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Quantum Optics
Leture at the Optics and Photonics Winter School
Poul Jessen
Brian Anderson
Rolf Binder
Jason Jones
Galina Khitrova
Miroslav Kolesik
Masud Mansuripur
Jerome Moloney
Ewan Wright
What is Quantum Optics?
– Not classical optics, but something more and different
– Science of non-classical light – Any science combining light and quantum mechanics
What is Light?
– Electromagnetic waves?
– Photons?
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Not so fast!
© Springer-Verlag 1995
Not so fast!
Anti-photon
W.E. Lamb, Jr.
Optical Sciences Center, Universityof Arizona, Tucson, AZ 85721, USA
Received: 23 July 1994/Accepted: 18 September 1994
Abstract. It should be apparent from the title of this
article that the author does not like the use of the word
"photon", which dates from 1926. In his view, there is no
such thing as a photon. Only a comedy of errors and
historical accidents led to its popularity among physicists
and optical scientists. I admit that the word is short and
convenient. Its use is also habit forming. Similarly, one
might find it convenient to speak of the "aether" or "vacuum" to stand for empty space, even if no such thing
existed. There are very good substitute words for
"photon", (e.g., "radiation" or "light"), and for "photonics" (e.g., "optics" or "quantum optics"). Similar objections are possible to use of the word "phonon", which
dates from 1932. Objects like electrons, neutrinos of finite
rest mass, or helium atoms can, under suitable conditions,
be considered to be particles, since their theories then have
viable non-relativistic and non-quantum limits. This paper outlines the main features of the quantum theory of
radiation and indicates how they can be used to treat
problems in quantum optics.
PACS: 12.20.-m; 42.50.-p
afterward, there was a population explosion of people
engaged in fundamental research and in very useful technical and commercial developments of lasers. QTR was
available, but not in a form convenient for the problems at
hand. The photon concepts as used by a high percentage
of the laser community have no scientific justification. It is
now about thirty-five years after the making of the first
laser. The sooner an appropriate reformulation of our
educational processes can be made, the better.
1 A short history of pre-photonic radiation
Modern optical theory [2] began with the works of Ch.
Huyghens and I. Newton near the end of the seventeenth
century. Huyghen's treatise on wave optics was published
in 1690. Newton's "Optiks', which appeared in 1704, dealt
with his corpuscular theory of light.
A decisive work in 1801 by T. Young, on the two-slit
diffraction pattern, showed that the wave version of optics
was much to be preferred over the corpuscular form.
However, so high was the prestige of I. Newton, that the
With all due resepect to W. Lamb, let us try again What is light?
– a wave?
– a stream of particles (photons)?
Take the question seriously
– test each hypothesis through experimentation! Key signature of wave behavior?
– Interference!
Double-slit experiment
Single-slit diffraction
Double-slit diffraction
Key signature of particle behavior?
Einstein: photo-electric effect
Electrons are released only for light with a frequency
! such that h! is greater than the work function of
w
the metal in question
But the quantum theory of electron excitation can
explain this based on classical electromagnetic fields,
so the photo-electric effect only confirms that
electrons are particles.
BS
Indivisibility
single
photon
input
A particle incident on a barrier is either transmitted or
reflected.
or
single
photon
input
Wave behavior
BS
M
single
photon
input
PD1
PD2
prob. of detection
PD1
!
BS
PD2
M
phase shifter
!
Particle behavior
prob. of detection
PD1
PD2
M
single
photon
input
BS
PD1
PD2
!
M
phase shifter
!
– Evidently a single photon can behave like a wave or a particle, depending on the experiment we do. This is what we know as
wave-particle duality.
– Does the photon “know” when it hits the first BS if we are doing a wave or particle experiment and then behaves accordingly?
– Wigner’s gedanken experiment: Delayed Choice!
Decide at random whether to put in the second BS only after the
photon has passed the first BS
Wave behavior
M
single
photon
input
BS
Particle behavior
BS
PD1
PD2
M
!
M
phase shifter
PD1
PD2
single
photon
input
BS
!
