Quantum Enhanced Precision Estimation of Transmission with Bright Squeezed Light

G. S. Atkinson,1, 2, ∗ E. J. Allen,1, † G. Ferranti,1 A. R. McMillan,1 and J. C. F. Matthews1, ‡

1Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering,University of Bristol, Tyndall Avenue, BS8 1FD, United Kingdom.

2Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering,University of Bristol, Tyndall Avenue, BS8 1FD, United Kingdom.

(Dated: February 25, 2021)

Squeezed light enables measurements with sensitivity beyond the quantum noise limit (QNL) for opticaltechniques such as spectroscopy, gravitational wave detection, magnetometry and imaging. Precision of a mea-surement — as quantified by the variance of repeated estimates — has also been enhanced beyond the QNLusing squeezed light. However, sub-QNL sensitivity is not sufficient to achieve sub-QNL precision. Further-more, demonstrations of sub-QNL precision in estimating transmission have been limited to picowatts of probepower. Here we demonstrate simultaneous enhancement of precision and sensitivity to beyond the QNL forestimating modulated transmission with a squeezed amplitude probe of 0.2 mW average (25 W peak) power,which is 8 orders of magnitude above the power limitations of previous sub-QNL precision measurements oftransmission. Our approach enables measurements that compete with the optical powers of current classicaltechniques, but have both improved precision and sensitivity beyond the classical limit.

Optical measurements are fundamentally limited by quan-tum fluctuations in the probe. The Poisson distributed pho-ton number n of coherent light – often used as a probe inclassical experiments – results in shot-noise, which representsthe quantum noise limit (QNL) in the precision of parame-ter estimation with classical resources [1]. Because the QNLscales with ∼ 1/

√n, longer measurements and higher inten-

sity can increase precision. We may also increase precisionwith more interaction between probe and sample via multiplepasses [2, 3] or optimising sample length [4]. However, therecan often exist restrictions on the total optical exposure, themeasurement time and sample properties [5]. By using non-classical light, the fluctuations in the probe can be reducedbelow the QNL, thus providing ‘sub-shot-noise’ precision perphoton [6]. The QNL defines the best precision achievablewithout the use of quantum correlations for a given apparatusand average photon number [7]. This is distinguished from thestandard quantum limit (SQL), which defines a measurement-independent limit to the precision that may be achieved usinga minimum uncertainty state of a given average photon num-ber, without quantum resources [8]. Because squeezed lightcan offer significant reduction in noise below the QNL [9],and can be generated with arbitrary intensity using coherentlaser light [10], it offers a practical approach for enhancingoptical techniques beyond classical limitations.

Precision in measuring a parameter can be quantified bythe inverse of the variance of corresponding measurement out-comes, and is bounded by the Fisher information according tothe Cramer-Rao bound [11]. By contrast, the sensitivity ofa measurement is the smallest possible signal that may be ob-served [12], and thus depends only on the signal-to-noise ratio(SNR). Photonic (definite photon number) quantum metrol-ogy uses photon counting to observe quantum correlations be-tween modes to reduce the noise of a measurement [13–15].Here one can attain a quantum advantage in both precision andsensitivity, by increasing the Fisher information and reducingthe optical noise floor. However, due to limitations in both

the maximum photon flux and detector saturation power, theprobe powers achievable are in practice O(106) photons de-tected per second (pW) [13, 14], which limits use to only casesthat are reliant on ultra low intensities. Homodyne detectionof squeezed vacuum has been used to estimate phase with sub-QNL precision [16, 17]. This is possible because measure-ment is performed away from low frequency technical noise,in a shot-noise limited bandwidth where squeezing reducesvacuum noise. However, as with photonic quantum metrol-ogy, strategies using squeezed vacuum for sub-QNL precisionhave also been restricted in maximum optical probe power.

Measurements using high power squeezed light can reachsub-QNL sensitivity in detection of phase modulation [6] andamplitude modulation (AM) [18]. This is because modula-tion introduces AC components in the detected signal, whichcan be made to coincide with a shot-noise-limited detectorbandwidth, while squeezed light reduces the optical noiserelative to the signal. The sensitivity of any frequency do-main measurement of an optical signal in a shot-noise lim-ited bandwidth may be improved by such techniques and thishas been demonstrated in a range of applications (e.g. [7, 19–30]). However, enhancing sensitivity is not a sufficient con-dition to enhance precision. When bright optical probes areused, the noise in the bright field often dominates over thevacuum noise, which prohibits the use of squeezed light forreaching precision beyond the QNL. For squeezed light toprovide a precision improvement in such a measurement, thevariance of the measured signal must be limited by opticalshot-noise. Here, we fulfil this condition and use bright ampli-tude squeezed light to measure modulated transmission withprecision beyond the QNL.

The parameter estimated in this work is the modulation in-dex δm = (P−P ′)/P , where P and P ′ are the maximum andminimum output power transmitted through a modulated loss(Fig. 1(e)) [1]. For modulation frequency Ω, sinusoidal AMgenerates optical sidebands at ±Ω from the carrier frequency.Upon photodetection, this leads to a single electronic side-

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Figure 1. Modelling enhanced precision measurement of δm, and experimental setup. (a-c) Plots of spectral noise power illustrating the effectof amplitude squeezing and modulation on a typical laser source, with the quadrature diagrams showing a coherent state defined by x, p at±Ω.(d) Theoretical model of the Fisher information per detected photon F ′(δm) for typical laser light which is quantum noise limited at Ω (solidline) and squeezed light (dashed lines): −1.6 dB and −2.6 dB are the measured and inferred generated squeezing levels in our experiment,−5.7 dB is amplitude squeezing previously achieved using an asymmetric Kerr interferometer [38] and −15 dB is the highest measuredsqueezing to date [9]. For each plot, M = 1, P = 0.2 mW, λ = 740 nm, η = 1, δm = 1× 10−4 and Var(R[H]) = 1× 10−5. (e) Schematicof the experiment and SEM image representative of the PCF structure. A pulsed laser at 740 nm propagates into the Sagnac interferometer forsqueezed state generation. A birefringent photonic crystal fibre (PCF) provides the nonlinear medium for Kerr squeezing. The electro-opticmodulator (EOM) combined with the polarising beamsplitter (PBS) are used to generate AM, which is measured on a spectrum analyser (SA).

band in the spectral noise power at frequency Ω that containsinformation about δm. Figure 1(a-c) illustrate the behaviourof the spectral noise power of an initial laser input (a), wherethe noise characteristics at Ω approximate that of a coherentstate |α〉 and so quantum noise dominates the variance of in-tensity. The light is subsequently squeezed in amplitude (b)and then modulated in amplitude (c). The insets illustrate theideal evolution of the state at ±Ω for an initial coherent state|α〉. The final state is amplitude squeezed with an averagephoton number of 〈n(±Ω)〉 = δm|α|2/2.

We derive an estimator for δm from the SNR of direct pho-todetection, similar to [18] which uses homodyne detection.For direct photodetection of AM in a shot-noise limited band-width around Ω, the SNR is given by δSNR = 〈ps〉/〈pn〉,where 〈ps〉 is the average signal component of the gener-ated electronic power at Ω, and 〈pn〉 is the average electronicpower generated from optical noise. In the limit of weakAM (δm 1) loss due to AM has a negligible effect on thesqueezing parameter Φ and the average optical power on themodulator output. Therefore, average measured photocurrentis expressed as i0 = qηP (1 − (δm/2))/~ω ≈ qηP/~ω, withelectron charge q, photodiode efficiency η, reduced Planckconstant ~ and carrier angular frequency ω. We then obtain

δSNR =〈ps〉〈pn〉

≈ δ2mi0

4qΦB, (1)

(see Appendix Section 1), whereB is the frequency resolutionbandwidth (RBW) of the noise spectrum, and corresponds tothe inverse of the integration time over which the spectrum is

measured. From Eq. (1), we define the estimator

δm =

√4qΦBδSNR

i0, (2)

where

δSNR =pΩ − pNpN − pE

, and i0 =qη〈P 〉~ω

. (3)

pΩ, pN and pE are the spectral noise powers of the electronicsideband, the optical noise floor and the electronic noise floorrespectively. 〈P 〉 is the average optical power output from themodulator, and both 〈P 〉 and pN may be pre-calibrated withhigh precision. The dependence of δm on the optical noise isthen contained in the measurement of pΩ.

For an input resistance of R to the measuring device (e.g.spectrum analyser or oscilloscope), we can define the powerof the electronic sideband as

pΩ = 2R|iΩ|2, (4)

where iΩ is the photocurrent in the frequency bin centered onΩ. By considering power fluctuations due to quantum opti-cal noise, low frequency classical optical noise, and electronicnoise, we find

Var(pΩ) = 〈p2Ω〉 − 〈pΩ〉2

≈ R2

M

[2qδ2

mi30ΦB + 4δ4

mi40Var(R[H])

+ 4q2δ2mi

20Var(R[N ])

](5)

3

(see Appendix Section 2). R[•] corresponds to the real part,His the DC component of the classical relative amplitude noisefrom the laser, N is the component of electronic noise in the±B/2 frequency interval around Ω, and M is a variance re-duction factor due to spectral averaging. The dependence ofVar(pΩ) onH is due to classical noise being transferred fromthe carrier to the optical sidebands upon modulation. We as-sume here that the variance of the optical noise due to theclassical intensity fluctuations scales quadratically with opti-cal power, as expected for technical laser noise [31]. To quan-tify any advantage in precision obtained by using squeezedlight, we compute the classical Fisher information on δm,F(δm). Since we use an amplitude squeezed state to performan amplitude measurement, the classical Fisher informationsaturates the quantum Cramer-Rao bound [32] — thereforeevaluating F(δm) bounds any quantum strategy. For our mea-surement strategy, and assuming α 1, δSNR is normallydistributed and we can define F(δSNR) according to [33]

F(δSNR) =1

Var(δSNR)=

(∂〈δSNR〉∂〈pΩ〉

)2

Var(pΩ)

−1

.

(6)F(δm) can be obtained from F(δSNR) by using [33]

F(δm) =

(∂δSNR∂δm

)2

F(δSNR). (7)

We find that Var(R[N ]) contributes negligibly to F(δm), andfrom Eq. (2-7), this leads to

F(δm) ≈M[

2qΦB

i0+ 4δ2

m Var(R[H])

]−1

(8)

The quantum advantage is then the ratio Q(δm) between thevalues of F(δm) for squeezed (Φ < 1) and coherent (Φ = 1)light. The variance of δm can be obtained by standard errorpropagation. We find

Var(δm) =

(∂〈δm〉∂〈pΩ〉

)2

Var(pΩ) =1

F(δm). (9)

Therefore, δm is an efficient estimator. We also find that, inthe limit of weak AM, 〈δm〉 = δm, meaning our estimator isunbiased.

The Fisher information per detected photon may be de-fined as F ′(δm) = F(δm)/〈N〉, where 〈N〉 = i0/qB isthe number of photons detected in the measurement timeB−1. Figure 1(d) illustrates the dependence of F ′(δm) onthe RBW for a typical laser source which is quantum noiselimited at Ω (solid black line) and various levels of squeez-ing (dashed lines), with all other parameters fixed. We findthat for higher RBWs, squeezing provides sub-QNL precisionin estimating δm. This can be seen from Eq. (8), since for2qΦB/i0 4δ2

m Var(R[H]), quantum noise limits the preci-sion of the measurement, and we find Q(δm)→ Qopt, where

Qopt =1

Φ. (10)

Because all information on δm is contained at modulation fre-quency Ω, this model suggests a practically achievable quan-tum advantage per detected photon.

For the measurement, we built a source of amplitudesqueezed light, based on [34]. 100 fs pulses with centralwavelength λ0 = 740 nm from a Spectra Physics Mai TaiTi:Sapphire laser are coupled into an asymmetric Sagnac in-terferometer, with a 90:10 splitting ratio beamsplitter (BS)(Fig. 1(e)). 14 m of photonic crystal fibre (PCF) provides astrong χ(3) nonlinearity in the interferometer. The fibre sam-ples used were originally fabricated for photon pair genera-tion work [35]. As the brighter 90% reflected pulses propa-gate through the PCF, they undergo self-phase modulation andbecome quadrature squeezed [10, 34]. These pulses interferewith the weaker (10%) counter-propagating pulses transmittedinitially at the BS — these provide a coherent displacement inphase space. This leads to amplitude squeezing on the outputof the interferometer [34]. The chosen central wavelength ofλ0 = 740 nm is close to the 730 nm zero-dispersion wave-length of the PCF, in order to minimise the spectral broad-ening, which enables optimal interference at the 90:10 BS.The zero-dispersion wavelength of the PCF may be tailored bythe fibre structure, making this approach applicable to a largerange of wavelengths. The average optical power of the outputstate is 0.2 mW, which equates to 25 W of peak power. Theamplitude squeezed light passes through a Thorlabs EO-AM-NR-C1 electro-optic modulator (EOM), that modulates polar-isation. A subsequent polarising beamsplitter (PBS) translatesthe polarisation modulation into a weak AM of depth δm, andgenerates optical sidebands at distance ±Ω from the carrierfrequency. The resulting state is measured with direct de-tection, by collecting all the light at photodiode PD1. Wecalibrate the shot-noise level using the balanced subtractionphotocurrent of PD1 and PD2. The balanced amplified pho-todetector used is a Thorlabs PDB440A(-AC). The spectralphotocurrent is analysed with a Rohde & Schwarz FPC1000spectrum analyser.

