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arXiv:cond-mat/0307421

Scattering length for composite bosons in the BCS-BEC crossover

P. Pieri, Giancarlo Calvanese Strinati

Published Jul 17, 2003• 0 citations
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Abstract

The present manuscript concerns the calculation of the boson-boson scattering length for the composite bosons that form as bound-fermion pairs in the strong-coupling limit of the BCS-BEC crossover. The material presented in this manuscript was already published as a part of a longer paper on the BCS-BEC crossover problem (P. Pieri and G.C. Strinati, Phys. Rev. B 61, 15370 (2000)). Given the recent experimental advances on the formation of ultracold bosonic molecules from a Fermi gas of atoms by a Feshbach resonance, the calculation of the boson-boson scattering length has now become of particular interest. The present short version of the above paper could thus be helpful to the scientific community working on ultracold atomic gases. Accordingly, the present manuscript is intended for circulation as a preprint only.

DOI
10.48550/arxiv.cond-mat/0307421

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•Jan 1, 2006•84 citations
The present manuscript concerns the calculation of the boson-boson scattering length for the composite bosons that form as bound-fermion pairs in the strong-coupling limit of the BCS-BEC crossover. The material presented in this manuscript was already published as a part of a longer paper on the BCS-BEC crossover problem (P. Pieri and G.C. Strinati, Phys. Rev. B 61, 15370 (2000)). Given the recent experimental advances on the formation of ultracold bosonic molecules from a Fermi gas of atoms by a Feshbach resonance, the calculation of the boson-boson scattering length has now become of particular interest. The present short version of the above paper could thus be helpful to the scientific community working on ultracold atomic gases. Accordingly, the present manuscript is intended for circulation as a preprint only.
Theoretical treatments of the BCS-BEC crossover need to provide as accurate as possible descriptions of the two regimes where the diluteness condition applies, either in terms of the constituent fermions (BCS limit) or of the composite bosons which form as bound-fermion pairs (BEC limit). This has to occur via a single fermionic theory that bridges across these two limiting representations. In this paper, we set up successive improvements of the fermionic theory, that result into composite bosons described at the level of either the Bogoliubov or the Popov approximations for pointlike bosons. This work bears on the recent experimental advances on the BCS-BEC crossover with trapped Fermi atoms, which show the need for accurate theoretical descriptions of the BEC side of the crossover.
We study Gaussian fluctuations of the zero-temperature attractive Fermi gas in the two-dimensional (2D) BCS-BEC crossover showing that they are crucial to get a reliable equation of state in the Bose-Einstein condensation (BEC) regime of composite bosons, bound states of fermionic pairs. A low-momentum expansion up to the fourth order of the quadratic action of the fluctuating pairing field gives an ultraviolent divergent contribution of the Gaussian fluctuations to the grand potential. Performing dimensional regularization we evaluate the effective coupling constant in the beyond-mean-field grand potential. Remarkably, in the BEC regime our grand potential gives exactly the Popov's equation of state of 2D interacting bosons, and allows us to identify the scattering length aB{a}_{B}aB​ of the interaction between composite bosons as aB=aF/(21/2e1/4)=0.551...aF{a}_{B}={a}_{F}/({2}^{1/2}{e}^{1/4})=0.551...{a}_{F}aB​=aF​/(21/2e1/4)=0.551...aF​, with aF{a}_{F}aF​ is the scattering length of fermions. Remarkably, the value from our analytical relationship between the two scattering lengths is in full agreement with that obtained by recent Monte Carlo calculations.
We study the zero-temperature grand potential of a three-dimensional superfluid made of ultracold fermionic alkali-metal atoms in the BCS-BEC crossover. In particular, we analyze the zero-point energy of both fermionic single-particle excitations and bosonic collective excitations. The bosonic elementary excitations, which are crucial to obtain a reliable equation of state in the Bose-Einstein condensate regime, are obtained with a low-momentum expansion up to the forth order of the quadratic (Gaussian) action of the fluctuating pairing field. By performing a cutoff regularization and renormalization of Gaussian fluctuations, we find that the scattering length aB{a}_{B}aB​ of composite bosons, bound states of fermionic pairs, is given by aB=(2/3)aF{a}_{B}=(2/3)\phantom{\rule{4pt}{0ex}}{a}_{F}aB​=(2/3)aF​, where aF{a}_{F}aF​ is the scattering length of fermions.
The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.
We show that an ansatz based on independent composite bosons [Phys. Rep. 463, 215 (2008)] accurately describes the condensate fraction of molecular Bose-Einstein condensates in ultracold Fermi gases. The entanglement between the fermionic constituents of a single Feshbach molecule then governs the many-particle statistics of the condensate, from the limit of strong interaction to close to unitarity. This result strengthens the role of entanglement as the indispensable driver of composite-boson behavior. The condensate fraction of fermion pairs at zero temperature that we compute matches excellently previous results obtained by means of fixed-node diffusion Monte Carlo methods and the Bogoliubov depletion approximation. This paves the way towards the exploration of the BEC-BCS crossover physics in mixtures of cold Fermi gases with an arbitrary number of fermion pairs as well as the implementation of Hong-Ou-Mandel--like interference experiments proposed within coboson theory.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning