Time-dependent perturbation theory

Expanding a state in unperturbed energy eigenstates with their free phases converts the time-dependent Schrodinger equation into coupled equations for slowly varying amplitudes. To first order, the perturbation on the right-hand side acts on the initial unperturbed state.

This is the time-dependent branch of perturbation theory; its corrections describe transition amplitudes.

An additional time-dependent operator in a quantum Hamiltonian. It may couple stationary states or shift their phases. It is the physical operator being treated, whereas time-dependent perturbation theory is a method for approximating the response.

The first-order response to a cosine drive contains both resonant and counter-rotating terms. The rotating-wave approximation removes the rapidly oscillating contribution only under near-resonant, weak-drive and appropriate time-scale assumptions. It does not remove the separate requirement of weak excitation when the initial amplitude is held fixed.

For $b>0$, the Fourier transform of $f$ is ${\int }_{-\infty }^{\infty }f(t)e^{ixt}dt=2b/(b^2+x^2)$. A weak interaction $D_{0}f(t)\cos \omega t$ between energy levels of gap $\hslash {\omega }_0$ gives final probability amplitude $-idb\{[b^2+({\omega }_0-\omega {)}^2{]}^{-1}+[b^2+({\omega }_0+\omega {)}^2{]}^{-1}\}/\hslash$, where $d$ is the transition matrix element. Only when $b,|{\omega }_0-\omega |\ll {\omega }_0$ does the single resonant lobe give an approximately squared Lorentzian line shape excitation profile. Its full width at half maximum is $2b\sqrt{\sqrt{2}-1}$ in angular frequency.

A uniform field $E(t)=E_0{e}^{-t/\tau }\Theta (t)$ couples through $V=-qEx$. Its first-order transition amplitude is $-q⟨f|x|i⟩\tilde{E}({\omega }_{fi})/(i\hslash )$, so the leading probability is $q^2|x_{fi}{|}^2{E}_0^2{\tau }^2/[{\hslash }^2(1+{\omega }_{fi}^2{\tau }^2)]$. A real pulse has equal spectral intensity at opposite frequencies; populations and dipole elements determine which upward or downward transitions are possible. Small peak amplitude does not remove the need to check total transition probability.

The step at the beginning of $E_0{e}^{-t/\tau }\Theta (t)$ retains high-frequency weight. For fixed nonzero transition frequency, $|\tilde{E}|\to E_0/|\omega |$ as $\tau \to \infty$. A slowly decaying pulse with an abrupt onset is therefore not an adiabatically switched perturbation. The diagonal accumulated phase can separately grow with duration.

A constant-amplitude drive lasting $T$ gives the Fourier transform in the title, where $\mathrm{sinc}x=\sin x/x$ and its value at zero is one. The transition probability in weak driving has a finite resonant maximum, first zeros at $\delta =\pm 2\pi /T$, and decreasing side lobes. Longer duration increases the peak as $T^2$ and narrows its width as $1/T$. The rotating-wave approximation discards a rapidly oscillating counterrotating term; it can have a small absolute error but a large relative error at the resonant term's zeros.

For a weak time-independent perturbation switched on at zero time, first-order time-dependent perturbation theory gives a transition probabilityThe frequency kernel has area $2\pi t$ and width of order $1/t$. Against a smoothly varying continuum of final states, it approaches $2\pi t\delta ({\omega }_f-{\omega }_i)$, giving the rate $2\pi |V_{fi}{|}^2\delta ({\omega }_f-{\omega }_i)/{\hslash }^2$. Using $E=\hslash \omega$ gives the energy form above. One integrates the Dirac delta function against the appropriate density of states; it is not an ordinary finite number for one isolated resonant level.

The time must be long enough to resolve the smooth continuum and short enough that the total transition probability remains small. For continuum variation scale $\Delta \omega$ and total rate $\Gamma$, the usual window is $(\Delta \omega {)}^{-1}\ll t\ll {\Gamma }^{-1}$. A discrete finite spectrum also limits this approximation through its recurrences.

For box-normalized plane waves in volume $\mathcal{V}$, a potential has matrix element $\tilde{W}({\mathbf{k}}_f-{\mathbf{k}}_i)/\mathcal{V}$. The final-state density per energy and solid angle is $\mathcal{V}m{k}_f/[(2\pi {)}^3{\hslash }^2]$. Dividing the golden-rule rate by incident flux $\hslash k_i/(m\mathcal{V})$ gives $(m^2/4{\pi }^2{\hslash }^4)(k_f/k_i)|\tilde{W}{|}^2$. Elastic scattering has $k_f=k_i$; the arbitrary box volume cancels.

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.

For a harmonic perturbation and continuum detuning $\Omega (k)$, first-order amplitudes containThe distributional limitselects energy-conserving continuum states and produces a transition probability linear in time.