The Yrast Line of a Rapidly Rotating Bose Gas: Gross-Pitaevskii Regime

x 2 1 + x 2 2 the distance from the axis of rotation, then the critical angular velocity is ω ⊥ . In a reference frame rotating with angular velocity Ω the Hamiltonian for one particle is

with L the component of the angular momentum along the rotation axis. It is convenient and instructive to complete the square and write (1) as

where A = mω ⊥ (x 2 , -x 1 , 0). In the rapidly rotating case, where 0 < ω ⊥ -Ω ≪ min{ω ⊥ , ω z }, it is natural to restrict the allowed wave functions to the ground state space of the first two terms in (2), which we denote by H, and this restriction will be made in this paper. The space H consists of functions in the lowest Landau level (LLL) for motion in the plane perpendicular to the axis of rotation, multiplied by a fixed Gaussian in the x 3 -direction. Apart from the irrelevant additive constants 2 ω ⊥ and ω 3 , the kinetic energy in H is simply ωL (3) * Electronic address: lieb@princeton.edu † Electronic address: rseiring@princeton.edu ‡ Electronic address: jakob.yngvason@univie.ac.at with ω = ω ⊥ -Ω > 0. Note that L is non-negative for functions in H. Its eigenvalues are 0, , 2 , . . . .

To characterize the functions in the space H, it is natural to introduce complex notation, z = x 1 + ix 2 . Functions in H are of the form

with f an analytic function. All the freedom is in f since the Gaussian is fixed. If the trapping potential in the x 3direction is not quadratic, the Gaussian in the x 3 -variable has to be replaced by the appropriate ground state wave function.

A fancy way of saying this is that our Hilbert space H consists of analytic functions on the complex plane C with inner product given by

where dz is short for dx 1 dx 2 . For simplicity we choose units such that m = = ω ⊥ = 1. The eigenfunction of the angular momentum L corresponding to the eigenvalue n is simply z n . In other words, L = z∂ z . We remark that the expectation value of r 2 = |z| 2 in this state is n + 1.

For a system of N bosons, the appropriate wave functions are analytic and symmetric functions of the bosons coordinates z 1 , . . . , z N . The Hilbert space is thus H N = ⊗ N symm H. The kinetic energy is simply ω times the total angular momentum.

In addition to the kinetic energy, there is pairwise interaction among the bosons. It is assumed to be short range compared to any other characteristic length in the system, and can be modeled by g 1≤i<j≤N δ(z i -z j ) for some coupling constant g > 0. Physically, this g is proportional to aω

where a is the scattering length of the three-dimensional interaction potential. On the full, original, Hilbert space ⊗ N symm L 2 (R 2 ), a δ-function as a repulsive interaction potential is meaningless. On the subspace H N the matrix elements of δ(z i -z j ) make perfect sense, however, and define a bounded operator δ ij . Using the analyticity of the wave functions this operator is easily shown to act as

which takes analytic functions into analytic functions. Its matrix elements in a two-particle function ψ(z 1 , z 2 ) are

The dimensional reduction from three to two dimensions for the N -body problem and the restriction to the LLL is, of course, only reasonable if the interaction energy per particle is much less than the energy gap 2 ω ⊥ between Landau levels and the gap ω 3 for the motion in the x 3 -direction. For a dilute gas the interaction energy per particle is of the order aρ where ρ is the average three-dimensional density [1]. Provided N g/ω is not small we can estimate ρ by noting that the effective radius R of the system can be obtained by equating

) ∼ N g/R 2 with the kinetic energy ωL ∼ ωR 2 . This gives R ∼ (N g/ω) 1/4 and the condition for the restriction to the LLL becomes

The physics of rapidly rotating ultracold Bose gases close to the LLL regime has been the subject of many theoretical and experimental investigations in recent years, starting with the papers [2,3,4,5,6]. The recent reviews [7,8,9,10] contain extensive lists of references on this subject. On the experimental side we mention in particular the papers [11,12,13] that report on experiments with rotational frequencies exceeding 99% of the trap frequency.

The discussion in the Introduction leads us to the following well-known model (see, e.g., [2,7,10]) for N bosons with repulsive short-range pairwise interactions:

It acts on analytic and symmetric functions of N variables z i ∈ C. The angular momentum operators are L i = z i ∂ zi , and δ ij acts as in (6). The parameters ω and g are assumed to be positive. The rigorous derivation of this model from the 3D Schrödinger equation for particles interacting with short, but finite range potentials will be presented elsewhere [14].

Note that the two terms in H commute with each other, and hence c

…(Full text truncated)…