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Electro-optic Control of Quantum Measurements

Benjamin Caird Buchler

A thesis submitted for the degree of Doctor of Philosophy of the The Australian National University

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Declaration

This thesis is an account of research undertaken between January 1998 and September 2001 at The Department of Physics, Faculty of Science, The Australian National University, Canberra, Australia.

Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university.

Benjamin C. Buchler September, 2001

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Acknowledgements

Firstly, I would like to thank my supervisors Hans Bachor and David McClelland. Together they have cultivated an excellent research environment and filled the labs with many talented people. In particular, the strong international collaborations that have been active during my PhD have contributed to a stimulating work environment, as well as opening many doors overseas.

Without a doubt, the most direct influence on my work has come from Ping Koy Lam, Malcolm Gray and Timothy Ralph. I deeply admire the careful and thorough approach Ping Koy has to experimental work. My experiments were laid on the very solid foundations which he built. Mal was not only a constant source of inspiration and new ideas, he was also largely responsible for designing the electronics required for our experiments. Without reliable electronics, many of the experiments in this thesis would have been impractical. My third supervisor, Tim, was whisked away to Queensland mid-PhD. I found his capacity to transform complex ideas into experimentally achievable goals to be quite remarkable, as was his patience when explaining these new schemes to me.

I have had the good fortune to work with many talented people, both inside and outside my own small research group. Daniel Shaddock, Bram Slagmolen, Warwick Bowen, Elanor Huntington, Jessica Lye, Cameron Fletcher and John Close have provided a constant supply of fresh ideas and fresh humour. This is always important for lab work which has the potential to be dull at times. I would specifically like to thank Dan Shaddock and Warwick Bowen for their help with the modulation-free SHG experiment and Elanor Huntington for assistance with the phase feedforward experiment. Most importantly, I am grateful for the timely arrival of Ulrik Andersen. Without him, the QND experiments would not have been so successful.

I would also like to extend my thanks to Friedrich K¨onig, Christine Silberhorn and Gerd Leuchs at the Universit¨at Erlangen. The time I spent in Erlangen added greatly to my appreciation of experimental quantum optics.

The support of the workshop staff has been tremendous. Often these are the people I turned to when, through my own lack of foresight, something had to be done yesterday. To Brett Brown, Paul MacNamara, Russell Koehne and Chris Woodland - thank you for all the last minute machining and repairing.

The administrative staff have also been invaluable. Whenever packages went missing or forms needed filling Zeta Hall, Jenny Willcoxson and Susan Maloney were always willing to lend a hand, or tell me what to do.

Outside work hours, there have been numerous friends, housemates and parents who have helped when the work was not going according to plan. Much of the time, they would not have been aware of the help they were giving, since it mostly consisted of putting up with some cranky moods which I could not adequately explain. For all those times -thanks and sorry.

Finally to Michelle: your companionship and love through it all has been important beyond measure.

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Abstract

The performance of optical measurement systems is ultimately limited by the quantum nature of light. In this thesis, two techniques for circumventing the standard quantum measurement limits are modelled and tested experimentally. These techniques are electro-optic control and the use of squeezed light.

An optical parametric amplifier is used to generate squeezing at 1064nm. The para-metric amplifier is pumped by the output of a second harmonic generation cavity, which in turn is pumped by a Nd:YAG laser. By using various frequency locking techniques, the quadrature phase of the squeezing is stabilised, therefore making our squeezed source suitable for long term measurements. The best recorded squeezing is 5.5dB (or 70%) below the standard quantum limit. The stability of our experiment makes it possible to perform a time domain measurement of photocurrent correlations due to squeezing. This technique allows direct visualisation of the quantum correlations caused by squeezed light. On the road to developing our squeezed source, methods of frequency locking optical cavities are investigated. In particular, the tilt locking method is tested on the second harmonic generation cavity used in the squeezing experiment. The standard method for locking this cavity involves the use of modulation sidebands, therefore leading to a noisy second harmonic wave. The modulation free tilt-locking method, which is based on spatial mode interference, is shown to be a reliable alternative.

In some cases, electro-optic control may be used to suppress quantum measurement noise. Electro-optic feedback is investigated as a method for suppressing radiation pressure noise in an optical cavity. Modelling shows that the ‘squashed’ light inside a feedback loop can reduce radiation pressure noise by a factor of two below the standard quantum limit. This result in then applied to a thermal noise detection system. The reduction in radiation pressure noise is shown to give improved thermal noise sensitivity, therefore proving that the modified noise properties of light inside a feedback loop can be used to reduce quantum measurement noise.

