⇦Chinese Physics Letters, 2020, Vol. 37, No. 7, Article code 076701 Quantum Droplets in a Mixture of Bose–Fermi Superfluids Jing-Bo Wang (汪景波)1, Jian-Song Pan (潘建松)2, Xiaoling Cui (崔晓玲)3,4*, and Wei Yi (易为)1,5* Affiliations 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China 2Department of Physics, National University of Singapore, Singapore 117542, Singapore 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 4Songshan Lake Materials Laboratory, Dongguan 523808, China 5CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei 230026, China Received 13 April 2020; accepted 13 May 2020; published online 21 June 2020 Supported by the National Natural Science Foundation of China under Grant Nos. 11974331, 11421092 and 11534014, and the National Key Research and Development Program of China under Grant Nos. 2016YFA0301700, 2017YFA0304100, 2018YFA0307600 and 2016YFA0300603.
*Corresponding author. Email: xlcui@iphy.ac.cn; wyiz@ustc.edu.cn Citation Text: Wang J B, Pan J S, Cui X L and Yi W 2020 Chin. Phys. Lett. 37 076701    Abstract We study the formation of quantum droplets in the mixture of a single-component Bose–Einstein condensate (BEC), and a two-species Fermi superfluid across a wide Feshbach resonance. With repulsive boson-boson and attractive boson-fermion interactions, we show that quantum droplets can be stabilized by attractive fermion-fermion interactions on the Bardeen–Cooper–Schrieffer (BCS) side of the resonance, and can also exist in the deep BEC regime under weak boson-fermion interactions. We map out the phase diagram for stable droplets with respect to the boson-boson and boson-fermion interactions, and discuss the role of different types of quantum fluctuations in the relevant regions of the BCS-BEC crossover. Our work reveals the impact of fermion pairing on the formation of quantum droplets in Bose–Fermi mixtures, and provides a useful guide for future experiments. DOI:10.1088/0256-307X/37/7/076701 PACS:67.60.-g, 03.75.Ss © 2020 Chinese Physics Society Article Text The recent experimental observation of self-bound quantum droplets in dipolar[1–6] or binary Bose–Einstein condensates (BECs)[7–11] has stimulated wide research interest.[12–20] Stabilized by a subtle balance between the attractive mean-field interactions and repulsive quantum fluctuations,[21,22] these exotic liquid-like states are a direct manifestation of quantum many-body effects, and open up the avenue of exploring gas-liquid transition in the quantum regime.[23] As a universal binding mechanism for the quantum droplet, Petrov pointed out in his seminal work[22] that, in a binary BEC, the mean-field inter-species attraction, which tends to collapse the BEC, is counteracted by the Lee–Huang–Yang (LHY) repulsion.[24] Based on such an understanding, quantum droplets are also predicted to exist at lower dimensions,[25–29] in spin-orbit-coupled BECs,[30] at finite temperatures,[23,31] and in photonic systems.[32,33] While most of the early theoretical studies focus on bosonic systems, people have also started to explore the existence of quantum droplets in Bose–Fermi mixtures,[34–38] where the coexistence of different statistics or distinct quasiparticles play an important role. For example, self-bound droplets exist in a mixture of BEC and non-interacting fermions,[37,38] with attractive boson-fermion interactions. In such a system, the large inter-species attraction is balanced by multiple repulsive contributions: the Fermi degenerate pressure, the LHY correction of the BEC, and the second-order correction to the boson-fermion interaction due to density fluctuations.