Highly efficient measurement of optical quadrature squeezing using a spatial light modulator controlled by machine learning

Jorge Amari; Junnosuke Takai; Takuya Hirano

doi:10.1364/OPTCON.484295

1. Introduction

Squeezed light is an essential resource for quantum information processing (QIP). For example, quantum computational advantage with a photonic processor was recently reported by Madsen et al. [1]. They used squeezed light from a pulsed optical parametric oscillator (OPO) to perform Gaussian boson sampling (GBS) using time-multiplexed and photon-number resolving architecture. Moreover, squeezed light is essential to realize measurement-based quantum computation (MBQC) [2,3]. A two-dimensional cluster state capable of 5,000 operational steps in 5-input mode and an ancillary state with Wigner negativity for measurement have been reported [4,5]. Squeezed light is also important for quantum sensing, such as quantum-enhanced stimulated Raman scattering microscopy [6], and for the test of basic quantum phenomena, such as the demonstration of the Einstein–Podolsky–Rosen paradox [7]. High squeezing is desired in many cases, e.g., squeezing of 4.5 dB is required to generate the two-dimensional cluster state [4].

There are three main methods to enhance the second-order nonlinear optical effect. The first is using an optical cavity to obtain an effectively long interaction length. The highest squeezing of 15.3 dB was realized using an optical cavity in 2016 [8]. However, the bandwidth becomes narrower as the enhancement becomes larger. Furthermore, a complex locking system is needed. The second method uses pulsed light to obtain high peak power. Since each pulse can be regarded as a single mode, we can naturally perform time-domain information processing. Using a light source with a higher repetition rate [9,10], it is possible to realize a faster quantum processor. The third method is using an optical waveguide to realize tight spatial confinement of light. By using an optical waveguide, it is possible to avoid gain-induced diffraction (GID) [11,12], and compact quantum devices can be realized through photonic integration technologies [13].

By combining the optical waveguide method with pulsed light, we should, in principle, be able to generate high-level and broadband-squeezed light with compactness and stability. In practice, the highest pulsed squeezing achieved using a waveguide is 5.0 dB [14], which is smaller than the 5.8 dB realized using a KTP bulk crystal in 1994 [15]. The obtained pulsed squeezing level of 5.0 dB was limited by the relatively small effective detection efficiency. To obtain high detection efficiency, it is necessary to improve the temporal and spatial mode matching between the squeezed light and the local oscillator (LO) in homodyne detection. It was shown that temporal mode matching could be improved to unity using temporally shaped LO pulses whose pulse width was shortened by parametric amplification pumped by pulsed light, but the spatial mode matching efficiency remains low (estimated to be 0.84) [14].

In this paper, we report the improvement of spatial mode matching using a spatial light modulator (SLM) whose optimal phase modulation distribution was automatically searched by machine learning. Since the SLM has 1024 $\times$ 1272 pixels with 8-bit resolution, fully optimizing a phase modulation distribution requires a search over a huge parameter space. We reduced the number of parameters by limiting the phase modulation distribution that can be expressed by simple functions such as a Gaussian function. We used Bayesian optimization to improve the visibility between the LO and probe light. We measured $-$5.88 dB squeezing generated by optical parametric amplification in a periodically poled LiNb$\rm {O_3}$ (PPLN) waveguide. To the best of our knowledge, this is the highest pulsed squeezing obtained to date, which updates the squeezing record of 5.8 dB reported by Kim et al. in 1994 [15].

2. Experimental setup

Figure 1 shows the experimental setup for the measurement of squeezed light. It is similar to the experimental setup for time-domain entanglement measurements [7]. The light source is a CW mode-locked laser with a wavelength of 1063 nm, a pulse width of 8.4 ps, and a pulse repetition rate of 86.5 MHz. The fundamental beam is injected into a bulk-type quasi-phase-matched KTiPO$\rm {P_4}$ (PPKTP) to generate horizontally polarized second harmonics with a wavelength of 531.5 nm. The temperature of the crystals is stabilized to the phase-matching temperature with a Peltier device.

Fig. 1. Schematic diagram of the experimental setup. (a) second-harmonic generation; (b) generation of temporally shaped LO by parametric amplification; (c) generation of vacuum squeezed light by parametric amplification; (d) SLM for controlling the spatial mode of LO; (e) homodyne detector for measurement of quadrature.

