• Molassa Doppler Cooling Limit pixel-art physics T-shirt, black tee, front view

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Molassa

The Doppler Cooling Limit Physics

\[k_B T_D = \frac{\hbar\Gamma}{2} \qquad (\text{two-level atom, best at } \delta = -\Gamma/2)\]
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Size guide S–5XL · inches

Molassa stands where six laser beams cross, and atoms freeze in her palm. Then they got colder than the rules allowed.

The Physics

In optical molasses, red-detuned laser beams from every direction slow a moving atom like friction, while random recoils from spontaneous emission heat it. For a simple two-level atom the balance sets a minimum temperature, the Doppler limit, best reached at detuning δ = −Γ/2. For sodium's D2 line that limit is about 235 μK.

k_B T_D = ħ Γ / 2

  • k_B: Boltzmann constant (J/K)
  • T_D: Doppler limit temperature (K)
  • ħ: reduced Planck constant (J·s)
  • Γ: natural linewidth of the transition (1/s)

In 1988 Lett, Phillips and co-workers measured sodium atoms at 43 ± 20 μK, far below the limit; Chu, Cohen-Tannoudji and Phillips shared the 1997 Nobel Prize in Physics.

Honest caveat: The Doppler limit is right for two-level atoms. Real atoms have many sublevels, and polarisation-gradient (Sisyphus) cooling beats it.

On the shirt

Front: Molassa, the pixel-art character, with the title. Back: the Doppler limit, the sodium numbers, the 1988 measurement, the Sisyphus explanation and the 1997 Nobel. Printed on a unisex tee.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

पैमाने के लिए: साइज़ M की चौड़ाई 20 in (50.80 cm) है, यानी लगभग 3.143 × 1034 प्लांक लंबाई (lP = 1.616255 × 10−35 m)। इंच-टेप ज़्यादा जल्दी काम करेगा।

Measurements in inches, ±1.5 in tolerance.

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The physics

I stand where six laser beams cross. Each beam is tuned a little below the atom's resonance, so a moving atom sees the beam it is running into Doppler-shifted closer to resonance and gets pushed back. Atoms wade through my light like flies through syrup: optical molasses.

But every scattered photon also gives a random kick. For a simple two-level atom the balance between that friction and the heating sets a floor, the Doppler limit k_B T_D = ħΓ/2, reached at detuning δ = −Γ/2. For sodium's D2 line that is about 235 microkelvin.

In 1988 Lett, Phillips and co-workers measured sodium atoms at 43 ± 20 microkelvin, far colder than the limit. The limit was right for two-level atoms; real atoms have many sublevels, and polarisation-gradient, or Sisyphus, cooling beats it. Chu, Cohen-Tannoudji and Phillips shared the 1997 Nobel Prize in Physics.

Equations

\[k_B T_D = \frac{\hbar\Gamma}{2} \qquad (\text{two-level atom, best at } \delta = -\Gamma/2)\]
\[\text{Na D2: } \Gamma = 2\pi\times9.795\ \mathrm{MHz} \ \Rightarrow\ T_D \approx 235\ \mu\mathrm{K}\]
SymbolMeaningUnit
\(k_B\) Boltzmann constant, 1.380 649 × 10⁻²³ J/K (exact) \(\mathrm{J/K}\)
\(T_D\) Doppler limit temperature \(\mathrm{K}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(\Gamma\) natural linewidth (decay rate) of the transition \(\mathrm{1/s}\)
\(\delta\) laser detuning from resonance \(\mathrm{1/s}\)

Sources

  1. Lett, Watts, Westbrook, Phillips, Gould & Metcalf (1988) Observation of Atoms Laser Cooled below the Doppler Limit, Phys. Rev. Lett. 61, 169
  2. Dalibard & Cohen-Tannoudji (1989) Laser cooling below the Doppler limit by polarization gradients: simple theoretical models, J. Opt. Soc. Am. B 6, 2023
  3. Steck, Sodium D Line Data
  4. Nobel Prize in Physics 1997, NobelPrize.org

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