[39,40] However, owing to the large contribution from Fermi pressure in the dilute limit, the stabilization of droplets requires a strong attractive boson-fermion interaction,[37] which is unfavorable experimentally, due to the inevitable atom loss in the strongly interacting regime. As a solution, it has been suggested that, by introducing synthetic spin-orbit coupling to modify low-energy, single-particle dispersion of the non-interacting Fermi gas,[35] the Fermi pressure can have more favorable scaling with the fermion density, such that droplets are stabilized under weaker boson-fermion interactions. Alternatively, the Fermi pressure can be lowered by the formation of Cooper pairs. In particular, when the two-component Fermi gas is close to a wide s-wave Feshbach resonance, the decrease in Fermi pressure becomes even more pronounced as the system is tuned away from the Bardeen–Cooper–Schrieffer (BCS) limit toward the resonance. The fate of droplets in such a system therefore sensitively depends on the interplay of the reduced Fermi pressure and quantum fluctuations, both of which can be quite different from those with a non-interacting Fermi gas. A key question is whether these competing factors inherent in the system can lead to experimentally favorable conditions for the observation of droplets in Bose–Fermi mixtures. In this work, we study the stability of quantum droplets in a mixture of BEC and Fermi pairing superfluids,[41,42] where the fermion-fermion interaction is tuned across a wide s-wave Feshbach resonance.[43] Given a fixed repulsive boson-boson interaction strength, we show that the critical attractive boson-fermion interaction strength to support quantum droplets can be further reduced by tuning the fermion-fermion interaction from BCS side toward resonance. This is facilitated by a modified boson-fermion fluctuation energy, which can turn from positive (repulsive) to negative (attractive) as the fermion-fermion interaction is tuned. This additional attractive contribution to the interaction energy relaxes the requirement on a large boson-fermion interaction, giving rise to a larger stability region of droplets when the system is tuned away from the weak-coupling BCS limit. In the BEC regime, the system can be considered as a mixture of a single-component BEC and a composite BEC of fermion dimers. We show that droplets can be stabilized by the LHY corrections of the two BECs, against the attractive boson-dimer interactions. Remarkably, droplets can be stabilized over a considerable region on the BEC side, and at sufficiently small boson-fermion interaction strengths which are readily achievable in current experiments. The Bose–Fermi superfluid mixture is described by the Hamiltonian $$\begin{align} H={}&\sum_{\boldsymbol k} \epsilon^{b}_{\boldsymbol k} b_{k}^† b_{k}+ \frac{U_{\rm b}}{2\,V}\sum_{{\boldsymbol k},{\boldsymbol k}',{\boldsymbol q}} b_{\boldsymbol k}^† b_{{\boldsymbol q}-{\boldsymbol k}}^† b_{{\boldsymbol q}-{\boldsymbol k}'} b_{{\boldsymbol k}'}\\ &+\sum_{{\boldsymbol k},\sigma}\epsilon^f_{\boldsymbol k} f_{k, \sigma}^† f_{k, \sigma}+\frac{U_{\rm f}}{V} \sum_{{\boldsymbol k}, {\boldsymbol k}',{\boldsymbol q}} f_{{\boldsymbol k}, \uparrow}^† f_{{\boldsymbol q}-{\boldsymbol k}, \downarrow}^† f_{{\boldsymbol q}-k', \downarrow} f_{{\boldsymbol k}', \uparrow}\\ &+\frac{U_{\rm b f}}{V}\sum_{{\boldsymbol k},{\boldsymbol k}',{\boldsymbol