Atomic physics

An atomic population is in statistical equilibrium when the total rate into each level equals the total rate out, subject to conservation of the total number of atoms or ions. The rates can include electron-impact transitions and spontaneous emission. This is weaker than thermodynamic equilibrium: a collisionally excited, radiatively decaying dilute gas generally does not have Boltzmann level populations.

A transition filling an inner atomic vacancy can transfer its released energy to another bound electron, ejecting it instead of emitting a photon. This provides an atomic relaxation channel following electron capture and ionization.

An electron ejected by the Auger effect. Its kinetic energy is determined mainly by the atomic levels involved, rather than directly by the energy of the radiation that created the original vacancy.

A normalized spectral line profile distributes a transition's emission or absorption over frequency, with $\int \phi (\nu )d\nu =1$ and units $\mathrm{H}{\mathrm{z}}^{-1}$. Thermal motion, finite excited-state lifetime and collisions can give different contributions to the shape. Normalization lets an optically thin integrated line measure the relevant column density without requiring the detailed profile.

The integrated absorption coefficient of an isolated spectral line is its line strength. For a mass coefficient write ${\kappa }_{\nu }=S{\varphi }_{\nu }$, where $\int {\varphi }_{\nu }d\nu =1$. The units of $S$ include the chosen spectral-coordinate unit; a strength integrated over wavenumber is not numerically interchangeable with one integrated over frequency. Changing normalized width at fixed strength redistributes opacity without changing its spectral integral.

A normalized collisional line model ${\kappa }_{\nu }=(S/\pi )\Delta /[(\nu -{\nu }_0{)}^2+{\Delta }^2]$, with integrated strength $S$ and half-width $\Delta$. The centre scales as $S/\Delta$ and far wings as $S\Delta /(\nu -{\nu }_0{)}^2$. In a dilute pressure-broadened gas $\Delta$ scales approximately with pressure.

A regime where optical-depth-one transmission occurs in the far wings of a strongly absorbing line. With a uniformly mixed absorber, $\Delta ={\Delta }_{s}p/p_s$ and $C=qS{p}_s/(2\pi g{\Delta }_s)\gg 1$, the radiative-wing optical depth is $\tau \simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This condition does not make pressure broadening valid in the arbitrarily rarefied upper atmosphere.

Gross atomic structure denotes the leading nonrelativistic Coulomb binding levels, before fine and hyperfine splittings. For a hydrogenic two-body system, the reduced mass controls the binding energy and characteristic separation. Fine structure is smaller by roughly ${\alpha }^2$; nuclear magnetic hyperfine effects have an additional electron-to-nuclear mass suppression.

Positronium is the electromagnetic bound state of an electron and a Positron. Their equal masses give reduced mass $m_e/2$, half the hydrogenic Rydberg binding scale and approximately twice the electron-proton separation of hydrogen. Its spin singlet and triplet differ in hyperfine energy; annihilation also affects their physical lifetimes and splitting.

For the dimensionless-spin Hamiltonian $A{\mathbf{S}}_e\cdot {\mathbf{S}}_p+2{\mu }_{B}B(S_{ez}-S_{pz})$, the two aligned-spin states remain at $A/4$. The two states of total spin projection zero mix and have the displayed energies. The parameter $4{\mu }_{B}B/A$ compares off-diagonal Zeeman mixing in the singlet-triplet basis with half the zero-field splitting. A crossover is gradual, not a level-crossing phase transition.

A many-electron atomic environment is replaced by a spherically averaged one-electron potential. Radial screening changes the Coulomb attraction and produces orbitals labelled by radial and angular quantum numbers. This spherical approximation and the separate self-consistency step in the Hartree method are related but distinct assumptions.

A transition driven by an oscillating magnetic field through $H^{\prime }=-\mathbf{\mu }\cdot \mathbf{B}$. The rank-one magnetic operator preserves parity and allows changes in total angular momentum of zero or one, excluding zero-to-zero. Its spherical components select changes in the angular momentum component of $0,\pm 1$. Hyperfine clock states can therefore couple even when each has zero first-order static magnetic moment.

A far-detuned laser produces an atomic AC Stark shift approximately $U=-\mathrm{Re}\alpha (\omega )|E_0{|}^2/4$. Its spatial gradient creates a conservative dipole force. For a suitable red-detuned ground-state atom, high intensity is an energy minimum and can confine several hyperfine states without relying on nonzero permanent magnetic moments. Photon scattering, differential light shifts and intensity noise remain limitations. This atomic use is more specific than a general dielectric-particle optical trap.

