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arXiv:2003.11887 · Physical Review A

Probabilistic hysteresis in an isolated quantum system: The microscopic onset of irreversibility from a quantum perspective

Ralf Bürkle, J. R. Anglin

Published Apr 14, 2020• 6 citations
arXivHTMLPDF

Abstract

Recently probabilistic hysteresis in isolated Hamiltonian systems of ultracold atoms has been studied in the limit of large particle numbers, where a semiclassical treatment is adequate. The origin of irreversibility in these sweep experiments, where a control parameter is slowly (adiabatically) tuned back and forth, turned out to be a passage back and forth across a separatrix (integrable case) or a passage in and out of a chaotic sea in phase space (chaotic case). Here we focus on the full quantum mechanical description of the integrable system and show how the semiclassical results emerge in the limit of large particle numbers. Instead of the crossing of a separatrix in phase space, where classical adiabaticity fails, the origin of irreversibility in the quantum system is a series of avoided crossings of the adiabatic energy levels: they become so close that already for modest particle numbers the change of the external parameter has to be unrealistically slow to reach the quantum adiabatic limit of perfectly reversible evolution. For a slow but finite sweep rate we find a broad regime where the quantum results agree with the semiclassical results, but only if besides the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ an initial ensemble of states is considered, with sufficient initial energy width. For a single initial energy eigenstate we find in contrast that the backward sweep reveals strong quantum effects even for very large particle numbers.

DOI
10.1103/physreva.101.042110

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Recently probabilistic hysteresis in isolated Hamiltonian systems of ultracold atoms has been studied in the limit of large particle numbers, where a semiclassical treatment is adequate. The origin of irreversibility in these sweep experiments, where a control parameter is slowly (adiabatically) tuned back and forth, turned out to be a passage back and forth across a separatrix (integrable case) or a passage in and out of a chaotic sea in phase space (chaotic case). Here we focus on the full quantum mechanical description of the integrable system and show how the semiclassical results emerge in the limit of large particle numbers. Instead of the crossing of a separatrix in phase space, where classical adiabaticity fails, the origin of irreversibility in the quantum system is a series of avoided crossings of the adiabatic energy levels: they become so close that already for modest particle numbers the change of the external parameter has to be unrealistically slow to reach the quantum adiabatic limit of perfectly reversible evolution. For a slow but finite sweep rate we find a broad regime where the quantum results agree with the semiclassical results, but only if besides the limit N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ an initial ensemble of states is considered, with sufficient initial energy width. For a single initial energy eigenstate we find in contrast that the backward sweep reveals strong quantum effects even for very large particle numbers.
Probabilistic hysteresis is a manifestation of irreversibility in a small dissipationless classical system [R. B\"urkle et al., Sci. Rep. 9, 14169 (2019)]: After a slow cyclic sweep of a control parameter, the probability that an initial microcanonical ensemble returns to the neighborhood of its initial energy is significantly below one. A similar phenomenon has recently been confirmed in a corresponding quantum system, when the particle number NNN is not too small. Quantum-classical correspondence has been found to be nontrivial in this case however; the rate at which the control parameter changes must not be extremely slow and the initial distribution of energies must not be too narrow. In this paper we directly compare the quantum and classical forms of probabilistic hysteresis by making use of the Husimi quantum-phase-space formalism. In particular, we demonstrate that the classical ergodization mechanism, which is a key ingredient in classical probabilistic hysteresis, can lead to a breakdown of quantum-classical correspondence rather than to quantum ergodization. Such quantum failure of ergodization leads to strong quantum effects on the long-term evolution even when the quantum corrections in the equations of motion, which are proportional to 1/N1/N1/N, would naively seem to be small. We also show, however, that quantum ergodization can be restored by averaging over energies, so that for not-too-narrow initial energy width and not-too-slow parameter change the classical results are recovered after all at large NNN. Finally, we show that the formal incommutability of the classical and adiabatic limits in our system, which is responsible for the breakdown of quantum-classical correspondence in the quasistatic limit, is due to macroscopic quantum tunneling through a large energetic barrier. This explains the extremely slow sweep rates needed to reach the quantum adiabatic limit that were reported in our previous work. The formal incommutability therefore has no consequences for any realistically slow sweeps unless NNN is quite small ($N\ensuremath{\lesssim}20$).
The microscopic onset of irreversibility is finally becoming an experimental subject. Recent experiments on microscopic open and even isolated systems have measured statistical properties associated with entropy production, and hysteresis-like phenomena have been seen in cold atom systems with dissipation (i.e. effectively open systems coupled to macroscopic reservoirs). Here we show how experiments on isolated systems of ultracold atoms can show dramatic irreversibility like cooking an egg. In our proposed experiments, a slow forward-and-back parameter sweep will sometimes fail to return the system close to its initial state. This probabilistic hysteresis is due to the same non-adiabatic spreading and ergodic mixing in phase space that explains macroscopic irreversibility, but realized without dynamical chaos; moreover this fundamental mechanism quantitatively determines the probability of return to the initial state as a function of tunable parameters in the proposed experiments. Matching the predicted curve of return probability will be a conclusive experimental demonstration of the microscopic onset of irreversibility.
Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system. Transformation to the rotating-axis basis reduces the problem to the integration of a set of coupled differential equations. In the adiabatic limit the integration is carried out by the method of successive approximations. For adiabatic parameter \ensuremath{\epsilon} of the order of the inverse, 1/N, of the number of quantum states with energy below the separatrix, the spread of the quantum adiabatic invariant is exponentially small in the inverse of the adiabaticity parameter. Surprisingly, the transition to this quantum adiabatic behavior occurs far below the minimum transition frequency of the system.
Quantum mechanically, a driving process is expected to be reversible in the quasistatic limit, also known as the adiabatic theorem. This statement stands in opposition to classical mechanics, where a mix of regular and chaotic dynamics implies irreversibility. A paradigm for demonstrating the signatures of chaos in quantum irreversibility is a sweep process whose objective is to transfer condensed bosons from a source orbital. We show that such a protocol is dominated by an interplay of adiabatic-shuttling and chaos-assisted depletion processes. The latter is implied by interaction terms that spoil the Bogoliubov integrability of the Hamiltonian. As the sweep rate is lowered, a crossover to a regime that is dominated by quantum fluctuations is encountered, featuring a breakdown of quantum-to-classical correspondence. The major aspects of this picture are not captured by the common two-orbital approximation, which implies failure of the familiar many-body Landau-Zener paradigm.
We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).
The pendulum is the simplest system having all the basic properties inherent in dynamic stochastic systems. In the present paper we investigate the pendulum with the aim to reveal the properties of a quantum analogue of dynamic stochasticity or, in other words, to obtain the basic properties of quantum chaos. It is shown that a periodic perturbation of the quantum pendulum (similarly to the classical one) in the neighborhood of the separatrix can bring about irreversible phenomena. As a result of recurrent passages between degenerate states, the system gets self-chaotized and passes from the pure state to the mixed one. Chaotization involves the states, the branch points of whose levels participate in a slow "drift" of the system along the Mathieu characteristics this "drift" being caused by a slowly changing variable field. Recurrent relations are obtained for populations of levels participating in the irreversible evolution process. It is shown that the entropy of the system first grows and, after reaching the equilibrium state, acquires a constant value.