Open Quantum Systems’ Decay across Time

The description of an open quantum system’s decay almost always requires several approximations so as to remain tractable. In this Letter, we first revisit the meaning, domain, and seeming contradictions of a few of the most widely used of such approximations: (semigroup) Markovianity, linear response theory, Wigner-Weisskopf approximation, and rotating-wave approximation. Second, we derive an effective time-dependent decay theory and corresponding generalized quantum regression relations for an open quantum system linearly coupled to an environment. This theory covers all timescales and subsumes the Markovian and linear-response results as limiting cases. Finally, we apply our theory to the phenomenon of quantum friction.

Figure 1

Semigroup Markovian quantum dynamics are incompatible with LRT. According to the QRT, semigroup Markovianity implies exponential decay of correlation functions and hence Lorentzian power spectra. LRT, however, requires non-Lorentzian spectra.

Figure 2

Decay of an initially excited open quantum system as produced by the semigroup Markovian rate ${\mathrm{\Gamma }}_{\mathrm{mkv}}$, second-order TCL rate ${\mathrm{\Gamma }}_{\mathrm{tcl}}$, LRT rate ${\mathrm{\Gamma }}_{\mathrm{lrt}}$, and product rate ${\mathrm{\Gamma }}_{\mathrm{pro}}$. The rates—including the sliding-window time average ${\overline{\mathrm{\Gamma }}}_{\mathrm{pro}}$ of the product rate—are shown in the top left. The time average merely serves to guide the eye. The $\mathrm{TCL}\times \mathrm{LRT}$ decay depicted on linear (upper right), log (lower left), and log-log (lower right) scales is the result of the original, nonaveraged, and rapidly oscillating rate ${\mathrm{\Gamma }}_{\mathrm{pro}}(t)$. $S_B$ and $S_D$ are chosen Drude-Lorentzian with zero detuning and ${\omega }_S+\delta {\omega }_{\mathrm{stat}}=100{\mathrm{\Gamma }}_{\mathrm{stat}}=10\gamma$. In all panels temporal regimes are indicated as color shading: short (red), intermediate (gray), and long (blue).