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Acta Physica Sinica

Coherent state phase estimation based on digital nonlinear phase-locked loop

Xu Han, Shuxin Chen, Hao Wu, Kun Chen, Lei Hong

Published Jan 1, 2019• 2 citations
Publisher

Abstract

The navigation mechanism of obtaining phase parameters based on quantum theory can break through the limitation of classical physical limit to navigation accuracy. In order to achieve the accurate estimation of the coherent state phase, it is usually assumed that the local oscillator phase must be orthogonal to the coherent state phase in the method of quantum homodyne detection. However, the coherent state phase is unknown and the hypothesis cannot be guaranteed to be correct in practice. In this paper we design a nonlinear phase-locked loop to solve the problem. Firstly, in order to obtain the Wigner distribution for a coherent state, we start with the Wigner distribution for the vacuum state and analyze the noise characteristics according to Wigner distribution of coherent state, then the output of homodyne detection is derived. Secondly, in order to avoid introducing errors in theory, caused by linearization and cope with the limiting requirement between local oscillator phase and coherent state phase in the phase tracking of coherent state, we design an orthogonal simplex cubature Kalman filter (OSCKF) algorithm to achieve the function of the nonlinear phase-locked loop. The algorithm converges by updating the state of the local oscillator phase multiple times, and then, the accurate coherent phase is obtained. Finally, according to the design of the phase-locked loop, we observe the data of homodyne detection and then verify the correctness of the OSCKF algorithm. The simulation results show that the OSCKF can converge to the real phase after observing 200 sampled data and the accuracy is higher than extended Kalmn filter (EKF) and cubature Kalman filter (CKF), and the real phase can be obtained under different local oscillator phase. In conclusion, the nonlinear phase-locked loop based on OSCKF algorithm breaks the limitation of traditional way in which the initial local oscillator phase is required to be orthogonal to the coherent state phase, and effectively avoid the linearization error and improve the anti-nonlinear ability. It is very significant in theory and application .

