Part III, Chapter 1

Bose-Einstein Condensation

When bosonic atoms are cooled below a critical temperature, they macroscopically occupy a single quantum state — a new state of matter.

1.1 Critical Temperature for BEC

Bose-Einstein condensation occurs when the thermal de Broglie wavelength λ_dBbecomes comparable to the interparticle spacing. The phase space density reaches nλ_dB³ ≥ 2.612... and atoms "pile up" in the ground state.

Derivation 1: BEC Critical Temperature

Step 1. For an ideal Bose gas, the total number of atoms in excited states is:

$$N_{\text{ex}} = \int_0^\infty \frac{g(\epsilon)\, d\epsilon}{e^{(\epsilon - \mu)/k_BT} - 1}$$

where g(ε) = (2π)(2m)³/² V ε¹/² / (4π²ℏ³) is the 3D density of states. At the onset of BEC, μ → 0.

Step 2. Setting N_ex = N (all atoms must be accommodated) with μ = 0:

$$N = V\left(\frac{mk_BT_c}{2\pi\hbar^2}\right)^{3/2} \zeta(3/2)$$

where ζ(3/2) ≈ 2.612 is the Riemann zeta function.

Step 3. Solving for T_c:

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

For &sup8;&sup7;Rb at n = 10¹&sup4; cm&supmin;³: T_c ≈ 170 nK. Equivalently, nλ_dB³ = ζ(3/2) ≈ 2.612 at the transition.

1.2 Condensate Fraction

Figure. Momentum distribution of a Bose gas. Above T_c (left): broad thermal (Gaussian) distribution. Below T_c (right): a sharp condensate peak at p=0 emerges atop the thermal background.

Below T_c, a macroscopic fraction of atoms occupies the ground state. The condensate fraction grows as temperature decreases.

Derivation 2: Condensate Fraction vs Temperature

Step 1. Below T_c, the number of atoms in excited states is:

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

Step 2. The condensate fraction is:

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

At T = 0, all atoms are in the condensate. For a harmonic trap, the exponent changes to 3 (instead of 3/2): N₀/N = 1 - (T/T_c)³.

1.3 Gross-Pitaevskii Equation

For a weakly interacting BEC, the macroscopic wave function ψ(r,t) obeys the Gross-Pitaevskii equation (GPE) — a nonlinear Schrödinger equation where the interaction is parameterized by the s-wave scattering length a.

Derivation 3: Gross-Pitaevskii Equation from Mean-Field Theory

Step 1. The many-body Hamiltonian for N interacting bosons is:

$$\hat{H} = \sum_i \left(-\frac{\hbar^2}{2m}\nabla_i^2 + V_{\text{ext}}(\mathbf{r}_i)\right) + \sum_{i<j} V(\mathbf{r}_i - \mathbf{r}_j)$$

Step 2. Assume all N atoms are in the same single-particle state φ(r): Ψ(r&sub1;,...,r_N) = ∏φ(r_i). Replace the two-body potential with a contact pseudopotential V(r) = gδ(r), where g = 4πℏ²a/m.

Step 3. Minimizing the energy functional E[ψ] = 〈Ψ|H|Ψ〉 subject to normalization gives:

$$i\hbar\frac{\partial\psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V_{\text{ext}} + g|\psi|^2\right)\psi$$

where ψ(r,t) = √N φ(r,t) is normalized to the total atom number. The nonlinear term g|ψ|² represents the mean-field interaction energy.

1.3.1 Thomas-Fermi Approximation

For large N (strong interactions), the kinetic energy is negligible compared to the interaction and potential energies. Dropping the kinetic term:

$$|\psi(\mathbf{r})|^2 = \frac{\mu - V_{\text{ext}}(\mathbf{r})}{g}, \qquad \text{for } V_{\text{ext}} < \mu$$

In a harmonic trap, this gives an inverted parabola density profile — the characteristic shape observed in BEC experiments via time-of-flight imaging.

