Nonlinear effects of colored nonstationary noise: Exact results

The master equation is derived for random systems under nonlinear time-dependent conditions. The (non-Markov) process is of such a type that with a time-dependent state transformation the dynamics can be modelled by a nonlinear but drift-free Langevin equation. The focus is on the statistical content of resulting master equation. The existence of stationary solutions and the quality of approximative results is discussed.     

Markovian master equation for a two-level atom in a strong field and/or a tailored reservoir

The master equation for a two-level atom driven by a strong classical field and damped into a tailored reservoir with nonflat density of modes is derived under the Born-Markov approximation. To derive the master equation, the dressing transformation on the atomic operators is performed first, and, next, the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The effects of a strong field and/or structured reservoir are seen as nonstandard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The master equation leads to the generalized optical Bloch equations that can easily be solved for the steady state and, together with the quantum regression theorem, allow for analytical expressions for the fluorescence, as well as absorption spectra.     

Some comments on nonequilibrium phase transitions in chemical systems

Two new approaches for investigating critical fluctuations near an instability point of unstable chemical models are proposed. The master equation approach is used. For a homogeneous system without the effect of diffusion, three single-component chemical systems exhibiting critical behavior are considered. The cumulant functions are expanded in a small parameter-the inverse size of the system-and singular perturbation solutions of the master equation are developed. Exponents describing the divergence of the second-order variance are found to be classical. For a system including diffusion effects, spatial correlations for a quasi-one-dimensional case are investigated by considering scale transformation behavior within the multivariate master equation formalism.This work was supported in part by NSF grants MPS-7411925 and CHE 76-05583.     

Steady state solutions of master equations without detailed balance

Steady state solutions of master equations with one variable are constructed. The method of solution is based on a transformation of the original equation for the probability into one for a slowly varying function. The method is of general applicability and is particularly useful in obtaining solutions in the case where detailed balance does not hold. Examples of such systems in chemical reaction models and the two photon laser are discussed.On leave from the University of Waikato, Hamilton, New Zealand     

Master Equation in Phase Space for a Spin in an Arbitrarily Directed Uniform External Field

The time evolution equation for the probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular, in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical paramagnet in an arbitrarily directed uniform external field.     

Master Equation in Phase Space for a Uniaxial Spin System

A master equation, for the time evolution of the quasi-probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for a uniaxial spin system subject to a magnetic field parallel to the axis of symmetry. This equation is obtained from the reduced density matrix evolution equation (assuming that the spin-bath coupling is weak and that the correlation time of the bath is so short that the stochastic process resulting from it is Markovian) by expressing it in terms of the inverse Wigner-Stratonovich transformation and evaluating the various commutators via the properties of polarization operators and spherical harmonics. The properties of this phase space master equation, resembling the Fokker-Planck equation, are investigated, leading to a finite series (in terms of the spherical harmonics) for its stationary solution, which is the equilibrium quasi-probability density function of spin “orientations” corresponding to the canonical density matrix and which may be expressed in closed form for a given spin number. Moreover, in the large spin limit, the master equation transforms to the classical Fokker-Planck equation describing the magnetization dynamics of a uniaxial paramagnet.     

Renormalization group transformation for the master equation of a kinetic Ising chain

We define and solve a dynamical real-space renormalization group transformation for the master equation of a kinetic Ising model in one dimension.The exactly solvable model we treat has the dynamical exponentz=2. We show that spin flip rates differing from Glauber's choice may have exponentsz differing from 2. A particular model which hasz=4 is presented.     

Solving non-Markovian open quantum systems with multi-channel reservoir coupling

We extend the non-Markovian quantum state diffusion (QSD) equation to open quantum systems which exhibit multi-channel coupling to a harmonic oscillator reservoir. Open quantum systems which have multi-channel reservoir coupling are those in which canonical transformation of reservoir modes cannot reduce the number of reservoir operators appearing in the interaction Hamiltonian to one. We show that the non-Markovian QSD equation for multi-channel reservoir coupling can, in some cases, lead to an exact master equation which we derive. We then derive the exact master equation for the three-level system in a vee-type configuration which has multi-channel reservoir coupling and give the analytical solution. Finally, we examine the evolution of the three-level vee-type system with generalized Ornstein–Uhlenbeck reservoir correlations numerically.     

Time-dependent probability density of statistical mechanics

We study fluctuations around nonequilibrium steady states of some model nonlinear chemical systems. A previous result of Nicolis and Prigogine states that the mean square fluctuation computed from a master equation in the space of internal states of the reacting species is identical to that calculated from Einstein's fluctuation formula. Our analysis of fluctuations based on that master equation leads with the assumption of local equilibrium to a result identical to that obtained from a master equation for the total concentration of the reacting species, which is different from the equilibrium (Einstein relation) result. Nicolis and Prigogine approximated one term in their master equation, and a discussion of this approximation is presented. The master equation without this approximation yields at equilibrium the result expected on the basis of Einstein's formula.Work supported in part by the National Science Foundation and Project SQUID, Office of Naval Research.     

Stationary momentum space solution of the Fokker-Planck equation for a simple model of a laser oscillator exhibiting spatial dispersion

Starting from a multimode hamiltonian for a system of radiating oscillators coupled with atomic reservoirs, the secular master equation for the radiation-density operator is calculated in the interaction picture after elimination of the atomic variables. Using the differential operator representation for coherent states this equation is transcribed into a multimode Fokker-Planck equation. The stationary solution in momentum space is given for the threshold region. Fourier transformation to configuration space results in a quasi-free energy formula for a laser oscillator exhibiting spatial dispersion.     

Diffusion equation for the quasi-incoherent motion of Frenkel excitons in molecular crystals

In a recent paper a master equation describing the quasi-incoherent motion of Frenkel excitons in molecular crystals has been derived within the Haken-Strobl model for the coupled coherent and incoherent motion of Frenkel excitons. Starting from this master equation and using only the translational symmetry of the crystal lattice, for crystals with one molecule in the unit cell a diffusion equation is derived. For crystals with several molecules in the unit cell instead of a diffusion equation one obtains a set of diffusion-like equations. These equations are solved explicitly for the case of two molecules in the unit cell and asymptotic expressions are discussed. It is shown that this asymptotic behaviour is again described by a diffusion equation.     

Theory of open systems I

A new method of treating open systems is presented. The normal treatment using the generalized master equation with the projection of Argyres and Kelley is meaningful only if the state of the reservoir never deviates appreciably from the reference state which appears in the projection. Otherwise, one must make at least a partial resummation of the perturbative expansion of the kernel of the generalized master equation. The present method avoids the introduction of a projection operator and allows us to overcome such resummation difficulties. It is based on an integrodifferential equation for the statistical operator of the composite system, which naturally provides a hierarchy of equations involving the statistical operator ?(t) of the open system and suitable quantities describing higher and higher order bath-system correlations. Treating the deviations of the bath from its initial equilibrium or stationary state as expansion parameters, one gets an approximation scheme, each step of which gives a closed system of equations for ?(t) and a suitable set of correlation quantities.Eliminating such quantities one obtains a closed linear integrodifferential equation for ?(t). The zeroth approximation in the deviations coincides with the Born approximation of the generalized master equation which uses the projection of Argyres and Kelley.On the other hand, even the first approximation is equivalent to the resummation of infinite contribution of the Born series of such a generalised master equation. When it is suitable, the concentration of the bath can also be used as an expansion parameter to handle the hierarchy.     

Spacetime locality of the BRST formalism

The spacetime locality of the BRST formalism is investigated. The analysis covers gauge theories with either closed or open algebras and is undertaken in the explicit context of the antifield formulation of the BRST theory. Under appropriate conditions, the homology of the Koszul-Tate differential modulo the spacetime exterior derivative is shown to be trivial in the space of non-integrated densities with positive antighost and pure ghost numbers. As a result: (i) the solution of the master equation can be taken to be a local functional; (ii) the gauge fixed action is also a local functional provided one takes the gauge fixing fermion to be a local functional as well; and (iii) the BRST transformation is local.     

Correspondence Between the Operational Element of an Amplitude Damping Harmonic Oscillator to the Kraus Operator of the Master Equation for Dissipation

We find the correspondence between the operational element of an amplitude damping harmonic oscillator and the Kraus operator solution to the master equation for dissipation. This reveals the equivalence between the two approaches to tackling the dissipation of oscillator-reservoir: one is solving the Kraus operator of the master equation, and the other is deriving the operational element of time evolution operator in the reservoir mode.     

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.     

Some comments on approximations to the master equation

A common strategy for approximating a master equation is to replace it by a diffusion-like equation. Many methods for deriving the form of such an equation have been suggested in the literature. We compare two of these in the light of an example in which the master equation can be solved exactly. One of these is the van KampenΩ-expansion, which generally does not give a useful approximation to the equilibrium solution, and the second is a technique which preserves the noise-free dynamics and gives the correct equilibrium solution. It is shown that the second moment calculated in the latter approximation is not an accurate one at short times. The difficulty is the restriction of the approximating equation to the diffusion form.     

Master equations with memory for systems driven by classical noise

A non-Markovian master equation is obtained for a two level atom driven by a phase noisy laser. The derivation is based on obtaining an equation for the density operator of the system averaged over the previous histories of the external noise. Averaging over the current value of the noise variable by means of the Zwanzig-Nakajima projection operator technique leads to a master equation with memory and a local-in-time master equation. The solutions to the resultant non-Markovian master equation, the structural properties of the equation, and the amenability of the equation to unravelling by the quantum trajectory method are all investigated.     

Master equation for the quasi-incoherent motion of Frenkel excitons

Starting from the Haken-Strobl-model for the coupled coherent and incoherent motion of Frenkel excitons, a master equation is derived describing the quasi-incoherent motion of these excitations. In contrast to previous derivations no use is made of special symmetries. Therefore this equation may also describe the energy transfer in materials with non-periodic structure, which play a role in biological systems. The master equation is solved for crystals with one and two molecules in the unit cell, and explicit expressions are given for the case of nearest neighbour interaction too. Furtheron, asymptotic forms of the solutions are discussed.     

重离子反应中质子与中子转移对超重元素合成的影响

研究了采用类弹碎片质量数为变量的一维(1D)主方程(ME)描述重离子熔合反应时的受限制条件, 建立了以类弹碎片中子和质子数为独立变量的二维(2D) ME, 并采用分步差分格式法数值解(2D) ME得到了熔合几率。 新的方法适用于研究任何弹靶组合系统, 给出了与实验符合更好的结果, 为寻求合成超重核的反应道提供了更为宽阔的选择空间。 同时也研究了弹靶的不同同位旋组合对复合核形成截面的影响。 The limitation of the one dimensional master equation with the mass number of project like fragment as the variable is discussed. A two dimensional master equation with neutron and proton numbers of the project like fragment as variables is set up, and solved numerically by means of a two step difference scheme to obtain fusion probability. The new equation is suitable for studying fusion reactions with any projectile target combination, and can better fit the experimental data, so that it can provide a large probability to choose the reaction system for synthesizing a certain superheavy nucleus. The influence of projectile target combinations with different isospins on the cross sections of compound nuclei is studied.