Quantum dynamics of a two-level system in a structured environment: Numerical study beyond perturbation theory

The dissipative quantum dynamics of a two-level system interacting with a structured reservoir consisting of a damped harmonic mode is investigated by means of the numerically exact path integral Monte Carlo method. This approach provides benchmark results in a broad range of parameter space, in particular, in those domains which are not accessible by approximate methods and alternative numerical schemes, that is, strong coupling between system and harmonic mode and from low to high temperatures. For low temperatures the numerical data are quantitatively in agreement with the noninteracting blip approximation only in the regimes of very weak and very strong coupling. It turns out that the entangled dynamics of the two-level system and the harmonic mode is relatively robust so that its signatures are observable up to relatively strong friction and high temperatures. Nonequilibrium initial preparations of the reservoir with respect to the initial state of the system give rise, for strong interactions, to a stepwise decay of the population, thus displaying coherent dynamics of the bath. The impact of an additional Ohmic bath coupled directly to the two-level system is studied as well, including the case where both reservoirs carry different temperatures.

Figure 1

Spectral densities of a damped harmonic mode (solid) and an Ohmic environment with a cutoff frequency ${\omega }_c$ (dashed line) as used in this study (in arbitrary units).

Figure 2

Population dynamics for the resonant case $\Omega =\Delta$ with $\Gamma /\Delta =0.0625$, coupling strength $\alpha =0.3$ ($p/\Delta {}^4=3\alpha /4$), and various temperatures $k_{\mathrm{B}}T/(\hslash \Delta )=0$, 0.5, 1, 2, and 10 (from bottom to top). In all figures the error bars representing the statistical errors are smaller than the symbols used, the dotted line indicates the equilibrium population, and $k_{\mathrm{B}}T/(\hslash \Delta )$ is simply denoted by $T$.

Figure 3

Same as Fig. 2, but for fixed temperature $T=0$ and various couplings $\alpha$.

Figure 4

Population dynamics in the resonant case according to the NIBA (dashed line) and the numerically exact PIMC (solid line) for $T=0$ and different values for $\alpha$; other parameters are as in Fig. 2.

Figure 5

Population dynamics for the off-resonant case $\Omega =0.5\Delta$, with $\Gamma /\Delta =0.0625$, $T=0$, and various $\alpha$.

Figure 6

Same as Fig. 5, but for $\Omega =1.5\Delta$.

Figure 7

Population dynamics for the resonant (circles), the blue-detuned $\Omega =1.5\Delta$ (triangles), and the red-detuned $\Omega =0.5\Delta$ (diamonds) case for weak coupling $\alpha =0.1$; other parameters are as in Fig. 5.

Figure 8

Same as Fig. 7, but for strong coupling $\alpha =0.5$.

Figure 9

Population dynamics in the presence of a damped harmonic oscillator bath with $\Omega =\Delta$, without (open symbols) and with (filled symbols) an additional Ohmic reservoir at $\alpha =0.1$ and $k_{\mathrm{B}}T/(\hslash \Delta )=0$, 2, and 10.

Figure 10

Same as Fig. 9, but with $\alpha =0.3$ and $k_{\mathrm{B}}T/(\hslash \Delta )=0$ and 10/3.

Figure 11

Population dynamics in the presence of a damped harmonic mode reservoir at $T=0$ and an additional Ohmic background at temperatures $k_{\mathrm{B}}T/(\hslash \Delta )=0$, 1, and 10 (filled symbols) with $\alpha =0.1$; the corresponding data for identical temperatures are depicted by open symbols.

Figure 12

Population dynamics for an equilibrium initial bath preparation [$\mathrm{\sigma ̄}=-1$; open (gray) symbols] and for a nonequilibrium preparation [$\mathrm{\sigma ̄}=0$; filled (black) symbols] at $T=0$ for $\alpha =0.1$ (circles) and $\alpha =0.5$ (diamonds).

Figure 13

Same as Fig. 12, but for strong coupling: $\alpha =1$ (circles) and $\alpha =2$ (diamonds).