q},\sigma} b_{{\boldsymbol q}-{\boldsymbol k}}^† f_{{\boldsymbol k}, \sigma}^† f_{{\boldsymbol q}-k', \sigma} b_{{\boldsymbol k}'},~~ \tag {1} \end{align} $$ where $\epsilon^{\rm b,f}_{k}=k^2/2m_{\rm b,f}$, $m_{\rm b}$ ($m_{\rm f}$) is the mass of boson (fermion) atoms, $b^†_{\boldsymbol k}$ ($b_{\boldsymbol k}$) creates (annihilates) a bosonic atom with momentum ${\boldsymbol k}$, $f^†_{{\boldsymbol k},\sigma}$ ($f_{{\boldsymbol k},\sigma}$) creates (annihilates) a fermionic atom with pseudo-spin $\sigma$ ($\sigma=\uparrow,\downarrow$) and momentum ${\boldsymbol k}$. The bare boson-boson interactions $g_{\rm b}$, the bare boson-fermion interaction, and the bare fermion-fermion interaction are related to the scattering lengths $a_{\rm b}$, $a_{\rm bf}$, and $a_{\rm f}$, respectively, through the renormalization relation $1/U_i=1/g_i-(1/V)\sum_{\boldsymbol k}m_i/k^2$ ($i$ = b, bf, f), where $g_i=4\pi a_i/m_i$, with $m_{\rm bf}=2m_{\rm b}m_{\rm f}/(m_{\rm b}+m_{\rm f})$ and $V$ the quantization volume. Here we take $\hbar=1$, and assume that the interaction strength between bosons and fermions is independent of fermion species. For a repulsively interacting BEC and attractive boson-fermion interaction, we have $g_{\rm b}>0$ and $g_{\rm bf} < 0$. On the BCS side of the Feshbach resonance, the bose-fermion interaction is dominated by collisions of Bogoliubov quasiparticles of the Bose and Fermi superfluids. Assuming small depletions of the superfluids, we rewrite the Hamiltonian as $$\begin{alignat}{1} H=E_{\rm b}+E_{\rm f}+\sum_{\boldsymbol k} \omega_{k} \alpha_{\boldsymbol k}^† \alpha_{\boldsymbol k}+\sum_{{\boldsymbol k}, \sigma} \xi_{k} \beta_{{\boldsymbol k}, \sigma}^† \beta_{{\boldsymbol k}, \sigma}+H_{\rm int},~~ \tag {2} \end{alignat} $$ where $\alpha_{\boldsymbol k}=u_{\boldsymbol k} b_{\boldsymbol k}+v_{\boldsymbol k}b^†_{-{\boldsymbol k}}$ ($\beta_{{\boldsymbol k},\sigma}=m_{\boldsymbol k}f_{-\eta_\sigma{\boldsymbol k},\sigma}+\eta_\sigma l_{\boldsymbol k}f^†_{\eta_\sigma {\boldsymbol k},\bar\sigma}$) is the annihilation operator for the boson (fermion) Bogoliubov quasiparticles, with dispersions $\omega_{\boldsymbol k}=\sqrt{\epsilon^b_{k}(\epsilon^b_k+2g_{\rm b}n_{\rm b})}$ and $\xi_{\boldsymbol k}=\sqrt{(\epsilon^f_k-\mu)^2+\varDelta^2}$, respectively. Here, the Bogoliubov coefficients $u_{\boldsymbol k}^2=1+v_{\boldsymbol k}^2=\frac{1}{2}\left(1+\frac{\epsilon^{b}_{k}+g_{\rm b} n_{\rm b}}{\omega_{\boldsymbol k}}\right)$ and $l_{\boldsymbol k}^{2}=1-m_{\boldsymbol k}^2=\frac{1}{2}\left(1-\frac{\epsilon^f_k-\mu}{\xi_{\boldsymbol k}}\right)$, $n_{\rm b}$ is the BEC density, $\mu$ and $\varDelta$ are respectively the chemical potential and pairing order parameter of the Fermi superfluid, $\eta_{\downarrow}=-\eta_{\uparrow}=1$, and $\bar\sigma$ represents the opposite spin to $\sigma$.
cpl-37-7-076701-fig1.png
Fig. 1. Functions (a) $\beta(\frac{a_{\rm b}}{a_{\rm f}})$ and (b) $f(\frac{m_{\rm b}}{m_{\rm f}},\frac{n_{\rm b}}{n_{\rm f}},\frac{a_{\rm b}}{a_{\rm f}})$ with increasing $a_{\rm b}/a_{\rm f}$ on the BCS side of the resonance.[44] We fix the parameters $a_{\rm bf}/a_{\rm b}=-10$, and $n_{\rm b}/n_{\rm f}=1$. In (b), the blue (red) curve corresponds to the mass ratio $m_{\rm b}/m_{\rm f}=1$ ($m_{\rm b}/m_{\rm f}=87/40$). Here $\beta_0=3(3\pi^2)^{2/3}/5m_{\rm f}$ and $f_0=f(z,1,\infty)$ are respectively proportional to the Fermi pressure and the second-order corrections due to boson-fermion interactions, in the absence of fermion pairing interactions.