Transfer of one or more electrons between a moving particle and its surroundings, changing the particle charge state.

Transfer of a system to an excited electronic state. Energy deposited into electrons need not become a permanent atomic displacement.

An LS coupling multiplet with specified total orbital angular momentum $L$ and total spin $S$, usually within a specified electron configuration. Its magnetic states have $M_L=-L,\dots ,L$ and $M_S=-S,\dots ,S$. Spin-orbit coupling splits it into atomic fine-structure levels.

An atomic term with total spin one half and spin multiplicity two. A doublet S term has only J=1/2, while a doublet with nonzero L can have J=L-1/2 and J=L+1/2. Spin multiplicity therefore need not equal the number of resolved spectral lines.

A level of an atomic term with specified total angular momentum $J$. Without an external field, its $2J+1$ magnetic states remain degenerate in a rotationally invariant Hamiltonian.

A monatomic ion with exactly one electron. Its nonrelativistic Coulomb potential energy levels scale as $Z^2$ at fixed principal quantum number, apart from reduced mass corrections.

In LS coupling, an atomic term symbol specifies total electronic spin $S$, total orbital angular momentum $L$, and total angular momentum $J$. The superscript $2S+1$ is the spin multiplicity; the letters $S,P,D,F,\dots$ denote $L=0,1,2,3,\dots$. Coupling gives $J=|L-S|,\dots ,L+S$, and a fixed-$J$ level has $2J+1$ magnetic substates before an external field resolves them. For example, ${}^{3}P$ has $S=L=1$ and levels $J=0,1,2$, containing $1+3+5=9$ states altogether. A term symbol does not itself specify the radial electron configuration.

A Rydberg atom has an electron excited to a high principal quantum number.

Rydberg blockade suppresses simultaneous excitation of nearby chemical atoms into strongly interacting Rydberg atom states. If the interaction shifts the doubly excited level by much more than the excitation linewidth and coherent drive scale, a pulse resonant for one isolated chemical atom becomes off-resonant when a neighbour is already excited. The resulting conditional dynamics can implement an entangling two-qubit gate for neutral-atom qubits.

A restriction on transitions between quantum states for a specified interaction. An electric-dipole selection rule is one example.

In the nonrelativistic electric-dipole approximation, the dipole operator does not act on spin, so transitions between different total-spin states vanish. Spin–orbit coupling can give such transitions weak intensity.

A quantum electrodynamics correction separates hydrogen atom levels such as $2s_{1/2}$ and $2p_{1/2}$ that coincide in the simplest model.

The Lamb shift measures an energy level displacement beyond a specified Dirac equation reference. In the point-proton, infinite-nuclear-mass Coulomb potential model, $2S_{1/2}$ and $2P_{1/2}$ share principal quantum number $n=2$ and total electronic angular momentum $j=1/2$, hence energy, despite differing orbital angular momentum values.^[ref]

The Lamb-Retherford experiment converts a small energy difference into a large detector change: prepare a long-lived quantum state, transfer population into a short-lived one, and read the surviving beam.

The zero-field energy levels here are averaged over hyperfine structure; $h$ is the Planck constant. Electron self-energy and vacuum polarization produce radiative corrections. Self-energy describes the bound electron's radiative interaction with its own electromagnetic field. Precision predictions also include nuclear recoil and proton charge radius effects; assigning recoil terms to the reference or Lamb shift is convention-dependent. A finite proton charge radius changes binding independently of radiative effects.^[ref]

A vacuum-electric field “jiggling” picture motivates one contribution, not every hydrogen atom energy correction.

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]

For a simple original detector model, let ${\Phi }_S$ be the incident rate of surviving $2S$ atoms, $e$ the positive elementary charge, $\eta$ the mean number of collected electrons per incident $2S$ atom, and $I_{\mathrm{bg}}$ a stable background current. Take current magnitudes as positive. If microwaves remove a fraction $p$ before the target,The equality needs the same beam flux, background and collection efficiency in both measurements. It is an ideal interpretation of a normalized current loss, not a claim that an actual galvanometer directly counts individual transitions. Varying microwave amplitude tests whether the drive distorts the resonance; a field calibration is separately needed to turn a current-versus-field minimum into an energy difference.

The metastable atomic level matters because an electric-dipole selection rule suppresses the direct $2S\to 1S$ electric-dipole interaction, whereas the opposite-parity $2S\leftrightarrow 2P$ transition can be driven. A long lifetime can store the prepared population until interrogation. After transfer to $2P$, a short lifetime supplies the loss mechanism and a broad natural linewidth. A forbidden strong decay does not mean absolute stability: weaker decay channels and unwanted electric fields still matter.

