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 ${\displaystyle H_{ab}={\frac {V_{ab}}{2}}\cos {(\omega t)}}$, 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 (${\displaystyle \omega _{cr}=-\omega }$), the effective detuning is ${\displaystyle \Delta \omega _{cr}=\omega +\omega _{0}}$ and the counter-rotating field adds a driving component perpendicular to the detuning, with amplitude ${\displaystyle \Omega _{cr}=|V_{ab}/\hbar |}$. The counter-rotating field effectively dresses the system, where we can define a new quantization axis slightly tilted from the original one, with an effective detuning

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

Therefore, the resonance frequency (${\displaystyle \omega _{res}}$) of the system dressed by the counter-rotating field is ${\displaystyle \Delta \omega _{eff}}$ away from our frame of reference, which is rotating at $-\omega$

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

and there are two solutions for ${\displaystyle \omega _{res}}$

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

and

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

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

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

Generalized Bloch-Siegert shift

As the Bloch-Siegert shift is caused by the off-resonant component of the oscillating field, the shift can be generalized to any non-resonant field of the form ${\displaystyle H_{or}={\frac {V_{or}}{2}}\cos {(\omega _{or}t)}}$ perturbing the spin. With a similar treatment as above, and invoking the RWA on the off-resonant field, the resulting Bloch-Siegert shift is

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

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.