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Physical Review A

Stochastic collective density-matrix approach for conditional spin squeezing

ZhiQing Zhang, Yuan Zhang, Haizhong Guo, Chongxin Shan, Gang Chen, Klaus Mølmer

Published May 14, 2025• 1 citation
Publisher

Abstract

In this article, we present a stochastic collective density-matrix approach to numerically study the conditional dynamics of large systems of identical particles in the presence of individual and collective dissipation, as described by stochastic master equations. To illustrate the application potential of the approach, we use it to study the conditional spin squeezing of hundreds of identical atoms in a bad optical cavity subject to homodyne detection. We demonstrate that the conditional spin squeezing can be vividly illustrated by a Gaussian-like distribution on the quantum states of the atomic ensemble, and we examine the influence of various processes, such as the individual atomic dissipation, on this distribution. In the future, the present approach can be used to reveal measurement effects of multiple-level atoms system, to explore generation of exotic nonclassical states, such as cat states, and to gauge approximate approaches, such as the stochastic mean-field approach.

DOI
10.1103/physreva.111.053715

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In this article, we present a stochastic collective density-matrix approach to numerically study the conditional dynamics of large systems of identical particles in the presence of individual and collective dissipation, as described by stochastic master equations. To illustrate the application potential of the approach, we use it to study the conditional spin squeezing of hundreds of identical atoms in a bad optical cavity subject to homodyne detection. We demonstrate that the conditional spin squeezing can be vividly illustrated by a Gaussian-like distribution on the quantum states of the atomic ensemble, and we examine the influence of various processes, such as the individual atomic dissipation, on this distribution. In the future, the present approach can be used to reveal measurement effects of multiple-level atoms system, to explore generation of exotic nonclassical states, such as cat states, and to gauge approximate approaches, such as the stochastic mean-field approach.
Stochastic master equations are often used to describe conditional spin squeezing of atomic ensemble, but are limited so far to the systems with few atoms due to the exponentially increased Hilbert space. In this article, we present an exact numerical solution of these equations for systems with identical atoms by mapping identical density matrix elements to a single quantity characterized by collective quantum numbers, and apply it to the system with hundred atoms in a bad cavity subject to a homodyne detection. We demonstrate that the spin squeezing can be vividly illustrated by the Gaussian-like distribution of the collective density matrix elements, and we examine the influence of the probe field strength and polarization, the detection efficiency, the spontaneous emission rate and the number of atoms. Our exact approach can play an important role in gauging the approximate approaches applied for systems with more atoms, such as Gaussian-state formalism and stochastic mean-field approach, and it permits also exploration of entanglement effects beyond these approaches.
Projective measurements of collective observables can be employed to herald the preparation of entangled states of quantum systems, and the resulting conditional dynamics is usually handled by stochastic master equation (SME) for small systems, and by an approximate Gaussian-state formalism for large systems. In this work, we present an alternative technique by developing a stochastic variant of cumulant mean-field theory, benchmark it against an exact stochastic collective density matrix approach by the simulations of hundreds of identical two-level atoms. More importantly, we demonstrate its full power by studying the conditional spin squeezing of thousands of three-level atoms coupled strongly with an optical cavity subject to individual decay and dephasing, and by simulating the experimental protocol to reveal formation and detection of the spin squeezed state. The proposed technique might be further extended to study more exotic quantum-measurement effects of large quantum systems, such as deterministic spin squeezing with quantum feedback, spin squeezing of optical clock transitions, and retrodictive spin squeezing by posterior measurements, and so on.
We describe an efficient numerical method for simulating the dynamics of interacting spin ensembles in the presence of dephasing and decay. The method builds on the discrete truncated Wigner approximation for isolated systems, which combines the mean-field dynamics of a spin ensemble with a Monte Carlo sampling of discrete initial spin values to account for quantum correlations. Here we show how this approach can be generalized for dissipative spin systems by replacing the deterministic mean-field evolution by a stochastic process, which describes the decay of coherences and populations while preserving the length of each spin. We demonstrate the application of this technique for simulating nonclassical spin-squeezing effects or the dynamics and steady states of cavity QED models with 105{10}^{5}105 interacting two-level systems. This opens up the possibility to perform accurate real-scale simulations of a diverse range of experiments in quantum optics or with solid-state spin ensembles under realistic laboratory conditions.
Abstract We analyze the generation of spin-squeezed states via coupling of three-level atoms to an optical cavity and continuous quantum measurement of the transmitted cavity field in order to monitor the evolution of the atomic ensemble. Using analytical treatment and microscopic simulations of the dynamics, we show that one can achieve significant spin squeezing, favorably scaling with the number of atoms N. However, contrary to some previous literature, we clarify that it is not possible to obtain Heisenberg scaling without the continuous feedback that is proposed in optimal approaches. In fact, in the adiabatic cavity removal approximation and large N limit, we find the scaling behavior N − 2 / 3 for spin squeezing and N − 1 / 3 for the corresponding protocol duration. These results can be obtained only by considering the curvature of the Bloch sphere, since linearizing the collective spin operators tangentially to its equator yields inaccurate predictions. With full simulations, we characterize how spin-squeezing generation depends on the system parameters and departs from the bad cavity regime, by gradually mixing with cavity-filling dynamics until metrological advantage is lost. Finally, we discuss the relevance of this spin-squeezing protocol to state-of-the-art optical clocks.
We describe an efficient numerical method for simulating the dynamics and steady states of collective spin systems in the presence of dephasing and decay. The method is based on the Schwinger boson representation of spin operators and uses an extension of the truncated Wigner approximation to map the exact open system dynamics onto stochastic differential equations for the corresponding phase space distribution. This approach is most effective in the limit of very large spin quantum numbers, where exact numerical simulations and other approximation methods are no longer applicable. We benchmark this numerical technique for known superradiant decay and spin-squeezing processes and illustrate its application for the simulation of non-equilibrium phase transitions in dissipative spin lattice models.
We describe an efficient numerical method for simulating the dynamics and steady states of collective spin systems in the presence of dephasing and decay. The method is based on the Schwinger boson representation of spin operators and uses an extension of the truncated Wigner approximation to map the exact open system dynamics onto stochastic differential equations for the corresponding phase space distribution. This approach is most effective in the limit of very large spin quantum numbers, where exact numerical simulations and other approximation methods are no longer applicable. We benchmark this numerical technique for known superradiant decay and spin-squeezing processes and illustrate its application for the simulation of non-equilibrium phase transitions in dissipative spin lattice models.