Master-equations for the study of decoherence

Different structures of master-equation used for the description of decoherence of a microsystem interacting through collisions with a surrounding environment are considered and compared. These results are connected to the general expression of the generator of a quantum dynamical semigroup in presence of translation invariance recently found by Holevo.Comment: 10 pages, latex, no figures, to appear in Int. J. Theor. Phy

Two Derivations of the Master Equation of Quantum Brownian Motion

Central to many discussion of decoherence is a master equation for the reduced density matrix of a massive particle experiencing scattering from its surrounding environment, such as that of Joos and Zeh. Such master equations enjoy a close relationship with spontaneous localization models, like the GRW model. This aim of this paper is to present two derivations of the master equation. The first derivation is a pedagogical model designed to illustrate the origins of the master equation as simply as possible, focusing on physical principles and without the complications of S-matrix theory. This derivation may serve as a useful tutorial example for students attempting to learn this subject area. The second is the opposite: a very general derivation using non-relativistic many body field theory. It reduces to the equation of the type given by Joos and Zeh in the one-particle sector, but correcting certain numerical factors which have recently become significant in connection with experimental tests of decoherence. This master equation also emphasizes the role of local number density as the ``preferred basis'' for decoherence in this model.Comment: 19 pages, RevTe

Atom Lasers, Coherent States, and Coherence:II. Maximally Robust Ensembles of Pure States

As discussed in Wiseman and Vaccaro [quant-ph/9906125], the stationary state of an optical or atom laser far above threshold is a mixture of coherent field states with random phase, or, equivalently, a Poissonian mixture of number states. We are interested in which, if either, of these descriptions of $\rho_{ss}$, is more natural. In the preceding paper we concentrated upon whether descriptions such as these are physically realizable (PR). In this paper we investigate another relevant aspect of these ensembles, their robustness. A robust ensemble is one for which the pure states that comprise it survive relatively unchanged for a long time under the system evolution. We determine numerically the most robust ensembles as a function of the parameters in the laser model: the self-energy $\chi$ of the bosons in the laser mode, and the excess phase noise $\nu$. We find that these most robust ensembles are PR ensembles, or similar to PR ensembles, for all values of these parameters. In the ideal laser limit ($\nu=\chi=0$), the most robust states are coherent states. As the phase noise $\nu$ or phase dispersion $\chi$ is increased, the most robust states become increasingly amplitude-squeezed. We find scaling laws for these states. As the phase diffusion or dispersion becomes so large that the laser output is no longer quantum coherent, the most robust states become so squeezed that they cease to have a well-defined coherent amplitude. That is, the quantum coherence of the laser output is manifest in the most robust PR states having a well-defined coherent amplitude. This lends support to the idea that robust PR ensembles are the most natural description of the state of the laser mode. It also has interesting implications for atom lasers in particular, for which phase dispersion due to self-interactions is expected to be large.Comment: 16 pages, 9 figures included. To be published in Phys. Rev. A, as Part II of a two-part paper. The original version of quant-ph/9906125 is shortly to be replaced by a new version which is Part I of the two-part paper. This paper (Part II) also contains some material from the original version of quant-ph/990612

The Post-Decoherence Density Matrix Propagator for Quantum Brownian Motion

Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the localization time, (\hbar / \gamma k T )^{\half}, where $\gamma$ is the dissipation and $T$ the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time, $\gamma^{-1}$. We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a completely different method, a result we previously obtained using the quantum state diffusion picture (Phys.Rev. D52, 7294 (1995)). We also go beyond this earlier result, in that we derive an explicit expression for the weighting of each phase space localized state in the approximately diagonal density matrix, as a function of the initial state. For sufficiently long times it is equal to the Wigner function, and we confirm that the Wigner function is positive for times greater than the localization time (multiplied by a number of order 1).Comment: 17 pages, plain Tex, submitted to Physical Review

Completely Positive Quantum Dissipation

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both position and momentum are expressed in terms of the collision cross-section.Comment: 8 pages, revtex, no figure

Simulations of Aerodynamic Damping for MEMS Resonators

Aerodynamic damping for MEMS resonators is studied based on the numerical solution of Boltzmann-ESBGK equation. A compact model is then developed based on numerical simulations for a wide range of Knudsen numbers. The damping predictions are compared with both Reynold equation based models and several sets of experimental data. It has been found that the structural damping is dominant at low pressures (high Knudsen numbers). For cases with small length-to-width ratios and large vibration amplitudes, the threedimensionality effects must be taken into account. Finally, an uncertainty quantification approach based on the probability transformation method has been applied to assess the influence of pressure and geometric uncertainties. The output probability density functions (PDF) of the damping ratio has been studied for various input PDF of beam geometry and ambient pressure

Positive Quantum Brownian Evolution

Using the independent oscillator model with an arbitrary system potential, we derive a quantum Brownian equation assuming a correlated total initial state. Although not of Lindblad form, the equation preserves positivity of the density operator on a restricted set of initial states

Non-Newtonian Couette-Poiseuille flow of a dilute gas

The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation is analytically solved for this Couette-Poiseuille flow to first order in the force and for arbitrary values of the Knudsen number associated with the shear rate. This allows us to investigate the influence of the external force on the non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille flow is analyzed when the shear-rate Knudsen number and the scaled force are of the same order and terms up to second order are retained. In this way, the transition from the bimodal temperature profile characteristic of the pure force-driven Poiseuille flow to the parabolic profile characteristic of the pure Couette flow through several intermediate stages in the Couette-Poiseuille flow are described. A critical comparison with the Navier-Stokes solution of the problem is carried out.Comment: 24 pages, 5 figures; v2: discussion on boundary conditions added; 10 additional references. Published in a special issue of the journal "Kinetic and Related Models" dedicated to the memory of Carlo Cercignan

Apparent wave function collapse caused by scattering

Some experimental implications of the recent progress on wave function collapse are calculated. Exact results are derived for the center-of-mass wave function collapse caused by random scatterings and applied to a range of specific examples. The results show that recently proposed experiments to measure the GRW effect are likely to fail, since the effect of naturally occurring scatterings is of the same form as the GRW effect but generally much stronger. The same goes for attempts to measure the collapse caused by quantum gravity as suggested by Hawking and others. The results also indicate that macroscopic systems tend to be found in states with (Delta-x)(Delta-p) = hbar/sqrt(2), but microscopic systems in highly tiltedly squeezed states with (Delta-x)(Delta-p) >> hbar.Comment: Final published version. 20 pages, Plain TeX, no figures. Online at http://astro.berkeley.edu/~max/collapse.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/collapse.html (faster from Europe) or from [email protected]

Diffusive limit for a quantum linear Boltzmann dynamics

In this article, I study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model I begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the gas particle scattering is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix that evolves according to a translation-covariant Lindblad equation. The main result is a proof that the particle's position distribution converges to a Gaussian under diffusive rescaling.Comment: 51 pages. I have restructured Sections 2-4 from the previous version and corrected an error in the proof of Proposition 7.