Transcript An introduction to Quantum Optics

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
» 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
19th century : Fresnel, Maxwell...
1900s : Planck, Einstein
1920s : Quantum mechanics
1950s : Quantum Electrodynamics
1960s : Quantum Optics
particle
wave
particle
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 :
pulses
light can be seen as a photon current
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 wave
Quantum 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)
(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
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
One photon interference
To MZ2
To MZ1
Ca beam
Grangier et al., Europhys. Lett 1 173(1986)
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
(1)
(2) and (4) give
(2)
(3)
(4)
which is not verified experimentally
the crystal does not produce classical particles
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.
magnet B
source
magnet A
A and B measure the spin of
particles 1 and 2 along a given
Time
axis.
B
A
Space
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
Bell inequalities (2)
a
a
source
c
b
A
c
b
B
Classical, hidden variable theory predicts
Sa
+
+
+
+
-
Sb
+
+
+
+
-
Sc
+
+
+
+
-
Sa
+
+
+
+
Sb
+
+
+
+
P(SaSb)+P(Sb Sc)+P(ScSa) = 1 + 2(P1+P8)  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 < 1!
“Bell inequalities” enable us to discriminate
Among the first experiments :
A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 91 (1982).
Sc
+
+
+
+
P1
P2
P3
P4
P5
P6
P7
P8
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
» only one works
» none works
» Vacuum fluctuations
» Path interferences
wave and particle
wave or particle
neither wave nor particle
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