1. Introduction

Optical parametric amplifiers (OPAs) can generate broadband squeezed light exceeding several THz [1,2], which is promising for various applications such as fast and time-resolved quantum information processing, broadband quantum-enhanced metrology, and research of ultrafast quantum phenomena [1,3–7]. In particular, squeezing of pulsed light [8] benefits from single-pass OPA because the peak power of pulses is strong enough to invoke nonlinear optical effects [9–11]. Single-pass OPA squeezing is also promising to realize quantum enhancements in biological imaging and spectroscopy with stimulated Raman scattering (SRS) and stimulated emission [12–14]. Since these methods utilize high-frequency lock-in detection [15–17], the wideband squeezing by the single-pass OPA is crucial. Note that broadband squeezing is not realized by a sub-threshold optical parametric oscillator (OPO), which is widely used to generate a continuous-wave (CW) squeezed vacuum, because its squeezing bandwidth is limited by the bandwidth of the cavity used, which reaches only several MHz to GHz [18–20].

One of the important technical challenges of the vacuum squeezing with a single-pass OPA is the precise phase control of squeezed vacuum with respect to the local oscillator (LO) light. When the phase error remains, the squeezed quadrature variance is mixed with that of the anti-squeezed quadrature, which results in a lower squeezing level [21,22]. For CW squeezing with an OPO, various schemes of phase control have been proposed as follows, while each scheme has a drawback when applied to squeezing with a single-pass OPA. In the noise locking method [23–25], the phase error signal of the LO phase is obtained by dithering the LO phase. However, the dithering inevitably mixes anti-squeezed quadrature with squeezed quadrature, leading to the degraded squeezing level and/or insufficient stability. In another method using a phase modulated probe beam [26,27], the phase error signal is obtained by picking up the probe beam from the optical path that contains squeezed vacuum. However, the beam picker causes optical loss in the squeezed vacuum and may degrade the squeezing level. The coherent control sideband (CCSB) method [28–30] uses frequency-shifted probe beam to realize precise phase locking, and is advantageous in achieving a high squeezing level because the phase error signal can be obtained without optical loss of the squeezed vacuum. However, the previous CCSB is only applicable to the squeezing by an OPO because the back-reflected light from the OPO is used to lock the CCSB light to OPO pump light, while such a back reflection does not exist in a single-pass OPA. Due to the absence of precise phase control, pulsed squeezed vacuum experiment is conducted on an anti-vibration optics table with a solid enclosure that prevents air movement [31] or does not use phase-locking to evaluate the generated squeezed state [32]. Note that not the phase locking of squeezed vacuum but the phase locking of bright squeezed state by single-pass OPA has been realized [12,33].

In this paper, we propose a phase-locking scheme based on CCSB for squeezing with a single-pass OPA. Specifically, by detecting the pump depletion by OPA, we can phase-lock the CCSB signal to the OPA pump light. In Section 2, we describe the principle of the proposed phase-locking scheme and discuss the feasibility of our method by considering the signal-to-noise ratio (SNR) of the error signal obtained from the pump depletion. We also demonstrate the proposed scheme on pulsed squeezing in Section 3. Section 4 compares the conventional CCSB, the proposed method, and some variations of the proposed method. Section 5 concludes this paper.

2. Principle of phase-locking scheme for squeezed vacuum generated by single-pass OPA

2.1 Apparatus and working principle of the phase-locking schemes

Figures 1(a) and (b) show the simplified schematics of the conventional CCSB scheme with an OPO [28,29] and the proposed scheme with an OPA, respectively. In both schemes, a laser generates coherent light at a frequency of $\omega$. It is split into two beams by a beam splitter (BS1). One beam goes into a second harmonic generation (SHG) crystal to generate SHG light at a frequency of $2\omega$ through the $\chi ^{(2)}$ nonlinearity of the crystal. This SHG light is used as a pump light for squeezing. After the light at $\omega$ is picked from the output of the SHG crystal, it is injected into an acousto-optic modulator (AOM) to generate a CCSB light at a frequency of $\omega + \Omega$. Then CCSB light is launched to an OPO through BS3 or an OPA through a dichroic mirror (DM) to generate an idler light and to amplify the CCSB light. A squeezed vacuum is generated in the OPO or the OPA by a degenerate parametric amplification. For the phase locking, two control loops are employed. The first one locks the CCSB signal to the pump light, and the second one locks the LO phase to the squeezed quadrature.

