Phase separation and collapse in Bose-Fermi mixtures with a Feshbach resonance
We consider a mixture of single-component bosonic and fermionic atoms with an interspecies interaction that is varied using a Feshbach resonance. By performing a mean-field analysis of a two-channel model, which describes both narrow and broad Feshbach resonances, we find an unexpectedly rich phase diagram at zero temperature: Bose-condensed and non-Bose-condensed phases form a variety of phase-separated states that are accompanied by both critical and tricritical points. We discuss the implications of our results for the experimentally observed collapse of Bose-Fermi mixtures on the attractive side of the Feshbach resonance, and we make predictions for future experiments on Bose-Fermi mixtures close to a Feshbach resonance.
The possibility of controlling the inter-atomic interaction strength via Feshbach resonances has recently played a pivotal role in investigating condensation phenomena in ultracold alkali gases. A prominent example is the realization of condensed pairs in two-component Fermi gases, where one has a crossover from the weakly-interacting BCS regime to a Bose-Einstein condensate (BEC) of diatomic molecules. Regal et al. (2004); Zwierlein et al. (2004) By tuning the two spin populations to be unequal, one can further explore quantum phase transitions and phase separation (see, e.g., Refs. Shin et al., 2006, Partridge et al., 2006 and references therein). Even richer scenarios are expected for heteronuclear resonances in Bose-Fermi mixtures, because one can in principle destroy the BEC by binding bosons and fermions into fermionic molecules. Powell et al. (2005) Moreover, a host of novel phenomena has been predicted, such as spatial separation of bosons and fermions induced by interspecies repulsive interactions, Mølmer (1998); Viverit et al. (2000) boson-mediated Cooper pairing, Heiselberg et al. (2000); Suzuki et al. (2008); Kalas et al. () density wave phases in optical lattices, Lewenstein et al. (2004) and the formation of polar molecules with dipolar interactions.
Following the recent detection of Feshbach resonances in Bose-Fermi mixtures, Stan et al. (2004); Inouye et al. (2004); Ferlaino et al. (2006); Deh et al. (2008) experiments have begun to explore how the behavior of the Rb-K system depends on the interspecies interaction strength. Ospelkaus et al. (2006a, b); Zaccanti et al. (2006); Modugno (); Zirbel et al. (2008) Thus far, repulsive interactions (generated by approaching the resonance from the side of positive scattering length ) have been observed to reduce the spatial overlap between fermions and the BEC. By contrast, attractive interactions can produce a sudden loss of atoms, which has been attributed to enhanced three-body recombination processes resulting from an unconstrained increase in the density. Current mean-field theories Mølmer (1998); Modugno et al. (2003); Chui et al. (2004); Adhikari (2004) predict that this total collapse of the mixture occurs above a critical density or interaction strength, where the mixture is dynamically unstable. However, these theories neglect any pairing between bosons and fermions, which we show plays a crucial role in the stability of the system.
In this paper, we address the possibility of pairing in Bose-Fermi mixtures by considering a two-channel model that explicitly includes fermionic molecules as an extra species of particle. Such a model encompasses both broad and narrow Feshbach resonances. We then determine the zero-temperature phase diagram for the Bose-Fermi mixture as a function of the interaction strengths and particle densities within a mean-field approximation. Sufficiently close to the resonance, we always find phase separation between a mixed BEC phase (where the BEC coexists with fermions) and either a pure BEC, mixed BEC, normal or vacuum phase. To our knowledge, this provides the first example of phase separation in a Bose-Fermi mixture with attractive interactions — e.g., He-He mixtures Graf et al. (1967) and polarized Fermi gases Shin et al. (2006); Partridge et al. (2006) require effectively repulsive interactions. We explain why our results are consistent with the collapse observed in current experiments and we discuss the best conditions for observing the predicted phase separation.
The paper is organized as follows: Section II describes the two-channel model of a Bose-Fermi mixture and how it is related to the single-channel theory. In Sec. III, we derive the zero temperature phase diagram of a homogeneous mixture. In Sec. IV we discuss its connection with current and future experiments and we then consider how the phase diagram will manifest itself in a trapped gas in Sec. IV.1. We present our conclusions in Sec. V.
Ii Two-channel model
We consider the two-channel Hamiltonian,
which describes a mixture of bosonic and fermionic atoms coupled to a fermionic closed channel molecule via the interaction , where is the three-dimensional volume. Setting , the kinetic terms are given by
where is the detuning from resonance and () is the chemical potential for the bosons (fermions). We also include the boson-boson interaction and the background boson-fermion interaction , with . Clearly, we always require if we want the Bose gas to be stable.
A key energy scale in our model is the width of the resonance
where is the absolute width of the resonance in terms of the magnetic field and is the difference in magnetic moments between the closed and open channels (which is of order the Bohr magneton). When compared to the boson condensation temperature, (with ) and the Fermi energy, , the width of the resonance defines both narrow, , and wide, , Feshbach resonances.
At zero temperature, the Hamiltonian (1) can be exactly diagonalized in the mean-field approximation, , implying that the two Fermi species dispersions, and , are now hybridized by the presence of the condensate:
This leads to the following expression for the grand-canonical free energy density :
When , the free energy is unbounded from below:
Therefore, a repulsive background interaction strength, , is required in order to have a stable solution. Later on, we will neglect in our calculations because, as we discuss in Sec. III, it will not affect the main features of the phase diagram provided it is sufficiently small.
Since experiments are performed at fixed densities, the chemical potentials and have to be determined from the total number of fermionic and bosonic atom densities,
Note that () includes fermions (bosons) bound into molecules. Despite the simplicity of our approach, we emphasize that our mean-field analysis will be quantitatively accurate in the case of narrow Feshbach resonances, Andreev et al. (2004) because Gaussian fluctuations beyond mean-field scale like . We also expect it to provide a qualitative description of the phase diagram for a broad Feshbach resonance as is the case in two-component Fermi gases — the phase diagram for polarized Fermi gases is qualitatively similar for broad and narrow Feshbach resonances. Parish et al. (2007)
ii.1 Connection to single-channel model
Before tackling the phase diagram, it is instructive to understand how our model connects with existing single-channel theories, where the closed channel molecule is ignored and we have Hamiltonian
with the effective Bose-Fermi interaction . Such a theory is expected to be appropriate for a broad Feshbach resonance, where there is only a small admixture of the closed channel molecule, which can therefore be neglected. However, a mean-field analysis of the single-channel model focuses on densities only and does not take account of pairing correlations. Within this approximation, the system becomes linearly unstable when: Viverit et al. (2000)
For the case of attractive interactions , one can easily show that the single-channel free energy is not bounded from below (). As a consequence, any state at finite density is metastable and, when Eq. (7) is satisfied, the system is unstable to a total collapse. Chui et al. (2004) On the other hand, when , the free energy is bounded and the instability is towards phase separation.
Now, we expect the two-channel model (1) to map to a single-channel Hamiltonian when we take the limit , , while holding fixed, because then the closed channel molecule effectively disappears from the problem while the scattering length is kept fixed. Indeed, we find that our two-channel mean-field theory (4) in the above limit is formally equivalent to the single-channel mean-field theory on the attractive side of the resonance with . Thus, in the low density regime we expect the two-channel theory to reduce to the single-channel theory. Therefore, we expect to have linear instabilities and first order transitions in the low-density region of the phase diagram. In addition, we do not expect our mean-field theory to include the pairing instabilities that have been shown to exist in the single-channel model, Kagan et al. (2004); Storozhenko et al. (2005) although we speculate that the narrow Feshbach resonance limit may capture the qualitative features of these pairing correlations.
Iii Phase Diagram
In the following, we will take the mass ratio to be as in Na-Li mixtures, because this atomic system has a narrow Feshbach resonance Stan et al. (2004); Gacesa et al. (2008) and a small repulsive . Thus we are likely to observe experimentally the phase-separated states we predict here. While the behavior of Na-Li mixtures is yet to be explored in detail, inter-species Feshbach resonances have already been identified experimentally. We emphasize that changing the mass ratio will only affect our results qualitatively and so our phase diagram will also be applicable to other atomic systems.
If one focuses on pairing only, it is clear that at the fixed density there is a transition as we cross the resonance: the BEC phase () will be completely depleted () by the binding of bosons into fermionic molecules. (Note that the formation of molecules can also involve a phase transition where the number of Fermi surfaces changes). This transition was shown to be continuous for the limiting case of vanishing coupling . Yabu et al. (2003) Assuming the transition remains second order for , as in Ref. Powell et al., 2005, the phase boundaries are found by solving . However, as we anticipated in Sec. II.1, in general we find that there can also be first order transitions between two BEC phases as well as between BEC and normal (N) phases. Before we explain the main features of the phase diagram, it is useful to examine how the second order transition becomes first order.
Precursors of first order transitions can be found in tricritical points, where and . Surprisingly, the mean-field energy can be expressed in terms of just three independent dimensionless parameters, and thus the phase transitions can be completely characterized by the dimensionless parameters
where . Referring to Fig. 1, in these units we find that for greater than the tetracritical values at which (open circles), there are lines of tricritical points [(blue) filled circles] where the transition changes from second to first order. Beyond the tetracritical points, the tricritical points are replaced by critical points [(red) filled squares]: At a critical point two equal energy minima of merge. In this case, an expansion in cannot be exploited any longer. However, this failure of the expansion is not so surprising because, when , the mean-field potential (4) is unbounded from below (). This non-analytic dependence on comes from the hybridized Fermi dispersions .
