Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mossbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscop...

Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.

https://iopscience.iop.org/article/10.1088/0034-4885/77/9/094001/pdf

Quantum non-Markovianity: characterization, quantification and detection

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Ito calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.

/pdf/an-introduction-to-quantum-filtering-40y9am2qpf.pdf

An Introduction to Quantum Filtering

We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the a

 t
 fulfill the q-commutation relations for some covariance function – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there
exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

q-Gaussian Processes: Non-commutative and Classical Aspects

Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.

/pdf/feedback-control-of-quantum-state-reduction-3cmrcla18a.pdf

Feedback control of quantum state reduction

Let ℬ be *-algebra with identity (usually it wil be a C*- or a W*-algebra). A quantum stochastic process over ℬ indexed by ℝ is defined by a triple {A, (jt)t∈ℝ, φ} where 


A is a *-algebra with identity.


jt : ℬ ↪A is an embedding (t∈ℝ).


φ is a State on A.

/pdf/quantum-stochastic-processes-441di813qz.pdf

Quantum Stochastic Processes

We show that, in the weak coupling limit, the laser model process converges weakly in the sense of the matrix elements to a quantum diffusion whose equation is explicitly obtained. We prove convergence, in the same sense, of the Heisenberg evolution of an observable of the system to the solution of a quantum Langevin equation. As a corollary of this result, via the quantum Feynman-Kac technique, one can recover previous results on the quantum master equation for reduced evolutions of open systems. When applied to some particular model (e.g. the free Boson gas) our results allow to interpret the Lamb shift as an Ito correction term and to express the pumping rates in terms of quantities related to the original Hamiltonian model.

/pdf/the-weak-coupling-limit-as-a-quantum-functional-central-13wqrdqclz.pdf

The weak coupling limit as a quantum functional central limit

We give a description of the reduced dynamics of an open quantum system whose initial state is (in general) correlated with the surroundings. The averaging method (whose mathematical theory was developed in another paper) is used to construct a series expansion of the reduced dynamics in powers of the ratio τ 1 /τ 0 of the characteristic relaxation times τ 1 and τ 0 of the surroundings and of the system respectively. To first order in τ 1 /τ 0 we obtain the well-known Markovian approximation which corresponds to the weak coupling limit. Starting from second order in τ 1 /τ 0 there appear various kinds of non-Markovian effects including initial slips which have attracted much attention in the recent literature. The relevance of these higher-order corrections stems from the fact that the characteristic time τ 1 cannot be much smaller than h kT for quantum surroundings at temperature T .

Non-markovian behavior in low-temperature damping: An application of the averaging method

The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM n with stationary faithful state π. If (ℳ, μ) is the quantum measure space in question (ℳ a von Neumann algebra and μ a faithful normal weight), then the semigroupe tL of transition operators on (M n , π) has generator

whereu is an arbitrary unitary element of the centraliser of (M n ⊗ℳ,φ⊗μ).

Alberto Frigerio

Papers

Quantum Stochastic Processes

The weak coupling limit as a quantum functional central limit

Non-markovian behavior in low-temperature damping: An application of the averaging method

Quantum Poisson processes and dilations of dynamical semigroups

An invitation to the weak coupling and the low density limit