Microscopic pairing theory of a binary Bose mixture with interspecies attractions: Bosonic BEC-BCS crossover and ultradilute low-dimensional quantum droplets

Ultradilute quantum droplets are an intriguing new state of matter, in which the attractive mean-field force can be balanced by the repulsive force from quantum fluctuations to avoid collapse. Here, we present a microscopic theory of ultradilute quantum droplets in three-, one-, and two-dimensional two-component Bose-Bose mixtures by generalizing the conventional Bogoliubov theory to include the bosonic pairing arising from the interspecies attraction. Our pairing theory is fully equivalent to a variational approach and hence gives an upper bound for the energy of quantum droplets. In three dimensions, we predict the existence of a strongly interacting Bose droplet at the crossover from Bose-Einstein condensates (BECs) to BCS superfluids and map out the bosonic BEC-BCS crossover phase diagram. In one dimension, we find that the energy of the one-dimensional Bose droplet calculated by the pairing theory is in excellent agreement with the latest diffusion Monte Carlo simulation [L. Parisi et al., Phys. Rev. Lett. 122, 105302 (2019)] for nearly all the interaction strengths at which quantum droplets exist. In two dimensions, we show that Bose droplets disappear and may turn into a solitonlike many-body bound state, when the interspecies attraction exceeds a critical value. Below the threshold, the pairing theory predicts more or less the same results as the Bogoliubov theory derived by D. S. Petrov and G. E. Astrakharchik [Phys. Rev. Lett. 117, 100401 (2016)]. The predicted energies from both theories are higher than the diffusion Monte Carlo results due to the weak interspecies attraction and the increasingly important role played by the beyond-Bogoliubov-approximation effect in two dimensions. Our pairing theory provides an ideal starting point to understand interesting ground-state properties of quantum droplets in various dimensions, including their shape and collective oscillations.

Figure 1

Three-dimensional energy per particle $E/N$ of a strongly interacting Bose droplet predicted by the pairing theory as a function of the density $n$ with increasing interspecies attraction from $a_{12}=-1.5a$ to $a_{12}=\mp \infty$ and, finally, to $a_{12}=+3.0a$, as indicated. The dashed line traces the equilibrium density. It ends at $a_{12}\simeq 3.6a$, where the strongly interacting Bose droplet disappears. The energy per particle $E/N$ is measured in units of ${10}^{-2}{\hslash }^2/\left(2m{a}^2\right)$, and the density $n$ is in units of ${10}^{-3}{a}^{-3}$.

Figure 2

Three-dimensional energy per particle $E/N$ as a function of the density $n$, near the threshold $a_{12}=+3.6a$.

Figure 3

(a) Three-dimensional minimum energy per particle $E_{min}/N$ as a function of the ratio $a/a_{12}$. The green dashed line shows half of the bound-state energy ${\epsilon }_B/2$ of a molecule or dimer formed by two bosons of different species. (b) The pairing order parameter ${\mathrm{\Delta }}_0$ (solid line) and the chemical potential $-\mu$ (dashed line) as a function of the ratio $a/a_{12}$. The minimum energy per particle $E_{min}/N$, the pairing order parameter ${\mathrm{\Delta }}_0$, and the chemical potential $-\mu$ are all measured in units of ${10}^{-2}{\hslash }^2/\left(2m{a}^2\right)$.

Figure 4

A sketch of the phase diagram of a binary Bose mixture across the bosonic BEC-BCS crossover, illustrated by the energy per particle $E/N$ at a positive and small intraspecies scattering length $0<n{a}^3\ll 1$.

Figure 5

The function $h_1\left(\alpha \right)$ (solid line) and its comparison to ${(1-\alpha )}^{3/2}$ (dashed line).