Another method of electro-optic control is electro-optic feedforward. This is also in-vestigated as a technique for manipulating quantum measurements. It is used to achieve noiseless amplification of a phase quadrature signal. The results clearly show that a feedforward loop is a phase sensitive amplifier that breaks the quantum limit for phase insensitive amplification. This experiment is the first demonstration of noiseless phase quadrature amplification.

Finally, feedforward is explored as a tool for improving the performance of quantum nondemolition measurements. Modelling shows that feedforward is an effective method of increasing signal transfer efficiency. Feedforward is also shown to work well in conjunction with meter squeezing. Together, meter squeezing and feedforward provide a comprehensive quantum nondemolition enhancement package. Using the squeezed light from our optical parametric amplifier, an experimental demonstration of the enhancement scheme is shown to achieve record signal transfer efficiency ofTs+Tm= 1.81.

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Contents

Declaration iii

Acknowledgements v

Abstract vii

Summary of symbols and acronyms xix

1 Introduction 1

2 The basics: theoretical quantum optics and experimental techniques 5

2.1 Quantum light . . . 5

2.1.1 States of light . . . 6

2.2 Sidebands and modulation of light . . . 8

2.2.1 Classical modulation of light . . . 8

2.2.2 Quantum sidebands . . . 10

2.3 Linearisation of the operators . . . 11

2.4 Multimode quantum optics: the noise spectra . . . 13

2.4.1 Power spectra and frequency domain uncertainty relations . . . 13

2.5 Experimental devices . . . 15

2.5.1 Beamsplitters . . . 15

2.5.2 Optical attenuation and measuring with a photodetector . . . 16

2.5.3 Balanced homodyne detection . . . 17

2.6 Open quantum systems and the quantum Langevin equation . . . 21

2.7 Optical cavities . . . 23

2.8 Cavity locking . . . 26

2.8.1 Modulation locking . . . 27

2.8.2 Spatial mode locking . . . 29

2.9 Locking homodyne detection . . . 30

2.10 Summary . . . 31

3 Phase signal amplification 33 3.1 Introduction . . . 33

3.2 Feedforward amplification . . . 36

3.3 Phase quadrature feedforward . . . 38

3.4 The experiment . . . 42

3.5 Conclusion . . . 45

4 Improving quantum nondemolition measurement 47 4.1 Introduction . . . 47

4.2 QND Criteria: an overview . . . 48

4.2.1 Signal transfer coefficients . . . 50

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x Contents

4.2.2 Conditional variance . . . 51

4.3 QND enhancement . . . 52

4.3.1 Pre-enhancement . . . 52

4.3.2 Post-enhancement with feedforward . . . 52

4.4 The sensitivity . . . 57

4.5 Conclusion . . . 61

5 Suppressing radiation pressure noise with feedback 63 5.1 Introduction . . . 63

5.2 Quantum feedback theory . . . 65

5.2.1 In-loop field properties . . . 67

5.2.2 Out-of-loop field properties . . . 67

5.3 Measurement of cavity detuning . . . 68

5.3.1 Fields exiting the cavity . . . 68

5.3.2 Cavity configurations . . . 69

5.3.3 Power scaling of the detuning signals . . . 71

5.4 Including feedback in the measurement model . . . 72

5.4.1 Feedback reduces photon number noise in a cavity . . . 74

5.4.2 What limits does this system beat? . . . 75

5.5 Detuning measurement via the cavity locking signal . . . 77

5.6 Thermal noise spectra . . . 80

5.7 Thermal noise detection models . . . 82

5.7.1 Impedance matched cavities with and without squashing. . . 84

5.7.2 Reflective single ended cavities . . . 84

5.8 Conclusions . . . 86

6 Second harmonic generation 87 6.1 Introduction . . . 87

6.2 Requirements of a frequency doubler . . . 87

6.2.1 Nonlinear susceptibility . . . 87

6.2.2 Phase matching . . . 88

6.3 Equations of cavity SHG . . . 91

6.4 Locking cavity SHG . . . 93

6.5 Experimental comparison of tilt and modulation locking . . . 96

6.5.1 Results . . . 97

6.5.2 Mechanical stability and optimisation . . . 101

6.6 Conclusion . . . 102

7 Squeezing from an optical parametric amplifier 103 7.1 Introduction . . . 103

7.2 OPA theory . . . 104

7.2.1 Classical parametric amplification . . . 104

7.2.2 Quantum behaviour of an OPA . . . 105

7.3 Experimental design . . . 107

7.3.1 The mode cleaner cavity . . . 107

7.3.2 The OPA cavity . . . 110

7.3.3 Locking the squeezing quadrature . . . 111

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Contents xi

7.4 Experimental results . . . 112

7.4.1 Regenerative gain . . . 112

7.4.2 Vacuum squeezing . . . 113

7.4.3 Locked squeezing results . . . 115

7.5 Correlation measurements . . . 116

7.5.1 A more detailed theory of correlation measurement . . . 118

7.6 Conclusions . . . 120

8 Quantum nondemolition experiments: realisation of enhancement 121 8.1 Introduction . . . 121

8.2 Experimental design . . . 122

8.3 QND results with the 50% beamsplitter . . . 122

8.3.1 The definition and measurement of the transfer coefficients. . . 123

8.3.2 Measurement of the conditional variance . . . 125

8.3.3 The addition of feedforward to the 50% beamsplitter . . . 126

8.3.4 Summary of the 50% beamsplitter results . . . 129

8.4 How ND is the QND? . . . 130

8.5 QND with a 92/8 beamsplitter . . . 130

8.5.1 Measurement of the transfer coefficients. . . 131

8.5.2 Conditional variance measurements . . . 133

8.5.3 Summary of the 92% beamsplitter results . . . 133

8.6 Comparison to other QND experiments . . . 135

8.7 Conclusion . . . 135

9 Conclusions and future possibilities 137

A Electronic noise 141

B Circuit designs 143

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List of Figures

1.1 Thesis structure . . . 2

2.1 The ‘ball on stick’ picture of quantum states . . . 8

2.2 Phase and amplitude modulation sidebands . . . 9

2.3 The sideband picture of quantum states . . . 11

2.4 A beamsplitter . . . 15

2.5 A model of an inefficient photodiode . . . 16

2.6 Self-homodyne detection . . . 18

2.7 Homodyne detection of the phase quadrature . . . 19

2.8 An empty optical cavity with various input and output fields. . . 23

2.9 Classical response of a cavity . . . 25

2.10 Noise filtering properties of an optical cavity . . . 26

2.11 General principle of cavity locking . . . 27

2.12 Pound-Drever-Hall locking . . . 28

2.13 Dither locking . . . 29

2.14 Spatial mode separation by a cavity . . . 29

2.15 Tilt locking signal arising from the interference of spatial modes . . . 30

3.1 Loss of signal-to-noise ratio due to attenuation . . . 33

3.2 Phase insensitive amplifier . . . 34

3.3 A general feedforward loop . . . 36

3.4 A phase quadrature feedforward loop . . . 38

3.5 Design of the phase feedforward experiment . . . 42

3.6 Input to the phase feedforward amplifier . . . 43

3.7 Output of the phase feedforward amplifier . . . 44

4.1 A general QND system . . . 49

4.2 Feedforward applied to a general QND system . . . 53

4.3 Effect of the gain phase in feedforward . . . 55

4.4 Feedforward applied to a squeezed light beamsplitter . . . 56

4.5 Feedforward applied to an OPA QND system . . . 57

4.6 A general QND scheme with feedforward and sensitivity correction . . . 58

4.7 Sensitivity as a function of squeezing . . . 59

4.8 QND performance with sensitivity correction . . . 61

5.1 An electro-optic feedback loop . . . 65

5.2 Cavity inside a feedback loop . . . 72

5.3 Effect of feedback on the intra-cavity noise as a function of the cavity mirror reflectivities . . . 75

5.4 Effect of laser power on thermal noise measurement . . . 82

5.5 Optimum laser power for different thermal noise measurement configurations 83 5.6 Thermal noise measurement with impedance matched cavities . . . 84

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xiv LIST OF FIGURES

5.7 Thermal noise measurement using a single ended cavity and ideal laser . . . 85

5.8 Thermal noise measurement using a single ended cavity and laser with clas-sical intensity noise . . . 85

6.1 Frequency doubling. . . 88

6.2 Three-wave mixing processes. . . 89

6.3 Harmonic conversion efficiency as a function of temperature and wavelength 90 6.4 The inputs and outputs of a quantum model for SHG and parametric am-plification . . . 91

6.5 Harmonic generation efficiency as a function of the input coupler reflectivity 93 6.6 Effect of modulation in a second harmonic generator . . . 94

