Tensor-network representation of excitations in Josephson junction arrays

Abstract

We present a nonperturbative tensor-network approach to the excitation spectra of superconducting circuits based on Josephson junction arrays. These arrays provide the large lumped inductances required for qubit designs, yet their intrinsically many-body nature is typically reduced to effective single-mode descriptions. Perturbative treatments attempt to include the collective array modes neglected in these approximations, but a fully nonperturbative analysis is challenging due to the many-body structure and the collective character of these modes. We overcome this difficulty using the DMRG-X algorithm, which extends tensor-network methods to excited states. Our key advance is a construction of trial states from the linearized mode structure, enabling direct computation of excitations, even in degenerate manifolds, which was previously inaccessible. Our results reveal significant deviations from, and allow us to improve upon, previous perturbative treatments in the regime of low array junction impedance.

Figures (3)

• Figure 1: (a) Lumped LC resonator obtained through a capacitively shunted superinductance. The fluxonium qubit circuit is obtained by closing the loop with a small shunt "black sheep" junction (in grey). (b) DMRG-X–computed frequencies of the chain modes (blue crosses) compared to perturbation theory (black circles), and cross-Kerr nonlinearities $\chi_{0k}$ between the $k^{\textit{th}}$ mode and the fundamental mode (orange crosses) compared to perturbation theory (green circles). The shaded area indicates the uncertainty of the DMRG-X results, while the pink dot-dashed line marks the plasma frequency of the junctions in the array. (c) Chain-site occupation $\langle \hat{b}_i^\dagger \hat{b}_i \rangle$ as a function of site index $i$ for the single-excitation states corresponding to the fundamental mode $\ket{\widetilde{1_0}}$ (solid line) and the first mode $\ket{\widetilde{1_1}}$ (dot-dashed line), and for the two-excitation state $\ket{\widetilde{1_0 1_1}}$ (dotted line).
• Figure 2: DMRG-X applied to the Fluxonium qubit. (a) Fluxonium energy spectrum as a function of the external flux $\varphi_{\mathrm{ext}}$ threading the circuit. The dashed gray lines show the effective-theory prediction from Ref. DiPaolo2021, while the dot-dashed pink line marks the plasma frequency of the junctions in the array. Blue markers denote the fluxonium state $\ket{\widetilde{1_\phi 0_2}}$ computed via DMRG, and orange and green markers indicate, respectively, the states $\ket{\widetilde{0_\phi 1_2}}$ and $\ket{\widetilde{1_\phi 1_2}}$ computed with DMRG-X. (b) Cross-Kerr interaction $\chi_{\phi 2}$ between the fluxonium qubit and the first even chain mode. Dots represent the nonperturbative DMRG-X results. The dashed gray line shows the perturbative prediction from Ref. Viola2015, while the orange dot-dashed line includes corrections to the chain-mode frequency SM. (c) Fidelity $\mathcal{F} = \braket{\psi_{\mathrm{trial}} | \tilde{\psi}}$ between the DMRG-X–computed eigenstate $\ket{\tilde{\psi}}$ and the trial state $\ket{\psi_{\mathrm{trial}}}$.
• Figure 3: Charge dispersion of the cross-Kerr $\chi_{\phi2}$ in a fluxonium based on a $40$-junction superinductance DiPaolo2021 while increasing the array junction impedance $z = Z/R_Q$ at half flux quantum $\Phi_{ext}/\Phi_0 = 0.5$. The circuit parameters are listed in \ref{['tab:device-parameters']}. (a) $\chi_{\phi2}$ between the fluxonium qubit and the second chain mode for a uniform gate charge in the array (solid lines) against the perturbative approach in the dashed line. (b) Fidelity $\mathcal{F} = \braket{\psi_\mathrm{trial}|\Tilde{\psi}}$ of the DMRG-X computed eigenstates $\ket{\Tilde{\psi}}$ and the trial states for $n_g = 0.25$.