Bose-Einstein Condensates: Solve the Gross-Pitaevskii Equation and Watch One Expand
Cool a gas of atoms to around a hundred billionths of a degree above absolute zero and something strange happens. Each atom behaves like a small wave, and as the gas gets colder those waves grow until they overlap. Below a critical temperature, most of the atoms drop into the same lowest-energy state and start moving as one. The whole cloud is then described by a single wave function, ψ. This is a Bose-Einstein condensate, or BEC.
Bose and Einstein predicted it in 1924 and 1925. It took until 1995 to make one: first at JILA in Colorado with rubidium atoms, then at MIT with sodium. Eric Cornell, Carl Wieman and Wolfgang Ketterle shared the 2001 Nobel Prize in Physics for it. Below you can make one yourself, at least on screen: the simulator solves the equation that describes a condensate, finds its ground state in a trap, then switches the trap off and lets it fly.
The Gross-Pitaevskii equation
A condensate obeys the Schrödinger equation with one extra term, added independently by Eugene Gross and Lev Pitaevskii in 1961:
iħ ∂ψ/∂t = [ −(ħ²/2m) ∇² + V(r) + g |ψ|² ] ψ
- −(ħ²/2m) ∇² is the kinetic energy. It resists squeezing: the smaller the cloud, the more it wants to spread out.
- V(r) is the trap, here a harmonic bowl made by magnetic fields or focused laser beams: V = ½ m (ω_x² x² + ω_y² y²).
- g |ψ|² is the interaction. Each atom feels a push proportional to the local density of the others. For rubidium-87 the atoms repel, with a strength set by the scattering length a, about 100 Bohr radii (5.3 nm).
The last term makes the equation nonlinear, so there is no simple formula for the answer. It has to be solved numerically, which is what the simulator does, live in your browser.
How to use the simulator
- Find the ground state. Whenever you change a setting, the simulator relaxes the wave function to its lowest-energy shape. It takes a second or two. The numbers below the picture fill in when it is done.
- Release the trap. The trap turns off and the cloud expands. This is “time of flight”, the standard way to look at a condensate: experiments let it expand for a few tens of milliseconds, then flash a laser at it and photograph its shadow. The main picture is that view from above.
- Watch the shape. The cuts on the right compare the simulated density with the Thomas-Fermi prediction, and the bottom plot tracks the aspect ratio during the flight.
- Show the phase. Turn on Show the phase as colour to see the wave function itself, not just the density. Pause a few milliseconds into the flight for the clearest view.
What to look for
1. A flat top instead of a bell curve. Without interactions (set the scattering length to 0), the ground state is the Gaussian of a quantum harmonic oscillator. With interactions, the atoms push each other outward until the push balances the trap everywhere. The density then takes the shape of the trap turned upside down, an inverted parabola: n(r) = (μ − V(r))/g. This is the Thomas-Fermi profile, and with 10,000 atoms the simulation and the dashed prediction almost coincide. Only the very edge is rounded off, over a distance about the size of the healing length.
2. The cloud turns itself sideways. In the elongated trap the condensate starts out four times longer along x than along y. After release it grows fastest along y, the direction it was squeezed most, and after about 8 ms it is round. By 30 ms it is almost twice as long along y as along x. A thermal cloud of ordinary hot atoms does not do this: its atoms fly off with random speeds that do not depend on direction, so it only ever becomes round (the dotted line). This flip of the aspect ratio was one of the first signs, in 1995, that a condensate had really formed.
Why does it flip? With interactions, the stored interaction energy turns into motion, and the steepest density slope, along the tight direction, gives the strongest push. Without interactions the reason is the uncertainty principle: the more tightly a wave is confined, the wider its spread of momentum. Try both and compare.
3. One wave, flowing outward. In the trap the phase is the same everywhere: one colour across the whole cloud. That is what it means for the atoms to share one wave function. After release, coloured rings appear. The atoms’ velocity is set by how fast the phase changes, v = (ħ/m)∇(phase), so closely spaced rings mean fast-moving atoms. The rings crowd together toward the edge and along y, exactly where the atoms are flying fastest.
4. Small condensates are different. Try the 300-atom preset. Interactions are now too weak to flatten the cloud, kinetic energy matters again, and the simple Thomas-Fermi theory no longer matches the full simulation.
The numbers
| Quantity | Formula (2D) | What it tells you |
|---|---|---|
| Interaction strength, g̃ | √(8π) N a / a_z | How strongly the atoms repel overall, without units. Large values mean the Thomas-Fermi picture applies. |
| Chemical potential, μ | ħ √(ω_x ω_y g̃ / π) | The energy needed to add one more atom. In temperature units it is a few tens of nanokelvin. |
| Thomas-Fermi radius | R_x = √(2μ / m ω_x²) | Where the density reaches zero. The tighter direction has the smaller radius. |
| Healing length, ξ | ħ / √(2mμ) | The shortest distance over which the density can change. It sets the size of a vortex core and the softness of the cloud’s edge. |
| Virial theorem | E_kin − E_trap + E_int = 0 | An exact relation for the ground state in 2D. The simulator uses it to check its own accuracy. |
How the simulation works
The wave function lives on a grid of up to 512 × 512 points. Each time step uses the split-step Fourier method: half a step with the trap and interaction energy, applied point by point; a full step with the kinetic energy, applied in momentum space after a fast Fourier transform; then the second half step. It is accurate and stable, and fast enough to run in a browser.
To find the ground state, the same equation is run in imaginary time. Replacing t with −iτ turns waves into decay: every excited component of the wave function fades away faster than the ground state, so whatever you start with relaxes to the lowest-energy shape. The simulator starts from the Thomas-Fermi guess and keeps going until the chemical potential stops changing.
Expansion is harder, because in 30 ms the cloud grows about twenty times larger along y and would run off any grid. The simulator therefore works in a frame that stretches along with the cloud, using the scaling solution worked out by Yvan Castin and Ralph Dum and by Yuri Kagan, Evgeny Surkov and Georgy Shlyapnikov in 1996. The transformation itself is exact; the full equation is still solved in the stretched frame, so any departure from simple scaling, as in the 300-atom case, shows up in the result. As checks, the expansion without interactions matches the exact analytic answer, the round trap grows exactly as √(1 + ω²t²), and the elongated trap matches Thomas-Fermi scaling theory within 1%.
What the model leaves out
- It is two-dimensional. The condensate is a flat pancake held in a tight vertical trap, a light sheet at 2 kHz, that stays on during the flight. Some 2D gas experiments work this way. In a full 3D release, the tight vertical direction expands fastest of all, by the same physics.
- It is at zero temperature. The Gross-Pitaevskii equation is a mean-field theory: every atom is in the condensate. Real clouds also have a thermal fraction and a small “quantum depletion” from the interactions.
- No gravity, no losses. Real clouds fall during time of flight and slowly lose atoms through three-body collisions. Neither changes the shape you see here.
Where optics comes in
Almost every step of a BEC experiment is done with light: laser cooling, optical dipole traps made from tightly focused beams, optical lattices, and the absorption imaging that takes the final picture. Those beams have to stay put. If a mirror drifts, the focus of a dipole trap moves, and the condensate starts sloshing back and forth or heats up. A lattice beam that wanders blurs the lattice, and an imaging beam that moves shifts every picture. Pointing stability over hours of data taking is not a detail in these labs; it decides whether an experiment works.
That is the problem Physiverse mounts are built for: monolithic flexure mounts that hold their angle through temperature changes and long experiments. See the product line.