We discuss the long-time relaxation of a qubit linearly coupled to a finite bath of $N$ spins (two-level systems, TLSs), with the interaction Hamiltonian in rotating wave approximation. We focus on the regime $N\gg 1$, assuming that the qubit–bath coupling is weak, that the range of spin frequencies is sufficiently broad, and that all the spins are initialized in the ground state. Despite the model being perfectly integrable, we make two interesting observations about the effective system relaxation. First, as one would expect, the qubit relaxes exponentially towards its zero-temperature state at a well characterized rate. Second, the bath spins, even when mutually coupled, do not relax towards a thermal distribution, but rather form a Lorentzian distribution peaked at the frequency of the initially excited qubit. This behaviour is captured by an analytical approximation that makes use of the property $N\gg 1$ to treat the TLS frequencies as a continuum and is confirmed by our numerical simulations.

我们讨论与有限浴线性耦合的量子位的长时间弛豫$氮$自旋（二能级系统，TLS），与旋转波近似中的哈密顿量相互作用。我们关注政权$氮\gg 1$，假设量子位-浴耦合很弱，自旋频率范围足够宽，并且所有自旋都在基态下初始化。尽管该模型是完全可积的，但我们对有效系统松弛做出了两个有趣的观察。首先，正如人们所期望的那样，量子位以良好表征的速率呈指数弛豫到零温度状态。其次，即使相互耦合，浴旋转也不会朝着热分布松弛，而是形成在最初激发的量子位的频率处达到峰值的洛伦兹分布。这种行为是通过利用该属性的分析近似来捕获的$氮\gg 1$将 TLS 频率视为连续体，并通过我们的数值模拟得到证实。