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Physical Chemistry Chemical Physics

The robustness of composite pulses elucidated by classical mechanics. II. The role of initial state distribution

Jonathan Berkheim, David J. Tannor

Published Jan 1, 2026• 0 citations
Publisher

Abstract

) pulse sequence, when there is a spread in initial conditions. The canonical framework enables us to assess the projected-area robustness of Levitt's pulse sequence, and we find that it is maintained to a great extent even when considering a spread of initial conditions. Nevertheless, by conducting a numerical optimization, we have identified several variants of Levitt's pulse sequence that produce a larger coherent population inversion when there is a spread in initial conditions. We then apply the methodology to constant-rotation pulses to give geometrical insight into their robustness and to contrast their mechanism with that of the distributed point-to-point dynamics.

DOI
10.1039/d6cp00875e

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) pulse sequence, when there is a spread in initial conditions. The canonical framework enables us to assess the projected-area robustness of Levitt's pulse sequence, and we find that it is maintained to a great extent even when considering a spread of initial conditions. Nevertheless, by conducting a numerical optimization, we have identified several variants of Levitt's pulse sequence that produce a larger coherent population inversion when there is a spread in initial conditions. We then apply the methodology to constant-rotation pulses to give geometrical insight into their robustness and to contrast their mechanism with that of the distributed point-to-point dynamics.
In nuclear magnetic resonance (NMR), Composite Pulses (CPs) are widely used to correct for pulse imperfections, e.g., RF field inhomogeneity and resonance offset. Although robust pulse sequences have been developed throughout the years, the imperfection of the initial state has not been widely discussed in the literature as an additional systematic error. In previous work, we developed a classical canonical framework to perform stability analysis and used this as a measure of CP robustness. In that work, a single initial condition was allowed to evolve under various pulse imperfections. The current work extends this approach to 2D2D2D distributions of initial conditions on the Bloch Sphere; the objective is to minimize the area in order to preserve coherence, while maximizing population inversion of the entire distribution. As a case study, we investigate Levitt's 90(x)180(y)90(x)90(x)180(y)90(x)90(x)180(y)90(x) pulse sequence, when there is a spread in initial conditions. The canonical framework enables us to assess the robustness of Levitt's pulse sequence, and we find that it is maintained to a great extent even when considering a spread of initial conditions. Nevertheless, by conducting a numerical optimization, we have identified several variants of Levitt's pulse sequence that produce a larger coherent population inversion when there is a spread in initial conditions.
Composite Pulses (CPs) are widely used in Nuclear Magnetic Resonance (NMR), optical spectroscopy, optimal control experiments and quantum computing to manipulate systems that are well-described by a two-level Hamiltonian. A careful design of these pulses can allow the refocusing of an ensemble at a desired state, even if the ensemble experiences imperfections in the magnitude of the external field or resonance offsets. Since the introduction of CPs, several theoretical justifications for their robustness have been suggested. In this work, we suggest another justification based on the classical mechanical concept of a stability matrix. The motion on the Bloch Sphere is mapped to a canonical system of coordinates and the focusing of an ensemble corresponds to caustics, or the vanishing of an appropriate stability matrix element in the canonical coordinates. Our approach highlights the directionality of the refocusing of the ensemble on the Bloch Sphere, revealing how different ensembles refocus along different directions. The approach also clarifies when CPs can induce a change in the width of the ensemble as opposed to simply a rotation of the axes. As a case study, we investigate the 90(x)180(y)90(x)90(x)180(y)90(x)90(x)180(y)90(x) CP introduced by Levitt, where the approach provides a new perspective into why this CP is effective.
In this work, we exploit the idea of composite pulses to achieve robust population inversion in a three-level quantum system. The scheme is based on the modulation of the coupling strength, while the other physical parameters remain unchanged. The composite pulses sequence is designed by vanishing high-order error terms, and can compensate the systematic errors to any desired order. In particular, this scheme keeps a good performance under the disturbance of waveform deformations. This trait ensures that population inversion can be nearly obtained even when the pulse sequence has a short jump delay. As an example, we employ the designed composite pulse sequence to prepare the WWW state in a robust manner in the superconducting circuits. The numerical results show that the fidelity can still maintain a high level in a decoherence environment.
