Continuous-variable optical quantum information processing, where quantum information is encoded in a traveling wave of light called a flying qubit, is a candidate for a practical quantum computer with high clock frequencies. Homodyne detectors for quadrature-phase amplitude measurements have been the major factor limiting the clock frequency. Here, we developed a real-time amplitude measurement method using a modular optical parametric amplifier (OPA) and a broadband balanced photodiode that is commercially used for coherent wavelength-division multiplexing telecommunication of the fifth-generation mobile communication systems (5G). The OPA amplifies one quadrature-phase component of the quantum-level signal to a loss-tolerant macroscopic level and suppresses the loss after the OPA from 92.4% to only 0.4%. This method was applied to a broadband squeezed vacuum measurement with a center wavelength of 1545.32 nm. In the time-domain measurement, the squeezing level of 5.1 ± 0.1 dB without loss correction was obtained by a real-time oscilloscope with a sampling rate of 160 GHz and an analog bandwidth of 63 GHz. The frequency-domain analysis also shows that a squeezing level of 5.2 ± 0.5 dB is obtained from DC to 43 GHz, which is limited by the balanced detector. This indicates that the proposed method can be easily broadened by using a broader bandwidth measurement instrument. By applying this method, not only can optical quantum computers with high clock frequencies be realized but also multi-core systems can be realized.
Research on continuous-variable optical quantum information processing (CVOQIP), in which quantum information is encoded in wave packets of an optical electromagnetic field, is aimed at realizing a fault-tolerant universal quantum computer.1–5 This form of processing has high carrier frequency, which can raise the upper limit of the clock frequency. In CVOQIP, measurement-based quantum computation (MBQC) is a reasonable way of performing large-scale quantum computations.6,7 MBQC uses a cluster state as a resource state, which is a special multipartite entangled state and contains a superposition of all input-output relations in quantum computation. The processor of an optical quantum computer consists of a time-domain multiplexed (TDM) cluster state6,7 and a homodyne measurement that measures the quadrature-phase amplitude of each wave packet. The TDM cluster state is created on the basis of a continuous-wave (CW) squeezed vacuum.8–11 A squeezed vacuum is generated by creating quantum correlations between photons contained within a given period.12 In a squeezed vacuum, the noise of the electromagnetic field at a particular quadrature-phase component is below the shot noise level.
The broader the squeezing bandwidth is, the shorter the quantum correlation time and the more wave packets can be “packed.” In other words, the upper limit of the clock frequency is the bandwidth of the squeezed vacuum. It has been reported that periodically poled lithium niobate (PPLN) waveguides can generate 6-dB of squeezing in a 6-THz bandwidth with a center wavelength of 1545 nm13 and 4-dB squeezing in a 25-THz bandwidth with a center wavelength of around 2000 nm.14 That is, the THz clock frequency can be realized in principle.
However, the bandwidth of the homodyne detectors and other devices is currently limited to about GHz as depicted in Fig. 1(a). To measure quantum light, the detection efficiency of photodiodes of the homodyne detector must be very high, but a photodiode specifically designed for this purpose will inevitably have a narrow bandwidth [Fig. 1(d)]. Realistically, the bandwidth is limited to a few hundred MHz15–17 or at best a few GHz.18,19
Recently, an optical parametric amplifier (OPA) assists to measure a broadband squeezed vacuum.14,20,21 OPAs act as phase-sensitive optical amplifiers that can amplify one quadrature-phase component without adding noise [a noise figure (NF) of 0 dB in principle22–24]. This is a significant difference from phase-insensitive optical amplifiers such as an erbium-doped fiber amplifier, which cannot cross the NF = 3 dB barrier.22 A quantum-level signal can be converted into a macroscopic level with a low-loss OPA with sufficiently high gain acting as an optical pre-amplifier.25 Consequently, optical losses and electrical noise are suppressed. However, the previous studies made only narrow bandwidth measurements or power measurements,14,20,21 while OPQC requires real-time broadband quadrature-phase amplitude measurements made by homodyne detectors. In other words, it is essential to obtain instantaneous values of the quadrature-phase amplitude measured in real time, as shown in the output of Fig. 1(f).
