Oscillatory Dynamics and Non-Markovian Memory in Dissipative Quantum Systems

The nonequilibrium dynamics of a small quantum system coupled to a dissipative environment is studied. We show that (i) the oscillatory dynamics close to a coherent-to-incoherent transition is significantly different from the one of the classical damped harmonic oscillator and that (ii) non-Markovian memory plays a prominent role in the time evolution after a quantum quench.

Figure 1

Time evolution of a spin-$1/2$ prepared in its up state and coupled to an Ohmic boson bath at $t=0$. The spin expectation value $|P(t)|$ is shown for different couplings $\alpha$. Dips correspond to zeros; to make those visible we use a log $y$ axis scale. Main panel: comparison of the numerical solutions of the FRG and RTRG equations as well as the analytical result Eqs. (2, 3, 4) for $\alpha =0.4682$. Right inset: the same as in the main panel but for various $\alpha$ (analytical expression and FRG for $\alpha =0.5318$ only; for this coupling the curves are barely distinguishable). The dynamics in the coherent regime is very different to that of the classical damped oscillator. Left inset: FRG data for $1-P(t)$ at short times (dots) compared to the NIBA prediction (dashed lines).

Figure 2

Spin expectation value $P(t)$ of a spin-$1/2$ with the initial time evolution given by the spin-boson Hamiltonian and a coupling from the incoherent regime ${\alpha }_i>1/2$. At $t=t_q$ it is switched to ${\alpha }_f<1/2$ from the coherent one. The scaling limit is realized by $\Delta /{\omega }_c=1/200$. Times are restricted to $t>t_q$; for $0<t<t_q$; see Fig. 1. Left inset: comparison of numerical FRG and RTRG data; the non-Markovian memory completely suppresses the coherent “oscillatory”behavior. It only survives for ${\alpha }_f$ smaller than a critical coupling ${\alpha }_c$; see the upper [numerical solution of FRG equations], central [numerical solution of RTRG equations], and lower [analytical result Eq. (10)] panels. Right inset: dependence of ${\alpha }_c$ on the time $T_K^i{t}_q$ the evolution was performed in the incoherent regime (FRG data). Two different definitions of ${\alpha }_c$ are compared. In the first ${\alpha }_c$ is defined as the ${\alpha }_f$ at which the first zero of $P(t)$ can be observed, in the second as the ${\alpha }_f$ at which the curve first shows a nonmonotonicity. The two values barely differ.

Figure 3

The same as in Fig. 2 but for quenches from the coherent to the incoherent regime. The spin expectation value $|P(t)|$ for ${\alpha }_i=0.4682$ and different quench times $t_q$ (indicated by the vertical dotted lines) is shown. At the respective $t_q$ the $x$ axis scale is switched from $T_K^{i}t$ to $T_K^{f}t$. Main panel: data of the numerical solutions of the FRG and RTRG flow equations. The thin dashed line is an exponential term with rate $T_K^f/2$ and a subleading power-law correction. Inset: FRG data with and without the memory term.