2.44)
⌘ =   dzB     B(z)2⇡  
The two quantities can be bounded from above. We start with ⌘N . Since
L(x)  1 everywhere and | N(x) 2|   0 everywhere, we can establish the following
bound on ⌘N : Z
dx
⌘N  | N(x)
2
| = 1. (2.45)
2⇡
Where the bound of 1 stems from the fact that  N(x) is square-normalized in
dx
2 . The condition that satisfies this bound is when  N(x) is much narrower than⇡
L(x), yielding strong frequency anti-correlation, which is the case we’re most
interested in.
The other factor, ⌘B, is maximized when  B(z) is constant over the region over
which it is defined. This can be shown via the Cauchy-Schwarz inequality.
Given square-integrable functions, defined on a domain with the inner-product
norm, the Cauchy-Schwarz inequality holds. Let g and h be such functions on such
a domain, D. The inequality can be stated as:
  Z     
2 Z Z
g(x)h⇤(x)dx  2 2     |g(x)| dx |h(x)| dx. (2.46)
D D D
45
Where we note that equality can be achieved when g(x) =  h(x) for   2 C.
This inequality can be straightforwardly applied to set a bound on ⌘B by
noting that  B(!) is square-normalized and restricted to a finite frequency range
[ !0,!0] for photons generated via SPDC. Then, considering some range, [ ⌦,⌦],
outside of which  B(x) = 0, we can write for g(z) = 1,
  Z ⌦  2 Z Z  dx  
⌦ dx ⌦ dx
⌘B ⌘   g(x) 
⇤ (x)    |g(x)
2
| dx 2| B(x)| . (2.47)
2⇡ B ⌦  ⌦ 2⇡  ⌦ 2⇡
Noting that  B(x) is square-normalized, this bound can be stated:
Z ⌦ dz 2⌦
⌘B  1 = . (2.48)
 ⌦ 2⇡ 2⇡
Beyond this, we can gather some insight into the form of  B(x) that saturates
this bound. Since the bound is saturated when  B(z) =  g(z) =  , and  B(z)
must be square-normalized, | B(z) 2 =   2| | | = ⇡/⌦ is maximal. In other words, a
rectangular spectral distribution over the region, [ ⌦,⌦] maximizes the quantity ⌘B
with a value ⌘B = ⌦/⇡ ! ⌘max = 2⌦/⇡.
We can get an intuitive interpretation of this result from the time-domain
picture. We can consider ⌘B the modulus-square of the Fourier transform of  B(x),
evaluated at time 0. For the optimal case, the time-domain function is found to be:
r
⌦
' (⌧) = e i!0⌧B sinc(⌦⌧) (2.49)
⇡
which is consistent with the bounds for ⌧ = 0. This can be thought of as
verification that ETPA is maximized if the pairs are correlated as tightly in time as
possible, which maximizes '(⌧ = 0).
46
Together, Eq. 2.45 and Eq. 2.48 yield the bound,
f 2⌦
⌘ = 2⌘N⌘
EPP
B =  (2.50)
Te ⇡
Given the heuristic nature of Eq. 2.29, the precise value of fEPP will depend
on how the entanglement time Te is quantified. For cases of interest, 'B(⌧) is a
simple, smooth function peaked around ⌧ = 0. In this case we can use a convenient
definition for the bandwidth, B, and temporal duration, T of a Fourier-transform
pair, f(!) and f̃(!).
Z
d! |f(!) 2| 1
B = =
2⇡ |f(! 2 2
Z max
| |f(!max)|
(2.51)
dt |f̃(t) 2| 1
T = =
2⇡ |f̃(t 2max| |f̃(tmax)|2
Under this definition of Te, for 'B(⌧max) = 'B(0), the bound is straightforward
with ⌘B = |'B(0) 2| = 1/Te, and fEPP  2. In principle, there can be cases in which
this definition is ill-posed, in which case detailed knowledge of the 'B(⌧) may be
necessary to predict the rate. However, this will always be constrained by the
bound described in Eq. 2.50. In practice, the entanglement time will be larger than
the lower bound Te   ⇡/⌦. Here we note that this parameterization of Te
corresponds to a value slightly larger than the FWHM of common functions2.
2Given a real, positive-definite, peak-normalized (i.e. F (!max) = G(tmax) = 1) functions F (!)
and G(t), we can define the bandwidths, B (in Hz) and T as:
Z
d!
B ⌘ F (!)
Z 2⇡ (2.52)
T ⌘ dtG(t)
These definitions have been used successfully in studies of spectral and temporal filtering of quan-
tum light [77]. They provide general approximations for the full-width for simple peak-normalized
functions such as Gaussians, Lorentzians, and sinc-squared functions, as illustrated in Table 1.
These definitions can be adapted to a square-normalized (complex) Fourier transform pair
47
 (2)
 e ⇡ 2 (2.54)
A0Te
Comparison of EPP and coherent-state TPA
A comparison with classical TPA of a coherent state with the same flux and
cross-sectional area with a Gaussian temporal envelope of duration Tc is useful to
compare the e ciencies of TPA and ETPA. This calculation yields a
flux-dependent quantum-enhancement factor, QEF under the assumption that the
classical pulse is spectrally narrower than the TPA linewidth:
✓ ◆
PEPP 1 T
QEF = F c⇡ (2.55)
P coh N T
F e
This QEF is an alternative way to report the benefit of ETPA, in contrast to
reporting an entangled two-photon absorption cross-section,  e. This QEF reduces
the ambiguity present in reporting  e values without careful consideration of the
experimental parameters under which it is measured.
The bounds on the TPA e ciency, Eq. 2.50, alongside the estimates of the
TPA e ciency, Eq. 2.54, provide of a solid theoretical backing for the argument
that large ETPA cross-sections cannot be explained by the solely by the e↵ects of
time-frequency entanglement of the interactions between the optical field and a
single molecule. Additional considerations such as NRP and RP pathways are
considered in [70]. In the next section, we consider ensemble e↵ects to understand
how inhomogeneous broadening and spatial propagation of a Gaussian beam e↵ect
the absolute rates of TPA and ETPA.
as follows: Z
d! |f(!) 2| 1
B ⌘ =
2⇡ |f(!max)|2 |f(!max)|2
Z (2.53)
f̃(t) 2| | 1
T ⌘ dt =
|f̃(t 2max)| |f̃(t 2max)|
48
TPA Gaussian Beam
Classical Gaussian-Beam TPA
So far, all of our calculations have assumed a single molecule illuminated by
some uniform field with cross-sectional area A0. However, this is not a good model
for realistic experimental configurations observing TPA. In particular, due to the
quadratic intensity dependence, a tight focus is desirable for e cient TPA. It is
worth briefly considering the spatial properties of Gaussian beam propagation, and
the e↵ect on the number of TPA events can be expected.
We want to calculate the rate of TPA for a Gaussian beam going through some
two-photon absorber. For the classical calculation we use a simple model that the
rate of 1- and 2-photon absorption (spatial rate of flux attenuation) are propor-
tional to the concentration, C, cross-sections,  (i) and flux density,  (r,', z):
d 
= (1) (2) 2 C     C    (2.56)
dz
To solve this we make the approximation that the pump beam is attenuated negli-
gibly, and assume there is no 1-photon component, setting  (1) = 0. So then:
Z Z
d  L/2
TPA =   dz = C (2)  2(r,', z) (2.57)
dz  L/2
For a Gaussian Beam of the form:
✓ ◆ ✓ ◆
w r2 2
E~
  kr
(r,', z) = E 00~" exp exp  i(kz +   (z)) (2.58)
w(z) w(z)2 2R(z)
where ~" is the polarization vector, w is the Gaussian beam waist (radius), w0 is the
radius of the beam waist at its smallest point,  (z) is the Guoy phase, and E0 is
the magnitude of the electric field.
49
The flux-density for this field can be written as:
✓ ◆
4   2r2
 (r,', z) = exp (2.59)
2⇡w(z)2 w(z)2
where   is the total photon flux.
For this case the number of TPA events can be written:
Z Z
L/2
TPA = dA C (2) 2(r,', z)
R2  L/2
Z Z 1 ✓ ◆ (2.60)16 2C (2) L/2 1 4r2 
= dz rdr exp
2⇡  L/2 w(z)4 0 w(z)2
R1
Noting that 0 r exp (
2
 r /a) dr = a/2, the radial integral evaluates to:
Z 1 ✓ ◆2
 4r 1 w(z)2
rdr exp = (2.61)
2
0 w(z) 2 4
With this Eq. 2.60 becomes:
 2
Z
C (2) L/2 1
dz (2.62)
⇡ 2 L/2 w(z)
Now for a Gaussian beam, w(z)2 = w20(1 + z
2/z2), where zr is the Rayleigh ranger
and zr = ⇡w20n/ . We can evaluate the integral:
Z
L/2 Z1 1 L/2 1
dz = dz
 L/2 w(z)2 w
2
0  L/2 1 + z
2/z2
✓ ◆ r (2.63)
zr L= 2 2arctanw0 2zr
Which finally yields:
✓ ◆ ✓ ◆
 2C (2) zr L 2 2C (2)n L2 2arctan = arctan (2.64)⇡ w0 2zr   2zr
There are a couple of interesting things to note. The first is that this quantity
50
is independent of w0 for L ! 1. This means that for classical TPA of an ideal
Gaussian beam, the focusing conditions do not directly a↵ect the ensemble rate
of TPA, given an arbitrarily long sample. Beyond this the vast majority of TPA
events occur close to the waist of the beam, which can be seen by comparing the
total possible TPA, proportional to arctan(1) = ⇡/2 with the TPA within the
range [ 2zr, 2zr]: proportional to arctan(
L=2zr
2 ) = ⇡/4. Which is to say that half ofzr
the total possible TPA for a Gaussian beam occurs in the interval [ 2zr, 2zr]
Entangled Two-Photon Absorption
To calculate the spatial properties for ETPA in a single mode, we first need to
adapt our ETPA calculations to this problem. For TPA of a single molecule at a
specific location we had defined the electric field operator as:
r Z
~ (+) ~! d!E (t) = ĉ(!)e i!t (2.65)
2"ncA0 2⇡
To generalize this to many molecules at many locations, we can write this in
terms of the spatial mode of the electric field explicitly:
r
~ Z! d!
E~ (+)(t, ~x) = µ(~x)ĉ(!)e i!t (2.66)
2"nc 2⇡
R
Where µ(~x) is the spatial mode of the entangled pairs, with d~x µ(~x) 2| | = 1.
q
From here, we see that we can straightforwardly replace 1 with µ(~x) for calcu-
A0
lating the TPA probability of a molecule located at position ~x. Then adopting the
TPA probability from previous work yields:
Z 0 Z Z ⌦ ↵
(2) 4 d! d! d!̃ c
†(!0)c†(! + !̃ 0  ! )c(!)c(!̃)
P (~x) =    fg|µ(~x)| Re (2.67)
2⇡ 2⇡ 2⇡  fg   i(!fg   !   !̃)
To calculate the expected number of events in a volume, we then integrate the
51
probability multiplied with the concentration or number density, ⇢N .
Z Z Z
L/2
TPA = dV ⇢N(~x)P (~x) / dz dA|µ(r,', z)
4
| (2.68)
 L/2
q ⇣ ⌘
2
If we take µ(r,', z) = 42 exp
 r , and assume ⇢ = C to be a con-
⇡w(z)2 w(z)2 N
stant concentration over the region [ L/2, L/2] then we can evaluate the total ex-
pected TPA as:
Z
4 2
Z
⇡ L/2 Z1 1
✓ ◆
4r2 
TPA / d' rdr exp
⇡2 0  L/2 w(z)4 0 w(z)2
Z Z
4 L/2 1 1 w(z)2 1 L/2 1
= 2⇡ = (2.69)
⇡2 4 L/2 w(z) 2 4 ⇡ w(z)2✓ ◆  L✓/2 ◆
2 zr L 2n L= 2 arctan = arctan⇡ w0 2zr   2zr
This leaves us with a total TPA rate of:
✓ ◆
TPA =  (2)
2n L
 fg arctan ⇥
⌦   2zZ r ↵
d!0
Z Z (2.70)
d! d!̃ c†(!0)c†(! + !̃ 0  ! )c(!)c(!̃)
Re
2⇡ 2⇡ 2⇡  fg   i(!fg   !   !̃)
From this we can see that as far as the spatial properties of ETPA in a single
spatial mode are concerned, such as those generated in a single-mode wave-guide,
the probability is the same as for classical excitation. For bulk SPDC sources where
spatial-spectral coupling is present alongside entanglement across spatial modes,
this description is insu cient, and is a direction of further research due to the dif-
ficulty modeling the spatial-spectral coupling alongside beam propagation. In our
experiments we focus tightly in order to minimize such e↵ects.
Generalization to Many Molecules
In the theory for TPA, we wrote things down in terms of a single molecule ho-
52
mogeneously broadened via Kubo dephasing theory. This accounts for linewidth
broadening via stochastic collisions, leading to the Lorentzian line shape we’re fa-
miliar with. However, a realistic treatment of many molecules in a solution must
take into account inhomogeneous broadening due to local environmental di↵erence
between many molecules. Here, we operate under the assumption that the pump is
much narrower than  fg, and that the marginal bandwidth is much broader than
 fg. For the following section we use the formalism developed in Chapter 4, which
is a more general case than low-gain SPDC, and reproduces the results exactly.
This yields the following formulas:
   
 (2)
Z
T  2    d!  
2
P fgcoh = f(!)g(!) 
A 20   + (!   2! )2   2⇡  
Z fg
fg 0
Z (2.71)
 (2)T d! d!̃  2
P fg 2 2incoh = (1 + ⇠) |g(!)| |g(!̃)| .
A0 2⇡ 2⇡  2 + (!fg   !   !̃)2fg
We can model inhomogeneous broadening as a distribution of various final
state linewidths, !fg, with the center of the distribution noted !̄fg, which are dis-
tributed according to a Gaussian distribution with variance,  2 . 3. Note that for
fg
homogeneously broadened distributions, by definition !fg = !̄fg, whereas in inho-
mogeneously broadened distributions, !fg is integrated over.
s  !
2⇡ (! !̄ )2   
P (!fg) =
fg fg
 2
exp . (2.72)
2 2
fg fg
Given this distribution of transition frequencies, we can include the probability of
having a given final-state transition frequency in our model. We’ll do this first for
R
3Note the normalization convention is d!2⇡P (!) = 1, for consistency in integrating over fre-
quency d!2⇡
53
the coherent contribution, then for the incoherent contribution:
Z
P inhomog
d!
= fgP (!
coh fg
)Pcoh(!fg)
2⇡
    s  !
 (2)
Z Z
T   d!  2 2⇡ d! (! !̄ )2  2   
=    f(!)g(!) 
fg exp fg fg fg⇥
A 2⇡    2 2⇡ 2 2  2 .0 + (! 2fg fg fg fg   2!0)
(2.73)
In the coherent case, the convolution of the homogeneous and inhomogeneous
distributions is a simple weighting factor independent of the PDC parameters, un-
der the current approximations. This convolution has the following properties: for
 fg    fg the TPA probability is reduced, since you have many instances where
!fg   2!0    fg. In this case, the molecular linewidth is no longer overlapped with
the pump spectrum. In contrast, when  fg ⌧  fg the probability is largely un-
changed, because all molecules fall in a regime where  2 /( 2 + (!   2! )2) ⇡ 1.
fg fg fg 0
For the incoherent contribution, the convolution is within the integral:
Z
P inhomog
d!
= fgP (! )P (! )
incoh fg incoh fg2⇡
(2) Z Z  T d! d!̃
= (1 + ⇠) |g(!) 2 2| |g(!̃)| ⇥ (2.74)
s A0 2⇡ 2⇡Z  !
2⇡ d! 2 2fg  (!fg   !̄  fg) fg
 2
exp .
2⇡ 2 2  2 + (!fg   !   !̃)2fg fg fg
For the incoherent contribution, when  fg ⌧  fg, the probability is largely
unchanged, as we would expect. On the other hand, if  fg    fg there are now
two regimes to think about. If  fg is narrower than the marginal PDC bandwidth,
then the probability is again largely unchanged. However, if  fg is broader than the
marginal PDC bandwidth, the probability is again reduced since some molecules
will have a transition frequency that falls entirely outside the incoherent PDC band-
width.
54
We recognize the Voigt profile, in Eq.2.734 and Eq.2.74:5. And will use this form
in the coming sections. E cient modelling of the Voigt profile can also be accom-
plished via the Tepper-Garćıa function.6
s Z  !
1 2⇡ d! 2 2 (!   !̄  
V (2! ,   ,  , !̄ ) = fg exp fg fg
) fg
0 fg fg fg
⇡ fg  2 2⇡ 2 2  2 + (!fg   2! )2fg fg fg 0
(2.75)
E↵ects of Broadening on Coherent Contribution
In some sense, the e↵ect of inhomogeneous broadening on the coherent con-
tribution is fairly simple, as can be seen in Fig 2.3. As mentioned above, for large
inhomogeneous broadening,  fg    fg, the TPA probability is greatly reduced, and
for minimal inhomogeneous broadening,  fg ⌧  fg the TPA probability is unaf-
fected.
However, in a real experiment, determining the homogeneous and inhomoge-
neous contribution to the linewidth is not trivial. For instance, if we estimate the
total linewidth as the quadrature sum of the inhomogeneous and homogeneous
linewidths, we can make predictions about the coherent contribution, holding  (2)
constant. As seen in Fig. 2.4, given the same total linewidth, the rate of TPA is
reduced as the portion of the linewidth due to inhomogeneous broadening is in-
creased.
An important caveat is that in the above scenario we held the cross-section,
 (2), constant. However, in our derivation,  (2) is dependent on the homogeneous
4Yielding:
 Z  (2) 2
P inhomog
  T    d!  
coh = A   f(!)g(!) 
  ⇥ ⇡ fgV (2!0,  fg, fg, !̄fg)
0 2⇡
5And
Z Z
inhomog  
(2)T d! d!̃
Pincoh = (1 + ⇠) |g(!)
