We explore the role of exceptional points and complex eigenvalues on the occurrence of the quantum Mpemba effect. To this end, we study a two-level driven dissipative system subjected to an oscillatory electric field and dissipative coupling with the environment. We find that both exceptional points and complex eigenvalues can lead to multiple quantum Mpemba effect. It occurs in an observable when time-evolved copies corresponding to two different initial conditions, one initially having higher observable value compared to the other and both relaxing towards the same steady state, intersect each other more than once during their relaxation process. Each of the intersections denotes a quantum Mpemba effect and marks the reversal of identities between the two copies, i.e., the copy with higher observable value before the intersection becomes the lower-valued copy (and vice versa) after the intersection. Such multiple intersections originate from additional algebraic time dependence at the exceptional points and due to oscillatory relaxation in the case of complex eigenvalues. We provide analytical results for quantum Mpemba effect in the density matrix in presence of coherence. Depending on the control parameters (drive and dissipation), observables such as energy, von Neumann entropy, temperature, etc., exhibit either single or multiple quantum Mpemba effect. However, the distance from steady state measured in terms of the Kullback-Leibler divergence shows only single quantum Mpemba effect although the corresponding speed gives rise to either single or multiple quantum Mpemba effect.
We explore the role of exceptional points and complex eigenvalues on the occurrence of the quantum Mpemba effect. To this end, we study a two-level driven dissipative system subjected to an oscillatory electric field and dissipative coupling with the environment. We find that both exceptional points and complex eigenvalues can lead to multiple quantum Mpemba effect. It occurs in an observable when time-evolved copies corresponding to two different initial conditions, one initially having higher observable value compared to the other and both relaxing towards the same steady state, intersect each other more than once during their relaxation process. Each of the intersections denotes a quantum Mpemba effect and marks the reversal of identities between the two copies, i.e., the copy with higher observable value before the intersection becomes the lower-valued copy (and vice versa) after the intersection. Such multiple intersections originate from additional algebraic time dependence at the exceptional points and due to oscillatory relaxation in the case of complex eigenvalues. We provide analytical results for quantum Mpemba effect in the density matrix in presence of coherence. Depending on the control parameters (drive and dissipation), observables such as energy, von Neumann entropy, temperature, etc., exhibit either single or multiple quantum Mpemba effect. However, the distance from steady state measured in terms of the Kullback-Leibler divergence shows only single quantum Mpemba effect although the corresponding speed gives rise to either single or multiple quantum Mpemba effect.
The quantum Mpemba effect (QMPE), an anomalous relaxation phenomenon, has been demonstrated in both closed and open Hermitian quantum systems. While some studies have linked the QMPE to Liouvillian exceptional points---non-Hermitian features emerged at the Liouvillian level---in open Hermitian quantum systems, it remains largely unexplored whether the QMPE can occur in intrinsic non-Hermitian systems, where non-Hermiticity is inherent at the Hamiltonian level. Here, we demonstrate unequivocally the occurrence of QMPE in experimentally realizable parity-time symmetric qubit systems immersed in a bosonic bath. Using established quantifiers for QMPE, we show numerically that the QMPE persists across parameter regimes both near and far from Hamiltonian and Liouvillian exceptional points, but disappears entirely when a Hermitian Hamiltonian is restored. Interestingly, neither Hamiltonian nor Liouvillian exceptional points demarcate the boundaries of the QMPE regime. To complement numerical results, we develop an analytical description based on a long-time approximation of the relaxation dynamics of quantifiers. This approach allows us to decipher the number of intersections between two dynamical trajectories of quantifier starting from two initial conditions in the validity regime of the long-time approximation, thereby providing additional information that delineates the parameter regimes supporting the genuine QMPE. We further demonstrate the robustness of QMPE against increasing the number of qubits and dephasing effect. Our findings not only broaden the scope of the QMPE but also suggest its intricate interplay with non-Hermitian features beyond exceptional points.
We demonstrate the quantum Mpemba effect in a quantum dot coupled to two reservoirs, described by the Anderson model. We show that the system temperatures starting from two different initial values (hot and cold) cross each other at finite time (and thereby reverse their identities; i.e., hot becomes cold and vice versa) to generate thermal quantum Mpemba effect. The slowest relaxation mode believed to play the dominating role in Mpemba effect in Markovian systems does not contribute to such anomalous relaxation in the present model. In this connection, our analytical result provides necessary condition for producing quantum Mpemba effect in the density matrix elements of the quantum dot, as a combined effect of the remaining relaxation modes.
