Speeding up Pontus-Mpemba effects via dynamical phase transitions

TL;DR

The paper establishes a link between dynamical phase transitions (DPTs) and quantum Mpemba effects (PME) in open quantum systems. Using a time-dependent self-consistent mean-field (SCMF) approach to a 1D lattice Gross-Neveu–type model with Lindblad dissipation, it identifies a long metastable region ${\cal M}$ preceding a DPT at time $t_*$, and shows that PME protocols can exploit detours around this region to accelerate relaxation to a target state. The main finding is that two-step PME protocols, which bypass the metastable DPT region by evolving through an auxiliary disordered phase, can outperform direct single-step protocols, with a robust speedup factor $\eta>1$ that is only weakly dependent on system size. The results suggest a general strategy to optimize quantum state preparation and cooling in open systems and may extend to higher-dimensional correlated models and various experimental platforms such as ultracold atoms and ion traps.

Abstract

We demonstrate that open quantum systems exhibiting dynamical phase transitions (DPTs) allow for efficient protocols implementing the Pontus-Mpemba effect. The relaxation speed-up toward a predesignated target state is tied to the existence of a long metastable time window preceding the DPT and can be exploited in applications to systematically optimize quantum protocols. As paradigmatic example for the connection between DPTs and quantum Mpemba effects, we study one-dimensional (1D) interacting lattice fermions corresponding to a dissipative variant of the Gross-Neveu (GN) model.

Figures (10)

• Figure 1: Phase diagram of the model (\ref{['s.2.1']}) in the $\mu$--$g$ plane for $J=1$, $k_B T=0.05$, and $\gamma=0.01$. The four points $P_i= (\mu_i, g_i)$ marked by stars correspond to $P_1=(0, 1.1)$, $P_2=(0.5,1.1)$, $P_3=(0.8,1.1)$, and $P_4=(0.5,0.9)$, respectively. Results were obtained from the steady-state limit of Eq. \ref{['tlme.2']}. The phases OP (blue), CP (orange), and DP (red) correspond to the ordered phase, crystal phase, and disordered phase, respectively; for details, see main text. Inset: Order parameter profile $m(x_j)$ at site $x_j=ja$ (with $a=1$), see Eq. \ref{['orderp']}, for the four points $P_i$ at system size $L=100$. The blue curve corresponds to $P_1$, the red curve to $P_4$, and the orange and green curves to $P_2$ and $P_3$, respectively.
• Figure 2: Color-scale plot for the time evolution of the lowest 21 Fourier modes $\hat{m}(\nu,t)$ of the order parameter $m_j$ in Eq. \ref{['orderp']} under parameter quenches between different regions of the phase diagram in Fig. \ref{['fig1']}. We use $L=100$ and $(\gamma,T)$ as in Fig. \ref{['fig1']}. Green arrows mark the critical time $t_*$ corresponding to DPTs. Red arrows mark the time scales for a relaxation crossover. Different panels correspond to (see main text for details): (a) Quench from CP to DP. (b) Quench between two states in the CP. (c) Quench from OP to CP. (d) Quench from OP to DP.
• Figure 3: PME for the GN model. Main panel: Dimensionless (normalized) order parameter distance $M(t)$ vs time $t$(in units of $1/J$), see Eq. \ref{['order_parameter_distance']}, computed from Eq. \ref{['tlme.2']} for two different protocols from ${\bf S}=P_2 \to {\bf F}=P_4$, see Fig. \ref{['fig1']}. Notice the semi-logarithmic scales. We use $L=100, \gamma=0.01,$ and $k_BT=0.05$. The blue curve corresponds to the single-step direct quench ${\bf S}\to {\bf F}$. The red curve corresponds to a two-step process, where the system first evolves along ${\bf S}\to {\bf A}=P_3$. At $t= 960$, the state I is reached. Now a second quench takes the system from ${\bf I}\to {\bf F}$. The orange-dashed curve is for the single-step protocol ${\bf S}\to {\bf A}$. Inset: Location of the parameter states $\{{\bf S},{\bf F},{\bf A},{\bf I}\}$ in the phase diagram, see Fig. \ref{['fig1']}. The black curve indicates the direct step ${\bf S}\to {\bf F}$, the dark red curve the two-step protocol ${\bf S}\to {\bf I}\to {\bf F}$. Note that the states along these trajectories are actually nonequilibrium states.
• Figure 4: Same as in Fig. \ref{['fig3']} but for ${\bf S}=P_1, {\bf F}=P_2$ and ${\bf A}=P_4$. The intermediate point I is reached at $t=200$ along the trajectory ${\bf S} \to {\bf A}$. Main panel: The blue curve shows the dynamics under the single-step protocol ${\bf S}\to {\bf F}$, the orange-dashed curve is for a single-step evolution ${\bf S}\to {\bf A}$. The red curve refers to the two-step protocol ${\bf S}\to {\bf I}\to {\bf F}$. Inset: Location of the parameter states $\{{\bf S},{\bf F},{\bf A},{\bf I}\}$ in the phase diagram, see Fig. \ref{['fig1']}. The black curve indicates the direct step ${\bf S}\to {\bf F}$, the dark red curve the two-step protocol ${\bf S}\to {\bf I}\to {\bf F}$. Note that the states along these trajectories are actually nonequilibrium states.
• Figure 5: PME speedup ratio $\eta$ vs system size $L$ for the protocol shown in the main panel of Fig. \ref{['fig3']}. For details, see main text.