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.
A Bose-Einstein condensate (BEC) is the fifth state of matter, predicted by Satyendra Nath Bose and Albert Einstein in 1924–1925 and first experimentally realised in 1995 by Cornell, Wieman, and Ketterle (2001 Nobel Prize). When bosons are cooled below a critical temperature T_c, a macroscopic fraction of particles collapses into the single lowest-energy quantum state, and all their wave functions merge into one coherent macroscopic matter wave described by the Gross-Pitaevskii (GP) equation: iℏ∂ψ/∂t = (−ℏ²∇²/2m + V(r) + g|ψ|²)ψ. The simulator solves this equation on a 128×128 grid using the split-step Fourier method, alternating a kinetic half-step in momentum space (via FFT) with a potential+interaction half-step in real space.
Adjust T/T_c to watch the condensate fraction N₀/N = 1 − (T/T_c)³ build up as the temperature falls below unity. Increase the interaction strength g to spread the density profile into a Thomas-Fermi parabolic shape. Turn on the stir slider to apply a rotating quadrupole perturbation that nucleates quantised vortices — topological defects where the condensate density drops to zero and the phase winds by 2π. Switch between density |ψ|² and phase arg(ψ) views to see the vortex structure directly.
What makes bosons special compared to fermions?
Bosons (integer-spin particles such as photons, helium-4 atoms, and rubidium-87 atoms) obey Bose-Einstein statistics and can occupy the same quantum state in unlimited numbers. Fermions (half-integer spin: electrons, protons, neutrons) obey the Pauli exclusion principle, which forbids two identical fermions from sharing the same state. BEC is therefore exclusive to bosons — fermions must pair up (as Cooper pairs) before they can condense, which is the mechanism behind superconductivity.
What is the critical temperature formula for BEC?
For an ideal Bose gas in 3D, T_c = (ℏ²/2πmk_B)(n/ζ(3/2))^(2/3), where n is the number density and ζ(3/2) ≈ 2.612. For rubidium-87 at the densities used in JILA (1995), this gives T_c ≈ 170 nanokelvin — 170 billionths of a degree above absolute zero. In a harmonic trap the condensate fraction scales as N₀/N = 1 − (T/T_c)³.
What is a quantum vortex and why is it quantised?
A quantum vortex is a topological defect in the condensate phase field where the phase θ(r) winds by a multiple of 2π around a point of zero density. Since the condensate velocity is v = (ℏ/m)∇θ, the circulation ∮v·dl around the vortex equals h/m per unit winding number — quantised in integer multiples. This is the quantum analogue of a tornado but with discretely allowed circulations, a hallmark of superfluidity.
Replacing real time t with −iτ (imaginary time) turns the time-evolution operator exp(−iHt/ℏ) into exp(−Hτ/ℏ), which exponentially damps all excited states relative to the ground state. After each imaginary-time step the wave function is renormalised, so the energy decays monotonically toward the ground state. This is computationally efficient compared to diagonalising the full Hamiltonian matrix.
A superfluid flows with zero viscosity. The condensate velocity v = (ℏ/m)∇θ is irrotational everywhere except at vortex cores, enabling frictionless flow below the Landau critical velocity. In dilute atomic gases BEC and superfluidity are essentially identical phenomena. In liquid helium-4, correlations reduce the condensate fraction to about 7% at T = 0 even though 100% of the fluid is superfluid — the two fractions differ due to the strong inter-atom interactions.
When the interaction term g|ψ|² dominates the kinetic energy (large N or large g), the kinetic term can be neglected. The GP equation then gives a simple algebraic solution: |ψ|² = max(0, (μ − V(r))/g), where μ is the chemical potential. The density profile becomes an inverted paraboloid — a Thomas-Fermi profile — which is a useful approximation for strongly interacting BECs and matches experiments well for large atom numbers.
Real BECs are trapped by magnetic fields (which confine atoms in weak-field-seeking states) or by focused laser beams (optical dipole traps, which confine atoms through the AC Stark effect regardless of spin). Laser cooling (Doppler and Sisyphus cooling) pre-cools the gas to microkelvin temperatures, and then forced evaporative cooling — removing the most energetic atoms — drives the cloud below T_c. The whole process takes a few seconds.
The interaction parameter g = 4πℏ²a_s/m, where a_s is the s-wave scattering length — the effective radius of inter-atom collisions at ultracold temperatures. For rubidium-87, a_s ≈ 5.8 nm (repulsive). Using magnetic Feshbach resonances, experimenters can tune a_s over several orders of magnitude from attractive (negative) to strongly repulsive (positive), allowing precise control of the condensate shape and stability.
BEC research has produced atom interferometers with sensitivity exceeding classical limits, used for precision measurement of gravity and rotation (inertial navigation). Vortex lattices in BECs are model systems for understanding type-II superconductors. Optical lattices (arrays of laser-induced potential wells) allow simulation of condensed-matter Hamiltonians — a realisation of Feynman's quantum simulator. The 2022 Nobel Prize was awarded for quantum information applications closely related to BEC control techniques.
The split-step method decomposes the GP time evolution into a kinetic step (the Laplacian, diagonal in momentum space) and a potential+interaction step (diagonal in real space). Each full step applies a half-step in real space, a full kinetic step via FFT into momentum space and back, then a second half-step in real space. This Strang splitting gives second-order accuracy in time and is unitary (norm-preserving) in the absence of imaginary-time damping.
In a true 2D system the Mermin-Wagner theorem forbids long-range order at finite temperature, so a conventional BEC cannot form. Instead a 2D Bose gas undergoes a Berezinskii-Kosterlitz-Thouless (BKT) transition: below a critical temperature vortex-antivortex pairs bind and the system becomes superfluid without true off-diagonal long-range order. The quasi-2D BEC in this simulator approximates the behaviour of a pancake-shaped trap commonly used in cold-atom experiments.