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Bulletin of the American Physical Society

Continuous monitoring of a quantum state beyond classical limits

Giorgio Colangelo, Ferran Martin Ciurana, R. J. Sewell, Morgan W. Mitchell

Published Jun 9, 2017• 0 citations
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Tracking the evolution of a quantum state beyond classical limits

Giorgio Colangelo, Ferran Martin Ciurana, R. J. Sewell +1 more

Summary form only given. Continuous monitoring or tracking of a quantum system is essential to high-sensitivity measurement of time-varying quantities. We estimate spin precession of a magnetically-sensitive atomic ensemble by measuring spin projectors Fαat different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. We show how to reduce this disturbance below δFα~ √N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state [1, 2] which allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics, surpassing classical limits in non-commuting observables for the first time [3]. Our method is close to practical application in high-performance atomic sensors, such as magnetometers and clocks, and is compatible with multi-pass and cavity build-up methods.

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Generalized master equations (GMEs)---time-local but generally neither trace-preserving nor Hermiticity-preserving---are convenient tools to compute properties of the environment of an open or continuously monitored quantum system. A two-sided master equation yields the fidelity and quantum Fisher information (QFI) of environment states, thereby setting fundamental limits for hypothesis testing and parameter estimation under continuous monitoring. For unmonitored noise or inefficient detection, the QFI of the detectable part of the environment may be obtained from a recently derived GME acting on multiple system replicas. Tilted master equations provide the full counting statistics of quantum jumps and diffusive measurements, enabling, e.g., studies of quantum thermodynamics beyond average values. Here we focus on bosonic linear systems, governed by a quadratic Hamiltonian and linear jump operators, whose dynamics preserves Gaussianity. For Gaussian initial operators, we recast a generic GME as a compact set of ordinary differential equations for the covariance matrix (a Riccati-type equation), first moments, and normalization. These equations can be integrated efficiently without Hilbert-space truncation, and admit analytical results in simple settings. We also provide specialized forms for fidelity/QFI and full counting statistics. We illustrate the formalism with a continuously monitored optical parametric oscillator, using it to determine sensitivity limits for frequency estimation and to benchmark Hasegawa's thermodynamic uncertainty relations.

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Summary form only given. Continuous monitoring or tracking of a quantum system is essential to high-sensitivity measurement of time-varying quantities. We estimate spin precession of a magnetically-sensitive atomic ensemble by measuring spin projectors Fαat different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. We show how to reduce this disturbance below δFα~ √N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state [1, 2] which allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics, surpassing classical limits in non-commuting observables for the first time [3]. Our method is close to practical application in high-performance atomic sensors, such as magnetometers and clocks, and is compatible with multi-pass and cavity build-up methods.
In this paper, we first consider the generation of mechanical squeezing in a dispersively or dissipatively coupled optomechanical system by continuously homodyning the output field of the optomechanical cavity. It is found that strong steady-state mechanical squeezing beyond the 3 dB limit can be achieved in both of the optomechanical systems. The properties of the squeezing are quite different for the two types of systems and the reasons for the differences are analyzed. We next consider the achievement of optomechanical steering in the dispersive or dissipative optomechanical system via continuously monitoring the position of the mechanical oscillator. It is revealed that the monitoring can lead the steady-state optomechanical entanglement to be enhanced considerably such that strong steerable correlations can be achieved between the mechanical oscillator and the cavity field. The effects of thermal phonons are studied and it is shown that the generated squeezing and steering are quite robust against the thermal fluctuations.
Generalized master equations (GMEs)---time-local but generally neither trace-preserving nor Hermiticity-preserving---are convenient tools to compute properties of the environment of an open or continuously monitored quantum system. A two-sided master equation yields the fidelity and quantum Fisher information (QFI) of environment states, thereby setting fundamental limits for hypothesis testing and parameter estimation under continuous monitoring. For unmonitored noise or inefficient detection, the QFI of the detectable part of the environment may be obtained from a recently derived GME acting on multiple system replicas. Tilted master equations provide the full counting statistics of quantum jumps and diffusive measurements, enabling, e.g., studies of quantum thermodynamics beyond average values. Here we focus on bosonic linear systems, governed by a quadratic Hamiltonian and linear jump operators, whose dynamics preserves Gaussianity. For Gaussian initial operators, we recast a generic GME as a compact set of ordinary differential equations for the covariance matrix (a Riccati-type equation), first moments, and normalization. These equations can be integrated efficiently without Hilbert-space truncation, and admit analytical results in simple settings. We also provide specialized forms for fidelity/QFI and full counting statistics. We illustrate the formalism with a continuously monitored optical parametric oscillator, using it to determine sensitivity limits for frequency estimation and to benchmark Hasegawa's thermodynamic uncertainty relations.