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AIP conference proceedings

Numerical simulation of free dissipative open quantum system and establishment of a formula for π

Souvik Agasti

Published Jan 1, 2020• 1 citation
Publisher

Abstract

We transform the system/reservoir coupling model into a one-dimensional semi-infinite discrete chain with nearest neighbor interaction through a unitary transformation, and, simulate the dynamics of free dissipative open quantum system. We investigate the consequences of such modeling, which is observed as finite size effect causing the recurrence of particle from the end of the chain. Afterwards, we determine a formula for π in terms of the matrix operational form, which indicates a robustness of the connection between quantum physics and basic mathematics.

DOI
10.1063/5.0001282

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We transform the system/reservoir coupling model into a one-dimensional semi-infinite discrete chain with nearest neighbor interaction through a unitary transformation, and, simulate the dynamics of free dissipative open quantum system. We investigate the consequences of such modeling, which is observed as finite size effect causing the recurrence of particle from the end of the chain. Afterwards, we determine a formula for π in terms of the matrix operational form, which indicates a robustness of the connection between quantum physics and basic mathematics.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
Abstract We transform the system/reservoir coupling model into a one-dimensional semi-infinite discrete chain through unitary transformation to simulate the open quantum system numerically with the help of time evolving block decimation (TEBD) algorithm. We apply the method to study the dynamics of dissipative systems. We also generate the thermal state of a multimode bath using minimally entangled typical thermal state (METTS) algorithm, and investigate the impact of the thermal bath on an empty system. For both cases, we give an extensive analysis of the impact of the modeling and simulation parameters, and compare the numerics with the analytics.
By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.
We investigate the mathematical structure of entropy-weighted quantum evolution within the framework of open quantum systems. Starting from a system–reservoir interaction model with weak entropy-coupled dynamics, we derive the reduced evolution of the system density matrix under the Born–Markov approximation. The resulting generatoris shown to be of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form, ensuring complete positivity and trace preservation. We further demonstrate that the associated dynamical semigroup admits both Stinespring dilation and Kraus operator representations, establishing that entropy-weighted reduced dynamics can be embedded into a fully unitary system environment framework. The entropy operator appearing in the effective description is identified with the positive-semidefinite dissipative structure induced by reservoir corre lation functions.The construction does not modify standard quantum mechanics but provides a parameterization of dissipative generators in terms of entropy-weighted couplings. This establishes a mathematically consis tent bridge between entropy-based dynamical formulations and conventional open quantum system theory.
Understanding the dynamics of open quantum systems is intrinsically important to control and utilize quantum hardware in various applications. The generic behavior of purely dissipative open quantum many-body systems with kkk-local, normal dissipation processes can be investigated using random matrix theory, revealing different decay timescales of observables determined by their complexity, as shown by Wang et al. [K. Wang et al., Phys. Rev. Lett. 124, 100604 (2020)]. The time evolution of an open quantum system is entirely described by the eigenvalue spectrum of the Lindbladian, in which the hierarchy of decay timescales is reflected in the formation of well-separated eigenvalue clusters. Here, we extend the analysis of this spectrum to the presence of unitary dynamics using a random matrix model. In the case of strong dissipation, the unitary dynamics can be treated perturbatively and the kkk locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features, which in turn implies the same robustness of the corresponding separation of decay timescales. We find the perturbative approach for the spectrum to yield an excellent agreement with exact-diagonalization results. For weak dissipation, the spectrum flows into clusters except for outlier eigenmodes, which we identify to be the local symmetries of the Hamiltonian. We support our analytic findings by numerical simulations and show that our generic model applies to a nonrandom Heisenberg chain.
Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir.