Cesium Fountain Clock: Atoms Tossed Upward to Define the Second

One Toss, Five Steps

A fountain clock runs the same ballistic cycle roughly once per second, forever. Each cycle has five acts.

• Collect and cool. Six laser beams, arranged as three orthogonal counter-propagating pairs and tuned a few linewidths below the cesium D2 line at 852.35 nm (6 ²S_1/2 → 6 ²P_3/2, natural linewidth Γ/2π ≈ 5.2 MHz), capture 10⁶–10⁸ atoms out of a room-temperature vapour or a slowed beam. Doppler cooling alone would stall at T_D = ħΓ/2k_B ≈ 125 µK; polarization-gradient (Sisyphus) cooling in the molasses pushes the ball down to ~1–2 µK, within a factor of ten of the 198 nK single-photon recoil limit.
• Launch. The three upward beams are detuned above the three downward beams by a few MHz, so the velocity at which an atom sees zero net Doppler shift is no longer zero. The whole ball is dragged upward; then every laser is switched off.
• Select a single state. A low-Q cavity plus a resonant push beam strips away everything except atoms in the single sublevel |F = 3, m_F = 0⟩, whose transition frequency is immune to magnetic field to first order.
• Interrogate twice with one cavity. The ball flies up through a cylindrical copper TE₀₁₁ cavity driven near 9,192,631,770 Hz, receives a π/2 pulse, coasts to apogee, falls back, and re-enters the same cavity ~0.5 s later for a second π/2 pulse.
• Read out. Below the cavity the atoms drop through probe beams and fluoresce; the ratio of F = 4 to F = 3 atoms is the answer, and it steers the microwave source.

Nothing holds the atoms during the flight. From launch to detection the only force acting on them is gravity; the cloud simply drifts apart at its own residual thermal velocity spread.

Moving Molasses: How You Throw an Atom Without Touching It

The launch is the elegant part. For one counter-propagating pair of beams whose frequencies differ by Δν, the standing wave — and the cooling force's zero — travels along the beam axis at

v_axis = λΔν/2

because an atom moving at that speed sees both beams Doppler-shifted back into equality. Real fountains use the (1,1,1) geometry, in which each beam makes an angle θ with the vertical such that cos θ = 1/√3, so the vertical launch speed is v = √3 λΔν/2. With λ = 852 nm and Δν = 5 MHz that is 3.7 m/s. Acousto-optic modulators impose the shift, so the launch velocity is set by a number in a synthesizer — repeatable cycle after cycle.

Ballistics then does the rest. A 4 m/s launch reaches h = v²/2g = 0.82 m and returns to the launch height in 2v/g = 0.82 s. Because the Ramsey cavity sits above the molasses region, the interval between the two crossings is somewhat shorter: for a cavity 20 cm below apogee the atoms cross at ~2 m/s and T = 2v_cav/g ≈ 0.4 s; real fountains run T ≈ 0.4–0.6 s, i.e. an apogee 20–45 cm above the cavity. During that flight the cloud is also expanding. At 1 µK the per-axis rms speed is √(k_BT/m) ≈ 8 mm/s, so a millimetre-scale ball grows to roughly a centimetre by detection — which is why cavity and detection apertures, not the initial atom number, usually set how many atoms survive the round trip. Typically only 10–30% do.

Ramsey's Separated Fields, With Gravity Supplying the Baseline

Norman Ramsey's 1949–1950 method of separated oscillatory fields — the work that won him a share of the 1989 Nobel Prize in Physics — replaced one long microwave interaction with two short ones separated by a long field-free drift. The first π/2 pulse puts each atom into an equal superposition of |F = 3, m_F = 0⟩ and |F = 4, m_F = 0⟩. During the drift the atomic superposition accumulates phase at the true atomic frequency ν₀ while the microwave source accumulates phase at ν. The second pulse converts the accumulated phase difference into a population difference:

P(F = 4) = ½[1 + cos(2π(ν − ν₀)T + Δφ)]

This is an interferometer in which the two arms are internal states rather than paths. Its fringes are spaced by 1/T and the central fringe has full width

Δν ≈ 1/(2T)

so T = 0.5 s gives a 1 Hz line on a 9.19 GHz carrier — a line quality factor Q = ν₀/Δν ≈ 1 × 10¹⁰. A thermal-beam clock with a 1 m cavity separation and 200 m/s atoms gets T ≈ 5–10 ms, a ~60–100 Hz line, and Q ≈ 10⁸. Gravity buys the fountain two orders of magnitude for free, in a machine whose interrogation region is shorter than the beam tube it replaced.

