In December 1900, Max Planck introduced a new, non-classical statistics to count energy quanta which he had postulated. This enabled him to derive the radiation formula of the heat spectrum on the basis of the two fundamental laws of thermodynamics [1]. In 1924, Satyandra Nath Bose gave a foundation for Planck's statistics, the photon statistics [2]. Applying it to a mono-atomic ideal quantum gas, Albert Einstein generalized this "Bose statistics" to quantum particles with non-zero rest mass [3]. For an ideal gas of massiv bosons he discovered an unprecedented thermodynamic phase transition. He drew an analogy to the condensation of vapour [4]. This Einstein condensation of bosons is now known as "Bose-Einstein condensation". There is no known record that Einstein ever raised the obvious question of a photon condensation [5].
According to Einstein's condensation hypothesis, condensed bosons are in the state of rest. However, the rest mass of photons is zero. Therefore, a photon condensation literally appears to lack substance. However, a subtle analysis shows that, in a grand-canonical ideal photon gas, a Bose-Einstein condensation is possible. The reason for this is that infinitely many photons of infinitesimally small energy do form a finite energy density in the condensate phase. [6, 7, 8].
70 years after Einstein's prognosis the experimental proof of a Bose-Einstein condensation was successful. In an ultra-cold gas of rubidium atoms, by a research group in Boulder, Colorado [9], and in an ultra-cold gas of sodium, by a group at MIT in Cambridge, Massachusetts [10]. In 2010, the group of Martin Weitz, University of Bonn, succeeded in realizing a photon condensation in an optical micro-resonator [11].
The technical potential of photon condensation includes a qualitatively new mechanism of photovoltaic energy conversion, the possibility to store electromagnetic energy, new options to realize coherent light sources [12, 13].
Eberhard E. Müller, General theory of Bose-Einstein condensation applied to an ideal quantum gas of photons in an optical microcavity
Physical Review A 100, 053837 (2019)
Max Planck: „Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum“, Verhandlungen der Deutschen Physikalischen Gesellschaft 2, (1900), S. 237-245.
Satyandra Nath Bose: "Plancks Gesetz und Lichtquantenhypothese". Zeitschrift für Physik 26 (1924) 178-181.
Albert Einstein: "Quantentheorie des einatomigen idealen Gases". Sitzungsberichte der Preussischen Akademie der Wissenschaften XXII, (1924), S. 261-167, Gesamtsitzung vom 10. Juli 1924.
Albert Einstein: "Quantentheorie des einatomigen idealen Gases". Zweite Abhandlung, Sitzungsberichte der Preussischen Akademie der Wissenschaften I, (1925), S. 3-14, Sitzung der physikalisch-mathematischen Klasse vom 8. Januar 1925.
Abraham Pais: "Raffiniert ist der Herrgott ..." Albert Einstein. Eine wissenschaftliche Biografie. Vieweg, Braunschweig, 1986. (Abschnitte 23c, 23d)
Eberhard E. Müller: "Bose-Einstein Condensation of Free Photons". In "Fundamental Aspects of Quantum Theory". Proceedings, Como, Villa Olmo, 1985, V. Gorini and A. Frigerio Eds., Plenum, New York, 1986.
Eberhard E. Müller: "Bose-Einstein Condensation of Free Photons in Thermal Equilibrium". Physica 139A (1986) 165-174.
Eberhard E. Müller: "Bose-Einstein Condensation in Dependence of the Mean Energy Density". Annals of Physics 184 (1988) 219-230
(The equations (27), (40) have to be corrected: lim sR(b, mR(b, u)) = (4/3) kbue(b, m(b, u)); the condensate does not contribute to the entropy density. Consequently section 3 is getting obsolet.)
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemann, E. A. Cornell: "Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor". Science 269 (1995) 198-201.
Davis, K. B., Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. "Bose-Einstein Condensation in a Gas of Sodium Atoms. Phys. Rev. Lett. 75 (1995) 3969-3973.
J. Klaers, J. Schmitt, F. Vewinger, M. Weitz: "Bose-Einstein Condensation of Photons in an optical microcavity". Nature 468 (2010) 545-548.
Eberhard Müller, US-Patent Nr. 4 809 292, "Method and Device to Transform Electromagnetic Waves", 1988.
Eberhard Müller, Europäisches Patent Nr. 1 114 424, "Vorrichtung und Verfahren zur kontrollierten Erzielung eines Photonenflusses zwischen Resonanzen eines elektromagnetischen Resonators", 2004