An ongoing theme in quantum physics is the interaction of small quantum systems with an environment. If that environment has many degrees of freedom and is weakly coupled, it can often be reasonable to treat its decohering effect on the small system using a ``memoryless,'' or Markovian description. This Colloquium shows that for many phenomena a more refined, non-Markovian, treatment is necessary. The suite of developing theoretical tools is reviewed, with which recent progress on this problem has been based.

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Colloquium: Non-Markovian dynamics in open quantum systems

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

A new quantum dynamic equation for excitation energy transfer is developed which can describe quantum coherent wavelike motion and incoherent hopping in a unified manner. The developed equation reduces to the conventional Redfield theory and Forster theory in their respective limits of validity. In the regime of coherent wavelike motion, the equation predicts several times longer lifetime of electronic coherence between chromophores than does the conventional Redfield equation. Furthermore, we show quantum coherent motion can be observed even when reorganization energy is large in comparison to intersite electronic coupling (the Forster incoherent regime). In the region of small reorganization energy, slow fluctuation sustains longer-lived coherent oscillation, whereas the Markov approximation in the Redfield framework causes infinitely fast fluctuation and then collapses the quantum coherence. In the region of large reorganization energy, sluggish dissipation of reorganization energy increases the time electronic excitation stays above an energy barrier separating chromophores and thus prolongs delocalization over the chromophores.

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Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: reduced hierarchy equation approach.

We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}

Generation of Non-classical Motional States of a Trapped Atom

The interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory. The latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time, and is written in a form feasible for perturbational analysis. Thus, the existing stochastic theory in which those cases mentioned above are discussed is equipped with a more tractable basic equation. Two problems discussed in the previous paper, i.e., the random frequency modulation of a quantal oscillator and the Brownian motion of a spin, are treated from the viewpoint of the stochastic theory without such explicit consideration of external reservoirs as was taken in the previous paper.

A generalized stochastic liouville equation. Non-Markovian versus memoryless master equations

Damping theoretical expansion formulas are given for basic equations of non-equilibrium systems: They are expressed in terms of “partial” and “ordered” cumulant functions. Validity and applicability of the method are examined with the use of a solvable model. The time-convolutionless formula is shown to be useful and systematic in obtaining reduced description of the systems.

Expansion Formulas in Nonequilibrium Statistical Mechanics

A new memoryless expression for the equation of motion for the reduced density matrix is derived. It is equivalent to that proposed by Tokuyama and Mori, but has a more convenient form for the application of the perturbational expansion method. The master equation derived from this form of equation in the first Born approximation is applied to two examples, the Brownian motion of a quantal oscillator and that of a spin. In both examples the master equation is rewritten into the coherent-state representation. A comparison is made with the stochastic theory of the spectral line shape given by Kubo, and it is shown that this theory of the line shape can be incorporated into the framework of the present theory.

Quantal master equation valid for any time scale

The method of projection operators, which plays an important role in the field of nonequilibrium statistical mechanics, has been established with the use of the Liouville--von Neumann equation for a density matrix to eliminate irrelevant information from a whole system. We formulate a unified and general projection operator method for dynamical variables. The main features of our formalism parallel those for the Liouville--von Neumann equation. (1) Two types of basic equations, time-convolution and time-convolutionless decompositions, are systematically obtained without specifying a projection operator. (2) Expansion formulas for both decompositions are also obtained. (3) Problems incorporating a time-dependent Liouville operator can be flexibly treated. We apply the formulas to problems in random frequency modulation and low field resonance. In conclusion, our formalism yields a more direct and easier means of determining the average time evolution of an operator than the one for the Liouville--von Neumann equation.

Unified projection operator formalism in nonequilibrium statistical mechanics.

A periodically driven linear system subject to multiplicative correlated noise is considered. It has been argued recently by several authors that such a simple system exhibits stochastic resonance. By introducing a general type of composite stochastic process, bridging two previously considered limiting cases of dichotomous and Gaussian noise, it is proved that, indeed, the amplitude of the average of the driven linear process at long times shows a pronounced maximum both as a function of the noise strength and as a function of the autocorrelation time. However, this kind of stochastic resonant behavior can be experimentally observable only in a special case where the initial phase of the external forcing is somehow fixed. Additional averaging over the uniform distribution of the initial random phase, inherent in most physical systems, leads to that the periodic output vanishes identically at long times. Moreover, the system response is typically defined in terms of the power spectrum rather than the amplitude of the average. The output signal given by the spectral density corresponding to the frequency of the external forcing is calculated via the long-time phase-averaged correlation function. It appears that the output signal simply diverges upon approaching the second moment instability point with increasing noise strength. No stochastic resonance is observed for any parameter settings. Interestingly, the resonancelike behavior of the system response as a function of the autocorrelation time is retained.

Fumiaki Shibata

Papers

A generalized stochastic liouville equation. Non-Markovian versus memoryless master equations

Expansion Formulas in Nonequilibrium Statistical Mechanics

Quantal master equation valid for any time scale

Unified projection operator formalism in nonequilibrium statistical mechanics.

Periodically driven linear system with multiplicative colored noise