Effective single-mode model of a binary boson mixture in the quantum droplet region

In a binary quantum droplet, the interspecies attraction dominates over the intraspecies repulsions and the mean-field energy is unstable. The mechanical stability is restored by the repulsive Lee-Huang-Yang (LHY) energy [D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)]. In the Bogoliubov theory of the binary quantum droplet, there are two branches of gapless excitations. The lower branch describes the phonon excitation and its energy is imaginary in the long-wavelength limit, implying dynamical instability. Recently, it was found that the phonon energy is renormalized by higher-order quantum fluctuations and the dynamical instability is removed [Q. Gu and L. Yin, Phys. Rev. B 102, 220503(R) (2020)]. In this paper, we study a binary quantum droplet in the path-integral formalism to construct an effective model with the correct phonon energy. By integrating out the upper excitation branch, we obtain an effective single-mode model describing density fluctuations, and derive the extended Gross-Pitaevskii equation (EGPE). In this approach, the LHY energy in the EGPE is purely positive without any assumption of neglecting the imaginary part. This effective single-mode model can be also used outside and close to the quantum droplet region such as in the LHY gas. Based on this effective model, we obtain the phase diagram of a uniform binary Bose system near the mean-field unstable point, which can be tested in experiments.

Figure 1

The one-loop diagrams of ${\mathrm{\Sigma }}_{11}$. Dashed lines with one-way arrow(s) denote the normal Green's function $G_{d^{*}d}$ while those with two head-to-head (tail-to-tail) arrows denote the anomalous Green's function $G_{d^{*}{d}^{*}}$ ($G_{dd}$). The external uncontracted solid half lines denote the ${\varphi }_c^{*}$ (outgoing) and ${\varphi }_c$ (incoming) boson fields.

Figure 2

The one-loop diagrams of ${\mathrm{\Sigma }}_{20}$. The notation is the same as in Fig. 1.

Figure 3

Total energy vs the total density for symmetric interactions $g_{aa}=g_{bb}=g$ and $\delta g/g=-0.05$. Our result is shown in the solid line. The dotted and dashed ones are from the original EGPE theory [9] and hydrodynamic method [37], respectively.

Figure 4

Phase diagram of the uniform ${}^{39}\mathrm{K}$ binary mixture near the mean-field unstable point at zero temperature. The unit of the vertical axis is given by $n_r=1.13\times {10}^{21}{\mathrm{m}}^{-3}$ which is a typical droplet density in experiments [6]. The system is a gas in region (I), a quantum liquid in region (II), with negative pressure and positive compressibility in region (III), and unstable with negative pressure and compressibility in region (IV). The system is a quantum droplet with zero pressure in the solid line, a LHY gas in the dashed-dotted line, and unstable with zero phonon speed in the dashed line. The upper axis is $\delta a=\delta g/4\pi$ in the unit of Bohr radius $a_0$. The data of the scattering lengths at various magnetic field are taken from Ref. [48].