Therefore, to summarize, we find that the first-order regime is confined by either tricritical or critical points within a region about the origin of the parameter space. Outside of this region, we only find continuous transitions. From the definition of , one can see that the ratio sets the energy scale of the problem and determines the chemical potentials at which the tricritical points are replaced by critical points in Fig. 1. As explained above, one expects first order transitions to appear in the limit . However, note also that one can access the regime of first order transitions even when the repulsion between bosons is large, by making , and sufficiently small — as a consequence we expect these results to apply in the low density limit close to resonance. We have checked that a small repulsive background interaction only brings small quantitative changes to this result by slightly reducing the size of the first order region.
While one can parameterize the entire phase diagram using the rescaled parameters , from the point of view of real systems, it makes more sense to consider the parameters . For a given experiment, is fixed by the typical boson-boson interaction strength and width of the resonance, while is determined by the Bose-Fermi scattering length . Therefore, the phase diagram can be plotted as a function of the chemical potentials (top row of Fig. 2). These slices correspond to planes in Fig. 1 given by . In order to be relevant to experiments on Na-Li mixtures, we use the scattering lengths and , where is the Bohr radius, and take the resonance width to be G. Gacesa et al. (2008) For these values of the parameters, we have and . However, the qualitative behavior of the phase diagram as depicted in Fig. 2 will apply for all . In this regime, first order transitions appear only in the finite interval around unitarity, . At the resonance, we find two critical points (and two accompanying critical endpoints, where first and second order transition meet), while moving far from the resonance in either direction, first one and then both critical points are replaced by tricritical points, until eventually for large enough detuning the first order transition region disappears altogether.
In terms of the phase diagram in density space (bottom row of Fig. 2), first order transitions imply phase separation (PS) between BEC and N (BEC and BEC) close to a tricritical (critical) point. Here, N satisfies when and when . Both BEC and N phases can be further characterized by the number of Fermi surfaces (FS). In the N region, the second-order boundaries between regions with different numbers of FS are defined by () and (). In the BEC phase instead one has to impose the condition . A special case of PS occurs when the N phase is the vacuum: Physically this is equivalent to a partial collapse of the system to higher densities. On the attractive side of the resonance, , and for small enough densities, , we expect to recover the results of the single-channel theory. In particular, close enough to the resonance and for small enough densities, the condition for linear instability becomes independent of the boson density, resembling Eq. (7).
At this point, a question that naturally arises is: why does one observe phase separation instead of the total collapse predicted by the single-channel mean-field theory? The answer is that the presence of closed-channel fermionic molecules stabilizes the large behavior and constrains unbounded increases in density, converting the total collapse into phase separation. One can intuitively understand this as follows: once the Bose-Fermi mixture becomes unstable, the resulting increase in density will likewise increase the population of (virtual) closed-channel fermionic molecules, which in turn will exert an increasing Fermi pressure. Eventually, this Fermi pressure will balance the negative pressure of the attractive interactions so that the gas becomes stable at a finite density. This scenario is perhaps most clearly illustrated close to unitarity, where phase separation takes the form of a partial collapse.
Within the PS region it is possible to distinguish a metastable and an unstable (or spinodal) region separated by a spinodal line, where minimum and maximum of merge. Inside the spinodal region, the dynamics of phase separation following a sudden quench proceeds via a linear instability (see, e.g., Ref. Lamacraft and Marchetti, 2008 and references therein), while it is characterized by nucleation in the metastable region. A calculation of the spinodal lines has also been carried out in Ref. Powell et al., 2005 at a fixed , although the possibility of phase separation was not considered.
In addition, Ref. Powell et al., 2005 suggests that, when , there is a smooth crossover at fixed density from the atomic to the molecular side of the resonance within the same BEC-1FS phase. We find that this crossover exists for sufficiently large densities, while phase separation intervenes at smaller densities and (Fig. 2). More generally, the size of the phase-separated region scales like for both and the molecular component of , while the condensed component of scales like .
Finally, we note that the narrow Feshbach resonance regime, , corresponds to densities or , therefore our mean-field treatment should be reasonable for the region of interest. Certainly, the unstable region is a robust feature of the overall phase diagram, because we can arbitrarily increase the size of it in density-space by decreasing or .