M
phase shifter
Wigners experiment was done in 2008
PRL 100, 220402 (2008)
week ending
6 JUNE 2008
PHYSICAL REVIEW LETTERS
Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons
Vincent Jacques,1 E Wu,1,2 Frédéric Grosshans,1 François Treussart,1 Philippe Grangier,3
Alain Aspect,3 and Jean-François Roch1,*
1
Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, UMR CNRS 8537, Cachan, France
2
State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai, China
3
Laboratoire Charles Fabry de l’Institut d’Optique, UMR CNRS 8501, Palaiseau, France
(Received 12 February 2008; published 3 June 2008)
We report an experimental test of quantum complementarity with single-photon pulses sent into a
Mach-Zehnder interferometer with an output beam splitter of adjustable reflection coefficient R. In
addition, the experiment is realized in Wheeler’s delayed-choice regime. Each randomly set value of R
allows us to observe interference with visibility V and to obtain incomplete which-path information
characterized by the distinguishability parameter D. Measured values of V and D are found to fulfill the
complementarity relation V 2 ! D2 " 1.
DOI: 10.1103/PhysRevLett.100.220402
As emphasized by Bohr [1], complementarity lies at the
heart of quantum mechanics. A celebrated example is the
illustration of wave-particle duality by considering single
particles in a two-path interferometer [2], where one chooses either to observe interference fringes, associated to a
wavelike behavior, or to know which path of the interferometer has been followed, according to a particlelike
behavior [3]. Although interference has been observed at
the individual particle level with electrons [4], neutrons
[5], atoms [6,7], molecules [8], only a few experiments
with massive particles have explicitly checked the mutual
exclusiveness of which-path information (WPI) and interference [9–13].
PACS numbers: 03.65.Ta, 42.50.Ar, 42.50.Xa
anced interferometer as used in some neutron interferometry experiments, one has partial WPI while keeping
interference with limited visibility [24]. The complementary quantities —WPI and interference visibility —could
then be partially determined simultaneously.
Consistent theoretical analysis of both schemes, independently published by Jaeger et al. [25] and by Englert
[26] leads to the inequality [27]
V 2 ! D2 " 1
(1)
which puts an upper bound to the maximum values of
simultaneously determined interference visibility V and
where V" is the half-wave voltage of the EOM. The
parameters ! and V" have been independently measured
for our experimental conditions and found equal to ! !
24 " 1# and V" ! 217 " 1 V at the wavelength # !
670 nm which is the emission peak of the negatively
charged N-V color center [32]. This allows R to vary
between 0 and 0.5 when VEOM is varied between 0 and
170 V.
When R ! 0, the VBS is equivalent to a perfectly transparent (or absent) beam splitter. Then, each ‘‘click’’ of one
of the two photodetectors (P1 or P2 ) placed on the output
EOM [see Eq. (2)]. In the laboratory framework, the random choice of VBS reflectivity is realized simultaneously
with the entering of the corresponding emitted photon into
the interferometer, ensuring the required spacelike separation [22].
As meaningful illustration of complementarity requires
the use of single particles, the quantum behavior of the
light field is first tested using the two output detectors
feeding single and coincidence counters with no voltage
applied to the EOM. The output beam splitter is then
absent, and each detector is therefore univocally associated
Wigners experiment was done in 2008
Clock
4.2 MHz
QRNG
VEOM
S
P
FIG. 1 (color online). Delayed-choice complementarity-test
experiment. A single-photon pulse is sent into a MachZehnder interferometer, composed of a 50=50 input beam splitter (BS) and a variable output beam splitter (VBS). The reflection coefficient is randomly set either to the null value or to an
adjustable value R, after the photon has entered the interferometer. The single-photon photodetectors P1 and P2 allow to record
both the interference and the WPI.
EOM
Path 1
Path 2
P2
WP
PBS
Φ
P1
VBS
FIG. 2 (color online). Variable output beam splitter (VBS)
implementation. The optical axis of the polarization beam splitter (PBS) and the polarization eigenstates of the Wollaston prism
(WP) are aligned, and make an angle ! with the optical axis of
the EOM. The voltage VEOM applied to the EOM is randomly
chosen accordingly to the output of a Quantum Random Number
Generator (QRNG), located at the output of the interferometer
and synchronized on the 4.2-MHz clock that triggers the singlephoton emission.
220402-2
1000
(c) R=0
Counts
tion beam splitter PBS of VBS with a piezoelectric actuator
(see Fig. 2). For each photon, we record the chosen configuration of the interferometer, the detection events, and
the actuator position. All raw data are saved in real time
and are processed only after a run is completed. The events
corresponding to each configuration of the interferometer
are finally sorted. For a given value R, the wavelike inforweek ending
mation
the
L REVIE
W L of
E Tthe
T Elight
R S field is obtained by measuring
visibility of the interference, predicted to be 6 JUNE 2008
1000
p!!!!!!!!!!!!!!!!!!!
