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Abstract
Long-term stable entanglement is a crucial aspect in the implementation of reliable quantum information processes. However, long-term continuous variable entanglement generation, especially in type-II non-degenerate optical parametric amplifier, has yet to be reported on. Here, we derive the relationship between entanglement and temperature fluctuations in the crystal of a type-II non-degenerate optical parametric amplifier, and propose a novel method for long-term stable entanglement generation by locking the temperature of the crystal. In the experiment, we obtain a 5.4 dB entanglement lasting two hours. The method holds promise in the generation of a truly usable above 10 dB entanglement and brings us closer to continuous-variable quantum information processing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Continuous-variable (CV) entanglement is attracting much attention because of its apparent usefulness in quantum information processing, such as quantum computation [1,2], quantum key distribution [3,4], quantum teleportation [5,6], quantum metrology [7–9].
Degenerate and non-degenerate optical parametric amplifiers (DOPAs and NOPAs), which employ type-I and type-II nonlinear crystals are two kinds of optical devices for successfully generating CV entanglement of optical field, to date, the maximum CV entanglement are around 10 dB and 8.4 dB, respectively [10,11].
Currently, the gravitational wave detectors GEO 600 is undergoing an upgrade. Permanent deployment has imposed further enhancements of the GEO 600 squeezed light laser. A (preferably high) squeezing factor must be provided continuously on timescales of hours to days [12–15]. Furthermore, long-term data acquisitions and information processing are required in practical quantum information protocols based on CV entanglement. Generating a long-term stable quantum entanglement is crucial if a reliable quantum information processing is to be implemented [16–18].
Using type-I DOPA to generate long-term stable squeezed light has been demonstrated in the laboratory [19–23]. However, there has been no report concerning the generation of long-term entanglement, especially in type-II NOPA. The generation of long-term stable squeezed light is used with a singly resonant DOPA, which has a broad phase matching condition and a wide full-width at half-maximum (FWHM) temperature of the parametric gain [24]. For type-I DOPA , the subharmonic fields are degenerate, so the change in crystal temperature has the same effect on the subharmonic fields. Compared with type-I DOPA, type-II NOPA is a compact source of CV entanglement, however, the subharmonic fields are non-degenerate. Within the type-II NOPA, the detuning of the cavity is determined by the difference in the refractive indices and constant of thermal expansion between the signal and idler fields when the crystal temperature changes. Therefore, type-II NOPA is highly sensitive to fluctuations in crystal temperature [25,26].
In this paper, we derive the entanglement as a function of the fluctuation in crystal temperature of a type-II NOPA and show that entanglement is significantly sensitive to the crystal temperature, thus motivating its strict control. The photothermal effect can be extremely fast compared when using temperature actuation external to the crystal [27–29]. McKenzie et al used the photothermal effect to achieve nonlinear phase matching locking in type-I DOPA [24], Pysher et al used the photothermal effect to phaselocking the signal fields emitted above threshold in a type-II NOPO [30]. We propose therefore a method, also exploiting this effect, to feedback the locking error signal of the crystal temperature by modulating the amplitude of the pump field. When passed through the crystal, the pump field has a frequency shift and a polarization that does not participate in the nonlinear process. Thus, a fast and accurate temperature actuation is achieved without adverse effect on the nonlinear interaction. Long-term stable entanglement of 5.4 dB lasting up to 2 hours is achieved via crystal temperature locking. This paper reports the first experimental generation of long-term stable CV entanglement.
2. Theory
The type-II NOPA device is pumped by a harmonic field and injected with two subharmonic fields with degenerate frequencies but orthogonal polarizations. The equations of motion for the intracavity fields with the rotating wave approximation are as follows [31,32],
Whereas the system in [31,32] is an optical parametric oscillator operating above the oscillation threshold and for which the two subharmonic fields have equal detuning, our system differs in that the doubly-resonant NOPA operates below the threshold. Hence, different detunings for the signal and idler modes may appear. The cavity length is locked by the signal field in the experiment and there is no detuning for the signal field. In consequence, we only consider detuning for the idler field.