2 2
| |g(!̃)| ⇡ fgV (! + !̃,  fg, fg, !̄fg)
A0 2⇡ 2⇡
6https://arxiv.org/pdf/astro-ph/0602124.pdf
55
1
0.8
0.6
0.4
0.2
0
10-2 10-1 100 101 102
fg/ fg
Figure 2.3. E↵ect of inhomogeneous broadening, as a function of  fg/ fg. We
see clearly for the case, fg    fg, the probability is strongly reduced, whereas for
 fg ⌧  fg, the probability asymptotically approaches the case without inhomoge-
neous broadening.
10-2
10-2.0 10-0.4 101.2
-4 0 0 010 10-1.6 100.0 1.60 0 10 0
10-1.2 0 10
0.4
0 10
2.0
0
10-0.8 100.8 ( 2 + 2 )-1/20 0 0 fg fg
1012 1014 1016
( 2 2 1/2fg+ fg)
Figure 2.4. E↵ect of TPA proqbability for a given homogeneous linewidth, with
a total linewidth estimated by  2 +  2 without adjusting for the di↵erence in
fg fg
 (2). Where 1 is the case with no inhomogeneous broadening   = 30 2 THz.⇡
56
Relative TPA probability
Relative TPA probability
102
10-2.0 10-0.4 101.20 0 0
10-1.6 0 10
0.0
0 10
1.6
0
10-1.2 0.40 10 0 10
2.0
0
100 10-0.8 0.8 2 2 -1/20 10 0 0( fg+ fg)
10-2
1012 1014 1016
( 2 2 1/2fg+ fg)
Figure 2.5. If the TPA cross-section is scaled by the homogeneous linewidth
in simulations, the prediction becomes that the dominant e↵ect is the e↵ective
linewidth to a good approximation
broadening,  fg, and in the canonical derivation this  fg is instead the density of
states. The definition of this a↵ects the prediction in the end relative to the TPA
cross-section
⇣ ⌘  ! 2 1  
 
X  
 (2) = 0
d d
~  
  ef ge    (2.76)
"nc 2 fg   !eg   !0  
e
With this in mind, if we scale the TPA cross-section by the homogeneous linewidth,
then the result is independent of the make-up of homogeneous and inhomogeneous
broadening, and is (approximately) only a function of the measured linewidth, as
shown in Fig.2.5. In Fig.2.4 and Fig.2.5, we’ve used the quadrature sum to estimate
p
the Voigt width. This can be better estimated by: fV ⇡ 0.5346f 2 2 7L+ 0.2166fL + fG.
Where fx are widths of the Voigt, Lorentzian, and Gaussian. This refinement has
no qualitative a↵ect on the conclusions.
Empirical Measurement
All of previous discussion has assumed that the true value of  (2) is known for
the transition of interest and that we’re making our predictions based on it. In real-
7Olivero, J. J.; R. L. Longbothum (February 1977). ”Empirical fits to the Voigt line width:
A brief review”. Journal of Quantitative Spectroscopy and Radiative Transfer. 17 (2): 233–236.
Bibcode:1977JQSRT..17..233O. doi:10.1016/0022-4073(77)90161-3. ISSN 0022-4073.
57
Relative TPA probability
ity,  (2) is an empirically determined quantity and is measured as a function of the
central frequency of the laser:  (2) =  (2)(!0).
In this section, rather than starting with theory based on homogeneous/inhomogeneous
linewidths, we assume the value of  (2) is determined via experiment, and adjust
our predictions accordingly.
Homogeneously Broadened
For the simplest case, in which  (2) is a single homogeneously broadened transition
(in the absence of inhomogeneous broadening) is measured with a monochromatic
laser, we can use Eq. 115 from [70]:
 2
Z
P =  (2) fg
1
f 2 2 dt|A(t)
4
| . (2.77)
  + (! 2
fg fg
  2!0) A0
In this case the frequency dependent  (2)(!0) can be written:
 2
 (2)(! ) =  (2) fg0 2 , (2.78)  + (!fg   2! )2fg 0
which is the quantity we measure in the lab using a monochromatic source. We also
recognize this quantity as appearing in Eq. 2.71, and can rewrite Pcoh as:
   
 (2)
Z
(! )T   d!  2
P 0  coh =   f(!)g(!)   (2.79)A0 2⇡
We can carry out the same calculation for Pincoh, which yields:
 (2)
Z Z
(!0)T d! d!̃  
2 + (!   2! )2
P = (1 + ⇠) fg
fg 0 2 2
incoh 2 |g(!)| |g(!̃)| . (2.80)A0 2⇡ 2⇡   + (!fg   !   !̃)2fg
Note that for !0 = !fg as is our usual assumption, nothing is changed.
58
Inhomogeneously Broadened
Repeating this procedure for the inhomogeneously broadened case, we first note
that the measurement made by a monochromatic laser in this case corresponds to:
Z
P =  (2)
1
f ⇡ 
4
fgV (2!0,  fg, fg, !̄fg) 2 dt|A(t)| (2.81)A0
Where now the measurement of  (2)(! ) made in the lab is:
in 0
 (2)(!0) =  
(2)⇡ fgV (2!0,  fg, fg,!fg). (2.82)in
Comparing this against Eq. 2.73, we see that the relation to the quantity measured
via monochromatic measurement is the same as in Eq. 2.79.
 (2)
 Z  
(!0)T   d!  2P inhom = in  
coh A   f(!)g(!)
   (2.83)
0 2⇡
and the predictions are the same regardless of any assumptions around the nature
of the linewidth, be it inhomogeneously broadened or not. The same holds for the
incoherent contribution.
(2) Z Z
inhomog   (!0)T d! d!̃ 2 2V (! + !̃,  fg,  , !̄P = (1 + ⇠) in g(!) g(!̃) fg fg
)
| | | | .
incoh A0 2⇡ 2⇡ V (2!0,  fg, fg, !̄fg)
(2.84)
Summary of Inhomogeneous Broadening
Two-photon absorption cross-sections are determined by empirical measure-
ments, in which the same homogeneous and inhomogeneous broadening is present
as in ETPA experiments. Because of this, the frequency dependent cross-section
will look the same in both experiments, and we can model the ETPA simply as a
homogeneously broadened experiment with linewidth matching the experimentally
59
measured linewidth.
Conclusions
In this chapter, we presented a summary of the derivation of the TPA proba-
bility for an arbitrary quantum state. We apply this to a realistic model for nar-
rowband TPA, for which analytical approximations are tractable. Within this ap-
proximation, it is further possible to write down strict bounds for the ETPA proba-
bility of a single molecule.
We also provided rigorous description of the spatial distribution of two-photon
absorption of entangled photons in a single Gaussian beam mode. It is worth not-
ing that this is not in general the case for entangled pairs generated via bulk SPDC.
This is in an area where further work would be useful to more accurately deter-
mine ETPA cross-sections, but is beyond the scope of this dissertation. In our ex-
periments, we avoid this complication by focusing tightly such that this becomes a
good approximation, due to the limits of our optical system.
Finally, we discuss the contribution of homogeneous and inhomogeneously
broadening in realistic ETPA experiments, demonstrating that the distinction is
not crucial, assuming the cross-section is itself determined experimentally.
Given these calculations, we can estimate the rates of ETPA for a given sys-
tem, assuming the relevant physical parameters are known. These predictions argue
that the enhancement of the two-photon absorption cross-section is well approx-
imated by the heuristic formula for  e from first predicted by Fei [44]. Based on
these predictions, the reported ETPA cross-sections are many orders of magnitude
than can be explained by the perturbative treatment of TPA. Beyond this, these
bound predict that ETPA signals from typical experimental configurations should
be undetectable.
60
The next chapter outlines experiments in which we attempt to experimentally
bound the ETPA cross-section by carefully designing an experiment in which sig-
nals of a given magnitude would be detectable. Based on the absence of a signal,
we can then determine an upper bound on the cross-section.
61
CHAPTER III
EXPERIMENTAL BOUNDS ON ETPA IN R6G
Introduction
In this chapter, we set out to establish experimental bounds on the e ciency
of ETPA given a carefully calibrated source of entangled photons and a well un-
derstood molecular dye. While we are unable to observe any signal with a bright
source of entangled photons, the absence of a signal with a calibrated detection
setup and source yields a bound on the ETPA cross-section. In addition to probing
TPA, we present experimental results comparing sum-frequency generation (SFG)
of entangled pairs and a CW diode laser. We use the same source and dispersion
control for both SFG and TPA experiments. The results, consistent with the the-
ory outlined in Ch 2, provide a limit on the quantum advantage in molecular TPA.
This chapter summarizes and expands on work published under the title, “Experi-
mental feasibility of molecular two-photon absorption with isolated time-frequency-
entangled photon pairs”, in Physical Review Research. [10]
Given the predictions for ETPA rates outlined in Ch. 2, signal rates are ex-
pected to be prohibitively low even for particularly well suited molecular dyes such
as Rhodamine 6G (R6G), which has excellent stability, high quantum e ciency,
moderate two-photon absorption cross-section, and is highly soluble, allowing for
very high sample concentrations. The experiments described in this chapter were
motivated by a set of experiments conducted by Tabakeav et al., which observed
a signal attributed to ETPA without verification of quadratic scaling [39]. After
the experiments described in this chapter were published, more recent work from
Tabakeav et al. observed quadratic scaling in a similar experiment at seemingly re-
duced rates [8].
62
The ETPA cross-sections for Rhodamine 6G pumped by 1064 nm entangled
photon pairs with bandwidth of 30 nm and 60 µm beam waist reported in [7] was
on the order of  e ⇡ 10 21 cm2. This disagrees strongly with estimates of  e from
(2)
the theoretical description developed in Ch. 2, which estimates it as  e = 2
  .
A0Te
The TPA cross-section of R6G, 9 GM , and the parameters described in their exper-
iment, A0 = 1.13 ⇥ 10 4 cm2, and Te = 10 13 s, allows us to estimate an expected
ETPA cross-section of  e ⇡ 1.6 10 32 cm2⇥ for the experimental configuration in
Tabakeav’s experiment.
Given the large discrepancies between reported and observed values, we set
out to replicate the experiments with R6G as a test system. In particular, our goal
was to control for dispersion, quadratic pair loss, and other linear e↵ects that could
mimic the observed signal. Quadratic scaling with pair loss is a detrimental e↵ect
that can lead to reduced strength of entangled two-photon interactions; however, it
is also a crucial tool for verifying the presence of a nonlinear, rather than linear, op-
tical interaction. As demonstrated nicely utilizing sum-frequency generation in [41],
evidence of a nonlinear interaction driven by entangled photons consists of linear
scaling in the pair rate and, crucially, quadratic scaling with attenuation or loss of
pairs.
Experiment
In this section we describe the experimental apparatus we used to bound the
e ciency of ETPA. In our experiment, we find that the bound on TPA in Rho-
damine 6G by time-frequency entangled photon pairs is several orders of magnitude
below previously reported values. The upper bound we can place on the enhance-
ment is no more than 2 ⇥ 106 times greater than what is predicted by the theory.
For comparison, many ETPA results, including [7] exceed theoretical expectations
by a factor of 1010 or more.
63
In what follows, we first describe the experimental setup and a study of TPA
scaling behavior of SFG with flux for both entangled pairs and classical light. We
then determine the expected TPA fluorescence signal from the molecular sample.
A comparison with the detection noise threshold allows us to establish an upper
bound on the enhancement of ETPA.
Experimental setup and overview
The experimental setup sketched in Fig. 3.1 has two parts: A light generation
block (panels a-b) and a detection block (c-e). A type-0 SPDC source, pumped by
a 532 nm CW laser provides time-frequency entangled photon pairs (EPP) cen-
tered around 1064 nm (panel a). In order to calibrate and align our collection ap-
paratus, we use a CW diode laser at 1064 nm (panel b). We use a continuous-wave
(CW) pumped SPDC source, as used in [7, 41]. The primary benefit to using a CW
pump to generate the entangled photons is that it maximizes the possible pair-
flux while remaining in the low-gain regime, while also maximizing the amount of
time-frequency entanglement attainable. A prism pulse compressor is employed to
compensate for second-order dispersion a↵ecting the photon pairs. Details on EPP
source and pulse compressor are described in subsequent sections.
In the first experiment (Fig. 3.1 panel c), we study SFG by isolated EPP to
confirm that all requirements for successful ETPA, i.e. dispersion compensation,
EPP collection, and focusing are met. This also helps constrain and calibrate ex-
pectations surrounding the e↵ective rates. In the second experiment (Fig. 3.1 panel
d), we carefully characterize and bound the pair flux in our focal volume, by cou-
pling into a single-mode fiber. The fiber coupled pairs are used to characterize the
pair rate and marginal spectrum of the PDC pairs. The diode laser is coupled into
the same fiber to ensure overlap between pairs and the alignment laser. Finally we
replace the fiber with our R6G sample (Fig. 3.1 panel e). This fiber coupling is de-
64
a) Solid State Laser Lithium Niobate
λ/2-Plate Crystal
532nm ND Tap-off
b) Diode Laser
1064nm
c)
Lithium Niobate FiberSP
    Single-photon
ND Crystal BP Detector (SPD)
d) Fiber e) Rhodamine
SPD
50:50 SPD Chopper
Beam Splitter SP PMT
Figure 3.1. For measurement of scaling behavior, either the entangled photon
pairs from a Lithium Niobate source (a), or coherent light from a diode laser (b)
are coupled through another Lithium Niobate crystal (c). The SFG is then coupled
into a single-mode fiber, which is connected to a single-photon detector. ND (Neu-
tral density) filters in front and behind the EPP source are used to compare scaling
behavior. Calibration of the fluorescence collection apparatus is performed by ob-
serving the two-photon absorption (e) from coherent light (b). We then replace the
molecular sample with a single mode fiber in the same place (d) to obtain a lower
bound on the number of photon pairs emitted by (a). PMT: Large-area photo mul-
tiplier tube, SP: Shortpass, BP: Bandpass
.
65
105 pump att., y=ax
PDC att., y=ax2
classical, y=ax2
1000
10
0.1
5 10 50 100 5001000
IR power [nW]
Figure 3.2. SFG counts at 532 nm as a function of IR power for three cases:
EPP, where power is varied in the parametric down-conversion pump beam (open
circles), EPP, where power is varied after pair generation (open triangles) and
a quasi-coherent state from a diode laser (solid diamonds). We show functions
corresponding to linear (solid line) and quadratic scaling (dashed and dotted), re-
spectively. Each data point was averaged over 180 s and had its dark count rate
subtracted. Vertical and horizontal error bars represent shot noise and measure-
ment accuracy of the optical power meter, respectively.
signed such that the R6G sample can be inserted without a↵ecting the focusing
lens or prior optics, ensuring that the beam overlap is not a↵ected.
SFG as Confirmation and Scaling Behavior
In the first experiment (panel c in Fig. 3.1), we demonstrate SFG with iso-
lated entangled pairs, as well as with a CW reference laser. After passing through
the prism compressor the light is focused into a nonlinear crystal identical to the
one which generates the PDC pairs. After the crystal, light driving SFG is filtered
out by short-pass and band-pass filters. Light generated via SFG is coupled into a
single-mode fiber and detected on an avalanche photo-diode (APD).
The SFG experiment verifies both linear scaling with entangled-pair flux, as
well as quadratic scaling with direct attenuation of the pairs after the SPDC crys-
tal. This constitutes solid evidence of an entangled nonlinear interaction, and is a
replication of the experiment demonstrated in [41]. These results are displayed on
a log-log plot in Fig. 3.2 alongside the quadratic scaling of SFG driven by the CW
IR laser. Adjustment of the prism compressor to maximize the SFG rates allows us
66
SFG counts [s-1]
to optimize the dispersion compensation in the experiment to o↵set the dispersion
introduced by the optical elements.
While this experiment isn’t calibrated in terms of absolute pair flux, it never-
theless demonstrates an enhancement in SFG driven by entangled pairs over SFG
driven by CW light at the same optical powers. Comparison of the lowest data
point for both experiments demonstrates a similar SFG rate, with around 25 times
less optical power for entangled pairs than classical IR light. Extrapolating the
classical e ciency to the 2 nW data point yields an expected SFG rate more than
500 times lower than that observed with the entangled pairs.
This constitutes a significant benefit of entangled pairs over CW excitation at
ultra-low flux, and is in agreeement with the enhancement expected from theory,
considering photon-number enhancement and time-frequency entanglement.
Bounds on TPA of Entangled Photon Pairs
After the entangled pair flux is calibrated—described in detail below—the fiber
is replaced with a 10mm-thick cuvette containing Rhodamine-6G in ethanol solu-
tion with a concentration of 2mM (Fig.3.1, panels d and e). The fiber assembly is
designed to be removed without a↵ecting the focusing lens or prior optics, ensur-
ing that alignment is una↵ected when the R6G sample is introduced. Fluorescence
emitted from the sample is focused by high-NA lens and collected on a large-area
photo-multiplier tube (PMT - Hamamatsu H7421-50 ) oriented at a 90-degree angle
from the incident light. Any scattered IR light is blocked using several short-pass
filters.
For completeness, we compared the performance of this geometry with a more
tightly focused backward-collection geometry with fluorescence imaged onto a pho-
ton counting avalanche photo-diode. While the detector e ciency and signal-to-
noise ratio improved in this setup, the ability to establish a lower bound on the ef-
67
105
104
1000
100
10
1
0.1
0.050.10 0.50 1 5 10 50
CW power [mW]
Figure 3.3. Fluorescence count rates from laser-driven TPA in Rhodamine-6G.
Counts rates were averaged over 5 seconds, and for 1800s for the lowest point. Hor-
izontal error bars reflect measurement accuracy of the optical power meter, vertical
error bars represent shot noise. Fit exponent is 1.972 ± 0.001. The blue shaded area
corresponds to fluorescence flux levels below detection threshold.
fective pair rate bounds was diminished due to poor coupling into the SM fiber and
this setup was unable to provide tighter bounds on the e ciency despite improved
detection.
To measure the rate of entangled pairs in the interaction volume, we place the
tip of a single-mode optical fiber (core diameter 4 µm Nufern 780HP) in place of
the Rhodamine sample (compare Fig. 3.1d). A 13mm-achromatic lens focuses the
beam into the sample/fiber in an epi-fluorescence geometry. While shorter focal
lengths can in theory increase collection e ciency, we found this focal length pro-
vided a good compromise between collection e ciency and coupling e ciency into
the optical fiber. The fiber-coupled pairs are split probabilsitically via a 50:50 fiber
beam splitter (Thorlabs TW930R5A2 ). Coincidence detection events are measured
with two superconducting nanowire single-photon detectors, with 80% detection
e ciency at 1064 nm. After correcting for detector e ciency and auxiliary atten-
uation necessary to prevent detector saturation, the lower bound on the entangled
photon pair rate within the focal volume for our experiment is 2.0( 0.2) 109± ⇥ Hz,
measured at a pump power of 1W.