Figure 2(a) shows the relative noise power traces of ampli-tude modulated squeezed light (−1.2 dB) and antisqueezedlight (2.7 dB) produced by the setup. The RBW is B =10 kHz, which is considerably wider than the linewidth of theoptical sidebands, measured to be < 1 Hz. The frequencyseparation of trace points in Fig. 2(a) is smaller than the RBWsince the trace is a result of multiple samples within eachRBW interval. This measurement demonstrates enhancedsensitivity detection of AM due to amplitude squeezing, asshown in [18]. The antisqueezed data in Fig. 2(a) is correctedfor the difference in optical power required to generate anti-squeezing and squeezing, by subtracting the difference in therespective shot-noise levels from this trace. The electronicnoise has also been subtracted from each trace.

From Eq. (9) we know that Var(δm) is proportional to Φand inversely proportional to 〈P 〉. However, the profile ofsqueezing with optical power is such that change in poweris negligible across the maximum observed squeezing range[−1.6, 2.7] dB in our setup, so here Var(δm) scales linearly

4

Figure 2. Observing a quantum advantage in parameter estimation using amplitude squeezed light. In each plot, the black dashed line representsthe QNL. (a) 10 MHz AM measured by direct detection. The red trace corresponds to −1.2 dB of squeezing, and the blue trace to 2.7 dB ofantisqueezing. (b) Measured quantum advantage in precision of experimentally estimated δm, Q(δm), for different squeezing Φ. The red linecorresponds toQopt. (c) MeasuredQ(δm) with varied RBW B, for an average −1.3 dB of squeezing.

with Φ. By fitting measured Var(δm) to a line, we infer mea-sured quantum advantage using

Q(δm) =Var(δm)QNL

Var(δm)Φ

, (11)

where Var(δm)QNL and Var(δm)Φ are variances of estimatesof δm, for coherent and squeezed light respectively.

Figure 2(b) shows measured Q(δm) for a range of squeez-ing Φ with fixed RBW B = 100 kHz. The value of δmhad a small experimental drift which varied between δm =[0.8, 1.0]×10−4 over the duration of the measurements. Eachsample of the spectral noise power corresponds to M ≈ 34,due to the video bandwidth filter, and every measurement ofVar(δm)Φ is taken from 50 samples of δm. Due to the highRBW, quantum noise limits the variance of δm, and the mea-surement saturates the optimal quantum bound Qopt givenby Eq. (10) (red curve). A quantum advantage of Q(δm) =1.44± 0.09 is observed with −1.6 dB of squeezing, in agree-ment with Qopt = 1.45.

By repeating this for a range of B and a fixed −1.3 dBof average squeezing, we plot in Fig. 2(c) the dependence ofQ(δm) on RBW. The red curve is a theoretical fitting calcu-lated using Eq. (8), with Var(R[H]) as a fitting parameter,which gives Var(R[H]) = 7 ± 1 × 10−6. We observe sub-QNL precision down to B = 100 Hz. The maximum quan-tum advantage observed here isQ(δm) = 1.34± 0.07, whichagain closely agrees with the the optimal Qopt = 1.35 forthe average squeezing parameter of Φ = 0.74. In Fig. 2(b)and Fig. 2(c), the error bars were derived from standard prop-agation of error calculations, with the data averaged over236 variance measurements. Reducing optical loss to mea-sure higher squeezing [9] enables greater enhancement inprecision. Accounting for detection efficiency (we measureηd = 0.84) and coupling efficiency between the interferome-ter output and the detector (ηopt = 0.81), we infer our max-imum measured squeezing value of −1.6 dB corresponds to−2.6 dB of generated amplitude squeezing.

We have demonstrated quantum enhanced precision param-eter estimation with bright squeezed amplitude light. Ourmodel shows the degree of precision enhancement is dictatedby the amount of squeezing, the RBW and the classical noiseon the generated sidebands. This exemplifies that for mea-surements of high power optical signals, sub-shot-noise sensi-tivity does not alone imply sub-shot-noise precision. We ver-ify our model with experiment, reporting up to a 44% quan-tum advantage in precision in the estimation of the modu-lation index, per detected photon. This general demonstra-tion motivates applications in areas such as spectroscopy [36]and imaging [37], where precision may determine the perfor-mance of the measurement, which can be improved by usingsqueezed light. We use amplitude squeezed light of 0.2 mWaverage optical power (25 W peak power) as a probe. Thispower is comparable to the photon dose required to induce aphotophobic response in living cells [39], therefore indicatingthis technique’s relevance to biological measurements.

Supporting data is available on request.

We thank J. Mueller, P. Mosley, A. Politi, J. Rarity, N.Samantaray, W. Wadsworth, S. Wollmann and W. Bowenfor helpful discussions. This work was supported by Quan-tIC - The UK Quantum Technology Hub in Quantum Imag-ing (EPSRC EP/T00097X/1). G.S.A. was supported bythe Quantum Engineering Centre for Doctoral Training (EP-SRC EP/L015730/1). E.J.A. acknowledges fellowship sup-port from EPSRC Doctoral Prize Fellowship (EP/R513179/1).J.C.F.M. acknowledges support from an EPSRC QuantumTechnology Fellowship (EP/M024385/1) and European Re-search Council starting grant ERC-2018-STG 803665.

All authors contributed to the initial design and conceptionof the experiment. G.S.A, E.J.A, G.F and A.R.M contributedto the experimental setup. G.S.A performed the data collec-tion. G.S.A, G.F and E.J.A contributed to the theoretical anal-ysis. All authors discussed the results and contributed to thepreparation of the manuscript.

