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Proposed Fermi-surface reservoir engineering and application to realizing unconventional Fermi superfluids in a driven-dissipative nonequilibrium Fermi gas

Taira Kawamura, Ryo Hanai, and Yoji Ohashi
Phys. Rev. A 106, 013311 – Published 12 July 2022
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  • INTRODUCTION
  • BCS THEORY OF NONEQUILIBRIUM SUPERFLUID…
  • QUANTUM KINETIC THEORY TO ASSESS THE…
  • NONEQUILIBRIUM SUPERFLUID STEADY STATES…
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We develop a theory to describe the dynamics of a driven-dissipative many-body Fermi system to pursue our proposal to realize exotic quantum states based on reservoir engineering. Our idea is to design the shape of a Fermi surface so as to have multiple Fermi edges by properly attaching multiple reservoirs with different chemical potentials to a fermionic system. These emerged edges give rise to additional scattering channels that can destabilize the system into unconventional states, which is exemplified in this work by considering a driven-dissipative attractively interacting Fermi gas. By formulating a quantum kinetic equation using the Nambu-Keldysh Green's function technique, we explore nonequilibrium steady states in this system and assess their stability. We find that, in addition to the Bardeen-–Cooper–-Schrieffer-type isotropic pairing state, a Fulde-Ferrell-type anisotropic superfluid state being accompanied by Cooper pairs with nonzero center-of-mass momentum exists as a stable solution, even in the absence of a magnetic Zeeman field. Our result implies a great potential of realizing quantum matter beyond the equilibrium paradigm by engineering the shape and topology of Fermi surfaces in both electronic and atomic systems.

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    • Received 2 November 2021
    • Revised 1 April 2022
    • Accepted 29 June 2022
    • Corrected 25 July 2022

    DOI:https://doi.org/10.1103/PhysRevA.106.013311

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Fermi surfaceFermionic condensatesSuperconductivity
    1. Physical Systems
    Nonequilibrium systems
    1. Techniques
    Nonequilibrium Green's function
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

    Corrections

    25 July 2022

    Correction: The previously published order of authors was presented incorrectly and has been fixed.

    Authors & Affiliations

    Taira Kawamura1,*, Ryo Hanai2, and Yoji Ohashi1

    • 1Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
    • 2Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea and Department of Physics, POSTECH, Pohang 37673, Korea

    • *tairakawa@keio.jp

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    Vol. 106, Iss. 1 — July 2022

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    • Figure 1
      Figure 1

      (a) Model nonequilibrium driven-dissipative two-component Fermi gas with an s-wave pairing interaction −U(<0). The main system is coupled with two reservoirs (α=L,R) with different chemical potentials μL=μ+δμ and μR=μ−δμ. Both the reservoirs consist of free fermions in the thermal equilibrium state at the environment temperature Tenv. f(ξpα) is the Fermi distribution function, where ξpα=ɛp−ωα is the kinetic energy, measured from ωα. Λα describes tunneling between the main system and the α reservoir. The pumping and decay of Fermi atoms by the two reservoirs bring about two edges at pF1=2mμR and pF2=2mμL in the Fermi momentum distribution np,σ in the main system (where σ=↑,↓ describe two atomic hyperfine states). (b) Expected types (A)–(D) of Cooper pairs, when the two edges work like two Fermi surfaces. (c) Ordinary (thermal equilibrium) Fulde-Ferrell (FF) pairing state under an external magnetic field.

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    • Figure 2
      Figure 2

      Summary of our main results in this paper. (a) Steady-state phase diagram of a driven-dissipative two-component Fermi gas, with respect to half the chemical potential difference δμ=[μL−μR]/2 between the two reservoirs, the damping rate γ caused by system-reservoir couplings, and the environment temperature Tenv (that are all scaled by the averaged chemical potential μ=[μL+μR]/2). This figure shows the weak-coupling case when (pFas)−1=−1 (where pF=2mμ) under the vanishing current condition Jnet=0. In the phase diagram, NBCS and NFF represent the nonequilibrium BCS (Q=0) and FF-like (Q≠0) states, where Q is the center-of-mass momentum of a Cooper pair. (b) The steady-state phase diagram at Tenv=0. (c) Hysteresis phenomenon in the regions II and III, shown in panel (b). In panel (c), we set Tenv=0 and γ→0+.

