Bose-Einstein condensation of photons from the thermodynamic limit to small photon numbers

References

• Pathria, R. Statistical Mechanics, 2nd ed.; Elsevier: Amsterdam, Netherlands, 1996.
   Google Scholar
• Penrose, O.; Onsager, L. Bose--Einstein Condensation and Liquid Helium. Phys. Rev. 1956, 104, 576–584. DOI: 10.1103/PhysRev.104.576.
   Web of Science ®Google Scholar
• Würfel, P. The Chemical Potential of Radiation. J. Phys. C: Solid State Phys. 1982, 15 (18), 3967.
   Google Scholar
• Kennard, E.H. On the Interaction of Radiation with Matter and on Fluorescent Exciting Power. Phys. Rev. 1926, 28, 672–683. DOI: 10.1103/PhysRev.28.672.
   Google Scholar
• Stepanov, B.I. A Universal Relation between the Absorption and Luminescence Spectra of Complex Molecules. Sov. Phys. Dokl. 1957, 2, 81.
   Google Scholar
• McCumber, D.E. Einstein Relations Connecting Broadband Emission and Absorption Spectra. Phys. Rev. 1964, 136, A954–A957. DOI: 10.1103/PhysRev.136.A954.
   Web of Science ®Google Scholar
• Nyman, R.A. Absorption and Fluorescence spectra of Rhodamine 6G. April, 2017. DOI: 10.5281/zenodo.569817.
   Google Scholar
• Schaefer, F., Ed. Dye Lasers; Springer-Verlag, Berlin, 1990.
   Google Scholar
• Coles, D.M.; Trichet, A.A.; Dolan, P.R.; Taylor, R.A.; Vallance, C.; Smith, J.M. Diffusion-driven Continuous-wave-pumped Organic Dye Lasers. Laser Photonics Rev. 2015, 9 (5), 538–544.
   Web of Science ®Google Scholar
• Carusotto, I.; Ciuti, C. Quantum Fluids of Light. Rev. Mod. Phys. 2013, 85, 299–366. DOI: 10.1103/RevModPhys.85.299.
   Web of Science ®Google Scholar
• Kavokin, A.; Baumberg, J.J.; Malpuech, G.; Laussy, F.P. Microcavities; OUP, Oxford, 2011; Vol. 16.
   Google Scholar
• Kasprzak, J.; Richard, M.; Kundermann, S.; Baas, A.; Jeambrun, P.; Keeling, J.; Marchetti, F.; Szymańska, M.; Andre, R.; Staehli, J.; et al. Bose--Einstein Condensation of Exciton Polaritons. Nature 2006, 443 (7110), 409–414.
   PubMed Web of Science ®Google Scholar
• Daskalakis, K.; Maier, S.; Murray, R.; Kéna-Cohen, S. Nonlinear Interactions in an Organic Polariton Condensate. Nat. Mater. 2014, 13, 271–278.
   PubMed Web of Science ®Google Scholar
• Plumhof, J.D.; Stöferle, T.; Mai, L.; Scherf, U.; Mahrt, R.F. Room-temperature Bose--Einstein Condensation of Cavity Exciton-polaritons in a Polymer. Nat. Mater. 2014, 13, 247–252.
   PubMed Web of Science ®Google Scholar
• Sun, C.; Jia, S.; Barsi, C.; Rica, S.; Picozzi, A.; Fleischer, J.W. Observation of the kinetic condensation of classical waves. Nat. Phys. 2012, 8 (6), 470–474.
   Web of Science ®Google Scholar
• Weill, R.; Levit, B.; Bekker, A.; Gat, O.; Fischer, B. Laser Light Condensate: Experimental Demonstration of Light-mode Condensation in Actively Mode Locked Laser. Optics express 2010, 18 (16), 16520–16525.
   PubMed Web of Science ®Google Scholar
• Hakala, T.K.; Moilanen, A.J.; Vakevainen, A.I.; Guo, R.; Martikainen, J.P.; Daskalakis, K.S.; Rekola, H.T.; Julku, A.; Torma, P. Bose--Einstein Condensation in a Plasmonic Lattice. arXiv:1706.01528 2017.
   Google Scholar
• Klaers, J.; Vewinger, F.; Weitz, M. Thermalization of a Two-dimensional Photonic Gas in a ‘White Wall’ Photon Box. Nat. Phys. 2010, 6, 512–515.