5

∗ george.atkinson@bristol.ac.uk† euan.allen@bristol.ac.uk‡ jonathan.matthews@bristol.ac.uk

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6

APPENDIX:QUANTUM ENHANCED PRECISION ESTIMATION OF TRANSMISSION WITH BRIGHT SQUEEZED LIGHT

1. Calculation of the Signal-to-Noise Ratio

Here we derive the expected value of the electronic power pΩ in the ±B/2 frequency interval centered on the modulationfrequency Ω, generated by the current i(t). We may write pΩ as

pΩ = 2R|iΩ|2, (12)

where R is the input resistance, iΩ =∫ Ω

i(ν)dν is the photocurrent in the frequency bin centered on Ω, using the notation∫ Ω ≡∫ Ω+B/2

Ω−B/2 , and the factor of 2 comes from the integration over positive and negative frequencies. It is important to note

that this differs from the standard definition of the power in a frequency band, pΩ = 2R∫ Ω |i(ν)|2dν [1]. The reason for the

definition used in Eq. (12) is that measuring devices such as spectrum analysers and oscilloscopes are fundamentally voltagedetectors, and therefore the displayed power level is computed from the voltage measured in a given frequency bin. This meansthat the integration of the photocurrent density effectively occurs before taking the absolute square. While the average value〈pΩ〉 does not significantly differ between these definitions, Var(pΩ) does. To keep the model consistent with the measurementsobtained by our spectrum analyser, we will use Eq. (12) as a definition for pΩ.

The amplitude of the optical field before modulation is applied may be written as A0(t) = [1 + ζ(t)]α0eiθ + a(t), where θ is

the phase of the classical field, a(t) describes the quantum amplitude fluctuations and ζ(t) is a stochastic noise function whichcorresponds to the low frequency classical noise of the laser. We have assumed a continuous-wave amplitude α0 for simplicity.The amplitude of the detected light after modulation may then be written as

A(t) = (Ψ0 + Ψm cos(2πΩt))([1 + ζ(t)]αeiθ + a(t)), (13)

where Ψ0 and Ψm are related to the modulation index by Ψ0 = 1 − δm/2 and Ψm = δm/2, and the detection efficiency η ismodelled as an additional loss before detection, such that α =

√ηα0. By making the assumption of large amplitude α 1 and

small modulation Ψm 1, we can approximate A(t) as

A(t) ≈ (Ψ0 + Ψm cos(2πΩt))[1 + ζ(t)]αeiθ + a(t) ≡ |α(t)|eiθ + a(t), (14)

where the effect of amplitude modulation on the quantum noise term has been neglected. The current at time t may then bewritten as

i(t) = q(A(t)†A(t) + ne(t)

)= q

(|α(t)|2 +

√2|α(t)|xθ(t) + a(t)†a(t) + ne(t)

), (15)

where we have defined the quadrature operator xθ(t) = 1√2[a(t)e−iθ+ a(t)†eiθ] and the electronic noise term ne(t) corresponds

to the number of electrons generated independently of the optical field. The component of the photocurrent at frequency ν isthen given by

i(ν) = q

[I(ν) +

√2

∫α(µ)xθ(ν − µ)dµ+

∫a(−µ)†a(ν − µ)dµ

], (16)

where∫≡∫∞−∞, and we have defined the unitary Fourier transforms

I(ν) =

∫ (|α(t)|2 + ne(t)

)e−2πiνtdt (17)

and

xθ(ν) =

∫xθ(t)e

−2πiνtdt =1√2

[a(−ν)†eiθ + a(ν)e−iθ

]. (18)

We also define a(ν) as the squeezed vacuum operator [40]

a(ν) = b(ν) cosh r(ν)− e2iθ(ν)b(−ν)† sinh r(ν), (19)

7

where b(ν) and b(ν)† are the bosonic creation and annihilation operators. The squeezing is defined such that r(ν) = r andθ(ν) = θ within the frequency bandwidth −Λ/2 ≤ ν ≤ Λ/2 (where Λ/2 > Ω) and r(ν) = 0 outside of this frequency range.The 2θ phase factor then orients the squeezing in the amplitude direction. By defining IΩ =

∫ ΩI(ν)dν we can write pΩ as

pΩ = 2q2R

[|IΩ|2 +

√2I∗Ω

∫ Ω ∫α(µ)xθ(ν − µ)dµdν + I∗Ω

∫ Ω ∫a(−µ)†a(ν − µ)dµdν

√2IΩ

∫ Ω ∫α(µ)∗xθ(ν − µ)†dµdν + 2

∫ Ω ∫ Ω ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)dµdµdνdν

+√

2

∫ Ω ∫ Ω ∫ ∫α(µ)∗xθ(ν − µ)†a(−µ)†a(ν − µ)dµdµdνdν + IΩ

∫ Ω ∫a(ν − µ)†a(−µ)dµdν

+√

2

∫ Ω ∫ Ω ∫ ∫α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)dµdµdνdν

+

∫ Ω ∫ Ω ∫ ∫a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)dµdµdνdν

], (20)

where (•)∗ denotes the complex conjugate. From Eq. (14), we can find the frequency dependence of the classical field amplitude:

α(ν) =

∫|α(t)|e−2πiνt = α

[Ψ0(δ(ν) + h(ν)) +

Ψm

2(δ(ν − Ω) + δ(ν + Ω) + h(ν − Ω) + h(ν + Ω))

], (21)

where h(ν) =∫ζ(t)e−2πiνtdt, and since classical noise is only observed at low frequencies (. 2 MHz), we can write for

example h(Ω) = 0. Equation (14) allows us to define I(ν) as

I(ν) = α2

[Ψ2

0

(δ(ν) + 2h(ν) +

∫h(µ)h(ν − µ)dµ

)+ Ψ0Ψm

(δ(Ω− ν) + δ(Ω + ν)

+ 2h(ν − Ω) + 2h(ν + Ω) +

∫h(µ)h(ν − µ− Ω)dµ+

∫h(µ)h(ν − µ+ Ω)dµ

)+

+Ψ2m

4

(δ(ν − 2Ω) + δ(ν + 2Ω) + 2δ(ν) + 2h(ν − 2Ω) + 2h(ν + 2Ω) + 4h(ν)

+

∫h(µ)h(ν − µ− 2Ω)dµ+

∫h(µ)h(ν − µ+ 2Ω)dµ+ 2

∫h(µ)h(ν − µ)dµ

) ]+ ne(ν). (22)

We can then find the expectation 〈pΩ〉 with respect to the random variables h(ν), ne(ν) and a(ν). Since these variables areindependent, the expectation value may be defined as 〈•〉 ≡ 〈〈〈0| • |0〉〉h(ν)〉ne(ν). To calculate this, we first compute

|IΩ|2 = Ψ20Ψ2

mα4

[1 + 4

∫ Ω

R[h(ν − Ω)]dν + 2

∫ Ω ∫R[h(µ)h(ν − µ− Ω)]dµdν + 4

∣∣∣∣∣∫ Ω

h(ν − Ω)dν

∣∣∣∣∣2

+ 4

∫ Ω ∫ Ω ∫R[h(ν − Ω)∗h(µ)h(ν − µ− Ω)]dµdνdν +

∣∣∣∣∣∫ Ω ∫

h(µ)h(ν − µ− Ω)dµdν

∣∣∣∣∣2 ]