Field linearization allows for an analytic solution of the noise properties of the quadratures. When NOPA is operated in the state of de-amplification, the correlation noise spectrum as a function of cavity detuning is expressed as
The detuning of the cavity $\Delta$ arising from changes in the optical path length $\Delta {L}$ of the signal and idler fields in the crystal is given by [24,33],
where $\upsilon _{FSR}$ denote the free spectral range (FSR) of the cavity, and $\lambda$ is the wavelength. The change in $\Delta {L}$ arising from two mechanisms, thermal expansion and refractive index change, in the following form [34],
Fig. 1. Entanglement as a function of crystal temperature fluctuation. The solid curve, dashed curve and dash-dotted curve indicate that the entanglement is 10 dB, 5.4 dB and 3 dB respectively when there is no crystal temperature fluctuation. The parameter values used are: $\lambda = 1080 nm$, $n = 1.7$, ${L_c} = 10 mm$, $\gamma = 6.0 \times {10^7}{s^{ - 1}}$, $\frac {{d{n_s}}}{{dT}} = 1.6 \times {10^{ - 5}}{K^{ - 1}}$, $\frac {{d{n_i}}}{{dT}} = 1.3 \times {10^{ - 5}}{K^{ - 1}}$, ${b_s} = 0.6 \times {10^{ - 6}}{K^{ - 1}}$, ${b_i} = 9 \times {10^{ - 6}}{K^{ - 1}}$, and $\sigma = 0.8$
In many experiments, the crystal temperature is maintained via an off-the-shelf temperature controller with a temperature sensor. Traditional temperature controllers have inherent disadvantages. First, these controllers, which have accuracies of approximately 1 mK, from previous analysis, cannot meet our experimental requirements (Fig. 1). Second, the external sensor reads out the external crystal temperature, rather than the temperature of the optical path through the crystal where the nonlinear interaction occurs [24].
As a temperature actuator for this nonlinear interaction region, the photothermal effect is extremely fast compared with the effect from temperature actuation applied externally to the crystal.
3. Experimental setup
Regarding the experimental setup (Fig. 2), a continuous-wave all-solid-state laser source emits both infrared at 1080 nm and green light at 540 nm. A part of the 1080 nm light is injected into the NOPA as a seed beam. The signal field (s-polarization) transmitted from the NOPA is demodulated to produce an error signal for the cavity length using the Pound-Drever-Hall technique [35,36]. The remainder is used as a local oscillator for homodyne detection. The 540 nm light with p-polarization is injected into the NOPA as the pump field. By employing the temperature feedback loop (TFL), a part of 540 nm light, referred to as the control field, is used to lock the crystal temperature.
Fig. 2. Schematic of the experimental setup. RF: Radio frequency source; PM: Phase modulator; PID: Proportional integral derivative controller; VCO: Voltage controlled oscillator; AOM: Acousto-optic modulator; PD: Photoelectric detector; TFL: temperature feedback loop; SA: spectrum analyzer; BHD: balanced homodyne detector; HWP: Half wave plate.
The NOPA consists of a doubly resonant standing wave cavity, having two mirrors (M1 and M2) of the same radius of curvature of 30 mm. The input mirror M1 is highly reflective (R>99.95 %) at 1080 nm and 540 nm, the output mirror M2 has a transmission of 5.4 % at 1080 nm and high transmittance at 540 nm. The measured finesse of the NOPA is 103, that is, the total extra loss of the NOPA is 0.6 %. An $\alpha$-cut type-II KTP crystal is housed in a Peltier-driven oven made of copper. The temperature of the crystal oven is actively controlled at the temperature for phase matching (about 61 °C), using a temperature controller (YG-2S-RA) having approximately 1 mK accuracy.
The TFL consists of a proportional integral derivative (PID) controller, a home-made voltage-controlled oscillator (VCO) and an acousto-optic modulator (AOM). The control field is frequency shifted (100 MHz) by the AOM to ensure that it does not participate in the nonlinear process. The change in phase of the idler field is more sensitive to the crystal temperature than the signal field. Thus, the locking error signal of the crystal temperature is derived from the reflected idler field (p-polarization) and is fed into the TFL. In our setup, the NOPO threshold is 400 mW, with a pump power of 280 mW and an injected seed beam of 5 mW. The maximum power of the control field is 150 mW.
4. Derivation of the crystal temperature locking error signal
Near the phase matching temperature, the cavity resonance conditions can be monitored using standard cavity readout techniques, and this readout is used to produce a locking error signal of the crystal temperature. The amplitude transmissivity $\Gamma (\Delta )$ and reflectivity $\Upsilon (\Delta )$ of the idler field are defined by two parameters [37];
Fig. 3. Transmitted power of the idler field and crystal temperature locking error signal as a function of temperature fluctuation obtained from theory and experiments: (a) and (b) transmission of the idler field, (c) and (d) associated crystal temperature locking error signal. Parameters values used are: $t_1=0.1 \%$, $t_2=5.4 \%$, absorption coefficient of KTP: 0.8 %/cm, thermal conductivity of KTP: 0.13 W/cm*K, spot size of control field: 40 $\mu m$, spot size of idler field: 56 $\mu m$. For other parameter settings, see Fig. 1.
5. Results
With the pump field injected into the NOPA, the relative phase of pump field and the seed field were swept with the PZT at 15 Hz. The parametric gain behavior was presented by measuring the intensity of the transmitted signal field from the NOPA normalized by the signal intensity without parametric gain (Fig. 4). A gain above and below 1 indicates parametric amplification and de-amplification, respectively. Without crystal temperature locking, the parametric gain drifted significantly and decreased from 12 to 7 during the unlocking period.
