Gross-Pitaevskii Solution Research Articles

Dipole modes of a trapped bosonic mixture: Fate of the sum-rule approach

We present a general discussion of the dipole modes of a heteronuclear bosonic mixture in a harmonic trap by comparing the prediction of the sum-rule approach with full Gross-Pitaevskii (GP) calculations. Yet in the range of interaction in which the mixture is miscible and stable at the mean field level, $g_{12}^2<g_{11}g_{22}$, we find that there are regimes where the sum rules displays significant deviations from the GP solution. We provide an interpretation for this behavior in terms of the Ehrenfest approach, which we show to be equivalent to the sum-rule approach for dipole excitations.

Breakup of rotating asymmetric quartic-quadratic trapped condensates

The threshold conditions for a rotating pancake-like asymmetric quartic-quadratic confined condensate to break in two localized fragments, as well as to produce giant vortex at the center within the vortex-pattern distributions, are investigated within the Thomas-Fermi (TF) approximation and exact numerical solution of the corresponding Gross-Pitaevskii (GP) formalism. By comparing the TF predictions with exact GP solutions, in our investigation with two different quartic-quadratic trap geometries, of particular relevance is to observe that the TF approach is not only very useful to display the averaged density distribution, but also quite realistic in establishing the critical rotational conditions for the breakup occurrence and possible giant-vortex formation. It provides almost exact results to define the contour of the condensate distribution, even for high rotating system, after the system split in two (still confined) clouds. The applicability of the Feynman rule to the vortex distribution (full-numerical GP solutions) is also being confirmed for these non-homogeneous asymmetric trap configurations. This study is expected to be relevant for manipulating the rotation and trap parameters in addition to Feshbach resonance techniques. It can also be helpful to define initial conditions for any further studies on dynamical evolution of vortex pattern distributions.

The derivation of the $$\mathbb {T}^{3}$$ T 3 energy-critical NLS from quantum many-body dynamics

We derive the 3D energy critical quintic NLS from quantum many-body dynamics with 3-body interaction in the T^3 (periodic) setting. Due to the known complexity of the energy critical setting, previous progress was limited in comparison to the 2-body interaction case yielding energy subcritical cubic NLS. Previously, the only result for the 3D energy critical case was HTX, which proved the uniqueness part of the argument in the case of small solutions. In the main part of this paper, we develop methods to prove the convergence of the BBGKY hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and separately, the uniqueness of large GP solutions. Since the trace estimate used in the previous proofs of convergence is the false sharp trace estimate in our setting, we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de Finetti theorem. For the large solution uniqueness argument, we discover the new HUFL (hierarchical uniform frequency localization) property for the GP hierarchy and use it to prove a new type of uniqueness theorem. The HUFL property reduces to a new statement even for NLS. With the help of CKSTT,IP which proved the global well-posedness for the quintic NLS, this new uniqueness theorem establishes global uniqueness.

Phase coherence and fragmentation in weakly interacting bosonic gases

We present a theory of measurement-induced interference for weakly interacting Bose-Einstein-condensed gases. The many-body state resulting from the evolution of an initial fragmented (Fock) state can be approximated as a continuous superposition of Gross-Pitaevskii (GP) states; the measurement breaks the initial phase symmetry, producing a distribution pattern corresponding to only one of the GP solutions. We discuss also analytically solvable models, such as two-mode on-chip adiabatic recombination and soliton generation in quasi-one-dimensional condensates.

Bosonic Molecules in Rotating Traps

We present a variational many-body wave function for repelling bosons in rotating traps, focusing on rotational frequencies that do not lead to restriction to the lowest Landau level. This wave function incorporates correlations beyond the Gross-Pitaevskii (GP) mean-field approximation, and it describes rotating boson molecules (RBMs) made of localized bosons that form polygonal-ring-like crystalline patterns in their intrinsic frame of reference. The RBMs exhibit characteristic periodic dependencies of the ground-state angular momenta on the number of bosons in the polygonal rings. For small numbers of neutral bosons, the RBM ground-state energies are found to be always lower than those of the corresponding GP solutions, in particular, in the regime of GP vortex formation.

Electron and boson clusters in confined geometries: symmetry breaking in quantum dots and harmonic traps.

We discuss the formation of crystalline electron clusters in semiconductor quantum dots and of crystalline patterns of neutral bosons in harmonic traps. In a first example, we use calculations for two electrons in an elliptic quantum dot to show that the electrons can localize and form a molecular dimer. The calculated singlet-triplet splitting (J) as a function of the magnetic field (B) agrees with cotunneling measurements with its behavior reflecting the effective dissociation of the dimer for large B. Knowledge of the dot shape and of J(B) allows determination of the degree of entanglement. In a second example, we study strongly repelling neutral bosons in two-dimensional harmonic traps. Going beyond the Gross-Pitaevskii (GP) mean-field approximation, we show that bosons can localize and form polygonal-ring-like crystalline patterns. The total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate.

Exact solution of two bosons in a trap potential: Transition to fragmentation

Fragmentation of a two-boson system in an external potential with contact interaction i.e., -function interaction is studied using a full variational treatment. The results are compared to those obtained by the standard mean-field method Gross-Pitaevskii equation and the more recently developed best mean-field method. The numerically exact calculations show a transition to fragmentation as a function of the bosonboson interaction strength. The Gross-Pitaevskii solution cannot describe fragmentation. The respective energy exhibits a bifurcation which can be viewed as a hint that fragmentation may take place in reality. The best mean-field approach, on the other hand, can predict the fragmentation of the system and give the correct physical description of the fragmented state. The Gross-Pitaevskii energy diverges as the boson-boson interaction goes to infinity, while the exact and the best mean-field energies saturate.

Crystalline Boson Phases in Harmonic Traps: Beyond the Gross-Pitaevskii Mean Field

Strongly-interacting bosons in two-dimensional harmonic traps are described through breaking of rotational symmetry at the Hartree-Fock level and subsequent symmetry restoration via projection techniques, thus incorporating correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form polygonal-ringlike crystalline patterns, both for a repulsive contact potential and a Coulomb interaction, as revealed via conditional-probability-distribution analysis. For neutral bosons, the total energy of the crystalline phase saturates in contrast to the GP solution, and its spatial extent becomes smaller than that of the GP condensate. For charged bosons, the total energy and dimensions approach the values of classical pointlike charges in their equilibrium configuration.