Bloch–Siegert shift

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).

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.

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 Felix Bloch and Arnold 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 ={\sqrt {(\Omega _{0})^{2}+(\omega -\omega _{0})^{2}}}

where $\Omega _{0}=|V_{ab}/2\hbar |$ is the on-resonance Rabi frequency.

Bloch–Siegert shift

Consider the effect due to the counter-rotating field. In the counter-rotating frame ( $\omega _{\mathrm {cr} }=-\omega$ ), the effective detuning is $\Delta \omega _{\mathrm {cr} }=\omega +\omega _{0}$ and the counter-rotating field adds a driving component perpendicular to the detuning, with equal amplitude $\Omega _{0}$ . 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 _{\mathrm {eff} }=\pm {\sqrt {\Omega _{0}^{2}+(\omega +\omega _{0})^{2}}}

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

\omega _{\mathrm {res} }+\omega =\pm {\sqrt {\Omega _{0}^{2}+(\omega +\omega _{0})^{2}}}

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

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

and

\omega _{\mathrm {res} }=-{\frac {1}{3}}\omega _{0}\left[1+{\frac {3}{4}}\left({\frac {\Omega _{0}}{\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 _{\mathrm {B-S} }={\frac {1}{4}}{\frac {\Omega _{0}^{2}}{\omega _{0}}}

The counter-rotating frequency gives rise to a population oscillation at $2\omega$ , with amplitude proportional to $(\Omega /\omega )$ , and phase that depends on the phase of the driving field.[1] Such Bloch–Siegert oscillation may become relevant in spin flipping operations at high rate. This effect can be suppressed by using an off-resonant Λ transition.[2]

AC-Stark shift

The AC-Stark shift is a similar shift in the resonance frequency, caused by a non-resonant field of the form $H_{\mathrm {or} }={\frac {V_{\mathrm {or} }}{2}}\cos {(\omega _{\mathrm {or} }t)}$ perturbing the spin. It can be derived using a similar treatment as above, invoking the RWA on the off-resonant field. The resulting AC-Stark shift is: $\delta \omega _{\mathrm {AC} }={\frac {1}{2}}{\frac {\Omega _{\mathrm {or} }^{2}}{(\omega _{0}-\omega _{\mathrm {or} })}}$ , with $\Omega _{or}=|V_{or}/2\hbar |$ .

References

Cardoso, George C.; et al. (23 June 2005). "In situ detection of the temporal and initial phase of the second harmonic of a microwave field via incoherent fluorescence". Physical Review A. 71 (6): 063408. arXiv:quant-ph/0410219. Bibcode:2005PhRvA..71f3408C. doi:10.1103/PhysRevA.71.063408.
Pradhan, Prabhakar (28 March 2009). "Suppression of error in qubit rotations due to Bloch–Siegert oscillation via the use of off-resonant Raman excitation". Journal of Physics B: Atomic, Molecular and Optical Physics. 42 (6): 065501. Bibcode:2009JPhB...42f5501P. doi:10.1088/0953-4075/42/6/065501. S2CID 15051122.

L. Allen and J. H. Eberly, Optical Resonance and Two-level Atoms, Dover Publications, 1987.

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[1] Cardoso, George C.; et al. (23 June 2005). "In situ detection of the temporal and initial phase of the second harmonic of a microwave field via incoherent fluorescence". Physical Review A. 71 (6): 063408. arXiv:quant-ph/0410219. Bibcode:2005PhRvA..71f3408C. doi:10.1103/PhysRevA.71.063408.

[2] Pradhan, Prabhakar (28 March 2009). "Suppression of error in qubit rotations due to Bloch–Siegert oscillation via the use of off-resonant Raman excitation". Journal of Physics B: Atomic, Molecular and Optical Physics. 42 (6): 065501. Bibcode:2009JPhB...42f5501P. doi:10.1088/0953-4075/42/6/065501. S2CID 15051122.