運用Green Function 研究阻尼JC模型之動力學

許寧翔

Published Jan 1, 2012• 0 citations

Abstract

In this thesis, we use three different methods to describe open quantum systems, in particular, to study the damped Jaynes-Cummings model at zero temperature. We begin with Liouville-von Neumann equation and Born approximation to obtain a time-convolution Master Equation. Then, we make Post-Markov approximation further and obtain a time-convolutionless Master Equation. The third method is based on Keldysh’s non-equilibrium Green function method plus Born approximation. Besides, we also solve exactly the damped Jaynes-Cummings model at zero temperature. We use these methods mentioned above to study the evolution of the probability of the excited state in damped Jaynes-Cummings model. We analyze the result of the exact solution to understand the effect of the coupling strength and spectrum width on the system. By compared the results obtained from the three different approaches with the exact solution, we find the validity of each approach. We also find that the non-equilibrium Green function approach is the most suitable one to study the dynamics of open systems, and may be extended to investigate the open systems dynamics in strong couplings and finite temperature in the future.

DOI: 10.6844/ncku.2012.01487

Related papers

Study of non-equilibrium Green's functions beyond Born approximation in open quantum systems

Katha Ganguly, Bijay Kumar Agarwalla

We provide a systematic approach to compute different kinds of non-equilibrium Green's functions for open quantum systems which are essentially two-point correlation functions in time. We reveal that the definition of Green's functions based on the Born approximation does not provide the correct results in the leading order of the system-bath coupling. We next provide a systematic correction term in Green's functions by going beyond the Born approximation and incorporating a finite correlation between the system and the bath. We primarily focus on two paradigmatic models of open quantum systems, namely, the dissipative Caldeira-Leggett model and the dissipative spin-boson model. We show that the inclusion of such a correction correctly reproduces the Kadanoff-Baym type equation for the so-called lesser or greater components of the Green's functions and provides the correct long-time result up to the first non-zero order of the system-bath coupling, satisfying the detailed balance condition in thermal equilibrium. We further extend our study to a system coupled to multiple reservoirs simultaneously that are maintained at different temperatures and obtain expressions for non-equilibrium steady-state energy current, once again correct up to the first non-zero order of the system-bath coupling.

arXiv:2309.03776Sep 7, 20230 citations

An Alternative Approach to Jaynes–Cummings Model with Dissipation at Finite Temperature

Kuang Le-Man, Chen Guanghong, Mo‐Lin Ge

An alternative approach to the Jaynes–Cummings model (JCM) with dissipation at a finite environmental temperature is presented in terms of a new master equation under Born–Markovian approximation. An analytic solution of the dissipation JCM is obtained. A variety of physical quantities of interest are calculated analytically. Dynamical properties of the atom and the field are investigated in some detail. It is shown that both cavity damping and environmental temperature strongly affect nonclassical effects in JCM, such as collapse and revivals of the atomic inversion, oscillations of the photon-number distribution, quadrature squeezing of the field and sub-Poissonian photon statistics.

Communications in Theoretical PhysicsSep 15, 19970 citations

An alternative approach to Jaynes-Cummings model with dissipation at finite temperature

Le‐Man Kuang, Guang‐Hong Chen, Mo‐Lin Ge +1 more

An alternative approach to the Jaynes-Cummings model (JCM) with dissipation at a finite enviromental temperature is presented in terms of a new master equation under Born-Markovian approximations. An analytic solution of the dissipation JCM is obtained. A variety of physical quantities of interest are calculated analytically. Dynamical properties of the atom and the field are investigated in some detail. It is shown that both cavity damping and environmental temperature strongly affect nonclassical effects in the JCM, such as collapse and revivals of the atomic inversion, oscillations of the photon-number distribution, quadrature squeezing of the field and sub-Poissonian photon statistics.

arXiv:cond-mat/9509094Sep 15, 19950 citations

Exact master equation for an open Jaynes–Cummings system

Akane Watanabe, Hiromichi Nakazato

A master equation for the Jaynes-Cummings (JC) system interacting with a bosonic reservoir through a conserved quantity is derived analytically without any approximation.The derivation is valid for any initial JC state that has no correlation with the initial reservoir at any temperature and yields a Gorini-Kossakowski-Lindblad-Sudarshan-type master equation with time-dependent coefficients.Several physical quantities of the open JC system are examined and the validity or limit of some of the standard approximations used to derive the master equation is discussed on the basis of the exact results obtained here.

Annals of PhysicsApr 27, 20223 citations

Markovian and non-Markovian master equations versus an exactly solvable model of a qubit in a cavity

Zihan Xia, Juan Garcia-Nila, Daniel A. Lidar

Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an analytical solution is available in the one-excitation subspace. We consider the non-Markovian time-convolutionless master equation up to the second (Redfield) and fourth orders as well as three types of Markovian master equations: the coarse-grained, cumulant, and standard rotating-wave approximation (RWA) Lindblad equations. We compare the exact solution to these master equations for three different spectral densities: impulse, Ohmic, and triangular. We demonstrate that the coarse-grained master equation outperforms the standard RWA-based Lindblad master equation for weak coupling or high qubit frequency (relative to the spectral density high-frequency cutoff ${\ensuremath{\omega}}_{c}$), where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by oscillatory, nondecaying behavior, the time-convolutionless approximation closely matches the exact solution for short evolution times (in units of ${\ensuremath{\omega}}_{c}^{\ensuremath{-}1}$) even outside the regime of validity of the Markovian approximations. For long evolution times, all master equations perform poorly, as quantified in terms of the trace-norm distance from the exact solution. The fourth-order time-convolutionless master equation achieves the top performance in all cases. Our results highlight the need for reliable approximation methods to describe open-system quantum dynamics beyond the short-time limit.

arXiv:2403.09944Physical Review AppliedJul 11, 202414 citations

Energy Conservation in Open Quantum Systems

Albertus J. J. M. de Klerk

The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.

Jan 1, 20180 citations