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Light-Matter Interactions
and Quantum Optics.
Jonathan Keeling
http://www.st-andrews.ac.uk/~jmjk/teaching/qo/
http://www.st-andrews.ac.uk/~jmjk/teaching/qo/http://www.st-andrews.ac.uk/~jmjk/teaching/qo/http://www.st-andrews.ac.uk/~jmjk/teaching/qo/http://www.st-andrews.ac.uk/~jmjk/teaching/qo/
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Contents
Contents iii
Introduction vii
1 Quantisation of electromagnetism 1
1.1 Revision: Lagrangian for electromagnetism . . . . . . . . .
2
1.2 Eliminating redundant variables . . . . . . . . . . . . . .
. . 3
1.3 Canonical quantisation; photon modes . . . . . . . . . . . .
5
1.4 Dipole approximation and coupling strength . . . . . . . . .
61.5 Further reading . . . . . . . . . . . . . . . . . . . . . . .
. . 8
2 Jaynes Cummings model 9
2.1 Derivation of Jaynes-Cummings model . . . . . . . . . . . .
92.2 Semiclassical limit . . . . . . . . . . . . . . . . . . . . .
. . 10
2.3 Quantum behaviour . . . . . . . . . . . . . . . . . . . . .
. 11
3 Decay of a two-level system 15
3.1 Many mode quantum model irreversible decay . . . . . .
15
3.2 Density matrix equation for relaxation of two-level system .
16
3.3 Effective decay rate . . . . . . . . . . . . . . . . . . . .
. . . 18
3.4 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 19
3.A WignerWeisskopf approach . . . . . . . . . . . . . . . . . .
20
4 Power broadening 21
4.1 Equations of motion with coherent driving . . . . . . . . .
. 21
4.2 Dephasing in addition to relaxation . . . . . . . . . . . .
. . 22
4.3 Power broadening of absorption . . . . . . . . . . . . . . .
. 25
4.4 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 27
5 Purcell Effect 29
5.1 The Purcell effect in a 1D model cavity . . . . . . . . . .
. 29
5.2 Weak to strong coupling via density matrices . . . . . . . .
335.3 Further reading . . . . . . . . . . . . . . . . . . . . . . .
. . 35
6 Cavity Quantum Electrodynamics 37
6.1 Optical transitions of atoms . . . . . . . . . . . . . . . .
. . 386.2 Microwave transitions of atoms . . . . . . . . . . . . .
. . . 42
6.3 Superconducting qubits in microwave resonators . . . . . .
44
iii
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iv CONTENTS
6.4 Quantum-dot excitons in semiconductor microstructures . .
46
6.5 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 50
6.A Transfer matrix approach to calculating reflectivity of a
Bragg
mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 50
7 Resonance Fluorescence 51
7.1 Spectrum of emission into a reservoir . . . . . . . . . . .
. . 51
7.2 Quantum regression theorem . . . . . . . . . . . . . . . .
52
7.3 Resonance fluorescence spectrum . . . . . . . . . . . . . .
. 54
7.4 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 56
8 Quantum stochastic methods 59
8.1 Quantum jump formalism . . . . . . . . . . . . . . . . . . .
59
8.2 Heisenberg-Langevin equations . . . . . . . . . . . . . . .
. 61
8.3 Fluctuation dissipation theorem . . . . . . . . . . . . . .
. . 638.4 Further reading . . . . . . . . . . . . . . . . . . . . .
. . . . 65
9 Quantum theory of the laser 67
9.1 Density matrix equation for a model laser system . . . . . .
67
9.2 Maxwell Bloch equations . . . . . . . . . . . . . . . . . .
. . 69
9.3 Reduction to photon density matrix equation . . . . . . . .
70
9.4 Properties of the laser rate equations . . . . . . . . . . .
. . 73
9.5 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 74
10 Laser fluctuations: linewidth and correlations 75
10.1 Laser linewidth . . . . . . . . . . . . . . . . . . . . . .
. . . 7510.2 Spontaneous emission and the parameter . . . . . . . .
. 78
10.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . .
. . . 79
10.4 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 80
10.A Derivation of off diagonal field density matrix equations .
. 81
11 Stronger coupling, micromasers, single atom lasers 83
11.1 Micromaser gain . . . . . . . . . . . . . . . . . . . . . .
. . 83
11.2 Micromaser and noise . . . . . . . . . . . . . . . . . . .
. . 85
11.3 Single atom lasers . . . . . . . . . . . . . . . . . . . .
. . . 87
11.4 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 89
12 Superradiance 91
12.1 Simple density matrix equation for collective emission . .
. 91
12.2 Beyond the simple model . . . . . . . . . . . . . . . . . .
. 96
12.3 Further reading . . . . . . . . . . . . . . . . . . . . . .
. . . 100
13 The Dicke model 101
13.1 Phase transitions, spontaneous superradiance . . . . . . .
. 101
13.2 No-go theorem: no vacuum instability . . . . . . . . . . .
. 103
13.3 Radiation in a box; restoring the phase transition . . . .
. . 104
13.4 Dynamic superradiance . . . . . . . . . . . . . . . . . . .
. 104
13.5 Further Reading . . . . . . . . . . . . . . . . . . . . . .
. . 106
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CONTENTS v
14 Three levels, and coherent control 10914.1 Semiclassical
introduction . . . . . . . . . . . . . . . . . . . 10914.2 Coherent
evolution alone; why does EIT occur . . . . . . . 113
14.3 Dark state polaritons . . . . . . . . . . . . . . . . . . .
. . . 11314.4 Further reading . . . . . . . . . . . . . . . . . . .
. . . . . . 116
A Problems for lectures 14 117
B Problems for lectures 59 121
C Problems for lectures 1014 123
Bibliography 127
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Introduction
The title quantum optics covers a large range of possible
courses, and sothis introduction intends to explain what this
course does and does not aimto provide. Regarding the negatives,
there are several things this coursedeliberately avoids:
It is not a course on quantum information theory. Some basic
notionsof coherent states and entanglement will be assumed, but
will not bethe focus.
It is not a course on relativistic gauge field theories; the
majorityof solid state physics does not require covariant
descriptions, and soit is generally not worth paying the price in
complexity of using amanifestly covariant formulation.
As far as possible, it is not a course on semiclassical
electromagnetism.While at times radiation will be treated
classically, this will generallybe for comparison to a full quantum
treatment, or where such anapproximation is valid (for at least
part of the radiation).
Regarding the positive aims of this course, they are: to discuss
how tomodel the quantum behaviour of coupled light and matter; to
introducesome simple models that can be used to describe such
systems; to dis-cuss methods for open quantum systems that arise
naturally in the contextof coupled light and matter; and to discuss
some of the more interestingphenomena which may arise for matter
coupled to light. Semiclassical be-haviour will be discussed in
some sections, both because an understanding
of semiclassical behaviour (i.e. classical radiation coupled to
quantum me-chanical matter) is useful to motivate what phenomena
might be expected;and also as comparison to the semiclassical case
is important to see whatnew physics arises from quantised
radiation.
The kind of quantum optical systems discussed will generally
consistof one or many few-level atoms coupled to one quantised
radiation fields.Realisations of such systems need not involve
excitations of real atoms, butcan instead be artificial atoms, i.e.
well defined quantum systems with dis-crete level spectra which
couple to the electromagnetic field. Such conceptstherefore apply
to a wide variety of systems, and a variety of character-istic
energies of electromagnetic radiation. Systems currently studied
ex-
perimentally include: real atomic transitions coupled to optical
cavities[1];
vii
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viii INTRODUCTION
Josephson junctions in microwave cavities (waveguides terminated
by re-flecting boundaries)[2, 3]; Rydberg atoms (atoms with very
high princi-ple quantum numbers, hence small differences of energy
levels) in GHz
cavities[4]; and solid state excitations, i.e. excitons or
trions localised inquantum dots, coupled to a variety of optical
frequency cavities, includ-ing simple dielectric contrast cavities,
photonic band gap materials, andwhispering gallery modes in
disks[5].
These different systems provide different opportunities for
control andmeasurement; in some cases one can probe the atomic
state, in some casesthe radiation state. To describe experimental
behaviour, one is in gen-eral interested in calculating a response
function, relating the expectedoutcome to the applied input.
However, to understand the predicted be-haviour, it is often
clearer to consider the evolution of quantum mechanicalstate; thus,
both response functions and wavefunctions will be discussed.
As such, the lectures will switch between Heisenberg and
Schrodinger pic-tures frequently according to which is most
appropriate. When consideringopen quantum systems, a variety of
different approaches; density matrixequations, Heisenberg-Langevin
equations and their semiclassical approxi-mations, again
corresponding to both Schrodinger and Heisenberg pictures.
The main part of this course will start with the simplest case
of a singletwo-level atom, and discuss this in the context of one
or many quantisedradiation modes. The techniques developed in this
will then be appliedto the problem of many two-level atoms, leading
to collective effects. Thetechniques of open quantum systems will
also be applied to describing las-ing, focussing on the more
quantum examples of micromasers and single
atom lasers. The end of the course will consider atoms beyond
the two-levelapproximation, illustrating what new physics may
arise. Separate to thismain discussion, the first two lectures
stand alone in discusing where thesimple models of coupled light
and matter used in the rest of the coursecome from, in terms of the
quantised theory of electromagnetism.
