RING: Rabi oscillations induced by nonresonant geometric drive

TL;DR

The paper addresses the challenge of achieving coherent qubit control without resonant energy exchange by introducing RING, a nonresonant geometric drive that induces complete Rabi oscillations when $\omega_d$ is much larger than the Larmor frequency $\omega_L$. Through Floquet theory and quasi-degenerate perturbation theory, it derives an effective 2×2 Floquet Hamiltonian $\mathcal{F}_{\rm eff} = \frac{\hbar}{2}(\Delta \sigma_z + \Omega_R \sigma_x)$ with $\Delta$ and $\Omega_R$ defined in terms of drive amplitudes and phases, and identifies the off-resonant resonance condition $\Delta=0$ yielding the RING frequency $\omega_{\rm RING}$. The authors demonstrate the mechanism in a minimal two-level system and in spin-orbit coupled quantum dots, where $\omega_{\rm RING}$ scales with system parameters and the Larmor frequency, enabling fast, high-frequency control. The work highlights practical advantages, including high-pass noise filtering and access to non-Abelian Berry phases in finite fields, and discusses potential experimental realizations across NV centers, semiconductor spins, and superconducting qubits, suggesting near-term tests of geometry-driven quantum control.

Abstract

Coherent control of two-level quantum systems is typically achieved using resonant driving fields, forming the basis for qubit operations. Here, we report a mechanism for inducing complete Rabi oscillations in monochromatically driven two-level quantum systems, when the drive frequency is much larger than the Larmor frequency of the qubit. This effect$\unicode{x2015}$Rabi oscillations induced by nonresonant geometric drive (RING)$\unicode{x2015}$requires that the control field is elliptical, enclosing a nonzero area per cycle. We illustrate the effect with numerical simulations, and provide an analytical understanding via a simple effective Hamiltonian obtained from Floquet theory and perturbation theory. We show that RING enables coherent oscillations without relying on resonant energy exchange, allows for high-pass noise filtering, provides access to non-Abelian phases in finite magnetic fields. We detail a realization in electrically driven spin-orbit qubits and argue that the RING mechanism enables amplification of the Rabi frequency using the same gate voltage amplitudes at higher drive frequencies. Our results broaden the landscape of quantum control techniques, by highlighting a pathway to achieving coherent oscillations under off-resonant driving conditions.

Figures (3)

• Figure 1: Rabi oscillation induced by nonresonant geometric drive in a two-level system. (a) Schematics of the driving scheme inducing RING in a two-level system, represented by a spin (sphere with arrow). The static component of the Zeeman field ($\omega_\textrm{L}$, red arrow) encloses a non-zero angle $\vartheta$ with the normal of the plane spanned by the drive fields ($\boldsymbol{\Omega}(t)$, blue arrow). (b) Excited-state population $p_\mathrm{e}$ as a function of time and drive frequency. Initial state is the ground state $\ket{\downarrow}$ of the static part of the Hamiltonian \ref{['eq:Hamiltonian']}, and the system is driven as specified by Eq. \ref{['eq:Hamiltonian']}. Red dashed line: drive frequency corresponding to full Rabi oscillation. Data is obtained via numerical solution of the Schrödinger equation see Appendix \ref{['app:nummeth']}. Drive strengths: $\Omega_{xz}/2\pi = \Omega_y/2\pi = 400$ MHz. Tilt angle: $\vartheta = \pi/8$. Larmor frequency: $\omega_\textrm{L}/2\pi = 25$ MHz. Phases: $\phi_{xz} = 0$ and $\phi_y = -\pi/2$. (c) Maximum excited state population as a function of Larmor and drive frequencies, with parameters same as for panel (b). For each point, $p_\mathrm{e}(t)$ was maximized in a time window of duration $T = 1000 \cdot 2\pi/\omega_\textrm{d}$. Besides the fundamental resonance at $\omega_\textrm{d} = \omega_\textrm{L}$ (and the fainter half-harmonic resonance at $\omega_\textrm{d} = \omega_\textrm{L}/2$), full population inversion occurs in the $\omega_\textrm{d} \gg \omega_\textrm{L}$ regime, with $\sim 1/\omega_\textrm{L}$ behavior. We have plotted the 'effective resonance' condition in Eq. \ref{['eq:ring_freq']} as a red dashed line. (d) Excited state population as a function of time and drive frequency, calculated analytically as in Eq. \ref{['eq:Rabioscillation']}, with the same parameter values as panel (c). We marked the frequency of full Rabi oscillations with red dashed line, using Eq. \ref{['eq:ring_freq']}.
• Figure 2: Structure of the Floquet matrix. The figure shows three adjacent Fourier sectors ($n = -1, 0, 1$) of the Floquet ladder, each containing two unperturbed eigenstates split by the Larmor frequency $\omega_\textrm{L}$. Arrows represent drive-induced couplings between states; their thickness reflects the magnitude of the matrix elements, which depends on the angle $\vartheta$. For $0 < \vartheta < \pi/2$, this thickness hierarchy is preserved, resulting in a shift of energy levels that reduces the effective level splitting. For $\vartheta > \pi/2$, the hierarchy reverses. Within $0 < \vartheta < \pi/2$, this structure allows for a 'effective resonance' in the $(\Omega, \omega_d)$ parameter space where Rabi oscillations are complete. Blue lines do not shift energy levels but enable coherent transitions; without them, Rabi oscillations vanish.
• Figure 3: Numerical and theoretical demonstration of the RING mechanism in electrically driven spin qubits. (a) Schematic figure of the electrically driven quantum dot using two electrodes. (b) Excited state population as a function of time and drive frequency, from numerical solution of Eq. \ref{['eq:Hringed']}. The parameters are set to the values: $E_0 = 4.4$ V/mm, $\vartheta = \pi/8$, $\alpha=6500$$\frac{\textrm{m}}{\textrm{s}}$, $\omega_0/2\pi = 140$ GHz, $\omega_\textrm{L}/2\pi = 30$ MHz, $m=0.05m_\textrm{e}$, where $m_\textrm{e}$ is the free electron mass. Red dashed line: drive frequency corresponding to full Rabi oscillation. (c) Maximum excited state population as a function of Larmor and drive frequencies, with the same parameters as in panel (b). Besides the resonant line, full population inversion occurs in the $\omega_\textrm{d} \gg \omega_\textrm{L}$ regime. We have plotted the 'effective resonance' condition in Eq. \ref{['eq:ringedfreq']} with red dashed line. (d) Excited state population as a function of time and drive frequency, calculated analytically, with the same parameter values as panel (b). We marked the frequency of full Rabi oscillations with red dashed line using Eq. \ref{['eq:ringedfreq']}.