Setting Experimental Bounds on Entangled Two-Photon
Absorption Cross Sections
byKristen M. Parzuchowski
B.S., Michigan State University, 2017
M.S., University of Colorado, 2020

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2023
Committee Members:
Ralph Jimenez, Co-Chair
Margaret M. Murnane, Co-Chair
Thomas R. Schibli
Adam Kaufman
Niels H. Damrauer

ii
Parzuchowski, Kristen M. (Ph.D., Physics)
Setting Experimental Bounds on Entangled Two-Photon Absorption Cross Sections
Thesis directed by Prof. Ralph Jimenez

Two-photon absorption (2PA) is widely used in microscopy for deep, sub-cellular imaging.
However, the efficiency of 2PA is limited by the properties of both the absorber and the excitation
light. Entangled photon pairs produced via spontaneous parametric down-conversion (SPDC) exhibit correlations in energy, time and space that may improve the excitation efficiency relative toa classical laser. The most significant improvement is expected at low photon flux where isolated
pairs interact with the absorber. In this regime, the rate of the entangled two-photon absorption(E2PA) process scales linearly with photon flux and the E2PA cross section (σE ).
Despite over a decade of publications claiming to measure huge σE that suggest a quantum
advantage exists of up to 10 orders of magnitude, in this thesis I will show strong evidence that
σE are several orders of magnitude lower than previously reported. First, we provide relevant
background information on nonlinear and quantum optics. Next, we discuss theoretical descriptionsof σE and review the large body of experimental work in the field. Afterwards, we discuss the four
experiments we designed to measure E2PA.
In the first and third experiments, we measure SPDC transmittance through samples of twophoton absorbers in room-temperature liquids. In our second experiment, we collect fluorescencefrom samples excited with SPDC. Despite the high sensitivity of the techniques, we could not
resolve a signal in any of the measurements. We set upper bounds on the σE of eight independent
absorbers that are up to five orders of magnitude lower than previously published σE .
The third experiment also served as a classical 2PA (C2PA) measurement system. We madeone-to-one comparisons between E2PA and C2PA to bound the quantum advantage. We derived
absolute C2PA cross sections that closely agreed with values already reported in literature. In the
fourth experiment, we designed a toluene-filled hollow-core-fiber platform for 2PA measurements.

iii
We measured C2PA down to 20 nanowatts, and expect to make further improvements. This
platform is at least 4-fold more sensitive than a standard cuvette-based technique and thus is ideal
for the next generation of E2PA measurements.

Dedication

To Nathan and KevinAcknowledgements

I am so grateful to have been surrounded by so many admirable individuals throughout mytime in graduate school. First I would like to thank my family for their unending support. Thankyou to my parents, Scott and Mary, for teaching me how to be a strong, independent individual.
Thank you to my brothers, Nathan and Kevin, for sparking my interest for science in the firstplace. Thank you to all of my family members whom there are way too many of to name. Thankyou to Benton for your encouragement and all the fun you bring to my life. Thank you to Bindy
for always being eager for a break from work. Thank you to my friends from all parts of my life –those who have stuck around since kindergarten to those I have only met this year. You make life
more fun! Thanks especially to Raegan, Rick, Dan, Heather, Lilah, Meeko, Sam, Mar and Scout.
To all my teachers – thank you for pushing me to be the best I can be. Thank you to myresearch advisor at Michigan State, Jaideep Singh, who taught me how to do research well andinspired me to keep doing it. Because of his guidance I sought out graduate schools throughout
the country and landed in Colorado – for this I am so grateful. Thank you to my research advisor
throughout graduate school, Ralph Jimenez. Thank you for setting me up with an awesome project
and team that has prepared me for wherever my career takes me. Under Ralph’s guidance I became
a confident researcher. Thank you to Marty Stevens for advising me and teaching me how to present
in plain English. Thank you to my committee members Margaret Murnane, Thomas Schibli, Adam
Kaufman and Niels Damrauer for your guidance.
Thank you to Marty, Mike Mazurek and Thomas Gerrits for teaching me about quantum
optics. Thanks to Alex Mikhaylov for teaching me about two-photon absorption and about working

vi
in a lab. Thanks to Alex, Mike and Marty for aligning and realigning optics with me. Thanks
to Mike and Marty for countless helpful discussions on experimental designs, data acquisition anddata analysis. Thank you to Charlie Camp, Daniel Lum and Meng-Chang Wu for lending yoursupport and advice from Gaithersburg.
Thank you to the Jimenez lab members past and present– Srijit, Alex, Ryan, Samantha,
Sheng, Josh, Emma, Nancy, Richard, Sushil, Alan, Miles, Daniel, Annika and Connor. Thanks to
Srijit, Alex, Ryan, Samantha, Sheng and Emma for the hikes, the shared meals, the Rec center
activities, etc. Thanks to Srijit for keeping me sane during the middle of graduate school. Thanks
to Alan, Miles and Daniel for continuing where I left off. I thank the incredible support staff at
JILA – without you I would not have been able to complete this research.
Thank you!

vii

Contents

Chapter
1 Introduction

1

1.1

Classical Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Spontaneous Parametric Down-conversion . . . . . . . . . . . . . . . . . . . . . . . .

6

1.5

Second-Order Coherence Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.6

Two-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Entangled Two-Photon Absorption

16

2.1

Publication Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3

Quantum Mechanical E2PA Cross Section . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

Probabilistic E2PA Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5

Experimental Literature Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6

Signatures of E2PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7

Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Entangled Photon Source and Characterization

31

3.1

Publication Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

viii
3.3

SPDC Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4

Characterizing Mean Photon Number . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5

Superconducting Nanowire Single-Photon Detectors . . . . . . . . . . . . . . . . . . 35

3.6

Time-of-Flight Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7

Entanglement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8

Characterizing Second-Order Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.9

Entanglement Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 An Introduction to Transmittance Measurements for Entangled Two-Photon Absorption

42

4.1

Publication Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4

Experimental Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5

Origin of Alignment-Mode Interference Pattern . . . . . . . . . . . . . . . . . . . . . 47

4.6

Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.7

Deriving a Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.8

Data Analysis Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.9

4.8.1

Case 1. Measuring sample and solvent without time delay . . . . . . . . . . . 53

4.8.2

Case 2. Measuring sample with time delay . . . . . . . . . . . . . . . . . . . . 54

4.8.3

Case 3. Measuring sample and solvent with time delay . . . . . . . . . . . . . 54

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Bounding Entangled Two-Photon Absorption Cross Sections with a Fluorescence-Based
Scheme58

5.1

Publication Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3

Experimental Setup and Characterization . . . . . . . . . . . . . . . . . . . . . . . . 59

ix
5.4

Fluorescence Collection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.5

Calculating C2PA and E2PA Cross Sections . . . . . . . . . . . . . . . . . . . . . . . 67

5.6

Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.7

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Bounding Entangled Two-Photon Absorption Cross Sections with Sensitive Transmittance
Measurements

81

6.1

Publication Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3

Operating Principle of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.5

Inferring 2PA Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.5.1

Estimating Detection Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.5.2

Count Renormalization with a Quadrant Photodiode . . . . . . . . . . . . . . 94

6.6

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.7

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 A Toluene-Filled Hollow-Core-Fiber Platform for Two-Photon Absorption Measurements 102
7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2

Light Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4

Calculating a C2PA Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.5

Experimental Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.6

Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.7

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

x
8 Conclusions and Outlook

117

Bibliography

120

Appendix
A Literature summary127

B Sample preparation details

131

C Detailed experimental diagram and parts list for Chapter 4132

D Data for Chapter 4135

E Detailed experimental diagram and parts list for Chapter 5136

F Alignment details for Chapter 5140

G Detailed experimental diagram and parts list for Chapter 6142

H Data for Chapter 6

145

I

151

Detailed experimental diagram and parts list for Chapter 7

J Fiber adapter technical drawings

154

xi

TablesTable
2.1

Brief summary of E2PA cross sections reported in literature . . . . . . . . . . . . . . 24

4.1

Comparison of data analysis case studies . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1

Summary of sample parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2

Summary of apparatus parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3

Summary of fluorescence measurement results . . . . . . . . . . . . . . . . . . . . . . 76

6.1

Summary of transmittance measurement results . . . . . . . . . . . . . . . . . . . . . 100

A.1 Summary of experimental data from E2PA publications in literature . . . . . . . . . 129
D.1 Coincidence rates measured in the experiment in Chapter 4 . . . . . . . . . . . . . . 135

xii

FiguresFigure1.1

Level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Schematic of joint spectral and temporal intensities of SPDC . . . . . . . . . . . . .

8

1.3

Hanbury Brown-Twiss interferometer and g (2) . . . . . . . . . . . . . . . . . . . . . . 11

2.1

Schematic of entanglement area and time . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2

Schematic of E2PA measurement schemes . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3

Compendium plot of E2PA results in literature . . . . . . . . . . . . . . . . . . . . . 26

3.1

Schematic of the SPDC source and detection schemes . . . . . . . . . . . . . . . . . 32

3.2

Time-of-flight spectrometer calibration curve . . . . . . . . . . . . . . . . . . . . . . 35

3.3

Measured joint spectral intensity and simulated joint temporal intensity . . . . . . . 37

3.4

Probabilistic estimate of the advantage of lossless dispersion compensation . . . . . . 39

4.1

Schematic of transmittance experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2

Measured joint spectral intensity and simulated joint temporal intensity . . . . . . . 45

4.3

Measured interference in the coincidence rate . . . . . . . . . . . . . . . . . . . . . . 46

4.4

Calculated interference in the coincidence rate . . . . . . . . . . . . . . . . . . . . . . 49

4.5

Measured coincidence rates with solvent and sample at zero and τ delays. . . . . . . 51

4.6

Schematic of notations used in data analysis case studies . . . . . . . . . . . . . . . . 53

5.1

Schematic of fluorescence experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

xiii
5.2

Illustration of the geometrical collection efficiency inside of the cuvette . . . . . . . . 63

5.3

Spectral overlap plots for fluorescence measurements . . . . . . . . . . . . . . . . . . 70

5.4

Results from classical and entangled two-photon excited fluorescence measurements . 75

6.1

Cartoon of the operating principle of the transmittance experiment . . . . . . . . . . 82

6.2

Schematic of transmittance experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.3

Measured joint spectral intensity and simulated joint temporal intensity . . . . . . . 86

6.4

Measured interference in the singles count rate . . . . . . . . . . . . . . . . . . . . . 88

6.5

Example dataset for a transmittance measurement . . . . . . . . . . . . . . . . . . . 97

6.6

Summary of transmittance measurement results . . . . . . . . . . . . . . . . . . . . . 98

7.1

Advantage of fiber-based measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.2

Index diagram for toluene-filled hollow-core fiber . . . . . . . . . . . . . . . . . . . . 104

7.3

Images of the fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4

Schematic of fluorescence experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.5

Scatter of alignment laser along fiber length . . . . . . . . . . . . . . . . . . . . . . . 111

7.6

Example dataset for a fluorescence measurement . . . . . . . . . . . . . . . . . . . . 114

7.7

Results from AF455 fluorescence measurements . . . . . . . . . . . . . . . . . . . . . 115

C.1 Detailed diagram of first transmittance experiment . . . . . . . . . . . . . . . . . . . 132
E.1 Detailed diagram of fluorescence experiment . . . . . . . . . . . . . . . . . . . . . . . 136
G.1 Detailed diagram of second transmittance experiment . . . . . . . . . . . . . . . . . 142
H.1 Transmittance dataset for pH11 water . . . . . . . . . . . . . . . . . . . . . . . . . . 146
H.2 Transmittance dataset for 1.9 × 10−4 mol L−1 fluorescein in pH 11 water . . . . . . . 146
H.3 Transmittance dataset for 9.4 × 10−3 mol L−1 fluorescein in pH 11 water . . . . . . . 146
H.4 Transmittance dataset for PBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
H.5 Transmittance dataset for 2.9 × 10−4 mol L−1 FAD in PBS . . . . . . . . . . . . . . 147

xiv
H.6 Transmittance dataset for 6.7 × 10−3 mol L−1 FAD in PBS. . . . . . . . . . . . . . . 147
H.7 Transmittance dataset for toluene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
H.8 Transmittance dataset for 1.65 × 10−3 mol L−1 ZnTPP in toluene. . . . . . . . . . . 147
H.9 Transmittance dataset for borate buffer . . . . . . . . . . . . . . . . . . . . . . . . . 148
H.10 Transmittance dataset for 8 × 10−6 mol L−1 qdot 605 in borate buffer . . . . . . . . 148
H.11 Transmittance dataset for methanol . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
H.12 Transmittance dataset for 1.8 × 10−4 mol L−1 Rh6G in methanol . . . . . . . . . . . 148
H.13 Transmittance dataset for 1.1 × 10−2 mol L−1 Rh6G in methanol . . . . . . . . . . . 149
H.14 Transmittance dataset for chloroform . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
H.15 Transmittance dataset for 7.3 × 10−4 mol L−1 9R-S in chloroform . . . . . . . . . . . 149
H.16 Transmittance dataset for 1.7 × 10−4 mol L−1 AF455 in toluene . . . . . . . . . . . . 149
H.17 Transmittance dataset for 4.0 × 10−3 mol L−1 AF455 in toluene . . . . . . . . . . . . 150
I.1

Detailed diagram of fiber experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

J.1

Technical drawing of fiber adapter clamp

J.2

Technical drawing of fiber adapter base . . . . . . . . . . . . . . . . . . . . . . . . . 156. . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter 1

Introduction

The field of nonlinear optics involves interactions of light with nonlinear media. Nonlinear
optical effects occur in certain materials when the intensity of light is high enough to induce sizable
nonlinear terms to the polarization of the material. The lowest order correction to the polarization is
expected to be comparable to the linear response when the intensity of the electric field approaches
that of atoms (1020 W m−2 ) [1].
The field of quantum optics covers the quantum mechanical properties of the individual
quanta of light, photons, and their interactions with matter. These two fields can delve into one
another, and we will see that this is the case when we consider the nonlinear processes of spontaneous
parametric down-conversion (SPDC) and entangled two-photon absorption (E2PA), both of which
concern single photon pairs and nonlinear materials. Although the former process uses the strong
field strengths typical of nonlinear optical phenomena, the latter pushes the boundaries of what is
and isn’t possible in nonlinear optics - using single photon pairs to seed a nonlinear process.
In this chapter we discuss the introductory material necessary for understanding this thesis.
First, we look at the theory of second harmonic generation (SHG) and discuss the factors, such asphase-matching, that alter the intensity of the generation process. We then turn to its time reverse
process, SPDC, and derive a wave function for the generated light and its intensity distribution.
We consider the second-order coherence function and how it can be used to characterize the photon
statistics of various sources including SPDC. Finally we discuss two-photon absorption (2PA) and
derive the cross section for both one and two-photon absorption.

2

1.1

Classical Nonlinear Optics
First let’s consider the classical Hamiltonian for the electromagnetic field
1
H=
2

Z

d3 r[E · D + B · H],(1.1)

where E, B, D = ϵ0 E+P, H = µB are the electric field, magnetic flux density, electric displacement
field and magnetizing field, respectively, ϵ0 , µ are the permittivity of free space and the permeabilityof the material, respectively, and P is the polarization of the material. This equation is the integral
over space of the electromagnetic energy density. Using Hamilton’s equations we derive Maxwell’sequations [2],
∇·D=0

(1.2)

∇·B=0

(1.3)

∂B
∂t
∂D
∇×H=−
,
∂t
∇×E=−

(1.4)
(1.5)

where we have assumed that there are no free charges or currents. From here, we can derive thewave equation for nonlinear media [1]∇2 E −

1 ∂ 2 NLn2 ∂ 2E=P .
c2 ∂t2
ϵ0 c2 ∂t2

(1.6)

where PNL is the nonlinear part of the polarization. The total polarization, P (t), of the material
can be expanded to include the nonlinear terms, [1]P(t) = P(1) (t) + PNL (t) = P(1) (t) + P(2) (t) + P(3) (t) + P(4) (t) + ...
= ϵ0 χ(1) E(t) + χ(2) E2 (t) + χ(3) E3 (t) + χ(4) E4 (t) + ... .

(1.7)

Here, χ(n) refers to the nth-order susceptibility of the medium and P (1) (t) is responsible for theordinary linear effect of a material to polarize in response to an applied electric field and thus
reduce the total electric field within the material. A wide range of nonlinear effects can occur fromthe higher order terms, and many can be more thoroughly understood using Eq. (1.6).

3

1.2

Second Harmonic Generation
Let’s consider the process of SHG, in which two photons of frequency ω are destroyed and aphoton of frequency 2ω is generated as shown in Fig. 1.1(a). We note that SHG is a special caseof the process sum-frequency generation (SFG), in which the two photons that are destroyed canhave non-degenerate frequencies.
In this section, our formalism is adapted from Ref. [1]. The polarization of the material isgenerated by two fields at frequency ω, thus we use the first term in the nonlinear polarization,
P (2) . We can assume that the electric field incident on the material is of the formEω (r, t) = Aω (r)ei(kω z−ωt) + c.c.,(1.8)

where Aω (r) is the amplitude of the field, kω is the wavenumber, and the field is propagating in
the z direction. We can then write the relevant polarization term as
P (2) = 2ϵ0 deff Eω (r, t) · Eω (r, t)= 2ϵ0 deff A2ω (r)(ei2(kz−ωt) + 1) + c.c.,

(1.9)

(2)

where deff is related to the second order susceptibility tensor χijk . The precise form of deff depends
on the polarization directions, propagation directions and frequencies of the optical waves and thesymmetry properties of the crystal. The first term is responsible for the SHG and the second for a
DC field known as optical rectification. We will drop this second term as it is not relevant for our
purposes. We will look for solutions to Eq. (1.6) of the form E2ω (r, t) = A2ω (r)ei(k2ω z−2ωt) + c.c..
Plugging Eq. (1.9) into Eq. (1.6), we arrive at∂2∂8deff ω 2 2 i∆kz
A
(z)
+
2ik
A
(z)
+
c.c.
=
−
Aω e
+ c.c.,
2ω
2ω
2ω
∂z 2
∂z
c2

(1.10)

where ∆k = 2kω − k2ω = 2ω(nω − n2ω )/c is the phase matching term. We assume we are working
in the undepleted pump regime where the amplitude Aω is independent of z. For A2ω , we assume
the slowly varying amplitude approximation, which implies that the first term is much smaller than

4

Figure 1.1: Level diagram of (a) second harmonic generation (SHG), (b) spontaneous parametricdown-conversion (SPDC) and (c) two-photon absorption (2PA). Frequencies ωp , 2ωp , ωs , ωi , ω1 , ω2and ωf refer to pump, second harmonic, signal, idler, nondegenerate frequency 1, nondegeneratefrequency 2 and fluorescence respectively. Solid and dashed lines refer to real and virtual energy
levels respectively. In SHG and SPDC energy is conserved. In 2PA, energy is lost through vibrational relaxation in both the excited and ground state (yellow wavy arrow).

the second. We drop the complex conjugate while maintaining the equality and are left with∂
4deff ω 2 i∆kz
A2ω (z) = i
A e.
∂zcn2ω ω

(1.11)

Next we perform the z integrals and arrive at
4deff ω 2 ei∆kL − 1.
A2ω (z) =A
cn2ω ω
∆k(1.12)

We can then solve for the intensity of the second harmonic using the time-averaged Poynting vector,
I2ω = 2n2ω ϵ0 c|A2ω |2 ,I2ω =8 (deff ωL)2 2
I sinc2 (∆kL/2).
c2 n2ω n2ω ϵ0 ω

(1.13)The second harmonic intensity scales quadratically with the pump intensity, which is expected
since two photons are needed to produce a second harmonic photon. The intensity is very sensitive
to the product of the phase matching parameter and the length of the crystal. For ∆k = 0 or
∆k << 1/L, the SHG efficiency is maximized and the intensity increases quadratically with thelength of the crystal.

5

1.3

Phase Matching
The phase matching parameter, ∆k, is important for all parametric nonlinear processes, sohere we discuss it more generally. This parameter describes the necessity for momentum conservation.
As the phase product ∆kL increases, the intensity of the field generated from the nonlinear
process decreases and then increases in an oscillatory manner. This can be understood as a phasewalk-off of the generated beam from the pump. After a certain length of crystal, L = Lc = 2/∆k,walk-off leads to power flowing back into the pump from the generated beam. If the goal is to
produce a sizeable signal, this walk-off should be minimized.
Perfect phase matching (∆k = 0) can be achieved by finding a material for which the indicesof refraction for all of the waves are equal. This is not easy to find, but can be done it multiple ways.
Typically the index of refraction of a material increases with frequency (normal dispersion), but it
may also decrease with frequency (anomalous dispersion). If the crystal is birefringent, the index
of refraction depends on the axis of the crystal. This birefringence can be used to phase match. For
certain wavelength and polarization combinations of the high frequency and low frequency fields,
the phase matching condition is met. In χ(2) processes, types 0, I and II phase matching refer tothe polarization of the high frequency photon being parallel (orthogonal) to both (neither), neither(both) and one (one) of the low frequency photons, respectively.
More variables can be added to the birefringent phase matching problem if we consider
temperature or angle tuning the crystal. The temperature affects the index of refraction andtuning it can change ∆k. For angle tuning, we can tilt the crystal and propagate the light along
an axis with an index of refraction that is some combination of the indices for the symmetry axesof the crystal.
Another way to phase match is called quasi-phasematching. In this case ∆k ̸= 0 for any setof bulk crystal parameters. Instead, the optical domain of the crystal is flipped periodically (onceper coherence length, Lc ) to walk the waves back into phase with one another before the generated

6
field intensity begins to decrease. This flipping of the optical domain effectively changes the signof χ(2) and the crystal is called a periodically poled crystal. If the crystal is periodically poled, it
results in a modification where ∆k → ∆k − 2π
Λ , where Λ = 2π/∆k is the poling period.

1.4

Spontaneous Parametric Down-conversion
Another nonlinear process called parametric down-conversion (PDC) is the time reverse of

SFG. In this process a pump photon is destroyed and two photons are created, a signal and an
idler photon, as shown in Fig. 1.1(b). When the process is seeded by a weak signal beam, it’sreferred to as stimulated PDC or parametric amplification. If there is no seed the process is called
spontaneous PDC (SPDC) or parametric fluorescence. Stimulated PDC can be described usingclassical nonlinear optics, as we showed for SHG, however SPDC, which is seeded only by thevacuum, requires a quantum description.
Here we have adapted the formalism from Ref .[3]. We begin using the classical Hamiltonian
for the electromagnetic field, Eq. (1.1). By inserting the expansion of the polarization (Eq. (1.7)),
we can split the Hamiltonian into a linear and nonlinear part, H = H L + H NL . The first term in
the nonlinear part takes the form
1
H NL = ϵ0
2

Z

(2)

d3 rχijl Ei (r, t)Ej (r, t)El (r, t),(1.14)(2)

where we have adopted Einstein notation, and χijl depends on the frequencies of pump, signal andidler. To reach the quantum description we must quantize the electromagnetic field. The electricfield functions E(r, t) are replaced by field observables Ê(r, t) which separate into a sum of positive,Ê + (r, t), and negative, Ê − (r, t), frequency contributions. The positive contribution takes the form
s
X
ℏω(k)
1
Ê + (r, t) = 1/2iâk,s (t)ϵϵk,s eik·r ,(1.15)2ϵ0Vk,swhere s indicates the polarization component, k indicates the wavevector, ϵ is a unit polarization
vector, â(â† ) is the annihilation (creation) operator, and V is the quantization volume.
Since the pump beam is bright enough to be treated classically, we leave the classical electricfield function. After making the field replacements for signal and idler fields, the Hamiltonian for7
the system is of the form
Ĥ = Ĥ L + Ĥ NL =

X
k,s

1
(2)
ℏω(k)(n̂k,s + 1/2) + ϵ0 χijl
2Z

d3 rEi (r, t)Êj (r, t)Êl (r, t),(1.16)

where n̂ is the number operator. In the nonlinear Hamiltonian multiple terms arise, but we keep
the terms necessary for SPDC,
1(2)
Ĥ NL = ϵ0 χijl
2

Z



d3 r Ei (r, t)Êj− (r, t)Êl− (r, t) + h.c. .

(1.17)

We can compute the wave function of the SPDC state by using first-order time-dependent perturbation theory, where in the interaction pictureZi t ′ NL ′dt H (t ) |Ψ(0)⟩,|Ψ(t)⟩ ≈ 1 −ℏ 0(1.18)

where |Ψ(0)⟩ = |01 02 ⟩ and 1 and 2 denote signal and idler photons. We make a few assumptions:that the pump is nearly monochromatic, the polarizations of signal and idler photons are fixed, thevolume of the crystal is large relative to the wavelength of light and the pump amplitude varies
slowly on the timescale of the interaction. Then we arrive at
Z Z|Ψ(t)⟩ ≈ C0 |01 02 ⟩ + C1
d3 k1 d3 k2 Φ(k1 , k2 )â†1 â†2 |01 02 ⟩,where C0 and C1 are constants, with C1 ∝ Lz

p

(1.19)

Ip where Lz is the crystal length in the z direction

and Ip is the pump intensity. We emphasize here that the intensity of the SPDC increases linearlywith pump intensity in contrast to SHG (Sec. 1.2). The joint transverse momentum [4] amplitudeis
Φ(k1 , k2 ) = N sinc∆kz Lz2ν(qp ),(1.20)

where N is a normalization constant, ∆kz = k1z + k2z − kpz is the phase matching parameter, ν isthe transverse pump momentum profile and qp is the projection of kp onto the transverse plane. If
we make a change of variables we can rewrite the function in terms of q1 and q2 , the projectionsof k1 and k2 onto the transverse plane,Φ(k1 , k2 ) = u (|q1 − q2 |) ν (|q1 + q2 |)


Lz λp
2
2
2
= N sinc|q1 − q2 | e−σp |q1 +q2 | .
8π(1.21)

8
Here λp is the pump wavelength and σp is the pump radius. We have made the assumption that thepump profile is Gaussian. We also note that, |q1 − q2 | = k1 sin(θ1 ) − k2 sin(θ2 ) and |q1 + q2 | = qp .
The function u(|q1 − q2 |) is set by phase matching and the function ν(|q1 + q2 |) is set by the pumpprofile. Here we note that this form of the joint transverse momentum amplitude illuminates that
it is not separable into a function that depends only on the transverse momentum of signal andanother for idler, rather the amplitude is separable into functions that depend on the difference or
sum of the transverse momenta of signal and idler. This shows that the two photons are entangled.

Figure 1.2: A schematic of a (a) joint spectral intensity (JSI) and (b) joint temporal intensity(JTI) of entangled photons produced via spontaneous parametric down-conversion. The diagonal
width ∆(ω1 + ω2 ) of the JSI is set by the bandwidth of the pump laser and ω1 + ω2 is equal to thepump laser frequency. The antidiagonal width ∆(t1 − t2 ) of the JTI is inversely proportional to the
bandwidth of the entangled photons [5]. This width is sometimes referred to as the entanglement
time, Te . Te can increase as the beam propagates due to the group velocity dispersion of the optics
the beam propagates through.

We can easily find the joint transverse momentum intensity, which is |Φ(k1 , k2 )|2 . If we write
the joint transverse momentum amplitude in terms of frequencies by making the substitutionsof ki → ωi ni /c, we end up with the joint spectral amplitude, Φ(ω1 , ω2 ), and can likewise solve
for the joint spectral intensity (JSI). A schematic of a JSI for SPDC is shown in Fig. 1.2(a).
We can write the joint transverse-momentum or spectral amplitude in space or time coordinates,
respectively, using a Fourier transform. In the space and time domain we will refer to these as thejoint transverse-position or temporal amplitudes. A schematic of a joint temporal intensity (JTI)9is shown in Fig. 1.2(b).
A number of useful quantities can be extracted from these various intensities. From the joint
transverse-position intensity, we can find the entanglement area Ae . The width of the 2-dimensional√
distribution of an idler photon’s position conditioned on the signal photon’s position is ∝ Ae . It
can be measured or calculated for a given transverse plane of the SPDC’s propagation, and is
known to be affected by optical elements. A lens, for example, performs a Fourier transform on
an imaged object [3, 4]. If the lens is placed one focal length in front of the crystal, the Fouriertransform of the SPDC beam at the crystal will lie at one focal length past the lens. This changes
the position correlations in the image plane to position anticorrelations in the Fourier plane. An
accurate comparison of measurement to calculation requires careful consideration of the optical
path. We can even estimate a “birth zone” [3] of the SPDC by considering the crystal plane. From
the JTI, we can determine the entanglement time, Te . This quantity is the width of the distributionof idler arrival times conditioned on signal arrival time. The values of Te and Ae are set by the
temporal and spatial second-order coherence functions, g (2) [6, 7].

1.5

Second-Order Coherence Function
An important quantity to characterize the photon statistics of a light source is the second-order coherence function. In this section we follow the formalism of Refs. [8, 9]. First consider the
first-order correlation function,
D
E
(1)
+
Gj (r1 , t1 ) = Ê−
(r
,
t
)
Ê
(r
,
t
)
,
1
1
1
1
j
j

(1.22)

where j indicates the mode of light that is measured at position r1 and time t1 . This function is
proportional to the counting rate of a photodetector. Next, consider a mode of light j measured
at position r1 and time t1 and another mode k measured at r2 and t2 , the rate at which thephotodetectors detect a coincidence is proportional to the second-order correlation function,
D
E
(2)
−
+
+
Gj,k (r1 , t1 ; r2 , t2 ) = Ê−
(r
,
t
)
Ê
(r
,
t
)
Ê
(r
,
t
)
Ê
(r
,
t
)
.
1 1
2 2
2 2
1 1
j
j
k
k

(1.23)

10
It’s useful to use a normalized version of these correlation functions, which at second order is written(2)(2)
gj,k (r1 , t1 ; r2 , t2 ) =Gj,k (r1 , t1 ; r2 , t2 )
(1)

(1)

Gj (r1 , t1 )Gk (r2 , t2 )
D
E
−
+
+
Ê−
j (r1 , t1 )Êk (r2 , t2 )Êk (r2 , t2 )Êj (r1 , t1 )
ED
E.
=D
+
−
+
Ê−
(r
,
t
)
Ê
(r
,
t
)
Ê
(r
,
t
)
Ê
(r
,
t
)
1 1
1 1
2 2
2 2
j
j
k
k

(1.24)

This function is referred to as the second-order coherence function in recognition of its ability todetermine if a photon source is coherent (|g (n) | = 1). Common factors can be canceled, and we canwrite this in terms of only creation and annihilation operators
DEâ†j (r1 , t1 )â†k (r2 , t2 )âk (r2 , t2 )âj (r1 , t1 )
(2)
E.
ED
gj,k (r1 , t1 ; r2 , t2 ) = Dâ†j (r1 , t1 )âj (r1 , t1 ) â†k (r2 , t2 )âk (r2 , t2 )(1.25)

Then, if the measurement is on a single mode (j = k) measured at the same place (r1 = r2 ) and if
the source is stationary (does not depend on exact values of t1 and t2 ), we can reduce this to theformg (2) (τ ) =â† (t)â† (t + τ )â(t + τ )â(t)⟨â† (t)â(t)⟩2,(1.26)

where τ = t2 − t1 . For the special case of τ = 0,
g (2) (0) =⟨n̂(t)(n̂(t) − 1)⟩.
⟨n̂(t)⟩2(1.27)

Depending on the optical setup, including the type of laser and detectors, there are different ways
in which we can measure and calculate g (2) (0) as described in Ref. [9]. In the case relevant for thisthesis of a pulsed source with photons detected using a Hanbury Brown-Twiss (HBT) interferometer(2)with “click” detectors as schematically shown in Fig. 1.3(a), we can measure γclick ,(2)

γclick [m] =Nc [m],
R1 R2 Trep Tint

(1.28)

where R1 and R2 are the singles count rates on detectors 1 and 2 and Nc [m] is the number ofcorrelation events recorded by timing electronics in the histogram bin centered at an integer mnumber of pulses with width set by the repetition period Trep for an experiment integration time
Tint . The quantity Nc [m] can be called the number of coincidences, and refers to the number of11

Figure 1.3: Schematic of (a) a Hanbury Brown-Twiss interferometer. Light is incident on abeamsplitter and detected at the two output ports on detectors 1 and 2. If detector 1 and detector
2 register photons within a short time window (coincidence) the difference in time (start-stop) isrecorded. This time is τ for the second order coherence function (g (2) (τ )) measurements. (b) The
second order coherence function for a coherent source, a thermal source and a single driven atomic
emitter [10].

times a click on detector 1 is followed by a click on detector 2 within the repetition period. The(2)

quantity γclick [0] equals g (2) [0] when the photon detection efficiency or the multi-photon generation
probability is very low [9].
In Fig. 1.3(b), the g (2) (τ ) values for sources demonstrating a range of photon statistics isshown. A value of g (2) (0) = 1 is consistent with a coherent beam of light from a laser, and demonstrates Poissonian photon statistics. For g (2) (0) < 1 the light shows sub-Poissonian photon statistics
and is consistent with a single-photon source. For g (2) (0) > 1 the light shows super-Poissonian photon statistics and is consistent with thermal light. Classical light fields are constrained to g (2) ≥ 1.
Signal or idler photons follow a thermal distribution with g (2) (0) = 2. SPDC has a g (2) (0) valuethat depends on the mean photon number. A degenerate, single-mode (in all degrees of freedom:polarization, spatial mode, spectral and temporal profiles) SPDC source can be modeled as a singlemode squeezed vacuum (SMSV), for which g (2) = 3 + 1/µ [11] where µ = â† â is the mean photon
number.

12

1.6

Two-Photon Absorption
Next we turn to a χ(3) nonlinear process: 2PA. In this process two photons are absorbed

by an absorber, such as an atom or molecule, and a transition is made from a ground to excitedstate as shown in Fig. 1.1(c). Unlike the parametric process’ SHG and SPDC, this process is
non-parametric. In a parametric process, the initial and final quantum states of the system are
identical. The only time that the system is not in this state is when it occupies a virtual state.
In a non-parametric process, the final state is a different and real quantum state of the system.
The part of the susceptibility involved in describing a parametric process is the real part and for
non-parametric is the imaginary part.
We can derive the cross section for 2PA in multiple ways, here we will opt to derive it using
second-order time-dependent perturbation theory for state amplitudes, as was done first by MariaGöppert-Mayer in 1931 [12]. We note that as pointed out in Ref. [5], this method considers only
the double quantum coherence (DQC) pathway’s contribution to 2PA, but the non-rephasing (NR)and rephasing (R) pathways also exist and may contribute especially if there are real intermediate
states close to resonance with the exciting field. These pathways can be found using a fourth-order
perturbative treatment of density matrices. Another method to derive the 2PA cross section is interms of χ(3) using energy considerations, see for example Ref. [13].
We will follow the derivation in Ref. [1] closely that considers an atom interacting with anelectric field. The Hamiltonian for this system can be written as
Ĥ = Ĥ0 + V̂ (t)ˆ= Ĥ0 + dE(t),(1.29)

where Ĥ0 is the free Hamiltonian for the atom, dˆ = −er̂ is the electric dipole moment operator, eis the charge of an electron, r̂ is the position operator andE(t) = Eeiωt + c.c.

