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Abstract
The strength of the quantum correlations of a continuous-variable entangled state is determined by several relative phases in the preparation, transmission, and detection processes of entangled states. In this paper, we report the first experimental and theoretical demonstrations of the precision of relative phases associated with the strength of quadrature correlations. Based on the interrelations of the relative phases, three precisely phase-locking methodologies are established: ultralow RAM control loops for the lengths and relative phases stabilization of the DOPAs, difference DC locking for the relative phase between the two squeezed beams, and DC-AC joint locking for the relative phases in BHDs. The phase-locking loops ensure the total phase noise to be 9.7±0.32/11.1±0.36 mrad. Finally, all the relative phase deviations are controlled to be in the range of −35 to 35 mrad, which enhances the correlations of the amplitude and phase quadratures to −11.1 and −11.3 dB. The entanglement also exhibits a broadband squeezing bandwidth up to 100 MHz. This paves a valuable resource for experimental realization and applications in quantum information and precision measurement.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Continuous variables (CV) quantum entanglement has been widely approved to be a valuable resource in quantum information networks [1–5], quantum computation [6,7], quantum communication [8–14], and quantum precision measurement [15–19]. In these applications, a strong entangled state has been proposed or applied to enhance the performance of the quantum protocols. For example, with $-10$ dB quantum correlations for the amplitude and phase quadratures, the secure distance of CV quantum key distribution (QKD) can be increased from 190 km (with $-3.5$ dB two-mode squeezing in the experiment) to 320 km under an excess noise of 0.1 and a modulation of 100 shot-noise units (SNU) [8]; the fidelity of quantum state transmission with quantum teleportation can be promoted from 0.7 ($-5.2$ dB squeezing) to 0.87 [11]; the non-classical sensitivity of simultaneous measurement of two non-commuting observables in quantum-dense metrology (QDM) can be improved from 6 dB ($-7$ dB squeezing) to 10 dB [16,20]; the signal-to-noise ratio (SNR) in multiplexing quantum communication can be improved from 7.8 dB ($-8$ dB squeezing) to 10 dB beyond the shot noise limit (SNL) [12].
Although the need for strong quantum correlations is enormous, it is a complex and tough task to implement an entanglement of more than 10 dB. To achieve the ultimate goal, squeezing factors, the balance of the beam splitter, channel losses, and multiple relative phases in the generation and detection processes of the entangled state must be globally considered. The first three influence factors had been detailly analyzed in an unbiased entangled state preparation with two single-resonant optical parametric amplifiers (OPAs) [21]. However, the multiple relative phases being associated with the squeezing angles are more critical for establishing a more robust entangled state.
Recently, the maximum quantum correlations had been fabricated with the optical parametric oscillator (OPO) [22] or OPA [21] technology. In preparation of the entangled state, two single-mode squeezed states are coupled on a 50/50 beam splitter (BS), which are also accompanied by at least five electronic servo control loops of the relative phases: the signal and pump beam of the OPO or OPA ${{\Phi }_{1}}$ (or ${{\Phi }_{2}}$), the two squeezed beams on the 50/50 BS ${{\varphi }_{S}}$, and the entangled beam and local oscillator ${{\varphi }_{A}}$ (${{\varphi }_{B}}$) in the balanced homodyne detection (BHD). For all of the relative phases, arbitrary deviation from the ideal phase angle will weaken the quantum correlations as well as introduce an asymmetric noise variance for the two quadrature components [21,23]. Moreover, the impact of phase angle deviation becomes acute as the optical loss being reduced to produce an entangled state of more than 10 dB. Currently, entangled quantum correlations have been experimentally or theoretically demonstrated without considering the influence and interrelations of phase angles. To achieve the requirements for stronger entangled states preparation, the precise and stable control of the phase angles is an essential prerequisite. Stable control of the squeezing angle was already feasible and had been demonstrated by using a coherent control technology with the OPO [24,25] or by reducing the residual amplitude modulation (RAM) [26,27] downstream the phase modulator before and after the OPA [28].
