Quantization of classical maps with tunable Ruelle-Pollicott resonances (original) (raw)

Localization of resonance eigenfunctions on quantum repellers

Physical review letters, 2009

We introduce a new phase space representation for open quantum systems. This is a very powerful tool to help advance in the study of the morphology of their eigenstates. We apply it to two different versions of a paradigmatic model, the baker map. This allows to show that the long-lived resonances are strongly scarred along the shortest periodic orbits that belong to the classical repeller. Moreover, the shape of the short-lived eigenstates is also analyzed. Finally, we apply an antiunitary symmetry measure to the resonances that permits to quantify their localization on the repeller. PACS numbers: 05.45.Mt, 03.65.Sq, 03.65.Yz

Quantum fingerprints of classical Ruelle-Pollicot resonances

2000

N-disk microwave billiards, which are representative of open quantum systems, are studied experimentally. The transmission spectrum yields the quantum resonances which are consistent with semiclassical calculations. The spectral autocorrelation of the quantum spectrum is shown to be determined by the classical Ruelle-Pollicot resonances, arising from the complex eigenvalues of the Perron-Frobenius operator. This work establishes a fundamental connection between quantum and classical correlations in open systems.

Classical resonances in quantum mechanics

Physical Review A, 1992

%e study the role of classical resonances in quantum systems subjected to a periodic external force. It is demonstrated that classical invariant vortex tubes determine the structure of Floquet states "captured" by resonances in the classical phase space. In addition, resonance quantum numbers are introduced. The analysis of simple model calculations leads to a qualitative description of nonperturbative phenomena relevant for the interaction of atoms or molecules with strong, short laser pulses.

Distribution of resonances for open quantum maps

2013

1.1. Statement of the results. In this paper we analyze simple models of classical chaotic open systems and of their quantizations. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller of the system. In a simplified

Relaxation and localization in interacting quantum maps

Journal of Statistical Physics, 1994

Quantum relaxation is studied in coupled quantum baker's maps. The classical systems are exactly solvable Kolmogorov systems (1) , for which the exponential decay to equilibrium is known. They model the fundamental processes of transport in classically chaotic phase space. The quantum systems, in the absence of global symmetry, show a marked saturation in the level of transport, as the suppression of diffusion in the quantum kicked rotor, and eigenfunction localization in the position basis. In the presence of a global symmetry we study another model that has classically an identical decay to equilibrium, but quantally shows resonant transport, no saturation and large fluctuations around equilibrium. We generalize the quantization to finite multibaker maps. As a byproduct we introduce some simple models of quantal tunneling between classically chaotic regions of phase space.

A Time-Dependent Theory of Quantum Resonances

Communications in Mathematical Physics, 1999

In this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues. We apply this theory to many-body Schrödinger operators and to the problem of quasiclassical tunneling.

Correspondence between classical and quantum resonances

Physical Review E, 2021

Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series toh = 0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limith → 0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.

The influence of classical resonances on quantum energy levels

The Journal of Chemical Physics, 1993

In this paper, we examine the behavior of the quantum energy levels of a coupled oscillator system as the zero-order frequencies are varied to carry the corresponding classical system through a resonance. We find that the levels exhibit a pattern that is characteristic of the resonance. This pattern consists of clusters of levels, each containing a number of curves that run roughly parallel to one another and a number of curves that undergo pairwise narrowly avoided crossings. Adiabatic switching calculations show that the "parallel" curves are associated with states within the classical resonance region, while the narrowly avoiding curves are associated with states that are outside this region. It is further shown that the curves describing resonance states are formed from zero-order nonresonant curves by the overlap of many avoided crossings. This reorganization of multiply intersecting lines into parallel curves reflects the classical reorganization of phase space at a resonance.

Intrinsic resonance representation of quantum mechanics

The Journal of Chemical Physics, 1997

The choice of basis states in quantum calculations can be influenced by several requirements, and sometimes a very natural basis suggests itself. However often one retreats to a ''merely complete'' basis, whose coefficients in the eigenstates carry little physical insight. We suggest here an optimal representation, based purely on classical mechanics. ''Hidden'' constants of the motion and good actions already known to the classical mechanics are thus incorporated into the basis, leaving the quantum effects to be isolated and included by small matrix diagonalizations. This simplifies the hierarchical structure of couplings between ''zero-order'' states. We present a ͑non-perturbative͒ method to obtain such a basis-state as solutions to a certain resonant Hamilton-Jacobi equation.

A Resonance Theory for Open Quantum Systems with Time-Dependent Dynamics

Journal of Statistical Physics, 2008

We develop a resonance theory to describe the evolution of open systems with time-dependent dynamics. Our approach is based on piecewise constant Hamiltonians: we represent the evolution on each constant bit using a recently developed dynamical resonance theory, and we piece them together to obtain the total evolution. The initial state corresponding to one time-interval with constant Hamiltonian is the final state of the system corresponding to the interval before. This results in a non-markovian dynamics. We find a representation of the dynamics in terms of resonance energies and resonance states associated to the Hamiltonians, valid for all times t ≥ 0 and for small (but fixed) interaction strengths. The representation has the form of a path integral over resonances. We present applications to a spin-fermion system, where the energy levels of the spin may undergo rather arbitrary crossings in the course of time. In particular, we find the probability for transition between ground-and excited state at all times.