" If things of sight such heavens be, what heavens are those we cannot see? "       Andrew Marvell

Introduction and references

The discovery of the particle-like behaviour of light leading to the invention of the term photon (ref. 17) at the turn of the twentieth century marks the starting point of the development of quantum theory. At that time many experiments designed to test the new theory remained in the realm of thought experiments. Due to the fast development of laser optics it is now possible to carry out experiments that enable us to actually visualize this strange quantum world. This site intends to give a non-technical, intuitive introduction to quantum optics of the light field.

Recent measurements are employed to illuminate abstract quantum mechanical concepts such as the uncertainty relation, the wave packet, quantum noise, Schroedingers Cat, quantum superpositions, Wigner functions, density matrices, etc. using the concrete example of the freely propagating light field. For an animated version of some of the most common quantum states of the light field, showing their experimentally measured quantum noise distribution and the corresponding motion of their wave packet, see the animation at the end, showing experimental data of the University of Konstanz including one-photon states by the group of Alexander Lvovsky (ref 15 and the link below). Recently added were calculations of states generated online, so you can work with

• Coherent States
• Squeezed States
• Thermal States and Squeezed Thermal States
• Coherent Superposition of two Coherent States (Schrödinger Cat State)
• Number States
• Coherent Superposition of a Number State with a Coherent State (Schrödinger Cat State by quantum optical catalysis)
• Coherent Superposition of two Number States
• Synthesis of a Coherent State by stepwise Addition of Number States
• Synthesis of a Fock State (Number State) by stepwise Addition of Coherent States
• Coherent Displacement of a Fock State

For further reading see:

1. E. Schroedinger, "Der stetige Übergang von der Mikro- zur Makromechanik", Die Naturwiss. 28, 665 (1926);

2. R. J. Glauber, "Coherent and incoherent states of the radiation field", Phys.Rev. 131 (1963);
P. Carruthers, M.M. Nieto, "Coherent states and the forced quantum oscillator", Am. J. Phys. 7, 537 (1965);

3. M. Freyberger, P. Bardroff, C. Leichtle, G. Schrade, and W.P. Schleich, "The art of measuring quantum states", Phys. World, Nov. 1997;

4. U. Leonhardt, Measuring the quantum state of light, Cambridge University Press, Cambridge 1997;

5. E.P. Wigner, "On the quantum correction for thermodynamic equilibrium,", Phys. Rev. A 40, 749, (1932);
M. Hillery, R.F. O'Connell, M.O. Scully, E.P. Wigner "Distribution functions in physics: Fundamentals", Physics Reports Volume 106, Issue 3, (1984);
W.P. Schleich, E. Mayr, D. Kraehmer, Quantum Optics in Phase Space, Wiley, Weinheim 1999;

6. D. F. Walls, "Squeezed states of light", Nature 306(1983);
E. H. Kennard, "Zur Quantenmechanik einfacher Bewegungstypen", Zeit. Phys. 44, 326 (1927);
D.F. Walls and G.J. Milburn, Quantum Optics, Springer Berlin 1994;
A.I. Lvovsky, "Squeezed Light", arXiv:1401.4118v2 [quant-ph], (2016); see also
M. Kizmann, T.L.M. Guedes, D.V. Seletskiy, A.S. Moskalenko, A. Leitenstorfer, G. Burkard, "Subcycle squeezing of light from a time flow perspective", Nature Physics 15, (2019);

7. R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, and J.F. Valley, "Observation of squeezing by four wave mixing in a cavity", Phys. Rev. Lett. 50, 2409 (1985);

8. L.A. Wu, H.J. Kimble, J.L. Hall and H. Wu, "Generation of squeezed states by parametric down conversion", Phys. Rev. Lett. 57, 691 (1986); see also
J. Bauchrowitz, T. Westphal, and R. Schnabel, "A graphical description of optical parametric generation of squeezed states of light ", Am. Journ. Phys. 81, (2013);

9. D.T. Smithey, M. Beck, M.G. Raymer, A. Faridani, "Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum", Phys. Rev. Lett. 70, 1244 (1993);

10. H.P. Yuen and V.W.S. Chan, "Noise in homodyne and heterodyne detection", Opt. Lett. 8, 177 (1983);

11. G. Breitenbach, S. Schiller, and J. Mlynek, "Measurement of the quantum states of squeezed light" , Nature, 387, 471 (1997);

12. G. Breitenbach and S. Schiller, "Homodyne tomography of classical and non-classical light", J. Mod. Opt. 44, 2207 (1997);
G. Breitenbach, F. Illuminati, S. Schiller, and J. Mlynek, "Broadband quantum state reconstruction: A spectrum of quantum states", Europhys. Lett.. 44, 192 (1998);
S. Schiller, G. Breitenbach, "Die Vermessung optischer Quantenzustände", Physikalische Blaetter, Mai 1999;

13. T. Felbinger, S. Schiller, and J. Mlynek, "Oscillation in 3-photon downconversion and generation of non-classical light", Phys. Rev. Lett. 80, 492 (1998);