(a) R=0.43
elation
V # 2 R!1 % R":
(3)
second
n ideal
The500
results, depicted in Fig. 3, show a reduction of V when
perfect
the randomly applied value of R decreases.
clelike
To test inequality (1), a value of the distinguishability D
usly in
is then0required, to qualitatively qualify the amount of WPI
e [32],
4π
which can be πextracted for2π
each value of3πR. We introduce
Phase
shift
Φ
(rad)
than 1,
the quantity D1 (resp. D2 ), associated to the WPI on path 1
1000path 2):
photon
(resp.
(b) R=0.05
D1 # jp!P1 ; path 1" % p!P2 ; path 1"j
(4)
single500
ndomly
correD2 # jp!P1 ; path 2" % p!P2 ; path 2"j
(5)
and R
he two
folwhere 0p!Pi ; path j" πis the probability
2π that the particle
3π
olarizalows path j and is detected
onshift
detector
Phase
Φ (rad)Pi . For a single
ctuator
particle
arriving
on
the
output
beam
splitter, one obtains
1000
n con"
"
(c) R=0
"
"
"
"
1
"
"
ts, and
% R"
D1 # D2 # "
:
(6)
"
"
"
"
"
"
al time
2
500
events
The distinguishability parameter D is finally defined as
ometer
[26]
e infor0
π
2π
3π
ng the
D # D1 &
D2 #
j1 Φ
%(rad)
2Rj:
(7)
Phase
shift
500
Counts
Counts
Counts
Wigners experiment was done in 2008
(3)
In order
to test
this relation,
we visibility
estimate Vthe
values in
of the
D1
FIG.
3 (color
online).
Interference
measured
and D2 by blocking
path of the
interferometer
and
delayed-choice
regime one
for different
values
of VEOM . (a)–
' 150 V N(R
#
0:43
and
V
#
(c)
correspond
to VEOM
measuring
the number
of detections
and
N
on
detectors
1
2
0
π
2π
Phase shift Φ (rad)
3π
FIG. 3 (color online). Interference visibility V measured in the
delayed-choice regime for different values of VEOM . (a)–
(c) correspond to VEOM ' 150 V (R # 0:43 and V #
93 $ 2%), VEOM ' 40 V (R # 0:05 and V # 42 $ 2%), and
VEOM # 0 (R # 0 and V # 0). Each point is recorded with
1.9 s acquisition time. Detectors dark counts, corresponding to
a rate of 60 s%1 for each, have been substracted to the data.
#
1 jN1 % N2 j
D1 #
is both
2 N1 &aN2particle
path 2 blocked
Light
and a(8)
wave at the same time.
#
1 jN1 % N2 j
D2 #
:
(9)
2 N1 & N2 path 1 blocked
What property we see depends on
These measurements are also performed in the delayedwhat
we described
decideabove.toWe
choice
regime,property
using the procedure
finally
obtain independent measurements of D and V for
measure.
different values of the reflection coefficient R, randomly
to the interferometer. The final results, depicted on
applied
2
Fig.This
4, lead to
& D2 # 0:97
0:03, close
to the
maxiisV totally
in $line
with
our
mal value permitted by inequality (1) even though each
general
quantum
quantity
varies from
0 to 1. theory.
The effects observed in this delayed-choice experiment
are in perfect agreement with quantum mechanics predictions. No change is observed between a so-called ‘‘normalchoice’’ experiment and the ‘‘delayed-choice’’ version. It
demonstrates that the complementarity principle cannot be
interpreted in a naive way, assuming that the photon at the
BTW, it works for ultracold atoms too!
We need a quantum theory of light where behaviors such as wave-particle duality is built in
– A formal procedure exists for obtaining a quantum theory from a known classical theory based on Lagrange – Hamilton formalism.
– One cannot “prove” that this procedure is correct. It is justified only
by repeated observation that quantum theories obtained in this fashion
“work”, in the sense that their predictions agree with experiment.
– This should not surprise us. Quantum Mechanics contains new physics
that is absent from Classical Mechanics and cannot be derived from it.
– An unfortunate consequence is that we often seem to pull ideas out of
thin air when we teach quantum mechanics. The problem gets worse
when we try to do things quickly, e. g., a 50 minute lecture on the quantum theory of light.
Quantization of the Electromagnetic Field
Starting point: Maxwell’s equations from classical Electromagnetism
1
! " E(r,t) = $ (r,t)
#0
! " B(r,t) = 0
'
! % E(r,t) = & B(r,t)
't
1 '
1
! % B(r,t) = 2 E(r,t) +
j(r,t)
2
c 't
# 0c
In Quantum Electrodynamics (QED)
the ME’s are still valid, but the fields
E and B become operators that depend
on space and time just like the
classical electric and magnetic fields.
For simplicity we consider empty space without charges or currents, and use
the two last Maxwell equations to derive a wave equation
$ 2 1 #2 '
&%! " c 2 #t 2 )( E(r,t) = 0
# !2
1 !2 &
% !z 2 " c 2 !t 2 ( E(z,t) = 0
$
'
paraxial waves propagating along the z-axis
Electromagnetic Field in a 1-Dimensional Cavity
Cavity of length L, cross section A, and volume V = L A, field E
= x E x
x
z
y
M
M
Any field inside the cavity can be expressed as a superposition of the cavity
normal modes, which are standing wave solutions to the wave equation with
nodes on the mirror surfaces.
!
E x (z,t) = " A j q j (t)sin(k j z) ,
j=1
! j j"
kj =
=
,
c
L
2! 2j m j
Aj =
" 0V
Here q j (t) are the time varying amplitudes for the different modes, and A j is
chosen so E x has units of electric field (V/m) and q j (t) has units of length (m). Each normal mode is an independent degree of freedom that can be
quantized by itself.
For mode number j we have electric and magnetic fields (the latter can be
found from E x using Maxwells equations)
E x( j ) (z,t) = A j q j (t)sin(k j z), By( j ) (z,t) =
Aj
d
!
!
q
(t)cos(k
z),
q
(t)
=
q j (t)
j
j
2 j
k jc
dt
Using classical E & M we can show that the energy of the field in mode j is
H = ! 0 " dr 3 ( E 2 + c 2 B 2 )
V
&
!0A L # 2
A 2j
2
2
2
2
=
dz A j q j (t) sin (k j z) + 2 q! j (t) cos (k j z) (
2 "0 %$
kj
'
=
1
1
m j) 2j q 2j + m j q! 2j
2
2
Writing the field energy in this form is highly suggestive. Harmonic Oscillator
1
1 2
2 2
H = m! x + mv
2
2
1
1 2
= m! 2 x 2 +
p
2
2m
EM field in mode j
Vs.
H=
1
1
m j! 2j q 2j + m j q! 2j
2
2
Quantum Theory
Ê x ! q j " q̂ j
x ! x̂ = x
p ! p̂ = "i!
[ x̂, p̂ ] = i!
d
dx
B̂y ! m j q! j " p̂ j
[ q̂ j , p̂ j ] = i!,
!" Ê x , B̂y #$ % 0
Postulate (leap of faith): EM field in a normal mode is a harmonic oscillator
e n u c le u
a c la s s ic
s
Quick review of the Quantum Harmonic Oscillator x
Standard paradigm: mass on a spring
! = K /m
n
e
K
!
!
Observables: x̂, p̂
Hamiltonian:!
Ĥ =
m
1
1 2
m! 2 x̂ 2 +
p̂
2
2m
origin
Energy eigenvalues and eigenfunctions:
N
En = !! (n + 1 / 2), no"t!
0e:
Ĥ! n (z) = En! n (z) "
V (x)
!
W e shou
ld regard E = !! (n + 1 / 2)
a c la s s ic
th is a s a
# $ x /2
al m o d e
! n (z) " e
H n ($ x)
l ofE t=h!!e(3 /a2) model
to m its e
!E = !"
H n : Hermite polynomial lf
E = !! (1 / 2)
W e can
ju s tif y th
is re
can
n
2
1
0
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Energy eigenfunctions (stationary states):
look at probability density ! n (x) 2
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Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable
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Absolute
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Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly
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the absolute
value of the eigenfunctions approaches the probability distribution of a classical particle
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quantum mechanics.
