Pontus-Mpemba effects

Abstract

Mpemba effects occur after a sudden quench of control parameters if for ''far'' (or ''hot'') initial states with respect to a final target state, the relaxation time toward the target state is shorter than for ''close'' (or ''cold'') initial states. Following a strategy of fishermen in Pontus described by Aristotle, we introduce the Pontus-Mpemba effect as a two-step protocol which includes the time needed for preparing the system in the ''far'' initial state that can now be an arbitrary nonequilibrium state. Our protocol needs no parameter distance concept and applies to general (classical or quantum) systems. We find that all possible Pontus-Mpemba effects fall into three classes and illustrate the theory for open Markovian two-state quantum systems.

Figures (4)

• Figure 1: Velocity field $\dot{\rho}$ for the Markovian dynamics of a two-level system constrained to the $r_1$-$r_2$ plane ($r_3=0$) of the Bloch vector ${\bf r}(t)=(r_1,r_2,r_3)^T$ with $|{\bf r}|\le 1$. Here $\dot{\rho}$ can be represented by ${\bf v}({\bf r})=\dot{\bf r}$, see Eq. \ref{['v_vec']}. Arrows and colors represent the direction and amplitude $(v=|{\bf v}|)$ of ${\bf v}({\bf r})$, respectively. In panel (a) [panel (b)], the steady state reached at long times is F [A], with the corresponding parameters in Eq. \ref{['lindblad_first']} specified in the End Matter. The PME protocol compares the direct process ${\bf S}\to {\bf F}$ along trajectory $\Gamma_{\rm SF}$ [blue curve in panel (a)] to the two-step process composed of (i) $\bf{S}\to {\bf I}$ along trajectory $\Gamma_{\rm SA}$ [green curve in panel (b)] and (ii) ${\bf I} \to {\bf F}$ along $\Gamma_{\rm IF}$ [yellow curve in panel (a)]. If the two-step process is faster, the PME occurs.
• Figure 2: Trace distance ${\cal D}_T (\rho^{({\rm F}\, ({\rm A}))}(t), \rho_{\rm F})$ vs time $t$ using different target (F) and auxiliary (A) states as environment for Markovian two-state systems. As in Fig. \ref{['fig1']}, blue (green) curves show the time evolution of ${\cal D}_T (\rho^{({\rm F} \,({\rm A}))}(t), \rho_{\rm F})$ along ${\bf S}\to {\bf F}({\bf A})$ under the influence of the respective environment, and yellow curves show ${\cal D}_T (\rho^{({\rm F})}(t), \rho_{\rm F})$ along ${\bf I}\to {\bf F}$ for different intermediate states I. Dotted vertical and horizontal lines are guides to the eyes only. For parameter values, see End Matter. (a): Example where ${\cal D}_T (\rho^{({\rm A})}(t), \rho_{\rm F})$ has a minimum and a crossing point with ${\cal D}_T (\rho^{({\rm F})}(t), \rho_{\rm F})$. For three states I, yellow curves show the trace distance along $\Gamma_{\rm IF}$. These three cases realize all three PME types. (b): Example where ${\cal D}_T (\rho^{({\rm A})}(t), \rho_{\rm F})$ has a minimum but no crossing point with ${\cal D}_T (\rho^{({\rm F})}(t), \rho_{\rm F})$. (c): Example for the case in Eq. \ref{['case-3']}, where no minimum in ${\cal D}_T (\rho^{({\rm A})}(t), \rho_{\rm F})$ exists.
• Figure A1: Velocity field profiles for a two-level system constrained to the $r_1$-$r_2$ plane as in Fig. \ref{['fig1']}, with the Kossakowski matrices \ref{['quantum_kossakowski']}. (a): Classical dissipative dynamics induced by the target bath without Hamiltonian contribution, ${\bf h}^{({\rm F})}=0$. (b): Strong quantum effects due to a large Hamiltonian contribution, ${\bf h}^{({\rm A})}=(0,0,-10)^T$, resulting in a spiral-type trajectory $\Gamma_{\rm SA}$.
• Figure A2: Trace distances ${\cal D}_T (\rho^{({\rm F}\, ({\rm A}))}(t), \rho_{\rm F})$ vs time $t$ as in Fig. \ref{['fig2']} but for the parameters in Fig. \ref{['figA1']}. The blue (green) curves show the time evolution of ${\cal D}_T (\rho^{({\rm F}\, ({\rm A}))}(t), \rho_{\rm F})$ along ${\bf S}\to {\bf F} ({\bf A})$ under the influence of the respective environment, and the yellow curve shows ${\cal D}_T (\rho^{({\rm F})}(t), \rho_{\rm F})$ along ${\bf I}\to {\bf F}$ for an intermediate state I. Dotted vertical and horizontal lines are guides to the eyes only.