14. D. Leibfried et al., "Quantum state of the motion of a trapped ion", Phys. Rev. Lett. 77, 4281 (1996);

15. A.I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, "Quantum state reconstruction of the single-photon Fock state", Phys. Rev. Lett. 87, (2002);
A.I. Lvovsky, and S.A. Babichev, "Synthesis and tomographic characterization of the displaced Fock state of light", Phys. Rev. A 66, (2002);
A.I. Lvovsky, and J. Mlynek, "Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics", Phys. Rev. Lett. 88, (2002);

16. T. Briant, P.F. Cohadon, M. Pinard & A. Heidmann, "Optical phase-space reconstruction of mirror motion at the attometer level", Eur. Phys. Journal D 22, 131 (2003);
I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, J. Mlynek, E. Steinsland, N. Blanc, and N.F. Rooij, "Interferometric measurements of the position of a macroscopic body: towards the standard quantum limit" , Phys. Rev. A, 59, 1038 (1999);
Pirkkalainen et al. "Squeezing of Quantum Noise of Motion in a Micromechanical Resonator." Phys. Rev. Lett. 115, 24, 243601 (2015).

17. Special issue Optics and Photonic News, "The nature of light: What is a photon?" ,OPN, (2003), see also
"A Nine Point Argument on: What is a Photon" by Mike Raymer,
"Experiments with single Photons" by Philippe Grangier, and
"Wie gross ist ein Photon?" by H.D.Zeh

18.A. Lvovsky, M. Raymer, Review article "Continuous-variable optical quantum-state tomography", Rev. Mod. Phys., 81, (2009)
V. D’Auria, A. Chiummo, M. De Laurentis, A. Porzio, S. Solimeno, and M.G.A. Paris "Tomographic characterization of OPO sources close to threshold" Optics Express Vol. 13 (2005)

19. S. Machida and Y. Yamamoto, "Observation of amplitude squeezing in a constant-current driven semiconductor laser" , Phys. Rev. Lett. 58, 1000-1003 (1987)

20. T. Coudreau, L. Vernac, A.Z. Khoury, G. Breitenbach, and E. Giacobino, "Quantum tomography of a laser beam interacting with cold atoms" , Europhys. Lett. 46, (1999)

21. G.Breitenbach, S. Schiller, and J. Mlynek, "81% Conversion Efficiency in Frequency-stable Continuous-Wave Parametric Oscillation", J. Opt. Soc. Am. B, 12, 2095 (1995)

22. T. P. Purdy, R. W. Peterson, and C. A. Regal, "Observation of radiation pressure shot noise on a macroscopic object", Science, 339:801, 2013;
T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, "Strong Optomechanical Squeezing of Light", prx.aps.org/abstract/PRX/v3/i3/e031012 (2013)
A.H. Safavi-Naeini1, S. Groblacher, J.T. Hill, J.Chan, M. Aspelmeyer, and O. Painter, "Squeezed light from a silicon micromechanical resonator", doi:10.1038/nature12307 (2013);

23. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, "Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency", Phys. Rev. Lett. 117, 110801 (2016)    see also The GEO600 project

24. A.S. Coelho, F.A.S. Barbosa, K.N. Cassemiro, M. Martinelli, A.S. Villar, and P. Nussenzveig "Analyzing the Gaussian character of the spectral quantum state of light via quantum noise measurements", arXiv:1502.01759v1 [quant-ph] (2015)
"Quantum state reconstruction of spectral field modes: Homodyne and resonator detection schemes", Phys. Rev. A 88, 052113 (2013)

25. H. Carmichael, M. Wolinsky, "Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions", Phys. Rev. Lett. 60, 1836 (1988);
P. Kinsler, P. Drummond, "Quantum Dynamics of the Parametric Oscillator", Phys. Rev. A 43, 6194 (1991);
R. Vyas and S. Singh, "Exact Quantum Distribution for Parametric Oscillators", Phys. Rev. Lett. 74, 2208 (1995);
see also The work at NIST to cat states of light link 1, and link 2,

Related Web sites:

Group of Mike Raymer, Eugene, Oregon

Ulf Leonhard, Weizmann Institute

Quantum state of the motion of a trapped ion, Dietrich Leibfried, NIST, Boulder, Colorado

Group of S. Haroche, M. Brune, ENS, Paris, Measurement of a cavity field Wigner function

Group of Andrew G. White at Univ. of Queensland, Quantum Optics and Quantum Computing

Group of S. Schiller, Düsseldorf, Quantum Optics and Relativity

Group of Antoine Heidmann, Quantum Optics and movable mirrors

Gravitational waves and quantum noise control, GEO600 Hannover

my dissertation (full text),   to my homepage

Vibrating strings and Lissajous curves. The classical treatment of harmonic oscillation.

Crystals and Interference. This site gives a short introduction to optics in non-isotropic media. The reader can experiment with a virtual polarization microscope, generating various kinds of interference patterns

Last modified: June 2020