Chemistry
in a harmonic potential with inverse square root singularities at the turning points.
theparticle
square ofCollege
the absolute
number, the square of the absolute value of the eigenfunctions approaches the probability distribution of number,
a classical
Physics value of the eigenfunctions approaches the probability distribution of a classical particle College Physics
in a harmonic potential
with inverse
square root singularities at the turning points.
in a harmonic potential with inverse square root singularities at the turning points.
General Chemistry
General
Chemistry
Mechanics
Mechanics
Physical Chemistry
Physical Chemistry
Mean position in state !
n (x)
Contributed by: Michael
Trott Mechanics
Quantum
Quantum Mechanics
Unique features:
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Quantum Physics
Quantum Physics
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Quantum fluctuations
Heisenberg uncertainty relation
Page 1 of 6
!xn =
Chemistry
College Phy
General Ch
Mechanics
Physical Ch
Quantum M
Quantum Ph
x̂ = 0
http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/
http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/
Page 1 of 6
Related T
!
n +1/ 2
m"
!x!p = !(n + 1 / 2)
Page 1 of 6
Creation and annihilation operators
Introduce dimensionless variables
Define
Can show that
X̂ = x̂ m! 2!,
P̂ = p̂
â = X̂ + iP̂
annihilation operator
â † = X̂ ! iP̂
creation operator
! n"1 # â ! n
! n+1 " â †! n
annihilation/creation of an excitation
â
†
â
2m!!
! n+1
!n
! n"1
â †â ! n = n ! n
(# of excitations)
What is a “phonon”?
A quantum of exitation in a Harmonic Oscillator Back to…
Harmonic Oscillator
1
1 2
2 2
H = m! x + mv
2
2
1
1 2
= m! 2 x 2 +
p
2
2m
EM field in mode j
Vs.
H=
1
1
m j! 2j q 2j + m j q! 2j
2
2
Quantum Theory
Ê x ! q j " q̂ j
x ! x̂ = x
p ! p̂ = "i!
[ x̂, p̂ ] = i!
d
dx
B̂y ! m j q! j " p̂ j
[ q̂ j , p̂ j ] = i!,
!" Ê x , B̂y #$ % 0
Postulate (leap of faith): EM field in a normal mode is a harmonic oscillator
x
QED paradigm: normal mode j in cavity z
y
Observables: q̂ (! Ê x ), p̂ (! B̂y )
Hamiltonian: Ĥ =
Energy eigenvalues and eigenfunctions:
En = !! (n + 1 / 2), n " 0
M
1
1 2
m! 2 q̂ 2 +
p̂
2
2m
Ĥ! n = En! n "
V (x)
En = !! (n + 1 / 2)
Number states
! n
No “wavefunction”, use
Dirac notation
! n " ! n
M
!E = !"
E1 = !! (3 / 2)
E0 = !! (1 / 2)
Creation and annihilation operators
Q̂ = q̂ m! 2!,
Introduce dimensionless variables
Define
Can show that
P̂ = p̂
â = Q̂ + iP̂
annihilation operator
â † = Q̂ ! iP̂
creation operator
! n"1 # â ! n
! n+1 " â ! n
†
annihilation/creation of an excitation
â
†
â
2m!!
! n+1
!n
! n"1
â †â ! n = n ! n
number states
What is a “photon”?
A quantum of exitation in a Normal Mode of the EM field
Photons as particles
Standing wave normal modes
! photons are delocalized in space
We can make superpositions of standing waves that correspond to wavepackets
& use these as our normal modes
! photons become localized in space
x
z
y
M
M
It is in this sense than we can talk about, e. g, a photon traveling along a specific
path in an interferometer, as in the first part of the lecture.
More about number states (Foch states):
Mean field in state
! n
Quantum fluctuations
Ê x ! q̂ = 0
!q̂n = q0 n + 1 / 2 "
(!Ê x )n = E0 n + 1 / 2
!q̂n=0 = q0 / 2 "
Vacuum fluctuations
(!Ê x )n=0 = E0 / 2
Does a laser emit light in a number state with a well defined number of photons?
Back to…
Harmonic Oscillator
1
1 2
2 2
H = m! x + mv
2
2
1
1 2
= m! 2 x 2 +
p
2
2m
EM field in mode j
Vs.
H=
1
1
m j! 2j q 2j + m j q! 2j
2
2
Quantum Theory
Ê x ! q j " q̂ j
x ! x̂ = x
p ! p̂ = "i!
