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Low-energy elementary excitations of a trapped Bose-condensed gas

Bogolyubov-De Gennes equations for the excitations of a Bose condensate in the Thomas-Fermi regime in harmonic traps of any asymmetry and introduce a classification of eigenstates. In the case of cylindrical symmetry we emphasize the presence of an accidental degeneracy in the excitation spectrum at certain values of the projection of orbital angular momentum on the symmetry axis and discuss possible consequences of the degeneracy in the context of new signatures of Bose-Einstein condensation.

The recent realization of Bose-Einstein condensation (BEC) in trapped alkali atom gases [1][2][3], followed by the second generation of experiments [4][5][6][7][8], has opened the possibility of investigating macroscopic quantum phenomena in these systems. For understanding the macroscopic quantum behavior of a trapped Bose-condensed gas especially important is the character of elementary excitations of the trapped condensate, which to a large extent is predetermined by the interaction between atoms. In dilute gases the interaction is primarily binary and is characterized by a single parameter, a, the s-wave scattering length. This allows one to develop a transparent theory which can be tested experimentally.
At present theoretical investigations of elementary excitations of trapped Bose condensates include analytical solutions for the spectrum of low-energy excitations in spherically symmetric harmonic traps in the Thomas-Fermi regime [9] and numerical analysis of the eigenfunctions and eigenenergies of the excitations in the traps of spherical and cylindrical symmetry [10][11][12][13][14]. In the latter case the eigenfrequencies of the lowest excitations, as those measured in the JILA [4,8] and MIT [7] experiments, have also been found analytically [9,15,16,13].
Most interesting are the low-energy excitations, i.e., the excitations with energies much smaller than the chemical potential (mean field interaction between particles), as they are essentially of collective character. Previous studies revealed that the eigenfrequencies of condensate oscillations are strongly different from those of a collisionless thermal gas [4,7,8], but are rather close to the frequencies of a thermal gas in the hydrodynamic regime [16,17]. In this paper we develop the method of finding analytical solutions of the Bogolyubov-De Gennes equations for the spectrum and wavefunctions of the condensate excitations in the Thomas-Fermi regime in harmonic traps of any type of asymmetry and introduce a classification of eigenstates. We analyse the structure of the excitation spectrum in the case of cylindrical symmetry and find an accidental degeneracy at certain values of the projection of orbital angular momentum on the symmetry axis. We address the question of how the accidental degeneracy can manifest itself, providing us with a clear distinction between the condensate oscillations and the oscillations of a classical gas in the hydrodynamic regime.
We consider a Bose-condensed gas in an external harmonic potential V (r) = M i ω 2 i r 2 i /2 with frequencies ω i and assume a pair potential of the atom-atom interaction of the form U (R) =Ũ δ(R), whereŨ = 4πh 2 a/M , a is the scattering length and M the atom mass. Then the grand canonical Hamiltonian of the system is written aŝ whereΨ(r) is the field operator of atoms, and µ the chemical potential. The field operator can be represented as a sum of the above-condensate part and the condensate wavefunction Ψ 0 = Ψ , which is a c-number: Ψ =Ψ ′ + Ψ 0 (see [18]). Assuming that the condensate density greatly exceeds the density of above-condensate particles we omit the terms proportional toΨ ′3 andΨ ′4 in Eq.(1) and write the Hamiltonian in the form The Gross-Pitaevskii equation for Ψ 0 normalized by the condition |Ψ 0 (r)| 2 dr = N 0 (N 0 is the number of particles in the condensate) follows directly fromĤ 0 in Eq.