Fig. 4. Parametric gain behavior as a function of time, normalized by the resonant power without parametric gain.
Next, the relative phase between the pump field and the seed field was locked in the de-amplification state, and the generated entanglement was measured using dual balanced homodyne detectors. From a comparison of the stability performance for entanglement with and without crystal temperature locking over a 10 minute duration (Fig. 5), we find that without crystal temperature locking, the amplitude-sum squeezing and phase-difference squeezing appear with a significant drift, whereas, with crystal temperature locking, they are both optimized. A significantly enhanced stability is present with crystal temperature locking.
Fig. 5. Comparison of stability performance for entanglement without and with the crystal temperature locking. (a) amplitude-sum squeezing. (b) phase-difference squeezing. The measurement parameters of spectrum analyzer: RBW: 100 kHz, VBW: 100 Hz. Analysis frequency: 3 MHz.
An assessment of the temperature stability was made by comparing the entanglement with and without crystal temperature locking. For the latter, the degree of entanglement produced a mean fluctuation of 1.53 dB, that is, the mean temperature fluctuation was 1.73 mK. For the former, the degree of entanglement yielded a mean fluctuation of 0.12 dB, which translates to a mean temperature fluctuation of 0.40 mK. The results were calculated at one-minute intervals using the data for entanglement versus temperature fluctuation (Fig. 1).
We next measured both the magnitude and phase of the transfer function of the TFL (Fig. 6). For larger value of the magnitude, a stronger noise suppression at the corresponding frequency. As seen in Fig. 5, with crystal temperature locking, the low frequency noise in the entanglement fluctuation is well suppressed. The control bandwidth of the TFL is approximately 17 Hz, which is the zero-point of the magnitude of the transfer function. The control bandwidth of the TFL is limited by the photothermal response bandwidth [24], this response of the crystal being approximately 200 Hz in our system. Normally, the electronic components in the TFL, such as the AOM, PD, and mixer contribute to the bandwidths each being above 1 MHz.
Fig. 6. Measured transfer function of the TFL. (a) Magnitude and (b) Phase of the transfer function.
To evaluate the effect of crystal temperature locking further, the long-term stability of the entanglement was recorded continuously for two hours. The measured amplitude-sum squeezing and phase-difference squeezing (Fig. 7) are 5.50 $\pm$ 0.09 dB and 5.40 $\pm$ 0.13 dB, respectively. The test run was terminated by losing lock of the mode cleaner in the 1080 nm light path. The mode cleaner is used to improve mode quality and stability of the injected signal of the NOPA and the local oscillator beams.
Fig. 7. Long-term entanglement recorded continuously for 2 hours. (a) amplitude-sum squeezing, (b) phase-difference squeezing. The measurement parameters of spectrum analyzer: RBW: 100 kHz, VBW: 100 Hz. Analysis frequency: 3 MHz.
The measured entanglement is affected by various inefficiencies in the experiment. The total detection efficiency is ${\eta _{total}} = {\eta _{prop}}{\eta _{phot}}\eta _{hd}^2 = 0.89 \pm 0.02$, where ${\eta _{prop}} = 0.97 \pm 0.02$, ${\eta _{phot}} = 0.96 \pm 0.02$, and ${\eta _{hd}} = 0.98 \pm 0.01$ are the measured propagation efficiency, the quantum efficiency of the photodiode, and the spatial overlap efficiency in the balanced homodyne detectors, respectively.
In future work, we look to decrease the intracavity loss of the NOPA and increase transmissivity of the output coupler of the NOPA, which promises to achieve a 10 dB entanglement, and as inferred by Fig. 1, the fluctuation of the degree of entanglement could be smaller than 0.4 dB using crystal temperature locking, when the mean temperature fluctuation is 0.40 mK. Crystal temperature locking can effectively improve the long-term stability performance of the entanglement generation system. Stable entanglement sources then become suitable for practical applications in quantum information science and technology.
6. Conclusion
We presented a detailed study of entanglement and crystal temperature fluctuation in type-II NOPA, and introduced a novel method employing crystal temperature locking to generate long-term entanglement. An experimental demonstration with crystal temperature locking was performed in which we obtained a substantial improvement in entanglement stability. Faster and more accurate temperature control was accomplished compared with current temperature controllers. A long-term stable entanglement source at 5.4 dB lasting up to 2 hours was achieved.
Long-term entanglement is a key resource in establishing quantum information processes that need long-term date acquisition and processing, such as quantum communication based on post-selection [40], gate sequencing for CV one-way quantum computation [2] and LIGO interferometry based on CV entanglement [9]. The method holds promise in applications to the generation of truly usable entanglement of above 10 dB, paving the way towards CV quantum information processing.
Funding
Ministry of Science and Technology of the People's Republic of China (2016Y-FA0301404); National Natural Science Foundation of China (11674205, 91536222); Program for OIT of Shanxi; Shanxi 1331 Project.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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