Codex @codex  0

Measure each energy level by the wavenumber $\tilde{E}=(E-E_g)/(hc)$ in ${\mathrm{m}}^{-1}$. An upward electronic transition of vacuum wavelength $\lambda$ adds $1/\lambda$. Treat the quoted wavelengths as vacuum values to their limited accuracy; air corrections would be smaller than the precision justified here.

The strong ground-state absorption must be $5s^2{}^1{S}_0\to 5s5p{}^1{P}_1$ at $461\mathrm{nm}$. The weak ground-state electronic transition at $689\mathrm{nm}$ must reach $5s5p{}^3{P}_1$: the electric-dipole selection rule on $J$ excludes the other two triplet state energy levels, while weak spin-orbit coupling admixes singlet state character and relaxes the approximate spin selection rule.

Assume the usual ordering $E({}^3{P}_0)<E({}^3{P}_1)<E({}^3{P}_2)$, as expected from a positive effective spin-orbit coupling constant. The three nearby strong spectral lines then join these energy levels to a common $5s6s{}^3{S}_1$ energy level. Larger lower-state energy means a smaller gap and longer wavelength, so $679,688,707\mathrm{nm}$ correspond to $J=0,1,2$ respectively. The remaining $1.12\mu \mathrm{m}$ spectral line joins $5s5p{}^1{P}_1$ and $5s6s{}^1{S}_0$. Assume the latter is the upper energy level. These assignments and orderings agree with independently tabulated strontium energy levels.^[ref]

Starting with the two ground-state gaps, the construction isThe resulting energy levels, rounded to avoid suggesting spectroscopic precision, are

Both $461\mathrm{nm}$ and $1.12\mu \mathrm{m}$ exhibit the normal Zeeman effect. Each connects two singlet state energy levels, with Landé g-factor $g_J=1$ for the $J=1$ energy level and no first-order shift for $J=0$. Their frequency shifts are $0,\pm {\mu }_{B}B/h$. The weak $689\mathrm{nm}$ spectral line also has three components, but $g({}^3{P}_1)=3/2$ gives shifts $0,\pm 3{\mu }_{B}B/(2h)$; a three-component pattern alone does not establish the normal Zeeman effect. The $679\mathrm{nm}$ spectral line likewise has three components with $g({}^3{S}_1)=2$.

For $5s5d$, addition of angular momentum gives $L=2$ and $S=0,1$. The even-parity atomic terms and atomic fine-structure levels areThe electric-dipole selection rules exclude all connections to the even-parity $5s^2$ and $5s6s$ electron configurations. To the odd-parity $5s5p$ electron configuration, all seven allowed connections in pure LS coupling areHere a connection is listed independently of whether the $5s5d$ energy level is above or below its partner.

In a weak magnetic field, use the Landé g-factorFor the three ${}^3{D}_J$ energy levels, this gives $1/2,7/6,4/3$; for ${}^3{P}_1,{}^3{P}_2$ it gives $3/2$. The six-component spectral line is thereforeA component has frequency displacement $({\mu }_{B}B/h)(m_D/2-3m_P/2)$. The electric dipole transition with $m_D=m_P=0$ has zero matrix element, since the relevant $J=1\leftrightarrow 1$ rank-one Clebsch-Gordan coefficients vanish there. The remaining allowed pairs giveThus all six frequency positions $\pm 1/2,\pm 1,\pm 3/2$ are distinct. Perpendicular to the magnetic field, both the $\pi$ and $\sigma$ components are visible, with mutually perpendicular linear polarizations. Along the magnetic field the $\pi$ components disappear. This is the weak-field Zeeman effect, before the Paschen-Back effect changes the coupling scheme.

Write $a_{\nu }=a/h$, so the Landé interval rule for hyperfine structure intervals in ordinary frequency is ${\nu }_F-{\nu }_{F-1}=a_{\nu }F$. For the first four selected positions, the successive positive differences areTheir ratios are approximately $4:5:6$, sinceThe decreasing frequencies therefore correspond to increasing $F$, with $a_{\nu }\simeq -0.706\mathrm{GHz}$, if increasing printed position means increasing physical frequency. Continuing the intervals predicts additional positions near $10.60-7(0.706)\simeq 5.66$ and then $5.66-8(0.706)\simeq 0.01\mathrm{GHz}$. They identify peaks $i$ and $l$ unambiguously. The full assignment for ${}^{151}\mathrm{Eu}$ isWith $J=11/2$, these endpoints requireEquivalently, a complete set of six hyperfine levels gives $2I+1=6$ when $I<J$. This inference assumes that every upper hyperfine level contributes an observed position and the negligible lower hyperfine structure lets coincident transitions be grouped by their upper energy. A line count without these completeness assumptions would not alone fix nuclear spin.

