INTRODUCTION:

To create a Fermi gas of atoms, experiments applied the same cooling technique as those used to achieve BEC in or . For Fermi gas of atoms, one has two stable alkali atoms and with an odd number of electrons, protons and neutrons. The first gas of fermionic atoms to enter the quantum degenerate regime was created at JLLA (DeMarco and Jin, 1999) using. The observation in this experiment was not a phase transition as in the case of Bose gas but rather the presence of more and more energy that would be expected classically as the Fermi gas was cooled below the Fermi temperature. Many more Fermi gas experiments using a variety of cooling techniques then followed (Trustcott et al, 2001; Oritiz and Dukelsky, 2005).

The next aim after the creation of a normal Fermi gas of atoms was to form a super fluid in a paired Fermi gas. In conventional superconductors s-wave pairing occurs between spin and spinelectrons. The hope was that s-wave pairing could similarity occur with the creation of two components atomic gas with an equal Fermi energy for each component. Such a two-component gas can be realized using an equal mixture of alkali atoms in two different hyperfine spin state. The idea was that BCS state would appear if the temperature of this two component gases were cold enough and the interaction between fermions attractive and large enough. However, for typical interactions the temperature required to reach a true BCS state were far too compared to achievable temperature to imagine creating cooper pairs. Stoof et al (1996) noted that the interaction between atoms were large compared to typical values of the scattering length as well as attractive bringing the BCS transition temperature closer to the realistic temperature (Combescot, 1999; Timmermans et al, 2001). It was then realized that a type of scattering resonance known as Feshbach resonance could allow arbitrary changes is the interaction strength. Theories were developed that explicitly treated the case where the interactions were enhanced by Feshbach resonance Ohasi and Grffin, 2002). In a theory, it was proposed that the BCS wave function was more generally applicable to the weakly interacting limit. As long as the chemical potential is found self consistently (as the interaction is increased the BCS ground state) can describe everything from cooper paring to the BEC of composite boson made up of two fermions. After nearly a century of and superconductor being considered as separate entities that experimental realization of super-fluid in BCS-BEC crossover regime would provide a physical link between both (Eagle, 1969). More recent interest in crossover theories has also come in response to the possibility that they could apply to high transition temperature superconductors. These super-conductors differ from normal superconductor both in their high transition temperature and the apparent pressure of the pseudo gap, which are both characteristics expected to be found in a Fermi gas in the crossover (Randeria, 1995; Chen et al, 2005).

In this study, the BCS-BEC crossover physics using BCS theory were investigated. BCS theory was originally applied in the limit where the interaction energy is extremely small compared to the Fermi energy. In this case, the chemical potential can be fixed at Fermi energy and further calculation becomes reasonable simple. Leggett ⁴⁷ pointed out that if the BCS gap equation is examined allowing to vary, the gap equation actually precisely the Schrodinger equation for diatomic molecule in the limit where dominates. The structure of the crossover theory originates with the work of Nozieres et al (1985) and Maxwell et al (2014).

Mathematical formula used in the evaluation of energy gap parameter and chemical potential:

Let us consider a homogeneous Fermi system in three dimensions in an equal mixture of two different states at temperature . BCS theory gap equation is written as

where

is the attractive interaction for scattering of fermions with momenta and to and. Then obtains the number equation

where is the total number of fermions in both states? To solve equation (1) in the BCS limit the standard approach is to assume that potential is constant at a value which implies that gap is constant as well i.e. In that case the gap equation becomes

Now one finds that this equation diverges. For BCS superconductor this problem is resolved because the interaction can be limited to within the Debye energy of Fermi energy. This is the result of the nature of the phonon mediated interaction between the electrons that gives rise to the attractive interaction. The simplification of BCS limit are that and that since the density of states is constant at the value . Then the gap equation becomes

Solving equation (3.4) gives the BCS results

To extend this calculation to the crossover in atomic system, one can no longer apply the cut off frequency. The solution to the problem in this case is nontrivial and requires a renormalization procedure. Randeria (1990)have obtained the result of such a procedure and obtained the renormalized gap equation

where the interaction is described by the s-wave scattering length and is the volume of the system. One can’t assume in the crossover. For this, one solves gap equation given in equation (3.6) and number equation (3.2) simultaneously for and gap parameter . One solves these parameter as a function of dimensionless parameter where Marini et al (1998) have done this analytically using the elliptical integrals. We have used to gap equation for BCS theory to calculate the parameter and. The evaluated values are shown in table (3 ) and (3) for a homogeneous Fermi gas at temperature T=0 as determined through Nozieres Schmitt Rink (NSR) theory (Noziers, 1985).

Beyond temperature T=0:

The phase transition temperature is an important parameter for superfluid system. In BCS-BEC crossover the transition temperature increases as the interaction in increased. It is lowest in the per-turbative BCS regime and highest in the BEC limit. In a homogeneous system in the BCS limit⁵¹.