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Lecture 1
Quantisation ofelectromagnetism in the
Coulomb gauge
Our aim is to write a theory of quantised radiation interacting
with quan-tised matter fields. Such a theory, e.g. the
Jaynes-Cummings model (seenext lecture) has an intuitive form:
HJ.C. =
kka
kak +i,k
izi + gi,k
+i ak + H.c.
. (1.1)
The operator ak creates a photon in the mode with wavevector k,
and sothis Hamiltonian describes a process where a two-level system
can changeits state with the associated emission or absorption of a
photon. The termka
kak then gives the total energy associated with occupation of
the mode
with energy k. While the rest of the course is dedicated to
studying suchmodels of coupled light-matter system, this lecture
will show the relationbetween such models and the classical
electromagnetism of Maxwells equa-tions.
To reach this destination, we will follow the route of canonical
quan-tisation; our first aim is therefore to write a Lagrangian in
terms of onlyrelevant variables. Relevant variables are those where
both the variable andits time derivative appear in the Lagrangian;
if the time derivative does notappear, then we cannot define the
canonically conjugate momentum, andso cannot enforce canonical
commutation relations. The simplest way ofwriting the Lagrangian
for electromagnetism contains irrelevant variables i.e. the
electric scalar potential and gauge of the vector potential A;that
irrelevant variables exist is due to the gauge invariance of the
theory.Since we are not worried about preserving manifest Lorentz
covariance,we are free to solve this problem in the simplest way
eliminating the
unnecessary variables.
1
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2 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM
1.1 Revision: Lagrangian for electromagnetism
To describe matter interacting with radiation, we wish to write
a La-
grangian whose equations of motion will reproduce Maxwells and
Lorentzsequations:
B = 0J + 00E E = /0 (1.2) B = 0 E = B (1.3)mr = q[E(r) + r
B(r)]. (1.4)
Equations (1.3) determine the structure of the fields, not their
dynamics,and are immediately satisfied by defining B = A and E = A.
Letus suggest the form of Lagrangian L that leads to Eq. (1.2) and
Eq. (1.4):
L =
1
2 mr2
+
02
dV
E2
c2
B2
+
q [r A(r) (r)] .(1.5)
Here, the fields E and B should be regarded as functionals of
and A.Note also that in order to be able to extract the Lorentz
force acting onindividual charges, the currents and charge
densities have been written as:
(r) =
q(r r), J(r) =
qr(r r). (1.6)
The identification of the Lorentz equation is simple:
d
dt
Lr
=d
dt[mr + qA(r)] = mr + q(r
)A(r) + q
tA(r)
=Lr
= q [r A(r) (r)]= q [(r ) A(r) + r ( A(r)) (r)] ,
thus one recovers the Lorentz equation,
mr = q
r [ A(r)] (r)
tA(r)
. (1.7)
Similarly, the equation that results from can be easily
extracted; since
L/ = 0, this becomes
L
= 0 E
q(r r) = 0. (1.8)
Finally, the equations for A are more complicated, requiring the
identity
A( A)2 = 2 ( A), (1.9)
which then gives:
d
dt
LA
= ddt
0E =LA
= 0c2 ( A) +
qr(r r),
(1.10)
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1.2. ELIMINATING REDUNDANT VARIABLES 3
which recovers the required Maxwell equation
0
d
dt
E =
1
0 B +
qr(r
r). (1.11)
Thus, the Lagrangian in Eq. (1.5), along with the definitions of
E and Bin terms of A and produce the required equations.
1.2 Eliminating redundant variables
As mentioned in the introduction, we must remove any variable
whosetime derivative does not appear in the Lagrangian, as one
cannot writethe required canonical commutation relations for such a
variable. It isclear from Eq. (1.5) that the electric scalar
potential is such a variable.
Since does not appear, it is also possible to eliminate directly
fromthe equation L/; using Eq. (1.8) and the definition of E, this
equationgives:
0 A 02 (r) = 0. (1.12)Rewriting this in Fourier space, one
has:
(k) =1
k2
(k)
0+ ik A(k)
. (1.13)
We can now try to insert this definition into the Lagrangian, to
eliminate. To do this, we wish to write E2 and B2 in terms of and
A; it is therefore
useful to start by writing
Ej(k) = ikj(k) + Aj(k) = i kj0k2
(k) +
jk kjkk
k2
Ak(k). (1.14)
This means that the electric field depends on the charge
density, and onthe transverse part of the vector potential, which
will be written:
Aj k =
jk kjkk
k2
Ak(k). (1.15)
The transverse1 part of the vector potential is by definition
orthogonal tothe wavevector k, and so the electric field is the sum
of two orthogonalvectors, and so:
|E(k)|2 = |A(k)|2 + 120k
2|(k)|2. (1.18)
1The combination:
jk(k) =
jk
kjkk
k2
, (1.16)
is the reciprocal space representation of the transverse delta
function; with appropriateregularisation[6, Complement AI], it can
be written in real space as:
jk(r) =
2
3(r)jk +
1
4r3
3rjrk
r2 jk
. (1.17)
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4 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM
Similarly, the squared magnetic field in reciprocal space is
given by:
|B(k)|2 = (k A(k)) (k A(k))
= k2
jk kjkkk2
Aj(k)Ak(k) = k2|A(k)|2. (1.19)
Thus, the field part of the Lagrangian becomes:
02
dV
E2 c2B2 = 1
0
d3k1
k2|(k)|2
+ 0
d3k|A(k)|2 c2k2|A(k)|2
. (1.20)
The notation
has been introduced to mean integration over
reciprocalhalf-space; since A(r) is real, A(k) = A(k), thus the two
half spacesare equivalent. This rewriting is important to avoid
introducing redundant
fields in the Lagrangian; the field is either specified by one
real variable atall points in real space, or by two real variables
at all points in reciprocalhalf-space. Similar substitution into
the coupling between fields and matter,written in momentum space
gives:
Lemmatter = 2
d3k [J(k) A(k) (k)(k)]
= 2
d3k
J(k) A(k) (k)
(k)
0k2 ik
k2 A(k)
.
(1.21)
This can be simplified by adding a total time derivative,
L L+ dF/dt;
such transformations do not affect the equations of motion,
since they addonly boundary terms to the action. If:
F = 2
d3k(k)
ik A(k)k2
, (1.22)
then
Lemmatter + dFdt
= 2
d3k
J(k) (k) ik
k2
A(k) |(k)|
2
0k2
.
(1.23)Then, using conservation of current, (k) + ik J(k) = 0,
one finally has:
Lemmatter + dFdt
= 2d3k J(k) A(k) |(k)|20k2
. (1.24)
Note that this set of manipulations, adding dF/dt has eliminated
the longi-tudinal part of the vector potential from the Lagrangian.
The form chosenfor F is such that this procedure is equivalent to a
gauge transformation;the chosen gauge is the Coulomb gauge. Putting
everything together, onehas:
Lcoulomb =
1
2mr
2 10
d3k1
k2|(k)|2
+ 0d3k |A(k)|2 c2k2|A(k)|2 + 2 J(k) A(k) . (1.25)
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1.3. CANONICAL QUANTISATION; PHOTON MODES 5
Thus, the final form has divided the interaction into a part
mediatedby transverse fields, described by A, and a static (and
non-retarded)Coulomb interaction. Importantly, there are no
irrelevant variables left
in Eq. (1.25) The Coulomb term can also be rewritten:
Vcoul =1
0
d3k1
k2|(k)|2 =
,
qq80|r r| . (1.26)
Note that since the Coulomb interaction is non-retarded, both
the Coulomband transverse parts of interaction must be included to
have retarded in-teractions between separated charges.
1.3 Canonical quantisation; photon modes
We have in Eq. (1.25) a Lagrangian which can now be treated via
canon-ical quantisation. Since only the transverse part of the
field A appears inEq. (1.25), we can drop the superscript label in
A from here on. To pro-ceed, we should first identify the canonical
momenta and the Hamiltonian,and then impose canonical commutation
relations. Thus,
p =Lcoulomb
r= mr + qA(r) (1.27)
(k) =Lcoulomb
A(k)= 0A(k). (1.28)
Then, constructing the Hamiltonian by H = i PiRi L, one finds:H
=
1
2m[p qA(r)]2 + Vcoul
+ 0
d3k
|(k)|220
+ c2k2|A(k)|2
. (1.29)
In order to quantise, it remains only to introduce commutation
relationsfor the canonically conjugate operators. Noting that A(r)
has only twoindependent components, because it is transverse, it is
easiest to writeits commutation relations in reciprocal space,
introducing directions ek,northogonal to k with n = 1, 2 ;
then:
[ri,, pj,] = iij (1.30)Aek,n(k), ek,n (k
)
= i(k k)nn . (1.31)
This concludes the quantisation of matter with electromagnetic
inter-actions in the Coulomb gauge. It is however instructive to
rewrite thetransverse part of the fields in terms of their normal
modes. The secondline of Eq. (1.29) has a clear similarity to the
harmonic oscillator, witha separate oscillator for each
polarisation and momentum. Rewriting innormal modes thus means
introducing the ladder operators:
ak,n =0
2ck
ckAek,n(k) +
i
0 ek,n(k)
, (1.32)
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6 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM
or, inverted one has:
A(r) =k
n=1,2
20kV en
ak,ne
ikr
+ a
k,neikr
. (1.33)
Inserting this into Eq. (1.29) gives the final form:
H =
1
2m[p qA(r)]2 +
k
n=1,2
kakak + Vcoul. (1.34)
1.4 Dipole approximation and coupling strength
Equation (1.34) applies to any set of point charges interacting
with the
electromagnetic field. In many common cases, one is interested
in dipoles,with pairs of opposite charges closely spaced, and much
larger distancesbetween the dipoles. In this case it is possible to
make the dipole approx-imation, by assuming that the
electromagnetic field does not vary acrossthe scale of one
dipole.
