Physics Today

Historical Notes on Feshbach and Shape Resonances

D. M. Eagles

Published Feb 1, 2005• 0 citations

Abstract

Daniel Kleppner’s Reference Frame piece implies incorrectly that the crossover between Bardeen-Cooper-Schrieffer and Bose–Einstein condensation theory was first studied by Anthony Leggett in 1980. Actually, my early work on this topic was published 11 years before then. 1 1. D. M. Eagles, Phys. Rev. 186, 456 (1969) https://doi.org/10.1103/PhysRev.186.456. In the mid-1980s, I and colleagues in Australia also found the first example 2 2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1. of a system that lay on the Bose-gas side of the BCS–Bose gas transition, in a ceramic sample of 3% zirconium-doped strontium titanate (SrTiO3), although with a pairing temperature much lower than that predicted for the 1969 model used for this material. In three dimensions there is a threshold in the coupling strength for the existence of Bose-gas superconductivity at very low carrier concentrations, and for strengths slightly above the threshold, the pair-binding energy is proportional to the square of the difference of the coupling from the threshold value. Thus, for example, a calculated coupling error that gives a value 5% above threshold instead of 1% above would make a factor of 25 error in the pair-binding energy.A later attempt 3 3. C. Andrikidis, R. J. Tainsh, D. M. Eagles, Physica B 165, 1517 (1990) https://doi.org/10.1016/S0921-4526(09)80344-X. to reach the Bose-gas regime in macrocrystalline samples of Zr-doped SrTiO3 with similar carrier concentrations to those in the ceramic sample mentioned indicated that some nonuniform state prevented the Bose-gas regime being reached. However, there is still much interesting physics awaiting study in this material.REFERENCESSection:ChooseTop of pageREFERENCES <<1. D. M. Eagles, Phys. Rev. 186, 456 (1969) https://doi.org/10.1103/PhysRev.186.456. Google ScholarCrossref, ISI, CAS2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 , Google ScholarCrossref, ISI, CAS D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1. , Google ScholarCrossref, ISI, CAS3. C. Andrikidis, R. J. Tainsh, D. M. Eagles, Physica B 165, 1517 (1990) https://doi.org/10.1016/S0921-4526(09)80344-X. Google ScholarCrossref© 2005 American Institute of Physics.

DOI: 10.1063/1.4796861

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BCS-to-BEC evolution details

D. M. Eagles

Again an article in Physics Today, (by Carlos Sá de melo October 2008, page 45), has incorrectly implied that Anthony Leggett was the first person to study the crossover from Bardeen-Cooper-Schrieffer to Bose–Einstein condensation. On page 47 of the article, it states that “a clear picture of the BCS-to-BEC evolution at zero temperature didn’t emerge until 1980, when Anthony Leggett realized that the physics could be captured by a simple description in real space of paired fermions with opposite spins.” Although the model I considered in my 1969 paper 1 1. D. M. Eagles, Phys. Rev. 186, 456 (1969). https://doi.org/10.1103/PhysRev.186.456 is slightly different from Leggett’s, figure 4 in my paper clearly shows regions where pairing without superconductivity occurs and where superconductivity is limited by the Bose- condensation temperature of pairs, and on page 458 I discuss a limit at which the diameter of pairs is small compared with the distance between them.I also disagree with a statement in the box on page 47 of Sá de Melo’s article that “the evolution from a Bardeen-Cooper-Schrieffer superfluid to a Bose–Einstein condensation superfluid cannot be studied in … superconductors.” At least in ceramic samples of SrTiO3 with 3% of the titanium replaced by zirconium, the transition has been studied by varying the carrier concentration via differing heat treatments to produce different concentrations of oxygen vacancies. 2 2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1 D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989). https://doi.org/10.1016/0921-4534(89)90466-8 It is possible that such a transition may be found in other superconducting semiconductors when people start to search for suitable materials. However, in three dimensions the pairing strength has to be above some threshold value to obtain the possibility of reaching the Bose-gas regime. Also, many authors think that the BEC regime occurs in underdoped cuprates, 3,4 3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005). https://doi.org/10.1016/j.physrep.2005.02.005 4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007). https://doi.org/10.1007/s10948-007-0258-z while the consensus is that overdoped samples are BCS-like.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. M. Eagles, Phys. Rev. 186, 456 (1969). https://doi.org/10.1103/PhysRev.186.456 , Google ScholarCrossref, ISI2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 , Google ScholarCrossref, ISI D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1 , , Google ScholarCrossref, ISI D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989). https://doi.org/10.1016/0921-4534(89)90466-8 , , Google ScholarCrossref, ISI3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005). https://doi.org/10.1016/j.physrep.2005.02.005 , Google ScholarCrossref, ISI4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007). https://doi.org/10.1007/s10948-007-0258-z , Google ScholarCrossref, ISI© 2009 American Institute of Physics.

Physics TodayJul 31, 20094 citations

BCS-to-BEC evolution details

Carlos A. R. Sá de Melo

Sá de Melo replies: I thank D. M. Eagles for his comments. My statement concerning the evolution from Bardeen-Cooper-Schrieffer to Bose–Einstein condensation superfluids was about clarity and not who was the first to propose the idea. Although I appreciate Eagles’s work, I still think that Anthony Leggett’s papers 1 are the clearest presentation on the topic up to 1980.In his very interesting book written in 1964, John Blatt describes the BEC theory of superconductivity and its relation to the BCS theory. 2 As he recounts, the possibility of pairing without superconductivity and Bose condensation of electron pairs at a lower temperature was suggested as early as 1946 by Richard Ogg Jr. In 1954 and subsequent years, Max Schafroth developed a firm theoretical framework for such pairing, but it was not supported by experimental evidence: No preformed pairs were found, and the BEC temperature was much too high for known superconductors. Blatt’s book thoroughly documents a clear connection between a variational, fixed-number ground state comprising antisymmetrized products of pair wavefunctions and the BCS equations that Eagles used in his 1969 paper. Eagles noticed that the BCS and number equations at zero temperature could be used to describe the evolution from BCS to BEC superconductivity in low-carrier-density thin films or bulk materials for fixed (and sufficiently strong) interaction and changing carrier density. However, Blatt’s main interest was in metals with high carrier densities. If one reads the full statement in the box entitled “Feshbach resonances” on page 47 of my article, it is clear that I am talking about changing interactions for fixed density and not changing density for fixed interactions. I do not think it is currently possible in any superconductor to control the interactions between fermions. Thus my statement is accurate.In many instances the change in carrier concentration in thin films or three-dimensional (bulk) systems is made via chemical doping, which may introduce disorder or distortions in the crystal structure and thus change the system in more than the desired way. I have serious doubts that ceramic samples of zirconium-doped strontium titanate described in Eagles’s letter indeed show the BCS-to-BEC evolution; the references provided do not offer clear evidence of such a phenomenon.Since cuprate superconductors are known to be quasi-2D and have d-wave pairing symmetry, the evolution from BCS to BEC superconductivity for sufficiently strong fixed interactions and changing carrier density would be accompanied by a topological quantum phase transition 3 and would not be a crossover as in the s-wave case. However, as much as I might want such a transition to be found for the cuprates, current experimental evidence shows that their normal state and superconducting properties are more complicated than simple BCS-to-BEC theories would predict.The beauty of the observation of the BCS-to-BEC evolution in ultracold atoms is that many of the materials difficulties encountered in standard condensed matter can be avoided and the tuning of the interaction for fixed low density can be controlled. However, one should keep an open mind that a similar realization may appear elsewhere. For instance, if carrier density could be controlled and changed continuously via electrostatic doping in good-quality thin films or at the surface of bulk low-carrier-density superconductors like some titanates or cuprates, then there is a good chance of observing the BCS-to-BEC evolution as a function of carrier density as envisioned by Eagles in 1969.REFERENCESSection:ChooseTop of pageREFERENCES << REFERENCES 1. A. J. Leggett, in Modern Trends in the Theory of Condensed Matter: Proceedings of the XVI Karpacz Winter School of Theoretical Physics, February 19-March 3, 1979, Karpacz, Poland, A. Pekalski, J. Przystawa, eds., Springer, New York (1980) Google Scholar A. J. Leggett, J. Phys. Colloq. 41, C7–19 (1980). https://doi.org/10.1051/jphyscol:1980704 , Google ScholarCrossref2. J. M. Blatt, Theory of Superconductivity, Academic Press, New York (1964). Google Scholar3. L. S. Borkowski, C. A. R. Sá de Melo, http://arxiv.org/abs/cond-mat/9810370v1 , Google Scholar R. D. Duncan, C. A. R. Sá de Melo, Phys. Rev. B 62, 9675 (2000). https://doi.org/10.1103/PhysRevB.62.9675 , Google ScholarCrossref, ISI© 2009 American Institute of Physics.

Physics TodayAug 1, 20090 citations

History of the BCS to Bose-Einstein Transition

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Historical Notes on Feshbach and Shape Resonances

A. R. P. Rau

In his interesting Reference Frame piece (Physics Today, August 2004, page 12), Daniel Kleppner relates Herman Feshbach’s reaction to the term “Feshbach resonance” and gives a capsule account of the topical and vital role this phenomenon plays in today’s explorations of cold collisions of fermions and bosons, and of condensates of atoms and molecules. Readers may be interested in the earlier history and why atomic physicists began to use a term that Feshbach himself dismissed as jargon.In the Los Alamos experiment of Howard Bryant’s group on photo-absorption by the negative ion of hydrogen, two prominent resonances were seen at photon energies of about 10.95 eV. This energy marks the position of the first excited state of the hydrogen atom above the single bound state of the negative ion.One narrow resonance occurs just below the threshold of that excitation, whereas the second, broad one occurs just above. Their physical mechanisms are different and, following the paradigm set by that historic experiment, it has become customary to refer to the two types of resonance, both common in atomic and molecular physics, as Feshbach and shape resonance, respectively. That is, a resonant coupling just below the relevant threshold, leading to narrow profiles, is distinguished from those lying just above the threshold to which they are most strongly coupled, and typically broad.The two types of resonances are most naturally viewed in state and coordinate space, respectively. Feshbach resonances are thought of in terms of the superposition of two quantum states, one with discrete and one with continuum character, leading to a temporary quasi-bound state that manifests itself as a resonance in that continuum channel. The independent and almost simultaneous work in nuclear physics by Feshbach using the language of projection operators and in atomic physics by Ugo Fano using the language of superposition of wave functions serves for their technical description.Shape resonances, on the other hand, may be associated with the shape of a trapping potential in real, coordinate space. Most often, the trapping potential is an angular momentum potential that combines with the internal one to create intervening barriers to low-energy particles. The resulting pictures often display “two-valley” potentials. In the case of the negative ion of hydrogen, such a barrier has been described in terms of the hyperspherical coordinates of that three-body system. Examples of such shape resonances abound in low-energy scattering of electrons from molecules.For more discussion and references, including to recent applications in mesoscopic condensed matter systems, see my pedagogical mini-review in Physica Scripta, volume 69, page C10, 2004.© 2005 American Institute of Physics.

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