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arXiv:2511.10364 · Physica Scripta

Microwave dressed states and vacuum fluctuations in a superconducting condensate

Anoop Dhillon, A. Hamed Majedi

Published Feb 11, 2026• 1 citation
arXivHTMLPDF

Abstract

Abstract Microwave dressed states are found to emerge within the superconducting condensate when coupled to a quantized electromagnetic field due to photon-Cooper pair entanglement. This renormalized energy separation exceeds the prediction of BCS theory, yielding an enhancement that depends on the number of photons and persists even in the presence of electromagnetic vacuum fluctuations. Our work introduces an equilibrium quantum model of microwave-enhanced superconductivity, representing a significant departure from the standard non-equilibrium Eliashberg theory. We further demonstrate that the superconducting condensate exerts a back-action on the electromagnetic field, suppressing electric field fluctuations, including those from the vacuum state. This result is consistent with Glauber and Lewenstein’s field quantization in dielectric media.

DOI
10.1088/1402-4896/ae44d8

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Abstract Microwave dressed states are found to emerge within the superconducting condensate when coupled to a quantized electromagnetic field due to photon-Cooper pair entanglement. This renormalized energy separation exceeds the prediction of BCS theory, yielding an enhancement that depends on the number of photons and persists even in the presence of electromagnetic vacuum fluctuations. Our work introduces an equilibrium quantum model of microwave-enhanced superconductivity, representing a significant departure from the standard non-equilibrium Eliashberg theory. We further demonstrate that the superconducting condensate exerts a back-action on the electromagnetic field, suppressing electric field fluctuations, including those from the vacuum state. This result is consistent with Glauber and Lewenstein’s field quantization in dielectric media.
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 nonequilibrium 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 nonequilibrium steady state 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.
The existence of vacuum fluctuations is one of the most important predictions of modern quantum field theory. In the vacuum state, fluctuations occurring at different frequencies are uncorrelated. However, if a parameter in the Lagrangian of the field is modulated by an external pump, vacuum fluctuations stimulate spontaneous downconversion processes, creating squeezing between modes symmetric with respect to half of the frequency of the pump. Here we show that by double parametric pumping of a superconducting microwave cavity, it is possible to generate another type of correlation, namely coherence between photons in separate frequency modes. The coherence correlations are tunable by the phases of the pumps and are established by a quantum fluctuation that stimulates the simultaneous creation of two photon pairs. Our analysis indicates that the origin of this vacuum-induced coherence is the absence of which-way information in the frequency space.
Superconducting current instability in thin rings, irradiated by a coherent microwave electromagnetic field, is studied. On the regular circular motion of the Cooper pairs in the supercurrent states the field imposes coherent oscillations of the condensate as a whole. The oscillating condensate should emit photons with the energies determined mainly by the discrete values of the superconducting ring energy. For the first time the probability of the supercurrent decay accompanied by the two- photon emission, is derived. The ring thickness is less than both the field skin-depth and the London penetration depth. Numerical results for the energy-angular distributions of the photons, as well as the waiting time of the stimulated two-photon transition depending on both the ring sizes and fluxoid number in the initial current state, are presented.
Recent experiments reporting cavity-vacuum-modified superconductivity raise a fundamental question: under what conditions can vacuum electromagnetic fluctuations increase a superconducting transition temperature? Starting from a Ginzburg--Landau theory minimally coupled to a quantized cavity mode, we derive the cavity-induced renormalization of the superconducting free energy. This correction comprises a positive diamagnetic contribution and a negative paramagnetic exchange contribution. We prove that, in a passive cavity, the latter cannot exceed the former, establishing a no-go theorem: within minimal cavity electrodynamics, vacuum fluctuations suppress, rather than enhance, superconductivity. We then identify two routes beyond this constraint, both involving additional collective degrees of freedom. In the collective-mode route, a cavity-active excitation amplifies the attractive paramagnetic contribution. In the competing-order route, the cavity weakens an order that competes with superconductivity, thereby indirectly enhancing superconductivity. Together, these results turn the no-go theorem into a practical design principle: cavity superconductivity enhancement requires an additional cavity-coupled material mode that either strengthens paramagnetic exchange or suppresses a competing order.
We measure longitudinal dressed states of a superconducting qubit, the single Cooper-pair box, and an intense microwave field. The dressed states represent the hybridization of the qubit and photon degrees of freedom and appear as avoided level crossings in the combined energy diagram. By embedding the circuit in an rf oscillator, we directly probe the dressed states. We measure their dressed gap as a function of photon number and microwave amplitude, finding good agreement with theory. In addition, we extract the relaxation and dephasing rates of these states.
In circuit quantum electrodynamics (circuit QED), an artificial 'circuit atom' can couple to a quantized microwave radiation much stronger than its real atomic counterpart. The celebrated quantum Rabi model describes the simplest interaction of a two-level system with a single-mode boson field. When the coupling is large enough, the bare multilevel structure of a realistic circuit atom cannot be ignored even if the circuit is strongly anharmonic. We explored this situation theoretically for flux (fluxonium) and charge (Cooper pair box) type multi-level circuits tuned to their respective flux/charge degeneracy points. We identified which spectral features of the quantum Rabi model survive and which are renormalized for large coupling. Despite significant renormalization of the low-energy spectrum in the fluxonium case, the key quantum Rabi feature—nearly-degenerate vacuum consisting of an atomic state entangled with a multi-photon field—appears in both types of circuits when the coupling is sufficiently large. Like in the quantum Rabi model, for very large couplings the entanglement spectrum is dominated by only two, nearly equal eigenvalues, in spite of the fact that a large number of bare atomic states are actually involved in the atom-resonator ground state. We interpret the emergence of the two-fold degeneracy of the vacuum of both circuits as an environmental suppression of flux/charge tunneling due to their dressing by virtual low-/high-impedance photons in the resonator. For flux tunneling, the dressing is nothing else than the shunting of a Josephson atom with a large capacitance of the resonator. Suppression of charge tunneling is a manifestation of the dynamical Coulomb blockade of transport in tunnel junctions connected to resistive leads.