The ground-state energy densities of the BEC and the Fermi superfluid, defined as $\mathcal{E}_{\rm b,f}=E_{\rm b,f}/V$, are respectively given as $$\begin{alignat}{1} \mathcal{E}_{\rm b}&=\frac{g_{\rm b}n_{\rm b}^2}{2}+g_{\rm LHY}n_{\rm b}^{5/2},~~ \tag {3} \end{alignat} $$ $$\begin{alignat}{1} \mathcal{E}_{\rm f}&=\frac{1}{V}\sum_{\boldsymbol k} \Big(\epsilon^f_k-\mu-\xi_{\boldsymbol k}+\frac{\varDelta^2}{2\xi_{\boldsymbol k}}\Big)+\mu n_{\rm f} :=\beta\Big(\frac{a_{\rm b}}{a_{\rm f}}\Big) n_{\rm f}^{5/3},~~ \tag {4} \end{alignat} $$ where the second term in Eq. (3) corresponds to the LHY correction, with $g_{\rm LHY}=64 /(15 \sqrt{\pi}) g_{\rm b} a_{\rm b}^{3/2}$, and $n_{\rm f}$ is the fermion density. The function $\beta(\frac{a_{\rm b}}{a_{\rm f}})$ reflects the reduced Fermi pressure in the presence of pairing superfluid, with $\beta(\frac{a_{\rm b}}{a_{\rm f}})$ approaching $\beta_0:=3(3\pi^2)^{2/3}/5m_{\rm f}$ in the BCS limit [see Fig. 1(a)],[44] with $\beta_0 n_{\rm f}^{5/3}$ being the Fermi pressure for a non-interacting two-component Fermi gas. Writing the interaction term in terms of quasiparticle operators $$\begin{align} &H_{\rm int}=g_{\rm b f} n_{\rm b} \sum_{{\boldsymbol k}, \sigma} l_{\boldsymbol k}^{2} \beta_{{\boldsymbol k}, \sigma} \beta_{{\boldsymbol k}, \sigma}^†\\ &+g_{\rm bf} \sqrt{\frac{n_{\rm b}}{V}} \sum_{{\boldsymbol k}, {\boldsymbol q}, \sigma, \bar{\sigma}}\left(u_{\boldsymbol k}+v_{\boldsymbol k}\right) m_{{\boldsymbol k}+{\boldsymbol q}} l_{{\boldsymbol q}} \alpha_{-{\boldsymbol k}}^† \beta_{{\boldsymbol k}+{\boldsymbol q}, \sigma}^† \beta_{-{\boldsymbol q}, \bar{\sigma}}^†,~~ \tag {5} \end{align} $$ we derive the mean-field boson-fermion interaction energy $g_{\rm b f}n_{\rm b} n_{\rm f}$, and its second-order energy correction $$\begin{align} \mathcal{E}^{(2)}&=g_{\rm b f}^{2} n_{\rm f} n_{\rm b} \sum_{\boldsymbol k} \frac{m_{\rm bf}}{k^{2}}\\ &-2g_{\rm b f}^{2} n_{\rm b} \sum_{{\boldsymbol k}, {\boldsymbol q}} \frac{(u_{\boldsymbol k}+v_{\boldsymbol k})^{2} l^2_{{\boldsymbol q}}m_{{\boldsymbol q}+{\boldsymbol k}}^2}{\omega_{\boldsymbol k}+\xi_{{\boldsymbol k}+{\boldsymbol q}}+\xi_{{\boldsymbol q}}}\\ &-2g_{\rm b f}^{2} n_{\rm b} \sum_{{\boldsymbol k}, {\boldsymbol q}} \frac{(u_{\boldsymbol k}+v_{\boldsymbol k})^{2} l_{{\boldsymbol q}}m_{{\boldsymbol q}} l_{{\boldsymbol q}+{\boldsymbol k}} m_{{\boldsymbol q}+{\boldsymbol k}}}{\omega_{\boldsymbol k}+\xi_{{\boldsymbol k}+{\boldsymbol q}}+\xi_{{\boldsymbol q}}}\\ &:=g_{\rm b f}^{2} n_{\rm b} n_{\rm f}^{\frac{4}{3}} f\Big(\frac{m_{\rm b}}{m_{\rm f}},\frac{n_{\rm b}}{n_{\rm f}},\frac{a_{\rm b}}{a_{\rm f}}\Big),~~ \tag {6} \end{align} $$ where the function $f(\frac{m_{\rm b}}{m_{\rm f}},\frac{n_{\rm b}}{n_{\rm f}},\frac{a_{\rm b}}{a_{\rm f}})$ reflects the magnitude of the second-order correction.[44] In Fig. 1(b), we see that $f(\frac{m_{\rm b}}{m_{\rm f}},\frac{n_{\rm b}}{n_{\rm f}},\frac{a_{\rm b}}{a_{\rm f}})$, being positive in the deep BCS regime, decreases as the fermion-fermion interaction is tuned toward resonance, and becomes negative in the strong-interaction regime on the BCS side of the resonance. Such a behavior signals a qualitative change in the second-order energy correction, from repulsive (in the BCS limit) to attractive (close to resonance), which significantly impacts the droplet formation, as we show later. We also note that the second-order corrections in Eq. (6) account for the scattering of BEC phonons with Bogoliubov quasiparticles of the Fermi superfluid. Such corrections originate from the intrinsic fluctuations in the system, which also contribute to induced interactions between fermions[45] as well as those between bosons.