 

Fig. 1. Schematics of the apparatus for conventional CCSB scheme (a) and proposed phase-locking scheme (b). BS, beam splitter; SHG, nonlinear crystal for second harmonic generation; OPA, nonlinear crystal for optical parametric amplification; AOM, acousto-optic modulator; PD, photodiode; FG, function generator; DM, dichroic mirror; RFSA, radio frequency spectrum analyzer. $\omega$ is the frequency of the laser, and $\Omega$ is the shifted frequency of the sideband, and $\phi _\mathrm {c}$ is the optical phase of signal light at the input of OPA crystal, and $\phi _\mathrm {LO}$ is the optical phase of LO light, and $\hat {\phi }$ is the estimated phase. In the conventional CCSB setup, the mechanism to control the OPO cavity length is omitted.

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The details of the first loop are as follows. In the conventional CCSB scheme, the phase of SHG light is controlled by using a signal from photodiode (PD1) that detects the beating of signal light and idler light at a frequency of $2\Omega$ (Fig. 1(a)). In the proposed method, the phase of the CCSB light is locked by controlling the phase of the driving signal of the AOM at a frequency of $\Omega$ based on the signal from PD1, which receives the residual pump light (Fig. 1(b)). At the output of the OPA crystal, we have the beating of amplified signal light and idler light at a frequency of $2\Omega$ like the conventional CCSB scheme. Because we inject stationary pump light and CCSB light into the OPA crystal, the total optical power at the output of the OPA crystal should be stationary as well. Thus, the power of residual pump light also has the AC component at a frequency of $2\Omega$. According to the calculation described in Appendix A.1, the AC signal from PD1 is expressed by

In the second loop, the other beam split by BS1 is used as LO light in the balanced homodyne detector (BHD) after the phase of LO light is controlled by a phase controller. The amplified CCSB light, idler light, and squeezed vacuum are launched to BHD. The LO light is combined by BS2 and its outputs are detected by two photodiodes (PD2 and PD3). According to the calculation described in Appendix A.2, the difference of photocurrents from PD2 and PD3 yields a heterodyne signal of amplified CCSB light and idler light, which is expressed as

2.2 Signal-to-noise ratio of the modulation in pump depletion

To confirm the feasibility of the proposed phase-locking scheme, here we calculate the SNR of the modulation in the pump depletion. We assume that the dominant noise source in the modulation signal is the shot noise due to the pump light. As described in Section A.1, the modulation $\tilde I'_\mathrm p (t)$ in the residual pump after the OPA crystal is expressed by Eq. (16). The power of modulation signal expressed in the number of photons is written by

Let us numerically calculate the SNR for a practical situation of squeezing. We assume that we use pump light whose wavelength is 421 nm and power $P_\mathrm {p}$ is 10.0 mW and a CCSB light whose power $P_\mathrm {c}$ is 1.0 µW, and that OPA gain is $G=10$. The single-photon power of pump light $\hbar \omega _\mathrm {pump}$ is $4.7\times 10^{-19}$ W. We assume that the bandwidth of modulation for phase-locking is 1 kHz. According to the Nyquist theorem, the time width $T$ can be derived by $T = 1/2B$, where $B$ is the considering bandwidth. Assigning these parameters into Eq. (5), we get $\mathrm {SNR}_\mathrm {pump} = 66.8$ dB. This SNR is high enough to conduct the proposed phase-locking scheme. Nonetheless, the thermal noise in the photodiode circuit and successive electrical amplifiers may reduce the theoretical SNR in the actual experiments.