1.4 Quantized Vortices

A BEC is a superfluid: it flows without friction. When rotated, it cannot support rigid-body rotation. Instead, angular momentum enters through quantized vortices — lines of zero density around which the phase winds by 2π.

Derivation 4: Quantized Circulation

Step 1. Write the condensate wave function as ψ = |ψ|e^iS. The superfluid velocity is:

$$\mathbf{v}_s = \frac{\hbar}{m}\nabla S$$

Step 2. The circulation around any closed path is:

$$\oint \mathbf{v}_s \cdot d\mathbf{l} = \frac{\hbar}{m}\oint \nabla S \cdot d\mathbf{l} = \frac{\hbar}{m}\Delta S = \frac{h}{m}\ell$$

where &ell; is an integer (the winding number) since ψ must be single-valued.

Step 3. Each vortex carries one quantum of circulation κ = h/m. For &sup8;&sup7;Rb: κ ≈ 7.6 × 10&supmin;&sup8; m²/s. At the vortex core (radius ≈ ξ = 1/√(8πna), the healing length), the density vanishes. Vortex lattices (Abrikosov lattices) form in rapidly rotating BECs.

1.5 The Atom Laser

An atom laser is a coherent beam of atoms outcoupled from a BEC — the matter-wave analog of an optical laser. Just as a laser emits coherent photons, an atom laser emits atoms all in the same quantum state.

Derivation 5: RF Output Coupling Rate

Step 1. Atoms in a magnetically trapped BEC (state |F, m_F〉) are transferred to an untrapped state by an RF pulse at the Larmor frequency:

$$\omega_{\text{RF}} = \frac{g_F \mu_B B(\mathbf{r})}{\hbar}$$

Step 2. The RF resonance condition selects a shell of constant |B| within the trap. The outcoupling rate from the condensate is:

$$\frac{dN}{dt} = -\Omega_{\text{RF}}^2 \int |\psi(\mathbf{r})|^2 \delta(\omega_{\text{RF}} - g_F\mu_B B(\mathbf{r})/\hbar)\, d^3r$$

Step 3. The outcoupled atoms fall under gravity, forming a coherent beam:

The atom laser beam inherits the coherence of the BEC. Its brightness (atoms per mode) exceeds thermal sources by many orders of magnitude. First demonstrated by Ketterle et al. (MIT, 1997) using pulsed RF output coupling.

1.6 Bogoliubov Excitations

The elementary excitations of a weakly interacting BEC are described by Bogoliubov theory. Linearizing the GPE around the ground state gives the excitation spectrum.

1.6.1 Bogoliubov Dispersion Relation

Writing ψ = (√n₀ + δψ)e^-iμt/ℏ and linearizing:

$$\epsilon(k) = \sqrt{\frac{\hbar^2 k^2}{2m}\left(\frac{\hbar^2 k^2}{2m} + 2gn_0\right)}$$

This interpolates between two regimes:

• Phonon regime (kξ << 1): ε ≈ ℏck, linear dispersion with speed of sound c = √(gn₀/m). This is the collective sound mode of the condensate.
• Free-particle regime (kξ >> 1): ε ≈ ℏ²k²/(2m) + gn₀, quadratic dispersion shifted by the mean-field energy.
• Healing length: ξ = 1/√(8πna) sets the crossover scale. For typical BEC parameters: ξ ≈ 0.1-1 μm.

1.6.2 Landau Critical Velocity

The linear phonon dispersion at low k implies a minimum velocity v_c for creating excitations:

$$v_c = \min_k \frac{\epsilon(k)}{\hbar k} = c = \sqrt{\frac{gn_0}{m}}$$

Objects moving through the BEC at v < v_c cannot create excitations and experience no drag — this is superfluidity. The Landau criterion has been verified by moving laser beams through BECs (Raman et al., 1999).

