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Anisotropic superfluidity in the two-species polar Fermi gas

Anisotropic superfluidity in the two-species polar Fermi gas

Anisotropic superfluidity in the two-species polar Fermi

PHYSICAL REVIEW A 82, 063624 (2010)Anisotropic superfluidity in the two-species polar Fermi gasRenyuan Liao and Joachim BrandInstitute for Advanced Study and Centre for Theoretical Chemistry and Physics, Massey University, Auckland 0632, New Zealand(Received 3 August 2010; revised manuscript received 20 October 2010; published 20 December 2010)We study the superfluid pairing in a two-species gas of heteronuclear fermionic molecules with equal density.The interplay of the isotropic s-wave interaction and anisotropic long-range dipolar interaction reveals richphysics. We find that the single-particle momentum distribution has a characteristic ellipsoidal shape that canbe reasonably represented by a deformation parameter α defined similarly to the normal phase. Interestingmomentum-dependent features of the order parameter are identified. We calculate the critical temperatures ofboth the singlet and triplet superfluids, suggesting a possible pairing symmetry transition by tuning the s-waveor dipolar interaction strength.DOI: 10.1103/PhysRevA.82.063624PACS number(s): 03.75.Ss, 67.85.Lm, 05.30.Fk, 74.20.FgI. INTRODUCTIONThe recent experimental realization and coherent control ofhigh phase-space density quantum gas of the polar molecules40 K 87 Rb [1–3] provides an excellent opportunity to study theeffects of anisotropic long-range dipole-dipole interactions.Theoretical proposals employing degenerate polar moleculesrange from the study of exotic quantum phases of matter [4,5]and quantum gas dynamics [6,7] to quantum simulations ofhighly correlated condensed matter systems [8] and schemesfor quantum information processing [9].Two fundamental properties of dipolar Fermi gases aresuperfluid pairing [10–13] and Fermi surface deformation[14–16], originating from the partially attractive nature of thedipolar interaction and anisotropic Fock exchange interaction.For dipolar Fermi gases with two hyperfine states, onecan tune not only the dipole-dipole interaction by a fastrotating orienting field [17], but also the s-wave interspeciesinteraction via a Feshbach resonance. Therefore, one expectsthat rich physics will emerge as a result of the interplay ofthe anisotropic long-range dipole interaction and short-ranges-wave interaction.In this work, we study BCS pairing by taking account ofthe Fock exchange term in a self-consistent way. We findthat the anisotropic nature of the dipolar interaction leads toan anisotropic momentum space distribution of the numberdensity and an anisotropic order parameter. We generalizethe definition of the deformation parameter introduced inRef. [14] to describe the anisotropic number distribution inthe pairing phase and find that it gives a good description.Interesting features of the order parameter in momentum spaceare revealed, manifesting fascinating consequences of thedipolar interaction. Competing effects of the contact s-waveinteraction and the dipolar interaction are identified in the studyof the transition temperature of the superfluid state, suggestingthe possibility of tuning the pairing symmetry by tuning thedipolar interaction.II. MODELWe consider a homogeneous gas of two species of fermionicheteronuclear molecules σ =↑ and ↓. For simplicity, wefurther assume that each species has the same mass, density,and dipole moment. The electric dipoles of the molecules withmoment d are oriented along the z axis by a sufficiently strongexternal electric field such that the spin-independent part ofthe electronic dipole-dipole interaction becomes V dd (q) =(4π/3)d 2 (3 cos 2 θ q − 1), with θ q being the angle betweenmomentum q and the direction of the z axis in which thedipoles are aligned. In addition, we assume that moleculesalso interact via a contact interaction with strength g. Thissystem is described by the following HamiltonianH − µn = ∑ kσ+ 12V(ɛ k − µ)c † kσ c kσ∑kpqσσ ′ V σσ ′(q)c † k+qσ c† p−qσ ′c pσ ′c kσ , (1)where µ is the chemical potential, n is the total number density,V is the volume, and ɛ k = k 2 /2m (where we have set ¯h = 1).The interaction potential V σσ ′(q) = gδ σ,−σ ′ + V dd (q) containsboth dipole-dipole and contact interactions. Anticipating theimportance of the Fock exchange term, we decouple theinteraction in all three channels [18]: direct channel, exchangechannel, and Cooper channel, resulting in the followingeffective mean-field HamiltonianH MF = ∑ ξ kσ c † kσ c kσkσ+ 1 ∑[ ∗ σ2′ σ (k)c −kσ ′c kσ + σ ′ σ (k)c † kσ c† −kσ ′]. (2)kσσ ′Here ξ kσ = ɛ k − µ + gn/2 + kσ , with self-consistent meanfields defined as kσ =− 1 ∑V dd (p − k)〈c pσ † Vc pσ 〉, (3)p σ ′ σ (k) = 1 ∑V σσ ′(k − p)〈c −kσ ′c kσ 〉. (4)VpSome comments are in order: The contact interaction affectsthe single-particle spectrum by shifting the chemical potential,which may be redefined as ˜µ = µ − gn/2. The self-energy kσ encodes the anisotropic dipolar contribution from the Fockexchange term to the dressed single-particle spectrum, whichjustifies our treatment. In addition, both parts of the interactioncontribute to the pairing field.1050-2947/2010/82(6)/063624(4) 063624-1 ©2010 The American Physical Society

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