Figure 6

One-dimensional thermodynamic potential ${\mathrm{\Psi }}_{\text{1D}}$, in units of ${\hslash }^2/\left(m{a}_{12}^3\right)={\varepsilon }_B{a}_{12}^{-1}$, as a function of the pairing parameter $\mathrm{\Delta }$ at different chemical potentials $\mu =0$ (solid line), $-0.8$ (dashed line), and $-0.9$ (dot-dashed line) and at $g_{12}=-0.75g$. Both $\mathrm{\Delta }$ and $\mu$ are measured in units of ${\varepsilon }_B$.

Figure 7

One-dimensional chemical potential $\mu$, the parameter $C$, and the pairing gap ${\mathrm{\Delta }}_0$, in units of ${\varepsilon }_B$, as a function of the total density $n$ (in units of ${\left|a\right|}^{-1}$) at $g_{12}=-0.75g$.

Figure 8

One-dimensional energy per particle as a function of the density at three interspecies interaction strengths $g_{12}/g=-0.60$ (black), $-0.75$ [red (light gray)], and $-0.90$ [blue (dark gray)]. Our pairing results (solid lines) are compared with the recent DMC data (symbols) [28], the $\mathrm{MF}+\mathrm{LHY}$ predictions (65) (dot-dashed lines), and the $\mathrm{MF}+\mathrm{LHY}$ results with Petrov's prescription (66) (dashed line, for only $g_{12}/g=-0.75$). The energy is in units of ${\varepsilon }_B/2$, and the density is in units of ${\left|a\right|}^{-1}$.

Figure 9

Minimum energy per particle $E_{min}/N$ or equilibrium chemical potential ${\mu }_{\text{eq}}$ in one dimension as a function of the interspecies interaction strength $g_{12}/g$, obtained from the pairing theory (solid line), the DMC simulation (circles) [28], and the $\mathrm{MF}+\mathrm{LHY}$ theory (65) (dot-dashed line). The energy is in units of ${\varepsilon }_B/2$.

Figure 10

Two-dimensional thermodynamic potential ${\mathrm{\Psi }}_{\text{2D}}$, in units of ${10}^{-6}{E}_T{a}_T^{-2}$, as a function of the pairing parameter $\mathrm{\Delta }$ at different chemical potentials $\mu =1.0$ (solid line), 0 (dashed line), and $-0.2$ (dot-dashed line) and at the interspecies interaction strength $ln(a_{12}/a)=5$. Both $\mathrm{\Delta }$ and $\mu$ are measured in units of ${10}^{-3}{E}_T$.

Figure 11

Two-dimensional chemical potential $\mu$, the parameter $C$, and the pairing gap ${\mathrm{\Delta }}_0$, in units of ${10}^{-3}{E}_T$, as a function of the total density $n$ (in units of ${10}^{-3}{a}^{-2}$) at $ln(a_{12}/a)=5$.

Figure 12

Two-dimensional energy per particle as a function of the density at three interspecies interaction strengths: $ln(a_{12}/a)=5$ in (a), $ln(a_{12}/a)=2$ in (b), and $ln(a_{12}/a)=1$ in the inset in (b). Our pairing results (solid lines) are compared with the $\mathrm{MF}+\mathrm{LHY}$ predictions with Petrov's theory (76) (dashed line). In (a), we show also the DMC data from Ref. [19]. The energy is in units of the interspecies binding energy $E_S$, and the density is in units of $a^{-2}$. We note that the scales of energy and density change by a factor of $\sim 10$ in the upper and lower panels.

Figure 13

One-dimensional energy per particle as a function of the density at the interspecies interaction strength $g_{12}=-0.75g$. Our pairing results [numerical, brown thick solid line, and analytical, black thin solid line; see Eq. (71)] are compared with the recent DMC data (symbols with error bars) [28], the $\mathrm{MF}+\mathrm{LHY}$ prediction (65) (dot-dashed lines), and the $\mathrm{MF}+\mathrm{LHY}$ results with Petrov's prescription (66) (dashed line). The energy is in units of ${\varepsilon }_B/2$, and the density is in units of ${\left|a\right|}^{-1}$.