by Matin Hallaji - University of Toronto T-Space · Matin Hallaji Doctor of Philosophy Graduate Department of Physics University of Toronto 2016 Weak value ampli cation (WVA) is a

Weak Value Amplification of a Post-Selected Single Photon

by

Matin Hallaji

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

© Copyright 2016 by Matin Hallaji

Abstract

Weak Value Amplification of a Post-Selected Single Photon

Matin Hallaji

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2016

Weak value amplification (WVA) is a measurement technique in which the effect of a pre- and

post-selected system on a weakly interacting probe is magnified. In this thesis, I present the first

experimental observation of WVA of a single photon. We observed that a signal photon - sent

through a polarization interferometer and post-selected by photodetection in the almost-dark

port - can act like eight photons. The effect of this single photon is measured as a nonlinear phase

shift on a separate laser beam. The interaction between the two is mediated by a sample of laser-

cooled 85Rb atoms. Electromagnetically induced transparency (EIT) is used to enhance the

nonlinearity and overcome resonant absorption. I believe this work to be the first demonstration

of WVA where a deterministic interaction is used to entangle two distinct optical systems. In

WVA, the amplification is contingent on discarding a large portion of the original data set.

While amplification increases measurement sensitivity, discarding data worsens it. Questioning

whether these competing effects conspire to improve or diminish measurement accuracy has

resulted recently in controversy. I address this question by calculating the maximum amount

of information achievable with the WVA technique. By comparing this information to that

achievable by the standard technique, where no post-selection is employed, I show that the

WVA technique can be advantageous under a certain class of noise models. Finally, I propose

a way to optimally apply the WVA technique.

ii

To my parents,

& Sahar,

for their unconditional love, support and sacrifices

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Acknowledgements

When I started graduate school, I had to make the tough decision of choosing between ex-

perimental and theoretical quantum optics. I chose experiment. Having no experience in

experimental physics (except a couple of undergraduate lab courses), the first couple of months

working in the lab were extremely difficult. But once I made the transition, I couldn’t be hap-

pier with the choice I made. Working long hours in the lab and the frustration that came with

it, although difficult to bear at times, shaped the person I am now. It taught me one important

lesson: don’t let your opinions bias you towards a desired outcome and always seek the truth.

Doing experiments comes with a reward that no amount of computer simulations and calcula-

tions on the paper can top: watching nature do what you predicted it would. The first time

I saw a cloud of atoms float on a sheet of light, the first time I observed off-resonance Raman

transition, the first time I saw a photon acting as if it was eight photons are the moments I

would never forget in my life.

I should start by thanking my supervisor, Aephraim Steinberg. The amount I learned from

him, his knowledge and superb physical intuition is incalculable. Without his help and guidance

I would have never been able to make the transition of becoming an experimentalist. During

my time in the lab, I had the chance to meet and work with so many amazing and brilliant

people. The first person I closely worked with was Chao Zhuang. He taught me a lot about the

apparatus and optical lattices. I had the privilege to work alongside with Greg Dmochoswki

on the LMI project. I also made some fantastic and dear friends in the physics departments:

Ramon Ramos, Nicolas Quesada and Shreyas Potnis. We shared a lot of moments together

and it saddens me deeply not to be able to see them everyday. My special thanks go to Josiah

Sinclair for his enormous help, support, contributions and many many interesting questions and

discussions. I should also thank Hugo Ferretti, Edwin Tham, David(D1) Schmid, David(D2)

Spierings, Ginelle Johnston, Dylan Mahler, Lee Rozema, Rockson Chang and Alex Hayat for

their helps, supports and priceless conversations.

Being an experimentalist comes with staying long nights in the lab working, debugging and,

if you are lucky, taking data. I consider myself very lucky to have spent most of those long nights

with an extremely smart person and a great friend, Amir Feizpour. None of the milestones we

achieved in the lab in the past two years would have been possible without Amir’s dedication

and hard work. He was an inspiration that kept me motivated to do the same.

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Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Disambiguation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Weak Value Amplification of Photon Number - Theory 5

2.1 Introduction to Weak Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Information versus Disturbance . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Weak Measurement and Weak Value Amplification . . . . . . . . . . . . . 8

2.2 Weak Value Amplification of a Post-Selected Photon . . . . . . . . . . . . . . . . 13

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Schrodinger picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Weak Value Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.4 Weak Values and PNR Detectors . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.5 Measuring the Amplified Effect of a Post-selected Single Photon . . . . . 30

2.2.6 Weak Values vs. Strong Values - ABL Rule . . . . . . . . . . . . . . . . . 31

2.3 Weak Measurement as a POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Apparatus 36

3.1 Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Trapping Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 Repumper Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.3 MOT setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.4 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.5 Remote Task Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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3.1.6 Maintenance and Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Probe and Coupling Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Signal Pulses, Photon Detection and Tagging . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.2 Photon Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.3 Gating the Photon Detection . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.4 Tagging the Shots with Successful Photon Detection . . . . . . . . . . . . 60

3.4 Polarization Interferometer and Tomography . . . . . . . . . . . . . . . . . . . . 61

3.5 Atom Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.1 Level Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.2 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5.3 Atom Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Experimental Results 68

4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Single Photon Cross-Phase Shift . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.2 XPS Versus Signal Polarization . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.3 Polarization Tomography on the Signal . . . . . . . . . . . . . . . . . . . 70

4.2 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 φclick and φno−click . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 φno−click versus |α|2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.3 Weak Value Amplification of a Single Post-selected Photon . . . . . . . . 73

4.2.4 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.5 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 On the Advantage of Weak Value Amplification 78

5.1 Fisher Information as a Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.1 Covariance Matrix and Fisher Information . . . . . . . . . . . . . . . . . . 79

5.1.2 Fisher Information versus Signal-to-Noise Ratio . . . . . . . . . . . . . . . 80

5.2 Measuring the Effect of a Single Photon . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.1 Gaussian White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.2 Noise with Time Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.3 Advantage of Post-selection without Weak-Value Amplification . . . . . . 87

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5.3 Experimental Proposal and Progress . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Fisher Information for Coherent States as the Signal . . . . . . . . . . . . 89

5.3.2 Experimental Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Outlook and Summary 92

6.1 Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Future of the current Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Revisiting Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4 Rydberg Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Weak Value Approximations 98

B Alignment Procedures 102

B.1 Polarization Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

B.2 Injection-Lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.3 Tapered Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.4 Polarization Maintaining Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.5 MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 107

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List of Tables

3.1 Typical Trapping parameters for MOT operation. TA running at 1A. . . . . . . 46

3.2 Typical Trapping parameters for molasses operation. TA running at 1A. . . . . 47

3.3 Typical Trapping parameters for imaging. TA running at 1A. . . . . . . . . . . 47

3.4 Typical repumper parameters for different operation. . . . . . . . . . . . . . . . 49

4.1 Polarization tomography on the signal . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Experimental parameters used in the WVA experiment. . . . . . . . . . . . . . . 72

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List of Figures

2.1 Overlap between two displaced states of the meter: (a) when the displacement

is smaller than the initial uncertainty of the meter, (b) when the displacement

is larger than the initial uncertainty of the meter. . . . . . . . . . . . . . . . . . 8

2.2 Demonstration of WVA as an interference between the meter states: (a) The

result of the interference between ψ+(x) and ψ−(x). The mean value of this

state is much larger than the displacement of either of ψ±(x). (b) The displaced

states ψ+(x) and ψ−(x). It can be seen that the two states are not resolvable.

The vertical lines demonstrated the mean value of each of the distributions. . . 11

2.3 Weak value versus δ. The insets show 12((1 + δ)ψ+(x)− (1− δ)ψ−(x)) for three

different values of δ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Schematic of the setup used to amplify the effect of a single photon. . . . . . . . 13

2.5 A real detector with efficiency η modeled as an ideal detector and a beam splitter

that only transmits a fraction η of the incoming light. In the case where the

incoming beam is a coherent state, the two “detected” and “undetected” modes

are independent of each other and the mean photon number inferred in the

detected mode does not affect the mean photon number in the undetected mode. 15

2.6 The difference between the inferred mean photon number for the “click” events

and |α|2 versus |α|2 for efficiency η = 0.2 (solid line.) The dashed line is when the

effect of dark-counts is added to the inferred photon numbers. The inset shows

the difference between nclick and nno−click versus |α|2, when no dark-counts are

considered (solid) and when the dark-counts are considered (dashed.) . . . . . . 16

2.7 Schematic of the WVA setup. The signal pulse |α〉s interacts with the probe

|β〉probe inside an interferometer. The two arm interact with the probe with

different strengths represented as per photon XPS’s of φ1 and φ2, where φ1 > φ2. 17

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2.8 Schematic of the WVA setup. The signal pulse |α〉s interacts with the probe

|β〉probe inside an interferometer. The two arm interact with the probe with

different strengths represented as per photon XPS of φ1 and φ2, where φ1 > φ2.

The effect of a detector with finite efficiency is modeled as a combination of a

beam splitter and a detector with 100% efficiency. . . . . . . . . . . . . . . . . . 24

2.9 Weak value of number of photons in the two arms of the interferometer versus

the post-selection parameter δ. For this plot, |α|2 and η are chosen to be 1 and

0.2 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 The weak value of number of photons in arm “1” of the interferometer versus

post-selection parameter, for post-selecting on different n’s at the PNR detector.

For this plot, |α|2 and η are chosen to be 1 and 0.2 respectively. . . . . . . . . . 29

2.11 The weak value of number of photons in arm “1” versus δ, calculated by consid-

ering higher photon terms,∑Pnn, and only post-selecting on a single photon,

n = 1. For this plot, |α|2 and η are chosen to be 1 and 0.5 for a) and 10 and 0.5

for b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.12 Comparison between the weak value of mean photon number versus the strong

value calculated using ABL rule. As expected the strong value remains un-

changed as the post-selection parameter changes whereas the weak value increases

as the post-selection parameter decreases. . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Schematic of a one-dimensional MOT setup. Top: Two coils in anti-Helmholtz

configuration create a magnetic field gradient. Two optical fields, with the same

circular polarization (opposite spins due to opposite propagation direction), are

sent through the center of the magnetic field. The two optical fields are red-

detuned from the J = 0→ J′ = 1 transition of the atoms placed near the center

of the magnetic field gradient. Bottom: the Zeeman shifted energy levels of an

atom in the magnetic field gradient. . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Schematic of the trapping beam setup. . . . . . . . . . . . . . . . . . . . . . . . 39

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3.3 Polarization spectroscopy setup. A circularly polarized pump saturates the

atoms, and a probe, which is vertically polarized, undergoes a polarization rota-

tion through the vapor. A combination of a HWP and a PBS projects the probe

in diagonal and anti-diagonal polarization basis. Two photo-diodes detect the

intensity of the probe in these basis. The subtracted signal is then used as the

error signal to feedback to the master laser for frequency stabilization. . . . . . 40

3.4 Polarization spectroscopy of F = 3→ F ′ transitions. The vertical lines show the

position of different transitions and crossover transitions. The blue circle shows

where the laser is locked to. The horizontal axis of the is detuning of the laser

from F = 3→ F ′ = 4 transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Schematic of AOM double-pass setup. . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6 Calibrations of the AOM double-pass setup. (a) Detuning of the laser from

F = 3→ F ′ = 4 transition versus the modulation voltage. (b) Output power of

the double-pass versus the modulation voltage. . . . . . . . . . . . . . . . . . . . 42

3.7 Schematic of injection-lock setup. The output of the double-pass is used to seed

a high power diode laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Measured power after the PM fiber versus amplitude modulation voltage of the

AOM, when TA running at 1.5A. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.9 Schematic of inside of the TA box, both top view and side view. . . . . . . . . . 46

3.10 Schematic of the setup to prepare the repumper beam. . . . . . . . . . . . . . . 48

3.11 Saturated absorption spectroscopy error signal for F = 2→ F ′ transitions. The

red circle indicates where the repumper laser is locked to. The transition from

the F = 2 ground state to the F ′ = 1, 2, 3 excites states as well as the crossovers

are marked on the signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.12 Calibration of output power of the PM fiber versus the modulation voltage. . . 49

3.13 D2-line of 85Rb atoms. The red arrow indicated the tapping beam and the blue

arrow indicated the repumper beam. . . . . . . . . . . . . . . . . . . . . . . . . 50

3.14 Schematic of optics used for combining the trapping and repumper beam and

capturing a MOT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.15 Power versus voltage read by Uberdetect for (a) trapping beam and (b) repumper

beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xi

3.16 Magnetic field measured at the center of one of the OP coils. The field is measured

along the coil axis, z, and the orthogonal directions x and y. . . . . . . . . . . . 53

3.17 Schematic of different stages used to convert the parameters defined in the Lab-

View software to analog signals used to control multiple devices. . . . . . . . . . 54

3.18 Schematic of the probe, the coupling and the signal beam. . . . . . . . . . . . . 56

3.19 Schematic of the signal pulses, SPCM and how the SPCM is gated. . . . . . . . 58

3.20 Probability of photon detection versus number of photons at the SPCM. From a

fit to the data, the efficiency of the SPCM is measured to be 70%. . . . . . . . . 59

3.21 Schematic of the tagging procedure. The result of the AND gate switches on

a 100MHz RF signal to an AOM upon a successful photon detection. The 1st

and 2nd diffraction order of the AOM are sent to the detector which detects the

probe. When a successful detection happens, this AOM sends a flash of light

to the detector which results in a short spike in the measured amplitude (and

phase.) A delay generator controls the relative delay between the expected XPS

and the position of the tag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.22 Schematic of the signal path. Dashed lines marked as 1 and 2 indicated the

beginning and the end of the polarization interferometer. . . . . . . . . . . . . . 62

3.23 Level scheme used for the WVA experiment. . . . . . . . . . . . . . . . . . . . . 63

3.24 Measured OD versus frequency of the probe for three situations: no external field,

no optical pumping with external field and with optical pumping and external

field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.25 The result of fitting the OD versus probe frequency to the weighted sum of five

shifted Lorentzian functions for (a) no optical pumping and (b) with optical

pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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4.1 Schematic of the experimental setup. Counter-propagating probe and signal

beams are focused down to a waist of 13µm inside a cloud of laser cooled 85Rb

atoms confined in a magneto-optical trap. A collimated coupling beam, prop-

agating in a direction orthogonal to that of the signal and probe, sets up EIT

for the probe. The signal beam is prepared in a linear polarization and after its

interaction with the probe beam is post-selected on a final polarization that is

almost orthogonal to the initial polarization. The probe and the signal beams

are separated from one another using a 10:90 beam splitters. . . . . . . . . . . . 69

4.2 XPS versus signal polarization. A quarter-wave plate is used to set the polar-

ization of the signal beam before the interaction. The maximum and minimum

measured phase shifts corresponds to the two circular polarizations σ+ and σ−.

The signal pulses used for this measurement contained around 600 photons. . . 70

4.3 Measured XPS for click events and no-clicks events. Red squares and green

circles are the measured phase shift conditioned on the SPCM not firing (no-

click) and firing (click) respectively. The horizontal axis shows the mean signal

photon number n and the post-selection parameter δ used for each case. . . . . 73

4.4 The measured phase shift for no-click cases versus the average photon number

in the signal beam |α|2 = n. A linear fit reveals a per-photon XPS of φ =

7.6± 0.3µrad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 XPS difference versus post-selection parameter. Red circles are measured phase

shift difference between click and no-click cases. Gray dashed line is a fit to

φ++φ−2 + φ+−φ−

2δ with φ++φ−2 assumed to be 7.6µrad and φ+ − φ− left as a free

parameter. The fit reveals a value of 11.5± 1.5µrad for φ+−φ−. The cyan solid

line is the full theoretical calculation with the numbers extracted from the two

fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Optimal weights obtained by the covariance matrix Cij = exp(−|i−j|/2.) versus

naive averaging when all data points are weighted equally. . . . . . . . . . . . . 80

5.2 Fisher information in WVT and ST versus post-selection probability. . . . . . . 82

5.3 Fisher information in WVT and ST versus post-selection probability. . . . . . . 83

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5.4 Calculated covariance matrix for (a) 10 points correlated on average and (b) no

time correlation. The noise for (a) is modeled as a slowly varying noise added to

a white noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.5 Fisher information in the standard technique and the weak value technique.

Figure (a) plots the fisher information when measurements are uncorrelated.

Figures (b),(c) and (d) are results for 0.8, 2 and 20 measurements per correlation

time. A total of 200 data points are used in the data set. The CM’s are calculated

by averaging over 10K trials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.6 Two noise traces, with long time correlations(top) and without time correlation

(bottom). The dots represents sampling from the noise with a fixed rate. The

red dots are a 20% random selection from them. . . . . . . . . . . . . . . . . . . 86

5.7 Fisher information in the standard technique, weak value technique, the rejected

subset and correlation between the post-selected subset and rejected subset for

various correlation times. The “optimal” corresponds to summing over the Fisher

information in WVT, rejected data and the correlations. Figure (a) plots the

fisher information when measurements are uncorrelated. Figures (b),(c) and (d)

are results for 0.8, 2 and 20 measurement shots per correlation time. . . . . . . . 88

5.8 The effect of post-selection without weak-value amplification on Fisher informa-

tion versus |α|2. The detector efficiency used for the plots are 100%, 20% and

1% for (a),(b) and (c) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.9 Fisher information for the WVT and ST versus number of correlated measure-

ments. We assumed δ = 0.26, |α|2 = 15 and η = 0.2. . . . . . . . . . . . . . . . . 90

5.10 Schematic of the setup to add slow phase-noise to the probe beam. . . . . . . . 91

6.1 Schematic of the improved level scheme. . . . . . . . . . . . . . . . . . . . . . . 93

B.1 Schematic of setup used to align the input polarization of the PM fiber. . . . . . 105

B.2 (a)Front (left) and side view (right) and (b)3D visualization of relative position

of MOT beams with the cuvette and magnetic field coils. . . . . . . . . . . . . . 106

xiv

Preface

The work presented in this thesis was done as part of the Light Matter Interaction (LMI)

project in the group of Prof. Steinberg. In particular, this weak value amplification project

is in close relation with another project, entitled “Observation of the nonlinear phase shift

due to single post-selected photons” (Nature Physics 11, 905-909) on which I was a principal

contributor and co-author, along with Amir Feizpour and Greg Dmochowski. Therefore, much

of the background to my work has been presented previously in the thesis of Amir Feizpour

(Nonlinear Optics at the Single Photon Level. PhD thesis, University of Toronto, Toronto,

2015.) In order to avoid repeating what has already been published, a few key elements of

this work, such as a full description of the nonlinearity induced by the atomic medium (the

so-called “N-scheme”) and detailed discussion of parts of the experimental setup, are just briefly

introduced; the reader can refer to the references mentioned above and cited later in my thesis

for more detail.

When I started my graduate studies in this group, the cold atom apparatus was shared

between a Lattice project, the goal of which was to study the quantum control techniques using

vibrational states of an optical lattice, and the LMI project. While I had been the lead on

the Lattice project, Amir Feizpour and Greg Dmochowski were setting up the EIT setup and

phase measurement apparatus. The work on the lattice project resulted in “Quantum control of

population transfer between vibrational states in an optical lattice” (arXiv:1510.09186) and my

work on the LMI project contributed to “Observation of EIT-enhanced cross-phase modulation

in the short-pulse regime” (arXiv:1506.07051). Once the lattice project was concluded, I com-

mitted myself fully to the LMI project and after redesigning and improving the apparatus, led

by Amir Feizpour on the “photon” side and me on the “atom” side, we succeeded in measuring

the nonlinear effect due to single post-selected photons. With Amir graduating I took the lead

on the LMI project and the results presented in Chapters 4 grow entirely out of the time during

xv

which I was leading the project. Below is the list of publications on which I’m an author or

co-author:

• Observation of Amplification of the Nonlinear Effect of a Post-Selected Single Photon Us-

ing Weak Measurement Matin Hallaji, Amir Feizpour, Greg Dmochowski, Josiah Sinclair

and Aephraim M. Steinberg (manuscript under preparation)

• Quantum control of population transfer between vibrational states in an optical lattice

Matin Hallaji, Chao Zhuang, Alex Hayat, Felix Motzoi, Botan Khani, Frank K. Wilhem

and Aephraim M. Steinberg, arXiv:1510.09186

• Observation of the nonlinear phase shift due to single post-selected photons Amir Feizpour,

Matin Hallaji, Greg Dmochowski and Aephraim M. Steinberg (Nature Physics 11, 905-909

(2015) , doi:10.1038/nphys3433)

• Optimal estimation, correlated noise, and weak value amplification Josiah Sinclair, Matin

Hallaji, Aephraim M. Steinberg, Jeff Tollaksen and Andrew N. Jordan (manuscript under

preparation)

• Observation of EIT-enhanced cross-phase modulation in the short-pulse regime Greg Dmo-

chowski, Amir Feizpour, Matin Hallaji, Chao Zhuang, Alex Hayat and Aephraim M.

Steinberg, arXiv:1506.07051

• Observing the Onset of Effective Mass Rockson Chang, Shreyas Potnis, Ramon Ramos,

Chao Zhuang, Matin Hallaji, Alex Hayat, Federico Duque-Gomez, John E. Sipe, Aephraim

M. Steinberg, Phys. Rev. Lett 112,170404 (2014)

xvi

Chapter 1

Introduction

1.1 Background

Quantum mechanics is one of the most successful theories to date. One important feature

of quantum mechanics is the way it treats measurements, where instead of deterministically

predicting the outcome of a particular measurement, it assigns probabilities to every possible

outcome. Understanding the nature of measurement in quantum mechanics remains an out-

standing problem[1, 2]. Due to its tremendous success, quantum mechanics has opened up the

field of quantum metrology, which promises to use the theory of quantum mechanics for de-

veloping novel and efficient techniques for parameter estimation and detecting physical effects

with high resolution.

In quantum mechanics, measurement of an observable of a system is accomplished by ob-

serving its effect on a measurement apparatus by entangling them via an interaction, e.g.

measuring the spin of a particle by entangling its spin to its center of mass degree of free-

dom using an inhomogeneous magnetic field[3]. To make the effect resolvable, one needs to

make the system and the meter interact strongly with one another. This resolvability, however,

comes at the cost of disturbing the system. In 1988, Aharonov, Albert and Vaidman (AAV)

introduced a new measurement technique in which the interaction between the system and the

meter is set to be very weak and the effect of the system is measured only if the system is

found to be in a particular final state after the measurement[4, 5, 6, 7]. In this approach, the

system’s disturbance is reduced at the cost of a reduction in the information acquired by the

measurement. The effect of the system on the probe, which is called the weak value, can then

be found by averaging the probe’s readout over many repetitions of the measurement. AAV

1

Chapter 1. Introduction 2

showed that in this type of measurements, the weak value depends equally on both the pre-

selected (initial) and the post-selected (final) states of the system. This feature of the weak

value makes it a powerful tool for exploring fundamental questions and paradoxes in quantum

mechanics[8, 9, 10, 11, 12, 13, 14, 15]. Interest in weak measurement for practical purposes has

grown significantly in past few years[16, 17, 18, 19, 20, 21]. Weak values can be outside of the

eigenvalue spectrum of the observable being measured, and can even be complex. Specifically, as

the overlap between the initial and final states becomes very small, the weak value can become

arbitrarily large 1. This feature, which is called weak value amplification (WVA), has been used

in metrology for parameter estimation and sensitive measurements[22, 23, 24, 25, 26, 27, 28].

Anomalous weak-values observed to date typically utilize two different degrees of freedom of a

single optical system as the “system” and the “probe”[22, 23] , and can therefore be explained

using Maxwell’s equations. Probabilistic weak interactions between two distinct optical systems

has also been demonstrated in [29, 30] where, although the interaction cannot be classically ex-

plained, the probabilistic nature of the interaction greatly reduces their usefulness to quantum

metrology. WVA with a deterministic interaction between two distinct physical systems has

previously only been observed in a transmon qubit system[13] but not, to the best of my knowl-

edge, in optical systems. Here, I present the first observation of weak value amplification of the

photon number in a signal beam. This is accomplished by measuring the nonlinear phase-shift

the signal writes on a probe beam via a deterministic weak interaction that is mediated by a

sample of cold atoms. We believe this work to be the first demonstration of WVA in which a

deterministic weak interaction is used to entangle two distinct optical systems.

Interest in the application of weak measurement to quantum metrology has grown signifi-

cantly in the past few years and debate over its usefulness is still ongoing. While proponents

argue in favor of weak measurement being superior to the standard technique2[31, 32, 33, 34,

35, 36] with experimental demonstrations in[37, 38], opponents claim that weak value tech-

nique is suboptimal[39, 40, 41, 42, 43]. Discussion on imaginary weak values or the effect of

technical noise can be found in[44, 45, 46, 47, 48]. At the core of this debate is the discussion

about quantifying the loss of information when measurement outcomes are rejected in weak

measurement. While it can be shown that in certain regimes the post-selected portion of the

data can contain nearly all the accessible information, some claim that weak measurement is in

1As long as the post-selection success is dominated by the overlap of pre- and post-selected states and notthe measurement back-action.

2I refer to the measurement without pre- and post-selection as the standard technique.


general a suboptimal technique. In [39], an optimal measurement technique is suggested, and

it is shown that this optimal technique is always superior to the WVA technique. But one can

argue that the suggested optimal measurement requires a large resource and is not feasible to

be performed experimentally[31]. Furthermore, one ought to assess the cost of implementing

a more complicated procedure against the additional information gained. In this thesis I will

study the performance of weak values under certain noise models and will show that in a certain

regime, the information in the post-selected data can be made to be the same as the information

in the optimal measurement technique.

The curious features of weak measurement have made it an attractive subject to be stud-

ied from many perspectives. Recently, the quantum mechanical nature of WVA has been

challenged[49] and attempts have been made to describe the effect classically based on mea-

surement disturbance. In their work[49], anomalous values are achieved by using a disturbance

model where the probability of disturbing the system depends on the outcome of the measure-

ment; even if the pre- and post-selected states of the systems are orthogonal, which means the

post-selection can only ever occur due to the measurement back-action. In our system, how-

ever, when the post-selection probability is dominated by the measurement disturbance (and

the ovelap between pre- and post-selected states is zero), no anomalous result is expected; hence

our results cannot be explained by the classical model given in the work mentioned[49].

In 2011, we proposed that WVA could be used to amplify the number of photons in a signal

pulse[44]. By sending a single photon through an interferometer and post-selecting on the rare

event of detecting the photon in the nearly-dark port of the interferometer, one can “amplify”

the mean photon number in each of the arms of the interferometer. By making a probe coherent

state interact with the split single photon in one of the arms, one can measure a nonlinear phase

shift equal to that which would be expected from several photons on the probe despite the fact

that the signal beam only ever contained a single photon. This thesis reports on experimental

observation of weak value amplification of photon numbers, as well as a detailed theoretical

study of the phenomenon and its applications.

1.2 Outline

In chapter 2, I present theoretical calculations of the weak value amplification of a post-selected

single photon. First, I will briefly introduce weak measurement and weak value amplification of


a single photon. I will then extend the discussion to an experiment where coherent states are

used instead of a single photon and comment on the effect of post-selection. A full quantum

mechanical calculation of the problem will be presented as well as an equivalent weak value

treatment. I will compare the weak value results to a similar experimental situation but with

strong interactions, and finally I will introduce a POVM picture of weak measurement.

In chapter 3 I will introduce the apparatus used for this experiment in detail. The key

components and elements used in the setup will be discussed alongside with maintenance and

diagnosis tips for the most important parts of the setup. This chapter is written with the goal

of it being a reference for students working on the apparatus in the future.

The main experimental results of the thesis are in presented in chapter 4. Calibration data

as well as a discussion of the main result will be provided. I will show how a single post-selected

photon can be made to act like 8 photons. I will comment on how this experiment outperformed

a similar former experiment we did on observing the effect of a single post-selected photon[50]

without weak value amplification.

In chapter 5 I will discuss the advantage of WVA in the presence of time-correlated noise. I

will discuss how the information is distributed between the post-selected data and the rejected

data as a function of various measurement parameters. I will show how weak measurement

and weak value amplification can outperform standard measurement techniques as the time

correlation in the noise increases.

Finally, in chapter 6, I discuss various changes that could be made to improve the current

setup as well as different future directions to be pursued.

1.3 Disambiguation

In this thesis, when discussing measurement and weak measurement theory, I refer to the mea-

surement apparatus as “probe” and the quantum system that the measurement is performed on

as “system”. When I discuss the experiment, the “system” is an optical pulse the mean photon

number of which is being measured. This pulse is referred to as “signal” in the experiment. The

“probe” is a coherent state which measures the number of photons in the “signal” (“system”)

which in the experiment is still called the “probe”.

Chapter 2

Weak Value Amplification of Photon

Number - Theory

In this chapter, I present the theory of amplifying the nonlinear effect of post-selected single

photons, using weak measurement. I first introduce weak measurement and weak value ampli-

fication and discuss how it can be used to amplify the number of photons in a signal beam.

I present exact calculation of this effect using the Schrodinger’s equation as well as the weak

value approach to the problem. A comparison between the weak values and the strong values

(expectation values from a strong measurement) and a POVM approach to weak measurement

will be discussed.

2.1 Introduction to Weak Measurement

In this section I will briefly introduce weak measurement and weak value. I first discuss the

relation between the information gain from a measurement and its disturbance. I then introduce

the weak measurement and weak value amplification.

2.1.1 Information versus Disturbance

In quantum mechanics, measuring an observable of a system is done by coupling the system

to a meter via a Hamiltonian so that the change in the state of the meter depends on the

value of the observable being measured. Then, by looking at the final state of the meter, one

gains information about the observable of interest. This way of treating measurements was

5

Chapter 2. Weak Value Amplification of Photon Number - Theory 6

introduced by von Neumann. The Hamiltonian describing the interaction between the meter

and system in such measurement is given by

H = ~χP .A (2.1)

where P is the momentum operator of the meter, A is the observable being measured of the

system and χ is the coupling strength between the meter and the system. This Hamiltonian

causes a displacement in the meter’s position proportional to the eigenvalues of observable A.

A strong measurement is when the coupling strength between the meter and the system is large

enough so that the displacement due to each eigenvalue of A is larger than the uncertainty in

the meter’s position. In this limit, displacements due to different eigenvalues are well resolved

and a single measurement reveals the information about the value of A. This information,

however, comes at the price of disturbing the coherences of the system. To illustrate this, let’s

consider a simple example where the observable A has only two eigenvalues α+ and α− with

eigenvectors |+〉 and |−〉, respectively. The meter’s position wavefunction is initialized to be

|ψ0〉. Let’s assume that the system is initially in the state |i〉 = a|+〉+ b|−〉. This corresponds

to an initial density matrix given by

ρi =

|a|

2 ab

ab |b|2

(2.2)

Here I use overbars to indicate complex conjugates. After the interaction the combined state

of the meter and system will be

|Ψf 〉 = exp(− i~~∫χP .Adt)|ψ0〉 ⊗ (a|+〉+ b|−〉)

=a|ψ+〉|+〉+ b|ψ−〉|−〉(2.3)

where |ψ+〉 and |ψ−〉 are the displaced meter states due to eigenvalues α+ and α−, respectively.

To see the effect of the measurement on the system, I trace over the meter’s Hilbert space.


Using position eigenstates |x〉 as a basis for the meter, we will have

Trmeter[|Ψf 〉〈Ψf |

]=

∫dx〈x|Ψf 〉〈Ψf |x〉

=

∫dx〈x|

[a|ψ+〉|+〉+ b|ψ−〉|−〉

][a〈ψ+|〈+|+ b〈ψ−|〈−|

]|x〉

=

∫dx[aψ+(x)|+〉+ bψ−(x)|−〉

][aψ+(x)〈+|+ bψ−(x)〈−|

].

(2.4)

Hence the density matrix of the system after the interaction becomes:

ρf =|a|2|+〉〈+|∫dx|ψ+(x)|2 + ab|+〉〈−|

∫dxψ+(x)ψ−(x)

ba|−〉〈+|∫dxψ−(x)ψ+(x) + |b|2|−〉〈−|

∫dx|ψ−(x)|2

(2.5)

Since the meter’s wavefunctions are assumed to be normalized we have∫dx|ψ±(x)|2 = 1. The

other two integrals are equal to the spatial overlap of the two displaced wavefunctions and can

be written as∫dxψ+(x)ψ−(x) = 〈ψ−|ψ+〉 and

∫dxψ−(x)ψ+(x) = 〈ψ+|ψ−〉. Therefore, the

final density matrix of the system can be written as

ρf =

|a|2 ab〈ψ−|ψ+〉ab〈ψ+|ψ−〉 |b|2

(2.6)

It’s evident that the coherences (off-diagonal terms in the density matrix) in the system are

reduced by a factor equal to the overlap between the two displaced meter states. If the two

displaced states are well resolved (have a very small overlap, figure 2.1(b)), then the system will

get completely decohered, but on the plus side, one measurement on the meter will reveal all

the information about the observable being measured. This is known as strong measurement.

On the other hand, if the two states have a large overlap, figure 2.1(a), the system will remain

coherent, but a measurement on the meter won’t give much information because the two meter

states are indistinguishable. In this regime, in order to be able to resolve the two displaced

states, one needs to repeat the measurement many times. At a first glance, the advantage of a

measurement without well resolved meter states may not be obvious, but, since the state of the

system is left almost undisturbed in these measurements, one could make use of the system after

it interacts with the meter. Specifically, one can initialize (pre-select) the system in a particular

state and after its interaction with the meter, can perform a projective measurement on the

system and condition reading the value of the meter only if a certain outcome is achieved (post-


1.0 0.5 0.0 0.5 1.0(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.0 0.5 0.0 0.5 1.0(b)

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2.1: Overlap between two displaced states of the meter: (a) when the displacement issmaller than the initial uncertainty of the meter, (b) when the displacement is larger than theinitial uncertainty of the meter.

selection.) This is known as weak measurement, which was introduced by Aharonov, Albert

and Vaidman[4]. We will discuss weak measurement in more detail in the next section.

2.1.2 Weak Measurement and Weak Value Amplification

Assume a system with initial state |i〉 that is interacting with a meter with initial state |ψ0〉 via

a Hamiltonian given by equation 2.1. After the interaction, the combined state of the system

and meter can be written as

|Ψf 〉 = exp(− i~~∫χP .Adt)|i〉|ψ0〉 = exp(−igP .A)|i〉|ψ0〉 (2.7)

where g =∫χdt determines the strength of the interaction. Assuming weak measurement,

g 1, one can expand the exponential as exp(−igP .A) = I − igP .A. By post-selecting the

system on final state |f〉, the state of the probe can be calculated to be

|ψf 〉 = 〈f |(I− igP .A)|i〉|ψ0〉

= (〈f |i〉 − igP 〈f |A|i〉)|ψ0〉

= 〈f |i〉(I− igP 〈f |A|i〉〈f |i〉 )|ψ0〉

= 〈f |i〉 exp(−igAwP )|ψ0〉

(2.8)


Where

Aw =〈f |A|i〉〈f |i〉 (2.9)

is defined as the weak value (for a discussion on the validity of the approximations made in

derivation of equation 2.9, see A.) Throughout the thesis, I will refer to the overlap of the initial

and final state as the post-selection parameter

δ ≡ 〈f |i〉, (2.10)

where δ 1 indicates post-selecting on an almost orthogonal state and δ = 1 indicates post-

selecting on the initial state1. The propagator exp(−igAwP ) displaces the state of the meter

|ψ0〉 by an amount equal to the weak value times g. So, if ψi(x) ∝ exp(−x2/2σ2), then after

the post-selection, the state of the meter becomes ψf (x) ∝ exp(−(x− gAw)2/2σ2) [4].

As mentioned in the previous section, for this technique to be valid, the disturbance of

measurement on the system should be negligible. To quantify this disturbance, we look at

the density matrix of the system after the interaction. This time I will use the momentum

eigenstates |p〉 as a basis for the meter

ρsystem =

∫dp〈p|e−igP A|ψ0〉|i〉〈i|〈ψ0|eigP A|p〉

=

∫dpe−igpA〈p|ψ0〉〈ψ0|p〉|i〉〈i|eigpA

=

∫dp|ψ(p)|2e−igpA|i〉〈i|eigpA,

(2.11)

where ψ(p) represents the momentum wavefunction of the meter. Equation 2.11 shows that for

each momentum component of the meter’s wavefunction p, the state of the system is displaced by

exp(−igpA). This disturbance can be made very small by choosing the momentum wavefunction

of the meter to be very narrow and centered around p = 0. Specifically, if ψ(p) = δ(p),

where δ represents the delta function, then the system will be completely undisturbed and

ρsystem = |i〉〈i|. Narrowing the momentum distribution, increases the position uncertainty of

the meter, which is the same criteria derived in the previous section. Therefore, for equation

2.9 to be a valid description of the measurement outcome, the measurement strength, g, and

the meter’s position uncertainty, σ, should be chosen so that the meter’s displacement due to

1In this thesis, I will assume the post-selection parameter is a real number.


different eigenvalues of A have large overlaps.

The weak value given in equation 2.9, has some curious features. Firstly, it is not necessarily

a real value. Also, as the overlap between the initial and final states of the system (〈f |i〉)decreases, the weak value can get (almost) arbitrarily large, which will be discussed later in

more detail. Weak values taking an imaginary value may be at odds with the nature of them

representing a measurement outcome, but as it will soon become clear, the imaginary part of

the weak value comes about due to the measurement backaction and the disturbance of the

system. By writing the weak value in terms of its real and imaginary part as Aw = AR + iAI ,

the final state of the probe, up to a normalization factor, becomes

ψf (x) = exp(−(x− gAw)2

2σ2) = exp(−(x− gAR − igAI)2

2σ2)

∝ exp(igAIσ2

x) exp(−(x− gAR)2

2σ2).

(2.12)

As it can be seen in equation 2.12, the position of the meter is displaced by an amount pro-

portional to the real part of the weak value. The imaginary part of the weak value acts as

exp(igAIσ2 x). Since this a position dependent phase on the meter’s wavefunction, it acts as a

displacement in momentum. Therefore, the momentum of the probe is shifted by an amount

equal to gAI/σ2. The Hamiltonian in equation 2.1, however, cannot cause a momentum shift

on the meter. This change of momentum is purely a consequence of post-selection. As shown

in equation 2.11, the backaction of the measurement on the system depends on the momentum

of the meter. The backaction due to a certain momentum component of the meter can make a

specific post-selection more probable. So when the system is post-selected in that specific final

states, the momentum distribution of the meter is biased towards the momentum components

which made that post-selection more likely. Therefore, by looking at the displaced momentum

state of the meter, one can learn about the backaction of the measurement on the system.

As seen in equation 2.9, as the overlap between the initial and final state approaches zero,

the weak value can become very large. It can even become larger than the largest eigenvalue

of the observable being measured. This property of the weak value is referred to as Weak

Value Amplification (WVA.) While it may seem to be at odds with the standard measurement

in quantum mechanics, where the outcome is bounded within the eigenvalues of the observ-

able, this amplification can be easily seen as an interference phenomenon between the meter’s

displaces states. To illustrate this, let’s assume that the system was initially prepared in an


equal superposition of eigenstates corresponding to two of the eigenvalues of the observable,

|i〉 = 1/√

2(|+〉 + |−〉). After the interaction, the meter’s state will be in a superposition of

the displaced states due to each of the eigenvalues. Post-selecting the system in an almost

orthogonal final state, |f〉 = 1/√

2((1 + δ)|+〉− (1− δ)|−〉), for post-selection parameter δ 1,

can cause the still coherent meter states to destructively interfere with each other. We will have

ψmeter = 〈f | 1√2

(ψ+(x)|+〉+ ψ−(x)|−〉)

=1

2((1 + δ)ψ+(x)− (1− δ)ψ−(x)).

(2.13)

The mean value of the resulting state can be much larger than the mean value of either of

the displaced meter states. In figure 2.2, an example of this phenomenon is demonstrated. In

figure 2.2(b), the two displaced meter states ψ±(x) are shown. In figure 2.2(a), the result of

12((1 + δ)ψ+(x)− (1− δ)ψ−(x)) is plotted. The vertical lines demonstrated the mean value of

each of the distributions. Note that δ = 〈f |i〉 is the overlap between the initial and final states

100 50 0 50 100(a)

0.02

0.00

0.02

0.04

0.06

0.08

20 10 0 10 20(b)

0.80

0.85

0.90

0.95

1.00

1.05

Figure 2.2: Demonstration of WVA as an interference between the meter states: (a) The resultof the interference between ψ+(x) and ψ−(x). The mean value of this state is much larger thanthe displacement of either of ψ±(x). (b) The displaced states ψ+(x) and ψ−(x). It can be seenthat the two states are not resolvable. The vertical lines demonstrated the mean value of eachof the distributions.

of the system, and the mean value of 12((1 + δ)ψ+(x)− (1− δ)ψ−(x)) depends on the value of

the parameter δ.

From equation 2.9, it may look like that the weak value can grow indefinitely large as

δ → 0. But the weak value starts to deviate from equation 2.9 when δ becomes so small that

the probability of post-selection is not given by δ2 anymore and instead is dominated by the


remaining, although tiny, backaction of the measurement. Figure 2.3 shows the weak value

versus δ. It can be seen that the weak value is inversely growing with δ up to a maximum

value. After that it start to become smaller as δ decreases. The insets of the plot show the

result of the interference between ψ±(x) for three different value of δ. The behavior of weak

values in figure 2.3 is the result of interference between two displaced Gaussian functions with

different weights, where the weight is set by the parameter δ. For δ = 0, the two Gaussian

functions have equal (in magnitude) but opposite weights and the destructive interference will

result in a distribution with zero mean (see the left inset of figure 2.3.) As δ increases, the

imbalance between the two interfering Gaussian functions will result in a distribution with a

mean that is larger than the mean of the original Gaussian functions (see the middle inset of

figure 2.3.) Further increase in δ, decreases the mean of the distribution resulting from the

interference between the two displaces Gaussians (the right inset.) For δ = 1, the two Gaussian

functions have equal wights and the constructive interference will also result in a distribution

with zero mean.

Figure 2.3: Weak value versus δ. The insets show 12((1 + δ)ψ+(x) − (1 − δ)ψ−(x)) for three

different values of δ.


2.2 Weak Value Amplification of a Post-Selected Photon

2.2.1 Introduction

WVA of a single photon

Here, I will show how weak measurement can be used to amplify the effect of a single photon.

Due to this amplification, the effect of a single photon on a separate probe beam can be similar

to the effect due to multiple photons[44]. A single-photon signal pulse is sent through an

interferometer as shown in figure 2.4. Once inside the interferometer, the state of the signal

can be written as |i〉 = 1/√

2(|0〉a|1〉b + |1〉a|0〉b). A coherent state probe weakly measures the

number of photons in path a 2 The interferometer is made a little imbalanced so that an ideal

single-photon detector, in (what used to be) the dark port of the interferometer, can post-select

the signal in |f〉 = 1/√

2((1 + δ)|0〉a|1〉b − (1 − δ)|1〉a|0〉b), with δ 1. With this pre- and

post-selection, the weak value of number of photons in arm a can be calculated to be

〈na〉w =〈f |a†aaa|i〉〈f |i〉 =

1

2+

1

2δ(2.14)

where δ = 〈f |i〉 1 is the overlap between the initial and final state of the signal. The

parameter δ is called post-selection parameter hereafter.

Imbalanced Beam-Splitter

a

b

|ipr

|1is

Nonlinear medium

Figure 2.4: Schematic of the setup used to amplify the effect of a single photon.

Without the post-selection, one would expect that the mean value of the photon number in

arm a to be na = 1/2. But with the post-selection, the effect of the photon is amplified by the

term 1/2δ. It is worth noting that the weak value of photon number in arm b can be shown to

2Later I will discuss how one can realize such photon measurements by using a photon-photon interaction ina nonlinear medium[51].


be 〈nb〉w = 1/2 − 1/2δ. Therefore, the total number of photons inside the interferometer will

be 〈na〉w + 〈nb〉w = 1; no extra photons are generated in the process of WVA. Note that the

presented formulas for the weak value are only valid in the limit δ 1.

Experimental realization of this amplification effect with single photons is extremely chal-

lenging. Single photon sources that are bright enough for this experiment to be done in a

feasible time are hard to set up and maintain. On top of that, losses in the path of the single

photon and inefficiencies in photon detection add to the difficulties of this experiment. These

experimental challenges can be overcome by using coherent states instead of single-photon fock

states for the signal. In what follows, I will show how doing post-selection (photon detection)

on a coherent state can result in an inferred additional photon in the mean value of photon

number, and then I will show how the effect of those additional photons can be amplified by

weak measurement.

The effect of post-selection on a coherent state

Here, I will show how by using a single-photon detector to detect the presence of a photon

in a coherent state signal pulse, in a certain limit, the inferred mean photon number of the

pulse, conditioned on successful detection, can increase by one photon. A careful analysis of

this problem using Bayesian statistics can be found in [52]. Here I will take a simple approach

to this problem by modeling the detector as the following. A single-photon detector with finite

efficiency η, can be modeled as an ideal detector with 100% efficiency preceded by a beam

splitter that transmits a fraction η of the incoming beam (“detected mode”) and reflects a

fraction 1− η of it (“undetected mode”), see figure 2.5

When the beam that is sent to the detector is a coherent state with mean photon number

n = |α|2, the two fields in the detected and undetected mode will also be coherent states |√ηα〉and |√1− ηα〉, respectively. Since the two coherent states in the two modes are independent of

each other, a photon detection on the detected mode wouldn’t change the inferred mean photon

number in the undetected mode. Therefore, the inferred mean photon number in the undetected

mode will always be nundetected = (1−η)|α|2. Now, since the detector after the the beam splitter

is assumed to be ideal, in the cases it doesn’t fire, with probability of Pno−click = e−η|α|2, the

inferred mean photon number in the detected mode will be zero. Therefore, when the detector

does not fire, referred to as “no-click” hereafter, the inferred mean photon number in the initial

coherent state will be nno−click = (1− η)|α|2 With the information above, we now calculate the


Real Detector

Ideal Detector

|p↵idetected

|p

1 ↵iundetected

|↵i

Figure 2.5: A real detector with efficiency η modeled as an ideal detector and a beam splitterthat only transmits a fraction η of the incoming light. In the case where the incoming beamis a coherent state, the two “detected” and “undetected” modes are independent of each otherand the mean photon number inferred in the detected mode does not affect the mean photonnumber in the undetected mode.

mean photon number in the cases when the detector fires, referred to as “click”. We know that

the mean photon number calculated from both “click” and “no-click” cases should be equal to

n = |α|2. Therefore, we have

n = Pclick × nclick + Pno−click × nno−click

|α|2 = (1− e−η|α|2)nclick + e−η|α|2[(1− η)|α|2]

⇒ nclick =1

1− e−η|α|2 ×[|α|2 − e−η|α|2 [(1− η)|α|2]

].

(2.15)

It is easy to show that in the limit where η|α|2 1, the inferred mean photon number for “click”

cases become nclick = |α|2 +1. In this limit, the inferred mean photon number in the undetected

mode is approximately nundetected = |α|2 and the mean photon number3 in the detected mode is

very close to 0. In the unlikely events when the detector fires, the best estimate of mean photon

number in the detected mode will be ndetected = 1. Therefore, the overall mean photon number

when detector fires is nclick = ndetected + nundetected = |α|2 + 1. Figure 2.6 plots the difference

between the inferred mean photon number nclick and |α|2 for η = 0.2. It is evident that for

low |α|2 this difference approaches 1. As |α|2 becomes larger, this difference becomes smaller

3not the inferred photon number


and approaches zero. This is due to the fact that for a large enough η|α|2, the higher photon

number terms in the coherent state in the detected mode become important and upon a photon

detection, the inferred mean photon number becomes ndetected = η|α|2 instead of simplify being

1. Therefore, the overall inferred mean photon number will be |α|2. In practice, there will

always be dark-counts, which cause the detector to fire when ideally it should not. The dark

counts degrade the effect of the post-selection, as shown in figure 2.6 with dashed line. A more

detailed study of the effect of the dark-counts can be found in [52].

0 5 10 15 20| |2

0.0

0.2

0.4

0.6

0.8

1.0

n clic

k|

|2

0 5 10 15 20| |2

01234

n clic

kn n

ocl

ick

Figure 2.6: The difference between the inferred mean photon number for the “click” events and|α|2 versus |α|2 for efficiency η = 0.2 (solid line.) The dashed line is when the effect of dark-counts is added to the inferred photon numbers. The inset shows the difference between nclickand nno−click versus |α|2, when no dark-counts are considered (solid) and when the dark-countsare considered (dashed.)

An experimental observation of the effect of this post-selected single photon was performed

in our group[50]. In the next section, I will show how this added photon by post-selection can

undergo weak value amplification.

2.2.2 Schrodinger picture

I will now extend the ideas presented in section 2.2 to when the signal pulse is a coherent state

instead of a single photon fock state. As shown in figure 2.7, the interaction between the signal

beam and the probe beam happens inside an interferometer. I will assume that the two arms

of the interferometer interact with the probe with different strengths. The propagator for the

interaction can therefore be written as

U = exp(− i~×∫H(t)dt) = exp(iφ1nprn1 + iφ2nprn2) (2.16)


where npr, n1 and n2 are the number operators for the probe field and the field in the two

arms of the interferometer, respectively. φ1 and φ2 represent the interaction strength between

the probe and the signal in the two arms of the interferometer. The interaction causes a phase

shift on the probe which is linear in the number of photons in the signal, and φ1 and φ2 are the

magnitude of this phase shift due to a single photon in each arm. Therefore, by measuring the

total cross-phase shift (XPS) of the probe, one can measure the mean photon number in the

signal4. We assume the initial combined state of the probe and signal, before the interferometer,


|ipr

Nonlinear medium

2

1

3

4

|↵is

Figure 2.7: Schematic of the WVA setup. The signal pulse |α〉s interacts with the probe|β〉probe inside an interferometer. The two arm interact with the probe with different strengthsrepresented as per photon XPS’s of φ1 and φ2, where φ1 > φ2.

is

|Ψi〉 = |α〉s|β〉pr. (2.17)

The creation operators for the signal pulse before the interferometer and the two modes of the

interferometer are related to one another by a†s = 1√2(a†2 − ia†1). Therefore, the state of the

signal inside the interferometer can be written as

|α〉s = exp(αa†s − h.c.)|vac〉

= exp(α√2

(−ia†1 + a†2)− h.c)|vac〉

= |−iα√2〉1|

α√2〉2.

(2.18)

4Note that probe beam also writes a nonlinear phase shift on the signal.


After the interaction the state of the combined system becomes

|Ψf 〉 = exp(iφ1nprn1 + iφ2nprn2)|−iα√2〉1|

α√2〉2|β〉pr

=∑

n

[exp(iφ1nprn1 + iφ2nprn2)

]|−iα√

2〉1|

α√2〉2[e−|β|2

2βn√n!|n〉pr

]

=∑

n

e−|β|2

2βn√n!

[exp(−inφ1n1)|−iα√

2〉1][

exp(−inφ2n2)| α√2〉2]|n〉pr

=∑

n

e−|β|2

2βn√n!|−iαe

inφ1

√2〉1|αeinφ2

√2〉2|n〉pr

(2.19)

Before I continue with calculating the nonlinear phase shift written on the probe (the measured

number of photons of the signal), I will introduce the criteria for the weakness of the interaction.

Weakness of the interaction

The interaction mediated by the propagator in equation 2.16 is considered weak if the state

after the interaction is almost indistinguishable from the initial state, 〈Ψi|Ψf 〉 ≈ 1. Therefore,

we have

〈Ψi|Ψf 〉 =[

1〈−iα√

2|2〈

α√2|pr〈β|

][∑

n

e−|β|2

2βn√n!|−iαe

inφ1

√2〉1|αeinφ2

√2〉2|n〉pr

]

=∑

n

e−|β|2

2βn√n!〈−iα√

2|−iαe

inφ1

√2〉1〈

α√2|αe

inφ2

√2〉2〈β|n〉pr.

(2.20)

Knowing 〈β|n〉pr = exp(−|β|2/2) βn√n!

and 〈α1|α2〉 = exp(− |α1|22 − |α2|2

2 + α1α2), equation 2.20

becomes

〈Ψi|Ψf 〉 =∑

n

e−|β|2 |β|2nn!

exp(−|α|2

2(1− einφ1)) exp(

−|α|22

(1− einφ2)). (2.21)

To simplify equation 2.21, I assume |β|2, which is the mean photon number in the probe, and

both φ1 and φ2 are sufficiently small, so that we have nφ1,2 1. It is worth noting that although

n is unbounded and can be arbitrarily large, |β|2 can be chosen so that the probability of having

an n-photon term that violates nφ1,2 1 is negligible. Under these assumption, exp(inφ1,2)


can be written as 1 + inφ1,2 for all probable n’s in the probe. Hence, equation 2.21 becomes

〈Ψi|Ψf 〉 =∑

n

e−|β|2 |β|2nn!

exp(in|α|2φ1

2) exp(

in|α|2φ2

2)

=∑

n

e−|β|2 |β|2nn!

exp(in|α|2φ)

=∑

n

e−|β|2 (|β|2 exp(i|α|2φ))n

n!

(2.22)

where φ = (φ1 + φ2)/2. Equation 2.22 can be simply written as

〈Ψi|Ψf 〉 = exp(−|β|2) exp(|β|2ei|α|2φ). (2.23)

Expanding ei|α|2φ to second order in φ yields

〈Ψi|Ψf 〉 = exp(i|α|2|β|2φ) exp(−|β|2|α|4φ2

2). (2.24)

From equation 2.24, it is evident that for the interaction between the probe and the signal to

be weak, 〈Ψi|Ψf 〉 ≈ 1, we should have

|β|2|α|2φ 1,

|α|2φ 2

β.

(2.25)

The first term in equation 2.25 can be seen as an overall phase and can be compensated for

(see [44].) The second term is the more interesting condition: |α|2φ is the average XPS written

on the probe, and 2/β is the minimum phase uncertainty of a coherent state with mean photon

number of |β|2, imposed by the number-phase uncertainty ∆φ∆n ≤ 1/2. Therefore, the second

condition in equation 2.25 indicates that for the interaction to be weak, the average XPS on the

probe should be much smaller than the initial phase uncertainty of the probe, which is same

condition mentioned in sections 2.1.1 and 2.1.2.

Expected XPS on the probe

I now continue from equation 2.47 to calculate the expected XPS written on the probe, when

a single photon is detected in the nearly-dark port of the interferometer. I model the imbal-

anced interferometer by writing the transformation matrix of the final beam splitter in the


interferometer as a1

a2

=

1+δ√2

i1−δ√2

i1−δ√2

1+δ√2

a3

a4

(2.26)

for δ 1, where the parameter δ quantifies the imbalance in of the interferometer. With this

transformation, we have

|−iαeinφ1

√2〉1|αeinφ2

√2〉2

=|−iαeinφ1

√2

(1 + δ√

2) +−iαeinφ2

√2

(1− δ√

2)〉3|−iαeinφ1

√2

(−i1− δ√2

) +αeinφ2

√2

(1 + δ√

2)〉4

=| − iαeinφ

2

[e−in

φ1−φ22 (1 + δ) + ein

φ1−φ22 (1− δ)

]〉3|αeinφ

2

[− e−in

φ1−φ22 (1− δ) + ein

φ1−φ22 (1 + δ)

]〉4

(2.27)

where φ = (φ1 + φ2)/2. I define (φ1 − φ2) ≡ ∆φ. If we again assume nφ1,2 1, equation 2.27

becomes

|−iαeinφ1

√2〉1|αeinφ2

√2〉2

=|−iαeinφ

√2

[(1 + i

n∆φ

2)(1 + δ) + (1− in∆φ

2)(1− δ)

]〉3

|αeinφ

√2

[− (1− in∆φ

2)(1− δ) + (1 + i

n∆φ

2)(1 + δ)

]〉4

=| − iαeinφ〉3|αeinφ(in∆φ

2+ δ)〉4.

(2.28)

Next, we assume n∆φ δ. The importance of this assumption will be further discussed later.

With this assumption, one can write

in∆φ

2+ δ ≈ δe in∆φ

2δ . (2.29)

Equation 2.28 then becomes

|−iαeinφ1

√2〉1|αeinφ2

√2〉2 = | − iαeinφ〉3|αδein(φ+ ∆φ

2δ)〉4 (2.30)

and the final state of the probe and signal, equation 2.47, will be

|Ψf 〉 =∑

n

e−|β|2

2β2

√n!| − iαeinφ〉3|αδein(φ+ ∆φ

2δ)〉4|n〉pr (2.31)


Now I calculate the expected XPS written on the probe, when the measurement is condi-

tioned on the single-photon detection in the dark port (mode “4” in figure 2.7.) For now I

assume a detector with 100% efficiency and later will discuss the effect of finite efficiency. For

αδ 1, the field in mode 4 of equation 2.30 can be written as

|αδein(φ+ ∆φ2δ

)〉4 = |0〉4 + αδein(φ+ ∆φ2δ

)|1〉4. (2.32)

Detecting a photon in the dark port is equivalent to post-selecting the field in mode 4 to be in

|1〉4. Therefore, upon post-selection, the combined state of the probe and signal, equation 2.31,

becomes

4〈1|Ψf 〉 = αδ∑

n

e−|β|2

2β2

√n!ein(φ+ ∆φ

2δ)|ineinφ〉3|n〉pr. (2.33)

This post-selection happens with probability Pps = |α|2δ2. After renormalizing 4〈1|Ψf 〉 and

writing the coherent state in mode 3 in terms of number states we will have

|Ψclick〉 =4〈1|Ψf 〉|αδ| =

∑

n,m

e−|β|2−|α|2

2

[βei(φ+∆φ/2)

]n√n!

[− iαeiφ

]m√m!

|m〉3|n〉pr

=∑

m

e−|α|2

2(iα)m√m!|m〉3|βei

[(m+1)φ+ ∆φ

2

]〉pr.

(2.34)

Therefore, for each m-photon term in mode 3, the XPS written on the probe is φm = (m+1)φ+

∆φ/2δ (for more information on operational definition of phase measurement see [53, 54].) I

define Pm = exp(−|α|2)×|α|2m/m! to be the probability of having m photons in mode 3. Then

the average XPS on the probe will be

Φclick =∑

m

Pm × φm = φ∑

m

mPm + (φ+∆φ

2δ)∑

m

Pm

Φclick = |α|2φ+ φ+∆φ

2δ.

(2.35)

It is evident, from equation 2.35, that when the detector fires, the XPS inversely grows

with the parameter δ. To compare the equation 2.35 with equation 2.14, we set φ2 = 0 in

equation 2.35, which is equivalent to only measuring the mean photon number in arm 1 in

the interferometer. With this assumption, we will have φ = φ1/2 and ∆φ = φ1. Therefore,


equation 2.35 becomes

Φclick =[ |α|2

2+

1

2+

1

2δ

]× φ1

⇒〈n1〉w =Φclick

φ1=|α|2

2+

1

2+

1

2δ,

(2.36)

which gives a similar result as equation 2.14, apart from the term |α|2/2. The added term is

due to the fact that we used a coherent state for the signal pulse. The other term, 12 + 1

2δ , is

the added photon due to post-selection and it’s amplified effect. To make this more clear, I will

investigate what the XPS on the probe would be, if the detector failed to fire, which is to say

we post-select on |0〉4 in equation 2.32. With this post-selection, equation 2.31 becomes

|Ψno−click〉 =4 〈0|Ψf 〉 =∑

n

e−|β|2

2βn√n!| − iαeinφ〉3|n〉pr

=∑

n,m

e−|β|2−|α|2

2βn√n!

(−iα)m√m!

eimnφ|m〉3|n〉pr

=∑

m

e−|α|2

2(−iα)m√

m!|m〉3|βeimφ〉pr.

(2.37)

Following the steps taken in derivation of equation 2.35, it is easy to show that the average

XPS when the detector fails to click is given by

Φno−click = |α|2φ. (2.38)

When the backaction dominates

In deriving the expected XPS’s for “click” and “no-click” cases, I made the assumption of

n∆φ δ, which led to equation 2.29. Here I will elaborate further on the meaning of this

assumption. In order to post-select on an unlikely final state, we made the interferometer

shown in figure 2.7 imbalanced. The amount by which the interferometer is imbalanced is

quantified by parameter δ, and the probability of post-selection happening, for an ideal detector,

is Pps = δ2|α|2 (under the δ 1 assumption.) What I have neglected here is the fact that

the post-selection could have happened due to the backaction of the measurement. Since the

two arms of the interferometer interact with the probe with different strength, the probe writes

different XPS’s on the two signal fields in the two arms of the interferometer. This difference

in phase shifts makes the interferometer inherently imbalance. For each n-photon term in the


probe, the difference between the two phase shifts is given by n∆φ. Therefore, if the post-

selection is dominated by the back-action of the measurement and not by the parameter δ,

when n∆φ > δ, one should not expect any weak-value amplification. To further illustrate this,

I will calculate the expected XPS on the probe in the extreme case where δ = 0, where the

post-selection only happens due to the back-action.

With δ = 0, equation 2.28 becomes |− iαeinφ〉3|αeinφ(in∆φ)/2〉4. Hence the combined state

of the probe and the signal after post-selecting on detecting a single photon in mode 4 becomes

|Ψps〉 =∑

n

e−|β|2

2βn√n!

(inα∆φ

2einφ)| − iαeinφ〉3|n〉pr (2.39)

It is easy to show that the relative phase between different n-photon terms in probe for each

m-photon term in the mode 3 is given by (m + 1)φ. Therefore, the expected XPS when the

post-selection is dominated by the measurement backaction is given by

Φδ=0 = (|α|2 + 1)φ. (2.40)

It can be seen that there is still one photon worth of extra XPS due to the post-selection, but

the amplification effect is absent.

It is worth noting that the wavefunction given in equation 2.39 is not normalized. The

normalization factor is given by

〈Ψps|Ψps〉 =∑

m,n

e−|β|2/2 β

m

√m!

(−imα∆φ

2e−imφ)3〈−iαeimφ|pr〈m|

e−|β|2/2 β

n

√n!

(inα∆φ

2einφ)| − iαeinφ〉3|n〉pr

=∑

m,n

e−|β|2 βmβn√

m!n!(nm|α|2 ∆φ2

4)ei(n−m)φ〈−iαeimφ| − iαeinφ〉3δn,m

=∑

n

e−|β|2 |β|2nn!

n2 |α|2∆φ2

4=|α|2∆φ2

4〈n2pr〉β

(2.41)

where 〈n2pr〉β is the expectation value of the square of the number operator of the probe field

given coherent state |β〉pr. This quantity is related to the variance of the photon number

distribution in the probe field. This normalization factor can also be seen as the probability

of the post-selection happening due to the backaction, and as discussed in section 2.1.2, this

probability depends on the variance of the photon number distribution of the probe field.


In this section, I used the Schrodinger equation to calculate the amplified effect of a post-

selected single photon on a probe in the limit where δ 1. In the next section, I will take an

equivalent approach, the weak value approach, to calculate the weak value of photon number

with arbitrary post-selection parameter. This approach will enable us to easily calculate the

expected XPS on a probe for every δ and to consider the effect of detector with finite efficiency.

2.2.3 Weak Value Approach

In this section, I will use the weak value formula given in equation 2.9 to calculate the ex-

pectation value of the mean photon number in the two arms of the interferometer, for all

post-selection parameters δ. I will also consider the effect of a detector with finite efficiency.

The operator being measured is n1(2) = a†1(2)a1(2), the photon number operator in each arm

of the interferometer. The initial state of the signal is given by |i〉 = |α〉s. The final state is

detecting a single photon at the single-photon detector, which is detecting a photon in mode

“5” in figure 2.8. The weak values are then defined as

〈n1(2)〉w =〈f |a†1(2)a1(2)|i〉

〈f |i〉 (2.42)

2

1

3

4

|ipr

|↵is


5

6

Ideal Detector

Detector with finite efficiency

Nonlinear medium

Figure 2.8: Schematic of the WVA setup. The signal pulse |α〉s interacts with the probe|β〉probe inside an interferometer. The two arm interact with the probe with different strengthsrepresented as per photon XPS of φ1 and φ2, where φ1 > φ2. The effect of a detector with finiteefficiency is modeled as a combination of a beam splitter and a detector with 100% efficiency.


I first model the imbalanced interferometer by writing the transformation matrix of the

imbalanced beam splitter as

a1

a2

=

cos θ sin θ

− sin θ cos θ

a3

a4

(2.43)

where the parameter θ ∈ [−3π/4,−π/4] determines the amount by which the beam splitter in

imbalanced. The parameter θ is related to the previously introduced post-selection parameter

δ by δ = (cos θ+sin θ)/√

2; θ = −3π/4 corresponds to δ = 1 (post-selecting on the initial state)

and θ = −π/4 corresponds to δ = 0 (post-selecting on a completely orthogonal state.) With

this model, the state of the signal after the interferometer can be written as

|α〉s = | α√2

(cos θ − sin θ)〉3|α√2

(cos θ + sin θ)〉4 (2.44)

Similar to figure 2.5, the detector in figure 2.8 can be modeled as an ideal detector proceeded

with a beam splitter that partially transmits the beam. This beam splitter can be modeled as

a4 = ra5 + ta6 (2.45)

with t2 = η and r2 = 1− η, η being the efficiency of the detector. With this model it’s easy to

show that the initial state of the signal can be written as

|i〉 = | α√2

(cos θ − sin θ)〉3|tα√2

(cos θ + sin θ)〉5|rα√2

(cos θ + sin θ)〉6 (2.46)

The final state is detecting a single photon at the detector

|f〉 = | α√2

(cos θ − sin θ)〉3|1〉5|rα√2

(cos θ + sin θ)〉6 (2.47)

Now, in order to calculate the weak values given in equation 2.42, we need to write n1

and n2 in terms of creation and annihilation operators in modes “3”,“5” and “6”. With some


algebra, it can be shown that

a†1a1 = cos2 θa†3a3 + r sin θ cos θa†3a6 + t sin θ cos θa†3a5

r sin θ cos θa†6a3 + r2 sin2 θa†6a6 + rt sin2 θa†6a5

t sin θ cos θa†5a3 + rt sin2 θa†5a6 + t2 sin2 θa†5a5

(2.48)

If we now define α3 = α√2(cos θ− sin θ), α5 = t α√

2(cos θ+ sin θ) and α6 = r α√

2(cos θ+ sin θ),

the weak value of number of photons in mode 1 becomes

〈f |a†1a1|i〉〈f |i〉 = cos2 θ|α3|2 + r sin θ cos θα3α6 + t sin θ cos θα3α5

+ r cos θ sin θα3α6 + r2 sin2 θ|α6|2 + rt sin2 θα6α5

+ t sin θ cos θα3

α5+ rt sin2 θ

α6

α5+ t2 sin2 θ

(2.49)

which can be simplified to

〈n1〉w =|α|2

2+

sin θ

cos θ + sin θ− η|α|2

2[sin θ(sin θ + cos θ)] (2.50)

A similar calculation reveals

〈n2〉w =|α|2

2+

cos θ


2[cos θ(sin θ + cos θ)] (2.51)

To see the relation between equations 2.50 and 2.51 and the weak value calculated in equa-

tion 2.36, I will first assume that η|α|2 1. Then, by expanding θ near θ0 = −π/4, and after

replacing θ with δ (δ = (cos θ + sin θ)/√

2), we arrive at

〈n1〉w =|α|2

2+

1

2+

1

2δ

〈n2〉w =|α|2

2+

1

2− 1

2δ

(2.52)

Assuming an XPS of φ1(2) for mode 1(2) of the interferometer, the expected XPS written on

the probe will be

Φclick = φ1〈n1〉w + φ2〈n2〉w = (|α|2 + 1)φ1 + φ2

2+φ1 − φ2

2δ(2.53)

which is exactly the same as what was calculated in equation 2.35, with φ = (φ1 + φ2)/2 and


∆φ = φ1 − φ2

It is worth noting that the total number of photons inside the interferometer is given by

〈n1〉w + 〈n2〉w = |α|2 − η|α|2δ2 + 1 (2.54)

where |α|2− η|α|2δ2 is sum of mean photon numbers in the undetected modes 3 and 6, and the

added photon is due to post-selection in mode 5, see figure 2.8. Figure 2.9 plots 〈n1〉w,〈n2〉wand 〈n1〉w + 〈n2〉w versus post-selection parameter δ for |α|2 = 1 and η = 0.2.

0.0 0.2 0.4 0.6 0.8 1.015

10

5

0

5

10

15

n

n1 w

n2 w

n1 w + n2 w

Figure 2.9: Weak value of number of photons in the two arms of the interferometer versus thepost-selection parameter δ. For this plot, |α|2 and η are chosen to be 1 and 0.2 respectively.

It is easy to show that equations 2.50 and 2.51 can be written in terms of the post-selection

parameter δ as

〈n1(2)〉w =|α|2

2+

1

2(1±

√1

δ2− 1)− η|α|2

2δ(δ ±

√1− δ2) (2.55)

So far I assumed that the post-selection is done by using a single-photon detector in port 4

of the interferometer in figure 2.8, which is to say we post-select on a single-photon fock state

in that mode. But in reality, the detectors used in a laboratory usually only detect the presence

of a photon and cannot distinguish between detecting a single photon and detecting multiple

photons. Therefore, for my treatment to be accurate, I need to assume that mean photon

number at the ideal detector in mode 5 is very low, so that detecting more than one photon is


extremely unlikely, |α|2ηδ2 1. In the next section, I will replace the single-photon detector

with a Photon-Number Resolving (PNR) detector[55] and will calculate the weak values when

the measurement is conditioned on detecting an n-photon state at the PNR.

2.2.4 Weak Values and PNR Detectors

In this section, I will calculate the weak value of photon number in the two arms of an interfer-

ometer as shown in figure 2.8, by post-selecting on detecting an n-photon state at the detector

in mode 5. As before, the initial state of the signal is given by equation 2.46. The final state

in which the signal is post-selected is given by

|f〉 = | α√2

(cos θ − sin θ)〉3|n〉5|rα√2

(cos θ + sin θ)〉6. (2.56)

Following a calculation similar to the one carried out in the previous section, the weak values

can be shown to be

〈n1〉w∣∣∣n

=|α|2

2+ n

sin θ


2[sin θ(sin θ + cos θ)] (2.57)

and

〈n2〉w∣∣∣n

=|α|2

2+ n

cos θ


2[cos θ(sin θ + cos θ)] (2.58)

These equations are very similar to equations 2.50 and 2.51, except that the inferred mean

photon number has increased by a factor of n. Most importantly, these equations show that the

effect of all n detected photons will be amplified due to the weak value amplification. Figure

2.10 shows the weak value of photon number in arm “1” of the interferometer (figure 2.8) versus

post-selection parameter δ. It can be seen that post-selecting on higher photon numbers at the

PNR detector results in larger amplification.

As mentioned in section 2.2.3, when using a single-photon detector instead of a PNR detec-

tor, we should either consider the effect of detecting higher photon numbers or limit ourselves to

the regime where the probability of detecting higher photons are low. Making the |α|2ηδ2 1

assumption, ensured the latter in previous section. Here I will investigate the effect of detecting

higher photon term on the weak value. For this, I only consider two detection outcomes at the

single-photon detector: the detector either detects the presence of a photon (“click”) or it does

not (“no-click”). The “no-click” case corresponds to 〈n1〉w∣∣∣n

for n = 0. Given a signal coherent


0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

n 1w

| n

n = 0n = 1n = 2n = 3n = 4

Figure 2.10: The weak value of number of photons in arm “1” of the interferometer versuspost-selection parameter, for post-selecting on different n’s at the PNR detector. For this plot,|α|2 and η are chosen to be 1 and 0.2 respectively.

state |α|2, efficiency η and post-selection parameter δ, the probability of a “no-click” outcome

is

Pno−click =∣∣〈0|√ηαδ〉

∣∣2 = e−η|α|2δ2

(2.59)

Therefore, the probability of a “click” outcome is given by

Pclick = 1− Pno−click = 1− e−η|α|2δ2(2.60)

The probability of each n-photon detection is given by

Pn =∣∣〈n|√ηαδ〉

∣∣2 = e−η|α|2δ2 (η|α|2δ2)n

n!(2.61)

Now we can calculate the effect higher photon detection events on the weak value of photon

number by

〈n1〉w∣∣∣click

=∑

n=1

Pn × 〈n1〉w∣∣click

Pclick(2.62)

Figure 2.11 compares the weak value of photon number in mode “1” in two cases: when the

higher photons terms are considered and when only the detecting a single photon is considered.

