Rabi resonance in spin systems: Theory and experiment

doi:10.1016/j.jmr.2014.02.014

Journal of Magnetic Resonance

Volume 242, May 2014, Pages 136-142

https://doi.org/10.1016/j.jmr.2014.02.014 Get rights and content

Highlights

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A secondary Rabi resonance is revealed under amplitude-modulated continuous wave excitation.
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Large Fourier components are observed at harmonics of the modulation frequency.
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Experiments agree with theoretical predictions derived from the Bloch equation.

Abstract

The response of a magnetic resonance spin system is predicted and experimentally verified for the particular case of a continuous wave amplitude modulated radiofrequency excitation. The experimental results demonstrate phenomena not previously observed in magnetic resonance systems, including a secondary resonance condition when the amplitude of the excitation equals the modulation frequency. This secondary resonance produces a relatively large steady state magnetisation with Fourier components at harmonics of the modulation frequency. Experiments are in excellent agreement with the theoretical prediction derived from the Bloch equations, which provides a sound theoretical framework for future developments in NMR spectroscopy and imaging.

Graphical abstract

Introduction

Excitation of magnetic resonance systems is typically achieved using short radiofrequency (RF) pulses with relatively high power in the order of kilowatts. Pulsed techniques predominantly replaced early continuous wave (CW) methods [1], [2], [3] due to their improved sensitivity and efficiency [4]. Recently, there has been renewed interest in CW techniques for magnetic resonance imaging (MRI) [5], [6], [7], largely motivated by the need to image samples with very short relaxation times. Continuous wave alternatives to the pulsed excitation paradigm are particularly advantageous for MRI, as opposed to spectroscopy, since the object is relatively large and thus high RF power is required to produce uniform flip angles using pulsed excitation methods.

In this work we demonstrate proof-of-concept measurements of the magnetisation during a CW excitation with an amplitude envelope modulated by a sinusoid. We measure the steady state magnetisation waveform and observe substantial frequency components at harmonics of the modulation frequency. Furthermore, we demonstrate that the steady state signal is maximum when the amplitude of the RF field equals the modulation frequency, establishing a secondary resonance condition that is analogous to resonance at the Larmor frequency.

Periodic modulation in NMR has been previously considered in numerous works. For example, Redfield observed a similar secondary resonance when the Rabi frequency of an RF field matched the frequency of an external magnetic field oscillating in the direction of the $B_{0}$ field [8]. Floquet theory was applied to systems consisting of an RF field with multiple frequencies in [9] and later generalised to solid-state NMR of rotating samples, where secondary resonances exist with respect to the sample spinning frequency, e.g. [10], [11], [12].

To our knowledge, the work presented here is the first experimental demonstration of such phenomena using an amplitude modulated RF field and provides a magnetic analogue of work in quantum optics. In the context of optics, Cappeller and Müller [13] considered a two-level atom excited with a sinusoidally varying phase and demonstrated a secondary resonance condition they termed ‘Rabi resonance’. Specifically, they showed an increase in the atom’s response when the rate of phase change is equal to the Rabi frequency. An amplitude modulated field was examined in [14] where the first six subharmonics of the Rabi frequency were observed experimentally. The second harmonic response to a phase modulated excitation has also been used as feedback to stabilise the intensity of an electromagnetic field [15], [16].

The novel magnetisation behaviour we demonstrate here arises from the nonlinear interaction of the RF field with the bulk magnetisation, accentuated by the use of an amplitude modulated CW pulse. Traditionally, pulsed techniques as well as continuous wave techniques such as stochastic MRI [5] and ‘sweep imaging with Fourier transformation’ [7] have assumed that the spin system can be treated in a linear time invariant framework. Indeed a focus of early work in spectroscopy was to ensure the linearity assumption remained valid to avoid distortion in the spectra, e.g. [17]. Although the system is approximately linear under certain conditions [18], it is our goal to investigate and exploit the nonlinear interactions.

The nonlinear features of the spin system are intrinsically interesting in their own right although we also envisage practical applications. For example, improved isolation between the transmitted and received signal can be achieved by transmitting at one frequency and receiving at a harmonic frequency. The two signals are separated in frequency, which allows digital or analog filters to extract only the signal of interest. This is particularly important in continuous wave NMR applications since the desired signal is orders of magnitude smaller than the transmitted RF signal [7].

A theoretical analysis and averaging solution of the Bloch equations were presented for a similar excitation in [19], [20]. The contribution here is twofold. First, we obtain experimental magnetic resonance data verifying the theoretical results. Secondly, we extend the analytic results to more general excitation and provide an explicit solution in terms of Bessel functions.

Section snippets

Theory

We consider a radiofrequency field oscillating at the Larmor frequency with an amplitude modulated envelope given by $ω_{e} (t) = ω_{1} (1 + α \cos (ω_{m} t)),$ where $ω_{1} = γ B_{1}, γ$ is the gyromagnetic ratio, $B_{1}$ is the amplitude of the RF field without modulation, $α$ is the modulation factor and $ω_{m}$ is the modulation frequency. The frequencies $ω_{1}$ and $ω_{m}$ defining the signal envelope are small compared to the Larmor frequency of the static field.

The starting point for the theoretical analysis of the response to amplitude

Experiments

The theory developed in Section 2 is validated by simulations and experiments. We use the following proof-of-concept experimental procedure to avoid hardware difficulties associated with simultaneous transmission and reception [7].

