Gherardini S, Giachetti G, Ruffo S, Trombettoni A
SISSA and INFN, I-34136 Trieste, Italy.
Department of Physics and Astronomy and LENS, University of Florence, I-50019 Sesto Fiorentino, Italy.
Phys Rev E. 2021 Sep;104(3-1):034114. doi: 10.1103/PhysRevE.104.034114.
We study the heat statistics of a multilevel N-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number M of measurements (M→∞). In this context, the conditions allowing for an infinite-temperature thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the nonequilibrium evolution of the system and its initial state. Exceptions to ITT, which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasicommuting) with the Hamiltonian of the quantum system or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces (N→∞), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits M→∞ and N→∞ matters: When N is fixed and M diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A nontrivial result is obtained fixing M/N^{2} where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.
我们研究了由一系列投影测量监测的多级(N)维量子系统的热统计。在大量（理想情况下为无限多）(M)次测量（(M→∞)）的热力学极限下，分析了热特征函数的晚期渐近性质。在此背景下，讨论了由量子系统的重复监测引起的允许无限温度热化（ITT）的条件。我们表明，ITT由一个对称随机矩阵的不动点确定，该矩阵对由测量序列产生的随机过程进行建模。这样的不动点与系统的非平衡演化及其初始状态无关。当中间测量的可观测量与量子系统的哈密顿量对易（或准对易）时，或者当测量之间的时间间隔小于或与系统能量尺度相当（量子芝诺 regime）时，会出现ITT的例外情况，我们将其称为部分热化。还给出了关于无限维希尔伯特空间（(N→∞)）极限的结果，该极限描述了具有离散谱的连续系统。我们表明极限(M→∞)和(N→∞)的顺序很重要：当(N)固定且(M)发散时，会出现ITT。在相反的情况下，系统变为经典，因此测量不再有效地改变系统状态。当固定(M/N^{2})时会得到一个非平凡的结果，此时会出现部分ITT。最后，给出了一个适用于旋转二维气体的部分热化示例。
