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 squeezing enhances measurement precision by using quantum-correlated “twin beams” to eliminate 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 intensities 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 randomly 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 temperature and laser detuning was determined. Relative intensity squeezing of 3 dB was demonstrated 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.

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 experiment 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.

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...... 38 4.5 Dependence of AOM efficiency on input rf power, beam polarisation and driving frequency. 39 4.6 Combination and isolation of beams using polarising beam-splitters...... 40 4.7 Absorption profiles indicating presence of isotopic impurity...... 42 4.8 Design of custom vapour cell heater...... 43 4.9 Differential noise measurement of probe and conjugate beam...... 44 4.10 Detector output voltage as a function of input beam power...... 46 4.11 Schematic of signal processing in the Rhode & Schwarz FSP7 spectrum analyser...... 48 4.12 Measured electrical noise floor power spectra...... 49

5.1 Measured electrical noise power of Ti:S laser compared to diﬀerential balanced beams. . . 52 5.2 Measured white-noise level for diﬀerential detection of beam with varying optical power. . 52 5.3 Measured noise on the probe beam...... 54 5.4 Power spectrum of rf signal generated by synthesizer...... 54 5.5 Intensity noise power for double-passed AOM beam...... 55 5.6 Peak four-wave mixing gain dependence on 2–photon detuning and pump power...... 56 5.7 Dependence of peak gain and detuning on pump power...... 56 5.8 Temperature dependence of peak gain...... 57

ix 5.9 Measured four-wave mixing noise power spectra...... 57 5.10 Approximate frequency response curve of PDB150A photodetector...... 58 5.11 Measured noise spectra compared to expected white-noise levels including gain roll off. . . 58 5.12 Measured noise spectra for probe and conjugate beams corrected for gain roll off...... 59 5.13 Measured gain and noise power scanning across four-wave mixing resonance...... 60 5.14 Measured four-wave mixing gains and associated relative intensity squeezing...... 62 5.15 Intrinsic gain and probe transmission coefficients obtained from gain profiles...... 63 5.16 Comparison of measured relative intensity noise to distributed gain/loss model prediction. 63

x Introduction 1

Experimentation at the frontiers of physics is continually pushing the bounds of measurement precision, in an ongoing quest to gain a better understanding of the natural world. Modern laser-based optical experiments are now routinely able to probe the “shot-noise limit” [1], where the experiments are limited by the intensity noise generated by quantum mechanical ﬂuctuations [2, 3]. Ultra-high sensitivity experiments such as gravitational wave detection [4, 5], precision measurement [6, 7], spectroscopy [8], and faint object imaging [9] will all beneﬁt greatly by circumventing the shot-noise limit. This goal can be achieved with an exotic type of light known as “squeezed states”.

High-precision experiments that probe atomic interactions through absorption of light are typically conducted in a differential measurement configuration (Figure 1.1). This arrangement reduces the influence of intensity fluctuations in the laser beam, which act to obscure experimental results. In this configuration a beam-splitter divides the incident beam in two; one beam is used to perform the experiment and the other to measure the time-varying intensity of the beam. By electronically subtracting the two signals, the intensity fluctuations of the laser are removed and the measurement precision is increased.

Detectors

Laser Experiment BS + -

Figure 1.1: Diﬀerential measurement with a “noisy” beam. The 50/50 beam-splitter (BS) produces two beams that allow the intensity ﬂuctuations to be electronically removed from the experimental measurement.

However, the two beams generated by the beam-splitter are not identical, resulting in some residual noise being observed. Each photon incident on the beam-splitter can only be observed by one detector, so each the photon must be deflected into one of the two paths, creating a small intensity increase in the arm the photon was deflected into (relative to the other arm). As the photons are deflected independently, the photon shot-noise on the two beams is uncorrelated. This shot-noise is measured by the detectors and produces a residual noise level known as the standard quantum limit (SQL).

Squeezed states of light are non-classical states of the electromagnetic field where the fluctuations in one measurement quadrature (typically either amplitude or phase) are smaller than those of a coherent state (such as that produced by a laser). These reductions occur at the expense of increasing the fluctuations in the other quadrature (Figure 1.2) in accordance with the Heisenberg uncertainty principle [10, 11]. However, experiments based on exploiting the squeezed quadrature will contain reduced fluctuations, resulting in lower noise and increased measurement precision.

As squeezed states are non-classical, they can only be generated from coherent (classical) laser light through a non-linear interaction. Although there are many diﬀerent approaches to the generation of squeezed light [12], the most common approaches are based on interactions with a non-linear medium

1 A. Coherent state B. Squeezed state

Figure 1.2: A graphical representation of the expected outcome of measuring the conjugate operators Xˆ and Yˆ, with associated uncertainties h∆Xˆi and h∆Yˆi. In a coherent state (A) the uncertainties are equal, but in an Xˆ-squeezed state (B) the uncertainty in Xˆ has been reduced at the expense of increasing the uncertainty in Yˆ.

such as a non-linear crystal or near-resonant atomic vapour. Coupling to an atomic medium is a convenient approach, as atomic resonances can be manipulated by other external fields (e.g. through other near-resonant fields or the Zeeman effect), leading to highly controllable and tuneable systems. Fur- thermore, many applications of squeezed light involve subsequent probing of atomic states, for which near-resonant light is essential.

One process that generates strong non-linearities in a medium is four-wave mixing, which is the interaction between four modes of the electromagnetic ﬁeld [13]. This process was used to experimentally generate the ﬁrst squeezed states of light in 1985 [14]. Only a modest 0.3 dB of noise reduction was observed in their system, as competing scattering processes limited the purity of the output state [15]. Since then, another technique based on optical parametric oscillators (OPOs) has become the conven- tional way to generate squeezed light [16], with the strongest squeezing recorded to date (−11.5±0.1 dB in January, 2010) produced by such a system [17].

However, these systems have significant intrinsic limitations. Optical parametric oscillators are based on parametric down conversion, which converts a single photon into two photons with equal total energy [13]. This is an inherently inefficient process and requires extremely high intensities to operate effectively. Such a system is usually coupled to an external cavity to enhance the non-linearity, increasing the conversion efficiency at the expense of mechanically coupling the system to the environment. This makes the system highly susceptible to environmental fluctuations, with mechanical instability introducing spurious noise at low Fourier frequencies.

Furthermore, as cavities only couple strongly to a single mode of the electromagnetic ﬁeld, squeezed light is usually only obtained for pure Gaussian (TEM00) laser beams. In practice, this means extra “mode cleaning” cavities are necessary to prevent noise build-up from the higher order (“unsqueezed”) modes. An important consequence of this is that producing multi-mode squeezed light for imaging applications requires each TEM component mode to be squeezed separately and then recombined [18].

These limitations have spurred exploration of alternate techniques, with recent work by the Lett group yielding a promising solution: relative intensity squeezing by four-wave mixing in atomic vapour [19]. Instead of creating a beam that has reduced noise in one quadrature, the Lett scheme produces two “twin beams” with their quadratures squeezed relative to each other, giving rise to quantum-correlated shot- noise. Measuring and subtracting the correlated beam intensities produces a signal with noise reduced below the SQL (Figure 1.3).

2 The system proposed by the Lett group is based on a cycle of far-detuned transitions in the “double-Λ” conﬁguration in a hot rubidium vapour. The non-linearity generated by this system is strong enough for no cavity enhancement to be required, making it substantially diﬀerent from earlier “twin beam” experiments based on OPOs [20, 21]. It therefore also inherently supports multi-mode beams and does not strongly couple to the environment. These features have recently been shown to extend the squeezed light generated in this way, both into the low frequency domain [22] and to the direct squeezing of masked images [23]: with up to 8.8 dB of squeezing measured experimentally.

Experiment Relative Laser intensity + - squeezer Sub-shot noise measurement Quantum-correlated beams

Figure 1.3: Relative intensity squeezing produces two “twin beams” which are have correlated shot noise. The shot noise measured in one arm cancels out some of the shot noise in the other, producing a sub-shot noise (squeezed) measurement.

What’s new: This thesis studies the production of relative intensity squeezed light through four-wave mixing in rubidium. Previously published theoretical and experimental works are expanded upon and extended to characterise the mixing process and predict the degree of squeezing generated. A novel theoretical framework capable of describing competing squeezing and absorption processes is developed, and used to quantify the degradation of squeezing by losses inherent in the mixing process. Four-wave mixing resonances are observed experimentally, and their noise spectra are characterised. Noise reduction of 3 dB below the shot-noise limit is demonstrated; in the ﬁrst independent conﬁrmation of squeezing in this system. Finally, experimental factors that limit measurement precision are considered and methods to circumvent them proposed.

Four-wave mixing 2

This chapter explores the process of the four-wave mixing in the “double–Λ” conﬁguration, and derives a consistent theoretical approach to modelling the system. A ﬁrst-principles analysis demonstrates how the system is capable of producing relative intensity squeezed light; a discussion of the experimental (“phase-matching”) conditions under which squeezed light can be created is also presented. Based on this, an explanation and evaluation of the capability of the system to produce multi-mode squeezed light is included and used to explain recently published experimental results.

2.1 Review of double–Λ four-wave mixing The process of four-wave mixing in the “double–Λ” energy level conﬁguration is described in this section, and a description of how it produces quantum correlated (squeezed) beams provided.

Four-wave mixing is a non-linear interaction between light and matter that permits the transfer of energy and momentum between four modes of the electric ﬁeld via interaction with a medium. Typically, a high intensity pump laser is used to induce a strong non-linear polarisation in the medium, which then causes the medium to interact with other ﬁeld modes provided certain “phase-matching” conditions are met (as discussed in §2.4).

The strength of non-linear interactions are quantiﬁed by the associated non-linear susceptibility tensors [24], with order one less than the number of waves involved in the interaction. In the absence of optical pumping, atomic vapours are homogeneous and isotropic, so the susceptibility tensor reduces to a single scalar value. For four-wave mixing interactions this is the third-order susceptibility χ(3). In an atomic vapour the susceptibility is strongest near resonance [25], so the most eﬃcient mixing will occur when the interacting modes are close to resonance with the medium. Rubidium-85 was chosen as the atomic species for this experiment because of its simple energy-level structure (Figure 2.1) and the applicability of near-resonant squeezed light in many rubidium based quantum- and atom-optics experiments [16].

Rubidium has two well-separated fine-structure transitions; from the ground state 5S 1/2 to the excited states 5P1/2 and 5P3/2, known as the D1 and D2 lines. The D1 line at 795 nm was chosen as the mixing transition for the simple hyperfine structure of the 5P1/2 excited state and evidence that the related phenomenon of coherent population trapping [26] is more efficient on this transition [27]. In principle, this scheme could operate with either isotope or transition.

The proposed four-wave mixing process exploits the hyperfine splitting of the ground state of rubidium to perform a four-step mixing cycle based on the “Λ” energy-level configuration (Figure 2.2) as first suggested by McCormick et al. [19]. The process is as follows: a high-intensity “pump” laser drives an off-resonant transition from the F =2 ground state to the virtual state |3i, from which a low-intensity “probe” beam stimulates a transition to the F = 3 ground state through stimulated Stokes (Raman) scattering. The pump then drives a second off-resonant transition to the virtual level |4i, resulting in spontaneous emission of a “conjugate” photon via anti-Stokes scattering. This returns the atom to the F = 2 ground state and closes the cycle. Coherence builds up between the ground states through electromagnetically-induced transparency, in a similar way to coherent population trapping [28]. Hence

5 121 MHz

63.4 MHz

29.3 MHz

362 MHz

line line 780 nm 795 nm

3.036 GHz

Figure 2.1: Hyperﬁne energy-level structure of lowest accessible states of 85Rb.

Probe Pump line (795 nm)

Pump

Conjugate

Figure 2.2: Proposed “double–Λ” four-wave mixing process. The hyperﬁne splitting of the 5P1/2 state is negligible compared to the laser detunings.

the process occurs as a coherent four-step cycle that converts two pump photons into a probe and a conjugate photon.

The output probe and conjugate photons are emitted in the same cyclical sequence and will therefore be inherently time-correlated, adding time-correlated shot-noise to the probe and conjugate beams. This enables the noise fluctuations in one beam (the “reference”) to be used to electronically subtract the shot-noise fluctuations from the other (the “signal” beam) and permits measurements to be made below the shot-noise limit. The probe and conjugate beams are therefore said to be relative intensity squeezed; that is, squeezed with respect to each other. The degree of noise reduction obtained in this way will now be quantified.

6 2.2 Operator model formalism Using a ﬁrst-principles approach, a quantum operator model of the process of four-wave mixing is developed in this section. This model provides ab initio predictions of the degree of squeezing produced by this system as well as a theoretical framework for further analysis.

The relative intensity squeezing experiment involves individual measurement of the intensities of the probe and conjugate beams, so the model needs to analyse the evolution of the relevant modes of the electromagnetic ﬁeld as they propagate through the medium and undergo mixing. Assuming that the energy lost to the environment during the mixing interaction (e.g. through momentum/energy transfer to the atoms) is negligible, the parametric approximation can be made and the mixing process can be modelled directly as energy transfer between the relevant EM ﬁeld modes. A simple three-mode Fock state model may therefore be used to describe the state of the pump, probe and conjugate beams.

The four-wave mixing process (Figure 2.2) is a cycle that annihilates a pump photon, creates a probe photon, annihilates another pump photon and then creates a conjugate photon. Labelling the annihilation operators of the probe, conjugate and pump bya ˆ, bˆ andc ˆ respectively and denoting the strength of the interaction by the (real) parameter β, the interaction Hamiltonian for this process is of the form1

Hˆ = i~ β bˆ†câˆ†cˆ + h.c. where h.c. denotes the Hermitian conjugate of the prior terms. Because the pump beam has very high power, it can be assumed that the intensity of the pump beam is not significantly changed by the mixing process. The initial and final states of the pump are then effectively the same coherent state |αpumpi.

This is known as the “undepleted pump” approximation, and asc ˆ |αpumpi = αpump |αpumpi it allows the substitutionc ˆ → αpump to be made, ˆ 2 ˆ† † H → i~ β αpump b aˆ + h.c.

2 Note that αpump ∝ Ipump, so the mixing strength is proportional to the pump intensity. Furthermore, the phase of αpump directly relates to the orientation of the measurement quadratures (see §A.2), so it may be choosen as real without loss of generality. Writing ξ as the proportionality constant, the Hamiltonian becomes Hˆ = i~ ξ bˆ†aˆ† + h.c. (2.1) The time-evolution operator corresponding to this Hamiltonian is

ˆ † † Uˆ (t) = e−iH t/~ = e−ξ(ˆabˆ−bˆ aˆ )t.

Supposing that the mixing interaction occurs over a timescale τ, the similarity transformation that describes the mixing process is † † Sˆ ≡ Uˆ (τ) = es(bˆ aˆ −aˆbˆ), (2.2) where s = ξτ is the “squeezing parameter” that quantifies the degree of mixing [29]. This operator is known in modern literature as the “two-mode squeezing operator” because it squeezes the modesa ˆ and 1 (s(â†)2−s∗aˆ2) bˆ together. It is often derived by analogy with the “single-mode” squeeze operator Sˆ 1 = e 2 , which was first discovered by Stoler in 1970 [30] and shown by Yuen [31] to produce states which had reduced quadrature fluctuations. The modern terminology of “squeezed states” is due to Hollen- horst [32].

The time-evolution of the ﬁelds within the medium is more easily observed by computing the Heisenberg picture time-evolution of the associated annihilation operators, daˆ i h i dbˆ i h i = Hˆ , aˆ = ξbˆ†, = Hˆ , bˆ = ξaˆ†. (2.3) dt ~ dt ~ 1Assuming perfect phase-matching, a requirement which will be revisited in §2.4.

7 These equations can be uncoupled by taking the second derivative and solving for the individual operators, d2aˆ dbˆ† = ξ = ξ2aˆ ⇒ aˆ(t) = cosh (ξt) aˆ + sinh (ξt) bˆ†. dt2 dt Supposing that the mixing interaction occurs over a timescale τ, the annihilation operators describing the state of the modes after mixing are [22]

aˆ → cosh(s)a ˆ + sinh(s) bˆ†, bˆ → cosh(s) bˆ + sinh(s)a ˆ†. (2.4)

Note that this process can also be conveniently expressed as a matrix transformation of the form        aˆ  cosh s sinh s  aˆ    →     , (2.5) bˆ† sinh s cosh s bˆ† and that cascading the squeezing transformation yields

† † † † † † s2(bˆ aˆ −aˆbˆ) s1(bˆ aˆ −aˆbˆ) (s1+s2)(bˆ aˆ −aˆbˆ) Sˆ (s2) Sˆ (s1) = e e = e = Sˆ (s1 + s2).

Hence the squeezing transform is “additive” in the parameter s.

These transformations enable the state of the beams after mixing to be calculated. The conjugate beam is unseeded (“vacuum fed”), so initially bˆ is the annihilation operator for the vacuum and hNˆ bi = 0. Conversely, the probe beam has power of order 100 µW, allowing the “bright beam” approximation hNˆ ai 1 to be made.

Applying the squeezing transformation to the number operators, the beams after mixing have

2 † 2 † hNˆ ai → cosh (s)haˆ aî + sinh (s) ' Ghaˆ aî, 2 † 2 † hNˆ bi → sinh (s)haˆ aî + sinh (s) ' (G − 1)haˆ aî, (2.6) † 2 † ⇒ hNˆ a + Nˆ ai → cosh(2s)haˆ aî + 2 sinh (s) ' (2G − 1)haˆ aî, † hNˆ a − Nˆ bi →haˆ aî, where G ≡ cosh2(s) is the intensity gain of the probe beam. Note that the probe and conjugate intensities are unbalanced, as although the mixing process adds the same power to both beams, the incident power of the probe beam is also transmitted. The expectation value hNˆ a − Nˆ bi is therefore non-zero and a net DC offset will be observed in the intensity difference signal.

These results will now be extended to consider ﬂuctuations in both the individual and relative intensities of the probe and conjugate beams.

2.3 Relative intensity squeezing As discussed above, every four-wave mixing cycle results in the production of a probe and conjugate photon, which are temporally correlated. It might therefore be expected that the intensity diﬀerence should manifest sub-shot noise statistics due to the shot-noise correlations between the two beams. This section quantiﬁes the noise reduction by calculating the relative intensity variance after mixing has occurred.

The number difference operator Nˆ a − Nˆ b describes the relative intensity fluctuations, and can be shown to be invariant under the squeezing transformation (2.2). Hence the relative intensity fluctuations after the squeezing process are exactly the same as the fluctuations before squeezing occurred,

† † † † Var Nˆ a − Nˆ b → Var aˆ aˆ − bˆ bˆ = Var aˆ aˆ = haˆ aˆi. SQZ

8 However, as more photons have been added to the beams, the total optical power in the experiment has been increased without increasing the relative noise on the beams. The standard quantum limit (SQL) is then the shot-noise which would be expected for a diﬀerential measurement made with equivalent total optical power (i.e. equal to the sum of probe and conjugate powers), namely

† Var Nˆ a − Nˆ b ≡ hNˆ a + Nˆ bi ' (2G − 1)haˆ aî. SQL Provided the input beams were originally shot-noise limited, this enables sub-shot noise measurements to be made. This is quantified by the “noise figure” or “degree of squeezing”, which is the ratio of the variance of the squeezed beams to the variance at the standard quantum limit, namely ˆ ˆ Var Na − Nb haˆ†aî 1 NF = SQZ ' = . (2.7) − h † i − Var Nˆ a − Nˆ b (2G 1) aˆ aˆ 2G 1 SQL

(dB) Typically the noise figure is quoted as the noise in decibels relative to the SQL, NF = 10 log10(NF). Since G > 1 for all real values of the squeezing parameter s, this system demonstrates noise reduction below the standard quantum limit. As the degree of mixing is increased, more correlated photons are generated in each beam, and this is reflected by further reduction in the noise figure. This implies that increasing the strength of the mixing improves the noise characteristics of the system without limit: a result that will be revisited later.

2.3.1 Amplified single-beam intensity fluctuations Although the relative intensity between probe and conjugate beams manifests reduced fluctuations compared to the SQL, the amplification process that generated these correlations is phase-sensitive and makes the individual beams very noisy. These individual fluctuations will be quantified in this section.

The variance in the number operator of one beam alone transforms under squeezing as

† † Var Nˆ a →Var (cosh(s)â + sinh(s)bˆ)(cosh(s)â + sinh(s)bˆ ) =Var cosh2(s)a ˆ†aˆ + cosh(s) sinh(s) (â†bˆ† + bâˆ) + sinh2(s) bˆbˆ† .

Expanding this variance into a covariance sum (§A.1) and eliminating relevant vacuum expectations, the only terms that do not vanish are

4 † 2 2 † † Var Nˆ a = cosh (s) Var aˆ aˆ + cosh (s) sinh (s) CoVar bˆ aˆ, aˆ bˆ .

The covariance term is easily evaluated as CoVar bˆ aˆ, aˆ†bˆ† = hbââˆ†bˆ†i − hbâîhaˆ†bˆ†i = haâˆ†bˆbˆ†i = h(1 + aˆ†aˆ)(1 + bˆ†bˆ)i ' haˆ†aî, which simplifies the number variance to

4 † 2 2 † Var Nˆ a = cosh (s) Var aˆ aˆ + cosh (s) sinh (s) haˆ aî = cosh2(s) cosh2(s) + sinh2(s) haˆ†aî = G (2G − 1) haˆ†aî. (2.8)

Since the power on the probe is hNˆ ai = GhNˆ 0i, the noise ﬁgure for the probe beam alone is

NF(probe) = 2G − 1. (2.9)

This corresponds to a linear increase in the noise on the probe beam as gain is increased, with a similar result applying to the conjugate beam. However, since the mixing process imparts these increased

9 ﬂuctuations to both the probe and conjugate beams, they cancel out in the relative intensity signal to produce an overall noise reduction in accordance with the earlier result.

