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The BCS-BEC crossover is a phenomenon exhibited by interacting fermion systems with tunable interactions that smoothly connects a state exhibiting Bardeen-Cooper-Schrieffer superconductivity to one exhibiting Bose Einstein condensation (BEC). On the BCS side of the crossover, the system is in a superconducting state composed of Cooper pairs, while on the BEC side the Fermions pair up, forming molecules which condense into a BEC.

Theoretical foundation
It was recognized in 1968 that the BCS ansatz could also be used to describe the BEC state. In 1980, Anthony Leggett and separately Philippe Nozières and Stephan Schmitt-Rink developed descriptions of the BCS-BEC crossover at zero temperature. The 1986 discovery of High-temperature superconductivity in the cuprates renewed interest in the BCS-BEC crossover phenomenon. It was thought this theory might be applicable, due to the short-coherence length and strong interactions observed in these systems. Work extending the understanding of the BCS-BEC crossover to finite temperature was performed by a number of researchers, including Mohit Randeria.

Besides these mean-field approaches, a BCS-BEC crossover has also been studied in the Fermi-Hubbard model with attractive interactions.

Mathematical Description
Consider a system with two species of fermions interacting with each other through contact interactions. The strength of the contact interactions is parameterized by the scattering length, $$a$$, and described by the pseudopotential $$V(r)=\frac{4\pi\hbar^2}{m}a\,\delta(r)$$,where $$m$$ is the mass of each particle and $$r$$ is the separation.

The BCS ansatz is given by
 * $$ |\Psi \rangle = \prod_k \left(u_k + v_k c^\dagger_{k\uparrow} c^\dagger_{-k \downarrow} \right) | 0 \rangle, $$

where $$u_k$$ and $$v_k$$ are variational parameters and $$c^\dagger_{k \sigma}$$ is the creation operator for a fermion with momentum $$k$$ and spin $$\sigma$$.

Zero tempereature
The variatonal parameters are determined by minimizing the expectation value of the energy of the system. They can be expressed in terms of the chemical potential, $$\mu$$, and the superconducting gap, $$\Delta$$.
 * $$u_k^2 = \frac{1}{2} \left(1 - \frac{\xi_k}{E_k} \right)$$
 * $$v_k^2 = \frac{1}{2} \left(1 + \frac{\xi_k}{E_k} \right),$$

where $$\xi_k = \epsilon_k \mu$$ for $$\epsilon_k$$ the dispersion relation and $$E_k = \sqrt{\xi_k^2 + \Delta^2}$$.

To determine the chemical potential and the superconducting gap, two equations must be solved simulatoneously
 * $$ n = 2 \int \frac{d^3k}{(2\pi)^3} v_k^2 $$
 * $$ -\frac{m}{4\pi \hbar^2 a} = \int \frac{d^3k}{(2\pi)^3} \left( \frac{1}{2 E_k} - \frac{1}{2\epsilon_k} \right).$$

The BCS limit
In the BCS limit, the chemical potential and superconducting gap are given by
 * $$ \mu = E_f$$
 * $$ \Delta = \frac{8}{e^2} e^{-\pi / 2k_f|a|},$$

where $$E_f$$ and $$k_f$$ are the Fermi energy and momentum respectively.

The BEC limit
In the BEC limit, the chemical potential and superconducting gap are given by
 * $$ \mu = -\frac{\hbar^2}{2ma^2} + \frac{\pi \hbar^2 a n}{m}$$
 * $$ \Delta = \sqrt{\frac{16}{3\pi}} \frac{E_f}{\sqrt{k_f a}}.$$

The chemical potential contribution is divided into two terms. The first reflects the binding energy of the fermion pairs, and the second is a mean-field shift due to interactions between fermion pairs (i.e. bosonic molecules).

Ultracold atoms
Realization of BCS-BEC crossover physics in ultracold atomic systems was enabled by the 2003 realization of moleculer Bose-Einstein condensates of lithium and potassium near a Feshbach resonance. Near the Feshbach resonance, the strength of the interaction between the fermionic atoms could be controlled by changing the strength of an applied magnetic field.

It was found that these systems exhibit extremely high superfluid transition temperatures at the Feshbach resonance. The critical temperature for a 3D gas is $$T_c = 0.16 T_f$$, which is a larger fraction of $$T_f$$ than is observed even in high-temperature superconductors.

This point is unique because the only length scale in the problem is that given by the interparticle spacing, and is frequently referred to as unitarity, or a unitary Fermi gas. The unitary Fermi gas also describes physics relevant to the quark-gluon plasma and neutron stars, and therefore these experimental studies are considered examples of quantum simulation. In the unitary Fermi gas, many thermodynamics quantities can be expressed as the product of universal constants multiplied by the Fermi energy. For example, the equation of state of the unitary Fermi gas is given by $$\mu = \xi E_f$$, where $$\mu$$ is the chemical potential of the gas, $$\xi$$ is the Bertsch parameter, and $$E_f$$ is the Fermi energy. The Bertsch parameter is the universal dimensionsless number mentioned before, and gives essentially all thermodynamic properties of the unitary Fermi gas.

Later experiments explored a variety of other properties of continuum Fermi gases in the BEC-BCS crossover, including studies of spin-imbalanced systems searching for FFLO superconductivity, exploration of pseudogap above the superfluid transition, and the thermodyanmic properties of these systems.

Exciton-polaritons
Exciton-polariton