A new perturbative expansion of the time evolution operator associated with a quantum system

Abstract

A novel expansion of the evolution operator associated with a -in general, time-dependent -perturbed quantum Hamiltonian is presented. It is shown that it has a wide range of possible realizations that can be fitted according to computational convenience or to satisfy specific requirements. As a remarkable example, the quantum Hamiltonian describing a laser-driven trapped ion is studied in detail.

Interaction-free evolution in the presence of time-dependent Hamiltonians

“…In general Eq. (1) is untractable and, except for some lucky cases [14][15][16], it requires special assumptions, such as for example the adiabatic one [17], or suitable approximations, like in the perturbative treatment [18,19]. Therefore, even the partial resolution of a class of timedependent problems in the presence of time-dependent Hamiltonians is of interest itself.…”

Perturbative Treatment of the Evolution Operator Associated with Raman Couplings

“…In order to study the class of dynamical problems associated with a quantum Hamiltonian of the form (1), one can fruitfully exploit a time-dependent perturbative method based on a suitable decomposition of the evolution operator [5] which is a generalization of the classical Magnus expansion [10].…”

“…A typical example is the case where the interaction picture Hamiltoniañ H(λ; t) is a almost periodic 1 operator-valued function of time, in particular, an operator-valued trigonometric polynomial with respect to the time variable. In this case, a remarkable gauge is fixed by the following tern of conditions (see [5]):…”

“…To this aim, we exploit a recently introduced perturbative method based on a suitable decomposition of the time evolution operator associated with a quantum Hamiltonian [5] (for the case of a time-independent Hamiltonian, see also [6,7,8]). In particular, we investigate the problem of determining the effective Hamiltonian -i.e.…”

Evanescent Wave Approximation for Non-Hermitian Hamiltonians

“…2 On the other hand, if one wants to catch some properties of a system or some aspects of its dynamical behaviour, it is often not necessary to consider the exact microscopic Hamiltonian model but one can construct effective Hamiltonian models that encode all the dynamical properties one wishes to study. Many techniques can be followed to construct effective Hamiltonian models, most of which are based on perturbation theory and adiabatic elimination [3][4][5][6][7] . A most useful tool in these derivations is the so-called rotating wave approximation (RWA), 8 consisting in removing some terms both on a physical ground (since they are not conserving the energy of the system) and at a mathematical level because of the appearance of fast phase factors in the interaction picture, implying negligible effects of the relevant terms in the dynamics of the system, especially when a coarse grained dynamic is to be evaluated.…”