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Abstract
We report an experimental demonstration of the continuous-variable Einstein-Podolsky-Rosen (EPR) paradox and quantum steering using a pulsed light source and nonlinear optical waveguides. In this pulsed light source experiment, time-domain measurements were performed in which one measured value of the quadrature phase amplitude of the light field was independently obtained for each pulse. This independence is useful for application to quantum information processing and fundamental physics. To realize time-domain measurements, in-house built homodyne detectors were used to detect individual pulses. In addition, to improve the temporal-mode matching between the local oscillator (LO) and entangled pulse, the duration of the LO pulse was shortened by single-pass optical parametric amplification. The product of the conditional variances of the quadrature amplitudes was 0.82 ± 0.09 < 1, which satisfies the condition for realization of the EPR paradox and quantum steering.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum entanglement is a quantum mechanical phenomenon that occurs when multiple quantum systems are correlated, and each system cannot be described separately. Strongly correlated entanglement is important for fundamental physics such as demonstration of the Einstein-Podolsky-Rosen (EPR) paradox [1] and quantum steering [2], as well as for application to quantum information processing. The EPR paradox is an argument regarding the contradiction between the completeness of quantum mechanics and local realism, i.e. the “realistic philosophy of most working scientists” [3]. In the original EPR gedanken experiment, two spatially separated quantum particles with position and momentum correlation were considered. According to quantum mechanics, it is possible to predict the outcome of a measurement of one particle by measuring either the position or momentum of the other particle beyond the precision allowed by the uncertainty relation. It was Schrödinger who introduced the term “steering” to denote the ability to affect the state of the other particle remotely. Wiseman et al. more recently gave an operational definition for steering and showed that steerable states are a strict subset of the entangled states [4]. To experimentally demonstrate the EPR paradox and steering, an ensemble of strongly entangled quantum systems must be generated and, in principle, independent measurements on the ensemble must be performed to evaluate the uncertainty of the conjugate observables.
Experimental demonstration of continuous-variable (CV)-EPR paradox and steering is a direct realization of the original EPR proposal and has been reported in a variety of physical systems, such as quadrature phase amplitude of a light field [5–9], position- and momentum-entangled photon pairs produced by spontaneous parametric down conversion [10] and four wave mixing [11], and recently quadratures of massive particles [12]. In particular, the quadrature phase amplitude of a light field was the first physical system that was used for demonstration of the EPR paradox [5], and has been extensively studied because the system is important for quantum information processing, especially quantum communication.
For quantum information processing, it is important to individually access each quantum state involved in the information manipulation [8, 13, 14]. In this context, characterization of the quantum correlation in the spectrum domain using a radio-frequency spectrum analyzer, as in the first experiment using a continuous wave (cw) light source [5], and using a pulsed light source [15] has difficulty in providing access to each individual quantum state [8]. In the time-domain measurement using cw light, the optical mode to be measured can be selected by a weight function that is multiplied with the continuous output of the homodyne detector and the output value multiplied for each time window is taken as a separate measurement result [16]. For each measurement to be independent, it is necessary that the selected modes are orthogonal to each other. In [16], Yoshikawa et al. reported independent measurements using a weight function that takes not only positive values but negative values. For the case of a pulsed light source, each pulsed mode is orthogonal to the others in the sense that the overlap between the spatio-temporal mode functions of different pulses is negligible and realization of independent measurements is easier from both a practical and fundamental perspective [8]. However, demonstration of the EPR paradox and steering using pulsed light in the time domain has not been reported to date.
One quadrature value from one pulse in the pulse train can be obtained using a pulsed light source if broadband squeezing and broadband, high-speed detectors are employed [17, 18]. Broadband squeezing can be generated by single-pass optical parametric amplification (OPA) utilizing χ(2) nonlinearity. A pulsed light source has high peak power, so that sufficient nonlinearity can be obtained by single-pass OPA without using an optical cavity, which effectively enhances the nonlinearity but limits the squeezing bandwidth. In addition, using an optical waveguide for OPA, squeezed states of light can be generated more efficiently thanks to the strong confinement of the waveguide, and the transverse mode of the light beam can be controlled, so that gain-induced diffraction [19], which distorts the phase front in the case of a bulk crystal, can thus be avoided [20,21]. The integration of optical circuits has attracted attention in recent years, and the use of optical waveguides is also important from this viewpoint [22–24].
