Quantum Mpemba effect from initial system-reservoir entanglement

Abstract

The Mpemba effect -- where hot systems cool faster than colder ones -- has intrigued both classical and quantum thermodynamics. As compared to classical systems, quantum systems add complexity due to quantum correlations. Recent works have explored anomalous relaxation and Mpemba-like effects in several quantum systems, considering isolated systems at zero temperature or open systems in contact with reservoirs under Markovian or non-Markovian dynamics. However, these models typically assume an initial unentangled system-bath state, overlooking the role of initial system-environment correlations. Here we propose a type of quantum Mpemba effect, distinct from the strong Mpemba effect, originating from initial system-bath entanglement solely. It is shown that the degree of initial entanglement significantly influences the early relaxation dynamics, with certain conditions causing backflow and retarded thermalization. As an example, we investigate the spontaneous emission of a two-level atom in a photonic waveguide at zero temperature, where an initial atom-photon entangled state results in delayed relaxation and pronounced Mpemba effect. These findings highlight the crucial role of quantum correlations in thermalization processes and open new avenues for identifying and engineering quantum Mpemba phenomena. Controlling relaxation dynamics through system-environment entanglement may have potential applications in quantum thermal machines, state initialization protocols, and quantum information processing, where precise control over thermalization is essential.

Figures (2)

• Figure 1: Schematic of the two kinds of quantum Mpemba effect arising (a) from different relaxation rates of states 1 and 2 (strong Mpemba effect), and (b) from an initial entangled system-bath state. In (a) the initial state of the composite system is the canonical product state $\rho(0)=\rho_S(0) \otimes \rho_B$, where $\rho_B$ is the Gibbs state of the bath at temperature $T$. The two different initial states 1 and 2 of the system, with density operators $\rho_S^{(1)}(0)$ and $\rho_S^{(2)}(0)$, display different decay rates. In (b) state 1 is the same as in (a), corresponding to an initial uncorrelated product state of the composite system, while 2 corresponds to an initial entangled system-bath state, as discussed in the main text.
• Figure 2: (a) Schematic of an array of coupled optical cavities with a two level atom placed in the resonator of index $l=0$. $J$ is the hopping rate of photons between adjacent resonators of the array. (b) Numerically-computed temporal behavior of the trace distance $D(\rho_S(t))$, which is equal to the survival probability $|c_a(t)|^2$ to find the atom in the excited state $|e \rangle$, corresponding to three different initial conditions $\rho(0)$ of the atom-photon field density operator in the single excitation sector and for parameter values $g_0/J=0.2$ and $\omega_0=\omega_c$. Curve 1 corresponds to the canonical initial condition $\rho(0)=|e \rangle \langle e| \otimes |0 \rangle \langle 0|$, with the atom initially in the excited state and the photon field in the vacuum state $|0 \rangle$. The decay curve $|c_a(t)|^2=|A(t)|^2$ is very well fitted by an exponential curve $\exp(- \Gamma t)$, with the spontaneous emission rate $\Gamma=g_0^2/J$ given by the Fermi golden rule. Curve 2 corresponds to the initial entangled atom-photon state $\rho(0)=| \psi(0) \rangle \otimes \langle \psi(0)|$, with $| \psi(0) \rangle$ given by Eq.(8) with $t_f=20/J$. Finally, curve 3 corresponds to the the initial entangled atom-photon state $\rho(0)=| \psi(0) \rangle \otimes \langle \psi(0)|$, with $| \psi(0) \rangle$ given by Eqs.(9) and (10) with $L=20$ The inset in (b) displays the three decay curves on a log time scale, clearly showing that the long-time decay behavior is the same for the three initial conditions.