Bose–Einstein Condensation | Felipe Taha Sant'Ana

Bose–Einstein condensation (BEC) is the macroscopic occupation of a single quantum state — a purely quantum phenomenon with no classical analogue. Predicted by Einstein in 1925 (building on Bose’s photon statistics), first observed in dilute atomic gases in 1995 (Cornell, Wieman, Ketterle — Nobel Prize 2001).

Quantum statistics: bosons and fermions

Quantum particles are indistinguishable. Under the exchange of two identical particles, the many-body wavefunction acquires a factor $e^{i\alpha}$. Physical states require $e^{2i\alpha} = 1$, so $e^{i\alpha} = \pm 1$:

Bosons ($+1$): integer spin; wavefunctions symmetric under exchange
Fermions ($-1$): half-integer spin; wavefunctions antisymmetric (Pauli exclusion)

The mean occupation of a single-particle state with energy $\epsilon_k$ in thermal equilibrium is

\begin{equation} \langle n_k \rangle = \frac{1}{e^{(\epsilon_k - \mu)/k_{\rm B}T} - 1} \quad \text{(Bose–Einstein)}, \end{equation}

where $\mu \leq \epsilon_{\min}$ is the chemical potential. As $\mu \to \epsilon_0^-$ (the ground-state energy), $\langle n_0 \rangle \to \infty$ — bosons can pile up without limit in the ground state.

The condensation transition

For a non-interacting ideal Bose gas of $N$ particles in volume $V$, the condensation temperature is found by requiring all excited states to saturate at their maximum occupation:

\begin{equation}\label{eq.Tc} T_c = \frac{2\pi\hbar^2}{m k_{\rm B}}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}, \end{equation}

where $n = N/V$ is the number density, $m$ the particle mass, and $\zeta(3/2) \approx 2.612$ is the Riemann zeta function. For $T < T_c$, a macroscopic fraction

\begin{equation} \frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2} \end{equation}

occupies the single-particle ground state. This fraction is the condensate.

The condition \eqref{eq.Tc} can be rewritten as $n\lambda_{\rm dB}^3 \gtrsim 2.612$, where $\lambda_{\rm dB} = h/\sqrt{2\pi m k_{\rm B} T}$ is the thermal de Broglie wavelength. BEC occurs when the quantum mechanical wave packets of adjacent atoms overlap.

The order parameter and spontaneous symmetry breaking

The condensate is described by a macroscopic wavefunction (order parameter)

\begin{equation} \Psi(\mathbf{r}, t) = \sqrt{n_0(\mathbf{r}, t)}\, e^{i\theta(\mathbf{r}, t)}, \end{equation}

where $$n_0 =

\Psi

^2$is the condensate density and$\theta$its phase. The appearance of a definite phase$\theta$signals the **spontaneous breaking** of the global$U(1)$symmetry$\Psi \to \Psi e^{i\alpha}$$. This is the same symmetry breaking that underlies superconductivity.

Gross–Pitaevskii equation

For a weakly interacting BEC (contact interactions with scattering length $a_s$, so $g = 4\pi\hbar^2 a_s/m$), the condensate dynamics obey the Gross–Pitaevskii equation:

\begin{equation}\label{eq.GP} i\hbar \frac{\partial\Psi}{\partial t} = \left(-\frac{\hbar^2\nabla^2}{2m} + V_{\rm ext} + g|\Psi|^2\right)\Psi. \end{equation}

This is a nonlinear Schrödinger equation. The $$g

\Psi

^2$term is the **mean-field interaction**: repulsive ($a_s > 0$) interactions stabilise the condensate against collapse; attractive ($a_s < 0$$) interactions lead to collapse above a critical atom number.

In a harmonic trap $V_{\rm ext} = \frac{1}{2}m\omega^2 r^2$, the Thomas–Fermi approximation (neglecting kinetic energy for large $N$) gives the inverted-parabola density profile

\begin{equation} n_0(r) = \frac{\mu - \frac{1}{2}m\omega^2 r^2}{g} \quad (r < R_{\rm TF}), \end{equation}

with Thomas–Fermi radius $R_{\rm TF} = \sqrt{2\mu/m\omega^2}$.

Superfluidity and quantised vortices

The superfluid velocity field is

\begin{equation} \mathbf{v}_s = \frac{\hbar}{m}\nabla\theta. \end{equation}

Since $\mathbf{v}_s = \nabla(\hbar\theta/m)$ is a gradient, the flow is irrotational: $\nabla\times\mathbf{v}_s = 0$ except at singularities. The circulation around any closed loop is quantised:

\begin{equation} \oint \mathbf{v}_s \cdot d\mathbf{l} = \frac{h}{m} \, n, \quad n \in \mathbb{Z}. \end{equation}

A single quantised vortex carries one quantum of circulation $h/m$ and a phase winding of $2\pi$ around its core, where $n_0 = 0$. An array of vortices on a rotating BEC forms an Abrikosov-like lattice — a direct analogue of type-II superconductors.

Bogoliubov spectrum

Linearising the GP equation around the uniform condensate $\Psi_0 = \sqrt{n_0}$ gives the Bogoliubov dispersion:

\begin{equation} E(k) = \hbar\sqrt{\frac{\hbar^2 k^4}{4m^2} + \frac{gn_0}{m}\, k^2}. \end{equation}

Two limits:

Long wavelengths ($k \to 0$): $E \approx \hbar c_s k$ — phonons propagating at the sound speed $c_s = \sqrt{gn_0/m}$. This linear dispersion is responsible for superfluidity (Landau criterion).
Short wavelengths ($k \to \infty$): $E \approx \hbar^2 k^2/2m$ — free-particle behaviour.

The crossover occurs at the healing length $\xi = \hbar/\sqrt{2mgn_0}$, the length scale over which the condensate heals around a vortex core or an impurity.

Experimental realisation

Cornell and Wieman (Boulder, 1995) achieved BEC in $^{87}$Rb at $T_c \approx 170\,$nK and $n \sim 10^{13}$ cm⁻³ using laser cooling followed by evaporative cooling in a magnetic trap. The condensate was identified by the characteristic bimodal momentum distribution: a narrow Thomas–Fermi peak (condensate) above a broad thermal cloud. Today BECs are routine tools for studying superfluidity, quantum simulation, atom interferometry, and analogue gravity.