2.3 Effect of error in $\phi _c$ on measured squeezing level

It is important to consider the case where $\phi _\mathrm {c} \neq 0$, because in practical situations, the $\phi _\mathrm {c}$ inevitably has finite fluctuation while the CCSB light is phase-locked. The non-zero $\phi _\mathrm {c}$ leads to the degradation of squeezing level, because the projection from anti-squeezed noise contaminates the squeezed shot noise.

Figure 2 shows the numerically calculated squeezing level as a function of the $\phi _\mathrm {c}$ and the OPA gain (see Appendix A.3 for the derivation). The horizontal axis is the squeezing level when $\phi _\mathrm {c}=0$, which is the same as the squeezing level calculated from the OPA gain. Each colored curve represents the condition to obtain the same squeezing level. We can see that the observed squeezing level decreases as the $\phi _c$ increases. Furthermore, there exists an upper bound of the $\phi _\mathrm {c}$ for the desired squeezing level.

 

Fig. 2. Squeezing level as a function of the $\phi _\mathrm {c}$ and the OPA gain.

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3. Demonstration on single-pass pulsed squeezing

3.1 Experimental setup

In this section, we experimentally demonstrate the proposed phase-locking scheme on pulsed squeezing by a single-pass OPA. Figure 3 shows a simplified schematic of the optical setup used. We used a picosecond Ti:sapphire (TiS) laser which generates 4-ps optical pulses at a repetition rate of $f_\mathrm {rep}=76.52$ MHz. The center wavelength of the laser was $843$ nm. It was split into two beams by PBS1. One beam went into an SHG crystal made of periodically-poled stoichiometric LiTaO$_3$ (PPSLT). The residual fundamental light after the SHG crystal was frequency-shifted using AOM2. This light was used as the CCSB light. The CCSB light was combined with the pump light using DM2 and injected into an OPA crystal. The details of the squeezing setup will be described elsewhere [34]. Briefly, we used a 10-mm-long PPSLT waveguide as the OPA crystal. It is designed for type-0 quasi-phase matching at a wavelength of 843 nm. The other light from PBS1 was frequency-shifted by AOM1 so that the light was used as LO. Function generators (FGs) provide the driving electronic signals for AOMs, which are phase-locked to $f_\mathrm {rep}$. The sideband frequency $f_\mathrm {CCSB} = \Omega / 2\pi$ was set to $f_\mathrm {rep} / 512 = 149.45$ kHz. The driving frequencies of AOM1 and AOM2 were set to $f_1 = f_\mathrm {rep} = 76.52$ MHz and $f_2 = f_\mathrm {rep} - f_\mathrm {CCSB} = 76.37055$ MHz, respectively. The frequency offset of AOMs should be the integral multiple of $f_\mathrm {rep}$ to get the signals at the frequency of exact $\Omega$ and $2\Omega$; otherwise, the frequency of signals would be different from $\Omega$ and $2\Omega$ due to the frequency difference of interacting combs in the OPA crystal. If this scheme is applied to CW squeezing, one can choose any frequency as long as it meets the specification of the AOM. The CCSB light and LO light had frequency offset of $f_1 - f_2 = f_\mathrm {CCSB} = 149.45$ kHz. The frequency of CCSB signal (i.e., 149.45 kHz) was determined considering that very low frequency degrades SNR of signals because of the intensity noise of the laser, while higher frequency leads to significant change in the angle of the output beam from AOM1 and causes optical misalignment when the frequency of LO light is changed from $\omega$ to $\omega - \Omega$. The wavefront of the LO light was shaped by a spatial light modulator (SLM) to improve the homodyne visibility. The amplified CCSB light, idler light, and squeezed vacuum from the OPA crystal were combined with LO light by PBS2. The BHD consists of PBS3, PD2, and PD3. The residual pump light was picked by DM3 from the output of the OPA crystal and detected by PD1. When we change the phase of a driving signal fed to AOM, the optical phase of the light passing through the AOM changes as well. The phases of CCSB light and LO light were adjusted by changing the phase of the electronic signal that each FG generates. The error signal for the optical phase is obtained by the downconversion of signals from the BHD and PD1, as described in section 2.1. Because the spectrum of pulsed laser is broad, the signals from the BHD and PD1 have unnecessary frequency components due to the interference between many frequency combs, typically at the frequencies of the integral multiple of $\omega _\mathrm {rep}$. The bandwidths of the photodiode circuits in BHD and PD1 and successive electrical amplifiers are limited so that these unnecessary signals are cut. We also added electrical lowpass filters before amplifiers to cut out these unnecessary signals. Since $\Omega$ and $\omega _\mathrm {rep}$ are far apart, such a lowpass filter can be designed easily.

