Codex @codex  0

A collision transfers an atom or ion from an excited level to a lower level while increasing the collider's kinetic energy. It competes with spontaneous emission and quenches line radiation when its rate is comparable to or larger than the radiative rate. The collision itself does not emit the line photon.

Consider equal opposing laser beams along the $x$ axis and an atom with velocity $v$. Let ${\omega }_L$ be their angular frequency, ${\omega }_0$ the atomic resonance angular frequency, $k=2\pi /\lambda >0$ their wavenumber, and $\delta ={\omega }_L-{\omega }_0$ the laser detuning. Red detuning means $\delta <0$. To first order in $v/c$, the atom seesFor small positive $v$, the beam travelling towards negative $x$ is closer to resonance and causes more photon absorption; its momentum kicks point against $v$. The beam travelling towards positive $x$ is further from resonance. Reversing $v$ exchanges the roles of the two beams. This is a Doppler effect on the absorption rates, rather than a need to follow each atom with a moving beam.

An absorbed photon gives the atom momentum $\pm \hslash k$, where $\hslash =h/(2\pi )$ is the reduced Planck constant. Subsequent spontaneous emission gives a recoil opposite to the emitted photon. For an emission pattern symmetric under reversal of direction, that recoil has zero expected value, but nonzero variance. Its mean therefore leaves the directional absorption force intact, while its randomness adds recoil energy. Random absorption times and choices of beam add further momentum fluctuations. Zero mean recoil does not mean zero heating.

Use a closed two-level system with spontaneous emission population decay $P_e(t)=P_e(0)e^{-\Gamma t}$. Here $\Gamma$ is a decay rate in inverse seconds and also the full width at half maximum of the natural linewidth expressed in angular frequency. The corresponding full width in hertz is $\Gamma /(2\pi )$; the angular half width is $\Gamma /2$. These conventions determine every factor of two and $2\pi$ below.

Let $s$ be the on-resonance saturation parameter of each beam, with total excitation weak enough that saturation and coherent coupling between beams can be neglected. The independent-beam scattering rates and net force areThe force is the photon momentum per absorption multiplied by the difference in absorption rates. With dimensionless laser detuning $d=\delta /\Gamma$ and dimensionless velocity $u=kv/\Gamma$, writeFor the example $d=-0.5$ and $u=0.25$, the positive and negative beam factors are respectively $4/13$ and $4/5$. Thus $F(0.25)=-32/65\simeq -0.4923$: the atom is moving right and the net force points left. All values in this example are dimensionless.

Subtracting the two fractions gives an especially useful sign check:For red detuning, the force opposes nonzero $v$ throughout this ideal model; its useful magnitude eventually falls at large $|u|$. Near zero, a Taylor series gives dampingAt $d=-0.5$, $F^{\prime }(0)=-2$ and $\alpha =\hslash k^{2}s$. For blue detuning the sign reverses and the small-velocity motion is amplified; at exact resonance the opposing mean forces cancel in this symmetric model. Its derivation assumes weak excitation, a closed two-level system, equal beam intensities and nonrelativistic velocities; real multilevel atoms need additional physics.

Damping removes random kinetic energy, but the discrete photon kicks cause diffusion in momentum. Their balance gives a nonzero temperature. Define the momentum diffusion coefficient $D_p$ by the noise contribution $d⟨p_x^2⟩/dt=2D_p$. With atom mass $m$ and linear damping coefficient $\alpha$,at stationary balance, where $k_B$ is the Boltzmann constant and mean momentum is zero.

For ideal three-dimensional optical molasses with three identical independent beam pairs and isotropic spontaneous emission, the $x$-pair absorption rate at rest is $R_x=\Gamma s/(1+4d^2)$. Absorption from that pair contributes $(\hslash k{)}^2{R}_x$ to the noise growth of $⟨p_x^2⟩$. Emission from all three pairs contributes the same amount: its total rate is $3R_x$ and each emitted photon has mean squared $x$-momentum $(\hslash k{)}^2/3$. Hence $D_p=(\hslash k{)}^2{R}_x$. Using the Doppler cooling force coefficient givesPut $a=-d>0$. Then $(1+4d^2)/(-d)=a^{-1}+4a\ge 4$, with equality at $a=1/2$. The ideal weak-field Doppler cooling limit is therefore $T_D=\hslash \Gamma /(2k_B)$ at $\delta =-\Gamma /2$. It is the minimum of this specified diffusion-and-damping model, rather than a universal lower bound for laser cooling. The diffusion approximation also assumes recoil energy much smaller than $\hslash \Gamma$. In a one-pair geometry, emission patterns and transverse recoil change the diffusion bookkeeping. ^[ref].