This paper theoretically and experimentally demonstrates the associations between the relative phases and quadrature correlations in the OPA process for the first time. By designing three phase-locking methodologies, the interrelations between the relative phases are destroyed and can be individually controlled to high precision, i.e., in the range of $-35$ to 35 mrad. This paves a way to experimental realization for the $-11.1$ and $-11.3$ dB correlations of the amplitude and phase quadratures, and its squeezing bandwidth can be up to 100 MHz.
2. Theoretical analysis for the interrelations of the phase angles
The entanglement [20,22] can be prepared by mixing two amplitude squeezed states ${{\hat {S}}_{1}}$ and ${{\hat {S}}_{2}}$ with relative phase ${{\varphi }_{S}}\textrm {=}\frac {\pi }{2}$ on a 50/50 beam splitter (BS), which outputs are two entangled modes A ($\hat {a}$) and B ($\hat {b}$)
The correlation variances of the entanglement can be given as the square of ${{i}_{A}}\pm {{i}_{B}}$
The squeezed angle ${{\theta }_{1,2}}$ is associated with the relative phase between the signal and pump beams ${{\Phi }_{1,2}}$ [28–30]. ${{\Phi }_{1,2}}\textrm {=}\pi$ corresponds to the de-amplification of the OPA. Here, ${{V}_{10}}\left ( X \right )$, ${{V}_{10}}\left ( Y \right )$ and ${{V}_{20}}\left ( X \right )$, ${{V}_{20}}\left ( Y \right )$ are the initial variances of the amplitude and phase quadratures of the two squeezed states [28,29,31–33]
Here $\nu$ is the full-width at half-maximum (FWHM) linewidth of the OPA, $f$ is the analysis frequency, and $\eta$ is the total detection efficiency. ${{P}_{1,2}}$ is the pump power, and ${{P}_{\textrm {th1,2}}}$ is the threshold pump power of the OPA. ${{\theta }_{\textrm {rms}}}$ is fluctuation in the squeezing angle and is referred to the root-mean-square (rms) of the total phase noise, which makes the amplified noise in orthogonal quadrature couple into the observed squeezed quadrature. Losses and phase noises finally codetermine the measured squeezing level.
Consequently, five phase angles (${{\Phi }_{1}}$, ${{\Phi }_{2}}$, ${{\varphi }_{S}}$, ${{\varphi }_{A}}$, ${{\varphi }_{B}}$) determine the final correlations measured by BHDs, and the upstream phase deviations are also accumulated in the downstream as showing in Fig. 1(a). To quantify the accumulated phase deviation effect, we redefine the relative phases as ${{\Phi }_{1,2}}\textrm {=}{{\Phi }_{10,20}}\textrm {+}\Delta {{\Phi }_{1,2}}$, ${{\varphi }_{S}}\textrm {=}{{\varphi }_{S0}}\textrm {+}\Delta {{\varphi }_{S}}$, and ${{\varphi }_{A,B}}\textrm {=}{{\varphi }_{A0,B0}}+\Delta {{\varphi }_{A,B}}$. As showing in Fig. 1(b), $\Delta {{\Phi }_{1,2}}$ results in a rotation of the squeezed angle (${{\theta }_{1,2}}\textrm {=}{{\theta }_{10,20}}\textrm {+}\Delta {{\theta }_{1,2}}$) during the preparation of squeezed state, meanwhile leads to an increased output power ${{P}_{\textrm {out1}}}$ of the OPA (operating on de-amplification), and the power ratio $R$ of ${{P}_{\textrm {out1}}}$ to the initial one ${{P}_{\textrm {out0}}}$ can be deduced as [34]
Fig. 1. Building blocks of the relative phase angles and the principle of phase deviation during entanglement generation: relationship of the relative phases for (a) the squeezed state, entangled state preparation and measurement, (b) squeezed state preparation, (c) entanglement preparation, (d) BHD measurement.