The expressions (2.7) and (2.9) demonstrate that relative intensity fluctuations are reduced at the expense of increasing fluctuations in the individual beams for G > 1 (and vice versa for G < 1). This is similar to the “which-way” Heisenberg uncertainty interpretation of the Mach–Zehnder interferometer, where fluctuations manifest primarily in either the individual paths (beam) or in the output superposition (difference) state.

2.3.2 Quadrature squeezing and the Heisenberg relation Squeezed light is traditionally studied by analysing the “quadratures” of the electric ﬁeld. This section brieﬂy derives the quadrature operators, relates them to the Heisenberg uncertainty relation, demonstrates how they are transformed by four-wave mixing and gives a physical interpretation of the results.

The electric ﬁeld operator corresponding to a single mode with wave-vector ~k and frequency ω at a given point in space and time is −i(~k·~r−ωt+ϕ) † i(~k·~r−ωt+ϕ) Eˆ(~r, t) = E~0 e aˆ + e aˆ , (2.10)

where E~0 contains the (complex) amplitude and polarisation of the field [33]. Separating the phase into real and imaginary parts, the electric field may be written as √ n o Eˆ = 2 E~0 cos ~k · ~r − ωt + ϕ Xˆ + sin ~k · ~r − ωt + ϕ Yˆ , (2.11) where Xˆ and Yˆ are the Hermitian operators Xˆ = √1 aˆ† + aˆ and Yˆ = √i aˆ† − aˆ , (2.12) 2 2 known as the “amplitude” and “phase” quadrature operators of the field. These operators are analogous to the position and momentum operators from the one-dimensional quantum harmonic oscillator, and obey the commutation relation h i h i nh i h io ˆ ˆ i † † i † † X, Y = 2 aˆ + aˆ, aˆ − aˆ = 2 aˆ , −aˆ + aˆ, aˆ = i.

Hence the two quadratures cannot be measured simultaneously and the Heisenberg uncertainty principle [10, 11] applies: ˆ ˆ 1 ˆ ˆ 1 h∆Xih∆Yi ≥ 2 |h[X, Y]i| = 2 . (2.13) Since the time-evolution of the field is represented by a linear combination of the expectation values of these two operators, they provide a convenient representation of the electric field and its fluctuations. A given state can be visualised in the “X–Y plane” as an ellipse whose position corresponds to the expectation value and size to the variance of each quadrature. The Heisenberg relation (2.13) therefore defines the minimum possible area of the ellipse. A quadrature-squeezed state manifests reduced fluctuations in that quadrature, and is represented by an ellipse that is “squeezed flat” in that direction (Figure 2.3). Quadratures can be measured experimentally in a process called homodyne detection, which has been studied extensively elsewhere (an in-depth discussion may be found in [16]).

The quadratures above can be generalised to correspond to measurement on an arbitrary axis rotated at angle θ to the Xˆ-axis as Qˆ(θ) = √1 eiθaˆ† + e−iθaˆ . (2.14) 2

10 A. B. C.

Figure 2.3: Uncertainty ellipses for a coherent state (A), amplitude-squeezed state (B) and phase-squeezed state (C). Each ellipse’s position represents its expectation value and its size the uncertainty in the respective quadratures.

The four-wave mixing squeezing process discussed above involves two beams and hence two modes. Using the subscripts “a” and “b” to denote the quadratures of the probe beam and conjugate respectively, the action of the squeezing transformation (2.4) is

Qˆ a(θ) → Qˆ a(θ) cosh(s) + Qˆ b(−θ) sinh(s), (2.15) with an analogous result for the conjugate. Note that the two quadratures have become linked, corresponding to the cross-correlations generated by the squeezing process. The sum/diﬀerence combinations of the amplitude and phase quadratures are then

±s ∓s Xâ ± Xˆb → e (Xâ ± Xˆb) and Yâ ± Yˆb → e (Yâ ± Yˆb), (2.16)

As the variances go as the square of the variable, these quadratures are (anti-)squeezed by a factor of 2s in variance (as quoted in [34]). The amplitude difference (sum) and phase sum (difference) quadratures form pairs of Heisenberg-uncertainty conjugate variables, so the amplitude difference and phase sum quadratures manifest reduced fluctuations at the expense of increased fluctuations in amplitude sum and phase difference.

The physical origin of this amplitude diﬀerence squeezing arises is the same as the relative intensity correlations discussed above. On the other hand, squeezing in phase sum arises from preferential phase- sensitive ampliﬁcation in accordance with the phase-matching condition ωprobe + ωconj = 2ωpump as described further below.

Direct measurement of these relative quadratures is a difficult task as the pump, probe and conjugate beams all have different frequencies. It is possible to perform balanced homodyne detection [16] in this system by constructing two adjacent squeezing configurations in the same vapour cell, and using one of these systems to generate the necessary “bright” local oscillator beams [23]. In this way, the amplitude difference and phase sum quadratures have been shown to exhibit −4.3 dB of squeezing, with anti- squeezing of 12 dB in the orthogonal quadratures. Note that the comparatively low degree of squeezing and high anti-squeezing arises from the increased losses, imperfect mode matching and phase instability inherent in the complexity of this approach. However, these measurements unambiguously demonstrate relative quadrature squeezing from this system, above and beyond classical intensity correlations.

2.4 Geometric phase-matching The phase-matching conditions are the experimental conditions that must be satisﬁed for eﬃcient four- wave mixing to occur, and require the waves to remain in phase throughout the mixing process. As discussed below in §2.4.1, they amounts to requiring conservation of energy and momentum across a

11 Conjugate Probe

4WM Pump

Figure 2.4: Geometry of four-wave mixing conﬁguration showing wave-vectors and angles. single four-wave mixing cycle. In this section, the phase-matching conditions will be analysed and the experimental circumstances that satisfy them determined. Deviations from perfect phase-matching will then be considered and a spatial bandwidth derived. The implications of such a bandwidth to multi-mode squeezing and imaging will then be discussed.

The conservation of energy and momentum equations for the double–Λ mixing process are

2ωpump − ωprobe − ωconj = 0 and 2~kpump − ~kprobe − ~kconj = 0, (2.17) where the negative signs arises from the direction of each transition (absorption or emission) in the process.

As the conjugate photons are produced by the mixing process, their energy and momenta are dictated by the phase-matching condition (i.e. only modes satisfying phase-matching will be populated). It is therefore important to determine the experimental conditions under which the phase-matching conditions can be met simultaneously and a conjugate beam produced. The above conditions indicate that the conjugate is generated with

ωconj = 2ωpump − ωprobe and ~kconj = 2~kpump − ~kprobe. (2.18)

The incidence angle between the pump and probe beams in the medium is deﬁned to be θ and the resulting angle between conjugate and pump is labelled ϕ (Figure 2.4). Taking components, the phase- matching condition can be rewritten as

kconj sin ϕ = (~kconj)y = 2(~kpump)y − (~kprobe)y = kprobe sin θ

kconj cos ϕ = (~kconj)x = 2(~kpump)x − (~kprobe)x = 2kpump − kprobe cos θ, where ki = |~ki|. Solving these equations simultaneously for the conjugate,

2 ~ 2 ~ 2 2 2 kconj = (kconj)x + (kconj)y = 4kpump + kprobe − 4kpumpkprobe cos θ kprobe and sin ϕ = sin θ. kconj

The magnitudes of the wave-vectors are related to the angular frequencies by the real-valued refractive index n through ki = ωini/c. As the perturbation to the refractive index in an atomic medium is small

(n ' 1) and the detuning between the conjugate and probe beams is also small (ωprobe − ωconj ≈ 6 GHz), the pump-conjugate angle is given by

ωprobenprobe sin ϕ = sin θ ' sin θ ⇒ ϕ ' θ. (2.19) ωconjnconj

12 Similarly rearranging for the pump-probe angle,

2 2 2 2 2 2 2 2 2 4kpump + kprobe − kconj 4npumpωpump + nprobeωprobe − nconjωconj cos(θ) = = 4kpump kprobe 4npumpnprobeωpumpωprobe 2 2 2 2 2 2 4npumpωpump + nprobeωprobe − nconj(2ωpump − ωprobe) = applying (2.18) 4npumpnprobeωpumpωprobe 2 2 2 2 2 2 2 4(npump − nconj)ωpump + (nprobe − nconj)ωprobe nconj = + 4npumpnprobeωpumpωprobe npumpnprobe 2 2 2 2 4(npump − nconj) + (nprobe − nconj) ' + 1 since ω ' ω and n ' 1. (2.20) 4 pump probe

θ2 Performing a Taylor expansion to second order in the angle, cos(θ) ' 1 − 2 , and the pump-probe angle is given by 2 1 2 2 2 θ ' 2 (5nconj − 4npump − nprobe). The real-valued refractive-index of the medium can be related to the real-valued electric susceptibility 2 χe by n ' 1 + χe [35]. The angle can then be written as

2 1 θ ≈ 2 (5χconj − 4χpump − χprobe). (2.21)

Note that in a vacuum χ = 0 and the beams must be co-propagating (θ = ϕ = 0) to satisfy phase- matching. In a medium such as a near-resonant atomic vapour however, chromatic dispersion occurs and the susceptibilities are both non-zero and frequency dependent. This means the terms do not cancel and a non-zero optimum phase-matching angle is obtained. Such an angle is necessary to spatially separate the beams after mixing has occurred and then perform intensity subtraction.

It is diﬃcult to tune the value of θ as overlap between the pump and probe within the cell must be maintained and the probe must remain incident on the detector to measure the intensity gain. However, it has been observed experimentally that a phase-matching angle of θ ≈ 7 ± 2 mrad permits strong four-wave mixing in the regime of interest [36], so a pump-probe angle of θ = 7 mrad was used in the experiment.

2.4.1 Phase matching from ﬁrst principles In this section, the phase-matching conditions will be derived from ﬁrst principles by augmenting the system Hamiltonian with the phase terms neglected from the earlier analysis (§2.2). This is demonstrated to result in a geometry-dependent spatial bandwidth, leading to “smoothing out” of the above phase-matching condition.

Using a similar approach to [37], each beam in the mixing process may be approximated as a plane wave of ﬁxed wave-vector and frequency. The electric ﬁeld of the ith beam can be expressed classically as

i(~ki·~r−ωit) E~i(~r, t) = E0,i ~i e , where E0,i is the complex amplitude and ~i the complex polarisation vector. Introducing this phase term into the interaction Hamiltonian (2.1) through canonical quantisation gives

~ ~ ~ ~ Hˆ ∝ e−i(kconj·~r−ωct)bˆ† ei(kpump·~r−ωpumpt)cˆ e−i(kprobe·~r−ωprobet)aˆ† ei(kpump·~r−ωpumpt)cˆ + h.c.

~ ~ ~ = ei(2kpump−kprobe−kconj)·~re−i(2ωpump−ωprobe−ωc) bˆ†cˆaˆ†cˆ + h.c.

= ei(∆~k ·~r − ∆ω t)bˆ†cˆaˆ†cˆ + h.c. (2.22)

13 where the parameters ∆~k ≡ 2~kpump − ~kprobe − ~kconj and ∆ω ≡ 2ωpump − ωprobe − ωconj are respectively called the phase and energy “mismatches”.

The total Hamiltonian for the system is obtained from the Hamiltonian density above by integrating over the entire interaction volume V . In the four-wave mixing geometry, the interaction volume corresponds to the overlap region between the pump and probe beams. The system Hamiltonian is therefore Z Z ~ H = Hˆ d3~r ∝ ei(∆k ·~r − ∆ω t)bˆ†cˆaˆ†cˆ d3~r + h.c. V ZV = e−i ∆ω t bˆ†cˆaˆ†cˆ ei∆~k ·~r d3~r + h.c. V Note that this integral amounts to a Fourier transform of the integration volume in the phase-mismatch variable ∆~k. The Hamiltonian may therefore be expressed as

† † H ∝ F V ∆~k · bˆ cˆaˆ cˆ + h.c.

For an approximately rectangular interaction region, the volume V can be represented by the rectangle function with side-lengths lx, ly and lz:  ! ! ! 0 for |x| > 1 x y z  2 V (x, y, z) = rect rect rect , where rect(x) =  lx ly lz  1 1 for |x| ≤ 2 .

The Fourier transform2 of the rectangle function is the sinc function,  !  x k L sin(πx)/πx for x , 0 F rect k = L sinc , where sinc(x) =  L 2π 1 for x = 0, so the Hamiltonian becomes         l (∆~k)  ly(∆~k)y  l (∆~k)  H ∝ χ(3)l l l sinc  x x  sinc   sinc  z z  bˆ†cˆaˆ†cˆ + h.c.  e x y z  2π   2π   2π 

Hence for a given set of wave-vectors, the phase-mismatch ∆~k acts to reduce the interaction strength ξ by a geometry-dependent amount. The mixing eﬃciency is therefore decreased for deviations from the ideally phase-matched case ∆~k = 0. However, the sinc function still permits strong gain over a range of ~k vectors, deﬁning a “spatial bandwidth” that increases as the interaction volume is reduced.

Note that decreasing the interaction volume also reduces the number of interacting atoms, and hence the peak gain. This is seen in the above expression through the explicit appearance of the lengths as weighting factors.

An analogous bandwidth also arises for the frequency mismatch. Including the phase term ei∆ω t in the Hamiltonian modiﬁes the Heisenberg equation of motion describing the squeezing process (2.3) to be

∂aˆ ∂2aˆ ∂aˆ Hˆ = ξ e−i∆ω tbˆ†aˆ† + h.c. ⇒ = ξe−i∆ωtbˆ† ⇒ = −i∆ω ξe−i∆ωtbˆ† + ξ2aˆ = −i∆ω + ξ2aˆ. ∂t ∂t2 ∂t This is a simple second order homogeneous linear diﬀerential equation, with solution

1 q ! 1 q ! − i ω t − i ω t † 2 ∆ 2 1 2 2 ∆ 2 1 2 ˆ aˆ(t) = e cosh ξ − 4 (∆ω) t aˆ + e sinh ξ − 4 (∆ω) t b . (2.23)

The phase pre-factor is cancelled out in the number operatora ˆ†aˆ, so for an interaction timescale τ this q 2 1 2 solution corresponds to squeezing with s = s0 − 4 (∆ω τ) , where s0 = ξτ is the degree of squeezing

2Using the non-unitary Fourier transform convention with normalization on the inverse transform.

14 for a perfectly phase-matched system. Hence there is also an associated frequency bandwidth for the process, which is inversely related to the interaction timescale τ.

Intuitively these two matching bandwidths interpreted as manifestations of the position-momentum and energy-time Heisenberg uncertainty relations, where conﬁning the photons to the interaction volume and timescale results in a momentum and energy spread.

2.4.2 Applications to multi-mode squeezing Recent experimental results have demonstrated both the correlation between corresponding sub-regions of the squeezed beams produced by this system [36] and the production of relative intensity squeezed images [23]. However, no physical explanation of the gain bandwidth or limitations in sub-region correlations have previously been provided. This section will extend the phase-matching analysis to provide just such an explanation.

The gain bandwidth derived above demonstrates that this system is intrinsically capable of simultaneously squeezing multiple spatial modes of the electric ﬁeld, allowing the production of squeezed images. This is in contrast to squeezing techniques which rely on cavity enhancement to increase the non-linearity (such as OPOs) at the expense of only interacting with the spatial mode that couples to the cavity. These schemes typically also require mode-cleaning cavities for single-mode optimisation, and complex experimental systems to operate in multi-mode conﬁgurations [38]. Twin beam techniques on the other hand, operate without a cavity and are therefore intrinsically capable of multi-mode interaction, as elaborated below.

An arbitrary mode of the EM-field can be expressed as a sum of plane-waves with varying amplitude over some range of k-vectors, and each mode can then undergo mixing in the medium. Provided these modes are within the gain bandwidth, conjugate photons will be generated in corresponding modes of the conjugate beam. Hence a multi-mode “copy” of the probe that manifests strongly correlated intensity fluctuations will be produced. Furthermore, the different spatial modes (k-vectors) are orthogonal and will be squeezed independently. So the probe and conjugate images will manifest squeezing over corresponding sub-regions. It should also be noted that the factor negative one between probe and conjugate wave-vectors in the phase-matching condition means that the conjugate image will be rotated 180◦ to the probe.

Squeezed image duplication could therefore be achieved by applying a spatial ﬁlter to a mask-encoded probe beam to strip out higher order modal information; this ensures it ﬁts within the gain bandwidth (Figure 2.5). The aperture width can be calibrated to clip the focussed beam and block modes extending beyond the gain bandwidth, to ensure the whole beam is duplicated in the conjugate.

Pump Translatable – S.A. Irises

Probe Rb Cell Conjugate

Mask Spatial Filter

Figure 2.5: Schematic for production and detection of squeezed images. The spatial ﬁlter eliminates higher order modes of the probe image and the translatable irises allow investigation of sub-region correlations.

15 In principle CCD cameras could be used to detect the images produced in this way. A spatial light modulator (SLM) could replace the mask to dynamically generate diﬀerent spatial distributions, enabling high-resolution multi-mode squeezed imaging of an object to be performed.

However, the phase mismatch also limits how small the correlated subregions are. Rewriting (2.18) to include the mismatch term, the conjugate momentum is given by

~kconj = 2~kpump − ~kprobe − ∆~k, where ∆~k has some spread associated with it. A given incident probe wave-vector ~k will therefore produce some span of conjugate vectors out, resulting in a superposition of modes at the detector. If the detection region is smaller than this matching bandwidth, “cross-talk” between the modes of the probe and conjugate will occur, giving rise to reduced correlations. This is exactly the behaviour observed experimentally [36]. However, the theoretical discussion presented above suggests that by controlling the interaction volume, smaller scale correlations can be obtained at the expense of reducing the gain.

2.5 Summary In this chapter, the double–Λ four-wave mixing system of McCormick et al [19] was analysed from ﬁrst principles to provide an understanding of relative intensity squeezing through four-wave mixing. The theoretical expressions for the squeezing operator and associated gain and noise ﬁgures found in the literature [30, 19], insight into the physical process was gained through their derivation. A theoretical framework has been created and will next be extended to include degradation of squeezing caused by optical losses.

Insight into the squeezing process was gained through derivation of theoretical expressions for the squeezing operator and associated gain and noise ﬁgures, which are stated without derivation in the literature [22]. This work provides a theoretical framework to be extended in the next chapter, to include the degradation of squeezing caused by optical losses.

The process and consequences of phase-matching were discussed from ﬁrst principles, and extended to tread deviations from ideal matching. The ﬁnite size of the interaction volume was shown to produce a spatial bandwidth, which permitted squeezing to occur over a range of incident ~k vectors but with reduced gain. A similar frequency bandwidth was also derived. The ability of the four-wave mixing system to produce squeezed images was considered; a limit on the size of the smallest correlated sub-regions is seen as a direct consequence of the spatial bandwidth, explaining previously published experimental work [23].

16 Optical losses 3

Optical losses are an unavoidable experimental reality. Whether arising from absorption, window reflec- tions, imperfect mirrors or finite detector efficiencies, some degree of attenuation will always be present. In a simple picture, optical losses can be thought of as randomly ejecting photons. If the beams consist of highly time-correlated photons, then this process will destroy some of the correlations and introduce relative fluctuations between the beams. It is therefore important to understand the loss mechanisms and quantify how these affect squeezing measurements.

This chapter studies the effects of different types of loss (namely absorption and detection efficiency) on the four-wave mixing squeezing system, and obtains new analytic expressions for the degree of squeezing capable of being produced by this system. The analytic results augment and extend a previously published numerical model [22], and was re-derived from first principles due to the scant details present in the published work. In particular, a novel method of evaluating relative intensity squeezing using matrices is presented.

3.1 Review of the optical loss formalism This section reviews the quantum operator description of optical losses, leading to a derivation of the beam-splitter transformation and the variance in the intensity of the output beam. This discussion is in- tended to complement standard treatments (such as [33]) which are traditionally based on an analysis of conservation of energy. Using a quantum operator approach provides more insight for further extensions of the model.

Optical losses can be interpreted as equivalent to passing a beam through an ideal (non-polarising) beam-splitter with an empty (“vacuum-fed”) port that causes some portion of the incident beam to be ejected. Modelling the action of the beam-splitter upon the state of the incident beam will therefore quantify the eﬀect of losses.

Consider an ideal beam-splitter with eﬃciency η (that transmits a fraction η of the incident beam and reﬂects a fraction 1 − η), combining two input modesa ˆ and bˆ to produce an output modec ˆ (Figure 3.1).

Input A Output C

Input B

Figure 3.1: Quantum operator view of a beam-splitter with transmission coeﬃcient η: input modes aˆ and bˆ mix to produce output mode cˆ. Optical loss on input A is modelled by leaving input B empty.

The beam-splitter is modelled in the Heisenberg picture by computing the time-evolution of the input beam annihilation operatora ˆ to obtain the output operatorc ˆ. In a loose sense, the beam-splitter can be

17 thought of as removing part of mode A and substituting in part of mode B, as described by the operator bˆ†aˆ. Denoting the interaction strength by ζ ∈ R (which will be related to η later), the interaction Hamiltonian is of the form

† † HˆBS = ~ζ aˆ bˆ + bˆ aˆ , (3.1) wherea ˆ and bˆ are the annihilation operators of the two input modes. The time-evolution of this Hamil- tonian in the Heisenberg picture leads to1

daˆ i h i h i = Hˆ , aˆ = iζ aˆ†bˆ + bˆ†aˆ, aˆ = −iζbˆ dt ~ d2aˆ dbˆ ⇒ = −iζ = −ζ2aˆ ⇒ aˆ(t) = cos (ζt) aˆ + sin (ζt) bˆ. (3.2) dt2 dt

Supposing that the interaction occurs over a timescale τ, the eﬃciency of the splitter may be deﬁned as η = cos2 (ζτ) to recover the “standard” quantum beam-splitter input/output mode relation [33],

√ p cˆ = η aˆ + 1 − η bˆ. (3.3)

If the input port B is left empty, then bˆ is an annihilation operator corresponding to the vacuum field and the output becomes a superposition of the input and vacuum. As first observed by Caves [2], this causes the input beam to interfere with the vacuum to produce “vacuum fluctuations” in the output. Even though the vacuum field contains no photons on average (hbˆ†bî = 0), its non-zero commutator adds a term to the output beam variance which describes these fluctuations.

