Effects of random localizing events on matter waves: formalism and examples

A formalism is introduced to describe a number of physical processes that may break down the coherence of a matter wave over a characteristic length scale l. In a second-quantized description, an appropriate master equation for a set of bosonic "modes" (such as atoms in a lattice, in a tight-binding approximation) is derived. Two kinds of "localizing processes" are discussed in some detail and shown to lead to master equations of this general form: spontaneous emission (more precisely, light scattering), and modulation by external random potentials. Some of the dynamical consequences of these processes are considered: in particular, it is shown that they generically lead to a damping of the motion of the matter-wave currents, and may also cause a "flattening" of the density distribution of a trapped condensate at rest.A formalism is introduced to describe a number of physical processes that may break down the coherence of a matter wave over a characteristic length scale l. In a second-quantized description, an appropriate master equation for a set of bosonic "modes" (such as atoms in a lattice, in a tight-binding approximation) is derived. Two kinds of "localizing processes" are discussed in some detail and shown to lead to master equations of this general form: spontaneous emission (more precisely, light scattering), and modulation by external random potentials. Some of the dynamical consequences of these processes are considered: in particular, it is shown that they generically lead to a damping of the motion of the matter-wave currents, and may also cause a "flattening" of the density distribution of a trapped condensate at rest.

arXiv:quant-ph/0508134Journal of Physics B Atomic Molecular and Optical PhysicsDec 5, 20051 citation

Propriétés statiques et dynamiques d'un condensat de Bose-Einstein dans un potentiel aléatoire

David Clément

Static and dynamical properties of atomic Bose-Einstein condensate in a random potential. The presence of disorder in systems with long-range coherence leads to fascinating effects such as the localization of waves and the appearance of novels quantum phases. While it is often difficult to isolate the effects of disorder in experiments, atomic Bose-Einstein condensates can be manipulated in a controlled random environment. In this report, several aspects of atomic Bose-Einstein condensates in a random potential induced by laser speckles are discussed. After presenting the realization and the characterization of the speckle potential we first study some properties of the trapped, disordered condensates, in particular the enhancement of large density modulations during time-of-flight. We show that this effect is not related to initial phase fluctuations but originates from small initial density modulations in the trap. Secondly, we report on collective modes of a disordered Bose-Einstein condensate. The absence of frequency shifts and the damping of the dipole mode in the presence of disorder are discussed. Then we present measurements of the velocity of sound in a disordered condensate and we show the absence of shift compared to the non-disordered case. At last, we report on the suppression of the 1D expansion of an interacting matter-wave in the presence of disorder in two different regimes. In a regime of strong disorder a detailed experimental analysis supports the proposed scenario of trapping where reflexions on a single modulation of the random potential and interactions play a crucial role. A different regime of weak disorder is then investigated theoretically and we show that Anderson localization can happen in this case under the conditions we establish. This scheme has been realized recently in experiments, leading to the observation of the Anderson localization of a matter-wave.Static and dynamical properties of atomic Bose-Einstein condensate in a random potential. The presence of disorder in systems with long-range coherence leads to fascinating effects such as the localization of waves and the appearance of novels quantum phases. While it is often difficult to isolate the effects of disorder in experiments, atomic Bose-Einstein condensates can be manipulated in a controlled random environment. In this report, several aspects of atomic Bose-Einstein condensates in a random potential induced by laser speckles are discussed. After presenting the realization and the characterization of the speckle potential we first study some properties of the trapped, disordered condensates, in particular the enhancement of large density modulations during time-of-flight. We show that this effect is not related to initial phase fluctuations but originates from small initial density modulations in the trap. Secondly, we report on collective modes of a disordered Bose-Einstein condensate. The absence of frequency shifts and the damping of the dipole mode in the presence of disorder are discussed. Then we present measurements of the velocity of sound in a disordered condensate and we show the absence of shift compared to the non-disordered case. At last, we report on the suppression of the 1D expansion of an interacting matter-wave in the presence of disorder in two different regimes. In a regime of strong disorder a detailed experimental analysis supports the proposed scenario of trapping where reflexions on a single modulation of the random potential and interactions play a crucial role. A different regime of weak disorder is then investigated theoretically and we show that Anderson localization can happen in this case under the conditions we establish. This scheme has been realized recently in experiments, leading to the observation of the Anderson localization of a matter-wave.