(1.30)

is a monochromatic electric field. We assume that the atomic wave function, ψ(r, t) obeys the

13
Schrödinger equation,iℏ∂ψ(r, t) = Ĥψ(r, t).
∂t(1.31)The solution for the wave function of this system can be expressed asXψ(r, t) =al (t)ψl (r, t)

l

=

X

al (t)ul (r)e−iωl t ,(1.32)lwhere ψl (r, t) are the eigenstates of the free Hamiltonian and al (t) is the time-dependent coefficient.
Inputting Eq. (1.32) into Eq. (1.31), multiplying by u∗m (r) and integrating over all space we findiℏX∂am (t) =al (t)Vml e−iωlm t ,∂t(1.33)lwhere ωlm = ωl − ωm andZVml =d3 ru∗m (r)V̂ ul (r).

(1.34)

Since Eq. (1.33) cannot be solved exactly, we instead perturbatively solve the equation by replacing
V with λV , where λ is the expansion parameter (which varies from 0 to 1 depending on the strengthof the perturbation), and expanding am (t) in powers of the interaction,
(1)
2 (2)
3 (3)
am (t) = a(0)m + λam + λ am + λ am + ...

(1.35)

Then we arrive at the generalized equation for any N ,iℏ∂ (N ) X (N −1)a=alVml e−iωlm t .
∂t m(1.36)lFirst, we consider the first-order process which corresponds to one-photon absorption (1PA). Before(0)

the laser is turned on we assume that the atom is in the ground (g) state, corresponding to ag = 1(0)

and al= 0 for all l ̸= g. Next we determine thatVmg = −dmg Ee−iωmg t + c.c.

(1.37)

The first term will correspond to 1PA, whereas the second to stimulated emission. We drop thesecond term as we are only interested in the absorption process. Then after integration over t we

14
arrive ata(1)m (t) =dmg E
(ei(ωmg −ω)t − 1).
ℏ(ωmg − ω)

(1.38)

We can describe the probability that the atom is in state m by
(1)
2
p(1)
m (t) = |am (t)| =|µmg E|2|dmg E|2 4sin2 ((ωmg − ω)t/2)=
f (t),ℏ2(ωmg − ω)2ℏ2(1.39)

where the function f (t) can be approximated by a Dirac delta function for long interaction times t,
f (t) = 2πtδ(ωmg − ω).

(1.40)

With this delta function the probability amplitude describes an infinitely narrow transition. To
approach a more realistic description, we can give the state m some finite width in frequency space.
We introduce the density of final states, ρf (ωmg ) where ρf (ωmg )dωmg is the probability that thetransition frequency lies between ωmg and ωmg + dωmg . Then we have
|dmg E|p(1)m (t) =
22Zdωmg δ(ωmg − ω)ρf (ωmg )

2πt

ℏ

=

|dmg E|22πtρf (ωmg = ω).
ℏ2(1.41)

Next we derive the one photon transition rate (Fermi’s golden rule)(1)
Rmg= p(1)
m (t)/t=|dmg E|22πρf (ωmg = ω).
ℏ2(1.42)

We can now solve for the purely atomic quantity, the 1PA cross section,
(1)
(1)
σmg
= Rmg
/ϕ =

|dmg |2 ωπρf (ωmg = ω),nϵ0 cℏ(1.43)

220cwhere ϕ = 2nϵℏω |E| is the photon flux. The 1PA cross section is quoted in units of cm , withtypical cross sections on the order of 10−16 cm2 . To derive the 2PA cross section, we follow theexact same procedure, but at second order in the interaction. One can show that the probabilityof being in state n is
(2)
2
p(2)
n (t) = |an (t)| =X dnm dmg E 2ℏ2 (ωmg − ω)m2

2πtρf (ωng = 2ω).

(1.44)

15
Then the two-photon transition rate can be written
(2)
(2)
2
Rng
= p(2)n (t)/t = |an (t)| =2

X dnm dmg E 2mℏ2 (ωmg − ω)

2πρf (ωng = 2ω).

(1.45)In later sections we will refer to this as R. Next, we define the 2PA cross section
(2)
(2)
σng
= 2Rng/ϕ2 =dnm dmg ωℏnϵ0 c(ωmg − ω)mX2πρf (ωng = 2ω).

(1.46)In later sections, we will refer to this quantity as σC . It’s useful to note that Eq. (1.46) is written
for degenerate 2PA, however the cross section can be derived for nondegenerate 2PA by adding asecond field to Eqn. (1.30). The factor of 2 in Eq. (1.46) differs from that shown in Ref. [1], but
agrees with the formalism of Refs. [14, 15]. It holds the units of photons per molecule and makes
the units for the cross section cm4 s photon−1 molecule−1 . This convention is used in this thesis
so that the cross section refers to the removal of two photons from the field and one molecule fromthe ground state. Typically σC are quoted in terms of GM (Goeppert-Mayer) units, where 1 GM
= 10−50 cm4 s photon−1 molecule−1 .
Typical values of σC for molecules are on the order of 1 to 100 GM [16, 17]. These values
require that a high flux laser (> 1025 photons cm−2 s−1 ) be used to observe an absorption signal.
Typically pulsed lasers meet these requirements by producing bursts of photons in a short time.
However, the shorter the pulse in time, the wider the bandwidth of the photons and thus the moredifficult it is for the sum of the energies of the photons to equal the transition energy. Amongthe many photons incident on the absorbing sample, only a small fraction (typically < 10−15 ) areabsorbed. Those that are absorbed arrived within the required short time window, which is set by
the virtual state lifetime (femtoseconds [18]), and arrived close enough to the absorber to interact
with it. Clearly, this excitation process is inefficient.

Chapter 2

Entangled Two-Photon Absorption

2.1

Publication Note
Parts of this chapter are adapted from:[19] Parzuchowski, K.M., Mikhaylov, A., Mazurek, M.D., Wilson, R.N., Lum, D.J., Gerrits, T.,Camp Jr, C.H., Stevens, M.J. and Jimenez, R., 2021. Setting bounds on entangled two-photon
absorption cross sections in common fluorophores. Physical Review Applied, 15 (4), p.044012.

2.2

Introduction
In the previous chapter we discussed spontaneous parametric down-conversion (SPDC) andtwo-photon absorption (2PA), and pointed out the inefficiency of the 2PA process. In this chapter
we discuss entangled 2PA (E2PA). This process refers to the excitation of a two-photon transition
using a light source of entangled photons produced via SPDC.
Despite the inefficiency of 2PA, the process is widely used in two-photon excitation microscopy(2PEM) for cellular imaging deep within biological tissue [20, 21]. Excitation of a two-photontransition occurs much more efficiently at the focus than outside of the focal volume, which leads
to a huge boost in resolution relative to a one-photon based technique. Fluorescence from therelaxation of the absorber is collected to reconstruct an image. 2PEM can only image about 1 mmdeep into living brain tissue. Imaging to greater depths would require a higher laser intensity that
perturbs the biological function of the tissue. Alternatively, a different light source that is more
efficient at excitation of 2PA could be used.

17
Numerous theoretical and experimental studies have investigated the possibility of enhancing
the efficiency of 2PA by exciting with nonclassical light [22, 11]. Photon pairs that are entangled in
the energy-time and position-momentum degrees of freedom can exhibit the strong temporal andspatial correlations needed for 2PA. Theoretical studies on simple model systems [23, 24, 6] havepredicted that using entangled photon pairs can lead to a significant “quantum advantage” in 2PA
rates. Here we define quantum advantage as the ratio of the minimum photon flux necessary toobserve classical 2PA (C2PA) to that for E2PA.
We will first look into the derivation of the E2PA cross section (σE ) as was first done by Fei
et al. [6] and discuss why a simpler definition is more useful for the experimental conditions usedin this thesis. We then present the simple probabilistic model for the cross section similar to thatshown by Fei et al. [6], and connect C2PA to E2PA. For a more comprehensive review of the theoryof E2PA we suggest the work of Drago and Sipe [25] and Raymer and coworkers [5, 26, 27] .
Afterwards, we will discuss the range of E2PA experiments in literature, emphasizing thatthe size and measurability of the cross sections is under debate. We also present the challenges ofreporting E2PA cross sections. Finally we will discuss the signatures of E2PA that serve as tools
for designing and conducting an E2PA measurement.

2.3

Quantum Mechanical E2PA Cross Section
To derive σE , we can again (Sec. 1.6) use second-order time-dependent perturbation theory.

We emphasize again that as mentioned in Raymer et al. [5], this derivation considers only the
double-quantum coherence pathway and leaves out other pathways that may contribute to 2PA
especially when the excitation field is near resonance with real intermediate states. Similar to
the theoretical derivations of SPDC, we must adopt a quantum description. The probability of
two-photon absorption can be written [28]

18

(2)

pf =

1
ℏ2

Z t2

Z t

1
= 4
ℏ

dt2
t0

t0

Z t

′

2

Z t′

2

′

t0

Z t2

Z t
dt2

dt1

dt2
t0

(+)

(+)

ˆ 2 )d(tˆ 1 )|i⟩⟨Ψf |Ê (t2 )Ê (t1 )|Ψi ⟩dt1 ⟨f |d(t
2
1
′

′

ˆ 2 )d(tˆ 1 )|i⟩dt1 ⟨i|dˆ† (t2 )dˆ† (t1 )d(t

t0
(−) ′
(−) ′
(+)
(+)
×⟨Ψf |Ê1 (t2 )Ê2 (t1 )Ê2 (t2 )Ê1 (t1 )|Ψf ⟩,
t0

(2.1)

where |i(f )⟩ denotes the initial (final) state wave function of the electron, |Ψi(f ) ⟩ denotes the initial(final) state wave function of the optical field. Here we have considered the case where the field
(+)

(+)

described by Ê1 (t1 ) interacts first and Ê2 (t2 ) interacts second [28]. We can take note thatthe probability (Eq. 2.1) consists of a material correlation function, M , and a second-order fieldcorrelation function, G(2) ,
Z t
Z t′
Z t
Z t2
2
1
′
′
′
′
′
′
(2)
pf = 4dt2
dt1
dt2
dt1 M (t2 , t1 , t2 , t1 )G(2) (t2 , t1 , t2 , t1 ).
ℏ t0
t0
t0
t0

(2.2)

By replacing the field correlation function with the product of classical field amplitudes, we canrecover Eq. (1.44). Evaluation of this equation for SPDC excitation shows that the field andmaterial parameters are inseparable, leading to a definition for σE that depends on both field
and absorber properties. It’s important to note here that the time ordering of the fields and dipole
moment operators results in E1 and E2 in Eq. (2.2) being associated to the first and second arriving
photons, respectively, of a frequency anticorrelated pair [29, 30]. The relevant G(2) function is thus
subtly different than the standard form in Eq. (1.23). The form of G(2) in Eq. 2.2 reduces to thestandard G(2) (0) when the linewidth of the absorber’s final state is significantly broader than the
bandwidth of the exciting field [29, 30, 26].
The probability of E2PA is known to depend linearly, rather than quadratically, on photon
flux [6, 24, 23] at low photon flux. The rate of E2PA contains both a linear component and theclassical quadratic component that dominates at high photon flux
1
RE = (σE ϕ + σC ϕ2 ).
2

(2.3)

Using Eq. (1.19), (2.3) and (2.2) one can derive the σE . Here we’ve taken the derivation from
Ref. [6] in the special case of a monochromatic pump and converted from natural to cgs units, used19
our convention for the factor of 2 in Eq. (2.3) and added in a density of final states, ρf ,πω1 ω2ρf (ωf i = ω1 + ω2 )2Ae Te (ℏnϵ0 c)2
)2
(
(j)
(j)
X
(j) 1 − exp[−iTe ∆2 − Te κj /2](j) 1 − exp[−iTe ∆1 − Te κj /2]
×
+ D12
,
D21
(j)
(j)
∆
−
iκ
/2
∆
−
iκ
/2
j
j
j
1
2
σE =

(2.4)

(j)

where Dkl = ⟨f |dk |j⟩⟨j|dl |i⟩ are the transition dipole matrix elements, k and l refer to photons(j)

1 and 2 or vice versa, ∆1(2) = ωj − ωi − ω1(2) is an energy mismatch parameter and κj is theintermediate state linewidth.
Large oscillations in σE as a function of Te (“entanglement induced two-photon transparencies” [6]) have been theoretically predicted for a few atoms and molecular fluorophores [31, 32, 33],
but have not been measured experimentally. For large values of Te κj the interference will wash
out and σE approaches the probabilistic model we present below (Eq. (2.11)) [6]. Furthermore,as the degrees of freedom of the absorber increases (i.e. for large molecules, absorbers that interact with eachother, absorbers that interact with the surrounding medium and vibronic coupling)
the amount of coherence between the photons and the absorber decreases, and σE is expected
to approach the classical limit [31]. The absorbers studied in this thesis are large molecules orquantum dots dissolved in room-temperature solvents, and thus likely can be approximated by theprobabilistic model.

2.4

Probabilistic E2PA Cross Section
Here we present a connection between a simple probabilistic theory describing E2PA andthe well-accepted description of C2PA. We consider a single-mode field with mean photon numberµ = â† â , the 2PA rate can be written [11, 34]
D 2 E
R = κ2 â† â2 = κ2 µ2 g (2) ,(2.5)

where κ2 (s−1 ) is a collection of constants quantifying the strength of the nonlinear interaction. It
has been demonstrated, for example, that thermal light (g (2) = 2) doubles the 2PA rate compared
to laser excitation (g (2) = 1) of the same intensity [35].

20
In the classical limit, the instantaneous 2PA rate for a single fluorophore can be written [15]1R = σC ϕ2 ,
2

(2.6)

where ϕ is the photon flux, with units of cm−2 s−1 .

The C2PA cross-section σC has unitsof GM, where 1 GM = 10−50 cm4 s. (In this section we omit photons, excitations and fluorophores/absorbers from the units of various quantities for brevity; in later chapters we include
them for clarity.)
For a pulsed laser (g (2) = 1) source with temporal and spatial mode set by the pulse duration
T (fs) and the beam area A (cm2 ), if we rewrite Eq. (2.5) in terms of the photon flux, ϕ = µ/(T A),
and substitute
κ2 =

σC
,
2T 2 A2(2.7)

we arrive at the classical limit in Eq. (2.6).
In contrast to laser light, SPDC produces photon pairs exhibiting correlations in energy,
time and space that can be tailored to enhance the rate and selectivity of 2PA [36, 37]. The energy
correlations between the signal and idler photons within a pair are set by conservation of energy
in the conversion of one pump photon to two down-converted photons and can be engineered tomatch the energy of a two-photon transition. Photon pair production is localized in space andtime [3, 38], allowing for excitation with photons that nearly simultaneously arrive in a localized
region of space.
A degenerate, single-mode (in all degrees of freedom: polarization, spatial mode, spectral
and temporal profiles) SPDC source can be modeled as a single-mode squeezed vacuum (SMSV),
for which g (2) = 3 + 1/µ [11]. Substituting this expression into Eq. (2.5) yields

R = κ2 µ + 3µ2 .

(2.8)

For a pulsed source, substituting ϕ and Eq. (2.7) into Eq. (2.8) gives
1
R = σC2ϕ
2+ 3ϕ .
TA

(2.9)

21

Figure 2.1: (a) Schematic of entanglement area, Ae , and beam area, A. Photons closely correlated
in space may have Ae smaller than A. (b) Schematic of entanglement time, Te , and the duration ofeither the pump pulse or the overall SPDC pulse, T . Photons closely correlated in time may have
Te smaller than T .

An alternative way to write Eq. (2.8) is [6]
RE =


1
σE ϕ + 3σC ϕ2 ,2(2.10)

where σE has units of cm2 and RE is the instantaneous E2PA rate. For a single-mode field, thetwo cross sections are related by σE = σC /(T A). At low photon flux, the first term dominates
and the E2PA process should scale linearly with ϕ [23, 24]. We note that it is more precise to saythat the E2PA rate is linear in photon number, as Eq. (2.8) shows. If pulse duration or spot size
were modified, the linear term should scale in the same way as the quadratic term, at least in the
single-mode case considered here. At high photon flux, where many photon pairs overlap in time,
the quadratic term dominates.
In a real experiment, the SPDC light typically occupies multiple modes and Eq. (2.8) does
not hold. In this case, the coefficient for the linear term could in principle be larger than thecoefficient for the quadratic term. Roughly speaking, if the two photons in a pair are more closely
correlated in time than the pump pulse duration, the interaction strength could be enhanced by
the factor T /Te , where Te is the entanglement time. Analogously, if the photons in a pair are moreclosely correlated in space than the beam size, the interaction strength could be modified by thefactor A/Ae , where Ae is the entanglement area. A schematic of these quantities is illustrated in22Fig. 2.1. Following this simple, probabilistic argument leads to the approximation
σE ≈

σC
.
Te Ae

(2.11)

Other than a factor of two difference due to our different definition of σE , this is the same approximation arrived at in Ref. [6], but following a different argument and making different assumptions.
To maximize the E2PA rate, Te and Ae should be as small as possible. For a large µ and a large
number of modes, g (2) →− 1 and the E2PA rate approaches the classical limit in Eq. (2.6).
Unlike σC , which depends only on wavelength for a particular molecular 2PA transition, thevalue of σE depends strongly on the properties of the excitation source and experiment. The valuesof Ae and Te evolve as the SPDC beam propagates through optics from the down-conversion crystal
to the sample [39, 40, 41, 42, 7, 43], and therefore depend on the details of the optical system used
to measure E2PA. Calculating σE for a given experimental geometry thus requires knowledge of
Ae and Te within the excitation volume. Clearly, these factors complicate the ability to compare
results from different experiments.

2.5

Experimental Literature Summary
There are broadly two experimental designs used for measuring E2PA as shown schematicallyin Fig. 2.2. The first is a transmittance-based experimental scheme, where a beam of entangled
photon pairs is sent through a sample and the transmittance (the fraction of photon pairs thatare transmitted through the sample) is measured. The second is a fluorescence-based experimental
scheme, where a beam of entangled photon pairs is sent through a sample and the fluorescence thatis emitted from the absorber is collected. Each of these techniques has their own unique advantages.
For transmittance measurements, one can use a sample that has a low quantum yield and one canuse a low photon pair flux. The latter of which is because the measured signal is a fractional changeof transmission and increasing the incident flux does not increase the value in the low flux regime.
Furthermore, the detectors must count single photons, thus low photon rates can be measured. For
fluorescence measurements, the desired signal is background free and the alignment of these setups

23

Figure 2.2: (a) Schematic of a transmittance-based measurement scheme. Photon pairs are generated by SPDC in a χ(2) nonlinear crystal and sent through a cuvette containing a solvent and 2PA
sample, then through a cuvette containing only the solvent. The ratio of photon counts detected forthe two measurements gives the transmittance. (b) Schematic of a fluorescence-based measurement
scheme. Photon pairs are generated by SPDC in a χ(2) nonlinear crystal and sent through a cuvettecontaining a solvent and 2PA sample. Fluorescence photons are counted on a detector angled at
90◦ relative to the beam propagation direction.

can be less sensitive. The latter of which is because the measured signal is not typically collected
onto a small detector, such as single-mode fiber-coupled detectors, far from the source, which ismore likely in a transmittance measurement.
The SPDC sources used in E2PA experiments can vary. The pump laser, for example, can becontinuous wave (CW) or pulsed. For a CW-pumped SPDC excitation source, the linear photonflux-dependent term in Eq. (2.10) will remain dominant over the quadratic term at higher average
fluxes than for that of a pulsed-pumped SPDC excitation source. This is because of the lower
likelihood of uncorrelated pairs arriving at a fluorophore at the same time. SPDC sources varyin phase matching types (Section 1.3) and thus the photons in a pair may have perpendicular orparallel polarizations. Furthermore, an experimental design may rotate the polarization of one
photon relative to its partner photon. Studies of C2PA [44, 45, 46] suggest that there is a small
difference in efficiency (by a factor of two or three) of excitation when the two photons haveperpendicular polarizations instead of parallel.
A summary of important parameters in selected E2PA reports is given in Table 2.1. Thesestudies represent experimental efforts from all three of the independent groups that have claimed to24
Table 2.1: Results and experimental parameters from selected E2PA studies. Cross sections (σCand σE ) are quoted at the corresponding excitation wavelength (λ). Cross sections and entanglement times (Te ) are found as described in the main text. We estimate the entanglement area (Aeste )required to explain the σE values based on Eq. (2.11).
Sample [Ref.]

λ
(nm)

σC
(GM)

σE
(10−19 cm2 fluorophore−1 )

Te
(fs)

Aest
e
−9
(10 µm2 )9R-S [47]
Rh6G [48]
RhB [49]
Tetraannulene [50]

800
1064
808
800

27.9
9.9 ± 1.5 [16]260 ± 40 [16]2960

2.02 − 2.69
0.0099 − 0.019
0.17 − 42
990

100
140
17
96

1.0 − 1.4
38 − 72
3.6 − 900
0.31

measure a signal from E2PA. Table 2.1 shows σC and σE values determined at several near infrared
wavelengths. In Ref. [47], pulsed-pumped type-II SPDC was generated to excite a sample with
1 − 25 × 106 photons s−1 (the beam waist is not specified) in transmittance- and fluorescence-based
E2PA schemes. For the studied 9R-S molecule, the σE values found using these two techniques
differ slightly from one another. The measurement uncertainty was estimated to be 9% and 12% fortransmittance- and fluorescence-based techniques. In Ref. [48], a continuous-wave (CW)-pumped
type-0 SPDC source was used for entangled two-photon excited fluorescence (E2PEF) measurements with an effective incident photon rate of 2 − 50 × 107 photons s−1 (beam waist of 60 µm).
A 100 times increase of the molar concentration led to a decrease in the measured σE value for
Rhodamine 6G (Rh6G) by a factor of two. The uncertainties on the measured σE were estimated tobe nearly 50%. A similar concentration dependence was observed in Ref. [49] using pulsed-pumped
type-II SPDC excitation with an incident pair rate of 50 − 7, 000 photon pairs s−1 (beam waistof 61 µm) in a transmittance-based scheme, where the concentration dependence of σE for Rhodamine B (RhB) was attributed to potential aggregation effects in the solutions. The uncertainties
on the published σE are ≈ 10%. In Ref. [50], a pulsed-pumped type-II SPDC source was used
to excite the tetraannulene sample with 1 − 25 × 106 photons s−1 in a transmittance-based E2PA
scheme. The measurement uncertainty was not estimated in this report. The values for σE of
990 × 10−19 cm2 fluorophore−1 for tetraannulene [50] and 0.0099 × 10−19 cm2 fluorophore−1 for
Rh6G [48] are the largest and smallest values, respectively, that have been reported. In all these25
reports the photon flux is not specified, except for an order-of-magnitude estimate in Ref. [49], andthe photon rate is not precisely defined.
No direct Te measurements were completed in the aforementioned reports. In the case ofRef. [48] we estimate Te based on the details provided by the authors, who estimated an effective
flux reduced to the fraction of photon pairs that have Te = 140 fs. In Refs. [47, 49, 50], the valueof Te was estimated at the output of the crystal, which in some cases can be orders of magnitude
smaller than the value at the sample’s position. This is especially true when the total group delay
dispersion (GDD) of the optics is large or when the bandwidth of the SPDC is large (Chapter 3
Section 3.7). The value of Te would be very sensitive to even small amounts of GDD in the lattercase.
The values of Ae are not specified in any of these reports. We estimate the entanglement
areas, Aeste , required to explain the results of these previous reports, based on the probabilistic−6 − 10−10 µm2 . This would require bothmodel, Eq. (2.11). The values of Aeste range from 10

photons within a pair to be confined to a region that is 10−6 − 10−10 times the diffraction limited
spot size. We have no evidence that this level of confinement is feasible [41, 42, 40, 7]. Thus, thesimple, probabilistic theory [6] used to derive Eq. (2.11) cannot explain these experimental results.
In addition to the inconsistencies of the results in Table 2.5 with theory, these reports have
only measured one signature of E2PA. As we will explain in the next Section, the signature thatthey measure is not unique to E2PA and thus the origin of the signal is unclear. Another campof researchers exist, including our researchers, that have not been able to resolve a signal due toE2PA, but have designed a sensitive experiment capable of measuring E2PA cross section upper
UB ).
bounds (σE

To the best of our knowledge, a complete summary of experimental results of E2PA is given
in Table A.1 and shown in Fig. 2.3. This plot shows σE (marked by various blue × symbols)
UB (marked by various orange ▼ symbols) values as a function of σ . From lightest toand σEC

darkest, the shades of blue represent the results from University of Geneva [48], University ofMichigan [47, 50, 51, 52, 53, 54, 55, 56] and Universidad de los Andes [49]. From lightest to darkest

26

Figure 2.3: E2PA cross sections (marked by various blue × symbols) and cross section upper bounds(marked by various orange ▼ symbols) reported throughout literature. The values are plotted asa function of σC . Each shade of blue or orange indicates results from a different organization as
mentioned in the main text. The data used for this plot can be found in Table A.1.

27
the shades of orange represent the results from Universidad Nacional Autónoma de México [57],California Institute of Technology [58], JILA/National Institute of Standards and Technology [59,
UB have values
19, 60, 61](the work presented in this thesis) and University of Oregon [62]. The σE

lower than the reported σE , signifying the stark disagreement between these two groups. The sizeof σE has yet to be narrowed down which is evident by the vertical axis extending 12 orders of
UB are recent, being published in the last 2 years, whereas the reports
magnitude. The reports of σE

of σE have spanned 16 years and are primarily being published by one group.
UB used for Refs. [61, 57, 58, 62] were not provided in those publications, but wereThe σEestimated in a similar manner to those used in our publications [59, 19] as shown in Appendix A.
We note that some groups chose not to discuss their results in terms of E2PA cross sections in part
because of the large variability of the value from one experiment to the next (Sec. 2.4). We use it
in this thesis to make orders-of-magnitude comparisons between the results from various groups.

2.6

Signatures of E2PA
Although numerous experiments in literature report that E2PA signals are large and mea-

surable, it is unclear what the origin of the measured signal is. In order to claim that a signal has
a particular origin, a unique signature (or preferably multiple signatures) of the process must be
identified. Here we discuss a number of signatures of E2PA, however this list is not exhaustive.
The most commonly known signature of E2PA is its linear dependence on excitation photon
flux at low photon flux (Eq. (2.10)). This signature is not unique to E2PA and is shared withmany other processes such as scattering, one photon absorption, or fluorescence from the coatingof an optic in the beam path. Nevertheless, this signature is typically the only signature of E2PA
measured by the groups who report σE .
Another signature of E2PA is the signal’s dependence on the time delay between photons in apair; as the time delay is scanned away from optimal overlap (zero time delay), the signal is expected
to decrease towards zero in accordance with the simultaneity requirement of 2PA. In a recent report,
Tabakaev et al. [48] observe such behavior in an E2PEF experiment. An interferometer was used28
before the sample to probabilistically and equally split photons and time delay half of the photon
pairs, while the other half traversed the same path. The resulting E2PEF as a function of the timedelay should consist of a constant signal due to the photon pairs that traveled the same path and avariable signal due to the photon pairs that traveled different paths. The signal at long time delays
should be half the signal at zero delay. Instead, Tabakaev et al. observe a signal that tends to zeroat large time delays. This unexpected result is pointed out by the authors of the study, but it has
yet to be explained. Some of the other reports [51, 63, 64] have included time-delay scans, but withdata reported at only a few delays.
Another signature of E2PA is the signal’s dependence on the presence of both photons in apair. If only one photon from each pair makes it to the two-photon absorber, the signal should
disappear. This test could be performed in a straightforward manner by adding a longpass or
shortpass filter centered at the pump wavelength (one must ensure that the alignment of the systemdoes not change). This was done in our publication [61] and was used as confirmation that theorigin of the signal was not E2PA. For type II SPDC, this can also be done by adding a polarizer tothe beam. Another way to check for this signature is to probabilistically and equally split photons
into two paths of an interferometer and block one arm of the interferometer, the signal should
reduce to a quarter of its original size.
One variation of this previous signature is to attenuate the SPDC beam and observe aquadratic decrease of the signal. This differs from attenuation of the pump beam, which would
cause the signal to scale linearly. This difference can be understood mathematically.
In the case of linear loss between the SPDC generation crystal and the sample, the 2PA rate
(Eq. (2.5)) is modified [9]D 2 E
R = κ2 T 2 â† â2 = κ2 T 2 µ2 g (2) ,(2.12)

where the linear loss has been modeled as a lossless beamsplitter with transmittance T . For
excitation with a pulsed single-mode SPDC source,
1
R = σC T 2
2




ϕxtal
2
+ 3ϕxtal ,TA(2.13)

29
where ϕxtal is the photon flux in the SPDC crystal (in our notation this is photon flux, not photon
pair flux). Rewriting in terms of the photon flux at the sample (ϕsample = T ϕxtal ) yields
RE =


1
σE T ϕsample + 3σC ϕ2sample .
2(2.14)To extract σE , the flux at the sample should be scaled by T . If the flux is adjusted by attenuating thepump power, the first term in RE should scale linearly with ϕsample . If the flux is instead adjusted
by attenuating the down-converted light, this term should scale quadratically with ϕsample . Thissignature should be present in either transmittance- or fluorescence-based measurement schemes.
This loss-scaling signature was demonstrated in up-conversion of down-conversion by Dayan etal. [65, 66], and can be used as a method to distinguish E2PA from other linear loss processes.

2.7

Thesis Overview
The overall objective of this thesis work is to experimentally investigate E2PA in molecules

dissolved in room temperature solvents. At the beginning of our research project we sought toreplicate some aspects of the experimental design of previous works where sensational claims weremade about the quantum advantage of E2PA. We soon found that we were not able to reproduce
the results from these reports and in fact were not able to measure a signal at all. Afterwards we
had to rethink our experimental design in order to increase our sensitivity. We designed sensitive
fluorescence and transmittance experiments, but still could not measure a signal. We were the firstto report upper bounds on σE , and we have done so for eight independent absorbers. Lastly, we
designed a new platform for more sensitive 2PA measurements.
In Chapter 3 we describe the SPDC source used for these various experiments. We detail thecapabilities of our systems and the methods used for characterizing this source.
In Chapter 4 we detail our first transmittance experiment. This setup includes an interferometer to test for time delay signatures of E2PA. We demonstrate how one could misinterpret
one-photon losses as a signature of E2PA, and go on to account for these losses. We set an upperbound on the E2PA cross section of the studied sample, which shows disagreement with results in30
literature. The sensitivity of this experiment was limited by residual interference.
In Chapter 5 we discuss our fluorescence experiment. In this experiment we use our maximum
SPDC flux and achieve our highest cross section sensitivity. We study both C2PA and E2PA in
the same setup under nearly identical excitation conditions. Our C2PA measurements for the six
studied samples are used to derive C2PA cross sections which all agree within a factor of two tothe values published in literature. For E2PA we cannot measure a signal above our noise floor, andset upper bounds on the cross sections.
In Chapter 6 we cover our second transmittance experiment. Here we take lessons from ourprevious setup and design a higher sensitivity experiment. We redesign the interferometer andimprove our data acquistion and analysis techniques to allow for long but stable measurements.
We achieve a sensitivity to 0.05% change in transmittance, which is a 20-fold improvement fromthe last experiment. However, we do not measure a signal for any of seven samples measured atvarious concentrations. We set upper bounds on the E2PA cross sections for these molecules.
In Chapter 7 we describe our toluene-filled hollow-core-fiber platform for fluorescence measurements. Here we confine the sample and light into a 37 cm-long, 5 µm diameter tubing. We
measure C2PA down to ≈ 20 nW excitation power. This is a four-fold improvement from the previous fluorescence experiment. We expect that we can use higher concentration samples to measure
C2PA at lower powers.
In Chapter 8 we conclude our work and discuss work that was not included in this thesis. In
another experiment, we found that the process hot-band absorption can mimic E2PA. This process
could be the source of the signals measured by other groups. We discuss future work including theuse of the fiber experiment for E2PA studies.

Chapter 3

Entangled Photon Source and Characterization

3.1

Publication Note
Parts of this chapter are adapted from:[59] Mikhaylov, A., Parzuchowski, K.M., Mazurek, M.D., Lum, D.J., Gerrits, T., Camp Jr, C.H.,
Stevens, M.J. and Jimenez, R., 2020, February. A comprehensive experimental system for measuring molecular two-photon absorption using an ultrafast entangled photon pair excitation source. In
Advanced Optical Techniques for Quantum Information, Sensing, and Metrology (Vol. 11295, pp.
48-61). SPIE.
and[19] Parzuchowski, K.M., Mikhaylov, A., Mazurek, M.D., Wilson, R.N., Lum, D.J., Gerrits, T.,Camp Jr, C.H., Stevens, M.J. and Jimenez, R., 2021. Setting bounds on entangled two-photon
absorption cross sections in common fluorophores. Physical Review Applied, 15 (4), p.044012.

3.2

Introduction
In this chapter we discuss the setup and characterization of the entangled photon source

used in the three E2PA measurements covered in this thesis. First we discuss the design of thesource. Next we discuss the multi-mode detection stage, where all spatial modes of the SPDC
are detected for the measurement and calibration of our mean photon number. Then we discuss
the single-mode detection stage, which is used to detect a single spatial mode of SPDC. Using this32
setup we can characterize the joint spectrum, the entanglement time and g (2) (0). Finally we discuss
entanglement area.

3.3

SPDC Source

Figure 3.1: Schematic of the SPDC source and detection stages. The 810 nm pump laser (red) is
frequency doubled (blue) in an SHG unit. A half-wave plate (HWP) and polarizer (pol) control
the power. The light is filtered (F) and focused into a type-0 ppKTP crystal to generate SPDC.
The pump light is filtered (F) out, and the SPDC is sent to either the multi-mode or singlemode detection stage. For multi-mode detection, all the light is focused onto a free-space-coupled
single-photon avalanche diode (SPAD). For single-mode detection, all the light is sent into an
Hanbury Brown-Twiss interferometer and launched into two single-mode fibers where it is detectedon superconducting nanowire single-photon detectors (SNSPDs).

A schematic of our photon pair source and detection stages is shown in Fig. 3.1. A pump laser
emits ≈ 110 fs pulses with a center wavelength of 810 nm (≈ 9 nm bandwidth) at a repetition rateof 8 × 107 pulses s−1 . The pulse duration and bandwidth is measured on a SwampOptics Grenouille8-50-USB. The laser is frequency doubled via second-harmonic generation (SHG) to produce 405 nm
light (≈ 3 nm bandwidth). A half-wave plate (HWP) and polarizer (pol) are used to control thepower. Any remaining 810 nm light is filtered out. The beam is focused with a 300 mm lens into a
type-0 periodically poled potassium titanyl phosphate (ppKTP) crystal to generate collinear SPDC
photon pairs at 810 nm. The crystal is temperature controlled at 30.00◦ C ± 0.01◦ C. The beam sizeat the focus in the crystal is approximately 30 µm. Filters are used to remove the remaining 405 nm
light. The part numbers for the components used in the experiments discussed in Chapters 4, 5, 6,
and 7 (including those shown in Fig. 3.1) are listed in Appendices C, E, G, and I.

33

3.4

Characterizing Mean Photon Number
In this section, we seek to characterize the mean photon number of the multimodal SPDC.

