Pairing and condensation in a resonant Bose-Fermi mixture
We study by diagrammatic means a Bose-Fermi mixture, with boson-fermion coupling tuned by a Fano-Feshbach resonance. For increasing coupling, the growing boson-fermion pairing correlations progressively reduce the boson condensation temperature and make it eventually vanish at a critical coupling. Such quantum critical point depends very weakly on the population imbalance and for vanishing boson densities coincides with that found for the polaron-molecule transition in a strongly imbalanced Fermi gas, thus bridging two quite distinct physical systems.
One of the main reasons behind the recent great interest in ultracold gases, is the possibility to reproduce physical systems relevant to other areas in physics, with a flexibility and a degree of tunability of physical parameters which is unimaginable in the original system of interest. The very same flexibility can also be used, however, to construct novel physical systems. A noticeable example of such novel systems, namely, resonant Bose-Fermi mixtures, will be here at issue. We will be interested in particular in the competition between boson-fermion pairing correlations and bosonic condensation, occuring in a mixture of single-component bosons and fermions when the boson-fermion pairing is made progressively stronger by means of a Fano-Feshbach resonance.
Initial theoretical studies of ultracold Bose-Fermi mixtures considered mainly non-resonant systems Viv00 (); Yi01 (); Alb02 (); Rot02 (); Lew04 (), where boson-fermion pairing is irrelevant, and studied, within mean-field like treatments, the tendency towards collapse or phase separation (as motivated by the first experimental results on non-resonant Bose-Fermi mixtures Mod02 ()). Bose-Fermi mixtures in the presence of a Fano-Feshbach resonance have been subsequently considered in Rad04 (); Kag04 (); Sto05 (); Dir05 (); Pow05 (); Avd06 (); Pol08 (); Riz08 (); Wat08 (); Mar08 (); Tit09 (), mostly for narrow resonances Rad04 (); Dir05 (); Pow05 (); Avd06 (); Mar08 (); Pol08 () and/or in specific contexts such as optical lattices, reduced dimensionality, zero temperature or vanishing densitiy of one component Kag04 (); Sto05 (); Riz08 (); Wat08 (); Tit09 ().
Most of current experiments on Bose-Fermi mixtures Gun06 (); Osp06 (); Osp06b (); Zir08 (); Ni08 () appear however to be closer to the case of a broad resonance, which is characterized by the smallness of the effective range parameter of the interaction potential with respect to both the average interparticle distance and the scattering length Sim05 (). Under these conditions, the resonant Bose-Fermi mixture is accurately described by a minimal Hamiltonian, made just by bosons and fermions mutually interacting via an attractive point-contact potential. This Hamiltonian is simpler and more “fundamental” in character than its counterpart for a narrow resonance because of the absence of any other parameter besides the strength of the boson-fermion interaction. In addition, its simplicity allows the implementation of more sophisticated many-body calculations (beyond mean-field).
We thus consider the following (grand-canonical) Hamiltonian:
Here , creates a particle of mass and chemical potential at spatial position , where indicates the boson and fermion atomic species, respectively, while is the bare strength of the contact interaction (we set throughout this paper). Ultraviolet divergences associated with the use of a contact interaction in (1) are eliminated, as for two-component Fermi gases Pie00 (), by expressing the bare interaction in terms of the boson-fermion scattering length via the (divergent) expression , where is the reduced mass.
The Hamiltonian (1) does not contain explicitly the boson-boson interaction. Provided it is repulsive and non-resonant, this interaction yields in fact a mean-field shift, which consists in a simple redefinition of the boson chemical potential (attractive boson-boson interactions would lead to mechanical instability and are excluded from our consideration). S-wave interaction between fermions is finally excluded by Pauli principle.
The effective coupling strength in the many-body system is determined by comparing the boson-fermion scattering length with the average interparticle distance (with , where and are the boson and fermion particle number density, respectively). In particular, we will use the same dimensionless coupling strength normally used for two-component Fermi gases, where the wave-vector (note that coincides with the noninteracting Fermi wave-vector only for ).
The expected behavior of the system is clear in the two opposite limits of the boson-fermion coupling. In the weak-coupling limit, where the scattering length is small and negative (such that ), the two components will behave essentially as ideal Bose and Fermi gases. Bosons will condense at , while fermions will fill out (at sufficienty low temperature) a Fermi sphere with radius , the chemical potentials and being modified with respect to their noninteracting values by the mean-field shifts , and , respectively. In the opposite strong-coupling limit where is small and positive (such that ), bosons will pair with fermions to form tightly bound fermionic molecules, with binding energy . In particular, in systems where , on which we focus in the present paper, all bosons will eventually pair with fermions into molecules, suppressing condensation completely.
In a complete analogy with the BCS-BEC crossover problem in a two-component Fermi gas Pie00 (), the choice of the self-energy diagrams will be guided by the criterion that a single set of diagrams should recover the correct physical description of the two above opposite limits. In the normal phase above the condensation critical temperature , to which we restrict in the present paper, this condition is met (as we shall see below) by the T-matrix choice of diagrams represented in Fig. 1, yielding the expressions:
for the bosonic and fermionic self-energies and , where and are bare bosonic and fermionic Green’s functions and is the many-body T-matrix:
In the above expressions , , , where and , are bosonic and fermionic Matsubara frequencies, respectively ( being the inverse temperature and integer), while and are the Fermi and Bose distribution functions and . The above self-energies determine finally the dressed Green’s functions entering the particle numbers equations:
Equations (26) fully determine the thermodynamic properties of the Bose-Fermi mixture in the normal phase. This phase is characterized by the condition ; upon lowering the temperature, condensation will then occur when .