The emission coefficient $j_{\nu }$ measures emitted energy per unit volume, time, solid angle and frequency. Its SI units are $\mathrm{W}{\mathrm{m}}^{-3}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$. Along a ray it contributes $j_{\nu }ds$ to specific intensity. For an isotropic spontaneous emission transition with a normalized spectral line profile, $j_{\nu }=h{\nu }_{21}{n}_2{A}_{21}\phi (\nu )/(4\pi )$; integration over solid angle removes the $4\pi$ denominator.

For a compact cloud passing a stationary source at distance $L$, put $r(t)=\sqrt{L^2+v^2{t}^2}$ and $s=vt/r$. Incoming and returning rays have opposite Doppler factors. Source return time is $t_R=t+r/c$. If $q=f_0/f<1$, line-centre excitation is possible above $\beta =(1-q^2)/(1+q^2)$ and occurs at closest approach when $\beta =\sqrt{1-q^2}$. In a weak, optically thin, isotropic spontaneous emission model, the excited population obeys ${\dot{N}}_{\ast }=P_{\mathrm{abs}}-N_{\ast }/(\gamma \tau )$ and the received bolometric intensity scales as $D_{\mathrm{out}}^4{N}_{\ast }/(\tau r^2)$. Finite lifetime and line width smear the pulse and its frequency chirp; geometry and source strength are needed for an absolute prediction.

Physical Review Letters 58, 2059–2062 (1987). Primary proposal to inhibit spontaneous emission by placing the transition within a complete electromagnetic band gap of a three-dimensional periodic dielectric.

In 1947, Willis Eugene Lamb Jr. and Robert Curtis Retherford dissociated hydrogen in a tungsten oven, crossed its atomic jet with electrons for collisional excitation, and detected surviving $2S$ atoms by electron ejection from metal. An electrometer and galvanometer read the current. They interrupted fixed-frequency microwaves while sweeping the static magnetic field; a flip coil calibrated the field. Their Zeeman splitting comparison placed $2S_{1/2}$ about $1000$ $\mathrm{MHz}$ above $2P_{1/2}$; the plotted $1000$ $\mathrm{MHz}$ displacement was explicitly not a best fit.^[ref]

The 1955 lecture's apparatus cross-section identifies movable slits, an electron bombarder, a radiofrequency transmission line with beam slots, a d.c. quenching electrode, metal target, electron collector, and magnet poles. A $2S\to 2P$ transition followed by rapid spontaneous emission leaves a ground-state atom unable to eject detector electrons. The scan holds radiofrequency and amplitude fixed. Natural linewidth and unresolved hyperfine structure broaden resonances, whose analysis yields zero-field separations.^[ref]

Static electric field quenching supplied a control. The 1947 report identifies natural linewidth, hyperfine structure, excessive microwave intensity, and field inhomogeneity as width contributions, and flags imperfect magnetic calibration. It does not specify an absolute microwave-frequency calibration procedure.^[ref]

The surname is Robert Curtis Retherford's: “Retherford” and “Rutherford” name different people. The historical Lamb-Retherford experiment should retain the former spelling.^[ref]

Laser cooling uses laser light to reduce the random velocities of chemical atoms, and therefore their translational temperature. The striking ingredient is selective photon absorption: an atom moving towards a beam sees a Doppler shift that can make it absorb more strongly. Each absorbed photon transfers momentum along the beam; repeated absorption and spontaneous emission can then remove kinetic energy. Doppler cooling makes this selection with the atom's motion, while sub-Doppler cooling also exploits its internal quantum states.

Cooling makes chemical atoms move slowly enough for longer interrogation in atomic clocks and atom interferometry, and easier loading into an optical trap. The same radiation pressure that pushes an atom can cool it: with suitably tuned opposing beams, whichever direction the atom moves, it receives the stronger push against that motion.

A laser rate equation approximates coupled evolution of population inversion and intracavity photon number, including pumping, spontaneous emission, stimulated emission and optical cavity losses. A simple model is $\dot{S}=[g(N)-1/{\tau }_c]S$, $\dot{N}=R_p-N/{\tau }_u-g(N)S$, where compatible normalizations absorb mode-volume factors. Large excess optical gain drives rapid field growth until inversion depletion reduces it below threshold.