The individual pulses are much shorter than T — a ~2 cm cavity crossed at ~2–3 m/s gives a ~6–10 ms interaction — so the sharp Ramsey fringes sit under a broad Rabi pedestal tens of hertz wide. Locking to the wrong fringe would be a 2 Hz error, so the servo is bootstrapped from the pedestal first.

Reading the Fringe and Closing the Loop

Detection is destructive and purely optical. Falling past the cavity, the cloud crosses a sheet of light resonant with the F = 4 → F′ = 5 cycling transition; each F = 4 atom scatters thousands of photons, and a photodiode collects fluorescence proportional to N₄. A push beam then blows those atoms away, a repumper transfers the F = 3 atoms up, and a second probe measures N₃. The clock uses the normalized quantity

P = N₄ / (N₃ + N₄)

which cancels shot-to-shot fluctuations in how many atoms were launched — the single most important trick for reaching quantum-projection-noise-limited operation.

The servo alternates the microwave frequency between the two half-maximum points of the central fringe, at ν₀ ± 1/(4T) ≈ ±0.5 Hz. The difference in P between successive cycles is an error signal that steers a synthesizer driven by a hydrogen maser or a cryogenic sapphire oscillator. The resulting instability follows

σ_y(τ) ≈ (1/πQ_line)√(T_c/Nτ)

With N ≈ 10⁶ detected atoms and a cycle time T_c ≈ 1.3 s this is a few × 10⁻¹⁴ at one second for the best fountains, and it averages down as τ^−1/2. Averaging down to the 10⁻¹⁶ level therefore takes of order a day in the best machines — and weeks in a fountain interrogated by a hydrogen maser, where the Dick effect degrades the short-term instability to the 10⁻¹³ level. Long continuous runs are exactly how a primary frequency standard is evaluated and reported to the BIPM for the calibration of International Atomic Time.

The Error Budget: Why the Last Two Digits Are Brutal

Since the 2019 revision of the SI, the second is defined by the unperturbed ground-state hyperfine frequency of a Cs-133 atom — at rest, at zero magnetic field, in the dark, at 0 K. Every real fountain therefore has to measure its own perturbations and subtract them. The dominant entries:

• Blackbody radiation shift. Room-temperature thermal photons AC-Stark-shift the clock transition by about −1.7 × 10⁻¹⁴ at 300 K, and the shift scales as T⁴. Its uncertainty is one of the largest entries in any room-temperature Cs fountain budget — the quadratic Zeeman shift below is a bigger correction, but the C-field can be mapped far more precisely than the effective wall temperature. NIST-F2 (2014) runs its flight tube near 80 K, where the shift collapses by (80/300)⁴ ≈ 0.5% to the 10⁻¹⁶ level — the main reason it improved on NIST-F1 by roughly a factor of three.
• Second-order Zeeman shift. A ~100 nT (1 mG) C-field inside layers of mu-metal defines the quantization axis and keeps the m_F = 0 clock transition resolved from its neighbours. The quadratic shift is 427.45 × 10⁸ B² Hz (B in tesla), or ~4 × 10⁻⁴ Hz — a fractional 4.6 × 10⁻¹⁴, applied as a correction after mapping the field with low-frequency Zeeman transitions.
• Cold-collision shift. Cesium's large ultracold scattering length makes the density-dependent shift unusually big, reaching 10⁻¹³ at high atom number. Fountains beat it by alternating between two well-controlled densities and extrapolating to zero.
• Distributed cavity phase. Power dissipated in the lossy cavity walls and lost through the endcap cutoffs forces a small travelling-wave component onto the standing wave, so the microwave phase varies across the aperture and the two-crossing cancellation is imperfect if the atoms sample different transverse positions going up and coming down. Evaluated by tilting the fountain and varying the microwave amplitude, it is a 10⁻¹⁷–10⁻¹⁶ entry.
• Gravitational redshift. A clock at NIST's Boulder site, 1650 m above the geoid, runs fast by gh/c² ≈ 1.8 × 10⁻¹³ — a thousand times the clock's own uncertainty. Every reported value is corrected to the geoid, so a fountain's accuracy is now partly limited by how well the local geopotential is known.

Also in the budget, and smaller: background-gas collisions at ~10⁻⁸ Pa, microwave leakage outside the cavity, Rabi and Ramsey pulling by neighbouring m_F lines, Majorana transitions in field gradients, and the second-order Doppler (time-dilation) shift, only ~2 × 10⁻¹⁷ at these launch speeds.

A Sixty-Year Lineage, From a Failure at MIT

Jerrold Zacharias proposed the fountain at MIT around 1953, hoping to select the slowest atoms out of the Maxwell–Boltzmann tail of a thermal cesium oven and toss them a metre upward. It failed: the expected flux of slow atoms was not there, because slow atoms leaving the source were scattered out by the faster ones behind them. The idea had to wait for a way to make slow atoms rather than filter for them.

