Interaction representation method for Markov master equations in quantum optics
Alexander Chebotarev, Julio Garcia, Roberto Quezada
备注：
12 pages, LaTeX2e. To be published in: Trends in Mathematics, Stochastic Analysis and Mathematical Physics, ANESTOC, Proc. of the Fourth International Workshop, Birkhauser, Boston, 2001
Conditions sufficient for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS.
The unital property of a minimal quantum dynamical semigroup implies the uniqueness of the solution of the corresponding Markov master equation in the class of quantum dynamical semigroups and, in the corresponding representation, it ensures preservation of the trace or unit operator. We recall that only in the unital case the formal generator of MME determines uniquely the corresponding QDS.
An easily solvable quantum master equation has long been sought that takes into account memory effects induced on the system by the bath, i.e., non-Markovian effects. We briefly review the Post-Markovian master equation (PMME), which is relatively easy to solve, and analyze a simple example where solutions obtained exhibit non-Markovianity. We apply the distinguishability measure introduced by Breuer et al., and we also explicitly analyze the divisibility of the associated quantum dynamical maps. We give a mathematical condition on the memory kernel used in the PMME that guarantees non-CP-divi…
We study the non-Markovian quantum interference phenomenon of a multi-state atomic system coupled to a bosonic dissipative environment by using the exact master equations derived in this paper. Two examples involving four-level systems with or without external control fields are investigated. Our results show that non-Markovian master equations are capable of exhibiting quantum interference phenomena in a temporal domain that has not fully explored before. In particular, we show that the environmental memory is instrumental in the onset of non-Markovian quantum interference pattern in differen…
Post-Markovian quantum master equations from classical environment fluctuations
在本文中，我们证明了两个常用的唯象后马尔可夫量子主方程可以推导出没有使用任何微扰近似。一个系统耦合到一个环境的特点是自经典的组态波动，后者服从马尔可夫动力学，定义了底层的物理模型。Shabani-Lidar方程[A. Shabani和D. A. Lidar，Phys.Rev.A71，020101（R）（2005）]及其相关的近似积分-微分核主方程是通过追踪两个不同的二分马尔可夫Lindblad动力学得到的，其中环境波动由附属系统考虑。此外，我们还找到了非马尔可夫系统动力学可以用测量轨迹系综来解释的条件，并给出了一种非马尔可夫量子跳跃方法。与最近的分析相反[L。Mazzola，E. M. Laine，H. P. Breuer，S. Maniscalco和J. Piilo，Phys.Rev.A81，062120（2010）]，我们还证明了这些主方程，即使具有指数记忆函数，也可能导致非马尔可夫效应，例如如果哈密顿系统不与耗散动力学耦合，则信息的环境到系统回流。
We construct a quantum Markovian Master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the Master equation to be extended beyond the adiabatic limit. The Non-Adiabatic Master Equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic Master equation, the NAME leads to coupled equations of motion for the population and cohe…
在本文中，我们研究了非马尔可夫动力学的量子比特通过比较两个广义主方程的记忆。在热浴的情况下，我们导出了最近在文献[A]中提出的后马尔可夫主方程的解。Shabani和D. A. Lidar，Phys. Rev. A { \bf 71}，020101（R）（2005）]，并且我们研究了指数衰减的存储器内核的动力学。我们比较了后马尔可夫主方程的解决方案与典型的记忆核主方程的解决方案。我们的研究结果导致一个新的物理解释的水库关联函数，并带来光的可用性与内存的系统考虑的主方程的限制。
Phenomenological memory-kernel master equations and time-dependent Markovian processes
具有记忆核的唯象主方程是否总是描述一个信息反向流动的非马尔可夫量子动力学？对系统过去状态的整合是非马尔可夫性的明确无误的标志吗？我们通过一个反例表明，情况并非总是如此。我们考虑两个常用的唯象积分-微分主方程描述的动力学自旋1/2的热浴。通过使用最近引入的度量来量化非马尔可夫性[H. P. Breuer，E.- M. Laine和J. Piilo，Phys. Rev. Lett. 103，210401（2009）]我们证明，只要方程保持其物理意义，非马尔可夫行为的关键特征不会出现在所考虑的内存内核主方程中。也就是说，没有从环境到开放系统的反向信息流。因此，在记忆核上的积分总是导致非马尔可夫动力学的假设，原来是脆弱的现象学近似。相反，所考虑的唯象方程能够描述与时间相关的马尔可夫过程相关的从系统到水库的时间相关的和单向的信息流。
Non-Markovian dissipative dynamics of two coupled qubits in independent reservoirs: a comparison between exact solutions and master equation approaches
Master equations describing open quantum dynamics are typically first order differential equations. When such dynamics brings the trajectories in state space of more than one initial state to the same point at finite instants in time, the generator of the corresponding master equation becomes singular. The first-order, time-local, homogeneous master equations then fail to describe the dynamics beyond the singular point. Retaining time-locality in the master equation necessitates a reformulation in terms of higher-order differential equations. We formulate a method to eliminate the divergent be…