+ Ψ0Ψmα2

[2

∫ Ω

R[ne(ν)]dν + 4

∫ Ω ∫ Ω

R[h(ν − Ω)∗ne(ν)]dνdν

+ 2

∫ Ω ∫ Ω ∫R[h(µ)∗h(ν − µ− Ω)∗ne(ν)]dµdνdν

]+

∣∣∣∣∣∫ Ω

ne(ν)dν

∣∣∣∣∣2

, (23)

where R[•] signifies the real part. Then, by evaluating the quantum part of the expectation value, we obtain the result:

〈pΩ〉 = 2q2R

[〈|IΩ|2〉+ Φ

∫ Ω ∫ Ω ∫〈α(µ)∗α(µ+ ν − ν)〉dµdνdν +BΛ

(Φ2

8+

1

8Φ2− 1

4

)]

= 2q2R

[α4

(Ψ2

0Ψ2m + 4Ψ2

0Ψ2m

∫ Ω

〈R[h(ν − Ω)]〉dν + 2Ψ20Ψ2

m

∫ Ω ∫〈R[h(µ)h(ν − µ− Ω)]〉dµdν

8

+ 4Ψ20Ψ2

m

⟨∣∣∣∣∣∫ Ω

h(ν − Ω)dν

∣∣∣∣∣2⟩

+ 4Ψ20Ψ2

m

∫ Ω ∫ Ω ∫〈R[h(ν − Ω)∗h(µ)h(ν − µ− Ω)]〉dµdνdν

+ Ψ20Ψ2

m

⟨∣∣∣∣∣∫ Ω ∫

h(µ)h(ν − µ− Ω)dµdν

∣∣∣∣∣2⟩ )

+ α2

(2Ψ0Ψm

∫ Ω

〈R[ne(ν)]〉dν

+ 4Ψ0Ψm

∫ Ω ∫ Ω

〈R[h(ν − Ω)∗ne(ν)]〉dνdν + 2Ψ0Ψm

∫ Ω ∫ Ω ∫〈R[h(µ)∗h(ν − µ− Ω)∗ne(ν)]〉 dµdνdν

+

(Ψ2

0 +Ψ2m

2

)ΦB + (2Ψ2

0 + Ψ2m)Φ

∫ Ω ∫ Ω

〈R[h(ν − ν)]〉dνdν

+

(Ψ2

0 +Ψ2m

2

)Φ

∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν − ν)〉dµdνdν

)+

⟨∣∣∣∣∣∫ Ω

ne(ν)dν

∣∣∣∣∣2⟩

+BΛ

(Φ2

8+

1

8Φ2− 1

4

) ], (24)

for the squeezing parameter Φ = e−2r. In Eq. (24), we have neglected terms involving the expectation value of the productof an odd number of creation or annihilation operators, and terms outside the domain of h(ν). By observing that α 1,〈|∫h(ν)dν|〉 1 and δm 1 for i0 ≈ qα2 we find

〈pΩ〉 ≈ R

i20δ2m

2+ 2qi0ΦB + 2q2

⟨∣∣∣∣∣∫ Ω

ne(ν)dν

∣∣∣∣∣2⟩ . (25)

Similarly, at a small frequency interval ∆f from Ω, we find that the electronic power of the optical noise floor and electronicnoise floor are respectively

〈pN 〉 ≈ R

2qi0ΦB + 2q2

⟨∣∣∣∣∣∫ Ω

ne(ν)dν

∣∣∣∣∣2⟩ and 〈pE〉 = R

2q2

⟨∣∣∣∣∣∫ Ω

ne(ν)dν

∣∣∣∣∣2⟩ . (26)

We then find that the optical signal-to-noise ratio δSNR may be expressed as

δSNR =〈pΩ〉 − 〈pN 〉〈pN 〉 − 〈pE〉

≈ i0δ2m

4qΦB. (27)

2. Calculation of the Variance of the Sideband Power

In order to calculate the variance Var(pΩ) = 〈p2Ω〉 − 〈pΩ〉2, an expression for 〈pΩ〉2 may be evaluated directly from Eq. (24)

to give

〈pΩ〉2 = 4q4R2

[〈|IΩ|2〉2 + (2Ψ2

0 + Ψ2m)〈|IΩ|2〉α2

(ΦB + 2Φ

∫ Ω ∫ Ω

〈R[h(ν − ν)]〉dνdν

+ Φ

∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν − ν)〉dµdνdν

)+ 〈|IΩ|2〉ΛB

(Φ2

4+

1

4Φ2− 1

2

)+

(Ψ4

0 + Ψ20Ψ2

m +Ψ4m

4

)α4

(Φ2B2 + 4Φ2B

∫ Ω ∫ Ω

〈R[h(ν − ν)]〉dνdν

+ 2Φ2B

∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν − ν)〉dµdνdν + 4Φ2

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν − ν)]〉〈R[h(ν − ν)]〉dνdνdνdν

+ 4Φ2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(ν − ν)]〉〈h(µ)∗h(µ+ ν − ν)〉dµdνdνdνdν

+ Φ2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫〈h(µ)∗h(µ+ ν − ν)〉〈h(µ)∗h(µ+ ν − ν)〉dµdµdνdνdνdν

)

9

+ (2Ψ20 + Ψ2

m)α2

(Φ2

8+

1

8Φ2− 1

4

)(ΦΛB2 + 2ΦΛB

∫ Ω ∫ Ω

〈R[h(ν − ν)]〉dνdν

+ ΦΛB

∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν − ν)〉dµdνdν

)+ Λ2B2

(Φ2

8+

1

8Φ2− 1

4

)2]. (28)

To find an expression for 〈p2Ω〉, we can again neglect terms where the expectation value of the quadrature operators vanishes,

giving

〈p2Ω〉 = 4q4R2

⟨|IΩ|4 + 4|IΩ|2

∫ Ω ∫ Ω ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)dµdµdνdν

+ 2I∗2Ω

∫ Ω ∫ Ω ∫ ∫α(µ)α(µ)xθ(ν − µ)xθ(ν − µ)dµdµdνdν

+ 2I2Ω

∫ Ω ∫ Ω ∫ ∫α(µ)∗α(µ)∗xθ(ν − µ)†xθ(ν − µ)†dµdµdνdν

+ 4|IΩ|2∫ Ω ∫ Ω ∫ ∫

α(µ)α(µ)∗xθ(ν − µ)xθ(ν − µ)†dµdµdνdν

+ 3|IΩ|2∫ Ω ∫ Ω ∫ ∫

a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)dµdµdνdν

+ |IΩ|2∫ Ω ∫ Ω ∫ ∫

a(−µ)†a(ν − µ)a(ν − µ)†a(−µ)dµdµdνdν

+ 2I∗Ω

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)α(µ)∗xθ(ν − µ)xθ(ν − µ)†a(−µ)†a(ν − µ)dµdµdµdνdνdν

