Exponentially Accelerated Approach to Stationarity in Markovian Open Quantum Systems through the Mpemba Effect

Photonic Mpemba effect

The Mpemba effect (ME) is the counterintuitive phenomenon in statistical physics for which a far-from-equilibrium state can relax toward equilibrium faster than a state closer to equilibrium. This effect has raised great curiosity for a long time and has been studied extensively in many classical and quantum systems. Here, it is shown that the Mpemba effect can be observed in optics as well. Specifically, the process of light diffusion in finite-sized photonic lattices under incoherent (dephasing) dynamics is considered. Rather surprisingly, it is shown that certain highly localized initial light distributions can diffuse faster than initial broadly delocalized distributions. The effect is illustrated by considering the random walk of optical pulses in fiber-based temporal mesh lattices, which should provide an experimentally accessible setup for the demonstration of the Mpemba effect in optics.

Observing the Quantum Mpemba Effect in Quantum Simulations

The nonequilibrium physics of many-body quantum systems harbors various unconventional phenomena. In this Letter, we experimentally investigate one of the most puzzling of these phenomena-the quantum Mpemba effect, where a tilted ferromagnet restores its symmetry more rapidly when it is farther from the symmetric state compared to when it is closer. We present the first experimental evidence of the occurrence of this effect in a trapped-ion quantum simulator. The symmetry breaking and restoration are monitored through entanglement asymmetry, probed via randomized measurements, and postprocessed using the classical shadows technique. Our findings are further substantiated by measuring the Frobenius distance between the experimental state and the stationary thermal symmetric theoretical state, offering direct evidence of subsystem thermalization.

Inverse Mpemba Effect Demonstrated on a Single Trapped Ion Qubit

The Mpemba effect is a counterintuitive phenomena in which a hot system reaches a cold temperature faster than a colder system, under otherwise identical conditions. Here, we propose a quantum analog of the Mpemba effect, on the simplest quantum system, a qubit. Specifically, we show it exhibits an inverse effect, in which a cold qubit reaches a hot temperature faster than a hot qubit. Furthermore, in our system a cold qubit can heat up exponentially faster, manifesting the strong version of the effect. This occurs only for sufficiently coherent systems, making this effect quantum mechanical, i.e., due to interference effects. We experimentally demonstrate our findings on a single ^{88}Sr^{+} trapped ion qubit. The existence of this anomalous relaxation effect in simple quantum systems reveals its fundamentality, and may have a role in designing and operating quantum information processing devices.

Microscopic Origin of the Quantum Mpemba Effect in Integrable Systems

The highly complicated nature of far from equilibrium systems can lead to a complete breakdown of the physical intuition developed in equilibrium. A famous example of this is the Mpemba effect, which states that nonequilibrium states may relax faster when they are further from equilibrium or, put another way, hot water can freeze faster than warm water. Despite possessing a storied history, the precise criteria and mechanisms underpinning this phenomenon are still not known. Here, we study a quantum version of the Mpemba effect that takes place in closed many-body systems with a U(1) conserved charge: in certain cases a more asymmetric initial configuration relaxes and restores the symmetry faster than a more symmetric one. In contrast to the classical case, we establish the criteria for this to occur in arbitrary integrable quantum systems using the recently introduced entanglement asymmetry. We describe the quantum Mpemba effect in such systems and relate the properties of the initial state, specifically its charge fluctuations, to the criteria for its occurrence. These criteria are expounded using exact analytic and numerical techniques in several examples, a free fermion model, the Rule 54 cellular automaton, and the Lieb-Liniger model.

Many-excitation removal of a transmon qubit using a single-junction quantum-circuit refrigerator and a two-tone microwave drive

Achieving fast and precise initialization of qubits is a critical requirement for the successful operation of quantum computers. The combination of engineered environments with all-microwave techniques has recently emerged as a promising approach for the reset of superconducting quantum devices. In this work, we experimentally demonstrate the utilization of a single-junction quantum-circuit refrigerator (QCR) for an expeditious removal of several excitations from a transmon qubit. The QCR is indirectly coupled to the transmon through a resonator in the dispersive regime, constituting a carefully engineered environmental spectrum for the transmon. Using single-shot readout, we observe excitation stabilization times down to roughly 500 ns, a 20-fold speedup with QCR and a simultaneous two-tone drive addressing the e-f and f0-g1 transitions of the system. Our results are obtained at a 48-mK fridge temperature and without postselection, fully capturing the advantage of the protocol for the short-time dynamics and the drive-induced detrimental asymptotic behavior in the presence of relatively hot other baths of the transmon. We validate our results with a detailed Liouvillian model truncated up to the three-excitation subspace, from which we estimate the performance of the protocol in optimized scenarios, such as cold transmon baths and fine-tuned driving frequencies. These results pave the way for optimized reset of quantum-electric devices using engineered environments and for dissipation-engineered state preparation.