68
Fluorescence Rate [s-1]
Next, we replace the fiber with a cuvette containing a 2mM solution of Rhodamine-
6G. We found this concentration to yield maximum detected fluorescence from
laser-driven TPA in the 90  geometry. In order to enable measurements that are
insensitive to slow-time drifts in background counts, we introduce an optical chop-
per with 50% duty cycle in the beam path prior to the sample cell. Fluorescence
measurements constitute the di↵erence in accumulated counts between open and
shut time intervals. Reported rates are adjusted to reflect the 50% duty cycle.
Using a CW 1064 nm infrared laser we measure the TPA-induced fluorescence
count rate on the PMT. The results are shown in Fig. 3.3. The lowest IR power
at which we observed a signal was 50 µW . At this power the measured fluores-
cence rate was 0.7(±0.1) Hz, which we define as the detection threshold for this
setup. This detection threshold is in good agreement with the expected 3  detec-
tion threshold over the duration of the 1800s measurement, given the dark rate of
30 Hz and 50% duty cycle of the optical chopper.
Comparison of TPA rates and SFG rates observed with our classical reference
laser show that SFG measurements are possible at roughly 4000 times lower flux
than TPA with the current setup (10 Hz TPA count rate was observed at roughly
200 µW classical excitation, whereas 10 Hz SFG was observed at roughly 50 nW ).
Given that the expected enhancement of both nonlinear processes is predicted by
the same four-frequency correlation function [53] (see also Eq. 4.48), the relative
enhancement between the two is expected to be similar.
After calibrating our system with CW measurements, we repeated the same
measurement over 1800 s with 2.0(±0.2) ⇥ 109 entangled pairs per second (gener-
ated by 1 W CW pump power). This did not produce a measurement above our
detection threshold. While longer measurements were also conducted without ob-
serving signal, we based our upper bound on these parameters, since they represent
the lowest classical flux for which we have a corresponding measurement.
69
In order to calculate bounds on the cross-section and enhancements from these
measurements we calibrate our detection setup using the classical TPA measure-
ment. We first extract the collection e ciency ⌘col by fitting the TPA fluorescence
rates from Fig. 3.3 with a quadratic function TPA = a · F 2 of the flux, in pho-
tons/s, (converted from IR power by: F = P ·  /hc ⌘ P/~! where P is optical IR
power,   is the laser wavelength, h is Planck’s constant and c is the vacuum speed
of light). The fit parameter a is related to the experimental parameters by:
C  (2)
Z
· dz
TPA = aF 2 = ⌘ ⌘   F 2laser col · det · · , (3.1)
⇡ w(z)2
where C denotes concentration,  (2) is the conventional TPA cross section, 9.4(±1.5)
GM [78],   = 0.8 is the fluorescence quantum yield, ⌘det = 10% is the detec-
tor quantum e ciency, and w(z) is the beam waist along the optical axis. The z-
R 2
integral evaulated for a Gaussian beam evaluates to dzw( )2 =
⇡ n
2 . The obtainedz  
collection e ciency is ⌘col = 1.9(±0.2)%, which is in reasonable agreement with
independent estimates for our setup. To account for the expected enhancement of
photon number correlations in EPP relative to classical, uncorrelated CW light,
we introduce a quantum enhancement factor QEF = B/F , where B = 10.6
THz denotes the marginal bandwidth of the entangled pairs (as measured by a
spectrometer) and F is twice the pair rate of 2.0( 0.2) 109± ⇥ Hz. This yields:
QEF = 2.7(±0.1)⇥ 103. Combining this QEF with Eq.3.1 gives us our TPA predic-
tion:
TPA = QEF TPA = 3.5( 0.3) 10 7EPP ⇥ laser ± ⇥ s
 1. (3.2)
Since we were unable to observe a measureable signal, the ratio between de-
tection threshold and expected fluorescence rate bounds any unexplained quantum
enhancement (E(bound)). This ratio is:
70
 1
E(bound)
0.7s
<   = 2.0(±0.7) 10
6
⇥ (3.3)
3.5⇥ 10 7
We can carry out the same analysis in terms of the ETPA cross-section, es-
timated as   = 2 (2)/A T , with  (2)e e e = 9GM [78], Te = 1/B = 94fs and
Ae = A0 = 1.26  7 2⇥ 10 cm , with the assumption that the entanglement area is
the size of the 4µm diameter fiber through which we confirm our pair flux, which
yields the estimate:   = 1.5 10 29 cm28e ⇥ .
Estimating a count rate is slightly complicated in this framework because the
beam diameter is not constant for a tightly focused beam. Nevertheless we can esti-
mate this as: Z
C dz
TPA e = F eA0/2 ⌘col · ⌘det ·   · , (3.4)⇡ w(z)2
Where   A = 2 (2)e 0 /Te. The complication in the calculation is due to the
implicit area dependence within the z integral, which accounts for the change in
beam area along the z axis as the beam moves through the focus. This calculation
assumes that Ae = ⇡w(z)2 within the area of interest. This yields an expected
rate of 3.5 ⇥ 10 7Hz, which is exactly the rate estimated via the QEF. By setting
TPA e = 0.7, which is our upper bound on the TPA rate and solving for  e we can
estimate our bound, which yields  e . 3⇥ 10 23cm2.
From the measurements presented we conclude that any enhancement of two-
photon absorption by time-frequency entangled photon pairs is orders of magni-
tude lower than previously reported. This supports our theoretical predictions in
described in Ch. 2[60, 79] that quantum enhancement is bounded by B/F , the ra-
tio of EPP bandwidth B to flux F . It also corroborates the independent results ob-
tained under di↵erent controlled experimental conditions by Parzuchowski et al. [9].
8We note that this corresponds to a larger estimate for  e than our comparison to Tabakeav’s
experiment due to the di↵erent focusing conditions. This is one of the major limitations of the use
of  e as discussed in Ch 2.
71
While our experimental design can rule out some adverse factors that might dimin-
ish ETPA, we include a more complete discussion in the section, Adverse E↵ects, in
this chapter.
Flux Calibration and source characterization
In this section we elaborate on the measurement used to characterize and bound
the pair flux used in the previous section(panel d in Fig. 3.1). This is achieved by
optimizing the two-photon flux collected by a single-mode fiber (SMF) (Nufern
780HP) placed at the beam focus. The fiber-coupled pairs are split using a 50:50
fiber beam-splitter (Thorlabs TW930R5A2 ) and detected on superconducting nanowire
single-photon detectors (SNSPDs) with a quantum e ciency of roughly 80% at
1064 nm (IDQuantique).
The fiber provides a small test volume to ensure that generated pairs are tightly
focused, with both arriving within the test volume. Because of this, coupling into
single-mode fiber provides a strict lower bound on entangled-pair flux within the
cross-section defined by the optical fiber. Additionally, coupling of the CW refer-
ence laser into the same single-mode fiber provides a stringent test of alignment
between the two beams at the location of the sample, ensuring proper collection ef-
ficiencies.
Since our setup is capable of generating pairs at rates much higher than can
measured using photon-counting detection, we characterize our pairs at low and
medium flux, and extrapolate these rates to the high flux case. In order to achieve
this we measure pairs over over a variety of pump power with subsequent attenua-
tion as necessary to avoid detector saturation. The number of pairs detected as a
function of pump power, after adjusting for the necessary attenuation, is shown in
Fig. 3.4, which we use to calibrate the pair-flux in our experiment. From this we
calculate a Klyshko e ciency of ⌘K = 16(±2)%, at our sample location, and use
72
1011
109
107
105
0.1 1 10 100 1000
Pump power [mW]
Figure 3.4. Photon pair coincidence rate as a function of pump power. The
measured rate is adjusted for the attenuation needed to prevent detector satura-
tion (open circles). Each data point was averaged over 5 seconds. The solid line
represents an exponential fit to the linear regime (open circles), the exponent is
1.02± 0.01. Points at higher power were calculated using relative pair rate observed
in the linear regime. and the observed count rates (solid triangles).
this to estimate our pair rate by multiplying the single count rates with ⌘K [80],
in the high-flux regime where direct measurement of pair rate is no-longer possi-
ble. This is preferable to measuring optical power and making the assumption that
there are exclusively correlated pairs in the beam, which is unrealistic in the pres-
ence of any loss. Obtaining a lower bound for the number of pairs in this way al-
lows us to account for any losses in the optical path up to that point, such as reflec-
tion losses on metal mirrors, which would not be possible by inferring the pair rate
from measured power at the SPDC wavelength.
Above 1W of SPDC pump power we observe a deviation from linear scaling,
which we attribute to photorefraction (degradation of crystal properties by laser-
induced charge migration) in the SPDC crystal. Due to this 1W is the maximum
power we utilize in this experiment. At this power we measure 2.0 9⇥ 10 pairs per
second. At this rate, the average separation of two pairs is roughly 500 ps, 3 orders
of magnitude larger than their correlation time of 100 fs, as estimated from the
measured EPP bandwidth.
The SPDC source uses a periodically poled, 10-mm-long magnesium oxide–doped
lithium niobate (PPLN) bulk crystal (Covesion MSHG1064-1.0-1.0 ) with a poling
73
Pair rate [s-1]
Figure 3.5. Marginal spectrum of the entangled photon pairs at various crystal
temperatures. Spectra were measured on a 500-m-long dispersive fiber time-of-
flight spectrometer, after passing through a 70 nm bandpass filter centered around
1055 nm.
period of 6.9 µm and a phase-matching temperature of 340 K. The type-0 process
is pumped with CW light from a diode-pumped solid-state laser at 532 nm (Co-
herent Verdi V-5 ). The forward-propagating (collinear) part of the SPDC mode
is collimated with an achromatic lens with a focal length of 100 mm. After filter-
ing with a narrowband 1064 nm bandpass filter (Thorlabs FB1070-10 ), the spatial
beam properties of the SPDC, including collimation and copropagation with the
1064nm laser beam, are verified using a CCD camera at various distances from the
collection lens.
The emitted SPDC, centered around 1064 nm, was coupled into a fiber optic
beam-splitter (Thorlabs TW930R5A2 ), with one output sent directly to the super-
conducting nanowire single-photon detectors, and the other sent through 1000 m
of optical fiber (Nufern 780HP). The first output of the beam-splitter serves as a
clock-signal for measuring the arrival time distribution of the other output. The
frequency-dependent dispersion in the optical fiber acts to spread the pulse in time,
to operate as a time-of-flight spectrometer analogous to that in [81, 82].
Figure 3.5 shows measured marginal spectrum of the entangled pairs. The
74
Periscope
Primary
Fold
P1/4
P2/3
Secondary
Fold
Figure 3.6. Schematic of the prism compressor. P1, P2, P3, P4: prisms in the
order the beam passes through them.
FWHM bandwidth extracted from a Gaussian fit is 40 nm (10.6 THz).
Dispersion compensation
Because of the broadband EPP spectrum, dispersion plays an important role
in this experiment. Even small amounts of dispersion in the PDC crystal, lenses,
and filters, which amount to roughly 4000 fs2, cause the photons within each pair
to lose temporal overlap. To avoid this, we compensate second-order dispersion in a
manner similar to the SFG experiments conducted by Dayan et al. [41]. Figure 3.6
shows a more detailed schematic of the employed prism compressor.
In the last step, the prism compressor is used to maximize the signal obtained
from SFG in the second crystal. SFG is the strongest at minimal dispersion. We
adjust it by changing the prism separation using the retroreflector at the secondary
fold. Around the maximum e ciency, we verify that changing prism separation
and changing insertion of the second prism (which is inherently stable in terms of
alignment) give the same result. We then lock prism separation and insertion in
that position before performing all the experiments described in this chapter. Fur-
ther details can be found in [10]. In addition, we vary the compressor setting for
the ETPA experiment to account for the dipsersion from the SFG crystal, with no
change in outcome.
75
Figure 3.7. Schematic of (a) the 90  collection geometry and (b) back-reflection
geometry. PMT: photomultiplier tube, SP: short-pass filter, APD avalanche photo-
diode.
Collection Geometry
We compared two di↵erent collection geometries for our experiments, summa-
rized in Fig. 3.7. The first utilized a large active area PMT, oriented at 90  to the
incoming beam. The second used a free-space APD in the epi-fluorescence geom-
etry, in which the focusing lens serves also as the collection lens for fluorescence
emitted in the backwards direction.
In the epi-fluorescence geometry, Fig. 3.7(b), backwards emitted fluorescence is
collected and collimated via the input focusing lens. A dichroic beam-splitter then
reflects the light from this setup to an APD, which had far superior dark count
rates and detection e ciency to the PMT used in the 90  geometry.
However, we found that collection e ciency was decreased and no significant
gain in signal collection could be achieved compared to the side collection geometry,
while backward collection was more sensitive to alignment. While collection e -
ciency could be increased by use of a shorter focal length focusing lens, the result-
ing reduction in overlap with the SM fiber, reduced our ability to confidently bound
the pair-flux. The main advantage of this geometry is that fluorescence emitted
close to the facet of the cuvette can be detected, even in the presence of strong flu-
orescence reabsorption by the dye that would otherwise prevent detection on the
side. Nevertheless, the backwards geometry proved useful for studying concentra-
tion e↵ects. A secondary benefit to the 90  geometry was the relative insensitivity
to alignment, which served as a redundancy to ensure no ETPA signal was missed.
76
4
6 10
a)
4
2
0
0 10 20 30 40 50 60 70 80
Concentration (mM)
1
2 mM
0.8 5 mM b)
10 mM
0.6 20 mM30 mM
0.4 40 mM50 mM
60 mM
0.2 70 mM
80 mM
0
-0.2 -0.1 0 0.1 0.2 0.3
Focus Sample Depth (mm)
0.8
c)
0.6
0.4
0.2
0 10 20 30 40 50 60 70 80
Concentration (mM)
Figure 3.8. Plots of (a) maximum collected TPA fluorescence as a function of con-
centration, (b) normalized TPA fluorescence intensity as a function of focal position
relative to the front face of the cuvette, and (c) decay of fluorescence intensity with
focal position, as quantified by a fit to an exponential decay e x/L, where x is the
distance into the sample, shown with 95% confidence intervals. All measurements
are performed at 10 mW CW laser excitation.
Concentration Dependence
At low concentrations, the rate of TPA and collected fluorescence scales lin-
early with the concentration. At very high concentrations, however, various e↵ects
reduce the fluorescence rate. Fluorescence quenching due to particle aggregation
and fluorescence reabsorption are two such e↵ects that place practical limitations
on the concentration that can be utilized for e cient fluorescence collection.
These e↵ects were investigated in the backwards collection geometry using a
high-NA short-focal-length (3 mm) aspheric lens (Thorlabs C330TME-B) and a free
space APD (CountBlue 10 ). The 3-mm lens serves as both the focusing lens for
the incident beam and the collection optic for the detection. The collimated TPA
fluorescence is reflected o↵ a dichroic beam-splitter and focused onto a free space
APD.
77
Decay Length (mm) Normalized Fluorescence (au)
TPA Fluorescence (cps)
This configuration reduces the e↵ect of reabsorption of fluorescence by the dye
by collecting at the front face of the sample volume and allows collection at concen-
trations limited mainly by aggregation quenching e↵ects. Despite this benefit, the
total collected TPA declines past a concentration of 20 mM , as seen in Fig. 3.8(a).
Figure 3.8(b) shows the normalized TPA fluorescence collected as the sample is
moved closer to the focusing lens. After a sharp increase, corresponding to the front
face of the sample overlapping the focal point, the collection e ciency begins to
decrease. While e↵ects such as increasing optical aberrations with focus depth can
contribute to this e↵ect, the concentration dependence seen in Fig. 3.8(c) suggests
reabsorption or pump depletion in the Rhodamine 6G solution. Although pump de-
pletion from TPA alone is negligible, other e↵ects such as scatter or absorption may
contribute to pump depletion. Due to increased reabsorption in the side collection
geometry, 2 mM was close to optimal for that experiment. These results also indi-
cate that while further optimization in sample concentration and geometry could be
achieved, these e↵ects would not achieve orders of magnitude increases in sensitiv-
ity.
Detection
While other methods have been proposed [83, 84], two primary methods have
been used to detect ETPA: di↵erential transmission and fluorescence detection.
While both are in theory capable of detecting TPA, they have certain advantages
and disadvantages.
There are two primary disadvantages to experiments which attempt to detect
a decrease in transmitted photon flux: First, all sources of linear loss are present in
the signal, requiring careful calibration and validation to rule out non-TPA related
78
signals. Second, shot-noise of the transmitted beam limits detection sensitivity.
SNRtransmission = p
Tref   Tsamp (3.5)
Var[Tref ] + Var[Tsamp]
Additionally, transmission experiments have typically utilized photon counting de-
tection, limiting the rate of pair detection to around 107 cps of entangled pairs.
Fluorescence detection on the other hand, is background free in the sense that
there is no inherent signal in the detection channel. Since detection can be sepa-
rated spatially from excitation by collection in the backwards direction or at 90 
to the beam. Additionally fluorescence from TPA without resonant intermediate
states is emitted at higher energy than the incident entangled photons, which can
be e ciently filtered spectrally, and rules out most linear e↵ects.
Beyond this, in the case of fluorescence detection shot-noise of the source does
not impact the ability to measure a signal, which is limited only by the dark rate of
the experimental apparatus, and the rate of detected counts.
R
SNRfluorescence = p signal (3.6)
Rsignal +Rdark
In contrast, the primary disadvantage to fluorescence detection is e ciency.
While there are many considerations that a↵ect the detection e ciency in a flu-
orescence experiment, the two most important are the quantum yield of the two-
photon absorbing molecule, and the optical collection geometry of the experiment.
The quantum yield of a molecule limits which molecules can be investigated, since
fluorescence detection is unsuitable for samples that decay non-radiatively.
Outside of this, the total system e ciency can be quite low, and requires care-
ful consideration to optimize. Collection e ciency in particular is limited by the
fact that the radiation is emitted isotropically, making collection di cult. Collec-
tion e ciencies above 10% are di cult to achieve outside of microscopy systems
79
utilizing high-NA oil-immersion objective lenses.
