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arXiv:2609.10062

Non-equilibrium dissipative stabilization of s- and d-wave superconductivity

Aleksey Lunkin, Yury Holubeu, Denis Golež, Zala Lenarčič

cond-mat.str-elcond-mat.supr-conquant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
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Abstract

Stabilizing superconductivity beyond its equilibrium regime has been one of the quests of nonequilibrium state engineering. We report the presence of nonthermal pairing gap in BCS superconductors coupled to two thermal baths, where one of the baths can be at a temperature well above the equilibrium $T_c$. Superconducting state is enabled by a highly non-equilibrium steady state stabilized by weak coupling to baths that break detailed balance condition. The steady-state is well described by a generalized Gibbs ensemble parametrized with chemical potentials for associated Bogoliubov quasiparticle. A strong, non-perturbative effect is reported for both $s$-wave and $d$-wave superconductors, and its possible realization in cavity-based experiments is discussed.

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11 pages, 7 figures

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Stabilizing superconductivity beyond its equilibrium regime has been one of the quests of nonequilibrium state engineering. We report the presence of nonthermal pairing gap in BCS superconductors coupled to two thermal baths, where one of the baths can be at a temperature well above the equilibrium TcT_cTc​. Superconducting state is enabled by a highly non-equilibrium steady state stabilized by weak coupling to baths that break detailed balance condition. The steady-state is well described by a generalized Gibbs ensemble parametrized with chemical potentials for associated Bogoliubov quasiparticle. A strong, non-perturbative effect is reported for both sss-wave and ddd-wave superconductors, and its possible realization in cavity-based experiments is discussed.
We show that the classical (mean-field) description of far from equilibrium superconductivity is exact in the thermodynamic limit for local observables but breaks down for global quantities, such as the entanglement entropy or Loschmidt echo. We do this by solving for and comparing exact quantum and exact classical long-time dynamics of a BCS superconductor with interaction strength inversely proportional to time and evaluating local observables explicitly. Mean field is exact for both normal and anomalous averages (superconducting order) in the thermodynamic limit. However, for anomalous expectation values, this limit does not commute with adiabatic and strong coupling limits and, as a consequence, their quantum fluctuations can be unusually strong. The long-time steady state of the system is a gapless superconductor whose superfluid properties are only accessible through energy resolved measurements. This state is nonthermal but conforms to an emergent generalized Gibbs ensemble. Our study clarifies the nature of symmetry-broken many-body states in and out of equilibrium and fills a crucial gap in the theory of time-dependent quantum integrability.
I analyze the low temperature limit of the BCS theory of s-wave single-band superconductors, when the attraction band may be asymmetric with respect to the chemical potential. I discuss equilibrium systems, taking consistently into account the variation of the energy and of the total number of particles with the populations of the quasiparticle energy levels. I show that the equation for the energy gap has two solutions, one of which is stable and the other one is metastable. When the chemical potential is the center of the attraction band (the standard BCS assumption), the energy gap in the stable solution is Δ0Δ_0Δ0​, whereas in the metastable one is Δ0/3Δ_0/3Δ0​/3. If the chemical potential is not in the center of the attraction band, then a quasiparticle imbalance appears. If the absolute value of the difference between the chemical potential and center of the attraction band is bigger than 2Δ02Δ_02Δ0​, then the superconducting energy gap cannot be formed. If the number of particles is conserved and the attraction band is asymmetric, then the stable solution is unphysical and only metastable solutions are realized.
It has recently been realized that the gap nodes of multiband superconductors that break time-reversal symmetry generically take the form of Fermi surfaces of Bogoliubov quasiparticles. However, these Fermi surfaces lead to a nonzero density of states (DOS) at the Fermi energy, which typically disfavors such superconducting states. It has thus not been clear whether they can be stable for reasonable pairing interactions or are in practice preempted by time-reversal-symmetric states with vanishing DOS. In this Letter, we show within BCS theory applied to a paradigmatic model that the time-reversal-symmetry-breaking states are indeed stabilized over broad parameter ranges at weak coupling. Moreover, we introduce a fast method that involves solving the inverse BCS gap equation, does not require iteration, does not suffer from convergence problems, and can handle metastable solutions.
The BCS model of an isolated superconductor initially prepared in a nonequilibrium state predicts the existence of interesting dynamical phenomena in the time-dependent order parameter as decaying oscillations, persistent oscillations, and overdamped dynamics. To make contact with real systems remains an open challenge as one needs to introduce dissipation due to the environment in a self-consistent computation. Here, we reach this goal with the use of the Keldysh formalism to treat the effect of a thermal bath. We show that, contrary to the dissipationless case, all dynamical phases reach the equilibrium order parameter in a characteristic time that depends on the coupling with the bath. Remarkably, as time evolves, the overdamped phase shows a fast crossover where the superconducting order parameter recovers to reach a state with a well-developed long range order that tends towards equilibrium with the damped Higgs mode oscillations. Our results provide a benchmark for the description of the dynamics of real out-of-equilibrium superconductors relevant for quantum technological applications.
We investigate the nonequilibrium steady states of a microwave-driven BCS superconductor in the presence of nonmagnetic impurity scattering and energy relaxation. Using the Floquet-Nambu-Keldysh Green's function method together with the self-consistent Born approximation, we self-consistently determine the superconducting order parameter in the microwave-driven steady state. We find that the nonequilibrium phase diagram depends strongly on the impurity-scattering strength. When impurity scattering is absent or weak, the steady-state order parameter is single-valued at each temperature and drive frequency. At intermediate scattering strengths, we find bistability between microwave-enhanced and microwave-suppressed superconducting states. As the impurity-scattering strength increases further, we instead find bistability between the superconducting and normal states. This evolution reflects competition between two effects of microwave driving: Eliashberg-type enhancement through the redistribution of thermally excited quasiparticles and suppression through pair-breaking-induced quasiparticle generation. In contrast to the robustness of equilibrium sss-wave superconductivity against nonmagnetic impurity scattering, our results demonstrate that nonmagnetic impurity scattering plays a central role in determining the character of nonequilibrium superconducting phase transitions under microwave driving.
We study the competing charge-density-wave and superconducting order in the attractive Hubbard model under a voltage bias, using steady-state nonequilibrium dynamical mean-field theory. We show that the charge-density wave is suppressed in a current-carrying nonequilibrium steady state. This effect is beyond a simple Joule-heating mechanism and a ``supercooled'' metallic state is stabilized at a nonequilibrium temperature lower than the equilibrium superconducting Tc{T}_{c}Tc​. On the other hand, a current-carrying superconducting state is dissipationless and thus not subject to the same nonthermal suppression, and can therefore nucleate out of the supercooled metal, e.g., in a resistive switching experiment. The fact that an electric current can change the relative stability of different phases compared to thermal equilibrium, even when a system appears locally thermal due to electron-electron scattering, provides a general perspective to control intertwined orders out of equilibrium.