Unifying Floquet theory of longitudinal and dispersive readout

TL;DR

This work unifies longitudinal and dispersive readout in circuit QED through a Floquet framework, showing that the AC Stark shift dictates the longitudinal coupling via the slope and the dispersive shift via the curvature of the driven Floquet spectrum. A Floquet-Schr"odinger-Wolff transformation derives an effective Hamiltonian that captures both readout channels, extending to multi-level systems and non-coherent cavity states. The theory explains when longitudinal readout outperforms dispersive readout, reveals how large drives can suppress readout via Floquet resonances, and provides compensation strategies applicable to transmon, fluxonium, and spin-hybrid devices. The results are supported by analytical calculations and numerical simulations, highlighting the practical advantages for fast, QND readout and the nuanced role of drive strength and detuning. This framework offers a cohesive description across adiabatic and diabatic regimes and across multiple qubit architectures, enabling optimized readout in complex cQED platforms.

Abstract

We devise a Floquet theory of longitudinal and dispersive readout in circuit QED. By studying qubits coupled to cavity photons and driven at the resonance frequency of the cavity $ω_{\rm r}$, we establish a universal connection between the qubit AC Stark shift and the longitudinal and dispersive coupling to photons. We find that the longitudinal coupling $g_\parallel$ is controlled by the slope of the AC Stark shift as function of the driving strength $A_{\rm q}$, while the dispersive shift $χ$ depends on its curvature. The two quantities become proportional to each other in the weak drive limit ($A_{\rm q}\rightarrow 0$). Our approach unifies the adiabatic limit ($ω_{\rm r}\rightarrow 0$) -- where $g_\parallel$ is generated by the static spectrum curvature (or quantum capacitance) -- with the diabatic one, where the static spectrum plays no role. We derive analytical results supported by exact numerical simulations. We apply them to superconducting and spin-hybrid cQED systems, showcasing the flexibility of faster-than-dispersive longitudinal readout.

Figures (9)

• Figure 1: Longitudinal readout of qubits. a) The qubit is driven at the resonator frequency $\omega_{\rm r}$, which experiences an effective $\sigma_z$-dependent drive. In the case of pure longitudinal coupling \ref{['eq:longitudinal']}, the cavity pointer states move in opposite direction depending on the qubit state $|\pm \rangle$. b) The transverse drive to the qubit couples different replicas (labeled by integers) of the Floquet quasi-energy spectrum, leading to an AC Stark shift. The emergent longitudinal qubit-photon coupling $g_\parallel$ is given by the slope of the AC Stark shift at the driving strength $A_{\rm q}$. The sign of $g_\parallel$ depends on whether $\omega_{\rm r}\gtrless \omega_{\rm q}$.
• Figure 2: a) Analytical time trajectories of $\langle a\rangle$ from the effective Hamiltonian \ref{['eq:effective_H_a']} for longitudinal (solid) and dispersive readout (dashed). The dots correspond to numerical simulations of Eq. \ref{['eq:ham']}. Longitudinal trajectories are obtained for $A_{\rm q}/\omega_{\rm q}=0.05$ and $\omega_{\rm r}/\omega_{\rm q}=1.1,1.15,1.5$ (thin to thick). For dispersive readout, we set $\omega_{\rm r}/\omega_{\rm q}=1.1$, to enforce the condition $\vert \chi^{(0)} \vert=\kappa/2$, with $g_\perp /\omega_{\rm q}=10^{-2}$ and $\kappa/\omega_{\rm q}=2\cdot 10^{-3}$. The cavity drive $A_{\rm r}$ is adjusted accordingly to match the longitudinal SNR at $t\rightarrow\infty$. Dispersive trajectories, for which $g_\parallel=0$, are expressed in the same unit as the longitudinal counterpart. b) Parametric increase and suppression of the SNR at time $t=0.5/\kappa$ as function of $A_{\rm q}$ for $\omega_{\rm r}/\omega_{\rm q}=0.9,1.1$ (vertical dashed lines in panel (c)). c) SNR for different values of $\omega_{\rm r}/\omega_{\rm q}$ and drive strength $A_{\rm q}$. The dashed line corresponds to the optimal dispersive case matching the longitudinal SNR in panel (a) for $A_{\rm q}/\omega_{\rm q}=0.05$. d) Deviations from unity of $g_\parallel/g_\parallel^{(0)}$ as function of $\omega_{\rm r}/\omega_{\rm q}$. e) Floquet spectrum as function of the renormalized drive $\mathcal{A}$ (assumed to be real) for $\omega_{\rm r }/\omega_{\rm q}=0.42$. The circle highlights avoided crossings between distant replicas at $A_{\rm q}\simeq0.5$, leading to the suppression of $g_\parallel$ in panel (d).
• Figure 3: Longitudinal readout of spin qubits. a) Floquet spectra as function of the renormalized drive $\mathcal{A}$ (assumed to be real). The qubit frequency $\omega_{\rm q}$ is set by the difference between the $\ket0$ and $\ket1$ qubit states for $A_{\rm q}=0$. The grey lines show the static limit $\omega_{\rm r}=0$, where $A_{\rm q}$ acts as a static detuning. b) Longitudinal splitting of the resonator pointer states depending on the initial state of the system for $\omega_{\rm r}/\omega_{\rm q}=1.4$ and $A_{\rm q}/\omega_{\rm q}=0.2$ (value corresponding to the dashed vertical lines in panel (a)). The dynamics given by $\bar{g}_\parallel$ can be compensated by an additional drive on the cavity (bottom panel). Dots correspond to numerical simulations, as in Fig. \ref{['fig:comparison']}a. c) SNR as in Fig. \ref{['fig:comparison']}c. d) Experimental setup. Model parameters (in units of $\Delta$): $t_{\text{sc}}=1$, $t_{\text{sf}}=1.3$, $g_\perp=2\cdot 10^{-2}$, $\kappa=2\cdot10^{-3}$, $\omega_{\rm q}=0.6$yu2023strong.
• Figure 4: Longitudinal readout of transmon and fluxonium qubits. a) Floquet spectra ($\mathcal{A}$ assumed to be real) compared to the static limit at $\omega_{\rm r}=0$ (gray thin lines). b) Evolution of the pointer states of the cavity in the presence of a compensation tone (analytical lines and numerical dots). The transmon is driven with $A_{\rm q}/\omega_{\rm q}=0.04$ and the fluxonium with $A_{\rm q}/\omega_{\rm q}=0.8$ (vertical dashed lines in panel (a)). c) SNR at $t=0.5/\kappa$ as in Fig. \ref{['fig:comparison']}c. Model parameters (in units of the Josephson Energy $E_J$): $g_\perp=3\cdot10^{-3}$, $\kappa=6.4\cdot 10^{-5}$, $E_C = 8.4\cdot 10^{-3}$, $\omega_{\rm q}=0.25$ for the transmon ($E_J/h=31.3$ GHz) ikonen2019qubit and $g_\perp=1\cdot10^{-2}$, $\kappa=1\cdot10^{-3}$, $E_C = 0.2$, $E_L=0.1$, $\Phi_{\rm ext}/\Phi_0 = 0.5$, $\omega_{\rm q}=0.03$ for the fluxonium ($E_J/h=5.57$ GHz) somoroff2023millisecond. Additional regimes for transmon and fluxonium are discussed in the SM SM.
• Figure S1: Floquet spectrum numerically computed considering $N_{\rm rep}=41$ replicas in the Floquet Hamiltonian \ref{['eq:floquet_matrix']} as function of the drive $\mathcal{A}$, that we chose to be real. We consider the case $\omega_{\rm r}<\omega_{\rm q}$ on the left and $\omega_{\rm r}>\omega_{\rm q}$ on the right, to highlight the change of curvature of the replicas. On the right side of each plot, we connect each quasi-energy $\varepsilon_j^{(n)}$ with its associated Floquet eigenstate $|u_j,n\rangle\rangle$.