We consider a system with just two charges: charge +q mass m1
atR + r/2 and q mass m2 at R r/2. If r where is a
characteristicwavelength of light, then one may assume that
A(R+r/2) A(Rr/2) A(R), and so the remaining coupling between
radiation and matter is ofthe form
HAp = q p1m1
p2
m2 A(R).
Then, in the case that one can write H0, a Hamiltonian for the
dipolewithout its coupling to radiation one can use p/m = x = i[H0,
x]/, thusgiving:
HAp = i q
[H0, r] A(R) (1.35)
If one considers a given basis of eigenstates |n for the free
Hamil-tonian H0 obeying H0|n = n|n one may then rewrite the above
in theform:
HAp = n,m
gk,nnm2
|nm|ak,neikR + ak,ne
ikR
(1.36)where we have introduced the coupling strengths gk,nnm.
These describe thetransitions between bare levels that are driven
by coupling to the photonfield, and are given by the expression
gk,nnm2
=q
20kVek,n n|[H0, r]|m = nmek,n dnm
20kV, (1.37)
where dnm = qn|r|m is the dipole matrix element and nm = n m.We
have assumed H0 commutes with the parity operator P : Px = x,and so
the coupling to radiation appears only in the off-diagonal terms
in
the two-level basis.
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1.4. DIPOLE APPROXIMATION AND COUPLING STRENGTH 7
Oscillator strength
The expression in Eq. (1.37) shows that the coupling strength
between
light and matter depends both on properties of the atom and the
radiationmodes. The relevant property of the radiation mode is the
field strength,describing the electric field (or vector potential)
associated with a singlephoton. The relevant propert of the atom is
a combination of its energyand dipole matrix element.
There is another characterisation of an atoms coupling to light,
whichis given by the atomic oscillator strength. We will here
restrict attentionto transitions involving coupling to electrons,
hence q = e, m = m0.
fnm =2m02
|dnm|2e2
nm. (1.38)
This dimensionless combination is important for several reasons,
which arediscussed partly here, and further elaborated in the
problems. The first isthat the oscillator strength describes the
relative contribution of a givenatomic transition to the atomic
susceptibility, as compared to an harmonicoscillator. The second is
that there is a sum rule which states
n fnm = 1
(at least for transitions involving only a single electron
changing state),implying that the dimensionless combination fnm
states how large the cou-pling is as compared to the maximum value
it could possibly take.
The susceptibility of a Lorentz oscillator is calculated by
finding P() =()0E() = ex(), where x is the displacement of the
oscillator, and theoscillator obeys:
mx =
m20
x
2mx
eE(t) (1.39)
It is easy to then decompose this into frequency components and
thus find
LO() =e2
m00
1
2 + 2i 20. (1.40)
To see the significance of the oscillator strength fnm, this
susceptibilityshould be compared to that corresponding to Eq.
(1.36). This means cal-culating:
P(t) =
dP()eit = |
mn
dmn|mn| (1.41)
in linear response theory, with H0 = m
m|m
m
|and the perturbation
H1 = i
nm
A(t)nmdnm|nm|, A(t) = i
dE()
e(it). (1.42)
Here, A(t) is a classical field corresponding to the probe field
E(t), and isan infinitessimal shift to ensure the field vanishes at
t . Exercise A1shows that the linear response found here evaluates
to:
() =e2
m00
nm
fnm1
( + i)2 (nm/)2 (1.43)
indicating that each transition contributes like a harmonic
oscillator, but
with a reduced strength given by fnm.
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8 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM
1.5 Further reading
The discussion of quantisation in the Coulomb gauge in this
chapter draws
heavily on the book by Cohen-Tannoudji et al. [6].
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Lecture 2
The Jaynes-Cummingsmodel
The aim of this lecture is to discuss the interaction between a
single two-level system and various different configurations of
photon fields. We startby introducing the Jaynes-Cummings model,
and then consider variousresults of this model. These will include
the interaction with a semiclassicallight field, and the behaviour
of a single quantised mode of light in aninitially coherent
state.
2.1 Derivation of Jaynes-Cummings model
At the end of the previous lecture, we began to discuss
approximations thatresult in simplified descriptions of coupled
lightmatter systems. In thissection, we continue this programme of
simplifications in order to reducethe model of coupled matter and
radiation modes to the Jaynes-Cummingsmodel, of a two-level atom
coupled to a single photon model. To effect thisreduction, three
further simplifications are required in addition to the
dipoleapproximation discussed previously: neglect of the A2 terms,
projection totwo-level systems, and the rotating wave
approximation.
Neglect of the A2 terms in expanding [p qA(r)]2 can be
justifiedin the limit of low density of atoms; considering only a
single radiationmode, the contribution of the A2 term can be
rewritten using Eq. (1.33)
as:HA2 =
N
V
q2
4m0
a + a
2. (2.1)
Thus, this term adds a self energy to the photon field, which
scales like thedensity of atoms. The relative importance of this
term can be estimatedby comparing it to the other term in the
Hamiltonian which is quadraticin the photon operators, ka
kak. Their ratio is given by:
N
V
q2
4m0(k)2 N
Va3B
Rydk
2(2.2)
thus, assuming particles are more dilute than their Bohr radius,
neglect
of HA2 is valid for frequencies of the order of the Rydberg for
the given
9
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10 LECTURE 2. JAYNES CUMMINGS MODEL
bound system of charges.To further simplify, one can now reduce
the number of states of the
atom that are considered; currently, there will be a spectrum of
eigen-
states of H0, and transitions are induced between these states
accordingto f|r|i. Then, restricting to only the two lowest atomic
levels and toa single radiation mode, one has a model of two-level
systems (describingmatter) coupled to bosonic modes (describing
radiation).
H =1
2
g(a + a)
g(a + a)
+ 0aa. (2.3)
The matrix notation represents the two levels of the atom. We
have intro-duced the energy splitting between the lowest two atomic
levels, and wehave used the notation g for the effective coupling
strength gab as definedin the previous lecture:
g2
= ek,n dba20kV
, (2.4)
The final approximation to be discussed here, the rotating wave
approx-imation, is appropriate when g 0. Considering g as a
perturbation,one can identify two terms:
co =g
2
0 a
a 0
, counter =
g
2
0 a
a 0
, (2.5)
where co, the co-rotating terms, conserve energy particle
number; andcounter do not. More formally, the effects of counter
give second orderperturbation terms like g2/(
0+), while
cogive the much larger g2/(
0). Including this approximation, one then derives the
Jaynes-Cummingsmodel:
H =1
2z +
g
2(a + a+) + 0a
a. (2.6)
where represent Pauli matrices.
2.2 Semiclassical limit
On applying a classical light field to a two-level system, we
can adaptthe Jaynes-Cummings model by neglecting the dynamics of
the photon
field, and replacing the photon creation and annihilation
operators withthe amplitude of the time-dependent classical field.
This gives the effectiveHamiltonian:
H =1
2
gei0t
gei0t
. (2.7)
We have made the rotating wave approximation, and hence only
includedthe positive/negative frequency components of the classical
field in the rais-ing/lowering operators for the two-level system.
To solve this problem, andfind the time-dependent state of the
two-level atom, it is convenient tomake a transformation to a
rotating frame. In general, if
U = eif(t) 0
0 eif(t)
, (2.8)
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2.3. QUANTUM BEHAVIOUR 11
then
H H =
H11 + f(t) H12e2if(t)
H21e2if(t) H22
f(t)
, (2.9)
thus, in this case we use f = t/2, giving:
H =1
2
g
g ( )
. (2.10)
Since this transformation affects only the phases of the
wavefunction, we canthen find the absorption probability by
considering the time averaged prob-ability of being in the upper
state. The eigenvalues of Eq. (2.8) are /2where =
( )2 + g22. Writing the eigenstates as (cos , sin )T,
and ( sin , cos )T corresponding respectively to the roots
(assumingtan() > 0). Substituting these forms, one finds the
condition: tan(2) =
g/( ). This form is sufficient to find the probability of
excitation. Ifthe two-level system begins in its ground state, then
the time-dependentstate can be written as:
= sin
cos sin
eit/2 + cos
sin cos
eit/2 (2.11)
Pex = ||2 = |2sin cos sin(t/2)|2 =
1 cos(t)2
(g)2
(g)2 + ( )2 .(2.12)
Thus, the probability of excitation oscillates at the Rabi
frequency =( )
2
+ (g)2
, and the amplitude of oscillation depends on detuning.On
resonance ( = ), one can engineer a state with || = || = 1/2,or a
state with || = 1 by applying a pulse with a duration (g)t = /2or
respectively.
2.3 Quantum behaviour
Let us now repeat this analysis in the case of a quantised mode
of radiation,starting with the Jaynes-Cummings model in the
rotating wave approxima-tion.
H =
1
2 ga
ga + 0aa. (2.13)Rabi oscillations in the Jaynes-Cummings
model
In the rotating wave approximation the total number of
excitations Nex =z + aa is a conserved quantity, thus if we start
in a number state of thelight field:
|n, = (a)nn!
|0, , (2.14)
then the general state at later times can always be written:
|(t) = (t) |n 1, + (t) |n, . (2.15)
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12 LECTURE 2. JAYNES CUMMINGS MODEL
Inserting this ansatz in the equation of motion gives
it (t)(t)
= n 1
2 1 + 1
2 ( ) g
n
gn ( ) (t)
(t) .
(2.16)After removing the common phase variation exp[i(n1/2)t[,
the generalsolution can be written in terms of the eigenvectors and
values of the matrixin Eq. (2.16), and reusing the results for the
semiclassical case, we have thestate
(t)(t)
= ei(n1/2)t
sin eit/2
cos sin
+ cos eit/2
sin cos
,
(2.17)and the excitation probability is thus:
Pex =
1 cos(t)2
g2
ng2n + ( )2 (2.18)
where we now have tan(2) = g
n/( ) and = ( )2 + g2n.Collapse and revival of Rabi
oscillations
Restricting to the resonant case = , let us discuss what happens
if theinitial state does not have a well defined excitation number.
Since the oscil-lation frequency depends on n, the various
components of the initial statewith different numbers of
excitations will oscillate at different frequencies.We can
therefore expect interference, washing out the signal. This is
indeed
seen, but in addition one sees revivals; the signal reappears at
much latertimes, when the phase difference between the
contributions of succesivenumber state components is an integer
multiple of 2.
Let us consider explicitly how the probability of being in the
excitedstate evolves for an initially coherent state
| = e||2/2 exp(a) |0, = e||2/2n
nn!
|n, . (2.19)
If resonant, the general result in Eq. (2.17) simplifies, as =
/4. Followingthe previous analysis, the state at any subsequent
time is given by:
| = e(||2+it)/2
n
(eit)nn!
cos
g
nt
2
|n, + sin
g
nt
2
|n 1,
. (2.20)
Thus, the probability of being in the excited atomic state,
traced over allpossible photon states, is given by
Pex =n
|n, ||2 = e||2n
||2nn!
sin2
g
nt
2
=1
2 1
2
e||2 n
||2n
n!
cos g
nt . (2.21)
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2.3. QUANTUM BEHAVIOUR 13
We consider the case 1. In this case, one may see that the
coefficients||2n/n! are sharply peaked around n ||2 by writing:
||2n
n! =1
2n exp[f(n)], f(n) = n ln(n) n n ln(||2), (2.22)By
differentiating f twice, one finds f(n) = ln
n/||2, f = 1/n, and
hence one may expand f(n) near its minimum at n = ||2 to
give:
f(n = ||2 + x) ||2 + x2
2||2 (2.23)
Using this in Eq. (2.21) gives the approximate result:
Pex 12
122||
2m
exp
m
2
2||2 + igt
||2 + m
, (2.24)
where it is assumed that ||2 is large enough that the limits of
the sum maybe taken to .
Three different timescales affect the behaviour of this sum.
Concentrat-ing on the peak, near m = 0, there is a fast oscillation
frequency g, de-scribing Rabi oscillations. Then, considering
whether the terms in the sumadd in phase or out of phase, there are
two time scales: collapse of oscilla-tions occurs when there is a
phase difference of 2 across the range of terms|m| < m
contributing to this sum, i.e. when gTcollapse(
||2 + m||) =. Taking m = || from the Gaussian factor, this
condition becomesgTcollapse 2. On the other hand, if the phase
difference between each suc-cessive term in the sum is 2, then they
will rephase, and a revival occurs,
with the condition gTrevival(||2 + 1 ||) = 2, giving gTrevival =
4||.
Thus, the characteristic timescales are given by:
Toscillation 1g|| , Tdecay
1
g, Trevival ||
g. (2.25)
and the associated behaviour is illustrated in Fig. 2.1.Let us
now consider how this behaviour may be approximated ana-
lytically. The simplest approach might be to replace the sum
over numberstates by integration, however this would inevitably
lose the revivals, whichare associated with the discreteness of the
sum allowing rephasing. Thus,it is necessary to take account of the
discreteness of the sum, and hence the
difference between a sum and an integral. For this, we make use
of Poissonsummation which is based on the result:m
(x m) =r
e2irm m
f(m) =r
dxe2irxf(x). (2.26)
This allows us to replace the summation by integration, at the
cost ofhaving a sum of final results, however in the current
context this is veryhelpful, as the final sum turns out to sum over
different revivals. Applyingthis formula, we may write Pex 12
[1
r {(r, ||, t)}] where
(r, ||, t) = dx
2||2
exp2irx x2
2
|
|2
+ igt|| +x
2
|
|
x2
8
|
|3
(2.27)
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14 LECTURE 2. JAYNES CUMMINGS MODEL
0 20 40 60 80 100 120 140 160 180 200
Time g
0
0.2
0.4
0.6
0.8
1
Excitation
probability
0 2 4 60
0.5
1
Figure 2.1: Collapse and revival of Rabi oscillations, plotted
for =
200.
Completing the square yields:
(r, ||, t) = eigt||2||2
dx exp
1
2||2
1 +igt
4||
x2 + i
2r +
gt
2||
x
=
eigt||2||2 exp||
2
2
(2r + gt/2
|
|)2
1 + igt/4||
dx exp
1
2||2
1 +igt
4||
x i||2 2r + gt/2||1 + igt/4||
2
=eigt||
1 + igt/4|| exp||
2
2
(2r + gt/2||)21 + igt/4||
. (2.28)
From this final expression one may directly read off: the
oscillation fre-quency g||; the characteristic time for the first
collapse 22/g (by takingr = 0 and assuming gt/|| is small); the
delay between successive revivals4||/g; and finally one may also
see that the revival at gt = 4r|| hasan increased width and
decreased height compared to the initial collapse,given by a
factor
1 + 2r2 as is visible in figure 2.1.
Further reading
The contents of most of this chapter is discussed in most
quantum opticsbooks. For example, it is discussed in Scully and
Zubairy [7], or Yamamotoand Imamoglu [8]. Further properties of
collapse and revival of Jaynes-Cummings oscillations are discussed
by Gea-Banacloche [9].
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Lecture 3
Two level systems coupledto many modes: Decay and
density matrix equations
The aim of this lecture is to introduce the behaviour of a
two-level systemcoupled to many photon modes, such that
irreversible decay now occurs.We will use the density matrix
description of the two-level system, andshow how coupling to a
reservoir of bath modes induces irreversible decay.The formalism we
develop here will be useful also when we later considerthe laser,
and its more quantum mechanical variants of single atom lasersand
micromasers.
3.1 Many mode quantum model irreversibledecay
In the previous lecture we discussed a two level system coupled
to a singlemode, considering both classical and quantum light. One
important dif-ference between these situations is that for
classical light, if the light fieldvanishes, then there can be no
transitions; in contrast, for the quantumdescription, spontaneous
emission also existed, in that the state |0, hy-bridises with the
state
|1,
, due to the +1 in the
n + 1 matrix elements.
We will now consider the effects of this +1 factor when we
couple a singletwo level system to a continuum of radiation
modes.
Our Hamiltonian will remain in the rotating wave approximation,
butno longer restricted to a single mode of light, and so we
have:
H =
2z +
n,k
kak,nak,n +
k
gk,n2
ak,n + H.c.
(3.1)
where k = ck and
gk,n
2=
ek,n dba20kV
. (3.2)
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16 LECTURE 3. DECAY OF A TWO-LEVEL SYSTEM
3.2 Density matrix equation for relaxation oftwo-level
system
We now discuss how the irreversible decay arising from the
coupling tomany photon modes may be described in the density matrix
approach. Weproceed by eliminating the the behaviour of the
continuum of radiationmodes in order to write a closed equation for
the density matrix of thesystem. If we define the system density
matrix as the result of tracing overall reservoir degrees of
freedom, we can write an equation of motion forthis quantity
as:
d
dtS = iTrR {[HSR , SR]} . (3.3)
in which S means state of the system, R the reservoir (in this
case, thecontinuum of photon modes), and TrR traces over the state
of the reservoir.
After removing the free time evolution of the reservoir and
system fields, theremaining system-reservoir Hamiltonian can be
written in the interactionpicture as:
HSR =k
gk2
ake
i(k)t + +akei(k)t
. (3.4)
In general Eq. (3.3) will be complicated to solve, as
interactions betweenthe system and reservoir lead to entanglement,
and so the full density ma-trix develops correlations. This means
that the time evolution of the systemwould depend on such
correlations, and thus on the history of system reser-
voir interactions. However this will simplify is we assume a
smooth densityof states for the bath, in which case it has a short
correlation (memory)time, and memory effects can be neglected. This
will allow us to make aMarkov approximation, meaning that the
evolution of the system dependsonly on its current state i.e.
neglecting memory effects of the interaction.
For a general interaction, there are a number of approximations
whichare required in order to reduce the above equation to a form
that is straight-forwardly soluble.