Consider an original illustrative resonance model, with all numerical settings chosen here. On one isolated branch approximate the transition frequency by $\nu (B)={\nu }_0+aB$, with known slope $a=20MHz/mT$ and drive frequency ${\nu }_d=1300\mathrm{MHz}$. Choose ${\nu }_0=1100\mathrm{MHz}$, hence resonance field $B_r=10\mathrm{mT}$. This local linear model omits hyperfine structure and nonlinear level mixing; it is not the historical hydrogen Zeeman splitting calculation.

Use a Lorentzian line shape for a weak-drive current dip,Choose baseline $C=1$, depth $A=0.25$ in arbitrary detector signal units, and full width at half maximum $w_{\nu }=100\mathrm{MHz}$, so $w_B=5\mathrm{mT}$. The half-depth points are $B_r\pm w_B/2$; this checks the factor of four. If an exponential decay with lifetime $\tau$ alone set this width, natural linewidth would give $w_{\nu }=1/(2\pi \tau )$ and $\tau \simeq 1.59\mathrm{ns}$. The chosen width is an explanatory input, not a measurement of a historical lifetime.

Generate 65 equally spaced fields from $2$ to $18\mathrm{mT}$ and add independent Gaussian noise with known standard deviation ${\sigma }_I=0.004$ signal units. A simultaneous nonlinear regression fit of $C,A,B_r,w_B$ gives $B_r\simeq 10.0007\mathrm{mT}$, $w_B\simeq 4.9305\mathrm{mT}$ and ${\nu }_0\simeq 1099.986\mathrm{MHz}$. With residual derivatives $J_{ik}=\partial I(B_i)/\partial {\theta }_k$ and $\theta =(C,A,B_r,w_B)$, the local covariance matrix estimate isIt gives centre standard deviation $0.01427\mathrm{mT}$, or $0.285\mathrm{MHz}$ in the inferred intercept. This uses the stipulated noise variance and a local linear approximation, rather than treating the width as the centre error. Repeated noisy samples can locate a broad line's centre much more closely than its width.

Include a common magnetic calibration offset with standard deviation $u_b=0.020\mathrm{mT}$, a fractional gain error $u_g=0.001$, and independent drive-frequency error $u_d=0.100\mathrm{MHz}$. To first order the inferred variance isHere $u$ denotes a standard uncertainty, not a frequency. The four contributions in $\mathrm{MHz}$ are $0.285$, $0.400$, $0.200$, and $0.100$; adding their squares gives $u({\nu }_0)\simeq 0.540\mathrm{MHz}$. The calibration offset shifts all sampled fields together, so increasing the sample count cannot average it away. Uncertainty in $a$, unresolved components, asymmetric line distortion, and errors in the assumed field dependence would require additional terms or a richer model. The historical inference requires the appropriate Zeeman splitting branches, rather than simply drawing a straight line through one resonance.

Zeeman splitting is the separation of energy levels or mode frequencies by a background magnetic field. In atomic physics, the magnetic field couples to a magnetic dipole moment, lifting quantum degeneracy. Transitions between the shifted quantum states can turn one spectral line into several. A split level and a split spectral line are different objects: the emitted photon measures the difference between two level energies, and the electric-dipole selection rule determines which differences appear.

The Stern-Gerlach experiment also probes magnetic dipole moments, through beam deflection in a magnetic field gradient. Here the readout is a spectral line.

The pattern makes an invisible magnetic field visible through spectroscopy. It also reveals angular momentum and magnetic dipole moments inside chemical atoms. Zeeman slowers exploit a changing resonance frequency to keep moving chemical atoms absorbing laser photons; a magneto-optical trap combines position-dependent shifts with electromagnetic polarization and laser cooling. Related mode shifts occur in Zeeman splitting of an antiferromagnetic spin wave.

The change of an energy level in a magnetic field. The displayed weak-field form applies to a hyperfine level with magnetic projection $m_F$, when $g_F$ is the corresponding Landé g-factor. A transition shift is the difference of the two level shifts divided by Planck constant. A clock transition can have zero first-order transition shift even when its two individual level slopes are nonzero. For $m_F=0$ hyperfine levels, the linear level shift vanishes and quadratic terms remain.

For an axially split $p$ multiplet with $p_z$ energy $E_0$ and equal $p_x,p_y$ energies $E_t$, a transverse $x$-directed magnetic field leaves $p_x$ unchanged and couples $p_y,p_z$ through the orbital angular momentum operator $L_x$. The matrix element has modulus ${\mu }_{B}B$, giving the two energy eigenvalues in the title. For ${\mu }_{B}B\ll |E_t-E_0|$, the orbital shifts are quadratic despite a linear Zeeman energy operator. A full quadratic-field calculation also includes the diamagnetic term; electron spin gives its own linear contribution.

