Questions: Bose-Einstein Condensation and Order Parameter

Question 1

A system of N = 10²³ bosons is cooled below Tc, and approximately 50% of all particles pile into the single ground state. A student says: 'This is just ordinary cooling — particles always settle into the lowest energy state when temperature drops.' What does this miss about Bose-Einstein condensation?

• A. The student is correct — BEC is simply the quantum version of classical particles cooling to the ground state
• B. BEC is distinct because a macroscopic number of particles share not just the same energy but the same quantum state, including the same phase, producing long-range phase coherence that has no classical analog and enables phenomena like superfluidity
• C. BEC differs from ordinary cooling only in requiring temperatures near absolute zero, whereas classical cooling can occur at room temperature
• D. BEC is distinct because the particles stop obeying the Bose-Einstein distribution below Tc and instead follow Maxwell-Boltzmann statistics

Answer: (B)

Classical cooling means particles occupy lower-energy states, but different particles have different phases — they are incoherent. In BEC, a macroscopic fraction (O(N)) of particles share the same single quantum state, including a definite quantum phase. This coherence is qualitatively new: it cannot be captured by any classical distribution. The ground state occupation goes from O(1) to O(N) — a qualitative transition, not a smooth continuation. This macroscopic phase coherence is what enables superfluidity, where the condensate flows without viscosity because scattering it requires disrupting the coherent quantum state of O(N) particles simultaneously.

Question 2

What role does spontaneous symmetry breaking play in Bose-Einstein condensation, and what symmetry is broken?

• A. Below Tc, the system spontaneously picks a definite phase φ for the condensate wavefunction ψ = √(n₀)e^(iφ), even though all choices of phase are equivalent — this selection of a specific phase from the continuous U(1) symmetry is the spontaneous breaking
• B. BEC breaks the translational symmetry of the gas, causing the particles to crystallize into a regular lattice
• C. The broken symmetry is time-reversal symmetry, because the condensate has a preferred direction of particle flow
• D. U(1) symmetry breaking below Tc means the total particle number N is no longer conserved in the condensate

Answer: (A)

The condensate wavefunction ψ = √(n₀)e^(iφ) is the order parameter. Above Tc, ψ = 0 (no definite phase). Below Tc, ψ ≠ 0 — the system selects a specific phase φ from a continuous circle of equivalent choices. This is spontaneous U(1) symmetry breaking: the Hamiltonian is symmetric under ψ → e^(iα)ψ (any global phase shift), but the ground state below Tc is not — it has a definite φ. This is the same abstract structure as ferromagnetism breaking rotational symmetry by choosing a preferred spin direction, but here applied to a quantum many-body system with a complex order parameter.

Question 3

The condensate order parameter ψ is zero above Tc and nonzero below Tc, with |ψ|² equal to the fraction of particles in the ground state.

• T. True
• F. False

Answer: True

The condensate wavefunction ψ = √(n₀)e^(iφ) is defined so that |ψ|² = n₀, the density of particles in the ground state. Above Tc, n₀ = O(1)/V → 0 in the thermodynamic limit, so ψ = 0 — the order parameter vanishes. Below Tc, a finite fraction N₀/N of all particles occupies the ground state, making n₀ = N₀/V a macroscopic density, so |ψ|² ≠ 0. The sharpness of this transition — from ψ = 0 to ψ ≠ 0 at Tc — identifies BEC as a genuine phase transition, not a smooth crossover.

Question 4

Bose-Einstein condensation occurs in any collection of bosons whenever they are cooled sufficiently, regardless of their density.

• T. True
• F. False

Answer: False

BEC requires that the thermal de Broglie wavelength λ_dB = h/√(2πmkT) become comparable to the interparticle spacing: nλ_dB³ ≈ 2.612. This means that both high enough density n and low enough temperature T are required for the condition to be met. At extremely low density, T would need to be so close to absolute zero that the condition may be experimentally unachievable. The famous 1995 experiments with ultracold atomic gases achieved BEC by combining laser cooling and evaporative cooling at densities around 10¹⁴ atoms/cm³ — a much lower density than ordinary gases, but achievable precisely because the temperature could be pushed to nanokelvin range.

Question 5

Explain why phase coherence — rather than mere macroscopic ground-state occupation — is the key to superfluidity in a Bose-Einstein condensate.

Model answer: When a macroscopic number of particles share the same quantum state, they share the same phase. A superfluid flows without viscosity because viscous flow requires momentum transfer to the walls — which means scattering the condensate into a different state. But scattering O(N) phase-coherent particles simultaneously requires a large energy barrier; small perturbations cannot disturb the condensate. Ground-state occupation alone (without phase coherence) would not produce this protection — what makes the condensate rigid to perturbation is the long-range order of the phase, which turns the condensate into a single macroscopic quantum object rather than a collection of individual ground-state particles.

This is the conceptual core connecting BEC to superfluidity. In normal matter, viscosity arises from particles scattering off each other and the container walls — momentum gets randomized. In a superfluid, the condensate flows as a coherent quantum entity: to scatter it requires changing the quantum state of all N₀ condensate particles simultaneously, costing O(N₀) energy. Below a critical velocity, no excitation is energetically available to do this, and the flow is dissipationless. The same phase-coherence argument underlies Cooper pair condensation in superconductors and the coherent state of photons in a laser.