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arXiv:1107.0369 · Physical Review A

Self-interfering matter-wave patterns generated by a moving laser obstacle in a two-dimensional Bose-Einstein condensate inside a power trap cut off by box potential boundaries

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

Published Sep 26, 2011• 15 citations
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Abstract

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.

DOI
10.1103/physreva.84.033634

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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.
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.
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.
Matter-wave interference mechanisms in one-dimensional Bose-Einstein condensates that allow for the controlled generation of dark soliton trains upon choosing suitable box-type initial configurations are described. First, the direct scattering problem for the defocusing nonlinear Schr\"odinger equation with nonzero boundary conditions and general box-type initial configurations is discussed, and expressions for the discrete spectrum corresponding to the dark soliton excitations generated by the dynamics are obtained. It is found that the size of the initial box directly affects the number, size and velocity of the solitons, while the initial phase determines the parity of the solutions. The analytical results obtained for the untrapped system are compared to those of numerical simulations of the Gross-Pitaevskii equation, both in the absence and in the presence of a harmonic trap. The numerical results bear out the analytical results with excellent agreement.
Dark solitons dynamically generated from a potential moving in a one-dimensional Bose-Einstein condensate are displayed. Based on numerical simulations of the Gross-Pitaevskii equation, we find that the moving obstacle successively emits a series of solitons which propagate at constant speeds. The dependence of soliton emission on the system parameters is examined. The formation mechanism of solitons is interpreted as interference between a diffusing wavepacket and the condensate background, enhanced by the nonlinear interactions. PACS numbers: 03.75.Mn, 03.75.Lm, 05.30.Jp