In this paper, an OPA amplified one quadrature-phase component of the light to be measured, then we detected its output with a balanced photodiode and a broadband amplifier. We measured a broadband squeezed vacuum with a center frequency of 194.0 THz (a wavelength of 1545.32 nm) generated by another OPA.13 We observed beyond 5-dB squeezing without any loss correction by a real-time oscilloscope with a sampling rate of 160 GHz and an analog bandwidth of 63 GHz.
The critical component of this work is a fiber-coupled, low-loss, and high-gain broadband OPA with a PPLN waveguide. Many of the other components are commonly used in digital coherent telecommunications in the fifth-generation mobile communication systems (5G). The 5G elements used in our demonstration were the balanced photodiodes and an electrical amplifier. In addition, as illustrated in Fig. 1(b), dividing the bandwidth of the squeezed vacuum using a dense wavelength division multiplexing (DWDM) coupler, as is done in WDM digital coherent communications, enables multiple independent quantum processors to be realized from a single light source.
The electromagnetic field of light with a carrier frequency of ω0 is represented by using annihilation and creation operators, and , excluding the oscillation term associated with the carrier frequency, as follows:
Here, these operators satisfy the commutation relation , where is the Kronecker delta function and are called the quadrature-phase amplitude operators. We often consider wavepackets of light defined by a temporal mode function f(t) as , and the quadrature-phase amplitude operators of this mode are defined as . Figure 1(c) shows examples of measurement results, which correspond to eigenvalues and of and , respectively.
where θ is the phase between the signal and LO lights, and it is assumed that the LO light is sufficiently strong and the quantum fluctuations are negligible. is the temporal mode function of the instruments, including the homodyne detector, amplifiers, filters, the oscilloscope, and so on. The Fourier transform of is related to the frequency response of the measurement system.
In contrast, broadband balanced detectors lift the bandwidth beyond GHz or even higher. Recently, GHz homodyne detectors built with silicon photonics technology18,19 have been demonstrated; one of them measured a broadband squeezed vacuum.18 Losses and electrical noises, however, drastically degrade the signal-to-noise ratio (SNR) of the homodyne system, which is critical for CVOQIP applications [Fig. 1(e)].
Here, we amplify the X component by using an OPA and measure the output with a homodyne detector, as depicted in Fig. 1(f). The output signal is represented as
where G is the gain of the OPA. Instead of de-amplification of the P component, the X component is amplified, as shown in Fig. 1(f). By setting θ = 0, the X component of the quadrature-phase amplitude can be measured. The ideal OPA acts as a phase-sensitive optical amplifier and does not add noise when amplifying the X component (see supplementary material IV B), thus suppressing the loss of the broadband homodyne detector.
Let the homodyne detector's and OPA's efficiencies be and , respectively. As shown in supplementary material II, the effective efficiency of the system can be written as
From this expression, if the loss of the OPA is small compared with the loss of the homodyne detector and the gain of the OPA is sufficiently high, , the OPA can suppress for the loss of the homodyne detector (also see supplementary material IV B). Optical losses, the detection efficiency of a broadband balanced photodiode, and electrical noise in the broadband measurement equipment would easily increase the total loss of the homodyne system to more than 90%, corresponds to . Consider, for example, a case where the loss is 90% (): if the OPA has a gain G of 35 dB and an efficiency of 79% (corresponds to 1-dB NF), the loss after the OPA can be improved from 90% to only 0.3%.
Figure 2(a) shows the experimental apparatus of our homodyne system. The fundamental light is from a continuous-wave fiber laser (NKT photonics, X15) at 1545.32 nm (not shown). The light to be measured is a broadband squeezed vacuum generated by a fiber pig-tailed OPA (OPA1 in the figure).13 The output fiber of OPA1 is connected to the input fiber of OPA2 (used as a phase-sensitive optical amplifier), which is excited by pump light with a wavelength of 773 nm. 10% of the OPA2 output is used for monitoring the output spectrum by an optical spectrum analyzer (OSA, Yokogawa AQ6370D) and locking the phase between the squeezed vacuum and the sub harmonic of the pump light by a feedback circuit (FBC1)21 (see supplementary material IV C). The remaining 90% is sent to a 50:50 fiber beam splitter, where it interferes with the local oscillator (LO) light and is received by a balanced photodiode (BD, , BPDV2150R). Delay lines (DLs) and variable optical attenuators (VOAs) are inserted in each path to adjust the optical path length and power balance. The two photodiodes are reverse biased by biasing circuits, and each photocurrent is monitored (not shown). The output of the balanced photodiode is amplified by a broadband amplifier (AMP, SHF, S807 B, 55-GHz bandwidth). The output is split by a broadband power splitter (SPL, Anritsu, W241A); then, one part is measured by a real-time oscilloscope (Keysight, DSO-Z 634A, 63-GHz bandwidth). The other part is monitored by an electrical spectrum analyzer (ESA, Keysight, EXA N9010B). The ESA is also used to generate an error signal to lock the phase between the LO light and the output of OPA2 by a feedback circuit (FBC 2) (see supplementary material IV C for details).
