Skip to main content
SpringerLink
Log in

Exploring the Transition from BCS to Unitarity Using Normal Modes: Energies, Entropies, Critical Temperatures and Excitation Frequencies

  • Published:
Journal of Low Temperature Physics Aims and scope Submit manuscript

Abstract

The transition from the weakly interacting BCS regime to the strongly interacting unitary regime is explored for ultracold trapped Fermi gases assuming a normal mode description of the gas instead of the conventional Cooper pairing. The Pauli principle is applied “on paper” using specific normal mode assignments. Energies, entropies, critical temperatures, and an excitation frequency are studied and compared with existing results in the literature. These normal modes have been derived analytically for N identical, confined particles from a first-order \(L=0\) group theoretic solution of a three-dimensional Hamiltonian with a general two-body interaction. In previous studies, normal modes were able to describe the unitary regime obtaining ground-state energies comparable to benchmark results and thermodynamic quantities in excellent agreement with experiment. As a precurser to this work, the behavior of the normal mode frequencies was investigated across the BCS to unitarity transition, and a microscopic basis of the large excitation gaps and universal behavior at unitarity was proposed. The current study now explores the ability of these normal mode frequencies to determine various properties along this transition, testing the ability of the microscopic dynamics of these normal modes to produce observable behavior away from unitarity. The results suggest that the physics of superfluidity can be described using normal modes across a wide range of interparticle interaction strengths and thus offers a clear microscopic alternative to the two-body pairing models commonly used to describe superfluidity along this transition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data Availability

The data generated or analyzed during this study are available from the author on reasonable request.

Code Availability

Not applicable.

References

  1. M. Greiner, C.A. Regal, D.S. Jin, Nature 426, 537 (2003)

    Article  ADS  Google Scholar 

  2. M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, S. Gupta, Z. Hadzibabic, W. Ketterle, Phys. Rev. Lett. 91, 250401 (2003)

    Article  ADS  Google Scholar 

  3. S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, R. Grimm, Science 302, 2101 (2003)

    Article  ADS  Google Scholar 

  4. J. Kinast, S.L. Hemmer, M.E. Gehm, A. Turlapov, J.E. Thomas, Phys. Rev. Lett. 92, 150402 (2004)

    Article  ADS  Google Scholar 

  5. T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S.J.J.M.F. Kokkelmans, C. Salomon, Phys. Rev. Lett. 93, 050401 (2004)

    Article  ADS  Google Scholar 

  6. C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. 92, 040403 (2004)

    Article  ADS  Google Scholar 

  7. M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman, W. Ketterle, Phys. Rev. Lett. 92, 120403 (2004)

    Article  ADS  Google Scholar 

  8. C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker-Denschlag, R. Grimm, Science 305, 1128 (2004)

    Article  ADS  Google Scholar 

  9. G.B. Partridge, K.E. Strecker, R.I. Kamar, M.W. Jack, R.G. Hulet, Phys. Rev. Lett. 95, 020404 (2005)

    Article  ADS  Google Scholar 

  10. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  11. A.J. Leggett, in Modern Trends in the Theory of Condensed Matter. Proceedings of the XVIth Karpacz Winter School of Theoretical Physics, Karpacz, Poland, pp. 13–27, Springer-Verlag, Berlin, (1980)

  12. A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001)

    Article  ADS  Google Scholar 

  13. D.M. Eagles, Phys. Rev. 186, 456 (1969)

    Article  ADS  Google Scholar 

  14. P. Nozieres, S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985)

    Article  ADS  Google Scholar 

  15. S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 80, 1215 (2008)

    Article  ADS  Google Scholar 

  16. M. Randeria, E. Taylor, Ann. Rev. Condens. Matter Phys. 5, 209 (2014)

    Article  ADS  Google Scholar 

  17. D.K. Watson, Phys. Rev. A 92, 013628 (2015)

    Article  ADS  Google Scholar 

  18. D.K. Watson, J. Phys. B 52, 205301 (2019)

    Article  ADS  Google Scholar 

  19. D.K. Watson, Phys. Rev. A 104, 033320 (2021)

    Article  ADS  Google Scholar 

  20. D.L. Rousseau, R.T. Bauman, S.P.S. Porto, J. Ramam Spect. 10, 253 (1981)

    Article  ADS  Google Scholar 

  21. K.D. Kokkolas, Class. Quantum Grav. 8, 2217 (1991)

    Article  ADS  Google Scholar 

  22. M.J. Clement, App J. 249, 746 (1981)

    ADS  Google Scholar 

  23. N. Zagar, J. Boyd, A. Kasaara, J. Tribbia, E. Kallen, H. Tanaka, and J.-I Yano, Bull. Am. Meteor. Soc. 97, (2016)

  24. S.C. Webb, Geophys. J. Int. 174, 542 (2008)

    Article  ADS  Google Scholar 

  25. B.V. Sanchez, J. Marine Geodesy 31, 181 (2008)

    Article  Google Scholar 

  26. J. Lee, K.T. Crampton, N. Tallarida, V.A. Apkarian, Nature 568, 78 (2019)

    Article  ADS  Google Scholar 

  27. L. Fortunato, EPJ Web Conf. 178, 02017 (2018)

    Article  Google Scholar 

  28. E.C. Dykeman, O.F. Sankey, J. Phys.: Condens. Matter 22, 423202 (2010)

    ADS  Google Scholar 

  29. M. Coughlin and J. Harms, arXiv:1406.1147v1 [gr-qc] (2014)

  30. R.M. Stratt, Acc. Chem. Res. 28, 201 (1995)

    Article  Google Scholar 

  31. C.R. McDonald, G. Orlando, J.W. Abraham, D. Hochstuhl, M. Bonitz, T. Brabec, Phys. Rev. Lett. 111, 256801 (2013)

    Article  ADS  Google Scholar 

  32. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999)

    Article  ADS  Google Scholar 

  33. D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998)

    Article  ADS  Google Scholar 

  34. H. Dong, W. Zhang, L. Zhou, Y. Ma, Sci. Rep. 5, 15848 (2015). https://doi.org/10.1038/srep15848

    Article  ADS  Google Scholar 

  35. H.C. Nagerl, C. Roos, H. Rohde, D. Leibfried, J. Eschner, F. Schmidt-Kaler, R. Blatt, Fortschr. Phys. 48, 623 (2000)

    Article  Google Scholar 

  36. B.A. McKinney, M. Dunn, D.K. Watson, J.G. Loeser, Ann. Phys. 310, 56 (2003)

    Article  ADS  Google Scholar 

  37. M. Dunn, D.K. Watson, J.G. Loeser, Ann. Phys. (NY) 321, 1939 (2006)

    Article  ADS  Google Scholar 

  38. W.B. Laing, M. Dunn, D.K. Watson, J. Math. Phys. 50, 062105 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  39. W.B. Laing, M. Dunn, D.K. Watson, Phys. Rev. A 74, 063605 (2006)

    Article  ADS  Google Scholar 

  40. W.B. Laing, D.W. Kelle, M. Dunn, D.K. Watson, J. Phys. A 42, 205307 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  41. M. Dunn, W.B. Laing, D. Toth, D.K. Watson, Phys. Rev. A 80, 062108 (2009)

    Article  ADS  Google Scholar 

  42. G. t’Hooft, Nucl. Phys. B72, 461(1974)

  43. G. t’Hooft, Nucl. Phys. B75 (1974)

  44. K.G. Wilson, Phys. Rev. Lett. 28, 548 (1972)

    Article  ADS  Google Scholar 

  45. K.G. Wilson, Rev. Mod. Phys. 55, 838 (1983)

    Article  Google Scholar 

  46. G. Chen, Z. Ding, C.-S. Lin, D. Herschbach, M.O. Scully, J. Math. Chem. 48, 791 (2010)

    Article  MathSciNet  Google Scholar 

  47. A. Svidzinsky, G. Chen, S. Chin, M. Kim, D. Ma, R. Murawski, A. Sergeev, M. Scully, D. Herschbach, Int. Rev. Phys. Chem. 27, 665 (2008)

    Article  Google Scholar 

  48. D.R. Herrick, J. Math. Phys. 16, 281 (1975)

    Article  ADS  Google Scholar 

  49. J.G. Loeser, J.H. Summerfield, A.L. Tan, Z. Zhang, J. Chem. Phys. 100, 5036 (1994)

    Article  ADS  Google Scholar 

  50. S. Kais, D.R. Herschbach, J. Chem. Phys. 100, 4367 (1994)

    Article  ADS  Google Scholar 

  51. J.G. Loeser, J. Chem. Phys. 86, 5635 (1987)

    Article  ADS  Google Scholar 

  52. D.R. Herschbach, J.G. Loeser, W.L. Virgo, J. Phys. Chem. A 121, 6336 (2017)

    Article  Google Scholar 

  53. W.L. Virgo, D.R. Herschbach, Chem. Phys. Lett. 634, 179 (2015)

    Article  ADS  Google Scholar 

  54. P.F. Loos, P.M.W. Gill, Phys. Rev. Lett. 105, 113001 (2010)

    Article  ADS  Google Scholar 

  55. A. Svidzinsky, M.O. Scully, D. Herschbach, Phys. Today 66, 33 (2014)

    Article  Google Scholar 

  56. S. Kais, T.C. Germann, D.R. Herschbach, J. Phys. Chem. 98, 11015 (1994)

    Article  Google Scholar 

  57. A. Svidzinsky, M.O. Scully, D.R. Herschbach, Phys. Rev. Lett. 95, 080401 (2005)

    Article  ADS  Google Scholar 

  58. A. Chatterjee, J. Math. Phys. 41, 2515 (2000)

    Article  MathSciNet  Google Scholar 

  59. Y.P. Varshni, Can. J. Phys. 71, 122 (1993)

    Article  ADS  Google Scholar 

  60. S.M. Sung, J.M. Rost, J. Phys. Chem. 97, 2479 (1993)

    Article  Google Scholar 

  61. S. Kais, P. Serra, Adv. Chem Phys. 125, 1 (2003)

    Google Scholar 

  62. V.S. Popov, A.V. Sergeev, Phys. Lett. A 172, 193 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  63. V.S. Popov, A.V. Sergeev, Phys. Lett. A 193, 165 (1994)

    Article  ADS  Google Scholar 

  64. S. Kais, D.R. Herschbach, J. Chem. Phys. 98, 3990 (1993)

    Article  ADS  Google Scholar 

  65. D.K. Watson, B.A. McKinney, Phys. Rev. A 59, 4091 (1999)

    Article  ADS  Google Scholar 

  66. B.A. McKinnney, D.K. Watson, Phys. Rev. A 65, 033604 (2002)

    Article  ADS  Google Scholar 

  67. B.A. McKinney, M. Dunn, D.K. Watson, Phys. Rev. A 69, 053611 (2004)

    Article  ADS  Google Scholar 

  68. M.Y. Veillette, D.E. Sheehy, L. Radzihovsky, Phys. Rev. A 75, 043614 (2007)

    Article  ADS  Google Scholar 

  69. Y. Nishida, D.T. Son, in The BCS-BEC Crossover and the Unitary Fermi Gas. ed. by W. Zwerger (Lecture Notes in Physics, Springer, 2011)

