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Journal of Physics B Atomic Molecular and Optical Physics

A strongly driven two-level atom revisited: Bloch - Siegert shift versus dynamic Stark splitting

Changjiang Wei, Andrew S. M. Windsor, Neil B. Manson

Published Nov 14, 1997• 29 citations
Publisher

Abstract

We calculate a probe response of a homogeneously broadened two-level transition in the presence of a strong pump. The derivation is carried out for a generalized interaction Hamiltonian, which gives a uniform account of both the Bloch - Siegert shift and dynamic Stark splitting. It is shown that the Bloch - Siegert shift can be viewed as an extreme situation of the dynamic Stark splitting, commonly known as light shift in the literature. Finally using a perturbative approach an analytical expression for the Bloch - Siegert shift is obtained, which is compared with the previous results.

DOI
10.1088/0953-4075/30/21/022

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We calculate a probe response of a homogeneously broadened two-level transition in the presence of a strong pump. The derivation is carried out for a generalized interaction Hamiltonian, which gives a uniform account of both the Bloch - Siegert shift and dynamic Stark splitting. It is shown that the Bloch - Siegert shift can be viewed as an extreme situation of the dynamic Stark splitting, commonly known as light shift in the literature. Finally using a perturbative approach an analytical expression for the Bloch - Siegert shift is obtained, which is compared with the previous results.
A superconducting qubit was driven in an ultrastrong fashion by an oscillatory microwave field, which was created by coupling via the nonlinear Josephson energy. The observed Stark shifts of the "atomic" levels are so pronounced that corrections even beyond the lowest-order Bloch-Siegert shift are needed to properly explain the measurements. The quasienergies of the dressed two-level system were probed by resonant absorption via a cavity, and the results are in agreement with a calculation based on the Floquet approach.
The dynamic Stark effect (Autler-Townes splitting and radiative shift) of a J = 0 ↔ J = 1 optical transition have been investigated. The barium atomic beam is illuminated by a strong pump beam, resonant or slightly non-resonant with the BaI resonance line (λ = 5 535 Å) ; the fluorescence induced by a weak probe beam is detected versus the probe frequency. With linear polarizations for the two beams one deals either with a two-level (parallel polarizations) or with a three-level system (perpendicular polarizations). In the latter case and for small detunings of the pump beam, the Autler-Townes doublet has been observed. For large detunings of the pump beam a single, shifted resonance has been obtained with both polarization settings. In the three-level case the observation of the passage from the Autler-Townes situation to the light-shift situation, is reported. The corresponding calculations are presented; a particular attention is paid to the case of large detunings for the pump beam (light-shift situation). The evolution calculated for the resonance curve (position and relative intensities of the peaks) when the relative polarization directions are changed is consistent with the experimental data.
Analytic expressions for the peak positions and pump frequency shift of equal intensity weak probe Autler-Townes transitions to a third level in the presence of a linearly polarized pump field are given. It is found that the presence of the third level affects the pump field dressing of the strongly excited two levels. A comparison of the Bloch-Siegert shift estimated by this weak probe technique with earlier methods suggests this condition provides a practical method for obtaining the Bloch-Siegert shift for three level systems where the pump field does not directly excite the probe transition. If the pump field off-resonantly drives the probe transition, additional light shifts occur which may dominate the Bloch-Siegert shift.
We apply a simple analytical method based on a unitary transformation to calculate the Bloch-Siegert (BS) shift over the entire driving-strength range. In quantitative comparison with the numerically exact BS shift obtained by Floquet formalism as well as the previous BS results, we confirm that our calculated results are not only accurate in the weak-driving regime but also correct in the strong-driving limit. In the intermediate strong-driving regime, the calculated values of the BS shift are nearly the same as the exact ones. It turns out that our calculation for the BS shift is beyond perturbation. Meanwhile, we demonstrate the signatures caused by the BS shift by monitoring the excited-state population and the probe-pump spectrum under the experiment accessible conditions. In particular, we find that when the driving frequency is fixed at the transition frequency of the system, the line shape of the probe-pump spectrum becomes asymmetric with the increase of the driving strength, which may be verified experimentally.
When trapped atoms are illuminated by weak lasers, off-resonant transitions cause shifts in the frequencies of the vibrational-sideband resonances. These frequency shifts may be understood in terms of Stark shifts of the individual levels or, as proposed here, as a vibrational Bloch-Siegert shift, an effect closely related to the usual (radio-frequency or optical) Bloch-Siegert shift and associated with rapidly oscillating terms when the rotating wave approximation is not made. Explicit analytic expressions are derived and compared to numerical results, and the similarities and differences between the usual and the vibrational Bloch-Siegert shifts are also spelled out.