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arXiv:2004.02232 · Physical review. B./Physical review. B

Bosonic representation of a Lipkin-Meshkov-Glick model with Markovian dissipation

Jan C. Louw, Michael Kästner, Johannes N. Kriel

Published Sep 23, 2020• 3 citations
arXivHTMLPDF

Abstract

We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems but takes into account leading-order finite-size corrections to the infinite-system result. As an application, we show that the dissipative Lipkin-Meshkov-Glick model equilibrates by passing through a continuum of thermal states with damped oscillations superimposed, until finally reaching an equilibrium state with a temperature that in general differs from the bath temperature. We discuss limitations of our analytic techniques by comparing to exact numerical results.

DOI
10.1103/physrevb.102.094430

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We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems but takes into account leading-order finite-size corrections to the infinite-system result. As an application, we show that the dissipative Lipkin-Meshkov-Glick model equilibrates by passing through a continuum of thermal states with damped oscillations superimposed, until finally reaching an equilibrium state with a temperature that in general differs from the bath temperature. We discuss limitations of our analytic techniques by comparing to exact numerical results.
We derive a Lindblad master equation that approximates the dynamics of a Lipkin-Meshkov-Glick (LMG) model weakly coupled to a bosonic bath. By studying the time evolution of operators under the adjoint master equation we prove that, for large system sizes, these operators attain their thermal equilibrium expectation values in the long-time limit, and we calculate the rate at which these values are approached. Integrability of the LMG model prevents thermalization in the absence of a bath, and our work provides an explicit proof that the bath indeed restores thermalization. Imposing thermalization on this otherwise nonthermalizing model outlines an avenue towards probing the unconventional thermodynamic properties predicted to occur in ultracold-atom-based realizations of the LMG model.
Motivated by recent prototypes of engineered atomic spin devices, we study a fully connected system of NNN spins 1/21/21/2, modeled by the Lipkin-Meshkov-Glick (LMG) model of a collective spin s=N/2s=N/2s=N/2 in the presence of Markovian dissipation processes. We determine and classify the different phases of the dissipative LMG model with Markovian dissipation, including the properties of the steady state and the dynamic behavior in the asymptotic long-time regime. Employing variational methods and a systematic approach based on the Holstein-Primakoff mapping, we determine the phase diagram and the spectral and steady-state properties of the Liouvillian by studying both the infinite-sss limit and 1/s1/s1/s corrections. Our approach reveals the existence of different kinds of dynamical phases and phase transitions, multistability, and regions where the dynamics is recurrent. We provide a classification of critical and noncritical Liouvillians according to their spectral and steady-state properties.
In a recent work, we have derived simple Lindblad-based equations for the thermalization of systems in contact with a thermal reservoir. Here, we apply these equations to the Lipkin-Meshkov-Glick model in contact with blackbody radiation and analyze the dipole matrix elements involved in the thermalization process. We find that the thermalization can be complete only if the density is sufficiently high, while, in the limit of low density, the system thermalizes partially, namely, within the Hilbert subspaces where the total spin has a fixed value. In this regime, and in the isotropic case, we evaluate the characteristic thermalization time analytically, and show that it diverges with the system size in correspondence with the critical points and inside the ferromagnetic region. Quite interestingly, at zero temperature the thermalization time diverges only quadratically with the system size, whereas quantum adiabatic algorithms, aimed at finding the ground state of the same system, imply a cubic divergence of the required adiabatic time.
ENGLISH ABSTRACT : Motivated by recent experiments, we study the time evolution of a two-species Bose-Einstein condensate which is coupled to a bosonic bath. For the particular condensate, unconventional thermodynamics have recently been predicted. To study these thermal properties we find the conditions under which this open quantum system thermalises—equilibrates to the Gibbs state describing the canonical ensemble. The condensate is mapped from its bosonic representation, describing N interacting bosons, onto a Schwinger spin representation, with spin angular momentum S = 2N. The corresponding Hamiltonian takes the form of a Lipkin-Meshkov-Glick (LMG) model. We find that the total system-bath Hamiltonian is too difficult to solve. Fortunately, in the case where the LMG model is only weakly coupled to a near-memoryless bath, we may derive an approximate differential equation describing the LMG model’s evolution. Upon further approximations this equation describes a quantum dynamical semigroup—preserving all the rules of quantum mechanics. The generator of this semigroup, known as the Lindbladian, is our main object of interest. Its nullspace is populated by the stationary state(s), which we wish to compare to the canonical ensemble. If the only state in this nullspace is the Gibbs state, then the system will thermalise to that state. For extensive bath temperatures we show that the Gibbs state is stationary in the thermodynamic limit N = S/2 → ∞. Numerically we find that this is the only stationary state, meaning the system thermalises. To prove thermalisation for the intensive temperature case we diagonalise the Lindbladian. To do this we first perform a Holstein-Primakoff (HP) mapping from spins onto bosons. Such a mapping may be approximated in the region of interest—close to the stationary state(s)—given one has knowledge of that regions expectation values, such as the magnetisation hSz ieq. The new approximated Lindbladian may then be diagonalised via a Bogoliubov transform. To find these expectation values we study the observables’ time evolution in the Heisenberg picture. We find that energy dissipation happens on a time scale purely characterised by the weak coupling. This allows us to set up an approximate differential equation for the energy which we solve, hence finding the equilibrium energy. Using this energy together with spin coherent states we are then able to find hSz ieq and hS 2 xieq. Performing the HP mapping in the region of interest then leaves a quadratic, in bosonic ladder operators, Lindbladian which we subsequently diagonalise with a Bogoliubov transform. The stationary states are found to be approximate Gibbs states. In the non-degenerate phase of the LMG model we find a single stationary state, thus proving thermalisation. In contrast, we find two (orthogonal) near stationary states in degenerate phase. Numerically we find, for finite system size, that these two states form an equal fraction of the true stationary state which is a Gibbs state. However the numerics further indicates that the spectral gap between this Gibbs state and its closest state tends to zero in the thermodynamic limit. As such we expect two (orthogonal) stationary states in the degenerate phase as S → ∞.
Abstract We present a systematic formalism for the computation of the density matrix of an N-state quantum system in the presence of classical noise or a coupling to the environment. In this formalism, the density matrix of the system is given as an expansion in the generators of the SU(N) group with real coefficients. This leads to a system of master equations. The parameters in these equations may be approximately expressed in terms of the components of the Redfield tensor, when the Born and Markov approximations are valid. The general form of the solution of the system of master equations is established. All relaxation and dephasing rates are then very simply expressed as eigenvalues of a certain matrix. This gives the formulation its simplicity and makes it uniquely suitable for numerical computation. The spectral representation of the components of the Redfield tensor is derived in the case when the environment is a harmonic oscillator bath in thermal equilibrium. Beyond the Born approximation, the decoherence of the system is determined by the Lindblad formula for the Liouvillian superoperator. The Lindblad formulae of some models of multi-state quantum systems are also presented.