Low-frequency phase measurement with high-frequency squeezing

doi:10.1364/OE.20.018173

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Low-frequency phase measurement with high-frequency squeezing

Zehui Zhai and Jiangrui Gao »View Author Affiliations

Optics Express, Vol. 20, Issue 16, pp. 18173-18179 (2012)
http://dx.doi.org/10.1364/OE.20.018173

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– Abstract
Introduction
Theoretical model
Conclusion
– References and links

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Abstract

Squeezed-state enhanced audio-frequency signal measurement is crucial for some special applications, such as gravitational wave detection. But generation of squeezed state of light at such frequency is more difficult than that at megahertz-frequency. In this paper we propose an experimental scheme to measure low-frequency phase signal with high-frequency squeezing. To utilize the high-frequency sidebands of the squeezed light, a two-frequency intense laser is applied in the interferometry instead of a single-frequency laser as usual. This technique is in the reach of modern quantum optics technology.

Introduction

The concept of radiation field squeezing [1

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

] has attracted much interest since its application to gravitational wave detection was proposed [2

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D Part. Fields 23(8), 1693–1708 (1981). [CrossRef]

]. This sort of radiation field was first generated with a four-wave mixing process by Slusher et al [3

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,” Phys. Rev. Lett. 55(22), 2409–2412 (1985). [CrossRef] [PubMed]

] and improved with an optical parametric oscillator (OPO) by Wu et al [4

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986). [CrossRef] [PubMed]

]. It has been useful in various quantum-enhanced measurement schemes, such as squeezed-light magnetometry [5

F. Wolfgramm, A. Cerè, F. A. Beduini, A. Predojević, M. Koschorreck, and M. W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett. 105(5), 053601 (2010). [CrossRef] [PubMed]

], displacement measurement [6

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88(20), 203601 (2002). [CrossRef] [PubMed]

], spectroscopy [7

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68(20), 3020–3023 (1992). [CrossRef] [PubMed]

], polarization measurement [8

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987). [CrossRef] [PubMed]

], phase measurement [9

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987). [CrossRef] [PubMed]

], and so on. Squeezed states of light are also vital ingredients for continuous-variable quantum communication and quantum computation [10

S. L. Braunstein and A. K. Pati, eds., Quantum Information with Continuous Variables (Kluwer Academic, 2003).

]. In some applications, such as gravitational wave detection, it is appealing to measure low-frequency signals with a squeezed light state [11

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1(8), 121 (2010). [CrossRef]

]. Low-frequency squeezing has attracted much interest in recent years as terrestrial gravitational wave detectors (GW-detectors) are approaching their shot noise limit, which has been successfully overcome with squeezing at audio frequencies very recently [12

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PankowM. A. PapaP. AjithP. PatelM. PedrazaP. PeirisL. PekowskyS. PennC. PeraltaA. PerrecaM. PhelpsM. PickenpackI. M. PintoM. PitkinH. J. PletschM. V. PlissiJ. PöldF. PostiglioneV. PredoiL. R. PriceM. PrijateljM. PrincipeS. PriviteraR. PrixL. ProkhorovO. PunckenV. QuetschkeF. J. RaabH. RadkinsP. RaffaiM. RakhmanovC. R. RametB. RankinsS. R. P. MohapatraV. RaymondK. RedwineC. M. ReedT. ReedS. ReidD. H. ReitzeR. RiesenK. RilesN. A. RobertsonC. RobinsonE. L. RobinsonS. RoddyC. RodriguezM. RodruckJ. RollinsJ. D. RomanoJ. H. RomieC. RöverS. RowanA. RüdigerK. RyanH. RyllP. SainathanM. SakoskyF. SalemiA. SamblowskiL. SammutL. Sancho de la JordanaV. SandbergS. SankarV. SannibaleL. SantamaríaI. Santiago-PrietoG. SantostasiB. S. SathyaprakashS. SatoP. R. SaulsonR. L. SavageR. SchillingS. SchlammingerR. SchnabelR. M. S. SchofieldB. SchulzB. F. SchutzP. SchwinbergJ. ScottS. M. ScottA. C. SearleF. SeifertD. SellersA. S. SenguptaA. SergeevD. A. ShaddockM. ShaltevB. ShapiroP. ShawhanD. H. ShoemakerA. 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WinklerC. C. WipfH. WittelA. G. WisemanG. WoanR. WooleyJ. WordenJ. YablonI. YakushinK. YamamotoH. YamamotoH. YangD. Yeaton-MasseyS. YoshidaP. YuM. ZanolinL. ZhangW. ZhangZ. ZhangC. ZhaoN. ZotovM. E. ZuckerJ. ZweizigThe LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7(12), 962–965 (2011). [CrossRef]

,13

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]. This experiment further enhanced confidence and passion in applying squeezed state to large scale GW-detectors. Actually great progress of generating this kind of squeezing has been made in recent years [14

W. P. Bowen, R. Schnabel, N. Treps, H.-A. Bachor, and P. K. Lam, “Recovery of continuous wave squeezing at low frequencies,” J. Opt. B Quantum Semiclassical Opt. 4(6), 421–424 (2002). [CrossRef]

–20

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys. 9(10), 371–378 (2007). [CrossRef]

