BCS to BEC Evolution

Home , BCS theory, Cooper pair, Superconductivity, Superfluidity

S¸.Furkan Ozt¨urk¨ September 30, 2019

Abstract This report presents a literature review and a conceptual understanding of the crossover from the Bardeen–Cooper–Schrieﬀer (BCS) state of weakly-correlated pairs of fermions to the Bose–Einstein condensation (BEC) of diatomic molecules in the atomic Fermi gas. In the context of ultracold Fermi gases, a BCS-BEC crossover means that by tuning the interaction strength, one goes from a BCS state to a BEC state without 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 and the molecules is that the molecules are localized in the real space, whereas the BCS pairs are made of two particles 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. Superfluidity is a collective behavior of atoms in which particles can flow coherently without any friction or dissipation of heat and superconductivity is basically a charged superfluidity. The first observation of superfluid behavior was made by Heike Kamerlingh Onnes in 1911, much before the development of quantum mechanics. He cooled a metallic sample of Hg below to 4.2 K, using liquid helium-4 and 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–Einstein condensation (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 of the zero particles are in the ground state at zero temperature, yet interactions in helium-4 are sufficiently strong to reduce the condensate fraction to 10% at zero temperature [2]. In 1946, Bogoliubov found a microscopic explanation for the superfluidity and put forward that weak repulsive interactions do not destroy the BEC state. More importantly he found that the formation of the BEC state does not guarantee superfluidity which depends on the existence of currents that flow without dissipation and requires correlations—interactions among bosons [6]. Also, in the same year, to account for the superconductivity of metal ammonia solutions, Richard Ogg 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 not supported by experimental evidence as no preformed pairs were observed and the BEC temperature was much too high for known superconductors [8]. Also, their theory had problems in dealing with real metals, because the pairs are hugely overlapping in real space, hardly describable as point bosons, and the anti-symmetry of 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 a pairing of electrons in the momentum space. BCS theory faced up to all these challenges and made quantitative predictions for the properties of superconductors [9]. Having got a successfully theory of superconductivity, the differences between the BCS and BEC have begun to be stressed. In 1969, Eagles studied the superconductivity in doped semiconductors like with a very low carrier density, where the attraction between electrons need not be 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 in real space of paired fermions with opposite spins for a dilute gas of fermions. Although this is mostly in the BCS limit, Leggett wanted to understand the extent to which some of its properties, such as the total angular momentum of the superfluid might be similar to that of a BEC of diatomic molecules [11]. In 1985, Nozieres and Rink (NSR), extended Leggett’s description to finite temperatures with finite ranged attraction near the critical temperature for superfluidity along with the evolution of the critical temperature [12]. The discovery of high-Tc cuprate superconductors in 1986 by Bednorz and Müller,created a new paradigm for superconductivity for which the BCS theory failed dramatically [13]. In 1993, motivated by the inapplicability of the BCS theory to cuprate superconductors, Melo, Randeria, and Engelbrecht extended the preliminary results of Leggett and NSR as an attempt to understand cuprates. They used a zero-ranged attraction characterized by the experimentally

1 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 achieved through chemical doping there was no control over the interactions until the application of Feshbach resonance in ultra-cold atoms. Using magnetically driven Feshbach resonances the tuning of interactions became possible in ultracold Fermi atoms. Following this, in 2003, three experimental groups simultaneously succeeded in producing BEC of bound fermion molecules in optical traps at ultracold temperatures. Two groups, Rudolf Grimm’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 came by the experimental detection of vortex lattice when the cloud of trapped fermions was ﬁrst rotated and then allowed to expand [18].

