Pi-phases in balanced fermionic superfluids on spin-dependent optical lattices

π-phases in balanced fermionic superfluids in spin-dependent optical lattices I. Zapata,1 B. Wunsch,2 N. T. Zinner,2 and E. Demler2 1Departamento de F´ısica de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain 2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA (Dated: June 26, 2022) We study a balanced two-component system of ultracold fermions in one dimension with attractive interactions and subject to a spin-dependent optical lattice potential of opposite sign for the two components. We find states with different types of modulated pairing order parameters which are conceptually similar to π-phases discussed for superconductor-ferromagnet heterostructures. Increasing the lattice depth induces sharp transitions between states of different parity. While the origin of the order paramter oscillations is similar to the FFLO phase for paired states with spin imbalance, the current system is intrinsically stable to phase separation. We discuss experimental requirements for creating and probing these novel phases. PACS numbers: 67.85.-d,03.75.Ss,71.10.Pm,74.45.+c One of the most intriguing examples of the interplay of superconductivity and magnetism is the Fulde-Ferrel- Larkin-Ovchinnikov (FFLO) phase, where Zeeman split- ting of the Fermi surfaces is expected to lead to spatial oscillations of the pairing amplitude. It is difficult to obtain such phases in superconductors, since the orbital effect of the magnetic field is typically much larger than the spin Zeeman splitting. Several proposals have been made, however they remain controversial [1]. For exam- ple, FFLO phase has been discussed in the context of heavy fermion CeCoIn5 superconductors [2, 3], but al- ternative interpretation in terms of competing magnetic order has also been given [4]. So far the only unam- biguous demonstration of FFLO-like physics has been achieved in heterostructures of ferromagnetic and super- conducting (F/SC) layers [5], where proximity coupling through ferromagnetic layers results in superconducting π-junctions (see [1] for a review). We note that π-phases arising from a different mechanism than FFLO have also been discussed for high-Tc cuprates [1, 6]. Recently, in cold atoms, there has been a large body of work, both experimental and theoretical, aimed at achieving FFLO states. The biggest difficulty is that FFLO phases are fragile and extremely susceptible to phase separation and the experimental situation remains unclear [7–12]. In this paper we propose a novel system of ultracold fermions in an optical lattice [13, 14] which can be used to observe FFLO type states with oscillat- ing pairing amplitude. The system which we discuss is somewhat similar to F/SC heterostructures and should be stable against phase separation. Our proposal relies on the ability to create spin dependent optical lattices and we find that beyond a certain critical strength of such optical potential, the superconducting pairing am- plitude becomes a sign changing function (we will refer to such states as as π-phases, see Fig. 2). The gap profile in the ground-state depends on the wavelength of the lattice, λ, the strength of the poten- tial, V0, and the interaction strength. In Fig. 1 we present 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 2 0 3 0 4 0 5 0 B C S F F 3π 2π π ∆m a x / E F k F λ V 0 / E F 0 . 0 6 0 . 1 2 0 . 1 7 0 . 2 3 0 . 2 9 0 . 3 4 B C S - F F FIG. 1: Phase diagram showing the emergence of π-phases for a spin-dependent lattice potential of wavelength λ and strength V0 for interaction strength g1DkF /EF = −2.04 and zero temperature. A gradient of colors gives ∆max := max|∆˜ m|, the largest Fourier component amplitude of the gap. The black lines indicate transitions from gap profiles with zero (BCS), two (π), four (2π) and six (3π) zero-crossings per unit cell. The dashed FF line and the BCS-FF arrow are from a Fulde-Ferrell calculation in the homogeneuos system and is explained in the text. the (V0, λ) phase diagram showing the transitions from constant gap to the π-phases with several zero-crossings per unit cell in the pairing amplitude. A color gradient gives the largest Fourier component amplitude of the gap (see below) and the black lines indicate the transitions. We clearly see π-phases occurring in a broad range of λ restricted from below only by the coherence length as we will discuss. The emergence of oscillations in the gap gives clear signatures in the Fourier transform. We will demonstrate how the rapid-ramp techniques can be used to observe these states in time-of-flight measurements. We also suggest ways to make spin dependent large wave- arXiv:0910.1803v3 [cond-mat.quant-gas] 28 Sep 2010 2 length lattice potentials in the high-field regime as is needed to access π-phases. The quasi-1D system we study is described by the ef- fective Hamiltonian [15] H −µ↓N↓−µ↑N↑= X σ=↑↓ Z dxΨ† σ(x)[−ℏ2 2m ∂2 ∂x2 + Vσ(x) −µσ]Ψσ(x) + g1D Z dxΨ† ↑(x)Ψ† ↓(x)Ψ↓(x)Ψ↑(x), (1) where g1D is the effective 1D coupling

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