Error correction model

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.

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 $\omega$ is identical to the spin's transition frequency ${\omega }_0$. 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 $H_{ab}=\frac{V_{ab}}{2}\cos (\omega t)$, 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 is

$\mathrm{\Omega }=\frac{1}{2}\sqrt{(|V_{ab}/\hslash |{)}^2+(\omega -{\omega }_0{)}^2}.$

Bloch-Siegert shift

Consider the effect due to the counter-rotating field. In the counter-rotating frame($-\omega$), the effective precession frequency is

${\mathrm{\Omega }}_{eff}=\frac{1}{2}\sqrt{(|V_{ab}/\hslash |{)}^2+(\omega +{\omega }_0{)}^2}.$

Then the resonance frequency is given by

$2\omega =\sqrt{(|V_{ab}/\hslash |{)}^2+(\omega +{\omega }_0{)}^2}$

and there are two solutions

$\omega ={\omega }_0[1+\frac{1}{4}{\left(\frac{V_{ab}}{\hslash {\omega }_0}\right)}^2]$

and

$\omega =-\frac{1}{3}{\omega }_0[1+\frac{3}{4}{\left(\frac{V_{ab}}{\hslash {\omega }_0}\right)}^2].$

The shift from the RWA of the first solution is dominant, and the correction to ${\omega }_0$ is known as the Bloch-Siegert shift:

$\delta {\omega }_{B-S}=\frac{1}{4}\frac{(V_{ab}{)}^2}{{\hslash }^2{\omega }_0}$

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.