An equal incoherent mixture within a fixed total-$J$ multiplet has density operator $\rho =P_J/(2J+1)$, where $P_J$ is its orthogonal projection. If the applied-field Hamiltonian operator is proportional to $L_z+2S_z$ and no spin-orbit interaction is retained, its energy eigenstates are the uncoupled states $|m_L,m_S⟩$. The Born rule gives their populations by the formula above, or equivalently by squared Clebsch-Gordan decomposition coefficients. Equal populations of coupled states need not give equal populations of magnetic energy eigenvalues.

For $L=1,S=1/2,J=3/2$, the nonextremal coupled states have weights $2/3$ and $1/3$ in their two uncoupled components. The five dimensionless magnetic energies $-2,-1,0,1,2$ then have populations $1/4,1/6,1/6,1/6,1/4$; the zero energy contains two independent states, each with population $1/12$. In a different, weak-field approximation with a dominant spin-orbit interaction, projecting into the isolated multiplet gives the Landé g-factor spectrum instead. The full operator and its projection must not be interchanged.

In August 1896, Pieter Zeeman observed broadened sodium spectral lines at Leiden, initially without resolved components. His lecture describes a slit and concave diffraction grating, then compares the broadening with the sodium doublet separation: about one thirtieth, roughly $0.2$ ångström. This supplied a spectral reference scale, not an absolute magnetic field calibration. Later cadmium photographs displayed resolved transverse triplets and axial doublets.^[ref]

The fuller report specifies a concave diffraction grating of ten-foot radius with 14,438 lines per inch, an adjustable slit and a micrometer eyepiece with vertical crosswire. Salt in a flame between electromagnet poles supplied the sodium spectral lines. Repeated switching broadened and restored them. To test changes in flame temperature or vapour density, Pieter Zeeman examined photon absorption through a heated porcelain tube with water-cooled glass windows. The arc lamp was displaced from the magnet and its light redirected by a mirror to avoid disturbing the arc. Rotating a narrower tube made the absorption spectral lines equally broad at different heights; the magnetic field still broadened their whole length.^[ref]

For the axial test, pierced electromagnet poles let the light travel along the magnetic field. A quarter-wave plate and polarizer selected one broadened edge; rotating the analyser by $90^{\circ }$ or reversing the magnet current exchanged the visible edge. The crosswire made its displacement detectable. This tested electromagnetic polarization as well as line width. For its rough estimate the report compared each side's widening with the sodium doublet separation, obtaining about one fortieth, and stated an approximate magnetic field intensity in historical units. These passages give no absolute field-calibration procedure; the inferred charge-to-mass ratio was explicitly a first approximation.^[ref]

Hendrik Antoon Lorentz's charged-oscillator theory connected the observed splitting to moving electric charge. In modern language, a line-position measurement needs a wavelength or frequency calibration and an independent magnetic field calibration before it determines a magnetic dipole moment. Resolving the individual components adds the transition information used in the worked example below.

Use the leading nonrelativistic magnetic coupling for an electronic level with total angular momentum operator $\mathbf{J}$, and choose the $z$ axis along a uniform external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$, $B\ge 0$. Write its magnetic quantum number as $m_J$, so $J_z|J,m_J⟩=\hslash m_J|J,m_J⟩$. There are $2J+1$ possibilities from $-J$ to $J$. With Landé g-factor $g_J$, the first-order shift isHere $e>0$ is the magnitude of the electron electric charge, $m_e$ its mass, $h$ the Planck constant, and ${\mu }_B$ the Bohr magneton. Both $g_J$ and $m_J$ are dimensionless; with $B$ measured in tesla and ${\mu }_B$ in joules per tesla, $\Delta E$ is in joules. The photon frequency shift $\Delta {\nu }_{ul}$ is in hertz, not angular frequency; indices $u,l$ denote upper and lower levels. NIST states this weak-field convention and its orbital/spin assumptions explicitly.^[ref]

The sign follows directly from magnetic dipole potential energy. Within this electronic $J$ multiplet, the projected magnetic dipole moment is $\mathbf{\mu }=-g_J{\mu }_B\mathbf{J}/\hslash$. Therefore $H_Z=-\mathbf{\mu }\cdot \mathbf{B}$ has the displayed positive coefficient. Subtracting the shifts of the two levels and using photon energy $h{\nu }_{ul}=E_u-E_l$ gives the transition formula. For the orbital case, ${\mu }_B/h=e/(4\pi m_e)$ gives the same leading frequency spacing as the classical charged oscillator. This explains the historical sensitivity to charge divided by mass; the quantum state derivation here is modern.

“Weak” compares interaction energies: mixing with neighbouring fine structure levels must be small, so $J$ remains a useful quantum number. The formula keeps the term linear in $B$ and neglects quadratic shifts. It also omits hyperfine structure; if that is resolved and controls the coupling, the appropriate total electronic-plus-nuclear angular momentum and splitting factor must replace $J,g_J$. Weak compared with fine structure does not automatically mean weak compared with hyperfine structure.

The Zeeman effect need not be diagonal between different $J$ multiplets. The familiar Landé g-factor gives the interaction projected into one isolated multiplet. Inter-multiplet matrix elements cause higher-order weak-field shifts and become important in the Paschen-Back effect.

The normal Zeeman effect is the simple three-frequency pattern obtained in the orbital limit. Assume negligible net spin, negligible hyperfine structure and quadratic shifts, and an allowed electric-dipole interaction between levels with orbital splitting factor one. A level with $J=0$ has only $m_J=0$ and no first-order shift. For the other levels, $g_J=1$ gives $\Delta E={\mu }_B{m}_{J}B$. Define $q=m_u-m_l$, rather than the final-minus-initial change of the emitting atom. The electric-dipole selection rule allows $q=0,\pm 1$, henceMany allowed transitions can share one frequency; a triplet need not mean exactly three transitions. The cleanest illustrative example is $J_u=1\to J_l=0$, with three upper quantum states and one unshifted lower quantum state. The $q=0$ component is called $\pi$, and $q=\pm 1$ components ${\sigma }^{\pm }$ with this stated convention. During emission the atom changes its magnetic quantum number by $-q$.

For an original numerical example choose $B=0.50\mathrm{T}$ and a zero-field vacuum wavelength ${\lambda }_0=500\mathrm{nm}$. Using ${\mu }_B\simeq 9.27401\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$ and $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ gives $\delta {\nu }_B\simeq 7.00\mathrm{GHz}$. The two outer components are separated by $2\delta {\nu }_B\simeq 14.0\mathrm{GHz}$. Differentiating the vacuum relation $\lambda =c/\nu$, with $c$ the speed of light, givesEach outer component lies about $0.00584\mathrm{nm}$ from the centre. Positive $q$ means higher frequency and shorter wavelength. Equal spacing in frequency is only approximately equal spacing in wavelength. These are calculated values for a chosen model, not historical measurements.

Viewed perpendicular to the magnetic field, the normal Zeeman effect has a $\pi$ component with linear polarization parallel to the magnetic field and two $\sigma$ components polarized perpendicular to it. Along the magnetic field, $\pi$ vanishes and the $\sigma$ pair has opposite circular polarizations. Pieter Zeeman described these geometry-dependent patterns in his lecture.^[ref]

The geometry can be derived from electric dipole radiation. Let the observation unit vector be $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, with $\theta$ measured from the magnetic field. Radiation retains only the transverse part of the oscillating electric dipole moment $\mathbf{d}$:For $\pi$, take $\mathbf{d}\parallel \hat{\mathbf{z}}$; its angular light intensity factor is ${\sin }^2\theta$, so it vanishes on axis. For ${\sigma }^{\pm }$, use normalized complex dipoles $(\hat{\mathbf{x}}\pm i\hat{\mathbf{y}})/\sqrt{2}$. Their angular factors are $(1+{\cos }^2\theta )/2$. Projection gives circles on axis, lines at $\theta =\pi /2$, and generally elliptical polarization between. These factors assume equal dipole magnitudes; actual line light intensities also depend on initial quantum state populations, transition strengths and collection geometry. No absolute handedness name is needed: it depends on the viewing and phase conventions.

A transverse polarizer can separate $\pi$ from $\sigma$ even before the spectral line components are fully resolved. Along the magnetic field, a quarter-wave plate followed by a polarizer converts the two opposite circular polarizations into distinguishable linear settings. This explains why polarization analysis carried information beyond a small increase in line width.

