We discuss the behavior of a quantum 1/2 spin coupled to a time dependent magnetic field, which can be quasiperiodic or random. For a quasiperiodic field, the time evolution of the system on a coarse grained space has autocorrelation, which does not decay, and positive topological entropy. The information complexity diverges as a stretched exponential, although the Shannon entropy is zero and there is no divergence of nearby orbits. A simple random process is introduced in order to reproduce the main qualitative features of the observed weakly complex behavior.
We define correlational (von Neumann) entropy for an individual quantum state of a system whose time-independent Hamiltonian contains random parameters and is treated as a member of a statistical ensemble. This entropy is representation independent, and can be calculated as a trace functional of the density matrix which describes the system in its interaction with the noise source. We analyze perturbation theory in order to show the evolution from the pure state to the mixed one. Exactly solvable examples illustrate the use of correlational entropy as a measure of the degree of complexity in comparison with other available suggestions such as basis-dependent information entropy. It is shown in particular that a harmonic oscillator in a uniform field of random strength comes to a quasithermal equilibrium; we discuss the relation between effective temperature and canonical equilibrium temperature. The notion of correlational entropy is applied to a realistic numerical calculation in the framework of the nuclear shell model. In this system, which reveals generic signatures of quantum chaos, correlational entropy and information entropy calculated in the mean field basis display similar qualitative behavior.
For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
We study the dynamics of nearest-neighbor entanglement for a one-dimensional spin chain with a nearest-neighbor time-dependent Heisenberg coupling
J(t) between the spins in the presence of a time-dependent external magnetic field
h(t) at zero and finite temperatures. We consider different forms of time dependence for the coupling and magnetic field: exponential, hyperbolic, and periodic. Solving the system numerically, we examined the system-size effect on the entanglement asymptotic value. It was found that, for a small system size, the entanglement starts to fluctuate within a short period of time after applying the time-dependent coupling. The period of time increases as the system size increases and disappears completely as the size goes to infinity. Testing the effect of the transition constant for an exponential or hyperbolic coupling showed a direct impact on the asymptotic value of the entanglement; the larger the constant is, the lower the asymptotic value and the more rapid decay of entanglement are, which confirms the nonergodic character of the system. We also found that, when
J(t) is periodic, the entanglement shows a periodic behavior with the same period, which disappears upon applying periodic magnetic field with the same frequency. Solving the case $J(t)=\ensuremath{\lambda}h(t)$, for constant $\ensuremath{\lambda}$, exactly, we showed that the time evolution and asymptotic value of entanglement are dictated solely by the parameter $\ensuremath{\lambda}=J/h$ rather than the individual values of
J and
h, not only when they are time independent and at zero temperature, but also when they are time dependent but proportional at zero and finite temperatures for all degrees of anisotropy.
The thermalization of one-dimensional quantum systems may be severely slowed by weak links, e.g., by the Griffiths regions that dominate the long-time dynamics of a system close to the many-body-localization transition. Here, the authors construct universal coarse-grained descriptions for entanglement production and for the dynamics of quantum operators in systems with weak links. They reveal unexpected differences between these processes (for example, entanglement growth can be slower than linear in time even when operators spread ballistically). The universal physics is explained using simple classical effective models. The authors' analysis of the spreading of quantum information and of conserved quantities shows that thermalization in the presence of weak links has a rich structure, featuring multiple dynamic length scales that diverge with different powers of time.
Driven many-body quantum systems where some parameter in the Hamiltonian is varied quasiperiodically in time may exhibit nonequilibrium steady states that are qualitatively different from their periodically driven counterparts. Here we consider a prototypical integrable spin system, the spin-
1/2 transverse field Ising model in one dimension, in a pulsed magnetic field. The time dependence of the field is taken to be quasiperiodic by choosing the pulses to be of two types that alternate according to a Fibonacci sequence. We show that a steady state emerges after an exponentially long time when local properties (or equivalently, reduced density matrices of subsystems with size much smaller than the full system) are considered. We use the temporal evolution of certain coarse-grained quantities in momentum space to understand this nonequilibrium steady state in more detail and show that unlike the previously known cases, this steady state is neither described by a periodic generalized Gibbs ensemble nor by an infinite temperature ensemble. Finally, we study a toy problem with a single two-level system driven by a Fibonacci sequence; this problem shows how sensitive the nature of the final steady state is to the different parameters.