+ 2I∗Ω

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)α(µ)∗xθ(ν − µ)†a(−µ)†a(ν − µ)xθ(ν − µ)dµdµdµdνdνdν

+ 2I∗Ω

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)a(−µ)†a(ν − µ)xθ(ν − µ)†xθ(ν − µ)dµdµdµdνdνdν

+ 2I∗Ω

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)a(−µ)†a(ν − µ)dµdµdµdνdνdν

+ 2IΩ

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†a(ν − µ)†a(−µ)xθ(ν − µ)dµdµdµdνdνdν

+ 2IΩ

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)xθ(ν − µ)†dµdµdµdνdνdν

+ 2IΩ

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)a(ν − µ)†a(−µ)dµdµdµdνdνdν

+ 2IΩ

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)†xθ(ν − µ)dµdµdµdνdνdν

+ 2I∗Ω

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)α(µ)xθ(ν − µ)a(ν − µ)†a(−µ)xθ(ν − µ)dµdµdµdνdνdν

+ 2I∗Ω

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)xθ(ν − µ)dµdµdµdνdνdν

+ 2IΩ

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)∗xθ(ν − µ)†xθ(ν − µ)†a(−µ)†a(ν − µ)dµdµdµdνdνdν

+ 2IΩ

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫α(µ)∗α(µ)∗xθ(ν − µ)†a(−µ)†a(ν − µ)xθ(ν − µ)†dµdµdµdνdνdν

+ 4

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)∗α(µ)α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)xθ(ν − µ)†xθ(ν − µ)dµdµdµdµdνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)dµdµdµdµdνdνdνdν

10

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)∗α(µ)a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)xθ(ν − µ)†xθ(ν − µ)dµdµdµdµdνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†a(−µ)†a(ν − µ)a(ν − µ)†a(−µ)xθ(ν − µ)dµdµdµdµdνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)∗α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)xθ(ν − µ)†a(−µ)†a(ν − µ)dµdµdµdµdνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)∗α(µ)∗xθ(ν − µ)†a(−µ)a(ν − µ)xθ(ν − µ)†a(−µ)†a(ν − µ)dµdµdµdµdνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫α(µ)α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)a(ν − µ)†a(−µ)xθ(ν − µ)dµdµdµdµdνdνdνdν

+

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫a(ν −µ)†a(−µ)a(−µ)†a(ν −µ)a(ν −µ)†a(−µ)a(−µ)†a(ν −µ)dµdµdµdµdνdνdνdν

⟩.

(29)

To explicitly evaluate 〈p2Ω〉 in the following, we calculate the integrals in Eq. (29) separately, using Eq. (21) and the commutation

relations of the bose operators with the expectation value taken on the vacuum state. Terms 2-5 of Eq. (29) respectively give:⟨4|IΩ|2

∫ Ω ∫ Ω ∫ ∫α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)dµdµdνdν

⟩

= (2Ψ20 + Ψ2

m)α2Φ

(〈|IΩ|2〉B + 2

∫ Ω ∫ Ω

〈|IΩ|2R[h(ν − ν)]〉dνdν

+

∫ Ω ∫ Ω ∫〈|IΩ|2h(µ)∗h(µ+ ν − ν)〉dµdνdν

), (30)

⟨2I∗2Ω

∫ Ω ∫ Ω ∫ ∫α(µ)α(µ)xθ(ν − µ)xθ(ν − µ)dµdµdνdν

⟩

=Ψ2m

4α2Φ

(〈I∗2Ω 〉B + 2

∫ Ω ∫ Ω

〈I∗2Ω h(ν + ν − 2Ω)〉dνdν

+

∫ Ω ∫ Ω ∫〈I∗2Ω h(µ− Ω)h(ν + ν − µ− Ω)〉dµdνdν

), (31)

⟨2I2

Ω

∫ Ω ∫ Ω ∫ ∫α(µ)∗α(µ)∗xθ(ν − µ)†xθ(ν − µ)†dµdµdνdν

⟩

=Ψ2m

4α2Φ

(〈I2

Ω〉B + 2

∫ Ω ∫ Ω

〈I2Ωh(ν + ν − 2Ω)∗〉dνdν

+

∫ Ω ∫ Ω ∫〈I2

Ωh(µ− Ω)∗h(ν + ν − µ− Ω)∗〉dµdνdν)

(32)

and⟨4|IΩ|2

∫ Ω ∫ Ω ∫ ∫α(µ)α(µ)∗xθ(ν − µ)xθ(ν − µ)†dµdµdνdν

⟩

= (2Ψ20 + Ψ2

m)α2Φ

(〈|IΩ|2〉B + 2

∫ Ω ∫ Ω

〈|IΩ|2R[h(ν − ν)]〉dνdν

+

∫ Ω ∫ Ω ∫〈|IΩ|2h(µ)∗h(µ+ ν − ν)〉dµdνdν

). (33)

11

The summation of terms 6 and 7 of Eq. (29) gives

⟨3|IΩ|2

∫ Ω ∫ Ω ∫ ∫a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)dµdµdνdν

⟩

+

⟨|IΩ|2

∫ Ω ∫ Ω ∫ ∫a(−µ)†a(ν − µ)a(ν − µ)†a(−µ)dµdµdνdν

⟩= 〈|IΩ|2〉ΛB

(Φ2

2+

1

2Φ2− 1

). (34)

It is also possible to combine terms 8-15 of Eq. (29) as follows:

2

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ [〈I∗Ωα(µ)α(µ)∗xθ(ν − µ)xθ(ν − µ)†a(−µ)†a(ν − µ)〉

+ 〈I∗Ωα(µ)α(µ)∗xθ(ν − µ)†a(−µ)†a(ν − µ)xθ(ν − µ)〉+ 〈I∗Ωα(µ)∗α(µ)a(−µ)†a(ν − µ)xθ(ν − µ)†xθ(ν − µ)〉+ 〈I∗Ωα(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)a(−µ)†a(ν − µ)〉+ 〈IΩα(µ)∗α(µ)xθ(ν − µ)†a(ν − µ)†a(−µ)xθ(ν − µ)〉+ 〈IΩα(µ)∗α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)xθ(ν − µ)†〉+ 〈IΩα(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)a(ν − µ)†a(−µ)〉

+ 〈IΩα(µ)∗α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)†xθ(ν − µ)〉]dµdµdµdνdνdν

= (4Φ2 − 2)

∫ Ω ∫ Ω ∫ Ω ∫〈R[IΩ]α(µ)α(µ+ ν − ν − ν)∗〉dµdνdνdν

= Ψ0Ψmα2(4Φ2 − 2)

(〈R[IΩ]〉B2 + 2

∫ Ω ∫ Ω ∫ Ω

〈R[IΩ]R[h(ν + ν − ν − Ω)]〉dνdνdν

+

∫ Ω ∫ Ω ∫ Ω ∫〈R[IΩ]h(µ− Ω)h(µ+ ν − ν − ν)∗〉dµdνdνdν

). (35)

Similarly, we find that terms 16-19 of Eq. (29) simplify as

2

∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ [〈I∗Ωα(µ)α(µ)xθ(ν − µ)a(ν − µ)†a(−µ)xθ(ν − µ)〉

+ 〈I∗Ωα(µ)α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)xθ(ν − µ)〉+ 〈IΩα(µ)∗α(µ)∗xθ(ν − µ)†xθ(ν − µ)†a(−µ)†a(ν − µ)〉

+ 〈IΩα(µ)∗α(µ)∗xθ(ν − µ)†a(−µ)†a(ν − µ)xθ(ν − µ)†〉]dµdµdµdνdνdν

= Φ2

∫ Ω ∫ Ω ∫ Ω ∫〈I∗Ωα(µ)α(µ+ ν − ν − ν)〉dµdνdνdν

+ Φ2

∫ Ω ∫ Ω ∫ Ω ∫〈IΩα(µ)∗α(µ+ ν − ν − ν)∗〉dµdνdνdν

= 2Ψ0Ψmα2Φ2

(〈R[IΩ]〉B2 + 2

∫ Ω ∫ Ω ∫ Ω

〈R[I∗Ωh(ν + ν − ν − Ω)]〉dνdνdν

+

∫ Ω ∫ Ω ∫ Ω ∫〈R[I∗Ωh(µ− Ω)h(µ+ ν − ν − ν)]〉dµdνdνdν

). (36)

We can write term 20 of Eq. (29) as

12

4

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫〈α(µ)∗α(µ)α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)xθ(ν − µ)†xθ(ν − µ)〉dµdµdµdµdνdνdνdν

= 4

⟨∣∣∣∣∣∫ Ω ∫

α(µ)xθ(ν − µ)dµdν

∣∣∣∣∣4⟩

= 4

⟨ ∣∣∣∣ Ψ0α

(∫ Ω

xθ(ν)dν +

∫ Ω ∫h(µ)xθ(ν − µ)dµdν

)+

Ψmα

2

( ∫ Ω

xθ(ν − Ω)dν +

∫ Ω

xθ(ν + Ω)dν

+

∫ Ω ∫h(µ− Ω)xθ(ν − µ)dµdν +

∫ Ω ∫h(µ+ Ω)xθ(ν − µ)dµdν

) ∣∣∣∣4 ⟩ . (37)

In the expansion of Eq. (37), many terms vanish due to both the restricted domain of h(ν) and the action of creation andannihilation operators on the vacuum. This allows Eq. (37) to be significantly simplified, leading to the result:

4

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫〈α(µ)∗α(µ)α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)xθ(ν − µ)†xθ(ν − µ)〉dµdµdµdµdνdνdνdν

= (4Ψ40 + 4Ψ2

0Ψ2m + Ψ4

m)α4Φ2

(1

2B2 + 2B

∫ Ω ∫ Ω

〈R[h(ν − ν)]〉dνdν

+

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν − ν)h(ν − ν)]〉dνdνdνdν +

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈h(ν − ν)∗h(ν − ν)〉dνdνdνdν

+B

∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν − ν)〉dµdνdν + 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(µ)h(µ+ ν − ν)∗h(ν − ν)]〉dµdνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫〈h(µ)∗h(µ+ ν − ν)h(µ)∗h(µ+ ν − ν)〉dµdµdνdνdνdν

)+

Ψ4m

8α4Φ2

(1

2B2 + 2B

∫ Ω ∫ Ω

〈R[h(ν + ν − 2Ω)]〉dνdν

+B

∫ Ω ∫ Ω ∫〈R[h(µ− Ω)h(ν + ν − µ− Ω)]〉dµdνdν + 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈h(ν + ν − 2Ω)∗h(ν + ν − 2Ω)〉dνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(µ− Ω)h(ν + ν − 2Ω)∗h(ν + ν − µ− Ω)]〉dµdνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫〈h(µ− Ω)∗h(µ− Ω)h(ν + ν − µ− Ω)∗h(ν + ν − µ− Ω)〉dµdµdνdνdνdν

). (38)

The summation of terms 21-24 of Eq. (29) can be written as

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫ [〈α(µ)∗α(µ)xθ(ν − µ)†xθ(ν − µ)a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)〉

+ 〈α(µ)∗α(µ)a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)xθ(ν − µ)†xθ(ν − µ)〉

+ 〈α(µ)∗α(µ)xθ(ν − µ)†a(−µ)†a(ν − µ)a(ν − µ)†a(−µ)xθ(ν − µ)〉

+ 〈α(µ)∗α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)xθ(ν − µ)†a(−µ)†a(ν − µ)〉]dµdµdµdµdνdνdνdν

=

(5Φ3

2− 2Φ +

1

2Φ

)∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈α(µ)∗α(µ+ ν + ν − ν − ν)〉dµdνdνdνdν

= (2Ψ20 + Ψ2

m)α2

(5Φ3

2− 2Φ +

1

2Φ

)(1

2B3 +

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν + ν − ν − ν)]〉dνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν + ν − ν − ν)〉dµdνdνdνdν

). (39)

13

Similarly, we can combine terms 25 and 26 of Eq. (29) to give

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫ [〈α(µ)∗α(µ)∗xθ(ν − µ)†a(−µ)a(ν − µ)xθ(ν − µ)†a(−µ)†a(ν − µ)〉

+ 〈α(µ)α(µ)a(ν − µ)†a(−µ)xθ(ν − µ)a(ν − µ)†a(−µ)xθ(ν − µ)〉]dµdµdµdµdνdνdνdν

= (2Ψ20 + Ψ2

m)α2(Φ3 − Φ

)( 1

2B3 +

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν + ν − ν − ν)]〉dνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(µ)h(µ+ ν + ν − ν − ν)]〉dµdνdνdνdν

