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 Incoming links: Laser cooling

Cold-atom measurement of the gravitational constant 2026-10-14

In 2014, Gabriele Rosi and colleagues used atom interferometry: two rubidium-87 clouds prepared by laser cooling in vacuum,

328 mm

apart, received shared three-pulse Raman transitions spaced by

160 ms

. Fluorescence readout and paired-fringe ellipse fitting yielded differential wave phase. Twenty-four tungsten alloy cylinders, about

516 kg

total, moved between C and F. Cloud subtraction rejected shared vibration; configuration subtraction isolated the source change.^[ref]

Figure 1.
Cold-atom measurement: two clouds and two source configurations
. Original conceptual scheme; selected cylinders represent surrounding arrays. Geometry is schematic.^[ref] **Details**.

Calibration used timed flight and light-sheet crossings; mirror tilt compensated Earth rotation. Wave vector reversal rejected unwanted shifts. Modeling included source Newtonian gravitational potential, atom trajectories, cloud distributions and moving supports. Source positioning, mass density variation and cloud geometry limited the reported combined relative standard uncertainty to

150

parts per million.^[ref]

To understand the subtraction without reproducing that full analysis, consider an ideal atom interferometer in uniform vertical acceleration during each pulse sequence. Let

k_{eff}

be its signed effective vertical wave vector component and

T

the pulse spacing. Its acceleration-dependent wave phase is

ϕ ≃ k_{eff} g T^{2}

; labels U and L denote upper and lower clouds. The double difference is then

S = (ϕ_{U} - ϕ_{L})_{C} - (ϕ_{U} - ϕ_{L})_{F} ≃ k_{eff} T^{2} [(g_{U} - g_{L})_{C} - (g_{U} - g_{L})_{F}] .

(1)

A background field identical in C and F cancels even if it differs between the two clouds. In a fixed-geometry, linear-response model, write

S = G K

, where

K

is calculated from the source mass density, calibrated pulse sequence and atom trajectories. Thus

G_{inferred} = S / K

. Measuring wave phase accurately is one part of the task; calculating

K

accurately is another. The real nonuniform field and finite clouds require the trajectory-dependent analysis described above rather than this uniform-field approximation.

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Doppler cooling limit 2026-10-14

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Damping removes random kinetic energy, but the discrete photon kicks cause diffusion in momentum. Their balance gives a nonzero temperature. Define the momentum diffusion coefficient

D_{p}

by the noise contribution

d ⟨ p_{x}^{2} ⟩ / d t = 2 D_{p}

. With atom mass

m

and linear damping coefficient

α

\frac{d ⟨ p _{x}^{2} ⟩}{d t} = - \frac{2 α}{m} ⟨ p_{x}^{2} ⟩ + 2 D_{p}, k_{B} T = \frac{⟨ p _{x}^{2} ⟩}{m} = \frac{D _{p}}{α}

(1)

at stationary balance, where

k_{B}

is the Boltzmann constant and mean momentum is zero.

For ideal three-dimensional optical molasses with three identical independent beam pairs and isotropic spontaneous emission, the

x

-pair absorption rate at rest is

R_{x} = Γ s / (1 + 4 d^{2})

. Absorption from that pair contributes

(ℏ k)^{2} R_{x}

to the noise growth of

⟨ p_{x}^{2} ⟩

. Emission from all three pairs contributes the same amount: its total rate is

3 R_{x}

and each emitted photon has mean squared

x

-momentum

(ℏ k)^{2} /3

. Hence

D_{p} = (ℏ k)^{2} R_{x}

. Using the Doppler cooling force coefficient gives

T (d) = \frac{ℏΓ}{8 k _{B}} \frac{1 + 4 d ^{2}}{- d}, d < 0.

(2)

Put

a = - d > 0

. Then

(1 + 4 d^{2}) / (- d) = a^{- 1} + 4 a \geq 4

, with equality at

a = 1/2

. The ideal weak-field Doppler cooling limit is therefore

T_{D} = ℏΓ/ (2 k_{B})

δ = - Γ/2

. It is the minimum of this specified diffusion-and-damping model, rather than a universal lower bound for laser cooling. The diffusion approximation also assumes recoil energy much smaller than