Mesoscopic superfluid to superconductor transition

TL;DR

This work develops and analyzes a mesoscopic Bose-Hubbard ring coupled to a single electromagnetic mode to realize and tomographically map the competition between superfluid, superconducting, fragmented, and Mott insulating phases. A spectrum-tomography approach reveals a valley-like energy landscape with Landau grooves supporting metastable flow states and Meissner-like grooves that modify the electromagnetic response. The SF–SC border is controlled by the generalized fine-structure constant $α$, while the Mott transition is governed by the quantum parameter $γ_L$, and the Meissner effect emerges as a Higgs-like mass for the EM mode, enhanced by electrostatic coupling. Entanglement and ergodicity measures correlate with the geometry of the accessible energy surface rather than simple chaoticity, providing a nuanced view of phase-space structure in mesoscopic light-matter circuits. Overall, the paper offers a minimal, numerically tractable framework for studying coupled SF/SC physics, Meissner screening, and chaos in a tunable quantum circuit, with potential implications for cavity QED implementations and mesoscopic superconducting devices.

Abstract

Spectrum tomography for the energy ($E$) of a ring-shaped Bose-Hubbard circuit is illustrated. There is an inter-particle interaction $U$ that controls superfluidity (SF) and the transition to the Mott Insulator (MI) regime. The circuit is coupled to an electromagnetic cavity mode of frequency $ω_0$, and the coupling is characterized by a generalized fine-structure-constant $α$ that controls the emergence of superconductivity (SC). The ${(U,α,ω_0,E)}$ diagram features SF and SC regions, a vast region of fragmented possibly chaotic states, and an MI regime for large $U$. The mesoscopic version of the Meissner effect and the Anderson-Higgs mechanism are discussed.

Figures (6)

• Figure 1: Born-Oppenheimer energies. The 91 energy levels of an $N=12$ trimer as function of $\phi$. From up to down, the model parameters are ${u=0,0,5}$ and ${\alpha=2,20,20}$, while $\omega_0=0.48$. The units of time here and in subsequent figures are chosen such that ${K=1}$. We use RGB color-code for the occupation as explained in the text. Most of the levels cannot be resolved, and therefore in the left panel we highlight representative energy levels: full condensation in the $k_0$ orbital (green); full condensation in the $k_{+}$ orbital (blue); and fragmented 20% 30% 50% occupation (grayish).
• Figure 2: The energy landscape: We consider $u{=}0$ trimer with ${N=12}$ particles coupled to an EM mode. The other parameters are ${\omega_0 = 0.48}$ and ${\alpha=2}$, while ${N_{osc}=150}$. The upper panels provide images for eigenstates of $\mathcal{H}_{osc}$Eq. (\ref{['eHosc']}) assuming that the particles are condensed in one of the three $k$ orbitals. Each row is the $\phi$ probability distribution. The rows are ordered by energy. The lower left panel combines the 3 upper upper panels. It would be the full spectrum if we had a single particle. For ${N=12}$ one has to combine the spectra that are associated with all possible occupations. The outcome (zoomed) is the middle lower panel. In the right lower panel we display the result for ${\alpha=1/2}$, where SC multi-stability is absent.
• Figure 4: The formation of SC multi-stability: (a) Each column is an image of the spectrum for a different value of $\alpha$. The energy levels (indexed by $\nu$) are RGB color-coded, reflecting orbital occupation as explained in the text. The interaction between the particles is weak (${u=0.01}$). The other parameters are ${\omega_0=0.48}$, and ${N=12}$ and ${N_{osc}=60}$. The states are in the range ${0<\nu<0.05\mathcal{N}}$, namely, the 277 lowest levels out of 5551 are displayed. (b) The dependence of the spectrum on $u$ for representative values of EM coupling, namely, ${\alpha=0.1,2,20}$. This set demonstrates the crossover from SF to SC, and how SF/SC is diminished due to $u$. (c) The information of panel b3 is displayed in a different way. Each point is positioned according to the normalized orbital occupations, and the color-code indicates $u$. As the interaction is increased multi-stability is diminished.
• Figure 6: Ergodicity and Entanglement: Left panels from top to botom: images of $\overline{\mathcal{M}}$ and $\mathcal{M}$ and $\mathcal{N}_{ent}$. The parameters are the same as in Fig.\ref{['fMSvsU']}. Right panels: scatter diagrams color-coded by $u$, for inspecting correlations between the 3 measures. The upper diagram allows to characterize the degree of ergodicity. For reference we plot a diagonal magenta line that indicates full ergodicity. The two other diagrams demonstrate that $\mathcal{N}_{ent}$ is more correlated with $\overline{\mathcal{M}}$ than with $\mathcal{M}$.
• Figure 7: Multi-stability versus mode frequency. The diagrams show the dependence of the color-coded spectrum on the mode frequency $\omega_0$. The 3 panels from top to bottom are: (a) RGB-coded occupation; (b) The normalized occupation $n_0$ of the zero momentum orbital; (c) The occupation $n_{osc}$ of the cavity mode. The other parameters are $\alpha=2$ and $u=3$, while $N=12$ and $N_{osc}=60$. As $\omega_0$ is increased the low states go through an SC region (where we have multi-stability); and an SF regime (where we have single-stability). Above the SC/SF region we have a fragmented regime (FR). To the right of the SC/SF/FR regimes we have an $n_0$ rainbow region where $n_{osc}=0,1,2$.