 

Fig. 3. Schematic of the optical setup used in the demonstration. PBS, polarizing beam splitter; HWP, half-wave plate; SLM, spatial light modulator. Black thin line indicate electronic signal paths and colored line indicate optical paths.

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3.2 Result

Figure 4(a) shows the RF powers of squeezed vacuum noise, shot noise, and thermal noise measured by an RSFA at a frequency of 37 MHz, which is close to the Nyquist frequency $f_\mathrm {rep}/2=38.26$ MHz of the optical pulses. The resolution bandwidth and video bandwidth were 1 MHz and 300 Hz, respectively. The optical power of LO light was 18.5 mW. At the input of the OPA crystal, the optical power of pump light and signal light was 10.6 mW and 1.28 mW, respectively, while very small portion of signal light was coupled to the OPA waveguide so as to avoid excessive pump depletion. As shown by the red line, the LO phase was successfully kept to the squeezed quadrature with the proposed scheme. Thanks to the single-pass setup, we were able to observe squeezing near the Nyquist frequency. Stable noise reduction of $-2.35$ dB was observed. The anti-squeezing level was estimated to be $7.23$ dB. The thermal noise level of the measurement was about $-57.4$ dBm, while the shot noise level was $-41.1$ dBm. Because the squeezing level observed during the LO phase scanning (blue line) was almost the same as the squeezing level observed under the phase-locking (red line), the phase noise was not the main factor of the limitation in the squeezing level in this experiment. The detail of the limiting factor will be described elsewhere [34]. To confirm the stability of phase-locking for a longer time, Fig. 4(b) shows the power of reduced shot noise for 60 s. The video bandwidth was 10 Hz. An average noise reduction of $-2.38$ dB was achieved. Although the actual SNR of the pump depletion signal was not measured during the experiment, we had enough SNR to apply our scheme.

 

Fig. 4. Results of the squeezing and phase-locking experiment for 500 ms (a) and 60 s (b). Balanced homodyne detection of the squeezed vacuum with LO phase scanning (blue line, squeezed noise), the shot noise (orange line), the thermal noise (green line), and the squeezed vacuum with proposed phase-locking (red line). The measurement frequency is 37 MHz. The resolution bandwidth was 1 MHz. The video bandwidth was 300 Hz in (a) and 10 Hz in (b).

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4. Discussion

We proposed and demonstrated that one of the two signals which are necessary for the CCSB method can be obtained by measuring the modulation in the pump depletion of the OPA process. Because the modulation signal has to be measured with the additional shot noise that originates from the residual pump light, the measurement of modulation in the pump depletion may be disadvantageous in SNR compared to the conventional CCSB method. Nevertheless, the SNR of the modulation signal is large enough as we described in Section 2.2.

