Low frequency phase signal measurement with high frequency squeezing

We calculate the utility of high-frequency squeezed-state enhanced two-frequency interferometry for low-frequency phase measurement. To use the high-frequency sidebands of the squeezed light, a two-frequency intense laser is used in the interferometry instead of a single-frequency laser as usual. We find that the readout signal can be contaminated by the high-frequency phase vibration, but this is easy to check and avoid. A proof-of-principle experiment is in the reach of modern quantum optics technology.


Introduction
The concept of radiation field squeezing [1] has attracted much interest since its application to gravitational wave detection was proposed [2]. This sort of radiation field was first generated with a four-wave mixing process by Slusher et al [3] and * Electronic address: zhzehui@sxu.edu.cn improved with an optical parametric oscillator (OPO) by Wu et al [4]. It has been useful in various quantum-enhanced measurement schemes, such as squeezed-light magnetometry [5], displacement measurement [6], spectroscopy [7], polarization measurement [8], phase measurement [9], and so on. Squeezed states of light are also vital ingredients for continuous-variable quantum communication and quantum computation [10]. In some applications, such as gravitational wave detection, it is appealing to measure low-frequency signals with a squeezed light state.
Low-frequency squeezing has attracted much interest in recent years as terrestrial gravitational wave detectors are approaching their shot noise limit, which can be overcome with squeezing at audio frequencies. Great progress of generating this kind of squeezing has been made in recent years [11][12][13][14][15][16][17]. 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 [13][14][15][16][17]. 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 [17]. An alternative way to conquer this difficulty is to use a 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-signal measurement. In 1987, Yurke et al [18] 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 s  , well away from the low-frequency technical noise.
In this study, we read low-frequency signals directly and calculate the SNR. We find that the readout signal can be contaminated by high-frequency phase vibrations, but this is easy to check. 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] shown in Fig where ( ) t  is a sum of the low-frequency cosine signal The frequency spacing is much larger than the measurement resolution bandwidth   . The quantum efficiency of photodiodes is supposed to be unity, so that the subtracted output photocurrent deduced from eqs.
where the quadrature fluctuations are defined as † bˆî By using the Wiener-Khinchine theorem, the power spectral density of the first term of eq.(5), i.e. the signal term, is where we use the approximation that  when and where ... en refers to the ensemble average, and is the Fourier transform of function .
is the quadrature variance of the squeezed state at frequency . To evaluate the SNR at frequency , we integrate the signal spectral density (eq. (6)) and noise spectral density (eq. (7)) in the frequency interval The integration of equ.(10) takes the quadrature variances to be uniform in the integral frequency interval. Expressions (9) and (11) show that the phase vibrations at frequencies and will contaminate the output photocurrent at frequency . Therefore, the photocurrent signal at may not faithfully reflect the phase vibration at . However, when the phase vibrations at and do not exist, the SNR of eq.(11) can be simplified to . (12) In this case, the photocurrent at frequency does reflect the phase vibration at and has sub-shot noise resolution if the quadrature variables of the squeezed state at It's interesting to note that in eq.(11) the signal appears at frequency and 1 w 1 2 w   while the noise at frequency 1 w   . 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 signal measurement. The subsequent photocurrent signal can be contaminated by higher-frequency phase vibrations, but this can be easily checked and avoided. 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. OPA: optical parametrical amplifier. BS1 and BS2: 50% beam splitters.