The Landé g-factor is the dimensionless electronic magnetic splitting factor relating a level's magnetic dipole moment to its total angular momentum. In the approximation where total orbital angular momentum $\mathbf{L}$ and total spin $\mathbf{S}$ couple to $\mathbf{J}=\mathbf{L}+\mathbf{S}$, take the orbital factor as one and the electron spin factor as $g_s\simeq 2$. For $J>0$,This follows by projecting $\mathbf{L}+g_s\mathbf{S}=\mathbf{J}+(g_s-1)\mathbf{S}$ onto $\mathbf{J}$ and using $2\mathbf{S}\cdot \mathbf{J}={\mathbf{J}}^2+{\mathbf{S}}^2-{\mathbf{L}}^2$. At $S=0$, it gives $g_J=1$; at $L=0$ and $J=S$, it gives $g_J=g_s$. A $J=0$ level instead has zero first-order electronic shift; the formula's zero denominator must not be used. This $g_J$ is distinct from the statistical weight of an atomic level $2J+1$, sometimes also written $g_J$.

For a free ion with one $f$ electron in its weak-field ground multiplet, $L=3,S=1/2,J=5/2$ and $g_J=6/7$. The six magnetic substates have adjacent spacing $g_J{\mu }_{B}B$ and full spread $5g_J{\mu }_{B}B$. A crystal field can change the applicable effective low-energy description.

The anomalous Zeeman effect is Zeeman splitting whose pattern is not the simple orbital triplet. Spin and spin-orbit interaction make the upper and lower Landé g-factors differ. The same weak-field formula still applies when its assumptions hold: “anomalous” does not mean nonlinear in the magnetic field.

For example, take an electronic transition with $L_u=1,S_u=1/2,J_u=1/2$ and $L_l=0,S_l=1/2,J_l=1/2$, neglecting hyperfine structure. With $g_s=2$, the formula above gives $g_u=2/3$ and $g_l=2$. The two allowed $q=0$ transitions have dimensionless shifts $g_u{m}_u-g_l{m}_l=\pm 2/3$, while the $q=\pm 1$ transitions give $\pm 4/3$. Thus there are four frequencies, even though each level contains only two magnetic quantum numbers. This is an independent worked deduction; it should not be substituted for the unresolved widths in the 1896 experiment.

Neglect hyperfine structure and use a magnetic field weak compared with the spin-orbit interaction. The sodium D doublet has transitions $3p{}^2{P}_{1/2}\to 3s{}^2{S}_{1/2}$ and $3p{}^2{P}_{3/2}\to 3s{}^2{S}_{1/2}$. Their Landé g-factors are $2/3$, $4/3$ and $2$ respectively. With $m_u,m_l$ denoting upper and lower magnetic labels, the photon shift isThe first transition has four allowed shifts ${\mu }_{B}B\{\pm 2/3,\pm 4/3\}$; the second has six, ${\mu }_{B}B\{\pm 1/3,\pm 1,\pm 5/3\}$. The most blue-shifted component of the second is $m_u=1/2\to m_l=-1/2$, with shift $5{\mu }_{B}B/3$. The apparently larger shift $m_u=3/2\to m_l=-1/2$ is excluded by the electric-dipole selection rule. The $3p{}^2{P}_{3/2}$ level lies above $3p{}^2{P}_{1/2}$, so in this weak-field model the $5/3$ component also has the shortest wavelength of the whole doublet. ^[ref]

For weak-field sodium D lines neglecting hyperfine structure, the smallest adjacent spectral spacing in the full electronic pattern is $s_E=2{\mu }_{B}B/3$. A drawing without a labelled energy axis can still estimate $B$ if it includes both unsplit D lines with known wavelengths: their separation calibrates $hc(1/{\lambda }_2-1/{\lambda }_1)$. Divide the drawn split spacing by the drawn unsplit separation before using that energy scale. Graphical line thickness, unequal drawn spacings and uncalibrated rendering limit precision; they do not justify exact numerical field values.

For the electronic ${}^2{S}_{1/2}\to {}^2{P}_{1/2}$ transition, neglect hyperfine structure and use weak-field LS coupling. The Landé g-factors are $g_g=2$ and $g_e=2/3$, so a transition from $m_g$ to $m_e$ shifts by $({\mu }_{B}B/h)(g_e{m}_e-g_g{m}_g)$.

With propagation perpendicular to the magnetic field, linear polarization parallel to the field drives the $\pi$ electric-dipole selection rule $\Delta m=0$. The two shifts are $\pm 2{\mu }_{B}B/(3h)$. Perpendicular linear polarization is an equal-amplitude combination of the two field-axis circular components, driving ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$ and shifts $\pm 4{\mu }_{B}B/(3h)$. Each geometry therefore selects a doublet, with twice the separation in the perpendicular-polarization case. Equal line heights within a pair require equal lower-state populations and an unsaturating probe; line positions do not require that extra population assumption.