  70. Y. Nishida, D.T. Son, Phys. Rev. Lett. 97, 050403 (2006)

    Article  ADS  Google Scholar 

  71. Y. Nishida, D.T. Son, Phys. Rev. A 75, 063617 (2007)

    Article  ADS  Google Scholar 

  72. P. Zhang, H.-C. Long, C.-S. Jia, Eur. Phys. J. Plus 131, 117 (2016)

    Article  Google Scholar 

  73. Y. Sun, G.-D. Zhang, C.-S. Jia, Chem. Phys. Lett. 636, 197 (2015)

    Article  ADS  Google Scholar 

  74. B. Roy, R. Roychoudhury, J. Phys. A 23, 3555 (1990)

    Article  ADS  Google Scholar 

  75. A. Gonzalez, G. Loyola, M. Moshinsky, Rev. Mex. Fis. 40, 12 (1994)

    Google Scholar 

  76. A. Chatterjee, Phys. Rep. 186, 249 (1990)

    Article  ADS  Google Scholar 

  77. E. Whitten, Nucl. Phys. B 149, 285 (1979)

    Article  ADS  Google Scholar 

  78. A. Gonzalez, Few Body Syst. 10, 43 (1991)

    Article  ADS  Google Scholar 

  79. E. Whitten, Phys. Today 33(7), 38 (1980)

    Article  Google Scholar 

  80. G. t’Hooft, “Large N”, arXiv:hep-th/0204069v1, (2002)

  81. C.J. Ahn, J. High Energy Phys. 125 (2011)

  82. C. Rim, W.I. Weisberger, Phys. Rev. Lett. 53, 965 (1984)

    Article  ADS  Google Scholar 

  83. C. Bender, S. Boettcher, Phys. Rev. D 51, 1875 (1995)

    Article  ADS  Google Scholar 

  84. M. Horacsu, K.D. Rothy, B. Schroer, Nucl. Phys. B 164, 333 (1980)

    Article  ADS  Google Scholar 

  85. E. Brezin, S.R. Wadia (eds.), The Large N Expansion in Quantum Field Theory and Statistical Physics (World Scientific Publ. Co., 1993)

  86. K. Capelle, L.N. Oliveira, Phys. Rev. B 73, 113111 (2006)

    Article  ADS  Google Scholar 

  87. W. Metzner, D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989)

    Article  ADS  Google Scholar 

  88. K.G. Wilson, J. Kogut, Phys. Rev. C 12, 75 (1974)

    Google Scholar 

  89. S. Boettcher, M. Moshe, Phys. Rev. Lett. 74, 2410 (1995)

    Article  ADS  Google Scholar 

  90. C.M. Bender, S. Goettcher, L.R. Mead, J. Math. Phys. 35, 368 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  91. D.K. Watson, Phys. Rev. A 93, 023622 (2016)

    Article  ADS  Google Scholar 

  92. D.K. Watson, Phys. Rev. A 96, 033601 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  93. Y.K. Liu, M. Christandl, F. Verstraete, Phys. Rev. Lett. 98, 110503 (2007)

    Article  ADS  Google Scholar 

  94. A. Montina, Phys. Rev. A 77, 022104 (2008)

    Article  ADS  Google Scholar 

  95. D.K. Watson, M. Dunn, Phys. Rev. Lett. 105, 020402 (2010)

    Article  ADS  Google Scholar 

  96. D.K. Watson, M. Dunn, J. Phys. B 45, 095002 (2012)

    Article  ADS  Google Scholar 

  97. W.B. Laing, M. Dunn, and D.K. Watson, EPAPS Document Number E-JMAPAQ-50-031904

  98. D.R. Herschbach, J. Avery, O. Goscinski (eds.), Dimensional Scaling in Chemical Physics (Kluwer, Dordrecht, 1992)

    Google Scholar 

  99. C. A. Tsipis, V. S. Popov, D. R. Herschbach, and J. Avery (eds.) New Methods in Quantum Theory, NATO Conference Book, Vol. 8. (Kluwer Academic, Dordrecht, Holland)

