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by The Oscillation PublicationMarch 5th, 2024

This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira;

(2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering;

(3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University.

- Introduction
- Model and Protocol
- Different Regimes in d˜− Γ˜ Plain and Multiple QMPE
- QMPE in Region (d): Second-order Ep
- QMPE in Region (a1): Oscillations
- Summary
- Appendix A: Eigenvectors of L in region (d)
- Appendix B: Eigenvectors of L in region (a1)
- Appendix C: QMPE in region (b): purely exponential relaxation
- Appendix D: QMPE in region (e): 3rd order EP
- References

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.

Memory effect describes the influence of past information on present events [1]. Such memory effects often lead to unexpected events. In physical systems, several forms of interesting memory effects have been observed in different contexts e.g. pulse-sign memory in charge-density wave conductors (memory of last input signal direction) [2, 3]; Kaiser effect in rocks and metals [4–6] and Mullins effect in rubbers [7–9] (both bearing memories of largest applied stress); Kovacs effect (memory of bath temperature that affects the thermalization process) [10–14]; aging and rejuvenation in spin glasses [15–17] and structural glasses [18–21], granular materials [22–25], dense suspensions [26] and active matter [27].

The Mpemba effect (MPE) was originally conceived as a counter-intuitive thermal phenomenon of faster cooling of hotter liquids, observed long ago [28] and lately rediscovered by Mpemba and Osborne [29]. The MPE thus bears the imprint of initial memory in a surprising way and constitutes a separate class of fascinating memory effects. Later, the MPE has been regarded as one of the typical anomalous relaxations where a system with initially higher observable (e.g. temperature, energy, entropy, distance function etc.) value can relax faster than its copy with initially lower observable value, when both of them decrease towards the same steady state value. In classical systems, MPE has been found in variety of classical systems including colloids [30, 31], granular fluids [32–38], optical resonators [39–41], inertial suspensions [42, 43], Markovian models [44–47], Langevin system [48] and many others [49–59].

In spite of considerable amount of works on MPE in the classical domain, only a few studies have been done on the quantum Mpemba effect (QMPE), such as quantum Ising model [60], Dicke model [61], few-level systems [62], V-type system [63], XXZ spin chain [64], single level quantum dot [65], XY spin chain [66] and free and interacting integrable systems [67]. In fact, only quite recently the quantum analogue of the original thermal MPE i.e. QMPE in temperature, has been theoretically revealed in a quantum dot [65] and the significant role of relaxation modes other than the slowest relaxation mode to create QMPE, has been shown [65]. However, the semiclassical approach taken in Ref. [65] naturally raises a question about the generality of QMPE in temperature and other observable, in presence of coherence in purely quantum mechanical systems. This constitutes one of the motivations of the present work.

Since the phenomenon of QMPE is a comparatively new area of research, it is important to explore the plausible connections of QMPE to other interesting features of quantum systems. In the present work, we would like to address some such connections. In particular, the study of exceptional points (EPs) and surfaces in non-Hermitian quantum systems and Liouvillian dynamics has caught a lot of attention in recent days [68–81] along with their applications in quantum optics [82, 83]. The occurrence of EPs is directly related to eigen-spectrum of a system and hence to its relaxation process. Therefore a natural curiosity would be to understand the role of EPs on the possibility of QMPE. In fact, the occurrence of double MPE (i.e. two intersections between trajectories) at an EP has already been pointed out for MPE in classical inertial suspensions [42]. Interestingly, the acceleration of relaxation process in open quantum systems by maximizing the gap between the zero mode and the slowest eigenmode at an EP, has recently been studied [84]. In general, such speed up is desired for several application purposes and therefore it has been a subject of immense interest in both classical and quantum systems [84–89]. Since QMPE allows faster relaxation of initially higher distant trajectories, it is important to find criteria for QMPE and to understand how to utilize QMPE to achieve faster approach to steady state. Another intriguing feature in non-Hermitian quantum mechanical systems is the appearance of complex eigenvalues [90–94]. It is noteworthy that the analysis of QMPE in the existing examples [60–63, 65] is based on purely exponential relaxation processes, neither the EP nor the oscillations have been considered yet. There is only an exception in Ref. [64] where more than one intersections between trajectories (two intersections to be precise, termed as revivals) has been observed in the entanglement entropy in the XXZ spin chain, although not discussed much and simply explained as a finite-size effect.

In this paper, we explore the following two topics: (i) the effect of EPs and complex eigenvalues on QMPE by analysing the Quantum Master Equation, and (ii) the existence of thermal QMPE in a purely quantum mechanical setting. We show that in the presence of EPs and complex eigenvalues, two time-evolved trajectories of an observable (e.g., temperature, energy) can intersect each other more than once before reaching the same final steady state. In the context of thermal QMPE, the first intersection implies that the initially hotter quantum system relaxes faster to intersect the initially colder quantum system and thereby reverses their roles, i.e. hotter becomes colder and vice versa. This can be followed by one or more such intersections so that the present hotter system can become colder again and so on. Similar observations have been obtained in average energy. Such multiple intersections of trajectories give rise to the remarkable phenomenon of multiple QMPE. To demonstrate multiple QMPE as a consequence of EP and complex eigenvalues, we study a two-level driven dissipative quantum system developed in Ref. [95], which we call Hatano’s model in this paper, as a typical and minimal model of open quantum systems. It is well known that many complicated quantum problems are mapped or reduced to two-level systems; needless to mention they are experimentally realizable (e.g., two internal electronic energy levels of an atom controlled by laser) and have applications in qubits, nuclear magnetic resonance, neutrino oscillations, etc. We also find thermal QMPE in presence of coherence, some cases remarkably with multiple intersections, following the same definition of temperature used in Ref. [65] . Importantly, we provide analytical results for the time evolution of the density matrix in the two-level driven dissipative system considered here. Consequently, we derive analytical criteria for QMPE in energy and in principle, one can analyze exactly (analytically and numerically) QMPE in any other observable.

The paper is arranged as follows. In Sec. II, we define the details of the model and the protocol used to study QMPE. The Sec. III describes different regions in the control parameter plane depending on the types of eigenvalues of the Lindbladian and the multiple QMPE is defined. In articular, we focus on regions with exceptional points and complex eigenvalues to discuss their effects on QMPE in the density matrix and observables such as energy, entropy, Kullback-Leibler (KL) divergence [96– 100] and temperature; by providing analytical results. In Sec. IV, we discuss QMPE in a region with second-order EPs. Section V demonstrates the effect of complex eigenvalues on the time evolution of the density matrix leading to multiple QMPE in the observables. We summarize the results and discuss possible future directions in Sec. VI. Explicit expressions of the eigenvectors used in the analyses in Secs. IV and V, are provided in Appendix A and Appendix B respectively. We discuss QMPE briefly in region of purely exponential relaxation in Appendix C and at a third-order EP in Appendix D.

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