en>Mogism: /* An example of ECM */Cleanup/Typo fixing, typo(s) fixed: relationsip → relationship (2) using AWB

2014-02-02T17:49:22Z

An example of ECM: Cleanup/Typo fixing, typo(s) fixed: relationsip → relationship (2) using AWB

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[[File:Rotating pot lid demonstrating blocksieger shift.gif|thumb|The pot lid is rotating around an axis along the surface of the table that is quickly rotating. This results in a secondary rotation which is perpendicular to the table. <br /><br />
This is equivalent to the Bloch-Siegert shift and can be seen by watching the motion of the red dot. ]]

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, 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 <math>\omega</math> is identical to the spin's transition frequency <math>\omega_0</math>. The RWA is, however, an approximation. In 1940 Bloch and Siegert showed that the dropped parts oscillating rapidly can give rise to a shift in the true resonance frequency of the dipoles.

==Rotating wave approximation==
In RWA, when the perturbation to the two level system is <math> H_{ab} = \frac{V_{ab}}{2} \cos{(\omega t)}</math>, a linearly polarized field is considered as a superposition of two circularly polarized fields of the same amplitude rotating in opposite directions with frequencies <math>\omega, -\omega</math>. Then, in the rotating frame(<math>\omega</math>), we can neglect the counter-rotating field and the [[Rabi frequency]] is
:<math>\Omega = \frac{1}{2} \sqrt{(|V_{ab}/\hbar |)^2 +(\omega -\omega_0)^2}.</math>

==Bloch-Siegert shift==
Consider the effect due to the counter-rotating field. In the counter-rotating frame(<math>-\omega</math>), the effective precession frequency is
:<math>\Omega_{eff} =\frac{1}{2} \sqrt{(|V_{ab}/\hbar |)^2 +(\omega +\omega_0)^2}.</math>
Then the resonance frequency is given by
:<math>2\omega = \sqrt{(|V_{ab}/\hbar |)^2 +(\omega +\omega_0)^2}</math>
and there are two solutions
:<math>\omega =\omega_0 \left[ 1 +\frac{1}{4} \left( \frac{V_{ab}}{\hbar \omega_0} \right)^2 \right]</math>
and
:<math>\omega =-\frac{1}{3} \omega_0 \left[ 1 +\frac{3}{4} \left( \frac{V_{ab}}{\hbar \omega_0} \right)^2 \right].</math>
The shift from the RWA of the first solution is dominant, and the correction to <math> \omega_0 </math> is known as the Bloch-Siegert shift:
:<math> \delta \omega_{B-S} =\frac{1}{4} \frac{(V_{ab})^2}{\hbar^2\omega_0}</math>

==References==
* J. J. Sakurai, Modern Quantum Mechanics, Revised Edition,1994.
* David J. Griffiths, Introduction to Quantum Mechanics, Second Edition, 2004.
* L. Allen and J. H. Eberly, Optical Resonance and Two-level Atoms, Dover Publications, 1987.

[[Category:Wave mechanics]]

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en>Mogism: /* An example of ECM */Cleanup/Typo fixing, typo(s) fixed: relationsip → relationship (2) using AWB