The relevant expectation values of the number operator are

† † hNˆ i = hcˆ cî = ηhaˆ aî = ηhNˆ 0i (3.4) p 2 hNˆ 2i = hcˆ†cˆcˆ†cî = h ηaˆ†aˆ + η(1 − η)(â†bˆ + bˆ†aˆ) + (1 − η)bˆ†bˆ i = η2h(â†aˆ)2i + η(1 − η)haˆ†bˆbˆ†aˆ + bˆ†aâˆ†bî = η2h(â†aˆ)2i + η(1 − η)haˆ†aî. (3.5)

Since hNi = ηhN0i, the transmission eﬃciency of the beam-splitter is indeed η and the variance in the output is

2 2 2 Var Nˆ = hNˆ i − hNˆ i = η Var Nˆ 0 + η(1 − η)hNˆ 0i. (3.6)

The terms in the expression above correspond to the original variance in the fraction of the beam that was transmitted (as per the identity (A.8) in the Appendix) and to extra ﬂuctuations arising from the random ejection of photons. This second contribution is often called “vacuum noise” (or “partition noise”) due to this interpretation as arising from “mixing” the beam with the vacuum on the beam-splitter.

3.2 Squeezing degradation caused by losses The eﬀects of optical losses upon the degree of squeezed light will now be explored. Using the beam- splitter relation derived previously, original expressions for the degree of squeezing after losses are obtained in both one- and two-beam scenarios. In particular, the role of unbalanced losses after four- wave mixing is studied and an optimum level of relative loss observed and quantiﬁed.

1Note thata ˆ and bˆ are operators that represent diﬀerent beams and therefore commute with each other.

18 3.2.1 Beam-splitter noise-ﬁgure Firstly, note that a beam that is initially shot-noise limited will remain shot-noise limited after losses, since 2 Var Nˆ 0 = hNˆ 0i ⇒ Var Nˆ = η hNˆ 0i + η(1 − η)hNˆ 0i = ηhNˆ 0i = hNˆ i. (3.7)

In other cases however, the variance of the output beam is altered. This is most easily identified by considering the noise figure, defined earlier as ratio of intensity variance to standard quantum limit. The effect of introducing losses with transmission factor η is

2 ! Var (N) η Var (N0) + η(1 − η)hN0i Var (N0) NF(BS) = = = 1 + η − 1 . (3.8) hNi ηhN0i hN0i This can be rewritten as NF(BS) − 1 = η (NF0 − 1), which can be seen to → 0 as η → 0. The utility of this result is to show that losses act to bring the variance of the output beam closer to the standard quantum limit (NF → 1), independent of whether the beam was initially squeezed or noisy.

Physical insight into this result can be obtained by thinking of the beam-splitter statistically as randomly ejecting photons from the incident beam (§A.4). When squeezed light is fed into the splitter, this destroys the quantum correlations and acts to increase the noise level. Conversely, random ejection of photons from noisy (that is, highly bunched) light reduces the degree of bunching and decreases the noise level towards the shot-noise limit.

3.2.2 Influence of losses on relative intensity squeezing In modelling the four-wave mixing squeezing system however, it is the relative intensity fluctuations which are of interest. The analysis above will therefore now be extended to quantify how losses cause decorrelation between the probe and conjugate beams, resulting in an original analytic expression for the level of relative intensity fluctuations in a four-wave mixing system followed by losses.

Squeezing by four-wave mixing relies on quantum correlations between the probe and conjugate beams to circumvent the shot-noise limit. Individually, the two beams manifest large intensity fluctuations (§2.3.1), so optical losses on either beam will decorrelate them, exposing their individual fluctuations and significantly increasing the relative intensity noise measured at the detector.

Consider a squeezing experiment with degree of squeezing s (§2.2) followed by optical losses, which result in a fraction ηa and ηb of the probe and conjugate beams being transmitted respectively (Figure 3.2).

Probe + Four-wave mixing Conjugate −

Squeezing Optical Losses Ideal Detection

Figure 3.2: Two-beam loss model describing squeezing followed by optical losses due to imperfect detection eﬃciency and optical components.

The relative intensity ﬂuctuations measured at the detector are Var Nˆ a − Nˆ b = Var Nˆ a − 2 CoVar Nˆ a, Nˆ b + Var Nˆ b .

19 The number variance of each beam due to losses is given by (3.6), and since the vacuum contributions do not correlate with the other modes, the covariance transforms as CoVar Nˆ a, Nˆ b → ηaηbCoVar Nˆ a, Nˆ b . (3.9)

Hence the relative intensity fluctuations after losses are related to the fluctuations before losses by ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ ˆ ˆ Var Na − Nb → ηaVar Na + ηa(1 − ηa)hNai + ηbVar Nb + ηb(1 − ηb)hNbi − 2ηaηb CoVar Na, Nb . (3.10) Defining the probe gain G = cosh2(s) as earlier, the variances in the individual beams (§2.3.1) are related to the incident probe beam (Nˆ 0) by

2 Var Nˆ a = G Var Nˆ 0 + G(G − 1)hNˆ 0i (3.11) 2 Var Nˆ b = (G − 1) Var Nˆ 0 + G(G − 1)hNˆ 0i.

The covariance of the squeezing process can be obtained by exploiting the invariance of the number diﬀerence operator under the squeezing transformation (§2.2), which implies that Var Nˆ 0 = Var Nˆ a − Nˆ b = Var Nˆ a + Var Nˆ b − 2 CoVar Nˆ a, Nˆ b n o ˆ ˆ 1 ˆ ˆ ˆ ⇒ CoVar Na, Nb = 2 Var N0 − Var Na − Var Nb .

Combining these results, the relative intensity variance of squeezed beams after losses is

n o2 Var Nˆ a − Nˆ b = ηaG − ηb (G − 1) Var Nˆ 0 + n o ηaG(ηaG − 2ηa + 1) + ηb (G − 1)(ηbG − ηb + 1) − 2ηaηbG (G − 1) hNˆ 0i. (3.12) For a shot-noise limited input beam Var Nˆ 0 = hNˆ 0i, this simpliﬁes to Var Nˆ a − Nˆ b 2 2 = 2G (ηa − ηb) + Gηa(1 − 2ηa) + (G − 1)ηb(1 − 2ηb) + 2Gηb(2ηa − ηb). (3.13) hNˆ 0i The applicable standard quantum limit for equal total optical power is Var Nˆ a − Nˆ b SQL hNˆ ai + hNˆ bi = = Gηa + (G − 1)ηb. (3.14) hNˆ 0i hNˆ 0i

The general result (3.13) has three special cases of interest: G = 1 ⇒ Var Nˆ a − Nˆ b = ηahNˆ 0i ⇒ NF = 1 (3.15) n o ηb = 0 ⇒ Var Nˆ a − Nˆ b = Gηa 1 + 2(G − 1)ηa hNˆ 0i ⇒ NF = 1 + 2(G − 1)ηa (3.16) n o η η = η ⇒ Var Nˆ − Nˆ = η (2G − 1)(1 − η) + η hNˆ i ⇒ NF = (1 − η) + . (3.17) a b a b 0 2G − 1 In the absence of squeezing (G = 1), no conjugate is generated and the probe remains shot-noise limited.

When the conjugate is completed absorbed (ηb = 0), lossy noise on the probe is observed, and the result of combining (2.9) with (3.8) is obtained. Finally, when the losses on the two beams are equal, the noise figure contains the lossless squeezing factor (2.7) weighted by the transmission efficiency, with an extra term for the introduced vacuum fluctuations.

Although the relative ﬂuctuations are increased by large losses, the system is particularly sensitive to unbalanced losses (ηa , ηb) due to the squared gain term. Hence increasing the gain will not necessarily reduce the measured noise in a given system (Figure 3.3).

20 –3

–4 Noise figure (dB) Noise figure

–5

–6

–7

–8

2 4 6 8 10 12 14 Gain, G

Figure 3.3: Predicted relative intensity squeezing in presence of losses, as gain is increased. Note that unbalanced losses cause noise to increase with gain above an optimum point. Solid lines show ηa < ηb and dashed lines ηa ≥ ηb.

It should be noted that however the general result is not symmetric in the transmission coeﬃcients so the minimum noise level does not arise for equal losses (Figure 3.4). Minimising the noise level for a given gain G and conjugate transmission ηb, the “optimum” probe transmission is

1 η = η − , a b 4(G − 1) where the factor (G − 1) arises due to the fraction of the transmitted probe beam containing the incident shot-noise which is not cancelled by the conjugate. The relative intensity ﬂuctuations are therefore ˆ ˆ Var Na − Nb G = 2ηb(1 − ηb)(G − 1) + ηb − , hNˆ 0i 4(G − 1) and the corresponding level of squeezing is

16(G − 1)2η2 − G NF = 1 − b . 8(G − 1)(2G − 1)ηb − 2G

Setting the transmission coeﬃcients in this way can actually reduce the relative intensity noise below the level corresponding to ideal detection (Figure 3.5).

This counter-intuitive result can be interpreted as reducing the influence of the noise present on the incident probe beam. Squeezing is achieved by four-wave mixing through adding correlated photons to the probe and conjugate beam, allowing the fluctuations to be subtracted. However, the incident probe beam carries shot noise which reaches the detector and remains uncancelled. Introducing a small degree of loss on the probe reduces this noise contribution, at the expense of destroying some of the correlations introduced by squeezing. Balancing these two effects creates an level of “optimal loss” at which the measured degree of squeezing is improved by a fraction of a dB.

21 Relative intensity noise dependence on losses (G=5) 2

0.2 0.4 0.6 0.8 1.0 Noise figure (dB) Noise figure

-2

-4

-6

-8

-10

Figure 3.4: Predicted relative intensity squeezing varying degree of optical losses (solid lines) compared to equal losses on both beams (dotted) and optimised loss ηb for a given ηa (dashed).

–4 B)

–6 Noise figure (d

–8

–10

Lossless result:

2 3 4 5 6 7 Gain

Figure 3.5: Predicted relative intensity squeezing for system with equal optical losses (dashed lines), and optimised probe loss (solid lines). Note that it is possible to improve over the lossless result (ηa = ηb = 1) by introducing a small degree of loss on the probe.

22 3.3 An interleaved gain/loss model The analysis presented thus far has considered the effect of losses occurring after mixing had been completed, which arise from optical component losses or imperfect detection efficiencies. However, the beams involved in the mixing process are necessarily near-resonant and are also absorbed as they travel through the vapour. The gain from four-wave mixing therefore competes with losses from absorption within the cell, with each process affecting the relative noise level. This section derives and details an interleaved gain/loss model capable of describing this phenomenon.

Four-wave mixing is a resonance-enhanced phenomenon, so the mixing eﬃciency (and hence squeezing) will be improved by bringing the laser beams closer to resonance and increasing the non-linear susceptibility. However, as detuning is reduced the Doppler-broadened absorption proﬁle increases the losses on the probe and conjugate beams and the degree of squeezing degraded.

Conversely, as the downward probe transition in the four-wave mixing process can be vacuum-seeded (becoming a spontaneous emission transition), the mixing process can squeeze vacuum input states. The vacuum modes introduced by losses will therefore become squeezed as they propagate through the cell and the correlations they degraded will be partially regenerated. The gain and loss processes are therefore intrinsically intertwined and must be modelled simultaneously.

The dynamics of these processes are investigated by splitting the passage of light through the cell into inﬁnitesimal regions of gain and loss (Figure 3.6). Each gain stage corresponds to squeezing by an incremental amount s and each loss stage resulting in partial transmission with incremental coeﬃcients ta and tb on the probe and conjugate respectively. By interleaving many such stages of gain and loss, the continuum dynamics of the four-wave mixing system are recovered.

SQZ ... SQZ ...

Gain Loss Gain Loss

Figure 3.6: Interleaved stages of consecutive gain and loss modelling competing mixing and absorption processes. The state of system after n stages of the model is represented by the operator pair (aˆn, bˆn).

Discretised models for competing gain/loss processes of this type were first proposed by Loudon [39] for single-mode propagation through a lossy medium. They were first applied by Caves and Crouch [40] to model single-mode squeezing losses in a parametric amplifier and by Jeffers, Imoto and Loudon [41] to model optical fibre-amplifiers. This was then extended to a two-mode numerical model for the double-Λ four-wave mixing system by McCormick et al [22]. Although the numerical model considered losses on the probe beam only, it was successfully employed to analyse experimental results, and optimise the pump detuning when keeping other parameters fixed.

To gain greater insight into the interaction between the competing processes, we now derive a fully analytical model for the relative intensity squeezing produced by this system. By including absorption losses on both beams as well as post-cell optical losses, the complete behaviour of the system can be analysed. We also demonstrate in §5.6 that the analytic expression can be easily inverted to extract the model parameters and ﬁt experimental measurements - a process that is both complex and computation- ally expensive in a multi-stage numerical model.

23 As previously, the evolution of the probe and conjugate modes are modelled via their ladder operators. The result of passing through a stage of squeezing and absorption corresponds to cascading the transformations of gain (2.4) and loss (3.3), giving q ˆ† 2 aˆn+1 = ta cosh(s)a ˆn + sinh(s) bn + 1 − ta cˆn+1, (3.18) q ˆ† ˆ† 2 ˆ† bn+1 = tb sinh(s)a ˆn + cosh(s) bn + 1 − tb dn+1,

th wherec ˆn and dˆn are the vacuum operators corresponding to the losses introduced at the n stage of the model. It should be noted that these vacuum modes are all orthogonal, as they correspond to the vacuum injected at diﬀerent points along the beam path (i.e. are diﬀerent spatial modes). Hence the vacuum noise contributions add incoherently as expected.

Since the squeezing parameter is additive and losses are multiplicative when combining transformations, the incremental coeﬃcients are related to the overall transmission coeﬃcients Ta and Tb and overall squeeze factor S by 1/2N 1/2N ta = Ta , tb = Tb and s = S/N, where N is the total number of discrete stages comprising the model.

In the absence of squeezing (S = 0) the transformation collapses directly into the optical loss result

(3.6), whereas in the absence of losses (Ta = Tb = 1) it becomes the squeezing transform (2.4).

Writing the distributed gain/loss transformation (3.18) in matrix form,

       p 2  aˆn  ta cosh s ta sinh s aˆn  1 − ta cˆn+1   +1 =     +  q  (3.19) ˆ†    ˆ†  2 ˆ†  bn+1 tb sinh s tb cosh s bn 1 − tb dn+1

This transformation can be applied recursively to eliminate the unknown intermediate operatorsa ˆi and ˆ† ˆ† bi , and obtain the state of the probe and conjugate beams (ˆaN and bN ) at the end of the vapour cell. Performing this recursion gives

     p 2    aˆ N  aˆ N−   1 − ta cˆN  ta cosh s ta sinh s   = A  1 +  q  where A =   ˆ†  ˆ†   2 ˆ†    bN bN−1 1 − tb dN tb sinh s tb cosh s

   p 2   p 2  aˆ N−   1 − ta cˆN   1 − ta cˆN−1  = A2  2 +  q  + A  q  ˆ†   2 ˆ†   2 ˆ†  bN−2 1 − tb dN 1 − tb dN−1

   p 2   p 2   p 2  aˆ N−   1 − ta cˆN   1 − ta cˆN−1   1 − ta cˆN−2  = A3  3 +  q  + A  q  + A2  q  ˆ†   2 ˆ†   2 ˆ†   2 ˆ†  bN−3 1 − tb dN 1 − tb dN−1 1 − tb dN−2 ···

   p 2   p 2   p 2  aˆ   1 − ta cˆN   1 − ta cˆ2   1 − ta cˆ1  = AN  0 +  q  + ··· + AN−2  q  + AN−1  q  (3.20) ˆ†  2 ˆ†   2 ˆ†  2 ˆ† b0 1 − tb dN 1 − tb d2 1 − tb d1

ˆ† The operatorsa ˆ N and bN may therefore be expressed as a sum of the incident probe and conjugate ˆ† ˆ† operators (â0 ≡ aˆ and b0 ≡ b ) and the 2N vacuum mode operators which model the losses on each ˆ† beam (thec î and di , for 1 ≤ i < N). ˆ† Using xi and yi to denote coefficients appearing ina ˆ N and bN respectively, this result can also be expressed as X ˆ† ˆ† ˆ† aˆ N = x1 aˆ + x2 b + x3 cˆ1 + x4 d1 + x5 cˆ2 + x6 d2 + · · · ≡ xi zî i (3.21) † † † † X bˆ = y1 aˆ + y2 bˆ + y3 cˆ1 + y4 dˆ + y5 cˆ2 + y6 dˆ + · · · ≡ yi zî, N 1 2 i

24 where the notational convenience operatorsz ˆi are deﬁned as

ˆ† ˆ† zˆ1 = aˆ, zˆ2 = b , zˆ2i+1 = cˆi andz ˆ2i+2 = di (1 ≤ i ≤ N).

This is a slightly unusual definition as when i is evenz î is a creation operator, but when i is oddz î is an annihilation operator. However, this choice simplifies the algebra in the following analysis.

3.3.1 Discrete stage calculation ˆ† Assuming that the coeﬃcients in the expansions ofa ˆ N and bN are known, an ab initio calculation of the relative intensity noise measured at the detector can be obtained. This section presents an original derivation leading to a novel expression for the variance (and hence degree of squeezing) in terms of the expansion coeﬃcients.

The operatorsz î are annihilation (creation) operators when i is odd (even), and each describes a distinct orthogonal mode. They therefore satisfy   hNii when i is odd [ˆz†, zˆ ] = (−1)i δ and hzˆ†zˆ i =  , (3.22) i j i j i i  hNii + 1 when i is even where hNii is the average number of photons occupying the modez î. All modes except the probe are † vacuum fed, so the number operator is only non-zero for the probe (i.e. hNˆ 1i = haˆ aî and hNˆ ii = 0 for i > 1).

After N stages of gain and loss, the relative intensity observed at the detector (deﬁned as Zˆ) is

ˆ † ˆ† ˆ † ˆ ˆ† Z ≡ aˆ N aˆ N − bN bN = aˆ N aˆ N − bN bN + 1, (3.23) ˆ ˆ† where the commutation relation [bN , bN ] = 1 follows from (3.22). Substituting in the mode expansion (3.21), the number diﬀerence may be written as X ˆ † Z = (xi x j − yiy j)z ˆi zˆ j + 1. (3.24) i, j

Thez î are mutually orthogonal, so they commute with each other and the variance sum relation (A.10) applies to give X ˆ † † Var Z = (xi x j − yiy j)(xk xl − ykyl) CoVar zî zˆ j, zˆkzˆl . (3.25) i, j,k,l Each covariance term in the sum is defined as

† † † † † † CoVar zî zˆ j, zˆkzˆl = hzî zˆ jzˆkzˆli − hzî zˆ jihzˆkzˆli.

Only one of the operators (ˆz1) corresponds to a non-vacuum mode, so the only possible non-zero expectations are those terms with matching creation and annihilation operators. There are therefore only three cases that need to be considered:

(A): i = j = k = l, (B): i = j , k = l, (C): i = l , j = k.

Case (A): When all the indices are the same, the covariance collapses directly into the number variance and † † † ˆ CoVar zî zî, zî zî = Var zî zî = Var Ni . Case (B): If i = j , k = l, then the covariance is between the number operators corresponding to different orthogonal (independent) modes, which is necessarily zero, † † ˆ ˆ CoVar zî zî, zˆkzˆk = CoVar Ni, Nk = 0.

25 Case (C): In the remaining case i = l , j = k,

† † † † † † CoVar zî zˆk, zˆkzî = hzî zˆkzˆkzîi − hzî zˆkihzˆkzîi.

† † As i , k, the corresponding operators commute, hzî zˆki = hzî ihzˆki, and as at least one of these modes † † must correspond to a vacuum state, hzî zˆki = hzî ihzˆki = 0. Hence the covariance is

† † † † † † ⇒ CoVar zî zˆk, zˆkzî = hzî zîzˆkzˆki = hzî zîihzˆkzˆki.

Combining these results, the variance in the number diﬀerence (3.25) simpliﬁes to

X 2 X 2 ˆ 2 2 ˆ † † Var Z = xi − yi Var Ni + xi x j − yiy j hzˆi zˆiihzˆ jzˆ j i. i i, j, i, j

All modes excepta ˆ are vacuum fed, so hNii = 0 for i > 1. In the bright beam approximation the probe † † † † beam has hN1i = haˆ aî 1, so haâˆ i = haˆ aî + 1 ≈ haˆ aî. Taking the higher order contributions from the commutators of the vacuum modes as negligible compared to the probe,

2 X X 2 ˆ 2 2 ˆ 2 † † † † Var Z ' x1 − y1 Var N1 + (xi x1 − yiy1) hzî zîihzˆ1zˆ1i + x1 x j − y1y j hzˆ1zˆ1ihzˆ jzˆ j i i>1 j>1 2 X X 2 2 ˆ 2 † † 2 † † ' x1 − y1 Var N1 + (x1 xi − y1yi) hzî zîihaˆ aî + (x1 xi − y1yi) haˆ aîhzîzî i i>1 i>1 2 22 † X 2 † † † = x1 − y1 Var aˆ aˆ + (x1 xi − y1yi) haˆ aî hzî zîi + hzîzî i i>1 2 22 † X 2 † = x1 − y1 Var aˆ aˆ + (x1 xi − y1yi) haˆ aî i>1

† † where (3.22) implies that hzî zîi + hzîzî i = 1 + 2hNii = 1 for i > 1 as hNii = 0. Sincea ˆ is fed with a coherent state, its variance is shot-noise limited (A.13), and the result simplifies to X 2 † 2 † Var Zˆ ' (x1 xi − y1yi) haˆ aî = x1~x − y1~y haˆ aî (3.26) i≥1 where ~x and ~y are vectors whose elements are xi and yi respectively.

