24-hour measurement of squeezed light
using automated stable fiber system

To accelerate research and development of quantum computation, it is desired to open prototypes and allow researchers to access them. Google and IONQ have developed a high-fidelity 11-qubit ion-trap quantum computing accessible via Google CloudEgan et al. (2021). IBM has released a superconducting 127-qubit quantum processor “Eagle” in 2021 on the IBM Cloudibm . Xanadu has conceived and built “Borealis”, a free-space-based quantum processor that can perform Gaussian boson samplingMadsen et al. (2022), and made it available on their Cloud platform. Since “Borealis” is a free-space system except for some parts of the system, pointing of the optical beams drifts, which is recovered by frequent manual alignments. If the system is constructed with a fiber system instead of a free-space system, the frequent alignments are removed. Thus the fiber system is advantageous when constructing a quantum computer operating long time.

In particular, we are aiming to build a fiber-based optical quantum computer which operates on continuous variables of light. Quantum computation based on continuous variables is gaining attentions. Recently, large-scale entanglements using time-domain multiplexing and frequency-domain multiplexing have been demonstratedRaussendorf and Briegel (2001); Menicucci et al. (2006); Menicucci (2011); Pysher et al. (2011); Yokoyama et al. (2013); Yoshikawa et al. (2016); Yang et al. (2016); Zhang et al. (2017); Larsen et al. (2019a); Asavanant et al. (2019); Pfister (2020). In time-domain multiplexing, large entanglement is generated by sending squeezed light to interferometers with asymmetric lengths of arms.

However, if the optical system is constructed with fiber, fiber-specific problems arise. Phase drifts and polarization fluctuations are caused by the ambient temperature changesTur, Boger, and Shaw (1995); Slavík et al. (2015); Lv et al. (2019). For phase drifts, an optical delay of 10 m in free-space systems causes typically a few wavelengths of phase drift, while that in the fiber system, if no measures are taken, induces several hundred wavelengths of phase drift. For polarization drifts, even if polarization-maintaining fibers are used, polarization extinction ratio (PER) of fiber components is finite, and thus concatenating these devices leads to polarization instability. Furthermore, power fluctuations are induced by the polarization fluctuations when combined with polarization-depend components. In addition, the coupling ratio of fiber beamsplitter is slightly dependent on the polarization and temperature. Since squeezed light is degraded by optical loss, all fiber components through which the squeezed light passes should have low optical loss.

Here, we solve the above issues by introducing stabilization devices, and successfully measure squeezed light stably for 24 hours. For phase drifts, we cover the entire system with a wind shield to passively suppress large phase drifts, and then introduce the fiber stretchers to actively suppress the phase drifts with feed-back controls. For polarization fluctuations, active polarization controllers are introduced together with polarizers. In order to generate the error signal for the polarization control, a polarization modulation method is employed. For power fluctuations, an power stabilization mechanism is introduced. For a fiber beamsplitter whose coupling ratio variation is critical, we control temperature of the beamsplitter and stabilize the coupling ratio without introducing significant optical loss. Automated alignment of the entire system is achieved by connecting the various stabilization mechanisms via a network and integrating their control program. Using the above system, we measure the squeezed light in 1545.3 nm generated by the optical parametric amplifier (OPA) with periodically poled lithium niobate (PPLN) waveguideKashiwazaki et al. (2020, 2023) at 2-minute intervals with automated alignment every 30 minutes. Squeezing levels with the average of $-4.42$ dB and the standard deviation of 0.08 dB are successfully measured over 24 hours. In this demonstration, the optical loss of squeezed light is avoided by placing the controllers in the paths where the squeezed light does not pass, except for the coupling ratio stabilization. On the other hand, in the case of a more complex interferometer built for optical quantum computers, there are situations where the polarization and the phase of squeezed light need to be controlled with low optical loss. In that case, the method proposed in Ref.Nakamura et al., 2023 can be available.

