Higgs gap modes in superconducting circuit quantisation

Abstract

We extend a recently developed projective circuit quantisation approach to incorporate superconducting Higgs modes associated to gap dynamics. This approach starts from a microscopic fermionic Hamiltonian for mesoscopic superconductors, and projects the system onto its low-energy "BCS" Hilbert space. We derive analytical results for the superconducting Higgs mass, "spring" constant, and oscillation frequency of the gap dynamics, which we validate numerically. We compute anharmonic corrections to the Higgs frequency for higher excitations of small superconducting islands, and compare our results to previous long-wavelength calculations.

Figures (5)

• Figure 1: The BCS potential (dashed), given by the diagonal part of the Hamiltonian function $\mathsf{V}_{\rm BCS}(\Delta)\equiv\mathsf{H}(\varphi=0;\Delta,\Delta)$. The energy is minimised at the BCS mean-field value $\Delta=\Delta_{\rm BCS}$, where $\mathsf{V}_{\rm BCS}=-E_{\rm min}=-n\mathcal{B} e^{-2/\lambda}$. The state at $\Delta=0$ is the zero-temperature Fermi sea.
• Figure 2: Normalised Fourier-basis overlap and Hamiltonian functions $\bar{\mathsf{W}}_\nu$ and $\bar{\mathsf{H}}_\nu$ at half-filling, with indicative contours shown. Along the diagonal $\Delta'=\Delta$ (dashed line), $\bar{\mathsf{H}}_\nu(\Delta,\Delta)$ is qualitatively the same as shown in \ref{['fig:Hdiag']}, with a minimum value of $-E_{\rm min}$ at $\Delta=\Delta_{\rm BCS}$. For illustrative purposes, we take $\nu=0$, $n=3\times10^5$ and $b=e^{1/\lambda}/2=10^3$.
• Figure 3: Illustrating the saddle-point structure in the ratio of the Hamiltonian and overlap functions $\mathsf{H}_\nu(\Delta,\Delta') / \mathsf{W}_\nu(\Delta,\Delta')$, around $\Delta=\Delta'=\Delta_{\rm BCS}$. Parameters are as in \ref{['fig:WHplot']}, with indicative contours shown.
• Figure 4: The lowest five Higgs eigenfunctions, ${\underline y}{}_{j}(\Delta)$, $j=0,..,4$, solved numerically on a discretised grid over $\Delta$, and superimposed on the BCS potential (dashed). Eigenfunctions are displaced along the energy axis by their eigenenergy $E_j$. Parameters are as in \ref{['fig:WHplot']}.
• Figure 5: The first excitation transition energy $E_{10}\equiv E_{1}-E_0$ computed numerically (circles) is in close agreement with the Higgs frequency $\omega_{\rm H}$ (solid line, \ref{['eqn:omegahiggs']}), as a function of the dimensionless bandwidth $b$; $E_{10}$ is essentially independent of system size $n$. We also show the anharmonicity, $\alpha_{210}=E_{10}-E_{21}$ (squares), for $n=10^5$ and $10^6$. Larger systems are less anharmonic. Dashed curves are the semi-analytic expression $15\,\alpha_{210}^{(3)}$, based on perturbative shifts.