]. It’s found that coherently controlling the phases of the experimental set-up while not to introduce extra noise is one of the keys to generate low-frequency squeezing [16

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Compact source of Einstein-Podolsky-Rosen entanglement and squeezing at very low noise frequencies,” Phys. Rev. A 70(4), 042315 (2004). [CrossRef]

–20

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys. 9(10), 371–378 (2007). [CrossRef]

]. In 2007, an unprecedented experiment of generating squeezed vacuum states with a noise power 6.5 dB below vacuum noise within the entire detection bandwidth of ground-based GW-detectors (10 Hz - 10 kHz) was demonstrated by using a sophisticated control scheme [20

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys. 9(10), 371–378 (2007). [CrossRef]

]. An alternative way to conquer this difficulty is to use two-frequency laser and broadband squeezing at higher frequency, which has been primarily used in many quantum optics laboratories, to enhance the signal-to-noise ratio (SNR) of an interferometer for lower-frequency phase measurement. When the carriers with frequencies other than central frequency of squeezed state appear in the interferometer, high-frequency squeezing sidebands will come into play. J. Gea-Banacloche and G. Leuchs [21

J. Gea-Banacloche and G. Leuchs, “Squeezed states for interferometric gravitational-wave detectors,” J. Mod. Opt. 34(6-7), 793–811 (1987). [CrossRef]

] theoretically showed that, in Michelson interferometers stabilized with phase modulation technique, broadband squeezing is needed. Almost at the same time, Yurke et al [22

B. Yurke, P. Grangier, and R. E. Slusher, “Squeezed-state enhanced two-frequency interferometry,” J. Opt. Soc. Am. B 4(10), 1677–1682 (1987). [CrossRef]

] proposed a squeezed-state enhanced two-frequency interferometer to perform sub-shot-noise measurement of low-frequency signals by reading the photocurrent at frequency _{$2 ν_{s}$}(frequency interval of the two-frequency laser), well away from the low-frequency technical noise. In this study, we read low-frequency signals directly and calculate the SNR. This technique can be straightforwardly extended to the other squeezing-enhanced measurement schemes mentioned above.

Theoretical model

We consider a squeezing-enhanced Mach-Zehnder interferometer as in Ref [9

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987). [CrossRef] [PubMed]

]. shown in Fig. 1 . The main idea is that by transferring the frequency of the intense laser to the frequencies of the entangled sidebands of the squeezed field, the low-frequency signal originating from the sidebands close to the intense laser frequencies can be detected with enhanced SNR for the entangled squeezing sidebands. The frequency relations of the usualscheme and our scheme are shown in Fig. 2 . Suppose the two-frequency laser and squeezed state have annihilation operators _{$\hat{A} = {\hat{a}}_{+} \exp [- i (ω_{0} + Ω) t] + {\hat{a}}_{-} \exp [- i (ω_{0} - Ω) t]$} and _{$\hat{B} = \hat{b} \exp (- i ω_{0} t)$} respectively, where _{$ω_{0}$} is the optical circular frequency and _$Ω$ is the circular frequency, at which squeezing occurs. With the mean-field approximation, the annihilation operators _{${\hat{a}}_{+}$}, _{${\hat{a}}_{-}$}and _$\hat{b}$ can be expressed as the sum of its mean field and fluctuation, i.e. _{${\hat{a}}_{i} = α_{i} + δ {\hat{a}}_{i} \begin{matrix} \end{matrix} (i = +, -)$}, and _{$\hat{b} = δ \hat{b}$}. Here, squeezing is supposed to be vacuum-squeezing, so that the corresponding mean field is zero. The mean amplitudes _{$α_{+} = α_{-} = α$} are real numbers by choosing a proper phase reference and are supposed to be equal. The average photon numbers in unit measurement time is _{$N = 2 α^{2} = P / ℏ ω_{0}$}, where _$P$ is the optical power of the intense two-frequency laser. With the relative phase _$φ$ at the first 50% beam splitter BS1 and relative phase _{$π / 2 + θ (t)$}at the second 50% beam splitter BS2 in the Mach-Zehnder interferometer, the annihilation operators of the various modes in Fig. 1 are related to each other via

\hat{C} = \frac{1}{\sqrt{2}} [\hat{A} + \hat{B} \exp (i φ)],

(1)

\hat{D} = \frac{1}{\sqrt{2}} [\hat{A} - \hat{B} \exp (i φ)],

(2)

and

\begin{array}{l} \hat{E} = \frac{1}{\sqrt{2}} [\hat{C} + \hat{D} \exp (i π / 2 + i θ (t))] \\ = \frac{1}{2} {[1 + \exp (i π / 2 + i θ (t))] \hat{A} + [1 - \exp (i π / 2 + i θ (t))] \hat{B} e^{i φ}}, \end{array}

(3)

\begin{array}{l} \hat{F} = \frac{1}{\sqrt{2}} [\hat{C} - \hat{D} \exp (i π / 2 + i θ (t))] \\ = \frac{1}{2} {[1 - \exp (i π / 2 + i θ (t))] \hat{A} + [1 + \exp (i π / 2 + i θ (t))] \hat{B} \exp (i φ)}, \end{array}

(4)

where _{$θ (t) = \sum_{s} θ_{w_{s}} \cos w_{s} t$} is a sum of the low-frequency cosine signal _{$θ_{w_{1}} \cos w_{1} t$} to be measured at frequency _{$w_{1}$} and some spurious signals which could be coupled to the photocurrent, where _{$θ_{w_{s}}$}is the signal amplitude at frequency _{$w_{s}$}. The frequency spacing is much larger than the measurement resolution bandwidth_{$Δ ω$}. The quantum efficiency of photodiodes is supposed to be unity, so that the subtracted output photocurrent deduced from Eqs. (1)–(4) is

\begin{array}{l} \hat{i} (t) = {\hat{F}}^{†} \hat{F} - {\hat{E}}^{†} \hat{E} \\ = N \sum_{s} θ_{w_{s}} \cos w_{s} t (1 + \cos 2 Ω t) + \sqrt{2 N} δ {\hat{X}}_{b}^{φ + π / 2} (t) \cos Ω t \\ = s (t) + n (t), \end{array}

(5)

where the quadrature fluctuations are defined as _{$δ {\hat{X}}_{b}^{φ} = δ \hat{b} e^{i φ} + δ {\hat{b}}^{†} e^{- i φ}$}, and the product terms of quadrature fluctuations and the terms multiplying _{$θ (t)$} and quadrature fluctuations are omitted for _{$θ < < 1$} and _{$δ X_{i}^{φ} < < α \begin{matrix} \end{matrix} (i = +, -, b)$}. We take the photocurrent duration as _$T$(_{$1 / T < < w_{s}$}), which is the reciprocal of measurement resolution bandwidth_{$Δ ω / 2 π$}. The photocurrent _{$i (t)$} is separated into a signal part _{$s (t)$} and noise part _{$n (t)$}, and the power spectral densities are calculated separately. By using the Wiener-Khinchine theorem, the power spectral density of the first term of Eq. (5), i.e. the signal term, is

\begin{array}{l} P_{s} (w) = \int_{- \infty}^{\infty} d τ e^{- i w τ} R (τ) = \frac{1}{T} \int_{- \infty}^{\infty} d τ e^{- i w τ} \int_{- T / 2}^{T / 2} d t s (t) s (t - τ) \\ = \frac{π N^{2}}{2} \sum_{s} (2 θ_{w_{s}}^{2} + θ_{w_{s}} θ_{w_{s} - 2 Ω} + θ_{w_{s}} θ_{w_{s} + 2 Ω}) [δ (w + w_{s}) + δ (w - w_{s})] \\ + \frac{π N^{2}}{4} \sum_{s} (θ_{w_{s}}^{2} + θ_{w_{s}} θ_{w_{s} + 4 Ω} + 2 θ_{w_{s}} θ_{w_{s} + 2 Ω}) [δ (w + w_{s} + 2 Ω) + δ (w - w_{s} - 2 Ω)] \\ + \frac{π N^{2}}{4} \sum_{s} (θ_{w_{s}}^{2} + θ_{w_{s}} θ_{w_{s} - 4 Ω} + 2 θ_{w_{s}} θ_{w_{s} - 2 Ω}) [δ (w - w_{s} + 2 Ω) + δ (w + w_{s} - 2 Ω)], \end{array}

(6)

where we use the approximation that _{$\sin ω T / (ω T) \approx 1$} when _{$ω = 0$}and _{$\sin ω T / (ω T) \approx 0$} when _{$ω \neq 0$}. Function _{$R (τ) = (1 / T) \int_{- T / 2}^{T / 2} d t s (t) s (t - τ)$} is the autocorrelation function of signal _{$s (t)$}. Function _{$δ (w - w_{1})$} is the Dirac delta function defined as _{$\int_{- \infty}^{\infty} {dte}^{-i w t} = 2 π δ (w)$}, and satisfies_{$\int_{w_{1} - Δ ω / 2}^{w_{1} + Δ ω / 2} d w δ (w - w_{1}) = 1$}. Also, _{$θ_{w_{s}} = θ_{- w_{s}}$} is used. The noise power spectral density of the second term on the other side of Eq. (5), i.e. the noise term, is

\begin{array}{l} P_{n} (w) = {〈 \frac{1}{T} \int_{- \infty}^{\infty} d τ e^{- i w τ} \int_{- T / 2}^{T / 2} d t n (t) n (t - τ) 〉}_{e n} \\ = {〈 {| F_{T} {n (t)} |}^{2} 〉}_{e n} / T \\ = \frac{N}{2} {〈 \int_{- T / 2}^{T / 2} d t δ X_{b}^{φ + π / 2} (t) (e^{- i (w - Ω) t} + e^{- i (w + Ω) t}) \int_{- T / 2}^{T / 2} d t^{'} δ X_{b}^{φ + π / 2} (t^{'}) (e^{i (w - Ω) t^{'}} + e^{i (w + Ω) t^{'}}) 〉}_{e n} / T \\ = \frac{N}{2} [V_{b}^{φ + π / 2} (w + Ω) + V_{b}^{φ + π / 2} (w - Ω)], \end{array}