We discuss values for various parameters that were measured during the fluorescence experimentdiscussed in Chapter 5. To characterize the SPDC mean photon number at the sample (µsample ),we direct all the SPDC light to the multi-mode detection stage shown in Fig. 3.1, which consistsof a silicon single-photon avalanche diode (SPAD). We measure the count rate at low photon flux,and correct for the SPAD dead time and efficiency, the photon statistics of the down-conversion
source, and the difference in optical losses between the two paths. We perform this procedure forthree low photon fluxes (where the SPAD is not saturated), and then extrapolate to the high flux
used in the fluorescence measurements. This high flux is the maximum flux we can produce fromour source due to the damage threshold of our crystal.
For a measured count rate Qmeas on the SPAD, the measured click probability per laser pulse
ismeas
Pclick
=

Qmeas
,
g

(3.1)

where g = 8 × 107 pulses s−1 is the pulse repetition rate. Assuming a non-paralyzing dead time,the dead-time-corrected click probability can be found using [67]corr=
Pclickmeas
Pclickmeas ,
1 − Ndead Pclick(3.2)

where Ndead is the number of laser pulses the SPAD is dead for following detection of a photon. We
measure the dead time of this SPAD as ≈ 52 ns, implying that Ndead = 4. For a pump power ofmeas = 0.055. The dead-time-corrected
50 µW, we measure Qmeas = 4.4 × 106 cnt s−1 , and hence Pclickcorr = 0.071. This is the per-pulse click probability we would have expected
click probability is Pclick

to measure in the absence of dead time.
Converting this click probability to mean photon number requires knowledge of the system
detection efficiency (ηSDE ) and the photon statistics of the SPDC source. These can be related

34
through the expression [68]corr
Pclick=∞X[1 − (1 − ηSDE )n ]P (n),

(3.3)

n=1

where P (n) represents the probability that a pulse of SPDC light contains n photons at the output ofthe down-conversion crystal. At a wavelength of 810 nm, we calculate 83% cumulative transmittanceof all the optics between the SPDC crystal and the SPAD based on manufacturer specifications.
The manufacturer-specified efficiency of the SPAD is 55%. Thus the system detection efficiency isηSDE ≈ 0.46. The photon number distribution of a SMSV can be written [11]n/2

P (n) =

µxtal n!
n 2
n2 ( 2 !) (1 + µxtal )(n+1)/2(3.4)

for n even and P (n) = 0 for n odd. Here µxtal is the mean photon number generated at the downcorr = 0.071 and η
conversion crystal. At 50 µW pump power (Pclick
SDE = 0.46), we solve Eqs. (3.3) and(3.4) for µxtal . These numbers are consistent with a SMSV with mean photon number µxtal = 0.22.
Although a SMSV can be a good first approximation to an SPDC source, our source emits
into many spatial and spectral modes, as evidenced in part by the joint spectral measurements in
Fig. 3.3(a). To approximate the many modes in our SPDC light, we perform a Bernoulli samplingof M equally populated SMSVs, each with mean photon number µxtal /M , assuming ηSDE = 0.46
for all modes. We substitute the resulting photon probability distribution P (n) into Eq. (3.3) for arange of M varying from 1 to 100, and find the resulting µxtal to only change in the range from 0.22
(one mode) to 0.21 (100 modes). Although we do not know exactly how many modes are present,
the number of modes does not significantly impact the resulting value of µxtal , so we use the valueof 0.22 for 50 µW pump power.
We repeat this procedure at two other low pump powers (75 and 100 µW), and extrapolate
the resulting linear fit to find µxtal ≈ 147 photons pulse−1 at the maximum pump power of 30 mW.
Correcting this for the 24% loss between the SPDC crystal and the sample yields µsample ≈ 112 photons pulse−1 . This mean photon number is used to calculate the peak photon flux at the sample
using Eq. (5.12).
We can estimate the number of SPDC spectral modes based on the ratio of the SPDC

35
to laser spectral widths, which gives ≈ 8 spectral modes. It follows that there are ≈ 14 photons mode−1 pulse−1 at the sample. For these operating conditions, at which many pairs arespectrally and temporally overlapped, we may expect a significant contribution to any measured
signal from the classical term in the E2PA rate (Eq. (2.3)).

3.5

Superconducting Nanowire Single-Photon Detectors
In the single-mode detection stage (Fig 3.1) the light is sent into an HBT interferometer(Fig. 1.3) and each arm is focused into a fiber held in a fiber launching stage. The 5 m-longsingle-mode fiber patch cables are connected to superconducting nanowire single-photon detectors(SNSPDs). The SNSPDs are cooled down to ≈ 4 K with a closed-cycle helium cryocooler. The
SNSPDs’ temperature is monitored. Bias and readout modules are used to extract the digitized
and amplified signals that are analyzed with a time tagger triggered to the laser. The singles andcoincidence rates are measured.

3.6

Time-of-Flight Spectrometer

Figure 3.2: Measured time of arrival (ns) of photons as a function of their wavelength (nm) for thetwo 500-m-long fibers

36
To build a time-of-flight fiber-based spectrometer [69], the 5-m-long patch cables in the singlemode detection stage (Fig. 3.1) are attached to 500-m-long fibers that then connect to the SNSPDs.
The fiber group delay dispersion causes the various frequency components of the photons to spreadout in time and thus get detected at different times. These arrival times can thus be associated to
the frequency of the light.
We characterized the fiber dispersion by launching the pulsed laser into the fiber, tuning the
wavelength in the range 660-1200 nm in 5 nm steps. The time of arrival of the laser (centeredat each wavelength) at the detector relative to the laser trigger pulse was recorded. We did themeasurement with the 5-m-long patch cables only and with the 5-m-long patch cables attached to
the long fibers. We subtracted the time of arrival for the short fiber from the time of arrival for thelong fiber. We plotted time of arrival as a function of wavelength in Fig. 3.2. Using the slopes ofthese curves in the 825-875 nm range, we can extract the dispersion parameter of the fiber, which
is −0.111 ± 0.004 and −0.115 ± 0.004 ns nm−1 km−1 in close agreement with the manufacturer’s
specifications at 850 nm of −0.106 ± 0.004 ns nm−1 km−1 . At the wavelength range of interest,785-835 nm, the slopes are both −0.055 ± 0.002 ns nm−1 .
To measure the joint spectral intensity (JSI), we direct all the SPDC to the single-mode
detection stage shown in Fig. 3.1 with the 500-m-long fibers attached. We operate at a low pump
power such that we expect up to one photon pair to be generated per pulse. The signal and idler
detection times for each coincidence are recorded. We accumulate coincidences until the shape of
the JSI has a large signal to noise ratio and the structure stops changing. Initially we have a plotof idler photon arrival times as a function of signal photon arrival times. We use the characterized
dispersion parameter of the fiber to convert the axes of the measured JSI from time of arrival towavelength. In Fig. 3.3(a) we show the JSI of the SPDC source without any bandpass filters. The
unfiltered SPDC source was used for the fluorescence experiment discussed in Chapter 5. In thetransmittance measurements discussed in Chapters 4 and 6, bandpass filters filtered the SPDC
to 9 and 11 nm FWHM respectively. The shape of the JSI indicates the expected wavelength
anticorrelation of SPDC. Taking the projection of the JSI on the vertical and horizontal axes (for

37
type-0 SPDC these can be called signal and idler projections, or vice versa) reveals several results,
see Fig.3.3(b). The vertical and horizontal projections are shown in red and blue, respectively, with
FWHM of 72 and 79 nm. The various features in the spectra and the detuning from degeneracy islikely a result of a combination of measurement artifacts, such as the spectral profiles of the optics.

Figure 3.3: (a) Measured joint spectral intensity (JSI) where λS,I are the signal and idler wavelengths. (b) The JSI is projected onto the horizontal axis and vertical axis showing the signal(blue) and idler (red) spectra respectively. The FWHM of the signal and idler spectra are 79 and
72 nm respectively. The overlap of the spectra is evident in the dark red region. (c) Calculated
joint temporal intensity (JTI) obtained through a discrete Fourier transform as described in themain text. (d) Projection onto the antidiagonal axis, tS -tI , of the JTI shown in (c) (blue) and fora transform-limited (β = 0 fs2 ) pulse (red). The FWHM of these projections are 1620 fs and 17 fs.

3.7

Entanglement TimeIn this section we discuss how we estimate the entanglement time Te of our SPDC source at the

sample. Here we consider the conditions in the fluorescence experiment as discussed in Chapter 5.

38
For the transmittance experiments discussed in Chapters 4 and 6, we discuss the estimation of theentanglement time for those experimental conditions within the chapters themselves.
The entanglement time is the width of the temporal g (2) function [6], which can be estimatedas the FWHM of the antidiagonal projection of the joint temporal intensity (JTI) distribution. We
do not measure the JTI directly; instead we calculate it based on our measured JSI. Computing
the JTI from the JSI requires knowledge of the spectral phase of the SPDC. We do not have ameasurement of this phase; instead we estimate the accumulated group delay dispersion (β) of thepulse from the center of the crystal to the center of the sample to be 3700 fs2 . Our most dispersive
elements are the polarizing beamsplitter and the ppKTP crystal. We set the joint spectral amplitude(JSA) in the frequency domain to the square root of the JSI in the frequency domain multiplied by√22
the phase factor due to β, JSA = JSIeiβ(ωS −ωP /2) /2 eiβ(ωI −ωP /2) /2 where ωS , ωI , and ωP are thefrequencies of the signal, idler and pump fields, respectively. In asserting this, we assume that the
SPDC is transform limited in the center of the crystal and that the only significant accumulated
phase factor is that due to β. We note that β for signal photons would be distinct from that foridler photons if the two were instead orthogonally polarized [39]. This is due to birefringence ofvarious optical elements. We perform a discrete Fourier transform on the JSA to obtain the joint
temporal amplitude (JTA). The magnitude squared of the JTA gives the JTI shown in Fig. 3.3(c).
The projection of the JTI onto the antidiagonal (tS -tI ) is shown in blue in Fig. 3.3(d).
We find that Te ≈ 1620 fs. This can be compared with a transform-limited (β = 0 fs2 ) pulse(Fig. 3.3(d) in red) that has Te ≈ 17 fs. The projections of the JTI onto horizontal and verticalaxes both have FWHMs of 1040 fs. This width is a good approximation for the pulse duration of
signal and idler beams because it is significantly larger than the pump pulse duration.
To make a probabilistic estimate of the advantage of lossless dispersion compensation or adispersion-free setup on the rate of E2PA in our experiment, we consider how dispersion affects the
SPDC’s fulfillment of the simultaneity requirement of 2PA. This simultaneity requirement assertsthat two photons must arrive at the absorber within a time window (∆t) set by the absorber’s
virtual state lifetime. For a dispersed pulse, fewer of the SPDC photon pairs arrive within this39

Figure 3.4: (a) The joint temporal intensity from Fig. 3.3(c) with a yellow filled-in region indicating
a 50 fs time window (see main text for explanation). (b) The ratio of the number of coincidences of
the transform-limited SPDC satisfying the relation |tS − tI | ≤ ∆t to that for the dispersed SPDC
(β = 3700 fs2 ) as a function of ∆t. The yellow bar indicates the 50 fs time window shown in (a).

time window. A precise calculation of the virtual state lifetime of these large molecular absorbers
is not feasible, however we can consider any arbitrary time window. In Fig. 3.4(a) the projectionof the JTI from Fig. 3.3(d) is shown with a reduced range along the horizontal axis, along with atime window ∆t = 50 fs indicated in yellow. The number of photon pairs of the transform limited
SPDC (β = 0 fs) which satisfy |tS − tI | ≤ ∆t divided by the number of photon pairs of the dispersed
SPDC (β = 3700 fs) which satisfy the same constraint (called the coincidence ratio) is shown in
Fig. 3.4(b) as a function of ∆t. A yellow bar indicates the 50 fs time window shown in Fig. 3.4(a).
For the smallest possible ∆t we can consider based on our resolution (1 fs), the coincidence ratiois 95. Thus a factor of 95 more photon pairs of the transform-limited SPDC satisfy |tS − tI | ≤ 1 fs
than for the dispersed SPDC. This implies that for a virtual state lifetime of 1 fs, lossless dispersion
compensation or a dispersion-free setup would at most improve our E2PA rate by a factor of 95. If
the virtual state lifetime is longer, the factor is smaller as indicated in Fig. 3.4(b). This probabilistic
analysis of the entanglement time’s effect on the rate of E2PA ignores quantum interference effects
predicted by more sophisticated theoretical models [6, 32, 33, 31]. The work on these models is
ongoing and needs more thorough study. However, as mentioned in Chapter 2, Section 2.3, theinterference effects are likely washed out for the absorbers studied in this thesis.

40
It is worth noting that we could reduce the Te of our source by reducing the SPDC bandwidthwith a bandpass filter that is narrower than the ≈ 76 nm width. This would consequently reduce
our photon flux. We do not try this for the fluorescence experiment as it seems unlikely that a decreased photon flux will increase our likelihood of measuring E2PA. However, for the transmittanceexperiments this is a useful strategy since we do not and cannot use our maximum SPDC flux. We
find that the Te values under these conditions are much smaller.

3.8

Characterizing Second-Order Coherence
To measure g (2) (0) we must use our single-mode detection stage (Fig. 3.1). The characteri-zation done here considers the low flux single-mode of photon pairs detected in the transmittance(2)

experiment presented in Chapter 6. For this section we look back at the parameter γclick written
out in Eq. 1.28 in terms of parameters measurable in our setup. First we show that γ (2) [0] ≈ g (2) [0].
The parameter g (2) [0] can be written out in terms of the per pulse photon probabilities ofthe measured source of photons [9], where P (n) is the probability that the source emits n photons,
g
=

(2)

P∞
[0] =

n=0 n(n − 1)P (n)P2[ ∞n=0 nP (n)]2P (2) + 6P (3) + 12P (4) + ...
.
[P (1) + 2P (2) + 3P (3) + ...]2(3.5)

For our pair source, there is no chance for emission of odd numbers of photons so P (1) = P (3) = 0
along with the even higher order photon numbers. We operate in a low flux regime where P (2) >>P (4) >> P (6), thus we only keep P (2) and arrive at
g (2) [0] ≈2P (2)
1=.
2(2P (2))
2P (2)(3.6)(2)

Similarly γclick [0] can be written out in terms of per pulse photon probabilities [9],
η1 η2 |R|2 |T |2 2P (2) + 6P (3)(1 − 21 η1 |R|2 − 12 η2 |T |2 ) + ...
(2)



,
γclick [0] =η1 |R|2 P (1) + 2P (2)(1 − 12 η1 |R|2 ) + ... η2 |T |2 P (1) + 2P (2)(1 − 21 η2 |T |2 ) + ...
(3.7)
where R and T are the reflection and transmittance efficiencies of the beamsplitter in the singlemode detection stage and η1 and η2 are the detection efficiencies for detectors 1 and 2. Again41
keeping only the P (2) terms,
2P (2)(2)

γclick [0] ≈4P (2)2 (1 − 12 η1 |R|2 )(1 − 12 η2 |T |2 )

.

(3.8)

Next we assume 21 η1 |R|2 << 1 and 21 η2 |T | << 1, which is true in our case, and arrive at(2)γclick [0] ≈1≈ g (2) [0].
2P (2)(3.9)

Looking back at Eq. 1.28, we measure the quantities R1 , R2 and Nc [0]. We use a pump laser withrepetition time, Trep = 12.5 ns and an integration time of 1 s. Typical values of R1 ,R2 and Nc [0](2)

are 1 × 105 cnt s−1 , 1 × 105 cnt s−1 and 6 × 103 cnt. This gives γclick = 48. This value should only
be considered accurate for the experiment discussed in Chapter 6. At high flux, such as that used
for the fluorescence measurements, g (2) would much smaller (approaching 1) since multiple pairs
can be overlapped in the same pulse, rendering the beam’s photon statistics similar to the photon
statistics of a laser.

3.9

Entanglement Area
Unfortunately, we do not have a direct measurement of entanglement area Ae ; we can only

estimate a range of values for Ae . The estimate of the lower bound is based on the diffraction
limit. We find no evidence that the two photons can be focused to a region significantly smaller
than that set by the diffraction limit. It has been shown [70, 71, 43, 42, 41, 38] that entangled
photons can be focused to a spot size that is a few-fold smaller, however we neglect these factors
here as they have a minor effect in the orders-of-magnitude comparisons we present. Thus, we set
the bound using a circular area with radius (r) set by the central wavelength of excitation (r ≈ λ),which is 2.1 µm2 . The estimate of the upper bound varies depending on the experiment. For thetransmittance experiment we set the upper bound to an elliptical area with diameters set by themeasured transverse FWHMs of an 810 nm laser backpropagated through the setup at its focus
in the sample. For the fluorescence experiment the diameters are set by the measured transverse
FWHMs of the SPDC beam at its focus in the sample.

Chapter 4

An Introduction to Transmittance Measurements for Entangled Two-Photon
Absorption

4.1

Publication Note
This chapter is adapted from:[59] Mikhaylov, A., Parzuchowski, K.M., Mazurek, M.D., Lum, D.J., Gerrits, T., Camp Jr, C.H.,
Stevens, M.J. and Jimenez, R., 2020, February. A comprehensive experimental system for measuring molecular two-photon absorption using an ultrafast entangled photon pair excitation source. In
Advanced Optical Techniques for Quantum Information, Sensing, and Metrology (Vol. 11295, pp.
48-61). SPIE.

4.2

Introduction
In this chapter, we study entangled two-photon absorption (E2PA) using a transmittance-

based scheme. We are motivated by the previous experimental works in literature that report
E2PA signals using simple transmittance measurement schemes such as that shown in Fig. 2.2.
We reiterate (Chapter 2 Section 2.6) that these previous publications only verified one signatureof E2PA, the linear dependence of the signal on photon flux, which is not unique to E2PA. Thissignature is consistent with many one-photon losses such as scattering, one-photon absorption,
fluorescence from an optical component or misalignment.
We design our experiment with the goal of reproducing the results from literature and improving on the simple design. Our design has the capability to test for a unique signature of E2PA:

43
it’s dependence on the time delay between photons in a pair (Chapter 2 Section 2.6). We introduce
control over the time delay using a Mach-Zehnder interferometer. We find that the interferometerallows us to distinguish E2PA from one-photon losses, but it requires careful consideration of the
one-photon losses.
Our two photon absorbing sample is Zinc-tetraphenylporphyrin (ZnTPP) dissolved in toluene.
ZnTPP is a commercially available compound for which E2PA cross section σE values have beenreported [53, 49]. It has been suggested that the value of σE = 2.37 × 10−17 cm2 for ZnTPP at
800 nm can be used as a reference standard for E2PA experiments [72], thus it’s an importantsample to study.
We first analyze our data using a simple method that doesn’t use the time-delayed data.
This technique is sensitive to one-photon losses and two-photon absorption. The derived changes
in transmittance can erroneously be used to derive an σE close to the values reported in literature.
When we switch to an analysis technique that uses the time-delayed data that is only sensitive to
two-photon absorption (2PA), we find that the changes in transmittance for ZnTPP are below aremeasurement sensitivity of 1%. Using this sensitivity we set an upper bound on the cross section,
UB , for ZnTPP of 1.7 × 10−19 cm2 . This is considerably lower than what has been reported. We
σEbelieve our measurement technique and analysis form a basis for conducting E2PA experiments ina more quantitative manner.

4.3

Experimental Setup
A schematic illustration of the experimental setup is presented in Fig. 4.1. Here we give a

brief overview of the setup, a thorough description of the components is given in Appendix C. It
consists of an SPDC pair source, an interferometer, a sample telescope and a single-mode detection
stage. The SPDC source is as described in Chapter 3 Section 3.1. We use a bandpass filter to filter
the SPDC bandwidth to ≈ 9 nm.
The collimated SPDC beam is steered into a Mach-Zehnder interferometer. A half-wave plate(HWP1) is set at the interferometer’s input to rotate the SPDC beam polarization such that the44

Figure 4.1: Schematic of the experimental setup. The 810 nm laser (dark red) is frequency
doubled at an SHG unit. The 405 nm beam (blue) is directed through a half-wave plate (HWP)
and polarizer (pol) to control the power. Any remaining 810 nm light is rejected with a set offilters and two dichroic mirrors (F). The beam is focused into a type-0 ppKTP crystal to generate
collinear degenerate SPDC at 810 nm (light red). The SPDC is collimated and the 405 nm pump is
blocked with a set of filters and two dichroic mirrors (F). HWP1 rotates the SPDC polarization forequal probabilistic splitting at a polarizing beamsplitter (PBS). The beams are directed througheither the stationary or delay arm of the interferometer. A retroreflector on a translation stage isused for time delay control in the delay arm. The beams are recombined at a PBS and focused into
a cuvette. The beam is collimated and directed into the single mode detection stage which consistsof HWP2 and a PBS to switch between detection modes.

photons are split probabilistically and equally at a polarizing beamsplitter (PBS). A retroreflector
is set on a motorized translation stage to adjust the length of the delay arm of the interferometer.
The beams traveling through the two arms are recombined at a PBS.
The SPDC is focused with a 50 mm lens into a cuvette that contains either a solvent ora solution of two-photon absorbing molecules dissolved in the solvent (sample). A 50 mm lenscollimates the light and the beam is sent into the single-mode detection stage described in Chapter 3
Section 3.5. Next, HWP2 and a PBS are used to switch between two detection modes: the alignment
mode (HWP2 at 22.5◦ ) and the E2PA measurement mode (HWP2 at 0◦ ). In the measurementmode, the polarization of the photons is not rotated at HWP2 and nearly all the light from thestationary arm goes to detector 1 and nearly all the light from the delay arm goes to detector 2. In
the alignment mode, the polarization of the photons is rotated 45◦ such that the light from eacharm has a nearly equal probability of going to either detector.
The sample preparation is done at described in Appendix B. We use the reported molar

45
extinction coefficient [73] of ZnTPP to prepare a sample of 1.0 × 10−4 mol L−1 ZnTPP in toluene.
A standard 1 cm spectroscopic quartz cuvette is used for all measurements.

4.4

Experimental Characterization

Figure 4.2: (a) Measured joint spectral intensity (JSI) where λS,I are the signal and idler wavelengths. (b) The JSI is projected onto the horizontal axis and vertical axis showing the signal (blue)
and idler (red) spectra respectively. The FWHM of the signal and idler spectra are both 8.7 nm,
which is set by an optical filter. The overlap of the spectra is evident in the dark red region. (c)Calculated joint temporal intensity (JTI) obtained through a discrete Fourier transform. (d) Projection onto the antidiagonal axis, tS -tI , of the JTI shown in (c) (blue) and for a transform-limited(β = 0 fs2 ) pulse (red). The FWHM of these projections are 213 fs and 208 fs.

We measure the joint spectral intensity (JSI) of our source using the time-of-fight spectrometer described in Chapter 3 Section 3.6. The JSI is shown in Fig. 4.2(a). The bandwidth is set bya bandpass filter placed after the SPDC source. The JSI projections onto vertical and horizontal
axes in Fig.4.2(b) are used to estimate the marginal FWHM bandwidths of the photons as 8.7 nm.

46
We estimate the total group delay dispersion (GDD) of our optical elements between the ppKTP
crystal and the center of the sample to be approximately 3100 fs2 at 810 nm. We use this GDD tosimulate the joint temporal intensity (JTI) using the procedure described in Chapter 3 Section 3.7.
The result is shown in Fig. 4.2(c). The FWHM of the projection of the JTI onto the antidiagonal
axis (Fig. 4.2(d)) is used to estimate the entanglement time as 213 fs. We note that this size of
GDD has a nearly negligible effect on our entanglement time due to the narrow bandwidth of ourfiltered SPDC. The pulse duration of the SPDC can be estimated by the projection of the JTI onto
the horizontal and vertical axes, which is ≈ 258 fs.

Figure 4.3: (a) The normalized coincidence rate as a function of time delay measured with toluenein the beam path for the alignment mode (black symbols) and E2PA measurement mode (redsymbols) configurations over a 500 fs delay range. (b-c) A 12 fs delay range of plot (a) centeredat zero delay for the alignment mode (b) and E2PA measurement mode (c). In (b) we observe
interference fringes whereas in (c) we observe low amplitude oscillations due to residual interference.
The rates are normalized to have average values close to 1 at large delay positions.

In the alignment mode configuration, interference occurs at the PBS in the single-mode
detection stage and can be measured in the coincidence rate between detectors 1 and 2. We plot
the alignment mode rate in black in Fig. 4.3(a) and (b). The pattern shows a dip as a functionof time delay that’s at a minimum (“zero delay”) at optimal overlap due to Hong-Ou-Mandel

47(HOM) interference. Fast-oscillating fringes are superimposed on the HOM dip due to NOONtype interference. We explain the interference pattern in the next section. We measure a fringe
visibility of ≈ 90%, which implies that the photons have a high overlap in all degrees of freedom(spatial, spectral, temporal and polarization mode) and thus are indistinguishable. In the E2PA
measurement mode, interference is suppressed as shown in red in Fig. 4.3(a) and (c). Due toimperfections of our polarization optics, a small amount of light from the stationary arm leaks todetector 2 and similarly from the delay arm to detector 1. This results in low amplitude residualinterference near zero delay.

4.5

Origin of Alignment-Mode Interference Pattern
When characterizing our setup in alignment mode, we measure the interferogram shown in

Fig. 4.3(a). Here we explain the structure of this interference pattern.
We begin by modeling the state of each photon pair produced in the SPDC crystal as (see,e.g., Ref. [74] for details):
ZZ
|ψ⟩ =

dω1 dω2 f (ω1 , ω2 )â†H,1 â†H,2 |0⟩ ,(4.1)

where the operator â†H,i is the creation operator that creates a horizontally-polarized photon withfrequency ωi . The function f (ω1 , ω2 ) defines the joint spectral amplitude (JSA) of the photon pair.
Because we are modeling light from a type-0 down-conversion process, we must ensure that the
JSA is symmetric, i.e. it must obey the relation f (ω1 , ω2 ) = f (ω2 , ω1 ).
To calculate the coincidence rate at the interferometer output as a function of time delay,
we first consider how light is transformed as it propagates through the interferometer. First, thelight travels through HWP1, which is set to an angle of 22.5◦ , which rotates horizontally polarized
light (indicated by subscript ‘H’) to diagonally polarized light. We represent this rotation with thefollowing transformation:
1 â†H,j → √ â†H,j + â†V,j ,
2

(4.2)

48
where j ∈ {1, 2}, and the ‘V’ subscript indicates vertical polarization. The light then enters theinterferometer, where the vertically-polarized component of the beam travels a fixed path length,
and the horizontally-polarized component is delayed by time τ , which is represented by multiplying
the âH,j operators by the phase term exp {−iωj τ }. The light is recombined at a PBS, then travels
through HWP2, which is also set to an angle of 22.5◦ . The horizontally polarized component of
the light is transformed according to Eq. (4.2), and the vertically polarized component undergoes
the transformation:
1 â†V,j → √ â†H,j − â†V,j .
2

(4.3)

Finally, the light is split at a PBS, where the vertical component travels to detector 1 andthe horizontal component travels to detector 2. The total transformation the light undergoes fromthe crystal through the last HWP can be summarized as:â†H,j →i1h †âH,j e−iωj τ + 1 + â†V,j e−iωj τ − 1 .
2

(4.4)

To calculate the coincidence rate, R(τ ), between detectors D1 and D2, we apply the transformation (4.4) to the SPDC state (4.1), and then find the magnitude of the terms that lead tocoincidences. The result is:
ZZ
R(τ ) ∝dω1 dω2 |f (ω1 , ω2 )|2 e−iω1 τ + 12

2

e−iω2 τ − 1 ,(4.5)

which we can can separate into two terms:
R = RHOM (τ ) + RNOON (τ ),

(4.6)

with
R

HOM

ZZ
(τ ) = 2dω1 dω2 |f (ω1 , ω2 )|2 [1 − cos(ω1 − ω2 )τ ] ,

(4.7)

and
RNOON (τ ) = 2ZZdω1 dω2 |f (ω1 , ω2 )|2 [1 + 2 cos ω1 τ − 2 cos ω2 τ − cos(ω1 + ω2 )τ ] .

(4.8)

49
The first term, RHOM , leads to a coincidence rate that is constant for large time delays, anddecreases to zero at delay τ = 0; this is HOM interference [75]. The second term, RNOON , describes
a NOON-type interferogram with fringes that oscillate at the sum of the two photon frequencies [76].
We complete the calculation by choosing a specific form of the JSA and evaluating the
integrals in Eqs. (4.7) and (4.8). For simplicity, we assume the pump laser has a Gaussian spectrumwith center frequency 2ω0 and width σp . We also assume the SPDC process is degenerate and thatthe photons have Gaussian spectra with width σ. We write the JSA as:(2σ 2 + σp2 )1/4
(ω1 − ω0 )2 (ω2 − ω0 )2 (ω1 + ω2 − 2ω0 )2
f (ω1 , ω2 ) =
exp −
.
−
−
4σ 2
4σ 2
4σp2
(2πσ 2 σp )1/2

(4.9)

The resulting coincidence rates are:
RHOM (τ ) =
11 − exp −σ 2 τ 2 ,2(4.10)

and1
RNOON (τ ) =2(σ 2 σp2 τ 2
1 − exp − 2
2σ + σp2)!
cos 2ω0 τ.

(4.11)The explicit form of Eqs. (4.10) and (4.11) demonstrates the HOM-type and NOON-type components of the interferogram.

Figure 4.4: Calculated coincidence rate as a function of interferometer time delay. (a) A 500 fs
range of the interferogram composed of fast-oscillating fringes superimposed on a HOM-type dip.
(b) A 12 fs range of the same graph to show the fringes, the period of ≈ 1.35 fs is consistent witha pump wavelength of 405 nm.

We can plot R(τ ) = RHOM (τ ) + RNOON (τ ) if we choose specific parameters for the JSA,
which we display in Fig. 4.4. For the pump we choose σp = 0.002 fs−1 and a center wavelength of

50
405 nm, which corresponds to a full width at half maximum (FWHM) of ≈ 0.2 nm. We choose a
center wavelength of 810 nm for the down-conversion, and σ = 0.01 fs−1 , which corresponds to asingle-photon spectrum with ≈ 6 nm bandwidth. With these parameters, the calculated coincidence
rate R(τ ) qualitatively approximates the measured interferogram in Fig. 4.3.

4.6

Experimental Procedures
After verifying the alignment of the setup, we rotate HWP2 to 0◦ to operate in the E2PA

measurement mode. Ideally, we would completely turn off any interference and measure zerointerference visibility, with the only observed change in coincidences as a function of time delay
due to E2PA. However, we measure residual interference. This artifact (linear interference fringes
superimposed on a possibly small remaining HOM dip) could be incorrectly attributed to E2PA. To
mitigate the effect of the residual interference, we record the coincidence rates over several fringes
in the vicinity of zero delay and average them. This averaging procedure is repeated for a delayof approximately -6.7 ps, corresponding to a path-length difference of 2 mm from zero delay; wedenote this position as “τ delay”. The uncertainty is estimated from the standard deviations in
the measured counts and is ≈ 1%. This defines the sensitivity limit in measuring changes in the
coincidence rates.
For the E2PA measurements we tune the 405 nm pump power from ≈ 60 µW to 11 mW using
the HWP; the coincidence rates are approximately 2 × 102 − 3.7 × 104 cnt s−1 and singles rates are
103 − 106 cnt s−1 . Measured coincidence rates are shown in Fig. 4.5 and given in Appendix D. We
record the coincidence rates for photon pairs transmitted through the solvent (blue symbols) andsubsequently through the sample (red symbols) at τ (a) and zero delays (b). All measurements arerepeated multiple times, from which standard deviations (relative) are estimated. Sample removal
and refill is done using a pipette and the cuvette is fixed in the beam path.

51

Figure 4.5: Coincidence rates measured in the experiment. The rates (in counts per second,cnt s−1 ) are measured as a function of the pump power (mW). The experiments were conducted forneat solvent (blue symbols) and sample (red symbols) at the (a) zero- and (b) τ -delay positions.
All rates were averaged according to the procedure described in Section 4.6.

4.7

Deriving a Cross Section
In this section, we describe how a measured change in transmittance due to E2PA can beused to estimate an E2PA cross section.
Consider a beam of photon pairs that is described by a flux ϕ. We can derive an expression
describing how ϕ changes while the beam propagates along the z-axis through a sample. The beam
enters the sample at z = 0. The sample’s length is L. Assuming the beam experiences only linear
losses while interacting with the sample that causes some attenuation in the flux characterized byκ, the change in the flux dϕ over some small distance dz can be writtendϕ= −ϕκ.
dz(4.12)

We make the approximation that the beam is collimated over the length L (Rayleigh range≫ L) and has constant intensity over its cross section. For our experimental conditions this is
not precise (the SPDC beam is not uniform and is focused into the sample); however, we ignore
this discrepancy and proceed with the simplest case. A more rigorous analysis (similar to what
has been developed for z-scan type measurements [15]) should be applied for a general treatment.
With the aforementioned assumptions integration of Eq. (4.12) becomes trivial. Assuming weak

52
absorption (κL ≪ 1), we find,
ϕout ≃ ϕin (1 − κL) .

(4.13)

We can introduce a relative change in the flux, δϕ, as
δϕ =

∆ϕ
.
ϕin

(4.14)

where ∆ϕ = ϕin − ϕout and ϕin(out) is the flux going into (out of) the sample. Using Eq. (4.13), thisbecomesδϕ ≃ϕin − ϕin (1 − κL)
= κL.
ϕin

(4.15)All of our experimental data is recorded as detected photon pair rates, R, rather than flux values.
Therefore, it is convenient to rewrite the equations in a slightly different way using R instead of ϕ.
Since ϕ differs from R only by a factor of 1/A where A is the beam area, it is straightforward toshow that,δϕ = δR =

∆R
≈ κL,
Rin

(4.16)

where ∆R = Rin − Rout and Rin(out) is the photon rate going into (out of) the sample. Since wefiber couple the light transmitted through the sample before it is detected, A is specific to thespatial mode coupled into fiber.
If the change in transmittance (δϕ or equivalently δR) is due to E2PA only, then κ = nσE ,where n is the number density of the sample (in molecules per cm3 ).
Then we find,
σE ≃

δR
,
nL

(4.17)

where σE and L are in units of cm2 molecule−1 and cm, respectively.

4.8

Data Analysis Case Studies
In this section we show how the measured coincidence rates are used to calculate a δR that is

only sensitive to E2PA rather than one-photon losses. We start with the most simplistic approach
adopted from previous reports (i.e. Refs. [53, 49]) and discuss the pitfalls of this method. We

53progressively add more complexity to show how our experimental setup and analysis can be used
to account for one-photon loss mechanisms.

Figure 4.6: A schematic showing notations used in the data analysis case studies. Each panel
corresponds to a different case considered in Section 4.8: (a) illustrates case 1, (b) illustrates case
2 and (c) illustrates case 3.

4.8.1

Case 1. Measuring sample and solvent without time delay

Here we start by using only data at zero delay. We assume that if there is a difference incoincidence rates measured with the solvent (R0solv ) and with the sample (R0samp ) (at zero delay),it’s due to E2PA. This situation is schematically depicted in Fig. 4.6(a). It shows coincidence rates
as a function of the time delay. All curves are not real data and not shown to scale. Under thisassumption, ∆R = R0solv − R0samp .
From Section 4.7, Eq. (4.16)
δR =∆R
Rsolv − R0samp
≃ 0
.
Rin
R0solv

We use the rates measured at different pump powers (Table D.1 in Appendix D) to calculate averageδR values. We find δR = 0.146 ± 0.010. This assumption ignores the likely scenario that switchinga solvent with a sample can introduce one-photon losses to the measurement. These losses could
be one-photon absorption, scattering or misalignment. In section 4.9, we will see that this way ofanalyzing the data leads to an overestimate of σE values.

