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