Owing to Eq.(4) the part of the Hamiltonian, linear in Ψ ′ (and not included in Eq.(2)), becomes equal to zero. The Hamiltonian (2) is bilinear in the operatorsΨ ′ ,Ψ ′ † and can be reduced to a diagonal form by using the Bogolyubov transformation generalized to an inhomogeneous case: [19]. Hereb ν ,b † ν are creation and annihilation operators of elementary excitations. The Hamiltonian takes the form (5) if the functions u ν , v ν satisfy the equations (Ψ 0 is taken real), and are normalized by the condition The Hamiltonian (5) does not contain the recently discussed term originating from the presence of the "momentum" operator of the condensate [20], since this term does not affect the elementary excitations. Eqs. (6), (7) and (4) represent a complete set of equations for finding the wavefunctions u ν , v ν and energies E ν of the excitations. We will discuss the case of repulsive (a>0) interparticle interaction in the Thomas-Fermi regime (µ ≈ n 0mŨ ≫hω i , n 0m is the maximum condensate density), where the presence of a small parameter (ω = i ω 1/3 i ) allows us to simplify the equations for the elementary excitations. First, we write Eqs. (6), (7) and (4) in terms of dimensionless eigenenergies ε ν = E ν /hω and coordinates y i = r i /l i , where l i = (2µ/mω 2 i ) 1/2 is the characteristic size of the condensate in the i-th direction: Here∆ = i (ω i /ω) 2 ∂ 2 /∂y 2 i , and y 2 = i y 2 i . With the dimensionless condensate densityn 0 = |Ψ 0 (r)| 2 /n 0m from Eq.(12), Eqs. (10) and (11) are reduced to the fourthorder differential equations for the functions f ν± = u ν ±v ν : Here we have omitted the index ν and written the terms proportional to ζ 2 separately. The low-energy excitations (E ≪ µ or εζ ≪ 1) are primarily localized inside the condensate spatial region. At characteristic distances from the condensate boundary we can omit all terms proportional to ζ 2 in Eqs. (13), (14) and use the Thomas-Fermi approximation for the condensate wavefunction (see [21,22]): following from Eq.(4) in which the kinetic energy term ζ 2∆ Ψ 0 is neglected. Then, using the substitution f ± (y) = C ± (1 − y 2 ) ± 1 2 W (y), we obtain the equation where the operatorĜ is given bŷ The relation between the normalization coefficients C + and C − follows from Eqs. (10), (11), (17) and (18): The solution (16) can be used in Eqs. (10) and (11) from the very beginning for finding the wavefunctions and spectrum of elementary excitations with energies E ν ≫hω i . However, for the excitations with energies comparable to the trap frequencies this would lead to an incorrect result. Moreover, such a procedure makes Eqs. (10) and (11) incompatible with each other. The physical reason is that the wavefunctions of such excitations vary over a distance comparable with the size of the condensate. Hence, the kinetic energy of the condensate, omitted in the derivation of Eq. (16), and the kinetic energy of the excitations are equally important. This is taken into account in our derivation of Eqs. (13) and (14), relying on the exact expression for Ψ 0 . In principle, the exact equations (13) and (14) can be used to obtain a systematic expansion of the excitation wavefunctions and energies in the ζ parameter.
In the case of spherical symmetry (ω i = ω = ω) the excitations are characterized by the orbital angular momentum l and its projection m. The solution of Eq. (17) has the form harmonic, x = y 2 , and the radial function P (x) is governed by a hypergeometric differential equation The solution of Eq. (20), convergent at x = 0, is the hypergeometric function which converges at x → 1 only when reduced to a polynomial. This immediately gives the energy spectrum: where n is a positive integer. The solutions of Eq.(20) are classical Jacobi polynomials P (l+1/2,0) n (1 − 2y 2 ) and, with the normalization conditions (8) and (19), we obtain where l c = (2µ/M ω 2 ) 1/2 is the size of the condensate. The spectrum (21) coincides with that found by Stringari [9] from the analysis of the density fluctuations in the hydrodynamic approach.
In the non-symmetric case with ω 1 = ω 2 = ω 3 , the operatorĜ is invariant under the inversion of any of the - There are three different sets of coefficients b i corresponding to the three eigenfrequencies Ω determined by the secular equation det[S] = 0 where S is the 3 × 3 matrix The coefficients b i are determined by the system of three linear equations. The first one is 3 i=1 b i + (Ω 2 /ω 2 ) = 0. The two other equations can be any of the three linearly dependent equations For cylindrically symmetric traps (ω 1 = ω 2 = ω ρ , ω 3 = ω z ) the projection of the orbital angular momentum on the z axis, m, is a conserved quantity. The eigenstates of the excitations can be labeled by the quantum numbers N , m, and the axial parity P z . The radial parity describing the behavior of the eigenfunctions with respect to simultaneous inversion of the two radial coordinates is [+] for even m, and [−] for odd m. The polynomials W can be represented in the form W = ρ |m| B nm (ρ, z) exp (imφ), where ρ and z are the dimensionless axial and radial coordinates, and B nm polynomials of power n = N − m. Each term of the polynomial B nm has the form z nz ρ nρ , where n z and n ρ are positive integers. The sum n z + n ρ takes the values 0, 2, ...n for even n, and 1, 3...n for odd n. The integer n ρ is even, n z being even for even n (P z = [+]), and odd for odd n (P z = [−]). The polynomials B nm (ρ, z) and eigenenergies E nm =hΩ nm can be found from the equation which follows directly from Eq. (17). The quantity β = ω z /ω ρ is the ratio of the axial to radial frequency.
The number of eigenmodes k at given m and n is determined by the quantum number n. Generally speaking, we obtain k = (n + 2)/2 for even n, and k = (n + 1)/2 for odd n. The eigenenergies can be found from the k-th order secular equation det[S] = 0, where S is the threediagonal k × k matrix and i, j = 1, 2...k. The coefficient q = 0 for even n, and q = 1 for odd n. For n = 0 (|m| ≥ 1) we have purely radial oscillations, with B 0m = const and Ω 0m = |m|ω ρ . The case n = 1 corresponds to the radial oscillations, in combination with the axial oscillations of the center of mass of the condensate. Here we have B 1m ∝ z and Ω 1m = |m|ω 2 ρ + ω 2 z . In both cases the coupling between the radial and axial motion is absent, and the condensate frequencies Ω 0m and Ω 1m are the same as those for a classical gas in the hydrodynamic regime (see [16,17]).
For n ≥ 2 the coupling between the radial and axial degrees of freedom becomes important, and the condensate oscillation frequencies will be different from the frequencies of a classical hydrodynamic gas. If n = 2 there are two coupled shape oscillations of the condensate with frequencies Ω ± 2m given by In the simplest case of m = 0 Eq.(26) gives the frequencies of the quadrupole shape oscillations of the condensate. The frequencies Ω ± 20 and the frequency of quadrupole radial oscillations Ω 02 were found in the hydrodynamic approach in [9]. They were also obtained in [15] by considering the condensate evolution under a weak modulation of the trap frequencies, in [16] from the Hamiltonian of the scaling dynamics, and in [13] on the basis of variational approach. The frequencies Ω 02 and Ω − 20 were measured in the JILA experiment [4] for β = √ 8 and calculated numerically for this trapping geometry in [10]. The frequencies Ω ± 20 were found in the MIT experiment [7] for β = 0.08.
For n = 3 we have two coupled shape oscillations which are now also coupled to the oscillations of the center of mass of the condensate. In this case we obtain Interestingly, for certain values of the projection of the orbital angular momentum, m, we find an accidental degeneracy in the spectrum of excitations. The simplest example concerns the frequencies Ω 1m = ω ρ |m| + β 2 and Ω 2m . As follows from Eq. (26), Ω − 2m = Ω 1m for the projection m satisfying the condition and, accordingly, integer β 2 ≥ 3. For β = √ 3 we have m = 0, i.e., the frequency of quadrupole shape oscillations Ω − 20 coincides with the frequency of axial oscillations Ω 10 . In the JILA trapping geometry, where β = √ 8, Eq.(28) gives |m| = 5.
Although the condensate frequencies Ω 2m do not significantly differ from those for a classical gas in the hydrodynamic regime (for m = 0 see [16]), the accidental degeneracy determined by Eq.(28) is characteristic only for the condensate. The presence of the accidental degeneracy can strongly influence the picture of the condensate oscillations. The coupling between the degenerate modes is provided by the interaction terms in the Hamiltonian (1), proportional toΨ ′3 andΨ ′4 and omitted above in the derivation of the Bogolyubov-De Gennes equations. Therefore, driving only one of the degenerate modes, it is feasible to expect the appearance of oscillations representing a superposition of the two modes. This phenomenon ensures a clear distinction between the condensate oscillations and the oscillations of a classical hydrodynamic gas and, hence, can be a signature of BEC for the gas in the hydrodynamic regime. E.L.S. and G.V.S. acknowledge the support from the Dutch Foundation FOM, from NWO (project 047-003.036), from INTAS, and from the Russian Foundation for Basic Studies. P.Ö, I.T. and M.W. acknowledge hospitality in the group of J. Mlynek. M.W. acknowledges financial support by the Deutsche Forschungsgemeinschaft. I. T. wishes to acknowledge the fellowship from the Alexander von Humboldt Foundation.