We assume the interaction is weak, so SR = S R + SR , whereSR
O(gk). Since S = TrR(SR), it is clear that TrR(SR) = 0,
however the result of TrR(HSRSR) may be non-zero. Such
corre-lations will be assumed to be small, but must be non-zero in
orderfor there to be any influence of the bath on the system. In
orderto take account of these small correlation terms, generated by
thecoupling, we use the formal solution of the density matrix
equationSR(t) = i
tdt[HSR(t
), SR(t)] to write:
d
dtS(t) = TrR
t0
dt
HSR(t),
HSR(t), SR(t
)
. (3.5)
We may now make the assumption of small correlations by
assumingthat SR(t
) = S(t)
R(t
), and that only the correlations generated
at linear order need be considered.
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3.2. DENSITY MATRIX EQUATION FOR RELAXATION OF
TWO-LEVEL SYSTEM 17
We further assume the reservoir to be large compared to the
system,and so unaffected by the evolution, thus R(t
) = R(0). Along withthe previous step, this is the Born
approximation.
Finally, if the spectrum of the bath is dense (i.e. the spacing
of energylevels is small), then the trace over bath modes of the
factors eikt
will lead to delta functions in times. This in effect means that
wemay replace S(t
) = S(t). This is the Markov approximation.
One thus has the final equation:
d
dtS(t) = TrR
t0
dt
HSR(t),
HSR(t), S(t) R(0)
. (3.6)
To evaluate the trace over the reservoir states, we need the
trace ofthe various possible combinations of reservoir operators
coming from HSR .
These will involve pairs of operators ak, ak. Using cyclic
permutations,these trace terms can be written as thermal average X
= TrR(RX). Forbulk photon modes, the relevant expectations are:
akak
=
akak
= 0,
akak
= kknk,
aka
k
= kk(nk + 1).
(3.7)In these expressions we have assumed nk = 0, i.e. we have
allowed fora thermal occupation of the photon modes. The results of
the previouslecture assumed an initial vacuum state, and thus nk =
0.
With these expectations, we can then write the equation of
motion forthe density matrix explicitly, writing k = exp[i(k
)(t
t)], to give:
d
dtS(t) =
k
gk2
2 t0
dt
+nkk + +(nk + 1)
k
s
+(nk + 1) + +nk (k + k)+s
+nk
k +
+(nk + 1)k
. (3.8)
In this expression, the first term in parentheses comes from the
expres-sion TrR [HSR(t)HSR(t
)SR ], the second comes from the conjugate pairTrR
[HSR(t)SRHSR(t
)] and t t [hence (k + k)], and the final termcomes from TrR
[SRHSR(t
)HSR(t)].At this point, we can now simplify the above expression
by restricting
to the case where there is a smooth density of states. We
define
(t t) =k
gk2
2 eik(tt) () = 2k
gk2
2 ( k). (3.9)Then, the restriction that is a smooth function
means that () does notsignificantly vary over the range , i.e. d/d
1. In this case (whichcorresponds to the Markov approximation,
meaning a flat effective densityof states) the integral over in Eq.
(3.23) becomes a delta function, sothat:
kgk
22
k= (t
t),
kgk
22 n
kk
= n(t
t). (3.10)
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18 LECTURE 3. DECAY OF A TWO-LEVEL SYSTEM
Here we have assumed that not only is the density of states
flat, but thatalso nk is sufficiently flat, that we might write an
averaged occupation n.Such an approximation is only valid at high
enough temperatures we
will discuss this in some detail in a few lectures time.
d
dtS(t) =
2n
+S 2+S + S+
2
(n + 1)
+S 2S+ + S+
(3.11)
The factor of 1/2 here comes from the regularised half integral
of the deltafunction. The two lines in this expression correspond
to stimulated absorp-tion (which exists only if n > 0), and
leads to excitation of the two-levelsystem, and to emission (both
stimulated and spontaneous). If n = 0, onlyemission survives, and
the results of the previous lecture are recovered for
the relaxation to equilibrium.One can see the behaviour from
this equation most clearly by writing
the equations for the diagonal components:
t = 2
n (2) 2
(n + 1) (2) = t (3.12)
Then, using = 1 , one may find the steady state probability
ofexcitation is given by:
0 = [n(1 ) (n + 1)] = n2n + 1
. (3.13)
This formula is as expected, such that if one uses the
Bose-Einstein occupa-tion function for the photons, n = [exp()1]1,
the excitation probabilityof the two-level system is the thermal
equilibrium expression.
3.3 Effective decay rate
We have not yet calculated an actual expression for the
effective decay rateappearing in the previous expression. Let us
now use the form of gk inEq. (3.2) to evaluate (). Writing the
wavevector in polar coordinates,with respect to the dipole matrix
element dab pointing along the z axis (seeFig. 3.1), we then
have:
() = 2V
(2)3
dd sin k2dk(ck )
2|dab|220ckV
n
(z ekn)2.(3.14)
From Fig. 3.1 it is clear that z ek2 = 0 and one may see that z
ek1 = sin . For concreteness, the k direction and polarisations can
beparametrised by the orthogonal set:
k =
sin cos sin sin
cos
, ek1 =
cos cos cos sin
sin
, ek2 =
sin
cos 0
.
(3.15)
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3.4. FURTHER READING 19
kz
e
e
2
1
Figure 3.1: Directions of polarisation vectors in polar
coordinates
Putting all the terms together, one finds:
() =2|dab|2
(2)220
d
d sin3
ckcdk
3c3(ck ) = 1
40
4|dab|2333c3
(3.16)This formula therefore gives the decay rate of an atom in
free space, asso-ciated with the strength of its dipole moment and
its characteristic energy.
It is useful to rewrite this decay rate in terms of the
oscillator strengthfab of the transition involved. Recalling the
definition offab, and continuingto use the shorthand = ab we
have
=
4
3
1
403c3
2e2
2m0fab =
1
3
e2
40c
2
m0c2fab (3.17)
Then, using the additional results:
=e2
40c, Ryd = 1
22m0c
2 (3.18)
where Ryd is the Rydberg energy and the fine structure constant,
onecan write the decay rate in the easily interpretable form:
=
1
33
Ryd fab. (3.19)
The smallness of is thus crucial in the fact that atomic lines
are welldefined (i.e. have linewidths smaller than their
separation).
3.4 Further reading
Once again, the contents of this chapter can be found in most
standard
quantum optics books, e.g.[7, 10].
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20 LECTURE 3. DECAY OF A TWO-LEVEL SYSTEM
3.A WignerWeisskopf approach
An alternative method to show the decay rate in simple
situations is to
consider solving the Schrodinger equation directly, to find the
probabilitythat the two-level system remains excited. To study
spontaneous emission,we start from the photon state |0, , in which
there are no photons. Thesubsequent state can be written as:
| = (t)|0 +k
k,n(t)|1k,n , (3.20)
where |1k,n contains a single photon in state k, n. Then,
writing theSchrodinger equations as:
it =
2
+k,n
gk,n
2
k,n, itk,n =
2
+ k k,n + gk,n2
, (3.21)
one may solve these equations by first writing = eit/2, k =
kei(/2k)t,
and then formally solving the equation for k, with the initial
conditionk(0) = 0, to give:
k,n = i gk,n2
tdtei(k)t
(t). (3.22)
Substituting this into the equation for gives:
t = t
dtk,n
gk,n22
ei(k)(t
t)
(t
)
= t
dt
d
2()ei()(t
t)(t) (3.23)
where we have used the same definition of as previously:
() = 2k,n
gk,n2
2 ( k,n) (3.24)We may then make the same Markov approximation as
before to yield
t = t
dt()(t t)(t) = ()2
(t) (3.25)
Thus, the probability of remaining in the excited state decays
exponentiallydue to spontaneous decay, with a decay rate () for the
probability. Theapproach used in this section is known as the
WignerWeisskopf approach.
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Lecture 4
Density matrix equationswith decay and coherent
driving
This lecture applies the framework of density matrix equations
introducedin the previous lecture in order to incorporate other
relaxation and decoher-ence processes, such as those arising due to
noise sources, or inhomogeneousbroadening in an ensemble of atoms.
We will then illustrate the use of thisapproach by discussing the
behaviour of a two-level system illuminated bya coherent state of
light, but able to radiate into a continuum of modes. Inthis
lecture we will study the steady state of this system, and in
subsequent
lectures we will look at the spectrum of emitted light, which
will requireknowing more about the full time-evolution of the
system.
4.1 Equations of motion with coherent driving
Before introducing extra contributions to dephasing, we begin by
applyingthe method developed in the previous lecture to consider
the combination ofcoherent driving and decay. We will assume from
hereon that the reservoirphoton modes are empty.1 The evolution of
the density matrix is thencontrolled by:
t = i [H, ] 2
+S 2S+ + S+
(4.1)
where H describes the coherent, Hamiltonian dynamics, which we
will taketo be given by:
H =
2z +
g
2eit+ + H.c.. (4.2)
1This limit is frequently relevant in cavity quantum
electrodynamics experiments,since the confined photon modes start
at energies of the order of 1eV
104K, so the
thermal population of such photon modes is negligible.
