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## An introduction to Quantum Optics

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### An introduction to Quantum Optics

T. Coudreau

Laboratoire Kastler Brossel, UMR CNRS 8552 et Université Pierre et Marie Curie, PARIS, France

also withPôle 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
- www.astro.lu.se/~dainis

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, Einstein particle
- 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 (1814-20) : Mathematical theory of diffraction and interferences
- Scalar wave
- Fresnel - Arago (1820-30) : polarization phenomena
- Transverse vectorial wave
- Faraday - Maxwell (1850-64) : light as an electromagnetic phenomena
- wave with with

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

("a candle burning at a distance slightly exceeding a mile”)

Photographic plate

Exposure time

"each photon then interferes only with itself”, Dirac

Quantum mechanics (~1925)

- Complete quantum theory of matter : energy levels, atomic collisions
- Atom-field interaction :

Classical electromagnetic waveQuantum atom

« Semi classical theory :

- Energy transfers only by units of
- Momentum transfers by units of

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 (1925-30)

- 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) : En
- contains 0 photons (quanta) : E0

(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").

QED remains a marginal theory (1930-47)

- 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…)

Toward new experiments

- Large success of quantum electrodynamics to predict properties of matter “in the presence of vacuum”.
- Agreement between theory and experiment 10-9
- Progress in optical techniques
- lasers
- better detectors
- non linear optics

Difference between wave and corpuscle

Wave

Continuous

Unlocalised, breakable

Photons

Discontinuous

Localised, unbreakable

A crucial experiment : the semitransparent plate

50% reflected

(1)

(2)

50%

transmitted

The plate does not cut the photon in two !

Experimental result

(1)

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

(2)

Prodution of a state

A single dipole (atom, ion…) emits a single photon at a time

Kimble, Dagenais and Mandel, Phys. Rev. Lett. 39 691 (1977)

First experimental proof of the particle nature of light

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

Hong, Ou and Mandel,

Phys. Rev. Lett. 59 2044 (1987)

No interference fringes : the crystal does not produce classical beams but

Value predicted by

classical theory

Perfect anticorrelations at zero phase shift

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

B

A

Space

magnet B

source

magnet A

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.

A and B measure the spin of particles 1 and 2 along a given axis.

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 ?

is there a “supertheory” (hidden variables) ?

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

a

source

c

b

c

b

Bell inequalities (2)B

A

Classical, hidden variable theory predicts

P(SaSb)+P(Sb Sc)+P(ScSa) = 1 + 2(P1+P8) 1

while Quantum Mechanics predicts :

P(SiSj) = cos2(60°) = 1/4 so that

P(SaSb)+P(Sb Sc)+P(ScSa) = 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

Weihs et al. performed an experiment using parametric down conversion and detectors 400 m apart

Weihs et al., Phys. Rev. Lett 81, 5039(1998)

A

B

Experimental result :

Non local correlations exist !

They do not allow superluminous transfer of information

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 work wave and particle
- only one works wave or particle
- none works neither 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)

Statistical properties of sources (2)

- 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 (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

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