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The fundamental and second-harmonic beams exiting from the PPKTP are split into two directions by a half-wave plate and a polarizing beam splitter. One of the two beams is sent to a PPLN optical waveguide (PPLN0) with a core diameter of 3 $\mu$m $\times$ 5 $\mu$m to generate a temporally shaped LO. We adjusted the relative phase between the fundamental beam and the second-harmonic beam (pump beam) using a piezoelectric actuator to maximize the parametric gain. The other beam is sent to another optical waveguide (PPLN1) for squeezed light generation by single-pass OPA after the fundamental beam is separated and removed by a Pellin–Broca prism. We also stabilized the temperature of these waveguides at phase-matching temperatures with Peltier devices. The generated squeezed pulse and shaped LO pulse are combined at the polarizing beam splitter (PBS) and then sent to a homodyne detector.

The most important difference compared to Ref. [7] is the addition of an SLM (Hamamatsu Photonics model X13138) after PPLN0. The reflection angle of the LO beam is set smaller than 2 degrees. The effective area size of the SLM is 12.8 mm $\times$ 16 mm. The phase modulation distribution of the SLM can be set by inputting a bitmap image to the SLM controller. The half beam splitter used in Ref. [7] to generate entanglement is removed. Other improvements over the previous setup are that the lens just after PPLN0 is replaced with a lens of the same model number as that after PPLN1, and a flip mirror is inserted before the homodyne detector to measure the visibility between the LO and the probe beam.

3. Improvement of the spatial-mode matching using machine learning

In this section, we describe the procedure and results of controlling the spatial mode of the LO beam using the SLM, whose phase modulation distribution is optimized with Bayesian optimization (BO). BO is a machine learning algorithm often used to find the optimal set of parameters that maximize a particular performance measure. It uses a stochastic model, typically a Gaussian process, to predict the behavior of the function to be optimized and then determines where to evaluate the function. The phase modulation distribution of the SLM is specified by a 1024 $\times$ 1272 matrix with 8 bits for each element (8 bit bitmap image of 1024 $\times$ 1272 pixels), as shown below:

M = \begin{pmatrix} M_{1, 1} & \cdots & M_{1, 1272}\\ \vdots & \ddots & \vdots \\ M_{1024, 1} & \cdots & M_{1024, 1272} \end{pmatrix}.

However, it is unrealistic to optimize all these 1,302,528 elements directly by BO since BO can only be used to optimize up to about ten parameters with our current experimental setup. We search only the phase modulation distribution represented by the matrix $M(\boldsymbol {\theta })$ parametrized by the parameter $\boldsymbol {\theta }$. In particular, we here report the result of optimization using the following function:

(1)$$\{M(\boldsymbol{\theta})\}_{x, y} = A \exp \left[ -\left(\frac{(x-\mu_x)^2}{\sigma_x^2} + \frac{(y-\mu_y)^2}{\sigma_y^2}\right) \right] + a_x x + a_y y + b.$$

We optimized five parameters, $\boldsymbol {\theta } = (A, \sigma _x, \sigma _y, a_x, a_y)$, by BO. The parameters $a_x$ and $a_y$ are linear phase gradient along $x$- and $y$- axis and $b$ is an offset. We determined the values of $(\mu _x, \mu _y)$, the pixel position corresponding to the center of the LO beam, by a separate measurement in which a spot-like phase distribution was applied to the SLM and a far-field beam profile is observed.

In the experiment, we measured the visibility between the LO and the probe beam as a benchmark for spatial mode matching between the LO and the squeezed beam. Because both the probe beam and the squeezed vacuum field exit from the same waveguide, these beams are expected to have a common spatial mode. When measuring the visibility, the polarization of the fundamental beam injected into the PPKTP was changed to vertical polarization to suppress the generation of second-harmonic beams, and residual weak second-harmonic beams injected into PPLN0 and PPLN1 were blocked. A flip mirror was inserted before the homodyne detector and the optical power after the PBS was measured by a photodiode with a photosensitive area of 10 mm $\times$ 10 mm and recorded by a USB data acquisition device. The powers of two beams were adjusted to be equal at the photodiode. The relative phase between the LO and the probe beam was swept with a piezoelectric actuator, and we calculated the visibility from the maximum and minimum voltage values.

