Based on a representation of the functional integral as the time evolution of the augmented density matrix we have worked out an implementation of the real-time path integral approach that is applicable to the dynamics of quantum dissipative systems with superohmic coupling to the environment. As a prototype for such a system we consider a laser-driven strongly confined semiconductor quantum dot coupled to acoustic phonons. First applications of this approach to quantum dot systems have already been published. Here, we provide a detailed description of the implementation, including a discussion of numerical issues and extend the formalism from two-level quantum dot models with a pure-dephasing type carrier-phonon coupling to the case of multiple electronic levels. The method allows for numerically exact calculations of the dot dynamics at strong dot-phonon and dot-laser coupling and at long times, usually inaccessible by other approaches. DOI: http://dx.doi.org/10.1103/PhysRevB.83.094303 Received 15 November 2010Published 16 March 2011©2011 American Physical Society

The dynamics of strongly confined laser driven semiconductor quantum dots coupled to phonons is studied theoretically by calculating the time evolution of the reduced density matrix using a numerical path integral method. We explore the cases of long pulses, strong dot-phonon and dot-laser coupling, and high temperatures, which, up to now, have been inaccessible. We find that the phonon-induced damping of Rabi rotations is a nonmonotonic function of the laser field that is increasing at low fields and decreasing at high fields. This results in a reappearance of Rabi rotations at high fields. This phenomenon is of a general nature which occurs for all temperatures and carrier-phonon coupling strengths.

A shape-dependent superconducting resonance can be expected when an energy level associated with the transverse motion in a wire passes through the Fermi surface. We show that the recently observed width-dependent increase of $T_c$ in Al and Sn nanowires is a consequence of this shape-resonance effect.

Quantum confinement of electrons in a high-quality metallic nanofilm results in the formation of quantum-well states. The band of single-electron states is split into a series of subbands moving in energy with changing film thickness. When the bottom of such a subband passes through the Fermi surface, a shape resonance appears leading to oscillations of the critical temperature $T_c$ as a function of film thickness. We present a quantitative description of recent experimental data on the film-thickness dependence of $T_c$ in Pb nanofilms with atomic-scale uniformity in thickness [Science 306, 1915 (2004)] through a numerical solution of the Bogoliubov–de Gennes equations.

We analyze the dependence of the basic superconducting quantities—the order parameter, energy gap, and critical temperature—on the size and shape of a nanoscale superconductor in the clean limit. The Bogoliubov–de Gennes equations are solved numerically for a metallic nanowire with rectangular cross section. This makes it possible to vary the confining geometry, passing from a nanowire to a nanofilm and testing the sensitivity of the basic superconductive characteristics on the transverse size and shape of the nanowire. Strong size and shape superconducting resonances are found that depend on the geometry of the sample.

We study the long-time dynamics and the stationary nonequilibrium state of an optically driven quantum dot coupled to acoustic phonons using numerically exact real-time path integrals and a fourth-order correlation expansion. By exploring wide ranges of temperatures, carrier-phonon and carrier-light coupling strengths, we characterize the stationary nonequilibrium state and compare the exact solution to known, approximatively derived results. It is found that analytical calculations tend to overestimate the influence of the carrier-phonon coupling, particularly at low temperatures and in the weak-coupling regime. The possibility of controlling the dot occupation in the stationary nonequilibrium by varying the laser detuning is discussed. A comparison between the numerical methods identifies the range of validity of the correlation expansion, which in the long-time limit is found to be surprisingly wide.