In this paper we derive the Lindblad and Redfield forms with and without secular approximation from the Born-Markov master equation for open quantum systems. The spectral correlation tensor of bath (the Fourier transform of the bath correlation function) and then the coefficients in the two forms of the master equation are reevaluated according to the scheme in Ref.[Phys. Rev. A 99, 022118 (2019)]. They are complex numbers rather than the real numbers getting from traditional simplified methods. The dynamics of two models [one is an open three-level quantum system model, and the other is the m…
We consider a general open system dynamics and we provide a recursive method to derive the associated non-Markovian master equation in a perturbative series. The approach relies on a momenta expansion of the open system evolution. Unlike previous perturbative approaches of this kind, the method presented in this paper provides a recursive definition of each perturbative term. Furthermore, we give an intuitive diagrammatic description of each term of the series, which pro- vides an useful analytical tool to build them and to derive their structure in terms of commutators and anticommutators. We…
Based on the recent experiments on the hybrid quantum system of a superconducting microwave cavity coupling strongly to an inhomogeneous broadening spin ensemble under an external driving field, we use the exact master equation approach to investigate its non-Markovian decoherence dynamics. Here the spin ensemble is made by negatively charged nitrogen-vacancy (NV) defects in diamond. Our exact master equation theory for open systems depicts the experimental decoherence results and reveals the mechanism how the decoherence induced by the inhomogeneous broadening is suppressed in the strong-coup…
Most future quantum devices, including quantum computers, require control that is broadband, meaning that the rate of change of the time-dependent Hamiltonian is as fast or faster than the dynamics it generates. In many areas of quantum physics, including quantum technology, one must include dissipation and decoherence induced by the environment. While Markovian master equations provide the only really efficient way to model these effects, these master equations are derived for constant Hamiltonians (or those with a discrete set of well-defined frequencies). In 2006, Alicky, Lidar, and Zanardi…
Critical assessment of two-qubit post-Markovian master equations
最近提出了一个后马尔可夫主方程来描述与记忆保持环境耦合的系统的演化[A. Shabani和D. A. Lidar，Phys.Rev.A71，020101（R）（2005）]。对于受适当选择的环境条件影响的单个量子位，相应的动力学总是合理和物理的。在这里，我们将这种情况扩展到两个量子位的情况，其中只有一个受到环境影响。我们展示了如何，尽管无辜的这样一个扩展，引入第二个量子比特应该做暨grano salis，以避免后果，如相关的动力学映射的正性的破坏。这暗示了在采用现象学衍生的模型来描述马尔可夫框架之外的进化时，必须小心。
We investigate a case of the Hu-Paz-Zhang master equation of the Caldeira-Leggett model without Lindblad form obtained in the weak-coupling limit up to the second-order perturbation. In our study, we use Gaussian initial states to be able to employ a sufficient and necessary condition, which can expose positivity violations of the density operator during the time evolution. We demonstrate that the evolution of the non-Markovian master equation has problems when the stationary solution is not a positive operator, i.e., does not have physical interpretation. We also show that solutions always re…
We introduce a classification scheme for the generators of open fermionic Gaussian dynamics. We simultaneously partition the dynamics along the following four lines: (1) unitary versus non-unitary, (2) active versus passive, (3) state-dependent versus state-independent, and (4) single-mode versus multi-mode. We find that only nine of these 16 types of dynamics are possible. Using this partition we discuss the consequences of imposing complete positivity on fermionic Gaussian dynamics. In particular, we show that completely positive dynamics must be either unitary (and so can be implemented wit…
Understanding the behaviour of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies. It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on…
Convolutionless and convolution master equations are the two mostly used physical descriptions of open quantum systems dynamics. We subject these equations to time deformations: local dilations and contractions of time scale. We prove that the convolutionless equation remains legitimate under any time deformation (results in a completely positive dynamical map) if and only if the original dynamics is completely positive divisible. Similarly, for a specific class of convolution master equations we show that uniform time dilations preserve positivity of the deformed map if the original map is po…
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to …