Despite the low collection e ciency, the background-free detection and insensi-
tivity to most linear e↵ects makes fluorescence detection preferable to transmission
measurements, especially in cases where the number of absorbed photons is a small
fraction of the overall intensity. Additionally, the robustness to spurious signals is a
major benefit for measuring very weak signals, as is the case for ETPA.
Note on Chopped Measurements
We implemented a chopped measurement in order to rule out slow time drifts
in average photon rate for long-duration measurements. We insert a chopper wheel
into the PDC beam with a 50% duty cycle, which alternatively blocks and trans-
mits the EPP beam. The chopper derives a reference signal with an infrared sensor,
which ensures the state (open/closed) can be accurately monitored. This reference
signal is monitored on the same time-to-digital converted responsible for monitoring
the photon detection events. This reference signal is used to sort detection events
between open and closed channels.
Comparison between open and closed cycles of the optical chopper gives us a
dark-subtracted measurement of the rate, which is insensitive to any noise that is
not synchronized to the chopper frequency. The down-side to this method is the
50% duty cycle. Due to this, we only measure half the rate we would otherwise.
However, this measurement rules out any noise source, temperature dependent dark
rate, or other environmental drift that is not correlated to the chopper frequency.
This is critical to rule out spurious signals for measurements where the measured
signal is below the dark rate.
The measurement also enables very simple statistical analysis and allows accu-
rate characterization of variances in both signal and dark channel, since the arrival
times of the photons are recorded rather than just the integrated rates. This allows
80
for construction of robust error rates and detection thresholds. In particular, we de-
fine our detection threshold as a 3  detection level. We assume that shot-noise on
p
the dark rate is  dark = dark, which is validated by the error rates on the dark
channel. While the error rates for bright TPA can vary from this substantially, er-
ror is dominated by the dark rates in signal starved experiments for which this esti-
mate is su cient.
To di↵erentiate our signal from noise, we subtract the accumulated counts on
our dark channel from the counts on our signal channel, TPA = Open   Closed.
q p
Simple error propagation on this yields that   2 2TPA =   +   = open+ closed.open closed
p
Given a dark rate of 10 Hz, and a 10,000 second measurement,  TPA is 100 10,
and the detectable count rate di↵erence at a 3  confidence level would be 2 ⇥ 3 ⇥
p
100 10. Where the factor of two results from the 50% duty cycle. This translates
p
to 6 10/100 ⇡ 0.18 Hz. For 3 Hz dark rate. This improves to approximately 0.105
Hz 3 -detection threshold for a 10,000 second measurement.
Simply put, the detection threshold decreases as the square-root of the acqui-
sition time. So given a 50% duty cycle, D dark rate in Hz, and measurement dura-
tion, T, the n  detection threshold is:
p
2n D
p (3.7)
T
Then the 5  detection threshold for our best detector is 17.4 s1/2. After 100 sec-
onds the threshold is 1.74cps, and after 104 s, or slightly less than 3 hours, it is
0.174 cps.
Adverse E↵ects
Performing an experiment in the absence of a detectable signal is prone to
many pitfalls. Here we attempt to compile a comprehensive list of such pitfalls,
along with our attempts to anticipate and mitigate them. This list is in parts com-
81
piled from Ref. [85] and [9] and amended with our own analysis of PDC experi-
ments.
Insu cient spatial overlap - If focusing into the molecular sample is insu -
ciently tight, or the waist is not inside the sample volume, there is a chance that
the two photons within a pair have poor spatial overlap. We address this by using
a single-mode fiber as a test volume for our actual photon flux. The pair rate is di-
rectly measured in coincidence.
Dispersion - Dispersion would cause the anti-correlated photons in a pair to
arrive at di↵erent times, and lose temporal overlap, preventing TPA. We estimate
the amount of dispersion from the specifications of all employed optics. This allows
us to design an appropriate prism compressor that compensates for second order
dispersion. We use the SFG experiment to verify successful dispersion compensa-
tion. The SMF experiment allows us to exclude any adverse e↵ects from alignment
caused by adjusting the amount of compression.
Competing single-photon processes - Scattering, pump light bleeding through
optical filters, single-photon absorption in the dye, solvent, optics or cuvette, just
to name a few, all share the same linear scaling behavior with loss. A linear signa-
ture alone is therefore not su cient to verify that TPA is the dominant process. It
is therefore recommended to verify that TPA and fluorescence rates scale quadrati-
cally with pair attenuation, as demonstrated in our SFG experiment.
Linear loss - The e↵ect of linear loss on photon pairs is that either photon can
be lost probabilistically. Because of this, one cannot infer the average number of
pairs from the average number of photons, or optical power. It is therefore neces-
sary to directly measure the number of pairs in coincidence. By characterizing the
pair rate at the molecular sample, we have ensured that these losses are taken into
consideration.
Detector saturation - Any and all of the signals one may wish to observe here
82
rely on a linear detector response. As the average duration between photon pairs
approaches the dead time of the detector, photons arriving in fast succession are
undercounted, and detector response to power becomes sublinear. Dead time of
common photon counting detectors are of the order of 50 ns for APDs, 10 ns for
PMTs, and as low as few nanoseconds for SNSPDs. Therefore, saturation e↵ects
usually start to appear at counts in excess of 107 Hz, and well below for coinci-
dences. We attenuate the PDC beam in order to avoid saturation, and then adjust
the measured coincidence rate for the amount of attenuation.
Insu cient collection e ciency - The most simple explanation for the absence
of a fluorescence signal this is insu cient collection e ciency, but is easily charac-
terized with classical TPA.
Asymmetric spectral detector response - Silicon-based detectors do not just suf-
fer from low quantum e ciency near the optical band gap of 1100 nm, but it is
also high asymmetric. This biases detection around 1064 nm towards the short-
wavelength half of the PDC spectrum, causing the unfortunate situation where
using count rates to align fiber coupling is not an option. There are three reme-
dies: Use a narrowband spectral filter to align on th center of the spectrum, align
directly on coincidences using a fiber beam splitter and two detectors, or use a dif-
ferent sort of detector with a more uniform response. The latter is the case when
using nanowire detectors: In the range of 1000   1100 nm, their quantum e ciency
only varies by few percent.
Reabsorption - Emission and absorption spectra of dyes such as Rhodamine
overlap slightly, causing some of the emitted fluorescence photons to be reabsorbed.
Higher dye concentration would increase the number of TPA events, but also in-
crease reabsorption of fluorescence photons. Using the fluorescence signal from clas-
sical TPA, we established that within the range of 0.2mmol/L to 20mmol/L, a con-
centration of 2mmol/L maximized detection for our collection geometry.
83
Aggregation - Molecular dyes have a tendency to aggregate at high concentra-
tion. Aggregation can cause a number of e↵ects, including fluorescence quenching,
and shifting of the energy spectrum.
Conclusions
In this chapter, we set experimental bounds on the e ciency of ETPA and at-
tempt to replicate experiments described in [7]. We demonstrate that our system is
capable of detecting TPA, and carefully characterized system e ciency by calibra-
tion with classical TPA. We consider the many adverse e↵ects which could lead to
diminished ETPA signal. In order to demonstrate an entangled nonlinear interac-
tion, we present SFG of our down-converted light source, which proves the ability
of our experimental system to observe nonlinear e↵ects of su cient magnitude, and
experiences the same benefits of time-frequency entanglement as TPA.
We are unable to observe any fluorescence from ETPA with our calibrated sys-
tem, bright source of entangled photons, and well-understood molecular dye. This
is in agreement with predictions from our theory described in Ch. 2 as well as ex-
periments from [9]. The upper bound we infer from our measurement is an ETPA
cross-section,  e . 3 ⇥ 10 23 cm2, which corresponds to a value not more than
2 ⇥ 106 times greater than what is predicted by the theory, accounting both for
photon-number and time-frequency enhancements. This is several order of magni-
tude below what was reported elsewhere. Since publication of the results presented
in this chapter, several other works have reported similar conclusions: [11, 12, 62].
84
CHAPTER IV
TPA OF BSV
Introduction
In this chapter, we describe a simplified model of high-gain, pulsed squeezed
vacuum, which we refer to as bright squeezed vacuum (BSV). This chapter reviews
and expands on results published in Phys. Rev. A under the title, “Theory of Two-
Photon Absorption with Broadband Squeezed Vacuum”[86]. We apply this model
to predict TPA rates, joint spectral intensity (JSI) measurements and SFG of BSV.
The motivation behind this is to demonstrate an experimental system capable of
observing two-photon absorption of entangled photons, and to study whether this
informs on the e ciency of a low-gain behavior of the same source. In Ch. 5, we
will utilize JSI and spectrally resolved SFG measurements at low and high gain to
validate this theoretical model.
The model we use assumes a CW pump in a single-mode nonlinear optical
waveguide, which generates squeezed vacuum in a single spatial mode valid for
both high and low gain. The squeezed vacuum is subsequently chopped into pulses.
This model has limitations especially in the regime of broadband, ultra-fast pulsed
lasers, however for narrowband pumped systems, in which the largest amount of
time-frequency entanglement is present, the model is realistic and describes the de-
sired features well.
This work reproduces many of the conclusions from previous work with bright
squeezed vacuum [53, 87, 88], in a simplified analytically tractable model. We relate
the predictions from this model to the heuristic equation cited by [44] (Eq. 1.4 in
Ch 1), and demonstrate the relationship to the degree of second order coherence,
which has been used to predict similar e↵ects.
85
One central result from this study is that in the high-gain regime, the compo-
nent that describes time-frequency entangled photon pairs at low gain is amplified
with nearly the same magnitude as the component that describes interactions be-
tween uncorrelated pairs within the pulse. This indicates the potential for achieving
high spectral and temporal resolution at high gain.
TPA of Chopped CW Squeezed Light (Phys Rev. A)
The model we described in this chapter follows closely the derivation in [86],
which in turn expands on a model for squeezed light in a waveguide from [89].
While a full description of broadband squeezed vacuum in terms of two-mode
squeezing between Schmidt-modes can be found in [22, 90, 91], our model uses a
simplified narrowband approximation. The high-gain results agree with the model
used in [87]. A separate long-pulse model is presented in [53].
The primary benefit of considering a narrowband CW pump is the ability to
express the results analytically without need for numerical calculations. Having a
tractable model easily relatable to experimental parameters, is critical to the
e↵ective estimation of interaction strengths for various experiments, where a full
numerical approach is impractical. In order to simulate the temporal dynamics of
narrowband pulses of squeezed light, we model the e↵ect of a shutter that opens
and closes after a time T . We assume that the shutter acts instantaneously and the
time duration it is open for is long compared to the coherence time of the squeezed
light. The results are valid in both low- and high-gain regimes, and allow us to
understand quantitatively how the two regimes merge at moderate gain. The model
enables us to describe the transition between gain regimes and calculate a
cross-over between the two that agree with the heuristic equation of [44].
The Heisenberg-picture two-mode squeezing transformation for collinear type-0
or type-I phase-matching and crystal length z described in [89], which transforms
86
the input field operators â(!) to output operators (!), is summarized as:
¯
b̂(!) = f(!)â(!) + g(!)â†(2!0   !). (4.1)
where,
 k(!)
f(!) = cosh[s(!)z]  i sinh[s(!)z],
2s(!)
(4.2)
 (!)
g(!) = i sinh[s(!)z],
s(!)
and  k(!) is the phase mismatch of wave numbers k(!)
 k(!) = kp   k(!)  k(2!0   2), (4.3)
This can be adapted to more rigorously model periodic poling as discussed in [86].
For collinear type-0 or type-1 the phase mismatch is approximated by
 k(!) k00⇡   , where k00 = @2[k(!)]/@!2 is the group-velocity dispersion. the
spectral gain coe cient is denoted as
p p
s(!) =  2   k(!)2/4 ⇡  2   2(!   !0)4, (4.4)
using the abbreviation  = k00/2. The gain coe cient is   = (!0/c) 
(2)
0 Ep0, where
Ep0 is the strength of the pump field and  
(2)
0 is the nonlinear coe cient of the
crystal and assumed to be independent of frequency in the region of interest.
The symmetries, s(2!0   !) = s(!), f(2!0   !) = f(!), g(2!0   !) = g(!) are
valid for type-0 and type-I phase-matching and are utilized in many subsequent
calculations.
We assume the initial state is the vacuum, and the low-gain results reproduce
the state derived in the Schrödinger picture in previous sections. This can be seen
87
1
Low Gain
High Gain
0.5
 b = 8.19 THz
 w = 8.17 THz
0
265 270 275 280 285 290 295 300
 (THz)
Figure 4.1. PDC spectra in the low- and high-gain regimes, with characteristic
widths w and b, respectively. In the low-gain regime, the spectrum is well approxi-
mated by sinc2[(! ! )2  0 ] and w is defined by its first zero crossing. In the high-gain
regime, the spectrum is well approximated by a super-Gaussian as in Eq. 4.9, and
b is defined by the e 1 crossing. Crystal length z = 0.01 m. Low gain:  z = 10 4,
high gain:  z = 10.
by considering the spectrum of the squeezed vacuum state:
hvac b̂†| (!)b̂(!0) 0|vaci = S(!)2⇡ (!   ! ), (4.5)
and     sinh[s(!)z]  2
S(!) = |g(!) 2 =  2|      (4.6)s(!)  
Plots of the spectrum in low and high gain for our experimental setup are
given in Fig. 4.1 The total photon flux can be calculated by the integrated
spectrum. At low gain, the spectrum is constant in bandwidth and the flux is
proportional to g(! ) 2| 0 | . At high gain, the spectral width changes slowly with
intensity, but the approximation F / |g(!0) 2| can nevertheless be useful.
Z
d!
F = |g(!) 2| (4.7)
2⇡
88
Normalized Intensity
For perfectly degenerate phasematching in the low-gain limit, (  ! 0), the
spectrum is sinc2-shaped and in the degenerate case its width can be parameterized
p
by the first zero-crossing w = ⇡/z(with units of (rad/s)). This FWHM is
⇡ 1.34w.
h i
S(!) ( z)2⇡ sinc2 (!   ! )20 z (4.8)
In the high-gain limit ( z   1) the spectrum becomes super-Gaussian, and in
the degenerate case its width can be parameterized by the width parameter
b = ( /2z)1/4 = ( z/⇡2)1/4w. The FWHM is ⇡ 1.82b.
✓ ◆
1 (!   ! )4
S(!) ⇡ e2 z exp 0  , (4.9)
4 b4
The growth of the total intensity is approximately exponential in gain and in
medium length at high gain, altered slightly by the bandwidth factor b.
Temporal Gating
To model pulsed BSV, we multiply the field in the time domain by an
open-closed shutter function W̃ (t), which equals 1 inside the window [ T/2, T/2]
and zero otherwise. In the frequency domain W (!) = T sinc[!T/2]. In the
frequency domain this product acts as a convolution, which we apply to our
Heisenberg picture operators:
Z
d!0
ĉ(!) = W (! 0  ! )b̂(!0)
Z 2⇡ Z (4.10)
d!0 0
f(!) W (! !0)â(!0
d!
⇡   ) + g(!) W (! !0)â†  (2! 00   ! ).
2⇡ 2⇡
Since our assumption is that the chopped window is long compared to the
coherence time of our squeezed light, W (!) is narrow compared to f(!) and g(!),
and we can pull these out of the integral. It is worth noting that this model does
not take into account e↵ects that come from an ultrashort pulse in the form of
89
increased overlap with the crystal’s phase-matching function, temporal pulse
walk-o↵, or other ultrafast e↵ects and is only valid in the limit of long pump pulses.
One limitation of this model even for long narrowband pump pulses is the presence
of nonuniform gain across the duration of more realistic Gaussian pulse envelopes.
This e↵ectively shortens the BSV pulse in time at high gain.
The form above motivates defining filtered creation and annihilation operators,
Z
d!00
Â(!) = W (! 00  ! )â(!00)
Z 2⇡ (4.11)
d!00
B̂(!) = W (! !00)â(2! !00    )
2⇡
Because W (!) is symmetric, B̂(!) = Â(2!0   !), and
ĉ(!) = f(!)Â(!) + g(!)Â†(2!0   !) (4.12)
The e↵ect of this transformation is that the commutator of the filtered
operators is found to be spectrally and temporally broadened. Denoting it by
[Â(!), Â†(!̃)] ⌘ D(!   !̃), where
Z
d!0
D(!   !̃) = W (! 0  ! )W (!̃ 0  ! ) = T sinc[(!   !̃)T/2] (4.13)
2⇡
R R
D(!) is normalized as D(0) = T , with d!2 D(!   !̃) = 1 and
d!
2 D(!
2
  !̃) = T ,
⇡ ⇡
and acts like a broadened delta function.
TPA of BSV
The TPA probability derived in Ch. 2 is valid for an arbitrary quantum state,
and we can apply it to the current case by noting that the initial state is the
vacuum, and replacing â(!) with the transformed operators ĉ(!). Recalling the
90
form of the TPA probability, can be written as:
Z Z Z
 (2)  d! d!̃ d!0fg C(4)(!0,! + !̃ !0  ,!, !̃)Pf = 2 + c.c. (4.14)A0 2⇡ 2⇡ 2⇡  fg   i(!fg   (! + !̃))
where C(4)(!a,! † †b,!c,!d) = hĉ (!a)ĉ (!b)ĉ(!c)ĉ(!d)i is the four frequency correlation
function. This is most easily evaluated by considering one half of the expression at
a time, with C(4) = h | i, and
⇣
| i = ĉ(!)ĉ(!̃) |vaci = f(!)g(!̃)Â(!)Â†(2!0   !̃) +
⌘ (4.15)
g(!)g(!̃)Â†(2!0   !)Â
†(2!0   !̃) |vaci
This can be broken into two orthogonal states |  cohi and incoh| i where
| i = |  coh incohi + | i . The incoherent contribution can be read o↵ as:
incoh † †
| i = g(!)g(!̃)Â (2!0   !)Â (2!0   !̃) |vaci (4.16)
To evaluate the coherent term, we first note that Â(!) |vaci = 0 and
Â(!)Â†(2!0   !̃) = D(2! †0   !   !̃) + Â (2!0   !̃)Â(!). With this in mind, we can
then write:
  coh| i = f(!)g(!̃)Â(!)Â†(2!0   !̃) |vaci
(4.17)
= f(!)g(!̃)D(2!0   !   !̃) |vaci
Paying close attention to the frequency variables, we can then write:
C(4) = g⇤(!0)f ⇤(! + !̃ 0  ! )f(!)g(!̃)D2(2!0   !   !̃) (4.18)coh
To calculate the coherent TPA probability, we evaluate the !̃ integral, treating
D2(2!0   !   !̃) inside the integral as a delta function (i.e., replacing ! + !̃ with
R
2!0), recalling that D2(!) = T . We also use the symmetry f(2!0   !) = f(!).
