We measure the power spectrum of the density distribution of a freely expanding two-dimensional (2D) degenerate Bose gas, where irregular density modulations gradually develop due to initial phase fluctuations in the sample. The spectrum has an oscillatory shape, where the peak positions are found to be independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the normalized spectral peak strength and observed to decrease at lower temperatures, confirming the thermal nature of the phase fluctuations. We investigate the relaxation dynamics of nonequilibrium states using the power spectrum. Free vortices are observed with ring-shaped density ripples in a perturbed sample after a long relaxation time.
We measure the power spectrum of the density distribution of a freely expanding two-dimensional (2D) degenerate Bose gas, where irregular density modulations gradually develop due to initial phase fluctuations in the sample. The spectrum has an oscillatory shape, where the peak positions are found to be independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the normalized spectral peak strength and observed to decrease at lower temperatures, confirming the thermal nature of the phase fluctuations. We investigate the relaxation dynamics of nonequilibrium states using the power spectrum. Free vortices are observed with ring-shaped density ripples in a perturbed sample after a long relaxation time.
We measure the power spectrum of density fluctuations emerging in a freely expanding quasi-two-dimensional (2D) degenerate Bose gas, and we investigate the scaling behavior of the spectrum for the expansion time. The power spectrum shows an oscillatory shape for long expansion times, where the spectral peak positions are observed to be shifted to lower spatial frequencies than the theoretical prediction for a noninteracting expansion case. We find the spectral peak positions in good agreement with the recent numerical simulation presented by Mazets [Phys. Rev. A 86, 055603 (2012)], where the atom-atom interactions are taken into account. We present a mean-field description of the interaction effect in the expansion dynamics and quantitatively account for the observed spectral peak shifts. The spectral shift is intrinsic to the free expansion of a quasi-2D Bose gas due to finite axial confinement. Finally, we investigate the defocusing effect in the power spectrum measurement.
Quantum gases are well isolated, highly controllable, and defect-free systems which can simulate many body quantum phenomena that have been studied in condensed matter systems.To study two-dimensional superfluid, we have developed an experimental apparatus that can produce Bose-Einstein condensates (BEC) of 23 Na atoms.The apparatus can generate a pure condensate of 10 7 atoms in an optically plugged magnetic quadrupole trap within 17 s.The Berezinskii-Kosterlitz-Thouless (BKT) theory provides a general framework of the superfluid phase transition in two dimension, which does not involve spontaneous symmetry breaking and emergence of an order parameter below a critical temperature.Instead, it is the formation of vortex pairs with opposite circulations below the critical temperature that mediates the superfluid phase transition.Recently, it has been demonstrated that degenerate Bose gases confined in highly oblate harmonic potentials undergo the BKT phase transition.This thesis focuses on our experimental research on thermal phase fluctuations (i.e., long-wavelength phonons and vortex pairs) in the superfluid i phase of trapped quasi-2D BECs.We have developed a quantitative probe to measure phase fluctuations in the 2D superfluid by free expansion, where the phase fluctuations in the sample were revealed as density modulations in the course of expansion.The power spectrum of the density fluctuations showed an oscillatory shape and the scaling behavior of the peak positions could be understood as a Talbot effect in matter waves.Employing this method, we demonstrated the thermal origin of phase fluctuations.We also investigated relaxation dynamics of nonequilibrium states of the quasi-2D system using the power spectrum.Thermally excited vortex pairs are the characteristic feature of the 2D superfluid.The quantized vortices are conventionally observed by a densitydepleted core after expanding a trapped sample.The method, however, cannot be applied to the 2D sample because the density modulations after free expansion lower the vortex core visibility.We enhanced the core visibility by radial compression of the sample before the expansion so the phonon modes in the 2D sample were relaxed in a 3D environment.Measuring vortex distributions, we revealed the pairing feature by spatial correlations of the vortex positions.We also studied BKT-BEC crossover phenomena in a finite-size sample trapped in a quasi-2D harmonic potential by investigating the vortex profiles at various temperatures.Condensates of atoms have an internal spin structure so they can host various kinds of topological excitations.The two-dimensional Skyrmion is one of the topological spin textures in the anti-ferromagnetic spinor condensate and ii we imprinted the structure using the magnetic field sweep technique.The 2D Skyrmion spin texture has a finite Berry curvature because of the non-coplanar spin configuration.We studied a geometric Hall effect, with condensates trapped in a harmonic potential with the Skyrmion spin texture.Under a linear driving of the spin texture, we observed a condensate dipole motion resonantly developed into a circular motion, which demonstrates the existence of an effective Lorentz force.
