Relative Intensity Squeezing by Four-Wave Mixing in Rubidium Martijn Jasperse Supervised by A/Prof Robert Scholten1 and Dr Lincoln Turner2 1 ARC Centre of Excellence for Coherent X-Ray Science, University of Melbourne, VIC 3010 2 School of Physics, Monash University, Clayton, VIC 3800 Submitted in total fulfilment of the requirements of the degree of Master of Philosophy School of Physics The University of Melbourne April 2010 Abstract This thesis is a theoretical and experimental study of the production of relative intensity squeezed light through four-wave mixing in a rubidium vapour. Relative intensity squeez- ing enhances measurement precision by using quantum-correlated “twin beams” to elimi- nate photon shot-noise. The “double-Λ” four-wave mixing process produces twin beams by stimulating a four-stage cyclical transition resulting in the emission of time-correlated “probe” and “conjugate” photons. Measuring and subtracting the corresponding beam in- tensities cancels the photon shot-noise, enabling measurements beyond the shot-noise limit. An ab initio analysis of the double-Λ scheme determined the experimental phase-matching conditions required to generate efficient mixing. Expressions for the expected level of squeezing were then derived. Deviations from perfect matching were considered and a spatial bandwidth for the mixing process was derived. This bandwidth was used to explain recent experiments obtaining multi-mode squeezed light from this system. Optical losses are an experimental inevitability that destroy quantum correlations by ran- domly ejecting photons. Expressions were derived to quantify the degradation of squeezing caused by losses. Sensitivity to unbalanced losses was exhibited and an optimum level of relative beam loss was observed. Near-resonant absorption within the vapour causes losses to compete with squeezing, so an interleaved gain/loss model was formulated to analyse the interplay of the two processes. A novel theoretical framework was developed and used to derive expressions for the level of squeezing produced in the presence of absorption. Four-wave mixing resonances were observed experimentally and the intensity noise spectra of the resulting beams were characterised. Gain dependence on beam power, cell tempera- ture and laser detuning was determined. Relative intensity squeezing of 3 dB was demon- strated and physical insight into the experimental results was gained through analysis with the theoretical model. Factors limiting the measured level of squeezing are discussed and design improvements proposed. i Declaration This is to certify that (i) the thesis comprises only my original work towards the MPhil, (ii) due acknowledgement has been made in the text to all other material used, (iii) the thesis is less than 50,000 words in length, exclusive of tables, maps, bibliographies and appendices Signature: Date: iii Acknowledgements A thesis is never a small undertaking: this work in particular very nearly gained the title of “the thesis that never was”. It was only through the support and encouragement of those around me that it become a reality. I am deeply thankful to those that have helped me along the way, and in particular to my supervisors Rob and Lincoln, whose support and encouragement pushed this project along. The guidance and eternal optimism provided by Rob kept me trying new things every time the exper- iment still wasn’t working. He was right; we got there in the end. It was Rob that introduced me to the wonderful world of atoms and lasers, and I will always be grateful to him for giving me a start in experimental science when I graduated my BSc as a wide-eyed mathematician. I would also like to acknowledge the financial support of the University of Melbourne, the Ernst and Grace Matthaei bequest and the ARC Centre of Excellent for Coherent X-Ray Science, who provided both for me personally and for the lab I worked in. This work would not have been possible otherwise. Thanks also to the mechanical and electrical workshops at the School of Physics for their respective assistance in the fabrication of the vapour cell heater and modification of the photodetector. “No man is an island”, so I’d like to thank everyone I worked with during my time at the University of Melbourne. In particular, the Atom Optics group and its honorary members: Simon Bell, Dave Sheludko, Sebastian Saliba, Mark Junker, Andy McCulloch, Lachlan Whitehead and Angela Torrance, as well as my fellow denizens in room 503. To the staff of The Potter, who cheerfully helped to convert my stipend into higher brain function, to the pod espresso machine, without which science on the 5th floor would not happen, and especially to Rob for keeping it stocked, I say thank you for facilitating my caffeine addiction. The original theory presented in this thesis would not have happened without the influence of my favourite organic suspension. Since it too has feelings, I’d also like to thank the Coherent 899 titanium-sapphire laser, whose unreli- ability could always be relied upon. The lessons you taught me in care and patience made me into the experimentalist I am today. For that matter, this work would not have been possible without the Verdi V-10, which would always go that one step further and give a full 10:5 W of blindingly brilliant green. Special thanks go to those brave souls who were cajoled into proofreading the drafts of this thesis: in particular to my father Jaap, who volunteered for the job, and to Lincoln, who tried to follow my maths. I’d also like to thank anyone who didn’t run away when I had a “simple” question, who humoured me when I said “I understand how it works now!”, or distracted me with an interesting question about their own work. And to my darling Катюша, who put up with me through this write-up, feigned interest at all the right moments, and kept me sane when I needed it most: words are not enough. But thank you all the same. Last, but nowhere near least, thank you to my incredible family for supporting me through all my crazy endeavours. Especially to my wonderful mother Patries, who has always helped me through the hard times. This is for you guys. v Table of Contents 1 Introduction 1 2 Four-wave mixing 5 2.1 Review of double–Λ four-wave mixing . 5 2.2 Operator model formalism . 7 2.3 Relative intensity squeezing . 8 2.3.1 Amplified single-beam intensity fluctuations . 9 2.3.2 Quadrature squeezing and the Heisenberg relation . 10 2.4 Geometric phase-matching . 11 2.4.1 Phase matching from first principles . 13 2.4.2 Applications to multi-mode squeezing . 15 2.5 Summary . 16 3 Optical losses 17 3.1 Review of the optical loss formalism . 17 3.2 Squeezing degradation caused by losses . 18 3.2.1 Beam-splitter noise-figure . 19 3.2.2 Influence of losses on relative intensity squeezing . 19 3.3 An interleaved gain/loss model . 23 3.3.1 Discrete stage calculation . 25 3.3.2 Infinitesimal expansion and analytic result . 27 3.3.3 Analytic expression for squeezing . 30 3.3.4 Impact of interleaved losses . 31 3.4 Summary . 33 4 Experimental design 35 4.1 Pump laser generation . 35 4.2 Probe laser generation . 37 4.2.1 Acousto-optic modulator . 37 4.2.2 Tunable rf source . 40 4.3 Beam alignment . 40 4.3.1 Beam waists . 41 4.4 Vapour cell . 42 4.4.1 Presence of isotopic impurity . 42 4.4.2 Vapour cell heater . 43 4.5 Relative intensity measurement . 44 4.5.1 Photodiode quantum efficiency . 45 4.5.2 Detector linearity . 46 4.5.3 Detection stage losses . 46 4.6 Noise spectrum measurement . 47 vii 4.6.1 Power spectrum analysis . 47 4.6.2 Background subtraction . 48 4.7 Summary . 49 5 Results and Analysis 51 5.1 Laser noise calibration . 51 5.2 Standard quantum limit measurement . 51 5.3 Classical probe noise analysis . 53 5.4 Four-wave mixing characterisation . 55 5.5 Relative intensity noise spectra . 56 5.6 Observations of noise reduction and squeezing . 61 6 Conclusions 65 References 67 A Statistical derivations 71 A.1 Statistical identities . 71 A.2 Coherent state variances . 71 A.3 Intensity shot noise derivation . 72 A.4 Statistical loss model . 73 B Distributed gain/loss model supplementals 75 B.1 Numerical algorithm code listing . 75 B.2 Parameter estimation . 76 viii List of Figures 1.1 Differential measurement with a “noisy” beam to remove classical intensity fluctuations. 1 1.2 A graphical representation coherent and squeezed states. 2 1.3 Relative intensity squeezing to improve measurement precision. 3 2.1 Hyperfine energy-level structure of lowest accessible states of 85Rb. 6 2.2 Proposed “double–Λ” four-wave mixing process. 6 2.3 Uncertainty ellipses for coherent and squeezed states. 11 2.4 Geometry of four-wave mixing configuration. 12 2.5 Schematic of production and detection of relative intensity squeezed images. 15 3.1 Quantum operator view of a beam-splitter. 17 3.2 Two-beam loss model. 19 3.3 Predicted squeezing in presence of losses for varying gain. 21 3.4 Predicted squeezing in presence of losses for varying losses. 22 3.5 Predicted squeezing for varying gain and loss. 22 3.6 Interleaved gain and loss model. 23 3.7 Normalised probe power predicted by gain/loss model. 31 3.8 Degree of squeezing predicted by distributed gain/loss model. 32 4.1 Schematic experiment to measure squeezing. 35 4.2 Retro-reflected saturation absorption spectroscopy configuration. 36 4.3 Energy levels and detunings of “double–Λ” system. 37 4.4 “Cat’s eye” double-pass AOM configuration.