In this work a two-particle irreducible (2PI) closed-time-path (CTP) effective action is used to describe the nonequilibrium dynamics of a Bose-Einstein condensate selectively loaded into every third site of a one-dimensional optical lattice. The motivation of this work is the recent experimental realization of this system. Patterned loading methods may be useful for quantum computing with trapped atoms. This system also serves to illustrate many basic issues in nonequilibrium quantum-field theory pertaining to the dynamics of quantum correlations and fluctuations which goes beyond the capability of a mean-field theory. By numerically evolving in time the initial-state configuration using the Bose-Hubbard Hamiltonian an exact quantum solution is available for this system in the case of few atoms and wells. One can also use it to test various approximate methods. Under the 2PI CTP scheme with this initial configuration, three different approximations are considered: (a) the Hartree-Fock-Bogoliubov (HFB) approximation, (b) the next-to-leading-order $1∕\mathcal{N}$ expansion of the 2PI effective action up to second order in the interaction strength, and (c) a second-order perturbative expansion in the interaction strength. We present detailed comparisons between these approximations and determine their range of validity by contrasting them with the exact many-body solution for a moderate number of atoms and wells. As a general feature we observe that because the second-order 2PI approximations include multiparticle scattering in a systematic way, they are able to capture damping effects exhibited in the exact solution, which a mean-field collisionless approach fails to produce. While the second-order approximations show a clear improvement over the HFB approximation, our numerical results show that they fail at late times, when interaction effects are significant.In this work a two-particle irreducible (2PI) closed-time-path (CTP) effective action is used to describe the nonequilibrium dynamics of a Bose-Einstein condensate selectively loaded into every third site of a one-dimensional optical lattice. The motivation of this work is the recent experimental realization of this system. Patterned loading methods may be useful for quantum computing with trapped atoms. This system also serves to illustrate many basic issues in nonequilibrium quantum-field theory pertaining to the dynamics of quantum correlations and fluctuations which goes beyond the capability of a mean-field theory. By numerically evolving in time the initial-state configuration using the Bose-Hubbard Hamiltonian an exact quantum solution is available for this system in the case of few atoms and wells. One can also use it to test various approximate methods. Under the 2PI CTP scheme with this initial configuration, three different approximations are considered: (a) the Hartree-Fock-Bogoliubov (HFB) approximation, (b) the next-to-leading-order $1∕\mathcal{N}$ expansion of the 2PI effective action up to second order in the interaction strength, and (c) a second-order perturbative expansion in the interaction strength. We present detailed comparisons between these approximations and determine their range of validity by contrasting them with the exact many-body solution for a moderate number of atoms and wells. As a general feature we observe that because the second-order 2PI approximations include multiparticle scattering in a systematic way, they are able to capture damping effects exhibited in the exact solution, which a mean-field collisionless approach fails to produce. While the second-order approximations show a clear improvement over the HFB approximation, our numerical results show that they fail at late times, when interaction effects are significant.

arXiv:cond-mat/0308305Physical Review AMar 22, 2004100 citations

BEC in a shaken optical lattice

E. Arimondo, Donatella Ciampini, Hans Lignier +2 more

Bulletin of the American Physical SocietyMay 22, 20090 citations

The Hartree-Fock-Bogoliubov Theory of Bose-Einstein Condensates

B. G. Wild

In this thesis we develop an orthogonalised Hartree-Fock-Bogoliubov (HFB) formalism that has a zero-energy excitation (in contrast with standard HFB). We demonstrate that this formalism satisfies the number and linear/angular momentum conservation laws (as does standard HFB). This formalism is applied to vortices in 2D Bose-Einstein condensates (BECs) in axially-symmetric harmonic traps, where we initially find solutions for on-axis vortices, determining the energy spectrum and hence the lowest core localised state (LCLS) energy. In the T=0 case we identify this with the anomalous mode which gives a zero excitation energy in the frame rotating at this frequency. For this reason the anomalous mode frequency was identified in the earlier literature with the precessional frequency for an off-axis vortex. However the LCLS energy is positive in the finite temperature case. Hence, associating this LCLS energy with the precessional frequency leads to the erroneous conclusion that the vortex precesses in the opposite direction to the T=0 case, which is clearly physically unreasonable. In order to address this problem, we derive an equation for the prediction of vortex precessional frequencies from the continuity equation, and use this equation solved self-consisently with the orthogonal HFB equations in the frame rotating at this predicted frequency to create off-axis vortices at pre-specified positions. Hence we are able to predict the precessional frequencies and show that these are consistent with the T=0 case, and are entirely unrelated to the LCLS energy. We also consider a two-state model and demonstrate that this model is insufficient for the description of single off-axis precessing vortices. We formulate a generalised multi-state model, using the normalisation conditions for the model to derive an equation predicting the precessional frequency of the vortex, and demonstrate equivalence with the continuity equation prediction. We use the time-dependent HFB equations to simulate creation of vortices by stirring the BEC by means of a Gaussian optical potential, finding very good agreement of the measured precessional frequencies of the vortices in stirred BECs with the predicted values. We find the existence of a critical stirring frequency for the creation of vortices in regions of appreciable superfluid density, in qualitative agreement with experiment. We then investigate the consequences of breaking rotational symmetry and find that breaking the axial symmetry of the harmonic trapping potential leads to loss of angular momentum, and hence to the decay of vortices. Finally we develop a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism in the Popov approximation to study the effect of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one-dimensional optical lattice.In this thesis we develop an orthogonalised Hartree-Fock-Bogoliubov (HFB) formalism that has a zero-energy excitation (in contrast with standard HFB). We demonstrate that this formalism satisfies the number and linear/angular momentum conservation laws (as does standard HFB). This formalism is applied to vortices in 2D Bose-Einstein condensates (BECs) in axially-symmetric harmonic traps, where we initially find solutions for on-axis vortices, determining the energy spectrum and hence the lowest core localised state (LCLS) energy. In the T=0 case we identify this with the anomalous mode which gives a zero excitation energy in the frame rotating at this frequency. For this reason the anomalous mode frequency was identified in the earlier literature with the precessional frequency for an off-axis vortex. However the LCLS energy is positive in the finite temperature case. Hence, associating this LCLS energy with the precessional frequency leads to the erroneous conclusion that the vortex precesses in the opposite direction to the T=0 case, which is clearly physically unreasonable. In order to address this problem, we derive an equation for the prediction of vortex precessional frequencies from the continuity equation, and use this equation solved self-consisently with the orthogonal HFB equations in the frame rotating at this predicted frequency to create off-axis vortices at pre-specified positions. Hence we are able to predict the precessional frequencies and show that these are consistent with the T=0 case, and are entirely unrelated to the LCLS energy. We also consider a two-state model and demonstrate that this model is insufficient for the description of single off-axis precessing vortices. We formulate a generalised multi-state model, using the normalisation conditions for the model to derive an equation predicting the precessional frequency of the vortex, and demonstrate equivalence with the continuity equation prediction. We use the time-dependent HFB equations to simulate creation of vortices by stirring the BEC by means of a Gaussian optical potential, finding very good agreement of the measured precessional frequencies of the vortices in stirred BECs with the predicted values. We find the existence of a critical stirring frequency for the creation of vortices in regions of appreciable superfluid density, in qualitative agreement with experiment. We then investigate the consequences of breaking rotational symmetry and find that breaking the axial symmetry of the harmonic trapping potential leads to loss of angular momentum, and hence to the decay of vortices. Finally we develop a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism in the Popov approximation to study the effect of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one-dimensional optical lattice.

Jan 1, 20112 citations

1∕\mathcal{N}

expansion of the 2PI effective action up to second order in the interaction strength, and (c) a second-order perturbative expansion in the interaction strength. We present detailed comparisons between these approximations and determine their range of validity by contrasting them with the exact many-body solution for a moderate number of atoms and wells. As a general feature we observe that because the second-order 2PI approximations include multiparticle scattering in a systematic way, they are able to capture damping effects exhibited in the exact solution, which a mean-field collisionless approach fails to produce. While the second-order approximations show a clear improvement over the HFB approximation, our numerical results show that they fail at late times, when interaction effects are significant.