Bose condensation in a three-dimensional harmonic trap ($k_B{T}_c=\hslash \bar{\omega }[N/\zeta (3){]}^{1/3}$)

Consider $N$ noninteracting bosons in one internal quantum state, at thermal equilibrium in the three-dimensional harmonic oscillator potentialHere $m$ is the mass, the positive ${\omega }_i$ are angular frequencies, and $\bar{\omega }$ is their geometric mean. Measure the one-particle energy $ϵ$ and chemical potential $\mu$ from the ground-state energy. The quantum harmonic oscillator levels then have $ϵ=\hslash ({\omega }_x{n}_x+{\omega }_y{n}_y+{\omega }_z{n}_z)$ with nonnegative integer $n_i$ and reduced Planck constant $\hslash =h/(2\pi )$.

When $ϵ\gg \hslash {\omega }_i$, counting lattice points can be approximated by the volume of a tetrahedron in the positive octant of $(n_x,n_y,n_z)$:This is a density of states, with units of inverse energy. The factor $1/6$ is the unit tetrahedron volume; differentiating produces the factor $1/2$. Treat the single ground state separately: its macroscopic population would be lost by replacing every level with a smooth density of states.

For $k_{B}T\gg \hslash {\omega }_i$, integrate the Bose-Einstein distribution over excited energy eigenstates. As $\mu$ approaches zero from below, their maximum population isHere $k_B$ is the Boltzmann constant and $\zeta$ the Riemann zeta function. A short proof uses $1/(e^x-1)={\sum }_{\ell \ge 1}{e}^{-\ell x}$ and ${\int }_0^{\infty }{x}^2{e}^{-\ell x}dx=2/{\ell }^3$. The nonnegative terms permit exchanging summation and integration by Tonelli's theorem, giving ${\int }_0^{\infty }{x}^2/(e^x-1)dx=2\zeta (3)$. Near zero the integrand is proportional to $x$, so its integral converges. That finite excited-state capacity is the mechanism of Bose-Einstein condensation.

Set $N_{\mathrm{ex}}^{\mathrm{sat}}(T_c)=N$. Particle conservation gives, to this continuum accuracy,Above $T_c$, the macroscopic condensate fraction is zero and the chemical potential adjusts below zero. This cubic power belongs to a three-dimensional harmonic trap; it is not the homogeneous three-dimensional ideal gas exponent. For $N=10^5$, $\bar{\omega }/(2\pi )=100\mathrm{Hz}$ and $\zeta (3)\simeq 1.202056903$, the model gives $T_c\simeq 209.5\mathrm{nK}$. At $T=0.8T_c\simeq 167.6\mathrm{nK}$, $N_0/N=0.488$, or approximately $48800$ condensed bosons. These are chosen parameters and calculated values.

Finite $N$ gives a smooth crossover rather than a sharp phase transition. The exact ideal gas sum over excited quantum harmonic oscillator levels differs from the continuum integral, especially when $k_{B}T$ is not large relative to every $\hslash {\omega }_i$. A sharp trapped-gas limit takes $N\to \infty$, $\bar{\omega }\to 0$ with $N{\bar{\omega }}^3$ fixed and fixed frequency ratios. Interactions also alter the critical temperature, condensate fraction and release profile; a dilute gas can have appreciable interaction energy compared with the trap level spacing. The cubic law is a useful baseline, not a fit guaranteed for either 1995 experiment.

The ideal isotropic-trap transition scale is $T_0=\zeta (3{)}^{-1/3}(\hslash \omega /k_B)N^{1/3}$. The numerical prefactor $\zeta (3{)}^{-1/3}\simeq 0.9405$ follows from the Bose integral ${\int }_0^{\infty }{x}^2/(e^x-1)dx=2\zeta (3)$.

At ground-relative chemical potential zero, the leading semiclassical excited population is $N_{\mathrm{ex}}=\zeta (3)(k_{B}T/\hslash \omega {)}^3$. Additional particles occupy the ground state once this capacity is reached. The continuum description requires many excited trap levels to be thermally accessible.

With the ground energy set to zero, the ideal-gas chemical potential approaches zero from below at condensation and remains there to leading thermodynamic order. Above onset it is determined by the trap density of states. The shape differs from that for a uniform gas.

The exact three-dimensional oscillator ground energy is $3\hslash \omega /2$ above the potential minimum. Subtracting it sets the condensate energy to zero. The commonly used $g(ϵ)={ϵ}^2/[2(\hslash \omega {)}^3]$ is the leading continuum term; it does not include the finite-level corrections from shifting the energy origin.

The condensate fraction is the population $N_0$ of a macroscopically occupied one-particle state divided by the total particle number $N$. For an interacting system, use a macroscopic eigenvalue of the one-particle density matrix, rather than assume that the relevant mode is an undeformed single-particle trap ground state. The condensate fraction is not generally the fraction measured as superfluid by a flow or rotation response.