Markovian master equations

We give a rigorous proof that under certain technical conditions the memory effects in a quantum-mechanical master equation become negligible in the weak coupling limit. This is sufficient to show that a number of open systems obey an exponential decay law in the weak coupling limit for a rescaled time variable. The theory is applied to a fairly general finite dimensional system weakly coupled to an infinite free heat bath.     

The Markovian master equations for unstable particles

The theory of Markovian master equations is applied to a certain model of unstable particles. The exponential decay law is obtained in the weak coupling limit. The connection to the method of one-parameter contracting semigroups on a single particle Hilbert space is given.     

On the relations between Markovian master equations and stochastic differential equations

We have investigated the relationship between Markovian master equations (m.e.) and the corresponding stochastic differential equations (s.d.e.) for closed systems, i.e., systems not subjected to external pumping. We show that the form of the fluctuations in the s.d.e., i.e., additive or multiplicative, depends upon the properties of the kernel of the m.e. and the range of the state space of the stochastic variable(s), i.e., bounded or unbounded. The knowledge of these two properties of the m.e. permits the determination of the way in which the fluctuations enter into the s.d.e. (i.e., additive or multiplicative) and the calculation of their statistics. Several examples are presented to illustrate the general theory.     

马尔科夫量子主方程的热和功: 概念, 涨落定理和计算

马尔科夫量子主方程的确立将近半个世纪,它们经常被用于不可逆热力学的研究。然而,在 过去相当长的一段时间里,这些研究几乎只关注了系综平均的结果,量子系统的随机热力学一直未 受到重视。由于过去二十年里各种经典和量子涨落定理的发现,这种状况才有了明显地改变。在本 文中,我们采用两种不同的策略系统总结了当前对马尔科夫量子主方程的随机的热和功的理解。策 略之一是把量子系统和其周围的热库当成是一个封闭的量子系统,在含时总哈密顿量的控制下,这 个系统发生幺正演化,对该复合系统做两次能量投影测量得到的能量本征值之差被定义为热和功。 另一种策略是首先把量子主方程分解成随机的量子跳跃轨迹,然后在轨迹上定义随机的热和功。我 们以尽可能详尽的方式介绍这些重要的物理概念、数学技术以及不同描述层次的涨落定理。我们也 用模型具体地说明这些结论。     

Maximally robust unravelings of quantum master equations

The stationary state of an open quantum system has infinitely many representations as an ensemble of pure states. We argue that the most natural ensemble is the most robust physically realizable ensemble. Robustness is quantified by the survival probability. Physical realization requires monitoring the environment to “unravel” the dissipative evolution.     

A fluctuation theorem for Floquet quantum master equations

We present a fluctuation theorem for Floquet quantum master equations. This is a detailed version of the famous Gallavotti–Cohen theorem. In contrast to the latter theorem, which involves the probability distribution of the total heat current, the former involves the joint probability distribution of positive and negative heat currents and can be used to derive the latter. A quantum two-level system driven by a periodic external field is used to verify this result.     

Time evolution of non-Hermitian quantum systems and generalized master equations

We inquire into the time evolution of quantum systems associated with pseudo-or quasi-Hermitian Hamiltonians. We obtain, in the pseudo-Hermitian case, a generalized Liouville-von Neumann equation for closed systems. We show that quantum systems with quasi-Hermitian Hamiltonians admit the proper interpretation in terms of open quantum system and derive a generalized Lindblad-Kossakowski equation. Finally, we extend such formalism to the study of decaying systems. Partially supported by PRIN “Sintesi”.     

On quantum Markovian processes

We try to justify the name quantum Markovian processes for Kossakowski dynamical semigroups. Next we give a proof that each nonconservative quantum Markovian process may be simply extended to a conservative one.     

Exact time evolution and second-order quantum master equations for two interacting qubits

Time evolution of two interacting qubits under the influence of thermal reservoirs is considered. When there is only one excitation in the whole system, an exact reduced dynamics can be obtained. The result is compared with those obtained by the time-convolutionless and time-convolution quantum master equations in the second order approximation.     

