We study the equilibration properties of isolated ergodic quantum systems initially prepared in a cat state, i.e a macroscopic quantum superposition of states. Our main result consists in showing that, even though decoherence is at work in the mean, there exists a remnant of the initial quantum coherences visible in the strength of the fluctuations of the steady state. We back-up our analysis with numerical results obtained on the XXX spin chain with a random field along the z-axis in the ergodic regime and find good qualitative and quantitative agreement with the theory. We also present and discuss a framework where equilibrium quantities can be computed from general statistical ensembles without relying on microscopic details about the initial state, akin to the eigenstate thermalization hypothesis.
We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bo…
We show that the physical mechanism for the equilibration of closed quantum systems is dephasing, and identify the energy scales that determine the equilibration timescale of a given observable. For realistic physical systems (e.g those with local Hamiltonians), our arguments imply timescales that do not increase with the system size, in contrast to previously known upper bounds. In particular we show that, for such Hamiltonians, the matrix representation of local observables in the energy basis is banded, and that this property is crucial in order to derive equilibration times that are non-ne…
We introduce structured random matrix ensembles, constructed to model many-body quantum systems with local interactions. These ensembles are employed to study equilibration of isolated many-body quantum systems, showing that rather complex matrix structures, well beyond Wigner's full or banded random matrices, are required to faithfully model equilibration times. Viewing the random matrices as connectivities of graphs, we analyse the resulting network of classical oscillators in Hilbert space with tools from network theory. One of these tools, called the maximum flow value, is found to be an e…
We study equilibration of an isolated quantum system by mapping it onto a network of classical oscillators in Hilbert space. By choosing a suitable basis for this mapping, the degree of locality of the quantum system reflects in the sparseness of the network. We derive a Lieb-Robinson bound on the speed of propagation across the classical network, which allows us to estimate the timescale at which the quantum system equilibrates. The bound contains a parameter that quantifies the degree of locality of the Hamiltonian and the observable. Locality was disregarded in earlier studies of equilibrat…
By using a physically-relevant and theory independent definition of measurement-based equilibration, we show quantitatively that equilibration is easier for quantum systems than for classical systems, in the situation where the initial state of the system is completely known (pure state). This shows that quantum equilibration is a fundamental, nigh unavoidable, aspect of physical systems, while classical equilibration relies on experimental ignorance. When the state is not completely known, a mixed state, this framework also shows quantum equilibration requires weaker conditions.
There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how far-from-equilibrium closed quantum system can behave as if relaxing to a stable equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. Alternatively, we consider here the equilibration of a molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration in this context i…
Equilibration to the steady state for a wide class of Luttinger liquid ballistic weakly linked tunnel contacts is extensively studied. Quantum fluctuations of tunnel current are considered in all orders in tunnel coupling and out of the equilibrium in the time domain. Especially, two important mathematical statements: Self-equilibration (SE-)theorem and Self-equilibration (SE-)lemma on the exact re-exponentiation of thermal average from the Keldysh-contour-ordered evolution operator for arbitrary weakly linked Luttinger liquid tunnel contact are proven. Demonstrated proof of SE-theorem and SE-…
Numerous pivotal concepts have been introduced to clarify the puzzle of relaxation and/or equilibration in closed quantum systems. All of these concepts rely in some way on specific conditions on Hamiltonians $H$, observables $A$, and initial states $ρ$ or combinations thereof. We numerically demonstrate and analytically argue that there is a multitude of pairs $H,A$ that meet said conditions for equilibration and generate some "typical" expectation value dynamics which means, $\langle A(t)\rangle \propto f(t)$ approximately holds for the vast majority of all initial states. Remarkably we find…
One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system siz…
Equilibration of isolated macroscopic quantum systems
我们研究了一个孤立的宏观量子系统的平衡，在这个意义上说，从一个稳定状态的偏差变得不可测量的小的绝大多数时间内任何足够大的时间间隔。主要的要求是，初始状态，可能远离平衡，表现出宏观人口最多一个能级和简并的能量本征值和能隙（能量本征值的差异）是不是非常大的多重性。我们的方法密切遵循并扩展了Short和Farrelly [2012 New J. Phys. 14 013063]最近的工作，特别是超越了有限维系统和大有效维度的领域。
The environment of an open quantum system is usually modelled as a large many-body quantum system. However, when an isolated quantum system itself is a many-body quantum system, the question of how large and complex it must be in order to generate internal equilibration is an open key-point in the literature. In this work, by monitoring the degree of equilibration of a single spin through its purity degradation, we are able to sense the chaotic behaviour of the generic spin chain to which it is coupled. Quite remarkably, this holds even in the case of extremely short spin chains composed of th…
For a quantum system to be captured by a stationary statistical ensemble, as is common in thermodynamics and statistical mechanics, it is necessary that it reaches some apparently stationary state in the first place. In this book chapter, we discuss the problem of equilibration and specifically provide insights into how long it takes to reach equilibrium in closed quantum systems. We first briefly discuss the connection of this problem with recent experiments and forthcoming quantum simulators. Then we provide a comprehensive discussion of equilibration from a heuristic point of view, with a f…
Equilibrium theormodynamics is characterized by two fundamental ideas: thermalisation--that systems approach a late time thermal state; and phase structure--that thermal states exhibit singular changes as various parameters characterizing the system are changed. We summarise recent progress that has established generalizations of these ideas to periodically driven, or Floquet, closed quantum systems. This has resulted in the discovery of entirely new phases which exist only out of equilibrium, such as the $π$-spin glass or Floquet time crystal.
We present a procedure to accelerate the relaxation of an open quantum system towards its equilibrium state. The control protocol, termed Shortcut to Equilibration, is obtained by reverse-engineering the non-adiabatic master equation. This is a non-unitary control task aimed at rapidly changing the entropy of the system. Such a protocol serves as a shortcut to an abrupt change in the Hamiltonian, i.e., a quench. As an example, we study the thermalization of a particle in a harmonic well. We observe that for short protocols there is a three orders of magnitude improvement in accuracy.
Understanding the dynamics of equilibration processes in quantum systems as well as their interplay with dissipation and fluctuation is a major challenge in quantum many-body theory. The timescales of such processes are investigated in collisions of atomic nuclei using fully microscopic approaches. Results from time-dependent Hartree-Fock (TDHF) and time-dependent random-phase approximation (TDRPA) calculations are compared for 13 systems over a broad range of energies. The timescale for full mass equilibration ($\sim2\times10^{-20}$s) is found to be much larger than timescales for neutron-to-…
We show that eigen-energies and energy eigenstates play different roles in the equilibration process of an isolated quantum system. Their roles are revealed numerically by exchanging the eigen-energies between an integrable model and a non-integrable model. We ?find that the structure of eigenenergies of a non-integrable model characterized by non-degeneracy ensures that quantum revival occurs rarely whereas the energy eigenstates of a non-integrable model suppress the fluctuations for the equilibrated quantum state. Our study is aided with a quantum entropy that describes how randomly a wave …
We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems we argue that the time-averaged expectation value of a local operator in any initial state is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium for all states with a given value of energy fluctuations. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at macroscopic level. In this way the deviation function establishes an explicit connection between the macrosc…
We characterize the conditions under which a multi-time quantum process with a finite temporal resolution can be approximately described by an equilibrium one. By providing a generalization of the notion of equilibration on average, where a system remains closed to a fixed equilibrium for most times, to one which can be operationally assessed at multiple times, we place an upper-bound on a new observable distinguishability measure comparing a multi-time process with a finite temporal resolution against a fixed equilibrium one. While the same conditions on single-time equilibration, such as a l…
We discuss the local equilibration of closed systems using the relative purity, a paradigmatic information-theoretic distinguishability measure that finds applications ranging from quantum metrology to quantum speed limits. First we obtain an upper bound on the average size of the fluctuations on the relative purity: it depends on the effective dimension resembling the bound obtained with the trace distance. Second, we investigate the dynamics of relative purity and its rate of change as a probe of the speed of fluctuations around the equilibrium. In turn, such speed captures the notion of how…
Finding equilibration times is a major unsolved problem in physics with few analytical results. Here we look at equilibration times for quantum gases of bosons and fermions in the regime of negligibly weak interactions, a setting which not only includes paradigmatic systems such as gases confined to boxes, but also Luttinger liquids and the free superfluid Hubbard model. To do this, we focus on two classes of measurements: (i) coarse-grained observables, such as the number of particles in a region of space, and (ii) few-mode measurements, such as phase correlators and correlation functions. We…
We consider the dynamics of local entropy and nearest neighbor mutual information of a 1-D lattice of qubits via the repeated application of nearest neighbor CNOT quantum gates. This is a quantum version of a cellular automaton. We analyze the entropy dynamics for different initial product states, both for open boundary conditions, periodic boundary conditions and we also consider the infinite lattice thermodynamic limit. The dynamics gives rise to fractal behavior, where we see the appearance of the Sierpinski triangle both for states in the computational basis and for operator dynamics in th…
Forgetfulness is a common feature of nature. Moreover, without forgetfulness, repeatability would be impossible. Despite this, small systems constantly leak information about their state to their surroundings, and quantum mechanics tells us that this information cannot be deleted, invariably returning to influence their future behaviour. How can physical nature be forgetful if it is forbidden to forget? The results within this thesis bridge this gap between what we see in the real world and what idealised physical theories say, explaining the emergence of forgetful processes from closed quantu…