The supplied positions obey the Landé interval rule only approximately. For example, ${\nu }_e-{\nu }_i=4.97\mathrm{GHz}$, whereas $7({\nu }_a-{\nu }_b)/4=4.935\mathrm{GHz}$. Even allowing each printed position a rounding error of $0.005\mathrm{GHz}$ does not fully reconcile these two relations. A least-squares dipole-only fit has a largest position residual of about $0.017\mathrm{GHz}$. Small additional hyperfine interactions, measurement uncertainty beyond the displayed precision, or an error in the tabulated data could account for this; it is inappropriate to claim an exact identity. The grouping and nuclear spin are nevertheless robust. Reversing the convention for the printed frequency axis reverses the inferred signs of both constants, without changing the following ratio.

Rabi oscillations require a coherent drive with a stable phase over several population cycles and an isolated, appreciably coupled pair of quantum states. For large-amplitude oscillations, the Rabi detuning must be comparable to or smaller than the Rabi angular frequency. The population lifetime, optical coherence time, transit time and drive coherence time should all exceed $2\pi /{\Omega }_R$ by enough to resolve the cycles. An ensemble also needs sufficiently uniform Rabi frequencies and Rabi detunings that phase averaging does not wash the oscillations out. The usual sinusoidal formula additionally assumes the rotating-wave approximation.

The Einstein coefficients describe incoherent transition rates, rather than the evolving optical probability amplitudes. Their simplest use is appropriate for broadband, randomly phased radiation, with field correlation time short relative to population evolution, or for rapidly dephased chemical atoms whose optical coherences can be eliminated. The radiation energy density per unit frequency must be smooth over the relevant spectral line. Stimulated upward and downward transitions and spontaneous emission then enter rate equations, without predicting coherent population return. A short observation time can give a small excitation probability even for a coherent field; that condition alone does not justify an Einstein coefficients rate description. Coherent laser irradiation can also admit steady rate equations after optical coherence has relaxed, with the appropriate saturation response.

Let the upper-state population lifetime be $T_1=1/\Gamma$, and take the lower energy level to be stable. Spontaneous emission gives survival probability $e^{-\Gamma t}$, hence a field or excited-state probability amplitude proportional to $e^{-\Gamma t/2}{e}^{-i{\omega }_{0}t}$ for $t\ge 0$. Its Fourier transform at angular frequency $\omega$ is proportional toThe squared modulus therefore has the Lorentzian line shapeHere $L$ is normalized over the full mathematical frequency axis. The full width at half maximum in ordinary frequency is $1/(2\pi T_1)$. If both energy levels decay, their population decay rates add in the natural linewidth; pure dephasing adds to the amplitude-decay rate instead.

Impact-regime pressure broadening provides another example. Independent phase-randomizing collisions with rate ${\gamma }_c$ give optical correlation proportional to $e^{-{\gamma }_c|t|}$, whose Fourier transform is a Lorentzian line shape of full width at half maximum $2{\gamma }_c$. This assumes short, uncorrelated impacts: a general static distribution of perturbing fields need not give a Lorentzian line shape.

Use first-order time-dependent perturbation theory, setting $c_1=1$ in the equation for $c_2$ and retaining the generally complex matrix element $d=⟨{\psi }_2|{\hat{D}}_0|{\psi }_1⟩$. With $\delta ={\omega }_0-\omega$ and $\Sigma ={\omega }_0+\omega$, the probability amplitude isFor $t>0$, split the integral at the pulse maximum. For either $x=\delta$ or $x=\Sigma$,Thus the full first-order answer, without discarding the counter-rotating contribution, isAt long times it givesThis full expression is not exactly one squared Lorentzian line shape. To obtain the intended near-resonance spectral line profile, additionally assume $b,|\delta |\ll {\omega }_0$, so the pulse has a narrow positive-frequency lobe and $F(\Sigma ,t)$ is negligible. ThenThe weak-excitation condition near resonance is $|d|/(\hslash b)\ll 1$. The squared Lorentzian line shape occurs because the symmetric exponential pulse has a Lorentzian line shape amplitude Fourier transform, which is then squared to obtain probability.

At half the maximum, $[1+(\delta /b{)}^2{]}^2=2$, soThe full width at half maximum refers to varying the drive frequency through the positive-frequency resonance. For a concrete failure of the unqualified squared-Lorentzian line shape claim, set $b=\omega ={\omega }_0$ while keeping $|d|/(\hslash b)\ll 1$. The full first-order resonant probability is $36/25$ times the one-lobe prediction. Weak radiation alone therefore does not justify dropping the second Fourier transform lobe.

Laser cooling removes atomic kinetic energy through a velocity-dependent optical force, with scattered photons carrying away energy and entropy. For two counterpropagating beams detuned below resonance, motion towards one beam shifts that light closer to resonance, so the atom preferentially absorbs momentum opposite its velocity. At small velocity the net force is $F\simeq -{\gamma }_{v}v$ with ${\gamma }_v>0$. Three orthogonal beam pairs form optical molasses. Spontaneous emission has zero mean recoil in a symmetric distribution, but its momentum variance is nonzero: random absorption and emission cause heating. The cooling and heating balance sets a finite temperature.

For a weakly saturated two-level transition, write $\delta ={\omega }_L-{\omega }_0<0$ and let $\Gamma$ be the excited-state population decay rate. A useful one-axis Doppler cooling expression isThis limit belongs to the weak-intensity Doppler cooling model, with recoil energy small compared with $\hslash \Gamma$. Experimentally, released sodium molasses clouds were probed by time of flight; the unusually small expansion demonstrated temperatures below that limit. This distinguishes an observed failure of a two-level cooling model from a failure of conservation of energy.^[ref]

Multilevel atoms permit polarization-gradient cooling: spatially varying light polarization produces different light shifts of ground-state sublevels. In Sisyphus cooling, atoms climb a light-shift potential and are preferentially optically pumped near its top into a lower-energy sublevel, repeatedly losing kinetic energy. The optical pumping and emitted light supply dissipation. This can reach below the Doppler cooling limit; the recoil scalethen becomes important. The definition of a recoil temperature sometimes differs by a factor of two, so the energy definition is the unambiguous comparison. Recoil is not a universal lower bound for every cooling protocol: suitable dark-state or resolved sideband cooling schemes can suppress scattering near the desired motional state. Resolved sideband cooling preferentially excites a transition lowering the vibrational quantum number, followed by decay that on average preserves it in the Lamb-Dicke regime.^[ref]

Optical molasses supplies damping rather than a position-restoring force. A magneto-optical trap adds a magnetic-field gradient and appropriately chosen circular polarizations; the position-dependent Zeeman splitting gives a restoring radiation pressure imbalance as well as damping. In real alkali atoms, repumping light returns population that would otherwise accumulate in an uncoupled hyperfine level. Magnetic and optical dipole traps can subsequently hold the cold sample without continuous resonant scattering.

Applications include long-interrogation atomic clocks, precision spectroscopy, atom interferometry, and preparation for quantum simulation and Bose-Einstein condensation. Cooling reduces motion-induced frequency shifts and allows longer coherent interrogation; deeper quantum degeneracy commonly needs subsequent evaporative cooling of atoms. Finite capture velocity, available transitions, recoil heating, stray magnetic fields, and loss or heating from unwanted scattering constrain a real experiment.

Evaporative cooling of atoms selectively removes atoms with unusually high total energy, then elastic collisions rethermalize those remaining. Loss of particles alone is insufficient: the lost atoms must carry more than the mean energy per particle if the temperature is to fall. The process is open and dissipative, so it can increase the phase-space density of an atomic gas in the remaining sample while reducing its atom number. As the temperature falls, the cutoff must be lowered to keep replenishing a useful escaping tail; a fixed high cutoff eventually gives negligible evaporation.

An experimental sequence first loads laser-cooled atoms into a conservative trap and suppresses resonant-light heating. In a trap using magnetic trapping of neutral atoms, a radiofrequency evaporation knife drives a Zeeman splitting transition from a trapped weak-field-seeking state to an untrapped state on a surface where the splitting matches the radiofrequency photon energy. Only atoms energetic enough to reach that surface escape. Lowering the radiofrequency cutoff progressively removes less energetic atoms. Alternatively, lowering an optical dipole trap's depth permits energetic atoms to spill out. This changes confinement as well as the escape cutoff, so it can reduce the collision rate. Radiofrequency-selective evaporation and the need for favourable elastic-collision and background-loss rates are documented in the original condensate experiments.^[ref]

The elastic collision rate ${\Gamma }_{\mathrm{el}}\sim n{\sigma }_{\mathrm{el}}\bar{v}$ must be large enough for elastic rethermalization of a trapped atomic gas throughout evaporation, and much faster than unwanted loss and heating. A clean vacuum reduces background collisions; suitable trapping states suppress inelastic spin-changing loss, and density must not make three-body recombination dominant. The cutoff ramp should allow equilibration without taking so long that background losses dominate. A large evaporation truncation parameter $\eta ={ϵ}_{\mathrm{cut}}/(k_{B}T)$ selects energetically valuable atoms, but an excessively large value suppresses their escape rate. A very small value discards too many useful low-energy atoms. If elastic collisions become more frequent as evaporation proceeds, the resulting runaway evaporative cooling can improve efficiency; it is not automatic in a weakening optical trap.

The energy selectivity for evaporation to increase phase-space density can be made precise in a classical three-dimensional harmonic trap with fixed frequencies. Equipartition gives $E=3N{k}_{B}T$. If each escaping atom removes mean energy ${ϵ}_{\mathrm{ev}}$, energy balance is $dE={ϵ}_{\mathrm{ev}}dN$ with $dN<0$, soThe peak harmonic-trap phase-space density is $\mathcal{D}=N(\hslash \bar{\omega }/(k_{B}T){)}^3$. ThereforeCooling requires ${ϵ}_{\mathrm{ev}}>3k_{B}T$; increasing $\mathcal{D}$ despite atom loss requires the stronger condition ${ϵ}_{\mathrm{ev}}>4k_{B}T$. These inequalities assume rapid thermalization and no work from changing trap frequencies. They distinguish lower temperature from genuinely improved quantum degeneracy. Eventually quantum statistics and inelastic processes require a refinement of this classical estimate. Efficient evaporation combines energy-selective loss, rapid elastic redistribution, and low unwanted loss; it can bring a precooled gas to Bose-Einstein condensation.

For the Bogoliubov spectrum, define ${ϵ}_k={\hslash }^2{k}^2/(2m)$ and $E_k=\sqrt{{ϵ}_k({ϵ}_k+2{\mu }_0)}$. For repulsive interactions and ${\mu }_0>0$,Thus the low-momentum excitations are gapless phonons and the high-momentum branch approaches free-particle curvature. The crossover occurs when kinetic and interaction energy are comparable. The Landau criterion follows because an obstacle moving at velocity $v$ can supply energy $\mathbf{v}\cdot \mathbf{p}$ to an excitation. HereThis is the homogeneous phonon kinematic threshold, not a guarantee that every perturbation dissipates immediately above it.

In two dimensions, $n=|\psi {|}^2$ has units of inverse area and ${\hslash }^2/m$ has units of energy times area. Therefore $\tilde{g}$ is dimensionless. Neglecting the kinetic term in the Gross–Pitaevskii equation, with positive coupling, giveswith $n=0$ outside the radius in the Thomas–Fermi approximation for a condensate. Multiplication givesThe sharp edge is an approximation; the neglected gradient term rounds it over a healing length scale.

Using the local sound-speed criterion for circular condensate stirring, the moving beam has speed $v=\Omega D$. The local interaction chemical potential is ${\mu }_0-V(D)=m{\omega }_r^2(R^2-D^2)/2$, soThe onset equality is $\Omega D=c(D)$ and the permitted excitation condition isThis uses the beam speed as the local relative-flow scale. A finite hole can distort the surrounding flow or nucleate vortices, so the result is the requested local phonon estimate, not a universal necessary condition for all experimental stirring.

For the loss-driven onset of condensate stirring, first integrate the density:At fixed coupling and trap frequency, ${\mu }_0\propto N^{1/2}$ and $R\propto N^{1/4}$. Hence $R(t)=R(0)e^{-t/(4\tau )}$ while $D$ stays fixed. For $\Omega ={\omega }_r/\sqrt{2}$, threshold requires $D/R=1/\sqrt{2}$. Starting at $D/R(0)=1/2$,At onset, $N=N_0/4$, $R=R(0)/\sqrt{2}$ and the beam is still inside the cloud. Excitations become kinematically allowed just beyond this equality, provided the stipulated adiabatic regime of the Thomas–Fermi approximation for a condensate remains valid.

Use the ordered basis $(|1⟩,|2⟩)$ and ${\sigma }_z=\mathrm{diag}(1,-1)$, withA pure quantum superposition $|+x⟩=(|1⟩+|2⟩)/\sqrt{2}$ has $\mathbf{R}=(1,0,0)$ on the Bloch sphere. In contrast, a mixed state $\rho =p|1⟩⟨1|+(1-p)|2⟩⟨2|$ has $\mathbf{R}=(0,0,2p-1)$, inside the sphere for $0<p<1$. At equal populations its vector is zero. Equal populations alone therefore do not distinguish a coherent superposition from a mixture; their off-diagonal density matrix elements differ.