In this case, if the error signal for ${{\varphi }_{S}}$ locking is extracted from the amplitude of the interference of the two squeezed beams, $\Delta {{\Phi }_{1,2}}$ induced power variation will lead to a phase deviation to ${{\varphi }_{S}}$ (Fig. 1(c)), because of the zero point of the error signal deviates from the middle of the interference fringe. Combining the equation of $R$ and the quantitative equation of the interference intensity, ${{\varphi }_{S}}$ can be modified as
If the initial power of the two squeezed beams is normalized to 1 (${{I}_{1}}\textrm {=}{{I}_{2}}$=1), then ${{\varphi }_{S}}$ is simplified to ${{\varphi }_{S}}=\textrm {arccos}\frac {1-{{R}^{2}}}{2R}$. With this methodology, the deviation of ${{\varphi }_{S}}$ related to $\Delta {{\Phi }_{1,2}}$ can be quantitatively demonstrated as describing in Fig. 2(a). For example, a 20 mrad phase deviation of ${{\Phi }_{1}}$ results in a 13 mrad deviation of ${{\varphi }_{S}}$, i.e., ${{\Phi }_{1}}\textrm {=}\pi \pm 0.02$ rad, ${{\varphi }_{S}}\textrm {=}\frac {\pi }{2}\textrm {+}0.013$ rad. Furthermore, as showing in Fig. 1(d), $\Delta {{\varphi }_{A,B}}$ will also introduce a squeezing angle rotation between the measurement base of the BHDs and entanglement quadratures.
Fig. 2. Demonstration that in relating the phase deviation to entanglement noise power is necessary for high degree entangled states generation. (a) $\Delta {{\varphi }_{S}}$ versus $\Delta {{\Phi }_{1,2}}$, (b) entanglement degree versus $\Delta {{\Phi }_{1,2}}$ with (blue line) and without (blue dotted line) the influence of $\Delta {{\Phi }_{1,2}}$ on $\Delta {{\varphi }_{S}}$, $\Delta {{\varphi }_{S}}$ (red line), $\Delta {{\varphi }_{A,B}}$ (green line), (c) $\Delta {{\varphi }_{A,B}}$ versus $\Delta {{\Phi }_{1,2}}$ for different entanglement degree with the influence of $\Delta {{\Phi }_{1,2}}$ on $\Delta {{\varphi }_{S}}$, (d) the optimal value of $\Delta {{\varphi }_{A,B}}$ (black prismatic) and maximum entanglement available (red circle) versus $\Delta {{\Phi }_{1,2}}$ with the influence of $\Delta {{\Phi }_{1,2}}$ on $\Delta {{\varphi }_{S}}$. The squeezing degree used to calculate the entanglement is $-12$ dB.
We comprehensively illustrate the influence of $\Delta {{\Phi }_{1,2}}$, $\Delta {{\varphi }_{S}}$, and $\Delta {{\varphi }_{A,B}}$ on entanglement correlations in Fig. 2(b)-(d) by simultaneous equations (5)-(10). Figure 2(b) shows the entanglement correlations noise power individually and independently relating to the deviations of the relative phases. It is clear from the figure that all the deviations of the relative phases excite phase angle rotation and contribute to the noise power growth, but we note that the dominant contributor to the magnitude of the noise power is the phase deviation of ${{\varphi }_{S}}$ (red line in Fig. 2(b)), while $\Delta {{\Phi }_{1,2}}$ will exacerbate this contribution in practical entangled states preparation (blue line in Fig. 2(b)).