The Mpemba effect refers to the anomalous relaxation of a quantum state that, despite being initially farther from equilibrium, relaxes faster than a closer counterpart. Detecting such a quantum Mpemba effect typically requires full knowledge of the quantum state during its time evolution, which is an experimentally challenging task since state tomography becomes exponentially difficult as system size increases. This poses a significant obstacle in studying Mpemba effect in complex many-body systems. In this work, we demonstrate that this limitation can be overcome by identifying suitable observables that signal rapid relaxation. Moreover, as long as the system equilibrates to a known unique steady-state, it is possible to fully detect the occurrence of quantum Mpemba effect just by measuring the observable for known state preparations. Our approach thus significantly reduces experimental complexity and offers a practical route for observing the quantum Mpemba effect in complex and extended multi-qubit setups.
An ancient and counterintuitive phenomenon known as the Mpemba effect (water can cool faster when initially heated up) showcases the critical role of initial conditions in relaxation processes. How to realize and utilize this effect for speeding up relaxation is an important but challenging task in purely quantum system till now. Here, we experimentally study the strong Mpemba effect in a single trapped ion system in which an exponentially accelerated relaxation in time is observed by preparing an optimal quantum initial state with no excitation of the slowest decaying mode. Also, we demonstrate that the condition of realizing such effect coincides with the Liouvillian exceptional point, featuring the coalescence of both the eigenvalues and the eigenmodes of the systems. Our work provides an efficient strategy to engineer the dynamics of open quantum system, and suggests a link unexplored yet between the Mpemba effect and the non-Hermitian physics. The Mpemba effect is an archetype for various anomalous relaxation phenomena. Here, the authors experimentally study a quantum version of the Mpemba effect in a single trapped ion system, where relaxation is exponentially accelerated by removing the excitation of the slowest decaying mode. This phenomenon, seen in Markovian open quantum systems containing Liouvillian exceptional points, indicates a link between the Mpemba effect and non-Hermitian physics.
We investigate the quantum Mpemba effect in a dissipative Dicke model, which consists of a spin-1/2 ensemble coupled to a bosonic mode, which in turn is coupled to a bosonic bath. We derive a sufficient criterion for occurrence of the quantum Mpemba effect, characterized by quantum coherence, in this model. We introduce the phenomenon of role reversal in the Mpemba effect, wherein changes in the system parameters invert the relaxation ordering of a given pair of initial states that exhibit the Mpemba effect, causing the faster-relaxing state to become slower and vice versa. We find the existence of role reversal in Mpemba effect for this Dicke model using different relaxation measures, including differential quantum coherence and entanglement, and trace distance, between the time-evolved and steady states.
The quantum Mpemba effect is a counterintuitive phenomenon in which a state initially farther from equilibrium relaxes more rapidly than one that starts nearer to equilibrium. In the context of finite-dimensional quantum batteries interacting with an environment, we introduce the notion of an ergotropic Mpemba crossing (EMC), defined by the intersection of ergotropy trajectories during the dynamics. For qubit batteries subjected to amplitude damping noise, we derive a condition for the occurrence of EMC in terms of the relative coherence of the initial states and fully characterize the region of state space that exhibits EMC with respect to a fixed reference state. Interestingly, our analysis reveals that under anisotropic Pauli noise, the emergence of EMC is jointly governed by the coherence and the energy of the initial states. To elucidate the physical origin of EMC, we decompose ergotropy into coherent and incoherent contributions and show that, in qubit systems, the coherent component plays a crucial role for EMC, an observation that strikingly does not extend to three-level batteries, providing the dimensional role in observing EMC. Further, by extending our analysis to non-Markovian environments, we demonstrate that, unlike the Markovian case, non-Markovian dynamics can give rise to multiple Mpemba crossings, with the total number of crossings always being odd. Also, we propose a set-up showing that EMC can be utilized to suppress the ergotropic loss, occurred due to dissipation. Moreover, analyzing the connection between the EMC and the conventional state Mpemba effect reveals that, for qubits, an EMC necessarily entails a state Mpemba crossing, while this correspondence breaks down for qutrits, where EMCs may arise without any state Mpemba crossing.