The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath
ϱB(t) and the system
ϱS(t) are written down. A formal solution of the equation for
ϱB(t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in
ϱS(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of
ϱB(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
We present equations of motion for the Su-Schrieffer-Heeger (SSH) Hamiltonian derived with the help of ansatz states similar to Davydov's so-called mod D 1 ) state for soliton dynamics in proteins. Such an ansatz state allows for quantum effects in the lattice and goes beyond previous calculations which mostly apply adiabatic models. In the most general case, called mod Phi 0 ), which is treated here in detail, we assume that the coherent-state amplitudes for the lattice depend on the site and the molecular orbital of the electrons. The equations of motion are derived from the Lagrangian of the system, a method which is equivalent to the time-dependent variational principle. In the resulting equations we find that, although the SSH Hamiltonian is a one-particle operator, indirect electron-electron interactions are present in the system which originate from the electron-phonon interactions. Inclusion of direct electron-electron interactions, will give insight into the interplay between electron-electron and electron-phonon interactions which can lead effectively to an attractive force between the electrons in systems other than polyacetylene, where bipolarons are known to be unstable. Further with our time-dependent wavefunction also vibrational details of absorption spectra can be computed. From the equations of motion several approximations can be derived. In a further approximation, mod Phi 2 ), the dependence of the coherent-state amplitudes on the lattice site is neglected. This mod Phi 2 ) ansatz state consists of a simple product of the electronic and the lattice wavefunctions; however, the electrons are not constrained to follow the lattice dynamics instantaneously as in the adiabatic case. Finally the classical adiabatic case is discussed on which soliton-dynamics simulations are usually based. Further we discuss how to include temperature effects in our model. Applications to soliton dynamics are discussed for the example of the mod Phi 2 ) model with emphasis on the dependence of the results on soliton width and temperature. We found that in contrast to results reported in the literature, where a similar ansatz is used, but only one electron is treated explicitly, the solitons remain stable also for small soliton widths. This indicates that the interactions of the electrons not occupying the soliton level with the lattice have a stabilizing effect on the soliton. Further our results indicate that the temperature model using random forces and dissipation terms to introduce temperature effects has to be applied with extreme care in this case due to the strong electron-lattice interactions.
Abstract A system of <?CDATA
N≫1?> N ≫ 1 interacting spinless quantum particles, described by a statistical operator F(t), is considered. A time-dependent projection operator formalism for a family of projectors, which select a statistical operator <?CDATA
FS(t)?> F S ( t ) for a group of S < N relevant particles by integration of the variables of the irrelevant N − S ‘environment’ particles, is presented. This formalism results in a nonlinear version of the inhomogeneous Nakajima–Zwanzig generalized master equation (GME) for the relevant part of the statistical operator F(t), which contains an undesirable initial condition term related to the irrelevant part of the statistical operator depending on N variables. By introducing an additional operator identity, the obtained nonlinear inhomogeneous Nakajima–Zwanzig equation is exactly converted into a homogeneous nonlinear equation, accounting for initial correlations in the kernel governing its evolution. Both of these equations are equivalent to the corresponding evolution equations for <?CDATA
FS(t)?> F S ( t ) and take into account the dynamics of the environment particles. The equations for <?CDATA
FS(t)?> F S ( t ) are specialized in the linear approximation for the particle density n and analyzed for a one-particle statistical operator <?CDATA
F1(t)?> F 1 ( t ) . It turns out that, in addition to the correlations caused by interparticle interaction, the irrelevant initial condition term in the inhomogeneous equation for <?CDATA
F1(t)?> F 1 ( t ) , defined by the two-particle correlations, also contains quantum correlations conditioned by the quantum particle statistics. The latter do not vanish with time and cannot be disregarded in the conventional way by applying e.g. Bogoliubov’s principle of weakening of initial correlations. In order to take initial quantum and other correlations (commonly discarded in an unconvincing way) into consideration, the obtained nonlinear homogeneous GME has been applied. Exact in the first approximation in n, we obtain a new homogeneous equation for <?CDATA
F1(t)?> F 1 ( t ) , which accounts for correlations and is valid for any time moment t. It is shown how this reversible-in-time equation changes at different timescales and converts into the irreversible quantum Boltzmann equation on the macroscopic timescale without the use of the conventional approximations.