Bose-Einstein Condensation | Physics

For bosons, the average occupation of a single-particle state of energy $\varepsilon$ is

$\langle n_\varepsilon \rangle = \frac{1}{e^{\beta(\varepsilon - \mu)} - 1}$

The chemical potential must satisfy $\mu \leq \varepsilon_0$ (the lowest single-particle energy) to prevent negative occupation numbers.

For a 3D free Bose gas with $\varepsilon = \hbar^2 k^2 / (2m)$The density of states is $g(\varepsilon) = (V/4\pi^2)(2m/\hbar^2)^{3/2}\sqrt{\varepsilon}$. The number of particles in excited states ($\varepsilon > 0$) is

$N_{\mathrm{ex} = \int_0^\infty \frac{g(\varepsilon)\, d\varepsilon}{e^{\beta \varepsilon} - 1} = V\left(\frac{mk_BT}{2\pi\hbar^2}\right)^{3/2}\,\zeta\!\left(\frac{3}{2}\right)}$

Where $\zeta(3/2) \approx 2.612$ is the Riemann zeta function.

Theorem 5.1 (BEC critical temperature). The maximum number of particles that can be accommodated in excited states is achieved at $\mu = 0$. When $N$ exceeds this maximum, the excess condenses into the ground state. The critical temperature is

$T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}$

Where $n = N/V$.

Proof. Setting $N = N_{\mathrm{ex}^{\max}}$ at $\mu = 0$ and solving for $T$:

$n = \left(\frac{mk_B T_c}{2\pi\hbar^2}\right)^{3/2}\,\zeta\!\left(\frac{3}{2}\right)$

$T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3} \qquad \blacksquare$

Below $T_c$, $\mu \approx 0$ and the condensate fraction is

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

This follows from $N_0 = N - N_{\mathrm{ex}}$ with $\mu = 0$:

$N_{\mathrm{ex} = N\left(\frac{T}{T_c}\right)^{3/2}}$

The energy below $T_c$:

$U = \int_0^\infty \frac{\varepsilon\, g(\varepsilon)\, d\varepsilon}{e^{\beta\varepsilon} - 1} = V\left(\frac{mk_BT}{2\pi\hbar^2}\right)^{3/2}\,(k_BT)\,\frac{3}{2}\,\zeta\!\left(\frac{5}{2}\right) \cdot \Gamma\!\left(\frac{5}{2}\right)$

$= \frac{3}{2}\,Nk_BT_c\,\zeta\!\left(\frac{5}{2}\right)\Big/\zeta\!\left(\frac{3}{2}\right)\,\left(\frac{T}{T_c}\right)^{5/2}$

The heat capacity:

$C_V = \frac{15}{4}\,Nk_B\,\zeta\!\left(\frac{5}{2}\right)\Big/\zeta\!\left(\frac{3}{2}\right)\,\left(\frac{T}{T_c}\right)^{3/2} \propto T^{3/2}$

This contrasts with the constant $C_V = \frac{3}{2}Nk_B$ above $T_c$ (equipartition). There is a cusp (discontinuity in the derivative) at $T_c$Characteristic of a phase transition.

Problem. Estimate $T_c$ for a gas of $N = 10^4$ rubidium-87 atoms confined in a harmonic trap with frequency $\omega_{\mathrm{ho} = 2\pi \times 100}$ Hz.

1. BEC is not a classical condensation: BEC is a purely quantum phenomenon driven by Bose statistics, not by interparticle interactions. An ideal Bose gas condenses, whereas a classical gas would not.
2. Finite-size effects: The critical temperature derived assumes the thermodynamic limit ($N \to \infty$, $V \to \infty$, $n$ fixed). For finite traps with $N \sim 10^4$, there are corrections of order $N^{-1/3}$.
3. Dimensionality matters: In 2D, the density of states is constant and the integral for $N_{\text{ex}}$ diverges at $\mu = 0$ only logarithmically. Strict BEC does not occur in 2D uniform gases (Mermin—Wagner—Hohenberg theorem).
4. Interactions modify $T_c$: Repulsive interactions slightly suppress $T_c$ relative to the ideal gas prediction. The shift is $\Delta T_c/T_c \propto (n^{1/3}a_s)$, where $a_s$ is the scattering length.

• Atom lasers: A BEC releases coherent matter waves, analogous to an optical laser. Coherence lengths exceeding 1 mm have been demonstrated.
• Precision measurement: BEC interferometry measures gravitational acceleration, rotations, and fundamental constants with extreme sensitivity.
• Superfluid helium: Liquid $^4$He below 2.17 K exhibits superfluidity, with approximately 10% of atoms in the condensate (strongly interacting, unlike the ideal gas model).
• Quantum simulation: Optical lattices loaded with BEC simulate the Hubbard model, enabling studies of quantum phase transitions.
• Slow light: Electromagnetically induced transparency in BEC reduces light speed to metres per second.

• Superconductivity: The BCS ground state is a condensate of Cooper pairs (composite bosons). The BCS—BEC crossover connects fermionic pairing to molecular BEC.
• Quantum field theory: BEC is an example of spontaneous symmetry breaking — the $U(1)$ phase symmetry of the matter field is broken, giving rise to a Goldstone mode (Bogoliubov phonon).
• Statistical mechanics: The BEC transition is a textbook example of a phase transition driven purely by statistics, requiring no interactions.

flowchart TD
    A[5_Bose Einstein Condensation] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Bose-Einstein condensation is the ultimate quantum overcrowding. When bosonic atoms are cooled below a critical temperature, their thermal de Broglie wavelengths overlap, and a macroscopic fraction collapses into the single lowest-energy quantum state. Unlike a classical gas, where particles are distinguishable and distribute across energies, bosons are indistinguishable and happily occupy the same state. The condensate fraction grows as temperature drops, like snow accumulating on the ground. BEC is a purely statistical phenomenon requiring no interparticle interactions, making it a textbook example of phase transition driven solely by quantum mechanics.

• Statistical Mechanics: Bose-Einstein statistics are derived from the grand canonical ensemble for particles with integer spin.

• The Grand Canonical Ensemble: The grand canonical ensemble is the natural framework for deriving Bose-Einstein and Fermi-Dirac distributions.

• Fermi Gas at Finite Temperature: The contrast between Bose and Fermi statistics leads to fundamentally different low-temperature behaviours.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

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