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Quantum Optics
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Introduction
Lasers
Interaction of Light With Matter
Field Quantization
Applications of Quantum Optics
M. S. Scully and M. O. Zubairy, Quantum Optics, Cambridge University Press, Cambridge (1997).
L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, NY (1975).
V. Vedral, Modern Foundations of Quantum Optics, Oxford University Press (2006).
M. Fox, An Introduction to Quantum Optics, Oxford University Press (2006).
W. Demtroder, Atoms, Molecules and Photons, Springer-Verlag, Berlin, Heidelberg (2006).
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Non-linear optics
Multiphoton processes
Interaction of light with matter II
Field quantization
Interaction of light with matter III
Saturation phenomena
Optical cooling and trapping of atoms
Quantum cryptography
Quantum computation
Entangled states and quantum teleportation
New trends in quantum optics
The optical Bloch equation
Optics
• Geometrical Optics (GO)
• Physical Optics (PO) - light is an electromagnetic wave. Contains GO as an
approximation
• Quantum Optics - light is quantized in chunks
of energy (photons). Contains PO (and hence
GO) as an approximation
Geometrical Optics
• in a homogeneous and uniform medium light
travels in a straight line
• the angle of incidence is the same as that of
reflection
• the law of refraction is governed by the sines
rule
• they can be derived from a more
fundamental principle - Fermat's principle
Fermat's principle
• Light travels such that the time is extremized
• in a homogeneous medium light travels in a
straight line - the speed of light is the same everywhere and
therefore a straight line - the shortest path between two points  the
shortest travel time
• same reasoning applies for the incidence and
reflection angles
• law of sines
refraction
Suppose that light is going from a medium of n =1 to
a medium of n
n = c/v
the time for travel from A to B
A
y1
n=1
qi
x
n
qr
The laws of GO can be derived
from Fermat's principle
y2
dd
B
• Newton believed that light is made up of
particles
• Hyugens believed that light is a wave - If
light is of particles collision of two beams will lead to some
interesting effect – does not happen
• interference - the key property that won the argument for
Huygens against Newton - demonstrated by Young’s "double slit"
experiment
http://www.whatthebleep.com/trailer/doubleslit.wm.low.html
Photoelectric effect
the energy of the ejected electrons proportional to the light frequency
ejection energy was independent of the total energy of illumination - the
interaction must be like that of a particle which gave all of its energy to the electron
Maxwell's equations
in vacuum

0
- permeability of free space
 0- permitivity of free space
E and B fields propagate at the speed of light
Maxwell - light is an electro-magnetic wave! – interference
Interference
• a light beam of wavelength l, passes through a single
slit of width a. A distance D after the slit we will obtain
a bright spot of diameter s
lD = s a
The Fraunhofer limit - the distance after which the light
starts to spread, a = s
D = a2/l
light starts to behave like a wave
For a 1 mm wide slit, l = 500 nm D = ?
At this D light behaves like a wave
How is GO reconciled with the fact that light is a
wave?
• a beam of light encounters two atoms
• the initial wavevector of light, k, also determines the
propagation direction
• suppose that the light changes its wave vector to k’ after
scattering
• What is the final amplitude?
the intensity is maximum,i.e., a straight line
What changes in QO?
• light is again composed of particles (photons),
behaving like waves - they interfere
both Newton and Huygens were right
basic properties of quantum behavior of light
• Mach-Zehnder interferometer
• beam-splitter transformation
The imaginary phase in front of c – when the light is reflected from a mirror at
90o it picks up a phase of
the light comes out only in one arm
Interaction free measurement
Detector 2
Detector 1
Absorber
if the photon is absorbed – neither of the detectors clicks
if the photon takes the other path – at the last bs it has an equal chance to be
transmitted or reflected – the two detectors click with equal frequencies - the
interference has been destroyed by the presence of the absorber
the presence of an absorber in path 5 can be detected, without the photon even
been absorbed by it - interaction free measurement!
Thus, if detector 2 clicks, then the photon has gone to path 6 - an obstacle in
path 5 else only detector 1 would be clicking
Elitzur-Vaidman bomb-testing problem
.Found. Phys 23, 987 (1993).
P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich
Phys. Rev. Lett. 74, 4763(1995).
