Dynamics of a Fermi system with resonant dissipation and dynamical detailed balance

The dissipative dynamics of a system of Fermions is described in the framework of a resonance model—the quantum master equation describes two-body correlations of the system with the environment particles. This equation, with microscopic coefficients depending on the exactly known two-body potential between the system and the environment particles, is discussed in comparison with other master equations, obtained on axiomatic grounds, or derived from a coupling with an environment of harmonic oscillators without altering the quantum conditions. The asymptotic solution is in accordance with the detailed balance principle, and with other generally accepted conditions satisfied during the whole time-evolution: Pauli master equations for the diagonal elements of the density matrix, and damped Bloch–Feynman equations for the non-diagonal ones, that we call dynamical detailed balance. For a harmonic oscillator coupled with the electromagnetic field through dipole interaction, a master equation with transition operators between successive levels is obtained. As an application, the decay width of a quantum logic gate is calculated.     

A soluble model for quantum mechanical dissipation

In order to derive the equations for dissipation and noise in a quantum mechanical system it is necessary to include the equations of motion of a suitably chosen bath interacting with the system. In this way the standard treatment arrives at an approximate master equation for the density matrix of the system, at the expense of a number ofad hoc assumptions. These assumptions are here scrutinized on the basis of an exactly soluble model. The conclusion is: the bath must obey certain specifications; the interaction must be weak; and the temperature must be so high that the interaction energy is within the classical domain rather than occurring in quanta. Some additional comments concerning dissipation in quantum mechanics are relegated to an appendix.     

Decoherence in atom–field interactions: A treatment using superoperator techniques

Decoherence is a subject of great importance in quantum mechanics, particularly in the fields of quantum optics, quantum information processing and quantum computing. Quantum computation relies heavily in the unitary character of each step carried out by a quantum computational device and this unitarity is affected by decoherence. An extensive study of master equations is therefore needed for a better understanding on how quantum information is processed when a system interacts with its environment. Master equations are usually studied by using Fokker–Planck and Langevin equations and not much attention has been given to the use of superoperator techniques. In this report we study in detail several approaches that lead to decoherence, for instance a variation of the Schrödinger equation that models decoherence as the system evolves through intrinsic mechanisms beyond conventional quantum mechanics rather than dissipative interaction with an environment. For the study of the dissipative interaction we use a correspondence principle approach. We solve the master equations for different physical systems, namely, Kerr and parametric down conversion. In the case of light-matter interaction we show that although dissipation destroys the quantumness of the field, information of the initial field may be obtained via the reconstruction of quasiprobability distribution functions.     

Anomalous isotropic luminescence from anthracene sandwich dimers

A classical statistical probability amplitude is introduced whose square modulus is the distribution function. This enables the analogy between classical statistical mechanics and quantum mechanics to be completed. The analogy is developed until quantum statistical derivations can be used in classical statistical mechanics. Two master equations are found: the classical equivalent of the Pauli Master Equation, and a generally valid master equation. Well-known classical equations are deduced from these in a special representation. Interference terms are found and discussed.     

Virtues and Limitations of Markovian Master Equations with a Time-Dependent Generator

A Markovian master equation with time-dependent generator is constructed that respects basic constraints of quantum mechanics, in particular the von Neumann conditions. For the case of a two-level system, Bloch equations with time-dependent parameters are obtained. Necessary conditions on the latter are formulated. By employing a time-local counterpart of the Nakajima–Zwanzig equation, we establish a relation with unitary dynamics. We also discuss the relation with the weak-coupling limit. On the basis of a uniqueness theorem, a standard form for the generator of time-local master equations is proposed. The Jaynes–Cummings model with atomic damping is solved. The solution explicitly demonstrates that reduced dynamics can be described by time-local master equations only on a finite time interval. This limitation is caused by divergencies in the generator. A limit of maximum entropy is presented that corroborates the foregoing statements. A second limiting case demonstrates that divergencies may even occur for small perturbations of the weak-coupling regime.     

On the generators of quantum dynamical semigroups

The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB() is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.     

Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes

The stochastic and quantum dynamics of open quantum systems interacting with stochastic perturbations in considered. The master equations for one time and multi-time correlation functions of such a system are derived to all orders in the interaction with the stochastic perturbations. The importance of the non-markovian character of such equations in the study of various problems in optical resonance is discussed. The simplified form of the non-markovian master equations in Born approximation is also given. It is shown that such non-markovian master equations in Born approximation are exact if there is only one random perturbation, of the telegraphic signal type, acting on the system. The master equations for the linear response functions of an open system interacting with stochastic perturbations are also derived. The non-markovian master equations for multitime correlations are used to study the behaviour of two level atoms interacting with fluctuating laser fields. Both amplitude and phase fluctuations are taken into account. Explicit results are presented for the spectrum of resonance fluorescence, absorption spectrum, photon antibunching effects etc. The calculations are done for arbitrary values of the relaxation parameters and intial conditions. In general the fluorescence spectrum is found to be asymmetric for off resonant fields.     

Nonlinear relaxation and fluctuations of damped quantum systems

The paper reexamines the treatment of irreversible quantum systems by master equations. Shortcomings of the conventional theory of quantum Markov processes pointed out by Talkner are analyzed. It is shown that a frequently used quantum regression hypothesis is not correct, in general. A new generalized master equation determining the relaxation to equilibrium is derived by means of time-dependent projection operator techniques. It is shown that this master equation also determines the time evolution of equilibrium correlations and response functions. The Markovian approximation is discussed, and a new type of Markovian limit, the Brownian motion limit, is introduced besides the weak coupling limit. The shortcomings of the conventional approach are resolved by deriving new formulae for the time evolution of the correlation and response functions of a quantum Markov process. The symmetries of the process are emphasized, and it is shown how the fluctuation-dissipation theorem and the detailed balance symmetry emerge from the master equation approach.     

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.     

Homogeneous Generalized Master Equations

It is shown that the method proposed in V. F. Los [J. Phys. A: Math. Gen. 34: 638–6403 (2001)], which allows for turning the inhomogeneous time-convolution generalized master equation (TC-GME) into homogeneous (while retaining initial correlations) time-convolution generalized master equation (TC-HGME) for the relevant part of a distribution function, is fully applicable to the quantum case and to the time-convolutionless GME (TCL-GME). It is demonstrated by rederiving the TC-HGME and showing that it works in both the classical and quantum physics cases. The time-convolutionless HGME (TCL-HGME) retaining initial correlations, which is formally the same for both the classical and quantum physics, has also been derived. Both the TC-HGME and TCL-HGME are exact equations applicable on any timescale and allow for consecutive treating the initial correlations and collisions on the equal footing. A new equation for a momentum distribution function retaining initial correlations has been obtained in the linear in the density of quantum particles approximation. Connection of this equation to the quantum Boltzmann equation is discussed.     

Dividing Quantum Channels

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.     

Loop quantum modified gravity and its cosmological application

A general nonperturvative loop quantization procedure for metric modified gravity is reviewed. As an example, this procedure is applied to scalar-tensor theories of gravity. The quantum kinematical framework of these theories is rigorously constructed. Both the Hamiltonian and master constraint operators are well defined and proposed to represent quantum dynamics of scalar-tensor theories. As an application to models, we set up the basic structure of loop quantum Brans-Dicke cosmology. The effective dynamical equations of loop quantum Brans-Dicke cosmology are also obtained, which lay a foundation for the phenomenological investigation to possible quantum gravity effects in cosmology.     

Deformed quantum harmonic oscillator with diffusion and dissipation

A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady-state solution of the equation for the density matrix in the number representation is obtained and the equilibrium energy of the deformed harmonic oscillator is calculated in the approximation of small deformation.     

On the description of noise in quantum systems

A formalism of probability operators which generalizes the notion of density operator is introduced into the theory of noisy quantum systems. The Markov property and the connexion between Heisenberg and Schrödinger picture for systems undergoing an irreversible change are discussed in detail. The probability-operator treatment of noise is related to the Langevin method discussed byLax through a generalized Einstein-relation. The master equation for the quantum mechanical oscillator with linear damping is written down in a Fokker-Planck-type approximation. By means of the Einstein-relation the coefficients in the Fokker-Planck-equation are related to the parameters in the phenomenological equations.     

Nakajima-Zwanzig and Time-Convolutionless Master Equations for a One-Qubit System in a Non-Markovian Layered Environment

Quantum operations, are completely positive (CP) and trace preserving (TP) maps on quantum states, and can be represented by operator-sum or Kraus representations. In this paper, we calculate operator-sum representation and master equation of one-qubit open quantum system in layered environment which is a generalized spin star model. The Nakajima-Zwanzig and time-convolutionless projection operators technique are applied for deriving the master equations. Finally, a simple example will be studied to consider the relation between completely positive maps and initial quantum correlation and show that vanishing quantum discord is not necessary for CP maps.     

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.