Energy flow in periodic thermodynamics

A key quantity characterizing a time-periodically forced quantum system coupled to a heat bath is the energy flowing in the steady state through the system into the bath, where it is dissipated. We derive a general expression which allows one to compute this energy dissipation rate for a heat bath consisting of a large number of harmonic oscillators and work out two analytically solvable model examples. In particular, we distinguish between genuine transitions effectuating a change of the systems's Floquet state and pseudotransitions preserving that state; the latter are shown to yield an important contribution to the total dissipation rate. Our results suggest possible driving-mediated heating and cooling schemes on the quantum level. They also indicate that a driven system does not necessarily occupy only a single Floquet state when in contact with a zero-temperature bath.

Figure 1

The ac Stark shift for the two-level system driven by a circularly polarized radiation field. Shown are those representatives of the quasienergies $\varepsilon$ which connect continuously to the bare energy levels $\pm \hslash {\omega }_0/2$. In the case of red detuning, when $\omega <{\omega }_0$, the two quasienergies repel each other with increasing scaled amplitude $\mu F/\left(\hslash {\omega }_0\right)$ (solid lines: $\omega /{\omega }_0=0.5$), whereas they approach each other and cross for blue detuning, when $\omega >{\omega }_0$ (dashed lines: $\omega /{\omega }_0=1.5$).

Figure 2

Population of the Floquet state labeled with a minus ($-$) according to Eqs. (82) and (83) for $\omega /{\omega }_0=1.5$ and scaled temperatures $k_{\mathrm{B}}T/\left(\hslash {\omega }_0\right)=0.1$ (solid line), $0.5$ (dotted), $1.0$ (dashed), and $5.0$ (double-dash-dotted).

Figure 3

Normalized energy dissipation rate $R^{\left(0\right)}=R/\left(\hslash {\omega }_0{\Gamma }_0\right)$ for the situations considered in Fig. 2, that is, for $\omega /{\omega }_0=1.5$ and scaled temperatures $k_{\mathrm{B}}T/\left(\hslash {\omega }_0\right)=0.1$ (solid line), $0.5$ (dotted), $1.0$ (dashed), and $5.0$ (double-dash-dotted).

Figure 4

Normalized energy dissipation rate $R^{\left(0\right)}=R/\left(\hslash {\omega }_0{\Gamma }_0\right)$ for the scaled temperature $k_{\mathrm{B}}T/\left(\hslash {\omega }_0\right)=1.0$ and scaled driving strengths $\mu F/\left(\hslash {\omega }_0\right)=0.25$ (solid line), $0.5$ (dashed), $0.75$ (dotted), $1.0$ (double-dash-dotted), and $1.25$ (dash-double-dotted).