The spin system is excited with an amplitude modulated RF field described by Eq. (1) for an initial duration of T (in the order of seconds). A circularly polarised RF field is generated using a single channel volume coil. The system takes 57 μs after the end of the

Results

The $T_{1}$ and $T_{2}$ relaxation times were measured with the method described in Section 3.1 to be 342 ms and 139 ms, respectively. These values were used in simulations and to evaluate theoretical expressions.

Fig. 1 displays the response of the spin system to the amplitude-modulated RF excitation described by Eq. (1) with $α = 1, ω_{1} = 100 Hz$ and $ω_{m} = 100 Hz$ . The measured y-component of the magnetisation (circles) are in excellent agreement with both the signal predicted by Eq. (25b) (solid line) and signal

Discussion

The existence of secondary Rabi resonance in magnetic spin systems has been experimentally demonstrated for the first time, providing a magnetic counterpart to similar experiments in quantum optics [13]. Our theoretical analysis describes both the transient and steady state components of the magnetisation response to amplitude modulated CW excitation.

The presence of the higher-order harmonics can be used to isolate the transmitted signal from the received signal. In principle, an amplitude

Conclusion

We have presented experimental results that demonstrate the response of a magnetic resonance system to a sinusoidally modulated RF field rotating at the Larmor frequency. The experiments demonstrate the steady state magnetisation is maximum when the RF amplitude matches the modulation frequency, establishing a Rabi resonance condition, akin to the condition demonstrated in nonlinear optics. Additionally, we have demonstrated that the magnetisation signal contains frequency components at

References (29)

H. Nilgens et al.
Hadamard NMR imaging with slice selection
Magn. Reson. Imaging
(1996)
A.J. Fagan et al.
Development of a 3-D, multi-nuclear continuous wave NMR imaging system
J. Magn. Reson.
(2005)
D. Idiyatullin et al.
Continuous SWIFT
J. Magn. Reson.
(2012)
M. Leskes et al.
Floquet theory in solid-state nuclear magnetic resonance
Progress Nucl. Magn. Reson. Spectrosc.
(2010)
Z. Gan et al.
Rotary resonance in multiple-quantum magic-angle spinning
Chem. Phys. Lett.
(2002)
M. Garwood et al.
The return of the frequency sweep: Designing adiabatic pulses for contempory NMR
J. Magn. Reson.
(2001)
B. Tahayori et al.
Revisiting excitation pattern design for magnetic resonance imaging through optimisation of the signal contrast efficiency
Biomed. Signal Process. Control
(2009)
F. Bloch
Nuclear Induction
Phys. Rev.
(1946)
F. Bloch et al.
The nuclear induction experiment
Phys. Rev.
(1946)
E.M. Purcell et al.
Resonance absorption by nuclear magnetic moments in a solid
Phys. Rev.
(1946)

R.R. Ernst et al.

Application of fourier transform spectroscopy to magnetic resonance

Rev. Sci. Instrum.

(1966)

A. Redfield

Nuclear magnetic resonance saturation and rotary saturation in solids

Phys. Rev.

(1955)

Y. Zur et al.

Multiphoton NMR spectroscopy on a spin system with I = 1/2

J. Chem. Phys.

(1983)

A. Schmidt et al.

The Floquet theory of nuclear magnetic resonance spectroscopy of single spins and dipolar coupled spin pairs in rotating solids

J. Chem. Phys.

(1992)

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The decay of Rabi oscillations provides direct information about coherence of electron spins. When observed in EPR experiments, it is often shortened by spatial inhomogeneity of the microwave field amplitude in a bulk sample. In order to suppress this undesired loss of coherence, we propose an additional dressing of spin states by a weak longitudinal continuous radiofrequency field. The Gaussian, cosine and linear distributions of the microwave amplitude is analyzed. Our calculations of the Rabi oscillations between the doubly dressed spin states show that for all these distributions the maximum suppression of the inhomogeneity-induced decoherence is achieved at the so-called Rabi resonance when the radio-field frequency is in resonance with the Rabi frequency of spins in the microwave field. The manifestations of such suppression in the published EPR experiments with the bichromatic driving are discussed. The realization of the Rabi resonance using the radiofrequency field could open new possibilities for separating the contributions of relaxation mechanisms from those due to the inhomogeneous driving in spin decoherence.
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Enhanced Transverse Nuclear Magnetization of an Atomic Co-Magnetometer at the Optimal Oscillating Magnetic Field
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View full text

Rabi resonance in spin systems: Theory and experiment

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Theory

Experiments

Results

Discussion

Conclusion

Magn. Reson. Imaging

J. Magn. Reson.

J. Magn. Reson.

Progress Nucl. Magn. Reson. Spectrosc.

Chem. Phys. Lett.

J. Magn. Reson.

Biomed. Signal Process. Control

Nuclear Induction

Phys. Rev.

The nuclear induction experiment

Phys. Rev.

Resonance absorption by nuclear magnetic moments in a solid

Phys. Rev.

Application of fourier transform spectroscopy to magnetic resonance

Rev. Sci. Instrum.

Nuclear magnetic resonance saturation and rotary saturation in solids

Phys. Rev.

Multiphoton NMR spectroscopy on a spin system with I = 1/2

J. Chem. Phys.

The Floquet theory of nuclear magnetic resonance spectroscopy of single spins and dipolar coupled spin pairs in rotating solids

J. Chem. Phys.