Detecting $\ensuremath{\pi}$-phase superfluids with $p$-wave symmetry in a quasi-one-dimensional optical lattice

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Detecting $π$ -phase superfluids with $p$ -wave symmetry in a quasi-one-dimensional optical lattice

Bo Liu, Xiaopeng Li, Randall G. Hulet, and W. Vincent Liu

Phys. Rev. A 94, 031602(R) – Published 2 September 2016

Abstract

We propose an experimental protocol to study $p$ -wave superfluidity in a spin-polarized cold Fermi gas tuned by an $s$ -wave Feshbach resonance. A crucial ingredient is to add a quasi-one-dimensional optical lattice and tune the fillings of two spins to the $s$ and $p$ band, respectively. The pairing order parameter is confirmed to inherit $p$ -wave symmetry in its center-of-mass motion. We find that it can further develop into a state of unexpected $π$ -phase modulation in a broad parameter regime. Experimental signatures are predicted in the momentum distributions, density of states, and spatial densities for a realistic experimental setup with a shallow trap. The $π$ -phase $p$ -wave superfluid is reminiscent of the $π$ state in superconductor-ferromagnet heterostructures but differs in symmetry and physical origin. The spatially varying phases of the superfluid gap provide an approach to synthetic magnetic fields for neutral atoms. It would represent another example of $p$ -wave pairing, first discovered in ${}^{3}{He}$ liquids.

Received 10 June 2015
Revised 26 January 2016

DOI:https://doi.org/10.1103/PhysRevA.94.031602

Physics Subject Headings (PhySH)

Cold gases in optical lattices Superfluidity

Fermi gases

Atomic, Molecular & Optical

Authors & Affiliations

Bo Liu^1,2, Xiaopeng Li³, Randall G. Hulet⁴, and W. Vincent Liu^1,2

¹Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
²Wilczek Quantum Center, Zhejiang University of Technology, Hangzhou 310023, China
³Condensed Matter Theory Center and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
⁴Department of Physics and Astronomy and Rice Quantum Institute, Rice University, Houston, Texas 77005, USA

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Vol. 94, Iss. 3 — September 2016

