Seventy Years to Ninety Nanokelvin: Research Notes on the First Bose-Einstein Condensate — Epoche C1
Research note, drafted before Thursday's journal club on Anderson et al. (1995), Science 269, 198. Question to settle before the meeting: why did Bose–Einstein condensation take seventy years to observe? The received answer — that the theory remained unclear or disputed — does not survive contact with the dates. The criterion was written down in 1925 and the last technical objection to it was withdrawn by 1938. What was missing was a refrigerator, and the size of what was missing can be stated exactly. All numbers below are reworked from scratch so that the arithmetic can be checked. 1924–1938: the theory is finished early Bose (1924) rederived Planck's radiation law by counting the quantum states of light without the classical assumption that photons are distinguishable. Einstein (1925) applied the same counting to a gas of massive atoms and drew a consequence Bose had no occasion to notice: below a certain temperature such a gas must place a macroscopic fraction of its atoms in the single lowest-energy quantum state. The mechanism is worth deriving, because the number that comes out of it governs everything below. In Bose statistics the mean occupation of a state at energy $\varepsilon$ is $1/(e^{(\varepsilon-\mu)/k_BT}-1)$, where $\mu$ is the chemical potential. Since occupations cannot be negative, $\mu$ cannot exceed the lowest energy, taken as zero. So the total number of atoms the excited states can hold is bounded by its value at $\mu = 0$. In three dimensions the density of states of a free particle in a box goes as $\varepsilon^{1/2}$, and the resulting integral $$N_{\text{exc}}^{\max}\;\propto\;\int_0^{\infty}\frac{\varepsilon^{1/2}}{e^{\varepsilon/k_BT}-1}\,\mathrm{d}\varepsilon$$ converges. Expanding the integrand as a geometric series in $e^{-\varepsilon/k_BT}$ and integrating term by term produces $\sum_{j\ge1} j^{-3/2}$, that is $\zeta(3/2)\approx 2.612$; the exponent $3/2$ traces directly to the $\varepsilon^{1/2}$ density of states, which is itself a consequence of three spatial dimensions. Collecting the constants gives the condition in its usual form, $$n\lambda_T^{3}\;\ge\;\zeta(3/2)\approx 2.612, \qquad \lambda_T=\frac{h}{\sqrt{2\pi m k_B T}},$$ where $n$ is the number density and $\lambda_T$ is the thermal de Broglie wavelength — the de Broglie wavelength of an atom carrying a typical thermal momentum, the factor $2\pi$ coming from the Gaussian integral in the partition function. The dimensionless product $n\lambda_T^3$ is called the phase-space density, and physically the criterion says that condensation begins when the wave packets of neighbouring atoms start to overlap. That the exponent matters can be checked by changing dimension: in a uniform two-dimensional gas the corresponding sum is $\sum j^{-1}$, which diverges, the excited states can absorb any number of atoms, and there is no condensation. The one substantial objection came from Uhlenbeck in 1927, who pointed out that a system of finitely many particles shows no sharp transition. He withdrew it by 1938 once the argument was properly formulated as a limit of large systems. The same year, London (1938) proposed that superfluid helium-4 — a dense, strongly interacting liquid, quite unlike Einstein's ideal gas — owes its transition to the same mechanism, and supported it with a number: inserting the density of liquid helium into the criterion gives a condensation temperature near $3\,\mathrm{K}$, against the measured lambda-point of $2.17\,\mathrm{K}$. Close enough to be suggestive, far enough to show that interactions matter. From 1938 the theory sat and waited. What the criterion demands Work the criterion through for rubidium-87, mass $m = 1.44\times10^{-25}\,\mathrm{kg}$, the species Anderson and colleagues used. The first constraint is that the sample must remain a gas. Three-body recombination — two atoms forming a molecule while a third carries away the released binding energy — is the process that turns a cold dense vapour into a solid, and since it requires three atoms to meet, its rate per atom scales as $n^2$. That quadratic scaling is what sets a ceiling: near $n = 10^{19}\,\mathrm{m^{-3}}$, which is $10^{13}\,\mathrm{cm^{-3}}$, the metastable gas survives for seconds, long enough to work with, while an order of magnitude more density costs two orders of magnitude in lifetime. With $n$ fixed, the criterion fixes the temperature. Requiring $n\lambda_T^3 = 2.612$ gives $\lambda_T = (2.612/n)^{1/3} \approx 0.64\,\mu\mathrm{m}$, and inverting the wavelength formula, $$T=\frac{h^{2}}{2\pi m k_B \lambda_T^{2}}\approx 9\times10^{-8}\,\mathrm{K},$$ roughly ninety nanokelvin. That single number explains the seventy years. No cryostat reaches it — the coldest bulk refrigeration, dilution refrigerators, stops three orders of magnitude higher — and in any case a wall at any temperature would adsorb the atoms on contact. The gas had to be cooled while touching nothing. The trade-off between the two constraints is unforgiving, and it is worth stating as a scaling. Holding $n\lambda_T^3$ fixed while $\lambda_T \propto T^{-1/2}$ gives $n \propto T^{3/2}$, or $T \propto n^{2/3}$: relaxing the density by a factor of ten to buy lifetime costs a factor of $10^{2/3}\approx 4.6$ in the temperature that must be reached. There is no dilute regime in which the problem becomes easy. Stage $T$ $n$ $n\lambda_T^3$ Room-temperature Rb vapour $300$ K $10^{13}\ \mathrm{cm}^{-3}$ $\sim 10^{-14}$ Magneto-optical trap (from 1987) $\sim 100\ \mu\mathrm{K}$ $10^{11}\ \mathrm{cm}^{-3}$ $\sim 7\times10^{-7}$ End of evaporation (June 1995) $\sim 9\times10^{-8}$ K $10^{13}\ \mathrm{cm}^{-3}$ $\approx 2.6$ 1985–1988: light reaches microkelvin, then stops The first stage of the ladder is optical. Chu and colleagues (1985) bathed atoms in six laser beams, opposed in pairs along three axes and tuned slightly below an atomic resonance. An atom moving towards one beam sees it Doppler-shifted upwards, closer to resonance, and therefore absor