Bose-Einstein Condensate & Superconductivity

Cool any gas and the atoms slow down — that part is familiar. But cool a gas of the right kind of atoms extraordinarily cold, and something with no everyday parallel happens: a huge fraction of the atoms suddenly stop being separate specks and start acting like one single thing, moving in perfect lockstep. Picture a stadium crowd: at room temperature everyone is milling about randomly; in a Bose-Einstein condensate it is as if every person in the stadium suddenly does the exact same motion at the exact same instant. That shared, single quantum state is the condensate.

Why cold makes atoms merge

Here is the key picture for a high-schooler: every atom is also a tiny wave, and the colder it gets, the wider that wave spreads. The width is the thermal de Broglie wavelength $\lambda_{\text{th}}$, and it grows as the temperature $T$ falls:

For a rubidium atom at about $170$ nK — that is $0.00000017$ degrees above absolute zero — this width works out to roughly $0.5\ \mu\text{m}$. The atoms in the trap sit only about $0.3\ \mu\text{m}$ apart, so each atom's wave is now wider than the gap to its neighbour. The waves overlap, the atoms can no longer be told apart, and they collapse into one shared state. Condensation happens precisely when the waves get this crowded, the condition $n\,\lambda_{\text{th}}^3 \approx 2.6$.

The precise version, and the leap to superconductors

Solving that crowding condition for temperature gives the critical temperature $T_c \propto n^{2/3}/m$ (full form in the next section), and below it the share of atoms in the condensate climbs as $N_0/N = 1-(T/T_c)^{3/2}$ — zero just above $T_c$, rising to everything at absolute zero. The deep idea then carries straight over to superconductivity: electrons are loners (fermions, no sharing allowed), but in a cold metal two of them link up through the vibrating lattice into a Cooper pair. A pair behaves like a boson, so the pairs can all condense into one state — and a current made of that single locked-together state flows with exactly zero resistance. The sliders let you drive both stories: Temperature and SC Tc set how far below the transition you are, and Magnetic field B tests how much field a superconductor will throw out before it gives up.

Try this in the sim above

In BEC Cooling mode, drag Temperature down past $T_c=170$ nK and watch the orange thermal cloud collapse into the glowing cyan condensate at the trap centre while the condensate fraction $N_0/N$ climbs toward 100%. Switch to Momentum Distribution and cool again: a sharp central spike grows out of the broad thermal hump — the famous bimodal "fingerprint" that won the 2001 Nobel Prize. Finally, in Meissner Effect mode set the temperature below the SC $T_c$, then raise $B$ and watch the field lines bend around the sample instead of through it — until $B$ gets large enough to destroy the superconducting state.

Bose-Einstein Condensation (BEC) and superconductivity (SC) are macroscopic quantum phenomena that emerge when integer-spin particles ('bosons') pile into a single quantum state below a critical temperature. In atoms, the bosons are dilute neutral atoms (like ⁸⁷Rb). In a superconductor, two electrons (fermions) pair up via phonon exchange to form a composite boson — the Cooper pair — which then condenses.

1. The de Broglie Thermal Wavelength

At temperature $T$, an atom of mass $m$ has thermal momentum $p \sim \sqrt{m k_B T}$, giving a wavelength:

BEC sets in when $\lambda_{\text{th}}$ becomes comparable to the inter-particle spacing $n^{-1/3}$ — i.e., when the wavefunctions overlap and the particles can no longer be distinguished.

2. BEC Critical Temperature

For an ideal gas of $N$ bosons in volume $V$, condensation begins when:

Solving for $T$ gives the critical temperature:

For ⁸⁷Rb at $n \approx 2.8\times 10^{13}$ cm⁻³, $T_c \approx 170$ nK — a hundred-millionth of a degree above absolute zero. Below $T_c$, the condensate fraction is:

3. The Macroscopic Wavefunction

All condensed atoms share a single wavefunction $\psi(\vec r,t)$ — typically the ground state of the trap. This wavefunction has a definite phase and obeys the Gross-Pitaevskii equation (for a weakly-interacting gas):

where $g = 4\pi\hbar^2 a_s/m$ is the contact interaction strength ($a_s$ is the s-wave scattering length).

4. BCS Theory of Superconductivity

Below $T_c$, electrons near the Fermi surface attract each other via virtual phonon exchange. Bardeen, Cooper, and Schrieffer (1957) showed that any net attractive interaction, however weak, makes the Fermi sea unstable to pair formation:

where $\omega_D$ is the Debye frequency and $N(0)V_0$ is the dimensionless coupling strength. The full temperature-dependent gap satisfies:

The BCS prediction $2\Delta(0)/(k_B T_c) = 3.528$ is a universal ratio for weak-coupling superconductors and agrees well with most conventional SCs.

5. Meissner Effect & London Penetration

A superconductor expels magnetic field from its interior — the Meissner effect. The field decays exponentially over the London penetration depth:

For Type II SCs, above a lower critical field $B_{c1}$, the field penetrates as quantised vortices each carrying flux $\Phi_0 = h/(2e) = 2.07\times 10^{-15}$ Wb (the 2e is the Cooper pair charge — direct evidence of pairing).

Symbol Table

Mapping to the simulation

The BEC mode shows $N$ atoms in a harmonic trap; their kinetic energy is sampled from a Maxwell-Boltzmann distribution at temperature $T$. As you lower $T$ below $T_c$ (drawn from the formula above), atoms condense into the trap ground state — they collapse to the centre and move slowly. Cooper-pair mode visualises the phonon-mediated attraction; Meissner mode shows field expulsion below $T_c$ with the London decay; Phase diagram tracks $B$-$T$ for Type II SCs with the upper/lower critical-field curves.