   Web of Science ®Google Scholar
• Klaers, J.; Schmitt, J.; Vewinger, F.; Weitz, M. Bose--Einstein Condensation of Photons in an Optical Microcavity. Nature 2010, 468, 545–548.
   PubMed Web of Science ®Google Scholar
• Anglin, J. Quantum Optics: Particles of Light. Nature 2010, 468 (7323), 517–518.
   PubMed Web of Science ®Google Scholar
• Rajan, R.; Ramesh Babu, P.; Senthilnathan, K. Photon Condensation: A New Paradigm for Bose--Einstein Condensation. Front. Phys. 2016, 11 (5), 110502. DOI: 10.1007/s11467-016-0568-3.
   Web of Science ®Google Scholar
• Klaers, J.; Schmitt, J.; Damm, T.; Vewinger, F.; Weitz, M. Bose--Einstein Condensation of Paraxial Light. Appl. Phys. B 2011, 105 (1), 17. DOI: 10.1007/s00340-011-4734-6.
   Web of Science ®Google Scholar
• Klaers, J. The thermalization, condensation and flickering of photons. J. Phys. B: At. Mol. Opt. Phys. 2014, 47 (24), 243001. http://stacks.iop.org/0953-4075/47/i=24/a=243001
   Web of Science ®Google Scholar
• Schmitt, J.; Damm, T.; Dung, D.; Vewinger, F.; Klaers, J.; Weitz, M. Bose–Einstein Condensation of Photons versus Lasing and Hanbury Brown-Twiss Measurements with a Condensate of Light. In Laser Spectroscopy: Proceedings of the XXII International Conference, World Scientific, 2016; pp. 85. https://arxiv.org/abs/1606.09015
   Google Scholar
• Proukakis, N.P.; Snoke, D.W.; Littlewood, P.B. Universal Themes of Bose--Einstein Condensation; Cambridge University Press, Cambridge, 2017.
   Google Scholar
• Klaers, J.; Weitz, M. Bose--Einstein Condensation of Photons and Grand-canonical Condensate Fluctuations. arXiv:1611.10286 2016.
   Google Scholar
• Schmitt, J.; Damm, T.; Vewinger, F.; Weitz, M.; Klaers, J. Thermalization of a Two-dimensional Photon Gas in a Polymeric Host Matrix. J. Phys. 2012, 14 (7), 075019. http://stacks.iop.org/1367-2630/14/i=7/a=075019
   Google Scholar
• Palatnik, A.; Tischler, Y.R. Solid-state Rhodamine 6G Microcavity Laser. IEEE Photonics Technol. Lett. 2016, 28 (17), 1823–1826.
   Web of Science ®Google Scholar
• Klaers, J.; Schmitt, J.; Damm, T.; Dung, D.; Vewinger, F.; Weitz, M. Bose--Einstein Condensation of Photons in a Microscopic Optical Resonator: Towards Photonic Lattices and Coupled Cavities. Proc. SPIE 8600 2013, 86000L-86000L--8. DOI: 10.1117/12.2001831.
   Google Scholar
• Dung, D.; Kurtscheid, C.; Damm, T.; Schmitt, J.; Vewinger, F.; Weitz, M.; Klaers, J. Variable Potentials for Thermalized Light and Coupled Condensates. Nat. Photon 2017, 11 (9), 565–569. Letter, DOI: 10.1038/nphoton.2017.139.
   Web of Science ®Google Scholar
• Schmitt, J.; Damm, T.; Dung, D.; Vewinger, F.; Klaers, J.; Weitz, M. Observation of Grand-canonical Number Statistics in a Photon Bose--Einstein Condensate. Phys. Rev. Lett. 2014, 112, 030401. DOI: 10.1103/PhysRevLett.112.030401.
   PubMed Web of Science ®Google Scholar
• Klaers, J.; Schmitt, J.; Damm, T.; Vewinger, F.; Weitz, M. Statistical Physics of Bose--Einstein-Condensed Light in a Dye Microcavity. Phys. Rev. Lett. 2012, 108, 160403. DOI: 10.1103/PhysRevLett.108.160403.
   PubMed Web of Science ®Google Scholar
• van der Wurff, E.C.I.; de Leeuw, A.W.; Duine, R.A.; Stoof, H.T.C. Interaction Effects on Number Fluctuations in a Bose--Einstein Condensate of Light. Phys. Rev. Lett. 2014, 113, 135301. DOI: 10.1103/PhysRevLett.113.135301.
   PubMed Web of Science ®Google Scholar
• Marelic, J.; Nyman, R.A. Experimental evidence for inhomogeneous pumping and energy-dependent effects in photon Bose--Einstein condensation. Phys. Rev. A 2015, 91, 033813. DOI: 10.1103/PhysRevA.91.033813.
   Web of Science ®Google Scholar
• Schmitt, J.; Damm, T.; Dung, D.; Vewinger, F.; Klaers, J.; Weitz, M. Thermalization Kinetics of Light: From Laser Dynamics to Equilibrium Condensation of Photons. Phys. Rev. A 2015, 92, 011602. DOI: 10.1103/PhysRevA.92.011602.
   Web of Science ®Google Scholar
• Damm, T.; Schmitt, J.; Liang, Q.; Dung, D.; Vewinger, F.; Weitz, M.; Klaers, J. Calorimetry of a Bose--Einstein-condensed Photon Gas. Nat. Commun. 2016, 7, 11340.
   PubMed Web of Science ®Google Scholar
• Marelic, J.; Zajiczek, L.F.; Hesten, H.J.; Leung, K.H.; Ong, E.Y.X.; Mintert, F.; Nyman, R.A. Spatiotemporal Coherence of Non-equilibrium Multimode Photon Condensates. J. Phys. 2016, 18 (10), 103012. http://stacks.iop.org/1367-2630/18/i=10/a=103012
   Google Scholar
• Schmitt, J.; Damm, T.; Dung, D.; Wahl, C.; Vewinger, F.; Klaers, J.; Weitz, M. Spontaneous Symmetry Breaking and Phase Coherence of a Photon Bose--Einstein Condensate Coupled to a Reservoir. Phys. Rev. Lett. 2016, 116, 033604. DOI: 10.1103/PhysRevLett.116.033604.
   PubMed Web of Science ®Google Scholar
• Marelic, J.; Walker, B.T.; Nyman, R.A. Phase-space Views into Dye-microcavity Thermalized and Condensed Photons. Phys. Rev. A 2016, 94, 063812. DOI: 10.1103/PhysRevA.94.063812.
   Web of Science ®Google Scholar
• Jagers, H. Polarization of a Photon Bose--Einstein Condensate, B.S. thesis, Utrecht University, Utrecht, Netherlands, 2016.
   Google Scholar
• Damm, T.; Dung, D.; Vewinger, F.; Weitz, M.; Schmitt, J. First-order Spatial Coherence Measurements in a Thermalized Two-dimensional Photonic Quantum Gas. Nat. Commun. 2017, 8, 158.
   PubMed Web of Science ®Google Scholar
• Kirton, P.; Keeling, J. Nonequilibrium Model of Photon Condensation. Phys. Rev. Lett. 2013, 111, 100404. DOI: 10.1103/PhysRevLett.111.100404.
   PubMed Web of Science ®Google Scholar
• Kirton, P.; Keeling, J. Thermalization and Breakdown of Thermalization in Photon Condensates. Phys. Rev. A 2015, 91, 033826. DOI: 10.1103/PhysRevA.91.033826.
   Web of Science ®Google Scholar
• Keeling, J.; Kirton, P. Spatial Dynamics, Thermalization, and Gain Clamping in a Photon Condensate. Phys. Rev. A 2016, 93, 013829. DOI: 10.1103/PhysRevA.93.013829.
   Web of Science ®Google Scholar
• Moodie, R.I.; Kirton, P.; Keeling, J. Polarization Dynamics in a Photon Bose-Einstein Condensate. Phys. Rev. A 2017, 96, 043844. DOI: 10.1103/PhysRevA.96.043844.
   Web of Science ®Google Scholar
• Hesten, H.J.; Mintert, F.; Nyman, R.A. Decondensation in Non-equilibrium Photonic Condensates: When Less is More. arXiv:1705.02173 2017. https://arxiv.org/abs/1705.02173
   Google Scholar
• Kopylov, W.; Radonjić, M.; Brandes, T.; Balaž, A.; Pelster, A. Dissipative Two-mode Tavis-Cummings Model with Time-delayed Feedback Control. Phys. Rev. A 2015, 92, 063832. DOI: 10.1103/PhysRevA.92.063832.
