The emergence of classical behavior and the origin of thermal equilibrium are two central problems in the foundations of physics. In the standard accounts, both phenomena are typically explained through interactions with an external environment: decoherence suppresses quantum interference, while coupling to a thermal bath drives relaxation toward equilibrium. Over the last decades, however, it has become clear that equilibration and thermalization can arise even in fully isolated quantum systems, in the operational sense that the expectation values of relevant observables remain close to equilibrium values for most of the time. Here, we ask whether the same intrinsic equilibration mechanism can also account for the emergence of classical behavior. Using rigorous bounds on equilibration in closed systems, we derive sufficient conditions under which a time-evolved pure state becomes, for most times, operationally indistinguishable from a classical mixture associated with a chosen physical property. We identify two complementary routes to such operational classicality: either the chosen property almost commutes with the system Hamiltonian or the observables used to probe the system lose access to the remaining coherence after equilibration. Our results show that classical behavior need not be confined to the energy basis and may emerge even when substantial coherence remains present in the equilibrium state. This establishes a direct connection between two foundational questions: the origin of thermalization in isolated quantum systems and the quantum-to-classical transition
Significant attention has been devoted to the problem of thermalization of observables in isolated quantum setups by individual eigenstates. Here, we address this issue from an open quantum system perspective, examining an isolated setup where a small system (specifically, a single fermionic level) is coupled to a macroscopic fermionic bath. We argue that in such a model, despite its full integrability, the system observables exhibit thermalization when the system-bath setup resides in a typical eigenstate of its Hamiltonian, a phenomenon known as weak eigenstate thermalization. This thermalization occurs unless it is suppressed by localization due to strong coupling. We further show that, following the quench of the system Hamiltonian, the system occupancy typically relaxes to the thermal value corresponding to the new Hamiltonian. Finally, we demonstrate that system thermalization also arises when the system is coupled to a bath that has been initialized in a typical eigenstate of its Hamiltonian. Our findings suggest that nonintegrability is not the sole driver of thermalization, highlighting the need for complementary approaches to fully understand the emergence of statistical mechanics.
The last decade has witnessed remarkable progress in our understanding of thermalization in isolated quantum systems. Combining the eigenstate thermalization hypothesis with quantum measurement theory, we extend the framework of quantum thermalization to open many-body systems. A generic many-body system subject to continuous observation is shown to thermalize at a single trajectory level. We show that the nonunitary nature of quantum measurement causes several unique thermalization mechanisms that are unseen in isolated systems. We present numerical evidence for our findings by applying our theory to specific models that can be experimentally realized in atom-cavity systems and with quantum gas microscopy. Our theory provides a general method to determine an effective temperature of quantum many-body systems subject to the Lindblad master equation and thus should be applicable to noisy dynamics or dissipative systems coupled to nonthermal Markovian environments as well as continuously monitored systems. Our work provides yet another insight into why thermodynamics emerges so universally.
The question of how classical systems approach thermal equilibrium is as old as the foundations of thermodynamics and statistical mechanics. How quantum systems decohere and thermalize is even more puzzling. This Colloquium provides an account of how the thermal equilibrium of a system is influenced by the presence of a thermal bath. It also gives a view of both classical and quantum aspects providing an understanding on the particularities of the quantum case. Moreover, a description of the challenges in the definition of heat from the perspective of fluctuating thermodynamical potentials is given. An old subject, perhaps, but definitely fundamental.
We study measures of decoherence and thermalization of a quantum system
S in the presence of a quantum environment (bath)
E. The entirety
S+E is prepared in a canonical-thermal state at a finite temperature; that is, the entirety is in a steady state. Both our numerical results and theoretical predictions show that measures of the decoherence and the thermalization of
S are generally finite, even in the thermodynamic limit, when the entirety
S+E is at finite temperature. Notably, applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider
S uncoupled from
E, but entangled with
E, to predict decoherence and thermalization measures of
S. This decoupling allows closed-form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of
S and of
E. Large-scale numerical results for both coupled and uncoupled entireties with up to 40 quantum spins support these findings.
We study the emergence of decoherent histories in isolated systems based on exact numerical integration of the Schrödinger equation for a Heisenberg chain. We reveal that the nature of the system, which we switch from (i) chaotic to (ii) interacting integrable to (iii) noninteracting integrable, strongly impacts decoherence of coarse spin observables. From a finite size scaling law we infer a strong exponential suppression of coherences for (i), a weak exponential suppression for (ii), and no exponential suppression for (iii) on a relevant short (nonequilibrium) timescale. Moreover, for longer times we find stronger decoherence for (i) but the opposite for (ii), hinting even at a possible power-law decay for (ii) at equilibrium timescales. This behavior is encoded in the multitime properties of the quantum histories and it can not be explained by environmentally induced decoherence. Our results suggest that chaoticity plays a crucial role in the emergence of classicality in finite size systems.