DOI
10.7498/aps.68.20181602

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The navigation mechanism of obtaining phase parameters based on quantum theory can break through the limitation of classical physical limit to navigation accuracy. In order to achieve the accurate estimation of the coherent state phase, it is usually assumed that the local oscillator phase must be orthogonal to the coherent state phase in the method of quantum homodyne detection. However, the coherent state phase is unknown and the hypothesis cannot be guaranteed to be correct in practice. In this paper we design a nonlinear phase-locked loop to solve the problem. Firstly, in order to obtain the Wigner distribution for a coherent state, we start with the Wigner distribution for the vacuum state and analyze the noise characteristics according to Wigner distribution of coherent state, then the output of homodyne detection is derived. Secondly, in order to avoid introducing errors in theory, caused by linearization and cope with the limiting requirement between local oscillator phase and coherent state phase in the phase tracking of coherent state, we design an orthogonal simplex cubature Kalman filter (OSCKF) algorithm to achieve the function of the nonlinear phase-locked loop. The algorithm converges by updating the state of the local oscillator phase multiple times, and then, the accurate coherent phase is obtained. Finally, according to the design of the phase-locked loop, we observe the data of homodyne detection and then verify the correctness of the OSCKF algorithm. The simulation results show that the OSCKF can converge to the real phase after observing 200 sampled data and the accuracy is higher than extended Kalmn filter (EKF) and cubature Kalman filter (CKF), and the real phase can be obtained under different local oscillator phase. In conclusion, the nonlinear phase-locked loop based on OSCKF algorithm breaks the limitation of traditional way in which the initial local oscillator phase is required to be orthogonal to the coherent state phase, and effectively avoid the linearization error and improve the anti-nonlinear ability. It is very significant in theory and application .
Abstract Optical phase tracking is an important technique for use in high-precision measurement applications, including optical frequency metrology and ground- or space-based gravitational wave observation, and coherent optical communications. When measuring fast-varying real-time signals, the response time limitations of the measurement system’s phase-locked loop cause the best operating point to be mismatched, and the measurement then becomes nonlinear. To make these measurements possible, this work proposes a time delay loop that theoretically enables optimal homodyne detection. When the time delay loop is combined with an extended Kalman filter, the estimated measurement accuracy is improved by 2.4 dB when tracking a fast-varying random signal with a velocity of 107 rad/s. This phase estimation improvement also increases as the interference angle deviates further from the optimal measurement point. The proposed method shows potential for use in real-time sensing and measurement applications.
The continuous-variable quantum key distribution with a local local-oscillator is extremely sensitive to the phase compensation noise caused by the inaccurate reference phase estimation. In order to overcome the influence of slow drift caused by the residual optical frequency difference, we utilize a vector Kalman filter to estimate and track the reference phase from pilots. The fast drift and slow drift are taken into account in the variation of the reference phase, so that phase estimation becomes more accurate than the conventional scalar Kalman filter. The mean square error of reference phase estimation is deduced theoretically, and the specified frame is designed to estimate the phase noise variance in real time. In simulations, the performances involving of the estimation accuracy, the actual excess noise and the secret key rate are compared between with four different phase estimation methods. The simulation results show that the vector Kalman filter is superior to the other methods, even though under a serious phase noise and a large optical frequency difference.
Employing quantum receivers to discriminate laser coherent states makes it feasible to achieve error probabilities below the standard quantum limit (SQL), particularly for photon-starved optical communication. However, the quantum receiver performance for phase shift keyed coherent states is severely degraded by random phase and intensity noise in realistic communication channels. This paper proposes a phase and intensity noise-tracking method based on the cubature Kalman filter (CKF) to improve the performance of time-resolving quantum receivers. The filter leverages the data collected during time-resolving state discrimination to perform real-time estimation and correction of time-varying phase and intensity noise. By conducting numerical simulations, we demonstrate that the proposed noise-tracking method enables the time-resolving quantum receiver to retain its superiority over the SQL while requiring significantly less computational effort than a Bayesian estimator. Additionally, the robustness against practical imperfections, including dead time and dark count rate of single photon detector (SPD), is shown through the simulations.
We propose a joint estimation scheme for fast, accurate, and robust frequency offset (FO) estimation along with phase estimation based on modified adaptive Kalman filter (MAKF). The scheme consists of three key modules: extend Kalman filter (EKF), lock detector, and FO cycle slip recovery. The EKF module estimates time-varying phase induced by both FO and laser phase noise. The lock detector module makes decision between acquisition mode and tracking mode and consequently sets the EKF tuning parameter in an adaptive manner. The third module can detect possible cycle slip in the case of large FO and make proper correction. Based on the simulation and experimental results, the proposed MAKF has shown excellent estimation performance featuring high accuracy, fast convergence, as well as the capability of cycle slip recovery.
We consider the continuous-time version of our recently proposed quantum theory of optical temporal phase and instantaneous frequency [M. Tsang et al., Phys. Rev. A 78, 053820 (2008)]. Using a state-variable approach to estimation, we design homodyne phase-locked loops that can measure the temporal phase with quantum-limited accuracy. We show that postprocessing can further improve the estimation performance if delay is allowed in the estimation. We also investigate the fundamental uncertainties in the simultaneous estimation of harmonic-oscillator position and momentum via continuous optical phase measurements from the classical estimation theory perspective. In the case of delayed estimation, we find that the inferred uncertainty product can drop below that allowed by the Heisenberg uncertainty relation. Although this result seems counterintuitive, we argue that it does not violate any basic principle of quantum mechanics.
In order to ensure the security of the local oscillator (LO) in continuous variable quantum key distribution (CV-QKD), the LO in CV-QKD based on the local LO (LLO) scheme no longer passes through the quantum channel. For the LLO scheme using homodyne detection, two orthogonal quadratures of reference pulse can only be measured sequentially, which, as a result, each signal pulse should be accompanied by a pair of identical reference pulses. However, due to the phase shift between two adjacent pulses, it is difficult to prepare dual reference pulses with the same phase relative to the LO. In this paper, we propose an improved method to get around this difficulty, which is demonstrated experimentally for a transmitter and a receiver connected by a 50 km single mode fiber. The phase noise measured in the experiment is 2.0×10−3rad2, which is comparable to the recent experimental results using heterodyne detection. This method simplifies the preparation of phase reference pulses and removes the obstacles for the application of homodyne detection in the LLO scheme. In addition, this method can also be used for the compensation of quadrature imbalance in heterodyne detection, then the performance of heterodyne detection can be improved.