In the Paschen-Back effect, the external magnetic field interaction becomes large compared with the spin-orbit interaction that holds $\mathbf{L}$ and $\mathbf{S}$ coupled. Then $J$ ceases to be a good approximate quantum number; $m_L,m_S$ become more useful, with leading electronic shifts $\Delta E\simeq {\mu }_{B}B(m_L+g_s{m}_S)$. In the ideal complete-decoupling limit an electric-dipole interaction preserves $m_S$, so its spin contribution cancels between the levels and orbital triplet spacing can reappear. Intermediate fields require mixing the coupled states. This is a change of coupling regime, distinct from the weak-field anomalous Zeeman effect; still larger fields require additional terms beyond this linear model.

At large but finite magnetic field, opposite-spin admixtures in approximate product eigenstates let nominally spin-forbidden electric-dipole channels borrow intensity. Their amplitudes scale as spin-orbit interaction energy divided by Zeeman energy; their intensities therefore vanish quadratically in that ratio.

For a one-electron P-to-S transition with nuclear spin zero, ten electric-dipole sublevel channels have finite strength at weak nonzero field. Six retain finite strength in the complete orbital-spin decoupling limit. Channel count is distinct from spectral frequency count: retaining first-order residual spin-orbit interaction energy gives five distinct frequencies, with two central channels degenerate.

For orbital angular momentum $\ell =1$ and spin $s=1/2$, considerThe quantum number $m_j=m_{\ell }+m_s$ is conserved. The stretched states with $m_j=\pm 3/2$ have energies $K\pm 2a$. The remaining eigenvalues follow from two $2\times 2$ blocks with off-diagonal entries $\sqrt{2}K$:The subscripts denote $m_j=\pm 1/2$. For $K>0$, the zero-field multiplets are $j=3/2$ at $K$ and $j=1/2$ at $-2K$. Their weak-field Zeeman splitting is $g_{j}a{m}_j$, with Landé g-factors $4/3$ and $2/3$, respectively. When $a\gg |K|$, the Paschen-Back effect gives approximate product states $(m_{\ell },m_s)$ and energies $a(m_{\ell }+2m_s)+2K{m}_{\ell }{m}_s$. Mixing within a fixed $m_j$ causes an avoided crossing; different $m_j$ sectors can cross because the Hamiltonian does not couple them.

Methods for preparing and probing chemical atoms in atomic physics.

State-selective field ionization distinguishes atomic quantum states through their different electric field ionization thresholds.

An ion trap confines an ion using electromagnetic fields, allowing repeated interrogation of the same particle. The name also covers related traps for an electron. A Paul trap uses an oscillating electric quadrupole field; a Penning trap combines a static electric quadrupole field with a static magnetic field. Confinement creates time to measure an energy level, frequency or magnetic dipole moment; it does not itself cool the particle or identify its internal quantum state.

The obstacle is Earnshaw's theorem. In a charge-free neighbourhood the static electric potential $\Phi$ obeys Laplace's equation, ${\nabla }^2\Phi =0$. For a particle of fixed nonzero electric charge $Q$, the potential energy $Q\Phi$ is also a harmonic function. The maximum principle for harmonic functions excludes a strict interior minimum. For a quadratic example, the Hessian matrix has zero trace; it cannot have three positive eigenvalues. A static purely electric field therefore cannot supply three-dimensional stable confinement of a free point electric charge. Time variation in a Paul trap, and the velocity-dependent Lorentz force in a Penning trap, change the assumptions of this argument.

An ion confined by an electromagnetic trap has both internal atomic states and quantized motional modes. Coherent internal-state control, state-dependent fluorescence and resolved-sideband operations allow these degrees of freedom to serve as quantum-information resources.

Trapped-ion quantum control prepares an ion's internal quantum state, couples it to quantized motion with laser pulses, and measures the result through state-dependent fluorescence. Its appeal is access to both a controllable two-level system and a quantum harmonic oscillator in one apparatus.

Confinement, preparation, control and readout solve different problems. The Paul trap's RF electric field confines the ion through its RF-averaged trapping potential; it still produces Paul trap micromotion. Doppler cooling and resolved sideband cooling reduce motional excitation. Optical pumping selects an internal quantum state. A coherent pulse then changes amplitudes, while electron shelving converts an internal population into a detectable stream of photons. A bright/dark count by itself does not reveal a quantum superposition's relative phase; an additional controlled rotation before detection can make that phase affect the measured population.

A quantum processor encoding qubits in long-lived internal states of confined ions. Coherent drives implement single-qubit rotations; shared vibrational modes can mediate entangling gates. Individual fluorescence measurements distinguish bright and dark qubit levels. Mode crowding and residual spin-motion entanglement complicate gates as a single chain grows.