Whereafter assuming a precise locking of ${{\varphi }_{S}}$ as a prerequisite, $\Delta {{\Phi }_{1,2}}$ and $\Delta {{\varphi }_{A,B}}$ are considered to demonstrate the relation of the relative phase deviation and entanglement noise power. As showing in Fig. 2(c), the deviation range of the relative phase should be more and more narrow with the degree of entanglement increasing. For example, both $\Delta {{\Phi }_{1,2}}$ and $\Delta {{\varphi }_{A,B}}$ should be precisely controlled to 0.1 mrad for 12 dB entanglement (red line in Fig. 2(c)), while relax to $+30$/$-20$ mrad for the 11.7 dB one. From Fig. 2(c), it also can be found an optimum $\Delta {{\varphi }_{A,B}}$ for a certain $\Delta {{\Phi }_{1,2}}$, and the results are shown in Fig. 2(d). Clearly, with the increasing of $\Delta {{\Phi }_{1,2}}$, the available maximum degree of the entanglement gradually decreases. Therefore, $\Delta {{\Phi }_{1,2}}$ and $\Delta {{\varphi }_{S}}$ are more harmful for high degree entanglement generation, which should involve stepping up more effort to suppress.
3. Experimental setup and result
The schematic of our experimental setup is illustrated in Fig. 3. The laser source is a continuous-wave single-frequency fiber laser with an output power of 2 W at 1550 nm wavelength (E15, $NKT$ $Photonics$). The mode cleaners (MCs) are employed to ensure the quadrature noises of the fundamental and second harmonic waves meet SNL above 5 MHz [30]. Here one of the 1550 nm MCs is omitted in Fig. 3. The fundamental wave serves as the seed beams of the degenerate OPAs (DOPAs) and local oscillators (LO, 10 mW) of the BHDs. Meanwhile, it is also applied to produce an up-conversion wave of 105 mW by the second harmonic generator (SHG), which is used as the pump laser of the DOPAs. Both of the DOPAs are semi-monolithic double-resonant cavities and their parameters are the same as Ref. [30], in which the pump beams are modulated with 42.5 MHz and used for cavity length sensing and locking. The outputs of the DOPAs are coupled on a 50/50 BS to prepare the entangled beams. The SHG has similar parameters with DOPA, except for a concave mirror with a transmissivity of 12$\%\pm$1.5$\%$ for 1550 nm and high transmission for 775 nm. The resonant conditions of the DOPAs and MCs are maintained by Pound-Drever-Hall (PDH) technique [35].
Fig. 3. Schematic of the entangled light source and feedback control loops for relative phase. MC, mode cleaner; SHG, second harmonic generator; DOPA, degenerate optical parametric amplifier; DBS, dichroic beam splitter; BS: beam splitter; EOM, electrooptical modulator; PD, photodetector; BHD, balanced homodyne detector.
The major concern of a robust entangled state in this work is the precise control of the relative phases as stating in section 2. To satisfy the requirement of ultralow phase deviation, we have developed an ultralow RAM control loop in previous researches [27,28,36], and upgrade the phase-locking technologies in this work are summarized in Table 1. The ultralow RAM control loops make sure the stable and precise locking of the cavities and ${{\Phi }_{1,2}}$. With this method, the squeezing phase noise could be controlled to 1.4$\pm$0.26 mrad in a double-resonant OPA process [30], and the zero-baseline offset of the error signals was reduced to ${+70}/{-50}\;$ ppm, about ${1}/{50}\;$ without the ultralow RAM design.
Table 1. Summary of the locking technique of the relative phase. See main text for further discussion.
Subsequently, 1$\%$ powers of the two entangled modes are leaked from the two BSs in the optical paths of the entangled beams. The two dc interference signals read by PD-${{\varphi }_{S}}$ are mutually subtracted to produce an error signal centered around zero. This zero-crossing can be directly used to produce an error signal for ${\pi }/{2}\;$ phase locking between the two squeezed beams [14]. It has an advantage to cancel the power fluctuations from the outputs of the DOPAs, and breaks the relevance between $R$ and ${{\varphi }_{S}}$.
Finally, to measure the maximum correlations of the entanglement quadratures, i.e., eliminate the deviation of the relative phase ${{\varphi }_{A,B}}$ as much as possible, we utilize a DC and AC joint locking technology for arbitrary phase control. The 47 MHz modulated sidebands on the seed beams are demodulated to produce the AC signals $-\sin \theta$ for BHD1 and BHD2, which can be used for 0 phase locking. The demodulated AC signal adds with the DC interference signal $\cos \theta$ (used for ${\pi }/{2}\;$ phase locking) exporting from the same BHD, then forms the final error signal of ${{\varphi }_{A,B}}$ as ${{\varepsilon }_{{{\varphi }_{A,B}}}}=-\sin \left ( \theta -{{\varphi }_{A,B}} \right )=k\cos \theta -\sin \theta$, where $k=\tan {{\varphi }_{A,B}}$. Therefore, the error signal of ${{\varphi }_{A,B}}$ can be calibrated by modifying the coefficient $k$, which can be expediently adjusted by the amplitude of the DC part. In our experiment, an attenuator acts as an amplitude adjuster for the DC signal, which is equivalent to modify the value of $k$. It can be found that ${{\varphi }_{A,B}}$ can be locked to 0$\sim {\pi }/{2}\;$ by changing the value of $k$ from 0 to $\infty$. This joint locking method is superior to the individual AC or DC locking technology, because it can conveniently calibrate the relative phases to the exact value by adjusting $k$ and promote us to read out the optimum quadrature correlations.
By optimizing the feedback control loops of the relative phases with the above methodologies, an entanglement, with quadrature amplitude and phase correlations of $-10.9\pm$0.2 dB and $-11.1\pm$0.2 dB at 5 MHz (Agilent N9020A), was directly observed by joint measurements of BHD1 and BHD2. Figure 4 presents the measured results at the pump power of 13.5 mW, and the detailed experimental parameters are also given in Table 3. All traces are normalized to the SNL corresponding to a 10 mW LO. Trace (I) corresponds to the SNL with entangled light blocked. Trace (II) and (III) are the variances of the quadrature amplitude and phase correlations. The electronic noise (Trace (IV)) of the BHD is 21 dB below SNL. Figure 4(a) shows the directly observed quadrature amplitude and phase correlations at 5 MHz with an inseparability criterion of the correlations of $\sqrt {V\left ( {{{\hat {X}}}_{\textrm {a}}}+{{{\hat {X}}}_{\textrm {b}}} \right )V\left ( {{{\hat {Y}}}_{\textrm {a}}}-{{{\hat {Y}}}_{\textrm {b}}} \right )}\textrm {=0}\textrm {.079}$. By subtracting the contribution of the electronic noise, the amplitude and phase correlations are amended to $-11.1$ dB and $-11.3$ dB. Figure 4(b) presents a broadband quantum noise correlation in the frequency range from 5 to 100 MHz. The quadrature amplitude and phase correlations are $-2.4\pm$0.2 dB and $-2.6\pm$0.2 dB at 100 MHz with an inseparability criterion of the correlations of $\sqrt {V\left ( {{{\hat {X}}}_{\textrm {a}}}+{{{\hat {X}}}_{\textrm {b}}} \right )V\left ( {{{\hat {Y}}}_{\textrm {a}}}-{{{\hat {Y}}}_{\textrm {b}}} \right )}\textrm {=0}\textrm {.56}$.
Fig. 4. Variances of the sum $V\left ( {{{\hat {X}}}_{\textrm {a}}}+{{{\hat {X}}}_{\textrm {b}}} \right )$ of the amplitude and difference $V\left ( {{{\hat {Y}}}_{\textrm {a}}}-{{{\hat {Y}}}_{\textrm {b}}} \right )$ of the phase quadratures. (a) Variances at analysis frequency of 5 MHz and (b) between 5 MHz and 100 MHz with a resolution bandwidth (RBW) of 300 kHz and a video bandwidth (VBW) of 200 Hz.