21
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22 LECTURE 4. POWER BROADENING
One may then write out the equations of motion for each
component ex-plicitly, giving
t = i 2 + g2 eit g2 eit 2 , (4.3)t = i
2
+g
2eit g
2eit
2
2. (4.4)
These two equations are sufficient, since the unit trace
property of a densitymatrix implies = 1 , and the off diagonal
elements are complexconjugate of one another by the Hermitian
structure of the density matrix.To solve these equations, it is
convenient to first go to a rotating frame,so that one writes = P
e
it, and secondly to introduce the inversion,Z = ( )/2. In terms
of these variables, one has:
tZ = i g2 (P P)
Z+ 12
(4.5)
tP = i( )P + i g2
2Z 2
P. (4.6)
To further simplify, we may separate the real and imaginary
parts of P =X+ iY, and write = for the detuning of the optical
pump, so thatone has the three coupled equations:
tX = Y 2
X (4.7)
tY = +X+ gZ 2 Y (4.8)
tZ = gY
Z+1
2
. (4.9)
These are Bloch equations for the Bloch vector parametrisation
of the den-sity matrix (in the rotating frame):
=
12 + Z X iY
X + iY 12 Z
. (4.10)
The three terms, ,g, correspond to: rotation about the Z axis
(i.e.
phase evolution); rotation about the X axis (i.e. excitation);
and causesthe length of the Bloch vector to shrink. Note that the
rotation rate aroundthe X axis is g, so a duration t = /g would
lead to the transition fromground to inverted state.
Before solving Eq. (4.74.9), we will first consider other kinds
of dephas-ing that may affect the density matrix evolution, leading
to a generalisedversion of these equations, allowing for both
relaxation and pure dephasing.
4.2 Dephasing in addition to relaxation
In addition to relaxation, where excitations are transfered from
the sys-
tem to the modes of the environment, pure dephasing is also
possible, in
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4.2. DEPHASING IN ADDITION TO RELAXATION 23
which the populations are unaffected but their coherence is
reduced. Inconsidering this kind of dephasing, it is useful to also
broaden our viewfrom considering a single two level system, to
describing measurements on
an ensemble of many two level systems. For the moment we will
assumethat even if there are many two level systems, they all act
independently.As such, the expectation of any measurement performed
on this ensemblecan be given by:
X =i
T r(iX) = T r(X), where =i
i. (4.11)
We thus have two types of decoherence one may consider:
Broadening within a two-level system, coming from time
dependentshifts of energies (and possibly also coupling strengths).
A variety of
sources of such terms exist; examples include collisional
broadeningof gaseous atoms, where the rare events in which atoms
approacheach other lead to shifts of atomic energies. In solid
state contexts,motions of charges nearby the artificial atom can
also lead to timedependent noise.
Inhomogeneous broadening, where the parameters of each
two-levelsystem vary, and so off-diagonal matrix elements,
involving time de-pendent coherences, are washed out in . This is
particularly an issuein solid state experiments, where the
two-level system energy may in-volve the dimensions of a fabricated
or self-assembled artificial atom,
and these dimensions may vary between different systems.
To describe both such effects together, we consider adding
noise,
+ (t). (4.12)
This noise will then lead to a decay of the off diagonal
correlations. To seethis, consider Eq. (4.6), with the above
replacement, and then perform agauge transform so as to remove the
explicit time dependence. One finds:
P(t) P(t)expi t
0(t)dt (4.13)
We will take (t) = 0, and two distinct limits for the
correlations of .
Fast noise limit
In order to extract a tractable model of time-dependent noise,
the simplestlimit to consider is that (t) has a white noise
spectrum, i.e. it is Gaussiancorrelated with:
(t)(t) = 20(t t). (4.14)This assumption is equivalent to
assuming that the noise is fast, i.e. thatany correlation timescale
it does possess is sufficiently short compared to
the dynamics of the system that it may be neglected.
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24 LECTURE 4. POWER BROADENING
With this assumption of instantaneous Gaussian correlations, one
maythen expand the exponential in Eq. (4.13) and write:
P(t) P(t)1 + it0
dt(t) 12!
t0
dtdt(t)(t) + . . .
=P(t)
1 20
2!
t0
dt +(20)
2
4!3
t0
dt2
+ . . .
= P(t)e0t
(4.15)
with the combinatoric factors coming from the number of possible
pairingsin the Wick decomposition of n.
This noise thus leads to an enhanced decoherence rate for the
off di-agonal terms, so that /2 should be replaced by 1/T2 = (/2) +
0 inthe density matrix equations. One may equivalently define the
relaxation
time 1/T1 = . In general T2 < T1, as dephasing is faster than
relaxation,however the factors of two in the above mean that the
strict inequality isT2 < 2T1, with equality holding when there
is only relaxation.
The Bloch equations are thus given by:
tX = Y 1T2
X (4.16)
tY = +X+ gZ 1T2
Y (4.17)
tZ = gY 1T1
Z+
1
2
. (4.18)
A more rigorous approach to deriving the effect of dephasing is
discussedin question A3, in analogy to the way we derived decay in
the density matrixformalism
Static limit (inhomogeneous broadening)
To describe inhomogeneous broadening, should represent the
system-dependent variation of energies, and should have no time
dependence. Thesame behaviour is also relevant if time-dependent
noise terms vary slowlyon the timescale of an experiment. (For
example, slow noise may arisefrom a nearby impurity have multiple
stable charging states, so that the
charge environment can take multiple different values; if the
charging anddischarging of this impurity is slow compared to the
experiment, this energyshift will effectively give inhomogeneous
broadening when averaged overmultiple experimental shots.)
For this case is static, but randomly distributed, and Eq.
(4.13) be-comes:
P(t) P(t)
dp()eit. (4.19)
If we consider a Lorentzian probability distribution for with
width 1,i.e. p() = 21/(
21 +
2), then one finds P(t) P(t) exp(1|t|). Alter-natively, a
Gaussian probability distribution leads to a Gaussian decay of
correlations.
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4.3. POWER BROADENING OF ABSORPTION 25
If one has both noise terms as in the previous section, and
inhomoge-neous broadening, one may then distinguish T2 as defined
above, and T
2 ,
where 1/T2 = (/2) + 0 + 1.
Distinguishing T2 and T2
Since both 0 and 1 lead to decay of the coherence functions, a
simpleexperiment measuring coherence will see the lifetime T2 .
However, thedynamics of each individual two level system remains
coherent for the longertimescale T2. Such extra coherence can
easily be seen in any sufficientlynon-linear measurement. An
example of how the difference of T2 and T
2
can be measured is given by photon echo, illustrated in Fig.
4.1.
X
Z
Y
g t= /2
X
Z
Y
X
Z
Y
g t=
X
Z
Y
Figure 4.1: Cartoon of photon-echo experiment, which
removesdephasing due to inhomogeneous broadening, leaving only
truedephasing rate to reduce intensity of revival.
From the Bloch equations, Eq. (4.164.18), it is clear that a
resonantpulse with duration gt = /2 brings the Bloch vector to the
equator.From there, inhomogeneous broadening means that (in a
rotating frame atthe mean frequency) the Bloch vectors for each
individual two-level systemstart to spread out. However, by
applying a second pulse with gt = ,a rotation about the X axis
means that whichever Bloch vector spreadat the fastest rate is now
furthest from the Y axis, and the subsequentevolution sees the
vectors re-converge. At the final time, the coherence ofthe
resulting pulse will have been reduced by 0 and , but 1 will
havehad no effect.
4.3 Power broadening of absorption
Having introduced the idea of extra dephasing terms, we may now
includethese in a re-examination of the question we started an
earlier lecture with what is the effect of shining classical light
on a two-level atom? However,we may now account for decay and
dephasing. Whereas previously theabsence of decay meant that the
result was Rabi oscillations, with decay asteady state is
eventually found.
Looking for steady state solutions of Eq. (4.164.18) one finds
from theequations for X and Z that:
Y = 1
gT1
Z+
1
2
, X = T2Y =T2gT1
Z+
1
2
. (4.20)
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26 LECTURE 4. POWER BROADENING
Substituting these into the equation for Y, and recalling Pex =
Z + 1/2,gives:
0 = Pex T2
gT1+
1
T2
1
gT1+ g
g
2(4.21)
After re-arranging, one then finds:
Pex =1
2
T1T2(g)2
(T2)2 + 1 + T1T2(g)2=
1
2
(T1/T2)(g)2
2 + [1 + T1T2(g)2] /T22(4.22)
The addition of damping to the equations of motion has thus had
anumber of important consequences:
It is responsible for a steady state existing at all (but note
that thissteady state is in a frame rotating at the pump frequency
, so the
physical polarisation is time dependent.)
It gives a finite width to the resonance, even at weak pump
powers in the Hamiltonian case, as g 0, the width of the
resonancepeak vanished.
It modifies the overall amplitude at resonance, by a factor
dependingon the power, such that at g 0, there is no response.
An important feature of this excitation probability is that the
linewidthdepends on the intensity of radiation. This is a
consequence of the nonlin-earity implicit in a two-level system.
Such dependence is not so surprising
given that in the Hamiltonian dynamics, the resonance width
already de-pended on the field strength. One may note that if one
considered an har-monic atomic spectrum, rather than a two-level
atom, no such broadeningwould be seen (see question A4).
0
0.1
0.2
0.3
0.4
0.5
-30 -20 -10 0 10 20 30
Pex
Figure 4.2: Power broadening of absorption probability
ofexcitation vs atom-photon detuning for increasing field
strength.
