Bose–Einstein Condensation in Harmonic Traps

Bose–Einstein condensation (BEC) represents a striking manifestation of quantum statistics at macroscopic scales. When a gas of bosonic particles is cooled below a critical temperature, a macroscopic fraction of the atoms occupies the lowest quantum state of the external potential, leading to long-range coherence and novel collective phenomena. The first realizations of BEC in dilute alkali gases in the mid-1990s opened an entire field of ultracold atomic physics, allowing experimentalists to create degenerate matter waves, study superfluidity in tailored potentials, and explore analogues of condensed-matter systems in highly controllable setups. Determining the transition temperature is a central task when designing BEC experiments or interpreting measurements, yet it requires juggling several physical constants and unit conversions. This calculator focuses on the widely used case of a three-dimensional harmonic trap, deriving the critical temperature T_c and related quantities from the atom number, mass, and trapping frequencies.

The theoretical foundation rests on the Bose distribution for noninteracting bosons in a potential V(r) = ½m(ω_x²x² + ω_y²y² + ω_z²z²). In the semiclassical approximation valid for large atom numbers, the density of states is proportional to E²/(ħ³ω_xω_yω_z). Integrating the Bose function over energy and equating the total number of particles to N yields the critical condition at which the chemical potential approaches the ground-state energy. The resulting expression for the transition temperature is

$k{B}_{}T{c}_{}=ħ\mathrm{\omega ̄}\left(\frac{N}{{\zeta }_{\left(}}\right){}^{\frac{1}{3}}$

where $\mathrm{\omega ̄}={{\omega }_x{\omega }_y{\omega }_z}^{\frac{1}{3}}$ is the geometric mean of the trap frequencies and ζ(3) ≈ 1.20206 is the Riemann zeta function. The factor of ħ converts the angular frequency to energy units, and k_B is Boltzmann's constant. Evaluating this expression yields the temperature at which the ground state begins to accumulate macroscopic population. In practice, interactions and finite-size effects shift the transition temperature slightly, but the formula provides an excellent starting point for dilute gases with weak interactions.

The phase-space density D = nλ³ offers another useful metric for assessing condensation. Here n is the peak number density and λ = h/√(2πmk_BT) is the thermal de Broglie wavelength. At the critical point for a harmonically trapped gas, the maximum density occurs at the trap center and can be related to N and the trap frequencies. The condition for BEC in the ideal gas is roughly D ≈ 2.612. The calculator computes both T_c and the phase-space density at an arbitrary temperature, allowing users to gauge how close a given experimental configuration is to condensation.

To use the calculator, input the total atom number N, the atomic mass in atomic mass units, and the trap frequencies ω_x, ω_y, and ω_z in hertz. The script converts the mass to kilograms, the frequencies to angular frequencies, and evaluates the geometric mean. It then applies the formula above to compute T_c in nanokelvin. For convenience, the result also appears in microkelvin. The phase-space density at T_c is fixed by the condensation criterion, but the calculator allows exploration of D at different temperatures by adjusting N or the trap frequencies.

BEC experiments span a broad range of species and trap geometries. Alkali atoms such as rubidium-87 and sodium-23 are common due to their convenient laser-cooling transitions. For Rb-87 with N = 10⁶ atoms in a trap with ω_x = ω_y = ω_z = 2π × 100 Hz, the critical temperature is approximately 280 nK. In contrast, heavier atoms or tighter traps shift T_c to higher values, easing the cooling requirements. Fermionic atoms, which obey the Pauli exclusion principle, do not undergo BEC; instead they form degenerate Fermi gases characterized by the Fermi temperature. Mixed Bose–Fermi systems enable studies of superfluid mixtures and polaron physics.

The table below presents sample critical temperatures for different atom numbers and trap frequencies using rubidium-87 as the species. These examples illustrate the scaling of T_c with N^1/3 and with the geometric mean of the trap frequencies.

N	ω (Hz)	Tc (nK)
1×10⁵	50	130
1×10⁶	100	280
5×10⁶	200	760

Although the ideal-gas formula ignores interactions, real gases exhibit mean-field shifts due to the s-wave scattering length a. Repulsive interactions typically lower T_c slightly, while attractive interactions can destabilize the condensate beyond a critical atom number. More sophisticated treatments incorporate these effects via Hartree–Fock or Monte Carlo methods. Nonetheless, the simple expression provides valuable intuition and is widely used to plan experiments. By combining it with estimates of cooling efficiency and trap depth, experimentalists can predict the parameters required to reach degeneracy.

Beyond its immediate practical use, the concept of BEC touches many areas of physics. Condensed-matter analogues include superconductivity, where Cooper pairs form a condensate of correlated electrons, and exciton–polariton condensates in semiconductors. In astrophysics, the idea of bosonic condensation has been applied to dark-matter models and to superfluid cores of neutron stars. Understanding the basic thermodynamics of trapped Bose gases lays the groundwork for these interdisciplinary connections.

In summary, this calculator offers a detailed yet accessible way to estimate the critical temperature for Bose–Einstein condensation in a harmonic trap. The accompanying discussion covers the derivation of the formula, the meaning of each parameter, and the broader context of ultracold quantum gases. By entering a few experimental numbers, users can immediately gauge the temperature scale at which quantum degeneracy emerges, facilitating the design and interpretation of cutting-edge BEC experiments.