Depletion of the Bose-Einstein condensate in Bose-Fermi mixtures

We describe the properties of a mixture of fermionic and bosonic atoms, as they are tuned across a Feshbach resonance associated with a fermionic molecular state. Provided the number of fermionic atoms exceeds the number of bosonic atoms, we argue that there is a critical detuning at which the Bose-Einstein condensate (BEC) is completely depleted. The phases on either side of this quantum phase transition can also be distinguished by the distinct Luttinger constraints on their Fermi surfaces. In both phases, the total volume enclosed by all Fermi surfaces is constrained by the total number of fermions. However, in the phase without the BEC, which has two Fermi surfaces, there is a second Luttinger constraint: the volume enclosed by one of the Fermi surfaces is constrained by the total number of bosons, so that the volumes enclosed by the two Fermi surfaces are separately conserved. The phase with the BEC may have one or two Fermi surfaces, but only their total volume is conserved. We obtain the phase diagram as a function of atomic parameters and temperature, and describe critical fluctuations in the vicinity of all transitions. We make quantitative predictions valid for the case of a narrow Feshbach resonance, but we expect the qualitative features we describe to be more generally applicable. As an aside, we point out intriguing connections between the BEC depletion transition and the transition to the fractionalized Fermi liquid in Kondo lattice models.

Figure 1

Phase boundary with detuning $\nu$ and temperature $T$, for fixed particle numbers $N_f∕N_b=1.11$. The dashed line has vanishing coupling and has been found with a purely classical analysis. The solid line has dimensionless coupling ${\gamma }^2∕T_0=2.5\times {10}^{-4}$ and has been determined using the mean-field theory of Sec. 4. For both, the condensed phase is on the left-hand side (for lower $T$) and labeled by $⟨b⟩\ne 0$.

Figure 2

Phase boundary with fermion number $N_f$ and detuning $\nu$, for three different temperatures. The coupling is ${\gamma }^2∕T_0=2.5\times {10}^{-4}$ and the masses are equal, $m^f=m^b$. The two phases are labeled as in Fig. 1, with the condensed phase favored for higher detuning, lower fermion number, and lower temperature.

Figure 3

The phase diagram at $T=0$ with dimensionless couplings (a) ${\gamma }^2∕T_0={10}^{-6}$, (b) ${\gamma }^2∕T_0=2.5\times {10}^{-4}$, and (c) ${\gamma }^2∕T_0=2.0\times {10}^{-2}$. The atomic masses have been taken to be equal, $m^f=m^b$, and the coupling between bosons is given by ${\lambda }^2{\left(m^b\right)}^3{T}_0=2\times {10}^{-3}$. The three distinct phases have, respectively, no Bose-Einstein condensate and two Fermi surfaces (labeled “2 FS, no BEC”), a condensate and two Fermi surfaces (“$2\mathrm{FS}+\mathrm{BEC}$”), and a condensate and a single Fermi surface (“$1\mathrm{FS}+\mathrm{BEC}$”). The dotted line indicates the fermion number at which Fig. 4 is plotted.

Figure 4

The effective mass $m^{\star }$ at the Fermi surface, with fermion number $N_f=0.1N_b$, coupling ${\gamma }^2∕T_0=2.5\times {10}^{-4}$, and equal atomic masses $m^f=m^b$. As can be seen from the dotted line in Fig. 3, these parameters give a phase with a single Fermi surface. This surface changes from having a molecular character, with $m^{\star }\simeq m^{\psi }$, to having an atomic character, $m^{\star }\simeq m^f$.

Figure 5

The Fermi wave numbers for the two mixed species of fermions, $\Psi$ and $F$, with coupling ${\gamma }^2∕T_0=2.5\times {10}^{-4}$ and equal atomic masses $m^f=m^b$. The solid lines have fermion number $N_f∕N_b=3∕2$, while the dashed lines have $N_f∕N_b=1∕2$. As can be seen in Fig. 3, the solid line goes between all three phases (at $\nu ∕T_0\simeq 0.25$ and $\nu ∕T_0\simeq 2.9$), while the dashed line goes from the phase with a single Fermi surface to that having two and back again (at $\nu ∕T_0\simeq -0.65$ and $\nu ∕T_0\simeq 1.3$). The wave numbers are measured in units of $k_f$, the Fermi wave number for free fermions with number $N_f$.

Figure 6

The phase diagram with the fermion chemical potential ${\mu }^f$ plotted on the vertical axis and the detuning $\nu$ on the horizontal axis. Both have been scaled to ${\mu }^{\psi }\text{−}\nu$, which is held fixed throughout the plot. The boundary between 2 FS, no BEC, and $1\mathrm{FS}+\mathrm{BEC}$ in Fig. 3 expands into a new phase, labeled “1 FS, no BEC,” within which there are only molecules, whose density is constant (both $N_f$ and $N_b$ remain fixed in the shaded region). The atomic masses are equal, $m^f=m^b$, and the couplings are ${\gamma }^2∕T_0=2.5\times {10}^{-4}$ and ${\lambda }^2{\left(m^b\right)}^3{T}_0=2\times {10}^{-3}$. (Since the boson density is not fixed in this plot, the value of $T_0$ appropriate to the phase 1 FS, no BEC has been used to define the dimensionless couplings.)

Figure 7

Two-loop corrections to particle numbers, shown as a fraction of the total particle numbers to lowest order. The masses of the particles and their total numbers are the same as in Fig. 1, $N_f∕N_b=1.11$. The dimensionless coupling is ${\gamma }^2∕T_0=2.5\times {10}^{-4}$. At each temperature, the corrections have been evaluated taking the detuning at its critical value, from Fig. 1.

Figure 8

The phase boundary, without (solid line) and with (dashed line) the correction to the detuning of Sec. 8. The solid line is the phase boundary as in Fig. 1, using the same parameters (and dimensionless coupling ${\gamma }^2∕T_0=2.5\times {10}^{-4}$). The dashed line includes the correction $\delta \nu$, from Eq. (8.18), found using a low-density approximation. For clarity, this correction has been exaggerated by a factor of 3. Since the correction is negligible for these parameters, the mean-field results of Secs. 4, 6 are sufficient.

Figure 9

The phase diagram at $T=0$, as in Fig. 3, with couplings ${\gamma }^2∕T_0=2.0\times {10}^{-2}$ and ${\lambda }^2{\left(m^b\right)}^3{T}_0=2\times {10}^{-3}$. The other parameters and the labels for the three phases are the same as in Fig. 3. The region where the phase is unstable, as determined in the Appendix, is indicated.