Low-energy spectrum of double-junction superconducting circuits in the Born-Oppenheimer approximation

Abstract

The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. We explore experimentally relevant parameter regimes and find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and how this leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.

Figures (7)

• Figure 1: The double-junction circuit as a single effective element. (a) Circuit diagram of the double-junction qubit with two Josephson junctions in series, shunted by a large capacitor. A low-energy qubit mode, $\phi$, and a high-energy internal mode, $\theta$, are shown with relevant circuit parameters. (b) Phase basis wavefunctions of the ground state (blue, bottom), the first qubit excitation (orange, middle) and the first internal mode excitation (red, top). Contour lines indicate the double-junction potential (Eq. \ref{['eq:H_qubit-internal']}). (c) An effective single-element representation of the double-junction circuit describing the qubit mode using the total Josephson energy $E_{J\Sigma}$ and junction asymmetry $\lambda$. (d) Appreciable higher harmonic coefficients extracted from the classical single-mode model (Eq. \ref{['eq:U_classical']}). (e) Qubit wavefunctions of the classical single-mode model (solid lines, effective potential energy in dark green) compared to the full two-mode model (dashed lines, first internal mode excitation in red), showing significant discrepancies of the classical model. The parameters are representative of realistic device parameters; $E_{J1}/h=E_{J2}/h=20\,GHz$, $E_C^\mathrm{int}/h=1.25\,GHz$ ($C_{J1} = C_{J2} \approx 8\,fF$), $E_C/h=200\,MHz$ ($C \approx 93\,fF$).
• Figure 2: Potential energy of (a) the double-junction circuit (Eq. \ref{['eq:H_qubit-internal']}) and (b) the simplified potential (Eq. \ref{['eq:simple_potential']}) for junction asymmetry $\lambda = 0.9$. The lines of minimal and maximal potential energy as a function of $\phi$ are shown in solid and dashed green, respectively. The simplified potential preserves the potential in the $\theta$-direction while the minima line is aligned to $\theta=0$. (c) The potential energy along the minimal (solid) and maximal (dashed) lines from (a-b) for different values of the junction asymmetry $\lambda$. (d) The relative numerical error $\delta_{ij}$ of low-energy levels between the original and simplified potentials from (a-b), showing the simplified potential reproduces the same low-energy spectrum as the original potential to high accuracy. We fix $E_C/h=200$ MHz, $\lambda E_{J\Sigma}/4 = 50E_C$ and $E_{J\Sigma}/E_C^\mathrm{int}=32$ as $\lambda$ is varied.
• Figure 3: Comparison of the numerical accuracy of the classical (Eq. \ref{['eq:U_classical']}) and the BO (Eq. \ref{['eq:U_BO']}) model. (a) Cumulative error $\Delta_j$ relative to the full two-mode model (Eq. \ref{['eq:potential_phi1_phi2']}) of the lowest $j$ qubit levels, as defined in the main text (Sec. \ref{['sec:BO_accuracy']}). The first excitation in the internal mode lies between the 5th an 6th qubit levels and the breakdown of the BO model is observed for the states above it (red region). (b-d) Cumulative error of the 3 lowest qubit levels ($j=3$) as a function of (b) the junction asymmetry $\lambda$, (c) the $E_{J\Sigma}/E_C^\mathrm{int}$ ratio, and (d) the capacitance asymmetry $k$. The cumulative error of the BO model remains below $>10^{-2}$ for most parameter choices and is roughly one to two orders of magnitudes less than the error of the classical model. Stars indicate points of identical configurations across all panels; $j=3$, $\lambda=0.95$, $E_{J\Sigma}/E_{C}^\mathrm{int}=32$, and $k=0$. Furthermore, $E_C/h=200$ MHz and $\lambda E_{J\Sigma}/4 = 50E_C$ are fixed for all configurations.
• Figure 4: Harmonics of the single-mode classical and BO potentials. (a) The Josephson harmonic coefficients of the single-mode potential energy in the classical (dashed lines) and BO (solid lines) approximations, showing slightly reduced harmonics for the BO model. Coefficients are scaled to the fundamental frequency coefficient. (b,c) Comparison between the classical (black, dashed), BO (black, solid) and correction (grey, solid) potentials for (b) $\lambda=0.5$ and (c) $\lambda=1$. The correction potential becomes increasingly significant for higher energies, consistent with giving a larger correction to higher energy levels. Parameters are $E_{C}/h = 200$ MHz, $\lambda E_{J\Sigma} / 4 = 50 E_C$, $E_{J\Sigma}/E_C^\mathrm{int} = 32$, and $k=0$ so $E_C^\mathrm{int}/h = 1.25$ GHz for $\lambda=1$ and $E_C^\mathrm{int}/h = 2.5$ GHz for $\lambda=0.5$.
• Figure 5: Numerically obtained charge dispersion of the qubit mediated by the internal mode. (a) Top: Charge dispersion of the internal mode ground state (brown) of the fast Hamiltonian (Eq. \ref{['eq:H_fast_int']} at $\phi=0$) showing peak-to-peak value $\varepsilon_0^\mathrm{int}$. Bottom: Charge dispersion of the qubit ground (blue) and first excited (orange) energies of the full two-mode model (Eq. \ref{['eq:potential_phi1_phi2']}) showing inherited dependence on the offset charge $N_g$ from the internal mode. Parameters used are $E_{J\Sigma}/E_C^\mathrm{int} = 8$ to illustrate pronounced $N_g$-dependence. (b) The dispersive potential (Eq. \ref{['eq:dispersion_potential']}) shown for different $E_{J\Sigma}/E_C^\mathrm{int}$ ratios, displaying exponential suppression for increasing ratios. (c) Peak-to-peak charge dispersion versus $E_{J\Sigma}/E_C^\mathrm{int}$ of the qubit (purple, solid) computed from the full two-mode model (Eq. \ref{['eq:potential_phi1_phi2']}) compared to our first-order model (green, dashed) using the dispersive potential (Eq. \ref{['eq:perturbation_theory']}). To evaluate $\varepsilon_{01}^\mathrm{model}$ we use eigenstates of the BO Hamiltonian (Eq. \ref{['eq:H_BO_final']}). We also compare to the charge dispersion of a transmon (grey), observing orders-of-magnitude less charge dispersion of the qubit mode. (d) Fourier coefficients of the dispersion potential $U_\mathrm{disp}$, showing suppression below $10^{-2}$ for $E_{J\Sigma}/E_C^\mathrm{int}\gtrsim30$. For all panels, $\lambda = 1$, $E_C/h = 200$ MHz, $E_{J\Sigma}/h = 40$ GHz.