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arXiv:1508.04401

Control of global properties in a closed many-body quantum system by means of a local switch

Maurizio Fagotti

Published Aug 18, 2015• 10 citations
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Abstract

We consider non-equilibrium time evolution after a quench of a global Hamiltonian parameter in systems described by Hamiltonians with local interactions. Within this background, we propose a protocol that allows to change global properties of the state by flipping a switch that modifies a local term of the Hamiltonian (creating a defect). A light-cone that separates two globally different regions originates from the switch. The expectation values of macroscopic observables, that is to say local observables that are spatially averaged within a subsystem, slowly approach new asymptotic values determined by the defect. The process is almost reversible: flipping again the switch produces a new light-cone with the two regions inverted. Finally, we test the protocol under repeated projective measurements. As explicit example we study the dynamics in a simple exactly solvable model but analogues descriptions apply also to generic models.

DOI
10.48550/arxiv.1508.04401

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We consider non-equilibrium time evolution after a quench of a global Hamiltonian parameter in systems described by Hamiltonians with local interactions. Within this background, we propose a protocol that allows to change global properties of the state by flipping a switch that modifies a local term of the Hamiltonian (creating a defect). A light-cone that separates two globally different regions originates from the switch. The expectation values of macroscopic observables, that is to say local observables that are spatially averaged within a subsystem, slowly approach new asymptotic values determined by the defect. The process is almost reversible: flipping again the switch produces a new light-cone with the two regions inverted. Finally, we test the protocol under repeated projective measurements. As explicit example we study the dynamics in a simple exactly solvable model but analogues descriptions apply also to generic models.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
Quench spectroscopy is a relatively new method which enables the investigation of spectral properties of many-body quantum systems by monitoring the out-of-equilibrium dynamics of real-space observables after a quench. So far the approach has been devised for global quenches or using local engineering of momentum-resolved excitations. Here, we extend the quench spectroscopy method to local quenches. We show that it allows us to extract quantitative information about global properties of the system, and in particular the elementary excitation spectrum. Using state-of-the-art numerical methods, we simulate the out-of-equilibrium dynamics of a variety of quantum systems following various local quench protocols and demonstrate a general scheme for designing an appropriate local quench protocol for any chosen model. We provide detailed examples of how the local quench protocol can be realized in realistic current generation experiments, including ultracold atomic gases and trapped ion systems.
We consider the nonequilibrium time evolution of a translationally invariant state under a Hamiltonian with a localized defect. We discern the situations where a light cone spreads out from the defect and separates the system into regions with macroscopically different properties. We identify the light cone and propose a procedure to obtain a (quasi)stationary state describing the late time dynamics of local observables. As an explicit example, we study the time evolution generated by the Hamiltonian of the transverse-field Ising chain with a local defect that cuts the interaction between two sites (a quench of the boundary conditions alongside a global quench). We solve the dynamics exactly and show that the late time properties can be obtained with the general method proposed.
We give a sufficient criterion that guarantees that a many-body quantum system can be controlled by properly manipulating the (local) Hamiltonian of one of its subsystems. The method can be applied to a wide range of systems: it does not depend on the details of the couplings but only on their associated topology. As a special case, we prove that Heisenberg and Affleck-Kennedy-Lieb-Tasaki chains can be controlled by operating on one of the spins at their ends. In principle, arbitrary quantum algorithms can be performed on such chains by acting on a single qubit.
The long-time dynamics of quantum systems, typically, but not always, results in a thermal steady state. The microscopic processes that lead to or circumvent this fate are of interest, since everyday experience tells us that not all spatial regions of a system heat up or cool down uniformly. This motivates the question: Under what conditions can one slow down or completely arrest thermalization locally? Is it possible to construct realistic Hamiltonians and initial states such that a local region is effectively insulated from the rest, or acts as a barrier between two or more regions? We answer this in the affirmative by outlining the conditions that govern the flow of energy and entropy between subsystems. Using these ideas we provide a representative example for how simple quantum few-body states can be used to engineer a ``thermal switch'' between interacting regions.
We study the dynamics arising from a double quantum quench where the parameters of a given Hamiltonian are abruptly changed from being in an equilibrium phase A to a different phase B and back ($\mathrm{A}\ensuremath{\rightarrow}\mathrm{B}\ensuremath{\rightarrow}\mathrm{A}$). As prototype models, we consider the (integrable) transverse Ising field as well as the (nonintegrable) ANNNI model. The return amplitude features nonanalyticities after the first quench through the equilibrium quantum critical point ($\mathrm{A}\ensuremath{\rightarrow}\mathrm{B}$), which is routinely taken as a signature of passing through a so-called dynamical quantum phase transition. We demonstrate that nonanalyticities after the second quench ($\mathrm{B}\ensuremath{\rightarrow}\mathrm{A}$) can be avoided and reestablished in a recurring manner upon increasing the time TTT spent in phase B. The system retains an infinite memory of its past state, and one has the intriguing opportunity to control at will whether or not dynamical quantum phase transitions appear after the second quench.