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Nonequilibrium strong-coupling theory for a driven-dissipative ultracold Fermi gas in the BCS-BEC crossover region

Taira Kawamura, Ryo Hanai, Daichi Kagamihara, Daisuke Inotani, and Yoji Ohashi
Phys. Rev. A 101, 013602 – Published 7 January 2020
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  • INTRODUCTION
  • FORMULATION
  • STRONG-COUPLING PROPERTIES OF A FERMI…
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We theoretically investigate strong-coupling properties of an ultracold Fermi gas in the BCS-BEC crossover regime in the nonequilibrium steady state, being coupled with two fermion baths. By developing a nonequilibrium strong-coupling theory based on the combined T-matrix approximation with the Keldysh Green's function technique, we show that the chemical potential bias applied by the two baths gives rise to the anomalous enhancement of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing fluctuations (although the system has no spin imbalance), resulting in the re-entrant behavior of the nonequilibrium superfluid phase transition in the Bardeen-Cooper-Schrieffer unitary regime. These pairing fluctuations are also found to anomalously enhance the pseudogap phenomenon. Since various nonequilibrium phenomena have recently been measured in ultracold Fermi gases, our nonequilibrium strong-coupling theory would be useful to catch up with this experimental development in this research field.

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    • Received 28 October 2019

    DOI:https://doi.org/10.1103/PhysRevA.101.013602

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    BCS-BEC crossoverFermi gases
    1. Physical Systems
    Nonequilibrium systemsStrongly correlated systems
    1. Techniques
    Nonequilibrium Green's function
    Atomic, Molecular & Optical

    Authors & Affiliations

    Taira Kawamura1,*, Ryo Hanai2,3, Daichi Kagamihara1, Daisuke Inotani4, and Yoji Ohashi1

    • 1Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
    • 2Department of Physics, Osaka University, Toyonaka 560-0043, Japan
    • 3James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
    • 4Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan

    • *tairakawa@keio.jp

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    Vol. 101, Iss. 1 — January 2020

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    • Figure 1
      Figure 1

      Model nonequilibrium driven-dissipative Fermi gas with a tunable s-wave pairing interaction −U(<0) associated with a Feshbach resonance. The nonequilibrium (main) system is coupled with (1) a pumping bath (L) with the chemical potential μL=μ+δμ and a coupling strength ΛL (which supplies Fermi atoms to the system), as well as (2) a decay bath (R) with μR=μ−δμ and ΛR (which absorbs Fermi atoms from the system). The baths are assumed to be free Fermi gases in the thermal equilibrium state at the temperature Tenv.

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    • Figure 2
      Figure 2

      Phase diagram of a nonequilibrium two-component Fermi gas with pumping and decay in terms of the temperature of the environments Tenv and the chemical potential bias δμ applied by the two baths. (See Fig. 1). (a) BCS regime (pFas)−1=−0.6. (b) Unitary limit (pFas)−1=0. Tenvc (solid line) is the superfluid phase transition temperature, and Tenv* (dashed line) is the pseudogap temperature, where the pseudogap appears in the densiy of state. (Note that the pseudogap is a crossover phenomenon, without being accompanied by any phase transition.) The regions labeled by “SF,” “N,” “PGBCS,” and “PGFFLO” correspond to the superfluid state, normal state, pseudogap regime where BCS-type (zero center-of-mass momentum) pairing fluctuations are dominant [5, 13, 34], and pseudogap regime where FFLO-type (finite center-of-mass momentum) pairing fluctuations are dominant, respectively. For the concrete criteria for determining these regimes, see Sec. 3b.

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    • Figure 3
      Figure 3

      (a) Dyson equation for 2×2 matrix Keldysh Green's function Ĝsys,σ (thick solid line) in the main system. The self-energy Σ̂NETMA,σ describes effects of pairing fluctuations in NETMA. (b) Particle-particle scattering matrix Γ̂ in NETMA. The wavy line is the pairing interaction −U. The Keldysh Green's function Ĝenv,σ (double solid line) involves effects of the two baths within the second-order Born approximation. (c) Dyson equation for Ĝenv,σ. The solid square denotes the tunneling Λα between the system and the α bath. D̂0,σα=L,R is the Keldysh Green's function in the α bath, given in Eq. (A4). Ĝ0,σ is the single-particle propagator in the initial thermal equilibrium state in Eq. (11). In dealing with this Dyson equation, we take the spatial average over the tunneling positions Riα and riα, to recover the translational invariance.

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    • Figure 4
      Figure 4

      Complex pole ν of the retarded particle-particle scattering matrix ΓR(q=0,ν) near Tenvc. We set (pFas)−1=−0.6 and δμ/ɛF=0.1.

