Real-Time Instantons in Complex-Driven Qubits

TL;DR

This work analyzes the quantum Rabi model under a PT-symmetric complex parametric drive $g(t)=g_0 e^{-i\omega_g t}$ to realize fast, deterministic qubit reset without dissipation. In a rotating frame, the counter-rotating terms dominate and the dynamics close under an SU(1,1) algebra, enabling a Wei–Norman factorization of the evolution operator that yields analytic qubit trajectories. The qubit’s $z$-component follows $\langle \sigma_z\rangle(t)=\langle \sigma_z\rangle(t_0)(1-2\tanh(\alpha t))$, mapping to a $\phi^4$ kink instanton with Euclidean action $S_E=4\alpha/3$, i.e. a Bloch-sphere instanton that deterministically tunnels from any initial state to the ground state. Numerical simulations over one million random initial states validate rapid, high-fidelity ground-state reset under the full Rabi dynamics with complex coupling, and the End Matter shows how an elliptically polarized drive with parameters $(\eta,\Phi)$ places the attractor at any Bloch point for arbitrary coherent-state engineering via deterministic tunneling.

Abstract

We consider the dynamics of the quantum Rabi model driven parametrically by a periodic modulation of a complex coupling. We show both analytically and numerically that instead of Rabi oscillations, this nonunitary coherent driving leads to a unidirectional instanton solution which mediates the rapid and deterministic one-way tunneling of any initial coherent state to the ground state, making the ground state a strong attractor in the quantum dynamics of the qubit. The timescale of this tunneling is shown to be inversely proportional to the effective resonant coupling, allowing for exceptionally fast, deterministic, and high-fidelity qubit reset through a purely coherent, PT-symmetric drive--without coupling to external dissipative baths, lossy resonators, or employing measurement-based feedback. Finally, we show how the drive can be engineered to place the strong attractor at any arbitrary point on the Bloch sphere.

Figures (2)

• Figure 1: Conceptual sketch of the qubit trajectories on the Bloch sphere. The red curved arrow shows the unitary Rabi path, a surface rotation that connects the excited $\ket{1}$ and ground $\ket{0}$ states through coherent oscillation. The blue dashed arrow shows the Bloch-instanton path, the straight-line trajectory generated by the PT-symmetric, non-Hermitian drive. In this case the azimuthal degree of freedom is suppressed, and the qubit follows an interior gradient-flow path directly through the Bloch ball—from the north to the south pole—realizing the real-time analog of a field-theoretic instanton.
• Figure 2: The dynamics of the z-component of the spin ($\langle \hat{\sigma}_z\rangle$) of a qubit under PT-symmetric parametric forcing. Black Line: analytical solution (Eq. \ref{['inst']}) for initial state $\hat{\sigma}_z(0)=1$ and $\langle \hat{a} \hat{a}^\dagger\rangle(0)=0$. Dark blue lines show the result of direct numerical simulations of the full quantum Rabi model, starting from one-million random initial conditions, which evenly sample the Bloch sphere, and which randomly samples the initial cavity Boson expectation within $[0,5]$. Light blue shading shows the bounds of quantum uncertainty. For an unknown initial state, the analytical curve represents an outer-bound estimate for the time to reset the unknown state coherently.