Regarding pulsed squeezing, temporal matching of the pulsed CCSB light, pump light, and LO light is important because such kind of mismatching not only reduces the measured squeezing level but also reduces the signal power of modulation in the pump depletion and BHD. Note that pulsed CCSB light and squeezed vacuum exist at different temporal positions due to the group velocity dispersion in the nonlinear crystal. This means that maximizing the signal from BHD does not necessarily maximize the squeezing level. We also remark that not only the careful adjustment of pulse delay but also the deliberate optical design is important to achieve a higher squeezing level [35].

4.1 Using the sum current of two PDs in BHD

Here we point out that it is possible to use the sum current of two PDs instead of using residual pump light for the phase-locking of CCSB light. Changing the Eq. (20) slightly, the sum current is expressed by

The AC component in this equation is similar to Eq. (16). Here, we derive the SNR of the sum current. The power of signal in the sum current based on the number of photons is written by

In the practical situation, the power of LO light is much stronger than the CCSB light. So we assume that $P_\mathrm {LO} \gg P_\mathrm {c}(2G-1)$. The SNR of the sum current can be evaluated by

Devide $\mathrm {SNR}_\mathrm {pump}$ by this SNR, we can evaluate the benefit of using sum current as

5. Conclusion

We have proposed the phase-locking scheme for squeezed vacuum generated by a single-pass OPA. We showed the feasibility of the proposed method by deriving SNR of the modulation signal in the residual pump light and experimentally demonstrated our scheme on pulsed squeezing. We also presented an intuitive explanation of the squeezing by a stretching effect in the phasor diagram of a sideband in the Appendix. We anticipate that the proposed scheme will be indispensable for applications of squeezing with single-pass OPA.

A. Appendix

A.1. Electric fields and phase detection of CCSB light

Here we introduce physical modeling and derive the phase detection protocol of CCSB light. For simplicity, we made the following assumptions: (1) Phase matching wavelength of SHG and OPA crystal is perfectly matched to the wavelength of the laser. (2) There is no optical loss and no electronic noise. (3) Spatial and temporal matching in the BHD is perfect, which is equivalent to the homodyne visibility of $100\%$. In the following part of this paper, the standard of the global optical phase is defined by the pump light at the input of the OPA crystal. We assume that the incident CCSB light $E_\mathrm {c}(t)$ at the input of the OPA crystal is given by

(11)$$E_c(t) = \sqrt{P_\mathrm{c}} e^{i((\omega + \Omega)t + \phi_\mathrm{c})},$$

where $P_\mathrm {c}$ denote the power of the incident CCSB light and $\phi _\mathrm {c}$ is the optical phase. In this paper, we define the optical power of a lightwave $P$ so that $N = PT/e$ holds, where $N$ is the number of photons, $e$ is the photon energy, and $T$ is the time duration under consideration.

We can derive the modulation in the residual pump light as follows. Let the OPA gain of the OPA crystal be $G$. Since we consider perfect phase matching, we can express the amplified CCSB light $E'_\mathrm {c}(t)$ and idler light $E'_\mathrm {i}(t)$ by

(12)$$\begin{aligned} E'_\mathrm{c}(t) & = \sqrt{G P_\mathrm{c}} e^{i((\omega + \Omega)t + \phi_\mathrm{c})}, \\ E'_\mathrm{i}(t) & = i\sqrt{(G-1) P_\mathrm{c}} e^{i((\omega - \Omega)t - \phi_\mathrm{c})}. \end{aligned}$$

Note that we have an imaginary symbol $i$ in the definition of $E'_\mathrm {i}(t)$ because we assumed that the optical phase of pump light and signal light were the same. The sum of optical power around the fundamental frequency $\omega$ at the output of the OPA, which consists of amplified CCSB light and idler light, is represented by

(13)$$|E'_\mathrm{c}(t) + E'_\mathrm{i}(t)|^2 = (2G-1)P_\mathrm{c} + 2P_\mathrm{c}\sqrt{G(G-1)}\sin(2\Omega t + 2\phi_\mathrm{c}).$$

The second term in Eq. (15) is the AC component at a frequency of $2\Omega$. Because we inject stationary pump light and CCSB light into the OPA crystal, the total optical power at the output of the OPA crystal should be stationary as well. Thus, the power of residual pump light also has the AC component at a frequency of $2\Omega$. Since the photon energy of pump light is twice as much as that of fundamental light, the amplitude of AC component in the power of residual pump light is one-half of that in Eq. (15). Consequently, considering the energy conservation, the AC component of the photocurrent of PD1 which receives the residual pump light $\tilde {I}'_\mathrm {p}(t)$ is expressed by

(14)$$\tilde{I}'_\mathrm{p}(t) ={-}P_\mathrm{c}\sqrt{G(G-1)}\sin(2\Omega t + 2\phi_\mathrm{c}).$$

This component is the modulation of the pump depletion and can be used for phase detection of CCSB light. The PD1 receives this modulation and downconversion of the photocurrent at a frequency of $2\Omega$ yields

(15)$$\begin{aligned} x &={-}P_\mathrm{c}\sqrt{G(G-1)}\sin(2\phi_\mathrm{c}), \\ y & ={-}P_\mathrm{c}\sqrt{G(G-1)}\cos(2\phi_\mathrm{c}), \end{aligned}$$

where $x$ and $y$ are the in-phase and quadrature component. We can derive the error signal of $\phi _\mathrm {c}$ from these components. Adjusting the phase of the AOM-driving signal generated by FG1 in Fig. 1(b), we can phase-lock the CCSB light.

A.2. Phase detection of LO light and locking to squeezed quadrature

In this section, we calculate the output signal of the BHD and derive the phase-locking protocol. To calculate the output signal from BHD, we introduce the LO light $E_\mathrm {LO}(t)$ by

(16)$$E_\mathrm{LO}(t) = \sqrt{P_\mathrm{LO}} e^{i(\omega t + \phi_\mathrm{LO})}.$$

After removing the residual pump light by DM, amplified CCSB signal, idler light, and squeezed vacuum are detected by BHD. Combining the equations in Eq. (13), the output signal of BHD is expressed by

(17)$$\begin{aligned} I_\mathrm{BHD}(t) & = \left| \frac{1}{\sqrt{2}} \left( E'_\mathrm{c}(t)+E'_\mathrm{i}(t)+E_\mathrm{LO}(t) \right) \right|^2 - \left| \frac{1}{\sqrt{2}} \left( E'_\mathrm{c}(t)+E'_\mathrm{i}(t)-E_\mathrm{LO}(t) \right) \right|^2 \\ &= 2\sqrt{P_\mathrm{LO}P_\mathrm{c}} \left( \sqrt{G}\cos(\Omega t + \phi_\mathrm{c} - \phi_\mathrm{LO})+\sqrt{G-1}\sin(\Omega t + \phi_\mathrm{c} + \phi_\mathrm{LO} ) \right). \end{aligned}$$

Since the number of photons in the squeezed vacuum is much less than that in the amplified CCSB signal light, idler light, and LO light, we ignore the contribution from the squeezed vacuum.

To describe the principle of phase-locking of LO light, we analyze the output of the BHD signal and derive the origin of squeezing. For simplicity, we assume that the phase of CCSB light $\phi _c$ is properly locked to 0 by the controller $C_1$ in Fig. 1(b), that is, $\phi _c = 0$ is supposed in the following discussion. The situation where $\phi _c \neq 0$ is discussed in Appendix A.3. After we phase-lock the CCSB light, the output of BHD becomes

(18)$$\begin{aligned} &I_\mathrm{BHD}(t) = 2\sqrt{P_\mathrm{LO}P_\mathrm{c}} \left( \sqrt{G}\cos(\Omega t + \phi_\mathrm{c} - \phi_\mathrm{LO})+\sqrt{G-1}\sin(\Omega t + \phi_\mathrm{c} + \phi_\mathrm{LO}) \right) \\ &\propto \cos(\Omega t)\left( \sqrt{G}\cos\phi_\mathrm{LO} + \sqrt{G-1}\sin\phi_\mathrm{LO} \right) + \sin(\Omega t)\left( \sqrt{G}\sin\phi_\mathrm{LO} + \sqrt{G-1}\cos\phi_\mathrm{LO} \right) \end{aligned}$$

Downconversion at a frequency of $\Omega$ yields

(19)$$\begin{aligned} x &= \sqrt{G}\cos\phi_\mathrm{LO} + \sqrt{G-1}\sin\phi_\mathrm{LO}, \\ y &= \sqrt{G}\sin\phi_\mathrm{LO} + \sqrt{G-1}\cos\phi_\mathrm{LO}, \end{aligned}$$

where $x$ and $y$ are the in-phase and quadrature components. These equations can be written in the following matrix form:

(20)$$\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{cc} \sqrt{G} & \sqrt{G-1} \\ \sqrt{G-1} & \sqrt{G} \end{array} \right) \left( \begin{array}{c} \cos\phi_\mathrm{LO} \\ \sin\phi_\mathrm{LO} \end{array} \right) =\mathbf{A}(G) \left( \begin{array}{c} \cos\phi_\mathrm{LO} \\ \sin\phi_\mathrm{LO} \end{array} \right).$$

By solving the eigenvalue equation of $\mathbf {A}(G)$, eigenvalues $\lambda _i$ and eigenvectors $\mathbf {v}_i$ of matrix $\mathbf {A}(G)$ are expressed by

(21)$$\begin{aligned} \lambda_1 &= \sqrt{G}+\sqrt{G-1} > 1, \mathbf{v}_1=(1,1)^\top \\ \lambda_2 &= \sqrt{G}-\sqrt{G-1} < 1, \mathbf{v}_2=({-}1,1)^\top. \end{aligned}$$

Equations (20) and (21) indicate that an ‘ellipse’ is generated in the $xy$ plane by OPA as depicted in Fig. 5 and this is essentially ‘squeezing’. A point in this $xy$ plane is a phasor, which consists of amplified CCSB light (frequency of $\omega + \Omega$) and idler light (frequency of $\omega - \Omega$). As the OPA gain $G$ increases, a more stretched ellipse is generated. The ellipse is the result of phase-sensitive amplification of OPA at a frequency of $\omega$, that is, the origin of squeezing and anti-squeezing. Note that the squeezed quadrature angle depends on the phase of the pump light. In the standard introductory description of quantum optics, the Wigner function of squeezed vacuum is stretched along the $y$ (or $p$) axis [36,37]. If we assume the phase of pump light as $\frac {\pi }{2}$, we get the stretched ellipse along the $y$-axis. In our derivation, we took the phase of pump light as zero, so the ellipse is rotated by 45-degree.

 

Fig. 5. Conceptual diagram of OPA. The left figure shows the phasor trace of CCSB light before OPA at a frequency of $\Omega$. The right figure shows the phasor trace of amplified CCSB light and idler after OPA.

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We can phase-lock the LO to squeezed or anti-squeezed quadrature from the error signal derived from the measured $x$ and $y$ components. When the LO phase $\phi _\mathrm {LO}$ is locked to $\frac {3}{4}\pi$ or $-\frac {1}{4}\pi$, squeezed quadrature is measured in the BHD. On the other hand, when $\phi _\mathrm {LO}$ is locked to $\frac {1}{4}\pi$ or $-\frac {3}{4}\pi$, anti-squeezed quadrature is measured.

However, in a practical situation, we may find squeezed quadrature at a different angle, because the electrical circuits such as photodetector circuits or downconversion circuits have a finite delay. These delays cause phase shifts in the modulation of the pump depletion or the output from the BHD. As a result, the OPA ellipse rotates under these delays, and the squeezed quadrature angle will be different from the ideal value. To cope with this problem, we need to sweep the LO phase and find the squeezed quadrature experimentally. The LO phase which corresponds to the squeezed quadrature is obtained when the amplitude of the output from BHD is smallest, that is, the measured $x^2+y^2$ value is smallest, during the LO phase is swept from $0$ to $2\pi$.

A.3. Rotation effect by non-zero $\phi _\mathrm {c}$

In this section, we derive $x$ and $y$, the in-phase and quadrature component, for arbitrary $\phi _\mathrm {c}$ and consider the effect of fluctuation in $\phi _\mathrm {c}$, and evaluate the performance degradation in the phase-locking of LO.

If we keep the $\phi _c$ in Eq. (20) and keep calculation in the same way as Eq. (22) and Eq. (24), the $x, y$ is expressed by

(22)$$\begin{aligned} x &= \sqrt{G}\cos(\phi_\mathrm{LO}-\phi_\mathrm{c}) + \sqrt{G-1}\sin(\phi_\mathrm{LO}+\phi_\mathrm{c}), \\ y &= \sqrt{G}\sin(\phi_\mathrm{LO}-\phi_\mathrm{c}) + \sqrt{G-1}\cos(\phi_\mathrm{LO}+\phi_\mathrm{c}), \end{aligned}$$

and these equations can be written in the following matrix form:

(23)$$\begin{aligned} \left( \begin{array}{c} x \\ y \end{array} \right) &= \left( \begin{array}{cc} \cos\phi_\mathrm{c} & \sin\phi_\mathrm{c} \\ -\sin\phi_\mathrm{c} & \cos\phi_\mathrm{c} \end{array} \right) \left( \begin{array}{cc} \sqrt{G} & \sqrt{G-1} \\ \sqrt{G-1} & \sqrt{G} \end{array} \right) \left( \begin{array}{c} \cos\phi_\mathrm{LO} \\ \sin\phi_\mathrm{LO} \end{array} \right) \\ &=\mathbf{R}(-\phi_\mathrm{c})\mathbf{A}(G) \left( \begin{array}{c} \cos\phi_\mathrm{LO} \\ \sin\phi_\mathrm{LO} \end{array} \right), \end{aligned}$$

where $\mathbf {R}$ represents rotation matrix. This equation shows that $\phi _\mathrm {c}$ rotates the generated ellipse in the $xy$ plane. This indicates that if the $\phi _\mathrm {c}$ has non-zero value, LO phase is not actually locked to the squeezed (or anti-squeezed) quadrature even when the $\phi _\mathrm {LO}$ is locked to $\frac {3}{4}\pi$ or $-\frac {1}{4}\pi$ ($\frac {1}{4}\pi$ or $-\frac {3}{4}\pi$). In particular, when we lock the LO phase to squeezed quadrature, the non-zero $\phi _\mathrm {c}$ leads to the degradation of squeezing level.

Here, we numerically calculate this degradation by solving the simultaneous equations of $y=-x$ and Eq. (23) for a given $\phi _\mathrm {c}$ to find the actual phase-locked $\phi _\mathrm {LO}$ value. We define $\theta = \phi _\mathrm {LO} - \frac {3}{4}\pi$ as the gap between the squeezed quadrature and actual phase-locked $\phi _\mathrm {LO}$. The observed squeezing level $S'$ under the degradation is derived by [21]

(24)$$S' = \frac{1}{S}\cos^2 \theta + S\sin^2\theta,$$

where $S$ denotes the original squeezing level. Note that the effect of negative $\phi _\mathrm {c}$ and its absolute $|\phi _\mathrm {c}|$ has the same effect. The observed squeezing level $S'$ as a function of OPA gain $G$ and $\phi _\mathrm {c}$ is shown in Fig. 2.

Funding

Core Research for Evolutional Science and Technology (JPMJCR1872).

Acknowledgments

This work is supported by JST CREST Grant Number JPMJCR1872, Japan.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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