6.7 Measurement of intensity modulation in the second harmonic due to cavity modulation . . . 95

6.8 Schematic of the second harmonic generation experiment . . . 96

6.9 Error signals for the second harmonic generation cavity . . . 97

6.10 Comparison of the second harmonic intensity noise spectra with tilt-locking and modulation locking . . . 98

6.11 Fluctuations in second harmonic power compared to the modulation and tilt error signals . . . 100

7.1 Regenerative gain of an optical parametric amplifier. . . 105

7.2 Detailed diagram of the OPA squeezing experiment. . . 108

7.3 Mode cleaning cavity . . . 109

7.4 The OPA crystal . . . 110

7.5 Measured regenerative gain of the OPA as a function of pump power. . . . 113

7.6 Scanned squeezed vacuum measurement . . . 114

7.7 Frequency spectrum of the squeezing . . . 115

7.8 Time stability of the squeezing . . . 116

7.9 Setup for measuring quantum correlations . . . 117

7.10 Correlation measurement results . . . 118

7.11 Intensity correlations between two entangled beams . . . 120

7.12 Phase correlations between two entangled beams . . . 120

8.1 Diagram of the QND experiment with a beamsplitter . . . 122

8.2 Signal transfer coefficient data for the 50% beamsplitter . . . 123

8.3 Meter transfer coefficient data for the 50% beamsplitter . . . 124

8.4 Squeezing used for the QND experiments . . . 126

8.5 Simplified diagram of the QND experiment with feedforward. . . 126

8.6 Response of the New Focus 4104 amplitude modulator . . . 127

8.7 Transfer coefficient data for the 50% beamsplitter with feedforward . . . 128

8.8 Conditional variance data for the 50% beamsplitter with feedforward . . . . 128

8.9 Comparison of theory with experimental results for the 50% beamsplitter . 129 8.10 Meter transfer coefficient data with the 92% beamsplitter . . . 131

8.11 Signal transfer coefficient data with the 92% beamsplitter . . . 131

8.12 Conditional variance measurement with the 92% beamsplitter and no feed-forward . . . 132

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LIST OF FIGURES xv

8.14 Conditional variance measurement with the 92% beamsplitter and

feedfor-ward used to correct signal gain . . . 132

8.15 Comparison of theory to experimental results for the 92% beamsplitter . . . 134

8.16 Summary of QND results compared with other published experiments . . . 135

A.1 Raw data for vacuum squeezing . . . 141

A.2 Vacuum squeezing data without electronic noise . . . 142

B.1 Low noise detector circuits . . . 143

B.2 Tilt detector circuit . . . 144

B.3 Piezo-electric servo circuit . . . 145

B.4 Laser servo circuit . . . 146

B.5 Temperature controller circuit: front panel . . . 147

B.6 Temperature controller circuit: PID . . . 148

B.7 Temperature controller circuit: Power supply . . . 148

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List of Tables

2.1 Comparison of some states of light you might meet in a quantum optics laboratory. . . 9 5.1 Summary of thermal noise detection models. . . 86 6.1 Stability parameters of the second harmonic generator locked using both

modulation and Tilt techniques. . . 99 8.1 QND results with a 50/50 beamsplitter. . . 129 8.2 QND results with a 92/8 beamsplitter. . . 134

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Summary of symbols and

acronyms

Term Definition

1: ∆ Standard deviation (square-root of variance, ∆2) 2: ∆ In chapter 5, cavity detuning.

α Classical cavity mode amplitude

ε Beamsplitter transmission

ηz Detector efficiency

κ Total decay rate of a cavity

κin Coupling rate of cavity input mirror

κout Coupling rate of cavity output mirror

κl Coupling rate of cavity loss

ω Frequency of sidebands relative to carrier Ω Frequency of carrier wave

Az (A†z) Annihilation (creation) operator of a travelling wave

¯

Az Classical amplitude of a travelling wave

a(a†) Annihilation (creation) operator of a cavity mode

Cs,m Correlation between sand m

cw Continuous wave

DC Direct current (Sometimes used more generally to mean low frequency)

Iz Photocurrent due toAz

OPA Optical parametric amplifier

n photon number operator QND Quantum nondemolition

QNL Quantum noise limit RBW Resolution bandwidth

RF Radio frequency

SHG Second harmonic generator SNR Signal-to-noise ratio

Ts Signal transfer coefficient

Tm Meter transfer coefficient

VBW Video bandwidth

Vz+ Amplitude noise spectrum of the fieldAz

Vz Phase noise spectrum of the fieldAz

Vs|m Conditional variance betweens andm Xz General quadrature ofAz

Xz+ Amplitude quadrature ofAz

Xz Phase quadrature of Az

z θ quadrature ofAz

References

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