随机量子主方程中的光谱带隙和中间带隙态
Spectral gaps and mid-gap states in random quantum master equations
摘要：
讨论了耦合噪声的混沌量子系统的衰变率。我们用随机$N \times N$厄米矩阵来模拟哈密顿量和系统-噪声耦合，并研究了所得Lindblad超算子的谱性质.我们考虑各种随机矩阵系综，并发现，对于所有这些渐近衰减率保持非零的热力学极限，即，超算子的谱被间隔为$N \rightarrow \infty$。随着耗散强度的增加，发生急剧的谱转变：对于弱耗散，主方程的非零特征值形成连续体;而对于强耗散，渐近衰减率是\emph{isolated eigenvalue}，即，与主方程的连续谱明显分离的“中间带隙状态”。对于有限的$N$，找到一个非常小的间隙的概率代数地消失，在系统大小的标度指数是广泛的，并且仅取决于随机矩阵的对称类和独立的衰变通道的数量。我们评论我们的结果的实验意义。
英文摘要：
We discuss the decay rates of chaotic quantum systems coupled to noise. We model both the Hamiltonian and the system-noise coupling by random $N \times N$ Hermitian matrices, and study the spectral properties of the resulting Lindblad superoperator. We consider various random-matrix ensembles, and find that for all of them the asymptotic decay rate remains nonzero in the thermodynamic limit, i.e., the spectrum of the superoperator is gapped as $N \rightarrow \infty$. A sharp spectral transition takes place as the dissipation strength is increased: for weak dissipation, the non-zero eigenvalues of the master equation form a continuum; whereas for strong dissipation, the asymptotic decay rate is an \emph{isolated eigenvalue}, i.e., a `mid-gap state' that is sharply separated from the continuous spectrum of the master equation. For finite $N$, the probability of finding a very small gap vanishes algebraically with a scaling exponent that is extensive in system size, and depends only on the symmetry class of the random matrices and the number of independent decay channels. We comment on experimental implications of our results.
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