Dynamical regimes of dissipative quantum systems

We reveal several distinct regimes of the relaxation dynamics of a small quantum system coupled to an environment within the plane of the dissipation strength and the reservoir temperature. This is achieved by discriminating between coherent dynamics with damped oscillatory behavior on all time scales, partially coherent behavior being nonmonotonic at intermediate times but monotonic at large ones, and purely monotonic incoherent decay. Surprisingly, elevated temperature can render the system “more coherent” by inducing a transition from the partially coherent to the coherent regime. This provides a refined view on the relaxation dynamics of open quantum systems.

Figure 1

Diagram showing the extend of the incoherent, asymptotically, and partially coherent sectors. The critical temperatures $T_{\mathrm{c}1}$ (lower branch) and $T_{\mathrm{c}2}$ (upper branch) obtained from the numerical solution of the RTRG equations (4) are shown as circles; solid lines correspond to the approximations of Eqs. (7) (lower) and (8) (upper). The insets exemplify the time evolution of the spin expectation value $P\left(t\right)$ in the different regimes for $\alpha =0.45$ and $T/T_{\mathrm{K}}=0.01$ (partially coherent), $0.3$ (asymptotically coherent), $0.59$ (incoherent), with the Kondo scale $T_{\mathrm{K}}$.

Figure 2

Time dependence of the spin expectation value $\left|P\right(t\left)\right|$ on a linear-logarithmic scale [(a) FRG, (b) RTRG]. Both approaches capture the transition from partially coherent to asymptotically coherent to incoherent dynamics when raising $T$.

Figure 3

Nonanalytical features of the propagator ${\Pi }_1\left(E\right)$ at coupling $\alpha =0.45$. Density plots of $|{\partial }_{\mathrm{Im}E}\mathrm{Re}{\Pi }_1+{\partial }_{\mathrm{Re}E}\mathrm{Im}{\Pi }_1|$ as a function of the complex variable $E$ are shown. Dashed horizontal lines indicate the zero temperature decay rates for the poles $T_{\mathrm{K}}$ and branching point $T_{\mathrm{K}}/2$. Vertical ones indicate the frequency $\pm \Omega$ of the poles at $T=0$ ($\Omega \approx \pi g{T}_{\mathrm{K}}$[3, 4, 5, 16]).