In the BEC limit ⁹¹, (8)

The BCS transition temperature can be extremely small due to the exponential dependence upon. If one puts the interaction strength in the alkali fermionic gas and a typical, is the Bohr radius.

which is completely inaccessible temperature in atomic system?

which can be accessible.

In the BCS limit paring and the phase transition to a superfluid state occur at the same temperature. However in the BEC limit this is not the case because the constituent fermions are very tightly bound pairs and can form far above.

DISCUSSIONS:

In this way, BCS-BEC crossover physics from BCS theory have studied during research period. There, theoretical formalism of Regal et al (2004) investigated in this study. In table and table, In this study, the evaluated result of the gap parameter and chemical potential as function of dimensionless parameter have determined through Nozieres Schmitt Rink (NSR) theory (Noziers, 1985) for BCS and BEC limit. There theoretical evaluated results appears that crossover occurs in a relatively small region of the parameter from. In table, the value and meaning of both and for the BCS and BEC limit were presented. The recent theoretical result (Oritiz and Dukelesky, 2005) also confirm such behaviour.

Table: An evaluated results of the gap parameter as a function of dimensionless parameter determined through Nozieres Schmitt Rink (NSR) theory (1985).


	BCS limit	BEC limit
2.5	2.85	0.00
2.0	2.53	0.00
1.5	2.10	0.00
1.0	1.82	0.00
0.5	1.02	0.00
0.0	0.00	0.00
-0.5	0.00	1.27
-1.0	0.00	1.13
-1.5	0.00	0.87
-2.0	0.00	0.42
-2.5	0.00	0.32
-3.0	0.00	0.00

Table : An evaluated results of the chemical potential as a function of dimensionless parameter determined through Nozieres Schmitt Rink (NSR) theory (Noziers, 1985)


	BCS Limit	BEC Limit
-2.5	1.122	0.00
-2.0	1.120	0.00
-1.5	1.116	0.00
-1.0	1.115	0.00
-0.5	1.115	0.00
0.0	1.115	0.00
0.5	0.00	-0.085
1.0	0.00	-1.052
1.5	0.00	-2.386
2.0	0.00	-3.458
2.5	0.00	-5.687
3.0	0.00	-6.432

Table : Crossover experiment have been performed with and. The regime corresponds to varying from through and to, where is the Bohr radius. =0.0529 nm. The value of both and are different in two limits.

BCS Limit	BEC Limit

is the gap parameter but its meaning is the excitation gap i.e. the smallest possible energy that can create a hole (remove fermions) in the superfluid in the BCS limit.

∆where and is positive when is positive (BCS limit), but becomes, when is negative.

REFERENCES:

1. DeMarco B and Jin DS (1999): Onset of Fermi degeneracy in a trapped atomic gas, Science 285, 1703.

2. Truscott AG, Stecker KE, Alexander WI, Partridge GB and Hulet RG (2001: Observation of Fermi pressure in a gas of trapped atoms, Science 291,2570.

3. Stoof HTC, Houbiers M, Sackett CA and Hulet RG (1996): Super-fluidity of spin polarized , Phys. Rev. Letter 76, 10.

4. Combescot R (1999): Trapped , A. high superfluid?, Phys. Rev. Letter (83): 37-66.

5. Timmermans E, Furuya K, Milonni PW and Kerman AK (2001): Prospect of creating composite Fermi –Bose superfluid, Phys. Letter, 228-285.

6. Ohashi Y and Griffin A (2002): BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance, Phys. Rev. Letter 89, 130402.

7. Nozieres P and Schmitt-Rink S (1985): Bose condensation in an attractive fermion gas; from weak to strong coupling superconductivity, J. Low temp., Phys. (JLTP) 59, 195.

8. Randeria M, Duan JM and Shieh LY (1990): Superconductivity in two-dimensional Fermi gas; Evolution from Cooper pairing to Bose condensation, Phys. Rev. B41, 327.

9. Maxwell (2014): physical Review 78(4) 471-477.

10. Randeria M (1995): In Bose-Einstein Condensation (edited by A Griffin, D. W. Snoke and S. Stringari). Cambridge Univ. Cambridge, UK, pp.355-392.

11. Regal CA, Greiner M and Jin DS (2004): Lifetime of molecule-atom mixtures near a Feshbach resonance in , Phys. Rev. Letter 92, 83-101.

12. Regal CA, Greiner M and Jin DS (2004): Observation of resonance condensation of fermionic atom pairs, Phys. Rev. Letter 92, 40-43.

13. Oritiz G and Dukelsky J (2005): Phys. Rev. A72, 043611 2005.

14. Marini M, Pistolesi F and Strinati GC (1998): Evaluation from BCS Super-conductivity to Bose condensation, Analytic results for the crossover in three dimensions, Eur, Phys. J. B1,151.

Received on 15.11.2020 Modified on 23.11.2020

Research J. Science and Tech. 2020; 12(4):302-306.

DOI: 10.5958/2349-2988.2020.00043.1