User:Looseyygooseyy/Fermionic condensate

A fermionic condensate is a superfluid state of matter formed by cooling a dilute gas of fermionic particles to ultracold temperatures. It is closely related to the Bose–Einstein condensate, a superfluid formed by bosonic atoms under similar conditions. In 2003, the first pure fermionic condensate was created using potassium-40 atoms by Deborah S. Jin and her team at the University of Colorado Boulder.

Fermion pairing
Historically, the most common way of pairing fermions together is through the creation of Cooper pairs. In 1957, L.N. Cooper showed that given a pair of electrons near a Fermi surface in the presence of an attractive potential at low temperatures, the paired state of electrons will have a lower energy than the Fermi energy. These pairs will exhibit the behavior of an integer spin particle, however, in an ensemble of fermions (like those found in an atomic gas) the Cooper pairs will spatially overlap and thus cannot be considered a composite boson. Another way of pairing fermions is to create a two-body bound state of fermionic particles. That is, given fermionic atoms in a weak attractive potential, one can form diatomic molecules and condense to a BEC below a certain critical temperature.

In an equilibrium state, there is an interest in what happens when the binding energy of the molecules becomes less than the Fermi energy- or equivalently- when the interaction energy of a Cooper pair state increases and approaches the Fermi energy. This quantum phase transition is known as the BSC-BEC crossover, the region at which pairs of fermions will exhibit properties of Cooper pairs and properties of diatomic molecules. In this state, it is thought possible to create a superconducting BEC from a dilute fermi gas.

Feshbach resonances
In experiments on ultracold gases, Feshbach resonances are used to explore many-body physics in the strongly interacting regime, when the scattering length $$a$$ exceeds the interparticle spacing; in particular for the study of the BCS-BEC crossover and the observation of fermion superfluidity A magnetically tuned Feshbach resonance can be described by a simple expression for the s-wave scattering length a as a function of the magnetic field B,


 * $$a=a_{bg}\left(1-\frac{\Delta}{B-B_0}\right)$$,

where $$a_{bg}$$ is the background scattering length, $$B_0$$ is the magnetic field strength where resonance occurs, and $$\Delta$$ is the resonance width. Given two fermions in a vacuum, pairing on the side with $$a > 0$$ can be understood as diatomic molecule formation, and superfluidity results from molecular Bose-Einstein condensation. Conversely, pairing with $$a < 0$$, the interaction is a many-body effect and the ground state of the system at zero temperature is a fermionic superfluid.

Feshbach resonances also prove useful in controlling the atom-atom interactions between alkali-metal atoms held in optical lattices. An analytic solution for two interacting atoms trapped in a harmonic trap can be modeled using a regularized delta-function potential. This approximation of the actual interaction potential is valid in the Wigner threshold regime. The eigenenergies for the relative motion in a symmetric harmonic trapping potential are given by


 * $$\frac{a}{\sigma} = \frac{1}{2}\frac{\Gamma(-E/2 + 1/4)}{\Gamma(-E/2 +3/4)}$$,

where $$\Gamma$$ is the Gamma function, the energy $$E$$ is in units of $$\hbar\omega$$, and $$\sigma$$ is the harmonic oscillator length.

Recent experiments on Fermi gases have focused on two-component spin mixtures of single isotopes like $${}^6$$Li or $${}^{40}$$K with resonant s-wave interactions (i.e. $$a \to\pm\infty$$), and can be performed without significant loss. For example, Deborah S. Jin's group cooled $${}^{40}$$K above the magnetic resonance and then swept across the Feshbach resonance to cool the sample to a molecular condensate. This is because unlike other alkali atoms, potassium isotopes cooled in the sub-Doppler regime are hindered by the presence of heating forces and photon reabsorption due to their excited state hyperfine splitting $$\Delta$$ being approximately equal to its natural linewidth $$\Gamma$$.

Characteristics of superfluidity
For simplicity, we consider a mixture of two types of fermions of the same isotope in different internal magnetic sublevels. The transition temperature of a dilute fermi gas is given by,

$$kT_c \approx 0.28 \epsilon_{F} e^{-1 / N(\epsilon_{F})|U_{0}|}$$.

Here, $$N(\epsilon_{F})|U_{0}|$$ is the interaction strength and $$\epsilon_{F}$$ is the fermi energy. This approximation was derived from the Lippmann–Schwinger equation, reducing contributions based on Fermionic thermal factors at temperatures near $$T_c$$ and accounting for induced interactions in a medium of fermions rather than in vacuo. The Hamiltonian for a interacting Fermi gas in the condensed state can be derived in part by using the Bogoliubov approximation on the excitation energies. Additionally, Hartree–Fock contributions are omitted from the Hamiltonian as they have no negligible influence on pairing in a dilute Fermi system.

Experimental Observations
When Eric Cornell and Carl Wieman produced a Bose–Einstein condensate from rubidium atoms in 1995, there naturally arose the prospect of creating a similar sort of condensate made from fermionic atoms, which would form a superfluid by the BCS mechanism. However, early calculations indicated that the temperature required for producing Cooper pairing in atoms would be too difficult to achieve. In 2001, Murray Holland at JILA suggested a way of bypassing this difficulty. He speculated that fermionic atoms could be coaxed into pairing up by subjecting them to a strong magnetic field.

In 2003, working on Holland's suggestion, Deborah Jin at JILA, Rudolf Grimm at the University of Innsbruck, and Wolfgang Ketterle at MIT managed to group fermionic atoms into forming molecular bosons, which were then able to undergo Bose condensation. On December 16, 2003, Jin produced the first fermionic condensate using methods for BEC formation and properties of BCS superconductivity by preparing the state using the highly controllable magnetic Feshbach resonances, cooling potassium-40 atoms to a temperature of 50 nanokelvin.

Current Research
Research lead by James Thompson at JILA and Anjun Chu, Dylan Young at the University of Colorado Boulder demonstrated the capability to emulate dynamical phases of superconductors and superfluids in optical cavities. While superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system’s parameters are abruptly changed, such as the cavity frequency and interactions mediated via the exchange of photons through the cavity.