Heisenberg picture operators in the stochastic wave function approach to open quantum systems

A fast simulation algorithm for the calculation of multitime correlation functions of open quantum systems is presented. It is demonstrated that any stochastic process which “unravels” the quantum Master equation can be used for the calculation of matrix elements of reduced Heisenberg picture operators, and thus for the calculation of multitime correlation functions, by extending the stochastic process to a doubled Hilbert space. The numerical performance of the stochastic simulation algorithm is investigated by means of a standard example. Received: 30 May 1997 / Revised: 4 November 1997 / Accepted: 7 November 1997     

The dynamical-quantization approach to open quantum systems

The dynamical-quantization approach to open quantum systems does consist in quantizing the Brownian motion starting directly from its stochastic dynamics under the framework of both Langevin and Fokker–Planck equations, without alluding to any model Hamiltonian. On the ground of this non-Hamiltonian quantization method, we can derive a non-Markovian Caldeira–Leggett quantum master equation as well as a non-Markovian quantum Smoluchowski equation. The former is solved for the case of a quantum Brownian particle in a gravitational field whilst the latter for a harmonic oscillator. In both physical situations, we come up with the existence of a non-equilibrium thermal quantum force and investigate its classical limit at high temperatures as well as its quantum limit at zero temperature. Further, as a physical application of our quantum Smoluchowski equation, we take up the tunneling phenomenon of a non-inertial quantum Brownian particle over a potential barrier. Lastly, we wish to point out, corroborating conclusions reached in our previous paper [A. O. Bolivar, Ann. Phys. 326 (2011) 1354], that the theoretical predictions in the present article uphold the view that our non-Hamiltonian quantum mechanics is able to capture novel features inherent in quantum Brownian motion, thereby overcoming shortcomings underlying the Caldeira–Leggett Hamiltonian model.     

A stochastic approach to simulations of fermionic systems

We introduce a Langevin equation approach to the analysis of fermionic theories. We find a Langevin type equation, depending on a parameter τ, such that its equilibrium distributions are those of the original fermionic system in the limit of τ going to zero. We explicitly treat a simple example, proving the exactness of the method in the limit τ → 0, and we estimate the error induced by the method at first order in τ.     

Berry phase in open quantum systems: a quantum Langevin equation approach

The evolution of a two level system with a slowly varying Hamiltonian, modeled as a spin 1/2 in a slowly varying magnetic field, and interacting with a quantum environment, modeled as a bath of harmonic oscillators is analyzed using a quantum Langevin approach. This allows to easily obtain the dissipation time and the correction to the Berry phase in the case of an adiabatic cyclic evolution.     

Density matrix renormalization group approach for many-body open quantum systems

The density matrix renormalization group (DMRG) approach is extended to complex-symmetric density matrices characteristic of many-body open quantum systems. Within the continuum shell model, we investigate the interplay between many-body configuration interaction and coupling to open channels in case of the unbound nucleus (7)He. It is shown that the extended DMRG procedure provides a highly accurate treatment of the coupling to the nonresonant scattering continuum.     

Steady-state currents through nanodevices: a scattering-states numerical renormalization-group approach to open quantum systems

We propose a numerical renormalization group (NRG) approach to steady-state currents through nanodevices. A discretization of the scattering-states continuum ensures the correct boundary condition for an open quantum system. We introduce two degenerate Wilson chains for current carrying left- and right-moving electrons reflecting time-reversal symmetry in the absence of a finite bias V. We employ the time-dependent NRG to evolve the known steady-state density operator for a noninteracting junction into the density operator of the fully interacting nanodevice at finite bias. We calculate the differential conductance as function of V, T, and the external magnetic field.     

Current in open quantum systems

We show that a dissipative current component is present in the dynamics generated by a Liouville-master equation, in addition to the usual component associated with Hamiltonian evolution. The dissipative component originates from coarse graining in time, implicit in a master equation, and needs to be included to preserve current continuity. We derive an explicit expression for the dissipative current in the context of the Markov approximation. Finally, we illustrate our approach with a simple numerical example, in which a quantum particle is coupled to a harmonic phonon bath and dissipation is described by the Pauli master equation.     

Exact solvability of open stochastic systems

It is shown that the boundary vectors in the matrix-product states approach to open stochastic diffusion processes are deformed coherent states of a deformed harmonicoscillator algebra. A unified deformed coherent states solution to the partially and totally asymmetric diffusion boundary problem is proposed and studied.     