On the standard quantum Brownian equation and an associated class of non-autonomous master equations

It is shown that the standard quantum Brownian equation (QBE) can violate positivity not only past the thermal correlation time, but at arbitrarily long times at high system frequencies. In an effort to improve the standard QBE, exact operator solutions are provided for a class of non-autonomous master equations. These exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations.     

Exact Fokker-Planck equations from stochastic master equations

A method for deriving exact Fokker-Planck equations from stochastic master equations by expanding the probability distribution in terms of Poisson distribution is given. It is applied to two non-linear chemical processes to obtain the steady state distributions.     

From convolutionless generalized master to finite-coupling Pauli master equations

Time-convolutionless Generalized Master Equations (TCL-GME) for probabilities of finding a system in a general state, irrespective of the state of the thermodynamic bath, are investigated. For general systems interacting with a genuine bath with a continuous spectrum described by time-independent system + bath Hamiltonians and after the thermodynamic limit for the bath, the long-time asymptotics of time-dependent coefficients can be taken as counterparts of Pauli-Master-Equation (PME) transfer rates. Here, within TCL-GME, asymptotics of these coefficients is calculated without resorting to any approximation. In the lowest order, these coefficients are known to turn to the usual Fermi Golden Rule transfer rates. Anyway, we argue that if the exact form of these coefficients has a long-time limit, this limit is inevitably equal to zero. This makes the possibility to derive standard markovian finite-coupling Pauli, rate or balance equations as long-time asymptotics to TCL-GME illusory. Invaluable discussions with Prof. P. Grigolini are also highly appreciated.     

Coherent-driving-assisted quantum speedup in Markovian channels

As is well known,the quantum evolution speed of quantum state can never be accelerated in the Markovian regime without any operators on the system.The Hamiltonian corrections induced by the action of coherent driving forces are often used to fight dissipative and decoherence mechanisms in experiments.For this reason,considering three noisy channels(the phase-flip channel,the amplitude damping channel and the depolarizing channel),we propose a scheme of speedup evolution of an open system by controlling an external unitary coherent driving operator on the system.It is shown that,in the presence of the coherent driving,no-speedup evolution can be transformed into quantum speedup evolution in the Markovian dynamics process.Additionally,under the fixed coherent driving strength in the above three noisy channels,the best way to achieve the most degree of quantum speedup for the system has been acquired by rotating the system with appropriate driving direction angles,respectively.Finally,we conclude that the reason for this acceleration is not the nonMarkovian dynamical behavior of the system but due to the oscillation of geometric distance between the initial state and the target final state.     

Asymptotic properties of markovian master equations

An asymptotic method of analysis of fluctuations in systems far from equilibrium is developed. A systematic singular perturbative expansion of the equation for the generating function is set up, using as smallness parameter the inverse of the size of the system. Static and time-dependent properties are analysed before, near and at a bifurcation point, both for homogeneous and inhomogeneous fluctuations. The connection with critical phenomena near equilibrium is also discussed.     

Open quantum Markovian systems and the microreversibility

The concept of microreversibility (detailed balance) as applied to open quantum Markovian systems is introduced. The properties of such systems obeying detailed balance are studied in detail. It is shown how microreversibility may be used to obtain the steady state solution for the density operator. Some properties of the eigenfunctions of the Liouville operator are discussed. The general formalism is applied to Pauli master equation and some other master equations describing the relaxation of oscillators, two-level atoms in contact with heat bath; parametric frequency conversion and a single mode laser.     

Langevin description of markovian integro-differential master equations

For a given master equation of a discontinuous irreversible Markov process, we present the derivation of stochastically equivalent Langevin equations in which the noise is either multiplicative white generalized Poisson noise or a spectrum of multiplicative white Poisson noise. In order to achieve this goal, we introduce two new stochastic integrals of the Ito type, which provide the corresponding interpretation of the Langevin equations. The relationship with other definitions for stochastic integrals is discussed. The results are elucidated by two examples of integro-master equations describing nonlinear relaxation.     

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

In addition to the well-known Landauer–Büttiker 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 full 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.