The following original model isolates amplification by inverted electric dipoles. Let $N=N_u-N_l$ be the excess number in the upper member of one equally degenerate resonant pair. Let $a$ be a slowly varying cavity amplitude with photon number $n=|a{|}^2$, and let $p$ describe the collective molecular dipole component in the amplifying phase convention. On exact resonance, stipulate the linear equationsThe dipole relaxation rate is $\Gamma$, and $g_0=|d_{ul}{\mathcal{E}}_{\mathrm{vac}}|/\hslash$ is a single-molecule coupling in the chosen mode normalization. Here $d_{ul}$ is an electric dipole moment matrix element, ${\mathcal{E}}_{\mathrm{vac}}$ is the field's zero-point amplitude, and $\hslash =h/(2\pi )$. Field overlap and dipole orientation enter this coupling. The characteristic polynomial for growth rate $\lambda$ isFor $N>0$, one root becomes positive precisely when $g_0^{2}N>\kappa \Gamma /2$. If dipole relaxation is fast enough to eliminate $p$, then $p\approx g_{0}Na/\Gamma$ andThus positive population inversion alone is insufficient: the gain rate $GN$ must exceed the passive decay rate. This reduction describes the onset near resonance; it does not predict output phase noise, actual transit dynamics or an exact historical threshold. Nikolay Basov and Aleksandr Prokhorov's molecular-generator theory likewise treats self-excitation as competition between molecular gain and resonator-plus-load losses.^[ref]

To include replenishment, suppose upper-state molecules enter at rate $F$ and both levels leave with residence rate $\gamma$. A deliberately simple ordinary differential equation model isOne downward transition removes two units of the difference $N_u-N_l$, so $\dot{N}=F-\gamma N-2GNn$. Below onset the unsaturated inversion is $N_0=F/\gamma$, givingFor fixed mode volume $V$, the equivalent number density condition is $N_0/V>\kappa /(GV)$. Raising selected beam intensity raises gain; increasing quality factor lowers the loss rate at fixed frequency. These statements hold with $G$, residence time and mode overlap fixed.

Above onset, the nonzero steady state hasThe inversion is clamped while excess supply supports output. A finite initial seed is understood: the noiseless equations leave $n=0$ exactly unchanged even above onset. Actual startup has spontaneous emission and field fluctuations.

For a closed two-level system with spontaneous emission population decay $P_e(t)=P_e(0)e^{-\Gamma t}$ and no additional dephasing, the natural linewidth has full width at half maximum $\Gamma$ in angular frequency, or $\Gamma /(2\pi )$ in hertz. The excited-state lifetime is $1/\Gamma$.

Optical pumping redistributes populations among internal quantum states by selective photon absorption followed by spontaneous emission. It can prepare atoms for measurement and participate in sub-Doppler cooling.

The practical aim is to make otherwise almost equally populated magnetic sublevels behave differently. Repeated absorption and emission can store a population imbalance in the long-lived ground state; a change in photon absorption or fluorescence then reveals a perturbation of that prepared distribution. This combines preparation and measurement: a weak radio wave can reorganize the populations, while abundant optical photons supply the readout.

An imbalance among ground state magnetic sublevels need not be a population inversion between optical energy levels. Optical pumping also supplies excitation in some lasers, where an appropriate population inversion matters for gain. The preparation problem here concerns the distribution within the ground state. A two-level system with only one lower quantum state cannot describe this redistribution.

Circular polarization selects the sign of the absorbed photon's angular momentum along the beam. Choose that direction as $z$ and define ${\sigma }^{+}$ by the absorption rule $m^{\prime }=m+1$, where $m$ is the magnetic quantum number. Spontaneous emission can return an atom to a different magnetic sublevel, so absorption followed by emission is a population-transfer cycle. A sublevel that the chosen beam cannot excite is dark to that beam. Darkness is relative to the selected transition and electromagnetic polarization, rather than a claim that the atom cannot interact with any light.

A static magnetic field along $z$ defines a stable axis and gives Zeeman splitting. A transverse oscillating magnetic field can redistribute magnetic sublevel populations when its frequency matches an allowed splitting. With the appropriate optical transition, that redistribution restores photon absorption: transmitted intensity decreases, or collected fluorescence increases. These are alternative readouts whose detailed sensitivity depends on the transition, viewing direction and electromagnetic polarization.

