Solution of the Master Equation in the Amplitude Damping Model Under the Action of Linear Resonance Force

The amplitude damping model master equations of density operators under the action of linear resonance force can be concisely solved by virtue of thermo entangled state representation and the technique of integration within an ordered product of operators. We obtain the infinitive operator-sum representation of density operators. This approach may also be effective to treat other master equations. Further, the evolution of the coherent state in this model is discussed. The results show that the coherent state maintains its original coherence character in the amplitude damping model under the action of linear resonance force.     

Solution of the Master Equation in Laser Process Under the Action of Linear Resonance Force

The master equation of density operator in laser process under action of linear resonance force can be concisely solved by virtue of thermo-entangled state representation and the technique of integration within an ordered product of operators. We obtain the infinitive operator-sum representation of density operator. As the application of this method, the evolution of thermal field and that of vacuum state in this model are discussed. The results show that thermal field maintains its original character, and vacuum state evolves thermal field in laser process.     

Using a New Disentangling Technique in the Solution of the Master Equation of a Driven Damped Harmonic Oscillator by Thermo Entangled States Representation

In this study we aim to solve the amplitude damping model master equation for a driven damped harmonic oscillator under the action of a classical force with arbitrary time dependence. We use thermo entangled state representation for the density operator, but to avoid a complicated disentangling process in such solutions, we introduce a simple and concise method to extract the density operator from its thermo entangled state representation. Whereas time evolution is a classical process, this method can be effectively used.

     

相位扩散通道中密度算符相关态的时间演化

利用热纠缠态的性质,对具有代表性的相位扩散主方程进行求解,得到关于密度算符的算符和表示形式,分析不同初始态下的密度算符的时间演化结果,发现在相位扩散通道下当初始态为粒子数态或热态时密度算符保持恒定,而当初始态为相干态时系统在发生相扩散的同时始终保持相干态特性不变.     

Infinitive Operator-Sum Representation for Damping in a Squeezed Heat Reservoir via the Thermo Entangled State Approach

Using the thermo entangled state approach, we successfully solve the master equation of a damped harmonic oscillator affected by a linear resonance force in a squeezed heat reservoir, and obtain the analytical evolution formula for the density operator in the infinitive Kraus operator-sum representation. Interestingly, the Kraus operators M_l,m,n,r and \(\mathfrak {M}_{l,m,n,r}^{\dag }\) are not Hermite conjugate, but they are still trace-preserving quantum operations because of the normalization condition. We also investigate the evolution for an initial coherent state for damping in a squeezed heat reservoir, which shows that the initial coherent state decays to a complex mixed state as a result of damping and thermal noise.     

Kraus operator solutions to a fermionic master equation describing a thermal bath and their matrix representation

We solve the fermionic master equation for a thermal bath to obtain its explicit Kraus operator solutions via the fermionic state approach. The normalization condition of the Kraus operators is proved. The matrix representation for these solutions is obtained, which is incongruous with the result in the book completed by Nielsen and Chuang [Quantum Computation and Quantum Information, Cambridge University Press, 2000]. As especial cases, we also present the Kraus operator solutions to master equations for describing the amplitude-decay model and the diffusion process at finite temperature.     

New Approach for Solving Master Equations in Quantum Optics and Quantum Statistics by Virtue of Thermo-Entangled State Representation

By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation <η |, which can arrange master equations of density operators ρ(t) in quantum statisticsas state-vector evolution equations due to the elegant properties of <η |. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of thedecay constant κ we find that the matrix element of ρ (t) at time t in <η| representation is proportional to that of the initial ρ₀ in the decayed entangled state <ηe^-κt| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ρ=∫(d²η /π)< η | ρ> D(η), which is different from all the previous known epresentations.     

New approach for deriving the exact time evolution of the density operator for a diffusive anharmonic oscillator and its Wigner distribution function

Using thermal entangled state representation,we solve the master equation of a diffusive anharmonic oscillator(AHO) to obtain the exact time evolution formula for the density operator in the infinitive operator-sum representation.We present a new evolution formula of the Wigner function(WF) for any initial state of the diffusive AHO by converting the WF calculation into an overlap between two pure states in an enlarged Fock space.It is found that this formula is very convenient in investigating the WF’s evolution of any known initial state.As applications,this formula is used to obtain the evolution of the WF for a coherent state and the evolution of the photon-number distribution of diffusive AHOs.     

Solving Master Equation for Two-Mode Density Matrices by Virtue of Thermal Entangled State Representation

We extend the approach of solving master equations for density matrices by projecting it onto the thermal entangled state representation (Hong-Yi Fan and Jun-Hua Chen, J. Phys. A35 (2002) 6873) to two-mode case. In this approach the two-photon master equations can be directly and conveniently converted into c-number partial differential equations. As an example, we solve the typical master equation for two-photon process in some limiting cases.     

Evolution law of a negative binomial state in an amplitude dissipative channel

For the first time we derive the evolution law of the negative binomial state In） （nI in an ampli-tude dissipative channel with a damping constant to. We find that after passing through the channel, the final state is still a negative binomial state, however the parameter γ evolves into The decay law of theaverage photon number is also obtained.     

New approach for deriving the exact time evolution of density operator for diffusive anharmonic oscillator and its Wigner distribution function

Using the thermal entangled state representation, we solve the master equation of a diffusive anharmonic oscillator (AHO) to obtain the exact time evolution formula for the density operator in the infinitive operator-sum representation. We present a new evolution formula of the Wigner function (WF) for any initial state of the diffusive AHO by converting the calculation of the WF to an overlap between two pure states in an enlarged Fock space. It is found that this formula brings us much convenience to investigate the WF's evolution of any known initial state. As applications, this formula is used to obtain the evolution of the WF for a coherent state and the evolution of the photon-number distribution of the diffusive AHO.     

A new kind of squeezed thermal state of continuum variables in degenerate case

In thermal field dynamics usually for every real field operator acting on a real space has an image acting on a fictitious space. In this work we construct a new kind of squeezed thermal state of continuum variables in which two real modes share one fictitious mode, this can be named the degenerate case. We also prove that this state-set makes up a new quantum-mechanical representation and show its application in solving some master equation.     

Infinite-Dimensional Kraus Operators for Describing a Thermal Channel with Self-Kerr Interaction

According to operator-sum representation theory, we have identified infinite-dimensional Kraus operators for describing a thermal channel with self-Kerr interaction after directly solving the corresponding master equation by virtue of thermo entangled state. Then we also prove in detail that Kraus operators hold the normalization. As an example, we exactly calculate the evolving result of a chaotic field in the thermal environment with the Kerr medium and find that the chaotic field evolves into a new chaotic field unaffected by the coupling factor with the Kerr medium.     

Diffusing opinions in bounded confidence processes

We study the effects of diffusing opinions on the Deffuant et al. model for continuous opinion dynamics. Individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside an interval centered around the present opinion. We show that diffusion induces an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion clusters are formed, although with some opinion spread inside them. If the diffusion jumps are not large, clusters coalesce, so that weak diffusion favors opinion consensus. A master equation for the process described above is presented. We find that the master equation and the Monte Carlo simulations do not always agree due to finite-size induced fluctuations. Using a linear stability analysis we can derive approximate conditions for the transition between opinion clusters and the disordered state. The linear stability analysis is compared with Monte Carlo simulations. Novel interesting phenomena are analyzed.     

Finite size effects in stochastically driven systems use of the Poisson representation

A method is presented to take into account finite size effects in a system under the influence of external noise. Inclusion of external noise in a master equation results in an effective master equation in which new transitions among states are possible. The steady state properties of a chemical systems are calculated using the Poisson representation of the master equation.     

Ket-Bra纠缠态方法研究含时外场中与热库耦合Qubit的演化

采用Ket-Bra纠缠态方法求解主方程, 研究了具有含时外场情况下单qubit和无相互作用的两qubit与热库耦合时的量子退相干、退纠缠现象. 对两qubit情形, 我们以共生纠缠度(concurrence)作为纠缠度量, 研究了其纠缠动力学演化过程. 研究表明即使系统内部不存在直接、间接的相互作用, 施加含时外场也能引起纠缠的震荡和复活, 这为通过施加控制外场抑制开放系统的退相干、退纠缠过程提供了理论支持.     

Dynamics of a three-level atom in a resonant light field

The paper is devoted to a consideration of the motion of a three-level atom in two resonant light waves. A kinetic equation of the Fokker-Planck type for the atomic distribution function is derived, which is valid when the recoil energy is small compared to the linewidths of the resonant transitions. The detailed behaviour of the radiation force and the diffusion tensor are studied numerically. The case of exact resonance and the nonresonant case are both considered. It is shown that a detuning from exact resonance results in a drastic decrease of the resonant light pressure force. For the detuning we determine the condition, under which an efficient action of the light pressure on a three-level atom takes place.     

Biased diffusion in a piecewise linear random potential

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.     

Influence of experimental resolution on the quantum-to-classical transition in the quartic oscillator

The quantum-classical transition problem is investigated for the quartic oscillator coupled to a thermal reservoir. We show for this model that the combination of relevant diffusion, classical action (represented by the amplitude of an initial coherent state) and the experimental uncertainties is necessary to achieve the classical regime. In order to study the role of limited resolutions of measurement apparatuses on the correspondence between the quantum and classical dynamics, we consider experimental errors due the preparation of the initial state of the quartic oscillator and the inaccuracies in the time measurements. A quantum break time depending on the diffusion constant, the amplitude of the initial coherent state and the inaccuracy of measurements is defined. We found, for this model, a regime where the increasing of diffusion does not anticipate classicality. In such regime, there is a minimum value for the classical action associated to classical behavior of the system.     

Evolution of the single-mode squeezed vacuum state in amplitude dissipative channel

Using the way of deriving infinitive sum representation of density operator as a solution to the master equation describing the amplitude dissipative channel by virtue of the entangled state representation, we show manifestly how the initial density operator of a single-mode squeezed vacuum state evolves into a definite mixed state which turns out to be a squeezed chaotic state with decreasing-squeezing and deeoherence. We investigate average photon number, photon statistics distributions for this mixed state.