We consider an analog of the internal Josephson effect in superfluid atomic Fermi-gases. Four different hyperfine states of the atoms are assumed to be trapped and to form two superfluids via the BCS-type of pairing. We show that Josephson oscillations can be realized by coupling the superfluids with two laser fields. Choosing the laser detunings in a suitable way leads to an asymmetric below-gap tunneling effect for which there exists no analogue in the context of solid-state superconductivity.PACS numbers: 05.30.Fk, Cooling of trapped gases of Fermionic atoms well below the Fermi temperature [1,2,3,4,5] makes it reasonable to anticipate the achievement of the predicted BCStransition [6,7,8,9]. The existence of a gap in the excitation spectrum of the superfluid Fermi-gas will be the first issue to address, and several methods for detecting it have already been proposed [10,11]. Trapped atomic Fermigases will allow to study and test fermion-fermion pairing theories in a tunable, controlled manner. For example the classic problem of the BCS-BEC crossover when the interparticle attraction varies [12] could be studied using the possibility of tuning the interatomic scattering length using Feshbach resonances [7,13]. Besides the standard superfluid phenomenology, gases of Cooperpaired atoms are expected to have properties which are specific to atomic gases only and not present, or not easily realizable, in metallic superconductors or Helium. For instance, the trapping potential has a major effect on the characteristic lengths of the superfluid Fermi-gas [14].In this paper we propose a way to investigate the Josephson effect in trapped superfluids of Fermionic atoms. We find a phenomenon that is unique to atomic Fermi-gases, namely an asymmetry in the Josephson currents corresponding to the "up" and "down" spin states. We assume that Fermionic atoms in four different hyperfine states (we label them |g , |g ′ , |e and |e ′ ) are trapped simultaneously in an optical trap -recently alloptical trapping and cooling below the degeneracy point of the two lowest hyperfine states of 6 Li has been demonstrated [4]. The s-wave scattering lengths are assumed to be large and negative between atoms in states |g and |g ′ , as well as between those in |e and |e ′ , and the chemical potentials µ g ≃ µ g ′ and µ e ≃ µ e ′ . For all other combinations of two atoms in different states the scattering length is assumed to be small and/or the chemical potentials unequal. This leads to the existence of two superfluids, one consisting of Cooper pairs of atoms in the states |g and |g ′ , and the other of |e -|e ′ pairs. The configuration is experimentally challenging, but in principle possible by the choice of right atoms and hyperfine states, adjusting the number of atoms, and tuning the scattering lengths in magnetic fields by using Feshbach resonances [7,13].The two superfluids are coupled by driving laserinduced transitions between the states |g and |e with the laser Rabi frequency Ω and detuning δ, and between the states |g ′ and |e ′ wit...
