We present explicit evaluations of quantum speed limit times pertinent to the Markovian dynamics of an open continuous-variable system. Specifically, we consider the standard setting of a cavity mode of the quantum radiation field weakly coupled to a thermal bosonic reservoir. The evolution of the field state is ruled by the quantum optical master equation, which is known to have an exact analytic solution. Starting from a pure input state, we employ two indicators of how different the initial and evolved states are, namely, the fidelity of evolution and the Hilbert-Schmidt distance of evolution. The former was introduced by del Campo et al. [Phys. Rev. Lett. 110, 050403 (2013)], who derived a time-independent speed limit for the evolution of a Markovian open system. We evaluate it for this field-reservoir setting, with an arbitrary input pure state of the field mode. The resultant formula is then specialized to the coherent and Fock states. On the other hand, we exploit an alternative approach that employs both indicators of evolution mentioned above. Their rates of change have the same upper bound, and consequently, provide a unique time-dependent quantum speed limit. It turns out that the associate quantum speed limit time built with the Hilbert-Schmidt metric is tighter than the fidelity-based one. As apposite applications, we investigate the damping of the coherent and Fock states by using the characteristic functions of the corresponding evolved states. General expressions of both the fidelity and the Hilbert-Schmidt distance of evolution are obtained and analyzed for these two classes of input states. In the case of a coherent state, we derive accurate formulas for their common speed limit and the pair of associate limit times. We also find exact expressions of the same quantities in the limiting case of thermalization of the vacuum state, as well as for dissipation of one- and two-photon states.

中文翻译：

---

马尔可夫玻色子环境中的量子演化速度

我们提出与开放连续变量系统的马尔可夫动力学有关的量子速度极限时间的显式评估。具体来说，我们考虑弱耦合到热硼储层的量子辐射场的腔模的标准设置。场态的演化由量子光学主方程决定，该方程具有精确的解析解。从纯输入状态开始，我们使用两个指标来指示初始状态和演化状态的差异，即演化的保真度和演化的希尔伯特-施密特距离。前者由del Campo等人介绍。[物理 牧师 110，050403（2013）]，他为马尔可夫开放系统的演化推导了与时间无关的速度极限。我们使用现场模式的任意输入纯状态对此现场储层设置进行评估。然后将生成的公式专用于相干态和Fock态。另一方面，我们采用了一种替代方法，该方法采用了上述两个进化指标。它们的变化率具有相同的上限，因此提供了唯一的时间依赖性量子速度极限。事实证明，使用希尔伯特-施密特（Hilbert-Schmidt）度量建立的关联量子速度限制时间比基于保真度的限制时间更严格。作为适当的应用，我们通过使用相应的演化态的特征函数来研究相干态和福克态的阻尼。获得并分析了这两种输入状态的保真度和希尔伯特-施密特进化距离的一般表达式。在相干状态的情况下，我们针对它们的共同速度极限和一对关联极限时间得出精确的公式。我们还发现了在真空状态热化的极限情况以及一光子状态和二光子状态的耗散情况下，相同数量的精确表达式。