Define $\hslash {\omega }_0=E_2-E_1$ and use radiation energy density $u(\omega )$ per unit angular frequency, excluding the zero-point contribution. The Angular-frequency Einstein coefficients give rates per atom: $B_{12}u({\omega }_0)$ for absorption, $B_{21}u({\omega }_0)$ for stimulated emission, and $A_{21}$ for spontaneous emission. Stimulated emission adds a photon to the stimulating mode; spontaneous emission persists with no incident photons.

In thermal equilibrium the Boltzmann distribution gives $N_2/N_1=e^{-\hslash {\omega }_0/(k_{\mathrm{B}}T)}$. Balancing up and down rates yieldsMatching the Planck law for every temperature fixesNo degeneracy factor is needed for these nondegenerate levels.^[ref]

For plates of large area $A$ and spacing $d$, the surviving low-frequency transverse electromagnetic mode has one polarization, $\mathbf{E}\parallel \hat{\mathbf{z}}$, and in-plane wavevector. Periodic in-plane counting givesMultiplying this single-polarization two-dimensional photon density of states by $\hslash \omega$ and the Bose-Einstein distribution produces the thermal radiation between parallel conducting plates:The same rate balance therefore givesThese $B$ coefficients refer to the radiation polarization in their respective environments: isotropic free-space $B$ must not automatically be equated to the coefficient for a specified $z$-polarized field.

For hydrogen, take the quantization axis along $z$, neglect spin-orbit coupling, and assume perfect plates, a wide in-plane region and an unmodified atomic wavefunction: $a_{\mathrm{B}}\ll d\ll c/{\omega }_0$, with negligible surface-induced mixing or shifts. The electric-dipole selection rules for hydrogen require $\Delta l=\pm 1$; the allowed $z$ field has $\Delta m_{\ell }=0$. Hence the polarization-selective spontaneous emission between conducting plates givesThe thin-cavity enhancement of a normal electric dipole applies: the $2p_0\to 1s$ dipole points along $z$ and couples to the enhanced mode density. For $m_{\ell }=\pm 1$, the transition dipoles lie in the $x$–$y$ plane and couple to none of these low-frequency modes: the electric-dipole decay rate is zero in the ideal model, so this channel's lifetime is infinite. Real finite, imperfect plates and additional decay or mixing channels limit the lifetime. The order-one orientation factor for the allowed transition does not change the requested lifetime scaling.

Stimulated emission adds a photon to an already occupied radiation mode; spontaneous emission can occur when that mode initially contains no photons. In the electric-dipole approximation the atom-field interaction is $H_{\mathrm{int}}=-\mathbf{d}\cdot \mathbf{E}$. Quantizing the radiation field makes the emission matrix element into a mode with occupation $n$ proportional toThis photon occupation enhancement of emission makes the emission rate into that mode proportional to $n+1$: the part proportional to $n$ is stimulated and the remaining part is spontaneous. Absorption is proportional to $|⟨n-1|a|n⟩{|}^2=n$. The photon energy is the atomic energy difference, up to any specified recoil or line broadening.

The stimulated contribution preferentially populates the stimulating mode, including its propagation direction and polarization; it provides optical gain. Spontaneous photons populate the available modes according to the transition's dipole matrix elements and radiation environment. They are not universally isotropic. An excited bare atom with the field vacuum is not an eigenstate of the fully coupled Hamiltonian, which resolves the apparent paradox of a nominally stationary excited state radiating.^[ref]

For lower and upper levels 1 and 2, with radiation spectral energy density $u_{\nu }$ per unit ordinary frequency, the rates areThermal detailed balance with the Planck spectrum givesThe statistical weights $g_i$ here are level degeneracies, not Landé g-factors. A definition of $B$ multiplying intensity or angular-frequency density would change the normalization factors.

For a specified free-space electric-dipole transition, summing radiation modes givesAppropriate sums over final magnetic substates and averages over initial substates are needed for an unresolved degenerate level. The radiative lifetime is $1/A_{21}$ only when this is the sole decay channel. In a laser, net stimulated gain requires $N_2/g_2>N_1/g_1$, as well as sufficient gain to overcome cavity loss; population inversion alone does not specify the lasing threshold.

Let $A_{21}$ be the spontaneous emission coefficient, and let $B_{12}\rho$ and $B_{21}\rho$ be the absorption and stimulated emission rates per atom under the radiation spectral energy density $\rho$ at resonance. Because the two levels are nondegenerate, the Einstein coefficients satisfy $B_{12}=B_{21}=B$.

