We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states are dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally, we present experimental realizations of multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.

Entanglement, a key feature of quantum mechanics, is a resource that allows the improvement of precision measurements beyond the conventional bound attainable by classical means. This results in the standard quantum limit, which is reached in today's best available sensors of various quantities such as time and position. Many of these sensors are interferometers in which the standard quantum limit can be overcome by using quantum-entangled states (in particular spin squeezed states) at the two input ports. Bose-Einstein condensates of ultracold atoms are considered good candidates to provide such states involving a large number of particles. Here we demonstrate spin squeezed states suitable for atomic interferometry by splitting a condensate into a few parts using a lattice potential. Site-resolved detection of the atoms allows the measurement of the atom number difference and relative phase, which are conjugate variables. The observed fluctuations imply entanglement between the particles, a resource that would allow a precision gain of 3.8 dB over the standard quantum limit for interferometric measurements.

We report on the generation, subsequent oscillation and interaction of a pair of matter-wave dark solitons. These are created by releasing a Bose-Einstein condensate from a double well potential into a harmonic trap in the crossover regime between one dimension and three dimensions. Multiple oscillations and collisions of the solitons are observed, in quantitative agreement with simulations of the Gross-Pitaevskii equation. An effective particle picture is developed and confirms that the deviation of the observed oscillation frequencies from the asymptotic prediction nu(z)/sqrt 2, where nu(z) is the longitudinal trapping frequency, results from the dimensionality of the system and the soliton interactions.

We present some experimental evidences of beyond mean field effects in an ultra-cold trapped Bose gas using two different trapping geometries: • We have studied the behavior of small samples of atoms (3000) in a very elongated trap. We observe the apparition of a quasi-condensate in the central region of the trap at low enough temperature. We show that a standard mean-field Hartree-Fock theory can not explain this phenomenon. We also show some measurements of the density fluctuations in the trap confirming the invalidity of a mean field approach 1 . • We have studied the behavior of two weakly linked condensates in a double-well trap 2 . When the tunneling energy is smaller than the interaction strength (Fock regime), a mean field description of the two condensates is not valid anymore. Studying the dynamic of the phase variance between the condensates, we have obtained some experimental evidences for beyond mean field behaviors. We analyze our results in the framework of the two mode approximation.

We discuss in detail the experimental investigation of thermally induced fluctuations of the relative phase between two weakly coupled Bose–Einstein condensates (BECs). In analogy to superconducting Josephson junctions, the weak coupling originates from a tunnelling process through a potential barrier which is obtained by trapping the condensates in an optical double-well potential. The observed fluctuations of the relative phase are in quantitative agreement with a many body two mode model at finite temperature. The agreement demonstrates the possibility of using the phase fluctuation measurements in a bosonic Josephson junction (BJJ) as a primary thermometer. This new method allows for measuring temperatures far below the critical temperature where standard methods based on time of flight measurements fail. We employ this new thermometer to probe the heat capacity of a degenerate Bose gas as a function of temperature.