[46] As a consideration of these induced interactions on top of Eq. (6) leads to double-counting of the fluctuation effect, in this work we have not included them explicitly. It follows from the above derivation that the ground-state energy density of the Bose–Fermi mixture is given by $$\begin{align} \mathcal{E}={}&\frac{g_{\rm b} n_{\rm b}^{2}}{2}+\beta\Big(\frac{a_{\rm b}}{a_{\rm f}}\Big) n_{\rm f}^{5/3}+g_{\rm b f} n_{\rm b} n_{\rm f}+g_{\rm LHY} n_{\rm b}^{5/2}\\ &+g_{\rm b f}^{2} n_{\rm b} n_{\rm f}^{\frac{4}{ 3}} f\Big(\frac{m_{\rm b}}{m_{\rm f}},\frac{n_{\rm b}}{n_{\rm f}},\frac{a_{\rm b}}{a_{\rm f}}\Big),~~ \tag {7} \end{align} $$ which is applicable on the BCS side of the resonance. In the deep BEC regime, however, the boson-fermion interaction involves breaking of composite bosons, which are higher-energy processes than the collision between Bose atoms and composite bosons formed by fermions.[47,48] It is therefore convenient to treat the system as a two-component BEC, with the ground-state energy density given by $$\begin{alignat}{1} \mathcal{E}={}&\frac{g_{\rm b}n_{\rm b}^2}{2}+\frac{g_dn_{\rm f}^2}{2}+g_{\rm bd}n_{\rm b}n_{\rm f}\\ &+\frac{8}{15\pi^2}m_{\rm b}^{3/2}(g_{\rm b}n_{\rm b})^{5/2} g\Big(\frac{2m_{\rm f}}{m_{\rm b}},\frac{g_dn_{\rm f}}{g_{\rm b}n_{\rm b}},\frac{g_{\rm bd}^2}{g_dg_{\rm b}}\Big).~~ \tag {8} \end{alignat} $$ Here $g_d=4\pi a_d/m_d$, $g_{\rm bd}=2\pi a_{\rm bd}/m_{\rm bd}$, where we take $a_d=0.6a_{\rm f}$,[49] $a_{\rm bd}=m_{\rm bd}a_{\rm bf}/m_{\rm bf}$,[47,48] $m_d=2m_{\rm f}$, $m_{\rm bd}=2m_{\rm b}m_{\rm f}/(m_{\rm b}+2m_{\rm f})$. The last term in Eq. (8) includes the LHY corrections for the two-component BEC.[44] With Eqs. (7) and (8), we can determine the existence and properties of the ground-state droplets through the conditions:[35] (i) $\mathcal{E} < 0,~ \mathcal{P}=0$; (ii) $\mu_{\rm b} \frac{\partial \mathcal{P}}{\partial n_{\rm f}}=\mu_{\rm f} \frac{\partial \mathcal{P}}{\partial n_{\rm b}}$; (iii) $\frac{\partial \mu_{\rm b}}{\partial n_{\rm b}}>0, ~\frac{\partial \mu_{\rm f}}{\partial n_{\rm f}}>0, ~\frac{\partial \mu_{\rm b}}{\partial n_{\rm b}} \frac{\partial \mu_{\rm f}}{\partial n_{\rm f}}>\left(\frac{\partial \mu_{\rm b}}{\partial n_{\rm f}}\right)^{2}$, where $\mu_{\rm b}=\frac{\partial \mathcal{E}}{\partial n_{\rm b}}$, $\mu_{\rm f}=\frac{\partial \mathcal{E}}{\partial n_{\rm f}}$, and the pressure $\mathcal{P}=n_{\rm b}\mu_{\rm b}+n_{\rm f}\mu_{\rm f}-\mathcal{E}$. We note that the chemical potential $\mu$ and the pairing order parameter $\varDelta$ of the Fermi superfluid are determined from the standard gap and number equations of the mean-field BCS-BEC crossover theory.[43] We emphasize from the outset that, while our approach provides an accurate description in the BCS- or BEC-limit, it should fail in the unitary region close to resonance, due to the sensitivity of droplets to higher-order quantum fluctuations that we neglect in our calculations. In the intermediate region, our approach should provide a qualitatively valid picture. We therefore mostly focus on the parameter range $|a_{\rm b}/a_{\rm f}|>0.5$ in the following.
