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 with Pôle Matériaux et Phénomènes
  Quantiques, Fédération de Recherche CNRS 2437 et
  Université Denis Diderot , PARIS, France

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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

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Outline

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

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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

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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...

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Some problems remain

• The spectral behaviour of black body radiation is
  not understood
• why the decrease at high frequency ?
• Position of spectral lines

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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

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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
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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
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Quantum mechanics (1925)

• 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

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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 ?

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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)

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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

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Consequence on atomic levels

• Excited levels of atoms are unstable

• Through a quadratic Stark effect, the vacuum
  fluctuations displace the excited levels ("Lamb
  shift").

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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)

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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

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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 !
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Experimental result
(1)
(2)

• 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

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

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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
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Particle interpretation
(1) (2) (3) (4)

• (2) and (4) give which is not
  verified experimentally
• the crystal does not produce classical particles

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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

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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) ?
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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
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Bell inequalities (2)
B
A
Classical, hidden variable theory
predicts P(Sa?Sb)P(Sb ?Sc)P(Sc?Sa) 1
2(P1P8) ? 1 while Quantum Mechanics predicts
P(Si?Sj) cos2(60) 1/4 so
that P(Sa?Sb)P(Sb ?Sc)P(Sc?Sa) 3/4 lt 1!
Bell inequalities enable us to
discriminate Among the first experiments A.
Aspect, P. Grangier, and G. Roger, Phys. Rev.
Lett. 49, 91 (1982).
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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
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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

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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)

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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)

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Statistical properties of sources (3)

• Laser field (stimulated emission inside an
  optical cavity)
• Poissonian distribution
• N photon state

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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 )
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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