An internal qubit uses two distinguishable electronic or hyperfine structure quantum states, denoted $|g⟩,|e⟩$. The ion's centre-of-mass motion has a different Hilbert space. In one approximately harmonic direction, a number state $|n⟩$ has energy $E_n=\hslash \nu (n+1/2)$, where $\nu$ denotes a motional angular frequency, not ordinary frequency. The tensor product of quantum systems gives $|g,n⟩$. Cooling towards $n=0$ prepares motion; it does not automatically prepare the internal qubit. Nor does the ground state mean a stationary point particle: zero-point energy and position spread remain.

For several ions, Coulomb forces couple their displacements into collective normal modes. Each selected normal mode has its own quantum harmonic oscillator number states, while each ion retains internal qubit states. A pulse addressing one ion can change a shared mode. This supplies a route to correlations between distant internal quantum states; it also makes residual motional excitation and unwanted excitation of other normal modes relevant. A single-ion pulse calculation alone establishes no experimental multi-ion quantum gate performance.

For normalized oscillator states and $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$, define $\eta =k\sqrt{\hslash /(2m\omega )}$. Normal ordering the displacement operator gives the displayed matrix element. A sudden momentum kick therefore creates a Poisson distribution of oscillator occupations with mean ${\eta }^2$. Its mean added motional energy is ${\eta }^2\hslash \omega ={\hslash }^2{k}^2/(2m)$, although most events retain $n=0$ when $\eta \ll 1$.

A transition between oscillator energy eigenstates that adds one quantum changes kinetic and potential expectation energies by $\hslash \omega /2$ each. A sudden unobserved momentum kick instead initially adds its energy only to kinetic motion and leaves the position distribution unchanged. Coherent superpositions distinguish these descriptions. Averaging over motional outcomes gives a recoil-scale total energy; time averaging or dephasing gives equal kinetic and potential shares. Broad-line transition rates need not be exactly equal to bare overlap probabilities if finite spectral weights matter.

The Lamb-Dicke regime means the wave-induced phase change across the occupied motional wavepacket is small. For one normal mode, let $x_0=\sqrt{\hslash /(2m\nu )}$ be the ground state's position standard deviation, and write $x=x_0(a+a^{†})$ using creation and annihilation operators. Here $m$ is mass and $\hslash$ the Planck constant divided by $2\pi$. The effective wave vector component $k_{\mathrm{eff}}$ is the single beam's component for a direct transition, or the difference of two beam components for a Raman transition. With dimensionless $\eta =|k_{\mathrm{eff}}|x_0$, the useful condition is $\eta \sqrt{2n+1}\ll 1$ over the populated number states, rather than merely $\eta \ll 1$.

Absorb the sign of $k_{\mathrm{eff}}$ into the pulse phase. Expanding the coupling factor givesThe constant term preserves $n$; the annihilation operator gives $a|n⟩=\sqrt{n}|n-1⟩$; the creation operator gives $a^{†}|n⟩=\sqrt{n+1}|n+1⟩$. The different square roots explain why removing a motional quantum becomes impossible at $n=0$, while adding one remains possible. Smallness of $\eta$ controls this spatial expansion; resolving the resulting transitions is a separate frequency requirement.

A motional sideband is a transition that changes internal excitation and a motional quantum number together. Define the internal energy gap as $\hslash {\omega }_0>0$. A carrier drives $|g,n⟩\leftrightarrow |e,n⟩$; the first red sideband drives $|g,n⟩\leftrightarrow |e,n-1⟩$; the first blue sideband drives $|g,n⟩\leftrightarrow |e,n+1⟩$. Their resonant drive angular frequencies follow by subtracting these energy levels. For a Raman transition, the drive is a beam-difference frequency with a sign fixed by the selected internal transition.

Choose a real carrier Rabi angular frequency $\Omega$ defined so that resonant carrier excitation is ${\sin }^2(\Omega t/2)$. In the Lamb-Dicke regime, isolated first-sideband Hamiltonians in a rotating frame can be written, after choosing the pulse phase, asThese formulas use a rotating-wave approximation: neglected carrier and other-sideband terms oscillate rapidly compared with the retained coupling. The drive's effective width and sideband coupling must be small compared with $\nu$; sufficient suppression of off-resonant carrier excitation must also be checked. The red interaction is a Jaynes-Cummings model with a motional quantum replacing a cavity photon. The blue interaction creates internal excitation and a motional quantum together; the external laser supplies energy. The driven-system rotating-frame Hamiltonian alone is not the total laboratory energy.