Abstract

We show that in some cases the coherent state can have a larger violation of the Leggett–Garg inequality (LGI) than the cat state by numerical calculations. To achieve this result, we consider the LGI of the cavity mode weakly coupled to a zero-temperature environment as a practical instance of the physical system. We assume that the bosonic mode undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of the violation of the inequality for both systems prepared initially in the coherent state \(|\alpha \rangle \) and the cat state \((|\alpha \rangle +|-\alpha \rangle )\). For the evaluation of the inequality, we choose the displaced parity operators characterized by a complex number \(\beta \). We look for the optimum parameter \(\beta \) that lets the upper bound of the inequality be maximum numerically. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the violation of the LGI in a specific range of three equally spaced measurement times (spacing \(\tau \)). Moreover, as we let \(\tau \) approach zero, the optimized parameter \(\beta \) diverges and the LGI reveals intense singularity.

Graphic Abstract

中文翻译：

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对相干和猫状态违反 Leggett-Garg 不等式

摘要

我们通过数值计算表明，在某些情况下，相干状态可能比猫状态更严重地违反 Leggett-Garg 不等式（LGI）。为了实现这一结果，我们将与零温度环境弱耦合的腔模式的 LGI 视为物理系统的一个实际实例。我们假设玻色子模式由于与环境的相互作用而发生耗散，但不受移相的影响。精确求解主方程，我们推导出了最初在相干态\(|\alpha \rangle \)和猫态\((|\alpha \rangle +|- \alpha \rangle )\). 为了评估不等式，我们选择以复数\(\beta \) 为特征的置换奇偶校验算子。我们寻找使不等式的上限在数值上最大的最佳参数\(\beta \)。与我们的预期相反，在三个等距测量时间（间距\(\tau \)）的特定范围内，相干态有时比 LGI 违反的上限表现出比猫态更强烈的量子质量。此外，当我们让\(\tau \)趋近于零时，优化参数\(\beta \)发散并且 LGI 显示出强烈的奇异性。

图形摘要