The quantum canonical ensemble in phase space

The density operator for a quantum system in thermal equilibrium with its environment depends on Planck's constant, as well as the temperature. At high temperatures, the Weyl representation, that is, the thermal Wigner function, becomes indistinguishable from the corresponding classical distribution in phase space, whereas the low temperature limit singles out the quantum ground state of the system's Hamiltonian. In all regimes, thermal averages of arbitrary observables are evaluated by integrals, as if the thermal Wigner function were a classical distribution. The extension of the semiclassical approximation for quantum propagators to an imaginary thermal time, bridges the complex intervening region between the high and the low temperature limit. This leads to a simple quantum correction to the classical high temperature regime, irrespective of whether the motion is regular or chaotic. A variant of the full semiclassical approximation with a real thermal time, though in a doubled phase space, avoids any search for particular trajectories in the evaluation of thermal averages. The double Hamiltonian substitutes the stable minimum of the original system's Hamiltonian by a saddle, which eliminates local periodic orbits from the stationary phase evaluation of the integrals for the partition function and thermal averages.Comment: 24 pages, 2 figure

Exact Markovian evolution of multicomponent quantum systems: phase space representations

The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function. It is here generalized for multiple components, so as to provide an explicit expression for the reduced density operator of any component, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P-function, which guarantees separability of the components. In the multicomponent context, a full dissipation matrix is defined, whereas its trace, equal to twice the previously derived dissipation coefficient, governs the rate at which the phase space volume of the argument of the Wigner function contracts, while those of the chord function expands. Examples of markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.Comment: 29 pages, 5 figure

Resonance-Assisted Tunneling

We present evidence that tunneling processes in near-integrable systems are enhanced due to the manifestation of nonlinear resonances and their respective island chains in phase space. A semiclassical description of this "resonance-assisted" mechanism is given, which is based on a local perturbative description of the dynamics in the vicinity of the resonances. As underlying picture, we obtain that the quantum state is coupled, via a succession of classically forbidden transitions across nonlinear resonances, to high excitations within the well, from where tunneling occurs with a rather large rate. The connection between this description and the complex classical structure of the underlying integrable dynamics is furthermore studied, giving ground to the general coherence of the description as well as guidelines for the identification of the dominant tunneling paths. The validity of this mechanism is demonstrated within the kicked Harper model, where good agreement between quantum and semiclassical (resonance-assisted) tunneling rates is found.Comment: 52 pages, 16 figures, submitted to Annals of Physic

Resonance-assisted tunneling in near-integrable systems

Dynamical tunneling between symmetry related invariant tori is studied in the near-integrable regime. Using the kicked Harper model as an illustration, we show that the exponential decay of the wave functions in the classically forbidden region is modified due to coupling processes that are mediated by classical resonances. This mechanism leads to a substantial deviation of the splitting between quasi-degenerate eigenvalues from the purely exponential decrease with 1 / hbar obtained for the integrable system. A simple semiclassical framework, which takes into account the effect of the resonance substructure on the KAM tori, allows to quantitatively reproduce the behavior of the eigenvalue splittings.Comment: 4 pages, 2 figures, gzipped tar file, to appear in Phys. Rev. Lett, text slightly condensed compared to first versio

A primer for resonant tunnelling

Resonant tunnelling is studied numerically and analytically with the help of a three-well quantum one-dimensional time-independent model. The simplest cases are considered where the three-well potential is polynomial or piecewise constant.Comment: accepted to EJP, 19 pages, 8 figure

Optimal dynamical characterization of entanglement

We show that, for experimentally relevant systems, there is an optimal measurement strategy to monitor the time evolution of entanglement under open system dynamics. This suggests an efficient, dynamical characterization of the entanglement of composite

The quantum canonical ensemble in phase space

The density operator for a quantum system in thermal equilibrium with its environment depends on Planck’s constant, as well as the temperature. At high temperatures, the Weyl representation, that is, the thermal Wigner function, becomes indistinguishable from the corresponding classical distribution in phase space, whereas the low temperature limit singles out the quantum ground state of the system’s Hamiltonian. In all regimes, thermal averages of arbitrary observables are evaluated by integrals, as if the thermal Wigner function were a classical distribution. The extension of the semiclassical approximation for quantum propagators to an imaginary thermal time, bridges the complex intervening region between the high and the low temperature limit. This leads to a simple quantum correction to the classical high temperature regime, irrespective of whether the motion is regular or chaotic. A variant of the full semiclassical approximation with a real thermal time, though in a doubled phase space, avoids any search for particular trajectories in the evaluation of thermal averages. The double Hamiltonian substitutes the stable minimum of the original system’s Hamiltonian by a saddle, which eliminates local periodic orbits from the stationary phase evaluation of the integrals for the partition function and thermal averages

Exact Markovian evolution of multicomponent quantum systems: phase space representations

The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function. It is here generalized for multiple components, so as to provide an explicit expression for the reduced density operator of any component, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P-function, which guarantees separability of the components. In the multicomponent context, a full dissipation matrix is defined, whereas its trace, equal to twice the previously derived dissipation coefficient, governs the rate at which the phase space volume of the argument of the Wigner function contracts, while those of the chord function expands. Examples of markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed