Parametric phase modulation in superconducting circuits

TL;DR

Problem: Conventional parametric modulation tunes coupling via amplitude, but this often induces time-averaged qubit-frequency shifts that complicate calibration in larger devices. Approach: Introduce a phase-modulation scheme using two simultaneous parametric flux pulses with a controllable relative phase $\delta\phi_p$ to realize a phase-tunable coupling $g_{\text{phase}}^n = C_{\phi} J_n(A)$. Findings: Demonstrated phase-controlled coupling for first-order sideband and parametric-resonance interactions at both sweet and off-sweet spots, with spectroscopy and population dynamics confirming tunability and suppressed frequency shifts. Significance: Compatible with existing tunable-coupler hardware, robust to pulse distortions, and scalable to higher-order sidebands, enabling flexible interaction Hamiltonians for quantum simulations and high-fidelity two-qubit gates.

Abstract

Parametric modulation, valued for its versatility, is widely employed in superconducting circuits for quantum simulations and high-fidelity two-qubit gates. Conventionally, the qubit coupling strength is determined by the amplitude of the parametric flux pulse, which affects the qubit parameters dramatically. In this paper, we propose and implement a phase-modulation scheme to tune the interaction strength via adjustment of the relative phase between the parametric flux pulses applied to two coupled qubits. We characterize this modulation for sideband couplings, at both sweet and off-sweet spots, achieving a broad range of coupling strengths, as confirmed by both population dynamics and spectroscopy methods. This approach enables phase-controlled modulation of coupling strength, providing a promising candidate for parametrically driven quantum simulations and gate operations.

Figures (12)

• Figure 1: Schematics of the experimental system and parametric phase modulation. (a) False-colored sketch of the superconducting circuit, showing the chip layout with four transmon qubits and four couplers. Qubits, $Q_1$ (blue) and $Q_2$ (red), along with the coupler $C$ (orange), are selected for the experiment. (b) Schematic of applied flux pulses. The dc flux bias and rf flux pulse (black lines) are delivered via dedicated on-chip lines. A dc flux biases the coupler $C$ to set a desired qubit-qubit coupling strength. Simultaneously, two parametric flux pulses are applied to $Q_1$ and $Q_2$ to induce time-varying qubit frequencies, thereby mediating the phase-controlled coupling. The oscillating blue (red) solid lines illustrate the instantaneous modulated frequencies of $Q_1$ ($Q_2$), while the dashed lines indicate their respective time-averaged frequencies.
• Figure 2: Demonstration of phase-modulated coupling and its suppressed frequency shifts. (a) Phase-modulated coupling strength for the first-order ($n=1$) sideband, $2g_{\mathrm{phase}}^1/2\pi$, achieved with dual parametric pulses and demonstrated at sweet and off-sweet spots. Experimental data points are shown for the sweet spot (fuchsia circles) and the off-sweet spot (teal rhombuses). Corresponding dashed lines represent fits to these datasets using Eq. \ref{['eq:gphase']}, rendered in distinct, high-contrast colors for clarity. The results from both operating conditions demonstrate that the relative parametric phase $\delta \phi_p$ effectively modulates the coupling strength. (b) Comparison of the induced qubit frequency shift required when tuning the coupling strength. Results from our phase-modulation method (fuchsia circles) are contrasted with those from conventional single-pulse amplitude modulation (teal rhombuses), highlighting the significant suppression of frequency shifts with our technique.
• Figure 3: Modulation of population oscillations between states $|10\rangle$ and $|01\rangle$ by the relative parametric phase $\delta\phi_p$. (a) Chevron pattern illustrating the population dynamics as a function of relative parametric phase $\delta\phi_p$ and evolution time. Oscillations at $\delta \phi_p =0.18\pi$ and $\delta \phi_p =1.22\pi$ (further detailed in panel (b)) are highlighted within the pattern. (b) Corresponding population oscillations versus evolution time at $\delta \phi_p =0.18\pi$ (fuchsia circles, fitted by brown solid line) and $\delta \phi_p =1.22\pi$ (teal stars, fitted by black dashed line). These traces demonstrate the phase-controlled modulation of the oscillation frequency, and thus the coupling strength.
• Figure 4: Spectroscopic observation of phase-modulated avoided crossings demonstrating tunable coupling. (a) Pulse sequence for the four-tone spectroscopy used to measure the $Q_1$ spectrum. (b) Measured spectrum of $Q_1$ under dual parametric pulses (on $Q_1$ and $Q_2$) at frequency $\omega_p/2\pi=70.8$ MHz, plotted as a function of varying parametric amplitude $\tilde{\Phi}_1$ on $Q_1$ (with $\tilde{\Phi}_2$ fixed at $0.13\Phi_0$). The inset shows an enlarged view of the avoided crossing corresponding to the first-order ($n=1$) sideband coupling, observed at $\tilde{\Phi}_1=0.08\Phi_0$. (c) The gap of the avoided crossing as a function of the relative parametric phase $\delta\phi_p$ between the dual parametric pulses. This demonstrates that $\delta\phi_p$ directly modulates the gap, i.e., the effective phase-tunable coupling strength $2g_{\text{phase}}^1/2\pi$.
• Figure 5: Three-tone spectroscopy of a parametrically modulated qubit, $Q_1$. (a) Spectrum of $Q_1$ at its sweet spot. A parametric pulse at a frequency of $\omega_{p1}/2\pi=79.2$ MHz is applied with varying amplitude $\tilde{\Phi}_1$. Only even-order sidebands are prominent, while odd-order sidebands are strongly suppressed. (b) Spectrum of $Q_1$ at an off-sweet spot ($\bar{\Phi}_1 = 0.064\Phi_0$). A parametric pulse at $\omega_{p1}/2\pi = 181.2$ MHz is applied with varying amplitude $\tilde{\Phi}_1$. Sidebands of all integer orders are observed.