(7)

where_{${〈 ... 〉}_{e n}$} refers to the ensemble average, and _{$F_{T} {n (t)} = n_{T} (w) = \int_{- T / 2}^{T / 2} d t n (t) e^{- i w t}$}is the Fourier transform of function _{$n (t)$}. _{$V_{b}^{φ + π / 2} (w + Ω)$}, defined as

V_{b}^{φ + π / 2} (w + Ω) = {〈 {| δ X_{b}^{φ + π / 2} (w + Ω) |}^{2} 〉}_{e n} / T = {〈 {| \int_{- T / 2}^{T / 2} d t δ X_{b}^{φ + π / 2} (t) e^{- i (w + Ω) t} |}^{2} 〉}_{e n} / T

(8)

is the quadrature variance of the squeezed state at frequency _{$w + Ω$}. To evaluate the SNR at frequency _{$w_{1}$}, we integrate the signal spectral density (Eq. (6)) and noise spectral density (Eq. (7)) in the frequency interval _{$[w_{1} - Δ ω / 2, \begin{matrix} \end{matrix} w_{1} + Δ ω / 2]$}, and take the ratio as the SNR:

\begin{array}{l} P_{s} (w_{1}) = \int_{w_{1} - Δ ω / 2}^{w_{1} + Δ ω / 2} \frac{d w}{2 π} P_{s} (w) \\ = N^{2} {(θ_{w_{1}} + θ_{w_{1} + 2 Ω} / 2 + θ_{w_{1} - 2 Ω} / 2)}^{2}, \end{array}

(9)

and

\begin{array}{l} P_{n} (w_{1}) = \int_{w_{1} - Δ ω / 2}^{w_{1} + Δ ω / 2} \frac{d w}{2 π} P_{n} (w) \\ = \frac{N Δ ω}{4 π} [V_{b}^{φ + π / 2} (w_{1} + Ω) + V_{b}^{φ + π / 2} (w_{1} - Ω)] . \end{array}

(10)

Therefore

S N R = \frac{P_{s} (w_{1})}{P_{n} (w_{1})} = \frac{2 T N {(θ_{w_{1}} + θ_{w_{1} + 2 Ω} / 2 + θ_{w_{1} - 2 Ω} / 2)}^{2}}{V_{b}^{φ + π / 2} (w_{1} + Ω) + V_{b}^{φ + π / 2} (w_{1} - Ω)} .

(11)

The integration of Eq. (10) takes the quadrature variances to be uniform in the integral frequency interval. Equations (9) and (11) show that the output photocurrent at frequency_{$w_{1}$} could be contaminated by the phase vibrations at frequencies _{$w_{1} - 2 Ω$} and _{$w_{1} + 2 Ω$}, and it may not faithfully reflect the phase signal at _{$w_{1}$}. However, frequency _$Ω$ is usually chosen to be at Megahertz so that the phase vibrations at _{$w_{1} - 2 Ω$} and _{$w_{1} + 2 Ω$} is rare in real experiment. The SNR of Eq. (11) can be simplified to

S N R = \frac{2 N T θ_{w_{1}}^{2}}{V_{b}^{φ + π / 2} (w_{1} + Ω) + V_{b}^{φ + π / 2} (w_{1} - Ω)} .

(12)

In this case, the photocurrent at frequency _{$w_{1}$} does reflect the phase vibration at _{$w_{1}$} and has sub-shot noise resolution if the quadrature variables of the squeezed state at the angle _{$φ + π / 2$} are squeezed at frequencies _{$w_{1} - Ω$} and _{$w_{1} + Ω$}.

Fig. 1 Squeezed-state enhanced Mach-Zehnder interferometer. OPA refers to optical parametrical amplifier. BS1 and BS2 refer to 50% beam splitters. Frequencies of the two-frequency laser are _{$ω_{0} + Ω$} and _{$ω_{0} - Ω$}. Dashed arrow refers to squeezing with center frequency _{$ω_{0}$} generated by OPA.

Fig. 2 Frequency relations of two-frequency intense laser, squeezing and signal sidebands for: (a) usual interferometer; (b) our scheme. _$w$ is the low frequency to be measured. The shadowed areas are entangled sidebands of the squeezed state. Long arrows are carrier frequencies of the intense laser and the short arrows are signal sidebands.

The physics of this scheme can be explained as follows. Instead of using a single-frequency intense laser at frequency _{$ω_{0}$} in the interferometry, we use an intense laser with the carrier frequencies _{$ω_{0} - Ω$} and _{$ω_{0} + Ω$}, with which the entangled upper and lower sidebands of vacuum squeezing interfere. Therefore, the low-frequency signal, which generates sidebands near the carrier fields, can be detected with sub-shot noise resolution, thanks to the entangled sidebands of squeezing. The terms with phase signals at frequencies _{$w_{1} - 2 Ω$} and _{$w_{1} + 2 Ω$} coupled to the SNR at frequency _{$w_{1}$} in Eq. (11) originate from the beating between one of the carriers and the sidebands at _{$w_{1} - 2 Ω$} and _{$w_{1} + 2 Ω$} of the other carrier. It’s interesting to note that in Eq. (11) the signal appears at frequency _{$w_{1}$} and _{$w_{1} \pm 2 Ω$} while the noise at frequency _{$w_{1} \pm Ω$}. This difference can be attributed to the fact that, in Mach-Zehnder interferometer, signal sidebands are generated around carrier fields while noise sidebands come from the injected squeezing.

Conclusion

We have calculated the utility of high-frequency squeezed-state enhanced two-frequency interferometry for low-frequency phase measurement. By means of a two-frequency laser interferometer, the higher-frequency sidebands of the squeezed state can be used to enhance the lower-frequency phase measurement. A proof-of-principle experiment is in the reach of modern quantum optics technology and is in progress in our laboratory. Moreover, this scheme is also useful for many other squeezing-enhanced measurement schemes, and also provides a method to generate low-frequency squeezing with high-frequency squeezing.

Acknowledgments

Zehui Zhai thanks Dabo Guo for helpful discussion. This work was supported in part by the National Science Foundation of China under Grant No. 11174189, No.60708010 and No.60978008, by the National Basic Research Program of China under Grant No.2010CB923102, and by the Open project of State Key Laboratory of Quantum Optics and Quantum Optics Devices under Grant No.200902.

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D. Ott, D. J. Ottaway, R. S. Ottens, H. Overmier, B. J. Owen, A. Page, Y. Pan, C. Pankow, M. A. Papa, P. Ajith, P. Patel, M. Pedraza, P. Peiris, L. Pekowsky, S. Penn, C. Peralta, A. Perreca, M. Phelps, M. Pickenpack, I. M. Pinto, M. Pitkin, H. J. Pletsch, M. V. Plissi, J. Pöld, F. Postiglione, V. Predoi, L. R. Price, M. Prijatelj, M. Principe, S. Privitera, R. Prix, L. Prokhorov, O. Puncken, V. Quetschke, F. J. Raab, H. Radkins, P. Raffai, M. Rakhmanov, C. R. Ramet, B. Rankins, S. R. P. Mohapatra, V. Raymond, K. Redwine, C. M. Reed, T. Reed, S. Reid, D. H. Reitze, R. Riesen, K. Riles, N. A. Robertson, C. Robinson, E. L. Robinson, S. Roddy, C. Rodriguez, M. Rodruck, J. Rollins, J. D. Romano, J. H. Romie, C. Röver, S. Rowan, A. Rüdiger, K. Ryan, H. Ryll, P. Sainathan, M. Sakosky, F. Salemi, A. Samblowski, L. Sammut, L. Sancho de la Jordana, V. Sandberg, S. Sankar, V. Sannibale, L. Santamaría, I. Santiago-Prieto, G. Santostasi, B. S. Sathyaprakash, S. Sato, P. R. Saulson, R. L. Savage, R. Schilling, S. Schlamminger, R. Schnabel, R. M. S. Schofield, B. Schulz, B. F. Schutz, P. Schwinberg, J. Scott, S. M. Scott, A. C. Searle, F. Seifert, D. Sellers, A. S. Sengupta, A. Sergeev, D. A. Shaddock, M. Shaltev, B. Shapiro, P. Shawhan, D. H. Shoemaker, A. Sibley, X. Siemens, D. Sigg, A. Singer, L. Singer, A. M. Sintes, G. Skelton, B. J. J. Slagmolen, J. Slutsky, R. J. E. Smith, J. R. Smith, M. R. Smith, N. D. Smith, K. Somiya, B. Sorazu, J. Soto, F. C. Speirits, A. J. Stein, E. Steinert, J. Steinlechner, S. Steinlechner, S. Steplewski, M. Stefszky, A. Stochino, R. Stone, K. A. Strain, S. Strigin, A. S. Stroeer, A. L. Stuver, T. Z. Summerscales, M. Sung, S. Susmithan, P. J. Sutton, D. Talukder, D. B. Tanner, S. P. Tarabrin, J. R. Taylor, R. Taylor, P. Thomas, K. A. Thorne, K. S. Thorne, E. Thrane, A. Thüring, C. Titsler, K. V. Tokmakov, C. Torres, C. I. Torrie, G. Traylor, M. Trias, K. Tseng, D. Ugolini, K. Urbanek, H. Vahlbruch, M. Vallisneri, A. A. van Veggel, S. Vass, R. Vaulin, A. Vecchio, J. Veitch, P. J. Veitch, C. Veltkamp, A. E. Villar, S. Vitale, C. Vorvick, S. P. Vyatchanin, A. Wade, S. J. Waldman, L. Wallace, Y. Wan, A. Wanner, X. Wang, Z. Wang, R. L. Ward, P. Wei, M. Weinert, A. J. Weinstein, R. Weiss, L. Wen, S. Wen, P. Wessels, M. West, T. Westphal, K. Wette, J. T. Whelan, S. E. Whitcomb, D. White, B. F. Whiting, C. Wilkinson, P. A. Willems, H. R. Williams, L. Williams, B. Willke, L. Winkelmann, W. Winkler, C. C. Wipf, H. Wittel, A. G. Wiseman, G. Woan, R. Wooley, J. Worden, J. Yablon, I. Yakushin, K. Yamamoto, H. Yamamoto, H. Yang, D. Yeaton-Massey, S. Yoshida, P. Yu, M. Zanolin, L. Zhang, W. Zhang, Z. Zhang, C. Zhao, N. Zotov, M. E. Zucker, and J. ZweizigJ. AbadieB. P. AbbottR. AbbottT. D. AbbottM. AbernathyC. AdamsR. AdhikariC. AffeldtB. AllenG. S. AllenE. Amador