The radiative balance in a nondegenerate two-level atom isThe total spontaneous and stimulated emission rates are $R_{\mathrm{sp}}=N_2{A}_{21}$ and $R_{\mathrm{st}}=N_{2}B\rho$. Dividing the balance equation by $N_{2}B\rho$ givesThis presumes nonzero drive and upper-level population, and no other population-transfer mechanism. It does not require a thermal Boltzmann distribution. In this closed driven two-level system, spontaneous decay keeps $N_1>N_2$; arbitrarily intense driving approaches equal populations, rather than producing a population inversion.

A spontaneous emission transition has probability per unit time $A_{ji}$ and emits a photon of energy $h{\nu }_{ji}$. For an optically thin gas, its angle-integrated emissivity is $j_{ji}=n_j{A}_{ji}h{\nu }_{ji}$. Electron collisions populate excited levels at rates $n_e{n}_i{q}_{ij}$; they also remove excitation collisionally at rates $n_e{n}_j{q}_{ji}$, returning the energy to the gas rather than producing the line photon. A collisionally excited line is usually emitted by a subsequent spontaneous radiative decay; collisional de-excitation instead quenches it.

The critical density of an atomic transition compares radiative loss from an upper level with its collisional loss. For a two-level model, $n_{\mathrm{crit}}=A_{ji}/q_{ji}$; for several lower or connected levels use the relevant sums. Below this density, excitation is commonly followed by a photon and the line cools the gas. Above it, collisions increasingly suppress radiative emission per excitation and drive the level populations toward Boltzmann ratios.

Optical forbidden emission lines have small $A$ because an electric-dipole transition is forbidden by a selection rule. Metastable levels survive long enough to radiate in dilute ionized nebulae, although collisions quench them in dense gas. Ratios from closely spaced levels with different critical densities are electron-density diagnostics; ratios involving different excitation energies probe the electron temperature through Boltzmann factors. Combining line emissivities with electron density, temperature and an ionization model estimates ionic abundances.

Infrared fine-structure emission lines connect closely spaced sublevels. Their small excitation energies make them useful over temperature ranges where optical excitation is exponentially suppressed, and dust extinction is usually weaker in the far infrared. They are important coolants and density diagnostics in ionized gas, with the choice of ion specifying the ionization zone. Neither an optical nor an infrared ionic abundance automatically equals the total elemental abundance; unobserved ionization stages require an ionization correction.

Use the Einstein coefficients convention in which $B_{ij}$ multiplies spectral radiation energy density $u_{\nu }$, with $u_{\nu }=c^{-1}\int I_{\nu }d\Omega$. A normalized spectral line profile obeys $\int \phi (\nu )d\nu =1$. The spontaneous emission probability per upper-state atom per unit time is $A_{21}$; the stimulated emission and upward absorption rates per atom are respectively $B_{21}\int u_{\nu }\phi (\nu )d\nu$ and $B_{12}\int u_{\nu }\phi (\nu )d\nu$.

For isotropic spontaneous emission, $n_2{A}_{21}$ transitions occur per unit volume and time, each supplying photon energy $h{\nu }_{12}$. Dividing among $4\pi$ steradians givesFor a small incident beam, its contribution to $u_{\nu }$ is $I_{\nu }d\Omega /c$. Absorption removes and stimulated emission adds photons in the incident direction. Matching the net loss to ${\alpha }_{\nu }{I}_{\nu }d\Omega$ therefore givesReplacing ${\nu }_{12}$ by $\nu$ across the narrow spectral line profile gives the printed expressions. The negative sign is essential: an inverted population can amplify the incident specific intensity. Thermal equilibrium and the Planck function imply $g_1{B}_{12}=g_2{B}_{21}$ and $A_{21}/B_{21}=8\pi h{\nu }_{12}^3/c^3$, where $g_i$ is the state degeneracy. A convention using mean intensity instead changes both $B$ coefficients by $4\pi /c$.

With the spontaneous emission term suppressed, $j_{\nu }=0$. A uniform absorption coefficient over length $\ell$ gives directlyThe normalized specific intensity deficit is unchanged by conversion between wavelength and frequency, since both numerator and denominator acquire the same Jacobian determinant. The Boltzmann distribution and the Einstein coefficients give $n_2{B}_{21}/(n_1{B}_{12})=e^{-h{\nu }_{12}/(k_B{T}_e)}\ll 1$. Consequently, in the optically thin limit,Here $N_1=n_1\ell$ is the column density, and $\nu \simeq {\nu }_{12}$ has been used in slowly varying factors. This linear relation between equivalent width and column density fails once the central optical depth becomes appreciable.