= Solution
A <two-photon transition> transfers population through two interactions with the electric-dipole operator. For <absorption> of two <photons>,
$$
E_f-E_i=\hbar(\omega_1+\omega_2).
$$
The intermediate atomic states enter a second-order amplitude rather than necessarily being populated. Suppressing field-normalization factors, its structure is
$$
\mathcal M_{fi}\propto\sum_r\left[
\frac{(\mathbf d_{fr}\cdot\boldsymbol\epsilon_2)(\mathbf d_{ri}\cdot\boldsymbol\epsilon_1)}
{E_i+\hbar\omega_1-E_r}
+
\frac{(\mathbf d_{fr}\cdot\boldsymbol\epsilon_1)(\mathbf d_{ri}\cdot\boldsymbol\epsilon_2)}
{E_i+\hbar\omega_2-E_r}
\right].
$$
Both time orderings contribute coherently. Close to a real intermediate resonance, linewidths and intermediate-state population must be treated explicitly; an off-resonant virtual-state approximation then ceases to be adequate. Well away from saturation the <absorption> rate scales as $I_1I_2$, or as $I^2$ for a single applied <frequency>.
Each <electric-dipole interaction> reverses <parity>, so two such interactions connect states of the same <parity>. The product operator contains angular tensor ranks $0,1,2$, subject to triangle and polarization <selection rules>. Under the <degenerate two-photon selection rule>, identical equal-frequency <photons> have no antisymmetric rank-one contribution cancels; ranks zero and two remain. This permits a $J=0\to0$ transition and excludes $J=0\to1$ in that degenerate E1-E1 case. It does not universally forbid every $\Delta J=1$ transition, because a rank-two tensor can also connect some such pairs. For a single orbital <electron>, the corresponding orbital changes are $\Delta L=0,\pm2$, with <spin> unchanged in the nonrelativistic electric-dipole approximation.
In <Doppler-free two-photon absorption>, counterpropagating equal-frequency <photons> give first-order Doppler cancellation when one <photon> is absorbed from each beam:
$$
\Delta_{\rm Doppler}=(\mathbf k_1+\mathbf k_2)\cdot\mathbf v\simeq0,\qquad
\Delta\mathbf p=\hbar(\mathbf k_1+\mathbf k_2)\simeq0.
$$
The <laser> <frequency> is half the transition <frequency>. Same-beam two-photon events can still produce a broader Doppler-shifted background; residual beam-angle errors, unequal frequencies, <AC Stark shifts> and finite transit time limit precision. Thus a narrow two-photon resonance requires control of beam geometry and <intensity>, not just high <laser> power.
A <stimulated Raman transition> is the absorption-emission version: the atom absorbs from one beam and emits into the other, transferring between two long-lived states through an off-resonant upper state. It instead obeys
$$
E_f-E_i=\hbar(\omega_1-\omega_2),\qquad
\Delta\mathbf p=\hbar(\mathbf k_1-\mathbf k_2).
$$
Counterpropagation now enhances <momentum> transfer rather than cancelling it. With single-photon <Hamiltonian (quantum mechanics)> <matrix elements> $\hbar\Omega_i/2$ and a common large angular detuning $\Delta$, eliminating the <excited state> gives
$$
|\Omega_{\rm eff}|=\frac{|\Omega_1\Omega_2|}{2|\Delta|},\qquad
\Gamma_{\rm sc}\sim\frac{\Gamma(|\Omega_1|^2+|\Omega_2|^2)}{4\Delta^2}.
$$
The two-photon detuning must include differential AC Stark and recoil shifts. Large detuning suppresses spontaneous scattering but also reduces coherent coupling at fixed optical power.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2016/paper-1-two-photon-paths.png]
{title=<two-photon absorption> and stimulated Raman transfer have different <energy> and <momentum> sums}
{description=Original level and photon-path schematics. A virtual intermediate line is not an additional stationary <eigenstate>; intermediate detunings are schematic, with no assumed atomic species or numerical level spacing.}
The coherent Raman route is useful for atomic beam splitters and mirrors. <Mark Kasevich> and <Steven Chu> demonstrated a laser-cooled sodium matter-wave interferometer using <stimulated Raman transitions>; its controlled internal-state changes accompanied separated external wave packets.<Atomic interferometry using stimulated Raman transitions by Mark Kasevich and Steven Chu>[Physical Review Letters 67, pp. 181–184 (1991), abstract and interferometer principle]{ref} Testing this interpretation requires scanning the two-photon detuning, controlling relative <laser> phase and checking output populations against pulse timing. \b[A two-photon process is defined by its coherent two-interaction pathway; absorption-absorption and absorption-emission have different resonance and recoil conditions.]
Back to article page
