Abstract
We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.
Highlights
The subject of thermalization in a closed quantum system of many interacting constituents, being on a crossroad of statistical physics, quantum mechanics, condensed matter and nuclear physics, attracts currently a great interest, both theoretical and experimental, see recent review papers [1,2,3]
It is known that the dynamics of wave function components in a closed system with a finite Hilbert space is subject to classical equations of motion and is quasiperiodic
It was qualitatively understood long ago [10] and formulated as a qualitative statement that all typical wave functions in the same energy region of such a complex system “look the same”. As it was indicated even much earlier in Statistical Physics by Landau and Lifshitz [11], the observables found for such quantum states are essentially the same as for the equilibrium statistical ensemble, just expressed in terms of energy rather than of temperature
Summary
The subject of thermalization in a closed quantum system of many interacting constituents, being on a crossroad of statistical physics, quantum mechanics, condensed matter and nuclear physics, attracts currently a great interest, both theoretical and experimental, see recent review papers [1,2,3]. Molecules (including biological), and atomic nuclei by their nature are systems of interacting quantum constituents In many cases, such as an isolated atom, nucleus, or molecule, the system is self-bound and lives in its intrinsic stationary state without any external heat bath prior to its use in an experiment. The exact solution of the many-body quantum problem in the finite orbital space reveals the internally developed chaotic behavior of stationary states and observables starting already at a moderately high excitation energy This behavior, with a smooth energy dependence, can be naturally translated into thermodynamic language with effective temperature and entropy. This work pursues a broad goal of understanding how non-stationary states and perturbations in a complex many-body systems evolve in time, what are the mechanisms for thermalization and decoherence, what role complex interactions and mean field play in this dynamics. This evolution is the road from a simple to compound state mixing many degrees of freedom in a quantum system
Published Version
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