An introduction to Quantum Optics T. Coudreau Laboratoire Kastler Brossel, UMR CNRS 8552 et Université Pierre et Marie Curie, PARIS, France also with Pôle Matériaux et Phénomènes Quantiques, Fédération de Recherche CNRS 2437 et Université Denis Diderot, PARIS, France

Why a course on quantum optics ? Quantum optics are concerned with the statistics of the electromagnetic field (variance, correlation functions …) The statistics give an idea on the nature of the source : thermal, poissonian... The statistics may give an idea on the basic properties of astrophysical sources »

Outline Historical approach » Electromagnetism » Planck and Einstein » Quantum Mechanics » Quantum Electrodynamics » Conclusive experiments Statistical properties of light Quantum optics with OPOs

Introduction Does light consist in waves or particles ? 17th century : Newton particle 19th century : Fresnel, Maxwell... wave 1900s : Planck, Einsteinparticle 1920s : Quantum mechanics 1950s : Quantum Electrodynamics 1960s : Quantum Optics

XIX th century Young (~1800) : interferences, a light wave can be added or substracted »Sinusoïdal wave Fresnel ( ) : Mathematical theory of diffraction and interferences »Scalar wave Fresnel - Arago ( ) : polarization phenomena »Transverse vectorial wave Faraday - Maxwell ( ) : light as an electromagnetic phenomena »wave withwith Everything is understood but...

Some problems remain The spectral behaviour of black body radiation is not understood : »why the decrease at high frequency ? Position of spectral lines

Some more problems... Photoelectric effect (Hertz and Hallwachs, 1887) »UV light removes charges on the surface while a visible light does not Planck : energy exchange occur with multiples of Bohr : atomic energy levels

Light is made of particles Light is made of unbreakable “quanta” of energy (Einstein 1905) This was later checked by Millikan The Compton effect (1923) The particle (“photon”) possesses a given momentum Photomultiplier : light can be seen as a photon current pulses

Interferences and photons Taylor (1909) : Young's slits with an attenuated source Exposure time "each photon then interferes only with itself”, Dirac ("a candle burning at a distance slightly exceeding a mile”) Photographic plate

Complete quantum theory of matter : energy levels, atomic collisions Atom-field interaction : Classical electromagnetic wave Quantum atom « Semi classical theory : »Energy transfers only by units of »Momentum transfers by units of Quantum mechanics (~1925)

Consequences of the semiclassical theory Photoelectric, Compton effects can be understood with a classical wave Pulses recorded in the photomultiplier are due to quantum jumps inside the material and not to the granular structure of light same for the photographic plate in Taylor ’s experiment Light remains a classical electromagnetic wave »Should Einstein be deprived of his (only) Nobel prize ? »And Compton ?

Quantum electrodynamics ( ) Quantum calculations are applied to light in the absence of matter In the case of a monochromatic light, the energy is quantified : » contains n photons (quanta) : E n »contains 0 photons (quanta) : E 0 (Vacuum, absence of radiation, fundamental state of the system)

Consequence on the electric field Existence of an Heisenberg inequality analogous to (for a monochromatic wave) Consequences »There is no null field at all moments (see “there is no particle at rest”) »The electromagnetic field in vacuum is not identically null The field is null only on average : existence of vacuum fluctuations

Consequence on atomic levels Excited levels of atoms are unstable Through a quadratic Stark effect, the vacuum fluctuations displace the excited levels ("Lamb shift").

Reasons 1) Problem of interpretation 2) Problem of formalism : many diverging quantities e.g. Vacuum energy : 3) Problem of "concurrence" : the more simple semiclassical theory gives (generally) the same results 2) was solved in 1947 (Feynman, Schwinger & Tomonaga) : QED serves as a base and model for all modern theoretical physics (elementary particles…) QED remains a marginal theory ( )

Toward new experiments Large success of quantum electrodynamics to predict properties of matter “in the presence of vacuum”. »Agreement between theory and experiment Progress in optical techniques »lasers »better detectors »non linear optics

Difference between wave and corpuscle A crucial experiment : the semitransparent plate Wave Continuous Unlocalised, breakable Photons Discontinuous Localised, unbreakable 50% reflected 50% transmitted (1) (2) The plate does not cut the photon in two !