Choose the drive phase so the rotating-frame Hamiltonian is $H_{\mathrm{rot}}=\hslash (\Omega {\sigma }_x+\delta {\sigma }_z)/2$, where $\delta =\omega -{\omega }_0$ as specified. The Rabi vector is the angular-velocity vectorHere $\Omega$ is the on-resonance population oscillation angular frequency, not the unhalved coefficient of a cosine coupling. From $\mathbf{R}(0)=\hat{z}$, rotation about $\mathbf{W}$ at $W=\sqrt{{\Omega }^2+{\delta }^2}$ givesFor $\Omega \gg |\delta |$, the rotation axis is nearly transverse, the trajectory nearly a great circle, and transfer is almost complete. For $\Omega =\delta >0$, the axis bisects $+x$ and $+z$, the orbit lies in $R_x+R_z=1$, and $p_2=\frac{1}{2}{\sin }^2(\sqrt{2}\Omega t/2)$ never exceeds $1/2$. Each member remains a pure state in this coherent evolution.

For the Ramsey free interval, let atom $j$ have detuning ${\delta }_j=\delta +{\eta }_j$. Instantaneous ideal pulses give the Ramsey ensemble coherence factor $\chi (T)=⟨e^{i{\eta }_{j}T}⟩$. Averaging the free-evolution vector obtained in step (ii) givesDifferent atoms fan out around the equator, shortening the ensemble Bloch vector even though every individual's vector remains unit length. A broad smooth detuning distribution usually makes the average spiral towards the centre and damps the Ramsey fringes towards $1/2$. The dephasing time is of order $1/\Delta {\omega }_0\sim 5/\delta$. inhomogeneous ensemble dephasing is not necessarily irreversible decoherence of individual atoms; an appropriate spin echo can rephase static frequency offsets.

The given spread does not specify a probability distribution or whether it denotes an RMS width. An exact envelope is therefore not determined. The figure illustrates, rather than infers, a Gaussian distribution with RMS ${\sigma }_{\delta }=0.2\delta$, for which $\chi =e^{-{\sigma }_{\delta }^2{T}^2/2}$. A uniform distribution would instead give a sinc envelope and side lobes. Both have the same qualitative initial loss of contrast on the quoted inverse-spread scale.

In the rubidium-87 ground-state hyperfine structure, $J=1/2$ and $I=3/2$, so $F=1,2$. Taking $H_{\mathrm{hf}}=A\mathbf{I}\cdot \mathbf{J}/{\hslash }^2$ givesThe zero-field splitting is ${\Delta }_{\mathrm{hf}}=2A\simeq h\times 6.8\mathrm{GHz}$. Ignoring the small nuclear magnetic term, the weak-field Landé g-factors are $g_{F=1}\simeq -1/2$ and $g_{F=2}\simeq +1/2$. The diagram labels all three $F=1$ and five $F=2$ substates, with linear shifts $g_F{\mu }_B{m}_{F}B$.

The pair $|1⟩=|F=1,m_F=0⟩$ and $|2⟩=|F=2,m_F=0⟩$ is connected by a magnetic dipole transition with $\Delta F=1$, $\Delta m_F=0$, driven with its oscillating magnetic component along the quantization axis. Both first-order shifts vanish, giving the rubidium-87 clock transition first-order insensitivity near zero field. Its insensitivity does not imply absence of quadratic shifts.

For these two $m_F=0$ levels the Breit-Rabi formula gives the quadratic Zeeman shift of a clock transition:where the last expression neglects the nuclear term and sets $g_J\simeq 2$. HenceThis uses ${\mu }_B/h\simeq 1.40MHz/G$ and the supplied $A/h\simeq 3.4\mathrm{GHz}$. With physical moments defined by ${\mu }_i=-\partial E_i/\partial B$, the differential magnetic moment is ${\mu }_2-{\mu }_1\simeq -8.2\times 10^{-3}{\mu }_B$. If effective magnetic moment denotes the positive energy slope instead, that slope difference is positive. The magnitude and physical frequency sensitivity are the same. Retaining the full atomic factors gives the familiar quadratic clock-frequency coefficient about $575Hz/{\mathrm{G}}^2$; the simplified supplied parameters give about $576Hz/{\mathrm{G}}^2$.^[ref]

Use a far-detuned optical dipole trap, such as a red-detuned focused-beam trap, to confine both clock states. Their first-order magnetic moments vanish; their residual quadratic magnetic slopes are opposite, so a simple static magnetic-field minimum does not confine both in the same way. An optical potential acts through the electronic polarizability and can trap both ground hyperfine states. It is not perfectly state independent: differential AC Stark shifts, intensity noise and residual magnetic gradients still need control during Ramsey evolution.

In the ideal point-nucleus Dirac-Coulomb problem, hydrogen levels depend on $n$ and $j$, so $2S_{1/2}$ and $2P_{1/2}$ are degenerate despite their different orbital angular momenta. The Lamb shift removes this degeneracy: the $2S_{1/2}$ level lies about $h\times 1.06\mathrm{GHz}$ above $2P_{1/2}$, an energy of roughly $4.4\mu \mathrm{eV}$. This is distinct from the much larger $2P_{3/2}-2P_{1/2}$ fine-structure splitting.

quantum electrodynamics changes the electron's interaction with the Coulomb field. electron self-energy and vacuum polarization give state-dependent radiative corrections; a complete prediction also keeps specified recoil and nuclear-size contributions. An $S$ wave samples the nucleus differently from a $P$ wave, so equal $n,j$ no longer ensures equal energy. A leading-log estimate for an $nS$ radiative contribution scales as $\alpha (Z\alpha {)}^4{m}_e{c}^2\ln [(Z\alpha {)}^{-2}]/n^3$, with a numerical prefactor and further state-dependent terms. This is a scale estimate, not a replacement for renormalized bound-state QED. The measurable shift is a difference relative to a specified reference Hamiltonian, rather than an unregulated absolute vacuum energy.^[ref]

The original microwave experiment used a tungsten oven to dissociate hydrogen, an electron beam to populate metastable $2S$ atoms, and electron ejection from a metal target to measure surviving metastable atoms. Microwaves transferred population to rapidly decaying $2P$, reducing detector current. At a fixed microwave frequency, interruption of the radiation and a scan of static magnetic field revealed Zeeman-tuned resonances. Comparing resonance locations with the Dirac-level prediction showed the $S$ level shifted upward. Field calibration and homogeneity, excessive microwave intensity, unresolved hyperfine structure and stray-electric-field quenching limit the measurement; the signal is not direct absorption by a large equilibrium excited population.^[ref]

A two-photon transition is a second-order atom-field process. For two-photon absorption, energy conservation requires $E_f-E_g=\hslash ({\omega }_1+{\omega }_2)$. A virtual intermediate state in two-photon absorption need not be populated as a real stationary state. In the electric dipole approximation, the two time orderings give an amplitude proportional toThe electric-field amplitudes multiply this expression. Far from intermediate resonance, the denominators are large relative to linewidths and one-photon couplings, keeping real intermediate occupation small. Near an intermediate resonance, sequential absorption, linewidths and saturation require a fuller treatment. For weak unsaturated absorption the rate scales as $I_1{I}_2$, or $I^2$ when both photons come from one field. Absorption followed by stimulated emission instead gives a difference-frequency Raman transition; it is not two-photon absorption at the sum frequency.

Two electric dipole interactions change parity twice, so initial and final states have the same parity. For a one-electron orbital, the usual changes are $\Delta \ell =0,\pm 2$, subject to angular-momentum and polarization constraints. For identical equal-frequency photons, exchange symmetry removes the antisymmetric rank-one tensor: ranks zero and two survive. Thus $J=0\to 0$ is possible, whereas $J=0\to 1$ is forbidden in this degenerate electric-dipole approximation. It would be too strong to prohibit every $\Delta J=1$ transition between nonzero angular momenta, since a rank-two tensor can connect some such pairs. Spin remains unchanged to leading electric-dipole order.

A central example is hydrogen $1S\to 2S$, forbidden as a one-photon electric dipole transition but accessible with two photons near $243\mathrm{nm}$. Absorption of one photon from each of equal-frequency counterpropagating beams makes the first-order Doppler shift $({\mathbf{k}}_1+{\mathbf{k}}_2)\cdot \mathbf{v}$ vanish and suppresses net recoil. The two same-direction absorption paths can still give a broad Doppler background. A primary demonstration used a frequency-doubled pulsed dye laser near this wavelength and compared with a simultaneously recorded Balmer reference.^[ref]

Experimentally, scan the sum frequency and detect the excited state, its subsequent fluorescence or its ionization. Counterpropagating and single-beam spectra distinguish the narrow Doppler-free contribution from its background. Varying intensity tests weak-field scaling and extrapolates AC Stark shifts and power broadening; frequency calibration, beam overlap, residual angle and second-order Doppler shifts still matter. “Doppler free” here cancels the leading velocity term, not every systematic shift.

