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 ( $\omega _{cr}=-\omega$ ), the effective detuning is $\Delta \omega _{cr}=\omega +\omega _{0}$ and the counter-rotating field adds a driving component perpendicular to the detuning, with amplitude $\Omega _{cr}=|V_{ab}/\hbar |$ . 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

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

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

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

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

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

and

\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 $H_{or}={\frac {V_{or}}{2}}\cos {(\omega _{or}t)}$ perturbing the spin. With a similar treatment as above, and invoking the RWA on the off-resonant field, the resulting Bloch-Siegert shift is

\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.

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