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    • Figure 5
      Figure 5

      Calculated superfluid phase transition temperature Tenvc in a two-component nonequilibrium Fermi gas, as a function of the interaction strength (pFas)−1 and the nonequilibrium parameter δμ. We set γ/ɛF=0.01. This value is also used in the following figures. ɛF=pF2/(2m) and TF are, respectively, the Fermi energy and the Fermi temperature in an assumed free Fermi gas with N=pF3/(3π2) fermions in the equilibrium state.

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    • Figure 6
      Figure 6

      Single-particle properties of a nonequilibrium Fermi gas in the BCS regime at (pFas)−1=−0.6. (a) Tenvc. (b) Single-particle density of states (DOS) ρ(ω). Each panel shows the result along the path (b1)–(b3) in panel (a). We fix Tenv in panels (b1) and (b3), and fix δμ in panel (b2). In these panels, we offset the results. The short horizontal line near each result is at ρ(ω)=0. (c) Intensity of single-particle spectral weight (SW) A(p,ω), normalized by ɛF−1. Each panel corresponds to the case at (c1)–(c5) in panel (a). The broad downward spectral structure “FF” in panel (c4) is associated with FFLO-type pairing fluctuations (see Sec. 3b4). [(d)–(f)] Same plots as panels (a)–(c), for (pFas)−1=0 (unitarity limit).

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    • Figure 7
      Figure 7

      Pseudogap temperature Tenv* (solid line) which is determined as the temperature below which a dip appears in DOS. The dotted line is Tenvc shown in Fig. 5.

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    • Figure 8
      Figure 8

      (a) Pseudogap temperature Tenv* at the unitarity [(pFas)−1=0]. (b) μ(Tenv*) as a function of the nonequilibrium parameter δμ. The dashed line is μ=δμ. (c) Atomic momentum distribution np,σenv(Tenv*) at (1)–(4) in panel (a).

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    • Figure 9
      Figure 9

      (a) Phase diagram in the BCS regime when (pFas)−1=−0.6. SF, superfluid state; N, normal state; PG, pseudogap regime. [(b)–(d)] −ReΓR(q,ν=2μ) in NETMA, as a function of the momentum |q|. Each panel shows the result along the path (b)–(d) shown in panel (a).

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    • Figure 10
      Figure 10

      Momentum distribution np,σenv of Fermi atoms, given in the first line in Eq. (14). We take (pFas)−1=−0.6, and the results are at (c2)–(c4) in Fig. 6. In the case of (c4), μ/ɛF=0.602 and δμ/ɛF=0.112, so that the Fermi momenta of the pumping bath and decay bath equal pFp=2m[μ+δμ]=0.847pF and pFd=2m[μ−δμ]=0.697pF, respectively.

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    • Figure 11
      Figure 11

      Single-particle spectral weight A(p,ω) (SW) in the static approximation. (a) Equation (47) is used, assuming strong FFLO-type pairing fluctuations around |qFF|/pF=0.2. A broad downward branch “FF” is consistent with Fig. 6. (b) Equation (44) is used, assuming strong BCS-type pairing fluctuations around q=0. In these model calculations, we set Δpg/ɛF=0.3, γ/ɛF=0.01, and use the same value of μ at (c4) in Fig. 6. The spectral intensity is normalized by ɛF−1.

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    • Figure 12
      Figure 12

      (a) Calculated photoemission spectrum (PES) L(p,ω) in a nonequilibrium Fermi gas, when (pFas)−1=−0.6. (a) shows PES in the PGFFLO regime at (c4) in Fig. 6. “FF” is related to the downward broad branch in Fig. 6. Panel (b) shows the result at the superfluid phase transition (c5) in Fig. 6, where pairing fluctuations are the strongest at q=0. The spectral intensity is normalized by [pF2ɛF]−1.

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    • Figure 13
      Figure 13

      Momentum distribution np,σ of Fermi atoms, given in Eq. (48). We set δμ/ɛF=0.1 and Tenv=0. The inset shows np,σenv, given in the first line in Eq. (14).

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    • Figure 14
      Figure 14

      (a) Calculated Tenvc in the BEC regime ((pFas)−1=0.6) where Tenvc does not exhibit the re-entrant behavior. (b) DOS ρ(ω). The left and right panels show the results along the paths (b1) and (b2) drawn in panel (a), respectively. In these panels, we offset the results. The short horizontal line near each result is at ρ(ω)=0. (c) SW A(p,ω). (d) PES L(p,ω). Panels (ci) and (di) (i=13) show the results at the position (i) in panel (a).

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    • Figure 15
      Figure 15

      Diagrammatic structure of NETMA self-energy Σ̂NETMA(n)(p,ω) in the last term in Fig. 3. pi±=(±pi+q/2,±ωi+ν/2), and ηα=1,2± is defined in Eq. (B7).

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