  100. J.G. Loeser, J. Chem. Phys. 86, 5635 (1987)

    Article  ADS  Google Scholar 

  101. S. Kais, D.R. Herschbach, J. Chem. Phys. 100, 4367 (1994)

    Article  ADS  Google Scholar 

  102. S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995)

    Article  ADS  Google Scholar 

  103. D.K. Watson, Ann. Phys. 419, 168219 (2020)

    Article  Google Scholar 

  104. S.K. Adhikari, Phys. Rev. A 79, 023611 (2009)

    Article  ADS  Google Scholar 

  105. X.-J. Liu, H. Hu, P.D. Drummond, Phys. Rev. Lett. 102, 160401 (2009)

    Article  ADS  Google Scholar 

  106. X.-J. Liu, H. Hu, P.D. Drummond, Phys. Rev. A 82, 023619 (2010)

    Article  ADS  Google Scholar 

  107. X.-J. Liu, H. Hu, P.D. Drummond, Phys. Rev. B 82, 054524 (2010)

    Article  ADS  Google Scholar 

  108. T. Grining, M. Tomza, M. Lesiuk, M. Przybytek, M. Musial, R. Moszynski, M. Lewenstein, P. Massignan, Phys. Rev. A 92, 061601(R) (2015)

    Article  ADS  Google Scholar 

  109. D. Blume, Physics 3, 74 (2010)

    Article  Google Scholar 

  110. D. Blume, Rep. Prog. Phys. 75, 046401 (2012)

    Article  ADS  Google Scholar 

  111. J. Levinsen, P. Massignan, S. Endo, M.M. Parish, J. Phys. B 50, 072001 (2017)

    Article  ADS  Google Scholar 

  112. J. Avery, D.Z. Goodson, D.R. Herschbach, Theor. Chim. Acta 81, 1 (1991)

    Article  Google Scholar 

  113. E.B. Wilson Jr., J. Chem. Phys. 9, 76 (1941)

    Article  ADS  Google Scholar 

  114. E.B. Wilson Jr., J.C. Decius, P.C. Cross, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra (McGraw-Hill, New York, 1955)