Experimental result But a very faint source does not produce a true one photon state : the beam is a superposition of different states, e.g. A faint source does not give a clear result (1) (2)

Prodution of a state A single dipole (atom, ion…) emits a single photon at a time Kimble, Dagenais and Mandel, Phys. Rev. Lett (1977) First experimental proof of the particle nature of light

One photon interference Grangier et al., Europhys. Lett 1 173(1986) Ca beam To MZ1 To MZ2

Non linear optics experiments With a pump at frequency  0, the crystal generates twin photons at frequencies  1 and  2. There is a perfect correlation between the two channels Furthermore, the system behaves as an efficient source of single photon states : the resulting light cannot be described by two classical waves emitted by a crystal described quantically

Interferences with twin beams No interference fringes : the crystal does not produce classical beams but Perfect anticorrelations at zero phase shift Hong, Ou and Mandel, Phys. Rev. Lett (1987) Value predicted by classical theory

Particle interpretation (2) and (4) give which is not verified experimentally the crystal does not produce classical particles (1) (2) (3) (4)

What have we learned ? Light can behave like a classical wave » Classical interferences Light can behave like a classical particle » One photon interferences Light can behave like a non classical state » Two photon interferences

Non Locality in Quantum Mechanics 1935 (A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935) ) : Einstein, Podolski and Rosen worry about the non-local character of quantum mechanics. Space Time A B A and B measure the spin of particles 1 and 2 along a given axis. is there a “supertheory” (hidden variables) ? source magnet A magnet B If the two observers choose the same axis, they get an opposite result but if they choose different axis, can they measure simultaneously orthogonal directions ?

Bell inequalities (1) 1965 (J. S. Bell, Physics 1, 195 (1965). ) : J.S Bell proposes a way to discriminate between a local hidden variables theory and quantum theory. One assumes that the experimental result depends on a “hidden variable” and on the magnets orientations but not on the other measurement : The classical probability to obtain a given result is given by While the quantum theory prediction is written

Bell inequalities (2) a b c source a b c A B Classical, hidden variable theory predicts P(S a  S b )+P(S b  S c )+P(S c  S a ) = 1 + 2(P 1 +P 8 )  1 while Quantum Mechanics predicts : P(S i  S j ) = cos 2 (60°) = 1/4 so that P(S a  S b )+P(S b  S c )+P(S c  S a ) = 3/4 < 1! “Bell inequalities” enable us to discriminate Among the first experiments : A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 91 (1982).

Non locality tests with non linear media Non local correlations exist ! They do not allow superluminous transfer of information A B Experimental result : Weihs et al. performed an experiment using parametric down conversion and detectors 400 m apart Weihs et al., Phys. Rev. Lett 81, 5039(1998)

QED : an accepted theory All measurement results (up to now) are in agreement with the predictions of quantum electrodynamics (including experiments of measurement and control of quantum fluctuations) No more mysteries the actual theory explains without ambiguity all phenomena but still "strange" behaviours Physical images » several may workwave and particle » only one workswave or particle » none worksneither wave nor particle » Vacuum fluctuations » Path interferences

Statistical properties of sources (1) Different sources, single atoms, nonlinear crystals, … are able to generate different types of fields. What should we study ? The statistical properties of the field The properties of statistical variables are described by Photon number probability distributions 2nd order moment : 2nd order coherence (1st order = interference)

Spontaneous emission by a single dipole (atom, ion, …) variance and photon number distribution : depend on pumping antibunching Spontaneous emission by an incoherent ensemble of dipoles (Thermal / chaotic light) bunching (Hanbury Brown & Twiss) Statistical properties of sources (2)

Statistical properties of sources (3) Laser field (stimulated emission inside an optical cavity) Poissonian distribution N photon state

Quantum correlations with an OPO At the output of an OPO, the signal and idler beams have quantum intensity correlations. Heidmann et al., Phys. Rev. Lett. 59, 2555 (1987) Result : 30 % noise reduction (now : over 85 %)

Conclusion No more mysteries QED explains without ambiguity all phenomena but still "strange" behaviours The results depend on the quantum state of the field – Vacuum – n photons – statistical mixture Statistical properties of light give an insight on the properties of the emitting object OPOs provide an efficient source of non classical light