Mpemba meets Newton: Exploring the Mpemba and Kovacs effects in the time-delayed cooling law

Despite extensive research, the fundamental physical mechanisms underlying the Mpemba effect, a phenomenon where a substance cools faster after initially being heated, remain elusive. Although historically linked with water, the Mpemba effect manifests across diverse systems, sparking heightened interest in Mpemba-like phenomena. Concurrently, the Kovacs effect, a memory phenomenon observed in materials such as polymers, involves rapid quenching and subsequent temperature changes, resulting in nonmonotonic relaxation behavior. This paper probes the intricacies of the Mpemba and Kovacs effects within the framework of the time-delayed Newton's law of cooling, recognized as a simplistic yet effective phenomenological model accommodating memory phenomena. This law allows for a nuanced comprehension of temperature variations, introducing a delay time (τ) and incorporating specific protocols for the thermal bath temperature, contingent on a defined waiting time (t_{w}). Remarkably, the relevant parameter space is two-dimensional (τ and t_{w}), with bath temperatures exerting no influence on the presence or absence of the Mpemba effect or on the relative strength of the Kovacs effect. The findings enhance our understanding of these memory phenomena, providing valuable insights applicable to researchers across diverse fields, ranging from physics to materials science.

Shortcuts of Freely Relaxing Systems Using Equilibrium Physical Observables

Many systems, when initially placed far from equilibrium, exhibit surprising behavior in their attempt to equilibrate. Striking examples are the Mpemba effect and the cooling-heating asymmetry. These anomalous behaviors can be exploited to shorten the time needed to cool down (or heat up) a system. Though, a strategy to design these effects in mesoscopic systems is missing. We bring forward a description that allows us to formulate such strategies, and, along the way, makes natural these paradoxical behaviors. In particular, we study the evolution of macroscopic physical observables of systems freely relaxing under the influence of one or two instantaneous thermal quenches. The two crucial ingredients in our approach are timescale separation and a nonmonotonic temperature evolution of an important state function. We argue that both are generic features near a first-order transition. Our theory is exemplified with the one-dimensional Ising model in a magnetic field using analytic results and numerical experiments.

Steady state engineering of a two-level system by the mixed-state inverse engineering scheme

The mixed-state inverse engineering scheme is a control scheme used for engineering the quantum state of a driven open quantum system from an initial steady state to a final steady state. In this paper, we present an analytical study of this scheme applied to the driven two-level model coupled to a heat reservoir. Typically, when the purity of the quantum state varies, incoherent control techniques are required for mixed-state engineering. However, we show that for both Markovian and non-Markovian dynamics, coherent control protocols can transfer the quantum state into the target state. This simplification comes at a cost, as the evolution of the quantum state must be limited to restricted conditions, resulting in special trajectories in its Hilbert space that connect the initial and target states.

Quantum Mpemba Effect in a Quantum Dot with Reservoirs

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.

Mpemba effect for a Brownian particle trapped in a single well potential

The Mpemba effect refers to the counterintuitive phenomenon of a hotter system equilibrating faster than a colder system when both are quenched to the same low temperature. For a Brownian particle trapped in a piecewise linear single well potential that is devoid of any other metastable minima, we show the existence of the Mpemba effect for a wide range of parameters through an exact solution. This result challenges the prevalent explanation of the Mpemba effect that requires the energy landscape to be rugged with multiple minima. We also demonstrate the existence of inverse and strong Mpemba effects.

Mpemba effect in driven granular gases: Role of distance measures

The Mpemba effect refers to the counterintuitive effect where a system which is initially further from the final steady state equilibrates faster than an identical system that is initially closer. The closeness to the final state is defined in terms of a distance measure. For driven granular systems, the Mpemba effect has been illustrated in terms of an ad hoc measure of mean kinetic energy as the distance function. In this paper, by studying four different distance measures based on the mean kinetic energies as well as velocity distribution, we show that the Mpemba effect depends on the definition of the measures.

Mpemba effect in a Langevin system: Population statistics, metastability, and other exact results

The Mpemba effect is a fingerprint of the anomalous relaxation phenomenon wherein an initially hotter system equilibrates faster than an initially colder system when both are quenched to the same low temperature. Experiments on a single colloidal particle trapped in a carefully shaped double well potential have demonstrated this effect recently [A. Kumar and J. Bechhoefer, Nature 584, 64 (2020)]. In a similar vein, here, we consider a piece-wise linear double well potential that allows us to demonstrate the Mpemba effect using an exact analysis based on the spectral decomposition of the corresponding Fokker-Planck equation. We elucidate the role of the metastable states in the energy landscape as well as the initial population statistics of the particles in showcasing the Mpemba effect. Crucially, our findings indicate that neither the metastability nor the asymmetry in the potential is a necessary or a sufficient condition for the Mpemba effect to be observed.

Relaxation Shortcuts through Boundary Coupling

When a hot system cools down faster than an equivalent cold one, it exhibits the Mpemba effect (ME). This counterintuitive phenomenon was observed in several systems including water, magnetic alloys, and polymers. In most experiments the system is coupled to the bath through its boundaries, but all theories so far assumed bulk coupling. Here we build a general framework to characterize anomalous relaxations through boundary coupling, and present two emblematic setups: a diffusing particle and an Ising antiferromagnet. In the latter, we show that the ME can survive even arbitrarily weak couplings.

Hyperacceleration of quantum thermalization dynamics by bypassing long-lived coherences: An analytical treatment

We develop a perturbative technique for solving Markovian quantum dissipative dynamics, with the perturbation parameter being a small gap in the eigenspectrum. As an example, we apply the technique and straightforwardly obtain analytically the dynamics of a three-level system with quasidegenerate excited states, where quantum coherences persist for very long times, proportional to the inverse of the energy splitting squared. We then show how to bypass this long-lived coherent dynamics and accelerate the relaxation to thermal equilibration in a hyper-exponential manner, a Markovian quantum-assisted Mpemba-like effect. This hyperacceleration of the equilibration process manifests if the initial state is carefully prepared, such that its coherences precisely store the amount of population relaxing from the initial condition to the equilibrium state. Our analytical method for solving quantum dissipative dynamics readily provides equilibration timescales, and as such it reveals how coherent and incoherent effects interlace in the dynamics. It further advises on how to accelerate relaxation processes, which is desirable when long-lived quantum coherences stagnate dynamics.

Symmetrized Liouvillian Gap in Markovian Open Quantum Systems

Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate toward the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of autocorrelation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always gives a correct upper bound on the decay of the autocorrelation function, but the standard Liouvillian gap does not.

Eigenvalue Crossing as a Phase Transition in Relaxation Dynamics

When a system's parameter is abruptly changed, a relaxation toward the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation operator results in a singularity in the dynamics analogous to a first-order equilibrium phase transition. While dynamical phase transitions are intrinsically hard to detect in nature, here we show how this kind of transition can be observed in an experimentally feasible four-state colloidal system. Finally, analytical proof of survival in the thermodynamic limit of a many body (1D Ising) model is provided.

Fast Functionalization with High Performance in the Autonomous Information Engine

Mandal and Jarzynski have proposed a fully autonomous information heat engine, consisting of a demon, a mass, and a memory register interacting with a thermal reservoir. This device converts thermal energy into mechanical work by writing information to a memory register or, conversely, erasing information by consuming mechanical work. Here, we derive a speed limit inequality between the relaxation time of state transformation and the distance between the initial and final distributions, where the combination of the dynamical activity and entropy production plays an important role. Such inequality provides a hint that a speed-performance trade-off relation exists between the relaxation time to a functional state and the average production. To obtain fast functionalization while maintaining the performance, we show that the relaxation dynamics of the information heat engine can be accelerated significantly by devising an optimal initial state of the demon. Our design principle is inspired by the so-called Mpemba effect, where water freezes faster when initially heated.

Anomalous Cooling and Overcooling of Active Colloids

The phenomenon that a system at a hot temperature cools faster than at a warm temperature, referred to as the Mpemba effect, has recently been realized for trapped colloids. Here, we investigate the cooling and heating process of a self-propelled active colloid using numerical simulations and theoretical calculations with a model that can be directly tested in experiments. Upon cooling, activity induces a Mpemba effect and the active particle transiently escapes an effective temperature description. At the end of the cooling process the notion of temperature is recovered and the system can exhibit even smaller temperatures than its final temperature, a surprising phenomenon which we refer to as activity-induced overcooling.