cpl-37-7-076701-fig2.png
Fig. 2. Energy-density contours on the $n_{\rm b}$–$n_{\rm f}$ plane, for (a)–(d) $m_{\rm b}/m_{\rm f}=1$ and (e)–(h) $m_{\rm b}/m_{\rm f}=87/40$. Contours in (a), (b), (e) and (f) are calculated for the BCS side using Eq. (7), with [(a),(e)] $a_{\rm b}/a_{\rm f}=-2$ and [(b),(f)] $a_{\rm b}/a_{\rm f}=-0.6$. Those in (c), (d), (g) and (h) are calculated using Eq. (8) for the BEC region, with [(c),(g)] $a_{\rm b}/a_{\rm f}=0.5$ and [(d),(h)] $a_{\rm b}/a_{\rm f}=1$. The white dashed lines represent zero-pressure ($\mathcal{P}=0$) contours and the red dots represent the minimum energy along the contours.
In Fig. 2, we show the calculated energy-density contours (color bar) on the plane of the boson and fermion densities ($n_{\rm f}$ and $n_{\rm b}$), with a large, attractive boson-fermion interaction ($a_{\rm bf}/a_{\rm b}=-10$) on either side of the resonance. Note that we use $a_{\rm b}$ and $2\pi/(m_{\rm b} a_{\rm b}^5)$ as the units for length and energy density, respectively. The zero-pressure contours are shown by the white dashed lines, and the red dots mark the energy minima along the corresponding contours, which determine the condition for stable droplets. Apparently, when the fermion-fermion interaction $a_{\rm b}/a_{\rm f}$ is tuned toward resonance, stable droplets on the BCS side feature increased densities, whereas those on the BEC side have decreasing fermion densities but slightly increased boson densities. While the dominance of attractive (repulsive) energy contributions leads to larger (smaller) densities, the tendencies discussed above reflect the different nature and behavior of quantum fluctuations in different regions of the BCS-BEC crossover.
cpl-37-7-076701-fig3.png
Fig. 3. Densities of stable droplets for [(a),(c)] $m_{\rm b}/m_{\rm f}=1$, and [(b),(d)] $m_{\rm b}/m_{\rm f}=87/40$, throughout the BCS-BEC crossover. The blue, red, and black lines respectively correspond to $a_{\rm bf}/a_{\rm b}=-18$, $a_{\rm bf}/a_{\rm b}=-14$, and $a_{\rm bf}/a_{\rm b}=-10$. The solid lines are calculated using Eq. (7), and the dashed lines are calculated using Eq. (8). The magenta dash-dotted lines indicate positions of $a_{\rm b}/a_{\rm f}=\pm 0.5$.
cpl-37-7-076701-fig4.png
Fig. 4. Boson and fermion densities of stable droplets with varying boson-fermi interactions $a_{\rm bf}/a_{\rm b}$, for [(a),(c)] $m_{\rm b}/m_{\rm f}=1$ and [(b),(d)] $m_{\rm b}/m_{\rm f}=87/40$, respectively. Both the BCS side $a_{\rm b}/a_{\rm f}=-0.5$ and BEC side $a_{\rm b}/a_{\rm f}=0.5$ are shown.
In Fig. 3, we explicitly plot the boson and fermion densities for the ground-state droplets as the fermion-fermion interaction is tuned. For both mass ratios $m_{\rm b}/m_{\rm f}=1$ [Figs. 3(a) and 3(c)] and $m_{\rm b}/m_{\rm f}=87/40$ [Figs. 3(b) and 3(d)], the boson and fermion densities monotonically increase toward resonance on the BCS side. This suggests that, when the fermion-fermion interaction is tuned toward resonance on the BCS side, the repulsive energy indeed becomes weaker, and droplets with a given density can thus be stabilized at a smaller attractive boson-fermion interaction strength, giving rise to a larger stability region in the BCS regime. However, considering atom-loss processes at large densities, the droplets are practically unstable very close to the resonance, once the densities become appreciable. By contrast, on the BEC side, the fermion density still decreases monotonically toward resonance, while the boson density first increases, but undergoes a rapid drop very close to resonance. We note that such a rapid drop in the boson density on the BEC side could be avoided when fermion-fermion fluctuations are taken into account, which should be attractive,[50] and significantly close to resonance. Our results thus suggest that, while droplets are stabilized over a considerable parameter region on either side of the resonance, their stability near the resonance sensitively depends on higher-order fluctuations. Furthermore, the stabilization of droplets on different sides of the resonance are due to the quantum fluctuations of different physical origin. On the BCS side, the dominant fluctuation energy comes from the second-order correction in the boson-fermion interaction, which becomes less repulsive and even negative close to resonance. However, in the deep BEC regime, it mainly comes from the repulsive LHY corrections. On the other hand, under a fixed fermion-fermion interaction on either side of the resonance, both fermion and boson densities of a stable droplet first increase then decrease as the attractive boson-fermion interaction becomes weaker (smaller $|a_{\rm bf}|$). Such a non-monotonic behavior is illustrated in Fig. 4, which suggests that droplets are destabilized by dominant repulsive energy contributions for sufficiently small boson-fermion interactions. While a similar situation occurs for a Bose–Fermi mixture with non-interacting Fermi gas, a key question here is whether droplets in a mixture of Bose–Fermi mixture can still survive under a weak boson-fermion interaction.
