Qubit dynamics driven by smooth pulses of finite duration

TL;DR

The paper addresses high-fidelity qubit control using smooth finite-duration pulses that start and end with linear ramps. It develops two analytic approximations based on a rotated LMSZ framework with an adiabatic middle, relating the dynamics to Weber's parabolic cylinder functions and their strong-coupling limits. The methods align well with IBM Quantum processor data across six pulse shapes, revealing reduced power broadening with $\Delta_{1/2}\propto\sqrt{\Omega_0}$ and suppressed sidebands ($P$ resonance satisfies $P=\sin^2(A/2)$ on resonance, where $A=\int_0^T \Omega(t)\,dt$). The integrated approximation offers a simple, parameter-free formula that captures the essential physics and matches the more detailed split approach, providing a practical tool for designing smooth, finite-duration pulses in superconducting qubits with high fidelity and minimal artefacts.

Abstract

We present a study of the dynamics of a qubit driven by a pulsed field of finite duration. The pulse shape starts and ends linearly in time. The most typical example of such a shape is the sine function between two of its nodes, but several other pulse shapes are also studied. All of them present smooth alternatives to the commonly used rectangular pulse shape, resulting in much weaker power broadening, much faster vanishing wings in the excitation line profile and hence much reduced sidebands. In the same time, such shapes with a well-defined finite duration do not suffer from the spurious effects arising when truncating a pulse of infinite duration, e.g. Gaussian. We derive two approximate analytic solutions which describe the ensuing quantum dynamics. Both approximations assume that the field changes linearly at the beginning and the end of the driving pulse, and adiabatically in between. The first approximation matches the linear and adiabatic parts at an appropriate instant of time and is expressed in terms of Weber's parabolic cylinder functions. The second, much simpler, approximation uses the asymptotics of the Weber function in order to replace it by simpler functions, and some additional transformations. Both approximations prove highly accurate when compared to experimental data obtained with two of the IBM Quantum processors. Both the greatly reduced power broadening and the greatly suppressed sidebands are observed for all pulse shapes, in a nearly complete agreement between theory and experiment.

Figures (8)

• Figure 1: Illustration of the assumptions used in the derivation of the split LMSZ-adiabatic approximation which assumes rotated-LMSZ evolution at $t \in [0, \lambda]\cup [1-\lambda,1]$ of the driving pulse and adiabatic evolution for $t \in [\lambda,1-\lambda]$.
• Figure 2: (Color online) Sech (red), exponential (blue) and Lorentzian$^2$ (brown) pulse envelopes with the same pulse width $\tau$. The dotted vertical lines show the truncation line scaled for the pulses with equal pulse width and duration, which is the case for most pulses used in this demonstration.
• Figure 3: (Color online) The observed (left) and numerical (right) excitation landscapes (transition probability against detuning and peak Rabi frequency) of rectangular (top) and sinusoidal (bottom) pulsed excitation enable us to compare the transition linewidth power expansion of the two models.
• Figure 4: (Color online) Measurements of off-resonant Rabi oscillations of a sine pulse, with the same width $\tau = 96.0$ ns as in Fig. \ref{['fig:broadening']}, at detuning $\Delta=-46.2$ MHz (blue crosses) fitted with the analytical expression for the sine model found using the integrated model (red dashed line), and with the numerical solution (green dotted line).
• Figure 5: (Color online) The measured transition line profile of the sinusoidal pulse envelope ($T=42.67$ ns) is plotted (blue hexagons) along with fits based on the split approximation (in red), and the integrated approximation (in green). The residuals of the two models are shown in the thin axes below the plot.