The parameters in figure 2.11a are chosen so that the assumption |α|2ηδ2 1 remains valid


0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

n 1w

Pn ×nn =1

(a)

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

n 1w

Pn ×nn =1

(b)

Figure 2.11: The weak value of number of photons in arm “1” versus δ, calculated by consideringhigher photon terms,

∑Pnn, and only post-selecting on a single photon, n = 1. For this plot,

|α|2 and η are chosen to be 1 and 0.5 for a) and 10 and 0.5 for b).

for most δ’s and it can be seen that the two methods don’t deviate from each other by much.

The difference, however, is more pronounced in 2.11b, where the parameters are chosen so that

the assumptions is only valid for δ 1. As it can be seen, in 2.11b, one would underestimate

the weak value when not considering the effect of detection of higher photon terms.

2.2.5 Measuring the Amplified Effect of a Post-selected Single Photon

The weak value of photon numbers in arm 1 and 2 of the interferometer for “click” and “no-

click” cases from equations 2.57 and 2.58 can be written as

〈n1(2)〉click =|α|2 + 1

2± 1

2δ

〈n1(2)〉no−click =|α|2

2

(2.63)

The term |α|2/2 is common in both “click” and “no-click” cases. Therefore, by subtracting

the measured effect of mean photon number when detector fires from when it doesn’t fire, one

can directly measure the effect of a single photon. If we assume a per photon XPS of φ1(2) for

arm 1(2) of the interferometer, looking at the difference between the measured XPS when the

detector fires, φclick, and the measured XPS when it doesn’t, φno−click, will result in

φclick − φno−click =φ1 + φ2

2+φ1 − φ2

2δ. (2.64)


Evidently, this quantity is independent of |α|2 and only shows the amplified the post-selected

photon.

2.2.6 Weak Values vs. Strong Values - ABL Rule

In this section, I will calculate the mean photon number in arm 1 of the interferometer, as shown

in figure 2.8, conditioned on post-selecting a single photon in arm 4, but this time assuming a

strong interaction between the signal and probe in arm 1. When the interaction is strong, the

two fields in arm 1 and 2 will no longer be coherent, hence no interference will happen at the

beam splitter that closes the interferometer. Because of this, the weak value formalism will no

longer be applicable and one should use the ABL rule to calculate the mean photon number in

arm 1.

The ABL rule, introduced by Aharonov, Bergmann and Lebowitz[56], evaluates the proba-

bility of a certain measurement outcome, n, when the measurement is conditioned on having a

certain initial state |i〉 and post-selecting on a certain final state |f〉

P (n|i, f) =|〈f |n〉〈n|i〉|2

∑

n

|〈f |n〉〈n|i〉|2(2.65)

Knowing the conditional probability of each outcome, the mean value can be calculated as

n =∑P (n|i, f)× n.

I will now calculate the mean value of number of photons in mode 1, using the ABL rule.

To make the calculations easier, I will limit the treatment to the case where the interferometer

is only slightly imbalanced, where the WVA occurs. Since the detected photon in port 4 could

have come from either of the arms of the interferometer, I should first calculate P (n1, n2|i, f),

which the conditional probability of having n1 photons in arm 1 and n2 photons in arm 2 given

the pre- and post-selection. For |i〉 = |α/√

2〉1|α/√

2〉2 and post-selecting on |f〉 = |α〉3|1〉4, one

can show

〈f |n1, n2〉1,2〈n1, n2|i〉 =

(α/√

2)2(n1+n2)

n1!n2!α

(− n1e

−δ + n2eδ). (2.66)

Knowing 〈f |n1, n2〉1,2〈n1, n2|i〉, one can numerically evaluate P (n1, n2|i, f) using equation 2.65.

Once P (n1, n2|i, f) is known, the photon number distribution in arm 1, conditioned on i and


f , can be calculated by

P (n1|i, f) =∑

n2

P0(n2)P (n1, n2|i, f), (2.67)

where P0(n2) is the initial photon number distribution in arm 2. Figure 2.12 plots the numerical

result of the mean photon number number using ABL rule for |α|2 = 2, and compares the result

with the weak values calculated for the same |α|2. As it can be seen, the weak value of the

mean photon number increases as δ gets smaller, whereas the mean value calculated using the

ABL rule remains constant.

0.00 0.05 0.10 0.15 0.20

0

2

4

6

8

10

12

14

n

ABL ruleWeak Value

Figure 2.12: Comparison between the weak value of mean photon number versus the strongvalue calculated using ABL rule. As expected the strong value remains unchanged as the post-selection parameter changes whereas the weak value increases as the post-selection parameterdecreases.

Figure 2.12 shows that if one were to measure the mean photon number in arm “1” of the

interferometer conditioned upon detecting a single photon in the nearly-dark port, the result

will be amplified if the measurement is done weakly. But if the measurement is performed

strongly, no anomalous result should be expected.

2.3 Weak Measurement as a POVM

Positive-Operator Valued Measure (POVM) is the most generalized formulation of measurement

in quantum mechanics. Often the probe and the system in a measurement belong to different

Hilbert spaces, and because of that a projective measurement on the probe cannot be written


as a projective measurement on the system. Every measurement in quantum mechanics can be

described by a POVM and weak measurement is no exception. Here I will present a POVM

treatment of the photon number measurement described in this chapter. For simplicity I make

the assumption that the measurement can only have two results and will limit the calculation

to real post-selection parameter and real weak values. A more detailed calculation of what

follows, for a different system, can be found in[57, 49].

As described in section 2.2.2, the propagator of the interaction, in the weak limit and when

only one arm interacts with the probe, can be written as

U = eiφ0nsnpr = I + iφ0nsnpr (2.68)

where φ0 1 is the per photon phase shift (describing the interaction strength) and ns and

npr are photon number operator for the signal and probe respectively. For a probe initialized

in a state |ψ〉pr, we define the Kraus operator for a phase measurement on the probe as

Kφ = 〈φ|eiφ0nsnpr |ψ〉pr

= 〈φ|ψ〉pr + iφ0ns〈φ|npr|ψ〉pr.(2.69)

With analogy to the relation between position and momentum operators x and p, one can

write the photon number operator in phase basis as 〈φ|npr|φ′〉 = −iδ(φ − φ′) ∂∂φ . Assuming a

Gaussian function for the initial wavefunction of the probe, ψpr(φ) = Nexp(φ2/2σ2), N being

the normalization factor, we will have

〈φ|npr|ψ〉pr = −i φσ2ψpr(φ) (2.70)

and equation 2.69 becomes

Kφ = (I + φ0nsφ

σ2)ψpr(φ). (2.71)

I will consider a coarse grained measurement so that for all φ ≤ 0 the readout is −1 and

for φ > 0 the readout is 1. The corresponding quantum operations on a signal described by


density matrix ρs can be written as

E−ρs =

∫ 0

−∞dφ KφρsK

†φ

E+ρs =

∫ ∞

0dφ KφρsK

†φ

(2.72)

Performing the integrals, to the first order in φ0, the operations become

E±ρs =1

2

[ρs ±

φ0

σ√

2(nsρs + ρsns)

]. (2.73)

The term (nsρs+ρsns) describes the measurement disturbance on the system. It is evident that

in the limit where no information is gained from the measurement, φ0 σ, the measurement

will not disturb the system. Defining measurement outcome O ∈ −1, 1 and C = φ0

σ√π

,

equation 2.73 can be written as

EOρs =1

2

[ρs +OC(nsρs + ρsns)

]. (2.74)

We are interested in finding the expectation value of O conditioned on pre-selecting the

system in state |i〉 and post-selecting the system in state |f〉. This expectation value is described

by

〈O〉 =∑

OOP (O

∣∣f, i) (2.75)

where P (O∣∣f, i) is the conditional probability of outcome O given the initial and final state of

the system. Using the identity P (O∣∣f, i) = P (O, f

∣∣i)/P (f∣∣i), equation 2.75 becomes

〈O〉 =∑

OO〈f |EO|i〉〈i|f〉|〈f |i〉|2

=∑

O

O2

|〈f |i〉|2 +OC[〈f |ns|i〉〈i|f〉+ 〈f |i〉〈i|ns|f〉]|〈f |i〉|2

=∑

O

O2

[1 +OC 〈f |ns|i〉〈f |i〉

].

(2.76)

Doing the summation over O, we find

〈O〉 = C〈f |ns|i〉〈f |i〉 . (2.77)


Hence the weak value of ns is defined as

〈ns〉w =〈f |ns|i〉〈f |i〉 = 〈O

C〉. (2.78)

The parameter C = φ0/σ√π can be thought of as a calibration factor that readjusts the outcome

of the coarse grained measurement of the probe for different measurement strength. A simple

example can illustrate the importance of the calibration factor C. Assume that we have built

a photon number measurement apparatus, with only two outcomes as mentioned above. For

calibration, we use this device to measure the number of photons in a signal pulse that contains

a single photon. If we get the outcome “1” 55% of the time and measure “-1” 45% of the time,

the expectation value of O will be 0.1. Therefore, in order for this measurement apparatus to

correctly evaluate the mean photon number in that pulse, we need to divide the mean value of

O by 0.1. With this calibration factor, if for a different pulse with unknown photon number we

measure 〈O〉 = 0.01, we know that the mean photon number is 0.1 and if we measure 〈O〉 = 0.5,

we know that the mean photon number is 5.

Chapter 3

Apparatus

As introduced in the previous chapter, in order to observe the amplified effect of a post-selected

single photon, one needs to send the signal through an interferometer. Once inside the in-

terferometer, the signal beam interacts with a coherent state probe via a nonlinear medium,

where the interaction strength is different in each arm. The effect of the signal on the probe

manifests itself as a cross-phase shift (XPS), which is linear in the number of photons in the

signal. After the interaction, a single-photon detector detects the presence of a single photon in

the nearly-dark port of the interferometer. The “darkness” of this port is controlled by using

an imbalanced beam splitter. The information about the number of signal photons inside the

interferometer is obtained by measuring the nonlinear phase shift written on the probe by the

signal, See figure 2.7. Therefore, the key elements for realizing this amplification effect can be

listed as

• A nonlinear medium

• An interferometer in which the signal beam in each arm interacts with the probe with

different strengths

• A beam splitter whose reflectance and transmittance can be precisely controlled

• A single photon detector

• A precise phase measurement apparatus

In practice, the nonlinear interaction between the signal and probe beam is mediated by a

sample of cold 85Rb atoms. Using electromagnetically induced transparency (EIT), we enhance

36

Chapter 3. Apparatus 37

the nonlinearity by using atomic resonance while eliminating absorption of the probe. The

two paths of the interferometer are made up of two orthogonal circular polarizations, which

overlap spatially. A certain choice of beam geometry and atomic level scheme ensures that

the interaction between the probe and the signal pulse is different for the two polarizations.

Combination of a quarter-wave plate (QWP), a half-wave plate (HWP) and a polarizer enables

us to “post-select” the polarization of the signal on an arbitrary final polarization. A Single-

Photon Counting Module (SPCM) is used to detect the presence of a single photon in the dark

port of the interferometer. The phase of the probe is measured using a beat-note interferometry

technique. In what follows in the rest of this chapter, all of these elements will be introduced

and discussed in detail.

3.1 Magneto-Optical Trap

The element used to mediate the interaction between the signal and the probe in this experiment

is a cloud of cold rubidium 85 atoms. Rubidium is an alkali metal that has one free electron

in the 5s orbital. Due to fine-structure splitting, the excited state 5p is split into a doublet

and the two transitions 52S1/2→ 52P1/2

and 52S1/2→ 52P3/2

, are referred to as the D1 line

(794.98nm) and the D2 line (780.24nm), respectively. Each of these transitions, in turn, have

additional hyperfine structure due to the coupling between the total angular momentum of

the electron and the angular momentum of the nucleus. The details of the hyperfine structure

depends on the isotope of interest, since different isotopes have nuclei with different spin. The

two naturally occurring isotopes of rubidium are 85Rb (stable) with nucleus angular momentum

of I = 5/7 and natural abundance of 72.2% and 87Rb (half-life of 49 billion years) with nucleus

angular momentum of I = 3/2 and natural abundance of 27.8%. For 85Rb, which is used for

this experiment, the hyperfine splitting splits the ground state into two ground states with total

angular momenta of F = 2 and F = 3. The excited state of the D2 line splits into 4 excited

states with total angular momenta of F ′ = 1 , 2 , 3 and 4, with lifetime of 26.23ns (decay rate

of Γ = 2π× 6.06 MHz.) In what follows I will show how these hyperfine structure of D2 line of

85Rb can be used for laser cooling and trapping as well as mediating the interaction between

the probe beam and the signal pulse (for more information about Rubidium atoms, see [58].)

Magneto-Optical Trapping (MOT)1 is a way of cooling and trapping atoms that makes

1also not very commonly called Zeeman Assisted Radiation Pressure Trap (ZARPT) or Zeeman assistedOptical Trap (ZOT)


use of a combination of an optical force and a magnetic field gradient. The scattering force

,+ ,I I-1 0 +1

z

E

J=0

J=1

J=0

J=1

mj = 0

mj = 0

mj = -1

mj = +1

+

B(z)

Figure 3.1: Schematic of a one-dimensional MOT setup. Top: Two coils in anti-Helmholtzconfiguration create a magnetic field gradient. Two optical fields, with the same circular polar-ization (opposite spins due to opposite propagation direction), are sent through the center ofthe magnetic field. The two optical fields are red-detuned from the J = 0 → J′ = 1 transitionof the atoms placed near the center of the magnetic field gradient. Bottom: the Zeeman shiftedenergy levels of an atom in the magnetic field gradient.

experienced by a cloud of atoms in a MOT is both velocity dependent (due to the Doppler

effect) and position dependent (due to the position dependent Zeeman shifts in the presence of

the magnetic field gradient.) Therefore, the atoms, which feel this force, can be seen as damped

harmonic oscillators. A certain choice of optical field intensity, detuning and magnetic field

gradient can result in cooling the atoms down to (usually) below their Doppler temperature

(defined as TDoppler = ~Γ/2kB, where Γ is the natural linewidth of the transition and kB is the

Boltzmann’s constant) and trapping them at the center of the magnetic field gradient. Figure

3.1 shows the schematic of a one-dimensional MOT setup.

The MOT setup used in our experiment has five main components: trapping beam, re-

pumper beam, magnetic field coils, optical pumping beam and the computer which controls the

tasks and atomic cycles. Each of these components will be discussed in detail in what follows

in this section.


3.1.1 Trapping Beam

The key component of the MOT setup is the trapping beam, which provides the radiation

pressure needed for laser cooling and optical molasses. Having a stable trapping beam in power

and frequency, enables maintaining a healthy MOT for a long period of time, which is a necessity

for long data runs. In designing a stable apparatus, one should avoid long free space optical

paths, specially in a lab that is not climate controlled. To achieve stability, we have designed

our trapping beam setup in a modular way with very short paths between each module. Figure

3.2 shows different modules used in the setup and their relative position.

Master LaserOI

AOM

H-O

I

TAAO

M

-

Master Laser

AOM Double Pass

Injection Lock

Spectroscopy

AOMSingle

Pass

Tapered Amp

Figure 3.2: Schematic of the trapping beam setup.

Master laser and Spectroscopy

The master laser used for the trapping beam is a New Focus 6013 Vortex which is an External

Cavity Diode Laser (ECDL.) The wavelength of the laser is centered at 780.24nm. This laser

is controlled by a laser driver and controller New Focus TLB 6000, which provides current

modulation, piezo modulation and temperature controller. The output power of the laser is

typically around 7.5mW (6mW after the Optical Isolator (OI) that is used immediately after


the laser to protect the laser from back-reflections.) A small portion of the output (100µW) is

branched off using a Polarizing Beam Splitter (PBS) and is sent to the polarization spectroscopy

setup. The error signal from the spectroscopy is sent to a homemade PID circuit which provides

a feedback to the current and piezo modulations to stabilize the frequency of the laser. Details

of the PID circuit and how to set the PID parameters can be found in [59]. Figure 3.3 shows

the schematic of the polarization spectroscopy setup.

Rb VaporCell

QWP

HWP

PBS

BSTo PID feedback

Photo Diode

100uWV-polarized

Probe Pump

20uWV-Polarized

80uWR or L-Polarized

Figure 3.3: Polarization spectroscopy setup. A circularly polarized pump saturates the atoms,and a probe, which is vertically polarized, undergoes a polarization rotation through the vapor.A combination of a HWP and a PBS projects the probe in diagonal and anti-diagonal polariza-tion basis. Two photo-diodes detect the intensity of the probe in these basis. The subtractedsignal is then used as the error signal to feedback to the master laser for frequency stabilization.

The master laser is locked to F = 3→ F ′ = 4 transition. Figure 3.4 shows the spectroscopy

signal for the transitions between the ground state F = 3 to the excited states F ′ = 2, 3 and 4

as well crossovers between the excited states. The blue circle shows where the master laser is

typically locked to (for more information about polarization spectroscopy, see[60, 61].)

The polarization spectroscopy setup is usually very stable and doesn’t drift significantly

within a few week time. Reduction in the size of the error signal or appearance of a Doppler

background are indicators that suggest the alignment of the spectroscopy setup is degrading,

which can result in an unstable lock. See B.1 for alignment procedure of the spectroscopy setup.

If after locking the laser, you observe that the error signal is oscillating on the scope, check the


X343 ! 3 3 ! 43 ! 2 X24X23

Figure 3.4: Polarization spectroscopy of F = 3 → F ′ transitions. The vertical lines show theposition of different transitions and crossover transitions. The blue circle shows where the laseris locked to. The horizontal axis of the is detuning of the laser from F = 3→ F ′ = 4 transition.

PID parameters of the locking circuit. See [59] on how to set the PID parameters.

Important note: Avoid placing magnetic elements, such as an optical isolator, close to the

Rubidium vapor cell when doing polarization spectroscopy setup. Polarization spectroscopy

works based on spin polarizing the atoms and external magnetic fields can degrade the signal.

AOM Double-Pass

As mentioned before, the master laser is locked to the F = 3→ F ′ = 4 transition. However, the

trapping beam is required to be about 20 MHz detuned from the F = 3→ F ′ = 4 transition for

best cooling and trapping performance. To finely adjust the frequency of the trapping beam,

we use an Acousto-Optic Modulator (AOM) in double-pass configuration and a second AOM

in single-pass configuration. Double-passing a beam through an AOM enables changing the

frequency of the beam without changing its alignment. Figure 3.5 shows the schematic of the

AOM double-pass.

The radio frequency (RF) by which the AOM is driven is set to be centered at -71MHz

(the minus sign indicates that the frequency of the diffracted beam is shifted down.) The RF

can be modulated by an external control voltage between -1V and 1V which corresponds to a

frequency range of 42MHz to 104MHz. Hence, the frequency of the output of the double-pass

can be changed in a rage of -208MHz to -84MHz. To have the ability to change the frequency

of the trapping beam around the F = 3 → F ′ = 4 transition, we use another AOM in single-


AOM

71MHz

Beam BlockBeam Block

Quarter-Wave Place (QWP)

PBS

4mW

0.2 to 1.9 mW

Figure 3.5: Schematic of AOM double-pass setup.

pass configuration, driven at a fixed frequency of +90 MHz.2 Calibration of detuning of the

trapping beam from F = 3→ F ′ = 4 transition and the output power of the double-pass versus

modulation voltage V are plotted in figure 3.6.

1.0 0.5 0.0 0.5Modulation Voltage (V)

140

120

100

80

60

40

20

0

20

Detu

ning

34

(MHz

)

(a)

1.0 0.5 0.0 0.5Modulation Voltage (V)

0.0

0.5

1.0

1.5

2.0Ou

tput

pow

er (m

W)

(b)

Figure 3.6: Calibrations of the AOM double-pass setup. (a) Detuning of the laser from F =3 → F ′ = 4 transition versus the modulation voltage. (b) Output power of the double-passversus the modulation voltage.

Alignment advice: If the double-pass is well aligned, the alignment of the output of the

double-pass should not change when varying the RF. Therefore, by monitoring the alignment

of beam at the output after some distance while frequency-modulating the RF, the goodness of

the alignment can be tested and improved.

Important notes:

• In setting up an AOM double-pass, never put the QWP before the AOM. The crystal

inside the AOM is highly birefringent. Its birefringence changes as the temperature of

2The AOM is located before the tapered amplifier.


the crystal changes. If the input polarization of the AOM is anything but horizontal or

vertical (with respect to the crystal inside the AOM), one would notice large polarization

drifts as the temperature of the crystal increases after few hours of operation.

• Since the output of the double-pass is used to seed an injection-locked laser (to be ex-

plained later), make sure that the other diffraction orders of the AOM don’t leak into

the output. They can cause a mode competition and instability of the injection-lock.

Specially for RF ranges where the output power is low.

Injection-Lock

The output power of the double-pass is too low to be used as the trapping beam or to seed

a tapered amplifier. Therefore, it is first used to seed an injection-locked diode laser. In this

technique, a seed light with low power is used to force a high power slave laser to emit light

with high power at exactly the frequency of the seed light. In our setup, as low as 100µW of

power (the output of the double-pass, considering the losses through its path to the slave diode

laser) is used to seed a diode laser with output power of 50 mW. Figure 3.7 shows the schematic

of the setup used for injection-locking.

Output PolarizerFaraday

Rotator

B

HWP1

PBS

Optical Isolatorw/o input polarizer

Diode Laser

Temperature- Controlled Housing

H D

DV

HWP2

0.2 to 1.8 mW

35 mW

2

1

Figure 3.7: Schematic of injection-lock setup. The output of the double-pass is used to seed ahigh power diode laser.

In order to separate the input beam from the output beam, a combination of PBS, HWP

and an OI with its input polarizer removed (referred to as H-OI) is used. The current running

through the slave diode laser and the temperature it’s kept at can be varied to ensure that the


frequency of the injection-locked laser is following the seed. To confirm that the slave laser is

following the seed laser, the frequency spectrum of the injection-locked laser is compared to that

of the seed light by sending them to a Fabry-Perot cavity with free spectral range of 150MHz.

The frequency spectra are measured by detecting the intensity of the output of the cavity while

scanning one of its mirrors. If the injection-locked laser is following the seed, one should see a

single peak in the spectra. Blocking the seed should make that single peak disappear and turn

the spectra into multiple random peaks. The quality of the injection-lock is very sensitive to

the input alignment. See B.2 for the alignment procedure.

Important note: Other diffraction orders of the AOM double-pass could leak into the seed light

for the injection-lock, and could win over the wanted seed to force the slave laser to follow them.

While scanning the current on the diode laser, if you notice that the diode can be injection-

locked to more than one frequency, check for leakage at the double-pass. If you cannot eliminate

the leakage, make sure you are locking the diode to the correct seed.

Tapered Amplifier

The last stage of trapping beam preparation is a Tapered Amplifier (TA) preceded by the

single-passed AOM mentioned before (See figure 3.2.) The TA amplifies the beam to the power

needed for a MOT and the AOM does the final adjustment of the frequency as well as giving

us a way to quickly switch the trapping beam on and off and to adjust its power. The RF

driving the AOM is amplitude modulated with a modulation voltage ranging between 0V to

1V, which corresponds to an RF amplitude of 0V to 20V. The 1st diffraction order of the AOM,

with maximum power of 20 mW is sent through a TA, while the 0th order is used for the optical

pumping and tagging light, which will be discussed later. The tapered amplifier is an SDL-

8630, with maximum power of 500mW. The typical output power of the TA, when run with

1A current, is around 200mW. By turning the current up to a maximum of 1.5A, the output

power can be >350mW. The output of the TA is then coupled to a Polarization Maintaining

(PM) single-mode Fiber, through which the beam is guided to the vacuum chamber. Figure

3.8 shows the calibration of the measured power after the PM fiber (the power used in trapping

beam) versus the amplitude modulation voltage.

The output power of the TA is extremely sensitive to the input alignment and polarization.

The input polarization is controlled using a HWP. The input alignment can be optimized by

overlapping the input beam with the Amplified Spontaneous Emission (ASE) light coming out


0.0 0.1 0.2 0.3 0.4 0.5 0.6Modulation Voltage (V)

0

20

40

60

80

100

120

140

Powe

r (m

W)

Figure 3.8: Measured power after the PM fiber versus amplitude modulation voltage of theAOM, when TA running at 1.5A.

of the input port of the TA. If however, you find that the improving the input alignment does

not improve the output power, or the output power is degrading with time, it indicates that

the alignment of the optical elements inside the TA box need to be improved. Figure 3.9 shows

the optical elements used inside the TA box. If you need to improve the alignment of these

elements, follow the steps in B.3 with extreme care. The optical elements inside the TA are

usually stable for a long time and the alignment procedure needs to be done once or twice a

year.