54
4.8.2

Case 2. Measuring sample with time delay

Next consider a case where only the sample is measured and the time delay is varied between
the zero- and τ -delay positions. Here we assume that the difference in the coincidence rates measured at τ and zero delays, Rτsamp and R0samp , respectively, is proportional to E2PA. This scenario
is depicted in Fig. 4.6(b). Under these assumptions, ∆R = Rτsamp − R0samp andδR =Rτsamp − R0samp∆R≃
Rin
RτsampThis technique avoids some of the pitfalls that render case 1 incorrect since the one-photon
losses that are associated with switching the sample with a solvent are not introduced. Those losses
are shown in Fig. 4.6(b) as Rsolv−samp .
From the measured rates we deduce δR = 0.006 ± 0.005. The first two lowest power points(lowest rates in Table D.1) are excluded from the calculation. Including these removed points
results in a large additional uncertainty in δR. One can see that in this case, the value of δR ismuch smaller than in case 1. This is because the largest one-photon losses contributing to the δR
derived in case 1 are removed in case 2. However, case 2 can only derive accurate δR values when
the one-photon losses in the experiment are the same at zero and τ delays. This assumption is
not correct, especially since we noted that there are residual interference effects evident at zerodelay. However, the delay-dependent one-photon losses are smaller or of similar magnitude to ourexperimental uncertainty as shown by the derived δR. In the next case study, we account for all
one-photon losses including the residual interference.

4.8.3

Case 3. Measuring sample and solvent with time delay

Here we account for all the one-photon losses by using both the sample and solvent measurements at both time delays. We assume that the residual interference effects are the same magnitude
for sample and solvent measurements. We denote the loss in the coincidence rate associated withresidual interference at zero delay as RHOM . Additionally, if there is some misalignment associatedwith replacing solvent with sample or due to moving the stage between the two delay positions, the55
corresponding losses in the coincidence rates are Rsolv−samp and Rstage , respectively. A schematic
describing this case is shown in Fig. 4.6(c), however Rstage is not shown.
In this case, the coincidence rates measured with the sample at the two delay positions areRτsamp = Rin − Rsolv−samp − Rstage ,R0samp = Rin − ∆R − RHOM − Rsolv−samp ,and for the solvent,Rτsolv = Rin − Rstage ,R0solv = Rin − RHOM .
Using these relations one can show∆R = (Rτsamp − R0samp ) − Rτsolv − R0solv .
Here, Rin ≃ R0solv . Then,
samp

(Rτ
∆R
δR ≃ solv =R0
− R0samp ) − Rτsolv − R0solv.
R0solvThus, δR is independent of Rsolv−samp , RHOM and Rstage . Using this analysis, δR = −0.001 ±0.004. Again, we excluded the lowest power points from the analysis to reduce the calculated
uncertainty. The average δR value is lower than the associated uncertainty, thus, we cannot assign
this change in transmittance to a measured signal in our experiment.

4.9

Results
Here we estimate σE values using Eq. (4.17). The derived σE values are listed in Table 4.1.

First we consider case 1 where we found δR = 0.146 ± 0.010. We can use this value to derive
σE = (2.43 ± 0.20) × 10−18 cm2 , however this change in transmittance may be due to one-photon
losses as discussed in Section 4.8.1. It’s useful to note that this σE value is within the rangeof σE values for ZnTPP from other reports where a very similar analysis procedure was used:
σE = 2.37 × 10−17 cm2 as reported in Ref. [53] and (1 − 10) × 10−18 cm2 as reported in Ref. [49].

56
For cases 2 and 3 we found δR = 0.006 ± 0.005 and δR = −0.001 ± 0.004, respectively.
These δR are effectively zero within the estimated uncertainty. Therefore, we did not use them tocalculate the corresponding σE values.
Let us consider a limiting case to estimate our experimental upper bound on σE . We call thissituation case 4 (Table 4.1). The smallest δR that can be measured reliably in our experiment is
UB ≈ 1.7 × 10−19 cm2 , which serves as an upper
about 0.01. Thus, we calculate from Eq. (4.17): σE

bound for ZnTPP.
Table 4.1: Summary of the changes in transmittance δR and calculated σE values for ZnTPP.
Cases 1-3 correspond to the 3 different analysis case studies, with case 3 illustrating the correct
analysis procedure. Case 4 illustrates a limiting situation used to place an upper bound on σE .
Case
1
2
3
4

4.10

δR

σE × 1019
cm2

0.146 ± 0.010
0.006 ± 0.005−0.001 ± 0.004
0.01

24.3 ± 1.6−−
1.7

Conclusions
In this chapter we presented our vision of a comprehensive transmittance–based scheme for

E2PA measurements. We performed a careful characterization of the experimental apparatus, andwe showed that the implementation of a time delay between the two photons was essential for a
one-photon loss insensitive transmittance measurement.
We used data analysis case studies to highlight the importance of using the time delay. If notime delay is implemented (case 1), then the calculated δR value includes spurious contributions
from scattering or other one-photon losses. The value of δR derived in case 2 is inherently freeof the measurable losses attributed to case 1, however it may include delay-dependent one-photon
losses. To eliminate all one-photon losses we applied the analysis presented in case 3. In cases 2
and 3 the derived δR values are significantly lower than the results obtained in case 1 (Table 4.1).

57
Thus, simplistic subtraction schemes like those used in case 1 are insufficient and can lead to a
many-orders-of-magnitude overestimate in σE values.
We illustrated this procedure in our pilot study of ZnTPP using 810 nm SPDC excitation.
We showed that if the time delay was not applied, one can erroneously use their measurements
to derive a σE value that is in agreement with published values in literature. The real changes intransmittance due to E2PA for ZnTPP were below our experimental sensitivity of ≈ 1%. Thus, weestimated ZnTPP’s σE ≤ 1.7 × 10−19 cm2 , which is significantly lower than previously publishedσE values.

Chapter 5

Bounding Entangled Two-Photon Absorption Cross Sections with a
Fluorescence-Based Scheme

5.1

Publication Note
This chapter is adapted from:[19] Parzuchowski, K.M., Mikhaylov, A., Mazurek, M.D., Wilson, R.N., Lum, D.J., Gerrits, T.,Camp Jr, C.H., Stevens, M.J. and Jimenez, R., 2021. Setting bounds on entangled two-photon
absorption cross sections in common fluorophores. Physical Review Applied, 15 (4), p.044012.

5.2

Introduction
In this chapter, we investigate entangled two-photon absorption (E2PA) using a fluorescence-

based measurement scheme. We are motivated to determine whether a large quantum advantage(QA) exists. Some reports suggest the QA is nearly 10 orders of magnitude [52, 54]. However, inthese reports it is unclear whether the signals are caused by E2PA or some other process as discussed
in Chapter 2 Section 2.5. Furthermore, the methodology in these reports has not advanced to thepoint where classical two-photon absorption (C2PA) and E2PA can be measured in the same setup,
and thus the excitation conditions can differ significantly leading to orders-of-magnitude error in
the QA quantification.
We present a method for measuring both entangled two-photon excited fluorescence (E2PEF)
and classical two-photon excited fluorescence (C2PEF) in one experimental setup. We characterize
our excitation sources, fluorescence collection system and samples to determine our 2PA cross-59
section sensitivity. The C2PEF measurements are used to derive C2PA cross sections, σC , for the six
studied fluorophores. The values strongly agree with previously reported C2PA cross sections [17,47, 77]. Although we do not detect measurable E2PEF signals for any of the six fluorophores,
we can bound the maximum efficiency of the E2PA process in each fluorophore by placing upperbounds on its E2PA cross section, σE . The C2PEF and E2PEF measurements are also used tobound the QA. Our established upper bounds on σE are up to four orders of magnitude lower than
the smallest published value of σE [48]. For two of the samples, the upper bounds on σE are fourand five orders of magnitude lower than the previously reported cross sections [48, 47].

5.3

Experimental Setup and Characterization
Here we give a brief overview of our experimental setup and characterizations. A thoroughdescription of the components is given in Appendix E. A schematic of the experimental setup is
shown in Fig. 5.1 and it consists of a laser source, a photon pair source, a single-mode detection
stage, a multi-mode detection stage and the two-photon excited fluorescence (2PEF) measurement
system. A pump laser emits ≈ 110 fs pulses with a center wavelength of 810 nm (≈ 9 nm bandwidth)
at a repetition rate of 8 × 107 pulses s−1 . The laser output is frequency doubled to produce 405 nm
light (≈ 3 nm bandwidth), of which 30 mW is used to produce SPDC for E2PA as described in
Chapter 3 Section 3.3. A small fraction of the 810 nm pump laser is routed around the nonlinear
crystals and used for C2PA.
We characterize the joint spectrum of the photon pairs with the time-of-flight fiber spectrometer using the single-mode detection stage as described in Chapter 3 Section 3.6. In thisconfiguration, the single-mode detection stage consists of a 50-50 fiber beamsplitter rather thana free-space beamsplitter. The SPDC is approximately degenerate and centered at 810 nm with≈ 76 nm bandwidth. We determine the entanglement time (Chapter 3 Section 3.7) using theestimated joint temporal intensity, which accounts for the approximately 3700 fs2 of dispersion accumulated by each photon pair before reaching the center of the cuvette. The value of Te at the
sample position is ≈ 1620 fs. Although Te is larger than in the ideal (dispersion-free) case, we make

60

Figure 5.1: Schematic of the experimental setup. The 810 nm laser (dark red) is split into twopaths; one path is used for C2PEF measurements and the other for E2PEF measurements. The
light in the C2PEF path is directed through a half-wave plate (HWP) and polarizing beamsplitter(PBS) to control the power of light directed to the 2PEF measurement system. The laser is opticallychopped (chopper) and focused into a sample. The fluorescence (green) is collected onto a photoncounting PMT and all scattered light rejected using filters (F). The PMT pulses are shaped andsent to a time tagger that is synchronized with the optical chopper. The light in the E2PEF path isfrequency doubled (blue) via second-harmonic generation (SHG) and focused into a type-0 ppKTP
crystal to generate collinear SPDC photon pairs at 810 nm (light red). Filters (F) are used toremove the remaining 405 nm light. To characterize the joint spectral intensity of the light, a flip
mirror directs the pairs into the time-of-flight spectrometer which uses the single-mode detection
stage. To characterize the absolute SPDC photon rate at the sample, a HWP and PBS direct thelight to a single-photon avalanche diode (SPAD) in the multi-mode detection stage. For E2PEF
measurements, the HWP is rotated to transmit all SPDC photons through the PBS; this lighttravels along the same path as the light used in the C2PEF measurements where it is focused into
a 2PA sample.

a probabilistic estimate that lossless dispersion compensation would at most increase the rate of
E2PA by a factor of 95 (Chapter 3 Section 3.7).
The SPDC photon rate is measured using the multi-mode detection stage. The optical system
was designed to minimize losses, thereby minimizing the number of unpaired photons focused
into the sample. Taking into account the single-photon detection rate, the SPAD dead time andefficiency, the photon statistics of the SPDC and the optical losses in our setup from the center of thecrystal to the center of the sample (≈ 24%), we estimate that ≈ 147 photons pulse−1 are generated
at the output of the crystal and ≈ 112 photons pulse−1 arrive at the sample while operating at our

61
maximum pump power (30 mW) (Chapter 3 Section 3.4).
We estimate that the value of Ae is in the range of 2.1−13, 700 µm2 as described in Chapter 3
Section 3.9. The value of Ae likely changes throughout the cuvette because the beam is not collimated. The range of values that Ae could take on at the edge of the cuvette is 2.1 − 2, 160, 000 µm2 .
This subtle point is taken into account in Section 5.5.
For C2PEF and E2PEF measurements, we use a polarizing beamsplitter (PBS) to combine
the SPDC and laser beams, and align them along the same path. The power of the laser beam
is controlled using a half-wave plate (HWP) in conjunction with the PBS, varying from 0.079 −10.5 µW. The beams are sent through an optical chopper, then focused in the center of a cuvette toa beam FWHM of ≈ 68 µm and ≈ 49 µm for the SPDC and laser beams respectively. For E2PEF
measurements, we block the laser beam, and for C2PEF measurements we block the SPDC beam.
The portion of the beam absorbed in the sample is partially re-emitted as fluorescence, which iscollected and focused onto a photon-counting photomultiplier tube (PMT). A combination of ashortpass and bandpass filter (selected for each fluorophore, see Section 5.5) in front of the PMT
reject scattered 810 nm and 405 nm light. The SPDC beam is found to have a larger divergencewithin the sample compared to the laser beam (Section 5.4). The divergence is taken into account byusing the characterization of the spatially dependent geometrical collection efficiency (Section 5.5).
The geometrical collection efficiency of the fluorescence collection system is characterized using
numerical simulations and 1PEF measurements (Section 5.4) and determined to be 15.4% and4.7% for a point source and line source (extending the length of the cuvette) of fluorescence,
respectively. The two beams are found to be displaced from each other in the sample by ≈ 5 µm
horizontally and vertically. As Section 5.4 and Appendix F explain, our experimental apparatus
is carefully designed and characterized to be robust against small changes in alignment like these.
The longitudinal displacement between the beams is compensated for.
The six fluorophores investigated in this study are the 1,3,5-triazine-based octupolar molecule“AF455” [78, 79] in toluene, Qdot ITK carboxyl quantum dot 605 (qdot 605) in borate buffer,
fluorescein in pH 11 water, the benzodithiophene derivative “9R-S” [47] in chloroform, rhodamine

62
590 (Rh6G) in methanol and coumarin 153 (C153) in toluene (details on sample preparation in
Appendix B). These samples are of particular interest because of their well-known and large valuesof σC at 810 nm (see Table 5.3). In addition, two of these samples (Rh6G and 9R-S) were studiedin previous reports of E2PA [48, 47].

5.4

Fluorescence Collection Efficiency
Initial characterization of the fluorescence system’s geometrical collection efficiency is per-formed using Zemax’s OpticStudio. The solvent, glass cuvette walls, four collection optics anddetector surface are modeled in the program. Using a merit function and an optimization algorithm, we find the ideal spacing of the optics.
A 2PA process can only occur if two photons are sufficiently spatially overlapped at a fluorophore, thus the rate of C2PA and E2PA depend on the focusing of the respective beams. For
C2PA, this is clearly evident through the quadratic photon flux dependence in the excitation rate
(Eq. (2.6)), where the photon flux depends inversely on the beam size. For E2PA, this spatial
dependence is hidden because the excitation rate depends linearly on photon flux (Eq. (2.10)) ina similar manner to 1PA (a beam-size-independent process). The spatial dependence is instead
included in the E2PA cross section (Eq. (2.11)), which depends inversely on the entanglement area.
In our experiment, the excitation beams are not collimated (see divergence of the SPDC
(green) and laser (blue) beams in Fig. 5.2(a)) and thus the excitation volume is a non-trivial shape.
Furthermore the E2PA beam does not have a constant entanglement area (or E2PA cross section)
which complicates the ability to perform an exact calculation of the E2PA cross section (or in ourcase an upper bound) for a given E2PA signal. To approximate this, we calculate the E2PA crosssection upper bound in the region with nearly constant entanglement area (Section 5.5). For thiscalculation and our calculation of the C2PA cross section (Section 5.5) it is critical to characterize
the collection efficiency of our system as a function of the origin of the fluorescence along the zdirection. Ideally we would also take into account the collection efficiency’s x- and y-dependence,
however as we discuss below, this is less critical to the final result.

63

Figure 5.2: Illustration of the geometrical collection efficiency inside of the cuvette and thelaser/SPDC beam overlap. (a) A cross section of the cuvette in the xz plane. The selected contours
show where the collection efficiency is constant, based on Zemax simulations. The magnitude of
the collection efficiency is scaled based on 1PEF measurements. The beam propagation is shownfor the laser beam (blue) and the SPDC beam (green). (b) A cross section of the cuvette in thexy plane. The magnified image on the right is a view of the center of the xy plane and shows
approximate beam FWHMs and overlap.

In Zemax, we simulate the collection efficiency as a function of the origin of the fluorescencewithin the cuvette volume. We model a point source of fluorescence that emits rays isotropicallyat some position in the cuvette. The number of those rays collected onto the detector are counted.
We systematically translate this source in all directions to trace out contour plots of collection
efficiency in the xz plane (centered in y) (Fig. 5.2(a)) and the xy plane (centered in z) (Fig. 5.2(b)).
In Fig. 5.2 we rescale the collection efficiency found through Zemax to match experimental values,
as discussed below. We find that the collection efficiency is slightly asymmetric in the x direction,collecting slightly better when the point source is displaced towards the PMT. We ignore this minor
asymmetry in the experiment and center the beams through the cuvette.
Although the spatial distributions of the excitation beams have some transverse extent,
Fig. 5.2 shows that transverse displacements from x = y = 0 must be large to significantly affectthe collection efficiency (> 102 µm at z = 0 and transverse displacements nearly negligible beyond

64|z| > 1 mm). Our excitation beams’ spatial distributions in the transverse directions are contained
within a region of nearly constant collection efficiency, thus in our calculations (Section 5.5) we
ignore the transverse spatial distribution of the excited fluorescence.
In Zemax, we simulate the total collection efficiency (κ) of a particular excitation volume forthe limiting cases of a uniform cylindrical excitation volume (50 µm diameter) that extends thelength of the cuvette (centered in x and y) and for that of a point source centered in the cuvette.
For the former, the collection efficiency is at a minimum for the system (for a uniform excitation
volume centered in the cuvette with 50 µm diameter), κmin = 6.1%, and for the latter, the collection
efficiency is at a maximum for the system, κmax = 20.2%
To a good approximation, the collection efficiency K (found using Zemax) as a functionof z (cm) fits to a complementary error function. This can be qualitatively understood by thesimilarity of the simulation of the collection efficiency as a function of z to a knife’s edge beamprofile measurement, which fits the same type of function. In both cases an intensity is measured
as a function of the placement of an object. This object alters the intensity passed to a detector.
Thus, the collection efficiency as a function of z takes the form
K(z) =

κmax
erfc (α(|z| − z0 )) ,
2

(5.1)

where κmax = 0.20, α = 2.8 cm−1 and z0 = 1.5 cm. These parameters are set by Zemax collection
efficiency simulations for the translation of a point source along the z direction (centered in x (cm)and y (cm)), and z = 0 is the center of the cuvette. The function K(z) is used to calculate theportions of an excitation volume extended along the z direction that contribute to the collected
fluorescence signal. Below we discuss our method to scale the collection efficiency as a function of′′z, K(z) (where indicates the experimental value rather than simulated), to fit the experimental
conditions.
′

We measure the minimum collection efficiency, κmin , of our fluorescence setup using a 1PEFbased technique. The results of this measurement are compared to the Zemax simulation of κmin′to scale the collection efficiency for experimental differences. As we mentioned, κmin characterizes

65
a system with a nearly uniform cylindrical excitation volume (50 µm diameter) that extends thelength of the cuvette, thus we use an excitation source which generates an excitation volume of thiskind.
In a similar manner to the treatment in Ref. [16], the measured 1PEF rate, F1 (cnt s−1 ), canbe described by
W ′F1 = N1 κminhνZλfγ(λ)Φ(λ)dλ,

(5.2)

λi

where N1 (excitations photon−1 ) is the number of excitations per photon, W (W) is the average
power incident on the sample, hν (J) is the average energy of an incident photon, λi,f (nm) are initialand final wavelengths chosen to integrate over the entire emission spectrum of the sample, γ(λ)(cnt photon−1 )] is the wavelength-dependent component transmission efficiency (detector, filters,
lenses and cuvette), and Φ(λ) (photons excitation−1 nm−1 ) is the differential quantum yield. A
R∞normalization of quantum yield is used such that 0 Φ(λ)dλ gives the value published in literature
for the total quantum yield of the fluorophore (Table 5.1).
The number of excitations per photon is found using
N1 = 1 − 10−OD ,(5.3)

where OD = ϵcl is the optical density of the sample at the excitation wavelength, ϵ (L mol−1 cm−1 )is the extinction coefficient of the sample at the central excitation wavelength, c (mol L−1 ) is themolar concentration of the sample and l (cm) is the cuvette length.
We estimate γ(λ),γ(λ) =

NY
i=1

Tfilteri (λ)

M
=3
Y
j=1

Tlensj (λ)Tcuvette (λ)

12
1 + Tcuvette(λ)Rsph.mirror (λ) QE(λ),
2

(5.4)

where T (λ) and R(λ) are the transmittance and reflectance of a given optic and QE(λ) is the PMT
quantum efficiency. The various manufacturers’ specifications are used to calculate γ(λ). Here, we
use one filter (bandpass) (F5) and thus N = 1. The laser, PMT quantum efficiency, fluorophoreabsorption and emission and filter transmittance spectra are shown in Fig 5.3(g)-(h).

66′

To calculate κmin , we input Eq. (5.3) into Eq. (5.2), and solve for the minimum collection
efficiency,
F1′

κmin =(1 − 10−OD )W/(hν)

λ
Rf

.

(5.5)

γ(λ)Φ(λ)dλλi

The excitation source is a CW 458 nm laser. The beam FWHM and Rayleigh range at thefocus on cam1 in Fig. E.1 is measured to be 15 µm and 1 mm respectively. This Rayleigh range
suggests that the beam size will be significantly larger at the edge of the 10 mm path length cuvettecompared to at the center. However, the beam has a small transverse spatial extent for all z (at|z| = l/2 the beam size is ≈ 4% of the cuvette width) relative to the collection efficiency contour
spacing in the transverse direction. Thus, the excitation volume can be approximated as a uniform
cylindrical volume that extends the length of the cuvette. Using a similar argument, although thebeam size is smaller than that used in the simulation (50 µm), the difference can be neglected basedon the relatively large spacing of the collection efficiency contours.
We use the samples Rh6G in ethanol and fluorescein in pH 11 water. The sample is prepared at a relatively low concentration (≈ 0.1 − 10 × 10−6 mol L−1 ) and the OD is measured
in a spectrophotometer. During measurements, the amount of power (W ) reaching the sample
is measured after lens L8 and varied using the ND wheel F6 after the output of the laser (seenotation in Fig. E.1). We first send the laser through the solvent to check that there is no signal
due to scattered light. Next, the laser is sent through the sample and a signal is measured. The
fluorescence signal is measured at six different excitation powers ranging from 10 - 150 nW.
Using the comparison of the experimentally determined and simulated minimum collection′′

efficiency, we rescale the maximum collection efficiency of the system, κmax = κmin /κmin × κmax ,′′which goes into the experimental K(z) (same as Eq. (5.1), but with κmax instead of κmax ). We′

measured an average κmin = 3.9±0.6% and 5.4±0.7% for Rh6G and fluorescein respectively. Using′

the average of these two, we find κmax = 15.4%.
We note that in fluorescence measurements, especially those performed at high sample concentrations, fluorescence self-absorption can reduce the measured signal. In our experiment, our67narrow cuvette width minimized this effect. Our measurements suggest that self-absorption is
negligible.

5.5

Calculating C2PA and E2PA Cross Sections
Here we describe the equations relevant for the calculation of the C2PA cross sections andE2PA cross-section upper bounds based on our measured data. First we describe the C2PEF signal
and the derivation of C2PA cross sections from the fit to our C2PEF data. Then we describe theE2PEF signal and the derivation of E2PA cross-section upper bounds based on our measurable
fluorescence lower bound.
The C2PEF signal, FC (cnt s−1 ), measured in our experiment can be described by
Zl/2
FC = g′Zλf

NC (z)K(z) dz−l/2

γ(λ)Φ(λ)dλ,

(5.6)

λi

where g (pulses s−1 ) is the pulse repetition rate, l (cm) is the cuvette path length, NC (z) (excitations cm−1 pulse−1 ) is the number of excitations per infinitesimal step dz (cm) along the cuvette′length per laser pulse, K(z) is the geometrical collection efficiency as a function of z (cm) as described in Eq. (5.1), γ(λ) (cnt photon−1 ) is the component transmission efficiency as described in
Eq. (5.4) (where here N = 2) and Φ(λ) (photon excitation−1 nm−1 ) is the differential fluorescenceR∞quantum yield. A proper normalization of quantum yield is used such that Φ = 0 Φ(λ)dλ gives
the value published in literature (Table 5.1) for the total quantum yield of the fluorophore. The
integration limits for the λ (nm) integral are set so that the integral spans over the entire emission
spectrum of the fluorophore. The laser, fluorophore emission, PMT quantum efficiency and filter
spectra for 2PEF measurements are shown in Fig 5.3(a)-(f).
The experimental conditions are such that ground state depletion and beam depletion are
negligible [15], thus we define NC (z) as
1
NC (z) = σC n21/2g
Z Z∞ Z∞−1/2g −∞ −∞ϕ(x, y, z, t)2 dxdydt,

(5.7)

68
where σC (1 GM = 10−50 cm4 s photon−1 fluorophore−1 ) is the C2PA cross section, n (fluorophores cm−3 ) is the number density of fluorophores and ϕ(x, y, z, t) (photons cm−2 s−1 ) is thephoton flux of the laser beam. The factor of 1/2 carries units of excitations per photons absorbed.
The dx (cm) and dy (cm) integrals extend over the entire beam and the dt (fs) integral extends
over the pulse repetition time (1/g). Equation (5.7), is related to the familiar phenomenological
C2PA excitation rate, R (excitations s−1 fluorophore−1 ), described in Eq. (2.6) by
1/2g
Z Z∞ Z∞NC (z) = n

Rdxdydt,(5.8)−1/2g −∞ −∞with R having implied dependence on x, y, z and t.
The temporal and transverse spatial profiles of the laser beam or SPDC beam can be approximated by Gaussian distributions. Assuming the laser is always on, ϕ(x, y, z, t) takes the form


 X


∞
x2
y2
(t + i/g)2
ϕ(x, y, z, t) = ϕ0 (z)Exp −4ln2+
Exp −4ln2,(5.9)∆x(z)2 ∆y(z)2τ2i=−∞where ϕ0 (z) (photons cm−2 s−1 ) is the peak photon flux as a function of z, τ (fs) is the FWHM
pulse duration and ∆x(z) (cm) and ∆y(z) (cm) are the FWHM beam widths. The photon flux has
z dependence because it is focused into the sample. The FWHM beam width in the x direction,for example, varies as a function of z as∆x(z) = ∆x0

p
1 + (z/zR )2 ,(5.10)

where ∆x0 (cm) is the beam FWHM at the focus and zR (cm) is the Rayleigh range.
We can define the average photon rate at the sample Q (photons s−1 ) in terms of the photon
flux ϕ(x, y, z, t),
1/2g
Z Z∞ Z∞Q=gϕ(x, y, 0, t)dxdydt =W,hν(5.11)−1/2g −∞ −∞where W (W) is the average laser or SPDC power and hν (J) is the average energy of an incident
photon. Here we have arbitrarily chosen to use the photon flux at z = 0. The peak photon flux,
ϕ0 (z), can be found from Eq. (5.11) by performing the integration of ϕ(x, y, 0, t) over x, y and t,√
W 4ln(2) 3/2
12 2µϕ0 (z) ==.
(5.12)hνπ∆x(z)∆y(z)gτ
T A(z)

69
The second equality emphasizes, in accordance with Chapter 2 Section 2.3, that the peak photon√
flux can be expressed as 2 2 multiplied by the mean photon number at the sample µ = Q/g(photons pulse−1 ) divided by the effective mode area, A(z) (cm2 ), and the effective pulse duration,p√
T = τ / 2ln(2) (fs). The factors of 2 and 2 scale the photon rate to the effective photon rate at
the location of the beam’s peak in space and time. The effective beam area as a function of z isfound through the x and y integration of the photon flux
A(z) =π∆x(z)∆y(z).
2ln(2)(5.13)

Using Eqs. (5.7), (5.9) and (5.12), and performing the integration over x, y and t, we canrewrite Eq. (5.6) in terms of the laser power

FC =

√


2

ln(2)
π

3/2

Zl/2

σC nW 2
τ g (hν)2

′

K(z)
dz
∆x(z)∆y(z)

−l/2

Zλf
γ(λ)Φ(λ)dλ.

(5.14)λi

To derive the C2PA cross section, we solve for σC in Eq. (5.14),
1
σC = √
2



π
ln(2)

3/2

τ g (hν)2
l/2
n
R
−l/2

FC /W 2′K(z) /(∆x(z)∆y(z))dz

λ
Rf

.

(5.15)

γ(λ)Φ(λ)dλλi

All the parameters in Eq. (5.15) are known for our measurements through experiments,
simulations and specifications. The parameter τ is measured using a SwampOptics Grenouille 8′50-USB, g and hν are specified by the laser manufacturer, n is measured (Appendix B), K(z)is determined through Zemax and experimental verification (Section 5.4), ∆x0 , ∆y0 and zR aremeasured (Appendix F), γ(λ) is calculated based on optics’ specifications (Section 5.4), Φ(λ) is
known from published measurements and FC /W 2 (cnt s−1 µW−2 ) is the fit to our experimental
C2PEF data. Table 5.1 shows values for sample specific parameters and Table 5.2 shows values forapparatus parameters general for all samples. The results of our C2PEF measurements produce
exp
the experimental C2PA cross sections (using Eq. (5.15)), σC, shown in Table 5.3.

If we assume the expected E2PEF signal depends only linearly on photon flux, we can estimate

70
T w o - P h o to n E x c ite d F lu o r e s c e n c e

2 0

L a s e r
E m
B P f ilt e r

A F 4 5 5

4 0

(b )

P M T Q E
S P f ilt e r

4 5
8

3 5

8

3 5

3 0

6

3 0

2 5

6

2 5
4

2 0

1 5
1 0
5
0

Q d o t 6 0 5

4 0

F ilte r O D /R e la tiv e In te n s ity

E ffic ie n c y ( % )4 5

P M T Q u a n tu m

(a )

4

1 5
2

1 0

2
5

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

0

0

9 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

0

9 0 0

(d )

2 0

4 5

F lu o r e s c e in

4 0

8

3 5

8

3 5

3 0

6

3 0

2 5

6

2 5
4

2 0

1 5
1 0
5
0

9 R -S

4 0

F ilte r O D /R e la tiv e In te n s ity

E ffic ie n c y ( % )4 5

P M T Q u a n tu m

(c )

4

1 5
2

1 0

2
5

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

0

9 0 0

0

5 0 0

6 0 0

7 0 0

8 0 0

0

9 0 0

2 0

4 5

R h 6 G

4 0

8

3 5

8

3 5

3 0

6

3 0

2 5

6

2 5
4

2 0

1 5
1 0
5
0

C 1 5 3

4 0

F ilte r O D /R e la tiv e In te n s ity

E ffic ie n c y ( % )

(f)
4 5

P M T Q u a n tu m

(e )

4 0 0

4

1 5
2

1 0

2
5

4 0 0

5 0 0

6 0 0

7 0 0

L a s e r
P M T Q E

A b s

8 0 0

0

0

9 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

0

9 0 0

O n e - P h o t o n E x c it e d F lu o r e s c e n c e

1 5

R h 6 G(h )E m
B P f ilt e r
6

3 5

3 0

F lu o r e s c e in

2 5

3 0

6

2 5

2 0

4

2 0

4

1 5

1 0

2

1 0

2
5

0

5
4 5 0

5 0 0

5 5 0

W a v e le n g t h ( n m )

6 0 0

0

0

4 5 0

5 0 0

5 5 0

6 0 0

0

F ilte r O D /R e la tiv e In te n s ity

E ffic ie n c y ( % )3 5

P M T Q u a n tu m(g )W a v e le n g t h ( n m )Figure 5.3: Spectral overlap summary for two-photon excited fluorescence (2PEF) measurements(a)-(f) of samples (a) AF455 in toluene, (b) qdot 605 in borate buffer, (c) fluorescein in pH 11 water, (d) 9R-S in chloroform, (e) Rh6G in methanol and (f) C153 in toluene and one-photon excited
fluorescence (1PEF) collection efficiency measurements (g)-(h) of samples (g) Rh6G in ethanol and(h) fluorescein in pH 11 water. For 2PEF, the laser (red), fluorophore emission (Em) (magenta),
PMT quantum efficiency (QE) (blue), bandpass (BP) filter (light green) and shortpass (SP) filter(orange) spectra are shown. The laser spectrum is measured using a USB4000 OceanOptics spectrometer. The SPDC spectrum is shown in Fig. 3.3. For 1PEF, the laser (indigo), fluorophoreabsorption (Abs) (light blue) and emission (Em) (magenta), PMT QE (blue) and BP filter (lightgreen) spectra are shown. The PMT QE is indicated along the left vertical axes, whereas all otherspectra use the right vertical axes. For the filters, the right vertical axes indicate the filter opticaldensity (OD), whereas for all other spectra the right vertical axes show a relative intensity. The
relative intensities of the laser, absorption and emission are normalized to the height of the peakfilter OD for the respective plot. The absorption and emission spectra are measured using a spectrophotometer and fluorometer, except for qdot 605 (data taken from ThermoFisher). The PMT
QE is taken from Hamamatsu specifications. All filter spectra are from the manufacturer, except
for the SP filter in the 350-550 nm range (spectrophotometer).

71
Table 5.1: Summary of sample parameters

Samplec × 106(mol L−1 )AF455
Qdot 605
Fluorescein
Rh6G
C153
9R-S

1100
8
1100
1500
1100
390

λ
Rf

Φ [Ref.](photon excitation−1 )γ(λ)Φ(λ)dλ/Φλi

0.67 [79]0.74 ± 0.04 [80]0.93 [81]0.90 [82]0.82 ± 0.04 [83]0.66 [47]0.05150.02850.07890.04840.05800.0157the E2PEF signal FE (cnt s−1 ) generated in our experiment as
ZzR

′

Zλf

K(z) dz

FE = gNE−zR

γ(λ)Φ(λ)dλ,(5.16)

λiwhere NE (excitations cm−1 pulse−1 ) is the number of excitations per infinitesimal step dz along
the cuvette length per laser pulse. The parameter NE is defined as
1
Q
NE = σE T n,2
g

(5.17)

where σE (cm2 fluorophore−1 ) is the E2PA cross section and T is the transmittance of the photons
through all the optics between the center of the crystal and the center of the sample. The parameter
T is included in NE but not NC because of the result found in Chapter 2 Section 2.6 (in this Section,
ϕ and Q are unlabeled but represent the values at the sample). As we mentioned in Section 5.4
and Chapter 2 Section 2.3, the dependence of the E2PA rate on the spatial overlap of photons iscontained in the cross section (unlike for C2PA) and thus a cross section is only valid for a beam of
constant entanglement area and thus size. Our SPDC beam is not collimated, instead we attempt
to compensate for the changing entanglement area by setting the limits of the z integral from −zR
to zR , which is the region that we expect the majority of a potential E2PEF signal to arise fromand should have fairly uniform entanglement area and time.

72
Table 5.2: Summary of apparatus parameters
Parameter
∆x0
∆y0
zR
τ
g
′
K(z)
Q
T
F LB

unit

Laser

SPDC

µm
µm
mm
fs
106 pulses s−1

49
49
5.1
111

51
84
0.4
1040
80

0.154
2 erfc (2.78(|z(cm)| − 1.51))
N/A
8.9×109

photons s−1

N/A

0.76

cnt s−1

0.22

We can rewrite Eq. (5.16) using Eq. (5.17),
1
FE = σE T Qn
2

ZzR

Zλf

′

K(z) dz−zR

γ(λ)Φ(λ)dλ.

(5.18)λi

To place an upper bound on the E2PA cross section we replace FE in Eq. (5.18) with the measurable
fluorescence lower bound F LB (cnt s−1 ) and solve for σE , which becomes the cross-section upper
UB (cm2 fluorophore−1 ),bound, σE

2F LB

UB
σE
=

T Qn

zRR
−zR

′

K(z) dz

λRf.

