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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.

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.