Ground state energy of mixture of Bose gases

Abstract

We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number N becomes large. In the dilute regime, when the interaction potentials have the length scale of order O(N −1 ), we show that the leading order of the ground state energy is captured correctly by the Gross-Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross-Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is O(1), we are able to verify Bogoliubov's approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaption to the multi-component setting is non-trivial in various respects and the analysis will be presented in details. Contents42 4.5. Validity of Bogoliubov correction 44 Appendix A. Quantum de Finetti Theorem 3.4 47 References 50 1 In the physical literature [34, 41, 33] a different convention is preferred for α ∈ {1, 2}, namely V (α) = c −1 α V (α) for mean-field regime, and V (α) = c 2 α V (α) (cα·) for Gross-Pitaevskii regime.

Lieb-Robinson bounds and growth of correlations in Bose mixtures

“…The rigorous analysis for interacting Bose mixtures has been developed in [18,26,1,4,16,20]. Other types of composite condensation (meaning, BEC with some sort of internal structure) analogous to condensate mixtures have been analysed mathematically in the case of pseudo-spinor condensates [17], spinor condensates [19], and fragmented condensates [5].…”

“…It is fairly standard that an ample selection of potentials U and V α may be be made above so as to realise H N1,N2 on H N1,N2,sym self-adjointly and also with lower bound growing at most linearly in the particle number (see [18,Sect. 2] and [16]), a condition that will be implicitly assumed here throughout.…”

Effects Beyond Center-of-Mass Separability in a Trapped Bosonic Mixture: Exact Results

“…An immediate study would be a comprehensive comparison between manybody and mean-field descriptions of demixing at the limit of an infinite number of particles. Therein, some properties, like the energy per particle and densities per particle, would exactly coincide and other properties, like variances per particle of many-particle observables and the overlap between the many-body and mean-field wavefunctions, can differ substantially [53][54][55][56][57][58][59][60][61][62][63][64][65]. For finite mixtures, the fragmentation [66] along the demixing pathway would be instrumental to follow.…”

Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger

“…This was improved and generalized much later in [204,285]. Later still it was realized that the trial function (5.12) actually does the job, with somewhat simpler computations [209,227]. We shall sketch only this latter estimate, and remark that the Dyson trial state giving the same energy as the more natural Jastrow one is remarkable, for it contains only special correlations.…”

“…We sketch the calculation with the Jastrow trial state, which is clearly bosonic. The details are in [227,Section 3.2]. As there we set A ≡ 0 for simplicity but the generalization is straightforward.…”