Bose–Einstein Condensation
Macroscopic occupation of the ground state and the experimental dilute-gas BEC.
Bosons obey the Bose–Einstein distribution
$$\langle n_i\rangle = \frac{1}{e^{(\varepsilon_i - \mu)/k_B T} - 1}.$$For a 3D ideal Bose gas of $N$ atoms, the chemical potential $\mu$ approaches the ground-state energy as $T$ drops. Below the critical temperature
$$k_B T_c = \frac{2\pi \hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}, \quad \zeta(3/2) \approx 2.612,$$a macroscopic fraction of atoms condenses into the lowest single-particle state:
$$\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}.$$The first dilute-gas BEC (Cornell, Wieman; Ketterle — 2001 Nobel) used laser cooling and evaporative cooling to bring ~$10^4$ rubidium atoms below ~$170$ nK. The condensate is described by a macroscopic wavefunction $\Psi(\mathbf r, t)$ satisfying the Gross–Pitaevskii equation:
$$i\hbar \partial_t \Psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V + g|\Psi|^2\right) \Psi.$$Properties: superfluidity, quantized vortices, interference patterns — quantum coherence at the macroscopic scale.