Iv Implications for experiment
The existence of three-body recombination in Bose-Fermi mixtures poses a major challenge to investigating experimentally the phase diagram we have described in Sec. III. In current experiments on Rb-K mixtures, the behavior of the mixture is dominated by the large attractive background interaction — one even achieves collapse far from the resonance by increasing the density. Ospelkaus et al. (2006b, a); Zaccanti et al. (2006) Even if one ignores the background scattering length, we find that the phase-separated states close to resonance, as in Fig. 2, evaluated for the parameters of Rb-K, involve such a large increase in density (becoming of order cm or more) that they will be destroyed by three-body recombination. Thus, we do not expect experiments on Rb-K mixtures to reveal the rich variety of phase-separated states we have predicted. On the other hand, a density of cm in Na-Li mixtures corresponds to and so the phase-separated state possesses densities of order cm, which should be easily accessible experimentally.
The dominant three-body process involves 1 fermionic atom and 2 bosonic atoms, with recombination rate away from resonance. D’Incao and Esry (2005) Thus, to minimize in the phase-separated state, we must reduce the densities and/or at which phase separation first appears. Assuming the bosons are mostly condensed and using the scalings described previously, we get and . Therefore, we must consider small , such as the Feshbach resonance for Na-Li mixtures, Stan et al. (2004) and perhaps small , although the dependence of on is sensitive to the precise power of , which in turn can depend on the width of the resonance. Petrov (2004) One might also worry about three-body losses resulting from collisions between the closed-channel molecule and the open-channel atoms. However, we point out that the ‘bare’ molecule-atom interaction is generally unrelated to the resonance-induced interaction in the open channel and is, thus, generally negligible. Therefore, any collisions between closed-channel molecules and atoms will have to be due to effective interactions resulting from higher-order processes that involve multiple scattering among the atoms in the open-channel. Such processes on the attractive side of the resonance should already be encompassed by our analysis of three-body collisions above.
On the repulsive side of the resonance, , the mean-field single channel theory predicts phase separation between atomic bosons and fermions induced by the repulsion. Viverit et al. (2000) One can instead access the regime of model (1) by starting from and sweeping through the resonance to the molecular side, . Indeed, while most of the experiments have focused on the repulsive Bose-Fermi interaction, fermionic molecules have been very recently produced. Ospelkaus et al. (2006c); Modugno (); Zirbel et al. (2008) However, these studies currently paint a pessimistic picture of the stability of these real (as opposed to virtual closed-channel) molecules with respect to three-body recombination.
iv.1 Trapped gases
In principle, one can detect phase separation using in situ absorption measurements. To extract the behavior of the trapped gas from the uniform phase diagram (Fig. 2), we use the local density approximation to convert the effects of the trap into spatially varying chemical potentials. As shown in Fig. 3, the spatial trajectory in a trap is then represented by the straight line in the phase diagram, assuming that the fermionic and bosonic atoms experience the same harmonic trapping potential . Zaccanti et al. (2006) Phase separation in a trap occurs when the first-order boundary between BEC and N is crossed, yielding a discontinuity in the density profiles. In this case, the central phase is always a BEC (with 1FS), while the surrounding phases can be either BEC or N (with different numbers of FS). In particular, the partially-collapsed gas will exhibit a sharp interface with the vacuum. The presence of density discontinuities will provide the primary signature for the phase separation predicted here.
In conclusion, we have analyzed the zero temperature phase diagram for a Bose-Fermi mixture with an interaction strength tunable via a Feshbach resonance. By making use of a two-channel model, we have found a finite region of phase separation around resonance that is bounded by either tricritical or critical points. Close to unitarity, phase separation takes the form of a partial collapse of the system, where phase separation occurs between a higher density mixed BEC phase (BEC coexisting with either atomic or molecular fermions) and the vacuum. Such a sudden increase in density implies a larger three-body recombination, and thus our phase-separated states may be a challenge to realize experimentally. Indeed, current experiments on Rb-K mixtures are dominated by total collapse in the case of attractive interactions. Ospelkaus et al. (2006b, a); Zaccanti et al. (2006) However, we have argued that mixtures with a small resonance width and a small repulsive background interaction, such as Na-Li mixtures, Stan et al. (2004), stand a better chance of realizing the phase diagram we predict.
Finally, we note that in this work we have neglected the possibility of fermionic superfluidity induced by density fluctuations of the bosonic condensate: For a spin polarized Fermi gas, boson-mediated -wave Suzuki et al. (2008) and -wave odd-frequency Kalas et al. () Cooper pairing have been recently analyzed for the single-channel model. In both cases it has been found that the conditions for Cooper pairing are favorable for a repulsive enough Bose-Fermi interaction strength . However, at least for Na-Li mixtures, we expect these phases to occur at densities much larger than the ones considered in the phase diagram of Fig. 2.
Acknowledgements.We are grateful to J. Chalker, V. Gurarie, P. B. Littlewood, and G. Modugno for stimulating discussions. FMM would like to acknowledge the financial support of the EPSRC. This research was supported in part by the National Science Foundation under Grant Numbers PHY05-51164, DMR-0645461 and DMR-0213706.
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