(5.19)

γ(λ)Φ(λ)dλλi

F LB is measured (Section 5.6), T is calculated based on the manufacturer’s specifications, Q is′measured (Chapter 3 Section 3.4) and the parameters n, K(z) , γ(λ) and Φ(λ) are found in themethods described for C2PEF. All parameters are listed in Tables 5.1 and 5.2.
In order to generate a curve for E2PEF as a function of the mean photon number (theFE
is solved for in Eq. (5.18) and a selected σE is used. This slope
diagonals in Fig. 5.4), the slope Q/g

is multiplied by the mean photon number on the horizontal axes.
The uncertainty on our C2PA cross sections and E2PA cross-section upper bounds are calculated by propagating the errors in all the measured and calculated parameters that go into either
Eq. (5.15) or Eq. (5.19). We multiply these values by the coverage factor (k = 2). The uncertainty
in C2PA cross sections is then ≈ 28% and for E2PA cross-section upper bounds ≈ 24%.

73

5.6

Data Acquisition
In this section we describe the details of data acquisition for C2PEF and E2PEF measure-ments. First we describe our fluorescence background subtraction method. Next we describe how
C2PEF measurements are performed. Afterwards we describe the choice of integration times forE2PEF measurements and how those measurements are performed. Lastly we describe how themeasured quantities, including the measurable fluorescence lower bound, are determined.
The laser and SPDC beams are optically chopped to perform on-the-fly background subtraction on the fluorescence signal. The timetagger histogram, which shows counts registered on the
PMT as a function of time, is separated into background and signal portions. The background
portion (chopper blade blocking beam) is subtracted from the signal portion (chopper blade passing beam). We calibrate this background subtraction method using a strong C2PEF signal. For≈ 5% of the measurement runtime the chopper blade is neither completely blocking nor passing
the beam; this portion of the measurement is discarded.
For C2PEF measurements, the laser power is controlled using a motorized half-wave plate(HWP3 in Fig. E.1). The power is measured (Thorlabs S130C power sensor and PM100D meter) by
flipping the sensor into the beam using a motorized flip mount that ensures repeatable positioning.
The power sensor and meter are compared with a calibrated photodiode to determine the correction
factor necessary for absolute power readings. At each power, 3 − 5 C2PEF measurements are
performed. The integration times at higher powers are 30 seconds and at lower powers are 30
minutes.
We characterize the stability of the fluorescence measurements using an Allan deviation
analysis, and base our measurement integration time for the E2PEF measurements on the result.
To do this, we place the 1.10 × 10−3 mol L−1 fluorescein sample in the cuvette, unshutter the laser
beam and measure the C2PEF signal every minute for one 14 hour period overnight and one 11
hour period during the day. We use this data to check the Allan deviation at various integration
times. The Allan deviation is found to have a minimum at 45 minutes integration time.

74
For E2PEF measurements, the SPDC pump laser power is set to 30 mW and monitored
periodically. Three E2PEF measurements are performed on each sample. These measurements
are each 45 minutes long. We also block the beam periodically and take a 45 minute backgroundmeasurement. We compare these measurements with those with the beam unblocked to look forsignificant changes in the signal. We find no changes.
The measurable fluorescence lower bound, F LB , is assigned by first checking the resultsof C2PEF measurements at low photon flux and then by measuring “zero signal” with SPDC
excitation. We measure C2PEF at rates as low as 0.38 ± 0.24 cnt s−1 that agree well with thequadratic fit of the C2PEF data measured at higher excitation flux. This sets our confidence insignals at least as low as 0.38 cnt s−1 . Next, we measure zero signal to determine what we should
expect in the absence of signal. To do this, we place the 1.10 × 10−3 mol L−1 fluorescein sample
in the cuvette, unshutter the SPDC beam and subsequently block the SPDC beam using black
aluminum foil tape (Thorlabs T205-1.0) placed after filters F3 in Fig. E.1. We then acquire datafor 405 minutes, or nine 45 minute measurements. The purpose of blocking the beam insteadof shuttering it is to serve as an additional check for scattered light entering the detector. The
fluorescein sample aids in this purpose by serving as a source that could be excited by either−1scattered or background light. The average of these measurements is 0.04+0.22−0.04 cnt s . It was clear

from these measurements that no stray signals enter the detector. From this, our F LB is set to0.22 cnt s−1 (2σ from zero) with ≈ 95% confidence. The value of F LB sets the vertical position of
the light green region in Fig. 5.4.

5.7

Results and Discussion
We measure C2PEF over a range of photon fluxes for all six fluorophores. We use the fit toour experimental data to derive C2PA cross sections (details in Section 5.5, values in Table 5.3).
The values strongly agree with the cross sections reported in literature. For all six fluorophores weare unable to discern an E2PEF signal. We use our measurable fluorescence lower bound to derive
upper bounds on the E2PA cross sections (details in Section 5.5, values in Table 5.3).

75
(a )

M e a n P h o to n N u m b e r ( p u ls e
1 0 2
4

1 0
F lu o r e s c e n c e ( c n t s -1 )

-1

1 0 3

1 0 4

1 0 5

1 0 6

)

1 0 3
1 0 2

-2

1 0

2 .

1 x

1 0

1 0 2

1 0

1 0 0

-2

1 0
1 0

1 8

1 9

1 0

1 0

2 0

2 1

1 0

2 2

1 0

1 0

1 0

8 x

4 .

2

1 0

cm
4

-2

2

1 0 -2

2 3

1 0

1 0

1 8

1 0

1 9

1 0

2 0

2 1

1 0

-2

cm
7

2 2

1 0

1 0 23

(d )
1 0 2

1 0 4

1 0 3

1 0 4

1 0 5

1 0 6

F lu o r e s c e in
F lu o r e s c e n c e ( c n t s -1 )

1 0 7

cm
3

-2

1 0 -1

(c )

1 0 3
1 0 2

1 0

-2

cm
3

1 0 7

1 0 4

1 .

0 x

1 0

5

-2

1 0

1 0 0

-2

1 0 4

1 0 5

1 0

2

-2

cm

2 .

-2

1 0

1 0

1 0 0
2

9 c m

1 0

0 x

5

-2

cm

2
-2

1 0 191 0 201 0 21

1 0 221 0 23

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

1 0 7

(e )

1 0 -2cm
4-2

1 0 -11 0 18

1 0 7

2

1 0 1

1 0 -1

1 0 6
2

2

1 0 2

cm
6

1 0 3

1 0 3

cm

2

1 0 1

1 0 2
9 R -S2

2

1 0 -2cm
8

1 0 181 0 191 0 201 0 211 0
1 0 221 0 23

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

1 0 7

(f)
4

1 0

R h 6 GF lu o r e s c e n c e ( c n t s -1 )

cm
1

1 0 0

1 0 -1
1 0 -2

-2

1 0

2

9 c m

)

1 0 6
2

1 0 1

cm
6

-2

1 0 5

2

2

1 0 1

1 0 4

1 0 3

cm
5

1 0 3

Q d o t 6 0 5
2

-2

1 0 2
4

1 0

cm
3

M e a n P h o to n N u m b e r ( p u ls e

(b )

1 0 7

2

A F 4 5 5E x c ita tio n
L a s e r
S P D C

-1

2

1 0

-2

3

cm

1 .

1 0

2 x

-2

5

cm

1 0

1 0 0

-2

6

1 0

1 0 201 0 21

-2

1 0

3

cm

1 .

-2

1 0 22P e a k P h o to n F lu x ( c m

-2

s

5

-2

6

cm

cm
2

1 0 -1
1 0 23

-1

1 0

1 0 0

cm
9

1 0

6 x

-2

2

1 0 1

cm

1 0 -11 0 19

2

1 0 2

2

1 0 18

2

1 0 3

2

1 0 1

1 0 -2C 1 5 3
2

1 0 3
1 0 2

1 0 4

)

1 0 -21 0

1 0 181 0 191 0 201 0 21-2cm
9

1 0 22P e a k P h o to n F lu x ( c m-2s

1 0 23-1)

Figure 5.4: Measured (blue data points), fit (blue solid lines) and calculated (black solid and purple
dashed lines) fluorescence signal (left vertical axis in cnt s−1 ) for (a) 1.10 × 10−3 mol L−1 AF455
in toluene, (b) 8 × 10−6 mol L−1 qdot 605 in borate buffer, (c) 1.10 × 10−3 mol L−1 fluoresceinin pH 11 water, (d) 3.90 × 10−4 mol L−1 9R-S in chloroform, (e) 1.50 × 10−3 mol L−1 Rh6G inmethanol and (f) 1.10 × 10−3 mol L−1 C153 in toluene. The bottom horizontal axis corresponds
to the peak photon flux (photons cm−2 s−1 ) of the coherent source (laser) (blue data points) or
SPDC source (red data points). On the upper horizontal axis we show the SPDC mean photon
number (photons pulse−1 ), which corresponds to the peak photon flux on the lower horizontal axis.
A signal below 0.22 cnt s−1 is indistinguishable from zero (green region). All E2PEF measurements
produce a null result. Solid diagonal black lines show the calculated count rate expected for various
potential σE in order-of-magnitude increments (σE noted along selected lines) for each fluorophore,assuming that the absorption rate is composed of only the linear photon-flux-dependent term. The
UB (noted in purple) for
purple dashed diagonal line represents the calculated count rate using σEeach sample.

76lit ), measured C2PA cross sections (σ exp ),Table 5.3: Summary of literature C2PA cross sections (σCC
UB ), estimates for the E2PA cross sections (σ est )measured E2PA cross-section upper bounds (σEE(based on Eq. (2.11) using Te = 1620 fs and Ae = 2.1 µm2 ) and measured quantum advantage
upper bounds (QAUB ).

Sample

exp
σC

lit
σC
[Ref.](GM)
AF455
Qdot 605
Fluorescein
9R-S
Rh6G
C153

350 ± 30 [17]27000 ± 8000 [77]21 ± 2 [17]27.9 [47]78 ± 7 [17]17 ± 2 [17]660 ± 180
46000 ± 13000
13 ± 422 ± 651 ± 1414 ± 4UB
est
σE
× 1025σE× 1030
2−1(cm fluorophore )2.1 ± 0.5
480 ± 120
1.0 ± 0.2
20 ± 61.2 ± 0.3
1.6 ± 0.4

190
14000
3.8
6.5
15
4.1

QAUB
410 ± 140730 ± 240
2000 ± 700
7000 ± 2000
1100 ± 400
2400 ± 800

Figure 5.4 shows measured fluorescence count rates as a function of peak photon flux for bothlaser (blue symbols) and SPDC (red symbols) excitation for all six fluorophores on log-log plots. For
all samples, we find the fit (blue line) to the C2PEF signal to have a quadratic power dependence(with exponents in the range 1.95 − 2.05); the signals are thus free of spurious events such as 1PEF
or scattered light. For AF455 (Fig. 5.4(a)), we measure C2PEF down to the lowest peak photon
flux of all the samples, 1.3 × 1021 photons cm−2 s−1 , which is only 620 times larger than our SPDC
peak photon flux. The C2PEF of fluorescein, 9R-S, Rh6G and C153 (Fig. 5.4(c)-(f)) is observed at
a minimum flux approximately a factor of 10 higher than for AF455 and qdot 605 (Fig. 5.4(a),(b)).
This minimum flux could be extended to lower values (but not as low as AF455 or qdot 605) if alonger integration time were used for the measurements. Fluorescence signals as low as 0.22 cnt s−1
should be measurable in our experiment (Section 5.6). We denote this measurable fluorescence lower
bound as F LB . A signal below this level is masked by the noise floor. As mentioned above, we do
not observe E2PEF for any of the studied samples. This is demonstrated by the SPDC excitation
data points (shown in red) lying below the noise floor (the green region in Fig. 5.4). For thesemeasurements we use 30 mW pump power, which is just below the damage threshold of the SPDC
crystal, to generate an SPDC peak photon flux of 2.1 × 1018 photons cm−2 s−1 .
The C2PEF measurements are averaged for each sample at each power. The E2PEF measurements are averaged for each sample. These averages are displayed on Fig. 5.4. The corresponding

77
vertical error bars are assigned in a systematic way. First, we compare the standard deviation of
the set of measurements to the sets’ uncertainty due to Poisson counting statistics. The larger ofthese two values is multiplied by two (coverage factor k = 2) and used for the vertical error bar.
The horizontal error bars correspond to the uncertainty in peak photon flux (bottom axis),
which is larger than the uncertainty in mean photon number (top axis). This larger uncertainty
arises because of the additional uncertainty in the beam size and pulse duration. The uncertainty
in the mean photon number, beam size and pulse duration is propagated to give an uncertainty inpeak photon flux. A coverage factor k = 2 is again used to achieve ≈ 95% confidence that the true
value lies within the bounds set by the error bars. We note that the conversion factor from mean
photon number to peak photon flux is different for the coherent source compared to the SPDC
source because of its shorter pulse duration and smaller beam size. The laser conversion factor
differs from that for SPDC by a factor of 16.7, thus Fig. 5.4 only shows the SPDC mean photon
flux on the top horizontal axes.
We use our experimental characterizations and the component of the E2PA rate that depends linearly on excitation flux to calculate E2PEF signals for various potential values of σE(Section 5.5). The results of these calculations are displayed as black diagonal lines in Fig. 5.4,
with the corresponding σE value noted along selected lines. The purple dashed diagonal line corresponds to the fluorescence signal calculated using the cross section that produces F LB at thepeak photon flux of our SPDC source. We denote this cross section the E2PA cross-section upper
UB . A summary of σ UB values is given in Table 5.3 and written in purple along the dashedbound, σE
E
UB of 1.0 ± 0.2 × 10−25 cm2 fluorophore−1 .
diagonal lines. The sample fluorescein has the lowest σE
UB for Rh6G, C153 and AF455 differ by less than or nearly a factor of two from
The values of σEthat for fluorescein. Many of the parameters for these four samples are similar in magnitude: concentration, quantum yield and the overlap of the emission spectra with the fluorescence collection
system’s transmittance spectrum (details on these parameters in Section 5.5). For 9R-S, the upperbound is one order of magnitude larger, which results from the poor overlap of the emission andsystem transmittance spectra. For qdot 605, the upper bound is a factor of 24 larger than for

78
9R-S. This is a result of poor spectral overlap, in addition to a sample concentration two orders ofmagnitude lower than that used for all other samples. As recommended by the supplier, we use theconcentration of qdot 605 as received to avoid compromising the chemical stability of the sample.
The upper bounds our measurements place on σE range from 10−25 to ≈ 5 × 10−23 cm2fluorophore−1 .

These are in stark contrast to the previously reported σE values of 10−21 −10−16 cm2 fluorophore−1 shown in Table 2.1 and Fig. 2.3. A particularly illuminating comparison can be made between our result and the published result for samples 9R-S and Rh6G. Using
the previously reported σE values, we estimate the expected E2PEF count rate in our setup. Assuming sample 9R-S has σE ≈ 2.4 × 10−19 cm2 fluorophore−1 [47], our calculations predict an
E2PEF signal of 2.6 × 104 cnt s−1 . For Rh6G, a value of σE ≈ 1.5 × 10−21 cm2 fluorophore−1 [48]predicts an E2PEF signal of 2.7 × 103 cnt s−1 . We note that Ref. [48] used 1064 nm excitation
whereas we excite at 810 nm, however, if σE follows the same dependence on excitation wavelength
as σC , 810 nm excitation should be more efficient by a factor of 7 [16]. In either case, we actually
measure a signal that is indistinguishable from zero, which is at least three to five orders of magnitude smaller than expected based on prior reports. We are able to reach such a high cross-section
sensitivity in part because of our relatively large incident SPDC photon rate (8.9×109 photons s−1 ).
Although Ae and Te likely vary between experiments, we have no reason to believe theseparameters alone differ by the many orders of magnitude required to explain this discrepancy.
However, because Ae and Te alter σE and because the role of these parameters is not completely
understood for E2PA in molecules, Ae and Te should be reported alongside σE values wheneverpossible. There are other experimental parameters that vary between experiments, such as pumplaser and SPDC spectral and temporal widths and the SPDC crystal characteristics. The effects ofthese differences are not well known and need more thorough study.
est . These
Table 5.3 also shows estimates of the E2PA cross section for the six fluorophores, σE
expestimates are based on the relation given in Eq. (2.11) using our derived σC
given in Table 5.3 andour estimates of Te and Ae specified in Section 5.3. We use the lower bound of Ae in this estimation
est could take on. Although we do not anticipate Eq. (2.11) to yield
to show the largest value σE

79
an exact result, this estimate can provide useful insight about σE values in a similar manner to
est values are three to fivethe estimates of ionization cross sections for atoms in Ref. [84]. These σEorders of magnitude below our established cross-section upper bounds. These estimates provide areference for the cross-section sensitivity necessary to observe E2PA.
We use our C2PEF and E2PEF results to determine an upper bound on the “quantum advantage” of 2PA (QAUB ). As previously mentioned (Chapter 2), we define the quantum advantage(QA) as the ratio of the minimum photon flux required to observe C2PA to that for E2PA. By
extrapolating our C2PEF fit to F LB for the sample AF455, for example, we determine that C2PEF
should be measurable down to 8.5 × 1020 photons cm−2 s−1 . E2PEF is not measurable at our
maximum SPDC photon flux, 2.1 × 1018 photons cm−2 s−1 , but might be measurable at a higher
photon flux. Thus, QAUB of 2PA for this sample is 410. Values of QAUB for all the samples(Table 5.3) range from 410 − 7000, in contrast with QA of nearly 1010 in previous reports [52, 54].
It is worth mentioning that the QA can be increased if Ae and Te are decreased while all otherexcitation parameters are held fixed, however a many-orders-of-magnitude increase is unlikely.
There are other publications in this field that support our findings. In particular, Ashkenazy
et al. [85] argued that using “typical” values of Ae (50 µm2 ) and Te (50 fs), they can estimate
σE ≈ 10−29 cm2 fluorophore−1 for metallic nanoparticles with a large C2PA cross section (σC ≈100 GM at 1050 nm). Cross sections of this size are in agreement with our established boundsof σE . Another interesting example is provided in the recent work by Li et al. [86] who used asingle setup to measure both C2PEF and squeezed-light 2PEF (SL2PEF) of the samples DCM
in dimethyl sulfoxide and fluorescein in pH 13 water. The squeezed light generated by four-wave
mixing in a Rubidium vapor cell was varied over the range of 1013 − 1016 photons s−1 (compare
to our ≈ 1010 SPDC photons s−1 ). The SL2PEF signals from DCM and fluorescein are factorsof ≈ 2.0 − 2.8 and ≈ 47 larger, respectively, than the C2PEF signals at the same excitation
flux. The authors did not report values for cross sections. However, these significant but modest
enhancements and the fact that measurements were performed with a squeezed light source thatprovides orders-of-magnitude higher photon rate than an SPDC source, are consistent with the80upper bounds established in this study.

5.8

Conclusions
In this chapter, we discussed important aspects of designing and implementing a fluorescence-

based E2PA measurement. We presented an experimental apparatus for measuring E2PEF andC2PEF in nearly identical experimental conditions. The results from C2PEF serve as a vitalreference point for the capability of our fluorescence system. Although we do not observe an
E2PEF signal, our results set upper bounds on σE of the six chosen fluorophores in the range of
10−25 − 5 × 10−23 cm2 fluorophore−1 . Two of these samples have published σE values that are fourand five orders of magnitude larger than the upper bounds we report.
We emphasize that σE depends on spatio-temporal properties of the excitation source, unlike
σC . Without knowing the entanglement area and entanglement time, there is significant ambiguity
in comparing cross sections measured in different experimental apparatuses. For our source, weestimated a range within which our entanglement area is constrained, 2.1 − 13, 700 µm2 , and weestimated the entanglement time, 1620 fs, based on our measured SPDC spectrum and estimatedgroup delay dispersion. While we had hoped to measure these quantities directly, in lieu of this we
made explicit the details of our setup and the assumptions that went into the estimation of thesequantities.
Our results differ significantly from previous E2PA publications using SPDC excitation.
Our evidence indicates that E2PA cross sections are orders of magnitude smaller than previously
claimed [51, 53, 63, 52, 50, 54, 47, 64, 55, 49, 48]. As we demonstrated in this report, the clarification of the inconsistencies in the field is underway. This is an important step forward in thequantification of the achievable “quantum advantage” and thus the merit of E2PA for spectroscopy
and imaging applications.

Chapter 6

Bounding Entangled Two-Photon Absorption Cross Sections with Sensitive
Transmittance Measurements

6.1

Publication Note
This chapter is an expansion of the work summarized in:[60] Mazurek, M.D., Parzuchowski, K.M., Mikhaylov, A., Nam, S.W., Camp, C.H., Gerrits, T.,
Jimenez, R. and Stevens, M.J., 2021, May. Bounding entangled two-photon absorption with sensitive transmittance measurements. In CLEO: QELS Fundamental Science (pp. FM3N-2). Optica
Publishing Group.

6.2

Introduction
In our first transmittance experiment (Chapter 4) we set upper bounds on entangled two-

photon absorption (E2PA) cross sections using a transmittance-based measurement scheme that
included an interferometer for a time delay. The sensitivity to changes in transmittance was atthe 1% level and was limited because of residual interference artifacts. In this experiment we
improve on that experiment to achieve a higher sensitivity. We incorporate a mirror mounted ona piezo-electric transducer in the delay arm of the interferometer for quick scans over interference
fringes, we add a quadrant detector to monitor and account for pump beam power and pointing
stability, we improve the data acquisition technique to allow for long but stable measurements andwe improve the data analysis technique. We achieve a sensitivity of 0.05%. We take measurementswith seven different molecules in solution at multiple concentrations, and observe no statistically82
significant change in transmittance. We set upper bounds on the E2PA cross sections in the range
10−22 − 10−19 cm2 absorber−1 .

6.3

Operating Principle of the ExperimentFig. 6.1 displays an illustration of our experimental setup, which we use to explain theoperating procedure of our measurement technique. A photon pair passes through an interferometerwith variable time delay. Varying the photons’ temporal overlap affects the rate of 2PA in thesample (Chapter 2 Section 2.6). By using polarizing beamsplitters we ensure that any light thatreaches detector D1 (D2 ) arrives via the stationary (delay) arm of the interferometer. (In reality,
due to experimental imperfections, a small fraction of the light that travels through the delay arm
may “leak-into” D1 . For simplicity, in this section we assume this fraction is negligible. However,we account for non-zero leakage in our final analysis, and we explain how in Section 6.5.)

Figure 6.1: Cartoon of our experimental setup. Photon pairs are created in the SPDC crystal, after
which they are probabilistically split via half-wave plate HWP1 and a polarizing beamsplitter (PBS)
into two paths through an interferometer before reaching a sample. The interferometer’s delay
arm D contains a moveable mirror which can add a time-delay τ relative to the interferometer’s
stationary arm, S. One-photon loss between the crystal and sample is represented by beamsplittersinside the interferometer with transmission probabilities T S and T D . If both photons arrive at the
sample, they will be absorbed via 2PA with probability pE2P A . Half-wave plate HWP2 is rotatedto ensure that all photons which traversed through the sample via the stationary arm will bedirected to detector D1 , and photons from the delay arm are sent towards D2 . One-photon loss
between the sample and detectors (including internal detector inefficiencies) is represented with thebeamsplitters with transmission efficiencies η1 and η2 .

The main advantage of this technique is that we can vary the 2PA rate without changing the83
contents of—and therefore the one-photon loss through—the cuvette. Furthermore, varying thetime delay does not affect light that traverses the stationary arm, and thus the total amount of
one-photon loss between the crystal and D1 should remain constant at all time delays. As a result,if we measure a delay dependence in the click rate at D1 , we can infer that 2PA has occurred.
To infer a specific 2PA rate from this delay-dependent measurement, we compare the expected
count rates at τ = 0 (where the 2PA rate is maximized) to those at a very large delay τ = T , (where
the 2PA rate is minimized). The probability that any individual photon reaches the cuvette via
the stationary arm is equal to the total transmission efficiency from the crystal to the cuvette,
S where R
S is the transmission
S = R
TXS
P BS TP BS is the reflection efficiency of the PBS and T

efficiency of all other optics between the crystal and the cuvette along the stationary arm path.
D =T
D whereSimilarly, a photon will reach the sample via the delay arm with probability TXS
P BS T

TP BS is the transmission efficiency of the PBS and T D is the transmission efficiency of all otheroptics between the crystal and the cuvette along the delay arm path. If two photons arrive at
the sample with negligible time delay between them, they will excite the sample via 2PA withprobability pE2P A . The one-photon transmission efficiency through the cuvette and to detector D1(D2 ) (and including the detector’s internal quantum efficiency) is η1 (η2 ). There are three paths
the photon pairs may take through the setup which can result in a click at D1 —both photons cantravel through the stationary arm, one photon can travel through each arm, or one photon may
travel through the stationary arm while the other is lost. Assuming the source emits N photon
pairs each second (and assuming the source is much more likely to emit a single pair than multiple
pairs at the same time), the average click rate at D1 is given by:
S1 (τ ) = N

h

S
TXS

2

(1 − pE2P A ) 2η1 − η12


(6.1)

 
S
D
S
S
D
+ 2TXS
TXS(1 − f (τ )pET P A ) η1 +2TXS
1 − TXS− TXS
η1 .
The function f (τ ) captures the time-dependence of the 2PA rate as a function of time delay: atzero delay f (0) = 1 and at large delay f (T ) = 0. The normalized change in click rate at these two
1 (0))
time delays (also called the change in transmittance) is ∆S = (S1 (TS1)−S. Using Eq. (6.1) and
(T )

84
rearranging, we find the two photon absorption probability, pE2P A as:
pE2P A =S η )∆S(2 − TXS
1.
SD∆STXS (2 − η1 ) + 2TXS

(6.2)

S and T D can be independently characterized, and the product T S η can be
The efficiencies TXS
XS
XS 1

measured using a version of Klyshko’s efficiency equation [87] (see Section 6.5.1 for details).
While pE2P A is defined as the probability of exciting one molecule in the sample if exactly
one pair of photons interacts with the sample, it can also be understood as the total 2PA rate thatwould occur if pairs reached the sample at an average rate of one pair per second. The E2PA rateis defined as
1
RE = σE Qnl,
2

(6.3)

where σE (cm2 molecule−1 ) is the E2PA cross section, Q (photons s−1 ) is the single photon rate
at the sample, n is the number density of the molecule in solution and l is the cuvette length.
Therefore, once pE2P A is characterized, we can solve for the E2PA cross section with:σE =pE2P A.
nl(6.4)

To first order, the measured change in click rate ∆S is linearly proportional to the 2PA probability
pE2P A and also to the E2PA cross section σE . Therefore, a sensitive measurement of ∆S canprovide a sensitive characterization of σE .

6.4

Experimental Setup
Our experimental setup consists of a photon-pair source, an interferometer, a sample anda single-mode detection stage (Fig. 6.2). Here we give a brief overview of the setup, a thoroughdescription of the components is given in Appendix G. The photon-pair source is as describedin Chapter 3, Section 3.3. In this experiment we pick off a small fraction of the 405 nm pumplight and measure it with a quadrant photodiode to monitor fluctuations in total beam power andpointing direction. Due to chromatic dispersion between the SPDC crystal and sample, wavelengthcomponents far from the central wavelength contribute minimally to 2PA—keeping them would

85
decrease the overall sensitivity of our measurement, therefore we filter the photon pairs so their
marginal spectrum has a full width at half maximum (FWHM) of about 12 nm.

Figure 6.2: Schematic of the experimental setup. The 810 nm laser (dark red) is frequency doubled(blue) via second-harmonic generation (SHG) and focused into a type-0 ppKTP crystal to generate
collinear SPDC photon pairs at 810 nm (light red). Filters (F) are used to remove the remaining
405 nm light. A half-wave plate (HWP1) rotates the polarization of the photons such that they
are split probabilistically and equally at a polarizing beamsplitter (PBS). Half of the light travels
through the delay arm of the interferometer. This arm has a mirror mounted on a piezo which itself
is mounted on a motorized delay stage for time delay control. The other half travels through thestationary arm. The light is recombined at a PBS and focused into a sample. The transmitted lightis collimated and detected at the single-mode detection stage. HWP2 switches between detection
modes.

We measure the joint spectral intensity (JSI) of our source using the time-of-fight spectrometer described in Chapter 3 Section 3.6. The result is shown in Fig. 6.3(a). The bandwidth is set bya bandpass filter placed after the SPDC source. The JSI projections onto vertical and horizontal
axes in Fig. 6.3(b) are used to estimate the marginal FWHM bandwidths of the photons, which are
11.1 nm and 12.8 nm. We estimate the total group delay dispersion (GDD) of our optical elements
between the ppKTP crystal and the center of the sample to be approximately 2800 fs2 at 810 nm.
We use this GDD to simulate the joint temporal intensity (JTI) using the procedure described in
Chapter 3 Section 3.7. The result is shown in Fig. 6.3(c). The FWHM of the projection of the
JTI onto the antidiagonal axis (Fig. 6.3(d)) is used to estimate the entanglement time, which is
153 fs. We note that this size of GDD has a small effect on our entanglement time due to thenarrow bandwidth of our filtered SPDC. The pulse duration of the SPDC can be estimated by the
projections of the JTI onto the horizontal and vertical axes and is ≈ 190 fs.

86

Figure 6.3: (a) Measured joint spectral intensity (JSI) where λS,I are the signal and idler wavelengths. (b) The JSI is projected onto the horizontal axis and vertical axis showing the signal(blue) and idler (red) spectra respectively. The FWHM of the signal and idler spectra are 11.1 and12.8 nm respectively, which is set by an optical filter. The overlap of the spectra is evident in thedark red region. (c) Calculated joint temporal intensity (JTI) obtained through a discrete Fouriertransform. (d) Projection onto the antidiagonal axis, tS -tI , of the JTI shown in (c) (blue) and fora transform-limited (β = 0 fs2 ) pulse (red). The FWHM of these projections are 153 fs and 131 fs.

After the source stage of the experiment, a half-wave plate (HWP1) rotates the photon pairs
to diagonal linear polarization, before a polarizing beamsplitter (PBS) probabilistically splits thepairs into two arms of a Michelson interferometer. The end mirror in the delay arm is mounted
on a piezo-electric transducer which sits on a delay stage, for fine and coarse control of delay. A
quarter-wave plate (QWP) in each interferometer arm rotates the polarization so that light fromboth arms is recombined into a single beam before it is focused with a 75 mm focal length lens into
a 10 mm long quartz cuvette containing the sample under test. The beam size at the focus andthe Rayleigh range is measured using an 810 nm diode laser back-propagated through the fiber of

87
the single-mode detection stage and all the optics to the sample position. The FWHM beam sizeis ≈ 59 µm and the Rayleigh range is ≈ 9.1 mm. The full set of samples we measure is summarizedin Table 6.1.
The light is collimated after the sample, and then travels through the single-mode detection
stage (Chapter 3 Section 3.3). Rotating half-wave plate HWP2 to 0◦ ensures that all light from thestationary (delay) arm is sent towards detector D1 (D2 ), and this is the configuration we measure
2PA in. We can rotate HWP2 to 22.5◦ to observe interference in the click rates at each detector, as
well as the coincidences between them (Fig. 6.4). These interference patterns also indicate the zerodelay position of our interferometer. Critically, the visibility of the interference pattern quantifies
the overlap of the beams from both interferometer arms—a high visibility indicates high temporaland spatial overlap. In Chapter 4 Section 4.5 we discussed how to model this interference pattern.
To characterize each sample, we compare D1 ’s click rate when measured at zero delay, andat a 4 mm delay (corresponding to 27 ps). We perform many measurements at each delay, andrepeatedly switch between them. Due to small alignment and manufacturing imperfections in thepolarization optics in our setup, a small amount of light from the delay arm leaks into detector D1 .
As a result, with HWP2 set to 0 degrees and near zero delay, we measure small residual interference
fringes in the click rate at D1 , with visibility on the order of ≈1%. Since these residual fringes canlead to a roughly 1% difference in the click rates at both delays, we average over them using two
methods. First, we constantly vary the position of the delay-arm mirror via a piezo-electric mountwhich is driven by a triangular voltage wave with amplitude set to average over five interference
fringes. Second, we implement further averaging by performing measurements at 14 different motor
positions around each delay value (which corresponds to averaging over roughly 2 fringes). Due
to differences in the delay-arm alignment at both delays, the amount of delay-arm light that iscoupled into D1 is different at the two delays. To measure the average leakage rate we also performbackground measurements at both delays, by physically blocking the interferometer’s stationary
arm with a shutter. Finally, to monitor drifts in the power and pointing-direction of the 405 nm
beam, we constantly measure the outputs of the quadrant photodiode. We measure each sample

88

Figure 6.4: Typical interference pattern measured with HWP2 set to an angle of 22.5◦ . Top panel
shows that the interferometer arms are considered balanced over a range of less than 0.1 mm ofstage delay, much less than the 2 mm of stage delay (4 mm total delay) that we move to for ‘largedelay’ measurements. The bottom panel displays individual interference fringes near zero delay,with a visibility V ≈ 98% — this high visibility implies that, inside the sample, light which has
arrived via the stationary interferometer arm has very high modal overlap with light that that has
arrived via the delay arm.

for a total of 34-65 hours corresponding to a total of roughly 1000-1900 independent measurements
at each delay.

6.5

Inferring 2PA Probability
In our experiment, we measure the parameter ∆S which is the relative change in the detection

rate at detector D1 as we move the interferometer from large delay to zero delay. Our task is to infera 2PA rate from a measurement of ∆S. In Section 6.3 we explain how to perform this inference for an89experiment which satisfies two ideal assumptions; first that all the light that travels the stationary(delay) arm ends up at detector D1 (D2 ) and second that the detectors have no background counts.
In this section we relax both of those assumptions and find a modified form of Eq. (6.2) which we
use to infer an upper bound on pE2P A from our measurement of ∆S.
To perform this inference, we begin by deriving an equation for the expected count rate S1
at detector D1 as a function of the transmission efficiencies through various sections of the setup.
For each photon pair generated at the crystal there are five distinct paths the pair can take which
may lead to a click at D1 . First, both photons could make it to the sample, and this can happen
one of three ways: either both photons traverse the stationary arm (which we label path SS), bothtraverse the delay arm (path Dτ Dτ ) or one photon traverses each arm (path SDτ ). Here the τsubscript on the delay arm labels if the interferometer is in the balanced (τ = 0) or unbalanced(τ = T ) position. If both photons reach the sample, they will make it through the sample withprobability (1 − ppath
E2P A ) — the path label allows us to account for the 2PA probability to dependon the specific path the two photons take before reaching the sample. Second, it is possible that
only one photon reaches the sample, which can happen one of two ways: one photon is lost and theother takes the stationary arm (path S) or one photon is lost and the other takes the delay arm(path Dτ ). When only one photon reaches the sample, no 2PA can occur. If one or two photons
are transmitted through the sample, D1 will click with a probability that depends on the number ofphotons (i.e. one or two) and the specific path the photon(s) travelled. After the sample, a single
photon will make D1 click with probability η1path .
To find an equation for the expected count rate at D1 , we start with equations for the probability that the pair of photons traverses each of the five paths explained in the above paragraph.