Polarization-Maintaining Fiber

The laser beam after the TA has the correct frequency and enough optical power to be used as

the trapping beam. To sent this beam to the vacuum chamber, where the MOT is captured,

we use a single-mode PM fiber. Two lenses are employed to match the spatial mode between

the output of the TA and the mode accepted in the fiber. An OI is used to prevent the beam

to reflect back into the TA. Overall, we manage to couple 40% of the output of the TA into the

single-mode PM fiber.

For the PM fiber to maintain the polarization, the input polarization must be set to be

linear and along either the fast or the slow axis of the PM fiber. If the polarization of the input

beam is not along either of the two, the polarization of the beam at the output of the fiber will

fluctuate dramatically. These polarization fluctuations depend on the tension on various parts

of the fiber, and any change in the position of the fiber will change the output polarization.

Therefore, it is extremely important to make sure the input polarization is set correctly. This


Top View:

Side View:

z

x

y

z

Input Lens Output lens Slit Cylindrical Lens

Semiconductor Chip

Figure 3.9: Schematic of inside of the TA box, both top view and side view.

change in polarization also depends on the frequency of the beam that is sent through the fiber.

One can take advantage of this dependence to set the input polarization. See B.4 for an easy

way to find the correct input polarization. The input polarization almost never changes and

once it is set correctly, it can be trusted for a long time. The efficiency of the coupling to the

fiber, however, needs to be optimized on a daily basis.

Typical Trapping Beam Parameters

The driver of the two AOM’s mentioned in previous sections are controlled by a computer,

which controls the tasks in the experiment using a Virtual Instrument (VI) program see 3.17.

An analogue voltage between -1V and 1V from the computer, with a controlled timing, sets the

RF frequency of the AOM double-pass (see figure 3.6a). Another analogue voltage, also with

controlled timing, between 0V and 1V sets the amplitude of the RF frequency which drives the

AOM single-pass.

AOM VI Voltage Corresponding Value

Double-Pass -0.45V 20MHz red-detunedfrom F = 3→ F ′ = 4

single-Pass 0.6V 70mW

Table 3.1: Typical Trapping parameters for MOT operation. TA running at 1A.



Double-Pass -0.2V → 0.1V 35MHz → 60MHz red-detunedfrom F = 3→ F ′ = 4

single-Pass 0.3V → 0.1V 10mW → 50mW

Table 3.2: Typical Trapping parameters for molasses operation. TA running at 1A.


Double-Pass -0.6V On Resonancewith F = 3→ F ′ = 4

single-Pass ≤ 0.6V 70mWshould be lowered if the cam-era is saturated

Table 3.3: Typical Trapping parameters for imaging. TA running at 1A.

Typically the trapping beam is used for three purposes: MOT, Molasses Cooling and Fluores-

cence Imaging. Tables 3.1, 3.2 and 3.3 show the typical parameters used for the MOT, molasses

and imaging stages respectively. The current to the TA is set to be at 1A for all these opera-

tions. note: Make sure the center frequency of the AOM drivers (71MHz for double-pass and

90 MHz for single-pass) are checked periodically. The center frequencies can drift with time.

3.1.2 Repumper Beam

The essence of laser cooling is to use a cycling transition so that the atoms can scatter multiple

photons to reduce their momentum. For the cooling to work, the electrons in 85Rb atoms,

should remain in the F = 3 ground state and scatter photons off of F ′ = 4. The F ′ = 4 excited

state cannot decay to the F = 2 ground state. However, the atoms can scatter a photon off

of the F ′ = 3 excited state, and the electron can decay to the F = 2 ground state. Once the

electron is in the F = 2 ground state, the cooling stops. To prevent the electrons to spend time

in the F = 2 ground state, a repumper beam is employed to pump the electrons out of F = 2

and keep this ground state empty. The frequency of the repumper beam is set to be around

20MHz red-detuned from F = 2→ F ′ = 3 transition.

Figure 3.10 shows the setup used for preparing the repumper beam. The laser used for

the repumper beam is a New focus Vortex II TLB 6913 with output power of 48mW and

a wavelength which is tunable around 780.24nm. The driver used for this laser is similar to


OI

-

HWP

Polarization-Maintaining Fiber

Rb CellErrorSignal

Laser

AO

M AO

M

46mW

22mW

48mW

0.3mW

Figure 3.10: Schematic of the setup to prepare the repumper beam.

the driver used for the trapping beam. After the OI, 300µW of the power is used to perform

saturated absorption spectroscopy.

2 ! 1

X12

2 ! 2

X13

X23

2 ! 3

Figure 3.11: Saturated absorption spectroscopy error signal for F = 2 → F ′ transitions. Thered circle indicates where the repumper laser is locked to. The transition from the F = 2 groundstate to the F ′ = 1, 2, 3 excites states as well as the crossovers are marked on the signal.

The result of the saturated spectroscopy for F = 2 → F ′ transitions are shown in figure

3.11. The laser is locked around 35MHz below F = 2→ F ′ = 3 transition.

Important Note: The red circle in figure 3.11 indicates where the laser is typically locked to.

To find the best place to lock the repumper laser, it’s best to change the offset of the lock while

measuring the density of the MOT, either by OD measurement or by fluorescence imaging after

molasses cooling. Lock the repumper where the highest density is achieved.


Two single-pass AOM’s with frequencies of +100 MHz and -84MHz are used to bring the

frequency of the repumper to be around 20MHz below the F = 2 → F ′ = 3 transition. After

the two AOM, 22mW of power is coupled into a PM fiber, similar to that of the trapping beam.

The coupling efficiency is typically 60%. A modulating voltage between 0V and 1V is used to

amplitude modulate the RF signal used to drive the -84MHz AOM single-pass. Figure 3.12

shows the calibration for the output power of the PM fiber versus the voltage used to set the

amplitude of the RF signal.

0.0 0.2 0.4 0.6 0.8 1.0Modulation Voltage (V)

0

2

4

6

8

10

12

14

16

Outp

ut p

ower

(mW

)

Figure 3.12: Calibration of output power of the PM fiber versus the modulation voltage.

Operation VI Voltage Corresponding Value

MOT 0.6V-0.8V 12mW to 14mW

Molasses 0.1V to 0.3V 1mW to 5mW

Imaging 0.8V 14mW

Table 3.4: Typical repumper parameters for different operation.

Typical repumper powers used for different operations are shown in 3.4.

3.1.3 MOT setup

In sections 3.1.1 and 3.1.2 the preparation of trapping beam and repumper beam were discussed.

Figure 3.13 shows the D2-line of 85Rb as well as were the trapping beam and repumper beams

are locked to. Figure 3.14 shows the schematic of how the two beams are combined as well as

the optics used for guiding the beams towards the center of the trap to capture a MOT.

Once the trapping beam leaves the fiber, out of a fiber to free-space adapter, it expands

for a distance of about 15cm. It is then collimated to have a beam size of about 1 inch


120M

Hz

63M

Hz

29M

HZ

3.03

5 G

Hz

790.

24nm

(384

TH

z)

1.77

0GH

z10

0MH

z

F = 2

52S1/2

F 0 = 3

F 0 = 2

F = 3

F 0 = 1

52P3/2

F 0 = 4

20M

Hz

20M

Hz

Figure 3.13: D2-line of 85Rb atoms. The red arrow indicated the tapping beam and the bluearrow indicated the repumper beam.

in diameter (1/e2) using an achromatic lens with focal length of 150mm. Using the same

technique, with a 45mm achromatic lens, the repumper beam is expanded to be about 1cm

in diameter. The polarization of the two beams are controlled using two HWP’s inside the

cage system (H1 and H2 in figure 3.14), so that the trapping beam and the repumper beam

are split with 2 : 1 power ration in the two outputs of the PBS1. The output with the higher

power is in turn split into two paths with 1 : 1 power ratio. The three resulting beams, with

the same power, are guided towards the center of the trap, which is defined by the magnetic

field gradient. Three QWPs (Q1,Q3 and Q5) are used change the polarizations of the beams to

circular polarizations. Three mirrors (M1,M3 and M5) are used to retro-reflect the MOT beams.

Another three QWPs (Q2,Q4 and Q6) are used to ensure that the retro-reflected beams have

the same circular polarization (opposite spin) as the incoming beam. Two magnetic coils, with

1.6A of current running through them in opposite directions (anti-Helmholtz configuration),

create a magnetic field gradient of around 20G/cm in the z-direction. Three set of Helmholtz

coils are used to compensate the stray magnetic fields down to a mG level. The density and the

temperature of the cloud heavily depends on the quality of the alignment of the MOT beams.

See B.5 for alignment procedure.


Half-Wave plate

Quarter-Wave plate

Lens

coil

Polarizing Beam Splitter

MOT

x

y

z

PBS2

H3 H1

H2

Q1Q1

Q2

Q3

Q4

Q5

Q6

Mirror

PBS1

M1

M2

M3

M4

M5

Repumper

Trapping

3

1 2

Figure 3.14: Schematic of optics used for combining the trapping and repumper beam andcapturing a MOT.

For convenient power measurements of the MOT beams, a cage-mountable detector is de-

signed and calibrated, which we named Uberdetect. It consists of a photodiode detector and

an ND (OD=3) filter. The calibration is done with a 5kΩ resistor in parallel with the photodi-

ode. Figure 3.15 shows the calibration results for the trapping beam and the repumper beam.

The results indicate a 3.8mW/mV power to voltage conversion for the trapping beam and a

0.7mW/mV conversion factor for the repumper beam. Different beam sizes account for the

difference between the two conversion factors.


0 5 10 15 20Voltage (mV)

0

10

20

30

40

50

60

70

80

Powe

r (m

W)

(a)

0 2 4 6 8 10 12 14 16 18Voltage (mV)

0

2

4

6

8

10

12

Powe

r (m

W)

(b)

Figure 3.15: Power versus voltage read by Uberdetect for (a) trapping beam and (b) repumperbeam.

3.1.4 Magnetic fields

Magnetic field gradient

The magnetic field gradient is generated by two coils, each with 200 turns and resistance of

1Ω, in an anti-Helmholtz configuration. The current in the coils is generated and controlled

by a Bipolar Operational Power (BOP) supplier made by KEPCO: model 20-10M. In order to

quickly switch the field gradient on and off, the BOP is operated in current-controlled mode.

In normal MOT operation, we run 1.6A current through the coils which generate a magnetic

field gradient of 20G/cm along the axis of the coils (10G/cm in the orthogonal directions.) The

BOP is remotely controlled by a computer and the current can be ramped up to 5A to obtain

higher field gradients. But it is advised not to maintain such high currents for a long time to

avoid over-heating the coils. The current generated by the BOP can be turned off in <500µs.

To eliminate fast oscillation of the current after switching on or off, a 3µF capacitor and a 67Ω

resistor are put in parallel with the output of the BOP.

Optical Pumping Coil

As mentioned before, three pairs of coils in Helmholtz configuration are used to compensate

stray magnetic fields. One of the three pairs is also used to generate a constant magnetic field

necessary for optical pumping, referred to as Optical Pumping (OP) coils. The currents for OP

coils are generated using another BOP made by KEPCO: model 36-12M. This BOP, which is


3 2 1 0 1 2 3Control Voltage (V)

6

4

2

0

2

4

6

B (G

)

Bz

Bx

By

Figure 3.16: Magnetic field measured at the center of one of the OP coils. The field is measuredalong the coil axis, z, and the orthogonal directions x and y.

remotely controlled by the computer, is also set to run in current-controlled mode with a 3µF

capacitor and a 67Ω resistor put in parallel with its output. The field generated by the coils

can be switched off in 240µs. Figure 3.16 plots the measured magnetic field at the center of one

of the OP coils in the direction of the axis of the coil, Bz as well as the orthogonal directions.

3.1.5 Remote Task Control

A typical experiment contains different stages and each stage requires certain experimental

parameters. It is important to be able to vary these parameters between different stages with

precise timing. Therefore, being able to control all of the experimental parameters remotely

and in an automated way is a crucial element for performing complicated experiments. Most

elements used in this experiment can be controlled using an analog voltage as a control signal.

These analog signals are generated by two National Instrument: PCI 6713 cards. Each card

has 8 analog output channels (16 in total) with voltage range of ±10V and update rate of

1MS/s. The precise timing of the experiment is controlled using the 20KHz clock of a National

Instrument: PCI 1200 digital card, which corresponds to a minimum time step of 50µs. The

digital card also outputs Transistor-to-Transistor Logic (TTL) signals with controllable delay,

which are used for triggering purposes.

The control values needed for an experiment along with their timing information (altogether

called a task) are set using National Instrument LabVIEW software, through Virtual Instru-

ments (VI’s.) Once a task is executed, the software writes the user-defined control values on the

PCI 6713 cards and the timing information on the PCI 1200 card. The analog card transfers


LabViewV

t1 t2 t3

PCI 1200

PCI6713 BNC 2110

Output to devices

Figure 3.17: Schematic of different stages used to convert the parameters defined in the LabViewsoftware to analog signals used to control multiple devices.

the information to an NI BNC 2110 device. The PCI 1200 converts the timing information

into a train of TTL’s that are spaced in accordance to the specified timings. These triggers

are then sent to the NI BNC 2110. Every time NI BNC 2110 receives a trigger, it updates

the output analog voltages. The analog outputs of the NI BNC 2110 are sent to the multiple

devices using coaxial cable with BNC connection. Figure 3.17 shows the schematic of how the

information from a VI get converted to analog outputs.

Important Note: The two analog cards can be timed using two different clocks, e.g. internal

clock of the PC and the clock on the digital card. The timing source of the cards can be

specified within the VI.

3.1.6 Maintenance and Diagnosis

To keep the setup in good condition, periodic checks and maintenance are necessary. Slow drifts

and fluctuations can lead to an unstable setup, if not corrected for as they happen. In this

section some maintenance details will be discussed and a diagnosis checklist will be provided.

Rubidium dispenser. To maintain the rubidium vapor pressure necessary for capturing a

MOT in the vacuum chamber, the temperature of the dispenser should be kept at around 37C

to 40C. This temperature should be readjusted at the beginning and the end of winter every

year. At the end of winter, the ventilation system in the lab is switched to cooling and the lab

gets colder. As a result, the temperature of the dispenser drops. This results in a lower vapor

pressure and one may notice that the MOT gets smaller and takes longer to load. Increasing the

voltage applied to the heating tapes should resolve this problem. At the beginning of winter,


the ventilation system is switched to heating and the temperature of the dispenser will rise. In

this case, the voltage applied to the heating tapes should be lowered to adjust the temperature

of the dispenser.

Running low on rubidium. The rubidium in the dispenser should be replaced once every

two to three years. Insufficient rubidium in the dispenser results in a smaller MOT and longer

loading times. Heating the dispenser harder will not resolve these symptoms.

Mode-hopping of the master laser. The trapping and repumper lasers are designed to have

a mode-hop free range of several GHz. While doing spectroscopy, you should be able to observe

all 4 hyperfine transitions of 85Rb and 87Rb. If you cannot see all 4 transitions, it indicates that

the laser is mode-hoping. Also, if you observe that the size of the error signal changes while you

scan through one transition, it usually indicates that the laser is multi-mode and it is about to

mode-hop. To find a mode-hop free region, change the current running through the diode and

if the problem persists, use temperature as a second nob to find a mode-hop free region.

Diagnosis Checklist: If you find that the MOT has disappeared and you have difficulty

getting a MOT, run through the following checklist

• 1) Make sure the magnetic field gradient is turned on and the VI is running.

• 2) Make sure the trapping laser is locked to the correct transition. The spectroscopy

signals for 85Rb F = 3 → F ′ = 2, 3 and 4 and 87Rb F = 3 → F ′ = 1, 2 and 3 are nearly

identical in shape can be confused with each other. Note that the frequency spacing

between the transitions in 87Rb are larger than the spacings in 85Rb.

• 3) Check the injection lock and make sure it is following the correct output order of the

double-pass.

• 4) Measure the repumper and trapping beam powers and make sure the power is balanced

among the three pairs of MOT beams.

• 5) Check the frequency of the RF signals driving the AOMs.

• 6) Use an IR-viewer to look for fluorescence in the cuvette.

• 7) Check the alignment of the MOT beam. Start with the back-reflected beams and make

sure they are overlapping with the incoming beam. Then make sure the three beams are

overlapping at the center of the cloud.


• 8) Check the polarization of the MOT beams.

3.2 Probe and Coupling Beams

In section 3.1, the MOT setup was discussed in detail. In this section, a brief description of

probe and coupling beam preparation will be provided. A more detailed description of this part

of the experiment can be found in [52]. Figure 3.18 shows the schematic of the probe beam,

the coupling beam and the signal pulses at the interaction region.

To single photon detector

To phase measurement detector

Probe

Coupling

Signal

MOT

13um focus10:90

Figure 3.18: Schematic of the probe, the coupling and the signal beam.

Probe

Preparation. The probe beams is composed of two frequency component: a component that is

on-resonance with the F = 2 → F ′ = 3 transition and an off-resonance component which is

tuned 100MHz above that transition. The probe beam is branched off of a master laser (New

Focus Vortex ) that is locked ≈30MHz red of the F = 2 → F ′ = 3 transition. By using an

AOM that is driven at +130MHz (double-passing at +65MHz) the off-resonance component

of the probe beam is generated. The RF signal driving this double-passed AOM is amplitude

modulated to switch the probe beam on and off. This off-resonance component is then sent

through another AOM at -100MHz. The 1st diffraction order of the AOM is used as the on-

resonance probe component. The RF signal driving this AOM is locked to a stable 10MHz clock.


The two beams are then combined on a beam splitter and are sent towards the interaction region

using a single-mode optical fiber. Typical probe power used in this experiment is 3 nW for the

on-resonance probe and about 20-30 nW of off-resonance probe.

Phase Measurement. After passing through the interaction region, the probe beam reflects

off of a 10:90 beam splitter and is then detected using a fast Avalanche Photodiode Detector

(APD.) The APD detects the 100MHz beating signal of the two components of the probe beam.

Any change in the phase and/or the amplitude of the on-resonance probe beam results in a

change in the phase and/or the amplitude of this 100MHz beating signal (This technique is

referred to as beat-note interferometry[62].) We use an IQ-demodulator, at 100MHz, to then

read off the phase (arctan( IQ)) and amplitude (

√I2 + Q2) of the beating signal. The same

stable 10MHz clock that was used for preparing the probe beam, is used to lock-in this 100MHz

demodulation. The resulting I and Q are then digitized with a sampling frequency of 1.5MHz.

To eliminate slow drifts in the phase measurement, the phase of the probe is averaged for 3

samples (200ns) before and 3 samples after when the XPS is expected to happen. This value

is then subtracted from the average of the probe phase in the interval when XPS is expected.

The resulting number is reported as the measured XPS. The probe beam contains about 2000

photons, which corresponds to 22mrad phase uncertainty (shot-noise.) We measure a single-

shot phase uncertainty of around 100mrad. We believe the source of the added uncertainty is

due to a combination of detector noise used to detect the beating signal and the electronics

afterwards. In order to measure the phase shift down to few µrad uncertainty level, we repeat

the measurement about half a billion times.

Coupling

To increase the interaction strength, the probe is tuned to be on resonance with F = 2→ F ′ = 3

transition. In order to eliminate absorption of the probe, a coupling beam is used to create

an electromagnetically-induced transparency (EIT) situation. EIT is a coherent effect in which

a destructive interference prevents the two involved laser beams from being absorbed by the

atoms[63, 64]. In order to create an EIT situation, the probe and the coupling lasers should

be phase-locked to one another. To achieve the phase-lock condition, the remainder portion of

the master laser used for preparing the probe beam is frequency-shifted down using an electro-

optic modulator (EOM) driven at 3GHz. It is then used to seed an injection-locked diode laser.

As a result, the second diode laser, called the coupling laser, is phase-locked to the master


laser. A combination of an AOM (single-passed at +103MHz) and fine tuning the frequency

by which the EOM is driven, sets the frequency of the coupling beam to be on resonance with

F = 3→ F ′ = 3 transition. The intensity of the probe and coupling beams are chosen so that

the width of the resulting EIT feature is measured to be 2MHz. The polarization of the probe

and coupling beams are set to be σ+ and π, respectively.

3.3 Signal Pulses, Photon Detection and Tagging

3.3.1 Preparation

To generate the signal pulses, 40mW of the total 55mW output power of the injection-locked

diode laser (coupling laser) is branched off and sent to an AOM double-pass at +80MHz. The

output of the double-pass is then sent through an AOM single-pass at +75MHz, which sets

the frequency of the signal beam to be around 18MHz blue-detuned from F = 3 → F ′ = 4

transition. By amplitude modulating the RF that drives the AOM single-pass we generate signal

pulses with 40-100ns duration. These pulses are then sent to the “Atoms Lab” (interaction

region) using a single-mode fibers.

AND

SPCM

Delay GeneratorDetector

ND Filter

Figure 3.19: Schematic of the signal pulses, SPCM and how the SPCM is gated.

3.3.2 Photon Detection

Before sending the signal pulses to the interaction region, they are sent through a PBS. Around

95% of the total power is transmitted and detected on a Thorlabs, PDA10A detector. The signal

from this detector is then used for gating the photon detections at a single-photon detector,

figure 3.19. The 5% remaining power is sent to the interaction region using a single-mode fiber.


An ND filter with eOD = 613± 12 is used to attenuate the signal pulses to very low intensities.

This way, signal pulses with very low mean photon numbers, < 0.2, can be prepared. After

passing through the interaction region, the signal pulses are collected in a multi-mode fiber with

30% overall collection efficiency. A Single-Photon Counting Module (SPCM) is used to detect

the presence of photons in the signal pulses. Figure 3.20 plots the probability of a click, defined

as number of photon detection events over the total number of pulses, versus mean photon

number per pulse. The data is fit to Pclick = 1− e−ηn where n is the mean photon number per

pulse and η is the efficiency. The efficiency of the SPCM is inferred to be 70%.

0 2 4 6 8 10 12 14 16Number of photons at the SPCM

0.0

0.2

0.4

0.6

0.8

1.0

P clic

k

Figure 3.20: Probability of photon detection versus number of photons at the SPCM. From afit to the data, the efficiency of the SPCM is measured to be 70%.

3.3.3 Gating the Photon Detection

Upon detecting a photon, the SPCM outputs a 30ns trigger. These triggers can be used to tag

the measured phases that correspond to successful photon detections. However, due to dark

counts, the SPCM sends triggers even when no signal pulse is present, which makes the tagging

procedure challenging. To eliminate these dark counts, we use a logic AND gate between the

output triggers of the SPCM and the output of the PDA detector that was mentioned before.

This is to say that we gate the SPCM with separately detected signal pulses. A delay generator

is used to correct for the time delay between the detected signals from the PDA and the triggers

from SPCM, see figure 3.19. The result of the AND gate is then used for tagging.

This technique removes all the dark counts that happen outside the time window in which

we expect to have a signal pulse. But a dark count can also happen during this time window.

The probability of getting a dark count in this time window depends on the temporal length


of the signal pulses. Any photon scattering event from the probe and the coupling beam has

a chance of ending up being detected at the SPCM through the signal path. Therefore, the

probability of a background photon detection also depends on the presence of the probe, the

coupling beam and the atoms in the interaction region. To measure the probability of a dark

count, we block the signal beams entering the interaction region and count the detection events

with the probe and coupling beam present. If we gate the SPCM with 100ns windows, we

observe that without atoms, we get <2% dark counts (without the atoms, there could not be

any scattering.) This number, however, increases to 10-12% when the atoms are switched on.

To decrease the dark counts, we gate the SPCM with 40ns time windows, which is the shortest

signal pulse the AOM can generate, and we observe that the dark counts reduce to 6%. To

observe the dark counts due to scattered photons, both the probe beam and the coupling beam

need to be present to maintain a certain population ratio in F = 2 and F = 3 ground states.

Therefore, in the absence of either of the two beams, the dark counts decrease to 2%.

It is worth mentioning that at first we tried to gate the SPCM with the electrical signals

that generate the optical signal pulses. To use those electrical signals, we had to send them

through 20m of coaxial cable. These long cables, however, caused broadening in the electrical

signals (due to the capacitance of the cable which is significant when the cable is long enough)

and back-reflection of the signal due to impedance mismatch between the AND gate and the

signal generator that generates the electrical signals. These imperfections could leak into the

measured beating signal of the probe beam through the AND gate and the tagging procedure.

