重复量子相互作用的弱耦合和连续极限
Weak Coupling and Continuous Limits for Repeated Quantum Interactions
摘要：
我们考虑一个量子系统与一个热浴接触，热浴由一个无限长的相同的子系统链组成，在逆温度$β$下处于热平衡。时间演化是离散的，并且使得在持续时间$τ$的每个时间步长上，参考系统通过强度$λ$的相互作用仅耦合到链的一个新元素。我们考虑三个渐近制度的参数$λ$和$τ$的小系统上的观测量的有效演变成为连续的适当的宏观时间尺度$T$和其发电机可以计算：弱耦合极限制度$λ\ra 0$，$τ=1$，制度$τ\ra 0$，$λ^2τ\ra 0$和临界情况$λ^2τ=1$，$τ\ra 0$。前两个制度是微扰的性质和有效的发电机，他们确定的是这样的，一个非平凡不变的子代数的观察自然出现。第三渐近制度超越了微扰制度，并提供了一个有效的动力学一般Lindblad生成器自然构造的相互作用哈密顿。相反，这个结果表明，人们可以附加到任何Lindblad生成器的重复量子相互作用模型，其渐近有效演化是由这个Lindblad算子生成的。
英文摘要：
We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical sub-systems at thermal equilibrium at inverse temperature $β$. The time evolution is discrete and such that over each time step of duration $τ$, the reference system is coupled to one new element of the chain only, by means of an interaction of strength $λ$. We consider three asymptotic regimes of the parameters $λ$ and $τ$ for which the effective evolution of observables on the small system becomes continuous over suitable macroscopic time scales $T$ and whose generator can be computed: the weak coupling limit regime $λ\ra 0$, $τ=1$, the regime $τ\ra 0$, $λ^2τ\ra 0$ and the critical case $λ^2τ=1$, $τ\ra 0$. The first two regimes are perturbative in nature and the effective generators they determine is such that a non-trivial invariant sub-algebra of observables naturally emerges. The third asymptotic regime goes beyond the perturbative regime and provides an effective dynamics governed by a general Lindblad generator naturally constructed from the interaction Hamiltonian. Conversely, this result shows that one can attach to any Lindblad generator a repeated quantum interactions model whose asymptotic effective evolution is generated by this Lindblad operator.
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