This problem, of a single atom coherently pumped, and
incoherently de-caying also shows interesting behaviour if we look
at its incoherent emission
spectrum. This is however something we will postpone until
lecture 7.
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4.4. FURTHER READING 27
4.4 Further reading
The discussion of the resonance fluorescence problem in
particular can be
found in chapter 4 of Meystre and Sargent III [10].
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Lecture 5
Two-level atom in a cavity:Purcell effect and Cavity
QED
In the previous few lectures we have discussed a single
two-level atom cou-pled either to a continuum of radiation modes,
or considering a cavity,which picks out a particular mode. In this
lecture we will consider morecarefully the situations in which a
cavity will pick out a single mode, bydescribing how the system
changes as one goes from no cavity, via a badcavity (which modifies
the field strength but does not pick out a singlemode) to a good
cavity, which can pick out a single mode.
To study this problem throughout this crossover, we will
consider a twolevel system coupled both to a cavity pseudo-mode
(which itself decays),and also coupled to non-cavity modes,
providing incoherent decay. We willshow how in this model, a bad
cavity can lead to either enhanced decay, anda good cavity can lead
to periodic exchange of energy between the cavityand the two-level
system, i.e. the Rabi oscillations we previously discussedfor a
perfect cavity.
Before discussing the crossover between bad and good cavities,
we firstinvestigate a toy model of a 1D cavity, to put into context
the meaning ofthe pseudo-mode description of the cavity mode.
5.1 The Purcell effect in a 1D model cavity
The aim of this section is to describe how a cavity modifies the
rate ofdecay of a two-level system, coupled to one-dimensional
radiation modes.Following the earlier discussion of
system-reservoir coupling, the decay ischaracterised by the
combination of reservoirs density of states and thecoupling of each
reservoir mode to the atom. This gave an expression forthe decay
rate of the form:
() = 2k
(
ck) gk22 . (5.1)
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30 LECTURE 5. PURCELL EFFECT
Recalling that the coupling strength gk depends on the mode
structure viagk Ek, where Ek is the field strength associated with
a single photon, wecan find the effect of the cavity on () by
determining the spatial profiles
of the modes of the full system, and inserting these into the
above sum.
a/2
L/2
Figure 5.1: Toy model of cavity, length a, in quantisation
volumeof length L.
We will consider a cavity of size a, embedded in a quantisation
volumeof length L, where we will take L later on. (See figure 5.1).
Our aimis to find the spatial profile of the electric field modes,
and thus to calculate
() k
( ck)|Ek(x = 0)|2. (5.2)
The electric field obeys the wave equation:
d2
dx2Ek(x) = k2r(x)Ek(x), (5.3)
where r(x) is the (spatially varying) dielectric constant, and k
is the
wavevector in vacuum. Both in the cavity, and outside, the modes
willbe appropriate combinations of plane waves. Restricting to
symmetric so-lutions (as antisymmetric solutions will vanish at x =
0), and matching theboundary at x = L/2, we may write:
Ek(x) =
f(x) Ak cos(kx) |x| < a2g(x) Bk sin
k|x| L2 |x| > a2 . (5.4)
As L , the normalisation condition will approach the simple
resultBk =
2/L, since the normalisation integral will be dominated by the
parts
outside the cavity. We now need to find the effect of the
imperfect barriers
on this mode in order to relate Ak, Bk, and thus find the
quantisationcondition specifying k, and the mode amplitude, Ek(x =
0) = Ak.
As a simple model, let us consider a varying dielectric index, =
0[1 +(|x| a/2)], giving the equation:
d2
dx2Ek(x) = k2
1 +
|x| a
2
Ek. (5.5)
Let us focus on x > 0; then, given the form of Ek(x) = (a/2
x)f(x) +(x a/2)g(x) with f = k2f and g = k2g, we may write:
d2
dx2Ek = k2Ek + 2x
a
2 g(x) f(x) + x a
2 [g(x) f(x)](5.6)
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5.1. THE PURCELL EFFECT IN A 1D MODEL CAVITY 31
Thus, to solve Eq. (5.5) we require that:
fa
2 = g
a
2 , g
a
2
f a
2 =
k2
4f
a
2+ g
a
2 (5.7)
Substituting the forms of f(x), g(x) into this equation, one may
eliminateAk, Bk to give the eigenvalue condition:
cos
kL
2
+
k
2cos
ka
2
sin
ka
2 kL
2
= 0. (5.8)
For finite L, this provides a restriction on the permissible
values of k. AsL , the permissible values of k become more dense,
such that, in thislimit, one may instead consider this equation as
depending on two separatevalues kL and ka; ifL a, it is possible to
significantly change kL withoutmodifying ka. Thus, we will instead
rewrite this equation as a condition
that specifies the value of kL given a fixed ka:
cos
kL
2
1 +
k
2sin
ka
2
cos
ka
2
= sin
kL
2
k
2cos2
ka
2
.
(5.9)Using this condition, one may then eliminate kL, and write
Ak in terms ofBk =
2/L and functions of ka, i.e.:
Ak =
2
L
11 + k sin(ka/2) cos(ka/2) + (k/2)2 cos2(ka/2)
(5.10)
If we define = k/2, then we may use this formula to give the
effectivedecay rate:
k =
c
=
0
1 + sin(ka) + 12 2(1 + cos(ka))
=0
1 + 2
2
+
1 +
2
4 cos(ka 0), tan 0 =
2
.
(5.11)
The form of this function in general is shown in Fig. 5.2. For
small k, theeffect of the cavity is weak (i.e. k 1), and so there
is little modificationcompared to the result without the cavity.
For larger k, there are sharppeaks, which can be described by an
approximately Lorentzian form (see
the inset of Fig. 5.2). This Lorentzian form can be found by
expandingEq. (5.11) near its peaks, which are at:
k0a = 0 + (2n + 1). (5.12)
We can then expand () near 0 = ck0. If 1, and if we may
neglectits k dependence, then we find:
0 1
1 + 2
2
22
1 + 2
2 2
4
1 12
a(0)
c
2
1
12 +
2a2
4c2 ( 0)2. (5.13)
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32 LECTURE 5. PURCELL EFFECT
0
5
10
15
20
25
0 1 2 3 4 5 6 7 8
()/0
/0
0.01
0.1
1
10
100
7.2 7.6 8
Figure 5.2: Effective decay rate vs frequency for the toy model
ofa 1D imperfect cavity. Inset compares the exact result, Eq.
(5.11)
to the Lorentzian approximation of Eq. (5.14)
Let us introduce the full width half maximum, , such that this
expressionbecomes:
0=
P(/2)2
( 0)2 + (/2)2 (5.14)
which leads to the definitions:
2=
2c
2a, P = 2 =
4c
a. (5.15)
Here 0 is the result for a 1D system in the absence of a cavity.
Thefactor P describes the maximum value of /0 at resonance. At
resonance,the presence of the cavity enhances the decay rate, and P
is the Purcellenhancement factor. It is helpful for comparison to
later results to rewritethis as:
P = 40
c
0a=
2
Q
a
, Q 0
(5.16)
where Q is the quality factor of the cavity mode. The
enhancement onresonance therefore depends both on the quality
factor, and on the modevolume.
Away from resonance, the decay rate decreases instead of
increasing,
since there are no modes with significant weight inside the
cavity; theminimum value of /0 occurs when k0a = 0 + 2n, and is
given by/0 = 1/
2 = 1/P, hence for this 1D case, P describes both the
enhance-ment on resonance, and the reduction off resonance.
As increases further, the Lorentzian density of states becomes
sharplypeaked, and so the Markovian approximation for decay of the
atom nolonger holds, instead one can use:
d
dt =
tdt(t)
d
2
0P(/2)2ei()(t
t)
( 0)2 + (/2)2
=
0P
4t dt(t)ei()(tt)(tt)/2. (5.17)
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5.2. WEAK TO STRONG COUPLING VIA DENSITY MATRICES 33
This form describes the behaviour one would see if the two-level
systemwere coupled to another degree of freedom, with frequency and
decayrate /2. Hence, we are led in this limit to introduce the
cavity pseudo-
mode as an extra dynamical degree of freedom. The density of
states impliesthat the cavity mode cannot be treated as a Markovian
system, it has anon-negligible memory time 1/, and so we must
consider its dynamics.
5.2 Weak to strong coupling via density matrices
This section considers the case in which a single cavity mode
must betreated beyond a Markovian approximation, by considering the
Jaynes-Cummings model, along with relaxation of both the two-level
system andthe cavity mode. The model is thus given by:
ddt
= i [H, ] 2
aa 2aa + aa
2
+ 2+ + + (5.18)
H =g
2
+a + a
+ z + 0a
a. (5.19)
One should note that in this equation, a is the creation
operator for apseudo-mode of the system, i.e. it does not describe
a true eigenmode; thetrue eigenmodes are instead superpositions of
modes inside and outside thecavity, just as in the previous
section. Because the pseudo-mode overlapswith a range of true
eigenmodes, the probability of remaining in the pseudo-mode will
decay (see question B1 for the 1D case). This coupling
betweencavity modes and modes outside the cavity can be described
in a Markovianapproximation, leading to the decay rate in the above
density matrixequation.
In addition to the decay of the pseudo-mode, we also include a
rate
describing decay of the two level system into modes other than
the cavitymode. In the one dimensional description considered
previously, such achannel must correspond to non radiative decay.