   Web of Science ®Google Scholar
• de Leeuw, A.W.; Stoof, H.T.C.; Duine, R.A. Schwinger-Keldysh theory for Bose--Einstein Condensation of Photons in a Dye-filled Optical Microcavity. Phys. Rev. A 2013, 88, 033829. DOI: 10.1103/PhysRevA.88.033829.
   Web of Science ®Google Scholar
• Chiocchetta, A.; Carusotto, I. Quantum Langevin Model for Nonequilibrium Condensation. Phys. Rev. A 2014, 90, 023633. DOI: 10.1103/PhysRevA.90.023633.
   Web of Science ®Google Scholar
• de Leeuw, A.W.; van der Wurff, E.C.I.; Duine, R.A.; Stoof, H.T.C. Phase Diffusion in a Bose--Einstein Condensate of Light. Phys. Rev. A 2014, 90, 043627. DOI: 10.1103/PhysRevA.90.043627.
   Web of Science ®Google Scholar
• de Leeuw, A.W.; Stoof, H.T.C.; Duine, R.A. Phase Fluctuations and First-order Correlation Functions of Dissipative Bose--Einstein Condensates. Phys. Rev. A 2014, 89, 053627. DOI: 10.1103/PhysRevA.89.053627.
   Web of Science ®Google Scholar
• de Leeuw, A.W.; Onishchenko, O.; Duine, R.A.; Stoof, H.T.C. Effects of Dissipation on the Superfluid Mott-insulator Transition of Photons. Phys. Rev. A 2015, 91, 033609. DOI: 10.1103/PhysRevA.91.033609.
   Web of Science ®Google Scholar
• Chiocchetta, A.; Gambassi, A.; Carusotto, I. Laser operation and Bose--Einstein Condensation: analogies and differences, arXiv:1503.02816. In Chapter in the Book: Universal Themes of Bose--Einstein Condensation, Proukakis, N.P., Snoke, D.W., LIttlewood, P.B. Eds.; Cambridge University Press: Cambridge, 2015; pp. 409–423.
   Google Scholar
• Snoke, D.; Girvin, S. Dynamics of Phase Coherence Onset in Bose Condensates of Photons by Incoherent Phonon Emission. J. Low Temp. Phys. 2013, 171 (1–2), 1–12. DOI: 10.1007/s10909-012-0854-6.
   Web of Science ®Google Scholar
• Lugiato, L.A.; Lefever, R. Spatial Dissipative Structures in Passive Optical Systems. Phys. Rev. Lett. 1987, 58, 2209–2211. DOI: 10.1103/PhysRevLett.58.2209.
   PubMed Web of Science ®Google Scholar
• Nyman, R.A.; Szymańska, M.H. Interactions in Dye-microcavity Photon Condensates and the Prospects for Their Observation. Phys. Rev. A 2014, 89, 033844. DOI: 10.1103/PhysRevA.89.033844.
   Web of Science ®Google Scholar
• Chiao, R.Y.; Boyce, J. Bogoliubov Dispersion Relation and the Possibility of Superfluidity for Weakly Interacting Photons in a Two-dimensional Photon Fluid. Phys. Rev. A 1999, 60, 4114–4121. DOI: 10.1103/PhysRevA.60.4114.
   Web of Science ®Google Scholar
• Chiao, R.Y. Bogoliubov Dispersion Relation for a ‘photon fluid’: Is this a Superfluid? Opt. Commun. 2000, 179 (1–6), 157–166. http://www.sciencedirect.com/science/article/pii/S003040189900615X
   Web of Science ®Google Scholar
• Bolda, E.L.; Chiao, R.Y.; Zurek, W.H. Dissipative Optical Flow in a Nonlinear Fabry-Pérot Cavity. Phys. Rev. Lett. 2001, 86, 416–419. DOI: 10.1103/PhysRevLett.86.416.