Given that we were to measure a signal that was 10,000 smaller that the noise, any added noise

and systematic could not be tolerated. To eliminate these problems, we decided to use a PDA

to detect the optical signal pulses after they have traveled the distance and use the detected

signal to gate the SPCM.

3.3.4 Tagging the Shots with Successful Photon Detection

In order to measure the phase shift due to “click” and “no-click” events separately, we need a

way to distinguish the two events; we need to tag the shots when “click” happens. The easiest

way to tag the shots with successful post-selection is to use a stable and fast clock to time-tag

the triggers from the SPCM and signal pulses. Unfortunately we were not in possession of

such a device. Therefore, we decided to we write the tags on the same demodulated signal

that contains the measured phase of the probe. This way, each measured shot contains the tag


RFSwitch

AOMVCO

Delay Generator Tagging Light

Probe

To Phase and AmplitudeMeasurement

XPS

Tags

Output ofAND gate

Figure 3.21: Schematic of the tagging procedure. The result of the AND gate switches on a100MHz RF signal to an AOM upon a successful photon detection. The 1st and 2nd diffractionorder of the AOM are sent to the detector which detects the probe. When a successful detectionhappens, this AOM sends a flash of light to the detector which results in a short spike in themeasured amplitude (and phase.) A delay generator controls the relative delay between theexpected XPS and the position of the tag.

information with it and no clocked time-tag is needed.

To tag the shots with successful photon detection, we use a short flash of light, that is

shone on the detector which detects the beating signal of the probe. Since the measurement

bandwidth used for the experiment is 2MHz centered at 100MHz, that short burst of light

should also have a 100MHz component. Therefore, the we use the 1st and the 2nd diffraction

orders of an AOM driven at 100MHz, to generate the flash of light. The RF to the AOM is

switched on and off using the output of the AND gate between the SPCM and the detected

signal pulses, so that the RF is only on when the output of the AND gate is high. The triggers

coming from the AND gate are delayed using a delay generator to precisely control the position

of tags with respect to the XPS. The amplitude of the tagging light is set to be high enough

for the tags to be single shot resolvable. Figure 3.21 shows the schematic of the tagging setup.

3.4 Polarization Interferometer and Tomography

To implement the weak value amplification of photon numbers, the signal pulse should be sent

through an interferometer. The interferometer used in this experiment is a polarization interfer-

ometer where the two paths of the interferometer are the two orthogonal circular polarizations


(right-handed and left-handed.) Since the two paths are spatially overlapping, this interferom-

eter is immune to many instabilities that occur in interferometers with non-overlapping paths.

Figure 3.22 shows the schematic of the signal path from before it is sent through the interac-

tion region to when it is collected in the multi-mode fiber for detection at the SPCM. The first

element inside the signal telescope is a polarizer to ensure the initial polarization of the signal

is linear. Well-calibrated QWP and HWP are used to control the initial polarization of the

signal. We set the QWP at 205 and the HWP at 79.

1

2

Signal

ProbeCoupling

To the SPCM

10:90 BS

Lens

Glan-ThompsonPolarizer

Half-Wave plateQuarter-Wave plate

Signal Telescope

Probe TelescopePolarizer

Figure 3.22: Schematic of the signal path. Dashed lines marked as 1 and 2 indicated thebeginning and the end of the polarization interferometer.

A linear polarization is an equal superposition of right-handed circular polarization, σ+,

and left-handed circular polarization, σ−. After the interaction region and before the signal is

detected at the SPCM, a combination of a QWP (fast axis at 62), a HWP (fast axis at 355 )

and a Glan-Thompson Polarizer (GTP) is used to project the output of the GTP to an arbitrary

polarization. This also gives us a tool to perform polarization tomography on the signal and

see how the polarization of the signal is affected during its propagation in the presence and

absence of the atoms, the probe beam and the coupling beam. Once the initial polarization of

the signal is well understood, the QWP and the HWP are set to project the output of the GTP

to the final polarization, corresponding to a certain post-selection parameter.

The polarization tomography is performed by measuring the signal power at the output

of the GTP, when projected in H-basis (QWP at 62 and HWP at 355), V-basis (QWP at


62 and HWP at 40), D-basis (QWP at 62 and HWP at 17.4) and R-basis (QWP at 62

and HWP at 332.5.) From the tomography result, we calculate the stokes parameters for the

initial polarization of the signal beam. Knowing the initial polarization, we calculate the final

polarization that corresponds to the desired post-selection parameter. Correct setting of the

QWP and the HWP projects the output of the GTP to the calculated final polarization.

3.5 Atom Preparation

3.5.1 Level Scheme

18 M

Hz

F=2

F=3

F'=1

F'=2

F'=3

F'=4

Probe

Coupling

Signal

Figure 3.23: Level scheme used for the WVA experiment.

A requirement for the WVA experiment is for the signal in the two paths to interact with

the probe with different strengths. Therefore, we should chose the level structure so that

the interaction of the probe with the σ+-polarized signal is different from its interaction with

σ−-polarized signal. Figure 3.23 shows the level scheme used for the WVA experiment. The

reason for this polarization dependence is the following: The size of the XPS is proportional


to Ω2sΩ

2pr/Ω

2C (in the low OD limit) where Ωi’s are the Rabi frequency of the signal, probe

and coupling beams. The Clebsch-Gordon (CG) coefficient for σ+ polarization of the probe

is largest for the F = 2,mF = 2 → F ′ = 3,mF ′ = 3 transition. Likewise, the CG coefficient

for π-polarized coupling beam is largest for the F = 3,mF = 3 → F ′ = 3,mF ′ = 3 transition.

Therefore, the Λ-system created between F = 2,mF = 2 and F = 3,mF = 3 ground states is the

dominant Λ-system3. One can easily show that for equal population distribution among F = 2

ground state, one should expect the XPS for the σ+ polarization of the signal to be around

3.6 times larger than the XPS due to σ− polarization. If one initializes all the population in

F = 2,mF = 2 ground state (via optical pumping), the ratio between the two XPS’s is expected

to be 28 (for more details see [52].)

3.5.2 Optical Pumping

0.0 0.2 0.4 0.6 0.8 1.0frequency (a.u.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

OD

B × ,OP ×fitB ,OP ×fitB ,OP fit

Figure 3.24: Measured OD versus frequency of the probe for three situations: no external field,no optical pumping with external field and with optical pumping and external field.

As mentioned in section 3.5.1, pumping all the population to be in F = 2,mF = 2 can

make the XPS for the σ+ polarization to be 28 time larger than that of the σ− polarization.

Therefore, we did optical pumping in F = 2 ground state to prepare the population in mF = 2.

Two optical beams were used for optical pumping: a pump beam and a repumper beam, which

we are going to call F3-repumper to avoid confusion with the repumper beam used for the MOT.

The pump beam was branched off of the repumper beam for the MOT. Using an AOM, the

frequency of the pump beam was set to be on resonance with F = 2 → F ′ = 2. F3-repumper

3The CG coefficient for F = 3,mF = 1 → F ′ = 3,mF ′ = 1 and F = 3,mF = −1 → F ′ = 3,mF ′ = −1 arethe same but the CG coefficient for F = 2,mF = −2 → F ′ = 3,mF ′ = −1 is 15 times smaller than the CGcoefficient for F = 2,mF = 2→ F ′ = 3,mF ′ = 3.


was branched off of the trapping beam and its frequency was set to be on resonance with

F = 3 → F ′ = 3. An external DC magnetic field of around 5G was created using the optical

pumping coils mentioned in 3.1.4. Around 200µW of pump and 1mW of F3-repumper were

combined at a beam splitter and were sent through the atoms along the DC magnetic field.

The polarization of both beams were set to be σ+. We did optical pumping for 200µs.

The see the result of the optical pumping, we measured the OD as seen by the probe beam,

with σ+ polarization, as we scanned its frequency. In the absence of the external magnetic field,

the OD versus frequency of the probe is expected to be a Lorentzian function. If, however, the

external magnetic field is turned on, the OD should have a much broader feature due to the

Zeeman shifts of the sub-levels of the F = 2 ground state. If all the atoms are pumped into

the desired ground state, in the presence of the external field, the OD versus frequency of the

probe should be a Lorentzian function with the original width (without the magnetic field)

but displaced in frequency. Figure 3.24 shows the result of the OD measurement versus probe

frequency for the three situations mentioned above.

0.0 0.2 0.4 0.6 0.8 1.0frequency (a.u.)

0.0

0.1

0.2

0.3

0.4

0.5

OD

datafitmF =2mF =1mF =0mF = 1mF = 2

(a)

0.0 0.2 0.4 0.6 0.8 1.0frequency (a.u.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

OD

datafitmF =2mF =1mF =0mF = 1mF = 2

(b)

Figure 3.25: The result of fitting the OD versus probe frequency to the weighted sum of fiveshifted Lorentzian functions for (a) no optical pumping and (b) with optical pumping.

Knowing the strength of the external magnetic field and the magnetic moment g-factor for

F = 2 ground state, we can fit the measured OD versus probe frequency in the presence of

the magnetic field to a weighted sum of five Lorentzian functions, each centered at a different

frequency. The weight for each Lorentzian reflects the population of the atoms in that particular

Zeeman sub-level. Figure 3.25 shows the result of this fit for the case of no optical pumping

and the case where optical pumping was performed. The fits suggest that initially we’d had

21% of the population in mF = 2, 43% in mF = 1, 19% in mF = 0, 16% in mF = −1 and no

detectable population in mF = −2. With optical pumping we measure 55% in mF = 2 and

45% in mF = 1 with no detectable population in the rest of the Zeeman sub-levels.


Note: We ended up not using optical pumping for the WVA experiment. Due to what we

believe is inhomogeneity of the external magnetic field, we couldn’t obtain an EIT situation

that was nearly good enough for the experiment to move forwards. Since time was of essence

and WVA could be done without optical pumping, we decided to move on with the experiment

and postpone a more detailed study of the optical pumping for later. The setup for optical

pumping has since been removed from the table due to equipment shortage for the tagging

setup.

3.5.3 Atom Cycle

Our measurement cycle has three parts: cooling for 20ms, population preparation for 0.5ms,

and free expansion for 1.5 ms. The cooling time can be as small as 6ms without any observable

effect in the density of the cloud. After the cooling stage, the population is mainly in F = 3

ground state. To pump the population to F = 2 ground state, We turn the repumper beam

off, lower the trapping beam power and increase its detuning for 0.5ms. This stage gives us

enough time to make sure the magnetic field gradient is also turned off. Then, we turn all the

MOT beams off and sent a trigger to the probe beam and coupling beam to turn them on. The

same trigger also triggers a function generator to start sending the signal pulses. A separate

trigger, which is simultaneous to the first trigger, sets the starting point for the IQ-analyzer to

take measurements. The measurement is done for 1.5ms during the free expansion of the cloud.

The first 300-400 µs of this time is used to ensure that the EIT is established and the initial

phase dynamics of the probe are suppressed. We then start sending the signal pulses every 2.4

µs. The collected data by the IQ-analyzer is sent to a computer via an Ethernet cable for the

extraction of the phase, the amplitude and the tags.

Terminology. We call each of the 1.5ms measurements a trace. A trace is divided in 2.4µs

intervals, each containing a signal (an XPS.) Each interval is called a shot. We take the data

in many parts each with several traces. Each part is referred to as a bin.

note: A task on the computer that controls the experiment (see section 3.17) consists of

the three mentioned parts, cooling, population preparation and free expansion, that is repeated

many times (usually 100 times but the number of repetitions can be modified.) Once the

repetitions are over, the computer generates another similar task and runs it again. Be advised

that the computer takes some time, typically around 10ms, to prepare in between each two

tasks. Therefore, there will be a delay between the triggers sent from the computer every time


the computer generates a new task.

Chapter 4

Experimental Results

In this chapter calibration results, details of the experiment and a discussion of the results

of the thesis will be provided. Here, I present clear data which demonstrates that a single

post-selected photon, using weak value amplification, can act like eight photons. I believe this

experiment to be the first ever weak value experiment that utilizes a deterministic interaction

between two distinct optical systems.

Figure 4.1 shows the schematic of the setup used for this experiment. (Details of the setup

can be found in chapter 3.) The nonlinear interaction between the signal and probe beams is

mediated by a sample of laser-cooled 85Rb atoms in a magneto-optical trap (MOT). Presence of

a coupling beam sets up electromagnetically induced transparency (EIT) for the probe beam.

By using EIT, the nonlinear interaction can be enhanced by tuning the probe beam to be on

resonance with the atoms while substantially reducing the accompanying absorption. As the

signal pulse passes through the medium, it causes an AC-Stark shift on the ground state which

the coupling beam addresses. Due to this energy shift, the probe and coupling beam will no

longer be on two-photon resonance and therefore, the probe beam sees a modified index of

refraction. As a result, the probe picks up a phase shift relative to what the phase would have

been in the absence of the signal pulse (Kerr effect[65, 66, 67, 68, 69, 70].) The change in the

index of refraction, and therefore, the phase shift of the probe, depends linearly on the number

of photons (intensity) of the signal pulse and the slope of the EIT feature at the frequency of

the probe. Therefore, measurement of the phase of the probe is equivalent to a non-demolition

photon number measurement[51, 71].

Terminology: I will refer to the the intrinsic per photon nonlinear phase shift of σ+-polarized

68

Chapter 4. Experimental Results 69

Figure 4.1: Schematic of the experimental setup. Counter-propagating probe and signal beamsare focused down to a waist of 13µm inside a cloud of laser cooled 85Rb atoms confined ina magneto-optical trap. A collimated coupling beam, propagating in a direction orthogonalto that of the signal and probe, sets up EIT for the probe. The signal beam is prepared ina linear polarization and after its interaction with the probe beam is post-selected on a finalpolarization that is almost orthogonal to the initial polarization. The probe and the signalbeams are separated from one another using a 10:90 beam splitters.

signal as φ+ and σ−-polarized signal as φ−. φclick refers to the average XPS on the probe

when the measurement is conditioned on the single-photon detector firing and φno−click is the

measured phase shift when the detector fails to fire. I will use “click” and “successful post-

selection” interchangeably.

4.1 Calibration

4.1.1 Single Photon Cross-Phase Shift

As a first calibration, we measured the nonlinear phase shift written on the probe as we varied

the mean photon number on the signal beam. The calibration was done using circularly po-

larized (σ+) signal pulses. We measured a per photon nonlinear phase shift of 13 ± 1.5µrad.

We also measured the effect of individual photons by post-selecting on a subsequent detection

of a photon after the interaction between the signal and the probe. We measured the effect of

individual post-selected photons to be 18± 4µrad. Details of these measurement can be found


in [50, 52].

4.1.2 XPS Versus Signal Polarization

150 200 250 300 350 400QWP Angle (Degrees)

0.51.01.52.02.53.03.54.04.55.0

XPS

(mra

d)

Figure 4.2: XPS versus signal polarization. A quarter-wave plate is used to set the polarizationof the signal beam before the interaction. The maximum and minimum measured phase shiftscorresponds to the two circular polarizations σ+ and σ−. The signal pulses used for thismeasurement contained around 600 photons.

As mentioned in section 3.4, for the WVA experiment to work, we need the two paths of

the polarization interferometer, which are σ+ and σ−, to interact with the probe with different

strengths. The polarization of the probe can be controlled before the interaction by using a

QWP. To calibrate the polarization dependence of the XPS, we measured the XPS written on

the probe by approximately 600 signal photons as we varied the polarization of the signal from

σ+ to linear to σ− by rotating the QWP. Figure 4.2 show the measured XPS versus the angle

of the QWP. The calibration shows that the measured XPS is largest for σ+ and smallest for

σ− polarization of the signal. For the WVA experiment, the angle of the QWP is set to be at

205, which corresponds to a linear polarization.

4.1.3 Polarization Tomography on the Signal

In section 3.4 I discussed how to perform polarization tomography on the signal beam. Under-

standing the initial polarization of the probe and minimizing the effect of the atoms is crucial


for post-selecting on the correct final polarization. Table 4.1 shows the result of the tomography

on the signal beam in various situations. The parameters S0, S1, S2 and S3 are the measured

stokes parameters of the signal polarization.

Atoms Probe Coupling S0 S1 S2 S3

off blocked blocked 10.1 -1.4 9.0 4.4

on blocked blocked 10.1 -1.4 9.0 4.5

on blocked on 10.2 -1.4 9.1 4.5

on 3nW blocked 5.7 -3.4 0.2 4.2

on 3nW on 9.9 -2.5 7.7 5.7

on 6nW on 8.7 -3 6 5.6

Table 4.1: Polarization tomography on the signal

The results indicate that in the absence of the coupling and the probe beam, the polarization

of the signal in the presence and the absence of the atoms remains the same. The reason is that

the population is initially mostly in the F = 2 ground state and the cloud is almost transparent

for the signal. Adding the coupling beam only helps with pumping the population out of the

F = 3 ground state and the polarization of the signal remains unchanged. On the other hand,

if the probe beam is present and the coupling beam is blocked, the polarization of the probe

dramatically changes and the signal gets absorbed by 44%. This is due to the fact that the

probe beam will pump the population out of the F = 2 ground state and the signal beam will

see a large OD. When the coupling beam and the probe beam (with 3nW of power) are both

present, the population will mostly remain in the F = 2 ground state and the polarization of

the signal won’t change by much and we observe less that 2% absorption. However if the probe

power is increased to 6nW, a larger fraction of the population will be in F = 3 ground state

and the signal will see a larger OD and will get absorbed by 14%.

A large population in the F = 3 ground state means a larger polarization rotation of the

signal. Therefore, any fluctuation in the OD will result in an unstable post-selection parameter.

The tomography results suggest that we need to turn the probe power down as much as possible.

Turning the probe power down, however, results in a larger phase-noise uncertainty which makes

the measurement of XPS harder. We chose the probe power to be 3nW for this experiment

because the polarization rotation of the signal with this probe power becomes almost negligible,

while the phase uncertainty remains low enough for the experiment to remain feasible to be

done. We measured shot-noise uncertainty for this probe power to be around 100mrad.


4.2 Measurement Results

4.2.1 φclick and φno−click

The experiment is performed using the measurement cycle introduced in 3.5.3. We analyze

the data by separating the measured shots into two groups of “click” and “no-click”, based on

the tags from the SPCM, see 3.3.4. The parameters were chosen in such a way to keep the

probability of a successful post-selection around 22-25%, 6% of which being background photon

detection. Table 4.2 shows the combination of parameters used for each data point along with

the total number of measurement used, Ntot, and the measured post-selection probability, Pclick.

In this table, n is the mean photon number for signal, η is the overall efficiency and δ is the

Data Point n δ η Ntot Pclick

1 95 0.10 0.2 42,111,000 31%

2 45 0.14 0.2 104,111,000 23%

3 20 0.22 0.2 102,380,000 25%

4 10 0.32 0.2 204,112,000 23%

5 40 1 0.03 374,443,000 20%

Table 4.2: Experimental parameters used in the WVA experiment.

post-selection parameter as defined in 2.10. As stated before, δ = 1 refers to post-selecting on

the initial state.

Figure 4.3 shows the results of the measured φclick and φno−click for various experimental

parameters. It can be seen that the measured phase shift for the “click” cases are always

larger than the measured phase shift for the “no-click” cases. The difference between the two

gets smaller as the post-selection parameter gets larger. In what follow, I will investigate the

dependence of this difference versus post-selection parameter in more detail.

4.2.2 φno−click versus |α|2

As a first step in analyzing the data, we extract the value of φ = φ++φ−2 . From equation 2.38,

we know φno−click should have a linear dependence on |α|2. The value of the slope of this linear

function is equal to φ.

Figure 4.4 plots the φno−click data versus |α|2 and the linear fit to the data. From the fit

we measure φ = 7.6± 0.3µrad. If we assume that the atomic population is equally distributed

between Zeeman sub-levels of F = 2 ground state, we expect φ+ = 3.6 × φ−, see 3.5.1. With


n

95 45 20 10 40

0.10 0.14 0.22 0.32 1.00

Figure 4.3: Measured XPS for click events and no-clicks events. Red squares and green circlesare the measured phase shift conditioned on the SPCM not firing (no-click) and firing (click)respectively. The horizontal axis shows the mean signal photon number n and the post-selectionparameter δ used for each case.

this assumption and the value measured for φ, we can infer φ+ and φ− to be 11.9±0.6µrad and

3.3±0.2µrad, respectively. However, in the next section I will show how this assumption is not

valid in our experiment.

4.2.3 Weak Value Amplification of a Single Post-selected Photon

In Section 2.2.5 (equation 2.64) we discussed how the quantity φclick − φno−click is independent

of |α|2 and directly shows the effect of a single post-selected photon. Figure 4.5 plots the

measured φclick − φno−click versus post-selection parameters δ, given in table 4.2. The plot

clearly shows that as δ becomes smaller, the effect of the post-selected single photon becomes

larger. For δ = 0.1, the smallest post-selection parameter used in our experiment, we measure

φclick − φno−click = 63± 18µrad. From a fit to φno−clicks versus |α|2 we measured a per photon

phase shift of φ = 7.6 ± 0.3µrad, which gives an amplification factor of of 8 ± 2. Hence, a

post-selected single photon can act like 8 photons.

To extract the value of φ+ and φ−, we fit the first 4 data points to the function φ++φ−2 +


0 20 40 60 80 100n

0

100

200

300

400

500

600

700

800

nocli

ck (

rad)

Figure 4.4: The measured phase shift for no-click cases versus the average photon number inthe signal beam |α|2 = n. A linear fit reveals a per-photon XPS of φ = 7.6± 0.3µrad.

φ+−φ−2δ . We use the measured value of 7.6µrad for φ++φ−

2 and leave φ+−φ− as a free parameter.

From the fit we infer φ+ − φ− to be 11.5± 1.5µrad. Knowing the sum and the difference of φ+

and φ−, we can infer their values to be 13±1µrad and 2±1µrad, respectively, which is different

from what one would have expected if the population was equally distributed in F = 2. The

values we measure for φ+ and φ− suggest that the the population is more likely to be in Zeeman

sub-levels with larger mF ’s. We believe that the probe beam is optically pumping the atoms

towards mF = 2 which results in φ+/φ− > 3.6.

To directly compare the effect of varying δ without changing the mean photon number, we

measure φclick − φno−click for δ = 0.14 and δ = 1, with almost the same mean photon number

(|α|2 = 45 and |α|2 = 40 respectively.) We measured 45.5 ± 13.0µrad for the former and

9 ± 10µrad for the latter. It is worth noting that for δ = 1 we do not expect to observe any

WVA effect and φclick − φno−click should be equal to φ.

4.2.4 Technical Details

Weakness Criteria. In derivation of weak value formulas in chapter 2, we had to make a few

assumptions to guarantee that the measurement is weak. First assumption was that |β|2×∆φ1, where |β|2 is the mean photon number in the probe beam and ∆φ = φ+ − φ−. Using 2000


0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80cli

ckno

click

(ra

d)

Datatheorytheory with 1intrinsic per photon phase shift

Figure 4.5: XPS difference versus post-selection parameter. Red circles are measured phaseshift difference between click and no-click cases. Gray dashed line is a fit to φ++φ−

2 + φ+−φ−2δ

with φ++φ−2 assumed to be 7.6µrad and φ+ − φ− left as a free parameter. The fit reveals a

value of 11.5± 1.5µrad for φ+ − φ−. The cyan solid line is the full theoretical calculation withthe numbers extracted from the two fits.

photons in the probe beam and ∆φ = 11µrad ensures that this condition is met. The most

important criteria for the weakness is given by equation 2.25 which says that the phase-shift

written on the probe should be smaller than the uncertainty in the phase of the probe. The

largest mean photon number used for the signal was |α|2 = 100, and φ = 7.6µrad. Therefore,

|α|2φ = 7.6× 10−4rad which is much smaller than 22mrad calculated phase uncertainty of the

probe. Experimentally, we measured 100mrad of phase uncertainty on the probe.

