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arXiv:1308.0058

Momentum density and phase maps of a two-dimensional trapped Bose-Einstein condensate excited by a red laser

Roger R. Sakhel, Asaad R. Sakhel, H. B. Ghassib

Published Jul 31, 2013• 0 citations
arXivHTMLPDF

Abstract

We investigate numerically the momentum density and phase maps $-$in coordinate and momentum space$-$ of a two dimensional Bose-Einstein condensate (BEC) excited by a moving red-detuned laser potential. The BEC is confined in a harmonic trap cutoff by hard walls. The system and excitation scheme are as in our previous work (Roger R. Sakhel {\it et al.} to appear in J. Low Temp. Phys. (2013)); but with twice the number of particles and interaction strength. We solve the time-dependent Gross-Pitaevskii equation numerically using the split-step Crank-Nicolson method in real time. It is demonstrated that the red-detuned laser has a phase-imprinting effect like a repulsive potential barrier. Signatures of excitations are extracted from the dynamics of the momentum densities and phase maps. Further, a new phase is defined in momentum space, which is used to reveal excitations. Therefore, phase maps in coordinate space and momentum space are compared for different BEC evolution times. We argue, that this momentum-space phase is especially important with regard to the studies of BEC momentum distributions. In addition, this work presents a new method of BEC interferometry and should contribute to the ongoing research in that field. One of our significant findings is the presence of substantial differences betwteen the momentum density obtained by a Fourier transform (FT) of the spatial density distribution and the one obtained from the modulus of the wavefunction in momentum space; the latter is obtained by a FT of the spatial wavefunction.

