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arXiv:1211.5760 · Physical Review E

Semiclassical theory of chaotic quantum resonances

Tobias Micklitz, Alexander Altland

Published Mar 29, 2013• 7 citations
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Abstract

States supported by chaotic open quantum systems fall into two categories: a majority showing instantaneous ballistic decay, and a set of quantum resonances of classically vanishing support in phase space. We present a theory describing these structures within a unified semiclassical framework. Emphasis is put on the quantum diffraction mechanism which introduces an element of probability and is crucial for the formation of resonances. Our main result is boundary conditions on the semiclassical propagation along system trajectories. Depending on whether the trajectory propagation time is shorter or longer than the Ehrenfest time, these conditions describe deterministic escape, or probabilistic quantum decay.

DOI
10.1103/physreve.87.032918

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States supported by chaotic open quantum systems fall into two categories: a majority showing instantaneous ballistic decay, and a set of quantum resonances of classically vanishing support in phase space. We present a theory describing these structures within a unified semiclassical framework. Emphasis is put on the quantum diffraction mechanism which introduces an element of probability and is crucial for the formation of resonances. Our main result is boundary conditions on the semiclassical propagation along system trajectories. Depending on whether the trajectory propagation time is shorter or longer than the Ehrenfest time, these conditions describe deterministic escape, or probabilistic quantum decay.
We address the decay in open chaotic quantum systems and calculate semiclassical corrections to the classical exponential decay. We confirm random matrix predictions and, going beyond, calculate Ehrenfest time effects. To support our results we perform extensive numerical simulations. Within our approach we show that certain (previously unnoticed) pairs of interfering, correlated classical trajectories are of vital importance. They also provide the dynamical mechanism for related phenomena such as photoionization and photodissociation, for which we compute cross-section correlations. Moreover, these orbits allow us to establish a semiclassical version of the continuity equation.
We study the quantum probability to survive in an open chaotic system in the framework of the van Vleck-Gutzwiller propagator and present the first such calculation that accounts for quantum interference effects. Specifically we calculate quantum deviations from the classical decay after the break time for both broken and preserved time-reversal symmetry. The source of these corrections is identified in interfering pairs of correlated classical trajectories. In our approach the quantized chaotic system is modelled by a quatum graph.
We consider quantum decay and photofragmentation processes in open chaotic systems in the semiclassical limit. We devise a semiclassical approach which allows us to consistently calculate quantum corrections to the classical decay to high order in an expansion in the inverse Heisenberg time. We present results for systems with and without time-reversal symmetry, as well as for the symplectic case, and extend recent results to nonlocalized initial states. We further analyze related photodissociation and photoionization phenomena and semiclassically compute cross-section correlations, including their Ehrenfest-time dependence.
We study the semiclassical Ehrenfest trajectories in open quantum systems. We first give a self-contained derivation of the Fokker--Planck equation that governs the time evolution of the mixing measure for a Gaussian mixture. Then, we embed the generalized Ehrenfest theorem recently obtained for open quantum systems into this phase-space picture to study the time evolution of the expectations of observables with respect to the Gaussian mixture. We show how the coherent and irreversible contributions are separated in the frictionless harmonic regime relevant to the semiclassical approximation. Our work provides a transparent phase-space interpretation of the emergence of classical trajectories in open quantum dynamics.
In the semiclassical limit of open ballistic quantum systems, we demonstrate the emergence of instantaneous decay modes guided by classical escape faster than the Ehrenfest time. The decay time of the associated quasibound states is smaller than the classical time of flight. The remaining long-lived quasibound states obey random-matrix statistics, renormalized in compliance with the recently proposed fractal Weyl law for open systems [W.T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)]. We validate our theory numerically for a model system, the open kicked rotator.
The semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.