There are previous works which demonstrate quantum optics experiments with fibers. In Ref. Takanashi et al., 2020, with a fiber packaged OPA, squeezed light is measured in fiber-based systems. In Ref. Larsen et al., 2019b, squeezed light from free-space optical parametric oscillator is injected to fiber interferometers to generate two-dimensional cluster states. However, in the above two demonstrations, the polarization of light and the coupling ratio of fiber beamsplitters are controlled manually or uncontrolled. To realize long-time stability, alignment needs to be automated. On the other hand, long-time squeezed light measurements have been reported for free-space systems without beam pointing alignmentShajilal et al. (2022). This previous experiment is working well because the system size of squeezed light measurement is relatively small. For more complex systems realizing optical quantum computer, fiber systems are advantageous as explained above, and thus our techniques introduced in this work can be applicable to such complex systems.

This paper is structured as follows. We explain squeezed light in Section II. We discuss why various stabilizations are needed in the experimental setup in Section III. We illustrate an overview of the experimental system in Section IV. We explain stabilization of optical power in Section V, phase lock in Section VI, polarization optimization in Section VII, and the coupling ratio stabilization of the fiber beamsplitter in Section VIII. We show the results of the 24-hour squeezed light measurement in Section IX. Section X is the conclusion.

The quadrature phase amplitudes $\hat{x}$ and $\hat{p}$ of the single-mode quantized field of light have uncertainty $\Delta(\hat{x})\Delta(\hat{p})\geq\hbar/2$, where $\hbar$ is the reduced Planck constant. The vacuum state is a minimum uncertainty state with $\Delta(\hat{x})=\Delta(\hat{p})=\sqrt{\hbar/2}$. A squeezed state has a quadrature phase amplitude where the fluctuation is smaller than that for a vacuum stateAndersen et al. (2016). For example, an $x$-squeezed state is squeezed as $\Delta(\hat{x})<\sqrt{\hbar/2}$ and instead anti-squeezed as $\Delta(\hat{p})>\sqrt{\hbar/2}$ to satisfy the uncertainty principle. Fault tolerant quantum computation can be achieved by using high-level squeezed statesMenicucci (2013); Fukui et al. (2018); Vahlbruch et al. (2016). An arbitrary quadrature phase amplitude $\hat{x}\cos\theta+\hat{p}\sin\theta$ can be measured by setting the phase of the local oscillator (LO) light to $\theta$ in an optical homodyne detection. A vacuum state has a quadrature distribution insensitive to $\theta$, while a squeezed state has that sensitive to $\theta$. Figure 1 (right) shows a typical quadrature distribution of a $x$-squeezed state, where the quadrature has the minimum variance at $\theta=0$, $\pi$, and the maximum variance at $\theta=\pi/2$, $3\pi/2$.

Figure 1 shows a simplified experimental setup for generating and measuring squeezed light. Here, the angular frequency of fundamental light is $\omega$. By injecting pump light with the angular frequency of $2\omega$ to a second-order nonlinear material, squeezed light is generated by parametric down conversion. In our demonstration here, squeezed light is generated by a PPLN waveguide OPA. In homodyne detection, squeezed light is combined with LO light at a 50:50 beamsplitter, and then is detected with two photodiodes in a homodyne detector, where the difference of the photocurrents shows the quadrature values. The squeezing and antisqueezing levels are obtained by comparing the variances with that of the vacuum state. The quadrature phase amplitude of the vacuum state is obtained by blocking the squeezed light and feeding only the LO light into the homodyne detector. Since the energy of a single photon corresponds to about 12,000 K in the telecommunication wavelength, thermal excitation is negligible at the room temperature and the vacuum state can be measured with this method.

It is necessary to lock the phase of the LO light depending on the measurement phase. For phase lock, coherent light of the fundamental wave is input to the OPA, and this light is referred to the probe light. Since the probe light undergoes phase-sensitive amplification which is in phase with the generated squeezed light, the relative phase between the probe light and the squeezed light can be locked. In homodyne detection, the interference signal between the probe light and LO light is locked, by which the relative phase between the squeezed light and the LO light is locked.

Although we build the experimental setup for the squeezed light measurement, the stabilization mechanics which are implemented in this system is easily extended to the more complicated system.

