In the presence of Lindblad decoherence, i.e., dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators $L_j^{}$ are pseudo-Hermitian, namely, $\eta L_j^{}{\eta }^{-1}=L_j^{†}$ with $\eta$ being a linear and positive-definite metric, and respect the $\mathcal{PT}$ symmetry as well. We demonstrate that the generalized density matrix ${\rho }_{\mathrm{G}}^{}\left(t\right)\equiv \rho \left(t\right)\eta$, instead of the normalized density matrix ${\rho }_{\mathrm{N}}^{}\left(t\right)\equiv \rho \left(t\right)/\mathrm{tr}\left[\rho \left(t\right)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.

中文翻译：

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具有Lindblad方程的PT对称非Hermitian哈密顿量的密度矩阵形式

在存在林德布拉德相干性的情况下，即由于与环境的相互作用而在开放量子系统中产生耗散效应的情况下，我们研究了非埃尔米特哈密顿人用林德布拉德方程描述的两级量子系统中本征态之间的跃迁概率，保留了奇偶时间反转（PT）对称性。首先，密度矩阵形式为$\mathcal{PT}$对称非Hermitian哈密顿系统的发展。显示林德运算符${\mathrm{大号}}_{Ĵ}^{}$ 是伪赫米特语，即 $\eta {\mathrm{大号}}_{Ĵ}^{}{\eta }^{-\mathrm{1个}}={\mathrm{大号}}_{Ĵ}^{†}$ 和 $\eta$ 是线性和正定度量，并尊重 $\mathcal{PT}$对称性。我们证明了广义密度矩阵${\rho }_{\mathrm{G}}^{}（Ť）\equiv \rho （Ť）\eta$，而不是归一化密度矩阵 ${\rho }_{\mathrm{ñ}}^{}（Ť）\equiv \rho （Ť）/\mathrm{TR}\left[\rho （Ť）\right]$应根据线性要求实施，以计算跃迁概率。其次，密度矩阵形式主义用于推导一般情况下的转移概率$\mathcal{PT}$对称非埃尔米特哈密顿量。在一些具体示例中，我们为过渡概率计算了紧凑的解析公式，并通过数字插图探索了它们的主要特征。在没有Lindblad反相干的情况下，我们还使用状态向量对当前结果和以前的结果进行了比较。