Hidden parameters in open-system evolution unveiled by geometric phase

We find a class of open-system models in which individual quantum trajectories may depend on parameters that are undetermined by the full open-system evolution. This dependence is imprinted in the geometric phase associated with such trajectories and persists after averaging. Our findings indicate a potential source of ambiguity in the quantum trajectory approach to open quantum systems.Comment: QSD analysis added; several stylistic changes; journal reference adde

Quantum Particle-Trajectories and Geometric Phase

"Particle"-trajectories are defined as integrable $dx_\mu dp^\mu = 0$ paths in projective space. Quantum states evolving on such trajectories, open or closed, do not delocalise in $(x, p)$ projection, the phase associated with the trajectories being related to the geometric (Berry) phase and the Classical Mechanics action. High Energy Physics properties of states evolving on "particle"-trajectories are discussed.Comment: 4 page

On geometric phases for quantum trajectories

A sequence of completely positive maps can be decomposed into quantum trajectories. The geometric phase or holonomy of such a trajectory is delineated. For nonpure initial states, it is shown that well-defined holonomies can be assigned by using Uhlmann's concept of parallel transport along the individual trajectories. We put forward an experimental realization of the geometric phase for a quantum trajectory in interferometry. We argue that the average over the phase factors for all quantum trajectories that build up a given open system evolution, fails to reflect the geometry of the open system evolution itself.Comment: Submitted to the Proceedings of the 13th CEWQO 2006 in Vienn

Conductance of Open Quantum Billiards and Classical Trajectories

We analyse the transport phenomena of 2D quantum billiards with convex boundary of different shape. The quantum mechanical analysis is performed by means of the poles of the S-matrix while the classical analysis is based on the motion of a free particle inside the cavity along trajectories with a different number of bounces at the boundary. The value of the conductance depends on the manner the leads are attached to the cavity. The Fourier transform of the transmission amplitudes is compared with the length of the classical paths. There is good agreement between classical and quantum mechanical results when the conductance is achieved mainly by special short-lived states such as whispering gallery modes (WGM) and bouncing ball modes (BBM). In these cases, also the localization of the wave functions agrees with the picture of the classical paths. The S-matrix is calculated classically and compared with the transmission coefficients of the quantum mechanical calculations for five modes in each lead. The number of modes coupled to the special states is effectively reduced.Comment: 19 pages, 6 figures (jpg), 2 table

Quantum trajectories and open many-body quantum systems

The study of open quantum systems has become increasingly important in the past years, as the ability to control quantum coherence on a single particle level has been developed in a wide variety of physical systems. In quantum optics, the study of open systems goes well beyond understanding the breakdown of quantum coherence. There, the coupling to the environment is sufficiently well understood that it can be manipulated to drive the system into desired quantum states, or to project the system onto known states via feedback in quantum measurements. Many mathematical frameworks have been developed to describe such systems, which for atomic, molecular, and optical (AMO) systems generally provide a very accurate description of the open quantum system on a microscopic level. In recent years, AMO systems including cold atomic and molecular gases and trapped ions have been applied heavily to the study of many-body physics, and it has become important to extend previous understanding of open system dynamics in single- and few-body systems to this many-body context. A key formalism that has already proven very useful in this context is the quantum trajectories technique. This was developed as a numerical tool for studying dynamics in open quantum systems, and falls within a broader framework of continuous measurement theory as a way to understand the dynamics of large classes of open quantum systems. We review the progress that has been made in studying open many-body systems in the AMO context, focussing on the application of ideas from quantum optics, and on the implementation and applications of quantum trajectories methods. Control over dissipative processes promises many further tools to prepare interesting and important states in strongly interacting systems, including the realisation of parameter regimes in quantum simulators that are inaccessible via current techniques.Comment: 66 pages, 29 figures, review article submitted to Advances in Physics - comments and suggestions are welcom

Open system dynamics with non-Markovian quantum trajectories

A non-Markovian stochastic Schroedinger equation for a quantum system coupled to an environment of harmonic oscillators is presented. Its solutions, when averaged over the noise, reproduce the standard reduced density operator without any approximation. We illustrate the power of this approach with several examples, including exponentially decaying bath correlations and extreme non-Markovian cases, where the `environment' consists of only a single oscillator. The latter case shows the decay and revival of a `Schroedinger cat' state. For strong coupling to a dissipative environment with memory, the asymptotic state can be reached in a finite time. Our description of open systems is compatible with different positions of the `Heisenberg cut' between system and environment.Comment: 4 pages RevTeX, 3 figure

Periodic orbits for classical particles having complex energy

This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.Comment: 12 pages, 9 figure

From quantum trajectories to classical orbits

Recently it has been shown that the evolution of open quantum systems may be ``unraveled'' into individual ``trajectories,'' providing powerful numerical and conceptual tools. In this letter we use quantum trajectories to study mesoscopic systems and their classical limit. We show that in this limit, Quantum Jump (QJ) trajectories approach a diffusive limit very similar to the Quantum State Diffusion (QSD) unraveling. The latter follows classical trajectories in the classical limit. Hence, both unravelings show the rise of classical orbits. This is true for both regular and chaotic systems (which exhibit strange attractors).Comment: 7 pages RevTeX 3.0 + 2 figures (postscript). Submitted to Physical Review Letter