In optical quantum computation, interference of squeezed light generates complicated entanglement structures. The polarizations must be matched and the relative phases must be locked to a specific direction at the interference points. If polarization and phase are not in the proper state, the quality of the entanglement is degraded. Even if the fiber components are made of polarization-maintaining fibers, connections between multiple fiber components shift the polarization state of the light from the polarization-maintaining axis. Birefringence variation caused by external disturbances fluctuates the polarization of light which is shifted from the polarization-maintaining axis. The polarization fluctuations also cause power fluctuations as the light passes through the polarization-dependent element. For example, fluctuations in LO power cause changes in the shot noise level in homodyne detections, which undermines the reliability of the experiment. Furthermore, changes in ambient temperature cause the fiber to expand and contract, which changes the optical path length, resulting in phase driftElezov et al. (2018). Besides, a fiber beamsplitter can be realized with a directional coupler, but its coupling ratio must be stable at an appropriate value.

In the squeezed light measurement experiments of this study shown in Fig. 1, the issues of polarization fluctuation, power fluctuation, phase drift, and coupling ratio fluctuation of the beamsplitter appear as follows, and stabilization mechanics are installed. Regarding polarization, if the polarizations of the squeezed light and the LO light are misaligned, the squeezing level is degraded. Squeezed light is generated by OPA, but the polarization of the squeezed light is parallel to the crystal axis of PPLN waveguide. For the polarization alignment, the probe light is input to the path of the squeezed light. The polarization of the probe light is adjusted to match the polarization of the squeezed light, and then the polarization of the LO light is adjusted to match the polarization of LO light. Polarization controls are implemented with devices that change polarization by applying pressure to the fiber and causing birefringence. In the case of the LO light, power fluctuations change the shot noise level of the homodyne detection. In the case of the probe light, they change the amplitude of the error signals of the various locks, resulting in misalignment of the locking points. Therefore, it is necessary to stabilize the power of the LO and probe light, and fiber pigtailed variable attenuators are used. As for the phase, if the phase lock is not able to cancel the phase drift, an anti-squeeze component will be mixed when measuring the squeezing level, resulting in the reduction of squeezing level. In this experiments, fiber stretchers are inserted for the phase lock between the pump light and the probe light and for that between the probe light and the LO light. The coupling ratio of the fiber beamsplitter used in this study for homodyne detection has severe hysteresis when the knob is rotated, making it difficult to adjust it manually. Futhermore, the polarization of the input light also change the coupling ratio. Therefore, the coupling ratio is controlled by temperature.

In this study, squeezed light is measured over 24 hours with the experimental system shown in Fig. 2. In order to ensure stable measurements, light power stabilization, phase lock, polarization optimization, and coupling ratio lock of the beamsplitter are performed, which are described in the following sections. In this section, we describe the entire setup of the experimental system.

The laser source consists of multiple devices. The seed laser is Koheras ADJUSTIK X15 (NKT Photonics), whose wavelength and power are 1545.3 nm and 15 mW, respectively. The preamplifier, which reduces relative power noise by feedback, is Koheras BOOSTIK Linecard (NKT Photonics) whose output power of 80 mW is then attenuated to 20 mW. The main amplifier is Koheras BOOSTIK (NKT Photonics) whose output power is 8 W. The second harmonic generator (SHG) is Koheras HARMONIK (NKT Photonics), which converts some part of 1545.3 nm light to 772.7 nm light. After the SHG, the 1545.3 nm light and the 772.7 nm light are emitted to free space, which are then coupled to fibers. The 1545.3 nm light is used as the probe and LO light and the 772.7 nm light as the pump light. Before the fiber coupling, motor-driven optical shutters are inserted for all light paths to block the light, which is utilized in the sequence of control explained later.

