Critical Mode Compression, Self-Kerr Duality, and Single-Mode-to-Droplet Instanton Crossover for Sub-Gap Scalar Fields in Josephson Metamaterials

We develop an exact theoretical and computational framework for trapping, controlling, and probing spatially localized sub-gap resonant modes of relativistic scalar fields in superconducting Circuit-QED metamaterials. Starting from a $(d+1)$-dimensional Klein-Gordon Lagrangian with a spatially graded effective mass profile $m_{\mathrm{eff}}^2(\mathbf{x}) = m^2 + V(\mathbf{x})$, we solve for the discrete bound-state spectrum ($\omega_n < m$) in 1D and 2D Gaussian potential wells, demonstrating agreement to $<0.06\%$ between exact sparse-matrix diagonalization and long-time symplectic lattice integration across both fundamental ($1s$) and dipole-excited ($1p$) sub-gap manifolds. We then construct an exact microscopic mapping between the relativistic scalar field and a transmission line of flux-tunable SQUID resonators, proving that the bound mode's Inverse Participation Ratio, $\mathrm{IPR}(V_0,\sigma) = \int |f_0(\mathbf{x})|^4 d^d x$, governs both the single-photon self-Kerr shift $\chi_{00} = -(E_C/\hbar)(\omega_{p,0}/\omega_0)^2\mathrm{IPR}_{\mathrm{lat}}$ (recovering the exact Koch transmon limit $-E_C/\hbar$ at single-site localization) and the effective anharmonicity $\lambda_{\mathrm{eff}} = \lambda\,\mathrm{IPR}$. Combining narrow-trap Dirac-delta asymptotics ($\mathrm{IPR} \propto \sigma$) with wide-trap anharmonic WKB expansions ($\mathrm{IPR} \propto \sigma^{-1/2}$), we prove the existence of a non-monotonic Critical Mode-Compression Resonance at $\sigma^* = 0.965\,m^{-1}$ ($V_0 = 0.80\,m^2$). In the non-perturbative tunneling regime—realized without Taylor-truncation artifacts in frustrated two-harmonic SQUID arrays or parametrically driven Kerr resonators—projecting the Euclidean action onto $f_0(x)$ yields an analytical collective-coordinate bounce $B_{\mathrm{SMCC}}(V_0,\sigma) = \frac{4\omega_0^3}{3\lambda\,\mathrm{IPR}}$ whose Pöschl-Teller ($\ell=2$) fluctuation determinant gives the exact Callan-Coleman decay rate $\Gamma_{\mathrm{SMCC}} = \frac{\omega_0}{\pi}\sqrt{6 B_{\mathrm{SMCC}}}\,e^{-B_{\mathrm{SMCC}}}$ and satisfies the exact one-loop self-Kerr duality $\delta\omega_0^{(1)}/\omega_0 = -1/B_{\mathrm{SMCC}}$. Finally, using the Nehari mountain-pass variational principle and spectral Petviashvili iteration, we prove a universal instanton hierarchy $B_{\infty} < B_{\mathrm{exact}}(V_0,\sigma) \le B_{\mathrm{bound}}(V_0,\sigma) \le B_{\mathrm{SMCC}}(V_0,\sigma)$, showing that $\mathbb{Z}_2$ parity symmetry keeps the single-mode bounce exact across the entire sub-gap manifold below the second even-mode threshold ($\sigma < \sigma_2 \approx 2.05\,m^{-1}$) before undergoing a dimensional crossover to a localized 2D Weinstein-Coleman droplet ($B_\infty = C_{2\mathrm{D}}(m^2-V_0)/\lambda$). We validate single-excitation sub-band trapping in a 5-qubit XY quantum circuit via Qiskit and provide concrete hardware parameters.