Laser cooling supplied it. In 1989 Mark Kasevich, Erling Riis and Steven Chu at Stanford demonstrated radio-frequency Ramsey spectroscopy in a laser-cooled sodium fountain, achieving linewidths far below any beam machine. André Clairon's group at LPTF (now SYRTE) at the Paris Observatory, with Christophe Salomon, saw the first Ramsey fringes in a cesium fountain in 1991 and built it into the first fountain frequency standard, running in 1994 and evaluated at ~3 × 10⁻¹⁵ in 1995 — already better than the best beam clocks. NIST-F1 followed in 1999, replacing NIST-7; NIST-F2, with its cryogenically shielded flight tube, was announced in April 2014 at ~1.1 × 10⁻¹⁶.

Roughly ten fountains worldwide now serve as primary frequency standards — SYRTE FO1/FO2/FOM, PTB CSF1 and CSF2, NPL-CsF2, INRIM IT-CsF2, NIM5, NICT and others — and their reports steer TAI. SYRTE's FO2 runs cesium and rubidium fountains in the same vacuum can, since the ⁸⁷Rb hyperfine frequency (6,834,682,610.904 312 Hz) is an accepted secondary representation of the second with a collision shift about fifty times smaller than cesium's.

Gravity is not negotiable in a fountain, which is why the space version looks different: PHARAO, the CNES-built cesium clock inside ESA's ACES payload flown to the ISS in April 2025, sends a slow (~5 cm/s) laser-cooled cesium beam through a Ramsey cavity with two interaction zones in microgravity to obtain a long T without any toss at all.

Four Things People Get Wrong

“The atoms are trapped while they are measured.” They are not, and they must not be. A magneto-optical or dipole trap would AC-Stark-shift the hyperfine transition by orders of magnitude more than the clock's uncertainty. Lasers and quadrupole are extinguished before launch; interrogation happens in free fall, in the dark, with only the ~100 nT bias field present.

“There are two cavities, one at each end.” A thermal-beam clock does, and pays for it: any phase difference between the two is indistinguishable from an atomic frequency offset, historically a 10⁻¹³-level systematic. The fountain crosses the same cavity, driven by the same field, twice, so that phase difference cancels to first order; only residual transverse phase structure survives.

“9,192,631,770 Hz is a measured value.” It was measured once, against ephemeris time, and then frozen: the 13th CGPM adopted it as the definition in 1967, the CIPM added “at rest at 0 K” in 1997, and the 2019 SI makes Δν_Cs a defining constant with zero uncertainty. A fountain does not measure the second; it realizes it, and its accuracy figure is the uncertainty in that realization.

“Optical clocks already define the second.” They do not. Strontium and ytterbium lattice clocks and Al⁺ ion clocks are already more than two orders of magnitude more accurate — ~8 × 10⁻¹⁹ for JILA's Sr lattice, 9.4 × 10⁻¹⁹ for NIST's Al⁺ quantum-logic clock — but they are secondary representations, and every optical frequency must still be tied back to cesium to be expressed in SI hertz. The CCTF roadmap targets a redefinition around 2030, contingent on independent optical species agreeing at the 10⁻¹⁸ level and on reliable intercontinental clock comparisons. Until then, the second is still a ball of cold cesium going up and coming down.

Cesium fountains against the beam clocks they replaced and the optical clocks that outperform them

Standard	Interrogation time T	Linewidth / line Q	Fractional accuracy
Thermal Cs beam (NIST-7, 1993–1999)	~5–10 ms across ~1 m at ~200 m/s	~60–100 Hz / ~10⁸	~5 × 10⁻¹⁵
Cs fountain (LPTF/SYRTE FO1, 1995)	~0.5 s ballistic	~1 Hz / ~10¹⁰	~3 × 10⁻¹⁵ (first evaluation)
Cs fountain (NIST-F1, 1999→)	~0.5 s ballistic	~1 Hz / ~10¹⁰	~3 × 10⁻¹⁶ (mature evaluations)
Cs fountain (NIST-F2, 2014)	~0.5 s, flight tube near 80 K	~1 Hz / ~10¹⁰	~1.1 × 10⁻¹⁶
Sr optical lattice (JILA, 2022–2024)	~1–10 s in a magic-wavelength lattice	~0.1–1 Hz / ~10¹⁵	~8 × 10⁻¹⁹ (secondary representation)
Al⁺ quantum-logic ion (NIST, 2019)	~0.15–1 s on a single ion	sub-Hz / ~10¹⁵	9.4 × 10⁻¹⁹ (does not define the second)