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The Bloch–Siegert shift is a phenomenon in quantum physics that becomes important for driven two-level systems when the driving gets strong (e.g. atoms driven by a strong laser drive or nuclear spins in
NMR Nuclear magnetic resonance (NMR) is a physical phenomenon in which atomic nucleus, nuclei in a strong constant magnetic field are disturbed by a weak oscillating magnetic field (in the near and far field, near field) and respond by producing ...
, driven by a strong oscillating magnetic field). When the rotating-wave approximation (RWA) is invoked, the resonance between the driving field and a pseudospin occurs when the field frequency \omega is identical to the spin's transition frequency \omega_0. The RWA is, however, an approximation. In 1940
Felix Bloch Felix Bloch (; ; 23 October 1905 – 10 September 1983) was a Swiss-American physicist who shared the 1952 Nobel Prize in Physics with Edward Mills Purcell "for their development of new methods for nuclear magnetic precision measurements and di ...
and Arnold Siegert showed that the dropped parts oscillating rapidly can give rise to a shift in the true resonance frequency of the dipoles. The Bloch–Siegert shift has been used for practical purposes in both NMR and
MRI Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to generate pictures of the anatomy and the physiological processes inside the body. MRI scanners use strong magnetic fields, magnetic field gradients, and rad ...
, including power calibration, image encoding, and magnetic field mapping.


Rotating wave approximation

In RWA, when the perturbation to the two level system is H_ = \frac \cos, a linearly polarized field is considered as a superposition of two circularly polarized fields of the same amplitude rotating in opposite directions with frequencies \omega, -\omega. Then, in the rotating frame(\omega), we can neglect the counter-rotating field and the
Rabi frequency The Rabi frequency is the frequency at which the Probability amplitude, probability amplitudes of two atomic electron transition, atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the transition dipole m ...
is :\Omega = \sqrt where \Omega_0 = , V_/2\hbar , is the on-resonance Rabi frequency.


Bloch–Siegert shift

Consider the effect due to the counter-rotating field. In the counter-rotating frame (\omega_\mathrm = -\omega), the effective detuning is \Delta\omega_\mathrm = \omega + \omega_0 and the counter-rotating field adds a driving component perpendicular to the detuning, with equal amplitude \Omega_0. The counter-rotating field effectively
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the system, where we can define a new quantization axis slightly tilted from the original one, with an effective detuning :\Delta\omega_\mathrm = \pm\sqrt Therefore, the resonance frequency (\omega_\mathrm) of the system dressed by the counter-rotating field is \Delta\omega_\mathrm away from our frame of reference, which is rotating at -\omega :\omega_\mathrm + \omega = \pm\sqrt and there are two solutions for \omega_\mathrm :\omega_\mathrm =\omega_0 \left 1 +\frac \left( \frac \right)^2 \right/math> and :\omega_\mathrm =-\frac \omega_0 \left 1 +\frac \left( \frac \right)^2 \right The shift from the RWA of the first solution is dominant, and the correction to \omega_0 is known as the Bloch–Siegert shift: : \delta \omega_\mathrm =\frac \frac The counter-rotating frequency gives rise to a population oscillation at 2\omega, with amplitude proportional to (\Omega/\omega), and phase that depends on the phase of the driving field. Such Bloch–Siegert oscillation may become relevant in spin flipping operations at high rate. This effect can be suppressed by using an off-resonant Λ transition.


Applications


NMR

When homonuclear
nuclear magnetic resonance decoupling Nuclear magnetic resonance decoupling (NMR decoupling for short) is a special method used in nuclear magnetic resonance (NMR) spectroscopy where a sample to be analyzed is irradiated at a certain frequency or frequency range to eliminate or part ...
is performed, Bloch–Siegert shifts may become significant due to the strength of the homonuclear decoupling field. Direct measurement of the homonuclear decoupling mean field strength can be achieved by measuring the resulting Bloch–Siegert shift.


MRI

The Bloch–Siegert shift is currently being widely investigated a potential encoding mechanism for MRI. The first significant use of the phenomenon in the MR imaging community was to perform mapping of the RF transmit field, by using the imaging system to measure the spatial phase accrual produced by an off-resonant RF pulse. Since then, it has been recognized that Bloch–Siegert shifts can be used in MRI sequences within imaging systems with a transmit field gradient to provide slice selection, phase encoding, and frequency encoding. The motivation for this research is to provide an alternative to conventional B_0 gradient encoding, which is currently used in clinical imaging systems but produces undesirable acoustic noise, peripheral nerve stimulation, and spatial design constraints.


AC-Stark shift

The AC-Stark shift is a similar shift in the resonance frequency, caused by a non-resonant field of the form H_\mathrm = \frac \cos perturbing the spin. It can be derived using a similar treatment as above, invoking the RWA on the off-resonant field. The resulting AC-Stark shift is: \delta \omega_\mathrm =\frac \frac, with \Omega_ = , V_/2\hbar , .


References

* L. Allen and J. H. Eberly, Optical Resonance and Two-level Atoms, Dover Publications, 1987. {{DEFAULTSORT:Bloch-Siegert shift Wave mechanics