The procedure to search the SLM parameters that maximize the visibility value by the BO is as follows:

1. The initial parameters are determined at random.
2. Our Python program generates a bitmap image according to Eq. (1) using the values of the parameters, outputs the image to the SLM controller, then the SLM modifies the LO beam profile.
3. The visibility is measured and the obtained value is input to the BO.
4. The BO outputs a new set of parameters.
5. Repeat step 2 to step 4.

Here, we describe the optimization process for the parameter $\boldsymbol {\theta }$ in step 4. Consider the case where the time step is $n$; let the parameters output by the BO so far be $\Theta _{n-1} = (\boldsymbol {\theta }_0, \boldsymbol {\theta }_1, \ldots, \boldsymbol {\theta }_{n-1})$ and let the visibility values measured when the bitmap image generated according to Eq. (1) with each of these parameters be $\boldsymbol {v}_{n-1} = (v_0, v_1, \ldots, v_{n-1})^{\top }$. $\boldsymbol {\theta }_0, v_0$ are the initial parameter and initial visibility, and $\boldsymbol {\theta }_t, v_t$ $(t \ge 1)$ are the parameter and visibility obtained at time step $t$.

The BO regresses the probability distribution of visibility:

(2)$$P\bigl(v|\boldsymbol{v}_{n-1}, \Theta_{n-1}\bigl) = \mathcal{N}\bigl(v|m_{n-1}(\boldsymbol{\theta}),\sigma^2_{n-1}(\boldsymbol{\theta})\bigl),$$

by Gaussian process regression, where $\mathcal {N}(x|m,\sigma ^2) = \frac {1}{\sqrt {2\pi \sigma ^2}} \rm {exp} \left (\frac {(x - m)^2}{2\sigma ^2}\right )$ is the normal distribution of the expected value $m$ with variance $\sigma ^2$. $m_{n-1}(\boldsymbol {\theta }), \sigma _{n-1}(\boldsymbol {\theta })$ are characterized as

(3)$$\begin{aligned}m_{n-1}(\boldsymbol{\theta}) = \boldsymbol{k}_{n-1}^{\top} C_{n-1} \boldsymbol{v}_{n-1}, \end{aligned}$$

(4)$$\begin{aligned}\sigma^2_{n-1}(\boldsymbol{\theta}) = k(\boldsymbol{\theta}, \boldsymbol{\theta}) - \boldsymbol{k}_{n-1}^{\top} C_{n-1} \boldsymbol{k}_{n-1}, \end{aligned}$$

by the kernel function $k(\boldsymbol {\theta }, \boldsymbol {\theta }')$, where $\boldsymbol {k}_{n-1} = (k(\boldsymbol {\theta }, \boldsymbol {\theta }_0), k(\boldsymbol {\theta }, \boldsymbol {\theta }_1), \ldots, k(\boldsymbol {\theta }, \boldsymbol {\theta }_{n-1}))^{\top }$ and $\{C_{n-1}\}_{i,j} = k(\boldsymbol {\theta }_i, \boldsymbol {\theta }_j) + \delta _{i,j} \beta ^{-1}$, with $\beta ^{-1}$ being the measurement noise. Thus, the inductive bias of the regression is determined by the kernel function. In this paper, we use the kernel function [16]

(5)$$k(\boldsymbol{\theta}, \boldsymbol{\theta}') = p_0 \exp \left( - \frac{p_1}{2} \| \boldsymbol{\theta} - \boldsymbol{\theta}' \| \right) + p_2 + p_3 \boldsymbol{\theta}^T \boldsymbol{\theta}',$$

where $p_0 = 1.0,\ p_1 = 5.0,\ p_2 = 0,$ and $p_3 = 0$.

Finally, from the regressed probability distribution of visibility, we derived the best guess parameters for measurement at the next time step. The best guess parameters should satisfy two conditions: the visibility values obtained are expected to be high and have low similarity to the parameters that have been already tested in order to reduce the unexplored parameter space. The function that returns a larger value if the parameters better satisfy the two conditions is called the acquisition function. The BO outputs a new parameter $\boldsymbol {\theta }$ for which the acquisition function is maximal by Latin hypercube sampling. In this paper, the acquisition function AF is

(6)$$\rm{AF}_{n-1}(\boldsymbol{\theta}) = m_{n-1}(\boldsymbol{\theta}) + \sqrt{log(2\delta^{{-}1})} \Bigl(\sqrt{\sigma_{n-1}^2(\boldsymbol{\theta}) + \gamma_{n-1}} - \sqrt{\gamma_{n-1}}\Bigl),$$

where $\gamma _{n-1} = \gamma _{n-2} + \sigma ^2(\boldsymbol {\theta }_{n-1})$, $\gamma _0 = 0$, $\delta = 10^{-6}$ [17], and the number of sampling is $10^6$.

We used a Python program for automated parameter optimization, including control of the data acquisition device, generation of phase modulation patterns (bitmap images) and output to the SLM controller. Using AMD Ryzen 7 4800U as a CPU, the first iteration took about 7 seconds and the 500th iteration took about 45 seconds in the current program, wheres the visibility measurement took about 3 seconds.

We found that the optimization results were different depending on the position of the lens just after PPLN0. We shifted the lens position 1.3 $\rm {\mu }$m, 2.7 $\rm {\mu }$m, 4.0 $\rm {\mu }$m, and 5.3$\rm {\mu }$m toward the waveguide relative to the position where the best visibility was obtained when the phase modulation distribution of the SLM is flat. Figure 2 shows the change in visibility over time as the parameter search by machine learning progresses for each lens position. A total of 500 data points were acquired at each lens position. The visibility values gradually increased as the machine learning searched for better parameters, though the value sometimes dropped as the machine learning searched unexplored parameters. We selected ten sets of parameters that gave top ten high visibility from each set of lens position data, and we measured the visibility with an oscilloscope while carefully adjusting the optical axis of the LO beam so that the visibility would increase for these 10 sets of parameters. The highest visibilities obtained for each lens position are shown in Fig. 3. The red solid line is the best visibility of 0.971 obtained with a flat SLM phase modulation distribution that was realized by applying the distortion correction pattern provided by the manufacturer. Table 1 lists the optimal five parameters of the phase modulation distribution at each lens position.

Fig. 2. Results of parameter search by machine learning at various lens positions. See Data File 1, Data File 2, Data File 3 and Data File 4 for underlying values [18–21].

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Fig. 3. Highest visibility value for each lens position. Blue points: visibility values measured when the SLM was set to the best guess phase modulation distribution at each lens position. Solid red line: best visibility obtained when no spatial phase modulation was applied to the LO with the SLM (lens position set at 1.6 $\rm {\mu }$m).

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Table 1. Optimal parameters at each lens position

View Table

Based on the results shown in Fig. 3, we inferred that the most suitable lens position is about 4.7$\rm {\mu }$m. We positioned the lens 4.7$\rm {\mu }$m toward the waveguide and searched for the optimal SLM phase modulation distributions by machine learning using the same procedure as before. The obtained best guess values were $(A, \sigma _x, \sigma _y, a_x, a_y) = (74,93,124,-0.01,-0.03)$. We next carefully adjusted the optical axis of the LO beam and $(A, \sigma _x, \sigma _y, a_x, a_y)$ to increase the visibility that was measured with an oscilloscope. We obtained the best visibility of 0.977 when the lens was at 4.7 $\rm {\mu }$m and $(A, \sigma _x, \sigma _y, a_x, a_y) = (70, 95, 120, -0.01, 0.01)$. These parameter values were used for the measurement of the squeezing. The spatial mode matching efficiency could also be improved further by considering more general phase distributions than explored in the present study.

4. Measurement of squeezing

In this section, we describe the setup and results of the squeezing measurements and then discuss the effective detection efficiency. In the squeezing measurements, we blocked the probe beam so that squeezed vacuum states were generated in PPLN1. The power of the pump beam for the pulse shaping, which is a second-harmonic beam injected into PPLN0, was 4.9 mW, and an amplification gain of over 10 dB was obtained, satisfying the conditions for sufficient pulse shaping [14]. We set the relative phase between the fundamental and pump beams to maximize amplification using a piezoelectric actuator. The power of the pump beam for squeezed light generation injected into PPLN1 was 2.8 mW. A homodyne detector output was connected to a spectrum analyzer and the noise power was measured with a center frequency of 5 MHz in zero-span mode, the resolution bandwidth of 1 MHz, and the video bandwidth of 100 Hz. We used photodiodes with a quantum efficiency of about 0.93 for the homodyne detector (Fermionics, FD150W).

