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arXiv:1911.08617 · Journal of Statistical Mechanics Theory and Experiment

Free bosons with a localized source

P. L. Krapivsky, Kirone Mallick, Dries Sels

Published Jun 1, 2020• 20 citations
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Abstract

Abstract We analyze the time evolution of an open quantum system driven by a localized source of bosons. We consider non-interacting identical bosons that are injected into a single lattice site and perform a continuous time quantum walks on a lattice. We show that the average number of bosons grows exponentially with time when the input rate exceeds a certain lattice-dependent critical value. Below the threshold, the growth is quadratic in time, which is still much faster than the naive linear in time growth. We compute the critical input rate for hyper-cubic lattices and find that it is positive in all dimensions d except for d = 2 where the critical input rate vanishes—the growth is always exponential in two dimensions. To understand the exponential growth, we construct an explicit microscopic Hamiltonian model which gives rise to the open system dynamics once the bath is traced out. Exponential growth is identified with a region of dynamic instability of the Hamiltonian system.

DOI
10.1088/1742-5468/ab8118

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Abstract We analyze the time evolution of an open quantum system driven by a localized source of bosons. We consider non-interacting identical bosons that are injected into a single lattice site and perform a continuous time quantum walks on a lattice. We show that the average number of bosons grows exponentially with time when the input rate exceeds a certain lattice-dependent critical value. Below the threshold, the growth is quadratic in time, which is still much faster than the naive linear in time growth. We compute the critical input rate for hyper-cubic lattices and find that it is positive in all dimensions d except for d = 2 where the critical input rate vanishes—the growth is always exponential in two dimensions. To understand the exponential growth, we construct an explicit microscopic Hamiltonian model which gives rise to the open system dynamics once the bath is traced out. Exponential growth is identified with a region of dynamic instability of the Hamiltonian system.
Abstract We study an open quantum system of free fermions on an infinite lattice coupled to a localized particle source. In the long time limit, the total number of fermions in the system increases linearly with growth rate dependent on the lattice geometry and dimensionality. We express the growth rate in terms of lattice Green functions and derive explicit formulae in one dimension and for the square lattice. The interplay between the dynamics and the coupling to the environment leads, in contrast to classical systems, to a non-monotonic dependence of the particle growth rate on the input rate. We show that for all lattices the particle growth rate is inversely proportional to the input rate when the latter becomes large. This is a manifestation of the quantum Zeno effect.
The aim of this work is to study the dynamics of quantum systems subjected to a localized fermionic source in the presence of bulk dephasing. We consider two classes of one-dimensional lattice systems: (i) a non-interacting lattice with nearest-neighbor and beyond, i.e., long-ranged (power-law) hopping, and (ii) a lattice that is interacting via short-range interactions modeled by a fermionic quartic Hamiltonian. We study the evolution of the local density profile ni(t)n_i(t)ni​(t) within the system and the growth of the total particle number N(t)N(t)N(t) in it. For case (i), we provide analytical insights into the dynamics of the nearest-neighbor model using an adiabatic approximation, which relies on assuming faster relaxation of coherences of the single particle density matrix. For case (ii), we perform numerical computations using the time-evolving block decimation (TEBD) algorithm and analyze the density profile and the growth exponent in N(t)N(t)N(t). Our detailed study reveals an interesting interplay between Hamiltonian dynamics and various environmentally induced mechanisms in open quantum systems, such as local source and bulk dephasing. It brings out rich dynamics, including universal dynamical scaling and anomalous behavior across various time scales and is of relevance to various quantum simulation platforms.
We study the quantum dynamics of filling an empty lattice of size LLL by connecting it locally with an equilibrium thermal bath that injects noninteracting bosons or fermions. We adopt four different approaches, namely, (i) direct exact numerics, (ii) Redfield equation, (iii) Lindblad equation, and (iv) quantum Langevin equation, which are unique in their ways for solving the time dynamics and the steady state. In this simple setup we investigate open quantum dynamics and subsequent approach to thermalization. The quantities of interest that we consider are the spatial density profile and the total number of bosons and fermions in the lattice. The spatial spread is ballistic in nature and the local occupation eventually settles down owing to equilibration. The ballistic spread of local density admits a universal scaling form. We show that this universality is only seen when the condition of detailed balance is satisfied by the baths. The difference between bosons and fermions shows up in the early time growth rate and the saturation values of the profile. The techniques developed here are applicable to systems in arbitrary dimensions and for arbitrary geometries.
We investigate the dynamics of full counting statistics (FCS) of the growth of the total number of fermions in a one-dimensional non-interacting lattice subjected to a localized particle gain (source) at one edge. The dynamics of the setup is modeled by the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) quantum master equation. We recast the counting problem that involves counting particle number at every lattice site to a problem where only the local injected site is involved. We employ the Schwinger-Keldysh path integral formalism within the GKSL framework and derive an analytical expression for the cumulant generating function at arbitrary times and obtain a Levitov-Lesovik type formula in the long-time limit, earlier derived for boundary driven setups in the steady-state. For clean lattices with either short- or long-range hopping, supporting single-particle delocalized eigenstates, we show that all cumulants grow linearly with time in the asymptotic regime. Our analytical results are in excellent agreement with direct numerical simulations. These findings establish a general framework for characterizing FCS and quantum fluctuations in driven open fermionic systems with localized particle injection.
We analyze time evolution of spread complexity (SC) in an isolated interacting quantum many-body system when it is subjected to a sudden quench. Characteristics features of the time evolution of the SC after the quench are analyzed for different timescales, both in integrable and chaotic models. For a short time after the quench, the SC shows universal quadratic growth, irrespective of the initial state or the nature of the Hamiltonian, with the timescale of this growth being determined by the local density of states. The characteristics of the SC in the next phase depend on the nature of the system, and we show that, depending on whether the survival probability of an initial state is Gaussian or exponential, the SC can continue to grow quadratically, or it can show linear growth. To understand the behavior of the SC at late times, we consider sudden quenches in two models, a full random matrix in the Gaussian orthogonal ensemble, and a spin-1/21/21/2 system with disorder. We observe that, for the full random matrix model and the chaotic phase of the spin-1/21/21/2 system, the SC shows linear growth at early times and saturation at late times. The full random matrix case shows a peak in the intermediate-time region, whereas this feature is less prominent in the spin-1/21/21/2 system, as we explain.
It is known that there are lattice models in which noninteracting particles get dynamically localized when periodic $\ensuremath{\delta}$-function kicks are applied with a particular strength. We use both numerical and analytical methods to study the effects of interactions in three different models in one dimension. The systems we have considered include spinless fermions with interactions between nearest-neighbor sites, the Hubbard model of spin-1/2 fermions, and the Bose-Hubbard model with on-site interactions. We derive effective Floquet Hamiltonians up to second order in the time period of kicking. Using these we show that interactions can give rise to a variety of interesting results such as two-body bound states in all three models and dispersionless few-particle bound states with more than two particles for spinless fermions and bosons. We substantiate these results by exact diagonalization and stroboscopic time evolution of systems with a few particles. We derive a pseudo-spin-1/2 limit of the Bose-Hubbard system in the thermodynamic limit and show that a special case of this has an exponentially large number of degenerate eigenstates of the effective Hamiltonian. Finally, we study the effect of changing the strength of the $\ensuremath{\delta}$-function kicks slightly away from perfect dynamical localization; we find that a single particle remains dynamically localized for a long time after which it moves ballistically.