Fiber components for the probe, the LO and the pump light are as follows. For the probe path, two acousto-optic modulators (AOM) SGTF200-1550-1P (Chongqing Smart science&Technology Development) are inserted. Then, for the probe and the LO light paths, a fiber-based variable optical attenuator for power stabilization, a fiber stretcher for phase control, a polarization controller are connected. In the pump light path, the light power is stabilized by a motorized waveplate and polarizing beamsplitter in free space. The probe light and the pump light are input to the OPA, which is a PPLN waveguide and packaged with fiber input and output. The PPLN waveguide inside the OPA is phase-matched with type 0. Only the component parallel to the crystal axis of the input pump light contributes to the generation of squeezed light at the PPLN waveguide. This means that polarization fluctuations of the pump light do not cause polarization fluctuations of the squeezed light. Therefore, for the pump light, only the power stabilization is sufficient, which is performed by a free-space motorized waveplate and a polarizing beamsplitter. In the demonstration in Section IX, the power of the pump light is set to be 300 mW.

In the homodyne detection, the probe light interferes with the LO light at a fiber beamsplitter with a coupling ratio of 50:50, and are sent to the homodyne detector. Since the utilized photodiodes are for the free-space light, the light is output to free space by a collimator and focused on the photodiode by a concave mirror with the radius of curvature of 100 mm. For all-fiber implementation, the photodiodes can be fiber pigtailed in the future. The photodiodes of the homodyne detector are high quantum efficiency InGaAs-OD@1550nm (Laser Components) with a measured quantum efficiency of 96%. The difference of photocurrents is converted to voltage signal by a transimpedance amplifier, which is composed of OPA847 (Texas Instruments) and 3 k$\Omega$ resistance. The signal from the homodyne detector is used for two purposes. The first is to generate a error signal for the phase lock, the polarization optimization, and the coupling ratio lock of the 50:50 beamsplitter. The second is to measure the squeezing level. The signal for the squeezing level measurement is high-pass filtered with the cutoff frequency of 15 MHz, and then is amplified by a factor of 15.

Because squeezed light is degraded by optical loss, optical components through which the squeezed light passes should have low optical loss. The fiber beamsplitters BS2 and BS-HD are 954P (Evanescent optics), which have loss of less than 0.1 dB. The fiber components are made of polarization maintaining fibers except for the polarization controller.

For the power stabilization, the phase lock, the polarization optimization and the coupling ratio stabilization of beam splitter, a number of field-programmable gate arrays (FPGAs), STEMlab 125-14 RedPitaya, are utilized, and they are controlled by a single computer via the socket communication. Multiple modulation and demodulation signals are required for phase lock and polarization optimization, which are synchronously generated by a direct digital synthesizer (AD9959, Analog Devices).

In Fig. 2, components which serve for power stabilization are colored in blue. To monitor the power fluctuation, some part of the light is picked up and fed back to the variable optical attenuator (NEW MMVOA-1-1550-P-8/125-SCSC-1-0.5, OZ Optics). To pick the light up, beamsplitters BS1-a, BS1-c (PMFC-1x2-1550-10/90-B-900-5-1-SC-P25, Opneti) with coupling ratio of 90:10 are used. To realize automated alignment, the light power needs to be stabilized at different orders of magnitude during alignment and measurement sequence. The target powers of the probe light are 1 $\mu$W for the OPA alignment, 100 $\mu$W for homodyne alignment, 100 nW for squeezed light measurement. The target powers of the LO light are 100 $\mu$W for alignment and 16 mW for measurement. For the low power, it is necessary to largely amplify weak light, and for the high power, it is necessary to reduce the amount of light to avoid the saturation of the PD. Therefore, the picked-up light is further divided by a 90:10 beamsplitter BS1-b, BS1-d, whose model number is the same as that of BS1-a and BS1-c. Then, the divided light goes to the two photo detectors (PDs), one for the high power and one for the low power. PD1, PD2, PD3 and PD4 contain a photodiode G8195-11 (Hamamatsu Photonics) and a transimpedance amplifier, where the transimpedance is 1 M$\Omega$, 10 k$\Omega$, 100 k$\Omega$ and 10 k$\Omega$, respectively. Before PD4, a 10 dB attenuator is inserted. Because the finite PER of the fiber components can cause the polarization change, the PER of BS1-a,b,c,d is ordered to be better than 25 dB. Without the power stabilization, the optical power drift was 15% at most, but with the power stabilization on, the standard deviation of the power fluctuation was 0.1%.

