## INTRODUCTION

Many applications in control of quantum systems involve the manipulation of a large ensemble by using the same control field. Often in practice, different members of the ensemble see variations in the experimental parameters that govern the dynamics. Magnetic resonance imaging (MRI) at high fields is an important case where the variation of an experimental parameter over the ensemble, mainly the radio-frequency (RF) field, has drastic consequences and raises challenging control problems. The irradiation of objects comparable or larger than the RF wavelength leads to destructive interferences and dielectric resonance effects which translate into RF inhomogeneity and thus dark zones on MRI images of human body parts such as the brain or pelvis(1, 2). Standard compensating pulse techniques such as adiabatic and composite pulses in general require too much energy to be implemented on a human subject(3). A simple composite pulse approach with a single coil(4) may however be implemented with a reasonable amount of energy, but still leaves room for some substantial improvement, and its performance quickly decreases as the spin resonance frequency deviates from the carrier frequency. Other *B*_{1} compensated 3D tailored RF pulses have also been reported in(5, 6), but rely on the small flip angle approximation or/and assume some particular *B*_{1} profile. Parallel transmission, namely Transmit SENSE(7–10) and RF shimming(3, 11), is another powerful method proposed for counteracting RF inhomogeneity at high fields. Although its feasibility has been verified experimentally, and is still the subject of stimulating and ongoing research, the method is costly, bulky, and time-consuming. It is costly mainly due to the fact that it requires (the purchase of) sophisticated hardware. It is bulky because it requires the designer to handle very large matrices, thus yielding long computation times. The time-consuming part, besides the computing part in the design process just mentioned above, arises from the N-fold transmit 3D *B*_{1} maps measurements (one for each channel) required for the algorithm to compute a solution, making an exam study even more lengthy for a patient. Finally, the nonlinearity of the Bloch equation tends to have an adverse effect on the performance of Transmit SENSE at large flip angles, despite recent improvements(12–15). As far as RF shimming is concerned, constraints of Maxwell's equations impose inevitable limits in performance when optimizing the RF field over a significant volume such as a human brain(3, 16) (but see(17) for an interesting combination of composite pulses with RF shimming).

In this article, we present an alternative method for uniformly exciting a volume of interest. The method is inspired from the strongly modulating pulses developed originally for nuclear magnetic resonance (NMR) quantum computing(18). Strongly modulating pulses initially fitted in the realm of “gate engineering,” where one is interested in finding an irradiation scheme that implements a target transformation with high fidelity no matter what the initial state of the system is. Soon after their creation, the design of these pulses was extended to account for RF inhomogeneity and their performance was measured and verified experimentally on numerous occasions(18, 19). As far as the external static *B*_{0} field inhomogeneity is concerned, it was simply too small to be a problem and the pulses turned out to be naturally robust with respect to small resonance frequency variations. This is not the case in magnetic resonance brain imaging because of the larger volumes involved and the varying magnetic susceptibilities, yielding a significant Larmor frequency dispersion that increases linearly with the external magnetic field. Although *B*_{1} and *B*_{0} inhomogeneities encountered in high-field MRI are much more severe than in liquid state NMR quantum computing, MRI often involves “state engineering” of a single spin (as opposed to gate engineering of coupled multispin systems). To achieve uniform excitation, indeed one must steer the equilibrium longitudinal state to a final one with a desired flip angle. As a result, the control problem is less demanding than in NMR quantum computing and hence relaxes some constraints on the optimization procedure, making the technique worth investigating for MRI needs.

However, the method reported in(18) is not directly applicable and needs important modifications. The first one is the cost function to minimize, where the gate engineering problem should be converted into the state engineering one, to achieve better performance. The goal here is to achieve a desired flip angle with minimum deviation with respect to the known *B*_{1} and *B*_{0} amplitudes. Hence in this work, the initial state is always assumed to be a z-magnetization state, limiting the demonstration to spoiled gradient echo sequences. Second, penalty functions are also adjusted to account for the different experimental limitations (time, power, frequency), and to minimize the specific absorption rate (SAR). Third, the search algorithm for the minimization procedure is now more sophisticated and combines a genetic algorithm (GA)(20) with the Nelder-Mead(21) direct search method. Although the work in(18, 19) uses the latter algorithm only, the use of a genetic algorithm increases the probability to find a global optimum, or at least generates a good start for the Nelder-Mead algorithm, thereby making the search faster. With the *B*_{1} and *B*_{0} inhomogeneity 2D histogram known, RF pulses whose durations are comparable or shorter than the transmit SENSE pulses(7–9) are returned in less than 30 s with a standard PC.

In the following section, we formulate the theory used to design the strongly modulating pulses. We use the exact Schr*ö*dinger, or Bloch, equation and account for *B*_{1} and *B*_{0} inhomogeneity by looping over their 2D histogram values. No magnetic field gradients during RF pulsing are assumed, making this method at the moment applicable only to nonselective 3D imaging. We proceed with some experimental results on two different human brains at 3 T. We then compare the strongly modulating pulses with adiabatic BIR4 pulses(22–24), also known to be robust against *B*_{1} and *B*_{0} inhomogeneity, in terms of performance and SAR. Finally, we discuss future work in prospect.