1.6.3 Quantum Depletion

Even at T = 0, quantum fluctuations (zero-point motion of Bogoliubov modes) deplete atoms from the condensate. The depletion fraction is:

$$\frac{N - N_0}{N} = \frac{8}{3\sqrt{\pi}}(na^3)^{1/2}$$

For dilute alkali BECs (na³ ~ 10&supmin;&sup6;), the depletion is ~1%. For liquid helium (na³ ~ 0.2), the depletion is ~90% — only ~10% of atoms are in the condensate.

1.7 Coherence and Interference

The macroscopic coherence of a BEC manifests spectacularly in matter-wave interference experiments.

1.7.1 Interference of Two BECs

Andrews et al. (MIT, 1997) observed high-contrast interference fringes when two independently prepared BECs were released and overlapped. The fringe spacing:

$$\lambda_{\text{fringe}} = \frac{ht}{md}$$

where d is the initial separation and t is the expansion time. This demonstrates that each BEC has a well-defined macroscopic phase, even though the relative phase between independently prepared condensates is random (it spontaneously appears upon measurement).

1.7.2 First-Order Coherence

The first-order correlation function g¹(r) = 〈&hat;ψ†(r')&hat;ψ(r' + r)〉/n measures the phase coherence. For a BEC below T_c:

• 3D BEC: g¹(r) → N₀/N as r → ∞ (off-diagonal long-range order, ODLRO)
• 2D quasi-condensate: g¹(r) decays algebraically as r^{-1/(n_sλ_T²)}(Berezinskii-Kosterlitz-Thouless phase)
• 1D gas: g¹(r) decays exponentially — no true long-range order (Mermin-Wagner theorem), but quasi-condensate behavior exists

1.7.3 Momentum Distribution and Time-of-Flight

Switching off the trap and allowing the cloud to expand ballistically (time-of-flight, TOF) maps the in-trap momentum distribution to a real-space density distribution. For a BEC, the narrow momentum distribution produces a sharp peak sitting on a broader thermal background — the iconic signature of BEC first photographed in 1995.

1.8 Spinor BECs and Multicomponent Condensates

When atoms are trapped in optical (non-magnetic) traps, their spin degree of freedom is liberated. A spin-F atom has 2F+1 magnetic sublevels, each of which can condense independently, forming a spinor BEC.

Spinor BEC Phenomena

• Spin domains: Different magnetic sublevels spatially separate, forming domains analogous to ferromagnetic domains
• Spin mixing dynamics: Coherent interconversion between spin states driven by spin-dependent interactions (e.g., 2|m_F=0〉 ↔ |+1〉 + |-1〉)
• Magnetic ordering: Ferromagnetic (F=1 &sup8;&sup7;Rb) or antiferromagnetic (F=1 ²³Na) ground states depending on the sign of spin-dependent interactions
• Topological defects: Spin textures, skyrmions, monopoles, and knots in the spinor order parameter

1.8.1 Two-Component BECs

Two hyperfine states of the same atom (or two different atomic species) form a binary mixture with coupled Gross-Pitaevskii equations. Depending on the inter- and intra-species scattering lengths, the mixture can be miscible (overlapping) or immiscible (phase-separated).

The miscibility condition is determined by:

$$g_{12}^2 < g_{11} g_{22} \quad \text{(miscible)}, \qquad g_{12}^2 > g_{11} g_{22} \quad \text{(immiscible)}$$

Two-component BECs serve as simulators for cosmological phase transitions (domain wall formation via the Kibble-Zurek mechanism) and as platforms for quantum magnetism.

Applications

Quantum Simulation

BECs in optical lattices simulate strongly correlated quantum systems — the superfluid-to-Mott insulator transition, Bose glass phases, and artificial gauge fields. This enables the study of condensed matter models that are computationally intractable.

Precision Measurement

BEC-based atom interferometers achieve extreme sensitivity for measuring gravity, rotations, and fundamental constants. Space-based BEC experiments (e.g., NASA CAL on the ISS) exploit microgravity for extended free-fall times.

Analog Gravity

Phonons in a BEC obey a wave equation analogous to a scalar field in curved spacetime. Supersonic flow of a BEC creates an acoustic analog of a black hole horizon, enabling laboratory studies of Hawking radiation (observed by Steinhauer, 2016).