    Google Scholar 

  115. M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, MA, 1962)

    Book  MATH  Google Scholar 

  116. See for example Ref. [114], Appendix XII, p. 347

  117. L.D. Landau, The theory of a Fermi liquid. Sov. Phys. 3, 920–925 (1956)

    MathSciNet  MATH  Google Scholar 

  118. L.D. Landau, Oscillations in a Fermi liquid. Sov. Phys. 5, 101–108 (1957)

    MathSciNet  MATH  Google Scholar 

  119. J. Carlson, S. Gandolfi, Phys. Rev. A 90, 011601(R) (2014)

    Article  ADS  Google Scholar 

  120. S.K. Adhikari, Phys. Rev. A 79, 023611 (2009)

    Article  ADS  Google Scholar 

  121. D. Blume, J. von Stecher, C.H. Greene, Phys. Rev. Lett. 99, 233201 (2007)

    Article  ADS  Google Scholar 

  122. S.Y. Chang, G.F. Bertsch, New J. Phys. 11, 023011 (2007)

    Google Scholar 

  123. T.D. Lee et al., Phys. Rev. 105, 1119 (1957)

    Article  ADS  Google Scholar 

  124. R. Jáuregui, R. Paredes, G. Toledo Sánchez, Phys. Rev. A 76, 011604(R) (2007)

    Article  ADS  Google Scholar 

  125. J.T. Stewart, J.P. Gaebler, C.A. Regal, D.S. Jin, Phys. Rev. Lett. 97, 220406 (2006)

    Article  ADS  Google Scholar 

  126. D.T. Son, arXiv:0707.1851 [cond-mat.other], (2007)

  127. Q. Chen, J. Stajic, K. Levin, Phys. Rev. Lett. 95, 260405 (2005)

    Article  ADS  Google Scholar 

  128. S. Nascimbene et al. Nature(London) 463, 1057 (2010)

  129. H. Hu, X.-J. Liu, P.D. Drummond, Phys. Rev. A 73, 023617 (2006)

    Article  ADS  Google Scholar 

  130. L. Luo, B. Clancy, J. Joseph, J. Kinast, J.E. Thomas, Phys. Rev. Lett. 98, 080402 (2007)

    Article  ADS  Google Scholar 

  131. L. Luo, J.E. Thomas, J. Low Temp. Phys. 154, 1 (2009)

    Article  ADS  Google Scholar 

  132. J. Kinast, A. Turlapov, J.E. Thomas, Q. Chen, J. Stajic, K. Levin, Science 307, 1296 (2005)

    Article  ADS  Google Scholar 

  133. M.J.H. Ku, A.T. Sommer, L.W. Cheuk, M.W. Zwierlein, Science 335(6068), 563–567 (2012)

    Article  ADS  Google Scholar 

  134. H. Hu, X.-J. Liu, P.D. Drummond, New J. Phys. 12, 063038 (2010)

    Article  ADS  Google Scholar 

  135. A. Bulgac, J.E. Drut, P. Magierski, Phys. Rev. Lett 99, 120401 (2007)

    Article  ADS  Google Scholar 

  136. E. Burovski, N. Prokof’ev, B. Svistunov, M. Troyer, New J. Phys. 8, 153 (2006)

    Article  ADS  Google Scholar 

  137. H. Hu, X.-J. Liu, P.D. Drummond, Phys. Rev. A 77, 061605(R) (2008)

    Article  ADS  Google Scholar 

  138. R. Haussmann, W. Zwerger, Phys. Rev. A 78, 063602 (2008)

    Article  ADS  Google Scholar 

  139. E. Burovski, N. Prokof’ev, B. Svistunov, M. Troyer, Phys. Rev. Lett. 96, 160402 (2006)

    Article  ADS  Google Scholar 

  140. A. Perali, P. Pieri, L. Pisani, G.C. Strinati, Phys. Rev. Lett. 92, 220404 (2004)

    Article  ADS  Google Scholar 

  141. L. Dell’Anna, S. Grava, Condens. Matter 6, 16 (2021)

    Article  Google Scholar 

  142. C.A.R. Sá de Melo, M. Randeria, J.R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993)

    Article  ADS  Google Scholar 

  143. J. Kinast, A. Turlapov, J.E. Thomas, Phys. Rev. A 70, 051401(R) (2004)

    Article  ADS  Google Scholar 

  144. R. Grimm, Proceedings of the International School of Physics “Enrico Fermi”, Vol. 164: Ultracold Fermi Gases, pp. 413–462, (2007)

  145. M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.H. Denschlag, R. Grimm, Phys. Rev. Lett. 92, 203201 (2004)

    Article  ADS  Google Scholar 

  146. Y.E. Kim, A.L. Zubarev, Phys. Rev. A 70, 033612 (2004)

    Article  ADS  Google Scholar 

  147. N. Manini, L. Salasnich, Phys Rev. A 71, 033625 (2005)

    Article  ADS  Google Scholar 

  148. H. Hu, A. Minguzzi, X.-J. Liu, M.P. Tosi, Phys. Rev. Lett. 93, 190403 (2004)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This research was funded by the National Science Foundation under Grant No. PHY-2011384.

Funding

I am grateful to the National Science Foundation for financial support under Grant No. PHY-2011384.

Author information

Authors and Affiliations

  1. Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK, 73019, USA

    D. K. Watson

Authors
  1. D. K. Watson
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

DKW, as the sole author, was responsible for the design of this research, the investigation, writing the original draft of the manuscript and editing the final draft.

Corresponding author

Correspondence to D. K. Watson.

Ethics declarations

Conflict of interest

The author has no conflicts of interests to declare that are relevant to the content of this article. The author has no relevant financial or non-financial interests to disclose. The author has no potential conflicts of interest.

Consent for Publication

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Watson, D.K. Exploring the Transition from BCS to Unitarity Using Normal Modes: Energies, Entropies, Critical Temperatures and Excitation Frequencies. J Low Temp Phys 212, 1–21 (2023). https://doi.org/10.1007/s10909-023-02966-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10909-023-02966-2

Keywords

Search

Navigation