non-Markovian Solution Research Articles

An Exact Algorithm to Make a Trade-Off between Cost and Probability in SSPs

In stochastic sequential decision problems, such as Stochastic Shortest Path Problems, the GUBS (Goal with UtilityBased Semantics) criterion considers a trade-off between probability-to-goal and cost-to-goal using a goal semantics based on Expected Utility Theory (EUT); in such a semantics, goal paths have priority over non-goal paths, but it implies neither the MAXPROB criterion nor the dual optimization criterion that finds the cheapest policy among the policies that maximize goal probability. Whereas evaluation criteria based on a sound theory such as EUT are desirable, optimal policies under GUBS are non-markovian. Non-markovian solutions are undesirable because there is not always a finite representation and even if it can be represented in a finite way, the representation may be too large to be stored. Here we define a special case of GUBS criterion that allows a finite representation to the optimal policy, the eGUBS criterion, where the cost utility function is exponencial. Considering this special case, we contribute with: (i) the proof that the eGUBSoptimal policy has a finite representation; (ii) the first exact algorithm to obtain finite optimal policies for the eGUBS criterion, and (iii) four strategies to find sub-optimal policies. We conduct experiments on one synthetic problem to evaluate each strategy. Although optimal solutions have a high memory cost, sub-optimal policies can save memory space with a small decrease in performance.

Nonperturbative Master Equation Solution of Central Spin Dephasing Dynamics

We solve the long-standing central spin problem for a general set of inhomogeneous bath couplings and a large class of initial bath states. We compute the time evolution of the coherence of a central spin coupled to a spin bath by resumming all orders of the time-convolutionless master equation, thus avoiding the need to assume weak coupling to the bath. The fully quantum, non-markovian solution is obtained in the large-bath limit and is valid up to a time scale set by the largest coupling constant. Our result captures the full decoherence of an electron spin qubit coupled to a nuclear spin bath in a GaAs quantum dot for experimentally relevant parameters. In addition, our solution is quite compact and can readily be used to make quantitative predictions for the decoherence process and to guide the design of nuclear state preparation protocols.

Non-Markovian dynamics of a qubit in a reservoir: different solutions of non-Markovian master equation

We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima—Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.

Random perturbations of bifurcation diagrams

Random perturbations of one dimensional bifurcation diagrams can exhibit qualitative behavior that is quite different from that of the unperturbed, deterministic situation. For Markov solutions of one dimensional random differential equations with bounded ergodic diffusion processes as perturbations, effects like disappearance of stationary Murkov solutions (‘break through’), slowing down, bistability, and random symmetry breaking can occur. These effects are partially the results of local considerations, but as the perturbation range increases, global dynamics can alter the picture as well. The results are obtained via the analysis of stationary solutions of degenerate Markov diffusion processes, of stationary, non-Markovian solutions of stochastic flows, and of Lyapunov exponents of stochastic flows with respect to steady states.

Influence of the counter-rotating terms on the spontaneous emission from a two-level hydrogenic atom: non-Markovian solution for atomic population-inversion expectation value and the radiative line shape

By using a new projection-operator method, developed recently by one of the present authors (J.S.), the counter-rotating terms are self-consistently included into the interaction of a two-level hydrogenic atom with a vacuum radiation field. A non-Markovian solution for the atomic population inversion showing new features is derived by a rigorous mathematical method. Moreover, the radiative line shape with some new characteristics is calculated within the scope of the model.

Non-Markovian solutions for probability amplitudes and dipole-moment expectation values in the case of spontaneous Lyman-α emission

By taking into account the counter-rotating terms, equations of motion for probability amplitudes are derived in the case of spontaneous Lyman-α emission. From these equations non-Markovian solutions for probability amplitudes as well as for dipole-moment expectation values are obtained. All the results are provided with accurate error estimates which are absolutely necessary in a rigorous mathematical treatment.

A Regularity Condition for Non-Markovian Solutions of Stochastic Differential Equations in the Plane

A Regularity Condition for Non-Markovian Solutions of Stochastic Differential Equations in the Plane

Comment on "Spontaneous and induced emission of soft bosons: Exact non-Markovian solution"

Comment on "Spontaneous and induced emission of soft bosons: Exact non-Markovian solution"

Non-Markovian effects on an anharmonic oscillator stochastically coupled to a thermal bath.

We present an approximation of a non-Markovian solution for the time evolution of the mean number of quanta in an anharmonic oscillator linearly and stochastically coupled to a thermal bath. A colored Gaussian bath is explicitly considered, and a quantum treatment of the bath correlation functions, based on the use of a relation between their real and imaginary parts, is applied. The behavior of the oscillator is characterized by a reduced set of (five) parameters referring to the oscillator, the bath, and their mutual interaction. A comparison is made between the prediction of the approach for the mean stationary number of quanta and its exact value, which enables us to analyze its accuracy. When the Markovian limit holds, an explicit expression for a parameter measuring the time scale for the relaxation of the oscillator is obtained in terms of the characteristic parameters of the model. Except for very high temperatures, the anharmonicity has a small influence on this relaxation time. A numerical study of the time dependence of the mean number of quanta is presented on the basis of the reported analytical solution. We find that the anharmonicity is not a significant parameter in the appearance nor in the increase of non-Markovian effects.

Spontaneous and induced emission of soft bosons: Exact non-Markovian solution.

In many applications it is assumed that the relaxation process can be described by master equation, $${\dot P_n} = - \sum\limits_k {\left( {{W_{nk}}{P_n} - {W_{kn}}{P_k}} \right).} $$ (1) Here, Pn is the probability that the system is in the state n once Wnk is a probability of transition (per unit time) from state n to state k. This equation is derived in the Markovian approximation (or in the Weisskopf- Wigner approximation for spontaneous radiation1). The necessary condition of this approximation is that the eigenfrequency ωmn has to be much larger than the transition rate Wmn (see, e.g., References 2 and 3). For spontaneous emission of phonons or photons, the transition rate ωmn is typically proportional to |ωmn|3. This means that the following condition $${W_{mn}}/|{\omega _{mn}}| \ll 1$$ (2) can be satisfied at very low frequencies, even for the degenerate levels with ωmn =0. On the other hand, at very low frequencies processes of induced emission and absorption of bosons may prevail over the spontaneous emission. The transition rate Wnm for typical one-boson processes has the form \({W_{nm}} = 2\pi /\bar n\sum\limits_k {|{B_k}{|^2}} \delta \left( {{E_n} - {E_m} \pm \bar n{\omega _k}} \right)\left( {{n_k} + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} \mp {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} \right)\) where Bk is the interaction parameter and nk is the number of bosons in the kth mode.

Markovian and non-Markovian solution of the generalized master equation for dynamics of particle transfer in a symmetric double-well potential

Dynamics of particle transfer in a symmetric double-well potential has been investigated. In restriction to two-level system, the convolution form of generalized master equation with retention of off-diagonal matrix elements has been solved on non-Markovian level as well in Markov approximation. It has been shown that subsystem dynamics crucially depends on interplay of three variables, correlation time of an environment, dephasing time and coherent transfer time. The character of dynamics then varies from damped oscillations through exponential decay and incoherent relaxation. It has been shown that on non-Markovian level, under some circumstances, the particle transfer is governed by the dynamics of slow environment.

Comment on first-passage times for processes driven by dichotomous fluctuations.

The recent derivation of the mean first-passage time for one-dimensional processes driven by additive dichotomous random processes [J. Masoliver, K. Lindenberg, and B. J. West, Phys. Rev. A 34, 2351 (1986)] can be extended to situations where the noise occurs multiplicatively and nonlinearly in the stochastic differential equation. For equations with Markovian dichotomous fluctuations in particular, this result allows for a complete and general probabilistic description of the statics and dynamics of the non-Markovian solution process in terms of stationary probability distributions and first-passage time statistics.