Analytic results can be obtained in the two opposite limits of the boson-fermion coupling details (). For weak-coupling , yielding and , in accordance with the expected mean-field shifts. For strong-coupling gets instead proportional to the molecular propagator:
For intermediate values of the boson-fermion coupling Eqs. (26) need to be solved numerically. The results for the condensation critical temperature vs the boson-fermion dimensionless coupling are presented in Fig. 2 for a mixture with at several values of density imbalance. footnotemasses () All curves start in weak-coupling from the corresponding noninteracting values, and decrease monotonically for increasing coupling due to the growing pairing correlations, which tend to deplete the zero momentum mode and distribute the bosons over a vast momentum region, as required to build the internal molecular wave-function. The critical temperature vanishes eventually at a critical coupling, corresponding to a quantum critical point which separates a phase with a condensate from a phase where molecular correlations are so strong to deplete the condensate completely.
Remarkably, the critical coupling value depends very weakly on the degree of density imbalance: all curves terminate in the narrow region . In this respect, it is interesting to consider the limit , where the critical coupling can be calculated independently by solving the problem of a single boson immersed in a Fermi sea. This is actually the same as a spin-down fermion immersed in a Fermi sea of spin-up fermions, since for a single particle the type of statistics is immaterial. The critical coupling reduces then to that for the polaron-to-molecule transition, recently studied in the context of strongly imbalanced two-component Fermi gases sch09 (); vei08 (); pro08 (); mas08 (); mor09 (); pun09 (); com09 (). In particular, the solution of our equations in this limit yields the critical coupling , in full agreement with the value (where ) reported in pun09 (); com09 () for the polaron-to-molecule transition, when treated at the same level of approximation footnote-a (); footnote-b (). The meeting of the properties of a Bose-Fermi mixture with those of a two-component Fermi mixture at this point of the phase diagram is quite remarkable, especially because, according to our results of Fig. 1, this “universal” point sets the scale for the quantum phase transition for all boson densities .
The chemical potentials and at vs. are reported in Fig. 3, for different values of the density imbalance. The two chemical potentials behave quite differently. The fermion chemical potential remains almost constant in the whole range of coupling considered; the boson chemical potential, on the other hand, diminishes quite rapidly with increasing coupling while depending little on the density imbalance. This different behavior results from the concurrence of several factors. For weak-coupling, the increasing (negative) mean-field shift of the fermion chemical potential for increasing coupling is partially compensated by the decrease of the temperature when moving along the critical line. On the molecular side, the fermion chemical potential is instead determined by the Fermi energy of the unpaired fermions plus a mean-field shift caused by interaction with molecules. Pauli repulsion makes this interaction repulsive details (); Kag04 () thus keeping the fermion chemical potential positive. The boson chemical potential, on the other hand, interpolates between the mean-field value in weak coupling and in strong coupling, as required by molecule formation.
Figure 4 reports the momentum distributions and (as obtained before momentum integration in Eqs. (5)-(6)) at , for a mixture with , at the coupling value (approximately at the quantum critical point). The two distributions are markedly different at low momenta, consistently with their different statistics, but coalesce into the same behavior just after the step in the fermion momentum distribution. This common behavior corresponds to the function (dashed line in Fig. 4), where is the normalized two-body internal wave function of the molecules, while the coefficient represents their density. In the present case , showing that a fraction of bosons remains unpaired but still does not condense even at such a low temperature. The extrapolation of these results at exactly zero temperature indicates the existence (in a coupling range starting right after the quantum critical point) of quite an unconventional Bose liquid, corresponding to the unpaired bosons, which do not condense even at zero temperature. The fraction of unpaired fermions is instead more conventional and consists in a Fermi liquid, which is responsible for the jump in the fermion momentum distribution. In particular, the position of the jump in Fig. 4 at , corresponds to an “enclosed” density of , in good numerical agreement with the value obtained from the Luttinger theorem lut60 () for the fraction () of unpaired fermions (using the value extracted independently above). Note finally that the number of unpaired bosons progressively decreases by increasing the coupling, as expected on physical grounds, reaching eventually a 100% conversion of bosons into molecules, as shown in the inset of Fig. 4, where the ratio is reported vs coupling at a constant temperature.
We wish to conclude finally by commenting on three-body losses and mechanical instabilities, that could prevent the experimental observation of the many-body physics described above. The choice of restricting our analysis to mixtures with a majority of fermions was precisely aimed at reducing the importance of these nuisances. Three-body losses are dominated by processes involving two bosons and one fermion, with a rate proportional to Zir08 (). They can thus be controlled by keeping sufficiently small with respect to . A predominance of fermions should also reduce the tendency to collapse, because of the stabilizing effect of Fermi pressure. As a matter of fact, we have verified that the compressibility matrix remained positive definite for all couplings, temperatures, and densities considered in our calculations. Apparently, pairing correlations act to protect the system from the mean field instabilities dominating the phase diagram of non-resonant Bose-Fermi mixtures. Resonant Bose-Fermi mixtures with a majority of fermions appear thus promising systems for the observation of a rich many-body physics.
Acknowledgements.We thank F. Palestini, A. Perali, and G.C. Strinati for a careful reading of the manuscript. Partial support by the Italian MIUR under Contract Cofin-2007 “Ultracold atoms and novel quantum phases” is acknowledged.
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