Thermal versus entropic Mpemba effect in molecular gases with nonlinear drag

Loosely speaking, the Mpemba effect appears when hotter systems cool sooner or, in a more abstract way, when systems further from equilibrium relax faster. In this paper, we investigate the Mpemba effect in a molecular gas with nonlinear drag, both analytically (by employing the tools of kinetic theory) and numerically (direct simulation Monte Carlo of the kinetic equation and event-driven molecular dynamics). The analysis is carried out via two alternative routes, recently considered in the literature: first, the kinetic or thermal route, in which the Mpemba effect is characterized by the crossing of the evolution curves of the kinetic temperature (average kinetic energy), and, second, the stochastic thermodynamics or entropic route, in which the Mpemba effect is characterized by the crossing of the distance to equilibrium in probability space. In general, a nonmutual correspondence between the thermal and entropic Mpemba effects is found, i.e., there may appear the thermal effect without its entropic counterpart or vice versa. Furthermore, a nontrivial overshoot with respect to equilibrium of the thermal relaxation makes it necessary to revise the usual definition of the thermal Mpemba effect, which is shown to be better described in terms of the relaxation of the local equilibrium distribution. Our theoretical framework, which involves an extended Sonine approximation in which not only the excess kurtosis but also the sixth cumulant is retained, gives an excellent account of the behavior observed in simulations.

Mpemba effect of a mean-field system: The phase transition time

The counterintuitive phenomenon-that an initially hotter water freezes faster than initially cooler water-is named the "Mpemba effect." Although it has been known for centuries, the underlying mechanism remains unclear. Recently, the Mpemba effect rekindled the interest of researchers since several studies identified that it might occur in some Markovian systems, and a general statistical-physical Mpemba effect framework was correspondingly proposed. In our previous study [Z.-Y. Yang and J.-X. Hou, Phys. Rev. E 101, 052106 (2020)10.1103/PhysRevE.101.052106], we observed the non-Markovian Mpemba effect in a mean-field system (MFS), where the Mpemba effect originates from the back-reaction of the thermal reservoir. Naturally, the phase transition time is the key to the occurrence of the Mpemba effect, which, however, has not been quantitatively described. Following the direction of previous work, this study rigorously derives the phase transition time under different conditions, and quantitatively describes the mechanism of the non-Markovian Mpemba effect in a MFS. In addition, the validation of our theory was further verified via the microcanonical Monte Carlo simulation. An accurate description of the underlying mechanism of our proposed MFS facilitates the generalization of the Mpemba effect framework in statistical physics and may benefit in answering the riddle of the century, the original Mpemba effect in water.

Slow growth of magnetic domains helps fast evolution routes for out-of-equilibrium dynamics

Cooling and heating faster a system is a crucial problem in science, technology, and industry. Indeed, choosing the best thermal protocol to reach a desired temperature or energy is not a trivial task. Noticeably, we find that the phase transitions may speed up thermalization in systems where there are no conserved quantities. In particular, we show that the slow growth of magnetic domains shortens the overall time that the system takes to reach a final desired state. To prove that statement, we use intensive numerical simulations of a prototypical many-body system, namely, the two-dimensional Ising model.

Relaxation-speed crossover in anharmonic potentials

In a recent Letter [A. Lapolla and A. Godec, Phys. Rev. Lett. 125, 110602 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.110602], thermal relaxation was observed to occur faster from cold to hot (heating) than from hot to cold (cooling). Here we show that overdamped diffusion in anharmonic potentials generically exhibits both faster heating and faster cooling, depending on the initial temperatures and on the potential's degree of anharmonicity. We draw a relaxation-speed phase diagram that localizes the different behaviors in parameter space. In addition to faster-heating and faster-cooling regions, we identify a crossover region in the phase diagram, where heating is initially slower but asymptotically faster than cooling. The structure of the phase diagram is robust against the inclusion of a confining, harmonic term in the potential as well as moderate changes of the measure used to define initially equidistant temperatures.

Kovacs Memory Effect with an Optically Levitated Nanoparticle

The understanding of the dynamics of nonequilibrium cooling and heating processes at the nanoscale is still an open problem. These processes can follow surprising relaxation paths due to, e.g., memory effects, which significantly alter the expected equilibration routes. The Kovacs effect can take place when a thermalization process is suddenly interrupted by a change of the bath temperature, leading to a nonmonotonic evolution of the energy of the system. Here, we demonstrate that the Kovacs effect can be observed in the thermalization of the center of mass motion of a levitated nanoparticle. The temperature is controlled during the experiment through an external source of white Gaussian noise that mimics an effective thermal bath at a temperature that can be changed faster than any relaxation time of the system. We describe our experiments in terms of the dynamics of a Brownian particle in a harmonic trap without any fitting parameter, suggesting that the Kovacs effect can appear in a large variety of systems.