In Fig. 2, components that serve for phase locks are colored in blue. There are two phase lock points: the parametric amplification of the OPA (the phase between the probe light and the pump light) and the homodyne detection (the phase between the probe light and the LO light) . The probe light is frequency-shifted by the AOMs before the OPA. The frequency is up-shifted by 200.0 MHz at the first AOM, and then down-shifted by 199.8 MHz at the second AOM, resulting in a total frequency shift of 200 kHz. At the OPA, the probe light is parametrically amplified depending on the phase between the probe light and the pump light. The parametric amplification generates $-200$ kHz detuned light from the 200 kHz detuned probe light, thus resulting in the a 400 kHz beat signal. The probe light is picked up by a fiber beamsplitter BS2 with a coupling ratio of 99.5:0.5, and the 400 kHz beat signal for the phase lock of the OPA is acquired with a PD5 using an avalanche photo diode (KPDEA007-T, Kyoto Semiconductor) followed by an amplification of $M$=10. At the homodyne detector, since the LO light is not frequency-shifted, a beat signal of 200 kHz is observed, which is used for the phase lock of the homodyne detection. The beat signals obtained by the PD5 and the homodyne detector are demodulated, and the error signals are fed back to the fiber stretchers (FST). The fiber stretchers have the following structure. A cube-shaped piezoelectric actuator PC4FL (Thorlabs) is sandwiched by two 25 mm-diameter semicircle metals, and a polarization-maintaining fiber is wound by 10 times around it. The dynamic range of the fiber stretcher is about 3 wavelengths when the applied voltage range is set to 70 V. If the input polarization state deviates from the polarization-maintaining axis of the polarization-maintaining fiber, the polarization changes depending to the pressure to the fiber. Therefore, the fiber stretcher is placed immediately after the linear polarizer. The entire experimental system is covered by an airtight windshield made of aluminum frames, aluminum plates, and polycarbonate plates to suppress severe phase drifts due to temperature changes. The phase lock precision of the OPA was about 2^∘ and that of the homodyne detection was about 0.3^∘.

In Fig. 2, components that serve for the polarization optimizations are colored in pink. First, the polarizations of the probe light before the OPA and the LO light are stabilized by linear polarizers (ILP-1550-900-1-0.5-SA-P30, Opneti) with the PER$>$30 dB. Note that since squeezed light is sensitive to optical loss, the polarization stabilization should not be performed by inserting a linear polarizer in the path of the squeezed light after the OPA. The polarizations of the probe light and the LO light after the linear polarizers are slightly changed unintentionally by the followed FST and BS1, which is critical in the squeezed light measurement. Therefore, we actively optimize the polarization using the fiber polarization controller PCD-M02 (LUNA), which can control polarization with three degrees of freedom.

The polarization optimization by feedback control is not a trivial problem. In the case of the LO, for instance, feedback signal which goes to the polarization controller can be adjusted so that the amplitude of the interference signal is maximized. To investigate whether the current polarization state is optimized, the polarization state needs to be shifted slightly and we need to check the decrease of the amplitude. A previous research implements the polarization optimization by random walkHuang (2009). However, this random walk method is vulnerable to light power fluctuations, which cause amplitude fluctuations of the interference signal. Even if the polarization state is already in the optimal state, the control system may judge that the polarization is not optimal due to the power fluctuations, and moves from the current polarization state.

To solve this problem, we introduce a polarization modulation method. By applying a modulation to the polarization and obtaining the error signal from the demodulation, it becomes possible to lock the polarization at the zero crosspoint of the error signal. For the phase-sensitive amplification in OPA, the polarization modulation is applied to the probe light. For the interference at BS-HD for homodyne detection, it is applied to the LO light.

Before explaining the beamsplitter coupling ratio lock, we refer to the structures of the beamsplitter. The beamsplitters are directional couplers, and light is exchanged between two fibers by evanescent coupling. Fiber beamsplitters are basically classified into two types: fused biconical tapered couplersKawasaki, Hill, and Lamont (1981); Pal, Chaudhuri, and Shenoy (2003) and side polished couplersParriaux, Gidon, and Kuznetsov (1981); Michel J.F and Herbert J. (1982). For the fused biconical tapered coupler, two fibers are twisted together and stretched on heating. In this method, the deformation of the cores results in the high optical loss. On the other hand, for the side-polished coupler, fibers are embedded in quartz substrates. Next, some part of the cladding is polished until it is removed to permit the evanescent coupling. Then the two pieces are contacted. In this method, the cores are not deformed, resulting in lower optical loss. To reduce the optical loss of squeezed light, we use side-polished fiber beamsplitters for the quantum light paths (BS2 and BS-HD in this demonstration).

