Shifts from a distant neighboring resonance

The line center of a resonance is affected by the presence of neighboring resonances due to quantum interference between the two resonant processes. Such shifts must be accounted for in high-precision measurements but are easily overlooked. In the present work, we develop an analytic formulation for the effect in the simplest case of a three-level atom. The shifts in this model system are large enough to be of concern for precision measurements.

Figure 1

The three-level atom. A single monochromatic applied field is nearly in resonance with the $|1\rangle \to |2\rangle$ transition but far off resonance with the $|1\rangle \to |3\rangle$ transition. Both state $|2\rangle$ and state $|3\rangle$ radiatively decay down to state $|1\rangle$. We consider the case in which the entire population starts in state $|1\rangle$ at $t=0$ and find the population of state $|2\rangle$ at time $t=T$. Interference effects between states $|2\rangle$ and $|3\rangle$ cause shifts in the resonant frequency for driving population to state $|2\rangle$.

Figure 2

HWHM for the two-level line shape. The circles represent the HWHM for the exact line shape of Eq. (3). These are in good agreement with the widths of Eq. (7) (dashed line) for sufficiently small $T$ and ${\Omega }_2$, where the perturbative result should be valid. For small ${\gamma }_{2}T$, the width of Eq. (5) (solid lines) agrees with Eq. (3) to larger ${\Omega }_2$. For large $T$, the width of Eq. (3) is well approximated by the steady-state solution of Eq. (4a) (dotted line).

Figure 3

The factor $f$ of Eq. (25) that determines the shift of the $|1\rangle \to |2\rangle$ transition due to the presence of state $|3\rangle$. The line shifts are obtained by least-squares fits of the three-level line shape of Eq. (13) to the symmetric two-level line shape of Eq. (3) with the center ${\Delta }_2$ floating in the fit. Note that $f$ starts at zero at $T=0$ and grows linearly as ${\gamma }_{2}T/5$ as predicted by Eq. (23) and that it reaches unity for large $T$, as predicted by Eq. (24).

Figure 4

Multiplicative factor for the ac Stark shift. The figure shows that the ac Stark shift must be multiplied by a factor that depends on Rabi frequency ${\Omega }_2$, on the lifetime ${\gamma }_2^{-1}$, on the interaction time $T$, and on the ratio of the energy separations: ${\omega }_{21}/{\omega }_{32}$. For long interaction times $T$, the shift goes to zero for ${\omega }_{21}/{\omega }_{32}=\infty$ and changes sign for finite ${\omega }_{21}/{\omega }_{32}$.

Figure 5

The $2{\hspace{0.5em}}^3{S}_1$ and $2{\hspace{0.5em}}^3{P}_J$ states of atomic helium in the presence of a dc magnetic field. The population is initially pumped to the $|1\rangle$ state. A laser field polarized in the direction of the magnetic field is nearly in resonance with the $|1\rangle \to |2\rangle$ transition. The off-resonant $|1\rangle \to |3\rangle$ transition causes a shift. Both states $|2\rangle$ and $|3\rangle$ decay with equal branching ratios to the $|1\rangle$ and $|0\rangle$ states.