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arXiv:1811.10281

Numerical Approach to the Evolution of the Spin-boson Systems and its Application on the Buck-Sukumar Model

Liu, Xueying, Xuezao Ren, Chen Wang, Gao Xianlong, Kelin Wang

Published Nov 26, 2018• 0 citations
arXivHTMLPDF

Abstract

We present a systematic numerical iteration approach to study the evolution properties of the spin-boson systems, which works well in whole coupling regime. This approach involves the evaluation of a set of coefficients for the formal expansion of the time-dependent Schrödinger equation $\vert t\rangle=e^{-i\hat{H}t}\vert t=0\rangle$ by expanding the initial state $\vert t=0\rangle$ in Fock space. The main advantage of this method is that this set of coefficients is unique for the Hamiltonian studied, which allows one to calculate the time evolution based on the different initial states. To complement our numerical calculations, the method is applied to the Buck-Sukumar model. Furthermore, we pointed out that, when the ground state energy of the model is unbounded and no ground state exists in a certain parameter space, the unstable time evolution of the physical quantities is the natural results. Furthermore, we test the performance of the numerical method to the Hamiltonian that use anti-Hermitian terms for modeling open quantum systems.

DOI
10.48550/arxiv.1811.10281

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We present a systematic numerical iteration approach to study the evolution properties of the spin-boson systems, which works well in whole coupling regime. This approach involves the evaluation of a set of coefficients for the formal expansion of the time-dependent Schrödinger equation ∣t⟩=e−iH^t∣t=0⟩\vert t\rangle=e^{-i\hat{H}t}\vert t=0\rangle∣t⟩=e−iH^t∣t=0⟩ by expanding the initial state ∣t=0⟩\vert t=0\rangle∣t=0⟩ in Fock space. The main advantage of this method is that this set of coefficients is unique for the Hamiltonian studied, which allows one to calculate the time evolution based on the different initial states. To complement our numerical calculations, the method is applied to the Buck-Sukumar model. Furthermore, we pointed out that, when the ground state energy of the model is unbounded and no ground state exists in a certain parameter space, the unstable time evolution of the physical quantities is the natural results. Furthermore, we test the performance of the numerical method to the Hamiltonian that use anti-Hermitian terms for modeling open quantum systems.
Abstract We study the evolution properties of spin-boson systems by a systematic numerical iteration approach, which performs well in the whole coupling regime. This approach evaluates a set of coefficients in the formal expression of the time-dependent Schrödinger equation by expanding the initial state in Fock space. This set of coefficients is unique for the spin-boson Hamiltonian studied, allowing one to calculate the time evolution from different initial states. To complement our numerical calculations, we apply the method to the Buck–Sukumar model. We find that when the ground-state energy of the model is unbounded and no ground state exists in a certain parameter space, the time evolution of the physical quantities is naturally unstable.
The feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spin-boson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration time-dependent Hartree context, is used to solve the time-dependent Schrödinger equations by explicitly including all the degrees of freedom (“system”+“bath”). Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of time-dependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.
Within the framework of the hierarchy equation of motion (HEOM), the quantum kinetic expansion (QKE) method of the spin-boson model is reformulated in the matrix representation. The equivalence between the two formulations (HEOM matrices and quantum operators) is numerically verified from the calculation of the time-integrated QKE rates. The matrix formulation of the QKE is extended to the system-bath factorized initial state. Following a one-to-one mapping between HEOM matrices and quantum operators, a quantum kinetic equation is rederived. The rate kernel is modified by an extra term following a systematic expansion over the site-site coupling. This modified QKE is numerically tested for its reliability by calculating the time-integrated rate and non-Markovian population kinetics. For an intermediate-to-strong dissipation strength and a large site-site coupling, the population transfer is found to be significantly different when the initial condition is changed from the local equilibrium to system-bath factorized state.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
We have developed in this paper a nonperturbative cluster-expansion strategy for generating the stochastically averaged time-evolution operator for quantum systems driven by Ornstein-Uhlenbeck (OU) colored noise. The method induces a boson mapping of the (real or complex) stochastic variable f, and interprets the stochastic average of a pair of variables f at two different times as the expectation value of the time-ordered product of the associated bosons with respect to the boson vacuum \ensuremath{\Vert}0B{0}_{\mathit{B}}0B​〉. The stochastic evolution is thus mapped onto a deterministic evolution in an expanded Fock space. The evolution of the system from the groups of state of interest is monitored using our recently developed nonperturbative time-dependent multireference coupled-cluster (TDMRCC) method. In this, the evolution operator U is written in a factorized form Uex{\mathit{U}}_{\mathrm{ex}}Uex​UM{\mathit{U}}_{\mathit{M}}UM​, where UM{\mathit{U}}_{\mathit{M}}UM​ evolves in the space of the starting functions (model space) and Uex{\mathit{U}}_{\mathrm{ex}}Uex​ brings in the virtual functions. Uex{\mathit{U}}_{\mathrm{ex}}Uex​ and UM{\mathit{U}}_{\mathit{M}}UM​ are both written as normal ordered exponentials involving cluster operators. In the present context, the TDMRCC method translates into one for generating the evolution operator UB{\mathit{U}}_{\mathit{B}}UB​ for the Hamiltonian containing the additional boson variables, and the stochastic averaging of UB{\mathit{U}}_{\mathit{B}}UB​ is realized as the expectation value 〈0B{0}_{\mathit{B}}0B​\ensuremath{\Vert}UB{\mathit{U}}_{\mathit{B}}UB​\ensuremath{\Vert}0B{0}_{\mathit{B}}0B​〉.We call the TDMRCC method involving the expanded Fock space a cluster cumulant method. We have analyzed the relation of the cluster cumulant approach with the Fox-Kubo operator cumulant expansion and the method of marginal averages involving the Fokker-Planck operator of the OU process. It has been shown that an order by order expansion of our equations in the power of stochastic coupling generates the perturbative cumulant results of Fox and Kubo. It is also demonstrated that the traditional Fokker-Planck method of marginal averages uses a Kubo-Schr\"odinger operator which is related to our boson mapped Hamiltonian by a transformation which converts the Fokker-Planck operator \ensuremath{\Gamma} to a manifestly Hermitian form. However, the method of marginal averages involves a linear expansion of UB{\mathit{U}}_{\mathit{B}}UB​ in terms of eigenfunctions of \ensuremath{\Gamma}, while the cluster cumulant method uses a cluster expansion, which is nonlinear in the expansion parameters. The former is thus analogous to a configuration-interaction expansion, while the latter is analogous to a coupled-cluster strategy. The method is illustrated by applying it to a simple yet nontrivial problem, viz., a harmonic oscillator linearly perturbed by a (real or complex) stochastic variable. Survivalities of both the ground and the first excited states have been considered for a wide range of coupling strengths and noise correlation times. Our method provides exact results in this case; these have been used as benchmarks for assessing performance of the second- and fourth-order cumulant results.
In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system-bath couplings or to special cases where specific numerical techniques become effective. Here we present a general and yet exact numerical approach that efficiently describes the time evolution of a quantum system coupled to a non-Markovian harmonic environment. Our method relies on expressing the system state and its propagator as a matrix product state and operator, respectively, and using a singular value decomposition to compress the description of the state as time evolves. We demonstrate the power and flexibility of our approach by numerically identifying the localisation transition of the Ohmic spin-boson model, and considering a model with widely separated environmental timescales arising for a pair of spins embedded in a common environment.