AMO Experiment Index

A silver deposit, a current–voltage trace, and a coincidence histogram reveal different aspects of quantum physics. This guide connects landmark AMO measurements to their physical interpretation: what was varied, what was detected, and which conclusion follows. The dates identify selected reports or developments, rather than the beginning of an entire field.

Helpful background. The AMO overview introduces the atomic, molecular, and optical systems compared here. The historical experiment index provides a broader chronology.

Landmark AMO experiments

Date	Experiment and measured signal	What to read next
1914	Franck–Hertz: collector current versus accelerating voltage	Discrete excitation energies
1922	Stern–Gerlach: spatial splitting of a silver beam	Magnetic-moment outcomes
1947	Lamb–Retherford: loss of metastable hydrogen at resonance	Hydrogen level splitting
1950–1955	Ramsey spectroscopy and a practical cesium clock: atomic resonance used to stabilize an oscillator	Atomic clocks
1954–1956	Hanbury Brown–Twiss: correlated detector currents	Intensity correlations
1975–1989	Laser-cooling proposals and atomic demonstrations: narrower velocity or motional distributions	Cooling mechanisms
1982–2017	Bell tests: joint outcomes at different analyzer settings	Bell experiments
1987	Hong–Ou–Mandel: output coincidences versus input delay	Two-photon interference
1995	Dilute-gas condensation: a new component in expanding-cloud images	Bose–Einstein condensation
2000s onward	Optical clocks: excitation fractions, frequency ratios, and systematic shifts	Optical frequency standards
2010	Molecular laser cooling: reduced transverse velocity spread of SrF	Cold molecules

Discrete Atomic Outcomes and Levels

Stern–Gerlach

An inhomogeneous magnetic field exerts a force on a magnetic dipole. In a one-dimensional schematic,

$$F_z \simeq \mu_z \frac{\partial B_z}{\partial z}.$$

Gerlach and Stern sent a collimated beam of neutral silver atoms through such a field and recorded the deposited beam on a plate. The beam split into distinct spatial components rather than merely broadening into a continuum. The robust detector-level statement is therefore about discrete deflections, given the field calibration and beam geometry.

The modern interpretation uses the valence-electron spin of ground-state silver. That interpretation is historically retrospective: the 1922 experiment preceded the discovery of electron spin. It is also incomplete to say that the device simply “measures spin.” The force couples to a magnetic moment, the spatial wave packets separate through unitary dynamics, and the downstream plate supplies an irreversible record. The canonical historical page reconstructs the original context; Stern–Gerlach Revisited develops the modern spin-analyzer model; and the reference card gives a short lookup.

What it supports: discrete outcomes for the magnetic-moment component in the prepared beam and apparatus.

What it does not show by itself: the full spin-$1/2$ algebra, state collapse as a unique dynamical law, or that a spin possessed a definite pre-existing value along every possible axis.

Franck–Hertz

Franck and Hertz accelerated electrons through mercury vapor and measured a collector current while varying the accelerating voltage. Repeated structures in the current appeared because electrons that reached a characteristic kinetic energy could lose energy in inelastic collisions, leaving too little kinetic energy to overcome the collector-region potential. The energy fingerprint is

$$e\,\Delta V \simeq \Delta E,$$

with corrections for contact potentials, the electron energy distribution, collisions, and tube geometry.

A voltage spacing near $4.9\,\mathrm{V}$ corresponds to an excitation energy near $4.9\,\mathrm{eV}$. The associated photon wavelength,

$$\lambda = \frac{hc}{\Delta E},$$

is about $253\,\mathrm{nm}$, consistent with a strong mercury resonance line. The agreement connects an electron-collision energy loss to optical spectroscopy. The current minima are not direct images of electrons jumping between Bohr orbits, and a single tube trace does not by itself identify every collision channel.

Use the Franck–Hertz historical page for apparatus logic and historical interpretation, or the compact reference card for lookup.