Introduction to Lasers
Laser - Light Amplification by Stimulated
Emission of Radiation
The invention and development of lasers paramount to the understanding of interaction
between light and matter
Normal modes in a cavity
Laser is a cavity with a
certain lasing medium
inside
bunch of atoms oscillating inside a box with
highly reflecting mirrors
What do Maxwell's equations tell us about the
radiation ?
light in a cavity cannot be a free propagating wave
the highly reflecting mirrors - electric field on the surface is very nearly zero
where l, n, m are integers and L is the cavity dimension in all three directions
the wavevector comes in each direction in discrete units of p/L
temporal dependence of the field
The number of states with the wavevector k lying in the interval (k, k+dk) is now
proportional to the surface of the sphere so that
This is a continuous number (cannot be true since the wavevector is discrete
units of p/L) and has incorrect dimensions
density of modes
density of modes including polarization is twice
higher
modes in a cavity are defined by their wavelengths
wavelengths can only assume certain sizes such that an integral number of half
wavelength is equal to the length of the cavity L= nl/2
- a consequence of the fact that the electric field has to disappear at the cavity
walls
Basic properties of lasers
• Directed - a beam of a diameter of a few
centimeters directed at the Moon surface would
generate a spot of a size of a few hundred meters
• Intense - intensities can easily reach 1010 W
• Monochromatic - The beam is nearly of one color
- frequency spread of 106 Hz, compared to the 1015
Hz frequency of light produced
• Coherent
• Short
- spatial and temporal coherence
In 1954 Townes in the USA and Basov and
Prokorov in Russia, suggested a method of
achieving lasing - using ammonia to produce
amplified Microwave radiation (MASER) - 1964
Nobel prize for Physics
In 1958 Townes and Schawlow calculated the
conditions to produce visible laser light
In 1960 the first LASER was demonstrated by
Maiman, using a Ruby crystal
Properties of light: blackbody radiation
heat propagates in a given medium: conduction and radiation
conduction is governed by a diffusion equation
rate of change of T a to T gradient
once the temperature is the same everywhere there is no
conduction
radiation is independent of temperature:
a body that reflects all radiation that falls on it – white body
a body that absorbs all the radiation that falls on it - blackbody
from Maxwell's equation we know that light is a
wave - a harmonic oscillator
what is the energy per oscillator in thermal
equilibrium?
statistical physics tells us that every independent
degree of freedom gets a energy of kT/2
the energy of HO of frequency w
energy density - Rayleigh and Jeans
the intensity becomes larger and larger the higher the frequency,
growing at a rate proportional the frequency square
Planck postulated that harmonic oscillators can have energies
only in the "packets" of
he knew from experiments the shape for blackbody radiation
curve and extracted a formula
it does not blow-up for large frequencies - approaches zero as
a way to derive this formula was to assume that the HO energies are quantized
To complete the derivation he assumed that the probability
to occupy the level with energy E is
The average energy is given by
N0 is the total number of osc.
The average energy per oscillator
Blackbody spectrum
there is nothing strange at large frequencies
the quantity I has to be multiplied by some frequency interval to obtain intensity
light is quantized
the total output intensity from a Black-body is obtained by integrating Planck's
expression
The value of the integral is p4/15 so that
Interaction of light with matter - Einstein's treatment
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Einstein thought about the interaction of light and matter in 1917, before the
advent of proper quantum mechanics in 1925
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what kind of information was available to him to attack the problem?