Atom Lasers and Interferometry

Coherent atom beams from atom lasers enable matter-wave interferometry with unprecedented sensitivity. Proposed applications include tests of the equivalence principle and detection of gravitational waves at low frequencies.

1.9 Exotic Bose-Einstein Condensates

BEC has been achieved in a remarkable variety of systems beyond alkali atoms.

Dipolar BEC (Cr, Dy, Er)

Chromium (Griesmaier et al., 2005), dysprosium, and erbium have large magnetic dipole moments (μ up to 10 μ_B). The long-range, anisotropic dipole-dipole interaction leads to anisotropic expansion, d-wave collapse, and self-bound quantum droplets — a new state of matter stabilized by quantum fluctuations.

Quantum Droplets

First observed by Kadau et al. (2016) in dysprosium. These are self-bound liquid-like droplets where mean-field attraction is balanced by the Lee-Huang-Yang quantum fluctuation correction. They exist without any external trapping potential and have been observed in both dipolar gases and Bose-Bose mixtures.

Supersolids

In 2019, three groups (Innsbruck, Stuttgart, Pisa) simultaneously observed supersolid behavior in dipolar BECs: a state that simultaneously breaks continuous translational symmetry (crystal) and gauge symmetry (superfluid). This elusive phase had been sought for decades in helium-4.

Photon BEC

Klaers et al. (Bonn, 2010) achieved BEC of photons in a dye-filled optical microcavity. The curved mirrors create a harmonic potential for 2D photons, and dye molecules provide thermalization. The critical photon number is ~77,000 at room temperature.

Exciton-Polariton BEC

In semiconductor microcavities, excitons coupled to cavity photons form polaritons — bosonic quasiparticles with extremely light effective mass (~10&supmin;&sup5; m_e). Polariton BEC occurs at relatively high temperatures (up to room temperature in organic microcavities).

BEC in Microgravity

NASA's Cold Atom Lab on the International Space Station creates BECs in microgravity, enabling longer free-fall expansion times and weaker trapping potentials. The MAIUS sounding rocket experiment achieved BEC during 6 minutes of weightlessness.

1.10 Low-Dimensional BEC Physics

Confining atoms tightly in one or two directions while leaving them free in the remaining direction(s) creates effectively 2D or 1D quantum gases with qualitatively different physics.

1.10.1 Two-Dimensional BEC and BKT Transition

In 2D, true long-range order is destroyed by thermal fluctuations (Mermin-Wagner theorem). Instead, a Berezinskii-Kosterlitz-Thouless (BKT) transition occurs: below T_BKT, vortex-antivortex pairs are bound, and the system has quasi-long-range order with algebraically decaying correlations. Above T_BKT, free vortices proliferate and superfluidity is destroyed. The BKT transition was observed in cold atoms by Hadzibabic et al. (ENS Paris, 2006).

1.10.2 One-Dimensional Bose Gas

In 1D, the Yang-Yang model provides exact thermodynamics. The key parameter is γ = mg/(ℏ²n_1D). For γ >> 1 (Tonks-Girardeau limit), bosons "fermionize" — their spatial distribution becomes identical to non-interacting fermions. Kinoshita et al. (Penn State, 2004) observed this crossover. The 1D Bose gas does not thermalize after a quantum quench (Kinoshita et al., 2006), demonstrating integrability.

Key Equations Summary

Critical Temperature (uniform):

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

Condensate Fraction:

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

Gross-Pitaevskii Equation:

$$i\hbar\frac{\partial\psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V_{\text{ext}} + g|\psi|^2\right)\psi$$

Bogoliubov Dispersion:

$$\epsilon(k) = \sqrt{\frac{\hbar^2 k^2}{2m}\left(\frac{\hbar^2 k^2}{2m} + 2gn_0\right)}$$

Quantized Circulation:

$$\oint \mathbf{v}_s \cdot d\mathbf{l} = \frac{h}{m}\ell$$

Speed of Sound:

$$c = \sqrt{gn_0/m}, \qquad g = 4\pi\hbar^2 a/m$$

Historical Context

1924-25 — Bose & Einstein: Predicted that an ideal gas of bosons would undergo a phase transition at low temperature, with macroscopic ground-state occupation.