Lamb–Retherford

The Dirac Coulomb spectrum makes the hydrogen $2S_{1/2}$ and $2P_{1/2}$ levels degenerate. Lamb and Retherford prepared metastable $2S$ hydrogen atoms, applied radio-frequency fields, and monitored the surviving metastable beam. Driving population into the short-lived $2P$ state caused rapid radiative decay and reduced the metastable count:

$$h\nu_{\mathrm{res}} \simeq E(2S_{1/2}) - E(2P_{1/2}),$$

after accounting for hyperfine structure, external fields, line shape, and apparatus calibration.

The resonance established a nonzero level interval. Quantum electrodynamics explains that interval through radiative and recoil corrections, but the detector did not directly observe a virtual photon, vacuum fluctuation, or electron self-energy. Those are elements of a successful theoretical organization of measured energies. The Lamb Shift Overview separates the experiment from the calculation and links onward to the QED treatment.

Optical Correlations and Two-Photon Interference

Hanbury Brown–Twiss

An HBT experiment correlates the outputs of two detectors. For stationary classical light, the normalized intensity correlation is

$$g_{12}^{(2)}(\tau) = \frac{ \left\langle I_1(t)I_2(t+\tau) \right\rangle }{ \left\langle I_1\right\rangle \left\langle I_2\right\rangle }.$$

In quantum photodetection, the corresponding expression uses normally ordered field operators. Two-detector correlations avoid the same-detector shot-noise term; finite response times still smear the measured curve.

The original stellar intensity interferometer extracted angular-source information from spatial intensity correlations at separated collectors. Laboratory versions demonstrated temporal photon bunching. Modern single-emitter measurements can instead show antibunching, $g^{(2)}(0)<g^{(2)}(\tau)$ near zero delay, after correcting for background and detector response.

The record is a pair of current traces or time tags. Converting those records to $g^{(2)}$ requires a normalization, bin width, live-time model, timing response, accidental-coincidence treatment, and uncertainty estimate. Classical wave theory explains thermal-light bunching; normally ordered quantum photodetection theory covers both classical and nonclassical fields. The canonical Hanbury Brown–Twiss page develops both descriptions.

Hong–Ou–Mandel

Hong–Ou–Mandel interference sends one photon into each input of a balanced beam splitter and scans their relative delay. For pure, otherwise ideal single-photon wave packets $\lvert\psi_1\rangle$ and $\lvert\psi_2\rangle$, the coincidence probability is

$$P_{\mathrm c}(0) = \frac{1}{2} \left( 1- \left| \langle\psi_1\vert\psi_2\rangle \right|^2 \right).$$

Perfect overlap suppresses coincidences because the two indistinguishable alternatives leading to one photon in each output interfere destructively. Changing arrival time, polarization, spectrum, or spatial mode reduces the overlap and restores coincidences. Loss, multiphoton emission, detector effects, and an unbalanced beam splitter change the measured visibility.

For independent mixed one-photon states, the ideal result becomes $P_{\mathrm c}=[1-\operatorname{Tr}(\rho_1\rho_2)]/2$. If both sources prepare the same density operator, the dip visibility relative to fully distinguishable photons equals $\operatorname{Tr}\rho^2$. Thus purity can be inferred with these additional assumptions, while a generic dip does not determine either input state.

The effect is not an attractive interaction between photons, and the photons need not meet at a localized point. A high visibility certifies overlap only within a stated source and measurement model; it does not alone establish source purity, entanglement, or indistinguishability in every unmeasured degree of freedom. See Beam Splitters for the canonical mode transformation.

HBT versus Hong–Ou–Mandel

Feature	Hanbury Brown–Twiss	Hong–Ou–Mandel
typical input	one field split between detectors, or one source viewed at two collectors	one excitation in each input mode
scan variable	detector delay or collector baseline	relative input delay or another distinguishability coordinate
central observable	normalized intensity correlation $g^{(2)}$	output coincidence probability or visibility
central question	how are detection events correlated within a field?	how well do two input wave packets overlap and interfere?
familiar feature	bunching, Poissonian behavior, or antibunching	coincidence dip at maximal overlap
not established automatically	full field state or a uniquely quantum origin for thermal bunching	full source purity, entanglement, or universal indistinguishability

Both use coincidence electronics, but the state preparation and inferred quantity differ. “A coincidence dip” is not enough information to identify which experiment was performed.

Laser Cooling Milestones

Laser cooling is a sequence of experimentally distinct achievements, not one temperature record. The basic Doppler force relies on velocity-dependent absorption followed by nearly isotropic spontaneous emission. For an ideal two-level transition in the low-intensity Doppler model, the characteristic limit is

$$T_{\mathrm D} = \frac{\hbar\Gamma}{2k_{\mathrm B}},$$

where $\Gamma$ is the excited-state population-decay rate in $\mathrm{s}^{-1}$ (also the natural angular-frequency linewidth). Multilevel structure, polarization gradients, recoil, confinement, and sideband resolution lead to other limits and mechanisms.

Milestone	System and control	Experimental signature	What changed
1975 proposals	counterpropagating, red-detuned light acting on moving atoms	theoretical velocity-dependent radiation pressure	established the Doppler-cooling principle
1978 trapped ions	bound ions repeatedly scattering near-resonant photons	narrowed fluorescence or motional response	demonstrated radiation-pressure cooling of trapped absorbers
1985 optical molasses	neutral sodium in three orthogonal beam pairs	long residence and reduced velocity spread	demonstrated three-dimensional viscous damping without conservative trapping
1987 magneto-optical trap	circularly polarized beams plus position-dependent Zeeman shifts in a magnetic-field gradient	localized fluorescent atom cloud	combined damping with a restoring force
1988 sub-Doppler atoms	polarization-gradient cooling of sodium	temperatures below the two-level Doppler prediction	exposed the role of multilevel optical pumping and light shifts
1989 motional ground state	trapped ion in the resolved-sideband regime	sideband asymmetry and suppressed motional excitation	reached near-zero-point motion in a selected trap mode
2010 molecular cooling	a molecule with engineered quasi-closed optical cycling	narrowed transverse velocity distribution of a SrF beam	extended direct optical cooling beyond atoms

Temperature is inferred from a model of motion, not read directly from fluorescence brightness. Common thermometry methods include time of flight, release and recapture, recoil-sensitive spectra, and red/blue motional sideband asymmetry. Each probes a distribution and has a validity regime.

Use Laser Cooling for the force picture, Doppler Cooling for the two-level limit, Sub-Doppler Cooling for polarization-gradient mechanisms, Magneto-Optical Traps for capture and confinement, and Cold Molecules for optical-cycle closure in molecules.

Bose–Einstein Condensation in Dilute Gases

The 1995 rubidium and sodium experiments combined laser cooling, evaporative cooling, magnetic trapping, resonant absorption imaging, and statistical modeling. A useful degeneracy parameter is the phase-space density

$$\mathcal D = n\lambda_{\mathrm{dB}}^3, \qquad \lambda_{\mathrm{dB}} = \frac{h}{\sqrt{2\pi m k_{\mathrm B}T}}.$$

The numerical transition criterion depends on geometry and interactions. For an ideal three-dimensional harmonic trap with geometric-mean angular frequency $\bar\omega$, the excited-state capacity is approximately

$$N_{\mathrm{ex}}^{\max} \simeq \zeta(3) \left( \frac{k_{\mathrm B}T}{\hbar\bar\omega} \right)^3.$$

It is therefore misleading to apply the homogeneous-gas criterion $n\lambda_{\mathrm{dB}}^3=\zeta(3/2)$ unchanged to every trapped cloud.

The first experiments found a narrow, high-density component in time-of-flight density images, onset near the predicted critical conditions, and anisotropic expansion consistent with the trapped condensate dynamics. Expansion maps the initial momentum distribution only when ballistic propagation is a good approximation; interactions can convert interaction energy into kinetic energy and change the image. Later interference, collective-mode, coherence, and correlation measurements probed complementary properties. A bimodal fit is powerful when the imaging response, expansion dynamics, finite-size effects, and interaction corrections are controlled; an unexplained narrow camera feature is not by itself proof of condensation.

The BEC Overview explains preparation and diagnosis in trapped atoms. The historical page connects the 1920s statistics to dilute-gas experiments, the many-body page derives ideal-gas condensation, and the reference card gives a compact milestone summary.

Atomic Clock Milestones

An atomic clock is a feedback system. An atom supplies a reproducible transition; a local oscillator interrogates it; a detector estimates an excitation probability; and a servo steers the oscillator.

Ramsey’s separated-oscillatory-field method narrowed the central spectroscopic feature without requiring one uninterrupted long pulse. Essen and Parry’s 1955 cesium apparatus established the feasibility of an atomic frequency standard. Cesium fountain clocks later lengthened interrogation times and controlled Doppler and collisional shifts. Optical ion and lattice clocks moved to much higher carrier frequencies while frequency combs made optical-to-microwave and optical-to-optical comparison practical.

Two basic figures of merit are

$$Q = \frac{\nu_0}{\Delta\nu}, \qquad y(t) = \frac{\nu(t)-\nu_0}{\nu_0}.$$

$Q$ describes spectroscopic resolving power. Frequency stability describes how fluctuations average with time, often through an Allan deviation. Accuracy, more precisely realized systematic uncertainty and bias relative to the definition or another standard, requires a shift budget. A narrow line alone guarantees neither stability nor accuracy.

As of this page’s review date, the SI second is defined by fixing the unperturbed ground-state hyperfine transition frequency of cesium-133 to exactly

$$\Delta\nu_{\mathrm{Cs}} = 9\,192\,631\,770\ \mathrm{Hz}.$$

Exactness belongs to the defining constant, not to every physical realization. Real clocks still evaluate blackbody, Zeeman, Stark, collisional, Doppler, gravitational, servo, and other shifts. Optical standards have surpassed cesium fountains in several performance measures, and the BIPM maintains a roadmap toward a possible redefinition of the second, potentially around 2030. That redefinition had not occurred at the 2026 review date.

Use Atomic Clocks for the measurement loop and timescale concepts, Optical Clocks for ion and lattice architectures, and Frequency Standards for stability statistics and uncertainty budgets.

Bell Tests Across AMO Platforms

For binary outcomes $A_x,B_y\in\{-1,+1\}$ at settings $x,y\in\{0,1\}$, define the CHSH combination

$$S = E_{00} + E_{01} + E_{10} - E_{11}.$$

Local hidden-variable models satisfying the CHSH assumptions obey

$$\left|S\right| \leq 2,$$

whereas quantum theory permits values up to $2\sqrt{2}$. A Bell experiment does not measure “nonlocality” with one detector. It accumulates joint outcomes for several setting pairs, defines the trial sample, estimates correlators, and evaluates a statistical test while controlling relevant loopholes.

Platform comparison

Experiment	Main achievement	Remaining qualification
Aspect–Dalibard–Roger, photons (1982)	polarization correlations with rapidly changing analyzers at separated detectors	inefficient detection required sampling assumptions; analyzer switching was periodic
Giustina and Shalm groups, photons (2015)	efficient detectors and fast random settings closed the major detection and locality loopholes together	trial definitions, setting independence, and statistical assumptions remain necessary
Rowe and colleagues, ions (2001)	efficient fluorescence readout closed the detection loophole	ions were micrometers apart; measurement events were not spacelike separated
Rosenfeld and colleagues, neutral atoms (2017)	event-ready spin measurements over $398\,\mathrm m$ with efficient readout and spacelike separation	heralding must define trials without depending on later settings or outcomes

The platform changes the engineering balance. Photons travel well and permit fast remote measurements but can be lost. Trapped particles offer high-efficiency readout but are harder to separate and interrogate within a spacelike window. Event-ready protocols define a trial through a herald generated before the measurement choices, reducing selection ambiguity.

