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). When the rotating wave approximation (RWA) is invoked, the resonance between the driving field and a pseudospin occurs when the field frequency formula_1 is identical to the spin's transition frequency formula_2. 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 formula_3, a linearly polarized field is considered as a superposition of two circularly polarized fields of the same amplitude rotating in opposite directions with frequencies formula_4. Then, in the rotating frame(formula_1), we can neglect the counter-rotating field and the Rabi frequency is where formula_7 is the on-resonance Rabi frequency. Bloch–Siegert shift. Consider the effect due to the counter-rotating field. In the counter-rotating frame (formula_8), the effective detuning is formula_9  and the counter-rotating field adds a driving component perpendicular to the detuning, with equal amplitude formula_10. 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 Therefore, the resonance frequency (formula_12) of the system dressed by the counter-rotating field is formula_13 away from our frame of reference, which is rotating at formula_14 and there are two solutions for formula_16 and The shift from the RWA of the first solution is dominant, and the correction to formula_19 is known as the Bloch–Siegert  shift: The counter-rotating frequency gives rise to a population oscillation at formula_21, with amplitude proportional to formula_22, and phase that depends on the phase of the driving field. 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. AC-Stark shift. The AC-Stark shift is a similar shift in the resonance frequency, caused by a non-resonant field of the form formula_23 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: formula_24, with formula_25.