Structural and dynamical aspects of avoided-crossing resonances in a three-level $\Lambda$ system

In a recent publication [Phys. Rev. A 79, 065602 (2009)], it was shown that an avoided-crossing resonance can be defined according to level-structural or dynamical criteria. We propose an experiment to observe the difference between the two definitions in a three-level $\Lambda$ system using microwave fields coupling hyperfine magnetic sublevels in alkali-metal atoms.

Figure 1

Simple Raman three-level setup with energy levels $|1\rangle$, $|2\rangle$, and $|3\rangle$, and detunings of the two lasers with respect to atomic transitions and coupling strengths (Rabi frequencies) ${\Omega }_1$ and ${\Omega }_2$. Initially, states $|1\rangle$ and $|3\rangle$ are assumed to be uncoupled, where ${\Omega }_p=0$. For the experimental determination of the different shifts, a coupling between states $|1\rangle$ and $|3\rangle$ becomes necessary, where ${\Omega }_p\ne 0$.

Figure 2

Bare (black solid line) and dressed (red dashed line, ${\Omega }_1={\Omega }_2=0.5{\delta }_2$) energy levels as a function of ${\delta }_1/{\delta }_2$.

Figure 3

Exact dynamical shift computed by numerically diagonalizing the full three-level Hamiltonian (1) (solid lines) as a function of the ratio between the coupling strengths. The approximate expression for the dynamical shift (9) is plotted with dotted lines.

Figure 4

(a) Exact energy splitting $\Delta \epsilon ={\epsilon }_3-{\epsilon }_2$ obtained by diagonalizing the three-level Hamiltonian (1) (black solid line), compared to the energy splitting obtained by identifying one of the maxima (in this case, the maxima around $\nu ~\Delta \epsilon$) of the $P_{|{\epsilon }_2\rangle \to |{\epsilon }_3\rangle }$ transition probability (red dashed line). In the $t\to \infty$, limit both lines converge (the calculation is for ${\delta }_{2}t/2\pi =125$). (b) Difference between the exact (probeless) position of the minimum splitting (structural resonance) and the structural resonance obtained from the probed system as a function of time (black solid line). This difference goes to zero in the long-time limit as expected. The red dashed line corresponds to the exact value of the dynamical shift ${\Delta }_D$, which is the precession required in the measurement of the structural resonance. The time marked by an x in (b) corresponds to $t=2\pi /{\Delta }_D$, the lower bound of the time required to resolve the dynamical shift [see Eq. (11)]. ${\Omega }_1=0.2{\delta }_2$ and ${\Omega }_2=0.5{\delta }_2$.