Background Counts. In section 3.3.3 we discussed how gating the SPCM with 40ns windows

reduced the background counts to about 6%. For every data point, the post-selection parameter,

detector efficiency and mean photon number in signal are set so that without background

photons, we expect the post-selection probability to be 19%. Table 4.2 shows the measured

probability of “click” for each data point. As you can see, expect for data point number 1 and

number 5, the rest agree very well with the expected background counts of 6% and true counts

of 19%. The reason background counts are much smaller for data point number 5 is that we

used an ND filter to reduced the overall collection efficiency of the detector, and as a result,

the background counts dropped to around 1-2 %. For data point 1, in the absence of the signal

pulses we measured the expected 6% dark counts and in the absence of probe (and presence of

the signal) we measured 20% true counts. But the presence of the two beams resulted in 30%

total counts. The reason for the discrepancy between the 30% measured background counts

for data point 1 and the expected 25% is a bit subtle. We believe that the main source of the


background photons is the scattered probe and coupling photons; if the probe sees a higher OD,

it will scatter more photons and as a result background counts will increase. For data point

1, we sent signal pulses with on average 95 photons. The ac-Stark shift due to 95 photons,

shifts the probe about 100KHz outside the transparency window and as a result the probe

sees a higher OD, and therefore, scatters more photons. The fact that these added background

photons only exist in the presence of the signal pulses makes them difficult to be further studied.

Stability Monitoring and Data Rejection Criteria. During the data collections, we continu-

ously monitored the OD and the “click” rates. The OD was monitored at two points in a trace:

once in between the shots when the probe was in the EIT (transparency OD), and once at the

end of the trace. The OD at the end of the trace was measured after turning the coupling beam

off.1 Typically we had transparency OD of 0.7-1 and final OD of 2.5-3. These three measured

quantities always fluctuate. We decided that when any one of the three deviated more than

10% from their initial value, we would stop the run and discard any measured XPS after that

point.

4.2.5 Discussion of the Results

I will now discuss the advantage of using the WVA over a similar experiment in which we

only used post-selection with no amplification[50]. In that experiment a combination of around

1 billion trials (300 million successful post-selection) were used to measure the XPS due to

single post-selected single photons with σ+ polarization. By looking at the difference between

the XPS measured for “click” and “no-click” events, we measured φ+ of 18 ± 4µrad. In this

experiment, where we use WVA technique, we used a total of around 830 million trials (200

million successful post-selection) to extract a φ+ of 13±1µrad. It is evident that the WVA

technique yielded a better signal-to-noise ratio. This advantage comes from two sources: first, by

doing a linear fit to the subset where the post-selection failed (“no-click” events), we measured

φ++φ−2 = 7.6±0.3µrad. The small uncertainty is due to using around 600 million measurement

and multiple photons for each data point, which is the first advantage. Second, we fit the

function 7.6 + φ+−φ−2δ to the measured φclick − φno−click for every δ. From the fit we inferred

φ+ − φ− to be 11.5± 1.5µrad. By combining this two results, we infer 13±1µrad. The second

advantage is due to the fact that the quantity φclick−φno−click gets amplified as δ gets smaller,

1Turning the coupling beam off removes the EIT condition on the probe and the measured OD would correpondto the “actual” OD of the cloud.


but since the probability of post-selection is proportional to δ2|α|2, one can maintain a certain

post-selection probability by increasing |α|2. Hence, the amplification is not accompanied by a

reduction in the size of “click” events, which would otherwise increase the uncertainty of the

post-selected data set. Due to this compensation, the SNR is improved.

Chapter 5

On the Advantage of Weak Value

Amplification

In this chapter, I will discuss the usefulness of weak measurement to the precision measure-

ment. I will introduce Fisher Information as a measure of the maximal information that could

be gained through measurement, and by using it as a figure of merit, I will show how weak

measurement performs compared to a standard measurement, where no post-selection is em-

ployed. I will focus on a specific noise model with long time correlation and demonstrate a

regime where weak value amplification can outperform a standard measurement.

The question of whether weak value amplification is useful or not, has gained a lot of

attention in the past few years. Finding a general and final answer to the question has proven

to be difficult and one can find arguments both in favor and against the usefulness of the

weak value technique (see the references in 1.1.) Some of the elements that contribute to

the difficulty of this question are difficulty in introducing a general enough noise model, using

optimal statistical techniques for data analysis, fair resource counting and consideration of

experimental complexities. Therefore, before I continue, I should clarify the context in which

I am comparing the weak value technique to the standard technique. In what follows in this

chapter, I will mostly consider weak value amplification of the nonlinear effect of a single

photon by sending it through an interferometer. For simplicity I will only consider the case

where only one of the paths of the interferometer interacts with the probe. The standard version

of this measurement would be to eliminate the interferometer and directly let the single photons

interact with the probe. No post-selection is employed in the standard technique. To make the

78

Chapter 5. On the Advantage of Weak Value Amplification 79

comparison between the two techniques fair, I assume that we have only N single photons to

perform the measurement with. The shot-noise of both techniques are assumed to be the same.

In the next section I will introduce the technique I will be using for data analysis.

5.1 Fisher Information as a Figure of Merit

A common problem in statistics is to know the minimum uncertainty that can be achieved for

measuring a quantity given a data set. In order to extract the maximal amount of information

(minimum uncertainty) from a data set, one needs to carefully study how correlated the data

is. When there are correlations (covariances) in a set of measurement results, not all the data

points have the same importance. Therefore, not all data points should be weighted equally

when trying to find the average of a quantity. Below, I will show how knowing the covariances

can help in finding the optimal weighting of different data points.

5.1.1 Covariance Matrix and Fisher Information

To find the optimal weights, which corresponds to an estimator with minimum uncertainty, one

should construct a matrix called the Covariance Matrix (CM). CM is a square matrix whose

size is equal to the size the data set. For a data set with ith element given by di ± σi, the

elements of the CM are calculated as Cij = 〈σiσj〉. The diagonal elements of the CM are equal

to the variance of the corresponding data point and the off diagonal elements are equal to the

covariance between two data points. Knowing the CM, the average of the data set can be found

by[31]

d =

∑i,j C−1

ij dj∑i,j C−1

ij

. (5.1)

Equation 5.1 reveals the optimal way of weighting the the data points in a data set. As an

example, figure 5.1 shows the calculated weights for 10 measurements by using a CM given

by Cij = exp(−|i − j|/2.) (exponentially decaying covariances.) As it can be seen, due to

correlation between the measurements, the initial and final measurement are given a larger

weight than the measurements in the middle, while the naive averaging would give each data

point a weight of 1/10.

From a maximum likelihood approach it can be shown the variance of a data set has a

lower bound called the Cramer-Rao Bound (CRB). It can be shown that the formula given in


0 2 4 6 8 10measurement index

0.00

0.05

0.10

0.15

0.20

0.25

0.30

weig

ht

optimalnaive

Figure 5.1: Optimal weights obtained by the covariance matrix Cij = exp(−|i − j|/2.) versusnaive averaging when all data points are weighted equally.

equation 5.1 is the unbiased estimator for the mean value that minimizes the variance. This

minimum variance can be calculated to be

Var(d)min =(∑

i,j

C−1ij

)−1(5.2)

One can formulate the CRB as Var(d) ≥ 1/I, where I is called the Fisher Information. There-

fore, given a CM, the Fisher information can be calculated as

I =∑

i,j

C−1ij . (5.3)

5.1.2 Fisher Information versus Signal-to-Noise Ratio

Often to quantify the information in a measurement, one would use the Signal-to-Noise Ratio

(SNR), which, as the name suggests, is the ratio between the size of the signal to the uncertainty

in the signal. Therefore, for a measurement result given by s ± σ1, the SNR is calculated as

SNR1 = s/σ1. If the uncertainty σ1 for this measurement is obtained by the technique given

by equation 5.1, then the Fisher information for this measurement is I1 = 1/σ21. If a different

approach was used to measure the same quantity, for example by amplifying the signal, and the

CM analysis of this second technique revealed A× s± σ2, where A is the amplification factor,


the SNR would have been SNR2 = A × s/σ2. However, for the second measurement, defining

the Fisher information as I2 = 1/σ22 is erroneous, because A×s is not an unbiased estimator for

the signal s. In order to correctly calculate the Fisher information, one should first make the

second measurement unbiased by writing it as s±σ2/A and the Fisher information can correctly

be evaluated to be I2 = A2/σ22. With the correct definition of the Fisher information, one can

see that I2/I1 =(SNR2/SNR1

)2. Therefore, to compare different measurement techniques, as

long as optimal estimators are used to calculate the SNR and unbiased estimators are used to

find the Fisher information, one can use either the SNR or the Fisher information to make the

comparison and get the same result.

5.2 Measuring the Effect of a Single Photon

In this section I will compare the Weak Value Technique (WVT) to the Standard Technique

(ST) for measuring the nonlinear effect of a single photon.1 I will use the Fisher information

as a figure of merit when doing the comparison. I will first compare the two in the case of

Gaussian white noise and then I will study the effect of adding correlations. At the end, I will

introduce an optimal weak value technique which also makes use of the rejected data and the

correlations between the post-selected data and the rejected data.

5.2.1 Gaussian White Noise

The nonlinear effect of a photon can be measured as a cross-phase shift (XPS) on a probe

coherent state, as studied in detail in chapter 2. In the ST, no post-selection is used and the

measured XPS on the probe will be given by φ0, the intrinsic per photon nonlinear phase shift.

If we assume the shot-noise of the phase measurement is given by σ0 and the measurement is

performed for N times, then the measured phase in the ST would be Sstd = φ0 ± σ0/√N . The

Fisher information can be found to be Istd = N/σ20.

In the WVT, the measured signal is given by 〈n〉w × φ0, where 〈n〉w is the weak value of

photon number. By a similar calculation to what resulted in equation 2.50, one can easily show

that

〈n〉w =1

cot θ + 1(5.4)

with probability of post-selection given by Pclick = (cos θ + sin θ)2/2 for θ ∈ [−3π/4,−π/4].

1It can be thought of as estimating φ0, the per photon nonlinear phase shift, or estimating χ(3).


Therefore, the weak value signal is given by Swv = φ0/(1 + cot θ) ± σ0/√PclickN . The Fisher

information can be shown to be Iwv = (N/2σ20)× sin2 θ.

0.0 0.2 0.4 0.6 0.8 1.0P

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 /

(N/

2 0)

weak valuerejectedstandardweak value + rejected

Figure 5.2: Fisher information in WVT and ST versus post-selection probability.

Figure 5.2 plots the Fisher information in the WVT and the ST versus post-selection param-

eter. As it can be seen, the Fisher information in the WVT, only gets half as large as the Fisher

information in the ST. The reason is that in ST, one photon is sent through the interaction

region whereas in the WVT, because of the interferometer, only half a photon is sent through

the interaction region. Therefore, the number of photons used in the ST is a factor of two larger

than the number of photons used in the WVT, which results in the Fisher information to be

twice as large. Second striking feature in figure 5.2 is that the maximum Fisher information

is achieved for probability of post-selection of 50% and not for when the the probability of

post-selection is much less than one, i.e. where WVA happens. To see the reason behind this,

we should look at the Fisher information contained in the rejected portion of the data. The

size of the rejected data set is given by (1− Pclick)N . We can find the size of the signal of the

rejected data the fact that Pclick × Swv + (1 − Pclick) × Srejected = φ0/2. Therefore, the Fisher

information of the the rejected data can be calculated to be Irejected = (N/2σ20)× cos2 θ(figure

5.2). It is easy to show that for P=50%, the rejected data set averages to zero, and as a result,

carries no information. Because of that, the information in the post-selected portion of the data

is maximized at P=50%.

The reason behind both of the features mentioned above, is that we set the system up in


such a way so that only one of the arms of the interferometer is interacting with the probe.

To illustrate this point better, let’s compare this system with a similar system in which the z-

component of a spin-12 particle is being measured. Figure 5.3 shows the result of the calculated

0.0 0.2 0.4 0.6 0.8 1.0P

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 /

(N/

2 0)

weak valuerejectedstandardweak value + rejected

Figure 5.3: Fisher information in WVT and ST versus post-selection probability.

Fisher information in the ST, the WVT and the rejected data for spin-12 particle. Evidently,

the Fisher information in WVT is maximized when the probability of post-selection vanishes

and is equal to the Fisher information in the ST. It is worth noting that for the weak value

of a single photon, if instead of measuring a signal that is proportional to the weak value of

the photon number in only of the arms of the interferometer, we were measuring the photon

number difference between the two arms - that is instead of measuring n1 we were measuring

n1 − n2 - the Fisher information versus post-selection probability would look exactly like what

is plotted in figure 5.3.

5.2.2 Noise with Time Correlation

In this section I will study the effect of noise with time correlation on the Fisher information

in the WVT and the ST. I will first introduce the method used for simulating the CM and

calculating the corresponding Fisher information. Next I will compare the Fisher information

in the WVT and the ST.


(a) (b)

Figure 5.4: Calculated covariance matrix for (a) 10 points correlated on average and (b) notime correlation. The noise for (a) is modeled as a slowly varying noise added to a white noise.

Method

To simulate a CM for a noise with certain correlation time and to find the Fisher information,

we take the following steps: First, we generate a noise array with a certain time correlation by

filtering a Gaussian white noise with a filter width that corresponds to the desired correlation

time. We create a matrix by multiplying the noise array by its transpose. This procedure is

then repeated over many trials and the final CM is calculated by averaging over all matrices

generated in each trial. Figure 5.4 shows two example of the simulated CM’s with and without

time correlations. Once the CM is calculated, we use equation 5.3 to calculate the Fisher

information.

To simulate the CM for WVT, in each trial we randomly select a subset of the noise. The

size of this subset is chosen to reflect the post-selection probability. The remainder of the noise

array constitutes the rejected subset. We then separately calculate the CM for the post-selected

group and rejected group.

At each trial, we also concatenate the post-selected noise array and the rejected noise array.

We also calculate a CM for the mixture of the two. This concatenated CM is used for the

optimal weak value technique. Note that the concatenated CM has two main diagonal blocks

that are equal to the post-selected CM and the rejected CM. The off-diagonal blocks of the

concatenated CM corresponds to the covariance between the post-selected data set and the

rejected data set.


WVT vs. ST

We now compare the Fisher information in the WVT to the Fisher information in the ST. The

Fisher information in the ST is calculated by summing over the elements of the inverse of CM

calculated above. The Fisher information in the WVT is calculates similarly by summing over

elements of the inverse of the post-selected CM. To make sure an unbiased estimator is used

for the WVT, the sum is multiplied by 〈nw〉2.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Fish

er In

form

ation

1e14weak valuestandard

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Fish

er In

form

ation


(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P

1.0

1.5

2.0

2.5

3.0

3.5

Fish

er In

form

ation


(c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P

0.2

0.4

0.6

0.8

1.0

Fish

er In

form

ation


(d)

Figure 5.5: Fisher information in the standard technique and the weak value technique. Figure(a) plots the fisher information when measurements are uncorrelated. Figures (b),(c) and (d)are results for 0.8, 2 and 20 measurements per correlation time. A total of 200 data points areused in the data set. The CM’s are calculated by averaging over 10K trials.

Figure 5.5 plots the simulation results for having 0, 0.8, 2 and 20 correlated measurements.

Evidently, as the number of correlated measurements increases, the Fisher information in both

the WVT and the ST decreases. Fisher information in the ST is more sensitive to correlations


and decreases faster than it does in the WVT. Therefore, as the correlation time increases, it

is advantageous to use the WVT over the ST. Also, it is worth noting that as the correlation

time increases, the maximum of the Fisher information in the WVT occurs at lower values

of post-selection parameter (P→0). Therefore, for long time correlations, not only the WVT

becomes superior over the ST, it is also advantageous to chose the post-selection parameter to

be in the weak value amplification regime.

To understand why post-selecting on a small portion of the data in the WVT is advantageous

in this regime, we should first note that the reason behind repeating a measurement to carefully

study the noise by sampling it as much as possible. But if the noise has time correlation,

sampling the noise multiple times within one time correlation will not better our understanding

of the noise, and one point within a correlation time is as good as many. To illustrate this

point, figure 5.6 shows two noise traces, one with long time correlations (top) and the other is

white noise (bottom.) The dots represent sampling both noises 100 times at a fixed rate. The

red dots are a random selection from these 100 measurements. It can be seen that in the case

of white noise, the randomly selected portion doesn’t sample the noise as well as the original

100 points. This loss in sampling is much less pronounced in the noise trace with long time

correlations. Hence, by sampling the noise once per correlation time and amplify the signal at

the same time, which is what weak value amplification naturally does, one would get a better

SNR (higher Fisher information.)

Figure 5.6: Two noise traces, with long time correlations(top) and without time correlation(bottom). The dots represents sampling from the noise with a fixed rate. The red dots are a20% random selection from them.


Optimal Weak Value Technique

In previous subsection, I ignored the role of the rejected data. I now calculate the Fisher

information that is left in the rejected portion of the data and in the correlation between

the post-selected data and the rejected data. To find the Fisher information, I first calculate

the inverse of the concatenated CM. Summing over the elements that correspond to the post-

selected data (and multiplying by 〈nw〉2 to make it unbiased) reveals the Fisher information

in the post-selected data. Note that this Fisher information is different from the one calcu-

lated in the previous subsection, because it is calculated from a different CM. Summing over

the elements that correspond to the rejected data and correlations (and multiplying them by

appropriate factors to make them unbiased) reveals the Fisher information in the rejected data

and the correlations. The total Fisher information can be found by summing over the individual

Fisher informations in each subset. I will show that the optimal weak value technique always

outperforms both the WVT and the ST.

Figure 5.7 plots the calculated Fisher information in the ST, the WVT, rejected dataset

and the correlations versus post-selection parameter, as the correlation time of the noise in-

creases. Evidently, as the time correlation becomes longer, the information in ST dramatically

decreases, whereas the maximum information in the optimal technique remains almost constant.

Therefore, the optimal weak value technique can eliminate the effect of the added correlations.

5.2.3 Advantage of Post-selection without Weak-Value Amplification

In this section, I will show that in a certain regime just doing post-selection, without weak value

amplification, can be advantageous over the standard technique. As an example, consider that

you want to measure the per photon phase shift of a nonlinear medium (measuring χ(3)) but

as a resource you have only coherent states with on average a tenth of a photon (|α|2 = 0.1.)

In the ST, the measured signal would be 0.1×φ0±σ0/√N . However, by using a single-photon

detector with 100% efficiency, one could condition the measurement on a successive photon

detection of the signal. The post-selection probability would be 10% and the measured signal

would be φ0±σ0/√

0.1×N . It can be seen that in this case the SNR has improved by a factor

of√

10 by just doing the post-selection.

Figure 5.8 plots the Fisher information in the ST and in post-selected data versus |α|2

for three different detector efficiency. The probability of post-selection is given by Pps = 1 −


0.0 0.2 0.4 0.6 0.8 1.0P

0

1

2

3

4

5

6Fi

sher

Info

rmati

on

1e13weak valuerejectedcorrelationsstandardOptimal

(a)

0.0 0.2 0.4 0.6 0.8 1.0P

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fish

er In

form

ation


(b)

0.0 0.2 0.4 0.6 0.8 1.0P

0.0

0.5

1.0

1.5

2.0

2.5

Fish

er In

form

ation


(c)

0.0 0.2 0.4 0.6 0.8 1.0P

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fish

er In

form

ation


(d)

Figure 5.7: Fisher information in the standard technique, weak value technique, the rejectedsubset and correlation between the post-selected subset and rejected subset for various corre-lation times. The “optimal” corresponds to summing over the Fisher information in WVT,rejected data and the correlations. Figure (a) plots the fisher information when measurementsare uncorrelated. Figures (b),(c) and (d) are results for 0.8, 2 and 20 measurement shots percorrelation time.

exp(−η|α|2) and the mean photon number conditioned on post-selection is given in equation

2.15. As it can be seen, in the case of a detector with 100% efficiency it is always advantageous

to use post-selection.2 As the efficiency of the detector decreases, the range of |α|2’s in which

post-selection is advantageous decreases. The noise is considered to be white in all these cases.

2for large |α|2, when Pps → 1, the two techniques will have the same Fisher Information.


0.0 0.2 0.4 0.6 0.8 1.0| |2

3.0

2.5

2.0

1.5

1.0

0.5

0.0

log(

/(N

/0

))

StandardPost-Selection

(a)

0.0 0.2 0.4 0.6 0.8 1.0| |2

3.0

2.5

2.0

1.5

1.0

0.5

0.0

log(

/(N

/0

))


(b)

0.0 0.2 0.4 0.6 0.8 1.0| |2

3.0

2.5

2.0

1.5

1.0

0.5

0.0

log(

/(N

/0

))


(c)

Figure 5.8: The effect of post-selection without weak-value amplification on Fisher informationversus |α|2. The detector efficiency used for the plots are 100%, 20% and 1% for (a),(b) and(c) respectively.

5.3 Experimental Proposal and Progress

In the previous section I showed how in the presence of noise with time correlation, weak value

amplification can be advantageous. In this section I will propose an experiment to demonstrate

this advantage. First I will extend this idea to the regime where a coherent state is used

instead of single photon. Then I will discuss how to generate a controlled time-correlated noise

experimentally. Finally I will report on the experimental progress made so far.

5.3.1 Fisher Information for Coherent States as the Signal

Here, I will calculate the Fisher information in the post-selected data in the experiment de-

scribed in chapter 4 and will compare it to the Fisher information in a similar experiment

without post-selection. For the weak value technique, we use the weak values given in equation

2.50. The signal for the ST is equal to |α|2 × φ0. Figure 5.9 plots the calculated Fisher infor-

mation for the two techniques as the number of measurements per correlation time increases. I

have used parameters δ = 0.26, |α|2 = 15 and η = 0.2 for this calculation. Evidently, when the

number of measurements per correlation time is very small (white noise regime) the ST outper-

forms the WVT. This is due to the fact that the amplification factor for the chosen parameter

is not large enough to compensate for the reduction in the size of the dataset. However, as the

number of correlated measurements increases, the Fisher information in the WVT is affected

less than the Fisher information in the ST. Therefore, in the regime where the measurements

are highly correlated, WVT becomes superior.


0.002 0.02 0.2 1.0 2.0 4.0 10.0 20.0 40.0 100.0Number of measurements per correlation time

12.0

12.5

13.0

13.5

14.0

14.5

log(

)

WVTST

Figure 5.9: Fisher information for the WVT and ST versus number of correlated measurements.We assumed δ = 0.26, |α|2 = 15 and η = 0.2.

5.3.2 Experimental Progress

In order to use our setup to demonstrate the advantage of the WVT over the ST, we need to add

noise with time correlations to our phase measurement. It is very important to add the noise

in such a way that it doesn’t get amplified when doing weak measurement. The added noise

should not cause further fluctuations in the interaction region and should not interfere with the

post-selection probability. Since the off-resonance portion of the probe does not interact with

any part of the setup and is only used as a reference for the probe beam, it is the perfect place

to add the slow noise. Adding slow phase noise to the off-resonance portion of the probe causes

slow drifts in the measured phase without interrupting the interaction between the signal and

the probe.

When we were first designing the phase measurement apparatus, we made it to be robust

to slow phase drifts. By doing reference measurements before and after when we the XPS is

expected, we filter out any slow phase drift. This, however, is an obstacle in front of adding slow

phase noise to the probe; the slow phase noise added to the probe would get completely filtered

out by our phase measurement. To get around this, we decided to only add a sudden phase jump

during the time we expect the XPS to occur. By slowly changing the magnitude of the jumps,

in accordance to a slow noise trace, we managed to add the slow noise drift to the measured

phase while keeping the measurement robust to all other slow phase drifts. Figure 5.10 shows

the schematic of the setup used to add slow phase noise. We have programmed the task control

computer so that the two analog cards controlling the setup are timed independently from each


RFSwitch

Phase-Shifter AOM

DCOffset

Slow Noise

Signal Gates

Gates Noise

10MHzReference

70MHz RF with Phase-Noise

Probe withPhase-Noise

Figure 5.10: Schematic of the setup to add slow phase-noise to the probe beam.

other, see section 3.17. While one of the cards control the experiment, the other card generates

a stream of slow noise. We can control the time correlation of this noise as well as its offset and

amplitude. We can choose between filtered Gaussian white noise or 1/f noise as well as different

modulation functions (square, saw-tooth, sinusoidal, etc.). We can also load noise from a saved

file. The generated noise is then gated using an RF switch to only be on for a 100ns every 2.4µs

(length of a shot.) An RF switch switches between the noise trance and a DC offset and uses a

100ns square modulation signal as the control. The DC offset is used to ensure that the gated

phase averages to zero. We employ a Phase-shifter circuit[72], to generate a 70MHz RF signal,

where the phase of the signal is controlled by the gated phase. A stable 10MHz references is

used to lock-in the RF to the phase measurement apparatus for demodulation purposes. The

RF drives an AOM, where the 1st order of the AOM, which contains the phase jumps, is used

as the off-resonance probe and the 0th order is used as the on-resonance probe. Therefore, the

phase of the beating signal of the on- and off-resonance probe will change in accordance to the

added slow noise. A delay in gating is used to ensure that the phase jumps happen exactly

when the XPS is expected. The apparatus mentioned above has been built and tested. The

Phase-shifter circuit is well calibrated and all the elements are in good shape for the experiment

to be carried out.3

3A sudden and unfortunate death of the turbo pump that maintains the vacuum in the chamber preventedus from moving forward and the experiment needed to be postponed until the vacuum problem is resolved.

