Cool bosons toward absolute zero: below the critical temperature Tc, a macroscopic fraction of particles collapses into the quantum ground state, forming a single coherent matter wave described by a macroscopic order parameter ψ(r,t).
GP equation:
iℏ dψ/dt = [-ℏ²/(2m) ∇² + V(r) + g|ψ|²] ψ
Trap: V(r) = ½ m ω² r²
Thomas-Fermi: μ = g |ψ|² + V(r)
Condensate fraction: N₀/N = 1 − (T/T⁽)³
Split-step half-step:
ψ* = exp(−i dt/2 [V+g|ψ|²]/ℏ) ψ (r-space)
ψ* = exp(−i dt ℏk²/2m) ψ* (k-space, via FFT)
ψ = exp(−i dt/2 [V+g|ψ*|²]/ℏ) ψ* (r-space)The first BEC was created at JILA in 1995 with rubidium-87 atoms cooled to 170 nanokelvin — 170 billionths of a degree above absolute zero. Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics for this achievement.
A Bose-Einstein Condensate (BEC) is a state of matter formed when bosons are cooled to temperatures near absolute zero. Below a critical temperature Tc, a macroscopic fraction of particles occupies the lowest quantum energy state, and all their quantum wave functions merge into a single coherent macroscopic matter wave described by one order parameter.
The Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation that governs BEC dynamics: iℏ dψ/dt = [−ℏ²/(2m) ∇² + V(r) + g|ψ|²] ψ. The nonlinear term g|ψ|² represents mean-field particle interactions, with g = 4πℏ²as/m proportional to the s-wave scattering length as.
The split-step Fourier method divides each time step into a kinetic half-step (applied in momentum space via FFT, where the Laplacian is diagonal as −k²) and a potential+interaction half-step (applied in real space). This operator-splitting gives second-order accuracy and efficiently handles the full GP nonlinearity without matrix diagonalisation.
Quantum vortices are topological defects where the condensate phase winds by multiples of 2π around a point of zero density. Unlike classical vortices, circulation is quantised in units of h/m. Under rotation, vortices arrange into Abrikosov-like triangular lattices and are a definitive signature of superfluidity.
Replacing real time t with −iτ turns the time-evolution operator into exp(−Hτ/ℏ), which exponentially damps all excited states relative to the ground state. Iterating and renormalising the wave function therefore converges to the GP ground state without solving an eigenvalue problem, making it computationally efficient.
Superfluidity is flow without viscosity. The condensate velocity field v = (ℏ/m) ∇θ is irrotational everywhere except at vortex cores, and the fluid resists perturbations below the Landau critical velocity. In dilute atomic gases BEC and superfluidity essentially coincide, though in liquid helium-4 the superfluid fraction is larger than the condensate fraction due to strong correlations.
For an ideal Bose gas in 3D: Tc = (ℏ²/2πmkB) (n/ζ(3/2))2/3, where ζ(3/2) ≈ 2.612. Real atomic BECs form below ~1 μK; the first Rb-87 BEC appeared at ~170 nK. In a harmonic trap the condensate fraction scales as N0/N ≈ 1 − (T/Tc)3.
For repulsive interactions (g > 0) the density profile broadens and in the Thomas-Fermi limit becomes an inverted paraboloid: n(r) = (μ − V(r))/g. For attractive interactions (g < 0) the cloud contracts and can collapse above a critical atom number. Strong repulsion also lowers the sound speed and stiffens the condensate against perturbations.
Satyendra Nath Bose (1924) and Albert Einstein (1925) predicted the condensation phenomenon theoretically. The first experimental BEC in a dilute atomic vapour was achieved in June 1995 by Eric Cornell and Carl Wieman (Rb-87 at JILA) and independently by Wolfgang Ketterle (Na at MIT). All three shared the 2001 Nobel Prize in Physics.
The simulation uses a 2D isotropic harmonic trap V(r) = ½ m ω² r², mimicking the magnetic or optical-dipole traps used in experiments. The trap frequency ω determines the natural length aho = √(ℏ/mω) and energy ℏω of the system. Increasing ω squeezes the condensate; decreasing it spreads the cloud and reduces the peak density.