Quantum turbulence and correlations in Bose-Einstein condensate collisions

We investigate numerically simulated collisions between experimentally realistic Bose-Einstein condensate wavepackets, within a regime where highly populated scattering haloes are formed. The theoretical basis for this work is the truncated Wigner method, for which we present a detailed derivation, paying particular attention to its validity regime for colliding condensates. This paper is an extension of our previous Letter [A. A. Norrie, R. J. Ballagh, and C. W. Gardiner, Phys. Rev. Lett. 94, 040401 (2005)] and we investigate both single-trajectory solutions, which reveal the presence of quantum turbulence in the scattering halo, and ensembles of trajectories, which we use to calculate quantum-mechanical correlation functions of the field

Bogoliubov dynamics of condensate collisions using the positive-P representation

We formulate the time-dependent Bogoliubov dynamics of colliding Bose-Einstein condensates in terms of a positive-P representation of the Bogoliubov field. We obtain stochastic evolution equations for the field which converge to the full Bogoliubov description as the number of realisations grows. The numerical effort grows linearly with the size of the computational lattice. We benchmark the efficiency and accuracy of our description against Wigner distribution and exact positive-P methods. We consider its regime of applicability, and show that it is the most efficient method in the common situation - when the total particle number in the system is insufficient for a truncated Wigner treatment.Comment: 9 pages. 5 figure

Three-body recombination of ultracold Bose gases using the truncated Wigner method

We apply the truncated Wigner method to the process of three-body recombination in ultracold Bose gases. We find that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be treated using a set of coupled stochastic differential equations that include diffusion terms, and can be simulated using known numerical methods. As an example we investigate the behaviour of a simple homogeneous Bose gas.Comment: Replaced paper same as original; correction to author list on cond-mat mad

Non-Markovian Open Quantum Systems: Input-Output Fields, Memory, Monitoring

Principles of monitoring non-Markovian open quantum systems are analyzed. We use the field representation of the environment (Gardiner and Collet, 1985) for the separation of its memory and detector part, respectively. We claim the system-plus-memory compound becomes Markovian, the detector part is tractable by standard Markovian monitoring. Because of non-Markovianity, only the mixed state of the system can be predicted, the pure state of the system can be retrodicted. We present the corresponding non-Markovian stochastic Schr\"odinger equation.Comment: 5 pages, 3 postscript figures; version with brief important improvement

ScotPID - a model of collaboration

ScotPID is a national personal development initiative in Scotland, with thirteen higher education institutions taking part in the development of case studies which enhance personal development planning for students. As a model of collaboration, ScotPID involves all stakeholders: each core project group is composed of an academic, IT support manager, careers service adviser and undergraduate student, with support from QAA Scotland. The case study is developed by the contribution of all of the members of the team. The strength of the ScotPID collaboration is the varied background of the team members. However, collaboration between the ScotPID teams should also be encouraged, to strengthen the inter-institutional approach further

On the optimal feedback control of linear quantum systems in the presence of thermal noise

We study the possibility of taking bosonic systems subject to quadratic Hamiltonians and a noisy thermal environment to non-classical stationary states by feedback loops based on weak measurements and conditioned linear driving. We derive general analytical upper bounds for the single mode squeezing and multimode entanglement at steady state, depending only on the Hamiltonian parameters and on the number of thermal excitations of the bath. Our findings show that, rather surprisingly, larger number of thermal excitations in the bath allow for larger steady-state squeezing and entanglement if the efficiency of the optimal continuous measurements conditioning the feedback loop is high enough. We also consider the performance of feedback strategies based on homodyne detection and show that, at variance with the optimal measurements, it degrades with increasing temperature.Comment: 10 pages, 2 figures. v2: minor changes to the letter; better explanation of the necessary and sufficient conditions to achieve the bounds (in the supplemental material); v3: title changed; comparison between optimal general-dyne strategy and homodyne strategy is discussed; supplemental material included in the manuscript and few references added. v4: published versio

Phase dynamics of a multimode Bose condensate controlled by decay

The relative phase between two uncoupled BE condensates tends to attain a specific value when the phase is measured. This can be done by observing their decay products in interference. We discuss exactly solvable models for this process in cases where competing observation channels drive the phases to different sets of values. We treat the case of two modes which both emit into the input ports of two beam splitters, and of a linear or circular chain of modes. In these latter cases, the transitivity of relative phase becomes an issue

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

Dissipation in a rotating frame: master equation, effective temperature and Lamb-shift

Motivated by recent realizations of microwave-driven nonlinear resonators in superconducting circuits, the impact of environmental degrees of freedom is analyzed as seen from a rotating frame. A system plus reservoir model is applied to consistently derive in the weak coupling limit the master equation for the reduced density in the moving frame and near the first bifurcation threshold. It turns out that additional interactions between momenta of system and bath appear which have been omitted in previous studies. Explicit expressions for the effective temperature and the Lamb-shift are given which for ohmic baths are in agreement with experimental findings, while for structured environments population inversion is predicted that may qualitatively explain recent observations.Comment: 7 pages, 5 figure

All-optical versus electro-optical quantum-limited feedback

All-optical feedback can be effected by putting the output of a source cavity through a Faraday isolator and into a second cavity which is coupled to the source cavity by a nonlinear crystal. If the driven cavity is heavily damped, then it can be adiabatically eliminated and a master equation or quantum Langevin equation derived for the first cavity alone. This is done for an input bath in an arbitrary state, and for an arbitrary nonlinear coupling. If the intercavity coupling involves only the intensity (or one quadrature) of the driven cavity, then the effect on the source cavity is identical to that which can be obtained from electro-optical feedback using direct (or homodyne) detection. If the coupling involves both quadratures, this equivalence no longer holds, and a coupling linear in the source amplitude can produce a nonclassical state in the source cavity. The analogous electro-optic scheme using heterodyne detection introduces extra noise which prevents the production of nonclassical light. Unlike the electro-optic case, the all-optical feedback loop has an output beam (reflected from the second cavity). We show that this may be squeezed, even if the source cavity remains in a classical state.Comment: 21 pages. This is an old (1994) paper, but one which I thought was worth posting because in addition to what is described in abstract it has: (1) the first formulation (to my knowledge) of quantum trajectories for an arbitrary (i.e. squeezed, thermal etc.) broadband bath; (2) the prediction of a periodic modification to the detuning and damping of an oscillator for the simplest sort of all-optical feedback (i.e. a mirror) as seen in the recent experiment "Forces between a Single Atom and Its Distant Mirror Image", P. Bushev et al, Phys. Rev. Lett. 92, 223602 (2004