90
These are:S 2
pSS = (TXS
) ,

(6.5)

D 2
pDτ Dτ = (TXS
) ,

(6.6)

S
D
pSDτ = 2TXS
TXS
,

(6.7)

S
S
D
pS = 2TXS(1 − TXS
− TXS
),

(6.8)

D
S
D
pDτ = 2TXS
(1 − TXS
− TXS
),

(6.9)

S (T D ) is the probability that a photon created at the crystal
where the transmission efficiency TXS
XS

will traverse the interferometer’s stationary (delay) arm and reach the sample. Because the pathfrom crystal to sample is in free-space, we expect small changes in alignment caused by moving the
D . Thus
delay arm to at most have a negligible effect on the delay-arm transmission efficiency TXS

we assume that the efficiency from the crystal to the sample via the delay arm is independent of
D without the τ subscript on thethe time delay in the interferometer, which is why we write TXS

path-label D.
Next, we write down the conditional probabilities that D1 will click, given the photon pairhas travelled a specific path:SS 2
p(click|SS) = (1 − pSS
E2P A )(2η1 − (η1 ) ),(6.10)

Dτ 2
Dτ
Dτ Dτ
p(click|Dτ Dτ ) = (1 − pE2P
A )(2η1 − (η1 ) ),(6.11)DτS
S Dττp(click|SDτ ) = (1 − pSD
E2P A )(η1 + η1 − η1 η1 ),(6.12)p(click|S) = η1S ,(6.13)p(click|Dτ ) = η1Dτ ,(6.14)

where the efficiency η1S (η1Dτ ) is the probability that a single photon that travels path S (Dτ ) andis transmitted through the sample will reach D1 and cause it to click. Because the light is coupled
into fiber between the sample and detector D1 , we expect that small alignment changes in the delay
arm may lead to large changes in the fiber-coupling efficiency, and so we allow the efficiency η1Dτto depend on the time delay in the interferometer.

91
We can use Eqs. (6.5)-(6.14) to find the desired equation for the expected rate of clicks at
D1 :
 S 2
S
S 2
S1 (τ ) = N (TXS
) (1 − pSS
E2P A )(2η1 − (η1 ) )
Dτ 2
Dτ Dτ
Dτ
D 2
+ (TXS) (1 − pE2P
A )(2η1 − (η1 ) )
Dτ
S Dτ
S
D
S
τ
+ 2TXS
TXS(1 − pSD
E2P A )(η1 + η1 − η1 η1 )SS
D+ 2TXS(1 − TXS
− TXS
)η1S

i
D
S
D
+ 2TXS(1 − TXS− TXS)η1Dτ + pBG ,(6.15)

where N is the pair-production rate at the source and we have introduced the parameter pBG torepresent the background and dark count rate on D1 . If we make the substitutions η1Dτ → 0 and
pBG → 0, we recover Eq. (6.1).
To estimate (and subtract) the rate of leakage from the delay arm into D1 , we also performmeasurements with a shutter blocking the stationary interferometer arm. In this configuration, we
S = 0, and measure
have TXS

h
Dτ 2
Dτ
Dτ Dτ
D 2
S1BG (τ ) = N (TXS
) (1 − pE2P
A )(2η1 − (η1 ) )i
DSD+ 2TXS(1 − TXS− TXS)η1Dτ + pBG .

(6.16)We then subtract our measurement of S1BG (τ ) from S1 (τ ) to find the corrected click rate at D1 :
 S 2
S
S 2
S1corr (τ ) = N (TXS
) (1 − pSS
E2P A )(2η1 − (η1 ) )
Dτ
S
D
S
S Dτ
τ
+ 2TXS
TXS(1 − pSD
E2P A )(η1 + η1 − η1 η1 )


S
S
D
+2TXS
(1 − TXS− TXS)η1S .

(6.17)

We calculate ∆S corr using the corrected click rates at D1 :∆S corr =S1corr (T ) − S1corr (0).
S1corr (T )(6.18)

In general, we expect the 2PA probability to be identical for the SS and Dτ Dτ paths, since in thesecases both photons travel identical paths between the crystal and sample and therefore should be92
perfectly overlapped in the sample. However, the modal overlap between the stationary and delay
Dτ Dτ
SS
τ
arms may not be 100%, and thus in general we expect pSD
E2P A < pE2P A = pE2P A . We can quantify

this overlap by looking at the visibility, V , of the interference pattern we measure when we rotateSS
τ
HWP2 to 22.5◦ , and we expect pSD
E2P A = V pE2P A . At large delay, τ = T , this visibility is zero,

which is why we expect no 2PA to occur if the pair traverses the SDT path. With this definition,
we can write ∆S as a function of pSS
E2P A , the near-zero-delay visibility, V , and the one-photontransmission and detection efficiencies throughout the setup:




D
η1 0
DT
D0
SS
D
+
(1
−
V
p
−
η
)η
TXS
V pSS
1
+
1
E2P A 1E2P A
η1S.
∆S corr =


1 S
D DT
S
TXS
(1 − pSS
E2P A )(1 − 2 η1 ) − 1 + 1 − TXS η1(6.19)

Finally, we can rearrange Eq. 6.19 to solve for pSS
E2P A , which is the main result of this section:
pSS
E2P A =S η S − T D η DT ) − η S T D (η D0 − η DT )∆S corr η1S (1 − 21 TXS
1
1 XS 1
1
XS 1
S η S (1 − 1 η S ) + V T D (η S + η D0 − η S η D0 )∆S corr TXS
1
1
1 1
XS 1
2 1

.

(6.20)

If we make the simplifying assumptions that η1Dτ = 0 and V = 1, we recover Eq. (6.2).
In the next section, we explain how to estimate the efficiencies η1S and η1Dτ from the countrates we measure on detectors D1 and D2 , as well as the coincidence rate between them. We areable to estimate pSS
E2P A by using Eq. (6.20) in conjunction with Eqs. (6.25) and (6.27) that we
present below.
SS
If there is no significant E2PA probability (i.e. pSS
E2P A ≈ 0), we can use pE2P A to upper
UB ). Similar to Eq. (6.4), we definebound the E2PA cross section (σE

UB
σE
=

pE2P A+ 2δσE ,nl(6.21)

where δσE is the propagated error from the Allan deviation of the dataset for the selected averaging
chunk size ( as discussed in Section 6.6) and the measurement of the concentration.

6.5.1

Estimating Detection Efficiencies

Eq. (6.20) above gives us the pSS
E2P A probability as a function of ∆S, the interference visibility
V , and the one-photon transmission and detection efficiencies through the setup. We have discussed

93
path
how to measure ∆S and V , and it is relatively straightforward to measure the TXS
efficiencies.

We can use a version of Klyshko’s method [87] to measure the detection efficiencies η1S and η1Dτ ,and in this section we describe how to do so. We start with η1S , and again we begin with equations
for the rate of clicks at detectors D1 , D2 , and coincidences between both detectors. In the limits
pE2P A << 1, η1Dτ << η1S , and η2S << η2Dτ , we can write our equation for the background-corrected
rate of clicks at D1 (Eq. (6.17)) as:
1
SS
S1corr (τ ) ≈ 2N η1S TXS(1 − η1S TXS).
2(6.22)

Similarly, the background-corrected rate at D2 is:1DD),
S2corr (τ ) ≈ 2N η2Dτ TXS(1 − η2Dτ TXS
2(6.23)and the coincidence rate between the two detectors is:S
D S Dτ
C(τ ) ≈ 2N TXS
TXS
η1 η2 .

(6.24)

We can solve the above system of equations for η1S , arriving at:
η1S =

1

C(τ )
S S corr (τ )
TXS
2



−1
1 C(τ )1C 2 (τ )1−
1−.
2 S1corr (τ )4 S1corr (τ )S2corr (τ )(6.25)

This equation allows us to accurately estimate η1S directly from the count rates measured in theexperiment.
To estimate the leakage efficiencies η1Dτ , we consider the coincidence rate C BG (τ ) we measure
when the stationary interferometer arm is blocked. In this case — ignoring background and/or darkcounts in the detectors — the only way that both detectors will click is if both photons travel tothe sample via the delay arm, are transmitted by the sample, and then one photon makes D1 click
while the other makes D2 click. Thus, in the limit pE2P A << 1, the rate is given by:
D 2 Dτ Dτ
C BG (τ ) ≈ 2N (TXS
) η1 η2 .

(6.26)

Using this equation and Eq. (6.24), we find:η1Dτ = η1SSTXSC BG (τ ).
Dτ C(τ )TXS(6.27)

94
6.5.2

Count Renormalization with a Quadrant Photodiode

Fluctuations in the power and pointing direction of the pump laser are one major sourceof noise in our experiment. Because we have to measure the count rates S1corr (0) and S1corr (T )at different times, this leads to extra noise in our measurement of ∆S. An important step in
our analysis procedure is normalizing the background-corrected count rates based on measured
fluctuations in the pump laser. In this section we explain how we perform this normalization.
We monitor fluctuations in the pump laser by directing a portion of the pump beam onto a
photodiode with an active area divided into four sections. When light is incident on the photodiode,a photocurrent is created by each of the four sections, proportional to the amount of light absorbedby that section. We can monitor three outputs from this photodiode: first, the sum of the signal
over all four sections, q sum , second, the difference between the signal from the bottom two sections
and the top two sections, q bt , and third the difference between the signal from the left and rightsections, q lr . The q sum signal is proportional to the total power incident on the photodiode, and theq bt and q lr signals monitor changes in the up-down and left-right position of the incident light. Thus,
we can monitor fluctuations in the pump laser power and pointing direction with this quadrant
photodiode.
We assume that the background-corrected singles rate at D1 , S1corr , can depend quadraticallyon these pump-laser fluctuations, via the relation:
S1corr (τ ) = β0τ + β1τ q sum + β2τ q bt + β3τ q lr + β4τ (q sum )2 + β5τ (q bt )2 + β6τ (q lr )2 ,(6.28)

for some unknown coefficients βiτ for i ∈ {0, 6}. These coefficients are unknown, and we allow themto depend on the interferometer time delay τ .
If we know the βiτ coefficients, we can renormalize the measured S1corr (τ ) rates using theabove model, and predict what rates we would have measured if the pump laser parameters wereidentical for the measurements at zero (τ = 0) and large (τ = T ) time delays. We achieve this goalby predicting the βiτ coefficients with a linear least-squares fit.
During our data acquisition, each independent ‘zero-delay’ or ‘large-delay’ measurement ac-

95tually consists of a set of measurements, performed at 14 different delay-arm motor positions. At
each of these 14 motor positions, we perform five successive, independent measurements of thecount rates and quadrant photodiode readings. We use one of these measurements (i.e. 20% of thedata) as ‘training data’ which we fit to Eq. (6.28) to estimate the βiτ coefficients. We use thesemodels to renormalize the other 80% of the data.
We perform our data analysis in chunks of m τ -delay measurements — this is how we average over m independent measurements and (using the Allan deviation as discussed in Section 6.6)
calculate the optimal averaging time. We begin by defining the matrix Qτtrain , which contains thequadrant photodetector readings for all the training data acquired in a chunk of m τ -delay measurements. Qτtrain has 14m rows (corresponding to the 14m independent training-data measurements
performed at τ -delay), and the entries of Qτtrain encode the quadrant photodiode readings:


sum
bt
q1lr (τ ) (q1sum (τ ))2 (q1bt (τ ))2(q1lr (τ ))2 
1 q1 (τ ) q1 (τ )


.

..
..
..
..
..
..
Qτtrain =  ..
.
.
.
.
.
.
.




sum
bt
lr
sum
2
bt
2
lr
2
1 q14m (τ ) q14m (τ ) q14m (τ ) (q14m (τ )) (q14m (τ )) (q14m (τ ))(6.29)We also define a vector representing all 14m training-data measurements of the backgroundcorrected singles rate in the measurement chunk:(S1corr (τ ))1
..
.


corr
⃗1,trainS(τ ) = (S1corr (τ ))14m




,



(6.30)

and a vector containing the model parameters:β⃗ τ =

T


β0τ

β1τ

β2τ

β3τ

β4τ

β5τ

β6τ

.

(6.31)

We perform a linear least-squares fit of our model to the training data by calculating:−1 τ⃗ corr (τ ).
β⃗ τ = (Qτtrain )T Qτtrain(Qtrain )T S
1,train(6.32)The remaining, (non-training) data from the chunk of m measurements can be arranged into
⃗ corr (τ ) — these are defined in a similar way as Qτ⃗ corrthe matrices Qτ and vectors S
1
train and S1,train (τ )

96
except they each have four times as many rows, because there is four times as much non-training
data as training data. Using our model, we can predict the count rates we would have measured
given the quadrant photodiode readings, using:⃗ pred (τ ) = Qτ β⃗ τ .
S
1(6.33)

We can also predict the mean rate we would have measured if the zero-delay and large-delay
measurements were performed with exactly the same pump laser parameters:
S1mean (τ ) =1 0(Q + QT ) β⃗ τ .
2(6.34)

Finally, we arrive at our definition for the renormalized, background-corrected count rate at D1 :⃗ corr⃗ norm (τ ) = S1 (τ ) S mean (τ ).
S
1⃗ pred (τ ) 1
S
1

6.6

(6.35)

Results
After acquiring data, our goal is to find an accurate estimate of ∆S. We want to analyze thedata in a way that is independent of as many systematic errors as possible. We begin by subtracting
the background count-rate measurements from the count rates measured at D1 . We notice a correlation between the quadrant detector readings and the count rates, implying that our experiment issensitive to instability of the pump beam. To account for this instability, we numerically fit a function of the quadrant detector outputs to the background-subtracted counts, and renormalize thecounts based on this fit—full details of this procedure are in Section 6.5.2. This procedure allows us
to account for and remove pump-laser noise that adds noise to the measured count rates. Finally,
we calculate the change in singles rates—∆S—from the background-subtracted, renormalized countrates at each delay.
We also perform an Allan deviation analysis on each dataset in order to estimate the uncertainty on our measurement of ∆S and to find the optimal total averaging time. To do this, we
begin by dividing the total dataset into a series of subsets each representing the same total numberof measurements. We apply the full analysis procedure from the above paragraph to each subset,

97

Figure 6.5: Example dataset, for 1.5 × 10−4 mol L−1 of ZnTPP in toluene. Plots 1-4 indicate theplots from left to right. Plot 1 shows count rates in D1 as a function of measurement number.
We plot the background-corrected count rates (corr) at both 0 ps and 27 ps interferometer delay.
We also plot the count rates after renormalizing (norm) to minimize the impact of pump laser
fluctuations which we monitor with the quadrant photodiode. Plot 2 shows the Allan deviationof ∆S as a function of the number of consecutive measurements we average over (i.e. the “chunksize”). The optimal averaging time (and corresponding minimum Allan deviation) is marked by
the red point. The blue trace is the Allan deviation when we calculate ∆S with the backgroundcorrected counts, and the orange is the measured Allan deviation when we calculate ∆S with thenormalized counts. Our normalization procedure reduces the Allan deviation by roughly a factorof 3, indicating that this procedure successfully removes some of the noise caused by pump-laser
fluctuations. Plot 3 shows the Allan deviation curve for our inference of pE2P A (which we inferfrom our measurement of ∆S). The optimal averaging time (and corresponding Allan deviation)
is marked by the red point. Plot 4 shows all measurements of ∆S and pE2P A performed on thissample. For this measurement, the optimal averaging time was roughly 1/9 of the total time wecollected data, and therefore we have nine independent measurements of both ∆S and pE2P A . For
this sample, there is no statistically significant indication that E2PA occurred.

resulting in a list of ∆S estimates from which we calculate an Allan deviation. We repeat thisanalysis for a variety of subset sizes. Because each sample was measured on a different day, andalso because the stability of our setup varied slightly each day, the optimal averaging time varies
from dataset to dataset.
We present an example dataset in Fig. 6.5. This data was taken with a 1.5 × 10−4 mol L−1
solution of Zinc-tetraphenylporphyrin (ZnTPP) in toluene. This dataset contains a total of 1842
independent measurements of the count rate at D1 at each time delay (Fig. 6.5, plot 1). Afterbackground-correcting, normalizing, and averaging these counts, we calculate ∆S for a variety ofdifferent averaging times. Fig. 6.5, plot 2 shows the Allan deviation of the ∆S values as a functionof averaging time, and we find the near optimal averaging time of 200 measurements gives an98Allan deviation of 0.0002 — we interpret this minimum Allan deviation as the uncertainty on our
measurement of ∆S. For each sample we investigate, we independently characterize the modal
overlap between the two interferometer arms with an interference measurement, and all of thesemeasurements returned a visibility (V) greater than 95%. We also independently characterize thelinear transmission efficiencies via each interferometer arm from the SPDC crystal to the cuvette,S
D = 25.9 ± 2%. Using these efficiencies and conservatively
and find TXS= 36.4 ± 2% and TXS

estimating V = 0.95, we use Eq. (6.20) (a corrected form of Eq. (6.2) that makes fewer assumptionsin its derivation) to infer the probability that any pair of photons that reaches the sample is absorbed
via E2PA, pE2P A . We estimate the uncertainty on our measurement of pE2P A by calculating theAllan deviation (Fig. 6.5, plot 3). Since this dataset has a total of 1842 measurements, and we
found it is near optimal to average over subsets of 200 measurements, this dataset contains nine
independent measurements of ∆S and pE2P A , and these are plotted in Fig. 6.5, plot 4. The
remaining 17 datasets are presented in Appendix H.
0.15
2σ

0.05
0.00
-0.05

4

AF455

0.17

0.73

0

9RS

11

0.18

Rh6G

0

0.008

Quantum dot

0

1.65

0.15

0

ZnTPP

6.7

0.29

0

FAD

0

Fluorescein

-0.15

9.4

-0.10

0.19

ΔS (%)

0.10

Concentration (10-3 mol L-1 )Figure 6.6: Summary of measured change in transmittance, ∆S, for all seven samples. The samples were measured at various concentrations, including some measurements of just the solvent(indicated with solution concentration set to 0). For all samples measured, we do not measure any
statistically significant change in the count rate at detector D1 when we vary the time delay in theinterferometer from 0 ps to 27 ps. The error bars are set by two times the Allan deviation (2σ for≈ 95% confidence that the real value lies within the error bars) for the selected averaging chunksize of each dataset.

In Fig. 6.6 we present the values of ∆S for all our datasets. We performed 18 independent data
runs for 18 independent samples. The samples consist of 1.9×10−4 mol L−1 and 9.4×10−3 mol L−1

99
fluorescein in pH 11 water, 2.9 × 10−4 mol L−1 and 6.7 × 10−3 mol L−1 flavin adenine dinucleotide(FAD) in phosphate buffered saline, 1.5×10−4 mol L−1 and 1.65×10−3 mol L−1 ZnTPP in toluene,
8 × 10−6 mol L−1 Qdot ITK carboxyl quantum dot 605 in borate buffer, 1.8 × 10−4 mol L−1 and1.1×10−3 mol L−1 rhodamine 6G (Rh6G) in methanol, 7.3×10−4 mol L−1 of the benzodithiophene
derivative “9R-S” in chloroform, 1.7 × 10−4 mol L−1 and 4.0 × 10−3 mol L−1 1, 3, 5-triazine-based
octupolar molecule “AF455” in toluene and each solvent. The details on sample preparation are
in Appendix B. The solvent data runs are listed with solution concentration set to zero. For eachdataset we find that the values of ∆S and pE2P A show no statistically significant difference fromzero.
In Table 6.1 we summarize the results for all our datasets. The samples, their C2PA crosssections from literature (σC ) and an estimate for their entangled two-photon absorption cross sectionest ) are shown. The estimate uses the probabilistic model (Eq. (2.11)). The measured sample(σE

concentrations (c), average detection efficiencies on detector 1 (η1S ), values of ∆S, pE2P A and upper
UB , found using Eq. 6.21) are listed. The σ UB values range
bounds on the E2PA cross section (σEEfrom 3.1 × 10−22 to 6.6 × 10−19 cm2 absorber−1 . These values are 5-8 orders of magnitude larger
est values, which is an estimate for the measurement sensitivity increase necessary tothan the σE

observe a signal.

6.7

Conclusion
In this chapter, we presented a sensitive transmittance-based E2PA measurement system.

We designed a measurement technique capable of detecting changes in transmittance due only to
E2PA, rather than one-photon loss mechanisms. We designed an interferometer to measure thedifference in the singles rate at detector 1 at zero delay and a very large delay. These two delay
positions represent states in which the E2PA rate is maximized and minimized (effectively zero),respectively. The one-photon loss between the SPDC crystal and detector 1 in these two states was
kept identical. We corrected the count rate for leakage of photons from the delay arm into detector
1 and normalized our counts based on the pointing and power stability of our pump laser. These20 [53]

ZnTPP

6.2

0.65

4.0 ± 1.2

est
σE
× 10292(cm absorber−1 )16 ± 46.8 ± 1.9210 ± 60

51 ± 14 [19]22 ± 6 [19]660 ± 180 [19]

Rh6G

9R-S

AF455

46000 ± 13000 [19] 14000 ± 4000

2.1 [54]

FAD

Qdot 605

13 ±4 [19]lit
σC
[Ref.](GM)

Fluorescein

Sample

η1S
(%)
5.99
6.17
6.40
6.30
6.69
7.73
7.29
6.55
6.89
7.30
5.05
6.08
5.43
4.90
5.02
7.08
5.04
6.41

c × 103(mol L−1 )00.190 ± 0.005
9.4 ± 0.2
00.29 ± 0.01
6.7 ± 0.5
00.15 ± 0.01
1.65 ± 0.10
00.0080 ± 0.0005
00.18 ± 0.01
11 ± 100.73 ± 0.04
0.17 ± 0.01
4.0 ± 0.2

0.054 ± 0.118
0.098 ± 0.118
0.096 ± 0.133−0.122 ± 0.106
0.070 ± 0.079−0.042 ± 0.118−0.018 ± 0.059−0.019 ± 0.159−0.146 ± 0.079−0.083 ± 0.159−0.113 ± 0.103
0.013 ± 0.079−0.111 ± 0.055

0.013 ± 0.030
0.024 ± 0.030
0.024 ± 0.034−0.030 ± 0.027
0.017 ± 0.020−0.012 ± 0.030−0.005 ± 0.015−0.004 ± 0.040−0.037 ± 0.020−0.022 ± 0.040−0.029 ± 0.026
0.003 ± 0.020−0.029 ± 0.014−0.072 ± 0.068−0.057 ± 0.118−0.211 ± 0.166
0.246 ± 0.130−0.099 ± 0.118−0.053 ± 0.042
0.063 ± 0.033−0.024 ± 0.030−0.018 ± 0.017−0.018 ± 0.030

pE2P A
(%)

∆S
(%)

13
0.98

2.5

29
0.31

660

25
2.4

19
0.90

44
0.42

UB
× 1021σE
2(cm absorber−1 )est ) (based on Eq. (2.11) using T =Table 6.1: Summary of literature C2PA cross sections (σC ), estimates for the E2PA cross sections (σEe2
153 fs and Ae = 2.1 µm ), measured sample concentrations (c), average detection efficiencies on detector 1 (η1S ), change in transmittance
UB ).
(∆S), probability that any pair that reaches the sample is absorbed via E2PA (pE2P A ) and E2PA cross-section upper bounds (σE

100

101
techniques provided a sensitivity to changes in transmittance ∆S of 0.02-0.05%.
We performed measurements on seven independent samples at multiple concentrations andmeasured no statistically significant change in transmittance. We set upper bounds on the E2PA
cross sections for the absorbers in the range 10−22 − 10−19 cm2 absorber−1 . For four of the studied
absorbers, E2PA cross sections have been published before [49, 53, 47, 54, 55, 48]. Our upperbounds are lower than these cross sections by factors in the range of ≈ 6 − 104 .
We find that transmittance measurements are less sensitive than fluorescence measurements.
After optimizing each technique for maximum sensitivity, our transmittance measurements upperbounds are typically three orders of magnitude higher than those for our fluorescence measurements (Chapter 5). The higher sensitivity is a result of the background-free nature of fluorescencemeasurements. This reduces the noise and thus the minimum signal one can detect. However,some advantages do exist for transmittance measurements as discussed in Chapter 2 Section 2.5.
Complicated variations of E2PA transmittance measurements could be designed to increase thesensitivity, however a many-orders-of-magnitude increase is not likely.

Chapter 7

A Toluene-Filled Hollow-Core-Fiber Platform for Two-Photon Absorption
Measurements

7.1

Introduction
In our first fluorescence experiment (Chapter 5) we measured two-photon excited fluorescence(2PEF) from a sample contained in a cuvette. We measured both classical two-photon absorption(C2PA) and entangled two-photon absorption (E2PA) in the same setup under nearly identicalconditions. In this chapter, we present a new platform for 2PEF measurements similar to that
presented in Ref. [88], which was used for C2PA, and in Ref. [89], which was used for Raman
spectroscopy. Here we confine a sample (two-photon absorbing molecules in solution) in a hollowcore fiber. We focus the excitation beam into the fiber to excite the two-photon absorbing molecules.
The molecules emit fluorescence and some of that light is guided back out of the fiber and can bedetected.
The benefit of this scheme is illustrated in Fig. 7.1. In the cuvette-based approach, light isfocused in free space to a waist w0 and expands quickly. The expansion is characterized by theRayleigh range zR . The parameter zR scales quadratically with w0 , thus if one wants to increase thebeam intensity at the focus by decreasing w0 , that intensity is maintained over a shorter distancezR . In contrast, in a fiber-based approach, light is focused into a fiber with waist w0 and thatbeam size is maintained over long lengths because of low-loss fiber confinement. A high intensityof light is advantageous for 2PA because two photons must be spatially and temporally overlapped
at a molecule in order for the absorption process to occur. Maintaining that intensity over long

103

Figure 7.1: The advantage of fiber confinement for 2PA measurements. Illustration of (a) a conventional cuvette-based and (b) a fiber-based measurement. In the cuvette-based approach, thelight is focused in free space to a√beam waist w0 . The Rayleigh range zR characterizes the distanceover which the beam size is ≤ 2w0 . When w0 is decreased so is zR , thus high intensity beams
can only be maintained over short distances. In the fiber-based approach, the light is focused into
fiber with waist w0 and maintained over long lengths because of low-loss fiber confinement.

lengths increases the likelihood that the two photons will excite a molecule because those photons
will encounter more molecules.
In this chapter, we discuss the conditions we selected for low-loss fiber confinement. Then, we
present our experimental setup. We then model our experiment in order to calculate a C2PA crosssection from our measurements. Afterwards, we present characterizations of the fiber platform anddiscuss our data acquisition technique. Finally, we present our results for the molecule AF455.

7.2Light Guidance
In this experiment we seek to confine both the sample, excitation photons and fluorescencephotons along the length of the fiber. In order to achieve broadband guidance, we designed ourfiber for index guidance. In index guidance, all light traveling in the core and incident onto thecore-cladding interface at an angle (with respect to the normal to the interface) equal to or greaterthan the critical angle, θc = arcsin(nclad /ncore ) where nclad and ncore are the indices of refraction of
the cladding and core of the fiber, is guided through the fiber because of total internal reflection.
The critical angle is only real if the index of the cladding is smaller than the index of the core. To

104achieve this criteria, we use a standard capillary tubing with a silica cladding and fill the hollow
core with toluene. The critical angle under these conditions is real for both the excitation andfluorescence wavelength regions.
1.54

Refractive Index

1.52
1.50

Core

1.48

Fundamental

1.46

HOM

1.44

Cladding

1.42
1.40

0.5

0.6
0.7
0.8
Wavelength (μm)

0.9Figure 7.2: Refractive index as a function of wavelength for the relevant media and first two modesof guided light. The toluene-filled core (orange) and silica cladding (blue) indices are shown. The
mode indices for the fundamental (green) and the first higher-order mode (HOM, red) are shown.
Additional modes can propagate through the filled fiber with mode indices filling the blue shaded
region between the HOM and cladding indices.

The refractive indices as a function of wavelength for the toluene core and silica cladding
are plotted in Fig. 7.2. The range of indices lying between the core and cladding allows for the
propagation of multiple modes of light. The effective index of refraction of the fundamental andfirst higher-order modes are shown and are calculated using Ref. [90]. We can calculate the numberof modes that can propagate along the fiber using the V-number,
V (λ) =πdλqn2core (λ) − n2clad (λ).

(7.1)

The number of modes that can propagate at a particular wavelength λ is then V (λ)2 /2. For theexcitation wavelength of 810 nm, ≈ 16 modes can propagate. For the fluorescence wavelengthsof AF455 in toluene, which is peaked at 451 nm, ≈ 80 modes can propagate. In the ideal case,all of the light would remain in the fundamental mode because this mode is less lossy, has lower
dispersion and has a Gaussian spatial profile. All of these characteristics will increase the likelihoodthat 2PA can occur.

105
In addition to toluene allowing broadband guidance of light in a fiber made of silica, toluene
also has a low absorption coefficient [91] at the excitation (0.0030 cm−1 ) and fluorescence (0.0039
cm−1 ) wavelengths. This is a necessary condition otherwise the long length of the fiber will add little
to no benefit since all the light would be absorbed after a short distance. For comparison, water has
an absorption coefficient that is about one order of magnitude larger at 810 nm (0.0209 cm−1 [91]).

7.3

Experimental Setup
To prepare the fiber, we first cut a clean facet on both ends of the fiber to minimize theloss that occurs when coupling light into fiber. We use a custom-built coil heater to remove about
20 mm of the polyamide coating from each end of the fiber. Next we cleave the fiber ends in theregions where the coatings have been removed. We inspect the ends under a digital microscope
to ensure that the cut looked smooth and to check for particle contamination. The fiber ends
are placed in a tubing sleeve to prevent breakage and to allow for simple attachment to the fiber
adapters in the setup. Images of the fiber under a digital microscope (Keyence VHX 7000) are
shown in Fig. 7.3(a)-(b).

Figure 7.3: Digital microscope images of one end of (a) the capillary and (b) the capillary inside of
the tubing sleeve. (c) sCMOS image of the 810 nm excitation laser guided through the capillary.
(d) EMCCD image of the fluorescence from AF455 guided out of the capillary. The last two images
are taken in the image plane of the fiber face. Some features of the tube and tubing sleeve arevisible in both images as described in Section 7.5.

The fiber and tubing sleeves are secured into the custom-built fiber adapters (see Appendix J
for technical drawings). One fiber adapter is connected with PEEK tubing to a syringe placed into

106
a syringe pump. This pump is used to fill the fiber. Both fiber adapters are connected with PEEK
tubing to valves that serve as drainage ports. The fiber adapters are fitted for fused silica optical
windows, which are sealed onto the front for coupling light into and out of fiber. These windows
also serve as view ports to check if fluid has flowed through the fiber.

Figure 7.4: Schematic of the experimental setup. The 810 nm laser is directed through a half-wave
plate (HWP), polarizer (pol) and neutral density filters (F) to control the power. The beam is thensent through a dichroic and focused into a toluene-filled hollow-core fiber held in two fiber adaptersat either end with fittings. Solvent or a solution is pumped into the fiber using a syringe and syringe
pump connected by PEEK tubing to one of the fiber adapters. Fluid can be drained out of thesystem by opening the valves connected to either fiber adapter. Light transmitted through the fiber
can be detected on a power meter or an sCMOS camera. A fraction of the fluorescence generated
inside the fiber is directed out of the fiber in the direction of the dichroic, where it is reflected andfocused onto the EMCCD camera. Scattered light is filtered out (F). The power before the fiber
can be measured with a power meter that flips in and out of the beam. A motorized shutter canbe used to block the beam and take a background measurement.

The microfluidic and optical setup is shown in Fig. 7.4. Here we give a brief overview of the
setup, a thorough description of the components is given in Appendix I. An 810 nm, 110 fs pulsed
laser is directed through a half-wave plate (HWP), polarizing beamsplitter (PBS) and neutral
density filters (F) for control over the power. The light is directed through a dichroic beamsplitter and focused into the 5-µm-diameter-core fiber. Light that is transmitted through the
37-cm-long fiber can be detected on a power meter or imaged on an scientific Complementary
Metal–Oxide–Semiconductor (sCMOS) camera. The power before the fiber can be measured witha power meter that flips into the beam path. Any fluorescence generated inside of the fiber andguided out in the direction opposite of the 810 nm beam is reflected at a dichroic beamsplitter

107
and focused onto an Electron-Multiplying Charge-Coupled Device (EMCCD) camera. Filters (F)remove any scattered 810 nm light. The sample AF455 is chosen for our measurements due to its
large C2PA cross section at 810 nm and its solubility in toluene. The details on sample preparation
are in Appendix B.