This yields:
91
   
 (2)
Z
T   fg  fg   d!  
2
Pcoh ⇡ 2 2   f(!)g(!)   (4.19)A0   + (!fg   2! )20 2⇡fg
Here we note that the broadened delta function served to enforce spectral
compression along ! + !̃.
To evaluate the incoherent contribution, we note that after applying the
commutator several times and paying close attention to the frequency variables,
D E
Â(!0)Â(! + !̃ !0)Â†  (!)Â†(!̃) can be written as (D2(!̃ !0) +D2(! !0    )).
This yields:
(4) ⇤ 0 ⇤ 0   0  C = g (! )g (! + !̃ ! )g(!)g(!̃) D2(!̃ ! ) +D2    (!   !0) (4.20)
incoh
Then, evaluating the !0 integral in Eq. 4.14 yields the incoherent TPA probability.
We note that for distinguishable photons such as those generated via Type-II
SPDC, the magnitude is half as large [86].
 (2)
Z Z
 fgT d! d!̃   2fg|g(!)| |g(!̃) 2|Pincoh ⇡ 2
A2 2
(4.21)
0 2⇡ 2⇡   + (!   !   !̃)
2
fg fg
From this result it is clear that the incoherent contribution does not benefit
from spectral compression, and for narrowband TPA linewidth  fg contributes
negligibly to the overall TPA rate. In contrast, when the TPA linewidth is broad,
the incoherent contribution is twice the overall magnitude of the coherent
contribution.
Inspecting these expressions, it is clear that Pincoh scales quadratically in the
photon flux g(!) 4| | , regardless of the gain  . In contrast, the coherent contribution
exhibits two separate behaviors. In the low gain regime |f(!)| ⇡ 1, and the
coherent contribution Pcoh scales linearly in the photon flux |g(!) 2| . In high-gain,
on the other hand, f(!) ⇡ g(!) and the coherent contribution also scales
92
quadratically. While this is not a new result, it is one worth emphasizing. Because
the coherent contribution is amplified at the same rate as the incoherent
contributions in the high gain limit, the time-frequency correlated portion of the
beam is present in similar magnitude to the uncorrelated portion. If this were not
the case, the coherent contribution would be overwhelmed by the incoherent at high
gain, and not contribute meaningfully to the final TPA probability.
This feature is crucial to our ability to assess low-gain TPA rates from the
comparison of high-gain BSV and classical TPA. If there were contributions that
did not scale in this way, rather remaining linear throughout, we would not be able
to conclude much about those features by measurements at high gain.
It is worth noting that the coherent and incoherent contributions play similar
roles in sum-frequency generation, as studied theoretically and experimentally in
[51, 53, 92], and the presence of the narrow correlated peak in SFG at high gain is
due to this feature. If the coherent contribution weren’t amplified in a way that
preserved the temporal and spectral correlations, this feature would disappear at
high gain.
In Fig. 4.2 we show plots of the coherent and incoherent contributions to the
TPA probability using Eqs. 4.19, 4.21 and parameters closely modelling our
experiment. In the low-gain regime the excitation probability is independent of
pulse duration T for fixed N. That is because the entangled photons arrive in tight
pairs regardless of the arrival times of each pair.
Also plotted in Fig. 4.2 (as the red dashed line) is the prediction for excitation
by a quasi-monochromatic coherent state, using Eq. (112) from [70]. In this case,
the TPA probability is: ✓ ◆
 (2) N 2
Pcoherent state = 2 T (4.22)A0 T
Where F = N/T . Comparison to the coherent contribution at high gain Eq.
4.19 reveals that the e ciency of BSV is equal to that of classical excitation for
93
10-6
A fg/2  = 0.048 THz B w/2  = 8.2 THz
w/2  = 8.2 THz T = 10 ps
T = 10 ps
10-8
10-10
Qcoh fg/2  = 0.48 THz
Qcla fg/2  = 0.16 THz
Qincoh fg/2  = 0.048 THz
C fg/2  = 0.048 THz D fg/2  = 0.048 THz
T = 10 ps w/2  = 8.2 THz
10-10
w/2  = 82 THz T = 10 ps
w/2  = 29 Thz T = 32 ps
w/2  = 8.2 THz T = 100 ps
101 102 103 104 101 102 103 104
Photons Per Pulse, N
Figure 4.2. Predicted mean number of molecules excited by TPA per pulse for
a final-state TPA linewidth that is much narrower than the BSV bandwidth but
broader than the e↵ective PDC pump bandwidth, from Eq. 4.19]. Realistic ex-
perimental parameters are as follows: 10 µm e↵ective beam radius (assumed col-
limated), 1 cm cuvette, and 10 mmol concentration of molecules assumed to have
9 GM TPA cross-section. TPA probability per molecule is evaluated from Eq.
4.19], assuming that twice the squeezed-light center frequency is resonant with
the two-photon transition. In each case the solid blue curve is the coherent ETPA
contribution, the dashed yellow curve is the incoherent ETPA contribution, and
the dotted red line is “classical” coherent-state TPA with a quasi-monochromatic
pulse duration T that matches that of the laser that pumps PDC process. Panel (a)
shows a representative case. A transition in scaling from linear in photon number
to quadratic is apparent at N ⇡ 125, which corresponds to a mean occupation of
one photon per temporal mode. (b) shows the e↵ect of changes in the absorber’s
linewidth, which a↵ects only the incoherent contribution. (c) shows the e↵ect of
increasing the low-gain bandwidth parameter w of the squeezed light, varied by
varying the crystal length z. The TPA e ciency of the coherent contribution is
increased in the low-gain regime and remains the same in the high-gain regime,
while the incoherent contribution decreases in e ciency. (d) shows the e↵ect of
increasing the time window T . In the low-gain regime the coherent contribution
remains unchanged; however, the high-gain e ciency is reduced by increasing T,
and the crossover to quadratic scaling occurs at a higher relative photon number.
Both incoherent and classical e ciency are reduced.
94
Expected Number of TPA Events
pulses of equal duration, T . This can be seen due to the fact that at high gain
f(!) = g(!), and both are real, in which case the square modulus of the integral in
Eq. 4.19 is simply F 2. A major conclusion of the present study is that after
consideration of the positive e↵ects of time-frequency correlations present at high
gain, TPA e ciency when driven by the coherent portion of BSV is the same as
when it is driven by a classical pulse of the same intensity and pulse duration.
However, as noted above, if the molecule has a broad TPA linewidth, the
incoherent contribution is a significant portion of the TPA probability and total
e ciency of BSV approaches a factor of 3 times as e cient as coherent excitation of
the same duration for arbitrarily broad TPA linewidth. As noted in Ch 1, the
four-frequency correlation function is closely related to the degree of second order
coherence, g(2)(0). The 3-fold increase in BSV e ciency over coherent excitation for
broad TPA resonances can be interpreted in the context of temporal fluctuations
given that g(2)(0) = 3 + 1/N̄ for squeezed vacuum, where N̄ is the number of modes
[46, 71, 93]. This can be approximated as N̄ = BT . A more complete discussion of
this correspondence is included in [86].
An important consideration in applying this theory to experimental results is
the cross-over between low and high gain. Knowing the high-gain e ciency of a
setup and the flux at which the cross-over occurs is su cient to predict rates at low
gain for the same experimental setup. From the discussion above, it can be inferred
that this should occur at around one photon per mode, but it is worthwhile to
derive this from our model.
In the low gain limit we can write the TPA probability as,
(2)
P low gain
  N 3
⇡ 2 w (4.23)coh A0 4⇡
R
where we approximated f(!) ⇡ 1, and evaluated the integral d!| 2 g(!)
2
| . The
⇡
95
incoherent contribution is negligible at low-gain. At high-gain, the coherent
contribution is simply given by Eq. 4.22. We can define the cross-over as the flux
when these two are equal:
✓ ◆
 (2)N 3  (2) N 2
2 w = 2 T (4.24)A0 4⇡ A0 T
Solving for N yields:
3w
Ncrossover = T ⇡ BT ⇡ N̄ (4.25)
4⇡
In agreement with the expectation, given the correspondence to the degree of
second order coherence, the cross-over occurs at roughly 1 photon per temporal
mode, BT . Where we have utilized the fact that the width parameter w is in rad/s
we note that 3/2⇥ w/2⇡ is a good approximation for B, the bandwidth in Hz.
It is worth noting that the dependence on the number of photons per temporal
modes BT can equivalently be expressed as F = 1/Te where Te = 1/B. In the CW
case, where the number of temporal modes of the pulse is not well-defined, this
interpretation is more clear.
Given this prediction, measurement of the TPA e ciency at high gain
alongside experimental confirmation of this cross-over is su cient to predict
low-gain ETPA e ciencies. Before we move on to experimental implementation of
these experiments, it is worth reviewing predictions for other measurements which
we can carry out in order to validate our model.
Joint Spectral Measurements
As discussed in Ch 1, the Joint Spectral Intensity can be written in terms of
⌦ ↵
the four-frequency correlation function: JSI(!, !̃) = ĉ†(!̃)ĉ†(!)ĉ(!)ĉ(!̃) . The
chopped model for BSV is described in the Heisenberg picture, and the initial state
is the vacuum. It is easiest to consider one half of the four-frequency correlation
96
function at a time, with ĉ(!) = f(!)Â(!) + g(!)Â†(2!0   !) from Eq. 4.12, and
recalling that Âi(!) |vaci = 0:
⇣
| i = ĉ(!)ĉ(!̃) |vaci = f(!)g(!̃)Â(!)Â†(2!0   !̃) +
⌘ (4.26)
g(!)g(!̃)Â†(2!0   !)Â
†(2!0   !̃) |vaci
As before, this can be broken into two orthogonal states |  cohi and   incoh| i
where   coh incoh| i = | i + | i . The incoherent contribution can be read o↵ as:
  incoh| i = g(!)g(!̃)Â†(2!0   !)Â
†(2!0   !̃) |vaci (4.27)
The coherent term,   coh| i can be written as, by recalling that Â(!)Â†(2!0  
!̃) = D(2!0   !   !̃) + Â†(2!0   !̃)Â(!), and Â(!) |vaci = 0
  coh| i = f(!)g(!̃)Â(!)Â†(2!0   !̃) |vaci
(4.28)
= f(!)g(!̃)D(2!0   !   !̃) |vaci
The full JSI is the sum of the incoherent and coherent contributions, cohh | i +
h  incoh| i . Where the coherent contribution forms an anti-diagonal line of the form:
    coh = f(!) 2h | i | | |g(!̃) 2| T 2sinc2 ((2!0   !   !̃)T/2) (4.29)
The incoherent contribution is equal to: (where for short !0 = 2!0   !)
D E
h |  incohi = |g(!) 2 2 0| |g(!̃)| Â(!̃ )Â(!0)Â†(!0)Â†(!̃0) (4.30)
We can simplify Â(!̃0)Â(!0)Â†(!0)Â†(!̃0) by applying the commutator several times,
97
which yields:
2 2        incohh | i = |g(!)| |g(!̃) T 2| 1 + sinc2((!   !̃)T/2) (4.31)
This takes the form of an uncorrelated background with a diagonal ridge. The full
JSI reads:
JSI(!, !̃) = |f(!) 2 2 2| |g(!̃)| D (2!0   !   !̃)+
⇣ ⌘ (4.32)
g(!) 2 g(!̃) 2 D2(0) +D2| | | | (!   !̃)
A few features stand out. At low gain this reproduces the JSI we expect, with
tight correlations along the anti-diagonal since in this limit |f(!)| ⇡ 1 and |g(!)|⌧
1. The limiting behavior at high gain, where f(!) ! g(!), is for each feature to
have the same relative height. Since they are summed together, the diagonal and
anti-diagonal features reach a relative height of 2, and the intersection reaches a
relative height of 3. Simulations can be seen with experimental data in Ch 5, Fig.
5.2.
Distinguishable PDC Joint Spectra
The same calculation can be repeated in the case of distinguishable pairs, such
as those generated via Type-II SPDC or generated via Type-0 SPDC in the spa-
tially distinguishable case. In this case, we introduce mode labels to our operators
as done in [86]:
ĉi(!) = f(!)Âi(!) + g(2!0   !)Â
†(2!0   !). (4.33)j
And ĉj(!) is defined equivalently, with mode labels reversed. The form of the
D E
JSI is slightly altered to accommodate the mode labels, JSI † †ij(!, !̃) = ĉ (!̃)ĉ (!)ĉi(!)ĉj(!̃) .j i
The calculation proceeds as before with | i = ĉ (!)ĉ
ij i j
(!̃) |vaci. The coherent con-
tribution is unchanged from the indistinguishable case outlined before:
  coh| i = f(!)g(!̃)Âi(!)Â
†(2!0   !̃) |vaci = f(!)g(!̃)D(2!0   !   !̃) |vaci (4.34)ij i
98
The incoherent contribution is slightly modified, due to the fact that distinguish-
D E
able modes commute: [Ai(!), Aj(!)] = 0 for i 6= j. Because of this, Â (!̃0)Â (!0
† 0
i j )Â (! )Â
†(!̃0)
D ED E j i
factors into the product of Â (!0)Â† 0j (! ) Âi(!̃0)Â
†(!̃0) . Applying the commuta-
j i
tor to each of these yields: h |  incohi = D2(0) |g(!) 2 2| |g(!̃)| , and the full JSI reads:
ij
JSI (!, !̃) = f(!) 2 g(!̃) 2 D2si | | | | (2!0   !   !̃) + |g(!)
2 g(!̃) 2D2| | | (0) (4.35)
Identical to the indistinguishable case, this reproduces the JSI we expect, with
tight correlations along the anti-diagonal. However, at high gain, the di↵use back-
ground term is still present with the same relative intensity, but the diagonal term
is absent. This is due to the distinguishability between the two channels, such that
no bunching in a given mode is present. Simulations and experimental data can be
seen in Ch 5, Fig. 5.4.
Same-Label Joint Spectra
We can also model the case in which we take a joint spectrum between coin-
cidences in the same channel, which we expect to be uncorrelated. A related e↵ect
can be seen in cross correlations between coincidences of separate sources in Type-
D E
II PDC [94]. In the case, the JSI takes the form: JSIii(!, !̃) = ĉ
†(!̃)ĉ†(!)ĉi(!)ĉi(!̃) ,i i
with:
⇣ ⌘⇣ ⌘
| i = f(!)Â †i(!) + g(!)Â (2!0   !) g(!̃)Â
†(2!0   !̃) |vaci (4.36)ii j j
In this case the coherent contribution vanishes,   coh| i = 0, since Âi(!), and Âj(2!ii 0 
!̃) commute. The incoherent contribution is the same as it was in the indistinguish-
able calculation with:   incoh| i = g(!)g(!̃)Âj(!)Â
†(2!0   !̃) |vaci. The di↵use back-ii j
99
ground is present as well as the diagonal term.
⇣ ⌘
JSIii(!, !̃) =
2 2 2
|g(!)| |g(!̃)| D (0) +D2(!   !̃) (4.37)
Unlike either of the previous cases, the JSI has no anti-diagonal feature, re-
flective of the fact that each photon’s frequency entangled pair is guaranteed to be
in the other channel. Because of this, there is no distinction between high and low
gain in relative intensities present in the JSI. The di↵use background term is still
present as well as the frequency bunching term which is observed along the diago-
nal. This is consistent with the fact that a signal or idler beam alone from SPDC is
thermal in nature.
Joint Spectrally Resolved Hong-Ou-Mandel Interference
Hong-Ou-Mandel (HOM) interference is a well studied two-photon interference
e↵ect. In the non-collinear configuration, we can measure the HOM interference
e↵ect as a function of wavelength and delay between signal and idler. To do so, we
model the recombination of signal and idler beams on a 50:50 beam-splitter, which
leads to the following operators in the two output arms:
1    
d̂ (!) = ei!⌧a p ĉi(!) + ĉs(!)
2
    (4.38)1
d̂ (!) = ei!⌧b p ĉi(!)  ĉs(!)
2
This can be straightforwardly modified for unequal splitting ratios and phase ef-
fects, as done in [34] for collinear, non-spectrally resolved experiments. As before,
we can write the JSI in terms of the four-frequency correlation function:
D E
JSIHOM(!, !̃) = d̂†(!̃)d̂†(!)d̂b(!)d̂a(!̃) (4.39)dadb a b
We’ll focus on the case where da = db, which models the case in which the
100
JSI is measured from coincidences in a single output port of the interferometer. As
before, we focus initially on the right half of the equation.
1⇣ ⌘⇣ ⌘
d̂b(!)d̂a(!̃) |vac = e
i!
i c
⌧ ĉi(!c) + ĉs(! ) e
i!d⌧
c ĉi(!d) + ĉs(!d) |vaci
1n
2
= ei!c⌧ei!d⌧ ĉi(!c)ĉ (! ) vac
I + ei! ⌧i d | i c ĉi(!c)ĉs(!d) |vac
II
i (4.40)
2 o
+ei!d⌧ ĉs(!c)ĉ (!
