Fluctuation-driven topological transition of binary condensates in optical lattices

We show the emergence of a third Goldstone mode in binary condensates at phase separation in quasi-one-dimensional (quasi-1D) optical lattices. We develop the coupled discrete nonlinear Schrödinger equations using Hartree-Fock-Bogoliubov theory with the Popov approximation in the Bose-Hubbard model to investigate the mode evolution at zero temperature, in particular, as the system is driven from the miscible to the immiscible phase. We demonstrate that the position exchange of the species in the ${}^{87}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ system is accompanied by a discontinuity in the excitation spectrum. Our results show that, in quasi-1D optical lattices, the presence of the fluctuations dramatically changes the geometry of the ground-state density profile of two-component Bose-Einstein condensates.

Figure 1

The evolution of the quasiparticle amplitudes corresponding to the ${}^{85}\mathrm{Rb}$ Kohn mode as the intraspecies interaction of ${}^{85}\mathrm{Rb}$ $\left(U_{22}\right)$ is decreased from $0.25E_R$ to $0.062E_R$. (a),(b) When $U_{22}\ge 0.18E_R$, the system is in the miscible phase and the Kohn mode $\left(l=1\right)$ has contributions from both the species. (c)–(e) When the system is on the verge of phase separation, then the Kohn mode of ${}^{85}\mathrm{Rb}$ goes soft. (f) At phase separation $U_{22}\le 0.065E_R$ the Kohn mode transforms into a Goldstone mode.

Figure 2

The evolution of the low-lying modes as a function of the intraspecies interaction of the ${}^{85}\mathrm{Rb}$ $\left(U_{22}\right)$ in the ${}^{87}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ TBEC held in quasi-1D optical lattices. Excitation spectrum (a) at zero temperature and (b) in the presence of quantum fluctuations. Here $U_{22}$ is in units of the recoil energy $E_R$.

Figure 3

The geometry of the condensate density profiles and its transition from the miscible to the immiscible regime. (a)–(c) The transition from the miscible to the sandwich profile for the ${}^{87}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ TBEC with change in the intraspecies interaction $U_{22}$ at $T=0$ K. The position exchange (c) in the sandwich profile occurs at $U_{11}=U_{22}=0.05E_R$. (d)–(f) show similar condensate density profiles for the Cs-Rb TBEC with change in the interspecies interaction $U_{12}$ at $T=0$ K. In this system the transition to the sandwich geometry occurs at $U_{12}^c=0.3E_R$.

Figure 4

The evolution of the energies of the low-lying modes as a function of the interspecies interaction $\left(U_{12}\right)$ in Cs-Rb TBEC held in a quasi-1D lattice potential. The excitation spectrum (a) at $T=0$ K, and (b) after including the quantum fluctuations. Here $U_{12}$ is in units of the recoil energy $E_R$.

Figure 5

The evolution of the quasiparticle amplitudes corresponding to the Kohn mode as the interspecies interaction is increased from $0.2E_R$ to $0.35E_R$ for a Cs-Rb TBEC in a quasi-1D lattice potential at $T=0$ K. (a)–(c) In the miscible regime, the Kohn mode has contributions from both species. (d)–(f) For $U_{12}>0.3E_R$ the Kohn mode of ${}^{87}\mathrm{Rb}$ goes soft, whereas that of ${}^{133}\mathrm{Cs}$ decreases in amplitude.

Figure 6

The evolution of the quasiparticle amplitudes corresponding to the Kohn mode for ${}^{87}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ TBEC in the presence of the fluctuations as the intraspecies interaction of ${}^{85}\mathrm{Rb}$ $\left(U_{22}\right)$ is decreased from $0.2E_R$ to $0.05E_R$. (a)–(e) The Kohn mode of ${}^{85}\mathrm{Rb}$ goes soft, whereas that of ${}^{87}\mathrm{Rb}$ decreases in amplitude and finally vanishes in (e). (f) The sloshing mode, which emerges after phase separation as the sandwich density profile transforms into a side-by-side profile.

Figure 7

The fluctuation-induced transition in the geometry of the total density profile (condensate + quantum fluctuations) of a TBEC at $T=0$ K in a quasi-1D lattice potential. (a)–(c) The transition in the ${}^{87}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ system from the miscible to the sandwich and finally to the side-by-side profile with change in the intraspecies interaction. (d), (e) The transition in the Cs-Rb TBEC from the miscible to the side-by-side profile with change in the interspecies interaction $U_{12}$. The geometry of the ground state of both systems in the immiscible regime is different from that at zero temperature in the absence of the fluctuations, Fig. 3.

Figure 8

The evolution of the quasiparticle amplitude corresponding to the Kohn mode for the Cs-Rb TBEC in the presence of fluctuations. (a)–(d) The Kohn mode evolves as the interspecies interaction is increased. (e),(f) It is transformed into a sloshing mode as the TBEC acquires the side-by-side density profile after phase separation.