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On the Reduction of a Problem in Magnetic Resonance

D. T. Pegg, G. W. Series

1973
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Proceedings of the Royal Society A
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T he p ro b lem in q u estio n is to find th e m o tio n o f a n a to m h a v in g a rb itra ry sp in J in a stro n g oscillating m ag n etic field a n d a s ta tic field w hose co m p o n e n t p e rp e n d ic u la r to th e o scillating field is sm all (case B). I t is show n how to re la te th is p ro b lem to th a t o f o rd in a ry m ag n etic resonance -a n a to m in a sm all oscillatin g field p e rp e n d ic u la r to a stro n g sta tic field (case A)b y a p p ly in g to each case a su
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... to each case a su ita b le tra n sfo rm a tio n . I n th e tra n sfo rm e d fram es th e d o m in a n t te rm s in th e e q u a tio n s o f m o tio n are e q u iv a le n t a n d su b sid iary te rm s are sim ilar, so t h a t re su lts w ell-know n in case A m a y be ta k e n over to case B. T he d o m in a n t te rm s in case B describe a se t o f resonances w hose line-shapes are giv en ex p licitly for a p a rtic u la r exam p le. S u b sid iary te rm s lead to sh ifts o f th e resonances a nalogous to th e B lo c h -S ie g e rt sh ift. T he leading te rm s in th ese sh ifts are given. T erm s o f h ig h er o rd er m a y be c a lc u la te d b y em p lo y in g a te c h n iq u e d u e to S hirley w hich is described. I n t r o d u c t i o n Among the problem s which have been extensively studied in th e semi-classical form ulation of quantum mechanics is th a t of a param agnetic atom in a configuration of fields suitable for stim ulating magnetic resonance, nam ely, a static field combined w ith a tim e-dependent field oscillating sinusoidally in the perpendicular plane. The oscillating field probes the system comprised of atom plus static field. F or m any purposes it is sufficient to consider only one rotating com ponent of the oscillating field, b u t the counter-rotating com ponent makes itself felt in distortions of th e main resonance (Bloch-Siegert effect) and in m ultiple-quantum resonances. The im port ance of these effects increases w ith the am plitude of the oscillating field. Their existence has draw n atten tio n to the interesting properties of a different system, nam ely, atom s plus oscillating field. A small transverse static field serves to explore this second system, the transverse field playing the role which the oscillating field itself plays in ordinary param agnetic resonance. This leads to the concept of 'dressed a to m s' (Cohen-Tannoudji & Haroche 1969; see also Allegrini & Arimondo 1971). Owing to the different symmetries of the two situations w ith respect to the timedependent fields, the analytical m ethods which have been developed for the first case m ay not be taken over directly to the second. In this work we show how to transform the two cases so th a t they become similar. The methods of approxim ation [ 281 ]

doi:10.1098/rspa.1973.0026
fatcat:hwulltj5ovgihow56wq7tywpie