The average number difference at the detector (DC intensity difference) is X ˆ † 2 2 † hZi = xi x j − yiy j hzî zˆ ji ' x1 − y1 haˆ aî. (3.27) i, j

Similarly, the total light power at the detector (and hence the standard quantum limit) is

† † † † X † haˆ aˆ N + bˆ bˆ N i = haˆ aˆ N i + hbˆ N bˆ i + 1 = xi x j + yiy j hzˆ zˆ ji + 1 N N N N i, j i 2 2 † ' x1 + y1 haˆ aˆi. (3.28)

The relative intensity squeezing noise ﬁgure is then † ˆ† ˆ PN 2 2 Var Var aˆ N aˆ N − bN bN (x xi − y yi) x1~x − y1~y (SQZ) ' i=1 1 1 . NF = = † † 2 2 = 2 2 (3.29) Var(SQL) ˆ ˆ x + y x + y haˆ N aˆ N + bN bN i 1 1 1 1

This result can be used to numerically predict the degree of squeezing for a given set of model parameters. The coeﬃcient vectors ~x and ~y can be calculated numerically by repeatedly applying the recursion relation, and then computing the variance sum as above.

26 3.3.2 Inﬁnitesimal expansion and analytic result

Revisiting the recursion result (3.20), analytic expressions for the coeﬃcients xi and yi will now be derived. Evaluating the variance sum (3.26) derived above in the continuum limit (N → ∞) using these expressions leads to a novel result that analytically predicts the squeezing produced by the gain/loss system.

The result of recursively applying the gain/loss transformation (3.20) can be rewritten as

    N  p 2  aˆ N   aˆ  X  1 − ta cî    = AN   + AN−i  q  (3.30) ˆ†  ˆ†  2 ˆ† bN b i=1 1 − tb di Similarly, the mode expansion (3.21) can be written in the same form,    ˆ† P ˆ†      N     aˆ N  x1aˆ + x2b + i x2i+1cî + x2i+2di  x1 x2  aˆ  X x2i+1 x2i+2 cî    =   =     +     . (3.31) ˆ†   ˆ† P ˆ†    ˆ†    ˆ† bN y1aˆ + y2b + i y2i+1cî + y2i+2di y1 y2 b i=1 y2i+1 y2i+2 di Equating the coefficients of the incident probe and conjugate (â and bˆ†) gives    N x x t cosh s t sinh s  1 2 N  a a    = A =   . (3.32) y1 y2 tb sinh s tb cosh s In the large N limit, the infinitesimal parameters may be asymptotically expanded as 2 S − 1 S ··· − 1 cosh(s) = cosh N = 1 2! N + = 1 O N2 3 S S − 1 S ··· S − 1 sinh(s) = sinh N = N 3! N + = N O N3 1/2N 1 1 1 ta = Ta = exp 2N log Ta = 1 + 2N log(Ta) + O N2 − 2 − 1 − 1 1 1 ta = 1 exp N log Ta = N log(Ta) + O N2 , 1 → ∞ where the O N2 terms will vanish as N . The matrix A may therefore be written as

   1 1 S 1  ta cosh(s) ta sinh(s) 1 + log(Ta) + O 2 + O 2  A =   =  2N N N N     S 1 1 1  tb sinh(s) tb cosh(s) N + O N2 1 + 2N log(Tb) + O N2  1  1  log(Ta) S  = 1ˆ +  2  + O 1 N  1  N2 S 2 log(Tb)  1     log Ta S  1 0 = 1ˆ + 1 A + O 1 where A =  2  and 1ˆ =   . N 0 N2 0  1    S 2 log Tb 0 1 Using (3.32), the coefficients of the probe and conjugate operators in the continuum limit are therefore   x x h iN  1 2 N 1ˆ 1 1   = lim A = lim + A0 + O 2 = exp(A0). (3.33)   →∞ →∞ N N y1 y2 N N This result can be used to analytically determine the theoretical beam power imbalance at the detector by (3.27), and the equivalent-power standard quantum limit by (3.28). ˆ† The coefficients of the vacuum operatorsc î and di will now be calculated analytically to determine the vacuum noise contributions in the infinitesimal stage approximation. Equating the ith terms of the corresponding sums in (3.20) and (3.31), the coefficients of the ith vacuum operators after N ≥ i applications of the gain/loss transformation are:

     p 2   p 2    x i x i  cî   1 − ta cî   1 − ta 0  cî   2 +1 2 +2   = AN−i  q  = AN−i  q       ˆ†  2 ˆ†  2  ˆ† y2i+1 y2i+2 di 1 − tb di 0 1 − tb di      p 2   x i   1 − t  x i   0  ⇒  2 +1 = AN−i  a and  2 +2 = AN−i  q  . (3.34)        2 y2i+1 0 y2i+2 1 − tb

27 There are 2N vacuum contributions in the variance sum (3.26), each contributing a term like:

  2      x2i+1 x2i+1  x1  − 2  −   −     (x1 x2i+1 y1y2i+1) =  x1 y1   = x1 y1   x2i+1 y2i+1    y2i+1  y2i+1 −y1

 p 2    1 − t  p  x1  − AN−i  a − 2 AN−i   ∗ = x1 y1   1 ta 0   ( ) 0 −y1     1 − t2 0  x  − N−i  a  N−i  1  = x1 y1 A   A   0 0 −y1 where the symmetry of A was applied at (∗) to eliminate the transpose. Summing over thesec ˆi vacuum contributions,

N N  2    X X 1 − t 0  x1  (x x − y y )2 = x −y AN−i  a  AN−i   1 2i+1 1 2i+1 1 1   −  i=1 i=1 0 0 y1  N  2     X 1 − t 0   x1  = x −y  AN−i  a  AN−i   (3.35) 1 1     −   i=1 0 0  y1

ˆ† The coeﬃcients x2i+2 and y2i+2 of the operator di similarly each contribute terms like

  2      x2i+2 0 0   x1  (x x − y y )2 =  x −y   = x −y AN−i   AN−i   . (3.36) 1 2i+2 1 2i+2  1 1   1 1  2    y2i+2  0 1 − tb −y1

Summing together all the contributions from all the vacuum modes, and expanding in the large N limit, the variance due to squeezed vacuum ﬂuctuations is

 N  2     X X 1 − t 0    x1  (x x − y y )2 = x −y  AN−i  a  AN−i   (3.37) 1 i 1 i 1 1   − 2  −  i>2  i=1 0 1 tb  y1  N−1       1 X − log Ta 0    x1  = x −y  Ai   Ai + O 1    (3.38) 1 1  N  −  N2  −   i=0 0 log Tb  y1 The bracketed sum is a ﬁnite geometric matrix series with N terms, which is deﬁned to be X,

N−1   1 X − log(Ta) 0  X = Ai TAi + O 1 and T =   , N N2  −  i=0 0 log(Tb) and obeys the relationship 1 N N − 1 AXA = X + N A TA T + O N2 . (3.39)

Expanding the left-hand side in the large N limit gives 1ˆ 1 1 1ˆ 1 1 1 1 1 AXA = + N A0 + O N2 X + N A0 + O N2 = X + N A0 X + N XA0 + O N2 . n o ⇒ − 1 1 AXA X = N A0 X + XA0 + O N2 Rearranging (3.39) then gives the result

N N N N 1 1ˆ 1 1ˆ 1 1 A0X + XA0 = A TA − T + O N = + N A0 T + N A0 − T + O N .

1 Taking the limit as N → ∞, the bracketed terms converge to the matrix exponential and the O N term tends to zero, yielding n o lim A0X + XA0 = exp(A0) T exp(A0) − T. (3.40) N→∞

28 Since the matrices A0 and T are known, this is a system of four simultaneous equations in four unknowns for the elements of X. Recasting in matrix form, the elements of X obey       log(T ) SS 0 [X] [B]  a   11  11  1       S log(Ta) + log(Tb) 0 S  [X]12 [B]12  2    =   ,  1       S 0 2 log(Ta) + log(Tb) S  [X]21 [B]21       0 SS log(Tb) [X]22 [B]22

where B ≡ eA0 T eA0 − T. Taking the inverse of the coeﬃcient matrix, the solution is

1 × 2 [X]  (log Ta + log Tb)(log Ta log Tb − 4S )  11 [X]   2 2     12 log(Tb)(log(Ta) + log(Tb)) − 4S −2S log(Tb) −2S log(Tb) 4S [B]11   =    . [X]21  −2S log(T ) 2 log(T ) log(T ) − 4S 2 4S 2 −2S log(T )  [B]     b a b a   12 [X]22  2 2     −2S log(Tb) 4S 2 log(Ta) log(Tb) − 4S −2S log(Ta)  [B]21  2 2   4S −2S log(Ta) −2S log(Ta) log(Ta)(log(Ta) + log(Tb)) − 4S [B]22

The result is an analytic (if somewhat technical) expression for the vacuum noise variance contributions.

Note that if Ta = Tb = 1, the matrix has a zero determinant and cannot be inverted. However, in this case

T = 0 giving X = 0 by deﬁnition. The same result is obtained by computing the limit of Ta, Tb → 1 in the general case above.

Rewriting (3.33) explicitly for the ﬁrst sets of coeﬃcients,         x1 1 x2 0   A0     A0     = e   and   = e   , y1 0 y2 1 the remaining terms in the variance summation can be written as         x1  x1  1  x1  − 2 −     − A0   A0   (x1 x1 y1y1) = x1 y1   x1 y1   = x1 y1 e   1 0 e   y1 −y1 0 −y1     1 0  x1  − A0   A0   = x1 y1 e   e   0 0 −y1         x2  x1  0  x1  − 2 −     − A0   A0   (x1 x2 y1y2) = x1 y1   x2 y2   = x1 y1 e   0 1 e   y2 −y1 1 −y1     0 0  x1  − A0   A0   = x1 y1 e   e   . 0 1 −y1

Combining all the variance contributions together, the intensity diﬀerence variance is therefore ˆ − ˆ         Var Na Nb X  1 0 0 0   x1  = (x x − y y )2 = x −y eA0   eA0 + eA0   eA0 + X   ˆ 1 i 1 i 1 1         hN0i i  0 0 0 1  −y1   n o  x1  − 2A0   = x1 y1 e + X   −y1       A 1 0  n 2A o 1 0  A 1 = 1 0 e 0   e 0 + X   e 0   . (3.41) 0 −1 0 −1 0

With the applicable standard quantum limit given by

    Var Nˆ a − Nˆ b SQL x1 1 x2 y2   e2A0   . = 1 + 1 = x1 y1   = 1 0   (3.42) hNˆ 0i y1 0

An extra degree of loss is now reintroduced to the model to include the eﬀects of optical losses occurring after the vapour cell. For a system with overall detection eﬃciency ηa on the probe and ηb on the

29 conjugate, the infinitesimal coefficients are modified by2     √ √ η 0  x  → → ⇒ − → − −  a   i xi ηa xi, yi ηb yi (x1 xi y1yi) (ηa x1 xi ηby1yi) = x1 y1     . 0 ηb yi

Two additional vacuum ﬂuctuation terms are also added to the variance sum,

√ p 2 √ p 2 2 2 ηa x1 1 − ηa + ηb y1 1 − ηb = ηa(1 − ηa)x1 + ηb(1 − ηb)y1     η (1 − η ) 0   x  −  a a   1  = x1 y1     . 0 ηb(1 − ηb) −y1

Including these extra factors in the earlier expressions then produces

  Var Nˆ a − Nˆ b SQL 1 A0 A0   = 1 0 e P e   hNˆ 0i 0       Var Nˆ a − Nˆ b ( ) SQZ 1 0  1 0  1 A0   2A0 1ˆ −   A0   = 1 0 e   P e P + PXP + ( P) P   e   , hNˆ 0i 0 −1 0 −1 0   η 0  a  where P =   (3.43) 0 ηb

The analytic calculation of squeezing in a competing gain/loss medium can now be obtained by taking the ratio of squeezed variance to standard quantum limit.

3.3.3 Analytic expression for squeezing Based on the analysis presented thus far, the noise ﬁgure (2.7) quantifying the degree of squeezing produced by an interleaved gain/loss system with imperfect detection eﬃciency is ( 2 2 2 2 2 2 2 3 NF = 8ξ sinh(2ξ) S TaTb sinh(2ξ) 2 log(Ta) ηaηb log (Tb) − 4S ηa + ηb + ηa 16S ηb log(TaTb) − 3ηa log (Tb)

2 p 2 Ta 2 +ηa log (Ta) log(Tb)(3ηa − 2ηb) + ηaξ TaTb 4S log(Tb) 2ηa log + log(Tb) + 4ηb log(Ta) log(Tb) − log (Ta) Tb Ta 2 4 2 2 2 2 2 + log(Ta) log(Tb) log log(TaTb) − S TaTb sinh (ξ) 2 log (Ta) log(Tb) −16S η + ηaηb − η − 3ηa log (Tb)(ηa + 2ηb) Tb a b 2 2 2 3 2 2 2 +4 log(Ta) log (Tb) 4S (ηa − ηb)(ηa + 3ηb) + ηaηb log (Tb) + 4 log (Ta) 4S (ηa + ηb) + ηa log (Tb)(2ηa + 3ηb) 2 2 2 4 2 3 + log(Tb) ηa log (Tb) − 16S ηb − ηa log (Ta) log(Tb)(3ηa + 4ηb) − 16S TaTbξ sinh (ξ) cosh(ξ) − log(Ta) log(Tb)×

2 2 2 2 2 2 2 2 2 8S ηa + ηb + ηa log (Ta)(2ηa + ηb) + ηa log (Tb) 16S ηb + log (Ta)(3ηa + 2ηb) + 8S log (Ta)(ηa − ηb)(ηa + ηb) 3 2 4 2 2 4 4 2 3 3 −ηaηb log(Ta) log (Tb) − ηa log (Tb) − 256S TaTbηaξ log(Tb) cosh (ξ) + 256S TaTbηaξ log(Tb) sinh(ξ) cosh (ξ)(ηa log(Tb) 4 p 2 2 2 2 p 2 − ηb log(Ta)) + 32ηaξ TaTb cosh (ξ) log(Tb) 8S ηa + log(Ta) log(TaTb) − 4S log(TaTb) + 2ξ TaTb sinh (ξ)× 4 2 2 2 3 2 2 −64S ηb log(TaTb) + log (Ta) log(Tb) 4S (2ηa + 1)(ηa + 4ηb) − ηa log (Tb) − ηa log (Ta) 4S + log (Tb) ) 2 3 4 2 2 2 2 4 4 +4S ηa(2ηa − 1) log (Tb) + log(Ta) 128S ηb + 4S log (Tb) −4ηa − 8ηaηb + ηa + 4ηb + ηa log (Tb) + ηa log (Ta) log(Tb) ,( ) 2 p 2 2 2 1 2 32ξ TaTb log(TaTb) log(Ta) log(Tb) − 4S S ηb sinh (ξ) + ηa 4 sinh(ξ)(log(Ta) − log(Tb)) + ξ cosh(ξ) , (3.44) q 1 2 − 2 where ξ = 4 16S + log(Ta) log(Tb) , S is the squeezing factor, Ta the probe transmission, Tb the conjugate transmission, with overall detection eﬃciency ηa in the probe arm and ηb in the conjugate arm.

Experimentally, the conjugate beam is very far-detuned from resonance (by more than 4 GHz) so its absorption is negligible. Hence Tb = 1 and taking the arm detection eﬃciencies to be equal ηa = ηb = η,

2Performing this substitution and adding the two extra variance terms is equivalent to applying the loss transformation (3.3) to each of the series expansions (3.21) and recalculating the relevant variance.

30 the general result simpliﬁes to

2S sinh2ξ p sinh2ξ log2T NF = 1 − η + η T a , (3.45) ξ cosh(2ξ + θ) a 2ξ3 cosh(2ξ + θ) where tanh θ = (log Ta −log Tb)/ξ. The correspond to the standard quantum limit, noise reduction due to intensity correlations generated by four-wave mixing, and the noise introduced by vacuum ﬂuctuations due to losses. The denominator describes the standard quantum limit, and its terms correspond to power increase due to gain from four-wave mixing and loss from absorption (since log Ta < 0).

Further considering the limit of no probe absorption, setting Ta = 1 gives ξ = S and the above result reduces to the earlier two-beam loss model in the case of equal eﬃciencies (3.17) demonstrating agreement between the two diﬀerent models.

3.3.4 Impact of interleaved losses The theoretical analysis presented above permits investigation of the inﬂuence of the competing gain and loss processes.

Consider a four-wave mixing system with intrinsic gain G = cosh2(S ), that is, probe gain G in the absence of absorption and transmission coeﬃcients Ta on the probe and Tb on the conjugate. If gain and loss were independent processes, the expected probe power out would be TaG. Comparing the predicted probe power out hNˆ ai to this estimate demonstrates the role of interleaved losses (Figure 3.7).

0.8 1.0 0.6

0.8 0.4

0.2 0.6 0.2 0.4 0.6 0.8 1.0 0.4 –0.2

–0.4 0.2 –0.6

0.2 0.4 0.6 0.8 1.0 –0.8

Figure 3.7: Normalised probe power predicted by gain/loss model in presence of absorption hNˆ ai relative to estimate TaG for diﬀerent values of intrinsic gain G.

In the case of no conjugate absorption (Tb = 1), interleaved four-wave mixing opposes losses and results in higher than expected transmission, peaking around Ta = 0.25. However, lower than expected transmission is observed in the presence of conjugate absorption when Ta > Tb. This is due to the interplay between the conjugate and probe modes in the gain process. Increases in the conjugate intensity causes the conjugate transition to be stimulated and produce more eﬃcient mixing. Conversely, conjugate absorption decreases the mixing eﬃciency reducing the probe gain.

Now considering the degree of squeezing produced by this system (Figure 3.8), quantitative agreement with previously published numerical results [22, Fig 4] is reached, and the same general trends as the two-beam loss model (§3.2.2) are observed. Increasing the intrinsic gain produces stronger squeezing for high transmission but increases the noise level for low transmission, with an optimum level of loss existing in-between. However, interleaving the gain and loss processes allows four-wave mixing to regenerate the correlations after losses have degraded them, producing higher levels of squeezing than in the case of losses after mixing is completed.

31 Figure 3.8: Degree of squeezing predicted by distributed gain/loss model in presence of probe absorption (top) and associated contour plot (bottom).

32 3.4 Summary Beginning with an original discussion on the quantum mechanical description of optical losses, the theoretical framework developed in the previous chapter was extended to model the inﬂuence of losses on the squeezed light generated by four-wave mixing. New analytic expressions were derived for the relative intensity variance measured at the detectors. The variance was shown to be highly sensitive to relative (i.e. unbalanced) losses, resulting in noise increases with increasing gain for highly unbalanced losses. It was observed that slightly higher losses on the probe than conjugate reduced the predicted noise by a fraction of a decibel. The optimum level of probe loss was derived for a given gain and conjugate loss and the physical meaning of these results discussed.

The loss model was then redeveloped as an interleaved gain/loss process, to model the competing effects of squeezing and absorption which occur within the vapour cell. The numerical model proposed in [22] was re-derived due to lack of details in the publication, and extended by considering the analytic limit of arbitrarily small stages of gain and loss. A matrix methods technique was developed to address the inclusion of infinitely many vacuum modes and to determine the variance introduced by absorption losses. This technique was used to derive an analytic expression for the level of squeezing produced by four-wave mixing in the presence of absorption, including a correction factor for the finite detection efficiency. The method employed to obtain this result could be easily applied to any system where interleaved losses are a competing process, adding a new avenue of analysis to the theoretical methods toolbox.

Experimental design 4

This chapter describes and discusses the experimental factors that must be taken into consideration in implementing the four-wave mixing system and measuring relative intensity correlations at the quantum limit. The details of the implementation are outlined and justiﬁed, including the production of the pump and probe lasers, beam alignment procedures, vapour cell type and conﬁguration, detection apparatus and spectral analysis procedures.

Schematically, in the experiment a high-power “pump” beam intersects a low-power “probe” beam at a small angle θ within a hot Rb vapour cell, causing them to interact via four-wave mixing to produce a conjugate beam (Figure 4.1). These probe and conjugate beams then manifest correlated intensity ﬂuc- tuations, which are obtained by individually measuring their respective intensities and subtracting the corresponding photocurrents to obtain the relative ﬂuctuations. These are then processed by a spectrum analyser to visualise the noise power spectrum.

Pump – S.A. Probe Heated Rb cell Conjugate

Figure 4.1: Schematic design of experiment to measure squeezing between probe (orange) and conjugate (purple) on the spectrum analyser (S.A.)

4.1 Pump laser generation The four-wave mixing scheme relies on a non-linear interaction between the pump and probe beams, using the atomic vapour as an interaction medium. A high-intensity pump beam is required to eﬃciently drive this process, and the Coherent 899 Ring Laser was chosen for this purpose.

The Coherent 899 is a titanium-sapphire (Ti:S) laser, which converts a high-power input laser into a stable, tuneable, low linewidth beam by pumping a titanium-doped sapphire crystal contained within a bow-tie cavity. The Ti:S was chosen both for its stability and versatility, as it produces constant output power with detuning, very low amplified spontaneous emission and high output beam quality. Furthermore, adjustment of the birefringent filter permits changing the output wavelength by tens of nanometers, enabling rough tuning between the two fine-structure ground-state transitions of Rb (the D1 line at 795 nm and the D2 line at 780 nm).

ACoherent Verdi V10 was used to provide up to 10.5 W of continuous 532 nm light to pump the Ti:S. Up to 800 mW of single-mode output power was generated at the desired wavelength of 795 nm, with 600 mW of pump power typically available for mixing. After tuning the etalons within the ICA and adjusting the feed-forward, a mode-hop free scan range of more than 20 GHz was obtained. This permitted continual scanning of the pump laser frequency across the entire region of interest for generating four-wave mixing.

35 As the experiment involved scanning across the four-wave mixing resonance instead of ﬁxating on a resonant feature, it was not necessary to keep the Ti:S frequency locked to an external reference. However, the 899 system contained an external Fabry-Perot cavity which generated a reference signal that could be used to check for cavity mode drift and hopping. This was important as the experiment typically involved detuning far from resonant features and mode hops due to drift were diﬃcult to otherwise detect.

The Ti:S was calibrated by fibre-coupling a fraction of its output beam to an Agilent 86140B optical spectrum analyser. The OSA was used to coarse-tune the laser to bring it close to resonance with one of the transitions of rubidium. Then a natural abundance mixed-isotope vapour cell containing 72% 85Rb and 28% 87Rb was used as a frequency reference by observing Doppler absorption of the beam. The cell was placed in a standard retro-reflected saturation absorption spectroscopy configuration (Figure 4.2) to eliminate Doppler broadening, and the separation between the characteristic “crossover” transitions [42] was used to calibrate the scan rate.

Rb cell

B. 87 Rb 85 Rb 85Rb 87Rb

Detuning (GHz) Intensity (arb. units) (arb. Intensity -1 0 1 2 3 4 5 6 7

Figure 4.2: Retro-reﬂected saturation absorption spectroscopy conﬁguration (A) and resulting transmission signal (B) with absorption dips corresponding to the two isotopes present. Overlapped beams shown separately for clarity.

Although the day-to-day stability of the Ti:S was typically good, it exhibited longer-term drift in its output power and scan range, which necessitated minor but frequent adjustment and recalibration. Un- fortunately, these adjustments required tweaking the cavity mirrors, which in turn changed the tilt of the cavity slightly and caused the output beam to drift in position and direction. Steering mirrors were placed directly after the Ti:S output to control this drift. Two irises (one placed near the laser, the other far from it) served as alignment markers to ensure day-to-day pointing stability and minimise drift on the down-stream optics.

A large feed-forward was necessary to obtain a large mode-hop range when scanning the laser frequency, which resulted in a power ramp (3%) during the scan. Power sensitive measurements were therefore carried out either in “zero-span” mode or calibrated by measuring the changing output power with a photodiode and digital oscilloscope. The Ti:S was operated in “free-running” mode, in which the cavity was only electronically locked to its internal etalons which allowed the frequency to be scanned. This introduced low-amplitude power ﬂuctuations to the output beam, but these were typically only observed in the ampliﬁed saturation absorption reference signal as noise.

Finally, the cavity was observed to produce an elliptical beam that was elongated in the horizontal axis (the direction perpendicular to the cavity axis). This was corrected with a cylindrical lens, and had to be taken into consideration when mode matching the beams in the vapour cell.

36 4.2 Probe laser generation The “double–Λ” four-wave mixing scheme operates on a sequence of two-step excitations between the F =2 and F =3 ground states of 85Rb via far-detuned virtual states (Figure 4.3). These Raman transitions are most probable when the detunings of the pump and probe lasers bring the two states into resonance. The high intensity of the pump beam induces a non-trivial AC-Stark shift that is compensated for with a “two-photon” detuning from the unperturbed F = 3 state, ∆2ph = ∆pump − ∆probe. The two-photon detuning also inﬂuences both the phase-matching condition and degree of undesired Raman scattering of the probe, so it is desirable to maintain explicit experimental control over this parameter.

Probe Pump

Pump line

Conjugate

Figure 4.3: Energy levels and associated transition detunings of “double–Λ” system.

85 The hyperfine splitting of Rb is ∆G = 3.036 GHz, so the probe beam needs to be generated with a stable (yet adjustable) detuning with respect to the pump beam around this value. Although this could be done by frequency-offset locking two lasers together, it is more convenient to produce the probe beam by externally modulating the pump beam at a given frequency, through exploitation of an interaction such as the acousto-optic effect. Modulation by an external reference frequency also provides excellent relative frequency stability between the pump and probe beams and introduces little extra noise provided a clean rf source is used to drive the process.

4.2.1 Acousto-optic modulator The probe beam was generated with an acousto-optic modulator (AOM). An AOM uses a piezoelectric transducer to produce a travelling sound wave within an internal crystal, resulting in a density grating. Light incident on the crystal can diffract off this grating to produce a deflected beam which is frequency shifted by the Doppler effect. This deflection allows easy isolation of the resulting probe beam from the undiffracted order. Furthermore, as the probe beam is weak by design (of order 100 µW) it is not important that the conversion process is highly efficient.

However, the degree of beam deflection is dependent on the driving frequency due to the diffraction process, so changing the input rf will change the beam alignment. To mitigate this effect, the AOM was placed in the “cat’s eye” double-pass configuration [43], so that the diffracted beam was reflected onto the modulator and was diffracted a second time to become counter-propagating with the incident beam (Figure 4.4). The linear polarisation of the light was rotated by 90◦ between the first and second passes to separate the diffracted beam using polarising-beam splitter (PBS) on the return trip.

37 +1518 MHz

Iris PBS Mirror

Iris AOM Input (+0 MHz) +1518 MHz Probe +3036 MHz

Figure 4.4: “Cat’s eye” double-pass AOM conﬁguration. Overlapped input (red) and probe (orange) beams shown separated for clarity.

Piezoelectric transducers have poor efficiency when operating at GHz frequencies, resulting in AOM diffraction efficiencies of only a few percent. The double-pass configuration therefore has the added advantage of halving the required modulation frequency to 1518 MHz, where transducers are more efficient.

The Brimrose GPF-1500-200 was chosen for these experiments and offered modulation at 1700 ± 500 MHz (3 dB down) with up to 34% transmission at 780 nm when driven by up to 1 W of rf. Higher power input rf generates larger amplitude sound waves within the crystal and results in stronger diffraction (Figure 4.5A). This interaction is also polarisation sensitive, with the polarisation parallel to the AOM axis diffracting more efficiently (Figure 4.5B) in a similar way to that expressed by Malus’ law.

The polarisation sensitivity has important consequences for the maximum achievable conversion efficiency through the double-pass, as the polarisation must be rotated by 90◦ before reaching the PBS, but the AOM only scatters one polarisation efficiently. If the input beam is initially linearly polarised at an 2 2 2 π angle θ to the AOM axis by (Figure 4.5B), the maximum double-pass efficiency is E cos (θ) cos (θ− 2 )= 1 2 2 4 E sin (2θ), where E is the efficiency of a single pass. At nominal specifications (E = 34%), this predicts 2.9% conversion to the output beam. Experimentally only 1% was obtained because of difficulty in mode matching on the second pass. Despite being very low, this conversion rate was acceptable as 10 mW split off the pump beam was sufficient to generate the 100 µW necessary for the probe.

As expected, the cat’s eye configuration was observed to mitigate the frequency-dependent deflection caused by the AOM. This permitted a wide range of driving frequencies without significantly reducing the overall double-pass efficiency (Figure 4.5C). In particular, less than a 5% drop in the probe power was observed over the experimental tuning range of 1500-1540 MHz.

However, when the Brimrose crystal was tilted to the large incidence angles necessary to maximise the diffraction efficiency, an internal reflection was produced that overlapped with the diffracted order. Be- cause this reflection was not frequency shifted, this caused the probe beam to contain several frequency components after being double-passed, only one of which would undergo the desired four-wave mixing. Other components would stimulate competing processes, contaminating the results.

This reflection became more prominent as the incidence angle was increased, reflecting up to 80% of the incident beam. Experimentally this appeared to correspond to an increase in diffraction efficiency, which made optimisation difficult. To overcome this, an infra-red CCD video camera was used to observe changes in the modal pattern of the beam and the diffracted order was passed through a hot (80◦C) pure 85Rb cell and its absorption spectrum observed. Due to the large Doppler width at this temperature, contamination was readily identified through unexpected absorption dips and alignment tweaked to eliminate the reflection.

38 A. Diffraction Efficiency Dependence on RF Power 35

10 Single-pass diffraction efficiency (%) efficiency Single-pass diffraction

18 20 22 24 26 28 30 Driving rf power (dBm)

B. Diffraction Efficiency Dependence on Polarisation Angle 20

10 Single-pass diffraction efficiency (%) efficiency Single-pass diffraction

20 40 60 80 Polarisation rotation angle, (°)

C. Diffraction Efficiency Dependence on Driving Frequency 100

60 Relative double-pass efficiency (%) double-pass efficiency Relative 50

20 1400 1450 1500 1550 1600 1650 1700 Driving rf frequency (MHz)

Figure 4.5: Dependence of single-pass AOM eﬃciency on input rf power (A) and beam polarisation (B), as well as the double-pass dependence on driving frequency in the “cat’s eye” conﬁguration (C). Dashed lines are a guide for the eye.

39 4.2.2 Tunable rf source To precisely control the two-photon detuning and minimise drift, a digital frequency synthesizer was used to generate the rf signal to drive the AOM. The frequency source chosen was an Analog Devices ADF4360-4 frequency synthesizer chip, which uses a digitally constrained voltage-controlled oscillator (VCO) to produce frequencies in the nominal range 1600 ± 150 MHz with an output power of up to −3 dBm.

The synthesizer was included in an Analog Devices evaluation board, which contained a communication interface between the controller and a PC, to load instructions via the serial port. A custom LabView interface was written to determine the counter and control register settings necessary to operate the board at the desired frequency, and to communicate with the synthesizer chip.

The spectrum of frequencies the ADF4360 is capable of producing is determined by its “channel width”, which is directly related to its frequency mode spacing. The output carrier frequency is then determined by integer logic on the control settings (as described in the “Circuit Description” of the manual [44]). Although reducing the channel width allows the output frequency to be set more accurately, it has a negative inﬂuence on noise characteristics (§5.3). In this experiment, the probe detuning ∆pr ≡ 2 fAOM − ∆G was desired to be tuned by several MHz, so it was not necessary to reduce the mode spacing to below 500 kHz.

A 44 dB Mini-Circuits ZHL-5W-2G rf amplifier was then used to drive the AOM with the low-power output from the ADF4360 frequency synthesizer. To prevent overloading the AOM, coaxial Mini- Circuits VAT-series attenuators were placed before the amplifier to control the net output power. Typ- ically the AOM was operated with an rf input power of +31 dBm, resulting in a slight temperature rise in the air-cooled device, but no observable negative effects on its operation due to overload.

4.3 Beam alignment By using linearly polarised beams, the four-wave mixing process produces a conjugate beam with the same polarisation as the probe but propagating in a diﬀerent direction (§2.4). The pump and probe beams can therefore be polarised in orthogonal directions, allowing the beams to be combined and separated on PBS cubes (Figure 4.6). Separate steering mirrors for each beam were used to set the incidence angle, which was determined by measuring the pump-probe beam separation an equal distance before and after the cell. A separation of 3.5 mm at 500 mm from the cell corresponded to the desired 7 mrad incidence angle.

Double-pass AOM Detection stage

+3.036 GHz

Conjugate Probe Ti:S laser Pump PBS PBS Beam-stop

Rb cell PBS

Steering mirrors

Figure 4.6: Combination and isolation of beams using polarising beam-splitters (PBS). Note that the ﬁnal PBS cannot completely separate the pump beam, causing some pump “leakage” to the detector.

40 In attempting to measure intensity correlations beyond the standard quantum limit, it is extremely important that the pump, probe and conjugate beams are isolated very well to prevent cross-contamination between the beams. As shown in §2.3.1, the probe and conjugate beams are very noisy, while the pump has completely uncorrelated shot-noise fluctuations. Cross-detection will therefore result in the obse- vation of super-Poissonian noise. Due to the small angle between the beams, their divergence after the cell and the overwhelming pump beam power, it is a technically challenging task to isolate the beams. In particular, polarisation optics do not operate with perfect efficiency and different polarising elements had to be investigated to determine the most effective approach.

Initially, an AR-coated Glan-Laser calcite polariser was used to reject the pump beam after the cell. The cube was mounted in a rotation stage, so that the ordinary axis could be rotated to give minimal transmission of the pump beam. The extinction ratio of the polarisers was very high (at least 104 : 1), resulting in very low leak-through of the pump but very high losses on the transmitted probe beam (at least 16%). This was primarily due to internal absorption by the calcite crystal (8%), improper rejection of the extraordinary polarisation into the ordinary ray, and reﬂection at the internal hypotenuse surface due to the large angle of incidence (approximately 30◦) and improper anti-reﬂection coating of the internal surface. Due to the sensitivity of squeezing to losses, the Glan-Laser polariser approach was unacceptable.

The polarisers were then replaced with standard dielectric stack polarising cubes with 1000 : 1 extinction ratio. These were physically rotated to minimise transmission of the probe through the PBS and hence minimise losses. Due to its high intensity, the pump “leakage” deﬂected towards the detectors was of similar power to the probe beam, but could be isolated using a large propagation distance and a tightly conﬁned iris. This permitted subsequent use of a short focal length lens to capture all of the transmitted probe light in the detector while tweaking the probe beam alignment, without being saturated by pump light.

Alignment of the steering optics was achieved by tuning the pump off-resonance and aligning the transmitted probe beam onto one port of the detector. Tuning the pump to the four-wave mixing resonance, a conjugate was produced and gain observed on the probe. The pump-beam alignment was then tweaked to maximise the probe gain without deflecting the transmitted probe away from the detector. Maximis- ing the gain without changing the detunings corresponds to increasing the mixing strength, and hence maximises the intensity of the conjugate. This could then be observed on an IR card and projected onto the other port of the photodiode to produce a differential signal. Further fine-tuning was performed by blocking one input port and tweaking the alignment mirror to maximise the photocurrent.

4.3.1 Beam waists As four-wave mixing is a non-linear process, its efficiency is increased by using higher intensity fields to stimulate the process. In order to obtain high intensity from the laser powers used, small beam waists were necessary. Telescopes for shrinking the beam waists and compensating the natural divergence of the Ti:S beam and residual divergence on the probe (caused by the double-pass AOM configuration) were obtained by using lens pairs.

A combination of f = 1000 mm and f = 500 mm lenses on the probe beam produced a 1/e2 beam radius of 375 µm at the cell, and an f = −200 mm, f = 220 mm lens pair on the pump generated a radius of 630 µm. These values were chosen to match published work [19] to ensure the intensity regime was sufficient to drive strong mixing. Beam waists were measured by capturing the beam profile with a Coherent BeamMaster knife-edge beam profiler and performing a Gaussian fit.

41 These lens pairs can also be used to ensure that the beams have a long Rayleigh length, and by focussing within the cell the beams have curvatures that are approximately planar. This allows better mode- matching between the beams and gives the smallest spread in k-vectors, which otherwise acts to “smooth out” the phase-matching and reduce the peak gain (§2.4.1). It also ensures the beams do not overlap due to their divergence after the cell which would cross-contaminate the beams.

4.4 Vapour cell The vapour cell used in this experiment was a 12 mm-long isotopically pure 85Rb cell manufactured by Triad Technologies. The short cell length was deliberately chosen to minimize the propagation distance within the cell and reduce the near-resonant absorption of the beams. The beams were overlapped over the entire length of the cell to ensure that mixing occurred throughout the medium. The windows of the cell were also AR-coated to further prevent losses, delivering 94% nett transmission at 795 nm.

Because the four-wave mixing process used to generate the squeezed light depends on the hyperfine structure of the mixing medium (which is different for the different isotopes of Rb) it was necessary to use an isotopically pure cell. Presence of the incorrect isotope lowers the concentration of the desired isotope at a given temperature, thereby reducing the number of atoms undergoing mixing and hence the overall level of squeezing, as well as introducing spurious absorption.

The 85Rb isotope was chosen both for its commercial availability and its more accessible ground-state hyperﬁne splitting of 3.0 GHz. The ground-state splitting of 87Rb is 6.8 GHz, and would either require a technically challenging 3.4 GHz AOM or two oﬀset-locked lasers.

4.4.1 Presence of isotopic impurity Despite using an isotopically pure 85Rb vapour cell, absorption dips corresponding to 87Rb were observed in the probe transmission spectrum when the vapour cell was hot (Figure 4.7). This demonstrates a small impurity of 87Rb within the cell, the concentration of which may be estimated from the known rubidium vapour pressure curves and the observed fractional absorption.

Comparison of Vapour Cell Absorption Profiles

Reference cell (22°C)

Pure 85Rb cell (160°C)

Figure 4.7: Absorption proﬁles indicating presence of 87Rb impurity in isotopically “pure” 85Rb cell. Dashed 87 lines indicate absorption dips corresponding to the hyperﬁne splitting of 5S 1/2 → 5P1/2 in Rb.

−σnL In a simple model for the absorption, the transmitted intensity is I = I0e , where σ is the cross-section of the transition, n the number density and L the cell length. Rearranging for the number concentration, ! 1 I n = − log . (4.1) Lσ I0

−13 2 In the case above, I/I0 ' 0.92, σ = 2.91 × 10 m [45] and the internal cell size is L = 5 mm, giving the density of 87Rb atoms as n ∼ 1013 m−3. Using the rubidium vapour pressure curves [46], the pressure

42 at this temperature is P = 0.97 Pa, giving a concentration in the order of 1020 m−3. The fraction of 87Rb in the cell is therefore only in the order of 10−7 but at this temperature it is suﬃcient to cause nearly 8% absorption of the probe.

Unfortunately, the absorption dip corresponding to 87Rb is close to the four-wave mixing resonance, resulting in spurious absorption of the beams. Although mitigated by high-intensity spectral hole-burning due to the small waist size, this will cause isotropic scattering of the pump light, some of which will enter the probe and conjugate detectors and introduce uncorrelated shot-noise.

4.4.2 Vapour cell heater For a given interaction volume (as deﬁned by beam size and vapour cell length) the strongest mixing and hence highest gain will be produced by interacting with the largest number of atoms possible. The melting point of rubidium is 39.3◦C, above which temperature the density of atoms within the vapour is controlled by the liquid-phase vapour pressure. Increasing the temperature of the vapour cell results in increasing the vapour pressure and hence the atomic density, generating more interactions and hence improving the overall gain. However, increasing the temperature expands the Doppler spread, resulting in absorption of the near-resonant beams that generates scatter and destroys the correlations. Tuneable (yet stable) control over the temperature of the cell is therefore important.

To achieve this goal, a custom copper oven was manufactured. The heat was provided by a 25 W, 5 Ω resistor thermally glued to a copper block surrounding the cell. The block provided a large mass for thermal stability and applied the heat uniformly around the cell. The current through the resistor was controlled using a 5 A-max variable power-supply, and thermal equilibrium between oven and its surroundings controlled its ﬁnal temperature. This was measured by a thermocouple attached to a small hole in the side of the copper block. Active temperature control was unnecessary because of the large thermal mass and isolation of the system.

The vapour pressure is determined by the temperature of the coldest point of the cell, around which rubidium will condense. It is therefore important to prevent thermal gradients within the oven and minimise its susceptibility to external temperature ﬂuctuations. To this end, several design modiﬁcations were made to the original oven to improve its thermal isolation (Figure 4.8).

Heating element Teflon-tipped grub screw

Copper block

Rb vapour cell Teflon washer

Aluminium foil cap Post

Figure 4.8: Final design of vapour cell heater: a resistive heating element generates heat, conducted to the vapour cell via a copper block, with several mechanisms employed to reduce heat loss.

Conduction through the vapour cell and oven mount to the optical table was minimised using a Teﬂon washer to separate the steel post from the copper oven. Large copper “wing” extensions were welded to either side of the window to trap hot air around cell windows. Aluminium foil caps were added to the

43 end-faces to minimise convection of air away from windows, with a small hole punched in the centre for the beams to travel through.

The entire system was then encased in a cardboard box, which further isolated the oven by minimising air convection to the rest of laboratory. In particular, this served to prevent air movement around the nipple of the vapour cell, which protruded beyond the copper block. Because it was situated next to the heating element and was otherwise surrounded by the heated copper, this was only a problem when cool drafts of air caused the cell to temporarily cool and the vapour to condense. This eﬀect was signiﬁcant. The four-wave mixing gain in a running experiment was observed to drop by a factor of 8 due to air drafts, before recovering as the cell reheated over a few seconds. Isolating external air moments using the box eliminated this problem.

4.5 Relative intensity measurement Measurement of the relative intensity signal was achieved with a “detection stage” (Figure 4.9). A small “pick-oﬀ” or “separation” mirror was used to isolate the conjugate beam from the pump and probe, and signiﬁcant care had to be taken to avoid cross-contamination. Typically this was observed as changing optical power (with an associated increase in measured noise power) when the pump beam passed through resonance. Lenses of focal length 150 mm were used to focus the beams into the active region of the photodiodes, which were shielded from ambient light with PVC tubing.

Probe M3

S.A. – Conjugate M3 L

M1 M2

Pump I M2 M1. Redirection mirror M2. Separation mirror I. Iris L. f = 150 mm lens Four-wave mixing M3. Alignment mirror

Figure 4.9: Diﬀerential noise measurement of probe and conjugate beam. Follows on from Figure 4.6.

To minimise the effect of unbalanced optical losses upon squeezing measurements (§3.2.2), it was necessary to use a high-efficiency balanced dual photodetector. The Thorlabs PDB150A was chosen for this purpose, and comprised two well-matched photodiodes combined with a low-noise, high-speed variable-gain differential transimpedance amplifier that produced a voltage proportional to the intensity difference between the two detected beams (termed the “rf output”).

The PDB150 detector also provides two SMA “monitor” connections which “directly” display the individual photodiode currents. However, connecting these ports to a oscilloscope was observed to capac- itively couple the photodiodes and produce incorrect readings, so they were not used. Individual beam powers were instead measured by blocking one port and observing the rf output corrected for the sign of the input used (positive for probe, negative for conjugate).

44 The photodetector provides multiple gain settings for its internal transimpedance amplifier, and different settings were used for different purposes. It is desirable to use as high a gain as possible, as increasing the gain both produces a larger signal and decreases the Johnson noise in the resistor. This simultaneously increases the noise power corresponding to the measured intensity and reduces the electric noise floor.

The detector had a maximum output voltage of ±5.8 V over a 1 MΩ termination, so saturation was a primary concern when using the detector to measure beam power on the oscilloscope. In this situation a gain of 104 V/A was used, which saturated at 940 µW optical input power.

In measuring the difference signal, the near-balancing of the probe and conjugate beam powers meant that saturation was not an issue. However, the signal gain of the transimpedance amplifier rolls off at a characteristic frequency depending on its gain setting, with larger gain settings limited to smaller signal bandwidths1. This bandwidth had to be significantly above the analysis frequency of 1 MHz, so noise measurements were carried out on the largest permissible gain setting of 105 V/A, which had a cut-off frequency (3 dB down) of 5 MHz.

4.5.1 Photodiode quantum efficiency Since imperfect detection efficiency results in further optical losses, the efficiency was estimated experimentally to quantify its effects and calibrate the theoretical model. The case of the PDB150A detector was opened to expose the pins of the photodiodes, and a 50 Ω oscilloscope probe was used to measure the photocurrent ipd generated by one of the diodes. A Newport 1918-C optical power meter was used to measure the beam power Ibeam, from which the responsivity was obtained as the ratio R = ipd/Ibeam. The effective quantum efficiency of the system is then η = (hc/λe)R, where hc/λ is the photon energy and e the elementary charge.

Unfortunately, ﬂuctuations and drift in the output power of the Ti:S produced large statistical uncertainties in the resulting value. To overcome this, the optical power and photocurrent had to be measured simultaneously. The Newport 1918-C provides a 3.5 mm mono audio jack for analog output, which was set to generate a 1 V output signal at the full-scale input intensity of 530.4 µW. A PBS cube and λ/2 waveplate were adjusted to produce two beams with equal intensities (balanced to within 1%), of which one was used to measure the intensity via the power meter and the other to generate a photocurrent within the detector.

The power and current could therefore be simultaneously measured and traces captured to average out the intensity fluctuations and determine the effective quantum efficiency. The alignment procedure was repeated several times to minimise the influence of misalignment and power imbalance. The photodiodes shipped with the PDB150A were found to have low quantum efficiency, and the detector was subsequently modified to use much higher efficiency Hamamatsu S3883 diodes. This was observed to produce a significant improvement (Table 4.1).

Photodiode Nominal R (A/W) Measured R (A/W) Efficiency η Thorlabs PDB150A 0.53 0.52 ± 0.02 81 ± 2% Hamamatsu S3883 0.58 0.61 ± 0.01 95.4 ± 0.5% Table 4.1: Comparison of effective photodiode responsivities (R) and associated effective quantum efficiencies (η) for different diodes within the PDB150A.

1Despite this observation, the PDB150A did not have a ﬁxed gain-bandwidth product.

45 4.5.2 Detector linearity The detector must be operated well within its linear regime to produce reliable noise measurements. Using the same apparatus set-up as above, the ampliﬁed rf output of the photodetector was recorded for various beam intensities (up to 1 mW) in the region of interest, for the two applicable gain settings. The resulting response curve (Figure 4.10) demonstrates the linearity of the detector in this regime despite changing the diodes, and provides a calibration between optical power and recorded rf voltage. As mentioned earlier, this allows the detector to be used as a power-meter by blocking one of its input ports.

PDB150A Voltage Calibration Curve 4.5

3.5 ) V (

u 3 o V

2.5

1.5 Detector rf voltage

1 104 V/A Gain 0.5 105 V/A Gain

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Optical power, P (mW)

Figure 4.10: Ampliﬁed (rf) detector output voltage as a function of input beam power using PDB150A with Hamamatsu S3883 diodes.

It should be noted that the gain of the transimpedance amplifier drops when connected to a low- impedance load. When connected to the oscilloscope on 50 Ω termination, the gain was measured to drop by a factor of 1.95 ± 0.01. This is close to the expected factor of 2 specified in the manual [47], and is significant for estimating expected noise levels on the 50 Ω terminated spectrum analyser.

4.5.3 Detection stage losses Despite all components being high-/anti-reflection coated, several components were measured to have significantly lower transmission than expected (e.g. 90% instead of > 98%) and had to be replaced. To prevent significant cascading of optical losses, a minimum transmission limit of 97% per component was enforced.

The optical transmission of each component in the beam-path after mixing was measured to determine the net optical losses of the system (Table 4.2). The optical components were measured to have a efficiency of 89.5 ± 0.5% which, when combined with the detector quantum efficiency (§4.5.1), gives a net detection efficiency of η = 85.3 ± 0.8%. This parameter is used for calibrating the theoretical squeezing models and inverting the experimental results to determine the level of squeezing at the cell.

46 Component Transmission Vapour cell (off-resonant) 94% Single vapour cell window 97% Polarising beam-splitter 97% Redirection mirror 98% Separation mirror 99% Lens 99% Alignment mirror 99% Overall beam path efficiency 89.5% Detector quantum efficiency 95.4% Calculated net efficiency 85.3% Table 4.2: Measured optical transmission of components in the apparatus detection stage. All measurement uncertainties are 0.5%, resulting in a 0.8% uncertainty in the calculated net efficiency.

4.6 Noise spectrum measurement The balanced differential photodetector produces a voltage proportional to the intensity difference between the two beams, which contains the relative intensity fluctuations as voltage fluctuations. Fourier- transforming this signal to the frequency domain then produces the noise spectrum. By the linearity of P 2 2 2 the detector response (ipd ∝ I), the measured electric power is ∝ V ∝ ipd ∝ I . Hence the electrical noise power measured at a given analysis frequency (within a particular bandwidth) corresponds to the relative intensity noise power at that frequency. When this drops below the value corresponding to the standard quantum limit, squeezing has been observed.

4.6.1 Power spectrum analysis Using the Wiener-Khintchine theorem, the power spectrum of a time-varying voltage V(t) is

P 1 2 1 ( f ) ≡ R F V(t) f ≡ R F V ? V f , where F denotes the non-unitary Fourier transform and ? the cross-correlation: Z∞ Z∞ −2πi f t ∗ F g(t) f ≡ g(t) e dt and (g ? h)(t) ≡ g (τ) h(t + τ) dτ. −∞ −∞ The normalisation factor R above is the termination resistance of the voltage measurement and gives the spectrum units of electrical power (W). The equality of the two deﬁnitions can be seen to follow from the Fourier transform cross-correlation theorem, F [g ? h] = F [g]∗F [h].

Hence the power spectrum is the electronic noise power measured at a given analysis frequency. Typ- ically the voltage signal is AC-coupled to make V(t) a square-integrable function to ensure the above integrals converge.

Power measurements can vary by several orders of magnitude across a spectrum, so noise powers are usually quoted in decibels relative to 1 W (dB) or 1 mW (dBm). The convention used to express this will be P(dB) P P(dBm) P(dB) ≡ 10 log10( ), ≡ + 30.

Because experiments are conducted over a ﬁnite measurement time, the analytic power spectrum cannot be obtained exactly. Instead, the noise power is measured over a given frequency range called the “resolution bandwidth” (RBW) of the measurement. Reducing the RBW of a measurement increases its accuracy at the expense of requiring a longer integration time.

When measuring and comparing white noise, the resolution bandwidth is arbitrary as it simply scales the power measured in each interval. Typically the largest bandwidth that keeps the signal signiﬁcantly

47 above the background (see below) and provides suﬃcient precision for the experiment is used. A 30 kHz resolution bandwidth was suﬃcient for most measurements.

The noise spectrum of the difference signal was measured experimentally with a Rhode &Schwarz FSP7 digital spectrum analyser. The spectrum analyser was operated in “FFT mode”, as shown schematically in Figure 4.11. The spectrum analyser heterodynes the input signal against a reference frequency to produce a low-frequency sideband that can be sampled by a digitizer. The digitizer was operated in “sampling” mode to record the instantaneous voltage at the sampling intervals. The results are buffered in memory and then Fast Fourier Transformed (FFT’d) and converted to a power spectrum. The spectrum is then low-pass filtered at the “video bandwidth” (VBW) to reduce the influence of high frequency noise being aliased down by the sampling process. The result is optionally linearly averaged over several repetitions and then displayed on the screen.

Reference Sampling Frequency Frequency

Input Signal Filters Digitizer Filters FFT Heterodyne

Figure 4.11: Schematic of signal processing in the Rhode & Schwarz FSP7 spectrum analyser.

When making noise measurements, the analyser was set to the analysis frequency of 1 MHz in “zero span” mode. This skips the buffering and FFT stages, and directly displays the measured power within the resolution bandwidth over time. This turns the spectrum analyser into a tuneable filter and detector that continually measures the noise power within the resolution band about the analysis frequency, allowing the laser frequency to be scanned and the effect of varying gain on noise power directly observed.

To calibrate the standard quantum limit for comparison, it was necessary to record the beam powers across a scan. The spectrum analyser and oscilloscope used to infer power were therefore operated in triggered mode using the X-scan output voltage from the Ti:S control box as the external input. The trigger delay between the two devices was calibrated using a sharp resonant feature.

Finally, as the FSP7 requires a pure AC input signal, a Mini-Circuits BLK-89+ coaxial SMA DC block was used to eliminate the DC contribution arising from the baseband intensity diﬀerence between the probe and conjugate (§2.3). This model DC-block was chosen as its bandwidth (0.1 MHz to 8 GHz) transmitted 1 MHz signals without attenuation.

4.6.2 Background subtraction The measured noise power results from many contributions such as shot-noise, Johnson (thermal) noise, electromagnetic interference and circuitry noise as well as the signal of interest. For a voltage signal arising from diﬀerent contributions, the power spectrum is P P P V(t) = V1(t) + V2(t) ⇒ ( f ) = 1( f ) + 2( f ) + 2 Re F V1 ? V2 f . Diﬀerent noise processes are independent and their contributions uncorrelated, so the cross-correlation term vanishes and the measured noise power from independent processes add together as P P P total( f ) = 1( f ) + 2( f ) + ··· . This implies that the noise power of the process of interest (in this case the electronic shot-noise corresponding to the intensity measurement) can be extracted by independently measuring the noise power

48 from the other contributions (the “background level”) and subtracting it from experimental results. Pro- vided the measured signal is suﬃciently above the background level (by at least 5 to 10 dB) this can be achieved without signiﬁcant loss of precision (known as “subtractive cancellation”).

Generally the measured noise depends on the spectral (analysis) frequency and measurement bandwidth and typically contains structure (Figure 4.12). Below 1.4 MHz the spectrum analyser noise dominates, until the analyser changes operational modes and its noise begins rolling off and amplifier noise begins to dominate. As indicated in the manual [47], the PDB150A amplifier on this gain setting generates increasing noise power for frequencies up to 50 MHz, above which it rolls off again rapidly.

Electronics Noise Floor Measurement -95

-100

-105

-110 Spectrum analyser noise -115

Noise power (dBm) Total detector noise

-120 0.1 0.2 0.3 0.4 0.5 0.7 1.0 2.0 3.0 4.0 5.0 7.0 10 20 Analysis frequency (MHz)

Figure 4.12: Measured electrical noise power for the spectrum analyser disconnected from the photodetector (blue) and the addition of ampliﬁer noise when connected to the detector on the 104 V/A gain setting (purple). RBW/VBW was 3/30 kHz.

This noise spectrum is called the “noise ﬂoor” of the measurement, as measurements of signals below this level cannot be easily made. The background has been subtracted from all spectra presented in the results.

4.7 Summary This chapter explained the experimental considerations which arise in implementing the four-wave mixing system. Production of the necessary laser beams was described and the use of an AOM in the cat’s eye configuration to generate the probe beam was recommended. A simple rf driver based on a frequency synthesizer was described and the diffraction efficiency of the AOM was studied to optimise its configuration. The experimentally obtained double-pass conversion efficiency of 1% was comparable to the theoretical maximum of 2.9%. The desirability of an isotopically pure vapour cell and the need for a heater were explained, and the design of a custom cell heater shown.

The quantum efficiency of the PDB150A dual balanced differential photodetector used to simultaneously measure the probe and conjugate intensities was determined and the importance of swapping its photodiodes for more efficient models explained. The response curve of the photodetector was obtained by measuring its output voltage as a function of beam power and was shown to be linear in the regime of interest. Optical losses due to components in the beam path were measured and the overall detection efficiency of the system obtained. Techniques for noise spectrum analysis were discussed and the process of background subtraction outlined, enabling acquisition of relative intensity noise spectra.

Results and Analysis 5

This chapter presents and discusses the results leading to the demonstration of noise reduction below the standard quantum limit. The noise spectrum of the Ti:S laser is analysed and shown to be near shot-noise limited. A similar characterisation of the probe beam illustrates large classical intensity ﬂuctuations which arise from electronic noise on the rf driving signal. Modifying the rf source reduces the intensity ﬂuctuations to an acceptable level.

Properties of the four-wave mixing resonance are investigated, including dependence of the peak gain on two-photon detuning, pump power and vapour cell temperature. Noise spectra of the probe and conjugate beams are characterised and a correction for gain roll oﬀ determined. Noise reduction by four-wave mixing is demonstrated, leading to 3 dB of observed squeezing. The analytic model is used to calculate the properties of the four-wave mixing resonance and determine the maximum level of squeezing for the given conﬁguration. Experimental factors impeding the observation of such noise reduction are outlined.

5.1 Laser noise calibration To generate squeezed light, it is desirable to start with a beam as close to shot-noise limited as possible. The Ti:S laser used in this experiment is a ring-cavity based system and is susceptible to noise at harmonics of the cavity mode spacing. In the Coherent 899, this mode spacing is 255 kHz. Many orders of these harmonics can be clearly observed in the noise spectrum of the directly measured laser intensity (Figure 5.1A).

The Ti:S spectrum is compared to an equal power differential measurement corresponding to the same beam being passed through a 50/50 PBS (Figure 5.1B) and the intensity difference between the two beams measured. These two beams should have the same classical correlations so only shot-noise should remain. A very flat noise spectrum was indeed observed, although noise peaks are still observed at low frequency. These are hypothesised to be due to the interplay of cavity dynamics with the polarisation of the output beam, which interact with the PBS and the differential signal; despite using an initial PBS to purify the polarisation of the Ti:S beam. Regardless, strong suppression (over 12 dB) of the higher mechanical resonances was still observed.

It is worth noting that any classical white-noise present in the Ti:S beam would be (at least partially) cancelled by differential measurement, resulting in a difference between the two measured white-noise levels. The two white-noise levels are observed to be the same within uncertainties, suggesting that the Ti:S beam is shot-noise limited in its white-noise regions – a hypothesis verified in the next section.

5.2 Standard quantum limit measurement The shot-noise level was calibrated using by measuring the electrical noise power using balanced beams at the “standard” analysis frequency of 1 MHz, where the noise spectrum is relatively ﬂat. The white noise level was measured for varying total optical power and found to produce a strongly linear relationship (Figure 5.2).

51 A. Electrical Noise Power Spectrum of 800 μW Beam -65

-70 Direct detection Balanced detection -75 RBW: 30 Hz -80 VBW: 100 Hz Differential balance: 99.14 ± 0.08 % White–noise level: -108.9 ± 0.9 dBm -85

Measured noise power (dBm) Measured -90

-95

-100

-105

-110

-115 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Analysis frequency (MHz)

B. Schematic of Balanced Beam Detection

Ti:S laser

– S.A. Reject

Figure 5.1: Measured electrical noise power (A) of the free-running Ti:S laser (blue) compared to diﬀerential detection (red) using balanced beams (B). The ﬁrst waveplate-PBS pair act to purify the transmitted beam polarisation, while the second pair are used to balance the beam powers to within 1%.

Standard Quantum Limit Calibration 100

60 Measured noise power (pW) Measured 50

Analysis Frequency: 1 MHz 20 RBW: 30 kHz VBW: 3 Hz 10 Gain: 105 V/A

0 0 50 100 150 200 250 300 350 400 Total optical power (μW)

Figure 5.2: Measured white-noise level for diﬀerential detection of beam with varying optical power.

52 The contributions to the measured noise power are electrical noise, shot noise and classical noise. The electrical noise is independent of intensity and produces a non-zero y-intercept in agreement with §4.6.2. √ Shot noise produces Poissonian ﬂuctuations ∆I ∝ I while classical noise is linear ∆I ∝ I. The measured electronic noise power which is proportional to the current squared, so shot-noise and classical noise will add linear and quadratic contributions1.

The measured noise power was fitted with the quadratic y = Ax2 + Bx + C, with the fitting coefficients and corresponding 2σ uncertainties given by

A = (0.5 ± 3.1) × 10−6 pW/ µW2 B = (246 ± 2) × 10−3 pW/ µW C = 2.11 ± 0.1 pW.

For a typical beam power P = 100 µW, the ratio of classical- to shot-noise power is at most AP/B < 0.14% (to within 2σ uncertainties), indicating that the classical noise is suppressed at least 25 dB below the shot-noise.

The classical noise contributions are therefore negligible and only shot-noise has been measured. Hence Figure 5.2 corresponds to the standard quantum limit. As these noise levels coincide with the white- noise level on the Ti:S beam, it may also be concluded that the Ti:S itself is shot-noise limited in its white-noise regions.

This calibration is used to experimentally compute the standard quantum limit (SQL) when investigating relative intensity squeezing across a four-wave mixing resonance. Since the gain varies across the resonance (as characterised in §5.4), the measured intensity (and hence the SQL) also changes with detuning. Each beam is individually super-Poissonian after mixing so it is not possible to directly switch between observing the SQL and squeezing. This is in contrast to measurements based on balanced homodyne detection, where the squeezed input may simply be blocked to recover the SQL. The beam intensities must therefore be individually measured at the detector (see §4.5.2) as the detuning is scanned to compute the SQL and hence determine the degree of squeezing.

5.3 Classical probe noise analysis Although the pump beam was shown to be near shot-noise limited, the power spectrum of the probe beam generated by the AOM was observed to have a signiﬁcantly higher white noise level than the diﬀerence signal and associated SQL value (Figure 5.3).

This is explained by the AOM imparting classical noise to the probe beam it generates. Even considering the AOM to be an ideal device with net diffraction efficiency ηaom, the time-dependence in the intensity of the output is: ∂I ∂I ∂η I = η I ⇒ out = η in + I aom . (5.1) out aom in ∂t aom ∂t in ∂t The first term describes the transmitted (Poissonian) fluctuations originally present in the beam, while the second term describes the fluctuations in the beam introduced by the fluctuating AOM efficiency.

To first order, the AOM efficiency (for a given configuration) can be approximated as only dependent on the rf power fed into the input,

∂ηaom ηaom ≈ η0 + (P − P0). (5.2) ∂P P0 Fluctuations in the driving rf power will therefore result in fluctuations in the diffraction efficiency of the AOM, and hence in the intensity of the beam it produces,

∂ηaom ∂ηaom ∂P ∂Iout ∂Iin ∂ηaom ∂P ≈ ⇒ ≈ ηaom + Iin. (5.3) ∂t ∂P ∂t ∂t ∂t ∂P ∂t P0 P0 1Note that the noise contributions are uncorrelated so the corresponding noise powers add linearly (§4.6.2).

53 Noise Power Spectrum of Probe Beam Direct detection -70 Balanced detection

-72 RBW / VBW: 10 kHz / 30 Hz -74

-76

Noise power (dBm) -78

-80 -82.3 ± 0.6 dBm -82

-84 3 dB

-86 -85.3 ± 0.5 dBm -88

-90 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Analysis frequency (MHz)

Figure 5.3: Measured noise on the probe beam as generated by the double-passed AOM driven at 1540 MHz. The balanced white-noise level (red dashed line) agrees with the SQL and is 3 dB below the direct detection level (blue dashed line). Noise peaks occur at harmonics of the Ti:S cavity mode spacing (255 kHz).

It is therefore pivotal to drive the AOM with a clean rf source to prevent introducing super-Poissonian noise on the probe beam.

The dynamics of the digital frequency synthesizer used to produce the driving rf signal were determined by its “channel width”, which describes the frequency spacing between the different modes it produces. Reducing the channel width allowed finer control over the carrier frequency but was observed to increase the rf noise and decrease stability (Figure 5.4). Furthermore, as the spurs occurred at harmonics of the channel spacing, it was important to choose a channel spacing co-prime with the analysis frequency. A 750 kHz channel spacing was used for most measurements, which suppressed the noise spurs by at least −58 dB relative to the carrier and allowed sufficient control over the probe frequency without introducing significant extra noise.

Measured Noise Power Spectra of AOM Driving Signals -10 Auto-channel 200 kHz channel -20 1 MHz channel Agilent N5181A Noise power (dBc) RBW / VBW: 100 Hz / 300 Hz -30

-40

-50

-60

-70

-80

-90 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency offset (kHz)

Figure 5.4: Power spectrum of rf signal generated by synthesizer for diﬀerent channel settings. Frequency is measured relative to desired value (1518 MHz) and power relative to the carrier. The Agilent N5181 was used for reference, but provided insuﬃcient power to drive the AOM.

