Stimulated Raman transitions via multiple atomic levels

We consider the stimulated Raman transition between two long-lived states via multiple intermediate states, such as between hyperfine ground states in the alkali-metal atoms. We present a concise treatment of the general, multilevel, off-resonant case, and we show how the lightshift emerges naturally in this approach. We illustrate our results by application to alkali-metal atoms and we make specific reference to cesium. We comment on some artifacts, due solely to the geometrical overlap of states, which are relevant to existing experiments.

Figure 1

A simple three-level “$\Lambda$” system, in which a Raman transition between states $|0\rangle$ and $|1\rangle$ is driven by “pump” and “Stokes” fields via intermediate state $|2\rangle$. For each field, the coupling strength and frequency are shown in parentheses, and frequencies are chosen to be near two-photon resonance: ${\omega }_{\mathrm{P}}=({\omega }_2-{\omega }_0)+\Delta$; ${\omega }_{\mathrm{S}}=({\omega }_1-{\omega }_0)+(\Delta +\delta )$. The single-photon detuning $\Delta$ is large compared with the couplings, $\left|\Delta \right|\gg {\Omega }_{\mathrm{P},\mathrm{S}}$, and, in this illustration, is negative: $\Delta <0$. The two-photon detuning $\delta$ is small compared with the separation between the ground states and, in this illustration, is also negative.

Figure 2

Illustration of the dependence of coupling strength described by Eqs. (27) (solid) and (28) (dashed). The values have physical meaning at integer and half-integer $m_{\mathrm{F}}$ only (dots); continuous lines are shown to guide the eye. A line is shown for each of several nuclear spins, beginning at $I=1/2$ and increasing in steps of one half out from the center (solid) and from bottom to top (dashed). The strongest coupling is between extremal $m_{\mathrm{F}}$ states by linear and circular polarisations.