1938 — London: Proposed that superfluidity in liquid &sup4;He is related to Bose-Einstein condensation, though strong interactions deplete the condensate to ~10%.

1961 — Gross & Pitaevskii: Independently derived the nonlinear mean-field equation describing the condensate wave function of weakly interacting bosons.

1995 — Cornell, Wieman & Ketterle: First achieved BEC in dilute alkali gases (&sup8;&sup7;Rb at JILA, ²³Na at MIT). Nobel Prize 2001.

1999 — Madison et al.: First observation of quantized vortices in a rotating BEC at ENS Paris.

2001 — Greiner et al.: Demonstrated the superfluid-to-Mott insulator quantum phase transition using BEC in an optical lattice.

Interactive Simulation

This simulation visualizes the BEC phase transition, condensate fraction, the Thomas-Fermi density profile, and a quantized vortex structure.

BEC: Phase Transition, Density Profile, and Vortices

Python

script.py98 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import zeta as riemann_zeta

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# Plot 1: Condensate fraction vs temperature
ax1 = axes[0, 0]
T_ratio = np.linspace(0, 1.5, 500)

# Uniform gas: N0/N = 1 - (T/Tc)^(3/2)
N0_uniform = np.where(T_ratio < 1, 1 - T_ratio**(3/2), 0)
# Harmonic trap: N0/N = 1 - (T/Tc)^3
N0_harmonic = np.where(T_ratio < 1, 1 - T_ratio**3, 0)

ax1.plot(T_ratio, N0_uniform, color="#a78bfa", linewidth=2, label="Uniform gas")
ax1.plot(T_ratio, N0_harmonic, color="#c084fc", linewidth=2, label="Harmonic trap")
ax1.axvline(x=1, color="gray", linestyle="--", alpha=0.5, label="Tc")
ax1.fill_between(T_ratio, N0_uniform, alpha=0.1, color="#a78bfa")
ax1.set_xlabel("T / Tc", color="white")
ax1.set_ylabel("Condensate fraction N0/N", color="white")
ax1.set_title("BEC Condensate Fraction", color="#c4b5fd", fontsize=14)
ax1.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax1.tick_params(colors="white")
for spine in ax1.spines.values():
    spine.set_color("#7c3aed")

# Plot 2: Thomas-Fermi density profile vs Gaussian (ideal gas)
ax2 = axes[0, 1]
r = np.linspace(-3, 3, 500)
R_TF = 2.0  # Thomas-Fermi radius

# Thomas-Fermi: inverted parabola
n_TF = np.maximum(1 - (r / R_TF)**2, 0)
# Gaussian (ideal gas ground state)
sigma = 1.0
n_gauss = np.exp(-r**2 / (2 * sigma**2))
n_gauss = n_gauss / np.max(n_gauss)

ax2.plot(r, n_TF, color="#a78bfa", linewidth=2, label="Thomas-Fermi (interacting)")
ax2.plot(r, n_gauss, color="#f97316", linewidth=2, linestyle="--", label="Gaussian (non-interacting)")
ax2.fill_between(r, n_TF, alpha=0.15, color="#a78bfa")
ax2.set_xlabel("Position r / a_ho", color="white")
ax2.set_ylabel("Density (normalized)", color="white")
ax2.set_title("BEC Density Profile", color="#c4b5fd", fontsize=14)
ax2.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax2.tick_params(colors="white")
for spine in ax2.spines.values():
    spine.set_color("#7c3aed")

# Plot 3: Vortex density profile (cross-section)
ax3 = axes[1, 0]
x = np.linspace(-5, 5, 400)
y = np.linspace(-5, 5, 400)
X, Y = np.meshgrid(x, y)
R = np.sqrt(X**2 + Y**2)