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Figure 1
(a) and (b) The single-particle energy spectrum of the lattice potential along the $k_{x}$ axis in the units of recoil energy $E_{r}$ for the lowest three bands through a plane-wave expansion calculation, when $V_{x} = 5 E_{r}, V_{y} = V_{z} = 18 E_{r}$ . Here we propose to fill spin $↓$ fermions to the $p$ band and spin $↑$ to the $s$ band, as shown in (b) and (a), respectively. (c) Zero-temperature phase diagram as a function of lattice filling and polarization when $t_{p} / t_{s} = 8, t_{p}^{'} / t_{s} = 0.05, t_{s}^{'} / t_{s} = 0.05$ , and $U / t_{s} = - 9$ . $p π SF$ and pFFLO stand for different modulated COM $p$ -wave superfluid states with the center-of-mass momentum of Cooper pairs located at $Q = (π / a, 0, 0)$ and $Q \neq (π / a, 0, 0)$ , respectively. NG-I refers to a normal gas (without pairing) where the $| s ↑ 〉$ band is fully filled while the $| p ↓ 〉$ band is partially filled. NG-II is another kind of normal state where the $| s ↑ 〉$ band is partially filled while the $| p ↓ 〉$ band is nearly empty. The gray area is forbidden due to the Fermi statistics constraint on the lattice filling. The gray line stands for the empty state. (d) Synthetic magnetic flux in a ladder system composed of two $π$ -phase superfluid chains (see Sec. S4 in the Supplemental Material [23]). An effective anticlockwise $π$ flux is generated for both spin $↑$ and $↓$ fermions. Here we choose $t_{s} = t_{p} = \tilde{t}, μ_{ν, σ} = 0$ , and interchain tunnelings are absent due to the large energy offset between neighboring sites along the $y$ direction. The “ $\pm$ ” in front of $\tilde{t}$ are for spin $↓$ and $↑$ , respectively.
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Figure 2
Prediction of spin-resolved density distribution in time of flight, in (a) for $| s ↑ 〉$ fermions, while in (b) for $| p ↓ 〉$ fermions. The dashed red and solid black lines show the density defined in the main text along the ${\tilde{k}}_{x}$ axis for superfluid and normal phases, respectively. The dashed-dotted lines show the intensities of the Wannier orbital functions $\propto | ϕ_{ν} ({\tilde{k}}_{x}) |^{2}$ for comparison. Other parameters are $n_{s ↑} = 0.5, n_{p ↓} = 0.5, t_{p} / t_{s} = 8, t_{p}^{'} / t_{s} = 0.05, t_{s}^{'} / t_{s} = 0.05$ , and $U / 2 t_{s} = - 12$ for superfluids or 0 for normal states. The time-of-flight densities measure the momentum distributions of the corresponding phases. The absence of sharp edges is a signature of the superfluid phase at zero temperature, which lacks a Fermi surface due to the opening of an energy gap.
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Figure 3
Top row shows occupied density of states (DOS) $ρ_{s ↑} (E)$ , (a) showing the finite-energy gap for the $π$ -phase superfluid state and (b) showing the midgap peak for pFFLO resulting from Andreev bound states. Bottom row show the axial density distributions of $| s ↑ 〉$ and $| p ↓ 〉$ fermions in momentum space for (c) $π$ phase and (d) pFFLO. The red and blue solid lines show $n_{s ↑}^{a} (k_{x})$ and $n_{p ↓}^{a} (k_{x})$ , respectively, while the blue dots show $n_{p ↓}^{a} (Q - k_{x})$ . See the main text for the definition of $ρ_{s ↑} (E)$ and $n_{ν σ}^{a} (k)$ . Since there is a large polarization for the pFFLO state in (b), which can be considered as an effective external magnetic field, it leads to a shift of the density of states. Therefore, the midgap peak in (b) is not at $E = 0$ . In (a) and (c), we choose $n_{s ↑} = 0.5$ and $n_{p ↓} = 0.5$ , while in (b) and (d), $n_{s ↑} = 0.52$ and $n_{p ↓} = 0.45$ . Other parameters are the same as in Fig. 1.
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Figure 4
(a)–(h) Shell structures with a background trapping potential. The superfluid gap $Δ$ and density profile $n_{s ↑}$ and $n_{p ↓}$ of $| s ↑ 〉$ and $| p ↓ 〉$ fermions are shown as a function of the coordinate $x$ , when $t_{p} / t_{s} = 8, t_{p}^{'} / t_{s} = 0.05, t_{s}^{'} / t_{s} = 0.05$ , and $U / t_{s} = - 9$ . The polarization $P = \frac{N_{s ↑} - N_{p ↓}}{N_{s ↑} + N_{p ↓}}$ is fixed at 0.3 and 0.03 for the first and second row, respectively. The frequency of the harmonic trap is chosen to be 120 Hz. (i) The occupied local density of states (LDOS) of the spin-down fermions in a $p$ -wave $π$ -phase superfluid. Other parameters are the same as in the second row. Here the spin-resolved LDOS is defined as $ρ_{ν σ}^{'} (E, x) = \frac{1}{2} \sum_{n} [| u_{n}^{ν σ} {(x) |}^{2} δ (E - ζ_{n} (x)) + | v_{n}^{ν σ} (x) |^{2} δ (E + ζ_{n} (x))]$ , where ${[u_{n}^{ν σ} (x), v_{n}^{ν σ} (x)]}^{T}$ is the local eigenvector corresponding to the eigenenergy $ζ_{n} (x)$ of the locally homogenous subsystem, when using the LDA.
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Physical Review A

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Detecting $π$ -phase superfluids with $p$ -wave symmetry in a quasi-one-dimensional optical lattice

Bo Liu, Xiaopeng Li, Randall G. Hulet, and W. Vincent Liu

Phys. Rev. A 94, 031602(R) – Published 2 September 2016

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Physical Review A

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Detecting π-phase superfluids with p-wave symmetry in a quasi-one-dimensional optical lattice

Bo Liu, Xiaopeng Li, Randall G. Hulet, and W. Vincent Liu

Phys. Rev. A 94, 031602(R) – Published 2 September 2016

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Detecting $π$ -phase superfluids with $p$ -wave symmetry in a quasi-one-dimensional optical lattice