Dephasing maps for quantum stochastic systems

Stochastic time evolution in a nonseparable and nonintegrable quantum system is manifested by rapid dephasing of gaussian wavepackets, whose topological distribution in the coordinate-momentum space defines its irregular regions, while wavepackets initiated in regular regions exhibit quasiperiodic evolution. A gradual transition from quasiperiodic to chaotic dynamics with increasing energy is observed.     

Completely mixing quantum open systems and quantum fractals

Departing from classical concepts of ergodic theory, formulated in terms of probability densities, measures describing the mixing behavior and the loss of information in quantum open systems are proposed. As application we discuss the chaotic outcomes of continuous measurement processes in the EEQT framework. Simultaneous measurement of four noncommuting spin components is shown to lead to a chaotic jumps on the quantum spin sphere and to generate specific fractal images of a nonlinear iterated function system.     

The stochastic quantum mechanics approach to the unification of relativity and quantum theory

The stochastic phase-space solution of the particle localizability problem in relativistic quantum mechanics is reviewed. It leads to relativistically covariant probability measures that give rise to covariant and conserved probability currents. The resulting particle propagators are used in the formulation of stochastic geometries underlying a concept of quantum spacetime that is operationally based on stochastically extended quantum test particles. The epistemological implications of the intrinsic stochasticity of such quantum spacetime frameworks for microcausality, the EPR paradox, etc., are discussed.Supported in part by NSERC Grant A5208.     

Density functional approach to quantum lattice systems

For quantum lattice systems, we consider the problem of characterizing the set of single-particle densities,, which come from the ground-state eigenspace of someN-particle Hamiltonian of the form whereH ₀ is a fixed, bounded operator representing the kinetic and interaction energies. We show that the conditions on are that it be strictly positive, properly normalized, and consistent with the Pauli principle. Our results are valid for both finite and infinite lattices and for either bosons or fermions. The Coulomb interaction may be included inH ₀ if the lattice dimension is 2. We also characterize those single-particle densities which come from the Gibbs states of such Hamiltonians at finite temperature. In addition to the conditions stated above, must satisfy a finite entropy condition.Research supported by the National Science Foundation under grant No. PHY-82-03669.Research supported by Office of Naval Research under grant No. 0014-80-G-0084.On leave from Department of Mathematics, University of Lowell, Massachusetts 01854.     

A theory of open systems based on stochastic differential equations

For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.     

Multiple-scale analysis of open quantum systems

In this work, we present a multiple-scale perturbation technique suitable for the study of open quantum systems, which is easy to implement and in few iterative steps allows us to find excellent approximate solutions. For any time-local quantum master equation, whether markovian or non-markovian, in Lindblad form or not, we give a general procedure to construct analytical approximations to the corresponding dynamical map and, consequently, to the temporal evolution of the density matrix. As a simple illustrative example of the implementation of the method, we study an atom-cavity system described by a dissipative Jaynes-Cummings model. Performing a multiple-scale analysis we obtain approximate analytical expressions for the strong and weak coupling regimes that allow us to identify characteristic time scales in the state of the physical system.     

Adiabatic quantum computation in open systems

We analyze the performance of adiabatic quantum computation (AQC) subject to decoherence. To this end, we introduce an inherently open-systems approach, based on a recent generalization of the adiabatic approximation. In contrast to closed systems, we show that a system may initially be in an adiabatic regime, but then undergo a transition to a regime where adiabaticity breaks down. As a consequence, the success of AQC depends sensitively on the competition between various pertinent rates, giving rise to optimality criteria.     

Phase transitions in open quantum systems

We consider the behavior of open quantum systems through the dependence of the coupling to one decay channel by introducing the coupling parameter alpha, which is proportional to the average degree of overlapping. Under critical conditions, a reorganization of the spectrum takes place that creates a bifurcation of the time scales with respect to the lifetimes of the resonance states. We derive analytically the conditions under which the reorganization process can be understood as a second-order phase transition and illustrate our results by numerical investigations. The conditions are fulfilled, e.g., for a uniform picket-fence level distribution with equal coupling of the states to the continuum. Energy dependencies within the system are included. We consider also the case of an unfolded Gaussian orthogonal ensemble and of a spectrum bounded from below. In all these cases, the reorganization of the spectrum occurs at the critical value alpha(crit) of the control parameter globally over the whole energy range of the spectrum. All states act cooperatively.     

Zeno dynamics for open quantum systems

In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics.