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 p（t） in quantum statistics as 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 the decay constant k we find that the matrix element of p（t） at time t in 〈η｜ representation is proportional to that of the initial po in the decayed entangled state （ηe^-kt｜ 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 = f（d^2η/π）（η｜ρ〉D（η）, which is different from all the previous known representations.

New approach to solving master equations of density operator for the Jaynes Cummings model with cavity damping

By introducing thermo-entangled state representation Ⅰη〉, which can map master equations of density operator in quantum statistics as state-vector evolution equations, and using ＂dissipative interaction picture＂ we solve the master equation of Jaynes-Cummings model with cavity damping. In addition we derive the Wigner function for density operator when the atom is initially in the up state Ⅰ↑〉 and the cavity mode is in coherent state.

Master equation describing the diffusion process for a coherent state

The evolution of a pure coherent state into a chaotic state is described very well by a master equation, as is validated via an examination of the coherent state＇s evolution during the diffusion process, fully utilizing the technique of integration within an ordered product （IWOP） of operators. The same equation also describes a limitation that maintains the coherence in a weak diffusion process, i.e., when the dissipation is very weak and the initial average photon number is large. This equation is dp/dt = -κ[a＋ap -a＋pa -apa＋ ＋ paa＋]. The physical difference between this diffusion equation and the better-known amplitude damping master equation is pointed out.

The population and decay evolution of a qubit under the time-convolutionless master equation

We consider the population and decay of a qubit under the electromagnetic environment. Employing the time- convolutionless master equation, we investigate the Markovian and non-Markovian behaviour of the corresponding perturbation expansion. The Jaynes-Cummings model on resonance is investigated. Some figures clearly show the different evolution behaviours. The reasons are interpreted in the paper.

Generalized Quantum Master Equation:A Tutorial Review and Recent Advances

The generalized quantum master equation(GQME)provides a general and exact approach for simulating the reduced dynamics in open quantum systems where a quantum system is embedded in a quantum environment.Dynamics of open quantum systems is important in excitation energy,charge,and quantum coherence transfer as well as reactive photochemistry.The system is usually chosen to be the interested degrees of freedom such as the electronicstates in light-harvesting molecules or tagged vibrational modes in a condensed-phase system.The environment is also called the bath,whose influence on the system has to be considered,and for instance can be described by the GQME formalisms using the projection operator technique.In this review,we provide a heuristic description of the development of two canonical forms of GQME,namely the time-convoluted Nakajima-Zwanzig form(NZ-GQME)and the time-convolutionless form(TCL-GQME).In the more popular NZ-GQME form,the memory kernel serves as the essential part that reflects the non-Markovian and non-perturbative effects,which gives formally exact dynamics of the reduced density matrix.We summarize several schemes to express the projection-based memory kernel of NZ-GQME in terms of projection-free time correlation function inputs that contain molecular information.In particular,the recently proposed modified GQME approach based on NZ-GQME partitions the Hamiltonian into a more general diagonal and off-diagonal parts.The projection-free inputs in the above-mentioned schemes expressed in terms of different system-dependent time correlation functions can be calculated via numerically exact or approximate dynamical methods.We hope this contribution would help lower the barrier of understanding the theoretical pillars for GQME-based quantum dynamics methods and also envisage that their combination with the quantum computing techniques will pave the way for solving complex problems related to quantum dynamics and quantum information that are currently intractable even with today’s state-of-the-art classical supercomputers.

Entropy and Entanglement in Master Equation of Effective Hamiltonian for Jaynes-Cummings Model

In this paper, we analytically solve the master equation for Jaynes-Cummings model in the dispersive regime including phase damping and the field is assumed to be initially in a superposition of coherent states. Using an established entanglement measure based on the negativity of the eigenvalues of the partially transposed density matrix we find a very strong sensitivity of the maximally generated entanglement to the amount of phase damping. Qualitatively this behavior is also reflected in alternative entanglement measures, but the quantitative agreement between different measures depends on the chosen noise model The phase decoherence for this model results in monotonic increase in the total entropy while the atomic sub-entropy keeps its periodic behaviour without any effect.

Solving nonlinear master equation describing quantum damping by virtue of the entangled state representation

This paper solves the newly constructed nonlinear master equation dρ/dt = κ[2f （N） aρ （1/f （N - 1））a^＋ -a^＋aρ- ρa^＋a], where f（N） is an operator-valued function of N = a^＋a, for describing amplitude damping channel, and derives the infinite operator sum representation of quasi-Kraus operators for the density operator. It also shows that in this nonlinear process the initial pure number state density operator will evolve into the binomial field （a mixed state） when f （N） = 1√N ＋ 1.

Short-Time Effects of Environment to Josephson Charge Qubit: A Master Equation Approach

In this paper we investigate an environmental effects to Josephson charge qubit (JCQ) when the environmentis taken as the Ohmic bath.At first,we derive the master equation from a JCQ-bath model.Then we investigate thecoefficients of the equations that describe the shift in frequency,diffusive,decoherence,and so on.It is shown that ourresult on decoherence agrees with experimental one very well as the time is short enough.

New approach for analysing master equations of generalized phase diffusion models in the entangled state representation

By virtue of the well-behaved properties of the bipartite entangled states representation, this paper analyse and solves some master equations for generalized phase diffusion models, which seems concise and effective. This method can also be applied to solve other master equations.

Shot noise in electron transport through a double quantum dot:a master equation approach

We study shot noise in tunneling current through a double quantum dot connected to two electric leads. We derive two master equations in the occupation-state basis and the eigenstate basis to describe the electron dynamics. The approach based on the occupation-state basis, despite being widely used in many previous studies, is valid only when the interdot coupling strength is much smaller than the energy difference between the two dots. In contrast, the calculations using the eigenstate basis are valid for an arbitrary interdot coupling. Using realistic model parameters, we demonstrate that the predicted currents and shot-noise properties from the two approaches are significantly different when the interdot coupling is not small. Furthermore, properties of the shot noise predicted using the eigenstate basis successfully reproduce qualitative features found in a recent experiment.

Long-time limit behavior of the solution to an atom's master equation

We study the long-time limit behavior of the solution to an atom＇s master equation. For the first time we derive that the probability of the atom being in the α-th （α = j ＋ 1 -jz, j is the angular momentum quantum number, jz is the z-component of angular momentum） state is {（1 - K/G）/[1 - （K/G）2j＋1]}（K/G）^α-1 as t → ＋∞, which coincides with the fact that when K/G 〉 1, the larger the a is, the larger the probability of the atom being in the α-th state （the lower excited state） is. We also consider the case for some possible generaizations of the atomic master equation.

A Stochastic Extension of Macroscopic Stability Criterion of Nonequilibrium Steady State in Chemical Reaction Systems Governed by Master Equation

By means of both the separation of the perturbation in accordance with characteristic parnmeters and the Kramers Moyal-expansion of the master equation, it is shown that the time derivative of the partial excess quantity of stochastic entropy due to the deviation from the most probable path is related to the responsibility of a system to the external macroscopic perturbations. This evolution rate of the partial excess stochastic entropy is equivalent to the partlal excess stochastic entropy production, as well as the stochastic excess entropy production rate based on the stochastic potential npproach. It appears also as an eqivalent quantity of the Gibbs excess entropy production for the Polsson distribution. The macroscopic stability of chemical reaction systems is dominnted by this new stochastic quantity when the local equilibrium thermodynamics is broken down .

Solving Master Equation for Two-Mode Density Matrices by Virtue of ThermalEntangled 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.

Generalized Master Equation for Space-Time Coupled Continuous Time Random Walk

The generalized master equation for the space-time coupled continuous time random walk is derived analytically, in which the space-time coupling is considered through the correlated function 9（t） ～ t^γ, 0 ≤ γ 〈 2, and the probability density function ω（t） of a particle＇s waiting time t follows a power law form for large t： ω（t） ～t^-（1＋α）, 0 〈 α 〈 1. The results indicate that the expressions of the generalized master equation are determined by the correlation exponent 7 and the long-tailed index α of the waiting time. Moreover, the diffusion results obtained from the generalized master equation are in accordance with the previous known results and the numerical simulation results.

On the quantum master equation under feedback control

The nature of the quantum trajectories, described by stochastic master equations, may be jump-like or diffusive, depending upon different measurement processes. There are many different unravelings corresponding to different types of stochastic master equations for a given master equation. In this paper, we study the relationship between the quantum stochastic master equations and the quantum master equations in the Markovian case under feedback control. We show that the corresponding unraveling no longer exists when we further consider feedback control besides measurement. It is due to the fact that the information gained by the measurement plays an important role in the control process. The master equation governing the evolution of ensemble average cannot be restored simply by eliminating the noise term unlike the case without a control term. By establishing a fundamental limit on performance of the master equation with feedback control, we demonstrate the differences between the stochastic master equation and the master equation via theoretical proof and simulation, and show the superiority of the stochastic master equation for feedback control.

On a tilted Liouville-master equation of open quantum systems

A tilted Liouville-master equation in Hilbert space is presented for Markovian open quantum systems.We demonstrate that it is the unraveling of the tilted quantum master equation.The latter is widely used in the analysis and calculations of stochastic thermodynamic quantities in quantum stochastic thermodynamics.

Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity

Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-five years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the field that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume （i.e., mesoscopic） in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation （CME）. Similar to the wavefnnction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved： multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic differential equations that offer alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand “complex behavior” and complexity theory, and from which important biological insight can be gained.

Number-resolved master equation approach to quantum transport under the self-consistent Born approximation

We construct a particle-number （n）-resolved master equation （ME） approach under the self-consistent Born approximation （SCBA） for quantum transport through mesoscopic systems. The formulation is essentially non-Markovian and incorporates the interplay of the multi-tunneling processes and many-body correlations. The proposed n-SCBA-ME goes beyond the scope of the Born- Markov master equation, being applicable to transport under small bias voltage, in non-Markovian regime and with strong Coulomb correlations. For steady state, it can recover not only the exact result of noninteracting transport under arbitrary voltages, but also the challenging nonequilibrium Kondo effect. Moreover, the n-SCBA-ME approach is efficient for the study of shot noise. We demonstrate the application by a couple of representative examples, including particularly the nonequilibrium Kondo system.

Master equation and runaway speed of the Francis turbine

The master equation of the Francis turbine is derived based on the combination of the angular momentum（Euler） and the energy laws. It relates the geometrical design of the impeller and the regulation settings（guide vane angle and rotational speed） to the discharge and the power output. The master equation, thus, enables the complete characteristics of a given Francis turbine to be easily computed. While applying the energy law, both the shock loss at the impeller inlet and the swirling loss at the impeller exit are taken into account. These are main losses which occur at both the partial load and the overloads and, thus, dominantly influence the characteristics of the Francis turbine. They also totally govern the discharge of the water through the impeller when the impeller is found in the standstill. The computations have been performed for the discharge, the hydraulic torque and the hydraulic efficiency. They were also compared with the available measurements on a model turbine. Excellent agreement has been achieved. The computations also enable the runaway speed of the Francis turbine and the related discharge to be determined as a function of the setting angle of the guide vanes.

Number-resolved master equation approach to quantum measurement and quantum transport

In addition to the well-known Landauer-Bfittiker scattering theory and the nonequilibrium Green＇s function technique for mesoscopic transports, an alternative （and very useful） scheme is quantum master equation approach. In this article, we review the particle-number （n）-resolved master equation （n-ME） approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and flfll counting statistics analysis. Moreover, we also review a newly developed master equation approach （and its n-resolved version） under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.