III IV
i d) |vaci + ĉs(!c)ĉs(!d) |vaci
This leaves us with four states that we recognize from previous calculations
multiplied by additional phase-factors, corresponding to delays in the interferome-
ter. Similar to the case analyzed in [34], we designate:
I = ei(!+!̃)⌧ | i , II = ei!⌧   , III = ei!̃⌧| i | i , IV = | i (4.41)
ii is si ss
Noting that II and III are the only non-orthogonal cross-terms, the full expression
becomes:
n o
JSIHOM
1
(!, !̃) = I 2| | + |II 2 + 2 2 ⇤ ⇤| |III| + |IV | + II III + III II (4.42)
dada 4
We can write the self-terms, ( 2|I| etc.), in terms of JSIs we had calculated previ-
ously:
I 2| | = JSI , II 2ii | | = JSI
2 2
is, |III| = JSIsi, |IV | = JSIss (4.43)
The cross-terms can be evaluated as:
II ⇤ III = ei(!̃ !)⌧ h | | i
is si
(4.44)
III II = e i(!̃ !)⌧⇤ h | | i
si is
Recalling that | i can be broken up into coherent and incoherent contributions
we analyze these separately to evaluate the cross-terms. First, we note that the co-
101
herent parts are equal:   coh =   coh =   coh| i | i | i . In contrast, the incoherent part can
is si
be written as:
D E
  incoh   incohh | | i = |g(!) 2| |g(!̃) 2| Âi(!̃
0)Â (!0)Â†(!0)Â†(!̃0) =
is si s i s
(4.45)
= g(!) 2| | |g(!̃) 2| D2(!   !̃)
Since As and Ai, commute these can be factored again, with hÂs(!0)Â
†(!̃0)i =
i
D(!   !̃). Putting this all together leaves:
( ⇣ ⌘
JSIHOM
1
(!, !̃) = |g(!) 2 2| |f(!̃)| 1 + cos((!   !̃)⌧) D2(2! 0  !   ! )+
dada 02
⇣ ⇣ ⌘ ⌘) (4.46)
g(!) 2 g(!̃) 2
1
| | | | D2(0) + 1 + cos((!   !̃)⌧) D2(!   !0)
2
We recognize this as the low-gain JSI with a diagonal cosine modulation across the
anti-diagonal. At high gain, the di↵use background grows in but is not modulated.
The anti-diagonal term / 1 + cos((!   !̃)⌧)D2(!   !0) is approximately constant in
magnitude for moderate delays.
Sum Frequency Generation
Having evaluated various forms of incoherent two-photon measurements, we
can now turn our attention to coherent nonlinear measurements. To model SFG,
the up-converted field in the perturbative approximation can be written as [53, 71,
94]: Z
d!
d̂sfg(!3) d̂
(in) (2)
⇡ !3)  iL    (!,!3)ĉ(!)ĉ(!3   !) (4.47)
2⇡
p
where   ⇡ ~2!30/16⇡"0c3A0n(2!0)n(!0)2.
As always, we can write the spectrum of the final field in terms of the expec-
D E
tation, SFG(! ) = d̂†3 (!3)d̂sfg(!3) . Noting that the term d̂(in)(!3) evaluates tosfg
zero when the input field (at the up-converted frequency) is the vacuum, this ex-
102
pression can be written as:
Z Z
d! d!0
SFG(!3) ⇡ (L  
(2))2  ⇤(!0,! !03   ) (!,!3   !)⇥
2⇡ 2⇡
⌦ ↵ (4.48)
ĉ†(! !0)ĉ†(!03   )ĉ(!)ĉ(!3   !)
We note the close correspondence with Eq. 2.249. Where the same four-frequency
correlation function (up to a change of variables) is found to determine the final
quantity. With the major di↵erences between the two being the fact that TPA is
not spectrally resolved, and the overlap function is determined by the molecular
resonances rather than the phase-matching of the SFG crystal.
Evaluating the four-frequency correlation function as before, we can write the
SFG spectrum in terms of coherent and incoherent parts. The coherent part is10:
  Z  d!  2
SFGcoh(!3) = (L  
(2))2D2(2!   ! )   0 3  (!,!3   !)f(!)g(!  3   !)  (4.49)2⇡
9Which can be written under the change of variables, ! = !3   !̃:
⌦ ↵
 (2)
Z Z Z
 fg d!0 d!̃ d!
† 0
3 â (! )â
†(! 03   ! )â(!3   !̃)â(!̃)
Pf = 2 + c.c.2A0 2⇡ 2⇡ 2⇡  fg   i(!fg   !3)
10Where we have used:
ĉ(!)ĉ(!3   !) |vaci = f(!)g(!3   !)D(2!0   !3) |vaci
103
And the incoherent part is11:
Z
2(L  (2) 2
d!
) | (!,! 2 23   !)| !|g(!)| |g(!
2
3   !)| (4.50)
2⇡
Where we have assumed the phase-matching function is symmetric in  (!, !̃) =
 (!̃,!). The end result being:
  Z   2
SFG(! ) (L  (2)
d!
3 ⇡ )
2D2(2!0   ! )    f(!   !)g(!) 3 3   +
Z 2⇡ (4.51)
2(L  (2) 2
d!
) |g(!) 2| |g(!3   !)
2
|
2⇡
We recognize the coherent term as the narrow spike, and the incoherent term
as the broad background, as also described in [53]. The narrow spike corresponds
to the e↵ective spectral compression seen in the case of ETPA. From this it is clear
that the integrated SFG spectrum has the same form as the TPA probability up to
the phasematching/lineshape considerations. This correspondence was also recog-
nized by Dayan [53]. Moreover, in the spatially single-mode picture it is clear that
the cross-over point will be the same as for TPA since the equations governing the
interaction strengths are identical.
11Here we have used:
Z Z 0
SFGincoh(! ) = (L  (2)
d! d!
3 )
2  ⇤(!0,! 03   ! ) (!,!3   !)⇥
2⇡ 2⇡
g⇤(!0)g⇤(!3   !
0)g(!3   !)g(!)D E
Â(2! !00   )Â(2!
0 † †
0   !3 + ! )Â (2!0   !3 + !)Â (2!0   !)
As well as:
D E
Â(2!0   !
0)Â(2! 0 †0   !3 + ! )Â (2!0   !3 + !)Â
†(2! 20   !) = D (!
0
  !) +D2(! !03     !)
104
The E↵ect of Dispersion
Up until now we have not treated dispersion, however dispersion represents
another probe of time-frequency correlations that we can access in the lab. We can
00 2
model dispersion by replacing ĉ(!) ĉ(!)eik /2(! ! )! 0 , and we can equivalently
make the same replacements for f(!) and g(!). Applying this to our equation for
the coherent BSV TPA probability, Eq. 4.19, we find:
   
 (2)T  Z  d! 00 2 P f(!)g(!)eik (! ! )⇡   0  coh 2   (4.52)A0 2⇡
which in general reduces the TPA e ciency. For the incoherent BSV TPA probabil-
ity, Eq 4.21 on the other hand, the replacement yields:
        00 2 2ik /2(!  )2    00! ik /2( 2! ! )  
 (2)
Z Z 0 0
   fg  g(!)e    g(!̃)e  
P 2 fg
T d! d!̃
coh ⇡ 2 2 (4.53)A 2⇡ 2⇡   + (!   !   !̃)20 fg fg
00 2
which has no e↵ect on the TPA probability, since g(!)eik /2(! !0) 2| | = |g(!) 2| .
It is worth noting that this is only true within the limitations of the model. For
instance the time-of-flight measurements disperse the pulse across several nanosec-
onds, which would clearly reduce the TPA e ciency of the incoherent contribution.
This contradiction stems from the approximations treating D(!) as a delta func-
tion. However, the model is valid for moderate amounts of dispersion resulting in a
temporal spread much less than the pulse duration.
We now turn our attention again to SFG, which can be written to include the
e↵ects of dispersion in terms of Eq. 4.51. To model broadband SFG phasematching,
we have assumed  (!,!3   !) = 1:
  Z  d! 200 2
SFG(! ) ⇡ (L   (2))2|D(2!   ! ) 2|    g(!)f(!)e
ik (! !0)   
3 0 3   +
Z 2⇡        (4.54)
(2) 2 d!   00 2(   )2  ik / ! !0   
2   00 2 2
2(L    ) g(!)e  g(! !)eik /2(!3 ! !0)  3    
2⇡
105
Here we see that the broad pedestal, which corresponds to the incoherent term,
is una↵ected by dispersion. On the other hand, the e ciency of the peak is sensi-
tive to dispersion. In general dispersion will reduce this e ciency and dispersion
compensation can increase this e ciency since the phase-matching function contain
dispersion from the SFG crystal itself.
Conclusions
In this chapter, we outline a simplified model of BSV for experiments in which
the pump is a narrowband pulse. Using this model, we predict TPA rates as well as
the outcomes of JSI measurements and spectrally resolved SFG. Within the approx-
imations made by this model, we verify the heuristic equations for TPA rate and
cross-over between linear and quadratic flux scaling for the case of SFG and TPA.
We also outline the relationship between the e ciencies of TPA for BSV and classi-
cal excitation in the frequency domain, reproducing a known result in terms of the
second order coherence function g(2).
The results from this chapter outline how the correlations from entangled pho-
tons persist in the high-gain regime, in agreement with the work from Dayan [53].
One of the central implications of this result is that characterization in the high
gain regime is su cient to predict low-gain e ciencies if the cross-over point is
known, which can be readily approximated as N = B/F . Because of this, in lieu
of low-gain TPA results, we can validate whether our model is capable of predicting
the correct enhancement due to time-frequency correlation by comparing the mea-
sured high gain TPA e ciency with that of classical TPA. Any further enhance-
ments would lead to greater e ciencies at high gain and a disagreement between
these two e ciencies.
Additionally, since the functional form of TPA and SFG are the same, and low-
gain SFG signals are readily observable, we can also test our prediction of the cross-
106
over point in SFG. This is an important test for two reasons, the first is that the
cross-over point is a direct consequence of the enhancement due to time-frequency
entanglement. The second is that if there were some low gain enhancement not de-
scribed by our theory which continues to scale linearly for all gain values (i.e., is
not observable at high gain), this enhancement would nevertheless skew the loca-
tion of the cross-over point.
A secondary result is that since time-frequency correlated component of these
fields persists at high gain, high temporal and spectral resolution utilizing BSV is
achievable. This can be seen by the e↵ects of dispersion on the coherent spike in
the SFG spectrum, as well as by the direct measurement of the SFG spectrum,
when distinguishable signal and idler beams are delayed relative to one another
prior to up-conversion.
In Ch 6, we carry out experiments to validate this model and see good agree-
ment with each of these predictions.
107
CHAPTER V
EXPERIMENTAL TPA OF BRIGHT SQUEEZED VACUUM
Introduction
In this chapter, we will apply the theoretical framework for two-photon absorp-
tion of low- and high-gain squeezed vacuum developed in the previous chapter to a
set of experiments designed to test some of the key predictions identified previously.
We demonstrate a first of its kind measurement of the joint-spectral intensities of
squeezed vacuum in the transition regime between low and high gain, for distin-
guishable and indistinguishable cases alongside spectrally resolved Hong-Ou-Mandel
interference.
While we are unable to observe TPA in the low-gain regime with our experi-
ment, we are able to confirm the previously known asymptotic e ciency limit that
TPA driven by coherent and incoherent contribution of BSV is roughly three times
as e cient as coherent excitation with the same pulse duration. This e↵ectively
limits the magnitude of the time-frequency enhancement.
We confirm that the crossover between linear and quadratic scaling occurs at
approximately 1 photon per mode by directly observing the cross-over in e cient
sum-frequency-generation with narrowband phasematching. This places limits on
the e↵ect of photon-number and time-frequency enhancement in SFG.
Finally, we are able to confirm that the time-frequency correlations of interest
persist in high gain, albeit alongside incoherent background terms. We do this in
two ways. First, by varying the degree of dispersion compensation present prior to
spectrally resolved SFG we observe the detrimental e↵ect of quadratic phase on the
sharp correlated spike in the sum-frequency signal, while the broad incoherent sig-
nal remains constant. Separately, by varying the time-delay between non-collinear
108
signal and idler beams, we are able to demonstrate simultaneously high resolution
in both time and frequency. This is strong validation that utilizing BSV to probe
the e↵ect of these correlations on the TPA e ciency is well motivated and remains
sensitive to correlation of this type.
Taken together, these results represent a solid understanding of the physics at
play and demonstrate that the predictions made in previous chapters agree with
the observable physics. Importantly, any major enhancement of TPA e ciency due
to time-frequency entanglement not captured by our theory would be reported by
these experiments. The consistency between our experiment and theoretical pre-
diction is the strong evidence that the orders-of-magnitude ETPA enhancement
reported elsewhere cannot be explained by time-frequency enhancement of two-
photon absorption.
BSV ETPA
Many studies have investigated SPDC primarily in the low-gain regime, which
can be roughly defined as the regime in which the process produces one or fewer
photon pairs per temporal mode of the PDC process. In this chapter, we instead
investigate the high-gain regime, in which many photon pairs per temporal mode
exist, leading to parametric gain. In this regime, we demonstrate key features of
the joint spectral intensity between any two photons within the BSV pulse. We
present experimental verification of key features and predictions of the model pre-
sented in the previous chapter and [86].
Two-Photon Absorption was also measured at rates consistent with experimen-
tal bounds on entangled two-photon absorption (ETPA) set in [9, 10].
We probe the model by various distinct measurements. We first measure joint
spectral intensities (JSIs) in the low- and high-gain regimes for both distinguish-
able and indistinguishable signal and idler beams. Our model correctly predicts the
109
relevant features of these measurements. We also use the distinguishable case to
measure spectrally resolved Hong-Ou-Mandel interference at low-gain.
We probe the temporal nature of the fields by investigating spectrally resolved
sum-frequency generation. Dispersion is utilized to probe the e↵ect of dispersion on
SFG e ciency. Separately, temporal delays between signal and idler beams prior
to SFG probe both time and frequency correlations. We compare the e ciency
of classical TPA and TPA driven by BSV, which agree well with predictions from
our model, a central result of our study. Finally, a high-e ciency, narrowband SFG
crystal is used to measure the crossover between low-gain linear scaling and high-
gain quadratic scaling regimes.
The sum of these results is strong evidence that this model is useful for the
case of interest, where a narrowband pump drives broadband down-conversion with
a high degree of time-frequency entanglement in the absence of spatial entangle-
ment. Moreover, we provide strong evidence that the model provides a useful start-
ing point to assess the feasibility and expected rates of an ETPA experiment in the
absence of low-gain signal, by confirming high-gain e ciencies relative to classical
TPA alongside confirmation of the low- to high-gain crossover rate in SFG.
Experimental Configurations
Squeezed vacuum was generated via Type-0 SPDC in a 10 mm Magnesium Ox-
ide Doped Periodically Polled Lithium Niobate (PPLN) crystal (Covesion MSHG1064-
1.0-1.0 ). The polling period was 6.93 µm, and the crystal temperature was set for
collinear and non-collinear phasematching as needed. Temperature was controlled
by a home-built oven and kept stable with closed-loop feedback control (Oxford In-
struments Mercury ITC ).
The pulsed pump laser is centered around a wavelength of 532 nm, and is ap-
proximately 8 ps in duration (Lumera Laser Hyper Rapid 50 ). The pump repetition
110
rate was varied between 100 kHz and 10 MHz as needed between experiments.
Pump intensity was controlled via a set of neutral density filters with fine-tuning
accomplished by a variable attenuator comprised of a  /2 waveplate and polariz-
ing beam splitter, after which a second  /2 waveplate rotated polarization to the
correct (V ) polarization for the crystal phasematching conditions. Residual funda-
mental harmonic from the pump was filtered via shortpass and notch filter Thorlabs
FESH0750, Newport FF01-540/80-25.
The pump pulse was focused by a 500 mm lens to a beam waist with a ra-
dius of 75 µm inside the crystal. Both pump beam and SPDC were collimated by
a 200 mm lens with anti-reflection coating for both pump and SPDC frequencies
(Edmund Optics YAR-BBAR 33-213 ). The pump was blocked by a set of three
long-pass filters (Thorlabs FELH0900, FELH0850, FELH0750 ), the first of which
was angled slightly to direct the reflected beam onto a beam-dump.
JSI Measurement
The joint spectral intensity (JSI) was measured by time-of-flight (TOF) spec-
trometry, similar to that in [81, 82, 94]. The principle behind the TOF measure-
ment is to utilize the fact that light of di↵erent frequencies travels through disper-
sive media at di↵erent group velocity, as determined by the frequency-dependent
refractive index, n(!), of the material. Given su cient dispersion, the frequency of
the light is mapped onto the time of arrival relative to some known time reference.
TOF measurements are typically used in photon-counting measurements, in which
the arrival time of single photons can be correlated to the laser clock.
In our experiment, the dispersion was generated via 500 m spools of single-
mode optical fiber (Nufern 780HP). Two 500 m fiber spools in series spread the
light from our PDC source out over approximately 6 ns. Our time-to-digital con-
verter (IDQuantique ID900 ) has 13 ps resolution. Our SNSPDs have a timing jitter
111
on the order of 35 ps. In order to generate a su ciently stable clock for the TOF
measurement, we derived a clock signal from a picko↵ of our pump laser, which was
coupled into a single mode fiber and detected on a photodiode (Thorlabs DET10A)
with 1.8 ns rising edge. The jitter was measured to be around 30 ps.
Frequency calibration of the TOF spectra was accomplished by inserting in-
terference filters with well known spectral properties at normal incidence into the
broadband SPDC beam at varying phasematching conditions. The sharp edges of
the interference provided several calibration points at which both time of arrival
and frequency were known. These calibration points were fit to a cubic polyno-
mial, and the resulting fit was used to calibrate further measurements. Due to the
spectrally flat response of the SNSPDs, no intensity calibration was needed. The
nonlinear mapping of frequency onto time-bins causes an e↵ective di↵erential in
frequency-bin width, however corrections due to this e↵ect are minor.
The full TOF measurement was achieved by sending the clock signal as well as
the counts from the SNSPDs to a time-to-digital converted, and recording coinci-
dence counts relative to the clock pulse. Measurement of both unheralded single-
channel spectra as well as joint spectra conditioned on coincidence events were
measured. For joint spectra a joint histogram of the coincidence events was gen-
erated with the frequency of one channel conditioned on the frequency of the other.
Experimental Results
Two cases were investigated; the first, which we refer to as the
non-distinguishable case, consists of Type-0 collinear SPDC. Down-converted pairs
were coupled into a Nufern HP780 single mode fiber, after which it passed through
two 500 m fiber spools for an e↵ective length of slightly over 1 km of dispersive
fiber. After the fiber spools the pairs were split via a broadband single-mode
beam-splitting fiber optimized for 1064 nm (Thorlabs TW1064R5F2B) and sent to
112
two separate superconducting nanowire single-photon detectors (SNSPDs).
