Driving a conventional superconductor with an appropriately tuned classical electromagnetic field can lead to an enhancement of superconductivity via a redistribution of the quasiparticles into a more favorable non-equilibrium distribution -a phenomenon known as the Eliashberg effect. Here we theoretically consider coupling a two-dimensional superconducting film to the quantized electromagnetic modes of a microwave resonator cavity. As in the classical Eliashberg case, we use a kinetic equation to study the effect of the fluctuating, dynamical electromagnetic field on the Bogoliubov quasiparticles. We find that when the photon and quasiparticle systems are out of thermal equilibrium, a redistribution of quasiparticles into a more favorable non-equilibrium steadystate occurs, thereby enhancing superconductivity in the sample. We predict that by tailoring the cavity environment (e.g. the photon occupation and spectral functions), enhancement can be observed in a variety of parameter regimes, offering a large degree of tunability.It has been known since the late 1960's that subjecting a superconductor to strong microwave radiation can lead to an enhancement of superconductivity [1,2]. The explanation of this was first provided by Eliashberg et. al. [3][4][5], who showed that the irradiation yields a nonthermal distribution of the Bogoliubov excitations with an effectively colder band edge. The degree of enhancement can be obtained by using standard BCS theory with a non-thermal quasiparticle distribution function. In the subsequent decades, Eliashberg's theoretical explanation for this effect has been extended and applied to a variety of other systems [6][7][8][9][10][11][12].In recent years there has been a renewed interest in non-equilibrium superconductivity motivated in-part by a number of "pump-probe" experiments which have found that materials subjected to intense THz pulses exhibit transient superconducting properties up to very high sample temperatures [13][14][15]. Understanding these transient states has led to a variety of theoretical models which go beyond the quasiparticle redistribution effect [16][17][18][19][20][21].All of these systems concern the interaction between quantum matter and a classical external field. Particularly interesting and novel however, is the effect that a fluctuating quantum gauge field has on quantum matter. Indeed, it has been a long-standing focus in the field of cavity-quantum-electrodynamics to realize the dynamical quantum nature of the electromagnetic field through the use of resonant electromagnetic cavities [22][23][24][25][26]. Recently there have been many advances in this area including the realization of exciton-polariton condensates [27,28], states formed from hybridizing cavity photons and semiconductor excitons. This paper extends some of these concepts to superconducting systems with an eye on cavity-induced Eliashberg-type enhancement of superconductivity. The central observation is that even in a non-equilibrium steady-state the BCS self-consistency equationc...

Weyl semimetals are a class of topological materials that exhibit a bulk Hall effect due to timereversal symmetry breaking. We show that for the idealized semi-infinite case, the Casimir force between two identical Weyl semimetals is repulsive at short range and attractive at long range. Considering plates of finite thickness, we can reduce the size of the long-range attraction even making it repulsive for all distances when thin enough. In the thin-film limit, we study the appearance of an attractive Casimir force at shorter distances due to the longitudinal conductivity. Magnetic field, thickness, and chemical potential provide tunable nobs for this effect, controlling the Casimir force: whether it is attractive or repulsive, the magnitude of the effect, and the positions and existence of a trap and antitrap. In 1948, Casimir [1] showed that quantum fluctuations in the electromagnetic field cause a force between two perfectly conducting, electrically neutral objects. This has since been extended to other materials [2,3]. Throughout this time, Casimir repulsion between two materials in vacuum has been a long sought after phenomenon [4,5]. There are principally four categories in which repulsion can be achieved: (i) modifying the dielectric of the intervening medium [4,6,7], (ii) pairing a dielectric object and a permeable object [5] (such as with metamaterials [8]), (iii) using different geometries [9][10][11], and (iv) breaking time-reversal symmetry [12,13]. In this paper, we are concerned with Casimir repulsion in identical time-reversal broken systems. Specifically, we will study how Weyl semimetals with time-reversal symmetry breaking can exhibit Casimir repulsion. The key ingredient to Casimir repulsion in this paper is the existence of a nonzero bulk Hall conductance σ xy = 0, σ xy = −σ yx [14].It is a general theorem that mirror symmetric objects without time-reversal symmetry breaking can only attract one another with the Casimir effect [15]. This is understood with the Lifshitz formula [2] where if we have two materials characterized by the two reflection matrices R 1 and R 2 and separated by a distance a in a parallel plate geometry, we havewhere the trace is a matrix trace and q z = √ ω 2 + k 2 . This integral generally yields an attractive force; however, if we break time reversal symmetry, obtaining antisymmetric off-diagonal terms in the reflection matrix R xy = −R yx there is the possibility of Casimir repulsion [16]. One candidate is a two-dimensional Hall material [12], and similarly, another is a topological insulator where the surface states have been gapped by a magnetic field [13,17]. A Hall conductance does not guarantee repulsion; longitudinal conductance can overwhelm any repulsion from the Hall effect (although the magnetic field FIG. 1. The setup we will consider here is two Weyl semimetals separated by a distance a in vacuum and with distance between Weyl cones 2b in k space (split in the z direction).can lead to interesting transitions [18]), and a Hall effect that is too strong c...

Following the recent success of realizing exciton-polariton condensates in cavities, we examine the hybridization of cavity photons with the closest analog of excitons within a superconductor, states called Bardasis-Schrieffer (BS) modes. Though BS modes do not typically couple directly to light, one can engineer a coupling with an externally imposed supercurrent, leading to the formation of hybridized Bardasis-Schrieffer-polariton states, which we obtain both via direct solution and through the derivation of an effective Hamiltonian picture for the model. These new excitations have nontrivial overlap with both the original photon states and d-wave superconducting fluctuations, implying that their condensation could produce a finite d-wave component of the superconducting order parameter -an s ± id superconducting state. arXiv:1807.06601v2 [cond-mat.mes-hall]