Here, we report a demonstration of the EPR paradox and quantum steering both in the frequency-domain and in the time domain measurement using pulsed light source. In the time domain measurement a single value for the field quadrature was acquired per pulse. We used single-pass OPA in periodically poled LiNbO3 (PPLN) waveguides to both generate squeezed states, and to shape the local oscillator (LO) pulses and improve the temporal mode matching efficiency between the entangled pulses and LO pulses [25, 26]. In the time-domain measurement, we realized independent measurements of adjacent pulses using in-house built low-noise high-speed homodyne detectors. The entangled pulses were characterized by calculating the covariance matrix from two conjugate quadrature values measured at two stations to obtain a product of conditional variance of 0.82 ± 0.09 < 1, where 1 is the critical value. To the best of our knowledge, this is the first demonstration of the EPR paradox and steering by independent time-domain measurement with pulsed light. We also evaluated the pulsed entanglement in the frequency domain and obtained a conditional variance of 0.93 ± 0.03 < 1 without correction for the amplifier noise.
2. Experimental setup
Figure 1 shows the experimental setup. It is similar to that used for our previous frequency-domain experiment [27], except for the use of a digital oscilloscope, which is necessary for the time-domain measurement. The laser source was a cw mode-locked Nd:YVO4 laser (Montfort M-PICO Nd:VAN Laser) operated at 1063 nm with a pulse duration of 9 ps and a pulse repetition rate of 86.6 MHz.
Fig. 1 Experimental setup. HWP: half-wave plate; QWP: quarter-wave plate; PZT: piezoelectric transducer; PBS: polarizing beam splitter; HBS: half beam splitter; HD: homodyne detector. Two photodetectors (PDs) above the PBSs are used to measure the visibility. This scheme was surrounded with 5 mm thick acrylic plates to protect from air movement.
The fundamental light is frequency doubled in a periodically poled (PP) KTiOPO4 (KTP) crystal to produce second-harmonic (SH) beams. The SH beams are directed to two experimental regions (pulse shaping region and entanglement generation region). In the pulse shaping region, the SH beam and a part of the fundamental light, which is unconverted to SH in PPKTP are focused into a PPLN waveguide (PPLNLO) to shorten the duration of the LO pulses by OPA [26]. In the entanglement generation region, only SH beams are directed to two PPLN waveguides (PPLN1 and PPLN2) to generate squeezed beams by OPA. The two generated squeezed beams are combined at the half beam splitter (HBS) and entangled beams are generated. The relative phase between two squeezed beams is set to π/2. The relative phase can be adjusted by changing the voltage of a piezoelectric transducer (PZT 3) before PPLN2. The generated entangled pulses and shaped LO pulses are combined at the polarizing beam splitter (PBS) on both Alice’s and Bob’s station, and then sent to two homodyne detectors (HDA and HDB). The signals from these homodyne detectors are both directed to a digital oscilloscope (Tektronics DPO7254) with high impedance connection for the time domain measurement and spectrum analyzers (Anritsu MS2830A (spectrum analyzer 1): for frequency-domain measurement; RIGOL DSA815 (spectrum analyzer 2): to monitor the noise level of entangled beams for time-domain measurement). The voltage signals from two homodyne detectors are recorded on different channels of the digital oscilloscope. In our experiment, the signals of HDA and HDB are coupled by a radio frequency (RF) 180° phase power combiner (R&K, HYB2CA) before the spectrum analyzers, and then sum correlation and difference correlation between Alice and Bob are measured.
3. Results and discussion
3.1. Characterization of homodyne detectors
In the time-domain measurement, which allows the quadrature phase amplitude of each pulse to be independently obtained, two homodyne detectors are required that can distinguish each pulse individually. Therefore, homodyne detectors that consist of a non-inverting amplification circuit with high-speed and high quantum efficiency photodiodes (Fermionics, FD150W) and a high-speed operational amplifier (Texas Instruments, OPA847) were constructed. Figures 2(a) and 2(b) show the results for the characterization of HDA and HDB, respectively. The performance of the homodyne detectors was evaluated in both the frequency domain and in the time domain.
Fig. 2 Evaluation of in-house built homodyne detectors in the time-domain measurement and in the frequency-domain measurement. (a) HDA. (b) HDB.
In the time-domain measurement, 1M samples were obtained from one measurement, which covers a time window of 100 μs and includes 8655 pulses. The sampling rate of the oscilloscope is 1G samples/s and the repetition rate of the laser source is 86.6 MHz; therefore, each pulse includes 115 or 116 samples. To divide recorded samples into independent pulses, an appropriate starting point should be selected for interval integration of the recorded samples. If the wrong starting point is selected, then a single pulse will be separated into two adjacent time windows. An appropriate starting point was selected in the same way as described in [13] and [18]. The correlation coefficient between two adjacent pulses was calculated while varying the starting point. The correlation coefficient takes a minimum at a certain starting point, which should be an appropriate starting point for interval integration of the recorded samples. From this starting point, 115 or 116 points per pulse were integrated and one quadrature phase amplitude was obtained, and this was repeated 8655 times. In Figs. 2(a) and 2(b), the horizontal axes indicate the photocurrent of one input port of a balanced homodyne detector, which is proportional to the input average power of the LO beam, and the vertical axes indicate the variance of voltage obtained from 8655 pulses. Black cross plots are the measured values and the red solid line is a linear fit from 0 mA to 0.7 mA in HDA (from 0 mA to 0.5 mA in HDB). The amplifier noise of the detector was not subtracted. In Fig. 2(a), the variance of the quadrature values deviates from linear at a photocurrent of 0.7 mA or more and in Fig. 2(b) the plots become gently saturated. In order to operate these homodyne detectors in linear regime, the photocurrent was 0.5 mA in HDA and 0.6 mA in HDB at the measurement of entanglement shown later. The insets in the upper left of both figures show the correlation coefficient between the nth and n + mth pulses at a PD current of 0.5 mA (in Fig. 2(a)) or 0.6 mA (in Fig. 2(b)). The correlation coefficients are smaller than 0.08 indicating that almost independent measurements are possible with these homodyne detectors. The insets in the lower right of both figures show the measured LO noise power spectrum with several input powers. The step around 30 MHz in the amplifier noise spectrum is an artifact due to the spectrum analyzer. The peaks at 86 MHz and 172 MHz are due to the harmonic frequency of the laser repetition rate. We can see that these detectors have a usable bandwidth up to 200 MHz.
3.2. Individual measurement of entanglement
To demonstrate EPR entanglement in the time-domain measurement, two spectrum analyzers were used to monitor the strength of the sum and difference correlations of quadrature amplitudes while performing the time-domain measurement. Here, the quadrature amplitudes X̂ and P̂ are defined by the creation and annihilation operators of an electromagnetic field, X̂ = â + ↠and P̂ = i(â − â†).
First, while monitoring the sum of the signals from the two homodyne detectors with a spectrum analyzer, the phase of LO beam on Bob’s side was adjusted by varying the voltage of the piezoelectric transducer (PZT 4), so that the noise variance of the sum of the signals became a minimum. The time-domain data at each channel of the oscilloscopes were then recorded, and the obtained quadrature values of pulse trains were treated as measurements of X̂A and X̂B. The pump beams input into PPLN1 and PPLN2 were then blocked, so that only the LO beams were input into the two homodyne detectors and the shot noise data were recorded. The pump beams were again input into PPLN1 and PPLN2, and then after checking that the LO phase was set to the phase with the minimum noise variance, the phase of the LO beam at Alice’s side shifted by π/2 by varying the voltage of PZT 5 and the phase of the LO beams at Bob’s side also shifted by π/2 by varying the voltage of PZT 4. After shifting both LO phases, the noise variance of the difference of the two homodyne detectors monitoring with the other spectrum analyzer was minimized by fine tuning the voltage of PZT 4. The quadrature values treated as P̂A and P̂B were recorded at this moment. In the end, the pump beams used for the generation of squeezed beam were again blocked, and the pulse trains of the shot noise were recorded again. We consider this series of measurements as one set of measurement and repeated this process ten times.
50 pulses picked up from obtained quadrature phase amplitudes as XA and XB are shown in Fig. 3(a), and as PA and PB are shown in Fig. 3(b). The correlation and anti-correlation in the measured quadrature values of the entangled pulse trains are clearly evident.
Fig. 3 50 quadrature values picked up from the measured entangled pulse train. (a) Quadrature values XA and XB. (b) Quadrature values PA and PB.
