Quantum kinetic theory VII: The influence of vapor dynamics on condensate growth

We extend earlier models of the growth of a Bose-Einstein condensate to include the full dynamical effects of the thermal cloud by numerically solving a modified quantum Boltzmann equation. We determine the regime in which the assumptions of the simple model are a reasonable approximation, and compare our new results with those that were earlier compared with experimental data. We find good agreement with our earlier modelling, except at higher condensate fractions, for which a significant speedup is found. We also investigate the effect of temperature on condensate growth, and find that this has a surprisingly small effect. The discrepancy between theory and experiment remains, since the speedup found in these computations does not occur in the parameter regime specified in the the experiment.Comment: Fourteen pages, TeX source with 11 figures. Changes : Extended section on formalism to include a derivation of the ergodic Boltzmann equation that we use, and a fuller explanation of the numerical methods. Explained more fully the possible errors with the experimental data. Added section detailing the source of possible errors in this formulation. Added comparison of our work with the manuscript cond-mat/0001323, and some analysis of the fits to the MIT growth curve

Number-Phase Wigner Representation for Efficient Stochastic Simulations

Phase-space representations based on coherent states (P, Q, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high dimensional quantum systems. However many problems using these techniques remain intractable over long integrations times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase space representations. This process requires an effective sampling of a non-classical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.Comment: 7 pages, 1 figur

Lower limit on the achievable temperature in resonator-based sideband cooling

A resonator can be effectively used as a cooler for another linear oscillator with a much smaller frequency. A huge cooling effect, which could be used to cool a mechanical oscillator below the energy of quantum fluctuations, has been predicted by several authors. However, here we show that there is a lower limit T* on the achievable temperature that was not considered in previous works and can be higher than the quantum limit in realistic experimental realizations. We also point out that the decay rate of the resonator, which previous studies stress should be small, must be larger than the decay rate of the cooled oscillator for effective cooling.Comment: 6 pages, 4 figures, uses psfra

The stochastic Gross-Pitaevskii equation II

We provide a derivation of a more accurate version of the stochastic Gross-Pitaevskii equation, as introduced by Gardiner et al. (J. Phys. B 35,1555,(2002). The derivation does not rely on the concept of local energy and momentum conservation, and is based on a quasi-classical Wigner function representation of a "high temperature" master equation for a Bose gas, which includes only modes below an energy cutoff E_R that are sufficiently highly occupied (the condensate band). The modes above this cutoff (the non-condensate band) are treated as being essentially thermalized. The interaction between these two bands, known as growth and scattering processes, provide noise and damping terms in the equation of motion for the condensate band, which we call the stochastic Gross-Pitaevskii equation. This approach is distinguished by the control of the approximations made in its derivation, and by the feasibility of its numerical implementation.Comment: 24 pages of LaTeX, one figur

Isolating intrinsic noise sources in a stochastic genetic switch

The stochastic mutual repressor model is analysed using perturbation methods. This simple model of a gene circuit consists of two genes and three promotor states. Either of the two protein products can dimerize, forming a repressor molecule that binds to the promotor of the other gene. When the repressor is bound to a promotor, the corresponding gene is not transcribed and no protein is produced. Either one of the promotors can be repressed at any given time or both can be unrepressed, leaving three possible promotor states. This model is analysed in its bistable regime in which the deterministic limit exhibits two stable fixed points and an unstable saddle, and the case of small noise is considered. On small time scales, the stochastic process fluctuates near one of the stable fixed points, and on large time scales, a metastable transition can occur, where fluctuations drive the system past the unstable saddle to the other stable fixed point. To explore how different intrinsic noise sources affect these transitions, fluctuations in protein production and degradation are eliminated, leaving fluctuations in the promotor state as the only source of noise in the system. Perturbation methods are then used to compute the stability landscape and the distribution of transition times, or first exit time density. To understand how protein noise affects the system, small magnitude fluctuations are added back into the process, and the stability landscape is compared to that of the process without protein noise. It is found that significant differences in the random process emerge in the presence of protein noise

Quantum Kinetic Theory VI: The Growth of a Bose-Einstein Condensate

A detailed analysis of the growth of a BEC is given, based on quantum kinetic theory, in which we take account of the evolution of the occupations of lower trap levels, and of the full Bose-Einstein formula for the occupations of higher trap levels, as well as the Bose stimulated direct transfer of atoms to the condensate level introduced by Gardiner et al. We find good agreement with experiment at higher temperatures, but at lower temperatures the experimentally observed growth rate is somewhat more rapid. We also confirm the picture of the ``kinetic'' region of evolution, introduced by Kagan et al., for the time up to the initiation of the condensate. The behavior after initiation essentially follows our original growth equation, but with a substantially increased rate coefficient. Our modelling of growth implicitly gives a model of the spatial shape of the condensate vapor system as the condensate grows, and thus provides an alternative to the present phenomenological fitting procedure, based on the sum of a zero-chemical potential vapor and a Thomas-Fermi shaped condensate. Our method may give substantially different results for condensate numbers and temperatures obtained from phenomentological fits, and indicates the need for more systematic investigation of the growth dynamics of the condensate from a supersaturated vapor.Comment: TeX source; 29 Pages including 26 PostScript figure

Number-Phase Wigner Representation for Scalable Stochastic Simulations of Controlled Quantum Systems

Simulation of conditional master equations is important to describe systems under continuous measurement and for the design of control strategies in quantum systems. For large bosonic systems, such as BEC and atom lasers, full quantum field simulations must rely on scalable stochastic methods whose convergence time is restricted by the use of representations based on coherent states. Here we show that typical measurements on atom-optical systems have a common form that allows for an efficient simulation using the number-phase Wigner (NPW) phase-space representation. We demonstrate that a stochastic method based on the NPW can converge over an order of magnitude longer and more precisely than its coherent equivalent. This opens the possibility of realistic simulations of controlled multi-mode quantum systems.Comment: 5 pages, 1 figur

Unraveling quantum dissipation in the frequency domain

We present a quantum Monte Carlo method for solving the evolution of an open quantum system. In our approach, the density operator evolution is unraveled in the frequency domain. Significant advantages of this approach arise when the frequency of each dissipative event conveys information about the state of the system.Comment: 4 pages, 4 Postscript figures, uses RevTe

Time-resolved noise of adiabatic quantum pumps

We investigate quantum-statistical correlation properties of a periodically driven mesoscopic scatterer on a time-scale shorter than the period of a drive. In this limit the intrinsic quantum fluctuations in the system of fermions are the main source of a noise. Nevertheless the effect of a slow periodic drive is clearly visible in a two-time current-current correlation function as a specific periodic in time modulation. In the limit of a strong drive such a modulation can change the sign of a current correlation function.Comment: 11 pages, 10 figure

Decoherence and the conditions for the classical control of quantum systems

We find the conditions for one quantum system to function as a classical controller of another quantum system: the controller must be an open system and rapidly diagonalised in the basis of the controller variable that is coupled to the controlled system. This causes decoherence in the controlled system that can be made small if the rate of diagonalisation is fast. We give a detailed example based on the quantum optomechanical control of a mechanical resonator. The resulting equations are similar in structure to recently proposed models for consistently combining quantum and classical stochastic dynamics