CeronD. AmariuteiR. S. AminS. B. AndersonW. G. AndersonK. AraiM. A. ArainM. C. ArayaS. M. AstonD. AtkinsonP. AufmuthC. AulbertB. E. AylottS. BabakP. BakerS. BallmerD. BarkerB. BarrP. BarrigaL. BarsottiM. A. BartonI. BartosR. BassiriM. BastarrikaJ. BatchJ. BauchrowitzB. BehnkeA. S. BellI. BelopolskiM. BenacquistaJ. M. BerlinerA. BertoliniJ. BetzwieserN. BeveridgeP. T. BeyersdorfI. A. BilenkoG. BillingsleyJ. BirchR. BiswasE. BlackJ. K. BlackburnL. BlackburnD. BlairB. BlandO. BockT. P. BodiyaC. BoganR. BondarescuR. BorkM. BornS. BoseP. R. BradyV. B. BraginskyJ. E. BrauJ. BreyerD. O. BridgesM. BrinkmannM. BritzgerA. F. BrooksD. A. BrownA. BrummittA. BuonannoJ. Burguet-CastellO. BurmeisterR. L. ByerL. CadonatiJ. B. CampP. CampsieJ. CannizzoK. CannonJ. CaoC. D. CapanoS. CarideS. CaudillM. CavagliáC. CepedaT. ChalermsongsakE. ChalkleyP. CharltonS. ChelkowskiY. ChenN. ChristensenH. ChoS. S. Y. ChuaS. ChungC. T. Y. ChungG. CianiF. ClaraD. E. ClarkJ. ClarkJ. H. ClaytonR. ConteD. CookT. R. CorbittM. CordierN. CornishA. CorsiC. A. CostaM. CoughlinP. CouvaresD. M. CowardD. C. CoyneJ. D. E. CreightonT. D. CreightonA. M. CruiseA. CummingL. CunninghamR. M. CutlerK. DahlS. L. DanilishinR. DannenbergK. DanzmannB. DaudertH. DavelozaG. DaviesE. J. DawT. DayangaD. DeBraJ. DegallaixT. DentV. DergachevR. DeRosaR. DeSalvoS. DhurandharJ. DiGuglielmoI. Di PalmaM. DíazF. DonovanK. L. DooleyS. DorsherR. W. P. DreverJ. C. DriggersZ. DuJ.-C. DumasS. DwyerT. EberleM. EdgarM. EdwardsA. EfflerP. EhrensR. EngelT. EtzelK. EvansM. EvansT. EvansM. FactourovichS. FairhurstY. FanB. F. FarrW. FarrD. FaziH. FehrmannD. FeldbaumL. S. FinnR. P. FisherM. FlaniganS. FoleyE. ForsiN. FotopoulosM. FredeM. FreiZ. FreiA. FreiseR. FreyT. T. FrickeD. FriedrichP. FritschelV. V. FrolovP. J. FuldaM. FyffeM. R. GanijaJ. GarciaJ. A. GarofoliR. GengL. Á. GergelyI. GholamiS. GhoshJ. A. GiaimeS. GiampanisK. D. GiardinaC. GillE. GoetzL. M. GogginG. GonzálezM. L. GorodetskyS. GoßlerC. GraefA. GrantS. GrasC. GrayN. GrayR. J. S. GreenhalghA. M. GretarssonR. GrossoH. GroteS. GrunewaldC. GuidoR. GuptaE. K. GustafsonR. GustafsonT. HaB. HageJ. M. HallamD. HammerG. HammondJ. HanksC. HannaJ. HansonJ. HarmsG. M. HarryI. W. HarryE. D. HarstadM. T. HartmanK. HaughianK. HayamaJ. HeefnerM. C. HeintzeM. A. HendryI. S. HengA. W. HeptonstallV. HerreraM. HewitsonS. HildD. HoakK. A. HodgeK. HoltT. HongS. HooperD. J. HoskenJ. HoughE. J. HowellB. HugheyT. Huynh-DinhS. HusaS. H. HuttnerD. R. IngramR. IntaT. IsogaiA. IvanovK. IzumiM. JacobsonH. JangW. W. JohnsonD. I. JonesG. JonesR. JonesL. JuP. KalmusV. KalogeraI. KamaretsosS. KandhasamyG. KangJ. B. KannerE. KatsavounidisW. KatzmanH. KauferK. KawabeS. KawamuraF. KawazoeW. KellsD. G. KeppelZ. KeresztesA. KhalaidovskiF. Y. KhaliliE. A. KhazanovB. KimC. KimD. KimH. KimK. KimN. KimY.-M. KimP. J. KingM. KinseyD. L. KinzelJ. S. KisselS. KlimenkoK. KokeyamaV. KondrashovR. KopparapuS. KorandaW. Z. KorthD. KozakV. KringelS. KrishnamurthyB. KrishnanG. KuehnR. KumarP. KweeP. K. LamM. LandryM. LangB. LantzN. LastzkaC. LawrieA. LazzariniP. LeaciC. H. LeeH. M. LeeN. LeindeckerJ. R. LeongI. LeonorJ. LiP. E. LindquistN. A. LockerbieD. LodhiaM. LormandJ. LuanM. LubinskiH. LückA. P. LundgrenE. MacdonaldB. MachenschalkM. MacInnisD. M. MacleodM. MageswaranK. MailandI. MandelV. MandicA. MarandiS. MárkaZ. MárkaA. MarkosyanE. MarosI. W. MartinR. M. MartinJ. N. MarxK. MasonF. MatichardL. MatoneR. A. MatznerN. MavalvalaG. MazzoloR. McCarthyD. E. McClellandS. C. McGuireG. McIntyreJ. McIverD. J. A. McKechanG. D. MeadorsM. MehmetT. MeierA. MelatosA. C. MelissinosG. MendellD. MenendezR. A. MercerS. MeshkovC. MessengerM. S. MeyerH. MiaoJ. MillerV. P. MitrofanovG. MitselmakherR. MittlemanO. MiyakawaB. MoeP. MoestaS. D. MohantyD. MoraruG. MorenoT. MoriK. MossaviC. M. Mow-LowryC. L. MuellerG. MuellerS. MukherjeeA. MullaveyH. Müller-EbhardtJ. MunchD. MurphyP. G. MurrayA. MytidisT. NashR. NawrodtV. NeculaJ. NelsonG. NewtonA. NishizawaD. NoltingL. NuttallJ. O’DellB. O’ReillyR. O’ShaughnessyE. OchsnerE. OelkerJ. J. OhS. H. OhG. H. OginR. G. OldenburgC. OsthelderC. D. OttD. J. OttawayR. S. OttensH. OvermierB. J. OwenA. PageY. PanC. PankowM. A. PapaP. AjithP. PatelM. PedrazaP. PeirisL. PekowskyS. PennC. PeraltaA. PerrecaM. PhelpsM. PickenpackI. M. PintoM. PitkinH. J. PletschM. V. PlissiJ. PöldF. PostiglioneV. PredoiL. R. PriceM. PrijateljM. PrincipeS. PriviteraR. PrixL. ProkhorovO. PunckenV. QuetschkeF. J. RaabH. RadkinsP. RaffaiM. RakhmanovC. R. RametB. RankinsS. R. P. MohapatraV. RaymondK. RedwineC. M. ReedT. ReedS. ReidD. H. ReitzeR. RiesenK. RilesN. A. RobertsonC. RobinsonE. L. RobinsonS. RoddyC. RodriguezM. RodruckJ. RollinsJ. D. RomanoJ. H. RomieC. RöverS. RowanA. RüdigerK. RyanH. RyllP. SainathanM. SakoskyF. SalemiA. SamblowskiL. SammutL. Sancho de la JordanaV. SandbergS. SankarV. SannibaleL. SantamaríaI. Santiago-PrietoG. SantostasiB. S. SathyaprakashS. SatoP. R. SaulsonR. L. SavageR. SchillingS. SchlammingerR. SchnabelR. M. S. SchofieldB. SchulzB. F. SchutzP. SchwinbergJ. ScottS. M. ScottA. C. SearleF. SeifertD. SellersA. S. SenguptaA. SergeevD. A. ShaddockM. ShaltevB. ShapiroP. ShawhanD. H. ShoemakerA. SibleyX. SiemensD. SiggA. SingerL. SingerA. M. SintesG. SkeltonB. J. J. SlagmolenJ. SlutskyR. J. E. SmithJ. R. SmithM. R. SmithN. D. SmithK. SomiyaB. SorazuJ. SotoF. C. SpeiritsA. J. SteinE. SteinertJ. SteinlechnerS. SteinlechnerS. SteplewskiM. StefszkyA. StochinoR. StoneK. A. StrainS. StriginA. S. StroeerA. L. StuverT. Z. SummerscalesM. SungS. SusmithanP. J. SuttonD. TalukderD. B. TannerS. P. TarabrinJ. R. TaylorR. TaylorP. ThomasK. A. ThorneK. S. ThorneE. ThraneA. ThüringC. TitslerK. V. TokmakovC. TorresC. I. TorrieG. TraylorM. TriasK. TsengD. UgoliniK. UrbanekH. VahlbruchM. VallisneriA. A. van VeggelS. VassR. VaulinA. VecchioJ. VeitchP. J. VeitchC. VeltkampA. E. VillarS. VitaleC. VorvickS. P. VyatchaninA. WadeS. J. WaldmanL. WallaceY. WanA. WannerX. WangZ. WangR. L. WardP. WeiM. WeinertA. J. WeinsteinR. WeissL. WenS. WenP. WesselsM. WestT. WestphalK. WetteJ. T. WhelanS. E. WhitcombD. WhiteB. F. WhitingC. WilkinsonP. A. WillemsH. R. WilliamsL. WilliamsB. WillkeL. WinkelmannW. WinklerC. C. WipfH. WittelA. G. WisemanG. WoanR. WooleyJ. WordenJ. YablonI. YakushinK. YamamotoH. YamamotoH. YangD. Yeaton-MasseyS. YoshidaP. YuM. ZanolinL. ZhangW. ZhangZ. ZhangC. ZhaoN. ZotovM. E. ZuckerJ. ZweizigThe LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7(12), 962–965 (2011). [CrossRef]
13.	H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Gr¨af, K. Danzmann, and R. Schnabel, “The GEO 600 squeezed light source,” Class. Quantum Gravity 27(8), 084027 (2010). [CrossRef]
14.	W. P. Bowen, R. Schnabel, N. Treps, H.-A. Bachor, and P. K. Lam, “Recovery of continuous wave squeezing at low frequencies,” J. Opt. B Quantum Semiclassical Opt. 4(6), 421–424 (2002). [CrossRef]
15.	R. Schnabel, H. Vahlbruch, A. Franzen, S. Chelkowski, N. Grosse, H.-A. Bachor, W. P. Bowen, P. K. Lam, and K. Danzmann, “Squeezed light at sideband frequencies below 100 kHz from a single OPA,” Opt. Commun. 240(1-3), 185–190 (2004). [CrossRef]
16.	J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Compact source of Einstein-Podolsky-Rosen entanglement and squeezing at very low noise frequencies,” Phys. Rev. A 70(4), 042315 (2004). [CrossRef]
17.	K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, “Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator,” Phys. Rev. A 72(4), 043819 (2005). [CrossRef]
18.	H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett. 97(1), 011101 (2006). [CrossRef] [PubMed]
19.	K. Goda, E. E. Mikhailov, O. Miyakawa, S. Saraf, S. Vass, A. Weinstein, and N. Mavalvala, “Generation of a stable low-frequency squeezed vacuum field with periodically poled KTiOPO₄ at 1064 nm,” Opt. Lett. 33(2), 92–94 (2008). [CrossRef] [PubMed]
20.	H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys. 9(10), 371–378 (2007). [CrossRef]
21.	J. Gea-Banacloche and G. Leuchs, “Squeezed states for interferometric gravitational-wave detectors,” J. Mod. Opt. 34(6-7), 793–811 (1987). [CrossRef]
22.	B. Yurke, P. Grangier, and R. E. Slusher, “Squeezed-state enhanced two-frequency interferometry,” J. Opt. Soc. Am. B 4(10), 1677–1682 (1987). [CrossRef]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(270.0270) Quantum optics : Quantum optics
(270.6570) Quantum optics : Squeezed states