Chapter 6

Outlook and Summary

In this chapter I will discuss the limitations of the current apparatus and possible routes to

overcoming them. These improvements primarily seek to increase the size of the nonlinearity.

A larger nonlinearity will open up the path to many experiments in quantum nonlinear optics,

which due to the small size of the current nonlinearity are experimentally inaccessible. I will

also discuss alternative experiments outside of quantum nonlinear optics that can be studied in

the future. In the last section, I will summarize the achievements of the thesis.

6.1 Possible Improvements

Alternative Level Scheme

As mentioned in section 5.2.2, the Fisher information in the post-selected data will be maximum

at very small post-selection parameter, if we were measuring n1 − n2 instead of just n1. To

perform this measurement, the two paths of the interferometer should interact with the probe

with the same strength but opposite signs, so that the XPS written on the probe would be

φ0 × (n1 − n2). The other advantage of having such interaction is that since φ1 = −φ2 = φ0,

the XPS conditioned on a successful post-selection (from equation 2.35) becomes φclick = φ0/δ

and the effect of the coherent state will be automatically canceled out. One way to realize such

interaction is by using the level scheme shown in figure 6.1. First, one needs to optically pump

all the population to F = 2,mF = 2 ground state. A combination of a σ+-polarized probe

and a π-polarized coupling beam creates the EIT. The signal is an equal superposition of σ+

and σ− polarization. Using a DC magnetic field, we can control the Zeeman shift between the

92

Chapter 6. Outlook and Summary 93

sub-levels of the F ′ = 4 excited state, and therefore, we can make σ+ polarization component of

the signal to be red-detuned and σ− component to be blue-detuned, which makes the ac-Stark

shift of he two polarizations to have opposite sign. By fine tuning the magnitude of δ1 and δ2

in accordance to the CG coefficients of the two transitions, we can ensure that the ac-Stark

shifts due to the two polarizations have the same magnitude. This will result in φ1 = −φ2, and

therefore, by using such level scheme, we can measure n1 − n2.

9/64" 3/8"

mF = 2 mF = 3 mF = 4

F 0 = 4

F = 3

F 0 = 3

F = 2

12

Figure 6.1: Schematic of the improved level scheme.

Filtering Background Photons

Eliminating the background photon detections will enable us to perform the experiment with

lower signal photon numbers and/or to use much smaller post-selection parameter to gain larger

amplification factors. Although gating the SPCM with smaller window can reduce the chance

of a background photon detection, but we are at the limit of the smallest time window we can

use. One way to reduce the background photons is to couple the signal to a single-mode fiber

instead of a multi-mode fiber. The downside, however, is that the collection efficiency of the

signal will also be reduced. Recently, D.T. Stack et al.[73] demonstrated that by using a vapor

cell containing 85Rb and a buffer gas, they could attenuate noise photons by up to two orders

of magnitude in their quantum memory scheme. A similar technique can be pursued to filter

the background photons in our setup.


Brighter Single Photon Sources

We are currently equipped with a single photon source that is compatible with Rb atom[74].

The best single photon rate we obtain from this source, however, is 100 per second. Observing

the effect of these single photons with our current setup needs continuously measuring for 10

days. Therefore, exploring alternative narrow-band single photon sources with high brightness

is necessary. One promising alternative sources uses four-wave-mixing in a atomic sample to

generate a narrow-band pair of photons[75, 76]. We are currently equipped with necessary

elements for generating photon pairs in atomic ensembles. Exploring this system might prove

it to be an easy and efficient alternative for the photon source we currently possess.

Different MOT Geometries and BEC

Higher optical densities could enhance the size of the nonlinearity. Although, due to group

velocity mismatch and signal absorption, one may not benefit from a high OD. Nonetheless, a

proper choice of level scheme, or using EIT schemes other than N-scheme[77, 78, 79], could put

us in a regime where the size of the nonlinearity grows with OD. However, due to the radiation

pressure from the scattered photons within the cloud, the density of a typical MOT cannot

surpass a certain amount. To obtain higher densities or OD’s, one need to explore different

MOT geometries[80, 81, 82, 83].

An alternative option is using a Bose Einstein Condensate (BEC) to mediate the interaction

between the probe and the signal. However, since the smallest photon scattering could destroy

the BEC, one needs to be careful when trying to obtain an EIT in a BEC without scattering

photons. Also, the BEC apparatus typically has a low duty cycle, therefore it is not suitable

for measurements where hundreds of millions of trials are needed. A BEC could be desirable if

one could obtain large enough nonlinearity to be close to single shot resolvability.

6.2 Future of the current Apparatus

In this section I discuss possible experiments that can be carried out in the near future with

small modifications to the current setup.


Imaginary Weak Values

In this thesis, I limited my discussions to real weak values. But as the formula for weak values

suggests, the weak value can be imaginary as well. In an experiment similar to what was carried

out in this thesis, imaginary weak values can be measured if one post-selects on a polarization

that has an imaginary overlap with the initial polarization, which will result in an imaginary

δ. The effect of the imaginary weak value can be observed as a shift in the amplitude of the

probe instead of its phase. Observing an imaginary weak value is fundamentally interesting as

well useful in quantum metrology[45, 46].

PNR detectors and Observing the Effect Higher Photon Detections

In section 2.2.4 we discussed the effect of using a PNR detector on the inferred photon number.

We also showed that post-selecting on an n-photon state in weak measurement results in an

amplification of all n photons. Therefore, using a PNR detector[55] or branching SPCMs with

beam splitters can be a possible experiment for the future to observe the effect of the n-photon

detections.

6.3 Revisiting Optical Lattices

The apparatus used in this thesis has the capability of carrying out optical lattice experiments.

Several coherent control, state manipulation and quantum chaos experiments were performed

using vibrational states of optical lattices on this setup[84, 85, 86]. Here we discuss two possible

experiments that could be done with optical lattices.

Rabi Oscillation and Breakdown of Rotating Wave Approximation

The coupling between different vibrational states of the optical lattice can be done by displacing

the lattice. A periodic displacement can act like a dipole coupling between adjacent vibrational

states, and one can observe Rabi oscillation. The Rabi frequency in our system can be made

large enough to be comparable with the energy difference between the two levels. Our sim-

ulations indicate that in this regime, the fast oscillating terms in the Hamiltonian cannot be

ignored and one should expect to see fast oscillations on top the expected Rabi oscillations. This

could be the first demonstration of breakdown of rotating wave approximation by observing

the fast population oscillations in real time.


Quantum Heat Engine

In 2010, Linden, Popescu and Skrzypczyk proposed a scheme to make the smallest possible self-

contained heat engine[87, 88]. Due to its need of three-body interactions and thermalization

of qubits, however, the scheme is difficult to be experimentally realized. Using a combination

of a one dimensional optical lattice and the internal electronic states of atom, one can tailor

an effective interaction between three “degrees of freedom” of a single atom. Thermalization

could be achieved by making an EIT dark state between two electronic states that constitute

the qubit, where the population ratio (temperature) can be controlled by adjusting the relative

power of the beams creating the EIT. Further research is needed to confirm the feasibility of

the proposed scheme. If implemented, it would be the first ever observation of the smallest

possible heat engine.

6.4 Rydberg Atoms

One way to enhance the nonlinearity at the single photon level, is to use exotic systems such as

cavity quantum electrodynamics[89], atoms inside hollow core fiber[90, 91, 92], atoms trapped

around a tapered nanofiber[93], a single atom strongly coupled to a micro-resonator[94, 95] and

very recently, using Rydberg atoms. Long-range strong interactions between two Rydberg atoms

and Rydberg blockade effect have been used to demonstrate strong nonlinearities[96, 97, 98, 99].

Transitioning to use Rydberg atoms is a natural next step for the future of this apparatus for

further studying of interaction between two single photons, and even generation of non-classical

states of light. Experiments involving Rydberg atoms distance themselves from the scheme pro-

posed by Monro and Nemoto[100, 101], in which the goal is to use relatively weak nonlinearities

and non-demolition measurement to perform a phase-gate. However, the possibility of obtain-

ing large enough nonlinearities using Rydberg atoms is still an open questions and experimental

investigation in needed to shed light on the subject[102, 103].

6.5 Summary

In this thesis, I reported on the experimental observation of amplifying the effect of a single

post-selected photon using weak measurement. I showed that a single photon, when prop-

erly pre- and post-selected, can write eight photons’ worth of phase shift on a coherent state


probe. I demonstrated that this technique, with comparable resources, resulted in a better

signal-to-noise ratio than a similar experiment which only utilized post-selection but no weak

value amplification. The presented experiment is the first ever demonstration of weak value

amplification using a deterministic weak interaction between two optical systems that cannot

be explained classically.

A detailed theoretical background of the experiment was also provided in this thesis. Using

the weak value approach, I calculated the weak value of photon number inside the two arms of

an interferometer versus the post-selection parameter. I also showed that the same result can

be derived from the Schrodinger equation. I studied the effect of detectors with and without

photon number resolvability and showed amplification of higher photon number states can be

achieved by using a photon number resolving detector. I also studied the effect of measurement

back-action on the expected weak values, and compared the weak value to what one would

expect to measure in a similar experiment but with strong interactions.

Finally, I used the Fisher information as a metric to compare weak measurement to a

standard measurement. I showed that in the white noise regime, the maximum information

weak values contain can be as large as the information in the standard measurement. By

using a noise model with time correlations, I demonstrated that as the number of correlated

measurements increases, the Fisher information in the standard technique decreases much faster

than it does in the weak value technique. An optimal weak value technique was introduced

in which one makes use of all the information in the weak value data set, the rejected data

set and the correlation between the two data sets. It was shown that the optimal weak value

technique is always superior. I proposed an experiment to demonstrate the advantage of weak

measurement and reported on the progress made towards its implementation.

Appendix A

Weak Value Approximations

The weak value formula was derived in equation 2.8. Two approximations were made in this

derivation: first, I approximated exp(−igP .A) as I− igP .A. Second, I wrote (I− ig 〈f |A|i〉〈f |i〉 )|ψ0〉as exp(−igAwP )|ψ0〉, where Aw is the weak value defined as 〈f |A|i〉/〈f |i〉. I now discuss the

conditions necessary for these approximations to be valid.

The combined state of the system and probe after the interaction can be written as

e−igP .A|i〉|ψ0〉 =∞∑

n=0

(−ig)n

n!PnAn|i〉|ψ0〉. (A.1)

After post-selecting the system in final state |f〉, the state of the probe becomes

|ψf 〉 =

∞∑

n=0

(−ig)n

n!Pn〈f |An|i〉|ψ0〉. (A.2)

Due to the post-selection, this state is not normalized. In section 2.1.2, I normalized the state

by the normalization factor 1/〈f |i〉, assuming that the probability of the post-selection is given

by |〈f |i〉|2. One can also find the normalization factor from equation A.2

N2 =

∞∑

n,m=0

〈ψ0|[(ig)m

m!Pm〈i|Am|f〉

][(−ig)n

n!Pn〈f |An|i〉

]|ψ0〉

=

∞∑

n,m=0

(−1)n(ig)m+n

m!n!〈i|Am|f〉〈f |An|i〉〈Pm+n〉ψ0 ,

(A.3)

where 〈P k〉ψ0 is the expectation value of P k with respect to the probe’s initial state |ψ0〉, and

98

Appendix A. Weak Value Approximations 99

1/N is the normalization factor. Equation A.3 can be written as

N2 = |〈f |i〉|2 − ig〈P 〉ψ0

(〈i|A|f〉〈f |i〉 − 〈f |A|i〉〈i|f〉

)

+ (ig)2〈P 2〉ψ0

(1

2〈i|A2|f〉〈f |i〉+

1

2〈f |A2|i〉〈i|f〉 − 〈i|A|f〉〈f |A|i〉

)+O(g3〈P 3〉ψ0),

(A.4)

which can be simplified to

N2 = |〈f |i〉|2[1 + 2g〈P 〉ψ0 Im(Aw)− g2〈P 2〉ψ0

(Re(A(2)

w )− |Aw|2)]

+O(g3〈P 3〉ψ0), (A.5)

where A(k)w is defined as 〈f |Ak|i〉/〈f |i〉. The same result was derived in [6]. Evidently the

normalization factor is given by 1/〈f |i〉 to the 0th order in g〈P 〉ψ0 . The first order will vanish

if one choses an initial probe state with zero mean momentum, 〈P 〉ψ0 = 0, or a post-selection

parameter which results in a real weak value. In this case, the deviation of |〈f |i〉|2 from equation

A.5 will be in the order of g2〈P 2〉ψ0 . For a probe state with zero mean momentum, 〈P 2〉ψ0 is

equal to the variance in the momentum distribution. Hence, the following condition ensures

that the deviation of |〈f |i〉|2 from equation A.5 would be vanishingly small

VarP 1

g2(Re(A(2)w )− |Aw|2)

. (A.6)

If the system is post-selected in the same state as the initial state, Re(A(2)w ) − |Aw|2) will be

the variance of A with respect to the initial state of the system, |i〉. Therefore, condition A.6

will become VarP 1/(g2VarA). This condition can more intuitively be understood as the

variance in the position of the probe should be much larger than g2VarA.

I now derive the condition necessary for validity of equation 2.8, assuming the condition

given in A.6 is satisfied. Using 1/〈f |i〉 as the normalization factor, the final state of the probe

after the post-selection (equation A.2) becomes

|ψf 〉 =

∞∑

n=0

(−ig)n

n!Pn〈f |An|i〉〈f |i〉 |ψ0〉. (A.7)

The final state of the probe in equation 2.8 was approximately written as

|ψ(a)f 〉 = e−igAwP |ψ0〉 =

∞∑

n=0

(−ig)n

n!PnAnw|ψ0〉. (A.8)


For |ψ(a)f 〉 to be a valid approximation we must have 〈ψf |ψ(a)

f 〉 ≈ 1. Therefore

〈ψf |ψ(a)f 〉 = 〈ψ0|

[ ∞∑

n=0

(ig)n

n!Pn〈i|An|f〉〈i|f〉

][ ∞∑

m=0

(−ig)m

m!PmAmw

]|ψ0〉

=∞∑

n,m=0

(−1)m(ig)m+n

m!n!〈Pm+n〉ψ0A

(n)w Amw .

(A.9)

It is easy to show that for m = n = 0, (−1)m (ig)m+n

m!n! 〈Pm+n〉ψ0A(n)w Amw = 1. Therefore, for

〈ψf |ψ(a)f 〉 to be approximately 1, all other terms in the summation should be vanishingly small.

I assume that the initial momentum wavefunction of the probe is a Gaussian function centered

at zero with a width given by σp. For this wavefunction, it is easy to show

〈PN 〉ψ0 =

(N − 1)!!σNp if N is even

0 if N is odd,

(A.10)

where (N − 1)!! = 1 × 3 × 5 × ... × (N − 1). Hence, for m + n = (2k + 1) , k = 1, 2, 3, ..., the

terms in the summation in equation A.9 are zero. For m+ n = 2k , k = 1, 2, 3, ..., we have

gm+n

m!n!〈Pm+n〉ψ0A

(n)w Amw =

gm+n(m+ n− 1)!!

m!n!σm+np A(n)

w Amw 1. (A.11)

Since (n − 1)!! < n!, for inequality A.11 to be true, we need to have gm+nσm+np A

(n)w Amw 1.

Therefore, the condition necessary for the validity of equation 2.8 is

σp 1

g ×(A

(n)w Amw

)( 1m+n

). (A.12)

Unless one can prove that the terms in the summation in equation A.9 approach zero faster

than a geometric series, the condition A.12 should be true for all m’s and n’s. No general

general statement can be given on that regard unless A(n)w is calculated as function of n for a

specific system.

In the original AAV paper [4], the condition for validity of weak values was presented as

σp maxn|〈f |i〉|

|〈f |An|i〉| 1n. (A.13)

However, no derivation for this derivation was presented. Later, in [5], the authors claimed that


the condition introduced by AAV was incorrect and instead introduced two other conditions.

One condition simply says σp 1/Aw and the second condition is

σp minn

∣∣∣ |〈f |A|i〉||〈f |An|i〉|∣∣∣1/(n−1)

. (A.14)

This condition was also not rigorously derived in [5].

Appendix B

Alignment Procedures

B.1 Polarization Spectroscopy

The following steps make the alignment of the polarization spectroscopy easier and ensure a

healthy signal. See figure 3.3 for reference.

• 1) Make sure that the probe and pump beams are overlapping inside the rubidium vapor

cell. You may find it easier to first overlap the beams and then to put the cell in place.

Also make sure the probe is well aligned with the two photo diodes

• 2) Block the beams hitting the photo diodes and look at the error signal on an oscilloscope

• 3) Turn the offset nob on the front of the PID circuit box to make sure the error signal is

zero

• 4) Block the pump beam.

• 5) Make sure the frequency of the laser is not resonant with any of the Rubidium transi-

tions and unblock the two beams hitting the two photo diodes

• 6) By adjusting the HWP before the PBS adjust the power balance of the two beams

hitting the photo diodes; if they are balanced, the error signal should be zero

• 7) Unblock the pump beam, vary the piezo voltage on the laser driver until you see a

transition. If you have followed the previous steps correctly, you should not be seeing any

Doppler background

102

Appendix B. Alignment Procedures 103

• 8) Once you find the wanted transition, rotate the QWP to adjust the slope and shape of

the wanted feature

B.2 Injection-Lock

Take the following steps to improve the alignment of the injection lock. The optical elements

mentioned below refer to figure 3.7.

• 1)Place the H-OI in front of the slave diode laser. Rotate the polarizer until the power

passing through the H-OI is maximized. This ensures that the polarizer is aligned to the

output polarization of the diode laser.

• 2)Rotate the HWP2 before the PBS until the output power, which reflects off of the PBS,

is maximized.

• 3)Rotate the HWP1 at the seed input to maximize the input transmission through the

PBS. If the first two steps are done correctly, the input should also be passing thought he

H-OI with maximum power.

• 4)overlap the input beam with the output of the diode laser at a point close to the diode

by adjusting the mirror 1.

• 5)Rotate the HWP2 for a small amount just to let a small portion of the output of the

diode laser to leak into the input port.

• 6)Using mirror 2, overlap this leaked portion with the input light at a point as close to

the input port as possible.

• 7)Repeat steps 4 and 6 until the two beams completely overlap at the two points.

• 8)Rotate HWP2 back to maximize the power in the output

• 9)Look at the output spectra at the Fabry Perot cavity and ensure that the slave laser is

following the seed by adjusting the current to the diode and its temperature

B.3 Tapered Amplifier

• 1)Make sure that the TA chip is running at low current (<0.2A) to avoid

exposing yourself to high optical power


• 2)Remove the cylindrical lens and the slit. You should unscrew the cage mounts holding

them in place and slide them out.

• 3)Align the input beam to the ASE coming out of the input port. Make the spatial mode

of the two beams are matching.

• 4)The input lens is mounted on a z-translation mount. While monitoring the output

power of the TA, gently move the lens in the z direction.

• 5)Realign the input beam and try to maximize the output power. If you see an improve-

ment, keep moving the input lens in the same direction. If the power is getting lower,

move the lens in the opposite direction. Repeat these two steps until the maximum output

power is achieved.

• 6)The output lens in mounted on a x,y,z-translation mount. Move it in the z direction to

collimate the beam in vertical direction (see figure 3.9 side view.)

• 7)Slide the slit back in place. The correct position for the slit is the point where the

horizontal component of the beam is focused (see figure 3.9 top view.) Move the output

lens in x and y direction if necessary. If the slid is in its correct position, you should not

be able to see any interference fringes along the path of the beam.

• 8)Slide the cylindrical lens back in place and move in z direction to collimate the beam

in horizontal direction.

B.4 Polarization Maintaining Fiber

• 1) Couple the light into the fiber. Make sure the input beam has linear polarization.

• 2) Place a HWP at the input and a second HWP, a polarizer and a photo-detector at the

output. See figure B.1.

• 3) Frequency modulate the input beam. Select the largest possible frequency range for

modulation. (Modulating the piezo of the Master Laser can modulate the frequency of

the beam in a several GHz range.)

• 4) You should see the power that is measured at the photo-detector to be modulated at the

same modulation frequency. Rotate the HWP at the output to maximize the amplitude


Frequency

time

Power

time

HWP HWPPM Fiber

Polarizer

Detector

Figure B.1: Schematic of setup used to align the input polarization of the PM fiber.

of this modulation. This indicated that the polarization inside the fiber is changing with

frequency and the input polarization is not right.

• 5) Rotate the HWP at the input to minimize the power modulation amplitude seen on

the photo-diode.

• 6) Rotate the HWP at the output to maximize the amplitude of the modulation. If the

changes are being made in the correct direction you should observe an overall decrease in

the amplitude of power modulation.

• 7) Repeat steps 5 and 6 until you don’t observe any power modulation at the output.

B.5 MOT

(The x,y and z direction and optical elements mentioned in what follows, refer to the directions

and corresponding elements indicated in figure 3.14)

• 1) Make sure the center of two coils are perfectly aligned in the z direction.

• 2) Mount the cage system and overlap the trapping beam and the repumper beam. The

fiber to free space adapters are mounted on xy-translation cage mounts. Use them to


x y

6cm

4.5cm

2.5cm

3.7cm

8.3cm

z

5.5cm

4cm

6cm

Coil

Cuvette

(a)

(b)

Figure B.2: (a)Front (left) and side view (right) and (b)3D visualization of relative position ofMOT beams with the cuvette and magnetic field coils.

ensure the repumper beam is perfectly at the center of the trapping beam.

• 3) Place two cage-mountable iris diaphragms in the outputs of the cage system.

• 4) Put PBS2 in place and make sure the reflected beam (path 1) is sent through the

center of the magnetic field gradient. You can measure where the beam is entering and

exiting the cuvette to ensure the beam is roughly passing through the center of the trap.

See figure B.2

• 5) Using the same measurements, align the paths 2 and 3. Now all beams in x, y and z

directions should be roughly overlapping at the center of the cuvette. (if the Rb pressure

inside the cuvette is high enough, you should be able to see them overlapping using an

IR-viewer.)

• 6) Place Q1, Q3 and Q5 in paths 1 , 2 and 3. Set the angles so that the polarization of

the beams in the three paths are circular. Important Note: Paths 1and 2 should have

the same circular polarization and path 3 should have the opposite circular polarization.

• 7) Place Q2, Q4 and Q6 in place. If the polarization in the paths are perfectly circular,

the angle of Q2, Q4 and Q6 should not be important. However, if the polarization

in those paths are elliptical, it is better to set the angle of Q2, Q4 and Q6 so that the

linear polarization component of the reflected light is orthogonal to the linear polarization


component of the incoming beam. This results in lower temperature when doing molasses

cooling.

• 8) Overlap the retro-reflected beams with the incoming beams.

• 9) Turn on the magnetic field gradient and open the iris diaphragms all the way. You

should be able to see the MOT now. (If you don’t, the problem might be that the circular

polarization setting doesn’t match the direction of the magnetic field gradient. Flip the

direction of the magnetic field by flipping the direction of the current. If the problem

persists, check the polarization setting.)

• 10) Once you see a cloud, start closing the iris diaphragms slowly until the MOT start to

disappear.

• 11) Improve the alignments of the three paths along with the retro-reflected beams to

improve the MOT.

• 12) Repeat steps 10 and 11 until the relative alignment of the three beams are improved.

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by Matin Hallaji - University of Toronto T-Space · Matin Hallaji Doctor of Philosophy Graduate Department of Physics University of Toronto 2016 Weak value ampli cation (WVA) is a