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Bose-Einstein Condensate 1995 Tee

\[\lambda_{\mathrm{dB}} = \frac{h}{\sqrt{2\pi m k_B T}}\]
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Meet The Condensate Queen, who gathers thousands of atoms into a single quantum state. Cool a gas of bosons until each atom's quantum wave is as wide as the gap between atoms, and a large fraction drops into the lowest state together: a Bose-Einstein condensate. In 1995 Eric Cornell and Carl Wieman at JILA did it first in a gas, with rubidium-87 at about 170 nanokelvin, and Wolfgang Ketterle at MIT followed with sodium. The three shared the 2001 Nobel Prize. The Condensate Queen on a pixel-art unisex tee; equations, symbols and sources in the physics panel below.

"Many atoms, one wave."

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Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

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

參考尺度：M 碼寬 20 in（50.80 cm），約為 3.143 × 1034 個普朗克長度（lP = 1.616255 × 10−35 m）。用捲尺量更快。

Measurements in inches, ±1.5 in tolerance.

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The physics

Cool atoms until their quantum waves are as wide as the spaces between them, and thousands of them fall into the same lowest-energy state and act as one. It was first done in a gas in 1995, at about 170 billionths of a degree above absolute zero.

The Condensate Queen does not rule her atoms. She waits until they cannot tell themselves apart.

Every atom is also a wave. At temperature T its thermal de Broglie wavelength, the first equation on this page, is λ_dB = h/√(2πmk_BT), where h is Planck's constant, m the atom's mass and k_B Boltzmann's constant. Hot atoms have tiny wavelengths and behave like separate billiard balls. Cool them and the waves spread out.

For bosons, particles that may share a quantum state, something sudden happens when the waves overlap. Einstein, extending Satyendra Nath Bose's counting of photons in 1924–25, showed that once the number density n reaches nλ_dB³ = ζ(3/2) ≈ 2.612, extra atoms can no longer fit in the excited states and pile into the single lowest one. For a uniform ideal gas that sets the critical temperature k_BT_c = (2πħ²/m)(n/ζ(3/2))^{2/3}; real experiments hold atoms in a magnetic trap, where k_BT_c ≈ 0.94 ħω̄N^{1/3}, with ω̄ the mean trap frequency and N the number of atoms.

Worked number: for rubidium-87 (m ≈ 1.443×10⁻²⁵ kg) at 170 nK, λ_dB = h/√(2πmk_BT) ≈ 0.45 μm. Condensation starts when the density reaches 2.612/λ_dB³ ≈ 2.8×10¹⁹ atoms per cubic metre, a gas roughly a million times thinner than air, held a fraction of a millionth of a degree above absolute zero.

Reaching that took laser cooling followed by evaporative cooling. In June 1995 Eric Cornell and Carl Wieman's team at JILA saw a sharp, narrow peak appear in the velocity distribution of rubidium atoms near 170 nK, the picture on the back of this shirt: a broad thermal cloud above T_c, a narrow spike on a broad pedestal below it. Wolfgang Ketterle's group at MIT produced a much larger sodium condensate months later. The three shared the 2001 Nobel Prize in Physics.

The Condensate Queen: "Above T_c you are all individuals. Below it, you are me."

Equations

\[\lambda_{\mathrm{dB}} = \frac{h}{\sqrt{2\pi m k_B T}}\]
\[n\,\lambda_{\mathrm{dB}}^{3} = \zeta(3/2) \approx 2.612\]
\[k_B T_c = \frac{2\pi\hbar^{2}}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\quad\text{(uniform gas)}\]
\[k_B T_c \approx 0.94\,\hbar\bar\omega\,N^{1/3}\quad\text{(harmonic trap)}\]
SymbolMeaningUnit
\(\lambda_{\mathrm{dB}}\) thermal de Broglie wavelength \(\mathrm{m}\)
\(h\) Planck constant \(\mathrm{J·s}\)
\(\hbar\) reduced Planck constant, h/2π \(\mathrm{J·s}\)
\(m\) mass of one atom \(\mathrm{kg}\)
\(k_B\) Boltzmann constant \(\mathrm{J/K}\)
\(T\) temperature \(\mathrm{K}\)
\(T_c\) critical temperature for condensation \(\mathrm{K}\)
\(n\) number density (peak density in a trap) \(\mathrm{m⁻³}\)
\(\zeta(3/2)\) Riemann zeta function at 3/2, ≈ 2.612 \(\mathrm{1}\)
\(\bar\omega\) geometric mean of the trap angular frequencies \(\mathrm{rad/s}\)
\(N\) number of atoms in the trap \(\mathrm{1}\)

Sources

  1. Anderson, Ensher, Matthews, Wieman & Cornell (1995) Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science 269, 198
  2. Davis et al. (1995) Bose-Einstein Condensation in a Gas of Sodium Atoms, PRL 75, 3969
  3. NobelPrize.org — The Nobel Prize in Physics 2001
  4. NIST CODATA 2022 — Boltzmann constant
  5. NIST CODATA 2022 — Planck constant

Source Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A. (1995). Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science 269, 198–201.doi.org/10.1126/science.269.5221.198

Field notes

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