Annales de PhysiqueJan 1, 20081 citation

THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES

Valeriy A. Brazhnyi, V. V. Konotop

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

arXiv:cond-mat/0405334Modern Physics Letters BJun 10, 2004446 citations

Anderson localization of a weakly interacting one-dimensional Bose gas

Tobias Paul, Mathias Albert, Peter Schlagheck +2 more

We consider the phase coherent transport of a quasi-one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length $L$ . Among the possible different types of flow we identified [T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale ${L}^{\ensuremath{\ast}}$ for the disordered region, above which most of the realizations of the random potential lead to time-dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.We consider the phase coherent transport of a quasi-one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length $L$. Among the possible different types of flow we identified [T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale ${L}^{\ensuremath{\ast}}$ for the disordered region, above which most of the realizations of the random potential lead to time-dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.

arXiv:0907.0098Physical Review ASep 17, 200940 citations

Matter waves from quantum sources in a force field

Tobias Kramer, Christian Bracher, M Kleber

Localized scattering phenomena may result in the formation of stationary matter waves originating from a compact region in physical space. Mathematically, such waves are advantageously expressed in terms of quantum sources that are introduced into the Schrödinger equation. The source formalism yields direct access to the scattering wavefunction, particle distribution and total current. As an example, we study emission from three-dimensional Gaussian sources into a homogeneous force field. This model describes the behaviour of an atom laser supplied by an ideal Bose-Einstein condensate under the influence of gravity. We predict a strong dependence of the beam profile on the condensate size and the presence of interference phenomena recently observed in photodetachment experiments.Localized scattering phenomena may result in the formation of stationary matter waves originating from a compact region in physical space. Mathematically, such waves are advantageously expressed in terms of quantum sources that are introduced into the Schrödinger equation. The source formalism yields direct access to the scattering wavefunction, particle distribution and total current. As an example, we study emission from three-dimensional Gaussian sources into a homogeneous force field. This model describes the behaviour of an atom laser supplied by an ideal Bose-Einstein condensate under the influence of gravity. We predict a strong dependence of the beam profile on the condensate size and the presence of interference phenomena recently observed in photodetachment experiments.

arXiv:quant-ph/0110155Journal of Physics A Mathematical and GeneralSep 24, 200240 citations

Exploring matter-wave dynamics with a Bose-Einstein condensate

R. Chang

Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both single-particle and many-body quantum phenomena. This thesis describes several experiments using a Bose-Einstein condensate of Rb-87 as a model system to study novel matter-wave effects that traditionally arise in vastly different systems, yet are difficult to access. We study the scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a sharp repulsive potential, two distinct transient scattering effects are observed: one due to the momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects lead to a redistribution of momenta, resulting in a rich interference pattern that may be used to reconstruct the single-particle wavefunction. In a second experiment, we study the response of a particle in a periodic potential to an applied force. By abruptly applying an external force to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial response of a particle in a periodic potential is in fact characterized by the bare mass, and only over timescales long compared to that of interband dynamics is the usual effective mass an appropriate description. This breakdown of the effective mass description on fast timescales is difficult to observe in traditional solid state systems due to their large bandgaps and fast timescale of interband dynamics. Both these experiments make use of the condensate's long coherence length, and the ability to shape and modulate the external potential on timescales fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both single-particle and many-body quantum phenomena. This thesis describes several experiments using a Bose-Einstein condensate of Rb-87 as a model system to study novel matter-wave effects that traditionally arise in vastly different systems, yet are difficult to access. We study the scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a sharp repulsive potential, two distinct transient scattering effects are observed: one due to the momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects lead to a redistribution of momenta, resulting in a rich interference pattern that may be used to reconstruct the single-particle wavefunction. In a second experiment, we study the response of a particle in a periodic potential to an applied force. By abruptly applying an external force to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial response of a particle in a periodic potential is in fact characterized by the bare mass, and only over timescales long compared to that of interband dynamics is the usual effective mass an appropriate description. This breakdown of the effective mass description on fast timescales is difficult to observe in traditional solid state systems due to their large bandgaps and fast timescale of interband dynamics. Both these experiments make use of the condensate's long coherence length, and the ability to shape and modulate the external potential on timescales fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.

Jan 8, 20141 citation

We consider the phase coherent transport of a quasi-one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length

L

. Among the possible different types of flow we identified [T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale ${L}^{\ensuremath{\ast}}$ for the disordered region, above which most of the realizations of the random potential lead to time-dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.