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Collective Excitations of a Two-Component Bose Condensate at Finite Temperature

We compare the collective modes for Bose-condensed systems with two degenerate components with and without spontaneous intercomponent coherence at finite temperature using the time-dependent Hartree-Fock approximation. We show that the interaction between the condensate and non-condensate in these two cases results in qualitatively different collective excitation spectra. We show that at zero temperature the single-particle excitations of the incoherent Bose condensate can be probed by intercomponent excitations.

Introduction. For a Bose-condensed system with two internal components, one of the fundamental questions is whether one can distinguish a system with spontaneous intercomponent coherence from one without it [1]. At zero temperature, the ground state energies calculated with the static Gross-Pitaevskii energy functional are the same for both cases [2]. The collective excitations calculated with the linearized time-dependent Gross-Pitaevskii (TDGP) equation are also identical for both cases. For example, the collective modes for a homogeneous two-component system are for both cases, as obtained from the linearized TDGP [3] or the Bogoliubov theory [4,5,6], where ε 0 q = 2 q 2 2m , ω 2 i = ε 0 q ε 0 q + 2g ii ρ i is the Bogoliubov mode for a single component system, g ii is the i-th intracomponent contact interaction, g 12 the intercomponent contact interaction, and ρ i is the density of i-th component with i = 1, 2. The excitations in Eq. (1) may be interpreted differently for the incoherent and coherent cases are different. Without intercomponent coherence, there are two macroscopically occupied states, a situation that has come to be known as a fragmented condensate [7]. In this case ω + may be interpreted as a density wave in which the two condensates move in phase, while ω − is a collective excitation associated with out of phase motion between the two condensates. With intercomponent coherence, there is only a single condensate. In this case ω + is interpreted as the superfluid mode, and ω − is a Goldstone mode corresponding to a spontaneously broken U(1) symmetry in the relative phase between the two components. As pointed out by Leggett [1], in the absence of particle exchange between the two components, no physical quantities can depend on the relative phase between the them. However, at finite temperatures, we will show that the coupling between the condensate and non-condensate particles provides a way of probing the nature of the condensates.
In this work, we compare the collective excitations for a two-component Bose-condensed system with and without intercomponent coherence at finite temperature using the time-dependent Hartree-Fock (TDHF) approximation, for the special case g 11 = g 22 > g 12 . Our principal findings are as follows. Without intercomponent coherence: (i) The motions of the two components and uncondensed particles give rise to gapless collective excitations. We find one in-phase (superfluid) mode. However, there can be two out-of-phase modes within some range of intercomponent interaction strengths at appropriate temperatures. This is illustrated in Fig. 1. (ii) The perturbative response to a field that allows intercomponent conversion has a resonance at frequencies given by the single-particle excitation energies. With intercomponent coherence, we find a gapless superfluid mode, and two "pseudo-spin waves", one gapless and the other gapped, as illustrated in Fig. 2. The gapped pseudo-spin wave has vanishing weight as temperature goes to zero. The differences in in these spectra allow one to distinquish a single condensate from a fragmented one.
Model. We consider the collective excitations in a twocomponent homogenous Bose-condensed atomic gas at finite temperatures k B T = 1/β. To simplify the physical picture, we assume the two internal states are degenerate and take the two-body interactions as contact forms with strength parameters g 11 = g 22 = g s = 4π 2 m a s and x the corresponding s-wave scattering length, and take g s > g x , so that the homogenous state is stable against phase separation. The system may be characterized by the gas parameter ρa 3 s , where ρ is the total particle density, and γ = g x /g s . Our unit of energy is E 0 = 2 2ma 2 s , where m the mass of the particles. Using a pseudo-spin language, the single-particle wavefunctions may be written as spinors with components ψ kλ ( r) = V − 1 2 e i k· r η λ , with V the volume of the system and η λ the spin state functions. In this work, the spinor index is denoted by Greek letters. For incoherent condensates, we write λ = 1, 2 with η 1 = 1 0 and η 2 = 0 1 the pseudospin states of the two condensates.
For coherent case, we write λ = ±, η + = 1 , and the condensation is assumed to be in the η + state. In both cases the macroscopically occupied states have zero momentum.
Method. We briefly outline our method here; details will be presented elsewhere. It is well known that to treat the dynamics of a condensate and thermally excited (non-condensate) particles in a fully consistent manner is theoretically challenging [8]. Recently we have shown this can be done within a TDHF approximation [9] by using a constrained grand canonical ensemble [10] in which the condensate particle number is fixed. We first determine the static Hartree-Fock (HF) ground state properties, including the chemical potential µ, the single-particle energies ε kα , the occupation numbers n kα , and the critical temperature β c . This involves defining a self energy Σ which is a functional of the densities ρ k1λ1, k2λ2 ≡< a † k1λ1 a k2λ2 >, with a kλ the boson destruction operator, and the condensate wavefunction(s) ψ 0λ . We then introduce a small perturbation δU , and assume Σ keeps its functional form with respect to the ρ's and ψ 0λ , which are now time-dependent. This results in self-consistent equations which can be solved in the linear response regime. This allows us to obtain equations for the response functions (0) , ω + ε k+ qµ − ε kλ χ k k ′ λµ,αβ ( q; ω) = (n k+ qµ − n kλ )δ αλ δ βµ δ k k ′ + (n k+ qµ − n kλ ) ss ′ ,λ1µ1, k1 Here s and s ′ denote the components of each spinor, and The Kronecker delta terms δ λc are 1 when λ is a condensate state, and 0 otherwise. Poles of the response functions correspond to collective excitations of the system.
