Shifts from a distant neighboring resonance for a four-level atom

In a recent paper [Phys. Rev. A 82, 052519 (2010)], the systematic shifts of a resonance due to quantum-mechanical interference from a distant neighboring resonance were derived. In that paper, the simplest three-level closed system was used to calculate analytic expressions for shifts in the resonant line centers. Here, we extend the analysis to the more relevant four-level system, which consists of two ground states and two excited states and which incorporates the physics of dark states. The shifts are shown to depend on the type of experiment performed and can be much larger than the shifts for the three-level system. The analytic formulas obtained are applied to the $2{}^{3}S$-to-$2{}^{3}P$ transitions in atomic helium, where significant shifts are found.

Figure 1

The four-level system considered in this paper. A single field of angular frequency $\omega$ couples $|1\rangle$ to $|2\rangle$ and to $|3\rangle$, both of which can decay to $|0\rangle$ and $|1\rangle$. We consider the case in which the field is nearly in resonance with the $|1\rangle \to |2\rangle$ transition and study the shifts in this resonance due to the distant $|1\rangle \to |3\rangle$ resonance. The figure shows state $|3\rangle$ above state $|2\rangle$ (i.e., ${\omega }_{32}>0$), but the results derived also apply for the case when state $|3\rangle$ is below state $|2\rangle$ (i.e., ${\omega }_{32}<0$).

Figure 2

Line shapes for the three-level system, with the $|2\rangle$ and $|0\rangle$ state populations as a function of applied frequency $\omega ={\omega }_{21}+{\Delta }_2$ shown in (a) and (b), respectively. These line shapes Eq. (7) are for ${\Omega }_2=0.1{\gamma }_2$ with ${\gamma }_{2\to 1}/{\gamma }_2=0.25$ (closed symbols) and 0.75 (open symbols). The solid lines are the approximate line shapes of Eqs. (10) and (11), which show good agreement at small ${\Omega }_{2}T$. The dotted lines are the approximate line shapes of Eqs. (14) and (15), which show good agreement with Eq. (7) at larger ${\Omega }_{2}T$.

Figure 3

Amplitudes and widths for the three-level line shapes for ${\Omega }_2/{\gamma }_2=1$ (squares), $1/10$ (diamonds), and $1/100$ (circles) for a range of ${\Omega }_{2}T$. The on-resonance amplitudes are shown in (a) and (b) for the $|2\rangle$ and $|0\rangle$ state populations. The full width at half maximum (FWHM) is shown in parts (c) and (d). For small ${\Omega }_{2}T$, the widths and amplitudes agree with Eqs. (10) and (11), which are represented by the solid lines. For larger ${\Omega }_{2}T$, they agree with Eqs. (14) and (15), shown here as dotted lines. The plots show both ${\gamma }_{2\to 1}/{\gamma }_2=0.25$ (closed symbols) and 0.75 (open symbols), but it is only for larger ${\Omega }_{2}T$ that the line shapes depend on this branching ratio, as shown by the bifurcated curves at the right of the plots. Data points and curves are discontinued at the right when the line shapes are no longer simple peaks. The heavy dashed lines in (d) show that, for small ${\Omega }_{2}T$, the width of the $|0\rangle$ state is approximated well by adding the natural width in quadrature with the width of Eq. (13).

Figure 4

Shifts due to quantum-mechanical interference with the distant $|1\rangle -\mathrm{to}-|3\rangle$ resonance for ${\Omega }_2/{\gamma }_2=1$ (squares), $1/10$ (diamonds), and $1/100$ (circles) for a range of ${\Omega }_{2}T$ for the (a) ${\rho }_{22}$ line shape and the (b) ${\rho }_{00}$ line shape. For small ${\Omega }_{2}T$, the shifts agree with Eqs. (16) and (17), which are represented by the solid lines. The heavy dashed line shows the approximations of Eqs. (21) and (23). For larger ${\Omega }_{2}T$, the shifts agree with Eqs. (25) and (26), shown here as dotted lines. The plots show both ${\gamma }_{2\to 1}/{\gamma }_2=0.25$ (closed symbols) and 0.75 (open symbols), but it is only for larger ${\Omega }_{2}T$ that the shifts depend on this branching ratio, as shown by the bifurcated curves at the right of the plots. Additionally, the shifts show a small dependence on ${\gamma }_{23\to 1}/{\gamma }_{23}$, which can be seen from further bifurcations at the right between ${\gamma }_{23\to 1}/{\gamma }_{23}=0.25$ (small plot symbols) and ${\gamma }_{23\to 1}/{\gamma }_{23}=0.75$ (large plot symbols). As with Fig. 3, the symbols and plot lines are discontinued to the right when the line shapes are no longer simple peaks.

Figure 5

The $n=2$ triplet states of helium. For atoms prepared in state $|1\rangle$ and a linearly polarized laser (as shown), the states $|0\rangle \text{--}|3\rangle$, which are shown, form a closed four-level system, such as that of Fig. 1. As in Fig. 1, the applied field is nearly resonant with the $|1\rangle -\mathrm{to}-|2\rangle$ transition, and interference with the far-off-resonant $|1\rangle -\mathrm{to}-|3\rangle$ transition causes small shifts. Since in this case, state $|3\rangle$ is below state $|2\rangle$, a negative value of ${\omega }_{32}$ must be used in the equations of Secs. 2 and IV.