Realization of a Strongly Interacting Bose-Fermi Mixture from a Two-Component Fermi Gas

We show the emergence of a strongly interacting Bose-Fermi mixture from a two-component Fermi mixture with population imbalance. By analyzing in situ density profiles of ${}^6\mathrm{Li}$ atoms in the BCS-BEC crossover regime, we identify a critical interaction strength, beyond which all minority atoms pair up with majority atoms and form a Bose condensate. This is the regime where the system can be effectively described as a boson-fermion mixture. We determine the dimer-fermion and dimer-dimer scattering lengths and beyond-mean-field contributions. Our study realizes a gedanken experiment of bosons immersed in a Fermi sea of one of their constituents, revealing the composite nature of the bosons.

Figure 1

Density profiles of imbalanced Fermi mixtures in a harmonic trap. The top row (a)–(e) shows the averaged column-density profiles for various magnetic fields [gray (green): majority, dark gray (blue): minority, black: difference]. The black dotted line is a zero-temperature Thomas-Fermi distribution fit to the majority wing ($r>R_{\downarrow }$). The middle row (f)–(j) and the bottom row (k)–(o) show the reconstructed three-dimensional density distributions and the spin polarizations obtained from the profiles in the top row. The critical polarizations ${\sigma }_c$ at the phase boundary $r=R_c$ are indicated by the right arrows. The values for $R_{\uparrow }$ (in $\mu \mathrm{m}$), $R_c/R_{\uparrow }$, and $R_{\downarrow }/R_{\uparrow }$ were respectively: for (a), (f), (k), 381, 0.33, 0.33; for (b), (g), (l), 380, 0.33, 0.33; for (c), (h), (m), 362, 0.35, 0.59; for (d), (i), (n), 371, 0.44, 0.72; for (e), (j), (o), 367, 0.41, 0.76. $T/T_{F0}\lesssim 0.05$, and $T_{F0}\approx 1.0\mu \mathrm{K}$ (see the text for definitions).

Figure 2

Emergence of a Bose-Fermi mixture in the phase diagram for a two-component Fermi gas. (a) The critical polarization ${\sigma }_c$ as a function of the interaction strength $1/k_{F\uparrow }a$ at the phase boundary. The open circle indicates the previously measured critical value on resonance, ${\sigma }_{c0}=0.36$ [2]. (b) $\kappa =(R_{\uparrow }^2-R_{\downarrow }^2)/(R_{\uparrow }^2-R_c^2)$. The solid (red) lines are (a) an exponential fit and (b) a linear fit to the data points. ${\sigma }_c=1$ and $\kappa =1$ (i.e., $R_{\downarrow }=R_c$) imply the absence of minority fermions in the normal phase. Each data point consists of 9 to 23 independent measurements, and the error bars indicate only the statistical uncertainty.

Figure 3

Characterization of a strongly interacting Bose-Fermi mixture. (a) The scattering length for dimer-fermion interactions $a_{\mathrm{bf}}$ and (b) for dimer-dimer interactions $a_{\mathrm{bb}}$ in units of the fermion-fermion scattering length $a$. Black solid circles were determined using mean-field theory and open (red) circles including the LHY correction for a strongly interacting Bose gas. The dashed lines indicate the calculated values $a_{\mathrm{bf}}=1.18a$ [4] and $a_{\mathrm{bb}}=0.6a$ [13]. Each data point represents 7 to 17 measurements, and the error bars indicate only the statistical uncertainty.

Figure 4

Observed profiles of strongly interacting Bose-Fermi mixtures compared to calculated profiles without any adjustable parameter. (a)–(c) Density profiles of bosonic dimers [dark gray (blue)] and unpaired excess fermions [gray (green)] for various magnetic fields. The numerically obtained density profiles (d)–(f) for bosons and (g)–(i) for fermion use $a_{\mathrm{bb}}=0.6a$ and $a_{\mathrm{bf}}=1.18a$ (dashed line: mean-field description, solid line: including the LHY correction). The horizontal dotted lines in (d)–(f) indicate the boson density corresponding to $1/k_{F,\mathrm{b}}a=1$. The values for $R_{\uparrow }$ (in $\mu \mathrm{m}$) and $R_c/R_{\uparrow }$ were respectively: for (a), 393 and 0.29; for (b), 381 and 0.33; for (c), 366 and 0.35.

Figure 5

Mean-field phase diagram of a dilute Bose-Fermi mixture. ${\overline{\mu }}_{\mathrm{f}}$ and ${\overline{\mu }}_{\mathrm{b}}$ are the chemical potentials for fermions and bosons in units of $\frac{{\hslash }^2}{2m_{\mathrm{f}}}(6{\pi }^2{\overline{n}}_{\mathrm{f}}{)}^{2/3}$ and $\frac{4\pi {\hslash }^2{a}_{\mathrm{bb}}}{m_{\mathrm{b}}}{\overline{n}}_{\mathrm{b}}$, respectively, where ${\overline{n}}_{\mathrm{f}}=\frac{9\pi }{2}\frac{a_{\mathrm{bb}}^3}{a_{\mathrm{bf}}^6}\frac{m_{\mathrm{b}}^3{m}_{\mathrm{f}}^3}{(m_{\mathrm{b}}+m_{\mathrm{f}}{)}^6}$ and ${\overline{n}}_{\mathrm{b}}=\frac{9\pi }{4}\frac{a_{\mathrm{bb}}^2}{a_{\mathrm{bf}}^5}\frac{m_{\mathrm{b}}^3{m}_{\mathrm{f}}^2}{(m_{\mathrm{b}}+m_{\mathrm{f}}{)}^5}$ (see Ref. 18). The thick lines indicate the phase transitions (solid line: first-order, dashed line: second-order). The thin lines represent typical cuts through the phase diagram realized in our trapped samples.