Controlling Collective Phenomena Via the Quantum State of Interaction-Mediators: Changing the Criticality of Photon-Mediated Superconductivity Via Fock States of Light

TL;DR

The paper develops the Mediator-Hilbert-Space Bethe-Salpeter (MHSBS) equation to study how the quantum state of the interaction mediator, including non-Gaussian states like Fock states, qualitatively alters two-body scattering and emergent collective phenomena in a cavity QED–style solid-state setting. By employing a real-time Keldysh formalism, it reveals an emergent Hilbert-space structure in the two-particle vertex that couples different mediator Fock sectors, reducing to the standard Bethe-Salpeter equation only for Gaussian mediators. In the concrete application to photon-mediated superconductivity, pure Fock states enhance pair correlations and produce photon-number–dependent critical exponents (e.g., $\gamma=n+1$, $\xi\propto\sqrt{n+1}$) while Gaussian statistics restore standard BCS exponents, with the critical temperature largely unchanged. The work suggests experimental platforms in solid-state cavities and ultracold atoms to actively control superconductivity and other collective phenomena via mediator-state engineering, and it highlights rich theoretical directions for extending universality and criticality analyses to include mediator Hilbert-space structure.

Abstract

How are two-body scattering and the resulting collective phenomena affected by preparing the mediator of interactions in different quantum states? This question has recently become experimentally relevant in a specific non-relativistic version of QED implemented within materials, where standard techniques of quantum optics are available for the preparation of desired quantum states of the photon mediating interactions between matter's constituents. We develop the necessary non-equilibrium approach for computing the vertex function and find that, in addition to the energy and momentum structure of the scattering, a further structure emerges which reflects the Hilbert-space distribution of the mediator's quantum state. This emergent structure becomes non-trivial for non-Gaussian quantum states of the mediator, and can dramatically affect scattering and collective phenomena. As a first application, we show that by preparing photons in pure Fock states one can enhance pair correlations, and even modify the criticality of the superconducting phase transition. Our results also reveal that the thermal mixture of Fock states regularises the strong pair correlations present in each of its components, yielding the standard Bardeen-Cooper-Schrieffer criticality. Besides the above QED platform, ultracold atomic mixtures are among the most promising candidates for the experimental implementation of these ideas.

Figures (8)

• Figure 1: Sketch of a possible scenario: an ensemble of particles within a cavity interacts via photons which are prepared in a thermal equilibrium state or in a pure photon number (Fock) state.
• Figure 2: Emergent scattering structure in the Mediator-Hilbert-Space Bethe-Salpeter (MHSBS) equation (see Eq. \ref{['eq:hierarchy_general']}). Schematic representation of the equation for the scattering vertex function $\Gamma^{n}$ between two Fermions (blue lines) mediated by a single-mode mediator Boson (wiggled line) initially prepared in a pure Fock state with occupation $n$. The properties of the Boson are encoded in the Green's function $D$, while the form factor of the Boson-Fermion coupling is $V^{(0)}$. Two Fermionic blue lines meet with the Bosonic wiggled line at the Yukawa vertex (given in Eq. \ref{['Yukawa']}) represented by black dots. The matrices $M$ and $M_{\rm res}$ encode the momentum, frequency, and causal structure of the scattering (with the symbol $\circ$ indicating the tensor product in all the corresponding spaces). In particular, $M_{\rm res}$ corresponds to scattering processes where a real mediator is fully absorbed/emitted by the Fermions. These processes are the ones depending on the quantum state of the mediator, and give rise to the emergent structure reflecting the Hilbert space of the latter. The vertex $\Gamma^{n}$ thus becomes coupled to vertices corresponding to smaller ($m<n$) Fock-space occupations $\Gamma^{m}$, with the coupling function $F_{n-m}$. The bottom row represents the form of the iterative solution of the MHSBS equation (see Eqs. \ref{['eq:hier_iterative']} and \ref{['gamma_n_itr']}).
• Figure 3: Superconducting critical behavior of electrons for different quantum states of the mediator photon in cavity QED. The MHSBS equation (see Fig. \ref{['fig:hierarchy']}) is solved in the pairing channel at the superconducting critical point. Two quantum states of the mediator photons are compared: a pure Fock state with $n$ photons, and a thermal mixture (Eq. \ref{['rho0_thermal']}) with average number of photons $\langle\hat{n}\rangle=n_B(T)$. Different critical properties are considered: the susceptibility exponent $\gamma$ (see Section \ref{['subsec:susceptibility']}), the correlation length $\xi$ and its critical exponent $\nu$ (see Section \ref{['subsec:correlation_length']}).
• Figure 4: The perturbative expansion of the intermediate vertex function $\Gamma(u)$, which takes the same form as in the thermal- equilibrium case abrikosov2012methods. The diagrams are arranged in powers of the coupling factor $V^{(0)}(\{k_i\},s)$, and are shown up to second order. Fermions are represented by solid blue lines ($G$ is the non-interacting Fermionic Green's function) while the mediator Bosons are represented by yellow wiggled lines. At first order, $\Gamma(u)$ is simply the bare interaction vertex $\Gamma(u)=V^{(0)}(\{k_i\},s) D(s,t,t';u)$. Similarly, in all higher order diagrams, the $u_s$ dependence is solely coming through the mediator GF $D(s,t,t';u)$, and the contribution from the mediator's initial state in each diagram can be computed analytically.
• Figure 5: Scattering structure in the standard Bethe-Salpeter (BS) equation (see Eq. \ref{['gamma_th_matrix']}). Schematic representation of the equation for the scattering vertex function $\Gamma^{\rm th}$ between two Fermions (blue lines) mediated by a mediator Boson (wiggled line) initially prepared in a Gaussian thermal state. The properties of the Boson are encoded in the Green's function $D$, while the form factor of the Boson-Fermion coupling is $V^{(0)}$. The matrix $M_{\rm th}$ encodes the momentum, frequency, and causal structure of the scattering (with the symbol $\circ$ indicating the tensor product in all these spaces). The scattering structure can be directly solved by inverting the corresponding matrix (see Eq. \ref{['gamma_th_matrix']}).