# Vortex with winding number l=1
xi = 0.5  # healing length
n_bulk = np.maximum(1 - R**2/16, 0)  # TF background
n_vortex = n_bulk * R**2 / (R**2 + xi**2)  # vortex core depletion
phase = np.arctan2(Y, X)

im3 = ax3.contourf(X, Y, n_vortex, levels=30, cmap="magma")
ax3.set_xlabel("x / xi", color="white")
ax3.set_ylabel("y / xi", color="white")
ax3.set_title("Vortex Density |psi|^2 (l=1)", color="#c4b5fd", fontsize=14)
ax3.set_aspect("equal")
ax3.tick_params(colors="white")
for spine in ax3.spines.values():
    spine.set_color("#7c3aed")

# Plot 4: Phase space density trajectory to BEC
ax4 = axes[1, 1]
# Typical experimental trajectory
stages = ["Oven", "Zeeman\nslower", "MOT", "Optical\nmolasses", "Magnetic\ntrap", "Evaporation", "BEC"]
psd = [1e-12, 1e-10, 1e-6, 1e-5, 1e-4, 1e-1, 2.612]
T_stages = [600, 0.5, 1e-4, 1e-5, 5e-5, 1e-7, 5e-8]

ax4.semilogy(range(len(stages)), psd, "o-", color="#a78bfa", linewidth=2, markersize=8)
ax4.axhline(y=2.612, color="#c084fc", linestyle="--", alpha=0.5, label="BEC threshold")
ax4.set_xticks(range(len(stages)))
ax4.set_xticklabels(stages, color="white", fontsize=8, rotation=30)
ax4.set_ylabel("Phase space density", color="white")
ax4.set_title("Road to BEC: Phase Space Density", color="#c4b5fd", fontsize=14)
ax4.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax4.tick_params(colors="white")
for spine in ax4.spines.values():
    spine.set_color("#7c3aed")

plt.tight_layout()
plt.savefig("output.png", dpi=150, bbox_inches="tight", facecolor="#0a0a0a")
plt.show()
print("Plot saved: BEC - condensate fraction, Thomas-Fermi profile, vortex, phase space trajectory.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Conceptual Questions

Q1: Is BEC a phase transition?

Yes — in the thermodynamic limit, BEC is a genuine phase transition (for 3D). The specific heat has a cusp at T_c. However, it is unique among phase transitions because it occurs even in the ideal (non-interacting) gas; interactions modify the details but are not required. In 2D, the transition is replaced by the BKT transition.

Q2: What is the difference between BEC and a laser?

Both involve macroscopic occupation of a single quantum mode. A laser has photons in a single electromagnetic mode (stimulated emission), while BEC has atoms in a single motional state. The atom laser (coherent output from a BEC) is the matter-wave analog of an optical laser.

Q3: Why does the Thomas-Fermi profile have sharp edges?

In the TF approximation, the density drops to zero where the external potential equals the chemical potential. The edge is sharp because kinetic energy is neglected. In reality, the density falls smoothly to zero over the healing length ξ — a thin surface layer.

Further Topics

BEC Frontiers

• Artificial gauge fields: Rotating BECs or Raman-dressed BECs experience effective magnetic fields, enabling simulation of the fractional quantum Hall effect and topological phases with neutral atoms.
• Solitons: Dark solitons (density dips) and bright solitons (self-bound wavepackets) are nonlinear excitations of the GPE. They have been created, observed, and tracked in BECs, testing nonlinear wave dynamics.
• Quantum turbulence: Tangled vortex lines in a BEC constitute quantum turbulence. Unlike classical turbulence, vorticity is quantized. The Kolmogorov energy spectrum has been observed in BEC turbulence experiments.
• BEC on atom chips: Miniaturized BEC experiments using microfabricated magnetic traps on chips enable portable quantum sensors, compact atom interferometers, and space-based experiments (CAL on ISS, BECCAL mission).