In addition to the low-loss demand, the coupling ratio of the beamsplitter at the homodyne detection (BS-HD) must be exactly 50:50 for the following reason. In homodyne detection, the LO light interferes with the squeezed light and are divided at BS-HD. Then the difference of two photocurrents is electrically amplified, and the residual fluctuation represents the quadrature phase amplitude. Because the power of the LO light is very high, unbalance of the two photocurrents caused by a slight deviation of the coupling ratio from 50:50 easily results in saturation of the homodyne detector. In addition, since the DC output of the homodyne detector is used for generating various error signals as shown in Fig. 2, we do not want to make the homodyne detector AC coupled to prevent saturation.

The coupling ratio is required to be 50:50 with an accuracy of less than 0.1% from the following experimental parameters. The LO power is 16 mW, the probe power is 1 $\mu$W. The difference of the two photocurrents is converted to voltage by a transimpedance amplifer with the feedback resistance of 3.03 k$\Omega$, and then further amplified by 15 times. The output voltage of the detector should be kept in the range from $-1$ V to 1 V to avoid saturation of the op-amp.

We use a fiber beamsplitter 905P (Evanescent Optics), whose coupling ratio can be changed by a micrometer. However, some obstacles are found for the stabilization of the coupling ratio. Firstly, the response of the coupling ratio has a hysteresis and a backlash for the micrometer displacement as shown in Fig. 7. Secondly, it takes about 6 minutes for the coupling ratio to be stable after the user takes his or her hand off the knob. Thirdly, for the beamsplitter used in this study, the coupling ratio has dependency on the temperature at a rate of 0.95%/^∘C around the coupling ratio of 50:50. Fourthly, the coupling ratio changes by about 2.9% depending on the polarization state of the input light. Therefore, it is hard to maintain the coupling ratio with the accuracy of 0.1%. We note that the above properties varies on the individuals.

To stabilize the coupling ratio of the beamsplitter, we use the temperature as a degree of freedom for control. We attach a Peltier element to the bottom of the beam splitter and a thermistor to the side of the beam splitter, and place them on a heat sink. As a first test, we controlled the temperature measured by the thermistor to be kept constant. After 24 hours, the coupling ratio varied by about 1% as shown in Fig. 8 (blue line), which was worse than the acceptable coupling ratio deviation of 0.1%. Therefore, we employ a method to stabilize the coupling ratio of the beam splitter by monitoring the drift of the average of the homodyne detection signal and by feeding it back to the Peltier element. As shown in Fig. 8 (red line), the coupling ratio fluctuation is locked with a standard deviation of 0.01% for 24 hours, which is sufficiently smaller than the target fluctuation of 0.1%. Although this method use temperature dependence of the fiber beam splitter, some individuals of fiber beam splitter have little temperature dependence. Therefore, it is necessary to select individuals which has a sufficient temperature dependence or to develop a fiber beam splitter with a structure that has enough temperature dependence.

Squeezed light is measured for 24 hours in the fiber system using various stabilization mechanisms explained above. As described in Appendix B, the alignments and the squeezed light measurements are repeated alternately in this experiment, with various controllers turned on and off in sequence. The spectra of the squeezing and antisqueezing levels in the range from 10 MHz to 200 MHz are acquired with resolution band width of 1 MHz and video band width of 10 Hz by a spectrum analyzer PXA Signal Analyzer N9030B (Keysight).

Figure 9(a) shows time variations of squeezing level and antisqueezing level at 40 MHz as blue and red points, respectively. As a reference, an example of the squeezing level and antisqueezing level are shown as sky blue and pink points, respectively, for the case where the polarization stabilizations and coupling ratio lock of the beamsplitter are deactivated. Note that the phase locks and the power stabilizations are active for obtaining the sky blue and pink points. The blue and red points are obtained every 2 minutes, and the sky blue and pink points every 11 minutes. The blue points show an average squeezing level of $-4.42$ dB with an extremely small standard deviation $\pm$0.08 dB over 24 hours. The red points show an average antisqueezing level of 7.85 dB with a standard deviation 0.13 dB. In contrast, for the sky blue points, the squeezing level is not kept as the time passes. These results ensure that stabilization controls work properly to keep the system stable. Note that squeezing and antisqueezing levels have some outliers because they are not measured properly for a following reason. When the phase drift exceeds the dynamic range of the fiber stretcher, the voltage applied to the fiber stretcher is reset and the phase is relocked. If squeezed light measurements coincide with the reset process, inappropriate squeezing or antisqueezing levels are obtained. This problem can be prevented by programming of the control system, but we did not implement it in this measurement. When the average and the standard deviation of the squeezing and antisqueezing levels are calculated, these outliers are omitted.

We can estimate the effective loss of the system from the squeezing level and antisqueezing level. In addition to the insertion losses of optical components, the polarization mismatches become effective losses for the squeezed light, which reduces the squeezing level. If the change of the effective loss of the system is small, we can verify that the polarization matching is stable during the measurement. Figure 9(b) shows the effective loss of the entire system estimated from the blue and red points in Fig. 9(a). From the squeezing level $sq$ and antisqueezing level $asq$ expressed in dB, we can evaluate the effective loss $L$ of the whole system from the following relations,

where $r\geq 0$ is the squeezing parameter. In the ideal case where a loss is absent, the squeezing and antisqueezing levels become symmetric ($sq=-asq$). In actual cases where a loss exists, a vacuum fluctuation is mixed to the squeezed or antisuqeezed quadrature, from which the above equations are derived. From the above equations, $L$ is expressed as,

The time variation of effective loss was calculated to be $27.0\pm 1.0$% as shown in Fig. 9(b). The fact that the loss fluctuation was only $\pm 1.0$% over 24 hours ensures that the control system is operating stably for a long period. The details of the 27% loss are as follows. The loss of the OPA is from 8% to 12%. Two connectors are used, where the loss of each connector is from 3% to 5%. The loss of BS2 is about 3%. The loss of BS-HD is about 0.5%. The loss of the homodyne detection device is about 4%. As an option, the connectors can be fused, by which the loss is reduced to 0.5% per one connection point. In this experiment, fusion splicing is not done for the reusability of the OPA.

Figure 10 shows a spectrogram of the squeezing and antisqueezing levels. The dots in Fig. 10 are due to the same reason of the phase re-locking as in Fig. 9. The line at 25 MHz is due to electric noise.

In this study, squeezed light with a squeezing level of $-4.42$ dB has been successfully generated and measured in a fiber-based experimental system with a very small standard deviation of $\pm 0.08$ dB for 24 hours. Squeezed light is the most fundamental quantum state of light in cluster quantum computationsRaussendorf and Briegel (2001); Menicucci et al. (2006); Menicucci (2011); Pysher et al. (2011); Yokoyama et al. (2013); Yoshikawa et al. (2016); Yang et al. (2016); Zhang et al. (2017); Larsen et al. (2019a); Asavanant et al. (2019); Pfister (2020) and loop quantum computationsTakeda and Furusawa (2017); Takeda, Takase, and Furusawa (2019). Based on the technology demonstrated in this paper, it will be possible to construct a system that is automatically maintained stable for long periods of time, enabling reliable operations of optical quantum computers offered as a cloud service.

This work was partly supported by Japan Science and Technology Agency (Moonshot R&D) Grants No. JPMJMS2064 and No. JPMJPR2254, Japan Society for the Promotion of Science KAKENHI Grants No. 18H05207 and Grants No. 20K15187, the UTokyo Foundation and donations from Nichia Corporation of Japan. T.N. acknowledges financial support from Forefront Physics and Mathematics Program to Drive Transformation (FoPM). M.E. acknowledges support from Research Foundation for Opto-Science and Technology.