[ x̂, p̂ ] = i!
d
dx
B̂y ! m j q! j " p̂ j
[ q̂ j , p̂ j ] = i!,
!" Ê x , B̂y #$ % 0
Postulate (leap of faith): EM field in a normal mode is a harmonic oscillator
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Number states are highly non-classical
– look at the probability density ! n (x) 2
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Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable
model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum
Absolute
value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable
Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly
solvable
Topics
Related Topics
number, the square ofRelated
the absolute
value of the eigenfunctions approaches the probability distribution of a classical particle
model of quantum
The ground state eigenfunction minimizes the uncertainty product. With increasing quantum Chemistry
model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing
quantum mechanics.
Chemistry
in a harmonic potential with inverse square root singularities at the turning points.
theparticle
square ofCollege
the absolute
number, the square of the absolute value of the eigenfunctions approaches the probability distribution of number,
a classical
Physics value of the eigenfunctions approaches the probability distribution of a classical particle College Physics
in a harmonic potential
with inverse
square root singularities at the turning points.
in a harmonic potential with inverse square root singularities at the turning points.
General Chemistry
General
Chemistry
Physical Chemistry
Physical Chemistry
Contributed by: Michael
Trott Mechanics
Quantum
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Harmonic Oscillator Eigenfunctions
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Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable
model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum
number, the square of the absolute value of the eigenfunctions approaches the probability distribution of a classical particle
in a harmonic potential with inverse square root singularities at the turning points.
Related Topics
Chemistry
College Physics
General Chemistry
Mechanics
Physical Chemistry
Quantum Mechanics
Contributed by: Michael Trott
http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/
Absolute value of the harmonic oscillator eigenfunctions. The harmonic oscillator is the most important exactly solvable
model of quantum mechanics. The ground state eigenfunction minimizes the uncertainty product. With increasing quantum
number, the square of the absolute value of the eigenfunctions approaches the probability distribution of a classical particle
in a harmonic potential with inverse square root singularities at the turning points.
Related Topics
Chemistry
College Physics
General Chemistry
Mechanics
Physical Chemistry
Quantum Mechanics
Contributed by: Michael Trott
http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/
Quantum Physics
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Physical Ch
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P
Harmonic Oscillator Eigenfunctions
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Quantum Ph
A quasi-classical state is a minimum-uncertainty oscillating wavepacket
http://demonstrations.wolfram.com/HarmonicOscillatorEigenfunctions/
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Quantum Mechanics
Contributed by: Michael Trott
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ground
state
X
We can make a quasi-classical state ! " (t) as a superposition of number states
! " (t) = e# "
2
i$ t n
(
"
e
)
2
% n n! ! n
!
X̂ " cos(# t),
P̂ " sin(# t)
A coherent state is the equivalent superposition of photon number states
! " (t) = e# "
2
i$ t n
(
"
e
)
2
% n n! ! n
Ê x ! Q̂ ! cos(" t),
&
P, By
coherent
state
B̂y ! P̂ ! sin(" t)
ground
state
Probability of detecting n photons
P(n) = e
!" 2
"
n!
2n
(shot noise)
Q, E x
An ideal laser comes very close to emitting a coherent state.
This is the closest we can come to a classical, monochromatic light field.
By
coherent
state
ground
state
Ex
Other Interesting Topics in Quantum Optics
– The quantum beam splitter (photons are bosons)
– Quantum theory of interferometers
– How to make number states, squeezed states, coherent states, etc.
– Two-level atoms in single-mode cavities (Jaynes-Cummings model)
– Generalizations of the Jaynes-Cummings model
– Excited atoms interacting with the vacuum, decoherence & decay
– Quantum theory of photodetection
– Quasi-probability distributions and non-classical light
Atoms and Photons:
Confessions of a Self-Admitted
Control Freak
Poul Jessen
Daniel Hemmer
Nathan Lysne
David Melchior
Enrique Montano
Hector Sosa-Martinez
Kyle Taylor
Light, Atoms and Quantum Control
Quantum Mechanics is our reigning theory of everything
- behavior of light and matter on scales from micro- to macroscopic ✓
Quantum Computing, Quantum Information Science
- quantum mechanics is a resource that allows us to do different things ✓
Central Challenge: Quantum Control
- how to make quantum systems do what we want, not what comes naturally ?
Experimental Setup – “NMR” w/Cold Atoms
Quantum Control for Fun and Science