Path-integral description of combined Hamiltonian and non-Hamiltonian dynamics in quantum dissipative systems

We present a numerical path-integral iteration scheme for the low-dimensional reduced density matrix of a time-dependent quantum dissipative system. Our approach simultaneously accounts for the combined action of a microscopically modeled pure-dephasing-type coupling to a continuum of harmonic oscillators representing, e.g., phonons, and further environmental interactions inducing non-Hamiltonian dynamics in the inner system represented, e.g., by Lindblad-type dissipation or relaxation. Our formulation of the path-integral method allows for a numerically exact treatment of the coupling to the oscillator modes and moreover is general enough to provide a natural way to include Markovian processes that are sufficiently described by rate equations. We apply this new formalism to a model of a single semiconductor quantum dot which includes the coupling to longitudinal acoustic phonons for two cases: (a) external laser excitation taking into account a phenomenological radiative decay of the excited dot state and (b) a coupling of the quantum dot to a single mode of an optical cavity taking into account cavity photon losses.

Figure 1

Time-dependent exciton occupation $C_X$ of a QD calculated by the path-integral formalism for constant driving with field strength $f=1.0{\mathrm{ps}}^{-1}$ for resonant [(a) and (b)] and detuned [(c) and (d)] excitation ($\mathrm{\Delta }=1.0$ meV) for different values of the radiative decay rate (see legend). Left column: without phonon interaction, right column: including phonon interaction at temperature $T=100$ K in addition to the radiative decay.

Figure 2

Time-dependent exciton occupation $C_X$ of a QD for off-resonant ($\mathrm{\Delta }=1.0$ meV) cw excitation for different field strengths (see legend) including radiative decay $\left(\gamma =0.05{\text{ps}}^{-1}\right)$ without the phonon-interaction (a) and including phonons at temperature $T=1$ K (b). (c) Stationary exciton occupation reached at long times when only the phonon interaction is present (green, dashed), when only the radiative decay is present (blue, dashed) and when both relaxation mechanisms are present (red, solid).

Figure 3

Comparison of the time-dependent exciton occupation of a QD for resonant (left) and detuned (right) cw excitation calculated by the path-integral method (red) and the weak coupling theory (blue). The radiative decay rate has been set to $\gamma =0.05{\mathrm{ps}}^{-1}$ and a relatively high field strength of $f=4.0{\mathrm{ps}}^{-1}$ was chosen.

Figure 4

Time-dependent occupation of the excited state $C_X$ of a QD inside a cavity for resonant (green) and off-resonant (red: $\mathrm{\Delta }=1.0$ meV, orange: $\mathrm{\Delta }=-1.0$ meV) coupling. The black dots show the results of a hybrid approach that is only applicable in the resonant case (see text). (Left) $T=1$ K. (Right) $T=100$ K. $\kappa =0.1{\mathrm{ps}}^{-1}$.