cpl-37-7-076701-fig5.png
Fig. 5. Phase diagram for stable droplets for the mass ratio (a) $m_{\rm b}/m_{\rm f}=1$, and (b) $m_{\rm b}/m_{\rm f}=87/40$. The shaded areas indicate the region $|a_{\rm b}/a_{\rm f}| < 0.5$, where our approach becomes unreliable. The phase boundaries (solid and dashed blue curves) are determined by the threshold $\max (n_{\rm b}a_{\rm b}^3,n_{\rm f}a_{\rm b}^3)=10^{-7}$.
To address the question, in Fig. 5, we show the phase diagrams for stable droplets on the $a_{\rm bf}/a_{\rm b}$–$a_{\rm b}/a_{\rm f}$ plane for different mass ratios. The phase boundaries between the liquid (stable) and gas (unstable) regions are determined by the small-density threshold $\max (n_{\rm b}a_{\rm b}^3,n_{\rm f}a_{\rm b}^3)=10^{-7}$, where densities for stable droplets become vanishingly small under a dominant repulsive energy contribution. On the BCS side, our phase diagrams clearly indicate the increased stability of droplets toward weaker boson-fermion interactions, when the fermions are tuned closer to resonance. This is a direct consequence of the decrease in the repulsive second-order boson-fermion fluctuation energy, as shown in Fig. 1(b). Furthermore, the critical boson-fermion interaction $a_{\rm bf}/a_{\rm b}$ for stable droplets remains small over a large region on the BEC side. For instance, with $a_{\rm b}/a_{\rm f}=2$, the droplets are stable for $a_{\rm bf}/a_{\rm b} < -0.45$ ($a_{\rm bf}/a_{\rm b} < -0.41$) with $m_{\rm b}/m_{\rm f}=1$ ($m_{\rm b}/m_{\rm f}=87/40$). Thus, an important message here is that stable droplets can be observed in a mixture of Bose–Fermi superfluids under a relatively weak boson-fermion interactions, either on the BCS side or, more dramatically, on the BEC side of the fermion-fermion Feshbach resonance. As a concrete example, we consider a mixture of $^{87}$Rb and $^{40}$K, close to the wide Feshbach resonance of $^{40}$K near $202$ G. Here the scattering lengths are: $95a_0$ between $^{87}$Rb and $^{87}$Rb atoms, $-261a_0$ between $^{87}$Rb and $^{40}$K, with $a_0$ the Bohr radius. We therefore have $a_{\rm bf}/a_{\rm b}\approx-2.75$, which should feature stable droplets on the BEC side of the resonance. Further, for a typical density $n_{\rm b}=5\times10^{-14}{\rm cm}^{-3}$, we have $n_{\rm b}a_{\rm b}^3\approx 10^{-5}$, which is on the order of stable densities shown in Fig. 3. Finally, it should be noted that we have neglected fluctuation energy of the fermion-fermion interaction for our calculation, as well as higher-order fluctuations. These contributions should ultimately determine the fate of droplets in the unitary region close to the Feshbach resonance. For example, the inclusion of the attractive, higher-order fermion-fermion fluctuations in the second-order correction of $g_{\rm bf}$[50] would provide a competition against the LHY corrections on the BEC side, thus further enhance droplet formation on the BEC side. However, the same contribution would further increase the density of stable droplets close to resonance on the BCS side, which would practically destabilize the system due to atom loss. These considerations suggest that droplet formation in the unitary region is still an interesting open question, which we leave to future studies. We thank Zengqiang Yu for helpful comments and discussion. 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