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    • Figure 3
      Figure 3

      Schematic energy band structure of our model. We measure the energy from the bottom (ɛp=0=0) of the energy band in the main system. We set μ≪ωα so that the reservoirs are huge compared to the main system.

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    • Figure 4
      Figure 4

      Self-energy corrections. (a) Σ̂int describes effects of the pairing interaction −U in the mean-field BCS approximation. The solid line is the dressed Nambu-Keldysh Green's function Ĝ in the main system. The wavy line is the pairing interaction −U, which is accompanied by the vertices τ±s⊗ηα± at both ends (where s=±), acting on the Nambu ⊗ Keldysh space. (b) Σ̂env describes effects of the system-reservoir couplings Λα=L,R in the second-order Born approximation. The dashed line is the Green's function D̂α=L,R in the α reservoir. The solid square represents the tunneling matrix Λα=L,R between the system and the α reservoir.

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    • Figure 5
      Figure 5

      (a) Nonequilibrium superfluid solutions of the gap equation (47) in the weak-coupling regime (pFaa)−1=−1) of a driven-dissipative Fermi gas. We set Tenv=0 and γ→+0, and impose the vanishing current condition in Eq. (49). Among the four mean-field solutions, NBCS and NIG (nonequilibrium interior gap state) are uniform superfluid states (Q=0). NFF and NFF′ are FF-like nonuniform states (Q≠0 and Q∥pz). The solid circle is at the BCS state in the thermal equilibrium case (δμ=0). (b) Calculated intensity of the pair amplitude of each state when δμ=0.145μ. The dotted line in each panel shows the position at p=pF=2mμ. We will show in Sec. 4b that NBCS and NFF are stable solutions, while NIG and NFF′ are unstable solutions.

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    • Figure 6
      Figure 6

      Calculated intensity of the pair amplitude and schematic pictures of pair-formation. (a) FF state. (b) NFF state. Fermions in the shaded regions (blocking regions) do not contribute to the pair formation.

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    • Figure 7
      Figure 7

      Same plots as Fig. 5 for nonzero environment temperatures Tenv.

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    • Figure 8
      Figure 8

      (a) Calculated δμ at the superfluid phase transition in the model driven-dissipative Fermi gas in Fig. 1. We take Tenv=0 and (pFas)−1=−1. The system exhibits the NFF (NBCS) superfluid instability on the solid (dashed) line. (b) Inverse particle-particle scattering vertex [χ(Q,ν=2μ)]−1 in Eq. (52), as a function of Q. Upper (lower) panel shows the result along the path (b1) [path (b2)] in (a).

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    • Figure 9
      Figure 9

      Calculated time evolution of the deviation |δΔ¯(q¯=0,t)| of the superfluid order parameter from the mean-field value. We set Tenv=0, γ=0.005μ, and δ|Δ¯(q¯=0,t=0)|=0.001μ. (a) δμ=0.05μ (region I). (b) δμ=0.13μ (region II). (c) δμ=0.145μ (region III). (d) δμ=0.16μ (region IV).

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    • Figure 10
      Figure 10

      Calculated time evolution of |δΔ¯(q¯,t)| in the region III (δμ=0.145μ), when |q¯|=0.001pF>0. We take Tenv=0, γ=0.005μ. (a) NBCS. (b) NIG. (c) NFF. (d) NFF′. In this figure, “q¯∥±Q” show the cases when q¯ is parallel and points to ±Q. Because NBCS and NIG are isotropic with Q=0, the results shown in panels (a) and (b) do not depend on the direction of q¯.

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    • Figure 11
      Figure 11

      Time evolution of the deviation |δΔ¯(q=0,t)| from the NBCS mean-field order parameter and effects of (a) environment temperature Tenv and (b) damping rate γ. We take δμ=0.05μ, and δ|Δ¯(q=0,t=0)|=0.001μ. In panels (a) and (b) we set γ→+0 and Tenv=0, respectively.

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