The data in Fig. 4(b) are fitted with the model of Eq. (5) (the black and blue dotted lines) [37], and it gains a total optical loss of 6.5$\%\pm$0.14$\%$, and a phase noise of 9.7$\pm$0.32 or 11.1$\pm$0.36 mrad for phase or amplitude quadrature, respectively. Table 2 lists all known sources of loss and phase noise in the entanglement preparation [31–33,38]. With one of the OPAs, the total phase noise for one-mode squeezing had been confirmed to be 1.4$\pm$0.26 mrad [30]. For phase noise measurement in DC-AC joint locking of the BHDs, the seed beam exits the OPA with the temperature of PPKTP far away from the phase matching condition, and the phase noise was determined to be 1.7$\pm$0.2 mrad. The phase noise between the two squeezed beams limits the noise to be 9.9$\pm$0.3 mrad, which is attributed to the lower signal-to-noise ratio for weak signal extraction in the photodetector. Finally, the total loss and phase noise codetermine the measured squeezing level. The impact of phase noise becomes more acute as the losses are reduced (Fig. 1 of [32]). Compared with the results of the fitted curves in Fig. 4(b), it concludes that the maximum squeezing can reach to -11.6 dB with 0 phase noise (the pink dotted line in Fig. 4(b)). Therefore, loss is the dominating limitation for the enhancement of the squeezing level, and should be furtherly reduced.
Table 2. Loss and Phase Noise Budget for our entangled light source.
With the results in Fig. 4(a) and Table 3, the phase-locking accuracy is calculated as showing in Fig. 5. When extracts the error signal of ${{\varphi }_{S}}$ from one of the entangled modes, our phase-locking technology controls $\Delta {{\varphi }_{A,B}}$ in the range of $-35$ to 80 mrad (Fig. 5(a)). When the difference of the two entangled modes is applied to lock the ${{\varphi }_{S}}$, it can be found that $\Delta {{\varphi }_{A,B}}$ is reduced to the range of $-35$ to 35 mrad (Fig. 5(b)), due to the elimination of the influence of $\Delta {{\Phi }_{1,2}}$ to $\Delta {{\varphi }_{S}}$. In the two cases, $\Delta {{\Phi }_{1,2}}$ always keeps in the locking range of -35 to 35 mrad. Therefore, the precise control of the relative phases in our entanglement light source is established.
Fig. 5. Simulated results of the phase deviation $\Delta {{\Phi }_{1,2}}$ and $\Delta {{\varphi }_{A,B}}$ of the $-11$ dB entanglement based on the experimental parameters in Table 3 and theoretical models in section 2. Extracting the error signal of ${{\varphi }_{S}}$ from (a) one of the entangled modes or (b) the difference of the two entangled modes.
Table 3. The experimental parameters used for the calculation of Fig. 5.
4. Conclusion
The experimental and theoretical analyses of the influence of relative phases for enhancing the correlations of entanglement were first demonstrated. The experimental results well agree with the theoretical ones, that take into account the internal relations of the relative phases (${{\Phi }_{1}}$, ${{\Phi }_{2}}$, ${{\varphi }_{S}}$, ${{\varphi }_{A}}$ and ${{\varphi }_{B}}$) during the preparation, transmission, and detection of entangled states. To ensure a high degree of entanglement, three precisely phase-locking technologies are established: ultralow RAM control loops for the lengths and relative phases of the DOPAs, difference DC locking for the relative phase between the two squeezed beams, and DC-AC joint locking for the relative phases in BHDs. The phase-locking loops ensure the total phase noise to be 9.7$\pm$0.32/11.1$\pm$0.36 mrad. Attributing to the accuracy control of the relative phases to the range of $-35$ to 35 mrad, the amplitude and phase correlations of the entangled state are enhanced to $-11.1$ and $-11.3$ dB, and also owns a broadband squeezing with bandwidth more than 100 MHz. We envision that the valuable entanglement resource will widen the applications of quantum information networks, quantum computation, quantum communication, and quantum precision measurement.
Funding
National Natural Science Foundation of China (62027821, 11654002, 11874250, 11804207, 11804206, 62035015); National Key Research and Development Program of China (2020YFC2200402); Key Research and Development Projects of Shanxi Province (201903D111001); Program for Outstanding Innovative Teams of Higher Learning Institutions of Shanxi; Program for Sanjin Scholar of Shanxi Province; Fund for Shanxi “1331 Project” Key Subjects Construction.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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