In the following experiments, the pump power of OPA2 was set to 1.2 W. The parametric gain G and NF of OPA2 were 35 and 1 dB, respectively (see supplementary material IV D and Ref. 24). The responsibility and the quantum efficiency of BD are 0.45 A/W and 37%, respectively. The measured loss after OPA2 was 92.4%, then from Eq. (7) the effective efficiency of the total homodyne system is 79%. Figure 2(b) shows the raw data of the homodyne detector. The black trace shows the electrical noise floor when all the light is blocked. The red trace shows the shot noise with the LO when the photocurrent of each photodiode is set to 3.0 mA, which corresponds to the LO power of 12.8 mW before the 50:50 fiber-based beam splitter. We acquired the output signal for 78.2 ns. Figure 2(c) shows fast Fourier transform (FFT) traces of the acquired data with different LO powers. These traces were calculated by averaging the results of 8192 measurements taken over a 0.8-ms period, and all the frequency response traces are averaged for 8192 traces. The traces show a flat frequency response up to 10 GHz. The 3-dB bandwidth of our system is 43 GHz of the BD, and even at that frequency, 20 dB of SNR is obtained when the photocurrent is 3.0 mA (red trace). All the traces are falloff beyond the frequency of 63 GHz, which is the edge of the oscilloscope's bandwidth.
Figure 3(a) shows the raw data corresponding to quadrature-phase amplitudes of the squeezed light of the anti-squeezing and squeezing components (left and right, respectively), when the pump for OPA1 is set to 438 mW. The center shows the shot noise when the pump for OPA1 is set to 0 mW. Figure 3(b) is a histogram based on the measurement results of Fig. 3(a). From the histogram, the calculated squeezing and anti-squeezing levels in the time-domain are 5.1 ± 0.1 and 14.2 ± 0.1 dB, respectively.
The FFTs of these data are shown in Fig. 3(c), and the normalized squeezing level calculated for several pump powers of OPA1 is shown in Fig. 3(d). From these figures, it can be seen that the homodyne system measured a 5.2-dB broadband squeezed vacuum from DC to 43 GHz. These results clearly show the measurement bandwidth of this study is not limited by the method but by the measurement instruments. Note that the peaks at 34 GHz are measurement artifacts caused by the oscilloscope.
Figure 3(e) shows the squeezing and anti-squeezing levels as a function of the pump power for OPA1. The occupied circles are calculated from the variance of time-domain data like in Fig. 3(a), and the error bars are estimated from 8192 datasets. For a pump power of 438 mW, a squeezing level of 5.2 ± 0.5 dB and an anti-squeezing level of 13.9 ± 0.1 dB are obtained. The dashed line is the fitting to the function
where ± represents anti-squeezing (+, blue) or squeezing (−, red), L represents the total effective loss of the system, including of OPA1 and the homodyne system, and a is a second-harmonic-generation coefficient in the unit of . The fitting result shows the system loss of 29%, including losses of OPA1, OPA2, optical propagation, detector, and the electrical noise of the instruments. This value limits the squeezing level that can be measured with this apparatus to 5.2 dB. Linearity is an important issue when parametric amplification is performed with high gain as in this experiment. In our case, the measured results on the anti-squeeze side of Fig. 3(e) are in good agreement with the theoretical line, indicating that parametric amplification is performed in the linear region in this experiment.
