BCS to BEC Evolution 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 con- densation (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 M¨uller,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 tempera- tures. 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 wave- function 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 con- densation. 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.