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arXiv:cond-mat/0011212 · Physical Review A

Rotating Bose gas with hard-core repulsion in a quasi-two-dimensional harmonic trap: Vortices in Bose-Einstein condensates

M. A. H. Ahsan, Narendra Kumar

Published May 31, 2001• 13 citations
arXivHTMLPDF

Abstract

We consider a gas of $N(<~15)$ Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\ensuremath{\Omega}$ about the z axis. Exact diagonalization of the $n\ifmmode\times\else\texttimes\fi{}n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum ${L}_{z},$ with $n\ensuremath{\sim}{10}^{5}$ (e.g., for ${L}_{z}=N=15, n=240782)$ was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})$ were determined. With the usual identification of $\ensuremath{\Omega}$ as the Lagrange multiplier associated with ${L}_{z}$ for a rotating system, the ${L}_{z}\ensuremath{-}\ensuremath{\Omega}$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.$ All the states in the regime $N>{L}_{z}>0$ are metastable. For ${L}_{z}>N,$ the ${L}_{z}$ values for the stable ground states generally increased with increasing critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\ensuremath{\Omega}}_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the ${L}_{z}=N$ single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $(\ensuremath{\sim}4)$ orders of magnitude in the moderately to the weakly interacting regime.

DOI
10.1103/physreva.64.013608

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Properties of rotating ultracold bosonic quantum gases

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•Dec 1, 2013•0 citations

Vortices and Persistent Currents in Rotating Bose Gases - a Diagonalization Approach

S. Bargi

In this thesis, I explore the behavior of rotating ultra-cold Bose gases, by diagonalizing the Hamiltonian. This method has the advantage of being exact in the limit of weak interactions, and thus is a useful complement to the more commonly used Gross-Pitaevskii (GP) equation, which relies on a mean-field approximation. It is generally known that an ultra-cold Bose gas shows some remarkable properties under rotation, such as quantized vortices and persistent currents, and these are one main reason why we choose to study rotation. Abstract A repulsively interacting Bose-Einstein condensate (BEC) in a harmonic trapping potential will form singly quantized vortices when rotated. Papers I-V are examples of different kinds of generalizations to this basic system, each showing its particular non-trivial new features. Abstract In Paper I we make the trapping potential weakly anharmonic, and confirm the existence of multiply quantized vortices in both repulsively and attractively interacting condensates, which were previously predicted in Gross-Pitaevskii calculations. Abstract In Papers II and III, we generalize to a two-component system in a harmonic trap. The vortices that form under rotation will in general be coreless, i.e. one component will form a vortex and rotate around the other component. In Paper II we give some analytical expressions for the dispersion relation and occupation numbers at low angular momenta, and in Paper III we show how multiply quantized vortices can be stable in these systems. Abstract Finally we investigate bosons in potentials which can support persistent currents, i.e. rotating states that do not decay in a finite amount of time. These are possible if there is an energy barrier separating the rotating state from the non-rotating ground state. In Paper IV, we have a two-component condensate in a one-dimensional potential and we calculate among other things the minimum interaction strength where persistent currents can occur. In Paper V, we study an experimentally relevant two-dimensional annulus. Again we find that persistent currents are possible above a certain interaction strength, and we find this strength as a function of the width of the annulus, as well as how it depends on the relative population of the two species.

•Jan 1, 2010•1 citation
We consider a gas of N(< 15)N(<~15)N(< 15) Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ about the z axis. Exact diagonalization of the n\ifmmode×\else\texttimes\finn\ifmmode\times\else\texttimes\fi{}nn\ifmmode×\else\texttimes\fin many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum Lz,{L}_{z},Lz​, with n\ensuremath∼105n\ensuremath{\sim}{10}^{5}n\ensuremath∼105 (e.g., for Lz=N=15,n=240782){L}_{z}=N=15, n=240782)Lz​=N=15,n=240782) was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix \ensuremathρ(r,r\ensuremath′)=\ensuremath∑\ensuremathμ\ensuremathλ\ensuremathμ\ensuremathχ\ensuremathμ∗(r)\ensuremathχ\ensuremathμ(r\ensuremath′)\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})\ensuremathρ(r,r\ensuremath′)=\ensuremath∑\ensuremathμ​\ensuremathλ\ensuremathμ​\ensuremathχ\ensuremathμ∗​(r)\ensuremathχ\ensuremathμ​(r\ensuremath′) were determined. With the usual identification of \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ as the Lagrange multiplier associated with Lz{L}_{z}Lz​ for a rotating system, the Lz\ensuremath−\ensuremathΩ{L}_{z}\ensuremath{-}\ensuremath{\Omega}Lz​\ensuremath−\ensuremathΩ phase diagram (or the stability line) was determined that gave a number of critical angular velocities \ensuremathΩci,i=1,2,3,…,{\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},\ensuremathΩci​,i=1,2,3,…, at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities \ensuremathΩci,i=1,2,3,….{\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.\ensuremathΩci​,i=1,2,3,…. All the states in the regime N>Lz>0N>{L}_{z}>0N>Lz​>0 are metastable. For Lz>N,{L}_{z}>N,Lz​>N, the Lz{L}_{z}Lz​ values for the stable ground states generally increased with increasing critical angular velocities \ensuremathΩci,{\ensuremath{\Omega}}_{\mathrm{ci}},\ensuremathΩci​, and the condensate was strongly depleted. The critical \ensuremathΩci{\ensuremath{\Omega}}_{\mathrm{ci}}\ensuremathΩci​ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the Lz=N{L}_{z}=NLz​=N single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several (\ensuremath∼4)(\ensuremath{\sim}4)(\ensuremath∼4) orders of magnitude in the moderately to the weakly interacting regime.
Abstract Using exact diagonalization, we investigate the many-body ground state for regular vortex patterns in a rotating Bose-condensed gas of N spinless particles, confined in a quasi-two-dimensional harmonic trap and interacting repulsively via finite-range Gaussian potential. The N -body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum L z , to obtain the lowest-energy eigenstate. Further, the internal structure of these eigenstates is analyzed by calculating the corresponding conditional probability distribution. Specifically, the quantum mechanically stable as well as unstable states in a co-rotating frame are examined in the moderately rotating regime corresponding to angular momenta 4 N ≤ L z < 5 N for N = 16 bosons. In response to externally impressed rotation, the patterns of singly quantized vortices are formed, shaping into canonical polygons with a central vortex at the trap center. The internal structure of unstable states reveals the mechanism of entry, nucleation and pattern formation of vortices with structural phase transition, as the condensate goes from one stable vortical state to the other. The stable polygonal vortex patterns having discrete p -fold rotational symmetry with p = 5 and p = 6 are observed. The hexagonal vortex pattern with p = 6 symmetry is a precursor to the triangular vortex lattice of singly quantized vortices in the thermodynamic limit. For unstable states, quantum melting of vortex patterns due to uncertainty in positions of individual vortices, is also briefly discussed.
We investigate the ground and low excited states of a rotating, weakly interacting Bose-Einstein condensed gas in a harmonic trap for a given angular momentum. Analytical results in various limits as well as numerical results are presented, and these are compared with those of previous studies. Within the mean-field approximation and for repulsive interaction between the atoms, we find that for very low values of the total angular momentum per particle, $L/\stackrel{\ensuremath{\rightarrow}}{N}0,$ where $L\ensuremath{\Elzxh}$ is the angular momentum and N is the total number of particles, the angular momentum is carried by quadrupole (∣m∣=2)(|m|=2)(∣m∣=2) surface modes. For L/N=1,L/N=1,L/N=1, a vortexlike state is formed and all the atoms occupy the m=1m=1m=1 state. For small negative values of $L/N\ensuremath{-}1,$ the states with m=0m=0m=0 and m=2m=2m=2 become populated, and for small positive values of $L/N\ensuremath{-}1,$ atoms in the states with m=5m=5m=5 and m=6m=6m=6 carry the additional angular momentum. In the whole region 0< L/N< 1,0<~L/N<~1,0< L/N< 1, we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of L/N.L/N.L/N. We have also found that an array of singly quantized vortices is formed as L/NL/NL/N increases. Finally, we have gone beyond the mean-field approximation and have calculated the energy of the lowest state up to order N for small negative values of $L/N\ensuremath{-}1,$ as well as the energy of the low-lying excited states.
We present exact diagonalization study on a system of 10≤N≤2410 \leq N \leq 2410≤N≤24 spinless bosons interacting via repulsive Gaussian potential, harmonically confined in xyxyxy-plane with an externally impressed rotation about the zzz-axis. The two-body interaction strength in the Gaussian potential is taken in the strongly interacting regime with values of interaction range in the regime 0≤σ≤10\leq σ\leq 10≤σ≤1. The diagonalization of the NNN-body Hamiltonian matrix, in subspaces of total angular momentum in the regime 0≤Lz≤4N0\le L_{z} \le 4N0≤Lz​≤4N corresponds to the filling fraction ν≲3.2ν\lesssim 3.2ν≲3.2 is carried out to obtain the variationally exact ground-state wavefunction and the corresponding eigenvalue. It is found that an increase in interaction range σσσ leads to (a) a systematic decrease in energy, (b) an increase in the critical angular velocity ΩciΩ_{c_{i}}Ωci​​ of the iii{th} vortex state and (c) an increase in the largest eigenvalue λ1λ_{1}λ1​, (condensate fraction), of one-particle reduced density matrix (OPRDM). The von Neumann entropy S1(Lz,σ)S_{1}\left(L_{z},σ\right)S1​(Lz​,σ), quantifying the quantum entanglement between the particles in the many-body ground state, is largely found to decrease with increase in σσσ. Crossings in von Neumann entropy for several of the angular momentum states are observed with variation in σσσ. The response of the Bose-Einstein condensate to rotation is examined through Lz(σ)−Ω(σ)L_{z}\left(σ\right)-Ω\left(σ\right)Lz​(σ)−Ω(σ) stability graph for several values of σσσ. A vortex state with larger plateau length on the Lz−ΩL_{z}-ΩLz​−Ω graph is considered to be more stable. The internal structure of the condensate, as depicted by the conditional probability distribution (CPD) in the body-fixed frame, exhibits characteristic features with interaction range σσσ. One such feature is the merging of the cores of the two-vortex state.
We consider a strongly interacting Bose-Einstein condensate in a spherical harmonic trap. The system is treated by applying a slave-boson representation for hard-core bosons. A renormalized Gross-Pitaevskii theory is derived for the condensate wave function that describes the dilute regime (like the conventional Gross-Pitaevskii theory) as well as the dense regime. We calculate the condensate density of a rotating condensate for both the vortex-free condensate and the condensate with a single vortex and determine the critical angular velocity for the formation of a stable vortex in a rotating trap.
In this dissertation, we study rotational properties of ultracold bosonic quantum gases in two trapped configurations, a quasi-two-dimensional gas in a harmonic trap and a quasi-one-dimensional gas in a toroidal trap. First, we investigate the effects of correlations on the properties of the ground state of the rotating harmonically-trapped Bose gas by adding Bogoliubov fluctuations to the mean-field ground state of an N-particle single-vortex system. We demonstrate that the fluctuation-induced correlations lower the energy compared to that of the mean-field ground state, that the vortex core is pushed slightly away from the center of the trap, and that an unstable mode with negative energy (for rotations slower than a critical frequency) emerges in the energy spectrum, thus, pointing to a better state for slow rotation. We construct mean-field ground states of 0-, 1-, and 2-vortex states as a function of rotation rate and determine the critical frequencies for transitions between these states, as well as the critical frequency for appearance of a metastable state with an off-center vortex and its image vortex in the evanescent tail of the cloud. Then, we show how the configuration-space form of the above-mentioned Bogoliubov ground-state wave function of a bosonic condensate with a single vortex in a harmonic trap can be described in terms of bosonic Jastrow correlations. We then generalize this result to study the effects of such correlations on a mean-field vortex lattice state and show that the included correlations lower the energy below that of the mean-field state. Although the reduction is relatively small, it is a precursor of the more general expected effect of correlations in describing the melting of the vortex lattice at a high angular momentum per particle. Finally, we study the stability and dynamics of an ultracold bosonic gas trapped in a toroidal geometry and driven by rotation in the absence of dissipation. We first delineate, via the Bogoliubov mode expansion, the regions of stability and the nature of instabilities of the system for both repulsive and attractive interaction strengths. To study the response of the system to variations in the rotation rate, we introduce a “disorder” potential, breaking the rotational symmetry. We demonstrate the breakdown of adiabaticity as the rotation rate is slowly varied and find forced tunneling between the eigenstates of the system. The nonadiabaticity is signaled by the appearance of a swallow-tail loop in the lowest-energy level, a general sign of hysteresis. Then, we show that this system is in one-to-one correspondence with a trapped gas in a double-well potential and thus exhibits macroscopic quantum self-trapping. Finally, we show that self-trapping is a direct manifestation of the behavior of the lowest-energy level.
In this thesis, I explore the behavior of rotating ultra-cold Bose gases, by diagonalizing the Hamiltonian. This method has the advantage of being exact in the limit of weak interactions, and thus is a useful complement to the more commonly used Gross-Pitaevskii (GP) equation, which relies on a mean-field approximation. It is generally known that an ultra-cold Bose gas shows some remarkable properties under rotation, such as quantized vortices and persistent currents, and these are one main reason why we choose to study rotation. Abstract A repulsively interacting Bose-Einstein condensate (BEC) in a harmonic trapping potential will form singly quantized vortices when rotated. Papers I-V are examples of different kinds of generalizations to this basic system, each showing its particular non-trivial new features. Abstract In Paper I we make the trapping potential weakly anharmonic, and confirm the existence of multiply quantized vortices in both repulsively and attractively interacting condensates, which were previously predicted in Gross-Pitaevskii calculations. Abstract In Papers II and III, we generalize to a two-component system in a harmonic trap. The vortices that form under rotation will in general be coreless, i.e. one component will form a vortex and rotate around the other component. In Paper II we give some analytical expressions for the dispersion relation and occupation numbers at low angular momenta, and in Paper III we show how multiply quantized vortices can be stable in these systems. Abstract Finally we investigate bosons in potentials which can support persistent currents, i.e. rotating states that do not decay in a finite amount of time. These are possible if there is an energy barrier separating the rotating state from the non-rotating ground state. In Paper IV, we have a two-component condensate in a one-dimensional potential and we calculate among other things the minimum interaction strength where persistent currents can occur. In Paper V, we study an experimentally relevant two-dimensional annulus. Again we find that persistent currents are possible above a certain interaction strength, and we find this strength as a function of the width of the annulus, as well as how it depends on the relative population of the two species.