Here we use perturbation techniques based on the averaging method to investigate Rabi oscillations in cw and pulse-driven two-level systems (TLS's). By going beyond the rotating-wave approximation, especifically to second-order in perturbation, we obtain the Bloch-Siegert shift of the TLS resonant frequency, in which the resonant frequency increases with the driving field amplitude. This frequency shift implies that short resonant
π-pulses in which the Rabi frequency is approximately 40\% or higher of the transition frequency do not achieve complete inversion in TLS's. Hence, guided by analytical results based on the averaging technique, we propose two methods for obtaining population inversions in the TLS driven by short
π-pulses: one with chirping and the other with pulse shaping and near resonance blue-shifted detuning. Both methods minimize dephasing due to the Bloch-Siegert shift, reduce the dependance of the excitation of the TLS on the pulse phase, and are very effective in achieving complete population inversions.
It is well known that area pulses make Rabi oscillation and chirped pulses in the adiabatic interaction regime induce complete population inversion of a two-state system. Here we show that chirped zero-area pulses could engineer an interplay between the adiabatic evolution and Rabi-like rotations. In a proof-of-principle experiment utilizing spectral chirping of femtosecond laser pulses with a resonant spectral hole, we demonstrate that the chirped zero-area pulses could induce, for example, complete population inversion and return of the cold rubidium atom two-state system. Experimental result agrees well with the theoretically considered overall dynamics, which could be approximately modeled to a Ramsey-like three-pulse interaction, where the
x and
z rotations are driven by the hole and the main pulse, respectively.
A theoretical nonperturbative study of the coherent excitation of a two-state system by N consecutive equally spaced identical pulses is presented. General relations between the evolution matrix elements in the cases of one and N pulses are obtained in a closed form. For pulse envelopes allowing analytical solutions, these relations enable analytical treatment of pulse-train excitation; for pulses that can only be treated numerically, they shorten the computations by a factor of N. The relations show that the multiple-pulse excitation of a two-state system can be considered as a quantum analog of the diffraction grating. The interaction dynamics substantially depends on the way the train is produced because the phase shift that is accumulated by the probability amplitudes due to the free evolution of the system during and between the pulses differs. In the limit of weak excitation, the results recover earlier conclusions based on perturbative treatments. Simple formulas are derived for the conditions for complete population inversion (CPI) and complete population return (CPR), which differ from the single-pulse ones. These general results are applied to four particular cases that allow analytical solutions: resonant, rectangular, Rosen-Zener, and Allen-Eberly pulses. A common feature in all these cases is that the number and the amplitude of oscillations in the state populations increase with the number of pulses, while their width decreases. For rectangular pulses, CPI is possible below a given value of the ratio between the detuning and the pulse area; this value increases nearly linearly with the number of pulses.For the Rosen-Zener model, CPI is found to be possible for two and more pulses, while it is impossible for a single non-resonant pulse. It is shown that for a given number of pulses there is an upper limit on the detuning for which CPI can be observed; this limit increases logarithmically with the number of pulses. For the Allen-Eberly model, CPR is established to be possible for more than one pulse, while it is impossible for a single pulse. The cases of even and odd number of pulses are shown to lead to substantially different results. The limitations imposed on the detuning by the conditions for CPR and CPI are derived and discussed.