BCS to BEC Evolution

S.Furkan Ozturk

September 30, 2019

Abstract

This report presents a literature review and a conceptual understanding of the crossover from theBardeen–Cooper–Schrieffer (BCS) state of weakly-correlated pairs of fermions to the Bose–Einstein con-densation (BEC) of diatomic molecules in the atomic Fermi gas. In the context of ultracold Fermi gases, aBCS-BEC crossover means that by tuning the interaction strength, one goes from a BCS state to a BEC statewithout encountering a phase transition. BEC state is a Bose-Einstein condensate of two-atom molecules(bound fermions), while the BCS state is made up of pair of atoms. The difference between the pairs andthe molecules is that the molecules are localized in the real space, whereas the BCS pairs are made of twoparticles with opposite momenta in the momentum space [1, 2].

1 History

In this section, I listed the progress of our understanding of superfluidity and superconductivity. Superfluidityis a collective behavior of atoms in which particles can flow coherently without any friction or dissipation ofheat and superconductivity is basically a charged superfluidity.

The first observation of superfluid behavior was made by Heike Kamerlingh Onnes in 1911, much before thedevelopment of quantum mechanics. He cooled a metallic sample of Hg below to 4.2 K, using liquid helium-4and found that the sample conducted electricity without dissipation which coined the term superconductivity[3]. Afterwards the superfluidity of bosonic liquid helium-4 was observed by Kapitza and Misener in 1938 [4].In the same year, Fritz London proposed a connection between the superfluidity of helium-4 and Bose–Einsteincondensation (BEC) [5]. In BEC, a finite condensate fraction of the total number of bosons occupies the lowest-energy (zero momentum) single-particle state at sufficiently low temperatures. For non-interacting bosons, all ofthe zero particles are in the ground state at zero temperature, yet interactions in helium-4 are sufficiently strongto reduce the condensate fraction to 10% at zero temperature [2]. In 1946, Bogoliubov found a microscopicexplanation for the superfluidity and put forward that weak repulsive interactions do not destroy the BECstate. More importantly he found that the formation of the BEC state does not guarantee superfluidity whichdepends on the existence of currents that flow without dissipation and requires correlations—interactions amongbosons [6]. Also, in the same year, to account for the superconductivity of metal ammonia solutions, RichardOgg proposed the possibility of pairing of electrons in real space such that they obey the bosonic statistics[7]. After that, in 1954, Max Schafroth developed a theoretical framework for such pairing, but it was notsupported by experimental evidence as no preformed pairs were observed and the BEC temperature was muchtoo high for known superconductors [8]. Also, their theory had problems in dealing with real metals, becausethe pairs are hugely overlapping in real space, hardly describable as point bosons, and the anti-symmetryof the electronic wavefunction was crucial to the problem of superconductivity. However, in 1957, Bardeen,Cooper, and Schrieffer proposed a theory of superconductivity known as the BCS theory and suggested apairing of electrons in the momentum space. BCS theory faced up to all these challenges and made quantitativepredictions for the properties of superconductors [9]. Having got a successfully theory of superconductivity, thedifferences between the BCS and BEC have begun to be stressed. In 1969, Eagles studied the superconductivityin doped semiconductors like with a very low carrier density, where the attraction between electrons need notbe small compared with the Fermi energy. This led to the first mean-field treatment of the BCS-BEC crossover[10]. Following this, in 1980, Leggett gave picture of the BCS-to-BEC evolution at T = 0 by a description inreal space of paired fermions with opposite spins for a dilute gas of fermions. Although this is mostly in theBCS limit, Leggett wanted to understand the extent to which some of its properties, such as the total angularmomentum of the superfluid might be similar to that of a BEC of diatomic molecules [11]. In 1985, Nozieres andRink (NSR), extended Leggett’s description to finite temperatures with finite ranged attraction near the criticaltemperature for superfluidity along with the evolution of the critical temperature [12]. The discovery of high-Tccuprate superconductors in 1986 by Bednorz and Muller, created a new paradigm for superconductivity forwhich the BCS theory failed dramatically [13]. In 1993, motivated by the inapplicability of the BCS theory tocuprate superconductors, Melo, Randeria, and Engelbrecht extended the preliminary results of Leggett and NSRas an attempt to understand cuprates. They used a zero-ranged attraction characterized by the experimentally

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measurable length scale as, the scattering length [14]. However, still an experimental realization of the BCS-to-BEC crossover had not been observed. Even though some tunability of the fermion density can be achievedthrough chemical doping there was no control over the interactions until the application of Feshbach resonancein ultra-cold atoms. Using magnetically driven Feshbach resonances the tuning of interactions became possiblein ultracold Fermi atoms. Following this, in 2003, three experimental groups simultaneously succeeded inproducing BEC of bound fermion molecules in optical traps at ultracold temperatures. Two groups, RudolfGrimm’s at the University of Innsbruck and Wolfgang Ketterle’s at MIT, used Li-6 atoms, and a third group,Deborah Jin’s at JILA, used K-40 [15, 16, 17]. And two years later, a direct evidence for the superfluidity cameby the experimental detection of vortex lattice when the cloud of trapped fermions was first rotated and thenallowed to expand [18].

2 BCS theory and Superconductivity

A theory of superconductivity formulated by John Bardeen, Leon Cooper, and Robert Schrieffer, explaining thephenomenon in which a current of electron pairs flows without resistance in certain materials at low tempera-tures. According to theory, superconductivity arises when a single negatively charged electron slightly distortsthe lattice of atoms in the superconductor, drawing toward it a small excess of positive charge creating anelectron-phonon pair. This excess, afterwards, attracts a second electron (electron-phonon interaction). It isthis weak and indirect attraction that binds the electrons together, into a Cooper pair in momentum space asshown in the figure below.

The Cooper pairs within the superconductor carry the current without any resistance, but why? Mathe-matically, because the Cooper pair is more stable than a single electron within the lattice, it experiences lessresistance. Physically, the Cooper pair is more resistant to vibrations within the lattice as the attraction to itspartner will keep it more stable. Hence, Cooper pairs move through the lattice relatively unaffected by thermalvibrations (phonons) below the critical temperature.

The existence of a single fermion pair, however, is not sufficient to describe the macroscopic behavior ofsuperconductors. It is necessary to invent a collective and correlated state in which many electron (fermion)pairs acting together to produce a zero-resistance state. For that, BCS theory proposes a many particle wave-function corresponding to largely overlapping fermion pairs with zero center of mass momentum, zero angularmomentum (s-wave), and zero total spin (singlet) [2]. Moreover, one of the most fundamental features of theBCS state is the existence of correlations between fermion pairs, which lead to an order parameter 1/kFas, forthe superconducting state. This order parameter is found to be directly related to the energy gap Eg in theelementary excitation spectrum. Because all relevant fermions participating in the ground state of an s-wavesuperconductor are paired, creating a single fermion excitation requires breaking a Cooper pair with a energycost. Thus the contribution of elementary excitations to the specific heat and other thermodynamic propertiesshows exponential behavior ∼ exp(−Eg/T ) at low temperature [1]. BCS theory successfully applied to manyexperimental results and explained the physics behind the conventional superconductors until the discovery ofcuprates.

3 From BCS to BEC superfluids

Even though the BCS theory became successfully and widely applicable to many phenomena, it is basically aweak attraction theory. A generalization of the BCS theory has been developed to include the strong attractionregime in which fermion pairs become tightly bound diatomic Bose molecules and undergo Bose–Einstein con-densation. BEC state is on the the strong attraction side of the phase space and is formed by the condensationof bound fermions in real space known as molecular bosons.

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The simplest conceptual picture of the BCS to BEC superfluidity evolution for s-wave (l = 0) pairing canbe constructed for low fermion densities and short-ranged interactions that are much smaller than the averageseparation between fermions [2]. The essence of the evolution at zero temperature is indicated in the ratio ofthe size of fermion pairs to the average separation between the fermions. In the BCS regime, the attraction isweak, the pairs are much larger than their average separation, and they overlap substantially in the momentumspace. In the BEC regime, the attraction is strong, the pairs are much smaller than their average separation,and they overlap very little in the real space. However the clear picture of evolution between these two regimesis understood at zero temperature, in 1980, when Leggett proposed a simple description in real space of pairedfermions with opposite spins. Leggett considered a zero-ranged attractive potential (contact interaction) betweenfermions and showed that when the attraction is weak, a BCS superfluid appeared, and when the attraction isstrong, a BEC superfluid emerged. And this result is generalized in 1993 by Melo, Randeria, and Engelbrechtby using a zero-ranged attraction potential characterized by the experimentally measurable length scale as, (swave scattering length). Also, since the natural momentum scale is Fermi wave-vector kF , the strength of theattractive interactions can be characterized by the dimensionless scattering parameter 1/kFas. This attractionstrength parameter changes sign from BCS to BEC side and becomes negative for attractive pairs and positivefor repulsive. That is why, in the BCS side pairs substantially overlap for negative as unlike the positive andrepulsive as of the BEC side. Therefore, the BCS weak-attraction limit is characterized 1/kFas � −1 whereasthe BEC strong-attraction limit, by 1/kFas � −1 and in between there exists the crossover region. As afunction of the order parameter 1/kFas the phase diagram for fermionic superfluids is given [2]:

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The phase diagram illustrates two important physical concepts of the BCS-to-BEC crossover. First, thenormal, non-condensed state for weak attractions is a Fermi liquid, that evolves smoothly into a molecularBose liquid without a phase transition. These two regimes are separated by a pair formation (or moleculardissociation) temperature Tpair characterized by chemical equilibrium between bound fermion pairs and unboundfermions. Second, Tpair and the critical temperature Tc are more or less the same in the BCS limit which meansthat pairs form and condense at the same temperature. On the other hand, in the BEC limit, fermion pairs(diatomic molecules) form first around Tpair and condense at the much lower temperature Tc = TBEC , wherephase coherence—condensation—is established. Furthermore, the chemical potential µ vanishes beyond theunitarity limit where as diverges. That line represents the actual separation between the BCS region, wherethe energy gap ∆k in the elementary excitation spectrum is related only to the order parameter 1/kFas andoccurs at finite momentum given and the BEC region, where the energy gap occurs at zero momentum and isrelated both to the chemical potential and to the order parameter [2]. Explicitly, in the BCS limit the chemical

potential is µ = EF and in the BEC limit µ = −(

1kF as

)2EF . And the energy gap is ∆ = e

−πkF |as|EF in the BCS

limit and ∆ = EF

√163π

1kF as

in the BEC limit. Plot of the chemical potential and energy gap is given below [1]:

It is the qualitative change in the elementary excitation spectrum at µ = 0 not the emergence of a boundstate at the unitarity limit, where as diverges that separates the BCS region from the BEC region. But howtune the interaction strength to cross the line of µ = 0 and realize the crossover?

3.1 Feshbach resonances

Feshbach resonances allow to tune the s-wave interaction by changing the scattering length by applying amagnetic field and are the essential tool to control the interaction between atoms in ultracold quantum gases[19]. In the context of scattering processes in many-body systems, the Feshbach resonance occurs when theenergy of a bound state of an interatomic potential is equal to the kinetic energy of a colliding pair of atoms.

The underlying requirement, shown schematically on the left, is that at zero magnetic field, the interatomicpotentials of two atoms in their ground state (the open channel) and in an excited state (the closed channel)be not too different in energy. The resonance, characterized by a divergence in the scattering length as,occurs when the energy difference ∆E between a bound state with energy Eres in the closed channel and theasymptotic threshold energy Eth of scattering states in the open channel is brought to zero by an applied

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external magnetic field B0. In short, a Feshbach resonance occurs when the bound molecular state in the closedchannel energetically approaches the scattering state in the open channel. Then even weak coupling can lead tostrong mixing between the two channels [19]. By this way the short range interaction strength which dependson the s-wave scattering length as can be tuned and the BCS-to-BEC crossover can be realized.

References

[1] C. Regal. Experimental realization of BCS-BEC crossover physics with a Fermi gas of atoms. PhD thesis,University of Colorado at Boulder, 2006.

[2] C. A. R. Sa de Melo. When fermions become bosons: Pairing in ultracold gases. Physics Today, 61:45–51,2008.

[3] H. Kamerlingh Onnes. Further experiments with liquid helium. C. On the change of electric resistance ofpure metals at very low temperatures etc. IV. The resistance of pure mercury at helium temperatures, pages261–263. Springer Netherlands, Dordrecht, 1991.

[4] P. Kapitza. Viscosity of liquid helium below the l-point. Nature, 141:74, Jan 1938.

[5] F. London. The l-phenomenon of liquid helium and the bose-einstein degeneracy. Nature, 141:643, Apr1938.

[6] N. Bogoliubov. On the theory of superfluidity. Journal of Physics, 11:23–32, 1947.

[7] Richard A. Ogg. Bose-einstein condensation of trapped electron pairs. phase separation and superconduc-tivity of metal-ammonia solutions. Phys. Rev., 69:243–244, Mar 1946.

[8] M. R. Schafroth. Theory of superconductivity. Phys. Rev., 96:1442–1442, Dec 1954.

[9] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of superconductivity. Phys. Rev., 108:1175–1204,Dec 1957.

[10] D. M. Eagles. Possible pairing without superconductivity at low carrier concentrations in bulk and thin-filmsuperconducting semiconductors. Phys. Rev., 186:456–463, Oct 1969.

[11] A. J. Leggett. Diatomic molecules and cooper pairs. In A. Pekalski and J. A. Przystawa, editors, LectureNotes in Physics, Berlin Springer Verlag, volume 115 of Lecture Notes in Physics, Berlin Springer Verlag,page 13, 1980.

[12] P. Nozieres and S. Schmitt-Rink. Bose condensation in an attractive fermion gas: From weak to strongcoupling superconductivity. Journal of Low Temperature Physics, 59(3):195–211, May 1985.

[13] J. G. Bednorz and K. A. Muller. Possible hightc superconductivity in the balacuo system. Zeitschrift furPhysik B Condensed Matter, 64(2):189–193, Jun 1986.

[14] C. A. R. Sa de Melo, Mohit Randeria, and Jan R. Engelbrecht. Crossover from bcs to bose superconductiv-ity: Transition temperature and time-dependent ginzburg-landau theory. Phys. Rev. Lett., 71:3202–3205,Nov 1993.

[15] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm.Bose-einstein condensation of molecules. Science, 302(5653):2101–2103, 2003.

[16] Markus Greiner, Cindy A. Regal, and Deborah S. Jin. Emergence of a molecular bose-einstein condensatefrom a fermi gas. Nature, 426:537, Nov 2003.

[17] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle.Observation of bose-einstein condensation of molecules. Phys. Rev. Lett., 91:250401, Dec 2003.

[18] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle. Vortices and superflu-idity in a strongly interacting fermi gas. Nature, 435:1047, Jun 2005. Article.

[19] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach resonances in ultracold gases.Rev. Mod. Phys., 82:1225–1286, Apr 2010.

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