Energy Exchange in Driven Open Quantum Systems at Strong Coupling

The time-dependent energy transfer in a driven quantum system strongly coupled to a heat bath is studied within an influence functional approach. Exact formal expressions for the statistics of energy dissipation into the different channels are derived. The general method is applied to the driven dissipative two-state system. It is shown that the energy flows obey a balance relation, and that, for strong coupling, the interaction may constitute the major dissipative channel. Results in analytic form are presented for the particular value $K=\frac{1}{2}$ of strong Ohmic dissipation. The energy flows show interesting behaviors including driving-induced coherences and quantum stochastic resonances. It is found that the general characteristics persists for $K$ near $\frac{1}{2}$.

Figure 1

A snapshot of ${\overline{E}}_{\mathrm{R}}(\overline{t})$ at time $\overline{t}\equiv 50/\gamma$ (solid curve) and ${\overline{E}}_{\mathrm{I}}$ (dashed curve) are plotted versus frequency $\omega$ in (a) and versus temperature $T$ in (b). The parameters are ${\epsilon }_0=10\gamma$, ${\epsilon }_1=5\gamma$, ${\omega }_c=5000\gamma$, plus $T=0.1\gamma$ in (a), and $\omega =5\gamma$ in (b). In the inset the constant-in-time contribution to the power $P_{\mathrm{R}}$ versus $\omega$ is plotted for ${\epsilon }_0=2{\epsilon }_1$ (solid blue line), ${\epsilon }_0={\epsilon }_1$ (dashed green line), and ${\epsilon }_0=0.4{\epsilon }_1$ (dotted black line). The arrows denote the positions of the ground and side frequencies in (a) and the quantum stochastic resonance in (b).

Figure 2

The absolute values of the amplitudes of the first harmonic $E_{R,1}$ (solid curve) and $E_{I,1}$ (dashed curve) are plotted versus driving frequency $\omega$. The parameters are the same as in Fig. 1. Both curves show resonances at ${\epsilon }_0/n$ (indicated by the arrows). Inset: phases of $E_{R,1}$ (solid curve) and $E_{I,1}$ (dashed curve) versus $\omega$. The two contributions are always in antiphase.