DOI
10.48550/arxiv.1308.0058

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We investigate numerically the momentum density and phase maps −-−in coordinate and momentum space−-− of a two dimensional Bose-Einstein condensate (BEC) excited by a moving red-detuned laser potential. The BEC is confined in a harmonic trap cutoff by hard walls. The system and excitation scheme are as in our previous work (Roger R. Sakhel {\it et al.} to appear in J. Low Temp. Phys. (2013)); but with twice the number of particles and interaction strength. We solve the time-dependent Gross-Pitaevskii equation numerically using the split-step Crank-Nicolson method in real time. It is demonstrated that the red-detuned laser has a phase-imprinting effect like a repulsive potential barrier. Signatures of excitations are extracted from the dynamics of the momentum densities and phase maps. Further, a new phase is defined in momentum space, which is used to reveal excitations. Therefore, phase maps in coordinate space and momentum space are compared for different BEC evolution times. We argue, that this momentum-space phase is especially important with regard to the studies of BEC momentum distributions. In addition, this work presents a new method of BEC interferometry and should contribute to the ongoing research in that field. One of our significant findings is the presence of substantial differences betwteen the momentum density obtained by a Fourier transform (FT) of the spatial density distribution and the one obtained from the modulus of the wavefunction in momentum space; the latter is obtained by a FT of the spatial wavefunction.
We explore the energy dynamics of a two dimensional (2D) trapped Bose-Einstein condensate (BEC) excited by a moving red-detuned laser potential (RDLP). The trap is a combination of a power-law (PL) potential cutoff by a hard-wall box potential (HWBP). It is found that by a restricted measurment of the energy within the boundaries of the HWBP, the energy demonstrates oscillations indicative of solitons. It is then demonstrated, that the geometry of the PL potential influences the lifetime of these oscillations, i.e., the lifetime of the ensuing solitons inside the HWBP. We argue that the energy dynamics in this setup are a good tool for measuring their lifetime. It is also found, that the condensate density dynamics display oscillatory patterns of a magnitude and order controlled by the velocity of the RDLP. A connection to oscillations in the chemical potential dynamics is also discussed. Essentially, we suggest future experiments for this kind of setup, which would measure the phonon energy dynamics to trace the lifetime of solitons.
We explore the energy dynamics of a two dimensional (2D) harmonically trapped Bose-Einstein condensate inside a box potential, excited by a moving red-detuned laser potential (RDLP). For an RDLP velocity vvv less than a critical value vcv_cvc​, energy oscillations are observed to begin simultaneously with the motion of the RDLP. For v>vcv > v_cv>vc​, these oscillations are delayed through a transient in the energy. At the end of the delay time tonst_{ons}tons​, the energy oscillations are regenerated again, and tonst_{ons}tons​ is found to depend on vvv through universal relations for two cases: one for an RDLP depth sufficient to break off a BEC fragment, another for a depth insufficient to split the BEC. In the case of splitting, tonst_{ons}tons​ exactly equals the time it takes the BEC fragment (dragged by the RDLP trough) to reach the hard wall of the box potential.
Abstract We investigate a quasi one-dimensional 87 R b Bose–Einstein condensate in a harmonic trap with an additional dimple trap (dT) in the center. Within a zero-temperature Gross–Pitaevskii mean-field description we provide a one-dimensional physical intuitive model, which we solve by both a time-independent variational approach and numerical calculations. With this we obtain at first equilibrium results for the emerging condensate wave function which reveal that a dimple trap potential induces a bump or a dip in case of a red- or a blue-detuned Gaussian laser beam, respectively. Afterwards, we investigate how this dT induced bump/dip-imprint upon the condensate wave function evolves for two quench scenarios. At first we consider the generic case that the harmonic confinement is released. During the resulting time-of-flight expansion it turns out that the dT induced bump in the condensate wave function remains present, whereas the dip starts decaying after a characteristic time scale which decreases with increasing blue-detuned dT depth. Secondly, once the red- or blue-detuned dT is switched off, we find that bright shock-waves or gray/dark bi-soliton trains emerge which oscillate within the harmonic confinement with a characteristic frequency.
We report the observation of highly energetic self-interfering matter-wave (SIMW) patterns generated by a moving obstacle in a two-dimensional Bose-Einstein condensate (BEC) inside a power trap cut off by hard-wall box potential boundaries. The obstacle initially excites circular dispersive waves radiating away from the center of the trap which are reflected from hard-wall box boundaries at the edges of the trap. The resulting interference between outgoing waves from the center of the trap and reflected waves from the box boundaries institutes, to the best of our knowledge, unprecedented SIMW patterns. For this purpose we simulated the time-dependent Gross-Pitaevskii equation using the split-step Crank-Nicolson method and the obstacle was modelled by a moving impenetrable Gaussian potential barrier. Various trapping geometries are considered in which the dynamics of the spatial and momentum density, as well as the energy, are considered. The momentum dynamics reveal an oscillatory behavior for the condensate fraction, indicative of excitations out of and de-excitations back into the condensate state. An oscillatory pattern for the energy dynamics reveals the presence of solitons in the system. Some vortex features are also obtained.
Summary form only given. Bose-Einstein condensates (BEC) of weakly interacting gases allow for the new regime of nonlinear atom optics. Of special interest are macroscopically excited states like dark solitons or vortices. Solitons, as particle like excitations of matter wave fields, provide a link from condensate physics to fluid mechanics, nonlinear optics and fundamental particle physics. Dark solitons in matter waves are characterized by a local density minimum and, a steep phase gradient of the wavefunction at the position of the minimum. We report on experimental and theoretical investigation of dark solitons in BECs of /sup 87/Rb which are produced by the method of phase imprinting. A highly anisotropic confining potential allows us to be close to the (quasi) 1D situation where dark solitons are expected to be dynamically stable. By monitoring the evolution of the density profile we observe the evolution of density minima travelling at a smaller velocity than the speed of sound. By comparison to analytical and numerical solutions of the 3D Gross-Pitaevskii equation for our experimental conditions we identify these density minima to be moving dark solitons. We have studied in detail the creation and the dynamics of dark solitons as a function of evolution time and imprinted phase.
We study the mean-field dynamics and the reduced-dimension character of two-mode Bose–Einstein condensates (BECs) in highly anisotropic traps. By means of perturbative techniques, we show that the tightly confined (transverse) degrees of freedom can be decoupled from the dynamical equations at the expense of introducing additional effective three-body, attractive, intra- and inter-mode interactions into the dynamics of the loosely confined (longitudinal) degrees of freedom. These effective interactions are mediated by changes in the transverse wave function. The perturbation theory is valid as long as the nonlinear scattering energy is small compared to the transverse energy scales. This approach leads to reduced-dimension mean-field equations that optimally describe the evolution of a two-mode condensate in general quasi-one-dimensional (1D) and quasi-two-dimensional geometries. We use this model to investigate the relative phase and density dynamics of a two-mode, cigar-shaped 87 Rb BEC. We study the relative-phase dynamics in the context of a nonlinear Ramsey interferometry scheme, which has recently been proposed as a novel platform for high-precision interferometry. Numerical integration of the coupled, time-dependent, three-dimensional, two-mode Gross–Pitaevskii equations for various atom numbers shows that this model gives a considerably more refined analytical account of the mean-field evolution than an idealized quasi-1D description.