Observation of coherent flux-charge interaction in a gate-tunable fluxonium

Technical Summary: Observation of Coherent Flux-Charge Interaction in a Gate-Tunable Fluxonium

Problem Statement
In superconducting quantum circuits, standard coupling mechanisms are limited to interactions between conjugate variables of the same type: capacitors couple charge to charge, and inductors couple flux to flux. There is no native two-terminal circuit element that directly couples the flux operator ($\hat{\Phi}$) and the charge operator ($\hat{Q}$), which are conjugate variables. The absence of a "flux-charge" coupling primitive limits the ability to engineer non-reciprocal devices, protected modes, and unconventional readout schemes that rely on mixing these quadratures. While the standard Josephson junction does not provide this interaction, voltage-tunable Josephson junctions based on semiconductors offer a potential pathway, as their energy depends on both the superconducting phase (flux) and the gate voltage (charge).

Methodology
The authors demonstrate this interaction using a hybrid superconductor-semiconductor fluxonium circuit. The device consists of a superinductor (formed by an Al/AlO$_x$ Josephson junction array) shunting a gate-tunable Josephson junction made from an InAs nanowire with epitaxially grown aluminum.

To isolate the flux-charge interaction from parasitic capacitive couplings, the researchers exploit parity selection rules inherent to the fluxonium spectrum at half-integer external flux ($\Phi_{ext} = \Phi_0/2$). At this specific flux bias:

1. The ground state $|0\rangle$ and the second excited state $|2\rangle$ both possess even parity.
2. The charge operator $\hat{n}$ is odd under parity reflection, making the matrix element $\langle 2|\hat{n}|0\rangle = 0$. Consequently, purely capacitive driving cannot induce transitions between these states.
3. The flux-charge interaction term, proportional to $\cos\hat{\phi}$, is even under parity reflection, allowing $\langle 2|\cos\hat{\phi}|0\rangle \neq 0$.

The experiment utilizes two distinct control lines: a standard drive line (capacitively coupled to the islands) and a gate line (modulating the critical current of the semiconductor junction). By comparing spectroscopy and Rabi oscillations driven through these two lines, the authors distinguish between standard charge-charge coupling and the desired flux-charge coupling.

Key Contributions and Results

1. Isolation of Flux-Charge Coupling:
   The team performed two-tone spectroscopy to measure the $|0\rangle \to |2\rangle$ transition. When driven through the capacitive drive line, the transition vanished exactly at $\Phi_{ext} = \Phi_0/2$ due to parity selection rules. However, when driven through the gate line, a clear signal was observed at the same flux point. This signal appeared only at working points where the junction energy $E_J$ was highly sensitive to the gate voltage ($\partial E_J/\partial V_G \neq 0$), confirming that the interaction is mediated by the parametric modulation of the junction energy.

2. The ChIVE Point (Charge-Insensitive with Variable-$E_J$):
   A critical challenge in utilizing gate-tunable junctions is that high sensitivity to gate voltage ($\partial E_J/\partial V_G$) typically leads to strong dephasing from charge noise. The authors identified a specific operating point, termed the "ChIVE point," where the transition frequency $f_{02}$ is first-order insensitive to the junction energy ($\partial f_{02}/\partial E_J = 0$), even though the junction energy itself remains sensitive to the gate voltage ($\partial E_J/\partial V_G \neq 0$).

   • At this point, the flux-charge coupling rate (proportional to $\partial E_J/\partial V_G$) is maximized.
   • Simultaneously, the dephasing rate (proportional to $\partial f_{02}/\partial V_G = \partial f_{02}/\partial E_J \cdot \partial E_J/\partial V_G$) is suppressed because $\partial f_{02}/\partial E_J = 0$.
     This allows for strong flux-charge coupling while maintaining first-order protection against gate-voltage noise.

3. Coherent Control:
   Using the ChIVE point, the authors performed time-domain Rabi measurements on the $|0\rangle \to |2\rangle$ transition. They demonstrated coherent Rabi oscillations driven exclusively through the gate line at $\Phi_{ext} = \Phi_0/2$. In contrast, identical measurements using the capacitive drive line showed no oscillations at this flux point. Numerical simulations of the driven dynamics, accounting for Stark shifts and parity selection rules, matched the experimental data, confirming that the coherent control was mediated by the flux-charge interaction.

Significance
The paper claims to have realized a circuit primitive that couples conjugate variables (flux and charge) directly, a capability previously unavailable in superconducting circuits. This interaction scales linearly with the driving amplitude while maintaining first-order insensitivity to charge noise via the ChIVE operating point.

The authors state that this work broadens the toolbox of superconducting circuits by providing a missing component for:

• Direct routes to non-reciprocity.
• Protected modes and error-correctable states.
• Unconventional readout methods.
• Cross-quadrature interactions necessary for squeezing and synthetic gauge fields.

While the current demonstration utilizes a classical drive to mediate the interaction, the authors note that inducing the charge oscillation via quantum fluctuations (e.g., from a transmon) would realize a fully quantum native flux-charge coupling, completing the set of conjugate-variable interactions in the superconducting toolbox.