# Bloch-Siegert Shift and its Kramers-Kronig Pair

## TL;DR

This paper reveals that the Bloch-Siegert shift in NMR originates from a complex susceptibility term with a measurable Kramers-Kronig pair, unifying the understanding of relaxation and shift phenomena through a novel theoretical and experimental approach.

## Contribution

It introduces a theoretical framework showing the Bloch-Siegert shift as part of a complex susceptibility with a Kramers-Kronig pair, verified by a new experimental method.

## Key findings

- The Bloch-Siegert shift has an associated absorptive Kramers-Kronig pair.
- Theoretical derivation uses finite time-propagation and quantum master equations.
- Experimental verification employs a novel refocussed nutation experiment.

## Abstract

We report that the Bloch-Siegert shift which appears in Nuclear Magnetic Resonance (NMR) spectroscopy can also be shown to originate as a part of a complex drive-induced second-order susceptibility term. The shift terms thus obtained are shown to have an absorptive Kramers-Kronig pair. The theoretical treatment involves a finite time-propagation of a nuclear spin-$1/2$ system and the spin-bearing molecule under the action of thermal fluctuations acting on the latter. The finite propagator is constructed to account for many instances of thermal fluctuations occurring in a time-scale during which the spin density matrix changes infinitesimally. Following an ensemble average, the resulting quantum master equation directly yields a finite time-nonlocal complex susceptibility term from the external drive, which is extremely small but measurable in solution-state NMR spectroscopy. The dispersive part of this susceptibility term originating from the non-resonant component of the external drive results in the Bloch-Siegert shift. We have verified experimentally the existence of the absorptive Kramers-Kronig pair of the second-order shift term, by using a novel refocussed nutation experiment. Our method provides a single approach to explain both relaxation phenomena as well as Bloch-Siegert effect, which have been treated using non-concurrent techniques in the past.