Mpemba effect in self-contained quantum refrigerators: Accelerated cooling

Abstract

We consider the qubit-qutrit model of self-contained quantum refrigerator and observe the quantum Mpemba effect in its cooling. In this system, the qutrit acts as the refrigerator while the qubit is to be cooled. The entire system is coupled to three bosonic heat baths, due to which the dynamics of the system is governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation. We investigate the Liouvillian that generates the dynamics of the system and find that it has a block diagonal form. The dynamics of each element of the system's density matrix can be determined by solving the dynamical equation of the corresponding block that contains it. We find that the steady state belongs to the block containing only the diagonal elements in the energy basis. We numerically solve for the steady state and investigate the steady-state cooling over a significant region of the parameter space. Moreover, we demonstrate the quantum Mpemba effect in the refrigerator: a Mpemba state obtained by applying a unitary on the equilibrium state of the system reaches the steady state faster than the equilibrium state, despite the Mpemba state being initially farther away from the steady state. The Mpemba state thus leads to an acceleration in cooling of the cold qubit. We also find that both local and global unitaries on the qubit-qutrit system can generate the Mpemba state. Finally, we study the effect of the system-bath couplings on the Mpemba effect.

Figures (7)

• Figure 1: Schematic diagram of self-contained refrigerator and Mpemba effect. (a) We present a schematic diagram of the qubit-qutrit self-contained refrigerator coupled to three baths. The circular object labeled as $A$ is the qubit that is to be cooled, with its energy of the excited level being $E_0$. The qutrit (labeled as $B$ and act as the spiral) is given by the elliptical object with excited energy levels given by $E_1$ and $E_2$. In both qubit and qutrit, the ground state is set at zero. The three squares are the three baths, with each labeled by its temperature. The cold bath at temperature $T_c$ couples the transition between the two levels of the qubit $A$. The hot bath at $T_h$ couples the transition between the middle level and ground level of the qutrit $B$, whereas the work bath at $T_w$ couples the transition between the highest level and ground level. We show the evolution of a distance-like quantity $D(\rho_{in}, \rho_{eq})$, which measures how far the system’s state is from the equilibrium steady state $\rho_{eq}$. Two initial states are considered: $\rho^{1}_{{in}}$, initially farther from equilibrium (shown in green), and $\rho^{2}_{{in}}$, initially closer to equilibrium (shown in red). Despite starting farther away, $\rho^{1}_{{in}}$ approaches the equilibrium state faster than $\rho^{2}_{{in}}$, illustrating the Mpemba effect. The trajectories intersect at the Mpemba time $t_M$, at which both states are equally distant from equilibrium. When $D$ reaches zero, the system has fully relaxed to the equilibrium state. The state $\rho^{1}_{{in}}$ equilibrates at time $t_1$, while $\rho^{2}_{{in}}$ equilibrates later, at time $t_2$, with $t_2 > t_1$.
• Figure 2: Generator of dynamics. We present a schematic showing the block-diagonal form of the Liouvillian of the self-contained qubit-qutrit refrigerator. The largest block $\Lambda^\text{diag}$ in yellow, corresponds to the dynamics of the diagonal elements of the density matrix in the energy eigenbasis and is given by Eq. \ref{['diagonal']}. The smaller four blocks in green, are of dimension $2\times2$ each and are given by Eq. \ref{['4blocks']}. The other elements (shown here in red circles) do not couple to any other element and their eigenvalues $\lambda_{ij}$ are given in Eq \ref{['liouv-eigenvals']}.
• Figure 3: Steady state of self-contained quantum refrigerator. (a) We present the change in the quantity $\Delta T = T_s - T^0_c$ with the variation in qubit-qutrit coupling $g$ and coupling between system and $h$ and $w$ bath with $\kappa_h = \kappa_w = \kappa$ and $\kappa_c = 10^{-3}$. Here, $T_s$ is the temperature of the qubit at the steady state whereas $T_c^0$ is the initial temperature of the qubit. Thus, $\Delta T$ measures the cooling capacity of the quantum refrigerator. (b) We present the variation of $\Delta T$ with the variation of $\kappa_c = \kappa_1$ and $\kappa_h = \kappa_2$ with $\kappa_w = 10^{-3}$ and $g = 0.5$. In both the cases, $E_0 = 0.7$, $E_1 = 1.0$ with $T_h = 3.0$ and $T_c = T_w = 1.0$, consequently $T_c^0 = 1$. The solid-black line denotes $\Delta T = 0$, beyond which the system no longer provides steady-state cooling.
• Figure 4: Mpemba effect in the absence of cold bath for dynamics due to $\Lambda_\text{diag}^y$. We consider the dynamics given by $\Lambda_y^\text{diag}$ (Eq. \ref{['diagonal-y']}), where we set $E = 1$, $T = 1$ and set the initial temperature of the cold qubit at $T_c(0) = 0.7$. The solid-blue line is the trace-distance of the state $\rho^y_0$ of the form of Eq. \ref{['init-state']}, while the dashed-orange line is the variation of the trace-distance of the Mpemba state $\rho_M^y$ from the steady state. The Mpemba state is obtained by applying the unitary $U^y = U_2^yU_1^y$ as given in Eq. \ref{['analytic-unitary']}. For the numerical demonstration we have set $g = 10^{-10}$ and $\kappa = \kappa_h = \kappa_w = 10^{-4}$. The inset magnifies the dynamics at the beginning, where we observe that the Mpemba state is initially farther away from the steady state that the thermal initial state.
• Figure 5: Mpemba effect in the absence of cold bath for dynamics due to the general $\Lambda_\text{diag}$. We present the trace distance $D(\tilde{\rho}_\text{th}^{AB}(t),\tau)$ in solid-blue and $D(\tilde{\rho}^{AB}_{1,\text{g}}(t),\tau)$ in dashed-orange line. Here $\tilde{\rho}_\text{th}^{AB}(t)$ and $\tilde{\rho}^{AB}_{1,\text{g}}(t)$ are the time-evolved states corresponding to the the initially thermal state and the Mpemba state respectively. In the inset we clearly observe that initially, $D(\tilde{\rho}_\text{th}^{AB}(0),\tau) < D(\tilde{\rho}^{AB}_{1,\text{g}}(0),\tau)$. The state $\tilde{\rho}^{AB}_{2,\text{l}}(0)$ demonstrates the Mpemba effect and reaches the steady state faster than the initially thermal state. Here we have considered the dynamics due to the general $\Lambda_\text{diag}$ (Eq. \ref{['diagonal']}), with $E_0 = 0.7$, $E_1 = 1.0$, $T_c(0) = 0.7$, $T_w = 1.0$, $T_h = 2.0$, $g =10^{-3}$, $\kappa_c = 0$ and $\kappa_h = \kappa_w = 10^{-4}$.