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arXiv:quant-ph/0108109

Time dependence of occupation numbers and thermalization time in closed chaotic many-body systems

V. V. Flambaum, Felix M. Izrailev

Published Aug 23, 2001• 3 citations
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Abstract

We study the time evolution of occupation numbers for interacting Fermi-particles in the situation when exact compound states are "chaotic". This situation is generic for highly excited many-particles states in heavy nuclei, complex atoms, quantum dots, spin systems and quantum computer models. Numerical data show perfect agreement with a simple theory for the onset of thermalization in close systems of interacting particles.

DOI
10.48550/arxiv.quant-ph/0108109

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We study the time evolution of occupation numbers for interacting Fermi-particles in the situation when exact compound states are "chaotic". This situation is generic for highly excited many-particles states in heavy nuclei, complex atoms, quantum dots, spin systems and quantum computer models. Numerical data show perfect agreement with a simple theory for the onset of thermalization in close systems of interacting particles.
We study the time evolution of an initially excited many-body state in a finite system of interacting Fermi particles in the situation when the interaction gives rise to the "chaotic" structure of compound states. This situation is generic for highly excited many-particle states in quantum systems such as heavy nuclei, complex atoms, quantum dots, spin systems, and quantum computers. For a strong interaction the leading term for the return probability W(t) has the form W(t) approximately exp(-Delta(2)(E)t(2)) with Delta(2)(E) as the variance of the strength function. The conventional exponential linear dependence W(t)=C exp(-Gammat) formally arises for a very larger time. However, the prefactor C turns out to be exponentially large, thus resulting in a strong difference from the conventional estimate for W(t).
Highly excited many-particle states in quantum systems (nuclei, atoms, quantum dots, spin systems, quantum computers) can be ‘chaotic’ superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is a result of the very high energy level density of many-body states which can be easily mixed by a residual interaction between particles. We consider the time dynamics of wave functions and increase of entropy in such chaotic systems. As an example, we present the time evolution in a closed quantum computer. A time scale for the entropy S ( t ) increase is t c ~&tgr; 0 /( n log 2 n ), where &tgr; 0 is the qubit ‘lifetime’, n is the number of qubits, S (0) = 0 and S ( t c )=1. At t _ t c the entropy is small: S ~ nt 2 J 2 log 2 (1/ t 2 J 2 ), where J is the inter-qubit interaction strength. At t > t c the number of ‘wrong’ states increases exponentially as 2 S ( t ) . Therefore, t c may be interpreted as a maximal time for operation of a quantum computer. At t >> t c the system entropy approaches that for chaotic eigenstates.
In this review the problem of statistical description of isolated quantum systems of interacting particles is discussed. Main attention is paid to a recently developed approach which is based on chaotic properties of compound states in the basis of non-interacting particles. In order to demonstrate the most important aspects of this approach, the matrix model of two-body random interaction between Fermi-particles has been used. Different problems have been considered such as the onset of chaos and statistical equilibrium, the relation between the structure of eigenstates and distribution of occupation numbers, the emergence of the Fermi-Dirac distribution in isolated systems of finite number of particles and many others. The application of the approach to dynamical systems with the classical limit is discussed as well.
The onset of thermalization in a closed system of randomly interacting bosons at the level of a single eigenstate is discussed. We focus on the emergence of Bose-Einstein distribution of single-particle occupation numbers, and we give a local criterion for thermalization dependent on the eigenstate energy. We show how to define the temperature of an eigenstate, provided that it has a chaotic structure in the basis defined by the single-particle states. The analytical expression for the eigenstate temperature as a function of both interparticle interaction and energy is complemented by numerical data.
Abstract A study of quantum-classical correspondence for the onset of thermalization in an isolated system of four interacting particles is presented. Provided that classical dynamics of the system is chaotic, thermalization is present and fits the expected statistical mechanics results. Quantum classical correspondence is established for the classical analogue of the single particle distributions of occupation numbers. The latter quantity is useful to test thermalization and it allows to obtain the microcanonical Boltzmann factor in the quantum and semi-classical cases. The correspondence between them and the statistical mechanics results are surprisingly good considering that we have only four interacting particles. The resulting Boltzmann factor for the occupation number distribution depends on the eigenfunction width, which is a semi-classical parameter explicitly calculated and consistent with the quantum one.
A theory of strongly interacting Fermi systems of a few particles is developed. At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual two-body interaction. This regime can be described as many-body quantum chaos. Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy. Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g. rare earths) and multicharged ions, molecules, clusters and quantum dots in solids. The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis. A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix. We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.