). (40)

The final term of Eq. (29) gives the result:∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫ ∫ ∫〈a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)a(ν − µ)†a(−µ)a(−µ)†a(ν − µ)〉dµdµdµdµdνdνdνdν

= B3Λ

(7Φ4

32− 3Φ2

8+

7

32Φ4− 3

8Φ2+

5

16

). (41)

By combining all the terms calculated for Eq. (29), we obtain the result for 〈p2Ω〉:

〈p2Ω〉 = 4q4R2

[〈|IΩ|4〉+ (4Ψ2

0 + 2Ψ2m)α2Φ

(〈|IΩ|2〉B + 2

∫ Ω ∫ Ω

〈|IΩ|2R[h(ν − ν)]〉dνdν

+

∫ Ω ∫ Ω ∫〈|IΩ|2h(µ)∗h(µ+ ν − ν)〉dµdνdν

)+

Ψ2m

2α2Φ

(〈R[I∗2Ω ]〉B + 2

∫ Ω ∫ Ω

〈R[I∗2Ω h(ν + ν − 2Ω)]〉dνdν

+

∫ Ω ∫ Ω ∫〈R[I∗2Ω h(µ− Ω)h(ν + ν − µ− Ω)]〉dµdνdν

)+ 〈|IΩ|2〉ΛB

(Φ2

2+

1

2Φ2− 1

)+ Ψ0Ψmα

2(4Φ2 − 2)

(〈R[IΩ]〉B2 + 2

∫ Ω ∫ Ω ∫ Ω

〈R[IΩ]R[h(ν + ν − ν − Ω)]〉dνdνdν

+

∫ Ω ∫ Ω ∫ Ω ∫〈R[IΩ]h(µ− Ω)h(µ+ ν − ν − ν)∗〉dµdνdνdν

)+ 2Ψ0Ψmα

2Φ2

(〈R[IΩ]〉B2 + 2

∫ Ω ∫ Ω ∫ Ω

〈R[I∗Ωh(ν + ν − ν − Ω)]〉dνdνdν

+

∫ Ω ∫ Ω ∫ Ω ∫〈R[I∗Ωh(µ− Ω)h(µ+ ν − ν − ν)]〉dµdνdνdν

)+ (4Ψ4

0 + 4Ψ20Ψ2

m + Ψ4m)α4Φ2

(1

2B2 + 2B

∫ Ω ∫ Ω

〈R[h(ν − ν)]〉dνdν

+

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν − ν)h(ν − ν)]〉dνdνdνdν +

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈h(ν − ν)∗h(ν − ν)〉dνdνdνdν

+B

∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν − ν)〉dµdνdν + 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(µ)h(µ+ ν − ν)∗h(ν − ν)]〉dµdνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫〈h(µ)∗h(µ+ ν − ν)h(µ)∗h(µ+ ν − ν)〉dµdµdνdνdνdν

)+

Ψ4m

8α4Φ2

(1

2B2 + 2B

∫ Ω ∫ Ω

〈R[h(ν + ν − 2Ω)]〉dνdν +B

∫ Ω ∫ Ω ∫〈R[h(µ− Ω)h(ν + ν − µ− Ω)]〉dµdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈h(ν + ν − 2Ω)∗h(ν + ν − 2Ω)〉dνdνdνdν

+ 2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(µ− Ω)h(ν + ν − 2Ω)∗h(ν + ν − µ− Ω)]〉dµdνdνdνdν

14

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫ ∫〈h(µ− Ω)∗h(µ− Ω)h(ν + ν − µ− Ω)∗h(ν + ν − µ− Ω)〉dµdµdνdνdνdν

)+ (2Ψ2

0 + Ψ2m)α2

(5Φ3

2− 2Φ +

1

2Φ

)(1

2B3 +

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν + ν − ν − ν)]〉dνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈h(µ)∗h(µ+ ν + ν − ν − ν)〉dµdνdνdνdν

)+ (2Ψ2

0 + Ψ2m)α2

(Φ3 − Φ

)( 1

2B3 +

∫ Ω ∫ Ω ∫ Ω ∫ Ω

〈R[h(ν + ν − ν − ν)]〉dνdνdνdν

+1

2

∫ Ω ∫ Ω ∫ Ω ∫ Ω ∫〈R[h(µ)h(µ+ ν + ν − ν − ν)]〉dµdνdνdνdν

)+B3Λ

(7Φ4

32− 3Φ2

8+

7

32Φ4− 3

8Φ2+

5

16

)]. (42)

In the expression for Var(pΩ) = 〈p2Ω〉 − 〈pΩ〉2, there is significant cancellation between 〈p2

Ω〉 and 〈pΩ〉2, and by taking theleading remaining terms we find that

Var(pΩ) = 〈p2Ω〉 − 〈pΩ〉2 ≈ 4q4R2

[〈|IΩ|4〉 − 〈|IΩ|2〉2 + 2〈|IΩ|2〉Ψ2

0α2ΦB

]≈ 4q4R2

[16Ψ4

0Ψ4mα

8

(∫ Ω ∫ Ω

〈R[h(ν − Ω)]R[h(ν − Ω)]〉dνdν −∫ Ω ∫ Ω

〈R[h(ν − Ω)]〉〈R[h(ν − Ω)]〉dνdν

)

+ 4Ψ20Ψ2

mα4

(∫ Ω ∫ Ω

〈R[ne(ν)]R[ne(ν)]〉dνdν −∫ Ω ∫ Ω

〈R[ne(ν)]〉〈R[ne(ν)]〉dνdν

)+ 2ΦBΨ4

0Ψ2mα

6

]. (43)

By associating H =∫ Ω

h(ν − Ω)dν =∫ B/2−B/2 h(ν)dν as the relative amplitude of the classical optical noise in the DC com-

ponent, N =∫ Ω

ne(ν)dν as the amplitude of the electronic noise in the ±B/2 frequency interval around Ω, and substitutingΨ0 ≈ 1, Ψm = δm/2 and i0 ≈ qηα2

0, we find that for M spectral averages,

Var(pΩ) ≈ R2

M

[2qδ2

mi30ΦB + 4δ4

mi40Var(R[H]) + 4q2δ2

mi20Var(R[N ])

]. (44)

From Eq. (44), an improvement in precision beyond the quantum noise limit may be obtained in the case that squeezing (Φ < 1)provides a significant reduction in Var(pΩ).