The second case we investigated, which we refer to as the distinguishable case,
was non-collinear Type-0 SPDC, achieved by lowering the temperature of the
SPDC crystal. The resulting change in the refractive index produces optimal
phase-matching for k-vectors which produces SPDC in a ‘ring-mode’, in which the
maximum of the spatial SPDC intensity distribution is ring-shaped. Correlated
pairs are generated on opposite sides of the ring of down-converted light due to
momentum conservation in the phase-matching considerations as described in Eq.
1.29. The flat edge of a D-shaped mirror was used to split the ring mode into signal
and idler beams, and the horizontal center of each beam was coupled into
single-mode fibers. Alignment was optimized by optimizing coincidences in the
low-gain regime in this configuration. This was split horizontally with a D-mirror
prior to coupling each half into separate SM fibers. These were each sent to one of
the two 500 m fiber spools and then coupled onto two SNSPDs.
In the distinguishable case two measurement configurations were used. The
first, as described above, looked at signal-idler pairs. The second, achieved by
inserting a 1064 nm beam-splitting fiber after the fiber spool in one arm, measured
joint spectra between photons from the same arm.
For the idler-idler JSI, one of the fibers from the distinguishable configuration
was split via a broadband single-mode beam-splitting fiber optimized for 1064 nm
and sent to two separate superconducting nanowire single-photon detectors
(SNSPDs).
When necessary, especially at high gain, neutral density filters were inserted to
reduce the rate of single events until the single-channel count rate was less than
100 kHz. This ensured that the TOF spectra were unbiased from pile-up artifacts.
113
Figure 5.1. Diagram of experimental setup for collinear joint spectral intensity
measurements. After generation of squeezed vacuum, light is coupled into a single
mode fiber. And sent to time-of-flight spectrometer consisting of approximatley
1 km of optical fiber prior to being split on a 50/50 fiber beam-splitter and directed
to SNSPDs
Collinear (Indistinguishable) Joint Spectra
Fig. 5.3 shows the experimental configuration in the collinear geometry. The
SPDC was coupled into a single-mode fiber via 8 mm B-coated aspheric lens
(Thorlabs C240TMD-B). The SPDC was then sent through approximately two
500 m fiber spools in addition to 100 m in addition to about 100 m auxiliary fiber.
After the fiber spools the SPDC was split via a broadband single-mode
beam-splitting fiber optimized for 1064 nm (Thorlabs TW1064R5F2B) and sent to
two separate superconducting nanowire single-photon detectors (SNSPDs).
Measurements of the JSI were made at 500 kHz repetition rate, and acquired over
600 s each. The average pump power was varied from 6.25 µW to 5 µW . Prior to
being coupled into the single mode fiber the SPDC intensity was attenuated to
below 105 cps by neutral density filters to avoid detector saturation12.
Fig 5.2 shows the results of the collinear JSI measurement at varying gain. The
low-gain joint spectrum is in agreement with typical measurements of the type [9,
95], showing tight anti-diagonal frequency correlation. The width of the correlation
is limited by the jitter of the TOF measurement. At moderate gain, a di↵use
background term and diagonal correlated feature is visible at low relative intensity.
12Detector saturation skews the measurement, as the first pair is always recorded, and subse-
quent pairs are missed. The resulting spectrum is skewed to the red.
114
Figure 5.2. Collinear Joint Spectral Intensities at varying gain with simulations.
a-c) Joint Spectral Measurements were taken with 6.25 µW , 400 µW , 5000 µW av-
erage pump power from low to high gain respectively. d-f) Simulated Joint Spetral
intensities. The resolution of the measurement is insu cient to resolve the anti-
diagonal width, which is smeared across several bins due to timing jitter.
115
At high gain, the intensity of the background term is large, and the diagonal
feature has the same intensity as the anti-diagonal.
These measurements are in good agreement with the theoretical predictions
from our model described in CH 4, Eq. 4.32. The corresponding predictions at low,
medium and high gain are also shown in Fig. 5.2, the predictions from the
simulations were broadened via convolution with a Gaussian kernel to account for
the a↵ect of timing jitter on the measurement. We note that the feature heights
don’t perfectly correspond to the the relative intensities predicted in the Eq. 4.32
due to the timing jitter. Because the narrow feature is spread across more bins
than it actually spans, the e↵ective intensity is reduced. Nevertheless, the relative
heights of the diagonal and anti-diagonal features are equal.
Since temporal correlations are averaged over in the JSI measurement there is
no large enhancement of the anti-diagonal feature relative to the background at
high gain. However, the presence of the feature does confirm that anti-diagonal
frequency correlation persists and that the coherent feature scales quadratically.
Similar measurement at high gain have been performed via cross-correlations
on a conventional grating-spectrometer by Maria Chekhova’s group [96, 97]. High
gain cross-correlations in frequency and momentum were also considered.
Non-collinear (Distinguishable) Joint Spectra
Fig. 5.3 shows the configuration for the non-collinear measurements. SPDC
was generated in a ring-mode by reducing the phase-matching temperature of the
crystal. The ring-mode is split vertically by a D-shaped mirror. Pairs along the
horizontal axis are coupled into two separate single-mode fibers, each of which are
coupled to separate 500 m fiber spools in addition to approximately 100 m of
auxiliary fiber before being directed to the SNSPDs. We note that the resolution of
the TOF measurement is reduced by about half due to the reduced fiber length.
116
Figure 5.3. Diagram of experimental setup for non-collinear joint spectral inten-
sity measurements. Squeezed vacuum is generated in a ring-mode, by reducing the
phase matching temperature of the crystal. The ring-mode is split vertically by
a D-shaped mirror. Pairs along the horizontal axis are coupled into two separate
single-mode fibers, each of which are coupled to separate 500 m fiber spools before
being directed to the SNSPDs.
Coupling lenses, repetition rate, pump power etc, and rate attenuation were all
identical to those reported in the collinear configuration.
Fig. 5.4, shows experimental results and simulations for the non-collinear
configuration. The low gain JSI is nearly identical to the indistinguishable case, at
slightly reduced resolution. Both moderate and high gain demonstrate the presence
of the broad background. Notably the diagonal bunching term is absent in both
experimental and simulated data. Similar to the collinear case, the simulations are
convolved with a Gaussian kernel to simulate measurement jitter. To the best of
our knowledge this is the first measurement of its kind showing these e↵ects at low
and high gain in frequency.
Slight asymmetries in signal and idler can be observed in the experimental
data. This is due to slight imperfections in the alignment of the beams into the
single-mode fiber. In addition, the measured spectra are broader than predicted in
the simulation. This can be explained due to spectral-spatial coupling in the real
experiment which was not considered by the model, which treats phasematching
identically to the collinear case13.
13Strictly speaking the use of mode-labels for the non-collinear case is an abuse of notation,
since the modes are post-selected on and have identical phasematching consideration.
117
Figure 5.4. Non-collinear joint spectral intensities at varying gain with simu-
lations. a-c) Measured Joint Spectral Measurements were taken with 6.25 µW ,
400 µW , 5000 µW average pump power from low to high gain respectively. d-f)
Simulated Joint Spetral intensities. The resolution of the measurement is insu -
cient to resolve the anti-diagonal width, which is smeared across several bins due to
timing jitter.
Figure 5.5. Non-collinear joint spectral intensities on a single channel a) Mea-
sured Joint Spectral Intensity was taken at 5000 µW average pump power. b) Sim-
ulated Joint Spetral intensities. Simulation in The resolution of the measurement is
insu cient to resolve the anti-diagonal width, which is smeared across several bins
due to timing jitter.
118
Figure 5.6. Diagram of experimental setup for non-collinear joint spectrally re-
solved HOM interference. Squeezed vacuum is generated in a ring-mode, by reduc-
ing the phase matching temperature of the crystal. The ring-mode is split vertically
by a D-shaped mirror. Pairs along the horizontal axis are split into two arms. The
reference arm reflected o↵ a pair of mirrors before arriving at the free-space beam-
splitter. The delay are is reflected o↵ a pair of mirrors on a delay stage before
arriving at the other input port of the beamsplitter. The recombined beams are
coupled into a single mode fiber. And sent to time-of-flight spectrometer consisting
of approximatley 1 km of optical fiber prior to being split on a 50/50 fiber beam-
splitter and directed to SNSPDs.
For completeness, the same configuration shown in Fig. 5.3 was used to
measure the JSI of a single channel, by inserting a single-mode fiber beam-splitter
after the fiber spool of one arm and sending the output ports of this beamsplitter
to the SNSPDs. In this configuration only high-gain measurements were considered
since the coincidence rates on a single channel at low-gain are prohibitively low.
The high gain measurement, taken at 5000 µW average pump power is shown in
Fig. 4.37. The same spectral broadening visible in the distinguishable case is
present in this case. In agreement with theoretical description only the background
and diagonally correlated bunching term are visible.
HOM Interference
Fig. 5.6, shows the configuration for the spectrally resolved HOM experiment.
The same non-collinear configuration as in the distinguishable case is used, but
instead of being coupled into separate fibers, the two arms are recombined on a
119
Figure 5.7. HOM interference, as a function of delay. The interference is Spec-
trally resolved at the indicated positions. While the HOM peak persists only for
80fs, the interference extends out much further.
50/50 beamsplitter (Thorlabs BSS11R14). The delay arm included a compensating
plate to account for the dispersion mismatch between the arms (Thorlabs
BCP45RP). The delay was scanned via computer controlled delay stage (Aerotek
ANT130L). After recombination on the beam-splitter the SPDC was coupled into a
single-mode fiber in the same configuration as the Indistinguishable JSI experiment.
HOM measurements were only conducted at low gain15.
The HOM experiment was conducted twice, both times in low-gain. Once with
200 nm steps, and once with 2 µm steps, which correspond to 1.3 fs and 13 fs of
delay respectively. Fig. 5.7 shows the results of the HOM experiment with 200 nm
step size. The bottom figure shows the integrated intensity of each measurement at
each delay, over a range of approximately 250 fs of delay. Above the integrated
trace a subset of the JSI measurements are highlighted above, with the
corresponding data-points indicated by circles on the integrated HOM trace.
The maximum of the integrated HOM interference can be seen to occur when
there is constructive interference across the entirety of the two-photon amplitude.
As the delay between the pairs is increased cosine interference fringes across the
14This is sold as a 30/70 beamsplitter, however for V-polarized light, the performance was
preferable to the 50/50 model from the same model family, and presented the best stock broad-
band 50/50 beamsplitter available.
15High gain measurements were also carried out, but the reduced contrast and prohibitively
long measurement times made the full experiment infeasible
120
Figure 5.8. Joint spectrally resolved HOM interference. Data was generated by
taking a JSI measurement at each delay vaule.
anti-diagonal can be seen, in good agreement with the predicted cos[(!   !̃)⌧ ]
dependence predicted in Eq. 4.46. The integrated HOM trace shows good visibility
with the central interference maximum reaching nearly 2, which indicates perfect
bunching due to HOM interference. At large delays ⌧   50 fs, the integrated trace
no longer shows evidence of interference. However, the spectrally resolved
measurements clearly demonstrate the presence of interference fringes. The cosine
fringes across the measurement reduce the overall intensity by a factor of 1/2 once
more than two symmetric minima are present.
The experiment is repeated with 2 µm steps. Fig. 5.8 shows a 3-dimensionl
rendering of the results. The y-axis shows the the time delay, with each x-z slice
representing an individual JSI measurement at that delay. Here the cos[(!   !̃)⌧ ]
dependence is clearly seen in both dimensions. At a particular frequency di↵erence,
121
Figure 5.9. HOM interference in terms of the frequency di↵erence. Gener-
ated from the same data as Fig. 5.8. The di↵erence of each detected coinci-
dence pair is histogramed. The shows very nice agreement with the prediction
of (1 + cos((!   !̃)⌧) dependence.
the fringes can be seen along the delay axis with frequency proportional to the time
di↵erence. Conversely at a given delay, fringes are seen along the
frequency-di↵erence axis. This demonstrates two-color HOM interference at
multiple frequencies simultaneously. While not visible in the diagram, interference
extends to the delay corresponding to the width of the pump pulse.
These measurements demonstrate that HOM interference occurs over the
entirety of the duration of the pump pulse, with oscillating components determined
by the frequency mismatch of the two interfering colors. The interference at the 0
frequency di↵erence recreates the integrated HOM interference for the pump pulse.
Fig. 5.9 shows the same results as Fig. 5.8, where only the frequency di↵erence
between each event was kept, rather than the full JSI. This demonstrates high
visibility for the individual fringes, and simplifies the picture.
In this section we demonstrate excellent agreement between model and the
behavior of our experiments in both low and high gain regimes. In the low gain
122
Figure 5.10. Diagram of experimental setup for SFG characterization. The
collinear SPDC is passed through a 90  periscope prior to a singly folded prism
compressor using two 1 inch SF-11 prisms, the delay between the prisms is ad-
justable. The returning beam is picked o↵ after passing through the 90  periscope
a second time, and sent to SFG crystals. The resulting upconverted light is coupled
into a fiber and characterized. Slightly di↵erent configurations are used for spec-
trally resolved and narrowband SFG experiments described in the text.
regime in particular we show that the model predicts the spectral correlation well.
The model also predicts the relative intensities of various correlated and
uncorrelated contributions well.
Scaling
In order to probe the spectral-temporal correlations present at high gain, we
utilize SFG in two separate experiments. The first experiment uses a second
identical 10 mm PPLN crystal for narrowband phasematching. This enables us to
investigate only the coherent contribution and simulate a narrow molecular
resonance. In this configuration we probe the transition from low-to high gain for
the coherent contribution. The second experiment uses a short crystal to
investigate spectral correlations at high gain.
Fig. 5.10 shows the experimental configuration for both of these experiments.
The collinear SPDC is passed through a 90  periscope prior to a singly folded prism
compressor using two 1-inch SF-11 prisms. The delay between the prisms is
adjustable. The returning beam is picked o↵ after passing through the 90 
periscope a second time and sent to the SFG stage. For the narrowband
experiment, SPDC is focused into a second identical PPLN crystal by a 200 mm
123
NIR AR-coated (B-coated) lens. The resulting upconverted light is collimated by a
200 mm visible AR-coated (A-coated) lens. The SPDC is blocked by two
interference filters (Thorlabs FESH0650, Semrock FF01-540/80-25 ) prior to being
coupled into single-mode fiber with 488nm cut-o↵ frequency and detected on an
APD (Laser-Components Count-10B).
In order to enable higher average powers at low relative squeezing gain, the
laser repetition rate was set to 10MHz for this experiment. Detection events from
the APD were measured on a time-to-digital converter (Qutools Qutau). In order to
enable ultra-low flux measurements, events were post-selected via software
time-gate with 2 ns window surrounding the arrival time. This resulted in a 50x
reduction in e↵ective dark rate, from 3Hz to 0.06 Hz. The software time-gate was
calibrated at high-gain with large count rates. SPDC power was measured via
coupling into multi-mode fiber and measurement on a fiber coupled power meter
(Thorlabs S150C ). While measurements were made below the stated minimum
power, linearity between the measured pump power and SPDC power in this regime
confirmed the measurements.
Fig. 5.11 shows the result of the experiment. The intensity of the SPDC was
altered over more than two-orders of magnitude. At low flux, corresponding to
10 pW -100 pW the scaling of the SFG rate is linear with the SPDC flux. Above
this the scaling transitions to quadratic. The data were fit to a linear quadratic
curve, indicated by the solid black line. The Dashed line corresponds to the linear
portion of the fit, and the dash-dotted line corresponds to the quadratic portion of
the fit. The fit was weighted by the inverse of the intensity to equalize the relative
errors. The cross-over between scaling regimes, seen as the intersection of the linear
and quadratic portion of the fit occurs at roughly 400 pW , which corresponds to
about 214 SPDC photons per pulse.
The number of modes in our state can be estimated as BT where B is the
124
101
100
10-1
101 102 103
PDC Flux (pW)
Figure 5.11. We demonstrate linear and quadratic scaling of SFG with PDC
power, when the pump is attenuated. The crossover point is near 400 pW at
10 MHz which corresponds to 214 photons per pulse, which is in good agreement
with single-mode predictions.
spectral bandwidth of the pulse in Hz, and T is the duration of the pulse. In our
experiment B is measured to be approximately 10 THz and the duration of our
pulse is about 8 ps. The resulting estimate is roughly 80 temporal modes. The
measured crossover agrees well with prediction, but is o↵ by a factor of roughly 2.5.
This is within the expectations given experimental uncertainties and
approximations given the mode estimate. Another source of disparity could be due
to the inherent spatial-spectral nature of our experiment.
The good agreement between the measured intensity of the crossover-point and
its prediction from our theory indicate that no major source of TPA e ciency
enhancement not captured by our theory is present.
Spectrally Resolved SFG
Utilizing a slightly modified experimental configuration, we probe the
spectrally resolved SFG spectrum generated via BSV. To do this, the 200 mm
125
SFG Flux (cps)
Under-Compressed
Optimized
Over-Compressed
500 520 540 560
480 490 500 510 520 530 540 550 560 570 580 590
wavelength (nm)
Figure 5.12. Spectrally resolved SFG of collinear BSV, with varying degrees of
dispersion compensation. The distance between prisms was adjusted to optimize
the coherent spike in the SFG spectrum and could be measured in real-time.
lenses in Fig. 5.10 were replaced by 100 mm with the same AR-coatings as before.
The PPLN crystal was also replaced with a 0.7 mm BBO crystal, cut for SFG near
800 nm and angle-tuned to achieve the appropriate phase-matching conditions at
1064 nm. The resulting SFG phase-matching bandwidth was broad enough to
achieve reasonably e cient phasematching across the entire SPDC bandwidth. The
single-mode fiber was replaced with a multi-mode fiber, and sent to a grating
spectrometer (Ocean Optics USB4F00977 ). The repetition rate was set to 100 kHz
for the remaining measurements.
Fig. 5.12 shows the result of adjusting the amount of dispersion compensation
in the prism compressor. As predicted in the previous chapter, the coherent peak is
sensitive to dispersion, whereas the incoherent pedestal is not. Dispersion
compensation could be monitored by the height of the peak and optimized in
real-time by adjusting the prism separation. The height of the peak is much greater
than the frequency correlations in the JSI would suggest. The enhancement of this
peak is an indication of the e↵ect of joint time-frequency correlations.
