Bloch-Siegert shift

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

$\Omega ={\frac {1}{2}}{\sqrt {(|V_{ab}/\hbar |)^{2}+(\omega -\omega _{0})^{2}}}.$

Bloch-Siegert shift

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

$\Omega _{eff}={\frac {1}{2}}{\sqrt {(|V_{ab}/\hbar |)^{2}+(\omega +\omega _{0})^{2}}}.$

Then the resonance frequency is given by

$2\omega ={\sqrt {(|V_{ab}/\hbar |)^{2}+(\omega +\omega _{0})^{2}}}$

and there are two solutions

$\omega =\omega _{0}\left[1+{\frac {1}{4}}\left({\frac {V_{ab}}{\hbar \omega _{0}}}\right)^{2}\right]$

and

$\omega =-{\frac {1}{3}}\omega _{0}\left[1+{\frac {3}{4}}\left({\frac {V_{ab}}{\hbar \omega _{0}}}\right)^{2}\right].$

The shift from the RWA of the first solution is dominant, and the correction to $\omega _{0}$ is known as the Bloch-Siegert shift:

$\delta \omega _{B-S}={\frac {1}{4}}{\frac {(V_{ab})^{2}}{\hbar ^{2}\omega _{0}}}$

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.