Optimal speed-up of multi-step Pontus-Mpemba protocols

Abstract

The classical Mpemba effect is the counterintuitive phenomenon where hotter water freezes faster than colder water due to the breakdown of Newton's law of cooling after a sudden temperature quench. The genuine nonequilibrium post-quench dynamics allows the system to evolve along effective shortcuts absent in the quasi-static regime. When the time needed for preparing the (classical or quantum) system in the hotter initial state is included, we encounter so-called Pontus-Mpemba effects. We here investigate multi-step Pontus-Mpemba protocols for open quantum systems whose dynamics is governed by time-inhomogeneous Lindblad master equations. In the limit of infinitely many steps, one arrives at continuous Pontus-Mpemba protocols. We study the crossover between the quasi-static and the sudden-quench regime, showing the presence of dynamically generated shortcuts achieved for time-dependent dissipation rates. Time-dependent rates can also cause non-Markovian behavior, highlighting the existence of rich dynamical regimes accessible beyond the Markovian framework.

Figures (5)

• Figure 1: Two-step Pontus-Mpemba effects in open two-level systems described by Eq. \ref{['eq:Lindblad_ME']}. (a) Trace distance $\mathcal{D}_T \left( \rho(t), \rho_\mathrm{F} \right)$ vs time $t$, comparing the direct (first system copy) quench protocol to two-step Pontus-Mpemba (second system copy) protocols. We use $|\mathbf{h}_\mathrm{S}|=1$ as energy unit. For both system copies, the initial state S is chosen as steady state, see Eq. \ref{['eq:steady-state']}, for $\mathbf{h}_\mathrm{S}=(0,0.998,0.062)$ and $(\gamma_{+},\gamma_{-},\gamma_{z})_{\rm S}=(0,0.2,0)$, while the final state F is fixed by $\mathbf{h}_\mathrm{F}=(0,-0.966,0.258)$ and the same rates, $(\gamma_+,\gamma_-,\gamma_{z})_{\rm F}=(0,0.2,0)$. For the second system copy, the auxiliary state A is chosen as steady state for $\mathbf{h}_\mathrm{A}=(0,2,2)$ and $(\gamma_+,\gamma_-,\gamma_z)_{\rm A}=(1,0,0)$. The trace distance for the first system copy is shown as red curve. The yellow curve shows the time dependence of ${\cal D}_T(\rho(t),\rho_{\rm F})$ for the second system copy under the attractor A, where blue, green, and black curves correspond to different choices for the intermediate time $t_{\rm I}$. The weak type-A Pontus-Mpemba effect is realized for $t_{\rm I}=t_\mathrm{I_{wA}}$, with the blue curve showing ${\cal D}_T(\rho(t),\rho_{\rm F})$ for $t>t_{\rm I}$. The weak type-B effect with $t_{\rm I}=t_\mathrm{I_{wB}}$ corresponds to the green curve for $t>t_{\rm I}$. Finally, the strong Pontus-Mpemba effect is achieved for $t_{\rm I}=t_\mathrm{I_{s}}$, with the black curve for $t>t_{\rm I}$. All trajectories reach the same final state F for $t\to \infty$. (b) Bloch ball representation of the system dynamics. The red curve represents the direct $\mathrm{S}\rightarrow \mathrm{F}$ trajectory of the Bloch vector, and the yellow curve shows the $\mathrm{S}\rightarrow \mathrm{A}$ trajectory for the second system copy. The blue curve connects $\mathrm{I_{wA}} \rightarrow \mathrm{F}$, corresponding to the blue curve in panel (a), realizing a weak type-A effect. In panels (b)--(d) the velocity field amplitude for the direct protocol is displayed on a color scale using a regular grid within the Bloch ball. The color bar is shown in panel (c). (c) Same as in (b) but for the green trajectory in (a), realizing the weak type-B effect. (d): Same as in (b) but for the black trajectory in (a), realizing the strong Pontus-Mpemba effect.
