Properties of a driven-dissipative non-equilibrium Fermi superfluid
Taira Kawamura1, Ryo Hanai2 and Yoji Ohashi1

1Department of Physics, Keio University
2James Franck Institute and Department of Physics, University of Chicago

Introduction non-equilibrium Fermi condensates✓ non-equilibrium superconductivity

Summary✓ exciton-polariton condensates

R, Matsunaga, et al. Science. 345, 6201 (2014)

K. Yoshioka, et al.
PRB 88, 041201(R) (2013)

K. O. Chong, et al.
Commun. Phys. 1, 25 (2018)

electronneutronelectronneutronWe also found that non-uniform Fulde-Ferrell like state canbe realized by the non-equilibrium effects. The stability
analysis of this state is our future work.

non-equilibrium + many body effectpair formation out of equilibrium

Formalism

We found that two different states can be realized as
non-equilibrium steady states. One of them is a gaplesssuperfluid state, which is not found in a thermal equilibrium
Fermi gas. However, this gapless state turned out to beunstable against superfluid fluctuations from the linear
stability analysis.

https://apatruno.wordpress.com/neutron-stars/

holeelectron

We studied superfluid states in a driven-dissipative Fermi gas.
By using Kadanoff-Baym equation, we derived a quantum
kinetic equation, which includes both non-equilibrium andmany-body effects. The fixed point of this kinetic equation
gives non-equilibrium steady states.

✓ neutron starAtomic momentum/energy distribution

Htot = Hsys + Henv + Hmix

✓ modelf‣ main systemfpumpingp, ω‣ environment systemdecayequilibrium

non-equilibrium(c.f.) electron energy distribution in a nanowire

main system
Λ L = ΛR = Λbath (p, ω‣ mixing termbath (δµ : non-equilibrium parameter
H. Pothier, S. Guéron, O. Birge, D. Esteve, and M.H. Devoret, PRL. 79, 18 (1997)✓ kinetic equation for the driven-dissipative Fermi gas

self-energy‣ Nambu lesser Green’s function
0G < (r1 t1 , r2 t2 ) = i @<latexit sha1_base64="l3G+aka6GdxkVWvXQ47FoJq6i6E=">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</latexit>h†" (r2 , t2 ) " (r1 , t1 )ih # (r2 , t2 ) " (r1 , t1 )ih††" (r2 , t2 ) " (r1 , t1 )i†# (r1 , t1 )i

h # (r2 , t2 )1Adiagonal : particle density⇥<latexit sha1_base64="IJF+ByHgmRca6/N3UXZD+HcIlCo=">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</latexit>

G1G<<G G1⇤(p, r, t) =d! ⇥ R⌃ ⌦ G<1 2⇡driving term

<G ⌦⌃A

Λ+<latexit sha1_base64="lRiNfifdid+QSQBf+3YaDA05y04=">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</latexit><latexit sha1_base64="ZvcNvpXBJz0EeP8j2Sdan+3zcUk=">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</latexit>UΛD̂<latexit sha1_base64="fNnvlIC2vzPvkncUceO8OSyOHA4=">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</latexit>

Hartree-Fock-Bogoliubov‣ Kadanoff-Baym equation

Z 1

environment effectsĜ=Σ

off-diagonal : pair amplitudeinteraction effects

R<<G ⌦⌃ +⌃ ⌦GA⇤2nd Born

non-equilibrium steady state @t G < = 0<latexit sha1_base64="PFLVO8pE2/MmIrNqGvPnPgRF+2w=">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</latexit>(p, !, r, t)

0 =collision termiU

X Z 1 d!
1 2⇡p<latexit sha1_base64="dFFnQmu25GRVr83V7LjmR76ZNIY=">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</latexit><G (p, !)12non-equilibrium gap equation

Result✓ non-equilibrium steady states✓ exotic pairing stateorder parameter

non-equilibrium T-matrix

0. 5

0. 5

order parameter

0.
0.15
0.1

(b)

non-equilibrium propagator including environment effects

0.10

Thouless criterion

0.15

0.

0.0

Cooper pair wave function

0.0

system-bath coupling(b) non-equilibrium distribution
(b)

p0.15

0.

0. 5

⇥ R

(q = Q, ⌫ = 0; T = Tc )<latexit sha1_base64="9uOMbxCAictkjs0b2RBTuSDw0LA=">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</latexit>

linear stability<latexit sha1_base64="7Yp4yyVo8QX8RAt4QmQnJVfe2Rs=">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</latexit>

0.1(q = 0, ⌫ = 0; T = Tc )<latexit sha1_base64="yA5IP18eapJGpTESgFea9ebrHag=">AAAD2XicjVK/bxMxFH7pAS3hR1NYkFgioqIiQeQrSEWgiAIDjG2atJVyaXTnOqkV23f1OanKKQMb6srAwARSB8SfwYLESof+CYixSCwMvHMuIKgg9ens7z1/n5/93gsiwWNDyGFuwjl1+szk1Nn8ufMXLk4XZi6txmFPU1anoQj1euDHTHDF6oYbwdYjzXwZCLYWdB+n+2t9pmMeqprZjVhT+h3F25z6Bl2twgNPsLZpeE98Kf2NxNOyWB3MeYFMtgcVctNTvQq5X6xVai27Rwc3PM07W6a5kdxykdEqlEiZ2FE8DtwMlCAbS+FM7iF4sAkhUOiBBAYKDGIBPsT4NcAFAhH6mpCgTyPidp/BAPKo7SGLIcNHbxfnDlqNzKvQTs+MrZpiFIG/RmURZskBeUeOyEfynnwhP/55VmLPSO+yi2sw1LKoNb13ZeX7WJXE1cDWb9V/FAGyT8bTiLuITZYDhXjHvk/aiAozmlh9hIwRSrOnkbMzRrVpOSpTjqyTqmWmG72c4irGaNive/ZtnNSOsc5ptdSYOhtow11bX47cyHrSytNhLvvPXh2t3KvOJtfJW/IVa/6GHJIPWHXV/0b3l1n1NeSxa92/e/Q4WJ0vu7fL88t3SouPsv6dgqtwDeawRxdgEZ7CEtQx7j58gs9w4DSc584LZ29Inchlmsvwx3Be/gSUiuBw</latexit>

chemical potential bias

0.0

ha p# ap," i0.05⇥ R(t) =(t)<latexit sha1_base64="FMsZ8HzZuXVEQquFw4Q2zLd1kRw=">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</latexit>⇤ 1⇤ 1=0=0

Cooper pair witha finite center-of-mass momentum

superfluid transition temperature

00.

(b) unstable(a) stablechemical potential bias

0.100

order parameter fluctuation

0.05

chemical potential bias

(a)

0.15

0.05

0.05
00(a)

0.

0. 2

0.0

0.04

00timeFulde-Ferrell like statenormal

uniform superfluid0.02

0.04

system-bath coupling

0.0p