Abstract A scheme is proposed for achieving robust population inversion in five‐level systems by means of composite pulses. An example of such a system consists of the magnetic sublevels with angular momenta and . Through elaborately constructing the relative phases of pulse pairs, the composite sequences perform well in suppressing the uncorrelated pulse area errors. In particular, the five pulse‐pair sequence possesses good robustness and a short evolution time. The composite sequences are further designed to compensate for a single type of pulse area errors to any desired order. This work provides a high‐efficiency way for robust quantum state manipulation in five‐level systems.
We introduce a method to rotate arbitrarily the excitation profile of universal broadband composite pulse sequences for robust high-fidelity population inversion. These pulses compensate deviations in any experimental parameter (e.g., pulse amplitude, pulse duration, detuning from resonance, Stark shifts, unwanted frequency chirp, etc.) and are applicable with any pulse shape. The rotation allows one to achieve higher-order robustness to any combination of pulse area and detuning errors at no additional cost. The latter can be particularly useful, e.g., when detuning errors are due to Stark shifts that are correlated with the power of the applied field. We demonstrate the efficiency and universality of these composite pulses by experimental implementation for rephasing of atomic coherences in a Pr3+:Y2SiO5{\text{Pr}}^{3+}\text{:}{\text{Y}}_{2}{\text{SiO}}_{5}Pr3+:Y2​SiO5​ crystal.
Population inversion from an initial quantum state to a desired quantum state represents an exciting new frontier for quantum manipulation; that frontier is endowed with a strong interdisciplinary character and connections to other scientific fields, including atomic, molecular, and optical physics, as well as in solid-state devices, for fundamental studies, nuclear magnetic resonance and other spectroscopic techniques, metrology, interferometry, optical control of chemical reactions or quantum-information applications. Two-level systems are ubiquitous in these areas, and the driving of a population inversion is an important operation. So far various schemes for population transfer to a target state from an initial state have been proposed in theory and implemented in experiment. To be useful, such methods must, of course, be reliable, fast, and robust. Most literatures on two-level models investigated population inversion, which could be modelled by infinite-time processes. In this paper, by using a composite adiabatic passage (CAP) technique—in which the single pulse driving the quantum transition is replaced by a sequence of pulses with well-defined control phases, we investigate the population inversion problem of two-level quantum system with a finite duration. We take a sinusoidally varying pulse model that continuously vanishes at the beginning and the end of the finite duration as an example. The high-fidelity population inversion of the two-level system with finite duration is achieved. We discuss the effects of both coupling strength and detuning strength on the transition probability. It is found that this protocol could suppress the oscillations in the transition probability and reduce the admissible error bellow the 10 -4 quantum computation benchmark. The fidelity can arrive at 1, even with simple three- and five-composite pulse sequences. The protocol combines the advantages of adiabatic passage and composite pulses techniques. By choosing the composite phases appropriately, a high-fidelity, fast, stable and extremely robust complete population inversion is achieved for the finite-time two-level model. The values of the composite phases are universal for they do not depend on the pulse shapes and the chirp as long as the latter satisfy the symmetry property. Furthermore, the classical Hamiltonian is applied to describe the dynamic properties of the quantum system and further verify the effectiveness and feasibility of the proposed CAP technique. The results we present here are general and in principle apply to any two- and multi-level quantum systems. The accuracy of the CAP technique and its robustness against parameter variations make the protocol suitable for ultrahigh-fidelity quantum manipulation. We believe that the protocol can be widely used in many different areas of science, ranging from the quantum information, the quantum chemical, and the quantum optics to the ultracold atomic and molecular physics.