• Figure 2: Continuous Pontus-Mpemba protocol for an open two-level system described by Eq. \ref{['eq:Lindblad_ME']} with the time-dependent rates \ref{['eq:gamma_time-dependent']}. (a) Trace distance $\mathcal{D}_T \left( \rho(t), \rho_\mathrm{F} \right)$ vs time $t$ for the direct protocol and for the continuous Pontus-Mpemba protocol. We use $|\mathbf{h}_\mathrm{S}|=1$ as energy unit. In both cases, the initial state S is the steady state \ref{['eq:steady-state']} corresponding to $\mathbf{h}_\mathrm{S}=(0.707,0.707,0)$ and $(\gamma_+,\gamma_-,\gamma_z)_{\rm S}=(0.5,0.1,0)$, while the final state F is determined by $\mathbf{h}_\mathrm{F}=\mathbf{h}_\mathrm{S}$ and $(\gamma_+,\gamma_-,\gamma_z)_{\rm F}=(0.01,0.05,0)$. The trace distance for the direct protocol ${\rm S}\rightarrow {\rm F}$ is shown as red curve. The green (blue) curve corresponds to the trace distance between S and F for a trajectory described by the time-dependent parameters in Eq. \ref{['eq:gamma_time-dependent']} with $\omega=0$ and $\kappa=0.035$ ($\kappa=0.2$). In all cases, the state F is reached for $t\to \infty$, and the system dynamics is Markovian (since $\omega=0$). The dashed line indicates the cutoff $\epsilon=10^{-4}$ for the trace distance. (b) Bloch ball representation of the system dynamics. The red curve represents the direct sudden-quench curve $\mathrm{S}\rightarrow \mathrm{F}$ for $\mathbf{r}(t)$, corresponding to the red curve in (a). Only the portion of the Bloch ball with $|\mathbf{r}|<0.25$ is shown. The velocity field amplitude for the F attractor is shown using color scales on a regular grid inside the Bloch ball; the color bar is shown in (c). (c) Same as (b) but for the blue curve in (a), realizing the continuous Pontus-Mpemba effect. The shown velocity field corresponds to the attractor S for the initial state. (d) Same as (c) but for the green curve in (a). Here the continuous Pontus-Mpemba effect does not take place.
• Figure 3: Comparison between direct and continuous Pontus-Mpemba protocols for open Markovian two-state systems. (a) Trace distance $\mathcal{D}_T \left( \rho(t), \rho_\mathrm{F} \right)$ vs time $t$. For both protocols, the initial state S is the steady state \ref{['eq:steady-state']} corresponding to $\mathbf{h}_\mathrm{S}=(0.183,0.183,-0.966)$ and $(\gamma_{+},\gamma_-,\gamma_z)_{\rm S}=(0.5,0.1,0),$ while the final state F is determined by $\mathbf{h}_\mathrm{F}=\mathbf{h}_\mathrm{S}$ and $(\gamma_{+},\gamma_-,\gamma_z)_{\rm F}=(0.1,0.5,0)$. The trace distance for the direct protocol ${\rm S}\rightarrow{\rm F}$ is shown as red curve; the dashed line is the cutoff $\epsilon$. The blue curve shows the trace distance between S and F for the time-dependent rates in Eq. \ref{['eq:gamma_time-dependent']} with $\kappa=0.6$ and $\omega=0.2$. Both protocols reach the same final state F for $t\to \infty$. Clearly, the continuous Pontus-Mpemba effect is realized, and a crossing of the trace distance curves takes place at $t\approx 13$. (b) Same as (a) but for $\kappa=0.4$ and $\omega=0.45$. The inconclusive regime is observed.
• Figure 4: Color-scale plots of the gain function $G(\kappa,\omega)$ in Eq. \ref{['gain_function']} for $\omega=0$, where $G>0$ indicates a speed-up under to the continuous Pontus-Mpemba protocol with respect to the direct sudden-quench protocol. Note that the dynamics is then always Markovian. (a)$G$ in the $\kappa$--$\theta$ plane, where $\theta$ is the angle between $\mathbf{h}_\mathrm{F}=\mathbf{h}_\mathrm{S}=(\sin{\theta},0,\cos{\theta})$ and the $z$-axis. The initial state S is determined by $(\gamma_{+},\gamma_-,\gamma_z)_\mathrm{S}=(0.75, 0.75, 0.75)$, and the final state F corresponds to $(\gamma_{+},\gamma_-,\gamma_z)_\mathrm{F}=(0.05, 0.1, 0.15).$(b) Same as (a) but for $(\gamma_{+},\gamma_-,\gamma_z)_\mathrm{S}=(0.5,0.1,0)$ and $(\gamma_{+},\gamma_-,\gamma_z)_\mathrm{F}=(0.1,0.5,0)$.
• Figure 5: Color-scale plots for the gain function $G(\kappa,\omega)$ in Eq. \ref{['gain_function']} for $\omega>0$ in Eq. \ref{['eq:gamma_time-dependent']}. Dashed lines separate Markovian (below) and non-Markovian (above) regimes, see Eq. \ref{['nMB']}. Gray areas correspond to the inconclusive regime. (a)$G$ in the $\kappa$--$\omega$ plane for $\mathbf{h}=\mathbf{h}_\mathrm{F}=\mathbf{h}_\mathrm{S}=(1,0,0)$ perpendicular to the $z$-axis. All other parameters are as in Fig. \ref{['fig:fig4']}(a). (b) Same as (a) but for $\mathbf{h}=(0,0,1)$ parallel to the $z$-axis. (Again, other parameters are as in Fig. \ref{['fig:fig4']}(a).)