he knew
about Planck's quantum assumption and the correct derivation of a blackbody's
spectrum
that atoms were also quantized and that electrons occupied stationary states
as in Bohr's atomic model
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no clue about how atoms interact with photons
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an atom can absorb a photon to move from a lower energy level to a higher
energy level – stimulated absorption
an atom can spontaneously emit a photon and jump from a higher energy level
to a lower one – spontaneous emission
he reckoned that at thermal equilibrium the rate of emission and absorption
have to be equalized
two level model
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stimulated absorption from the 1 state
and spontaneous emission from state 2
if an atom is left for long enough in an excited state it will
naturally (spontaneously) emit energy
The emission rate - A21 (known as Einstein's A coefficient),
so that the total number of atoms spontaneously emitting
per unit of time is A21N2 (N2 is the No. of atoms in level 2)
The rate of absorption is also proportional to the density of
radiation u(w12) so that the total number of atoms absorbed
per unit of time is u(w12)B12N1 (B12 is the Einstein B
coefficient and N1 is the No. of atoms in the level 1)
In equilibrium the two rates are equal (by definition)
• the two level atom is sitting in the wall of the blackbody
cavity - it has to produce the same kind of radiation that exists
inside since the walls and the radiation are in equilibrium
• the two expression for radiation density coincide only at small
temperature or high frequency
• from the correspondence at low T
• right relationship between spontaneous emission and
stimulated absorption even though we made a mistake, - we
haven't been able to reproduce the right Planck radiation
formula
The correct rate equation
• we need to write a new detailed balance that includes
another process to obtain the right blackbody formula
• stimulated emission
is the number of atoms that emit in a stimulated
fashion per unit of time
Einstein concluded that stimulated emission must also exist
• the equilibrium condition implies dN2/dt = 0
B21 = B12 - the rate of stimulated emission and absorption are equal to
each other
what about the actual values of A and B?
Einstein was unable to say anything about them - he lacked a more
precise quantum mechanical formulation
Saturation density
Suppose that the density of light is made larger and larger
The atoms would follow this increase for some time
absorbing more and more, but would ultimately reach
their maximum capacity when all atoms become excited
At this point light would just continue to propagate through
material without being absorbed by the atoms as they
are saturated
The saturation radiative density, WS, is defined so that the
rate of spontaneous and stimulated emissions are equal
similar to
• A two-level atom therefore either emits or absorbs light - total
energy is in this case conserved
• How about the momentum conservation?
in stimulated emission the light is emitted in the same direction
as the absorbed light - the total net momentum transfer is zero
for an initially stationary atom - since the momentum has to be
conserved - spontaneously emitted light cannot be in any
particular direction
for example a gas of atoms - suppose that spontaneous
emission was directed in a particular way - then you would
expect the gas to drift in a particular direction (i.e., the centre
of mass would be moving) - this never happens in reality spontaneous emission has to be random - uniformly distributed
over the 4p solid angle centered on the atom
important consequences in laser cooling of atoms
Optical Excitation of Two-Level Atoms
rate equation
the radiation density <W> has two independent components
the thermal black body density <WT> and the external density <WE> (the latter
did not exist in Einstein's treatment)
at room temperature, blackbody radiation will be much smaller than from an
external source  blackbody radiation is absent
suppose that all the population is initially in the ground state (i.e., state 1), then
by solving the rate equation
for short times
the number of excited atoms increases
linearly with time
for times long enough, the number of atoms approaches its steady state value
Steady State
from the rate equation
Steady state rates for emission and absorption
when the laser is initially turned on, the population in the excited state
starts to increase linearly, finally reaching its steady state value
in the steady state the populations do not change any more and the total
amount of energy stored in the atoms is given by
once the laser action stops, these atoms release this energy through
spontaneous emission
Lifetime and amplification
• three different processes involved in the interaction between
a two level atom and light
• we may get the wrong impression that stimulated processes
are continuous in time, while the spontaneous emission is an
abrupt
• this is not correct – spontaneous emission is also continuous
the rate equation that we had previously
without external field, <W> = 0
The lifetime is
i.e., inversely proportional to the rate of spontaneous
emission. The population decreases exponentially
Amplification criterion
• for amplification the rate of stimulated emission should be
much bigger than for spontaneous emission
• Let's look at two different regimes: the microwave and the
visible light at the room temperature (T = 300K)
for = 0.1 m, the right hand side is close to 0
for = 500 nm it is huge, e100
Masers are possible
Lasers are impossible!
Population inversion
• the ultimate condition for amplification is population inversion
between two levels, i.e., N2 > N1
• in thermal inversion equilibrium population inversion is
impossible as the weights of states go as
so that
N2 is always less populated than N1
• However, the steady state rate for N2 is given by
always less then N/2 (it approaches this limit for high intensities as
the population inversion is impossible not only in thermal equilibrium, but also under
the presence of an external coherent source - independent of the frequency of
radiation - wrong conclusion