2 BCS theory and Superconductivity

A theory of superconductivity formulated by John Bardeen, Leon Cooper, and Robert Schrieffer, explaining the phenomenon in which a current of electron pairs flows without resistance in certain materials at low temperatures. According to theory, superconductivity arises when a single negatively charged electron slightly distorts the lattice of atoms in the superconductor, drawing toward it a small excess of positive charge creating an electron-phonon pair. This excess, afterwards, attracts a second electron (electron-phonon interaction). It is this weak and indirect attraction that binds the electrons together, into a Cooper pair in momentum space as shown 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 less resistance. Physically, the Cooper pair is more resistant to vibrations within the lattice as the attraction to its partner will keep it more stable. Hence, Cooper pairs move through the lattice relatively unaffected by thermal vibrations (phonons) below the critical temperature. The existence of a single fermion pair, however, is not sufficient to describe the macroscopic behavior of superconductors. 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 wavefunction corresponding to largely overlapping fermion pairs with zero center of mass momentum, zero angular momentum (s-wave), and zero total spin (singlet) [2]. Moreover, one of the most fundamental features of the BCS state is the existence of correlations between fermion pairs, which lead to an order parameter 1/kF as, for the superconducting state. This order parameter is found to be directly related to the energy gap Eg in the elementary excitation spectrum. Because all relevant fermions participating in the ground state of an s-wave superconductor are paired, creating a single fermion excitation requires breaking a Cooper pair with a energy cost. Thus the contribution of elementary excitations to the specific heat and other thermodynamic properties shows exponential behavior ∼ exp(−Eg/T ) at low temperature [1]. BCS theory successfully applied to many experimental results and explained the physics behind the conventional superconductors until the discovery of cuprates.

3 From BCS to BEC superﬂuids

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

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

3 The phase diagram illustrates two important physical concepts of the BCS-to-BEC crossover. First, the normal, non-condensed state for weak attractions is a Fermi liquid, that evolves smoothly into a molecular Bose liquid without a phase transition. These two regimes are separated by a pair formation (or molecular dissociation) temperature Tpair characterized by chemical equilibrium between bound fermion pairs and unbound fermions. Second, Tpair and the critical temperature Tc are more or less the same in the BCS limit which means that 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 , where phase coherence—condensation—is established. Furthermore, the chemical potential µ vanishes beyond the unitarity limit where as diverges. That line represents the actual separation between the BCS region, where the energy gap ∆k in the elementary excitation spectrum is related only to the order parameter 1/kF as and occurs at ﬁnite momentum given and the BEC region, where the energy gap occurs at zero momentum and is related both to the chemical potential and to the order parameter [2]. Explicitly, in the BCS limit the chemical 2 −π 1 k |a | potential is µ = EF and in the BEC limit µ = − EF . And the energy gap is ∆ = e F s EF in the BCS kF as q 16 1 limit and ∆ = EF in the BEC limit. Plot of the chemical potential and energy gap is given below [1]: 3π kF as

It is the qualitative change in the elementary excitation spectrum at µ = 0 not the emergence of a bound state at the unitarity limit, where as diverges that separates the BCS region from the BEC region. But how tune 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 a magnetic 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 the energy 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 interatomic potentials 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 the asymptotic threshold energy Eth of scattering states in the open channel is brought to zero by an applied

4 external magnetic ﬁeld B0. In short, a Feshbach resonance occurs when the bound molecular state in the closed channel energetically approaches the scattering state in the open channel. Then even weak coupling can lead to strong mixing between the two channels [19]. By this way the short range interaction strength which depends on 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. S´ade 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 of pure metals at very low temperatures etc. IV. The resistance of pure mercury at helium temperatures, pages 261–263. 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, Apr 1938. [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 superconductivity 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-ﬁlm superconducting semiconductors. Phys. Rev., 186:456–463, Oct 1969. [11] A. J. Leggett. Diatomic molecules and cooper pairs. In A. P¸ekalski and J. A. Przystawa, editors, Lecture Notes in Physics, Berlin Springer Verlag, volume 115 of Lecture Notes in Physics, Berlin Springer Verlag, page 13, 1980.

[12] P. Nozières and S. Schmitt-Rink. Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. Journal of Low Temperature Physics, 59(3):195–211, May 1985. [13] J. G. Bednorz and K. A. Müller. Possible hightc superconductivity in the balacuo system. Zeitschrift für Physik B Condensed Matter, 64(2):189–193, Jun 1986. [14] C. A. R. Sáde Melo, Mohit Randeria, and Jan R. Engelbrecht. Crossover from bcs to bose superconductivity: 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 condensate from 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.