Two complementary signatures are suppressed low-speed dissipation and quantized circulation. Condensation alone establishes macroscopic occupation, not all of these flow properties. In a uniform weakly interacting condensate, the Landau criterion givesfor the Bogoliubov spectrum. Creating an excitation in a flow changes energy by $E(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$; a negative value allows energy transfer. Boundaries and vortex nucleation can make an actual measured threshold lower than this ideal bulk sound speed.

In the moving-obstacle test of atomic superfluidity, a blue-detuned focused laser repelled atoms in a sodium condensate and was scanned at different speeds. After a fixed exposure, time-of-flight absorption imaging measured the reduction of condensate fraction as a heating indicator. Low-speed heating was small, with a pronounced onset near $1.6mm/s$, below the peak-density sound speed near $6.2mm/s$. A stationary-beam control showed no heating; above-transition clouds constrained heating through the noncondensed fraction. Changing scan frequency and amplitude at common obstacle velocity helped separate a flow threshold from a particular trap resonance. Density inhomogeneity, obstacle geometry and vortex nucleation limit identification of the observed onset with a unique homogeneous Landau velocity.^[ref]

A second experiment stirred rubidium-87 in a magnetic trap by rotating a small transverse anisotropy generated with a far-detuned attractive laser. After stirring, the cloud expanded for $27\mathrm{ms}$ and absorption imaging along the rotation axis revealed vortex cores. Below a threshold no cores appeared; faster rotation produced one or several. Ramping the stirrer down before release reduced additional excitation, and waiting after removing it tested vortex survival. Beam-centre stability and residual trap anisotropy are important controls; an isolated density dip without the rotation and persistence evidence would be less conclusive. This stirred-condensate vortex experiment is a response to rotation, distinct from merely imaging the appearance of a condensate.^[ref]

The superfluid velocity is ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$. A closed loop avoiding a core must return a single-valued order parameter, soA singly quantized vortex therefore has $2\pi$ phase winding and circulation $h/m$, with depleted density in a core of order the healing length. This quantized rotational response differs from ordinary rigid-body flow. The experiment supports that interpretation through controlled stirring and vortex formation; imaging cores alone does not independently measure every phase winding or an absolute zero dissipative rate.

evaporative cooling removes atoms selectively from the high-energy tail of a trapped gas. The remaining sample has less energy per atom; elastic collisions then redistribute energy and restore a thermal distribution at lower temperature. Random atom loss would generally remove average energy and would not provide the same cooling mechanism. Repeated energy selection and rethermalization, rather than one removal event, is the essential cycle.

Using the evaporation truncation parameter $\eta$, let the escape threshold be ${ϵ}_{\mathrm{cut}}=\eta k_{B}T$. As the gas cools, a fixed threshold is populated by fewer atoms, so evaporation stalls. In forced evaporative cooling the threshold is lowered during the sequence. In a magnetic trap, a radiofrequency evaporation knife flips trapped weak-field-seeking atoms into untrapped spin states where the Zeeman splitting matches the photon energy. Energetic atoms reach that resonance surface; sweeping the radiofrequency lowers the energy knife. In an optical dipole trap, lowering the trap depth offers a different implementation, although it usually also weakens confinement.

A simple evaporative cooling energy balance shows when atom loss is useful. For a classical gas in a three-dimensional harmonic trap of fixed frequencies, equipartition gives $E=3N{k}_{B}T$, counting both kinetic and potential energy. Suppose escaping atoms remove average energy ${ϵ}_{\mathrm{ev}}=(\eta +\kappa )k_{B}T$, with $\kappa$ a distribution-dependent excess above the threshold. For $dN<0$,Thus preferential removal cools when the removed energy exceeds $3k_{B}T$. More stringently, cooling must raise the phase-space density of an atomic gas, not merely lower the temperature:For this harmonic-trap phase-space density, with fixed geometric-mean trap frequency $\bar{\omega }$, $d\ln \mathcal{D}/d\ln N=1-3\alpha$. Therefore phase-space density rises during atom loss if $\alpha >1/3$. The numerical factors here belong to the stated harmonic-trap model, not every evaporative apparatus.

There must be enough elastic collisions to repopulate the tail: ${\Gamma }_{\mathrm{el}}\sim n\sigma \bar{v}$ should exceed damaging loss and heating rates, and the ramp should allow rethermalization. Increasing density can improve elastic collisions but also enhances inelastic and three-body losses. Very rapid removal can leave a nonequilibrium cloud; very slow removal exposes it to background-gas losses. Lowering an optical trap can reduce density and collision rate, so a falling temperature is not by itself evidence of efficient evaporation.

The dilute-gas condensation experiments made this tradeoff concrete. Eric A. Cornell and Carl Wieman's account explains how selective escape and contraction of a trapped cloud raise phase-space density, while the need for rapid two-body equilibration and slow three-body loss restricts usable density. It also describes expansion imaging as an experimental diagnostic.^[ref] Useful controls compare different knife ramps and hold times, measure both atom number and expansion temperature, and check trap-frequency calibration. Pure adiabatic expansion can cool without increasing phase-space density. Successful evaporation pays an atom-number cost to increase the remaining gas's phase-space occupancy.

A two-photon transition transfers population through two interactions with the electric-dipole operator. For absorption of two photons,The intermediate atomic states enter a second-order amplitude rather than necessarily being populated. Suppressing field-normalization factors, its structure isBoth 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 $I_1{I}_2$, 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 $J=0\to 0$ transition and excludes $J=0\to 1$ 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 $\Delta L=0,\pm 2$, 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: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 obeysCounterpropagation now enhances momentum transfer rather than cancelling it. With single-photon Hamiltonian matrix elements $\hslash {\Omega }_i/2$ and a common large angular detuning $\Delta$, eliminating the excited state givesThe 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.

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.^[ref] Testing this interpretation requires scanning the two-photon detuning, controlling relative laser phase and checking output populations against pulse timing. A two-photon process is defined by its coherent two-interaction pathway; absorption-absorption and absorption-emission have different resonance and recoil conditions.

At a carbon-13 resonance frequency of $75\mathrm{MHz}$, the three resolved NMR coupling constants areEach is the spacing of adjacent lines within its own quartet. The lower panel of the original coupling diagram in 1(b)(ii) marks the corresponding carbon–fluorine interactions. The bond superscripts count bonds along the connecting path; they are not additional numerical factors multiplying $J$.

Use the D3h molecular point group for phosphorus pentafluoride, with $z$ along the two axial P–F bonds. For each symmetry operation, only atoms left in place contribute to the character of the Cartesian displacement representation; each contributes the trace of the corresponding $3\times 3$ spatial matrix. In the class order $E,2C_3,3C_2^{\prime },{\sigma }_h,2S_3,3{\sigma }_v$ this givesFor example a $C_2^{\prime }$ fixes P and one equatorial F, and each fixed atom has Cartesian trace $-1$, giving $-2$. reflection in the equatorial plane fixes four atoms, each with trace $1$, giving $4$. Reduction with the character table givesSubtract translations $E^{\prime }+A_2^{\prime \prime }$ and rotations $A_2^{\prime }+E^{\prime \prime }$ to obtainIts dimension is $2+6+2+2=12=3(6)-6$, as required for a nonlinear six-atom molecule. Thus there are eight symmetry-distinct fundamental frequencies in the harmonic picture, with each E mode doubly degenerate. This analysis concerns vibrations about one trigonal-bipyramidal minimum; it does not treat large-amplitude rearrangements between minima.

Let $T_{v^{\prime },v^{\prime \prime }}=T_e+G^{\prime }(v^{\prime })-G^{\prime \prime }(v^{\prime \prime })$. For a fixed row, moving one column to the right therefore givesThus adjacent-column entries decrease by the lower-state vibrational spacing, independent of the upper-state row in the ideal model. If the positive difference is instead defined as left minus right, it is ${\tilde{\omega }}_e^{\prime \prime }-2{\tilde{\omega }}_e^{\prime \prime }{x}_e^{\prime \prime }(v^{\prime \prime }+1)$. Stating the subtraction direction prevents an artificial negative vibrational frequency.

Taking the Hessian of the potential with respect to mass-weighted displacements givesThe equations of motion are $\ddot{\mathbf{q}}=-D_0\mathbf{q}$, so eigenvalues are squared angular frequencies. Normalized eigenvectors and frequencies areBoth vectors are in the q basis, and $\mu =m_a{m}_b/(m_a+m_b)$. The translation corresponds to equal physical displacements $\delta y_a=\delta y_b$. The stretch has $m_a\delta y_a+m_b\delta y_b=0$ and opposite physical displacements with magnitude ratio $|\delta y_a/\delta y_b|=m_b/m_a$. Ordinary frequencies are $\nu =\omega /(2\pi )$; spectroscopic wavenumbers are $\omega /(2\pi c)$.