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 21, 2011
Revised Manuscript: December 18, 2011
Manuscript Accepted: January 27, 2012
Published: July 23, 2012

Citation
Zehui Zhai and Jiangrui Gao, "Low-frequency phase measurement with high-frequency squeezing," Opt. Express 20, 18173-18179 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18173

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Abadie, J.
Abbott, B. P.
Abbott, R.
Abbott, T. D.
Abernathy, M.
Adams, C.
Adhikari, R.
Affeldt, C.
Ajith, P.
Allen, B.
Allen, G. S.
Amador Ceron, E.
Amariutei, D.
Amin, R. S.
Andersen, U.
Anderson, S. B.
Anderson, W. G.
Arai, K.
Arain, M. A.
Araya, M. C.
Aston, S. M.
Atkinson, D.
Aufmuth, P.
Aulbert, C.
Aylott, B. E.
Babak, S.
Bachor, H.-A.
Baker, P.
Ballmer, S.
Barker, D.
Barr, B.
Barriga, P.
Barsotti, L.
Barton, M. A.
Bartos, I.
Bassiri, R.
Bastarrika, M.
Batch, J.
Bauchrowitz, J.
Beduini, F. A.
Behnke, B.
Bell, A. S.
Belopolski, I.
Benacquista, M.
Berliner, J. M.
Bertolini, A.
Betzwieser, J.
Beveridge, N.
Beyersdorf, P. T.
Bilenko, I. A.
Billingsley, G.
Birch, J.
Biswas, R.
Black, E.
Blackburn, J. K.
Blackburn, L.
Blair, D.
Bland, B.
Bock, O.
Bodiya, T. P.
Bogan, C.
Bondarescu, R.
Bork, R.
Born, M.
Bose, S.
Bowen, W. P.
Brady, P. R.
Braginsky, V. B.
Brau, J. E.
Breyer, J.
Bridges, D. O.
Brinkmann, M.
Britzger, M.
Brooks, A. F.
Brown, D. A.
Brummitt, A.
Buchler, B.
Buonanno, A.
Burguet-Castell, J.
Burmeister, O.
Byer, R. L.
Cadonati, L.
Camp, J. B.
Campsie, P.
Cannizzo, J.
Cannon, K.
Cao, J.
Capano, C. D.
Caride, S.
Carri, J.
Caudill, S.
Cavagliá, M.
Caves, C. M.
Cepeda, C.
Cerè, A.
Chalermsongsak, T.
Chalkley, E.
Charlton, P.
Chelkowski, S.
Chen, Y.
Cho, H.
Christensen, N.
Chua, S. S. Y.
Chung, C. T. Y.
Chung, S.
Ciani, G.
Clara, F.
Clark, D. E.
Clark, J.

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Low-frequency phase measurement with high-frequency squeezing

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