The uncertainty product of an out-of-equilibrium many-particle system

In the present work we show, analytically and numerically, that the variance of many-particle operators and their uncertainty product for an out-of-equilibrium Bose-Einstein condensate (BEC) can deviate from the outcome of the time-dependent Gross-Pitaevskii dynamics, even in the limit of infinite number of particles and at constant interaction parameter when the system becomes 100% condensed. We demonstrate our finding on the dynamics of the center-of-mass position--momentum uncertainty product of a freely expanding as well as of a trapped BEC. This time-dependent many-body phenomenon is explained by the existence of time-dependent correlations which manifest themselves in the system's reduced two-body density matrix used to evaluate the uncertainty product. Our work demonstrates that one has to use a many-body propagation theory to describe an out-of-equilibrium BEC, even in the infinite particle limit.Comment: 26 pages, 5 figure

Zoo of quantum phases and excitations of cold bosonic atoms in optical lattices

Quantum phases and phase transitions of weakly- to strongly-interacting bosonic atoms in deep to shallow optical lattices are described by a {\it single multi-orbital mean-field approach in real space}. For weakly-interacting bosons in 1D, the critical value of the superfluid to Mott insulator (MI) transition found is in excellent agreement with {\it many-body} treatments of the Bose-Hubbard model. For strongly-interacting bosons, (i) additional MI phases appear, for which two (or more) atoms residing in {\it each site} undergo a Tonks-Girardeau-like transition and localize and (ii) on-site excitation becomes the excitation lowest in energy. Experimental implications are discussed.Comment: 12 pages, 3 figure

Accurate multi-boson long-time dynamics in triple-well periodic traps

To solve the many-boson Schr\"odinger equation we utilize the Multiconfigurational time-dependent Hartree method for bosons (MCTDHB). To be able to attack larger systems and/or to propagate the solution for longer times, we implement a parallel version of the MCTDHB method thereby realizing the recently proposed [Streltsov {\it et al.} arXiv:0910.2577v1] novel idea how to construct efficiently the result of the action of the Hamiltonian on a bosonic state vector. We study the real-space dynamics of repulsive bosonic systems made of N=12, 51 and 3003 bosons in triple-well periodic potentials. The ground state of this system is three-fold fragmented. By suddenly strongly distorting the trap potential, the system performs complex many-body quantum dynamics. At long times it reveals a tendency to an oscillatory behavior around a threefold fragmented state. These oscillations are strongly suppressed and damped by quantum depletions. In spite of the richness of the observed dynamics, the three time-adaptive orbitals of MCTDHB(M=3) are capable to describe the many-boson quantum dynamics of the system for short and intermediate times. For longer times, however, more self-consistent time-adaptive orbitals are needed to correctly describe the non-equilibrium many-body physics. The convergence of the MCTDHB($M$) method with the number $M$ of self-consistent time-dependent orbitals used is demonstrated.Comment: 37 pages, 7 figure

Time-dependent multi-orbital mean-field for fragmented Bose-Einstein condensates

The evolution of Bose-Einstein condensates is usually described by the famous time-dependent Gross-Pitaevskii equation, which assumes all bosons to reside in a single time-dependent orbital. In the present work we address the evolution of fragmented condensates, for which two (or more) orbitals are occupied, and derive a corresponding time-dependent multi-orbital mean-field theory. We call our theory TDMF($n$), where $n$ stands for the number of evolving fragments. Working equations for a general two-body interaction between the bosons are explicitly presented along with an illustrative numerical example.Comment: 16 pages, 1 figur

Formation of dynamical Schr\"odinger cats in low-dimensional ultracold attractive Bose gases

Dynamical Schr\"odinger cats can be formed when a one-dimensional attractive Bose-gas cloud is scattered off a potential barrier. Once formed, these objects are stable in time. The phenomenon and its mechanism -- transformation of kinetic energy to internal energy of the scattered atomic cloud -- are obtained by solving the time-dependent many-boson Schr\"odinger equation. Implications are discussed.Comment: 11 pages, 3 figure

Build-up of coherence between initially-independent subsystems: The case of Bose-Einstein condensates

When initially-independent subsystems are made to contact, {\it coherence} can develop due to interaction between them. We exemplify and demonstrate this paradigm through several scenarios of two initially-independent Bose-Einstein condensates which are allowed to collide. The build-up of coherence depends strongly on time, interaction strength and other parameters of each condensate. Implications are discussed.Comment: 11 pages, 3 figure

Uncertainty product of an out-of-equilibrium Bose-Einstein condensate

The variance and uncertainty product of the position and momentum many-particle operators of structureless bosons interacting by a long-range inter-particle interaction and trapped in a single-well potential are investigated. In the first example, of an out-of-equilibrium interaction-quench scenario, it is found that, despite the system being fully condensed, already when a fraction of a particle is depleted differences with respect to the mean-field quantities emerge. In the second example, of the pathway from condensation to fragmentation of the ground state, we find out that, although the cloud's density broadens while the system's fragments, the position variance actually decreases, the momentum variance increases, and the uncertainty product is not a monotonous function but has a maximum. Implication are briefly discussed.Comment: 14 pages, 3 figure