The amplifier noise level was $-$79.27 dBm and the shot noise level was $-$69.90 dBm. We measured a squeezing level of $-$75.78 dBm and an anti-squeezing level of $-$53.17 dBm. Figure 4 shows the result of the measurement of squeezing. The amplifier noise was subtracted. We measured $-$5.88$\pm$0.04 dB squeezing and +16.73$\pm$0.02 dB anti-squeezing. The estimated effective detection efficiency was 0.753 and the squeezing parameter was 2.06 from the squeezing level $S_+$ and anti-squeeze level $S_-$ using the following equation:

(7)$$S_{{\pm}} = \eta \exp{({\pm} 2r)} + 1 - \eta,$$

where $\eta$ is the effective detection efficiency and $r$ is the squeezing parameter [22]. The squeezing parameter represents the degree of deamplification and amplification of the vacuum noise: the quadrature variances become $e^{\pm 2r}$ if $\eta =1$. The effective detection efficiency should be the product of the waveguide transmissivity of 0.89, detector quantum efficiency of 0.93, optical transmissivity after the waveguide of 0.98, temporal mode-matching efficiency of 1.0 [14] and spatial mode matching efficiency of 0.95 (= $0.977^2$), and is calculated to be 0.77, which is close to the estimated value from the squeezing levels. This estimation shows that the waveguide loss and photodiode inefficiency are two main factors that limit the present squeezing level. Further improvements should be feasible by using a low-loss optical waveguide and high-efficiency photodiode. A waveguide with a transmission loss of 0.1 dB/cm [23] and photodiode with quantum efficiency of 0.995 [8] have been reported. With these state-of-the-art devices, 10 dB of squeezing would be possible.

Fig. 4. Squeezing with SLM. Solid black line: normalized shot noise level; Solid blue line: squeezing level; Solid red line: anti-squeezing level; Solid green line: noise level when the LO phase was swept by a 200 mHz sawtooth wave (using piezoelectric actuator).

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5. Conclusion

In summary, we demonstrated improved visibility between the probe and LO with an SLM programmatically controlled by machine learning. We used a temporally shaped LO generated by optical parametric amplification to improve temporal mode matching between squeezed light and the LO and control the spatial mode of the LO with the SLM to improve spatial mode matching between the squeezed light and the LO. As a result, we detected pulsed 5.88$\pm 0.04$ dB squeezed light generated by optical parametric amplification in a PPLN waveguide. To the best of our knowledge, this is the highest squeezing obtained for pulse squeezers. This achievement will lead to the realization of compact and efficient quantum information technology.

Funding

Japan Society for the Promotion of Science (19K03703).

Acknowledgements

We wish to thank Prof. K. Shibata for his advice and support for the experiment. JA and JT would like to thank Prof. K. Shin for teaching them machine learning. This work is supported by a Grant-in-Aid for Scientific Research (C) (Grant No. 19K03703).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Data File 1 Ref. [18], Data File 2 Ref. [19], Data File 3 Ref. [20], Data File 4 Ref. [21]

References

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Name	Description
Data File 1	Figure 2 (a)
Data File 2	Figure 2 (b)
Data File 3	Figure 2 (c)
Data File 4	Figure 2 (d)

Lens position	A	$σ_{x}$	$σ_{y}$	$a_{x}$	$a_{y}$
1.3 $μ$ m	24	211	207	$-$ 0.07	0.14
2.7 $μ$ m	38	78	152	0.01	$-$ 0.07
4.0 $μ$ m	63	103	252	0.01	$-$ 0.09
5.3 $μ$ m	84	105	145	0.09	$-$ 0.09

Highly efficient measurement of optical quadrature squeezing using a spatial light modulator controlled by machine learning

Abstract

1. Introduction

2. Experimental setup

3. Improvement of the spatial-mode matching using machine learning

4. Measurement of squeezing

5. Conclusion

Funding

Acknowledgements

Disclosures

Data availability

References

Supplementary Material (4)

Data availability

Cited By

Figures (4)

Tables (1)

Equations (8)

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