Coherent Excited States in Superconductors due to a Microwave Field

Abstract

We describe theoretically the depairing effect of a microwave field on diffusive s-wave superconductors. The ground state of the superconductor is altered qualitatively in analogy to the depairing due to a dc current. In contrast to dc depairing, the density of states acquires, for microwaves with frequency ω 0 , steps at multiples of the photon energy Δ AE nℏω 0 and shows an exponential-like tail in the subgap regime. We show that this ac depairing explains the measured frequency shift of a superconducting resonator with microwave power at low temperatures. DOI: 10.1103/PhysRevLett.117.047002 How is the superconducting state modified by a current, i.e., when the condensate is moving? The answer to this question is well known for the case of a dc current flowing in a superconducting wire. For a dc current, the Cooper pairs gain a finite momentum which leads to the suppression of the superconducting properties of the wire [1,2]. The modulus of the superconducting order parameter Δ is reduced and the sharp BCS singularity near the gap is smeared. This depairing effect of a current or of a magnetic field was studied theoretically soon after the creation of the microscopic theory of superconductivity [3]. The moving superconducting condensate has been called a coherent excited state generated by the momentum displacement operator ρ q ¼ P n expðiq · r n Þ by Anderson [4] as part of the explanation of the Meissner effect from the original form of the BCS theory. The momentum displacement operator, when applied to the BCS ground state, creates excited pairs of electrons k 1 , k 2 with the momentum pairing k 1 þ k 2 ¼ q instead of zero [1,[4][5][6]. This momentum displacement q ¼ jqj corresponds to a superfluid drift velocity v s ¼ ℏq=m, where m is the electron mass. In the Green's function technique, it is possible to introduce the superfluid velocity in a gauge-invariant way, v s ∝ ½∇φ − ð2e=ℏÞA , where φ is the phase of the superconductor, e the electron charge, and A the vector potential of the electromagnetic field. The equivalence of depairing due to an electric current and due to a magnetic field is well established, both theoretically [7] and experimentally [2], using thin and narrow superconducting wires with a uniform current density. The theory of depairing by a dc current was reformulated, using the Usadel equations [8], for diffusive films with an elastic scattering length much smaller than the BCS coherence length [9]. The results of this theory [9] were confirmed experimentally by Romijn et al. However, a general theory for depairing by a microwave field, a time-dependent vector potential A, has not been formulated. In current experimental research, there are many cases in which a superconductor is used at very low temperatures, T=T c ≪ 1, where the density of quasiparticles is very low and the response of the superconductor is dominated by the response of the superfluid. At higher temperatures, it is well known that microwave radiation can be absorbed by quasiparticles, leading to a nonequilibrium di...

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