7.4

Calculating a C2PA Cross Section
In a similar manner to that shown in Chapter 5 Section 5.5, we can model the C2PEF signal

detected on the camera, FC (cnt s−1 ), as
Z λf

Z l

γ(z, λ)κ(λ)Φ(λ)dλdz,

NC (z)

FC = g
0

(7.2)

λi

where g (pulses s−1 ) is the pulse repetition rate, l (cm) is the length of the fiber, NC (z) (excitations cm−1 pulse−1 ) is the number of excitations per infinitesimal length of fiber dz (cm) per pulse,
λi and λf (nm) are wavelengths chosen such that the integral extends over the entire emission spectrum of the sample, γ(z, λ) (cnt photon−1 ) is the component transmission efficiency, κ(λ) is thecollection efficiency and Φ(λ) (photon excitation−1 nm−1 ) is the differential fluorescence quantumR∞yield. A proper normalization of quantum yield is used such that Φ = 0 Φ(λ)dλ gives the value
published in literature for the total quantum yield of the fluorophore.
From this equation, we find that it is advantageous to increase the length of the fiber for
all z such that NC (z)γ(z, λ) > 0. This product never drops below zero, but can be zero if thephotons are dispersed in time far enough that two photons are never temporally overlapped at that
position z, or if the photon loss along the length l is 100% for either the excitation photons or the
fluorescence photons, or some combination of both leading to a negligible product NC (z)γ(z, λ) ≈ 0.
In this experiment we try to make l long enough that NC (l)γ(l, λ) ≈ 0, so that we can achieve thehighest signal at a given photon flux. We would like the sensitivity improvement relative to ourprior fluorescence technique (Chapter 5) to be as high as possible.
Here we define the collection efficiency as the fraction of light that can be collected by thefiber and directed out in the direction of the detector. Here we assume that the fluorescence is108emitted isotropically and that the fraction of that light collected can be described by the solid angleof a cone with apex angle 2(90◦ − θc ), where θc is the critical angle (Section 7.2). Then we canwrite,

 2

ncore (λ) − n2clad (λ)1κ(λ) =
1 − cos arcsin,
2ncore (λ)(7.3)

where ncore (λ) and nclad (λ) are the indices of refraction of the core and cladding materials (toluene
and silica) at the wavelength of the fluorescence. The component transmission efficiency describes
the transmission efficiency of fluorescence from its point of generation z (cm) to the detector, under
the assumption that it is directed out of the fiber. It takes into account all the loss mechanisms fromvarious optical components and the media the light propagates through. We define this quantityasγ(z, λ) = Exp (−(asol (λ) + ϵsam (λ)c + µ(λ))z) Twindow (λ)Tlens (λ)Rdichroic (λ)Tlens (λ)Tfilter (λ)QE(λ),(7.4)
where asol (λ) (cm−1 ) is the absorption coefficient of the solvent, ϵsam (λ) (L mol−1 cm−1 ) is theextinction coefficient of the sample, c (L mol−1 ) is the concentration of the sample, µ(λ) (cm−1 )is the scattering coefficient of the fiber, T describes the transmittance of an optic, R describes thereflectance of an optic and QE (cnt photon−1 ) describes the quantum efficiency of the camera.
We describe the number of excitations per infinitesimal length dz per pulse as
1
NC (z) = σC n2

Z 1/2g Z ∞ Z ∞
−1/2g

−∞

ϕ(x, y, z, t)2 dxdydt

(7.5)

−∞

where σC (1 GM = 10−50 cm4 s photon−1 fluorophore−1 ) is the C2PA cross section, n (fluorophores cm−3 ) is the number density of the fluorophores and ϕ(x, y, z, t) (photons cm−2 s−1 )is the photon flux of the laser beam. The factor of 1/2 carries units of excitations per photons
absorbed. The temporal and transverse spatial profiles of the laser beam are approximated by
Gaussian distributions. The transverse spatial profile of the light inside the fiber will not fit aGaussian distribution if it is confined into the higher-order modes of the fiber, but in this calculation we make a few approximations based on the assumption that all the light is in the fundamental

109
mode. Assuming the laser is always on, ϕ(x, y, z, t) takes the form
 X

 2


∞
x
y2
(t + i/g)2
ϕ(x, y, z, t) = ϕ0 (z)Exp −4ln2+
Exp −4ln2,
τ (z)2∆x20 ∆y02

(7.6)

i=−∞

where ϕ0 (z) (photons cm−2 s−1 ) is the peak photon flux as a function of z, τ (z) (fs) is the FWHM
pulse duration and ∆x0 (cm) and ∆y0 (cm) are the FWHM beam widths. The peak photon flux
has z dependence because of loss and dispersion in the fiber. We can define the average photon
rate Q(z) in terms of the photon flux,
Z 1/2g Z ∞ Z ∞ϕ(x, y, z, t)dxdydt =Q(z) = g
−1/2g

−∞

−∞

W (z)
hν

(7.7)

where W (z) (W) is the average power of the beam as a function of z and hν (J) is the averageenergy of an incident photon. We can also write the power in a form to show its dependence on
propagation losses in fiber,
W (z) = W0 Exp (−(asol (λe ) + ϵsam (λe )c + µ(λe ))z)

(7.8)

where W0 (W) is the average power at z = 0, λe = 810 nm and ϵsam (λe ) ≈ 0. The parameter W0
is related to the power measured at the input of the fiber Win (W) by W0 = ηC Win where ηC is
the coupling efficiency into fiber. The power measured at the output of the fiber Wout is a good
estimate for W (l). We can solve for the ϕ0 (z) using Eq. 7.7 by integration of the flux over x, y andt,
ϕ0 (z) =4ln(2)
π

3/2

W (z)
.
hνg∆x0 ∆y0 τ (z)(7.9)

The pulse duration has z dependence because of dispersion and can be described byqτ (z) = τ04 + (4ln2)2 (D0 + βz)2 /τ0(7.10)

where D0 is the group delay dispersion (GDD) accumulated by the pulse before the fiber and β isthe total group velocity dispersion (GVD) of the fiber.
Now we can rewrite Eq. (7.2) using these equations as


Z l
Z
√
ln(2) 3/2 σC nW02
Exp(−2(asol (λe ) + µ(λe ))z) λf
FC = 2γ(z, λ)κ(λ)Φ(λ)dλdz.
πg(hν)2 ∆x∆y 0
τ (z)λi(7.11)

110
Then we can solve for the C2PA cross section
1
σC = √
2



π
ln(2)

3/2

g(hν)2 ∆x∆y
FC /W02.
RlRλnExp(−2(asol (λe ) + µ(λe ))z)/τ (z) f γ(z, λ)κ(λ)Φ(λ)dλdz
0λi(7.12)All the parameters in Eq. 7.12 are known through experiments or simulations. The parameter τ0 ismeasured using a SwampOptics Grenouille 8-50-USB, D0 is measured, β is estimated using Comsol
Multiphysics Simulation software, g and hν are specified by the laser manufacturer, n is measured,k(λ) is calculated, ∆x0 and ∆y0 are estimated, asol (λ) is from literature [91], µ(λ) is determinedfrom scattering measurements (Section 7.5), Φ(λ) is known from published measurements [79]and FC /W02 (cnt s−1 µW−2 ) is the fit to our experimental C2PEF data. The parameter γ(λ) is
calculated based on optics’ specifications, our scattering measurements and the published extinction
coefficient and spectra of AF455 [17].

7.5Experimental Characterization
In this section we discuss the experimental characterizations we performed in order to derivea C2PA cross section using Eq. (7.12). First, we discuss the transmission efficiency of the fiber.
Then, we discuss the characterization of propagation losses of the fiber by imaging fiber scatter.
Afterwards, we present images of the excitation and fluorescence light at the fiber facet for identification of the mode content of the fiber. Finally, we discuss our characterization of the setup’s
dispersion.
We maximize the transmission of the 810 nm light through the fiber by using mirrors to walkthe beam and adjusting the position of the focusing lens before the fiber. We measure the power
before and after the fiber, and the ratio of the power transmitted out to the power sent in is whatout
we call our transmission efficiency ( WWin ). This efficiency accounts for coupling, scattering andabsorption losses. The best alignment resulted in about > 55% transmission efficiency. However,during our measurements the transmission efficiency was ≈ 40% but varied from dataset to dataset.
We characterize the loss of light along the length of the fiber by imaging the scattering. The

111
intensity of the scattering is proportional to the power W (z), and thus should exponentially decay
according to Eq. (7.8) with λe replaced with the wavelength of the light propagating through thefiber.

Figure 7.5: An image of the scatter from the alignment laser (inset) and a plot of the integrated
intensity (blue) of the scatter as a function of fiber length. The intensity of the scatter is higher
at the ends due to secondary reflections off the fiber tubing sleeves and fittings. These regions are
removed from the plotted data. The fiber is not visible inside of the fittings and fiber adapters.
The integrated intensity is fit (orange) to an exponential decay.

With only the solvent in the fiber (c = 0), we send a 458 nm alignment laser through the fiberand image the scattering on a smart phone camera using long exposure settings as shown in Fig. 7.5.
The image is integrated vertically to find an integrated intensity as a function of fiber length. The
intensity is fit to the exponential in Eq. (7.8) to derive asol (458 nm) + µ(458 nm) = 0.0093 cm−1 .
This corresponds to a loss of 96.9% of the light along the length of the fiber. We take the samemeasurement with 750 nm (a visible wavelength close to our excitation wavelength) light and thedata does not fit to an exponential. We believe that asol (750 nm) + µ(750 nm) is not large enough tomeasure this way. As mentioned in Section 7.2, the values of asol (λ) are known. From publications
asol (750 nm) = 0.0036 cm−1 and asol (810 nm) = 0.0030 cm−1 . If we set µ(750 nm) = µ(810 nm) = 0,
we find that these absorption coefficients correspond to a loss of 12.5% and 10.5%, respectively,
outof the light along the length of the fiber. Considering with our best alignment W
Win > 55%,

112
and that some fraction of that light is lost due to the coupling efficiency ηC in addition to the
10.5% absorption loss, we find it reasonable to estimate µ(810 nm) ≈ 0. This low scattering loss is
consistent with the light primarily occupying the fundamental mode of the fiber.
We image the 810 nm light at the output of the fiber using two lenses and the sCMOS
camera, as shown in Fig. 7.3(c). In this image, the bright larger and smaller concentric circles
are the imaged fiber cladding outer diameter and the outline of the core modes, respectively. The
larger elliptical shape is likely formed by light guided through the inside of the tubing sleeve as canbe seen by comparison to the digital microscope image in Fig. 7.3(b). Thus, we find that the lightoccupies some lower intensity cladding and tubing sleeve modes in addition to the core modes.
The fluorescence is collected at the front of the fiber using the lens optimized to focus 810 nm
light into the fiber, which roughly collimates the visible fluorescence. The light is reflected at thedichroic beamsplitter, and another lens focuses the image plane of the fiber facet onto the EMCCD,
as shown in Fig. 7.3(d). In this image, the high intensity bright spot is from fluorescence guided
through the core. Light forms an elliptical ring around this bright spot, and is likely from lightguided through the core of the tubing sleeve as can be seen by comparison to the digital microscope
image in Fig. 7.3(b). A larger circular ring of light surrounds that elliptical ring, and is likely fromlight guided through the cladding of the tubing sleeve. Thus, there are many modes occupied by
the fluorescence as predicted by the calculation of a large number of modes in Section 7.2.
To measure the GDD D0 accumulated by the pulse as it propagates through the opticalsetup, we use the GDD tuning function of the Chameleon Discovery laser. We increase the dispersion compensation in increments and take C2PA measurements at each step. The step with the
maximum C2PA signal corresponds to the value at which the GDD is optimally compensated at
the input of the fiber. We measure D0 ≈ 1300 fs2 , which agrees with an independent calculationof the GDD in the setup. To determine the fiber GVD β, a Comsol simulation is used to model
the filled fiber. The low-order modes are solved for. For the fundamental mode, β is identical toβ of toluene, thus there are no contributions from waveguide dispersion. However, for higher ordermodes it’s likely that β is affected by waveguide dispersion. These higher-order modes are not113considered in the calculations of Section 7.4.

7.6

Data Acquisition
In the Section we discuss the camera settings used for data acquistion, the characterizationof camera baseline and dark count rates, the operations that go into measuring a single frame ofdata and the conversion of a camera signal to a detected fluorescence rate.
To determine which camera settings were optimal, we wrote a script to calculate the signal tonoise ratio (SNR) of our measurements at various EMCCD settings based on our expected signal
levels. We determined the optimum camera settings to be: electron multiplying (EM) output
amplifier, EM gain set to 30, preamplifier set to 1, 1 MHz horizontal shift rate, 10 s integration
time and 24x24 binned pixels (a superpixel). Although the integration time could be increased
further to increase the SNR, we found that clock-induced charge (CIC) occurred more frequently
at those integration times. Any CIC was removed from the data. To speed up frame readout on
the EMCCD, a pixel region of interest (ROI) is selected and only those pixels are read out. To
select a ROI, the fluorescence was imaged at a relatively high power and the region with significant
photon counts was selected.
Before datasets are acquired, the baseline and dark count rates of the camera are characterized. Both are characterized with the built-in camera shutter closed. The baseline measurements
are taken at the minimum integration time of the camera. The dark rate measurements are taken at
the 10 s measurement integration time. From each of these characterization datasets we calculate
an average value and uncertainty using an Allan deviation analysis like that shown in Fig. 7.6 forthe fluorescence rate. A typical baseline measurement in our ROI yields 570.4 ± 1.0 ADU pixel−1
and a typical dark rate measurement in our ROI yields 2.09 ± 0.05 electrons s−1 pixel−1 .
Each measurement frame consists of a background and a signal measurement. The difference
between a background and a signal measurement being that the shutter (Fig. 7.4) is closed during
a background measurement. The power meter after the fiber is used to measure the power before
each signal measurement and is used for plotting the power dependence of the signal (Fig. 7.7). The

114
number of measurements done at each power varied. As the power decreased, the signal decreased
and thus we performed more measurements to lower the uncertainty.
To calculate the fluorescence rate FC in cnt s−1 detected by the camera, the pixels of interest
are integrated over to determine the camera signal N (ADU). We converted N to a count rateusing,
FC =

NS
,
GT

(7.13)

where S (electrons ADU−1 ) is the CCD sensitivity for the selected output amplifier and preamplifier,
G (electrons cnt−1 ) is the EM gain and T (s) is the integration time.

7.7

Results

Figure 7.6: Example dataset for 21.8 nW excitation power. (a) The average background subtracted
image is shown for the 7x7 superpixel region of interest. (b) The fluorescence count rate calculated
from each frame’s background subtracted imaged is plotted in blue vs frame number. Averaging
in chunks of 373 frames is shown in red which gives a photon rate of 0.98 cnt s−1 . (c) The Allan
deviationof the photon rate is calculated as a function of chunks of frames used to average. A√
1/ N (where N is the number of frames) line is used to guide the eye to the best number offrames to average. The value used in the analysis is indicated by a red data point, corresponding
to averaging chunks of 373 frames with an Allan deviation of 0.52 cnt s−1 . This dataset consists of
2547 frames, which took about 12 hours to run.

An example dataset is shown in Fig. 7.6 for the lowest W0 of 21.8 nW. This dataset was
acquired over ≈ 12 hours. In Fig. 7.6(a), the background-subtracted image, which is averaged overthe duration of the dataset, is shown for the 7x7 superpixel ROI. In Fig. 7.6(b) the fluorescence
rate extracted from each frame’s background-subtracted image is plotted in blue as a function of

115
frame number. In Fig. 7.6(c) the Allan deviation of the fluorescence rate is plotted for different
numbers of frames averaged over. A near optimum number of frames to average of 373 is selectedwhich corresponds to an Allan deviation of 0.52 cnt s−1 . We average the data in chunks of 373
frames and plot the result in red in Fig. 7.6(b). These measurements have an average value of0.98 cnt s−1 .
The results of our fluorescence measurements for a sample of 2 × 10−5 mol L−1 AF455 intoluene are plotted in Fig. 7.7. Here we show measured fluorescence count rates as a function of thepower transmitted out of the fiber. The fit to the data is a linear regression performed on a log-log
scale. The slope of the fit is 2.05, which confirms the two-photon origin of the signal. The lowest
data point is measured at a power out Win of 19.5 nW, which corresponds to an initial power W0
of 21.8 nW.

Figure 7.7: Fluorescence rate measured on the detector as a function of the power transmitted outof the fiber for a sample of 2 × 10−5 mol L−1 AF455 in toluene. The fit to the data has a slope of
2.05.

Using Eq. 7.12, we derive a C2PA cross section for AF455 of 170 GM. This result is 2-4
fold lower than cross sections reported in literature [17, 19]. We suspect that our cross section islower due to two assumptions in our calculations that rely on all the light being confined to the116fundamental mode of the fiber. In reality there are 16 modes which can propagate through thefiber and a fraction of the light is likely occupying those higher order modes as shown in Fig. 7.3(c).
The photons in those higher-order modes have a decreased likelihood to be overlapped spatiallyand temporally at a molecule due to the complex spatial profiles and the increased group velocity
dispersion.

7.8

Conclusions
We designed a toluene-filled hollow-core-fiber platform for two-photon excited fluorescencemeasurements. We used the results of our measurements and the characterizations we performed on
the experimental setup to derive a C2PA cross section for AF455. The cross section agrees withina factor of 2 and 4 to the values of published cross sections. We believe that using a quantitative
characterization of the mode composition of the excitation laser inside of the fiber would improve
our calculation. Nevertheless, the close agreement to literature cross sections is strong evidence
that our fiber platform can be used to derive absolute values of two-photon cross sections.
We measured C2PEF down to 21.8 nW excitation power. This is a 4-fold improvementfrom the sensitivity of the cuvette-based measurement system presented in Chapter 5. We believethat this is an underestimate of the system capabilities. Higher concentration samples should bemeasurable using this platform and will likely increase the signal strength. Measurements at an≈ 2 × 10−4 mol L−1 concentration level are feasible and would presumably lead to an additional
3-fold sensitivity improvement.
We minimized the losses in the system and optimized the alignment for 40% transmission
efficiency through the fiber rendering this platform ideal for E2PA studies, in which loss is detrimental to the measurement sensitivity (Chapter 2 Section 2.6). Our future studies will implement
entangled photon excitation with the goal of quantifying the first E2PA cross section, rather thanan upper bound.

Chapter 8

Conclusions and Outlook

In this thesis we presented three sensitive experiments designed and used for entangled twophoton absorption (E2PA). Two are transmittance schemes and one is a fluorescence scheme. In
all three cases, we were unable to resolve an E2PA signal because of its very small magnitudeunder excitation by typical values of spontaneous parametric down-conversion (SPDC) photon
fluxes. We were the first to set upper bounds on E2PA cross sections, and we did so for a wide
variety of absorbers - eight independent absorbers in room temperature liquids. Observing E2PA
is challenging due to the nonlinear nature of 2PA and SPDC. The former requires strong spatial,
temporal and spectral correlations between two photons while the latter occurs with a low likelihood(≈ 10−7 ) set by the material of the crystal and the stringent phase-matching conditions. In ourwork, our maximum SPDC photon rate, which was limited by the damage threshold of the nonlinear
crystal, is ≈ 1010 photons s−1 (a few nanowatts). For comparison, the expected classical twophoton absorption (C2PA) rate at a few nanowatts of excitation power (in a standard scheme)
is about 1 molecule excited every few minutes – which is not detectable. Further complications
arise in the fact that the E2PA signal at this maximum photon flux is already expected to havesome contribution from the classical quadratic-photon-flux-dependent term (Eq. (2.3)). This is theregime where photon pairs are overlapped (mean photon number ≈ 150 photons pulse−1 ) and aphoton from one pair and another from a separate pair can excite a two-photon transition. In thequadratic regime, the quantum advantage of E2PA is not expected to be large.
The results of these experiments are unsurprising considering the recent theory [5, 27, 26, 25]

118
and experiments [62, 57, 58] that have predicted or measured similar results. There exists theother camp of researchers whose results disagree with ours and motivated our investigation in thefirst place. However, their results do not prove the origin of the signal that they measure. In a
paper of ours that is not covered in this thesis [61], we designed an experiment to measure E2PA,
but instead measured a signal from hot-band absorption (HBA) that mimicked the behavior of
E2PA. We distinguished between the two processes by testing for signatures of E2PA, such as
the quadratic dependence of the signal on attenuation of the SPDC and by filtering the SPDC
(Chapter 2, Section 2.6). The signals measured by other groups could be HBA, scattering [58] orsome other one-photon loss mechanism.
In our efforts to measure E2PA we developed C2PA measurement systems. We made one-toone comparisons between E2PA and C2PA to place upper bounds on the quantum advantage on theorder of 102 − 103 in the studied photon flux regime. Our cuvette-based fluorescence measurement
system was used to derive absolute values of C2PA cross sections that agreed within a factor of two
to literature cross sections. Our most recent C2PA system is the toluene-filled hollow-core fiberpresented in Chapter 7. This system is capable of measuring C2PA at nanowatt excitation powers,
thus making it ideal to attempt an E2PA measurement with.
The absolute magnitude of E2PA cross sections has yet to be narrowed down experimentally,
as evidenced by Fig. 2.3. A few routes exist to get to that point. The first is to use waveguides
such as our hollow-core fiber system. Another route is to use recent theory to guide the choiceof experimental studies. Atomic systems may show promise due to the energy anticorrelations of
SPDC being ideal for tuning to resonance with a narrow atomic transition. These simple atomic
systems are nice starting points because they are theoretically well understood. Once an E2PA
signal is measured, a wide range of parameters can be tuned to try to improve the signal.
Going forward, the perspective around the use of E2PA is shifting. It’s clear that if there
is a measurable quantum advantage of E2PA, it’s much smaller than earlier experimental reports
had thought. All the work that goes into designing a sensitive experiment to simply attempt tomeasure an E2PA signal indicates that using E2PA for microscopy is likely not going to be fruitful.

119
The use of E2PA for applications like sensing and spectroscopy is possible, and leads to a more
realistic and constructive ideology.

Bibliography[1] Robert W. Boyd. Nonlinear Optics, Third Edition. Academic Press, 2008.
[2] Daniel Adam Steck. Quantum and Atom Optics. Open Publication, 2007.
[3] J. Schneeloch and J. C. Howell. Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone. J. Opt, 18(5):053501,
April 2016.
[4] Vatshal Srivastav, Natalia Herrera Valencia, Saroch Leedumrongwatthanakun, Will McCutcheon, and Mehul Malik. Characterizing and tailoring spatial correlations in multimodeparametric down-conversion. Phys. Rev. Appl., 18(5):054006, November 2022.
[5] Michael G. Raymer, Tiemo Landes, and Andrew H. Marcus. Entangled two-photon absorption
by atoms and molecules: A quantum optics tutorial. J. Chem. Phys., 155(8):081501, August
2021.
[6] Hong-Bing Fei, B. M. Jost, S. Popescu, B. E. A. Saleh, and M. C. Teich. Entanglement-induced
two-photon transparency. Phys. Rev. Lett, 78(9):1679, March 1997.
[7] Bradley M. Jost, Alexander V. Sergienko, Ayman F. Abouraddy, Bahaa E. A. Saleh, andMalvin C. Teich. Spatial correlations of spontaneously down-converted photon pairs detectedwith a single-photon-sensitive ccd camera. Opt. Express, 3(2):81–88, July 1998.
[8] Roy J. Glauber. The quantum theory of optical coherence. Phys. Rev., 130(6):2529, June
1963.
[9] M. J. Stevens. Single-Photon Generation and Detection, chapter Photon Statistics, Measurements, and Measurements Tools. Academic Press, October 2013.
[10] Christopher C. Gerry and Peter L. Knight. Introductory Quantum Optics. Cambridge University Press, 2005.
[11] L. Gilles and P. L. Knight. Two-photon absorption and nonclassical states of light. Phys. Rev.
A, 48:1582, August 1993.
[12] Maria Göppert-Mayer. Elementary processes with two quantum transitions. Ann. Phys.,401(3):273—-294, 1931.
[13] Peter Cronstrand. Quantum Chemical Calculations of Nonlinear Optical Absorption. PhD
thesis, Royal Institute of Technology, 2004.

121[14] W. Martin McClain and Robert A. Harris. Excited States, chapter Two-Photon Molecular
Spectroscopy in Liquids and Gases. Academic Press, 1977.
[15] Mariacristina Rumi and Joseph W. Perry. Two-photon absorption: an overview of measurements and principles. Adv. Opt. Photonics, 2(4):451–518, December 2010.
[16] Nikolay S. Makarov, Mikhail Drobizhev, and Aleksander Rebane. Two-photon absorption
standards in the 550–1600 nm excitation wavelength range. Opt. Express, 16(6):4029–4047,
March 2008.
[17] S. de Reguardati, J. Pahapill, A. Mikhaylov, Y. Stepanenko, and A. Rebane. High-accuracy
reference standards for two-photon absorption in the 680-1050 nm wavelength range. Opt.
Express, 24(8):9053–9066, May 2016.
[18] Robert R. Birge and Brian M. Pierce. Semiclassical time-dependent theory of two-photon
spectroscopy. the effect of dephasing in the virtual level on the two-photon excitation spectrumof isotachysterol. Int. J. Quantum Chem., 29(4):639–656, April 1986.
[19] Kristen M. Parzuchowski, Alexander Mikhaylov, Michael D. Mazurek, Ryan N. Wilson,
Daniel J. Lum, Thomas Gerrits, Jr. Charles H. Camp, Martin J. Stevens, and Ralph Jimenez.
Setting bounds on entangled two-photon absorption cross sections in common fluorophores.
Phys. Rev. Appl., 15(4):044012, April 2021.
[20] W. Denk, J. H. Strickler, and W. W. Webb. Two-photon laser scanning fluorescence microscopy. Science, 248(4951):73–75, April 1990.
[21] W. R. Zipfel, R. M. Williams, and W. W. Webb. Nonlinear magic: multiphoton microscopy
in the biosciences. Nat. Biotechnol, 21:1369–1377, October 2003.
[22] Konstantin E. Dorfman, Frank Schlawin, and Shaul Mukamel. Nonlinear optical signals andspectroscopy with quantum light. Rev. Mod. Phys, 88(4):045008, December 2016.
[23] J. Javanainen and P. L. Gould. Linear intensity dependence of a two-photon transition rate.
Phys. Rev. A, 41(9):5088, May 1990.
[24] J. Gea-Banacloche. Two-photon absorption of nonclassical light. Phys. Rev. Lett, 62(14):1603,
April 1989.
[25] C. Drago and J. E. Sipe. Aspects of two-photon absorption of squeezed light: The continuouswave limit. Phys. Rev. A, 106(2):023115, August 2022.
[26] Tiemo Landes, Michael G. Raymer, Markus Allgaier, Sofiane Merkouche, Brian J. Smith, andAndrew H. Marcus. Quantifying the enhancement of two-photon absorption due to spectraltemporal entanglement. Opt. Express, 29(13):20022–20033, June 2021.
[27] Michael G. Raymer and Tiemo Landes. Theory of two-photon absorption with broadband
squeezed vacuum. Phys. Rev. A, 106(1):013717, July 2022.
[28] Jr. Jan Peřina, Bahaa E. A. Saleh, and Malvin C. Teich. Multiphoton absorption cross section
and virtual-state spectroscopy for the entangled n-photon state. Phys. Rev. A, 57(5):3972,
May 1998.

122[29] B. R. Mollow. Two-photon absorption and field correlation functions. Phys. Rev, 175(5):1555,
1968.
[30] G. S. Agarwal. Field-correlation effects in multiphoton absorption processes. Phys. Rev. A,
1:1445, May 1970.
[31] G. Kang, K. N. Avanaki, M. A. Mosquera, R. K. Burdick, J. P. Villabona-Monsalve, T. Goodson III, and G. C. Schatz. Efficient modeling of organic chromophores for entangled two-photon
absorption. J. Am. Chem. Soc, 142(23):10446–10458, May 2020.
[32] Jun Kojima and Quang-Viet Nguyen. Entangled biphoton virtual-state spectroscopy of thea2 σ + –x2 π system of OH. Chem. Phys. Lett, 396(4–6):323–328, August 2004.
[33] Ryan K. Burdick, Oleg Varnavski, Andrew Molina, Leslie Upton, Paul Zimmerman, andTheodore Goodson III. Predicting and controlling entangled two-photon absorption in diatomic molecules. J. Phys. Chem, 122(41):8198–8212, September 2018.
[34] Heinz P. Weber. Two-photon-absorption laws for coherent and incoherent radiation. IEEE J.
Quantum Electron, 7(5):189–195, May 1971.
[35] A. Jechow, M. Seefeldt, H. Kurzke, A. Heuer, and R. Menzel. Enhanced two-photon excited
fluorescence from imaging agents using true thermal light. Nat. Photonics, 7:973–976, October
2013.
[36] F. Schlawin, K. E. Dorfman, and S. Mukamel. Entangled two-photon absorption spectroscopy.
Acc. Chem. Res, 51(9):2207–2214, September 2018.
[37] Roberto de J. León-Montiel, J. Svozilı́k, J. P. Torres, and A. B. U’Ren. Temperature-controlled
entangled-photon absorption spectroscopy. Phys. Rev. Lett, 123(2):023601, July 2019.
[38] E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd. Influence of pump coherence on
the quantum properties of spontaneous parametric down-conversion. Phys. Scr, 93(8):084001,
July 2018.
[39] Alejandra Valencia, Maria V. Chekhova, Alexei Trifonov, and Yanhua Shih. Entangled twophoton wave packet in a dispersive medium. Phys. Rev. Lett, 88(18):183601, April 2002.
[40] T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih.
Two-photon geometric optics. Phys. Rev. A, 53(4):2804, April 1996.
[41] Wuhong Zhang, Robert Fickler, Enno Giese, Lixiang Chen, and Robert W. Boyd. Influence ofpump coherence on the generation of position-momentum entanglement in optical parametricdown-conversion. Opt. Express, 27(15):20745–20753, July 2019.
[42] M.P. Edgar, D.S. Tasca, F. Izdebski, R.E. Warburton, J. Leach, M. Agnew, G.S. Buller, R.W.
Boyd, and M.J. Padgett. Imaging high-dimensional spatial entanglement with a camera. Nat.
Commun, 3(984), August 2012.
[43] Manuel Unternährer, Bänz Bessire, Leonardo Gasparini, Matteo Perenzoni, and André Stefanov. Super-resolution quantum imaging at the Heisenberg limit. Optica, 5(9):1150–154,
September 2018.

123[44] W. M. McClain. Polarization of two-photon excited fluorescence. J. Chem. Phys, 58(1):324,
January 1973.
[45] Joseph R. Lakowicz, Ignacy Gryczynski, Henryk Malak, and Zygmunt Gryczynski. Two-color
two-photon excitation of fluorescence. Photochem. Photobiol, 64(4):632–635, June 1996.
[46] Alexandra Rapaport, Ferenc Szipöcs, and Michael Bass. Dependence of two-photonabsorption-excited fluorescence on the angle between the linear polarizations of two intersecting
beams. Appl. Phys. Lett, 82(26):4642, April 2003.
[47] A. Eshun, Z. Cai, M. Awies, L. Yu, and T. Goodson III. Investigations of thienoacene molecules
for classical and entangled two-photon absorption. J. Phys. Chem. A, 122(41):8167–8182,
September 2018.
[48] D. Tabakaev, M. Montagnese, G. Haack, L. Bonacina, J.-P. Wolf, H. Zbinden, and R. T. Thew.
Energy-time-entangled two-photon molecular absorption. Phys. Rev. A, 103(3):033701, March2021.
[49] J. P. Villabona-Monsalve, O. Calderón-Losada, M. Nuñez Portela, and A. Valencia. Entangled two photon absorption cross section on the 808 nm region for the common dyes Zinc
tetraphenylporphyrin and rhodamine B. J. Phys. Chem. A, 121(41):7869–7875, September
2017.
[50] A. R. Guzman, M. R. Harpham, Ö. Süzer, M. M. Haley, and T. G. Goodson III. Spatial
control of entangled two-photon absorption with organic chromophores. J. Am. Chem. Soc,
132(23):7840–7841, May 2010.
[51] Dong-Ik Lee and T. Goodson. Entangled photon absorption in an organic porphyrin dendrimer.
J. Phys. Chem. B, 110(51):25582––25585, December 2006.
[52] M. R. Harpham, Ö. Süzer, C. Ma, P. Bäuerle, and T. Goodson III. Thiophene dendrimers as
entangled photon sensor materials. J. Am. Chem. Soc, 131(3):973–979, January 2009.
[53] L. Upton, M. Harpham, O. Suzer, M. Richter, S. Mukamel, and T. Goodson III. Optically excited entangled states in organic molecules illuminate the dark. J. Phys. Chem. Lett,
4(12):2046–2052, May 2013.
[54] J. P. Villabona-Monsalve, O. Varnavski, B. A. Palfey, and T. Goodson III. Two-photon
excitation of flavins and flavoproteins with classical and quantum light. J. Am. Chem. Soc,
140(44):14562–14566, October 2018.
[55] J. P. Villabona-Monsalve, R. K. Burdick, and T. Goodson III. Measurements of entangled
two-photon absorption in organic molecules with cw-pumped type-I spontaneous parametricdown-conversion. J. Phys. Chem. C, 124(44):24526––24532, October 2020.
[56] Juan P. Villabona-Monsalve, Nikolai A. Tcyrulnikov, Emmaline R. Lorenzo, Nicole LaBine,
Ryan Burdick, Matthew D. Krzyaniak, Ryan M. Young, Michael R. Wasielewski, andTheodore Goodson III. Two-photon absorption in electron donor–acceptor dyads and triads using classical and entangled photons: Potential systems for photon-to-spin quantum
transduction. J. Phys. Chem. C, 126(14):6334–6343, March 2022.

124[57] Samuel Corona-Aquino, Omar Calderón-Losada, Mayte Y. Li-Gómez, Hector Cruz-Ramirez,
Violeta Álvarez Venicio, Marı́a del Pilar Carreón-Castro, Roberto de J. León-Montiel, andAlfred B. U’Ren. Experimental study of the validity of entangled two-photon absorption
measurements in organic compounds. J. Phys. Chem. A, 126(14):2185–2195, April 2022.
[58] Bryce P. Hickam, Manni He, Nathan Harper, Szilard Szoke, and Scott K. Cushing. Singlephoton scattering can account for the discrepancies among entangled two-photon measurement
techniques. J. Phys. Chem. Lett., 13(22):4934–4940, May 2022.
[59] Alexander Mikhaylov, Kristen M. Parzuchowski, Michael D. Mazurek, Daniel J. Lum, Thomas
Gerrits, Charles H. Camp, Martin J. Stevens, and Ralph Jimenez. A comprehensive experimental system for measuring molecular two-photon absorption using an ultrafast entangled
photon pair excitation source. In Adv. Opt. Tech. for Quantum Information, Sensing, andMetrology, volume 11295. International Society for Optics and Photonics, SPIE, February
2020.
[60] Michael D. Mazurek, Kristen M. Parzuchowski, Alexander Mikhaylov, Sae Woo Nam,Charles H. Camp, Thomas Gerrits, Ralph Jimenez, and Martin J. Stevens. Bounding entangled two-photon absorption with sensitive transmittance measurements. In Conf. on Lasers
and Electro-Optics, page FM3N.2. Optica Publishing Group, February 2021.
[61] Alexander Mikhaylov, Ryan N. Wilson, Kristen M. Parzuchowski, Michael D. Mazurek, Charles
H. Camp Jr., Martin J. Stevens, and Ralph Jimenez. Hot-band absorption can mimic entangled
two-photon absorption. J. Phys. Chem. Lett., 13(6):1489–1493, February 2022.
[62] Tiemo Landes, Markus Allgaier, Sofiane Merkouche, Brian J. Smith, Andrew H. Marcus, andMichael G. Raymer. Experimental feasibility of molecular two-photon absorption with isolated
time-frequency-entangled photon pairs. Phys. Rev. Res., 3(3):033154, August 2021.
[63] O. Varnavski, B. Pinsky, and T. Goodson III. Entangled photon excited fluorescence in organic
materials: An ultrafast coincidence detector. J. Phys. Chem. Lett, 8(3):388–393, January 2017.
[64] O. Varnavski and T. Goodson III. Two-photon fluorescence microscopy at extremely low
excitation intensity: The power of quantum correlations. J. Am. Chem. Soc, 142(30):12966–
12975, July 2020.
[65] B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg. Nonlinear interactions with an ultrahighflux of broadband entangled photons. Phys. Rev. Lett, 94(4):043602, February 2005.
[66] Barak Dayan. Theory of two-photon interactions with broadband down-converted light andentangled photons. Phys. Rev. A, 76(4):043813, October 2007.
[67] L. Neri, S. Tudisco, F. Musumeci, A. Scordino, G. Fallica, M. Mazzillo, and M. Zimbone. Note:
Dead time causes and correction method for single photon avalanche diode devices. Rev. Sci.
Instrum, 81(8):0086102, August 2010.
[68] Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn.
Linear optical quantum computing with photonic qubits. Rev. Mod. Phys, 79(1):135, January
2007.

