Electric-dipole interaction

In the long-wavelength approximation, a particle of charge $q$ in a spatially uniform electric field has perturbationIts transition amplitudes are controlled by matrix elements of the position operator.

The electric dipole operator is the charge-weighted position operator of a system. It is odd under parity and transforms as a rank-one spherical tensor operator. In the nonrelativistic electric-dipole approximation it acts on spatial coordinates without directly changing spin, explaining the parity, angular-momentum and approximate spin selection rules.

A parity eigenstate has zero expectation of an odd coordinate, hence no diagonal electric-dipole coupling to a uniform oscillatory field. Stationarity alone does not prove this: an oscillator displaced by $x_c$ has $⟨x⟩=x_c$. Degenerate subspaces or a parity-breaking static field require a separate symmetry analysis.

A detuned optical drive still admixes a decaying excited state. Its spontaneous scattering rate is of order $\Gamma |\Omega {|}^2/(4{\Delta }^2)$ for weak far-detuned driving with coupling $\hslash \Omega /2$. Increasing detuning at fixed optical power reduces both unwanted scattering and useful coherent coupling.

An electric dipole transition is a transition driven, in the long-wavelength approximation, by the electric-dipole interaction $H_{\mathrm{int}}=-\mathbf{d}\cdot \mathbf{E}(t)$. Its leading amplitude is proportional to a matrix element of the electric dipole moment between the initial and final states. A vanishing element gives an electric-dipole selection rule; a nonzero element alone does not ensure a large probability, since the drive frequency, duration and strength also matter. For a charged quantum harmonic oscillator driven along its axis, $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ gives the single-quantum rule $\Delta n=\pm 1$.

A coherent transition involving two atom-field interactions. two-photon absorption uses a sum of photon energies; stimulated absorption-emission uses their difference. A second-order intermediate contribution need not represent real population in a stationary atomic state.

absorption of two photons whose summed energy matches an atomic transition. In the weak-field regime its rate scales with the product of the two intensities. Two electric-dipole interactions preserve the initial parity.

An intermediate contribution to a second-order transition amplitude need not be an occupied, on-shell stationary atomic state. Both time orderings must be summed with their energy denominators. Large detuning can suppress real intermediate occupation while preserving a coherent two-photon coupling. Near resonance, finite linewidth, saturation and sequential population transfer must be treated explicitly.

Absorbing one photon from each of two opposing equal-frequency beams cancels the first-order Doppler shift through their nearly zero summed wavevector. Absorbing both photons from the same beam does not share this cancellation and can create a broad background.

For identical equal-frequency electric-dipole photons, exchange symmetry removes the antisymmetric rank-one transition tensor. Ranks zero and two survive. A zero-to-zero angular-momentum transition can occur, while zero-to-one is forbidden in this approximation; the rank-two triangle condition still permits some nonzero-to-nonzero transitions with a change of one.

The matrix element of the electric dipole moment operator between two quantum states. Electric-dipole transition strength is proportional to its squared magnitude; molecular symmetry can force every Cartesian component to vanish.

For a localized charge distribution in an external field varying little across its extent, expand the external scalar potential about a reference point $\mathbf{R}$. The constant term couples to total charge; the first spatially varying term gives the electric-dipole interaction $-\mathbf{d}\cdot \mathbf{E}(\mathbf{R},t)$, where $\mathbf{d}$ is the electric dipole moment. Higher spatial multipoles are neglected.

For a passing charge at minimum distance $b$, the atomic extent must be small compared with $b$. For a plane electromagnetic wave, the analogous long-wavelength condition is $ka\ll 1$, as in the electric-dipole approximation for radiation. A short interaction time and a spatially uniform field across the atom are different assumptions; neither implies the other.

Because the position operator has odd parity, an electric-dipole matrix element vanishes between states of the same parity. For orbital-angular-momentum eigenstates, the full rule is $\Delta \ell =\pm 1$ and $\Delta m=0,\pm 1$, with the field polarization selecting the allowed change in $m$.

For $0<x<w$, normalized states ${\psi }_n=\sqrt{2/w}\sin [(n+1)\pi x/w]$ have centre-reflection eigenvalue $(-1{)}^n$. Since $x-w/2$ is odd, off-diagonal electric-dipole matrix elements vanish between equal-parity states. Ground state n=0 cannot reach n=2 at first order; an initial n=1 can reach even labels including n=0. This does not impose an all-orders prohibition: two dipole interactions can restore the original parity. The general nonzero elements follow from position matrix elements in a symmetric infinite square well after shifting the origin.

For an inversion-symmetric system, electric-dipole transitions require opposite parities. Transitions between d-shell terms of the same gerade parity can acquire weak vibronic or symmetry-breaking intensity.