   PubMed Web of Science ®Google Scholar
• Vyas, V.M.; Panigrahi, P.K.; Banerji, J. A Scheme to Observe Universal Breathing Mode and Berezinskii-Kosterlitz-Thouless Phase Transition in a Two-dimensional Photon Gas. Phys. Lett. A 2014, 378 (20), 1434–1437. http://www.sciencedirect.com/science/article/pii/S0375960114002266
   Web of Science ®Google Scholar
• de Leeuw, A.W.; van der Wurff, E.C.I.; Duine, R.A.; van Oosten, D.; Stoof, H.T.C. Theory for Bose--Einstein Condensation of Light in Nanofabricated Semiconductor Microcavities. Phys. Rev. A 2016, 94, 013615. DOI: 10.1103/PhysRevA.94.013615.
   Web of Science ®Google Scholar
• Calvanese Strinati, M.; Conti, C. Bose--Einstein Condensation of Photons with Nonlocal Nonlinearity in a Dye-doped Graded-index Microcavity. Phys. Rev. A 2014, 90, 043853. DOI: 10.1103/PhysRevA.90.043853.
   Web of Science ®Google Scholar
• Kruchkov, A. Bose--Einstein Condensation of Light in a Cavity. Phys. Rev. A 2014, 89, 033862. DOI: 10.1103/PhysRevA.89.033862.
   Web of Science ®Google Scholar
• Kruchkov, A.J. One-dimensional Bose--Einstein Condensation of Photons in a Microtube. Phys. Rev. A 2016, 93, 043817. DOI: 10.1103/PhysRevA.93.043817.
   Web of Science ®Google Scholar
• Cheng, Z. Bose--Einstein Condensation of Ideal Photons in a One-dimensional Barrel Cavity. Phys. Rev. A 2016, 93, 023829. DOI: 10.1103/PhysRevA.93.023829.
   Web of Science ®Google Scholar
• Sob’yanin, D.N. Hierarchical Maximum Entropy Principle for Generalized Superstatistical Systems and Bose--Einstein Condensation of Light. Phys. Rev. E 2012, 85, 061120. DOI: 10.1103/PhysRevE.85.061120.
   Web of Science ®Google Scholar
• Sob’yanin, D.N. Bose--Einstein Condensation of Light: General theory. Phys. Rev. E 2013, 88, 022132. DOI: 10.1103/PhysRevE.88.022132.
   Web of Science ®Google Scholar
• Weiss, C.; Tempere, J. Grand-canonical Condensate Fluctuations in Weakly Interacting Bose--Einstein Condensates of Light. Phys. Rev. E 2016, 94, 042124. DOI: 10.1103/PhysRevE.94.042124.
   PubMed Web of Science ®Google Scholar
• Zannetti, M. The Grand Canonical Catastrophe as an Instance of Condensation of Fluctuations. EPL (Europhys. Lett.) 2015, 111 (2), 20004. http://stacks.iop.org/0295-5075/111/i=2/a=20004
   Google Scholar
• Weitz, M.; Klaers, J.; Vewinger, F. Optomechanical Generation of a Photonic Bose--Einstein Condensate. Phys. Rev. A 2013, 88, 045601. DOI: 10.1103/PhysRevA.88.045601.
   Web of Science ®Google Scholar
• Wahl, C.; Brausemann, R.; Schmitt, J.; Vewinger, F.; Christopoulos, S.; Weitz, M. Absorption Spectroscopy of Xenon and Ethylene-Noble Gas Mixtures at High Pressure: Towards Bose--Einstein Condensation of Vacuum Ultraviolet Photons. Appl. Phys. B 2016, 122 (12), 296.
   Web of Science ®Google Scholar
• Weill, R.; Bekker, A.; Levit, B.; Zhurahov, M.; Fischer, B. Breaking Two Laser Axioms: Lasing Without Inversion and Thermal Equilibrium. arXiv:1607.01681 2016.
   Google Scholar
• Zel’Dovich, Y.B.; Levich, E. Bose Condensation and Shock Waves in Photon Spectra. Sov. Phys. JETP 1969, 28 (11), 1287.
   Google Scholar
• Mendonça, J.T.; Terças, H. Bose--Einstein Condensation of Photons in a Plasma. arXiv:1704.05981 2017.
