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arXiv:1306.1171 · Physical Review A

Postadiabatic Hamiltonian for low-energy excitations in a slowly-time-dependent BCS-BEC crossover

Bernhard M. Breid, J. R. Anglin

Published Sep 16, 2013• 2 citations
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Abstract

We develop a Hamiltonian that describes the time-dependent formation of a molecular Bose-Einstein condensate from a Bardeen-Cooper-Schrieffer state of fermionic atoms as a result of slowly sweeping through a Feshbach resonance. In contrast to many other calculations in the field, our Hamiltonian includes the leading postadiabatic effects that arise because the crossover proceeds at a nonzero sweep rate. We apply a path-integral approach and a stationary phase approximation for the molecular $\mathbf{k}=\mathbf{0}$ background, which is a good approximation for narrow resonances [see, e.g., Diehl and Wetterich, Phys. Rev. A 73, 033615 (2006) as well as Diehl, Gies, Pawlowski, and Wetterich, Phys. Rev. A 76, 053627 (2007)]. We use two-body adiabatic approximations to solve the atomic evolution within this background. The dynamics of the $\mathbf{k}\ensuremath{\ne}\mathbf{0}$ molecular modes is solved within a dilute gas approximation and by mapping it onto a purely bosonic Hamiltonian. Our main result is a postadiabatic effective Hamiltonian in terms of the instantaneous bosonic (Anderson-)Bogoliubov modes, which holds throughout the whole resonance, as long as the Feshbach sweep is slow enough to avoid breaking Cooper pairs.

DOI
10.1103/physreva.88.033617

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We develop a Hamiltonian that describes the time-dependent formation of a molecular Bose-Einstein condensate from a Bardeen-Cooper-Schrieffer state of fermionic atoms as a result of slowly sweeping through a Feshbach resonance. In contrast to many other calculations in the field, our Hamiltonian includes the leading postadiabatic effects that arise because the crossover proceeds at a nonzero sweep rate. We apply a path-integral approach and a stationary phase approximation for the molecular k=0\mathbf{k}=\mathbf{0}k=0 background, which is a good approximation for narrow resonances [see, e.g., Diehl and Wetterich, Phys. Rev. A 73, 033615 (2006) as well as Diehl, Gies, Pawlowski, and Wetterich, Phys. Rev. A 76, 053627 (2007)]. We use two-body adiabatic approximations to solve the atomic evolution within this background. The dynamics of the k\ensuremath≠0\mathbf{k}\ensuremath{\ne}\mathbf{0}k\ensuremath=0 molecular modes is solved within a dilute gas approximation and by mapping it onto a purely bosonic Hamiltonian. Our main result is a postadiabatic effective Hamiltonian in terms of the instantaneous bosonic (Anderson-)Bogoliubov modes, which holds throughout the whole resonance, as long as the Feshbach sweep is slow enough to avoid breaking Cooper pairs.
We describe the time-dependent formation of a molecular Bose-Einstein condensate from a BCS state of fermionic atoms as a result of slow sweeping through a Feshbach resonance. We apply a path integral approach for the molecules, and use two-body adiabatic approximations to solve the atomic evolution in the presence of the classical molecular fields, obtaining an effective action for the molecules. In the narrow resonance limit, the problem becomes semiclassical, and we discuss the growth of the molecular condensate in the saddle point approximation. Considering this time-dependent process as an analogue of the cosmological Zurek scenario, we compare the way condensate growth is driven in this rigorous theory with its phenomenological description via time-dependent Ginzburg-Landau theory.
We investigate the Bose-Einstein condensation of fermionic pairs in a uniform two-component Fermi gas, obtaining an explicit formula for the condensate density as a function of the chemical potential and the energy gap. We analyze the condensate fraction in the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of weakly interacting Cooper pairs to the Bose-Einstein condensate of molecular dimers. By using the local-density approximation we study confined Fermi vapors of alkali-metal atoms for which there is experimental evidence of condensation also on the BCS side of the Feshbach resonance. Our theoretical results are in agreement with these experimental data and give the behavior of the condensate on both sides of the Feshbach resonance at zero temperature.
The continuous crossover between a Bardeen-Cooper-Schrieffer (BCS)-type superfluid of fermion pairs and a Bose-Einstein condensate (BEC) of tightly bound bosonic molecules can be attributed to the spontaneous breaking of global U(1)U(1)U(1) gauge symmetry which underlies both quantum condensation phenomena. Recently much attention has been paid to this problem, since Feshbach resonances allow for an experimental implementation of crossover physics in cold fermion gases. The strong interactions close to resonance call for an analysis beyond Mean Field Theory. We develop a systematic functional integral approach for the description of this phenomenon. Starting from a Yukawa-type atom-molecule model, a symmetry analysis allows to both construct the equation of state and to classify the thermodynamic phases in a unified way. The onset of superfluidity is signalled by the emergence of a massless Goldstone mode associated with the broken continuous U(1) symmetry. Beyond Mean Field, we include fluctuations of the molecule field self-consistently via the solution of suitable Schwinger-Dyson equations. The phase diagram is computed, and a variety of universal features are established. A new form of crossover from an exactly solvable narrow resonance limit to broad resonances or pointlike interactions is found. At low temperature our results agree well with quantum Monte Carlo simulations and recent experiments. Our approach is further developed in the frame of functional renormalization group equations. While the effective bosonic theory in the BEC regime shows the characteristics of a Bogoliubov theory for small temperatures, the phase transition is of second order.
We apply the formulation developed in a recent paper [Y. Ohashi and A. Griffin, Phys. Rev. A 72, 013601 (2005)] for single-particle excitations in the BCS-BEC crossover to the case of a broad Feshbach resonance. At T=0T=0T=0, we solve the Bogoliubov--de Gennes coupled equations taking into account a Bose condensate of bound states (molecules). In the case of a broad resonance, the density profile n(r)n(r)n(r), as well as the profile of the superfluid order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$, are spatially spread out to the Thomas-Fermi radius, even in the crossover region. This order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$ suppresses the effects of low-energy Andreev bound states on the rf tunneling current. As a result, the peak energy in the rf spectrum is found to occur at an energy equal to the superfluid order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r=0)$ at the center of the trap, in contrast to the case of a narrow resonance, and in agreement with recent measurements. The local density approximation is found to give a good approximation for the rf-tunneling spectrum.
We extend our previous investigations of fermionic condensation in broad Feshbach resonances by using the two-channel model developed for narrow Feshbach resonances. We investigate two crossovers: the Bardeen-Cooper-Schrieffer--Bose-Einstein condensation (BCS-BEC) crossover by changing the sss-wave scattering length and the crossover from a narrow to a broad resonance by changing the atom-molecule coupling. At zero temperature we analyze, as a function of both atom-molecule coupling and sss-wave scattering length, the chemical potential, the energy gap, and the condensate fraction of atoms. In particular, we predict the contribution of Cooper pairs and preformed molecules to the total condensate density along the two crossovers.
We present the random phase approximation (RPA) theory of the Bose-Einstein-condensation-Bardeen-Cooper-Schrieffer crossover in an atomic Fermi gas near a Feshbach resonance that includes the relevant two-body atomic physics exactly. This allows us to determine the probability for the dressed molecules in the Bose-Einstein condensate to be in the closed channel of the Feshbach resonance and to compare with the recent experiments of Partridge et al. [95, 020404 (2005)10.1103/PhysRevLett.95.020404] with , who have measured the same quantity.