Charging Effects in the Inductively Shunted Josephson Junction

The choice of impedance used to shunt a Josephson junction determines if the charge transferred through the circuit is quantized: a capacitive shunt renders the charge discrete, whereas an inductive shunt gives continuous charge. This discrepancy leads to a paradox in the limit of large inductances $L$. We show that while the energy spectra of the capacitively and inductively shunted junction are vastly different, their high-frequency responses become identical for large $L$. Inductive shunting thus opens the possibility to observe charging effects unimpeded by charge noise.

Figure 1

Examples of dispersively shunted Josephson junctions. (a) Island-based devices, like the Cooper pair box, are obtained by shunting a Josephson junction capacitively and are typically operated by coupling to charge. (b) Loop-based devices like the one-junction flux or phase qubit use an inductive shunt of the Josephson junction and are commonly operated by coupling to flux. (c) For large inductances as in the fluxonium device, both charge and flux coupling play a role. (d) According to Thévenin’s theorem, both circuits (a) and (b) represent specific realizations of a Josephson junction shunted by an impedance $Z(\omega )$. Taking the limit of large inductance should smoothly connect the realization (c) to (a).

Figure 2

Energy spectra for the inductively shunted Josephson junction at zero flux. The progression (a)–(c) shows energy spectra for $E_J/E_C=2.5$ and decreasing inductive energy $E_L$. Energies are given in units of the plasma oscillation frequency $\hslash {\omega }_p=\sqrt{8E_J{E}_C}$. For small $E_L$, the spectrum exhibits regions with significantly different level spacings. (d) Comparison with the CPB bands for the same $E_J/E_C$ ratio reveals that regions of small (large) level spacings coincide with band (gap) regions in the CPB.

Figure 3

Metaplasmon and persistent-current states for large inductances. Panel (a) shows the potential in $\phi$ space (cyan), the corresponding bent CPB bands (orange), and examples of wave functions at zero flux: black, ground-state wave function (lowest $s=0$ metaplasmon state); blue, $s=0$ persistent-current states; green, lowest $s=1$ metaplasmon state; yellow, $s=2$ metaplasmon state. Localization in $\phi$ space differs characteristically, with metaplasmon states being centered at $\phi =0$ and delocalized, while persistent-current states are symmetric and antisymmetric superpositions of wave functions localized towards the edges of the parabolically deformed bands. (b) Corresponding CPB spectrum for comparison. Using the same color coding, (c) and (d) show the spectrum overlaid by the CPB bands and the flux dependence of energy levels, respectively. Metaplasmon states (independent of flux) and persistent-current states (sensitive to flux) are easily distinguished. For the parameters chosen here, $E_J/E_C=2.5$ and $E_L/E_C={10}^{-3}$, interband coupling becomes significant at higher energies (levels in gray), leading to avoided crossings between $s=1$ persistent-current states and $s=2$ metaplasmon states.

Figure 4

Matrix elements relevant for excitation when coupling the junction charge to an ac voltage. The magnitude of the matrix element for a transition to final state $|j⟩$ is shown as a function of the final state energy $E_j$, in (a) for $E_L/E_C={10}^{-3}$ starting in the ground state and in (b) for $E_L/E_C={10}^{-4}$ for several initial states. The prominent gap in the matrix elements demonstrates the strong suppression of transitions from metaplasmon states to persistent-current states. For transitions among metaplasmon states, the matrix elements reach a distinct maximum for the transition occurring in the corresponding CPB without the inductive shunt (vertical dashed lines). Lowering $E_L$ makes the peak narrower, and leads to a selection rule excluding all but the CPB-allowed transitions.