It is worth noting that the height and width of the peak are limited by the
resolution of the spectrometer which is about 2 nm, due to this the contrast of the
126
SFG Intensity
Figure 5.13. Diagram of experimental setup for direct measurement of spectral-
temporal correlations. Non-collinear SPDC is recombined in a HOM configuration,
as in Fig. 5.7. After recombination on the beam-splitter, SPDC is sent through an
SFG setup. The resulting up-converted light is coupled into a single-mode fiber,
and sent to a high-resolution spectrometer.
peak in this measurement is strongly reduced. The width of the peak is expected to
reflect the spectrum of the pump pulse which we estimate as 0.4 nm, which is well
below the resolution of the spectrometer.
Subsequent dispersion optimized measurements replaced on a single-mode fiber
coupled high-resolution spectrometer demonstrate peak-to-pedestal ratios
approaching 80 were achieved (not shown), which is in good agreement with the
number of modes in the state (Horiba iHR 320, 1200g/mm). Due to the stringent
alignment requirements into single-mode fiber, the compressor position was not
modified in this configuration.
Direct measurement of spectral-temporal correlations
We also probe the e↵ect of temporal correlations directly in the noncollinear
configuration. Fig. 5.13 describes the experimental apparatus. After traversing the
same HOM interferometer used for the HOM experiments, the recombined beam is
sent through a broadband SFG setup, utilizing the same broadband phase-matching
configuration described previously, using a 0.7 mm BBO. After up-conversion, the
remaining SPDC light is blocked by interference filters (Thorlabs FESH0650,
Semrock FF01-540/80-25 ). The up-converted light is coupled into a single-mode
127
Figure 5.14. Spectrally resolved SFG of BSV generated in the distinguishable
configuration, with temporal delay scanned between signal and idler beams.
fiber with 488 nm cut-o↵ frequency, and sent to a high-resolution spectrometer
(Horiba iHR 320, 1200g/mm).
We are able to measure the correlation time, in which the coherent
contribution is e ciently up-converted, by delaying one arm relative to the other.
Fig. 5.14 shows the SFG spectrum when the delay is scanned. The time window in
which the coherent contribution is up-converted is around 200 fs in width. This is
longer than measured in the HOM experiment, which is to be expected since
dispersion is expected to broaden the temporal coherence. The frequency width is
around 0.3 THz, which is in good agreement with the width of the pump spectrum.
The product of these two is approximately 0.06, which is well below  ⌫ t & 1,
which defines the Fourier relationship expected for separable two-photon states [24,
27].
It is worth noting that the FWHM width and the variance of the distribution
128
Figure 5.15. Schematic of experimental apparatus for comparison of classical
TPA and TPA driven by BSV. To ensure spatial overlap and compare solely time-
frequency entanglement in a single Gaussian mode, both BSV and classical refer-
ence (not shown) are coupled into the same 5 mm optical fiber. The light driving
TPA is reflected o↵ a dichroic beam-splitter(DM: Thorlabs DMSP900 ), and fo-
cused onto the R6G sample cell via 3mm aspheric lens(Thorlabs C330TMD-B).
The sample cell consists of a standard 1 cm quartz cuvette, with 1 mm side width.
Fluorescence collected in the epi-fluorescence geometry is collimated in the back-
wards direction and passes through the DM, before passing through a filter stack
() and being coupled into a 50µm core multi-mode fiber, which is detected on a
fiber coupled APD(Laser-Components Count-10B). The output port of the SMF
filter is connectorized which allows for direct characterization of the light coupled
into the fiber. Alignment of BSV onto the fiber is accomplished by optimizing on
coincidences at low gain, for which JSIs are measured to ensure the correct spectral
characteristics of the transmitted BSV.
are strongly disparate, due to the narrow tall spike, and the broad background
distribution. In fact, while the spike enables both high time and frequency
resolution due to its strong relative contrast, the product of the widths of the
incoherent term is much greater than 1, since it is broad in both time and
frequency.
The vertical lines across the figure are fringes, which we attribute to imperfect
splitting of the two modes in the interferometer.
Comparison of BSV and Classical TPA
While SFG comparisons are convenient due to their relatively strong
interaction and mode-selectivity, experiments utilizing the molecular samples in
question are necessary to rule out unexpected dynamics in the TPA test system.
129
While no ETPA fluorescence was observed in the low-gain regime, comparison of
BSV and classical excitation can be an indication if any residual enhancement not
predicted by the theory is present.
To test this, we measured the e ciency of TPA driven by the approximately
10 ps duration, 1064 nm central wavelength pulsed pump laser and compared this
to the e ciency of TPA driven by our BSV. Because the classical reference used the
same laser as is used to generate the pump which in turn generates the BSV,
perfect spectral agreement is guaranteed. Similarity between the duration of the
pulses is also ensured, however due to cascaded nonlinearities, the duration of the
BSV pulse was shorter in duration than the classical pulse. The SFG which creates
p
the 532 nm pump pulse shortens the pulse by a factor of 2, due to non-uniform
SFG conversion e ciency across the pulse. After this, the nonuniform gain
generating BSV e↵ectively shortens the duration of the BSV pulse. The duration of
both BSV and classical pulses are measured via autocorrelation to account for
temporal e↵ects in the comparison.
Fig. 5.15 shows the experimental configuration used for the comparison. To
ensure di↵erences in spatial mode did not contribute to this measurement, we
filtered the BSV and classical laser light through the same short 5 mm long optical
fiber (Nufern 780HP). The short fiber was connectorized to enable direct
characterization of the coincidence rate and JSI of the coupled light. This ensures
that the correct spectrum and correlated pairs are coupled into the fiber prior to
the experiment. TPA is probed in the epi-fluorescence geometry. After reflection o↵
of a dichroic mirror (DM-Thorlabs DMSP900 ), the light is focused by a 3 mm
aspheric lens (Thorlabs C330TMD-B) onto the sample cell, which consists of a
standard 1 cm Quartz cuvette with 1 mm walls. The backwards-emitted
fluorescence is collimated and passes through the DM and a set of filter stacks prior
to being coupled into a 50 µm multi-mode optical fiber, and detected on an APD
130
(Laser-Components Count-10B). TPA was verified by scaling measurements as well
as by confirmation of the fluorescence spectrum and lifetime of the detected light
for both TPA and BSV.
If the TPA dye had a narrow linewidth, such that only the coherent
contribution had significant overlap, we would expect the e ciency of our classical
excitation to match precisely the excitation of our BSV light, as discussed in Ch 4.
However, due to the broad absorption linewidth of Rhodamine 6G, the incoherent
contribution has significant overlap, and the expected TPA e ciency is between 1
and 3 times greater than for classical TPA, depending on the overlap between the
lineshape and the incoherent contribution, where an e ciency of 3 is attained in
the limit of uniform TPA e ciency across entire incoherent contribution and in the
absence of reduction of TPA e ciency of the coherent contribution due to
dispersion.
The duration of the 1064 nm pulse was measured to be 9.2 ps via intensity
autocorrelation. The duration of the BSV pulse was measured to be 2.5 ps by the
same measurement. This yields an expected e ciency increase of a factor of 3.6 due
to the di↵erence in pulse durations. Given this adjustment, the e ciency of TPA
driven by BSV was 1.78 times as e cient as the classical TPA. This is in
reasonable agreement with the factor of 3 considering the finite spectral absorption
bandwidth of Rhodamine 6G and the presence of dispersion in the optical fiber.
The duration of the BSV pulse was a↵ected by the presence of nonlinear gain
in the SPDC process at high gain. Because the pulse driving the BSV is Gaussian
rather than uniform, the gain is stronger at the peak of the pulse than the
shoulders. The e↵ect of this is to e↵ectively shorten the temporal envelope of the
BSV pulse. This is one of the major shortcomings of the chopped CW model of
BSV for realistic Gaussian pulses.
The relative similarity in e ciency of TPA is in good agreement with
131
4 BSV Pre = 2755.22x
1.97
10 BSV Post = 2764.13x1.94
Coherent = 433.54x2.00
102
100
10-1 100
Power ( W)
Figure 5.16. Comparison of BSV TPA intensity and classical TPA reveals pure
quadratic scaling in attenuation prior to generation and post generation, as well as
a ratio of approximately 6.4 in raw e ciency between the two. Both BSV and clas-
sical reference beam were coupled into the same 5 mm SM optical fiber (780HP)
with power measured after the fiber to ensure spatial mode matching, and evaluate
only frequency correlations. The Duration of the Pulsed 1064 nm pulse was mea-
sured to be 9.2 ps in duration via intensity autocorrelation. The duration of the
BSV pulse was measured to be 2.5 ps by the same measurement.
132
TPA Counts (Hz)
theoretical predictions and rules out any significant enhancement in TPA by
time-frequency entangled pairs at high gain not captured by the theoretical
description.
These experiments are a good indication that no signal is expected at the
low-gain, especially after comparison with relative rates of SFG and TPA.
Nevertheless, we also conducted a second series of tests with both pulsed and CW
SPDC sources, where alignment was ensured via pulsed excitation in the high-gain
regime.
A slightly altered experimental configuration was used for these experiments.
No spatial mode-filtering was applied, and a 2 cm PPLN crystal was used to
increase rates and more closely align with the experiments described in [8]. In the
pulsed, high-gain regime at 100 kHz, TPA was readily observable. As expected,
quadratic scaling was observed as BSV intensity was decreased down to the
detection threshold. Similarly, increasing the repetition rate but keeping the
average power steady at 150 nW resulted in linear decrease in rates, consistent with
quadratic per-pulse scaling. At higher than 5 Mhz repetition rates, we were no
longer able to produce 150 nW , due to the decreased gain in the SPDC crystal.
We were able to generate about 150nW average SPDC power at 1W CW
excitation power. Using this source aligned to the BSV source and utilizing the
chopper setup described in Ch 3 we measured no observable signal. This
experimental apparatus had the benefit of lower dark rates, 3 Hz, and higher
detection e ciencies than the experiment described in Ch 3, due to the use of the
APD rather than the PMT. No measured signal was observable over the course of
an 1800 s experiment. The absolute characterization of this experiment is still
outstanding, but to date we have not been able to replicate the results in [8],
despite lower noise rates and otherwise almost identical experimental
configurations.
133
Conclusions
In this chapter, we validated many of the predictions from Ch 4 for both low-
and high-gain squeezed vacuum. Joint spectral intensity measurements were con-
ducted in both regimes, showing good agreement with the spectral correlations
predicted for the measurements. These measurements are to our knowledge the
first of their kind to utilize a TOF spectrometer to characterize broad-band bright
squeezed vacuum in both regimes and in collinear and non-collinear configurations.
HOM-interference in the non-collinear configuration at low-gain shows good agree-
ment with the predictions, and further validates the model for incoherent two-photon
measurements.
By directly observing the cross-over from linear to quadratic scaling of sum-
frequency-generation with incident SPDC flux, we confirm that the cross-over be-
tween linear and quadratic scaling occurs at an intensity of approximately 1 photon
per mode. This ensures that high-gain TPA e ciency measurements can accurately
predict low-gain behavior, since the e ciency can be accurately extrapolated.
Utilizing classical TPA alongside TPA driven by BSV, we are able to confirm
the previously known asymptotic e ciency limit of TPA driven by BSV relative to
classical TPA to within a factor of 2. This e↵ectively limits the magnitude of the
time-frequency enhancements to the TPA e ciency by entangled photons that scale
in the same manner as the frequency correlations present in SFG and JSI measure-
ments.
Finally, we are able to confirm that the time-frequency correlations of interest
persist in high gain, albeit alongside incoherent background terms. We first vary
the degree of dispersion compensation, in spectrally resolved measurements of SFG
driven by BSV. In doing so, we observe the detrimental e↵ect of quadratic phase on
the central spike of the SFG spectrum, which is driven by the coherent portion of
134
the BSV beam. While doing so, the broad incoherent background of the SFG spec-
trum remains constant. Separately, by varying the time-delay between non-collinear
signal and idler beams, we are able to demonstrate time resolution down to 200 fs
in the central spike of around 0.3 THz in frequency. This corresponds to a time-
bandwidth product well below 1, demonstrating simultaneously high resolution in
both time and frequency. This is strong validation that utilizing BSV to probe the
e↵ect of these correlations is well motivated, and remains sensitive to correlation of
this type.
Taken together, these results demonstrate that the models we developed en-
capsulate the relevant physics to describe ETPA and the nonlinear e↵ects of time-
frequency correlations in BSV. Importantly, any major enhancement due to time-
frequency entanglement not captured by our theory would be reported by these
experiments. The consistency between our experiment and theoretical prediction
is the strongest evidence to date that the orders of magnitude ETPA enhancement
reported elsewhere is not due to time-frequency correlation, but rather another spu-
rious linear or nonlinear process.
135
CHAPTER VI
SUMMARY AND CONCLUSION
Future Work
My hope from this work is to guide the further exploration and critical
evaluation of experimental work investigating time-frequency entangled photon
pairs for metrology. The results from other experimental groups, while lacking some
of the experimental verification steps we’d like to see, seem to unambiguously
indicate the presence of some as-of-yet fully explained phenomenon. Plausible
explanations include hot-band absorption [62] and other scattering phenomena [11].
However, even these do not fully explain the results seen in [7, 8]. Conversely, a
comparison to a broad classical pulse of similar bandwidth would also be instructive
to rule out frequency dependent e↵ects, which may be only visible due to the large
bandwidth of most PDC sources.
In particular, a standardized set of validation experiments for ETPA
measurements is desirable. Demonstration of both linear and quadratic scaling
should be a minimum for ETPA experiments. Especially such experiments claiming
ETPA cross-sections on the order of 10 18 cm2 should have ample SNR to
demonstrate the e↵ect over at least an order of magnitude of flux scaling. While
quadratic scaling alone does not unambiguously indicate ETPA, it at least indicates
some type of nonlinearity in the experiment.
Another impact I’d like to see from our work is an increased focus on proper
characterization of the source in experiments to enable better understanding of the
observed e↵ect, as well as more controlled comparisons across various
implementations of experiments. For instance, measuring an entangled two-photon
cross-section is not well grounded without thorough characterization of
136
entanglement time and area. The latter includes an understanding of the classical
beam parameters, as well as the propagation of the spatial correlation along the
optical system. Another crucial piece is the standardization of reference samples for
the measured e↵ects. Due to the di culty in synthesis of some of the molecular
systems used in [1–3] and others, experimental replication is limited to samples that
are commercially available or easily synthesized. Dyes such as Rhodamine 6G,
Rhodamine B, ZnTPP make good candidates.
Additional theoretical work examining possible nonlinear e↵ects not
encompassed by ETPA with larger interaction strengths would also be useful, in
line with the explorations conducted in [11, 98].
While some work has been done in atomic systems [50, 51], further work in
these simpler model systems would be instructive to the true ability of
time-frequency entanglement to enhance TPA rates in narrowband systems. One
di culty in such systems is the presence of resonant intermediate states, so more
analysis into resonant intermediate states would likely need to be carried out.
Further investigation into the spatial propagation and whether an e↵ectively
increased focal volume plays a role in amplifying the measured signals above
expectation is also warranted. This is alluded to in the inclusion of the
entanglement area in the entangled two-photon absorption cross-section. However,
beyond careful characterization of the entanglement area at the focus of the beam,
the propagation of these correlations through the optical setup is important. The
reason for this is that in extended systems, there are many molecules or atoms
displaced in each spatial direction from the focus of the beam. Interaction strengths
for these are typically assumed to vary based on Gaussian beam propagation,
however the nature of the spatial correlations is more complex than this, and it is
unclear the degree to which spatial e↵ects could increase e↵ective nonlinear
interactions strengths averaged over an entire sample.
137
Alongside study of such e↵ects for spatially multimode bulk SPDC sources, the
use of waveguide SPDC sources, which constrain the generation of time-frequency
entangled photons to a single nearly Gaussian spatial mode would aid the
characterization and control of ETPA experiments, ruling out spatial e↵ects in the
observed signals. This would rule out one major source of ambiguity between
experiments.
As indicated by our work utilizing bright squeezed vacuum, simultaneously
high temporal and spectral resolution is achievable using such a source. Further
theoretical investigations into the use of spectral-temporal correlation in high-gain
squeezed vacuum states for enhanced metrology and nonlinear spectroscopy are
warranted, since these overcome the primary limitation presented by the use of
time-frequency entangled photon pairs, while retaining some of the
spectral-temporal correlations present in the low-gain regime. In order for schemes
based around BSV to be successful, careful considerations into the nature of the
background signal is needed, since selectivity in either the detection, (e.g.,
spectrally resolved SFG), or in the probed interactions (e.g., narrow
phase-matching or TPA linewidth), is needed for high contrast measurements. In
the absence of these, the integrated magnitude of the incoherent signal is large
reducing the overall contrast of the desired coherent signal.
Final thoughts
While many questions still exist about the exact e↵ects observed in some of the
experiments discussed in this work, we have demonstrated a solid theoretical
framework capable of explaining the e↵ects that we have been able to observe.
These disagree in strength by many orders of magnitude with the e↵ects observed
elsewhere [5, 8]. Experimental validation of these models in our lab has shown that,
when in the observable regime, the magnitude of these interactions agrees well with
138
our predictions. Similarly, e↵ects we expect to be unobservable are, indeed,
unobservable. A comparison to SFG with an interaction strength several
orders-of-magnitude larger than that of TPA, in which we are able to observe
time-frequency entanglement enhanced signals, further validates our theoretical
work.
Direct observation of TPA rates in BSV confirms our that our theoretical
description is able to predict the relative strength of TPA via classical excitation
and excitation by BSV. The presence of many orders of magnitude enhanced
entangled photon signals not predicted by our theory would alter this relationship.
So we take this agreement to indicate that the necessary dynamics have been
included in our model. Further experiments investigating the transition to the
low-gain regime in SFG convincingly demonstrate that the cross-over occurs at the
pair flux we would expect, further validating our model’s ability to predict the
magnitude of entangled two-photon interactions, since a transition to linear scaling
at high values would indicate a stronger interaction than predicted at low gain.
Finally, experiments utilizing SFG driven by BSV demonstrate that
time-frequency correlations persist at high-gain, in the presence of a background
term of similar magnitude. This indicates that careful experimental designs could
utilize the enhanced time-frequency resolution present in broadband BSV for
nonlinear spectroscopic techniques.
139
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