54 The classical ﬂuctuations imparted from the AOM are linear in the beam intensity and are therefore super-Poissonian, with the corresponding electric noise power quadratic in beam intensity (see §5.2). Although the quadratic term dominates for larger beam powers, for the 200 kHz channel the noise power is completely dominated by the shot-noise contribution in the regime of interest (below 100 µW).

Technical Probe Beam Noise 3

2.5 Noise power (pW)

1.5

Auto-channel 1 200 kHz channel SQL

RBW: 100 Hz 0.5 0 50 100 150 200 250 300 350 Probe beam power (μW) Measurement region

Figure 5.5: Intensity noise power for double-passed AOM beam. Note that for beam powers ≤ 75 µW the beam is essentially shot noise limited. Dashed lines represent quadratic ﬁts.

The performance of the Analog Devices A4360 was compared to an Agilent N5181A frequency synthesizer. Although the N5181A demonstrated superior noise suppression of −72 dBc at its spurs, the carrier power generated was only −26.5 dBm. After amplification, the rf power driving the AOM was only 17 dBm, resulting in a very low diffraction efficiency (Figure 4.5A) that had to be compensated for with higher optical input power. However, because of the intensity weighting in (5.3) and the slope of the efficiency curve, the probe beam was observed to manifest significantly higher intensity noise than with the ADF4360, even though the rf signal was cleaner.

Although further ampliﬁers could have been used to boost the carrier power of the N5181A synthesizer, the ADF4360 operated suﬃciently well in the region of interest after correcting the channel width. If higher beam powers are desired for use in applications, this consideration must be revisited.

5.4 Four-wave mixing characterisation The four-wave mixing strength depends on many experimental parameters. It was optimised by adjusting the beam alignment to ﬁx the pump-probe angle θ and then tuning the one- and two-photon detunings to maximise the observed gain. Experiments were performed by scanning the Ti:S frequency (corresponding to the 1–photon detuning), while the 2–photon detuning was ﬁxed by the AOM driving frequency. Due to the large parameter space, only peak values were recorded when optimising the gain.

The 2–photon detuning was varied while scanning the pump to investigate the interaction between these eﬀects and a clear optimum observed (Figure 5.6). Increasing the pump power was found to linearly increase both the peak gain and shift the resonant frequency (Figure 5.7). These observations can be 2 explained by the weighting factor Epump ∝ Ipump in the mixing strength (see §2.2) due to the pump stimulating two of the four transitions, and a linear increase in the AC Stark shift of the virtual level |3i with increasing power.

55 Gain Dependence on AOM Frequency 14 490 mW pump

12 330 mW pump Probe gain Probe

0 10 20 30 40 50 60 70 80 90 2-Photon detuning, (MHz)

Figure 5.6: Peak four-wave mixing gain measured for 100 µW probe beam when varying 2–photon detuning and pump power. Dashed lines are guides for the eye.

A. Gain Dependence on Pump Power B. Optimum Detuning Dependence on Pump Power 18 55

16 50

45 14 40 12 35

10 30

Probe gain Probe 8 25

20 6 15 4

Optimum 2-photon detuning, (MHz) 10

2 5 100 150 200 250 300 350 400 450 100 150 200 250 300 350 400 450 Pump power (mW) Pump power (mW)

Figure 5.7: Dependence of peak gain (A) and detuning (B) on pump power.

The gain was also expected to be highly dependent on the vapour cell temperature due to the competing eﬀects of Doppler-broadened absorption and increased mixing strength. Increasing the temperature increases the vapour pressure and enables mixing with more atoms; but it also increases the Doppler width causing absorption of the probe (Figure 5.8). This was seen to lead to an optimum temperature of 125 ◦C, used for most experiments.

5.5 Relative intensity noise spectra The noise spectra of the beams generated by the four-wave mixing interaction were then investigated to observe the correlated fluctuations. Noise spectra for the individual probe and conjugate beams as well as their intensity difference in a low gain (G = 2.4) system were captured and compared (Figure 5.9). Note the amplified fluctuations present at low frequencies and the roll off at high frequencies, with a white-noise region between them. The difference signal shows significant noise cancellation and approaches the SQL in white region, however the noise level increases above approximately 2 MHz.

The decrease in probe and conjugate noise power is due to a roll oﬀ in the ampliﬁer gain at higher frequencies (Figure 5.10). A simple power law model G ∼ A f B was used to quantify this behaviour,

56 Gain Dependence on Vapour Cell Temperature 2.5 2.4

2.3 Probe gain Probe

2.2

2.1

1.9

1.8

1.7

1.6

1.5

1.4 110 115 120 125 130 135 Cell temperature (°C)

Figure 5.8: Temperature dependence of peak gain showing interplay between increasing number density and Doppler absorption. Two-photon detuning was ∆2ph = 6 MHz with beam powers Ppump = 570 mW and Pprobe = 13 µW.

Four-Wave Mixing Noise Power Spectra (530 μW Total)

–85 Probe Conjugate Difference SQL

–90 RBW / VBW: 3 kHz / 30 kHz

–95 Noise power (dBm)

–100

–105 0.1 0.2 0.3 0.4 0.5 0.7 1.0 2.0 3.0 4.0 5.0 7.0 10 20 Analysis frequency (MHz)

Figure 5.9: Measured noise power spectra comparing individual probe and conjugate beams to the diﬀerence signal and standard quantum limit. Dashed lines are a guide for the eye. and was calibrated to the data presented in the manual. The gain is related to the power spectrum by

P 2 2 P(dB) ∝ G ipd ⇒ = 20 log10(G) + 20 log10(ipd) + (const), which was observed to agree well with measured results (Figure 5.11). Hence the intensity noise- spectrum can be recovered by simply subtracting the logarithm of the gain from the logarithm of the noise power. This was observed to extend the white-noise region to nearly 20 MHz (Figure 5.12A), where the decrease in detector gain and increase in ampliﬁer noise make the signal and background contributions comparable in power.

The diﬀerence signal also demonstrates clear classical noise peaks. These noise peaks arise from power imbalances between probe and conjugate beams, which cause the classical ﬂuctuations to remain uncancelled. The noise due to imbalance between the beams can be estimated by inverting the individual power spectra to obtain the corresponding photocurrents, q ∝ P ipd( f ) signal( f ).

57 Frequency Response of PDB150A Transimpedance Amplifier 0

–10 Relative gain (dB) Relative

–20

–30

–40

0.1 0.5 1.0 2.0 5.0 10 20 50 100 200 Frequency (MHz)

Figure 5.10: Approximate frequency response curve of PDB150A photodetector on the 105 gain setting. Estimated using the calibration curve provided by the manufacturer (inset, reproduced from [47]).

Probe and Conjugate Noise Spectra –92

–94

–96 Measured noise (dBm) Measured –98

–100

–102 0.5 0.7 1.0 2.0 3.0 4.0 5.0 7.0 10 20 Analysis frequency (MHz)

Figure 5.11: Measured probe (blue) and conjugate (purple) noise spectra overlaid with expected white-noise levels including gain roll oﬀ (orange).

For perfectly correlated probe and conjugate beams with diﬀerent intensities, the currents would be perfectly correlated and the noise power would subtract coherently. The corresponding “coherent” difference signal would have power spectrum

2 − ⇒ P P1/2 − P1/2 icoherent = iprobe iconj coherent = probe conj . (5.4)

This provides a limit on the power spectrum of the intensity diﬀerence. Comparing the measured noise diﬀerence to this coherent limit (Figure 5.12B), the indicated noise peaks are shown to be entirely due to the classical power imbalance. These noise peaks can only be eliminated by reducing the power imbalance between the beams.

It should also be noted that once corrected for the gain roll off, the difference signal demonstrates clear increase in power with frequency (noise roll up). There are two explanations for this behaviour, and the observed results likely arise from a combination of the two effects.

The ﬁrst explanation is a time-delay between the two beams. It has been previously noted that this double-Λ system is capable of producing long time-delays, with relative delay between the two beams in the order of 8 ns [48]. Due to the high intensity of the pump beam, cross-phase modulation occurs between the mixing beams causing the pump beam to change the refractive index of the probe and conjugate beams diﬀerently, resulting in slower propagation of one beam.

58 A. Gain Corrected Noise Spectra for Probe and Conjugate –75 Probe –80 Conjugate

–85

–90

Noise power (dBm) –95

–100 0.1 0.2 0.3 0.4 0.5 0.7 1.0 2.0 3.0 4.0 5.0 7.0 10 20 Analysis frequency (MHz)

B. Corrected Difference Spectrum and Coherent Estimate Probe –85 Difference Coherent limit –90

–95

–100

–105 Noise power (dBm) –110

–115 0.1 0.2 0.3 0.4 0.5 0.7 1.0 2.0 3.0 4.0 5.0 7.0 10 16 20 Analysis frequency (MHz)

C. Corrected Difference Noise Power Relative to Coherent Estimate 20

0 Noise difference (dB) Noise difference

0.1 0.2 0.3 0.4 0.5 0.7 1.0 2.0 3.0 4.0 5.0 7.0 10 20 Analysis frequency (MHz)

Figure 5.12: Measured noise spectra for probe and conjugate beams corrected for gain roll oﬀ (A). Gain- corrected diﬀerence signal compared to probe beam and coherent limit (B) and noise power above coherent limit (C) showing noise peaks are due to intensity imbalance. RBW/VBW was 3/30 kHz.

Considering the probe and conjugates to be perfectly correlated with a time delay τ, the detector volt- ages are related by Vconj(t) = AVprobe(t − τ) where 0 ≤ A ≤ 1 models the intensity diﬀerence. The corresponding electronic noise power spectrum is then

P 2 2 diﬀ = F [Vprobe(t) − Vconj(t)] = F [Vprobe(t)] − AF [Vprobe(t − τ)] 2πi f τ 2 2 P = (1 − Ae )F [Vprobe(t)] = 1 + A − 2A cos(2π f τ) probe.

This predicts an increase in noise power up to an analysis frequency of f = 1/2τ ∼ 60 MHz for τ ∼ 8 ns. Fitting this simple model to the diﬀerence signal given the gain-corrected noise spectra yields a relative delay of τ ≈ 14 ns, which is within the range of delays observed to be produced by this system [48]. In principle this delay can be compensated for by introducing an optical delay line to the applicable arm, and doing so is known to improve the level of squeezing by a fraction of a decibel [49].

The second eﬀect is the frequency gain bandwidth of the process (§2.4.1). This is controlled by the time-scale of the interaction, which is in turn given by the Raman bandwidth of the process. The pump power is 400 mW, giving Ωpump = 500 MHz, and as the conjugate emission is the bandwidth limiting

59 transition, the applicable 1-photon detuning is ∆1ph ' 4 GHz. The Raman bandwidth [48] is therefore

2 Ωpump ∆R = ∼ 16 MHz, 4∆1ph which agrees well with the observed increase in noise power (Figure 5.12B).

Noise measurements were then made by setting the spectrum analyser to zero-span mode at an analysis frequency of 1 MHz with RBW/VBW of 3/30 kHz. To compare the measured relative intensity noise with the standard quantum limit, noise traces were captured as the pump laser frequency (one-photon detuning) was scanned and compared to the standard quantum limit using the calibration in §5.2. Typical gain proﬁles and noise spectra are shown in Figure 5.13.

A. Beam Powers Generated by Four-Wave Mixing

3.0 Probe 2.5 Conjugate 2.0 Difference

1.5

1.0

0.5

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 -0.5 One-photon detuning (GHz) Beam power relative to incident probe Beam power relative -1.0

B. Four-Wave Mixing Measured Relative Intensity Noise -60

-65

-70

-75

Noise power (dBm) Standard quantum limit -80 Four-wave mixing

-85 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 One-photon detuning (GHz)

Figure 5.13: Measured probe and conjugate beam powers (A) and corresponding relative intensity noise compared to SQL (B) as one-photon detuning is scanned across four-wave mixing resonance.

Scanning the detuning brings the probe laser into resonance with the hyperfine transitions of the D1 line and large Doppler-broadened absorption profiles are observed. Two clear four-wave mixing resonances can be seen around pump detunings of 3.7 GHz and 5.2 GHz above the 5S 1/2, F=3 → 5P1/2 resonance. Competing absorption processes cause the probe to be absorbed on the left resonance and the conjugate to be absorbed on the right resonance, leading to the significantly mismatched beam powers seen in the difference signal. Note that balanced beam losses would produce a difference signal with the same power as the incident probe.

The associated noise spectrum (Figure 5.13B) demonstrates several important features. Firstly, the hyperﬁne transitions of 5S 1/2 → 5P1/2 are observed at detunings of 0 GHz and 3.0 GHz. Although the probe is completely Doppler absorbed, a ﬂat noise level of −77 dBm is recorded in the power spectrum. This is due to scattered pump light reaching the detector and raising the background noise level across

60 the entire scan. Eliminating this would require improved isolation of the beams after mixing by using longer propagation distances and more tightly conﬁned irises as discussed in §4.5.

Secondly, for detunings around 1.5 GHz and above 6.0 GHz the probe beam is transmitted without gain or loss. The measured noise level coincides with the standard quantum limit, indicating that the classical noise on the probe beam at this power is negligible and the beam is shot-noise limited.

Thirdly, neither of the two four-wave mixing resonances demonstrate sub-shot noise ﬂuctuations due to the mismatched beam probe and conjugate powers. In particular, the ﬁrst resonance around 3.7 GHz shows strong probe absorption leading to noise levels several decibels above the SQL. The second resonance at 5.2 GHz demonstrates a small amount of absorption on the conjugate leading to closer power matching, which reduces the measured relative intensity noise, but fails to fall below the SQL in this instance.

These results are typical of the four-wave mixing system when improperly optimised, but provide insight into the processes occurring.

5.6 Observations of noise reduction and squeezing With an understanding of how four-wave mixing affects the noise spectrum of the beams generated by the process, and of the available parameter space, the system was optimised to produce relative intensity squeezed light. Peak gains of up to 40 were observed when operating very close to resonance, but relative intensity fluctuations significantly above the shot noise limit were measured because of strong competing absorption. The system was therefore tweaked to produce relatively low gain (G ∼ 2) at frequencies beyond the Doppler absorption profile.

Figure 5.14A demonstrates the intensity noise measured at 1 MHz for one such resonance, obtained ◦ with 480 mW pump power, cell temperature 120 C and two-photon detuning ∆2ph = 8 MHz. Strong gain is observed and near balancing of probe and conjugate beam powers over the broad resonance about a detuning of 1 GHz, indicating low absorption. The measured relative intensity noise power (Figure 5.14B) is approximately flat between the hyperfine resonances, corresponding to the transmitted intensity fluctuations in the probe beam.

As discussed in §4.6.2, leakage of the pump light raises the background noise level and may be quantified by measuring the noise when the probe is fully absorbed. This background level is 5.5 pW, which combined with the shot-noise of 11 pW present on the 35 µW incident probe, leads to an expected flat noise level of 16 pW between the hyperfine transitions. Experimentally 20 pW was recorded because of unbalanced beam losses increasing the noise level.

Computing the measured intensity noise relative to the standard quantum limit shows a peak noise reduction of 3 dB (Figure 5.14C). Subtracting the scattered pump noise from this measurement suggests a noise level of −4.1 dB relative to the SQL could be obtained by eliminating cross-detection of the pump beam.

The theoretical model derived earlier (§3.3) can be used to determine a limit on the experimentally achievable degree of squeezing. From the measured probe and conjugate gain curves, the intrinsic gain

G and probe loss Ta parameters can be computed across the scan (Figure 5.15), and the competition between gain and loss illustrated. Feeding these coeﬃcients back into the analytic model produces the expected level of squeezing (Figure 5.16), demonstrating that the maximum squeezing observable from this resonance is −5.6 dB .

61 A. Measured Probe and Conjugate Gain 180 160 Probe 140 Conjugate 120 Total 100 Difference 80 60 Beam power (μW) 40 20 0 0 0.5 1 1.5 2 2.5 3 Detuning (GHz) B. Measured Relative Intensity Noise 50

40 Computed SQL Measured noise 30

20 Noise power (pW)

0 0 0.5 1 1.5 2 2.5 3 Detuning (GHz) C. Measured Squeezing 12

4 –3.0 dB

2 Noise figure (dB) Noise figure 0

–2

–4 0 0.5 1 1.5 2 2.5 3 Detuning (GHz)

Figure 5.14: Measured probe and conjugate gain (A), relative intensity noise (B) and corresponding noise ﬁgure (C) demonstrating −3 dB of squeezing. RBW/VBW was 3/30 kHz.

The measured noise level is signiﬁcantly above the theoretical estimate produced by the distributed gain/loss model. In addition to the reasons described above, inverting the gain proﬁles for the model parameters assumes that gain occurs throughout the cell. If the pump and probe beams are not completely overlapped, there will be regions of decreased mixing where the intensity is reduced. Not only does this make the gain spatially varying, but it increases the strength of absorption relative to squeezing and increases the measured noise level. This implies that further noise reduction could be achieved by more careful alignment and mode-matching of the pump and probe beams within the vapour cell.

However, both measured and predicted squeezing values fall far short of the −8.8 dB obtained by the Lett group in this system [22]. Although limited by the technical diﬃculties described above, this dis- crepancy principally arises from diﬃculties in optimising for noise reduction. Sensitivity to interplay between incidence angle, beam detuning and power means that maximising the observed gain does not maximise the degree of squeezing (see e.g. [50, Fig 3]). The procedure employed to search parameter space (§4.3) therefore produces a sub-optimal combination necessitating an alternative approach. Since the SQL changes with gain (§5.2) this is a non-trivial optimisation problem, and a comprehensive parameter search was beyond the scope of this project.

62 A. Measured Probe and Conjugate Gain

Probe gain 2.5 Conjugate gain Difference

2.0

1.5

1.0

0.5

0.5 1.0 1.5 2.0 2.5 Detuning (GHz)

B. Computed Gain and Transmission Parameters

Mixing gain 2.5 Probe transmission

2.0

1.5

1.0

0.5

0.5 1.0 1.5 2.0 2.5 Detuning (GHz)

Figure 5.15: Measured probe and conjugate beam gain (A) and corresponding mixing gain and probe transmission coeﬃcients (B) across four-wave mixing resonance.

A. Measured Noise Power Compared to Theoretical Estimate 40 Standard quantum limit 35 Measured noise Model prediction 30

25 Noise power (pW)

0.5 1.0 1.5 2.0 2.5 Detuning (GHz) B. Measured Squeezing Compared to Theoretical Estimate

Noise figure (dB) Noise figure Detuning (GHz)

0.5 1.0 1.5 2.0 2.5

–2

–4 Measured noise Model prediction

–6

Figure 5.16: Comparison of measured relative intensity noise to theoretical estimate (A) and comparison of corresponding squeezing values (B). Theoretical curves produced by distributed gain/loss model with parameters obtained from measured probe and conjugate gains. RBW/VBW was 3/30 kHz.

Conclusions 6

This thesis has presented a theoretical and experimental investigation of the emerging technique of relative intensity squeezed light production by four-wave mixing in a hot rubidium vapour. By combining a theoretical understanding and analysis of the underlying processes with an evaluation of the experimental complications, a thorough understanding of the capabilities of the system was obtained. This work therefore comprises a broad knowledge base valuable to further research that seeks to exploit relative intensity squeezed light for precision measurement.

A ﬁrst-principles analysis of the double–Λ four-wave mixing scheme demonstrated that it can produce relative intensity squeezed light, and predicted the level of noise reduction. The phase-matching conditions were analysed and an expression for the pump-probe angle corresponding to ideal matching derived. Deviations away from ideal matching were considered and a spatial bandwidth which permitted gain over a range of incident wave-vectors was observed. The production of squeezed images by four-wave mixing with correlated sub-regions was considered, and a limit on the size of the smallest correlated sub-regions observed as a consequence of the spatial bandwidth.

The degradation of relative intensity squeezed light by optical losses was also evaluated. Novel analytic expressions predicting the measured relative intensity noise were derived and seen to be highly sensitive to unbalanced losses. However, slightly higher losses on the probe than on conjugate was shown to reduce the predicted noise by a fraction of a decibel, even below ideal detection in some cases. An interleaved gain and loss model was developed to analyse the competing eﬀects of squeezing and absorption. A novel technique was developed to consider the continuum limit, and applied to derive an expression for the level of squeezing produced in competition with absorption. The matrix methods approach created for this purpose can be applied to any system where losses are a competing process, making it a potentially powerful theoretical tool.

Four-wave mixing resonances were experimentally observed, and the intensity noise spectra of the resulting beams were characterised. The dependence of mixing gain on beam power, cell temperature and laser detuning were determined and explained physically. Relative intensity squeezing of 3 dB below the standard quantum limit was demonstrated. The mixing resonance was analysed using the theoretical model and the maximum possible squeezing evaluated. Finally, the experimental factors limiting the observed level of squeezing were discussed, and modiﬁcations that would improve the degree of measured squeezing suggested.

In conclusion, the production of relative intensity squeezed light through four-wave mixing in an atomic vapour is a straightforward and powerful technique. The observation of strong squeezing in this system demonstrates its capability to push measurement boundaries. The near-resonant beams it produces combined with its inherently multi-mode nature makes it ideally suited for low-light atom imaging and quantum-information processing. It therefore provides an extremely versatile experimental technique for performing measurements beyond the standard quantum limit.

References

[1] P. C. D. Hobbs. Ultrasensitive laser measurements without tears. Applied Optics, 36(4):903–920, 1997.

[2] C. M. Caves. Quantum-mechanical radiation-pressure ﬂuctuations in an interferometer. Physical Review Letters, 45(2):75–79, 1980.

[3] B. L. Schumaker. Noise in homodyne detection. Optics Letters, 9(5):189–191, 1984.

[4] C. M. Caves. Quantum-mechanical noise in an interferometer. Physical Review D, 23(8):1693– 1708, 1981.