Figure 4 shows the correlation diagrams obtained from one series of measurements. We calculated the covariance matrix from the obtained quadrature values XA, XB, PA, and PB from one series of measurements. The calculated covariance matrix from ten measurements is shown in Eq. (1). Each element of the covariance matrix is an average value of ten measurements.
Fig. 4 Correlation diagrams of 8655 quadrature values obtained from individual measurements of the entangled pulse trains. The blue plots indicate the shot noise and the red plots indicate the entanglement. (a) Quadrature values XA and XB. (b) Quadrature values PA and PB. For this measurement, the output pump powers of PPLN, PPLN2 and PPLNLO were 2.0 mW, 2.0 mW, and 5.7 mW, respectively.
The strength of the entanglement correlation was calculated using the covariance matrices obtained from ten measurements. The EPR-Reid criterion [3] was used to confirm the demonstration of EPR paradox and quantum steering:
by evaluating the strength of the entanglement correlation (where denotes the conditional variance of Bob’s measured quadrature values conditioned by Alice’s measurement results [3].). If the strength of the entanglement correlation (i.e. the product of conditional variances of X̂ and P̂) seems to violate Heisenberg’s uncertainty relationship, Alice can predict Bob’s measurement results exceeded the fundamental limit for a separable state. The violation also means a stronger correlation than that is allowed by the local hidden-state model of Bob, and is a demonstration of quantum steering [28]. Thus we can confirm that not only EPR paradox but also steering has been experimentally demonstrated [3].The EPR-Reid value ε2 was calculated from the obtained covariance matrices and the average value for ten measurements was taken. The obtained EPR-Reid value was
which satisfies the EPR-Reid criterion. In this measurement, the scaling factors gX and gP, which can be adjusted to minimize were calculated from covariance matrix for the best linear estimate to minimize the conditional variances [29]: gX = 〈ΔXBΔXA〉/〈Δ2XA〉 = 0.91 ± 0.03, gP = −〈ΔPBΔPA〉/〈Δ2PA〉 = 0.92 ± 0.09.After the time-domain measurements, we also evaluated CV-EPR entanglement in the frequency domain using spectrum analyzer 1. Figure 5 shows the results of the frequency-domain measurement. Unlike the former experiment reported in [27], the amplifier noise was not subtracted from the measurement data. The blue plots indicate the shot noise variance. The red plots represent the noise variance of entangled beams with the LO phase scanned by sweeping the voltage of PZT 4 while the voltage of PZT 5 was kept constant. The voltage of PZT 4 was swept with a 1 Hz triangular wave and the lag between the calculations and the measured plots for the time range in 0.4–0.5 s is caused by turn back of piezo. The yellow plots represent the noise variance of the entangled beams with both LO phases were unchanged. From these yellow plots, the stability of our experimental setup is evident because the LO phase can be kept stable for at least 0.5 s. This stability is due to the setup being surrounded by acrylic plates to protect from air movement.
Fig. 5 Measured noise variance of (a) X and (b) P quadrature values for the entangled beams and the LO beams from the spectrum analyzer. The traces were recorded with a center frequency of 5 MHz, a resolution bandwidth of 1 MHz and a video bandwidth of 100 Hz. The detector amplification noise (i.e. the dark noise) was 9.5 dB below the shot noise of X and 9.3 dB below the shot noise of P. It is conceivable that the difference of these shot noise levels is due to the incomplete symmetry of the RF power combiner.
The EPR-Reid criterion (Eq. (3)) was evaluated without any optimization (i.e. gX = 1, gP = 1), unlike the evaluation of the time-domain measurement. The results achieved 〈Δ2(XA − XB)〉 = −3.19 ± 0.06 dB and 〈Δ2(PA + PB)〉 = −3.17 ± 0.06 dB below the shot noise level. From these results, the estimated the EPR-Reid criterion is
Therefore, the EPR-Reid criterion was also satisfied in the frequency-domain measurement.The measured strengths of the quantum correlation of the entanglement and the squeezing level are limited by imperfect detection efficiencies [30]. The calculated noise variances of the sum and difference quadratures from the following equation are shown as solid lines in Fig. 5.
