Quench dynamics of a dissipative quantum system: A renormalization group study

We study dissipation in a small quantum system coupled to an environment held in thermodynamic equilibrium. The relaxation dynamics of a system subject to an abrupt quench in the parameters of the underlying Hamiltonian is investigated using two complementary renormalization group approaches. The methods are applied to the Ohmic spin-boson model close to the coherent-to-incoherent transition. In particular, the role of non-Markovian memory for the relaxation before and after the quench of the spin-boson coupling and the Zeeman splitting of the up and down spin is investigated.

Figure 1

(a) Diagrammatic view of a quench propagator corresponding to Eq. (19) (time representation) or Eq. (22) (Laplace representation). (b) Diagrammatic series for a quench kernel; the first diagram is expressed in Eq. (24). The thick horizontal lines are propagators of a local system; diamonds are quench vertices; single circles depict tunneling vertices, while doubled circles represent an interaction vertex; the green lines connecting circles correspond to reservoir contractions.

Figure 2

Diagrammatic representation of the self-energy flow equation. The dot indicates the derivative w.r.t. $\Lambda$ and the slanted line is the single-scale propagator $S^{\Lambda }$.

Figure 3

Quantum-dot model considered within FRG.

Figure 4

Power-law scaling of $1-P\left(t\right)$ for $T_{K}t\ll 1$ in the relaxation protocol. The data are produced using the numerical solution of the full FRG flow equations. Symbols represent the time dependence of the logarithmic derivative for different values of $g$. Horizontal dashed lines show the prediction $1+g$ of Eq. (58) for the log-derivative (for the definition of this, see the main text). For $T_{K}t⪅10T_K/{\omega }_c$ band effects start to matter and deviations from power-law scaling appear. The arrows indicate the $g$ dependent $10T_K/{\omega }_c$ (with ${\omega }_c/t_h=2\times {10}^2$).

Figure 5

The nonanalyticities of the propagator ${[E+i{\Gamma }_1\left(E\right)]}^{-1}$ in Eq. (34) as a function of the complex variable $E$ for positive coupling $g>0$. The main branch cut (thick solid line) and two poles (circles) with attached second order in $g$ branch cuts (thick dashed lines) are shown. The thin dashed line shows the integration contour $\mathcal{C}$ of Eq. (34).

Figure 6

$\left|P\right(t\left)\right|$ for all time scales. For $g>0$ the competition of the two Eqs. (75) and (79) manifest in the sign change of $P\left(t\right)$, appearing as a dip on the logarithmic scale. The partially coherent ($g>0$, red squares) and incoherent ($g<0$, purple crosses) dynamics are compared.

Figure 7

Time evolution of the spin expectation value $P\left(t\right)$ in the quench protocol. The coupling is quenched from $g_i>0$ to $g_f=-g_i<0$. The time of the quench $t_q$ is varied as indicated by the vertical dotted lines. If $t_q$ is smaller than the time $t^{*}$ of the first zero of $P\left(t\right)$ in the absence of the quench, the non-Markovian memory transfers the nonmonotonic behavior to the time regime after the quench in which the time evolution is performed with a negative coupling constant. Note that the time axis is chosen to be $T_K^{i}t$ before the quench and $T_K^{f}t$ after it. Since $T_K^i$ and $T_K^f$ can differ by orders of magnitude scaling, the time axis in this way is physically meaningful. Upper panel: $P\left(t\right)$ obtained from the numerical solution of the FRG (symbols) and RTRG (lines) flow equations. Lower panel: the analytical RTRG result (89). Curves are shown only for times $T_K^f(t-t_q)\gtrsim 1$ for which this result is applicable.

Figure 8

Time evolution of the spin expectation value $P\left(t\right)$ in the quench protocol obtained from the numerical solution of the FRG flow equations. Here, the coupling is quenched from $g_i<0$ to $g_f=-g_i>0$. Note that the time axis is chosen to be $T_K^{i}t$ before the quench and $T_K^{f}t$ after it. Different lines correspond to different quench times $t_q$ increasing from left to right. Upper panel: $g_f<g_c$, that is, the time evolution even after the quench is monotonic even though it is performed with a positive coupling constant. The non-Markovian memory heavily affects the dynamics after the quench. The qualitative behavior is independent of $t_q$. Lower panel: $g_f>g_c$; nonmonotonic behavior is found after the quench. This is again independent of $t_q$.

Figure 9

Time evolution of the spin expectation value $P\left(t\right)$ after a quench of the Zeeman splitting from zero to $\epsilon /T_K=5$ obtained within FRG. The stationary value $P_{\mathrm{stat}}\ne 0$ was subtracted and the data are presented on a linear-log scale. The behavior for systems with different couplings [$g=0.0636$ (dashed red line) and $g=0.1910$ (dotted-dashed blue line)] does not significantly differ from the noninteracting case (black solid line). The inset shows $\left|P\right(t\left)\right|$ before the quench.

Figure 10

Time evolution of the spin expectation value $P\left(t\right)$ after a quench of the Zeeman splitting from $\epsilon /T_K=10$ to zero for different $g$ (lines) obtained within FRG. For comparison, $\left|P\right(t\left)\right|$ of the relaxation protocol is shown (symbols). Inset: $P\left(t\right)$ before the quench at $T_K{t}_q=22.5$.

Figure 11

Time dependence of the left (upper panel) and right (lower panel) currents in a two-lead setup with a bias voltage $V/T_K=10$ applied symmetrically across the dot. At time $T_K{t}_q=11.75$, that is, after reaching the stationary state with respect to the initial preparation, the level position $\epsilon$ is abruptly lowered from $10T_K$ to that being resonant with the left lead $\epsilon -V/2=0$. Black solid lines show the noninteracting case, while blue dashed lines are FRG data obtained for $g=0.3184$. Insets: Time evolution of the left and right currents with $\epsilon -V/2=0$ initially and no parameter quench at a later time.