Quantum theory of the Josephson junction between finite islands

Abstract

Superconducting circuits comprising Josephson junctions have spurred significant research activity due to their promise to realize scalable quantum computers. Effective Hamiltonians for these systems have traditionally been derived assuming the junction connects superconducting islands of infinite size. We derive a quantized Hamiltonian for a Josephson junction between finite-sized islands and predict measurable corrections to the qubit frequency and charge susceptibility to test the theory.

Figures (5)

• Figure 1: The circuit diagram depicts two superconducting islands (dashed lines) separated by a Josephson junction with Josephson energy $E_\text{J}$ and capacitive charging energy $E_\text{C}$. A voltage proportional to the (dimensionless) offset charge $n_\text{g}$ is applied via capacitive coupling, where the constant of proportionality and all other device parameters are related to the geometric and material properties of the islands via the predictions in maldonado2025mesoscopic. We formulate a quantum theory for the population imbalance $\hat{n} = (\hat{c}^{\dagger}_\mathrm{L}\hat{c}^{}_\mathrm{L} - \hat{c}^{\dagger}_\mathrm{R}\hat{c}^{}_\mathrm{R} )/2$ in a circuit with $N = (\hat{c}^{\dagger}_\mathrm{L}\hat{c}^{}_\mathrm{L} + \hat{c}^{\dagger}_\mathrm{R}\hat{c}^{}_\mathrm{R} )/2$ bosons per island, where $\hat{c}^{}_\text{L}$ and $\hat{c}^{}_\text{R}$ denote the annihilation operators for bosons in the left and right islands.
• Figure 2: Energy eigenvalues $E_k$ (first three levels, $k=0,1,2$) of the finite-island Hamiltonian [Eq. \ref{['eq:quantum_H']}] vs. offset charge $n_\text{g}$ are plotted as solid colored curves for a range of $E_\text{J}/E_\text{C}$ ratios (labels to the right of each row) and total number of bosons $2N$ (labels at the top of each column). The $N\rightarrow \infty$ limit of each band is plotted as a dotted curve of matching color. For $\abs{n_\text{g}}\ll N$, the infinite-island spectrum is recovered. For $\abs{n_\text{g}}\gg N$, anharmonicity is lost, since $E_k \approx n_\text{g}^2 + 2\abs{n_\text{g}}k$ asymptotically. Changing $2N$ by $\pm 1$ yields parity switches: half-integer shifts in the local extrema of each band [e.g., compare (b) and (c)]. For $E_\text{J}/E_\text{C}\ll 1$ [see (a)-(d)], the first band gap [Eq. \ref{['eq:qubit_frequency']}] takes the form of an ellipse [solid black curve of energy $E_\text{C}/4 \pm \hbar\omega_\text{q}/2$ in the inset of (a)].
• Figure 3: The expected population imbalance $\langle \hat{n}\rangle = \bra{\psi_0(n_\text{g})}\hat{n}\ket{\psi_0(n_\text{g})}$ in the ground state $\ket{\psi_0(n_\text{g})}$ of the finite-island Hamiltonian [Eq. \ref{['eq:quantum_H']}] vs. offset charge $n_\text{g}$ is plotted as a solid blue curve in (a) with $2N=10$, $E_\text{J}/E_\text{C}=0.2$. The $N\rightarrow \infty$ limit of $\langle \hat{n} \rangle$ is plotted as a dotted blue curve. For $\abs{n_\text{g}}\ll N$, the infinite-island expectation is recovered. For $\abs{n_\text{g}}\gg N$, the staircase saturates at $\langle\hat{n}\rangle\approx \pm N$. The corresponding (dimensionless) charge susceptibility $d\langle\hat{n}\rangle/dn_\text{g}$ vs. offset charge $n_\text{g}$ is plotted in (b). For $E_\text{J}/E_\text{C}\ll 1$, the magnitude of the peaks occurring at $n_\text{g}\in \{-N+1/2,...,N-1/2\}$ are given by Eq. \ref{['eq:susceptibility']} (solid black curve).
• Figure 4: Panel (a) depicts the numerical verification of Eq. \ref{['eq:first_order_freq']} (dashed curve) by a representative example in the regime $1\ll E_\mathrm{J}/E_\mathrm{C} \ll N^2$ (solid line) obtained via numerical diagonalization of Eq. \ref{['eq:quantum_H']}. Panel (b) depicts the curvature of the charge dispersion at zero offset charge for a range of $N$, each converging to the analytical result (dashed line) for sufficiently large $E_\mathrm{J}/E_\mathrm{C}$. Parity switching of the curvature is emphasized.
• Figure 5: Panel (a) depicts the numerical verification of Eq. \ref{['eq:first_order_susceptibility']} (derivative of the dashed curve) by a representative example in the regime $1\ll E_\mathrm{J}/E_\mathrm{C} \ll N^2$ (solid curve) obtained via numerical diagonalization of Eq. \ref{['eq:quantum_H']}. While better agreement can be found for larger values of $N^2 E_\mathrm{C}/E_\mathrm{J}$, this example highlights the ability of perturbative corrections to capture the saturation occurring at $\abs{n_\mathrm{g}}\gtrsim N$. Panel (b) depicts the curvature of the charge susceptibility at zero offset charge for a range of $N$, each converging to the analytical result (dashed curve) for sufficiently large $E_\mathrm{J}/E_\mathrm{C}$. Parity switching of the curvature is emphasized.