Ultracold quantum gas · Cooper pairing

BEC–BCS Crossover

How do loose Cooper pairs turn smoothly into tightly bound molecules that condense? By shrinking steadily. As the attraction grows, huge overlapping Cooper pairs tighten into small composite molecules that behave roughly like bosons and share one phase. Superfluid order persists through the whole BCS-to-BEC crossover, with no jump along the way.

At low temperature, fermions of opposite spin form glowing pairs. Sweep the attraction from the BCS side through unitarity to the BEC side: the huge, overlapping Cooper pairs shrink into composite bosonic molecules, lock their phases and gather into a condensate at the center.

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 made of two fermions, not two ordinary bosons. Only the tightly bound pairs on the BEC side behave approximately like composite bosons. Pairing and condensation are separate steps: pairs can form first, and long-range phase coherence sets in at a lower temperature.

Physics tutorial

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

BackgroundIdentical fermions cannot share a single-particle state, but fermions with opposite spins can form pair correlations when they attract each other. With weak attraction those correlations are enormous Cooper pairs; with strong attraction they become tightly bound diatomic molecules.

Why it mattersThe crossover puts BCS pairing in superconductors and Bose–Einstein condensation in one continuous picture, and it separates forming pairs from condensing them.

Start with the essentials

Focus question: As pairs shrink from larger than the spacing between particles to smaller than it, how does the superfluid order stay continuous?
One-sentence intuition: On the BEC side, elementary bosons do not pair up. Two fermions first form a composite molecule that acts roughly like a boson, and these molecules then share a common 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 is a superposition of empty and occupied by a pair with opposite momenta and opposite spins. As a result, 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 the condensate density. Long-range agreement of $\phi$ gives 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 fermionic atoms in two spin components. 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 modelA bond means a local two-body correlation has formed. Condensation also needs many pairs to share one macroscopic phase. On 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. On the BEC side, tight two-particle molecules gather near the center.
02
Separate pairing from condensation
Choose Molecular BEC and gradually raise $T/T_F$ .
What to observe: The phase wavefronts and the central condensate fade first, but some short pair bonds remain. Preformed pairs can survive above the superfluid transition.
03
Look at each level of order separately
Turn off Cooper-pair amplitude and Condensate phase field, then turn them back on one at a time.
What to observe: The bonds show local two-body correlation. The coherent wavefronts and synchronized pulses show global many-body phase order.

BEC–BCS Crossover

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

Core mathematical model

Crossover control parameter

BCS paired state

Condensate order parameter

Common difficulties

A Cooper pair is not two bosons

Paired fraction is not condensate fraction

Run the experiment

Cross the full interaction range at low temperature

Separate pairing from condensation

Look at each level of order separately

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The evidence for dilute-gas condensation

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