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Physica A Statistical Mechanics and its Applications

Generalization of Kramers formula for open quantum systems

Yu. V. Palchikov, G. G. Adamian, N. V. Antonenko, W. Scheid

Published Dec 1, 2002• 17 citations
Publisher
DOI
10.1016/s0378-4371(02)01219-0

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Kramers' theorem ensures double degeneracy in the energy spectrum of a time-reversal symmetric fermionic system with half-integer total spin. Here, we are trying to go beyond the closed system and discuss Kramers' degeneracy in open systems out of equilibrium. A natural way to extend the Kramers' degeneracy in open quantum systems is by the degeneracy of different spins' spectra together with the vanishing interspin spectrum. We find the violation of Kramers' degeneracy in time-reversal symmetric open quantum systems is locked with whether the system reaches thermal equilibrium. After a general coupling to an environment in a time-reversal symmetry-preserving way, the Kramers doublet experiences an energy splitting at a short time and then a recovery process. We verified the violation and revival of Kramers' degeneracy in a concrete model of interacting fermions and we find Kramers' degeneracy is restored after the local thermalization time. By contrast, for time-reversal symmetry $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}}^{2}=1$, we find that although there is a violation and revival of spectral degeneracy for different spins, the inter-spin spectral function is always nonzero. We also prove that the degeneracy in spectral function protected by unitary symmetry can always be maintained.
Quantum diffusion equations with transport coefficients explicitly depending on time are derived from the generalized non-Markovian Langevin equations. The asymptotic behavior of the friction and diffusion coefficients is investigated in the case of the FC and RWA couplings between the collective and internal subsystems. An asymptotic expression is obtained for the propagator of the density matrix of the open quantum system with the general quadratic Hamiltonian, linearly coupled (in coordinate and momentum) to internal degrees of freedom. The effect of different sets of transport coefficients on the decoherence and decay rate of the metastable state is investigated using the master equation for the reduced density matrix of open quantum systems. The developed approach is used to study the capture of the projectile nucleus by the target nucleus at energies near the Coulomb barrier. Capture cross sections in asymmetric reactions are well described with allowance for the calculated capture probabilities. Particular cases where dissipation favors penetration through the potential barrier are found. The generalized Kramers formula for the quasi-stationary decay rate of the quantum metastable systems is analytically derived.
On the basis of a master equation for the reduced density matrix of open quantum systems, we study the influence of time-dependent friction and diffusion coefficients on the decay rate from a potential well and the capture probability into a potential well. Taking into account the mixed diffusion coefficient Dqp, the quasistationary decay rates are compared with the analytically derived Kramers-type formulas for different temperatures and frictions. The diffusion coefficients supplying the purity of states are derived for a non-Markovian dynamics.