Ultracold quantum gas · Cooper pairing

BEC–BCS Crossover

Opposite-spin fermions form luminous pairs at low temperature. Sweep the attraction from the BCS side through unitarity and into the BEC side: huge overlapping Cooper pairs tighten into composite bosonic molecules, lock their phases, and collect into a central condensate.

Many-body regimeUnitary Fermi gas

Paired fraction

N_{\mathrm{pair}}/N

$0.82$

Condensate fraction

N_0/N

$0.65$

Pair size $k_F\xi_{\mathrm{pair}}=1.50$

Pairing gap $\Delta=0.55\,E_F$

Critical temperature $T_c=0.35\,T_F$

A Cooper pair is not a pair of ordinary bosons. It is built from two fermions; only tightly bound pairs on the BEC side behave approximately as composite bosons. Pair formation and condensation are also distinct: pairs can exist before lower temperature establishes long-range phase coherence.

Physics tutorial

The BEC–BCS crossover: two limits of the same fermion pair

BackgroundIdentical fermions cannot occupy one single-particle state, yet opposite-spin fermions can develop pair correlations through attraction. In the weak-coupling limit those correlations are enormous Cooper pairs; in strong coupling they become tightly bound diatomic molecules.

Why it mattersThe crossover places BCS pairing in superconductors and Bose–Einstein condensation inside one continuous framework, while separating pair formation from condensation.

Start with the essentials

Focus question: As pair size shrinks from larger than the interparticle spacing to smaller than it, how can superfluid order remain continuous?
One-sentence intuition: The BEC side does not pair elementary bosons. Two fermions first form a composite molecule that behaves approximately as a boson, and those molecules then establish a shared phase.

Core mathematical model

Crossover control parameter

\gamma=\frac{1}{k_Fa}

$\gamma<0$ is the weakly attractive BCS side, $\gamma=0$ is unitarity, and $\gamma>0$ enters the BEC side with a two-body bound state.

BCS paired state

\lvert\mathrm{BCS}\rangle=\prod_{\mathbf k}\left(u_{\mathbf k}+v_{\mathbf k}c^\dagger_{\mathbf k\uparrow}c^\dagger_{-\mathbf k\downarrow}\right)\lvert0\rangle

Each momentum mode superposes an empty state with an opposite-momentum, opposite-spin pair, so many Cooper pairs overlap strongly in real space.

Condensate order parameter

\Psi_{\mathrm{pair}}(\mathbf r)=\sqrt{n_0(\mathbf r)}e^{i\phi(\mathbf r)}

$n_0$ is condensate density, while long-range agreement of $\phi$ supplies superfluid phase coherence. Pairing alone does not guarantee condensation.

Common difficulties

A Cooper pair is not two bosons

Typical misconceptionBecause the endpoint is a BEC, both particles shown must already be bosons.

Better mental modelThe cyan and magenta particles are two spin components of fermionic atoms. Under strong attraction, a tightly bound fermion pair behaves approximately as one composite boson.

Paired fraction is not condensate fraction

Typical misconceptionAs soon as a bond appears between two particles, the system has condensed.

Better mental modelPairing signals a local two-body correlation. Condensation also requires macroscopic phase agreement among many pairs. With heating, the coherent core can vanish while some tightly bound pairs survive.

Run the experiment

01
Cross the full interaction range at low temperature
Hold $T/T_F$ low and sweep $1/(k_Fa)$ slowly from negative to positive.
What to observe: Long overlapping bonds dominate the BCS side; the gap and coherence peak near unitarity; tight two-particle molecules collect near the center on the BEC side.
02
Separate pairing from condensation
Choose Molecular BEC and gradually raise $T/T_F$ .
What to observe: The phase wavefronts and central condensate fade first, while some short pair bonds remain: preformed pairs can survive above the superfluid transition.
03
Isolate the two levels of order
Turn off Cooper-pair amplitude and condensate phase field, then restore them one at a time.
What to observe: Bonds encode local two-body correlation; coherent wavefronts and synchronized pulses encode global many-body phase order.