A statistically significant violation excludes the tested class of local hidden-variable explanations under the stated assumptions. It does not enable faster-than-light signaling, prove that every conceivable hidden- variable theory is impossible, or determine a unique interpretation of quantum mechanics. For the theorem and its assumptions, use Bell’s Theorem and the CHSH Inequality. For the experimental arc, use Bell-Inequality Experiments and Loophole-Free Bell Tests.

Exercises

1. Franck–Hertz energy scale

Successive features in a mercury Franck–Hertz trace are separated by $4.90\,\mathrm V$. Estimate the excitation energy in electronvolts and the vacuum wavelength of a photon with that energy. Use $hc=1239.841984\,\mathrm{eV\,nm}$.

Solution

For a singly charged electron,

$$\Delta E \simeq e\Delta V = 4.90\,\mathrm{eV}.$$

The corresponding wavelength is

$$\lambda = \frac{1239.841984\,\mathrm{eV\,nm}} {4.90\,\mathrm{eV}} \simeq 253\,\mathrm{nm}.$$

This conversion checks the energy scale against optical spectroscopy. A precision analysis must additionally treat contact potentials, voltage calibration, electron-energy spread, and the relevant mercury line.

2. Identifying condensation

A trapped gas develops a narrow central feature in an absorption image after time of flight. List three checks needed before interpreting it as a condensate and one complementary measurement that strengthens the interpretation.

Solution

Useful checks include:

1. calibrating imaging saturation, resolution, optical depth, and camera response;
2. fitting a thermal-plus-condensed distribution with the appropriate trap geometry and expansion model; and
3. verifying onset versus atom number and temperature near the predicted critical phase-space density while checking finite-size and interaction effects.

A complementary test could measure matter-wave interference, first-order coherence, collective-mode frequencies, anisotropic expansion, or correlations. No single item replaces a complete consistency analysis.

3. Clock quality factor

An optical clock interrogates a transition at $\nu_0=429\,\mathrm{THz}$ and resolves a $1.00\,\mathrm{Hz}$ FWHM feature. Find $Q$. Explain why this number does not give either the clock’s fractional accuracy or its one-second stability.

Solution

The quality factor is

$$Q = \frac{429\times10^{12}\,\mathrm{Hz}} {1.00\,\mathrm{Hz}} = 4.29\times10^{14}.$$

$Q$ describes the carrier frequency relative to a spectroscopic width. Short-term stability also depends on signal-to-noise ratio, atom number, cycle time, local-oscillator noise, dead time, and servo design. Accuracy requires a systematic-shift evaluation and uncertainty budget. Neither is fixed by linewidth alone.

4. Comparing Bell platforms

Classify the principal experimental strength and unresolved limitation in each case: high-efficiency detection of two ions a few micrometers apart; fast polarization measurements of distant photons with low detector efficiency; and event-ready measurements of neutral atoms $398\,\mathrm m$ apart with efficient readout.

Solution

The nearby-ion experiment strongly addresses the detection loophole through efficient state readout, but the measurement events are not spacelike separated, so it does not close the locality loophole.

The distant-photon experiment can enforce strong spacelike separation and rapid setting changes, but low detection efficiency introduces a fair- sampling or detection-loophole concern unless the protocol and efficiency threshold address it.

The event-ready neutral-atom experiment combines a heralded trial, high-efficiency readout, and distant, spacelike-separated choices and detections. Its conclusion is still conditional on the trial definition, setting-independence premise, causal timing, instrument model, and statistical test.

Further reading

The AMO bibliography recommends textbooks for the theory behind these measurements. The original reports below are useful for the experimental conditions and historical claims.

References

• M. H. Anderson et al., “Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor,” Science 269, 198–201 (1995), doi:10.1126/science.269.5221.198.
• A. Aspect, J. Dalibard, and G. Roger, “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,” Physical Review Letters 49, 1804–1807 (1982), doi:10.1103/PhysRevLett.49.1804.
• Bureau International des Poids et Mesures, “SI base unit: second” and “Roadmap to the redefinition of the second”, accessed 14 October 2026.
• K. B. Davis et al., “Bose–Einstein Condensation in a Gas of Sodium Atoms,” Physical Review Letters 75, 3969–3973 (1995), doi:10.1103/PhysRevLett.75.3969.
• F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser Cooling to the Zero-Point Energy of Motion,” Physical Review Letters 62, 403–406 (1989), doi:10.1103/PhysRevLett.62.403.
• L. Essen and J. V. L. Parry, “An Atomic Standard of Frequency and Time Interval: A Cæsium Resonator,” Nature 176, 280–282 (1955), doi:10.1038/176280a0.
• J. Franck and G. Hertz, “Über Zusammenstöße zwischen Elektronen und den Molekülen des Quecksilberdampfes und die Ionisierungsspannung desselben,” Verhandlungen der Deutschen Physikalischen Gesellschaft 16, 457–467 (1914), original paper.
• W. Gerlach and O. Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld,” Zeitschrift für Physik 9, 349–352 (1922), doi:10.1007/BF01326983.
• M. Giustina et al., “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Physical Review Letters 115, 250401 (2015), doi:10.1103/PhysRevLett.115.250401.
• R. Hanbury Brown and R. Q. Twiss, “A New Type of Interferometer for Use in Radio Astronomy,” Philosophical Magazine 45, 663–682 (1954), doi:10.1080/14786440708520475.
• R. Hanbury Brown and R. Q. Twiss, “Correlation between Photons in Two Coherent Beams of Light,” Nature 177, 27–29 (1956), doi:10.1038/177027a0. Here “coherent” refers to mutual coherence of the split beams, not a Glauber coherent-state preparation.
• T. W. Hänsch and A. L. Schawlow, “Cooling of Gases by Laser Radiation,” Optics Communications 13, 68–69 (1975), doi:10.1016/0030-4018(75)90159-5.
• C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of Subpicosecond Time Intervals between Two Photons by Interference,” Physical Review Letters 59, 2044–2046 (1987), doi:10.1103/PhysRevLett.59.2044.
• W. E. Lamb Jr. and R. C. Retherford, “Fine Structure of the Hydrogen Atom by a Microwave Method,” Physical Review 72, 241–243 (1947), doi:10.1103/PhysRev.72.241.
• P. D. Lett et al., “Observation of Atoms Laser Cooled below the Doppler Limit,” Physical Review Letters 61, 169–172 (1988), doi:10.1103/PhysRevLett.61.169.
• N. F. Ramsey, “A Molecular Beam Resonance Method with Separated Oscillating Fields,” Physical Review 78, 695–699 (1950), doi:10.1103/PhysRev.78.695.
• W. Rosenfeld et al., “Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes,” Physical Review Letters 119, 010402 (2017), doi:10.1103/PhysRevLett.119.010402.
• M. A. Rowe et al., “Experimental Violation of a Bell’s Inequality with Efficient Detection,” Nature 409, 791–794 (2001), doi:10.1038/35057215.
• L. K. Shalm et al., “Strong Loophole-Free Test of Local Realism,” Physical Review Letters 115, 250402 (2015), doi:10.1103/PhysRevLett.115.250402.
• E. S. Shuman, J. F. Barry, and D. DeMille, “Laser Cooling of a Diatomic Molecule,” Nature 467, 820–823 (2010), doi:10.1038/nature09443.