Equation (5) shows the effect of intercomponent interaction and the interaction between the condensate and non-condensate on the collective excitations. The plus and minus signs in Eq. (5) can be interpreted as resulting from the in-phase and out-of-phase motion of the two components, respectively. We find only one in-phase (gapless) mode as a solution, which reduces to ω + in Eq.
(1) in the zero temperature limit. Thus the interaction between the condensate and non-condensate has little effect on the in-phase motion. However, this interaction has a striking effect on the out-of-phase motion. Within a range g x,1 < g x < g x,2 , the values g x,1 and g x,2 depending on the gas parameter ρa 3 and temperature T , the system can support a new out-of-phase mode for small q. This is illustrated Fig. 1 for ρa 3 = 10 −4 and β/β c = 3, where it can be seen that 1 2 ℑ [χ 1111 ( q; ω) − χ 2211 ( q; ω)] carries an extra resonance. In order to see why this occurs, we show the polarization functions P c and P n in Fig. 3. For ε 0 q < 2g s ρ 0 , P c ( q; ω) is a monotonic function of ω and approaches 1/g s with zero slope as ω → 0, while P n ( q; ω) is non-monotonic at small ω. The combination g s (2 − γ)P ( q; ω) is then non-monotonic at small ω, so for certain ranges of γ, Eq. 5 is satisfied twice. This is shown in the inset in Fig. 3.
The function χ 21,12 ( q; ω) supports a collective mode [11,12,13], corresponding to an excitation in which the "flavor" of a particle changes. From Eq. (2) one finds where P c 12 ( q; ω) = 2ρ 0 ε 0 q + g s ρ 0 At zero temperature the collective mode requires exciting a particle from one of the condensates to an excited state of the other component, so χ 21,12 has a pole at the single particle excitation spectrum, ω = ε q − µ = ε 0 q + g s ρ 0 . Thus, this provides a way to probe the single-particle spectrum. We note that, since χ 21,12 is the response to a "flavor-changing" perturbation, this collective mode is not found in the TDGP approximation when species exchange is absent from the unperturbed Hamiltonian. It is apparent that our TDHF approach is able to capture a broader variety of excitations for this system.
We now turn to the coherent case. The system can condense into any spinor given by η(θ) = 1 √ 2 η 1 + e iθ η 2 , resulting in a single condensate. We choose the system condensing into η + = η(θ = 0), which spontaneously breaks the U(1) symmetry. Therefore, we would expect at least two gapless modes: a superfluid density mode and a Goldstone mode (a pseudo-spin wave) for the spontaneous U(1) symmetry breaking.
The density wave may be found from the poles of χ ++ ( q; ω) = k1 k2;αβ χ k1 k2 ++,αβ ( q; ω) and χ −− ( q; ω) = where g p = g s + g x , P ++ ( q; ω) = P ++ c ( q; ω) + P ++ n ( q; ω), P −− ( q; ω) = P n −− ( q; ω) with P ++ c ( q; ω) = 2ρ0ε q ω 2 −(ε 0 q ) 2 +gpρ0ε 0 q and P αβ n ( q; ω) = 1 Ω ′ k n q+ kα −n kβ ω+ε q+ kα −ε kβ , where the prime excludes the momentum of the condensate modes. As in the single component case, the noncondensate is too dilute to sustain a propagating second sound mode within this approximation. The resulting response and superfluid mode are thus qualitatively similar to the single component case. By contrast, the non-condensate component has a dramatic effect on the pseudo-spin waves. In order to see this, we define χ k1 ±∓ ( q; ω) = k2αβ χ k1 k2 ±∓,αβ ( q; ω) and split these into condensate and non-condensate parts: where the prime on the sum excludes the momentum of the condensate mode. In terms of these quantities, Eq. (2) yields n ( q; ω) g m P +− n ( q; ω) 1 − g s P +− n ( q; ω) g m P +− n ( q; ω) g m P −+ n ( q; ω) −g s P −+ n ( q; ω) g m P −+ n ( q; ω) 1 − g s P −+ n ( q; ω) where g m = g s − g x . Eq. 9 yields two spin waves, as shown in Fig. 2, one of which is gapless and originates from the U(1) symmetry breaking of the relative phase of the two internal states. At T = 0, this gapless mode is reduce to ω − given by Eq. (1). The other spin wave has a gap as q → 0 given by Here ρ + and ρ − are the non-condensate density in spinor η + and η − , respectively. The gapped spin wave, unlike the gapped flavor-changing mode obtained from Eq. (6) in the incoherent case, disappears as T → 0 (i.e., its weight in the response functions vanishes), which can be see directly from Eq. (9) since both P n +− and P n −+ are zero in this limit.
In the above calculations we have discussed the special case of interaction strengths g 11 = g 22 . For g 11 = g 22 but g 2 12 < g 11 g 22 so that the homogeneous state is still stable against phase separation, we find for the incoherent case results which are qualitatively the same. The coherent system, by contrast, is more complicated. In particular one finds that the spinors of the non-condensate modes are neither parallel nor anti-parallel to the condensate mode spinor. This leads to the response functions in Eq. 2 all being coupled together, so a classification of the modes as flavor-changing and flavor-preserving is no longer possible. Nevertheless, one still finds two gapless linear modes and one gapped mode as we found in the g 11 = g 22 case. Details of this more complicated situation will be presented elsewhere.
In conclusion, we have compared the collective excitations of two-component Bose-condensed systems with and without coherence at finite temperature. The coupling between the condensate and non-condensate depends on whether there is intercomponent coherence, and the two identical dispersions given by Eq. (1) at zero temperature evolve to quite different structures. We also have shown that, at zero temperature the singleparticle excitations in the incoherent case can be probed by flavor-changing excitations, which can not be done in the coherent case. This work was supported by the NSF Grant No. DMR0454699.