where , , , and . Here we assume ϕA is a linear function of time. For the calculation of the sum correlation, we set ϕb = π − ϕa and for the difference correlation, ϕb = −ϕa. In this experiment, the total detection efficiency of Alice’s side was ηa = ηqaηmaηoa = 0.68 and that of Bob’s side was ηb = ηqbηmbηob = 0.66, where ηqa(= 0.91) and ηqb(= 0.91) are the detector quantum efficiencies, ηma (square of visibility between entanglement and LO beams of Alice’s side = 0.872 = 0.76) and ηmb (square of visibility between entanglement and LO beams of Bob’s side = 0.862 = 0.74) are the mode-matching efficiencies between the entangled beams and LO beams, and ηoa (= 0.98) and ηob (= 0.98) are the transmittances of the optical components from the HBS to the homodyne detectors. 〈Δ2x̂1〉, 〈Δ2p̂1〉, 〈Δ2x̂2〉, and 〈Δ2p̂2〉 are characterized by the squeezing parameters r1 and r2, and effective optical losses in the interferometer ξ1 and ξ2. The squeezing parameters r1 (= 1.20) and r2 (= 1.20) are obtained from separate measurements of squeezing and antisqueezing levels. ξ1 and ξ2 are obtained from the transmissivity of the waveguide ξw = 0.89, the transmissivity of the optical components of the interferometer ξo1(= 0.99) and ξo2(= 0.99), the mode-matching efficiency between two squeezed beams, which is square of visibility between two squeezed beams ξv (= 0.982 = 0.96). ξ1 = ξwξo1ξv = 0.85, ξ1 = ξwξo1ξv = 0.85 were then calculated. The quantum efficiencies of photo-diodes were measured by a power meter with calibration uncertainty of 2 %. The values of visibility were estimated by measuring the visibility of the correspondence path using the probe beams shown in the right lower part of Fig. 1. For example, the visibility between two squeezed beams was evaluated by measuring the visibility between the output probe beams from PPLN1 and PPLN2.The measured noise levels (the red plots in Fig. 5) agree well with the calculated noise levels using Eq. (5) and measured shot noise and amplifier noise levels (the solid lines). The amplitude of calculated noise levels are slightly smaller than that of the measured levels. This may due to the uncertainty of values of efficiencies used for the calculation. In the entanglement measurements, we performed very careful alignment so that the amplitude of measured noise levels became larger; the optical parameter of the squeezed beams were not exactly the same as that of the probe beams. The mode-matching efficiencies ηma and ηmb should depend on both temporal and spatial mode matching. Nevertheless, for the calculation without considering the temporal mode-matching efficiencies, the experimental results are slightly better than calculations. This suggests that the temporal mode-matching efficiency becomes unity due to pulse shaping. In this work, the parametric amplification gain for shaping the duration of the LO pulse was approximately 15 dB. This parametric gain for the pulse shaping should be sufficient to improve the temporal mode-matching efficiency [26]. The degree of measured entanglement in the present experiment is mainly limited by imperfect spatial mode matching efficiencies between the entangled beams and the LO beams. The improvement of the mode matching efficiency, for example using a spatial light modulator, is an important challenge in the future.
4. Conclusion
We have demonstrated the EPR paradox and steering in a CV regime. Time-domain measurements were performed to obtain single quadrature values per pulse and each measurement was independent. High-speed and broadband homodyne detectors were constructed to distinguish each pulse individually and realize independent measurements. Furthermore, to resolve the difficulty of temporal mode mismatching between the LO pulse and the entangled pulse, the duration of the LO pulse was shortened using OPA. The product of conditional variances calculated from the measured quadrature values of entangled pulses was 0.82 ± 0.09, which satisfied the EPR-Reid criterion for the first time with pulsed light source and waveguides to our knowledge.
Funding
ImPACT Program of the Council of Science, Technology and Innovation (the Cabinet Office, Government of Japan); Cross-ministerial Strategic Innovation Promotion Program (SIP) (Council for Science, Technology and Innovation (CSTI)); JST, PRESTO (JPMJPR17G3); JSPS KAKENHI (19K03703).
Acknowledgments
The authors gratefully acknowledge contributions from Yun Zhang, Naoyuki Hashiyama, Akane Koshio, Akito Ohshiro, and Ryuhi Okubo. Y. E. acknowledges support from the Leading Initiative for Excellent Young Researchers (LEADER).
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