We investigate theoretically the evolution of the two-point density correlation function of a low-dimensional ultracold Bose gas after release from a tight transverse confinement. In the course of expansion thermal and quantum fluctuations present in the trapped systems transform into density fluctuations. For the case of free ballistic expansion relevant to current experiments, we present simple analytical relations between the spectrum of ``density ripples'' and the correlation functions of the original confined systems. We analyze several physical regimes, including weakly and strongly interacting one-dimensional (1D) Bose gases and two-dimensional (2D) Bose gases below the Berezinskii-Kosterlitz-Thouless (BKT) transition. For weakly interacting 1D Bose gases, we obtain an explicit analytical expression for the spectrum of density ripples which can be used for thermometry. For 2D Bose gases below the BKT transition, we show that for sufficiently long expansion times the spectrum of the density ripples has a self-similar shape controlled only by the exponent of the first-order correlation function. This exponent can be extracted by analyzing the evolution of the spectrum of density ripples as a function of the expansion time.
The physical properties of a homogeneous system at the thermodynamic equilibrium are strongly constrained by its dimensionality. In this respect, the two dimensional Bose gas constitutes a particularly interesting system: while it is impossible to observe a long range order at non-zero temperatures, there exists nevertheless a phase transition to a superfluid state at low temperatures. Furthermore, owing to the reduced dimensionality, the equation of state of the weakly interacting two-dimensional Bose gas is scale invariant. In this thesis, we present an experimental study of the two-dimensional Bose gas. We measure its equation of state with two different methods, and find a good agreement with analytic and numerical predictions. These results confirm as well the scale invariance of the system. Furthermore, one of the methods allows for a determination of the equation of state with a single adjustable parameter. We then characterize the superfluid response of the system., by showing evidence for a dissipationless response of the system to a moving perturbation. Finally, we analyze the fluctuations dynamics of the 2D Bose gas, which confirms both the suppression of density fluctuations in the superfluid phase, and the dominating contribution of phonons to the phase fluctuations.
Motivated by the MIT experiment [Gorlitz {\em et al.}, Phys. Rev. Lett. {\bf 87}, 130402 (2001)], we analytically study the effect of density and phase fluctuations on various observables in a quasi one-dimensional degenerate Bose gases. Quantizing the Gross-Pitaevskii Hamiltonian and diagonalize it in terms of the normal modes associated with the density and phase fluctuations of a quasi-one dimensional Bose gas. We calculate dynamic structure factor S(q,ω) from low-energy condensate density fluctuations and find that there are multiple peaks in S(q,ω) for a given momentum q due to the discrete energy spectrum. These multiple peaks can be resolved by a two-photon Bragg pulse with a long duration which transfer the momentum to the system. We calculate the momentum transferred $ P_z(t)$ by using the phase-density representation of the Bose order parameter. We also calculate the single-particle density matrix, phase coherence length, and momentum distribution by taking care of the phase fluctuations upto fourth-order term as well as the density fluctuations. Our studies on coherence properties shows that 1D Bose gases of MIT experiment do not form a true condensate, but it can be obtained by a moderate changes of the current experimental parameters.
Expansion of a degenerate Bose gas released from a pancakelike trap is numerically simulated under the assumption of separation of the motion in the plane of the loose initial trapping and the motion in the direction of the initial tight trapping. The initial conditions for the phase fluctuations are generated using the extension to the two-dimensional case of the description of the phase noise by the Ornstein-Uhlenbeck stochastic process. The numerical simulations, taking into account both the finite size of the two-dimensional system and the atomic interactions, which cannot be neglected on the early stage of expansion, did not reproduce the scaling law for the peaks in the density fluctuation spectra experimentally observed by Choi, Seo, Kwon, and Shin [Phys. Rev. Lett. 109, 125301 (2012)]. The latter experimental results may thus require an explanation beyond our current assumptions.