We study one- and two-dimensional systems of two interacting particles in a time dependent harmonic potential. In a case of one-dimensional geometry a frequency of the potential varies periodically, while in the two-dimen- sional~case the harmonic potential rotates with a constant angular velocity. We show that depending on the driving frequency the distance between the particles can either explode or stay bound. Repulsive interaction can prevent the explosion, which seems quite counter-intuitive. Our work is related to Ecole Normale Supérieure experiment and shows that the effect found there is purely classical.
We review a progress in understanding of statistical properties of a quantum degenerate Bose gas. We show how the Maxwell demon ensemble helps to compute fluctuations of the Bose-Einstein condensate of an ideal Bose gas according to the microcanonical ensemble. Then, we review a method of measuring these fluctuations. Using a soluble model of interacting Bose gas we also stress the importance of higher-order correlation functions. Finally, we review our novel computational method of studying an interacting Bose gas near its critical temperature.
We discuss our numerical studies of the low energy excitations of trapped Bose condensates using a Bogoliubov-Hartree treatment. In the zero temperature limit, the lowest few excitation frequencies calculated within the Bogoliubov approximation agree well with the experimental data. Finite temperature results obtained using the Popov approximation display qualitative differences from the experimental data close to the critical temperature region. Details of our numerical approach are presented and comparison with other results is discussed.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for a detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactly soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives a quite accurate description of the system in the limit of weak interactions.
A Lagrangian based method is used to derive an analytical model for studying the dynamics of matter-wave bright soliton created in a harmonic potential which is attractive in the transverse direction and expulsive in the longitudinal direction. By means of sech trial functions and a Ritz optimization procedure, evolution equations are constructed for width, amplitude and nonlinear frequency chirp of the propagating soliton of the atomic condensate. Our equation for the width is an exact agreement with that of Carr and Castin, obtained by more detailed analysis. In agreement with the experiment of Paris group, the expulsive potential is found to cause the soliton to explode for N|a_s|=0.98, N being the number of atoms in the condensate and a_s -- the scattering length of the atom-atom interaction.
We consider the superfluid phase transition that arises when a Feshbach resonance pairing occurs in a dilute Fermi gas. This is related to the phenomenon of superconductivity described by the seminal Bardeen-Cooper-Schrieffer theory. In superconductivity, the phase transition is caused by a coupling between pairs of electrons within the medium. This coupling is perturbative and leads to a critical temperature T_c which is small compared to the Fermi temperature T_F. Even high-T_c superconductors typically have a critical temperature which is two orders of magnitude below T_F. Here we describe a resonance pairing mechanism in a quantum degenerate gas of potassium (^{40} K) atoms which leads to superfluidity in a novel regime - a regime that promises the unique opportunity to experimentally study the cross-over from the Bardeen-Cooper-Schrieffer phase of weakly-coupled fermions to the Bose-Einstein condensate of strongly-bound composite bosons. We find that the transition to a superfluid phase is possible at the high critical temperature of about 0.5 T_F. It should be straightforward to verify this prediction, since these temperatures can already be achieved experimentally.
As research in quantum optics has advanced, so too has our ability to precisely tailor the quantum state of a system. Indeed, techniques for quantum state preparation have become sufficiently advanced that an entire subfield has appeared which has been given the name "quantum control". Parallel to these advances have been other striking developments in quantum optics, in particular, laser cooling and trapping of neutral atoms. In this paper we describe some of the recent advancements in laser cooling, particularly in our laboratories, and point out that laser cooling and trapping is also realizing an important form of quantum control. In laser cooling, instead of exercising control over the internal quantum state of an atom or molecule or a laser field, we are instead controlling a complementary set of degrees of freedom: those of the external coordinates of the atom.
A simple picture describes the results of recent treatments of partially-condensed, dilute, trapped Bose gases at temperature T>0. The condensate wave function is nearly identical to that of a T=0 condensate with the same number of condensate atoms, N_{0}. The cloud of non-condensed atoms is described by the statistical mechanics of an ideal Bose gas in the combined potentials of the magnetic trap and the cloud-condensate interaction. We provide a physical motivation for this result, show how it emerges in the Hartree-Fock-Bogoliubov-Popov approximation, and explore some of its implications for future experiments.
This paper presents fundamental principles, characteristics, and limitations of various experimental methods of cooling and trapping of neutral atoms by laser light and magnetic fields. In addition to surveying the experimental techniques, basic properties of quantum degenerate gases are discussed with particular emphasis on the Bose-Einstein condensate. We also present main parameters and expected characteristics of the first Polish Bose-Einstein condensate apparatus built in the National Laboratory of Atomic, Molecular, and Optical Physics in Toruń, Poland.
We study an ultracold dilute gas of bosonic atoms in an optical lattice induced by intersecting laser beams. As a first approximation we neglect confining potential and atom-atom interactions. In this case the Gross-Pitaevskij equation reduces to simple Mathieu equation. Upon choosing periodic boundary conditions this problem has well known periodic solution. This simple picture allows to demonstrate localization of the wave packet and formation of the band structure. We calculate spectrum of the excited states as a function of the strength of modulating potential and using a standard adiabaticity criterion we predict the most efficient way to ramp up optical lattice, without higher state excitation. Finally, we discuss the influence of the atom-atom interaction (nonlinearity) on the adiabaticity of the process.
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