[5] K. Goda, O. Miyakawa, E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. Weinstein, and N. Mavalvala. A quantum-enhanced prototype gravitational-wave detector. Nature Physics, 4(6):472–476, 2008.

[6] M. Xiao, L.-A. Wu, and H. J. Kimble. Precision measurement beyond the shot-noise limit. Physi- cal Review Letters, 59(3):278–281, 1987.

[7] I. V. Sokolov and M. I. Kolobov. Squeezed-light source for superresolving microscopy. Optics Letters, 29(7):703–705, 2004.

[8] E. S. Polzik, J. Carri, and H. J. Kimble. Spectroscopy with squeezed light. Physical Review Letters, 68(20):3020–3023, 1992.

[9] M. I. Kolobov and P. Kumar. Sub-shot-noise microscopy: imaging of faint phase objects with squeezed light. Optics Letters, 18(11):849–851, 1993.

[10] W. Heisenberg. Uber¨ den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift f¨urPhysik A, 43(3):172–198, 1927.

[11] H. P. Robertson. The uncertainty principle. Physical Review, 34(1):163–164, 1929.

[12] L. Davidovich. Sub-poissonian processes in quantum optics. Reviews of Modern Physics, 68(1):127–173, 1996.

[13] Y. R. Shen. The principles of non-linear optics. John Wiley and Sons, Inc., 1984.

[14] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley. Observation of squeezed states generated by four-wave mixing in an optical cavity. Physical Review Letters, 55(22):2409– 2412, 1985.

[15] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley. Squeezed states in optical cavities: A spontaneous-emission-noise limit. Physical Review A, 31(5):3512–3515, 1985.

[16] H. A. Bachor and T. C. Ralph. A Guide to Experiments in Quantum Optics. Wiley-VCH Verlag GmbH & Co, 2004.

67 [17] M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel. Observation of squeezed states with strong photon-number oscillations. Phys. Rev. A, 81(1):013814, Jan 2010.

[18] G. Hetet, O. Glockl, K. A. Pilypas, C. C. Harb, B. C. Buchler, H.-A. Bachor, and P. K. Lam. Squeezed light for bandwidth-limited atom optics experiments at the rubidium D1 line. Journal of Physics B, 40(1):221–226, 2007.

[19] C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett. Strong relative intensity squeezing by four-wave mixing in rubidium vapor. Optics Letters, 32(2):178–180, 2007.

[20] A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy. Observation of quantum noise reduction on twin laser beams. Physical Review Letters, 59(22):2555–2557, 1987.

[21] S. Reynaud, C. Fabre, and E. Giacobino. Quantum ﬂuctuations in a two-mode parametric oscillator. Journal of the Optical Society of America B, 4(10):1520–1524, 1987.

[22] C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett. Strong low-frequency quantum correlations from a four-wave-mixing ampliﬁer. Physical Review A, 78(4):043816, 2008.

[23] V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett. Entangled images from four-wave mixing. Science, 321(5888):544, 2008.

[24] R. W. Boyd. Nonlinear Optics. Academic Press, second edition, 2003.

[25] M. O. Scully and M. S. Zubairy. Quantum optics. Cambridge University Press, 1997.

[26] T. T. Grove, M. S. Shahriar, P. R. Hemmer, P. Kumar, V. S. Sudarshanam, and M. Cronin-Golomb. Distortion-free gain and noise correlation in sodium vapor with four-wave mixing and coherent population trapping. Optics Letters, 22(11):769–771, 1997.

[27] M. Stahler,¨ R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev, and V. Yudin. 85 Coherent population trapping resonances in thermal Rb vapor: D1 versus D2 line excitation. Optics letters, 27(16):1472–1474, 2002.

[28] M. D. Lukin, P. R. Hemmer, M. Lo¨ﬄer, and M. O. Scully. Resonant enhancement of parametric processes via radiative interference and induced coherence. Physical Review Letters, 81(13):2675– 2678, 1998.

[29] C. C. Gerry and P. L. Knight. Introductory Quantum Optics. Cambridge University Press, 2005.

[30] D. Stoler. Equivalence classes of minimum uncertainty packets. Physical Review D, 1(12):3217– 3219, 1970.

[31] H. P. Yuen. Two-photon coherent states of the radiation ﬁeld. Physical Review A, 13(6):2226– 2243, 1976.

[32] J. N. Hollenhorst. Quantum limits on resonant-mass gravitational-radiation detectors. Physical Review D, 19(6):1669–1679, 1979.

[33] R. Loudon. The Quantum Theory of Light. Oxford University Press, 2003.

[34] R. C. Pooser, V. Boyer, A. M. Marino, and P. D. Lett. Squeezed light and entangled images from four-wave-mixing in hot rubidium vapor. arXiv:0811.1243v1 [quant-ph], Nov 2008.

[35] R. H. Good. Classical Electromagnetism. Saunders College Publishing, 1999.

68 [36] V. Boyer, A. M. Marino, and P. D. Lett. Generation of spatially broadband twin beams for quantum imaging. Physical Review Letters, 100(14):143601, 2008.

[37] Z. Y. Ou, L. J. Wang, and L. Mandel. Vacuum eﬀects on interference in two-photon down conversion. Physical Review A, 40(3):1428–1435, 1989.

[38] M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb. Tools for multimode quantum information: Modulation, detection, and spatial quantum correlations. Physical Review Letters, 98(8):083602, 2007.

[39] R. Loudon. Theory of noise accumulation in linear optical-ampliﬁer chains. IEEE J. Quantum Elect., 21(7):766 – 773, jul 1985.

[40] C. M. Caves and D. D. Crouch. Quantum wideband traveling-wave analysis of a degenerate parametric ampliﬁer. Journal of the Optical Society of America B, 4(10):1535–1545, 1987.

[41] J. R. Jeﬀers, N. Imoto, and R. Loudon. Quantum optics of travelling-wave attenuators and ampli- ﬁers. Physical Review A, 47(4):3346–3359, 1993.

[42] D. A. Smith and I. G. Hughes. The role of hyperﬁne pumping in multilevel systems exhibiting saturated absorption. Americal Journal of Physics, 72(5):631–637, 2004.

[43] E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jeﬀerts. Double-pass acousto-optic modulator system. Review of Scientiﬁc Instruments, 76(6):063112, 2005.

[44] Analog Devices Corporation. ADF4360-4: Integrated Synthesizer and VCO, 2004. Avail- able online at http://www.analog.com/static/imported-files/data_sheets/ADF4360-4.pdf. Retrieved March 2008.

[45] H. J. Metcalf and P. van der Straten. Laser Cooling and Trapping. Springer-Verlag New York, 1999.

[46] D. A. Steck. Rubidium 85 D line data, May 2008. Available at http://steck.us/alkalidata/. Retrieved July 2008.

[47] Thorlabs. PDB100 Series Balanced Ampliﬁed Photodetectors, April 2007. Available online at http://www.thorlabs.com/Thorcat/12400/12405-D02.pdf. Retrieved October 2007.

[48] V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett. Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor. Physical Review Letters, 99(14):143601, 2007.

[49] A. M. Marino, V. Boyer, and P. D. Lett. Violation of the Cauchy-Schwarz inequality in the macro- scopic regime. Physical Review Letters, 100:233601, 2008.

[50] C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett. Relative-intensity squeezing at audio frequencies using four-wave mixing in an atomic vapor. arXiv: quant-ph/0703111v1, Mar 2007.

[51] P. C. D. Hobbs. Building Electro-Optical Systems: Making It All Work. Wiley-Interscience, 2000.

[52] L. Mandel. Image ﬂuctuations in cascade intensiﬁers. British Journal of Applied Physics, 10:233– 234, 1959.

[53] R. E. Burgess. Homophase and heterophase ﬂuctuations in semiconducting crystals. Discussions of the Faraday Society, 28:151–158, 1959.

Statistical derivations A

This appendix contains the statistical identities and standard results used throughout this thesis. Note that in many instances, care must be taken in applying standard statistical identities to operators due to their commutators. A statistical loss model is also included, as an alternate and more physically intuitive derivation of the optical loss variance equation (3.6).

A.1 Statistical identities The notation hXi is used to denote the expected value (or “ensemble average”) of the operator X in a given state, with the associated variance and covariance deﬁned by:

Var (X) ≡ h(X − hXi)2i = hX2i − hXi2 (A.1) CoVar (X, Y) ≡ h(X − hXi)(Y − hYi)i = hXYi − hXihYi. (A.2)

Taking account of commutators, the following identities then follow immediately.

Var (X) ≡ CoVar (X, X) (A.3) CoVar (X, Y) = CoVar (Y, X) + h[X, Y]i (A.4) ⇒ Var (X ± Y) = Var (X) + Var (Y) ± (2 CoVar (X, Y) − h[X, Y]i). (A.5)

Furthermore, the following scaling laws apply for constants α, β:

hαX + βi = αhXi + β (A.6)

CoVar (α1X1 + β1, α2X2 + β2) = α1α2 CoVar (X1, X2) (A.7) ⇒ Var (αX + β) = α2Var (X) . (A.8)

Note that the variance relations do not apply if the constant α models a statistical process (such as random losses), in which case the variance must be calculated formally (see §A.4).

In the case of several variables {Xi} which are mutually orthogonal, the commutators in the above expressions disappear, simplifying the expected value and variance of the operator sum to:

P P h i Xii = i hXii (A.9) P P Var i Xi = i, j CoVar Xi, X j . (A.10)

A.2 Coherent state variances Coherent states are the quantum analogue of classical plane waves, and are deﬁned as eigenstates of the photon annihilation operatora ˆ. The coherent state |αi obeysa ˆ |αi = α |αi where α ∈ C, and can be expressed as a power series in terms of the number eigenstates as:

∞ n − 1 |α|2 X α |αi = e 2 √ |ni . (A.11) n=0 n!

71 The average number of photons populating this mode is: haˆ†aî = hα| aˆ† aˆ |αi = α∗α = |α|2, (A.12) with corresponding number variance: Var aˆ†aˆ = haˆ†aâˆ†aî − haˆ†aî2 = h(â†)2aˆ2i + haˆ†[â, aˆ†]âi − haˆ†aî2 = (α∗)2α2 + haˆ†aî − |α|4 = haˆ†aî = |α|2. (A.13)

The expectation value of the quadrature operator (§2.3.2), which describes the electric field in terms of its phase θ, in a coherent state is: hQˆ(θ)i = √1 heiθaˆ† + e−iθaî = √1 eiθhaˆ†i + e−iθhaî = √1 eiθα∗ + e−iθα √2 2 2 = 2 |α| cos (θ − ϕ) where α = |α| eiϕ. (A.14)

In particular, the amplitude and phase quadrature expectation values are: √ √ hXˆi ≡ hQˆ(0)i = 2 |α| cos(φ) = 2 Re(α) , √ √ ˆ ˆ π hYi ≡ hQ( 2 )i = 2 |α| sin(φ) = 2 Im(α) .

The complex eigenvalue α of a given coherent state can therefore be written as the composition of the n o two quadrature expectation values as α = √1 hXî + ihYî . 2 The variance in a given quadrature measurement is: ˆ 1 iθ † −iθ Var Q(θ) = 2 Var e aˆ + e aˆ 1 2iθ † † † −2iθ = 2 e Var aˆ + 2 CoVar aˆ , aˆ − h[â , aˆ]i + e Var (aˆ) † 1 † 1 = CoVar aˆ , aˆ − 2 h[â , aˆ]i = 2 . (A.15) where the variance and covariance terms vanish due to the coherent state being an eigenstate ofa ˆ, and the canonical commutator between ladder operators is [â, aˆ†] = 1. Hence the variance is independent of which quadrature θ is being measured, and a coherent state is said to manifest equal fluctuations in all quadratures.

A.3 Intensity shot noise derivation The precise measurement of small intensity changes in laser beams is pivotal to many experiments in optics, where fluctuations in the incident beam intensity act to obscure the results being obtained. Technical noise arising from mechanical oscillations, finite efficiencies and imperfect equipment are often the dominant source of noise in any experiment, and a wealth of experimental techniques exist to mitigate these problems [51]. However, there is a fundamental limit to how precise a measurement of classical intensity can be made.

Consider a laser beam with angular frequency ω and average power P, measured over a ﬁnite time interval τ during which N photons are observed. Each photon carries energy ~ω so the measured power is therefore (number of photons) × (photon energy) N~ω P = = . (A.16) (measurement interval) τ Using the identity Var (ξX) = ξ2Var (X) for constant ξ (see §A.1), the variance in the measured power over a ﬁxed measurement time τ is !2 ~ω Var (P) = Var (N) . (A.17) τ

72 Ideally, a laser generates a coherent state |αi. In this state, the variance in the observed number of photons is equal to the mean number of photons, Var (N) = hNi (§A.2). Hence the variance in the measured power is !2 ! ~ω ~ω Var (P) = hNi = P = 2 ~ω P ∆f ∝ P. (A.18) τ τ where ∆f ≡ 1/2τ is the bandwidth of the measurement, also called the Nyquist frequency.

The corresponding fractional uncertainty in the measured laser power is √ r ∆P Var (P) 2 ~ω ∆f 1 = = ∝ √ , (A.19) P P P P which is the classic “one-over-root-N” uncertainty relation that arises for Poisson processes such as photon counting.

This result is known as the shot-noise limit of the measurement. Since it arises directly from quan- tised detection of the electromagnetic ﬁeld, it describes the most precise measurement possible using a coherent state.

It is also important to note that (A.19) contains no explicit dependence on time, so the Fourier transform of the intensity signal is independent of the analysis frequency. The corresponding power spectrum therefore contains fluctuations of equal noise power level at all frequencies, a phenomenon known as “white” noise. This is in contrast to other sources of noise such as mechanical resonances, which generate noise at specific resonant frequencies, and “flicker” noise (also called 1/f or “pink” noise), which decreases in power at higher frequencies. Hence it is not possible to reduce the degree of shot- noise fluctuations by performing measurements at a carefully selected analysis frequency.

The only way to reduce the fractional noise power in a given bandwidth ∆f using a coherent state is to increase the laser power. In some low-light experiments this is not feasible, so to overcome the shot noise limit we must use something other than a coherent state for the measurement – we require a squeezed mode of the electromagnetic ﬁeld.

A.4 Statistical loss model Optical losses were previously analysed in §3.1 by analysing the quantum beam-splitter and obtaining a transformation for the annihilation operator describing the beam. The variance of the output beam intensity was then shown to contain a vacuum term which arose from its commutator. Historically, this result was obtained instead by considering losses to be a random statistical process which ejected photons from the incident beam and computing the variance of this process. This analysis is presented here as an alternate explanation of losses, to both augment the earlier discussion and provide further physical insight into its meaning.

The loss mechanism is modelled as a random process which operates on each photon independently and has a ﬁxed probability η of transmission. For a given number M of incident photons, the number of photons transmitted by the medium therefore follows a binomial distribution ! M P(Transmit N of M) = ηN (1 − η)M−N . (A.20) N However, the number of photons incident on the medium follows its own distribution. The distribution of photons transmitted by the medium is therefore a binomial selection from the number distribution of the incident beam. Thus the probability of measuring N photons is X∞ P(Transmit N photons) = P(Start with M photons) × P(Transmit N of M) (A.21) M=N

73 P(Start with M photons) = | hM | ψi |2 (where |ψi is the initial beam state). (A.22)

The expected number of photons transmitted hNi can then be related to the initial expected number of photons hN0i by solving (A.21) to obtain X∞ hNi ≡ M × P(Transmit M photons) = ηhN0i. (A.23) M=0 Similarly, the variance of the number of photons transmitted by the medium is, by deﬁnition X∞ Var (N) ≡ (M − hNi)2P(Measure M photons). (A.24) M=0 As ﬁrst shown by Mandel [52], this expression has a simple closed form in the case of the binomial selection process (A.21) 2 Var (N) = η Var (N0) + η(1 − η)hN0i. (A.25)

This result is known as the “Burgess variance theorem” (after [53]). The ﬁrst term in (A.25) is the attenuated variance corresponding to the part of the original distribution that was transmitted, whilst the second term accounts for ﬂuctuations arising from the random ejection of photons from the beam. This second part is known generally as “partition noise”, because it arises from the partitioning of photons into those that are transmitted and those that are not.

74 Distributed gain/loss model supplementals B

B.1 Numerical algorithm code listing The listing below contains the algorithm used to implement the ﬁnite stage distributed gain/loss model of

§3.3, written in Matlab. The (2N +4)×1 matrices A and B contain the coeﬃcients xi and yi respectively, with the extra two elements used to model optical losses introduced by the detector stage.

1 function snr = calc_sqz(N,G,T1,T2,TD) 2 %%%%%%%%%%%%%%%%%%%%%%%%%% calc_sqz(N,G,T1,T2,TD) %%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Computes the expected intensity difference squeezing for a two-mode 4 % squeezing system with intrinsic internal gain G and distributed transmission 5 % factor T1 on the probe beam and T2 on the conjugate. 6 % The net detection efficiency after the media (including losses) is TD. 7 % Note: TD can be a scalar or 2-element vector for different arm efficiencies 8 % 9 % Returns the "noise to signal ratio" - that is, the squeezed noise divided 10 % by the classical noise limit (i.e. the number of photons) 11 % 12 % Calculation is performed using a differential gain/loss system whereby 13 % consecutive stages of gain corresponding to squeezing and loss 14 % corresponding to absorption are combined. This is an approximation 15 % of the number of continuous vacuum modes coupled to the system, and 16 % becomes exact in the limit N->infinity (see Jeffers’ treatise on losses) 17 % 18 % The parameter N is the number of stages to use - increase N for a 19 % more accurate result at the expense of computation time and memory 20 % usage (going up as 2N). 21 % 22 % Valid in the "bright beam" regime for the probe and a vacuum injected 23 % (unseeded) conjugate 24 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 25 snr = 1; % default value 26 27 if nargin < 2, G = 1; end 28 if nargin < 3, T1 = 1; end 29 if nargin < 4, T2 = 1; end 30 if nargin < 5, TD = 1; end 31 32 % check that there’s something to do 33 if G == 1, return , end 34 35 % compute incremental squeeze parameter from overall gain 36 ds = acosh(sqrt(G)) / N; % s is additive 37 % compute squeeze coefficients 38 a = cosh(ds); 39 b = sinh(ds); 40 % compute incremental loss coefficient 41 t = sqrt([T1 T2].ˆ(1/N)); % t is multiplicative 42 r = sqrt(1-t.ˆ2);

75 43 44 % generate coefficient matrix 45 A = zeros(2*N+4,1); B = A; 46 A(1) = 1; B(2) = 1; 47 48 % compute incremental solution 49 for i=1:N 50 A0 = A; B0 = B; 51 % expand operators for gain then loss 52 A = t(1)*(a*A0+b*B0); 53 B = t(2)*(b*A0+a*B0); 54 % entry (i+2) in the vector is the coupling to the i’th vacuum mode 55 A(2*i+1) = r(1); B(2*i+2) = r(2); 56 end 57 58 % apply detection efficiency losses 59 TD = TD.*[1 1]; t = sqrt(TD); r = sqrt(1-TD); 60 A = A*t(1); B = B*t(2); 61 % couples both probe and conj to vacuum 62 A(2*N+3) = r(1); B(2*N+4) = r(2); 63 64 % compute signal to noise ratio - now in tasty compact formula! 65 % in the bright beam approximation, this is the squeezing to classical noise ratio 66 pwr = (A(1)ˆ2+B(1)ˆ2); 67 if pwr == 0, return , end 68 snr = sum((A(1)*A-B(1)*B).ˆ2)/pwr;

Listing B.1: Gain/loss algorithm implemented in MATLAB

B.2 Parameter estimation The analytic results of §3.3.2 may also be inverted to infer values for the model parameters from experimental measurements. In particular, the ratios of detected probe and conjugate powers to the input power 2 2 can be easily measured experimentally, and correspond to the model coeﬃcients x1 and y1 respectively.

The model has three formal parameters: the squeeze factor S , probe transmission Ta and conjugate

transmission Tb. In practice, the conjugate is suﬃciently far-detuned (approximately 4 GHz) from res-

onance that its absorption within the cell is negligible. The conjugate transmission is therefore Tb = 1, and the two remaining parameters can be obtained by solving the matrix exponential to obtain the probe and conjugate power,       x 1 1/4 4 ξ cosh(ξ) + log(T ) sinh(ξ)  1   Ta  a    = exp(A0)   =   . y1 0 4 ξ 4 S sinh(ζ)

1 p 2 2 where ξ = 4 16S + (log Ta) . Including a correction factor η for ﬁnite optical losses, the observed probe and conjugate gains are

!2 √ p log T sinh(ξ) η T S 2 sinh(ξ)2 x2 = η T cosh(ξ) + a , y2 = a . 1 a 4ξ 1 ξ2

These equations can be simultaneously solved numerically for the two unknowns, with an initial estimate provided by the ﬁrst approximation

! 2 y x x2 ∼ η T cosh2(S ) y2 ∼ η sinh2(S ) ⇒ S ∼ asinh √1 T ∼ 1 . 1 a 1 η a 2 η + y1

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s: JASPERSE, MARTIJN

Title: Relative intensity squeezing: by four-wave mixing in rubidium

Date: 2010

Citation: Jasperse, M. (2010). Relative intensity squeezing: by four-wave mixing in rubidium. Masters Research thesis , School of Physics, The University of Melbourne.

Persistent Link: http://hdl.handle.net/11343/35487

File Description: Relative intensity squeezing: by four-wave mixing in rubidium

Terms and Conditions: Terms and Conditions: Copyright in works deposited in Minerva Access is retained by the copyright owner. The work may not be altered without permission from the copyright owner. Readers may only download, print and save electronic copies of whole works for their own personal non-commercial use. Any use that exceeds these limits requires permission from the copyright owner. Attribution is essential when quoting or paraphrasing from these works.