Quadratic Quantum Hamiltonians revisited

We construct the integrals of motion for several models of the quantum damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the time-dependent Schroedinger equation with variable quadratic Hamiltonians. An extension of Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the c...

The notion of Loschmidt echo (also called "quantum fidelity") has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of "quantum chaos", quantum computation and decoherence. In this paper, we study this quantity semiclassically (as $\hbar \to 0$), taking as reference quantum states the usual coherent...

Bosonic quadratic Hamiltonians, often called Bogoliubov Hamiltonians, play an important role in the theory of many-boson systems where they arise in a natural way as an approximation to the full many-body problem. In this note we would like to give an overview of recent advances in the study of bosonic quadratic Hamiltonians. In particular, we relate the reported results to what can be called the time-dependent diagonalization problem.

We use coherent states as a time-dependent variational ansatz for a semiclassical treatment of the dynamics of anharmonic quantum oscillators. In this approach the square variance of the Hamiltonian within coherent states is of particular interest. This quantity turns out to have natural interpretation with respect to time-dependent solutions of the semiclassical equations of motion. Moreover, our approach allows for an estimate of the decoherence time of a classical object d...

This review is intended for readers who want to have a quick understanding on the theoretical underpinnings of coherent states and squeezed states which are conventionally generated from the prototype harmonic oscillator but not always restricting to it. Noting that the treatments of building up such states have a long history, we collected the important ingredients and reproduced them from a fresh perspective but refrained from delving into detailed derivation of each topic....

It is well known that the Gaussian wave packet dynamics can be written in terms of Hamilton equations in the extended phase space that is twice as large as in the corresponding classical system. We construct several generalizations of this approach that include non-Gausssian wave packets. These generalizations lead to the further extension of the phase space while retaining the Hamilton structure of the equations of motion. We compare the Gaussian dynamics with these non-Gaus...

The Loschmidt Echo M(t) (defined as the squared overlap of wave packets evolving with two slightly different Hamiltonians) is a measure of quantum reversibility. We investigate its behavior for classically quasi-integrable systems. A dominant regime emerges where M(t) ~ t^{-alpha} with alpha=3d/2 depending solely on the dimension d of the system. This power law decay is faster than the result ~ t^{-d} for the decay of classical phase space densities.

A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground ...

A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can find a unitary transformation such that the transformed quadratic Hamiltonian is diagonal but still written in terms of the original position and momentum observables. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.

We address the time decay of the Loschmidt echo, measuring sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, alike the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral prope...