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arXiv:cond-mat/0609484 · Physical Review A

Lowest-Landau-level description of a Bose-Einstein condensate in a rapidly rotating anisotropic trap

Alexander L. Fetter

Published Jan 18, 2007• 45 citations
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Abstract

A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional trap can be described with the lowest Landau-level set of states. In this case, the condensate wave function $\ensuremath{\psi}(x,y)$ is a Gaussian function of ${r}^{2}={x}^{2}+{y}^{2}$, multiplied by an analytic function $P(z)$ of the single complex variable $z=x+iy$; the zeros of $P(z)$ denote the positions of the vortices. Here, a similar description is used for a rapidly rotating anisotropic two-dimensional trap with arbitrary anisotropy $({\ensuremath{\omega}}_{x}∕{\ensuremath{\omega}}_{y}\ensuremath{\leqslant}1)$. The corresponding condensate wave function $\ensuremath{\psi}(x,y)$ has the form of a complex anisotropic Gaussian with a phase proportional to $xy$, multiplied by an analytic function $P(\ensuremath{\zeta})$, where $\ensuremath{\zeta}\ensuremath{\propto}x+i{\ensuremath{\beta}}_{\ensuremath{-}}y$ and $0\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. The zeros of $P(\ensuremath{\zeta})$ again fix the locations of the vortices. Within the set of lowest Landau-level states at zero temperature, an anisotropic parabolic density profile provides an absolute minimum for the energy, with the vortex density decreasing slowly and anisotropically away from the trap center.

DOI
10.1103/physreva.75.013620

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A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional trap can be described with the lowest Landau-level set of states. In this case, the condensate wave function \ensuremathψ(x,y)\ensuremath{\psi}(x,y)\ensuremathψ(x,y) is a Gaussian function of r2=x2+y2{r}^{2}={x}^{2}+{y}^{2}r2=x2+y2, multiplied by an analytic function P(z)P(z)P(z) of the single complex variable z=x+iyz=x+iyz=x+iy; the zeros of P(z)P(z)P(z) denote the positions of the vortices. Here, a similar description is used for a rapidly rotating anisotropic two-dimensional trap with arbitrary anisotropy (\ensuremathωx∕\ensuremathωy\ensuremath⩽1)({\ensuremath{\omega}}_{x}∕{\ensuremath{\omega}}_{y}\ensuremath{\leqslant}1)(\ensuremathωx​∕\ensuremathωy​\ensuremath⩽1). The corresponding condensate wave function \ensuremathψ(x,y)\ensuremath{\psi}(x,y)\ensuremathψ(x,y) has the form of a complex anisotropic Gaussian with a phase proportional to xyxyxy, multiplied by an analytic function P(\ensuremathζ)P(\ensuremath{\zeta})P(\ensuremathζ), where \ensuremathζ\ensuremath∝x+i\ensuremathβ\ensuremath−y\ensuremath{\zeta}\ensuremath{\propto}x+i{\ensuremath{\beta}}_{\ensuremath{-}}y\ensuremathζ\ensuremath∝x+i\ensuremathβ\ensuremath−​y and 0\ensuremath⩽\ensuremathβ\ensuremath−\ensuremath⩽10\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}10\ensuremath⩽\ensuremathβ\ensuremath−​\ensuremath⩽1 is a real parameter that depends on the trap anisotropy and the rotation frequency. The zeros of P(\ensuremathζ)P(\ensuremath{\zeta})P(\ensuremathζ) again fix the locations of the vortices. Within the set of lowest Landau-level states at zero temperature, an anisotropic parabolic density profile provides an absolute minimum for the energy, with the vortex density decreasing slowly and anisotropically away from the trap center.
A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional harmonic trap can be described with the lowest Landau-level set of single-particle states. The condensate wave function $\ensuremath{\psi}(x,y)$ is a Gaussian $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{r}^{2}/2)$, multiplied by an analytic function f(z)f(z)f(z) of the complex variable z=x+iyz=x+iyz=x+iy. The criterion for a quantum phase transition to a non-superfluid correlated many-body state is usually expressed in terms of the ratio of the number of particles to the number of vortices. Here a similar description applies to a rapidly rotating nonsymmetric two-dimensional trap with arbitrary quadratic anisotropy (${\ensuremath{\omega}}_{x}^{2}<{\ensuremath{\omega}}_{y}^{2}$). The corresponding condensate wave function $\ensuremath{\psi}(x,y)$ is a complex anisotropic Gaussian with a phase proportional to xy\mathit{xy}xy, multiplied by an analytic function f(z)f(z)f(z), where $z=x+i{\ensuremath{\beta}}_{\ensuremath{-}}y$ is a stretched complex variable and $0\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. Both in the mean-field Thomas-Fermi approximation and in the mean-field lowest Landau level approximation with many visible vortices, an anisotropic parabolic density profile minimizes the energy. An elongated condensate grows along the soft trap direction yet ultimately shrinks along the tight trap direction. The criterion for the quantum phase transition to a correlated state is generalized (111) in terms of N/LzN/{L}_{z}N/Lz​, which suggests that a nonsymmetric trap should make it easier to observe this transition, or (222) in terms of a ``fragmented'' correlated state, which suggests that a nonsymmetric trap should make it harder to observe this transition. An alternative scenario involves a crossover to a quasi one-dimensional condensate without visible vortices, as suggested by Aftalion et al., Phys. Rev. A 79, 011603(R) (2009).
We develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest ppp-wave state of a particle in a harmonic trap with frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.
We found an analytical solution for the vortex structure in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level approximation. This solution is exact in the limit of a large number of vortices and is obtained for the case of a condensate in a anisotropic harmonic potential. The solution describes as limiting cases both a triangle vortex lattice in the symmetric potential trap and a quasi-one-dimensional structure of vortex rows in an asymmetric case, when the rotation frequency is very close to the lower trapping potential frequency. The shape of the density profile is found to be close to the Thomas-Fermi inverted paraboloid form, except in the vicinity of edges of a condensate cloud.
We study the global density profile of a rapidly rotating Bose-Einstein condensate in a harmonic trap with transverse frequency ω⊥ω_{\perp}ω⊥​. By introducing an additional variational degree of freedom to the lowest Landau level wave function, we demonstrate that with increasing strength of the interparticle interaction, the global density profile changes from a Gaussian to the inverted parabolic one characteristic of Thomas-Fermi theory. The criterion for the lowest Landau level wave function to be a good approximation for the global structure is that the mean field energy be small compared with ℏω⊥/Nv\hbar ω_{\perp}/N_{\rm v}ℏω⊥​/Nv​, where NvN_{\rm v}Nv​ is the number of vortices in the cloud. This condition is more stringent than the requirement that the mean field energy be small compared with ℏω⊥\hbar ω_{\perp}ℏω⊥​ which is necessary for the lowest Landau level wave function to be a good approximation to the local structure. Our results show that the lowest Landau level wave function is inappropriate for the global structure of the system realized in recent experiments even though this wave function can describe the local structure well.
In a rapidly rotating Bose-Einstein condensate (BEC) the condensate wave function is analogous to that of a type II superconductor close to the critical magnetic field Hc2{H}_{c2}Hc2​. This wave function is a superposition of the lowest Landau levels (LLL) of a charged particle in a magnetic field and corresponds to a vortex lattice. This leads to the inverted-parabola density profile of the BEC in the parabolic trap similar to slower rotation outside the LLL regime. Then one can apply the continuum theory of the Tkachenko modes (transverse sound modes in the vortex lattice) suggested for slower rotation, but with another shear modulus also known for type II superconductors at Hc2{H}_{c2}Hc2​. The theory yields a simple expression for the Tkachenko eigenfrequencies of the BEC cloud. The comparison with experiment shows that the experimental conditions have not yet reached the LLL regime.
Motivated by recent experiments, we model the dynamics of a condensed Bose gas in a rotating anisotropic trap, where the equations of motion are analogous to those of charged particles in a magnetic field. As the rotation rate is ramped from zero to the trapping frequency, the condensate stretches along one direction and is squeezed along another, becoming long and thin. When the trap anisotropy is slowly switched off on a particular timescale, the condensate is left in the lowest Landau level. We use a time-dependent variational approach to quantify these dynamics and give intuitive arguments about the structure of the condensate wave function. This preparation of a lowest Landau level condensate can be an important first step in realizing bosonic analogs of quantum Hall states.