In order to confirm the performance of the suppressing losses after OPA2, we inserted a VOA [not shown in Fig. 2(a)] after OPA2 to worsen and measured the squeezing level as a function of the loss as shown in Fig. 3(f). Note that the measurements in Fig. 3(f) are not broadband measurements by the oscilloscope but rather narrowband measurements made with ESA at a sideband frequency of 100 MHz, and we did not lock the LO phase (see supplementary material VII). The dashed and solid lines are calculated traces when G = 0 and 35 dB. The markers with error bars represent the measured values. It is clear from this figure that OPA2 with G = 35 dB effectively suppresses the loss after OPA2 from 92.4% to 0.4%, and the system can measure the original squeezing level.
In summary, we have experimentally demonstrated high-detection efficiency (79%) and the broadband (DC to 43 GHz) homodyne system using an OPA module and commercially available 5G components. The OPA suppresses the losses after OPA2 from 92.4% to only 0.4%. With this system, we performed real-time measurements of the quadrature-phase amplitude of broadband squeezed vacuum in the time domain, obtaining a squeezing level of 5.2 ± 0.5 dB without loss correction. This value exceeds one of the thresholds (4.5 dB) for cluster state generation,10 indicating that this method can be applied to quantum calculations. Except for the handling of non-Gaussian quantum states,4,32 the necessary elements for optical quantum computation are LO phase switching and feedforward systems, which are parts of classical information processing. Even at present, there are electro-optical modulators and electronics above 40 GHz in bandwidth as 5G technology,33,34 and when “Beyond 5G” or 6G technologies35 appear in the future, the speed of classical information processing will be further increased. This means that the broadband homodyne detector developed in this study will make it feasible for the clock frequency of optical quantum computers to exceed 40 GHz.
We conclude this paper by proposing a multi-core optical quantum processor by effectively using the full-bandwidth of squeezed vacuum with our homodyne system. Quantum entanglement between two frequency sidebands is the essential feature of a squeezed vacuum. In this study, we measured the quantum correlation in the ±43 GHz bandwidth around a frequency of 194.0 THz, as shown in Fig. 1(a). Therefore, by dividing the optical spectrum into frequency-band pairs with DWDM couplers (e.g., with a frequency spacing GHz) and applying our homodyne system for each as illustrated in Fig. 1(b), a multi-core optical quantum processor can be realized, wherein the operating clock of each processor exceeds 40 GHz. As well, optical frequency comb technology can be used to prepare phase-coherent LO beams for each homodyne system.36–38 This research shows that CVOQIP is highly compatible with mature 5G technology.
See the supplementary material for the complete experimental apparatus, including the details of OPAs, phase-locking methods, and the evaluation of the system's stability.
The authors acknowledge support from the UTokyo Foundation and donations from Nichia Corporation of Japan. T.Y. acknowledges support from the Advanced Leading Graduate Course for Photon Science (ALPS). M.E. acknowledges support from the Research Foundation for Opto-Science and Technology. This work was partly supported by the Japan Science and Technology Agency (No. JPMJMS2064) and the Japan Society for the Promotion of Science KAKENHI (Nos. 18H05207 and 20K15187). The authors acknowledge Dr. Hiroshi Yamazaki for supporting the setup of the broadband detection system and Mr. Takeru Ebihara for fruitful discussion.
Conflict of Interest
The authors have no conflicts to disclose.
Asuka Inoue, Naoto Takanashi, and Takeshi Umeki conceived and planned the project. Asuka Inoue, Takahiro Kashiwazaki, Taichi Yamashima, and Takeshi Umeki designed and constructed the experimental apparatus and acquired the data. Takahiro Kashiwazaki, Takushi Kazama, Koji Enbutsu, Kei Watanabe, and Takeshi Umeki developed the low-noise and high-gain OPA module. Asuka Inoue, Takahiro Kashiwazaki, and Mamoru Endo analyzed the data and wrote the manuscript with assistance from all other coauthors. Akira Furusawa supervised the project.
Asuka Inoue: Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Akira Furusawa: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Takahiro Kashiwazaki: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Taichi Yamashima: Data curation (equal). Naoto Takanashi: Conceptualization (equal); Writing – review & editing (equal). Takushi Kazama: Resources (equal). Koji Enbutsu: Resources (equal). Kei Watanabe: Resources (equal). Takeshi Umeki: Supervision (equal); Writing – review & editing (equal). Mamoru Endo: Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.