   Google Scholar
• Marcos, D.; Tomadin, A.; Diehl, S.; Rabl, P. Photon Condensation in Circuit Quantum Electrodynamics by Engineered Dissipation, New. J. Phys. 2012, 14 (5), 055005. http://stacks.iop.org/1367-2630/14/i=5/a=055005
   Google Scholar
• Fani, M.; Naderi, M.H. Thermalization and Bose--Einstein Condensation of a Photon Gas in a Multimode Hybrid Atom-membrane Optomechanical Microcavity. J. Opt. Soc. Am. B 2016, 33 (6), 1242–1250. http://josab.osa.org/abstract.cfm?URI=josab-33-6-1242
   Web of Science ®Google Scholar
• Fani, M.; Naderi, M.H. Coherent Coupling Between an Optomechanical Membrane and an Interacting Photon Bose--Einstein Condensate. J. Mod. Opt. 2017, 1–14. DOI: 10.1080/09500340.2017.1311957.
   Web of Science ®Google Scholar
• Zhang, X.; Yin, M.; Liang, W. Position-dependent Oscillated Decay of a Two-level Atom Immersed in a Two-dimensional Photon Fluid. Opt. Commun. 2016, 359, 44–48. http://www.sciencedirect.com/science/article/pii/S0030401815301450
   Web of Science ®Google Scholar
• Zhang, J.J.; Yuan, J.H.; Zhang, J.P.; Cheng, Z. Temperature Dependence of Atomic Decay Rate Induced by the BEC of Photons. Phys. E 2012, 45, 177–182.
   Google Scholar
• Chiocchetta, A.; Larré, P.É.; Carusotto, I. Thermalization and Bose--Einstein Condensation of Quantum Light in Bulk Nonlinear Media. EPL (Europhys. Lett.) 2016, 115 (2), 24002.
   Google Scholar
• De Martini, F.; Jacobovitz, G.R. Anomalous Spontaneous-Stimulated-Decay Phase Transition and Zero-Threshold Laser Action in a Microscopic Cavity. Phys. Rev. Lett. 1988, 60, 1711–1714. DOI: 10.1103/PhysRevLett.60.1711.
   PubMed Web of Science ®Google Scholar
• Yokoyama, H.; Brorson, S.D. Rate Equation Analysis of Microcavity Lasers. J. Appl. Phys. 1989, 66 (10), 4801–4805. DOI: 10.1063/1.343793.
   Web of Science ®Google Scholar
• Yokoyama, H.; Suzuki, M.; Nambu, Y. Spontaneous Emission and Laser Oscillation Properties of Microcavities Containing a Dye Solution. Appl. Phys. Lett. 1991, 58 (23), 2598–2600. DOI: 10.1063/1.104834.
   Web of Science ®Google Scholar
• Yokoyama, H. Physics and Device Applications of Optical Microcavities. Science 1992, 256 (5053), 66–70. http://science.sciencemag.org/content/256/5053/66
   PubMed Web of Science ®Google Scholar
• De Martini, F.; Cairo, F.; Mataloni, P.; Verzegnassi, F. Thresholdless Microlaser. Phys. Rev. A 1992, 46, 4220–4233. DOI: 10.1103/PhysRevA.46.4220.
   PubMed Web of Science ®Google Scholar
• Björk, G.; Karlsson, A.; Yamamoto, Y. Definition of a Laser Threshold. Phys. Rev. A 1994, 50, 1675–1680. DOI: 10.1103/PhysRevA.50.1675.
   PubMed Web of Science ®Google Scholar
• Rice, P.R.; Carmichael, H.J. Photon Statistics of a cavity-QED Laser: A Comment on the Laser-Phase-Transition Analogy. Phys. Rev. A 1994, 50, 4318–4329. DOI: 10.1103/PhysRevA.50.4318.
   PubMed Web of Science ®Google Scholar
• Dolan, P.R.; Hughes, G.M.; Grazioso, F.; Patton, B.R.; Smith, J.M. Femtoliter Tunable Optical Cavity Arrays. Opt. Lett. 2010, 35 (21), 3556–3558. http://ol.osa.org/abstract.cfm?URI=ol-35-21-3556
   PubMed Web of Science ®Google Scholar
• Palatnik, A.; Aviv, H.; Tischler, Y.R. Microcavity Laser Based on a Single Molecule Thick High Gain Layer. ACS Nano 2017, 11 (5), 4514–4520. DOI: 10.1021/acsnano.6b08092.
   PubMed Web of Science ®Google Scholar