125
[69] T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. Woo Nam,
R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. Zijlstra, T. Klapwijk, and V. Zwiller. Generation of degenerate, factorizable, pulsed squeezed light at telecom
wavelengths. Opt. Express, 19(24):24434–24447, November 2011.
[70] J. Stöhr. Overcoming the diffraction limit by multi-photon interference: a tutorial. Adv. Opt.
Photonics, 11(1):215–313, November 2018.
[71] L. A. Rozema, J. D. Bateman, D. H. Mahler, R. Okamoto, A. Feizpour, A. Hayat, andA. M. Steinberg. Scalable spatial superresolution using entangled photons. Phys. Rev. Lett,
112(22):223602, June 2014.
[72] Ryan K. Burdick, Juan P. Villabona-Monsalve, George A. Mashour, and Theodore Goodson
III. Modern anesthetic ethers demonstrate quantum interactions with entangled photons. Sci.
Rep., 9(11351), August 2019.
[73] Graham H. Barnett, Mervyn F. Hudson, and Kevin M. Smith.
Concerning mesotetraphenylporphyrin purification. J. Chem. Soc., Perkin Trans. 1, pages 1401–1403, 1975.
[74] Yoon-Ho Kim and Warren P. Grice. Generation of pulsed polarization-entangled two-photon
state via temporal and spectral engineering. J. of Mod. Opt., 49:2309–2323, July 2002.
[75] C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of subpicosecond time intervals between
two photons by interference. Phys. Rev. Lett., 59:2044–, November 1987.
[76] Keiichi Edamatsu, Ryosuke Shimizu, and Tadashi Itoh. Measurement of the photonic de broglie
wavelength of entangled photon pairs generated by spontaneous parametric down-conversion.
Phys. Rev. Lett., 89:213601, November 2002.
[77] Till T. Meiling, Piotr J. Cywiński, and Hans-Gerd Löhmannsröben. Two-photon excitation
fluorescence spectroscopy of quantum dots: Photophysical properties and application in bioassays. J. Phys. Chem. C, 122(17):9641–9647, April 2018.
[78] Ramamurthi Kannan, Guang S. He, Tzu-Chau Lin, Paras N. Prasad, Richard A. Vaia, andLoon-Seng Tan. Toward highly active two-photon absorbing liquids. synthesis and characterization of 1,3,5-triazine-based octupolar molecules. Chem. Mater, 16(1):185–194, November
2004.
[79] J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, LoonSeng Tan, and P. A. Fleitz. Understanding the one-photon photophysical properties of a
two-photon absorbing chromophore. J. Phys. Chem. A, 108(26):5514–5520, June 2004.
[80] A. K. Gaigalas, P. DeRose, L. Wang, and Yu-Zhong Zhang. Optical properties of CdSe/ZnS
nanocrystals. J. Res. Natl. Inst. Stand. Technol, 119:610–628, December 2014.
[81] M. M. Martin and L. Lindqvist. The pH dependence of fluorescein fluorescence. J. Lumin,10(6):381–390, March 1975.
[82] A. Penzkofer and W. Leupacher. Fluorescence behaviour of highly concentrated rhodamine
6G solutions. J. Lumin, 37(2):61–72, May 1987.

126[83] R. Królicki, W. Jarzȩba, M. Mostafavi, and I. Lampre. Preferential solvation of coumarin 153- the role of hydrogen bonding. J. Phys. Chem. A, 106(9):1708–1713, February 2002.
[84] G. Mainfray and C. Manus. Multiphoton ionization of atoms, chapter 2. Academic Press,
1984.
[85] Ariel Ashkenazy, Kai Wang, Manuel Unternährer, Dror Fixler, and André Stefanov. Estimationof the rate of entangled-photon-pair interaction with metallic nanoparticles based on classicallight second-harmonic generation measurements. J. Phys. B, 52(14):145401, June 2019.
[86] T. Li, F. Li, C. Altuzarra, A. Classen, and G. S. Agarwal. Squeezed light induced two-photon
absorption fluorescence of fluorescein biomarkers. Appl. Phys. Lett, 116(25):254001, June 2020.
[87] D N Klyshko. Use of two-photon light for absolute calibration of photoelectric detectors. Sov.
J. of Quantum Electron., 10(9):1112, September 1980.
[88] Gareth O. S. Williams, Tijmen G. Euser, Jochen Arlt, Philip St.J. Russell, and Anita C.
Jones. Taking two-photon excitation to exceptional path-lengths in photonic crystal fiber.
ACS Photonics, 1(9):790–793, August 2014.
[89] Di Yan, Jürgen Popp, Mathias W. Pletz, and Torsten Frosch. Highly sensitive broadband
raman sensing of antibiotics in step-index hollow-core photonic crystal fibers. ACS Photonics,
4(1):138–145, January 2017.
[90] John M Fini. Microstructure fibres for optical sensing in gases and liquids. Meas. Sci. Technol.,15(6):1120, May 2004.
[91] S. Kedenburg, M. Vieweg, T. Gissibl, and H. Giessen. Linear refractive index and absorption
measurements of nonlinear optical liquids in the visible and near-infrared spectral region. Opt.
Mater. Express, 2(11):1588–1611, November 2012.
[92] Aliaksandr Karotki, Mikhal Drobizhev, Mikalai Kruk, Charles Spangler, Erik Nickel,
Nugzar Mamardashvili, and Aleksander Rebane. Enhancement of two-photon absorption in
tetrapyrrolic compounds. J. Opt. Soc. Am. B, 20(2):321–332, February 2003.
[93] M. D. Young, J. J. Field, K. E. Sheetz, R. A. Bartels, and J. Squier. A pragmatic guide tomultiphoton microscope design. Adv. Opt. and Photonics, 7(2):276–378, May 2015.

Appendix A

Literature summary

In Table A.1, a summary of data from all (to the best of our knowledge) experimental E2PA
UB ) arereports in literature is shown. E2PA cross sections (σE ) or cross section upper bounds (σElisted for various studied absorbers. Abbreviated name are sometimes shown for the absorbers.
Here we list more formal names for the absorbers, however the reports should be referenced for amore thorough description. The abbreviated absorbers are:
– FAD = flavin adenine dinucleotide– ZnTPP = Zinc tetraphenylporphyrin– Qdot 605 = Qdot ITK carboxyl quantum dot 605– Rh6G = rhodamine 6G– AF455 = 1,3,5-triazine-based octupolar molecule “Air Force 455”
– RhB = rhodamine B– Dyad 1 = 9-(N-piperidinyl)perylene-3,4-dicarboximide (6PMI) linked to naphthalene-(1,4:5,8)-bis(dicarboximide) (NDI)– Dyad 2 = 6PMI linked to NDI via a phenyl spacer– Triad 1 = dyad 1 with the additional link of 6PMI to a tetrathiafulvalene (TTF) donor– Triad 2 = dyad 2 with the additional link of 6PMI to a TTF donor– LDS 798 = styryl 11

128– C153 = coumarin 153– OM82C = nitrogen centered tolane dendrimer– FMN = flavin mononucleotide– 9R = 9-Ringed benzodithiophene compound– 9R-X = 9-Ringed benzodithiophene derivative with central atom X– XT = thiophene dendrimers with X thiophene units– H2 TPP = tetraphenylporphyrin
UB , however the information in their
A few of the references in the table did not report a σE
UB . Here we discuss how we calculated these values.
report could be used to estimate σE

In Ref. [62], the authors designed a sensitive fluorescence measurement technique, but could
UB in their experiment using a
not measure a fluorescence signal due to E2PA. We can estimate σE

version of Eq. (5.19). From their report we find the values for the measurable fluorescence lower
bound F LB = 0.7 s−1 , the concentration c = 2 × 10−3 mol L−1 (n ∝ cNA , where NA is Avogadro’s
number), the cuvette length l = 1 cm, the quantum yield Φ = 0.8, the detection efficiency γ =0.1, the collection efficiency κ = 0.019. We also find the SPDC photon pair rate at the sample
2.0 × 109 photons s−1 , which is a lower bound on the photon rate because of fiber coupling. Thisvalue intrinsically accounts for loss in the setup and effectively gives T Q = 4.0 × 109 photons s−1 .
We then solve for the upper bound using,
UB
σE
=

2F LB
,
T QnκlγΦ

(A.1)

UB = 1.9 × 10−25 cm2 absorber−1 .
and derive σE

In our work, Ref. [61], a sensitive fluorescence measurement was implemented for two samples.
UB
The provided information in the paper gives the values for parameters necessary to calculate σEusing Eq. (A.1). For the 3.0 × 10−4 mol L−1 LDS798 sample, a signal was measured due to hotband absorption (HBA). We can use the HBA signal as lower bound for the signal that could be
detected due to E2PA in this setup: F LB ≈ 33 cnt s−1 for 1.0 µW SPDC excitation. The other

129
UB )
Table A.1: A summary of all E2PA cross sections (σE ) and cross section upper bounds (σEpublished in literature prior to 2023. For each report, the publication date, group, and excitation
wavelength is listed. Each reported absorber is listed with its C2PA cross section (σE ) and E2PA
UB ). References marked with a * did not reportcross sections (σE ) or cross section upper bounds (σEUB , but we calculated it based on their experimental details as discussed in this Appendix.
σE
Date [Ref.]
MM/DD/YYThis work(Chapter 6)Group

JILA/NIST

Wavelength Absorber
(nm)

810

04/06/22 [57] * UNAM

810

05/29/22 [58] * Caltech

812

03/31/22 [56]810

Univ. Michigan

02/07/22 [61] * JILA/NIST

1064

08/13/21 [62] * Univ. Oregon

1064

04/06/21 [19]JILA/NIST

810

03/01/21 [48]Univ. Geneva

1064

10/26/20 [55]Univ. Michigan

810

02/28/20 [59]JILA/NIST

810

10/22/18 [54]Univ. Michigan

800

09/25/18 [47]Univ. Michigan

800

09/21/17 [49]Uniandes808

05/21/13 [53]Univ. Michigan

800

05/24/10 [50]Univ. Michigan

800

01/05/09 [52]Univ. Michigan

800

12/07/06 [51]Univ. Michigan

800

σC [Ref.](GM)

UB × 1024
σE × 1018
σE(cm2 absorber−1 ) (cm2 absorber−1 )Fluorescein
13 ± 4 [19]FAD
2.1 [54]
ZnTPP
20 [53]
Qdot 605
46000 ± 13000 [19]Rh6G
51 ± 14 [19]9R-S
22 ± 6 [19]AF455
660 ± 180 [19]RhB
260 ± 39 [16]ZnTPP
20 [53]
Rh6G
51 ± 14 [19]Dyad 1
4.06 ± 0.41
0.038 ± 0.007
Dyad 20.60 ± 0.05
0.19 ± 0.02
Triad 1
3.82 ± 0.21
0.092 ± 0.011
Triad 20.82 ± 0.14
0.071 ± 0.004
Rh6G
9.9 ± 2.8
LDS 798
220 ± 60
Rh6G
9.9 ± 2.8 [61]AF455
660 ± 180
Qdot 605
46000 ± 13000
Fluorescein
13 ± 49R-S
22 ± 6
Rh6G
51 ± 14C153
14 ± 4Rh6G
9.9 ± 2.8 [61]0.0019 ± 0.0009
Bisannulene
150 [50]
4.50 ± 0.58
OM82C
370 [53]
3.08 ± 0.06
FAD
2.1 [54]0.979 ± 0.009
ZnTPP
20 [53]0.754 ± 0.010
FMN
1.6 [54]0.038 ± 0.002
ZnTPP
20 [53]
FAD
2.1
0.29
FMN
1.6
1.4
9R
29.5
0.810 ± 0.097
9R-N
82.5
0.525 ± 0.0063
9R-S
27.9
0.202 ± 0.024
9R-Se
9.01
0.164 ± 0.020
ZnTPP
20 [53]42 ± 5.2
RhB
260 ± 39 [16]4.2 ± 0.34
OM82C
370
23.3
ZnTPP
20
23.7
42T
620
12.7
90T
1130
20.8
Bisannulene
150
32
Triannulene
1650
81
Tetraannulene 2960
99
6T
6
0.13
18T
230
0.71
42T
620
2.6
90T
1130
5.9
H2 TPP
10 [92]
10

420
900
2400
660000
310
2500
980
1700
18000
2000 ± 1000

0.00027 ± 0.00006
4.7 ± 1.1
0.19
0.21 ± 0.05
48 ± 120.10 ± 0.02
2.0 ± 0.6
0.12 ± 0.03
0.16 ± 0.04

170000130
parameters are the component transmission efficiency (which includes the detection efficiency aswell as the transmission efficiency of other optical components such as filters) γ = 0.025, thequantum yield ϕ = 0.054, the loss between the crystal and the sample T = 0.26, the length of thesample l = 1 cm and the collection efficiency κ = 0.042. For the 1.1 × 10−3 mol L−1 sample of
Rh6G, γ = 0.082, ϕ = 0.9 and no signal was detected, thus we use the sensitivity of the detection
UB = 4.7 × 10−24 cm2 absorber−1 for LDS798 andsystem F LB = 0.5 cnt s−1 . Then we derive σE
UB = 2.7 × 10−28 cm2 absorber−1 for Rh6G.
σE

In Ref. [58], the authors present a sensitive transmittance and fluorescence measurement
system, but instead of measuring E2PA, they measure a scattering signal from Rh6G with crosssection (2 ± 1) × 10−21 cm2 absorber−1 . The authors were not able to filter out the scattering to
UB value as well.
achieve a higher sensitivity, thus this cross section serves as a σE

In Ref. [57] the authors designed a sensitive transmittance experiment with the goal of making
their experiment insensitive to one-photon losses. The authors derived Eq. (10),
σE =

Γ
,
clNA

(A.2)

where Γ is the measured one-photon loss insensitive change in transmittance, found using singles andcoincidence rates measured through sample and solvent. The authors used multiple measurement
techniques of differing levels of complexity, including a time-delay technique. For all measurement
UB , we estimate upper
techniques and for two samples of varying concentrations, Γ ≈ 0. To set a σE

bounds on Γ from their measurements. We use their measurements for the highest concentrationsof their samples, for RhB c = 5.8 × 10−2 mol L−1 and for ZnTPP c = 1.4 × 10−3 mol L−1 .
Then for all measurement techniques Γ ≤ 0.060 for RhB and Γ ≤ 0.015 for ZnTPP. We plugthese upper bounds for Γ into Eq. (A.2) using the length of the sample l = 1 cm, and derive
UB = 1.7 × 10−21 cm2 absorber−1 for RhB and σ UB = 1.8 × 10−20 cm2 absorber−1 for ZnTPP.
σE
E

Appendix B

Sample preparation details

The “AF455” fluorophore [78, 79] is provided by Drs. T. Loon-Seng Tan and T. Cooper fromthe Air Force Research Laboratory. Flavin adenine dinucleotide (FAD), Zinc tetraphenylporphyrin(ZnTPP), fluorescein, rhodamine 590 (6G) and coumarin 153 (540A) are ordered from SigmaAldrich and used as received. Qdot ITK Carboxyl Quantum dot 605 (qdot 605) in borate buffer
is ordered from ThermoFisher, stored at 4◦ C and only used for six months after receiving. The
thienoacene fluorophore “9R-S” [47] is provided by Prof. T. Goodson from the University ofMichigan. Various solvents are used to prepare the samples including phosphate buffered saline(PBS, prepared), toluene (≥ 99.98%), pH 11 water (Hydrion pH 11 buffer capsule in distilled water),methanol (≥ 99.9%), ethanol (≥ 99.5%), chloroform (≥ 99.9%) and borate buffer (prepared). The
concentration and absorption/emission spectra of all samples (except qdot 605) are checked using a
UV-VIS-NIR spectrophotometer (Agilent Cary 5000 Scan) and a fluorometer (Horiba Fluorolog-3
FL3-222). The absorption and emission spectra are compared with published spectra to ensure thesamples are not contaminated or degraded.

Appendix C

Detailed experimental diagram and parts list for Chapter 4

Figure C.1: Detailed diagram of our experimental setup. See main text for abbreviation definitions
and part numbers.

In Fig. C.1, we show a detailed diagram of our setup with labeled parts. We list the partnumbers below.

Main source– Laser source = Coherent Chameleon Discovery– SHG = APE HarmoniXX SHG
Pair source– HWP1 = zero-order half-wave plate 405 nm (Thorlabs WPH05M-405)– Pol = glan laser calcite polarizer (Thorlabs GL10-A)

133– F1 = dichroic mirrors (3 x 10Q20BB.1 and TLM-400-45S-1025), interference bandpass
filters (2 x Thorlabs FBH405-10, 1 x Thorlabs FB405-10, 1 x Semrock FF01-405/10-25)
and colored glass filter (Thorlabs FGB37M)– L1 = 300 mm focal length lens (Thorlabs LA4579-A)– ppKTP crystal (Raicol Crystals Ltd., type-0 SHG, AR coated, 3.425 µm poling period,
10 mm long)– crystal temperature controller (Covesion PV10) set to 30.00◦ C ± 0.01◦ C– F2 = interference longpass filters (Semrock BLP01-442R-25, BLP01-633R-25 and 3 x FF01496/LP)– L2 = 175 mm focal length lens (KPX103AR.16)– F3 = dichroic mirrors (2 x ARO MR6040) and bandpass filters (Thorlabs FBH800-40,
FBH810-10)
Interferometer– HWP2 = zero-order half-wave plate 808 nm (Thorlabs WPH10M-808)– PBS = polarizing beam splitting cube (Thorlabs PBS12-780-HP)– hollow roof prism mirror (Thorlabs HR1015-AG)– Newport PM10069 motorized translation stage– Newport PM500-C precision motion controller– PBS = polarizing beam splitting cube (Thorlabs PBS12-780-HP)– HWP3 = half-wave plate 800 nm (Tower Optical)
Sample telescope– L3 = 50 mm focal length lens (Thorlabs LA1131-B)– 1 cm UV quartz sample cuvette (Starna, 23-Q-10)

134– L4 = 50 mm focal length lens (Thorlabs LA1131-B)
Single-mode detection stage– PBS = polarizing beam splitting cube (Thorlabs PBS12-780-HP)– L5 and L6 = 12.7 mm focal length achromatic doublet (Thorlabs AC064-013-B)– SMF = 2 x 5 m-long patch cables (Thorlabs 780HP), optional 2 x 500 m-long single mode
fiber (Nufern 780-OCT)– SNSPDs = superconducting nanowire single-photon detectors (Quantum Opus, LLC, Opus
One, optimized for the 850-1200 nm wavelength region) with a detection efficiency of ≈ 75%
at 810 nm– closed-cycle helium cryocooler (Sumitomo HC-4E2)– temperature monitor (SIM 922)
– detector bias and readout modules (Quantum Opus, LLC, QO-SIM-CRYO)– time tagger = picosecond event timer and time-correlated single photon counting system(PicoQuant HydraHarp 400)

Appendix D

Data for Chapter 4

Coincidence rate data recorded in our experiments (illustrated in Fig. 4.5) and used in theanalysis section are presented in Table D.1 for neat solvent and sample at zero and τ delays. All
rates are presented at 10 different 405 nm pump powers ranging from ≈ 60 µW to 10.9 mW. The
rates are listed in counts per second (cnt s−1 ). The rates are averaged near τ and zero delays,marked as Rτ and R0 , respectively. Measurements at each pump power are repeated multiple
times, from which averages and standard deviations (used as the uncertainties) are calculated.
Table D.1: Coincidence rates measured with solvent (solv) and sample (samp) at zero (0) and τ
delays.
Pump Power(mW)

Rτsolv(cnt s−1 )Rτsamp(cnt s−1 )R0solv(cnt s−1 )R0samp(cnt s−1 )

0.06
0.10
0.45
1.08
2.15
3.25
4.72
6.53
8.60
10.9

198 ± 8306 ± 61491 ± 35
3670 ± 60
6960 ± 100
11200 ± 100
16350 ± 120
22670 ± 130
29850 ± 9037780 ± 200

163 ± 6258 ± 91240 ± 60
3120 ± 70
59960 ± 120
9640 ± 14014080 ± 140
19530 ± 310
25830 ± 310
32790 ± 380

195 ± 5300 ± 51481 ± 233650 ± 50
6920 ± 7011140 ± 70
16230 ± 110
22520 ± 100
29670 ± 140
37520 ± 190

165 ± 5252 ± 61249 ± 39
3103 ± 48
5880 ± 110
9590 ± 170
14010 ± 230
19340 ± 330
25540 ± 300
32570 ± 270

Appendix E

Detailed experimental diagram and parts list for Chapter 5

Figure E.1: Detailed diagram of our experimental setup. See main text for abbreviation definitions
and part numbers.

In Fig. E.1, we show a detailed diagram of our setup with labeled parts. We list the partnumbers below.

Main source– Laser source = Coherent Chameleon Discovery– SHG = APE HarmoniXX SHG
Pair source– HWP1 = zero-order half-wave plate 405 nm (Thorlabs WPH05M-405)

137– Pol = glan laser calcite polarizer (Thorlabs GL10-A)– F1 = dichroic mirrors (3 x 10Q20BB.1 and TLM-400-45S-1025), interference bandpass
filters (2 x Thorlabs FBH405-10, 1 x Thorlabs FB405-10, 1 x Semrock FF01-405/10-25)
and colored glass filter (Thorlabs FGB37M)– L1 = 300 mm focal length lens (Thorlabs LA4579-A)– ppKTP crystal (Raicol Crystals Ltd., type-0 SHG, AR coated, 3.425 µm poling period,
10 mm long)– crystal temperature controller (Covesion PV10) set to 30.00◦ C ± 0.01◦ C– F2 = interference longpass filters (Semrock BLP01-442R-25, BLP01-633R-25 and 3 x FF01496/LP)– L2 = 200 mm focal length lens (Thorlabs LA1979-B)– F3 = dichroic mirrors (2 x ARO MR6040) and interference longpass filter (Thorlabs
FELH0700)
Single-mode detection stage– L3 = 12.7 mm focal length achromatic doublet (Thorlabs AC064-013-B)– fiber beamsplitter (Thorlabs FC830-5OB-FC)– SMF = 2 x 500 m-long single mode fiber (Nufern 780-OCT)– SNSPDs = superconducting nanowire single-photon detectors (Quantum Opus, LLC, Opus
One, optimized for the 850-1200 nm wavelength region) with a detection efficiency of ≈ 75%
at 810 nm– closed-cycle helium cryocooler (Sumitomo HC-4E2)– temperature monitor (SIM 922)
– detector bias and readout modules (Quantum Opus, LLC, QO-SIM-CRYO)

138– time tagger = picosecond event timer and time-correlated single photon counting system(PicoQuant HydraHarp 400)
Multi-mode detection stage– L4 = 50.2 mm focal length lens (Newport KPX082AR.16)– SPAD = single-photon avalanche diode (PerkinElmer SPCM-AQR-14)– counter = timer/counter/anaylzer (Tektronix FCA3103)
2PEF measurements– optical chopper head and controller (New Focus 3501 Optical Chopper)– L8 = 50 mm focal length lens (Thorlabs LA1131-B)– UV quartz sample cuvette with 2 mm width × 10 mm path length (FireFlySci, 1FLUV2),
the narrow width is chosen to reduce fluorescence self-absorption in the sample– machined cuvette holder designed for stability and low footprint to bring optics close toexcitation volume– L9, L10, L11 = Collection Optic with High Numerical Aperture (COHNA) lens system [93]– F5 = shortpass filter (Semrock FF01-758/SP-25) and sample-dependent bandpass filter(AF455 and C153 - Semrock FF02-470/100-25, qdot 605 and 9R-S - Chroma ET610/75m,
fluorescein and Rh6G - Semrock FF01-535/150-25) (filter spectra is shown in Fig. 5.3)
– SM = spherical mirror with 15 mm focal length, 35 mm diameter (Edmund Optics, #43467)– PMT = photon-counting metal package photomultiplier tube (Hamamatsu H10682-210)– thermoelectric cooler (TEC) (CP40336) to cool PMT to 5◦ C
– CPU cooler (Rosewell PB120) for heat sink of TEC– time tagger = picosecond event timer and time-correlated single photon counting system(PicoQuant HydraHarp 400)

139
Beam alignment and characterization (details on use in Appendix F)– L7 = 50 mm focal length lens (Thorlabs LA1131-B)– L12 = 50 mm focal length lens (Thorlabs LA1131-B)– L13 = 62.9 mm focal length lens (Newport KPX085AR.16)– cam1 = UI-3590LE-C-HQ camera– cam2 = Thorlabs UI-224XSE camera
1PA source– 458 nm source = OBIS 458 LX– F6 = neutral density (ND) filter wheel (Thorlabs)
Other parts– HWP2 = zero-order half-wave plate 808 nm (Thorlabs WPH10M-808)– HWP3 = half-wave plate 800 nm (Tower Optical)– L5 = 88.3 mm focal length lens (Newport KPX091AR.16)– L6 = 75 mm focal length lens (Newport KPC037AR.16)– PBS = polarizing beam splitting cube (Thorlabs PBS122)– F4 = longpass interference filter (FELH0700)

Appendix F

Alignment details for Chapter 5

A telescope (L5 and L6) is used to resize the laser beam close to the SPDC beam size at theirfoci in the sample. The alignment of the beam into the sample is checked using two cameras (cam1and cam2). Lenses L7 and L8 are placed approximately the same distance from the flip mirror,
enabling a view on cam1 of the beams at and near their foci in the cuvette. With this camera, wecheck the alignment of the beams through alignment irises, and measure beam size, Rayleigh range
and overlap of the laser and SPDC beams. Two lenses after the sample (L12 and L13) collimate
and focus either beam onto cam2. With this camera, we verify that the beams remain overlapped
after passage through the cuvette, are centered along the x-direction inside of the cuvette andpropagate nearly perpendicular to the cuvette walls they are incident on. To check that the beams
are centered in the x-direction, we first use a translation stage to translate the cuvette along thisaxis and observe on the camera when the beams’ strike the walls of the cuvette. We translate themicrometer to the midpoint of the locations of the wall striking events. To check that the beams
propagate perpendicular to the walls they are incident on, we ensure that adding the cuvette does
not displace the beams in the x and y-directions significantly.
A typical transverse spatial overlap of the two beams at their foci (cam1) in the sample isshown in Fig. 5.2(b). The centers of the laser and SPDC beam are displaced from one anotherby ≈ 5 µm vertically and horizontally. Zemax simulations (Fig. 5.2) indicate that displacementsof this magnitude have no effect on the collection efficiency. The beams’ centers on cam2 are also
overlapped within ≈ 5 µm vertically and horizontally. The beam overlap is checked regularly. To

141initially align the beams in the z-direction, the lens L8 is placed roughly one focal length awayfrom the center of the cuvette.
The COHNA lens system (L9, L10 and L11) and filters (F5) are contained within a 25.4 mmdiameter lens tube. The spacing of the optics in the lens tube is based on Ref. [93]. The COHNA lenssystem (and filters) and the spherical mirror (SM) are each placed on a three-axis stage and initiallyaligned in the three directions based on the optimal spacings found using Zemax’s OpticStudio.
To optimize the alignment of the system (COHNA, SM and lens L8), we first adjust lens L8 to
optimize collected C2PEF, which ensures that the excitation volume is centered with respect tothe collection optics. Next, the COHNA lens system and the spherical mirror are each adjusted tomaximize the collection of C2PEF. This process is iterated until the collection efficiency is optimal.
A CW 458 nm source excites 1PEF in the sample to aid in the characterization of the geometricalcollection efficiency (Section 5.4).
The alignment procedure using C2PEF optimizes the alignment of the system for C2PEF.
For E2PEF, the alignment (only lens L8) must be slightly altered because we observe (on cam1) a
shift between the foci of the laser and SPDC beam in the z-direction of ≈ 500 µm. We compensate
for this by shifting lens L8 so that either beam’s focus is in the center of the cuvette prior tomeasurements.

Appendix G

Detailed experimental diagram and parts list for Chapter 6

Figure G.1: Detailed diagram of our experimental setup. See main text for abbreviation definitions
and part numbers.

In Fig. G.1, we show a detailed diagram of our setup with labeled parts. We list the partnumbers below.

Main source– Laser source = Coherent Chameleon Discovery– SHG = APE HarmoniXX SHG
Pair source– HWP1 = zero-order half-wave plate 405 nm (Thorlabs WPH05M-405)– Pol = glan laser calcite polarizer (Thorlabs GL10-A)

143– F1 = dichroic mirrors (3 x 10Q20BB.1 and TLM-400-45S-1025), interference bandpass
filters (2 x Thorlabs FBH405-10, 2 x Semrock FF01-405/10-25) and colored glass filter(Thorlabs FGB37M)– Quad detector = quadrant photodiode (First Sensor QP50-6-18u-SD2)– L1 = 300 mm focal length lens (Thorlabs LA4579-A)– ppKTP crystal (Raicol Crystals Ltd., type-0 SHG, AR coated, 3.425 µm poling period,
10 mm long)– crystal temperature controller (Covesion PV10) set to 30.00◦ C ± 0.01◦ C– F2 = interference longpass filters (Semrock BLP01-442R-25, BLP01-633R-25 and 3 x FF01496/LP)– L2 = 200 mm focal length lens (Thorlabs LA1979-B)– F3 = dichroic mirrors (2 x ARO MR6040) and bandpass filter (Semrock FF01-810-10)
Interferometer– HWP2 = zero-order half-wave plate 808 nm (Thorlabs WPH10M-808)– PBS = polarizing beam splitting cube (Thorlabs PBS12-780-HP)– QWP1 = quarter-wave plate (Tower Achromatic Waveplate A-12.7-A-250-B-2)– QWP2 = quarter-wave plate (Thorlabs AQWP05M-980)– shutter = optical beam shutter (Thorlabs SH05)– piezoelectric Optic Mount (Thorlabs KC1-P)– motorized actuator (Thorlabs Z825B)
Sample telescope– L3 = 75 mm focal length lens (Thorlabs LA1608-B)– 1 cm UV quartz sample cuvette (Starna, 23-Q-10)

144– L4 = 75 mm focal length lens (Thorlabs LA1608-B)
Single-mode detection stage– HWP3 = half-wave plate 800 nm (Tower Optical)– PBS = polarizing beam splitting cube (Thorlabs PBS12-780-HP)– L5 and L6 = 12.7 mm focal length achromatic doublet (Thorlabs AC064-013-B)– SMF = 2 x 5 m-long patch cables (Thorlabs 780HP) optional 2 x 500 m-long single mode
fiber (Nufern 780-OCT)– SNSPDs = superconducting nanowire single-photon detectors (Quantum Opus, LLC, Opus
One, optimized for the 850-1200 nm wavelength region) with a detection efficiency of ≈ 75%
at 810 nm– closed-cycle helium cryocooler (Sumitomo HC-4E2)– temperature monitor (SIM 922)
– detector bias and readout modules (Quantum Opus, LLC, QO-SIM-CRYO)– time tagger = picosecond event timer and time-correlated single photon counting system(PicoQuant HydraHarp 400)

Appendix H

Data for Chapter 6

The plots that follow show the datasets for each of 17 independent measurements (1 additional
measurement shown in Chapter 6). Here I describe the layout of the plots which is common to alldatasets. Plots 1-4 indicate the plots from left to right. Plot 1 shows count rates on D1 as a functionof measurement number. We plot the background-corrected (corr) count rates at both 0 ps and27 ps interferometer delay. We also plot the count rates after renormalizing (norm) to minimize theimpact of pump laser fluctuations which we monitor with the quadrant photodiode. Plot 2 displays
the Allan deviation of ∆S as a function of the number of consecutive measurements we average over(i.e. the “chunk size”). The optimal averaging time (and corresponding minimum Allan deviation)
is marked by the red point. The blue trace is the Allan deviation when we calculate ∆S with thebackground-corrected counts, and the orange is the Allan deviation when we calculate ∆S with
the normalized counts. Our normalization procedure reduces the Allan deviation, indicating that
this procedure successfully removes some of the noise caused by pump-laser fluctuations. Plot 3
displays the Allan deviation curve for our inference of pE2P A (which we infer from our measurementof ∆S). The optimal averaging time (and corresponding Allan deviation) is marked by the red point.
Plot 4 shows all measurements of ∆S and pE2P A performed on the sample as a function of themeasurement chunk number.

146

Figure H.1: Dataset for pH 11 water.

Figure H.2: Dataset for 1.9 × 10−4 mol L−1 fluorescein in pH 11 water.

Figure H.3: Dataset for 9.4 × 10−3 mol L−1 fluorescein in pH 11 water.

Figure H.4: Dataset for PBS.

147

Figure H.5: Dataset for 2.9 × 10−4 mol L−1 FAD in PBS.

Figure H.6: Dataset for 6.7 × 10−3 mol L−1 FAD in PBS.

Figure H.7: Dataset for toluene.

Figure H.8: Dataset for 1.65 × 10−3 mol L−1 ZnTPP in toluene.

148

Figure H.9: Dataset for borate buffer.

Figure H.10: Dataset for 8 × 10−6 mol L−1 qdot 605 in borate buffer.

Figure H.11: Dataset for methanol.

Figure H.12: Dataset for 1.8 × 10−4 mol L−1 Rh6G in methanol.

149

Figure H.13: Dataset for 1.1 × 10−2 mol L−1 Rh6G in methanol.

Figure H.14: Dataset for chloroform.

Figure H.15: Dataset for 7.3 × 10−4 mol L−1 9R-S in chloroform.

Figure H.16: Dataset for 1.7 × 10−4 mol L−1 AF455 in toluene.

150

Figure H.17: Dataset for 4.0 × 10−3 mol L−1 AF455 in toluene.

Appendix I

Detailed experimental diagram and parts list for Chapter 7

Figure I.1: Detailed diagram of our experimental setup. See main text for abbreviation definitions
and part numbers.

In Fig. I.1, we show a detailed diagram of our setup with labeled parts. We list the partnumbers below.

Optical components for 810 nm– Laser source = Coherent Chameleon Discovery– HWP = zero-order half-wave plate 808 nm (Thorlabs WPH10M-808)– Pol = polarizing beam splitting cube (Thorlabs PBS12-780-HP)– L1 = 200 mm focal length lens (Newport KBX076AR.16)– L2 = 175 mm focal length lens (Thorlabs KBX073AR.16)– F1 = neutral density filters (Thorlabs ND02A, ND20A, NE06A, NE30A, NDUV10A)

152– dichroic = dichroic beamsplitter (Semrock FF738-FDi01-t3)– shutter = optical beam shutter (Thorlabs SH05)– K-cube solenoid controller (Thorlabs KSC101)– power sensor 1 = photodiode sensor (Thorlabs S130C)– power meter 1 (Thorlabs PM100D)– motorized flip mount (Thorlabs MFF101)– L3 = 10 mm focal length aspheric lens (Thorlabs ASL1210)– 1 mm thick fused silica window (Edmund 84-449)– fiber = silica capillary tubing (Molex 1068150002)– 1 mm thick fused silica window (Edmund 84-449)– power sensor 2 = photodiode sensor (Ophir PD300R-UV)– power meter 2 (Ophir Starbright)– L4 = 8 mm focal length aspheric lens (Thorlabs C240TMD)– L5 = 125 mm focal length lens (Newport KPX097AR.16)– sCMOS = Andor Marana back-illuminated sCMOS (MARANA-4BV6U-99BE)
Additional components for ≈ 450 nm– 458 nm source = OBIS 458 LX– F2 = neutral density filters (Thorlabs NE40A, NE05A)– L6 = 125 mm focal length lens (Newport KPX097AR.14)– F3 = filters (Semrock FF01-750/SP and FF02-470/100)– EMCCD = Andor iXon Ultra 888 EMCCD (DU-888U3-CS0-#BV)
Microfluidics

153– KDS 230 Legacy syringe pump
– 2.5 ml stainless steel syringe (KDS 780801)– PEEK tubing (Idex 1576)– one-way valves (Idex P-732A)– custom-built fiber adapters (Appendix J)– fitting (Idex F-125 Microtight)– tubing sleeve (Idex F-181 Microtight)– fitting (Idex F-126SX Microtight)– AppleRubber o-ring (Viton with inner diameter of 3.56 mm and width of 0.5 mm)

Appendix J

Fiber adapter technical drawings

Here we show the technical drawings for the fiber adapter base and clamp. This adapter isdesigned to fit an o-ring (Viton with inner diameter of 3.56 mm and width of 0.5 mm) and window(fused silica with diameter of 5 mm and width of 1 mm) in between the base and the clamp. The
adapter is used with three fittings, one for connecting to fiber (Idex F-125) and two for connecting
to PEEK tubing (Idex F-126SX).

155

Figure J.1: Technical drawing of fiber adapter clamp. Hidden dimensions are shown as dashedlines. Dimensions are shown in inches.

156

Figure J.2: Technical drawing of fiber adapter base. Hidden dimensions are shown as dashed lines.
Dimensions are shown in inches.