However, in three dimen-sions, if the cavity is not spherical, then
decay into non-cavity directionsis possible and is described by .
In the three dimensional case, one mayconsider = (1
/4), depending on solid angle the cavity encloses.
For the experimental systems discussed below, this is generally
the case,due to the need to have access to insert atoms into the
cavity. More gener-ally describes the possibility of relaxation
into any mode other than thecavity mode; in solid state contexts,
other possible reservoirs often exist(e.g phonons, other
quasiparticle excitations etc).
Let us solve the equation of motion, Eq. (5.18), starting in an
initiallyexcited state | , 0. From this initial state, the only
possible other sub-sequent states are | , 1 and | , 0. As the last
of these states cannotevolve into anything else it may be ignored,
and we can write a closed setof equations for three elements of the
density matrix:
PA = 0,0, PB = 1,1, CAB = 0,1 (5.20)
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34 LECTURE 5. PURCELL EFFECT
By taking appropriate matrix elements, one finds:
d
dt
PA =
i
g
2
(CAB
CAB)
PA (5.21)
d
dtPB = i g
2(CAB CAB) PB (5.22)
d
dtCAB = i( 0)CAB + i g
2(PA PB) +
2CAB (5.23)
One may further simplify these equations by noting that
PAPB|CAB|2 = 0is conserved, and thus writing PA = ||2, PB = ||2,
CAB = . Substi-tuting this leads to the simpler equations:
d
dt
=
/2 ig/2
ig/2 i
/2
(5.24)
where we have written = 0 . In general this leads to two
frequenciesfor decaying oscillations of the excitation
probability:
i
2
i+ i
2
+
g2
4= 0. (5.25)
Bad cavity limit Purcell effect in 3D
In the limit of a bad cavity, i.e. g, , the two frequencies
correspondingto the above equation correspond separately to decay
of the excited two-level system and decay of photons. Because these
are on very different
timescales, one may treat the coupling perturbatively writing =
i/2 +, where at O() the determinant equation becomes:
i
i
2+
2
+
g2
4= 0, (5.26)
and since this bad cavity limit gives:
= i2
+
g2
2 + 42
+
g2
2 + 42. (5.27)
This describes a cavity-enhanced decay rate, eff = +cav, where
the cav-
ity decay rate describes the Purcell effect, as discussed above.
On resonancewe have:
cav = 4g
2
2 1
(5.28)
and recalling the earlier 3D results:
=30|dab|230c3
,g
2
2 = 0|dab|220V
(5.29)
(where we have assumed resonance, so 0 = ), one may write:
cav = P = 4
3c3
2V 20
1
=
3
42 Q3
V
. (5.30)
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5.3. FURTHER READING 35
This closely resembles the one dimensional result before; the
Purcell en-hancement depends both on the quality factor, and also
on the mode vol-ume. An estimate of the minimum decay rate off
resonance in this case
yields cav,min = cav,max
20
2 = 3
1621Q
3V
. However, if the back-
ground decay rate also exists, this off-resonant reduction is
not as rele-vant as the on-resonant enhancement.
Strong coupling Rabi oscillations
If the cavity is sufficiently good, then rather than decay, the
excitationprobability oscillates. In the resonant case of = 0, one
can write thegeneral solution for the oscillation frequency as:
= i( + )
4 g2
4 4
2
. (5.31)
In order that Rabi oscillations exist, it is thus necessary that
g > ()/2.In order that they should be visible before decay has
significantly reducedthe amplitude, it is necessary that one has g
> , . In this limit, one hasstrong coupling, and decay is indeed
very strongly modified.
5.3 Further reading
For a discussion of the Purcell effect, and the crossover
between weak and
strong coupling, see e.g. Meystre and Sargent III [10].
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Lecture 6
Features of various of CavityQED systems
Having introduced all of the concepts required to understand a
single atomcoupled to radiation in a cavity, this lecture now
discusses a number ofsystems which can realise this situation. The
aim here is to understandwhat parameter ranges can physically be
achieved in different systems, tounderstand the advantages and
disadvantages of the different systems, andto understand the
physical origin of the various coupling strengths anddecay
rates.
A summary of the characteristic values is given in table 6.1.
The discus-sion below explains the origin and significance of these
numbers. In eachcase there are several stages to consider: the
nature of the two-level sys-tem, and the energies, decay rates, and
coupling strengths of the transitioninvolved. In addition, one must
also consider what designs of cavities arecompatible with the
transition and nature of the two-level system.
System Atom[1, 11] Atom[4] SC qubit[3] Exciton[5]Optical
Microwave Microwave Optical
/2 1 MHz 1 kHz a 30 MHz 30 GHz/2 6 MHz 30 Hz 3 MHz b 0.1 GHzg/2
10 MHz 50 kHz 100 MHz 100 GHz
/2 390 THz 50 GHz 10 GHz 400 THz3/V 105 101 /a = 1 101
Q 108 108 104 104
Other tmax tflight 100s 1/T2 3MHz 1/T2 3GHzaRecent work reaches
10Hz[12]bNot well known, dominated by cavity induced decay
Table 6.1: Characteristic energy scales for different cavity
QEDrealisations, as discussed in the text.
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38 LECTURE 6. CAVITY QUANTUM ELECTRODYNAMICS
6.1 Optical transitions of atoms
We start by considering the conceptually simplest case, in which
the two
level system is a transition between two low lying levels of a
real atom.Although real atoms have many levels, the level spacing
is anharmonic(E Ryd/n2), hence it is possible for the cavity to be
near to resonanceonly for a single transition between the ground
and excited states. This isonly possible because the decay rate / f
3(/Ryd), so that the naturallinewidth is much smaller than the
transition. Such transitions involve lightat optical frequencies,
and so require cavities designed to produce highreflectivity at
these frequencies.
Atomic energy levels; fine structure, hyperfine structureand
linewidths
Considering optically allowed transitions m = 0, 1, l = 1, the
sim-plest spectra will correspond to alkali atoms, involving s p
transitions.However, to properly understand these transitions, one
must consider boththe fine structure, and the hyperfine
structure.
Fine structure arises due to the spin-orbit coupling:
ESO =1
2m2c2
1
r
dV
dr
S L (6.1)
The order of magnitude of this expression is:
ESO 1mc2
2
2ma2B
e2
40aB Ryd2
mc2 2Ryd. (6.2)
The power of 2 indicates that this is much smaller than the
principalelectronic level splitting, but still bigger than the
linewidth, hence it canbe resolved. The spin orbit term increases
with atomic number due tothe larger potential gradient that exists
at small radii within the screeningcloud. Since the term produces
an LS coupling, its eigenstates are given bythe quantum numbers of
|L|, |S|, |J| and one other commuting observable,e.g. Jz. Thus, for
alkali atoms, the s state is not split, but the p state issplit
into J = 1/2, 3/2 states.
Hyperfine splitting arises due to coupling between the
electronic angularmomentum J and the nuclear angular momentum I.
The coupling I Jmeans that rather than using Jz, Iz to label
eigenstates, one should considereigenstates of |F|, Fz with F = J +
I. A level scheme for 87Rb is shownin Fig. 6.1, based on values in
Steck [13]. 87Rb is a boson, as it has anodd number of electrons
and an odd number of nucleons. Its ground statenuclear spin is |I|
= 3/2, hence the |J| = 3/2 state splits into four levels|F| = 0 . .
. 3. The notation F = 3 indicates an excited electronic state
withhyperfine state F = 3, whereas F = 2 indicates a ground
electronic statewith hyperfine state F = 2.
The transition between the fine structure level S1/2
P3/2 is known as
the D2 line. The dominant transition within this line is the F =
3 F =
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6.1. OPTICAL TRANSITIONS OF ATOMS 39
F=1
0.5GHz
F=0
F=1
F=2
F=3
F=2
F=1
F=21/2S
P1/2
3/2P
1.6eV=390THz=780nm
7GHz
7THz
266MHz
Figure 6.1: Level scheme for 87Rb (bosonic), showing fine
andhyperfine structure. For further details, see[13]
2. This has a dipole matrix element dD2 = 1.3eA. Using the
transitionenergy ED2 = 1.6eV, one finds fD2 = 0.7, so this
transition is close to
saturating the oscillator sum rule. Using this transition
strength, the aboveestimate of decay rate gives:
13
0.7 11373
1.6eV13.6eV
350THz 3MHz (6.3)
This estimate is smaller than the actual width, = 38MHz because
it doesnot properly include summation over possible spin states.
However, thislinewidth is rather small, and so it is possible to
resolve even hyperfinestructure, given the scales in Fig. 6.1. To
have a true two-level system, onehas two choices: either to avoid
any stray magnetic fields, and rely on thefact that the cavity mode
polarisation will couple to a particular superpo-
sition of the 2F + 1 = 7 levels labelled by mF = 3 . . . 3.
Alternatively,this remaining degeneracy could be split by a
magnetic field.
Optical cavities
From the above considerations, in order to have a cavity couple
to a an indi-vidual hyperfine level transition, the cavity
linewidth must also be smallerthan the hyperfine splitting, i.e
10MHz. Given the transition frequencyof E = 390THz, this implies a
cavity quality factor of Q 108 is required.
One may also consider the quality factor required for Purcell
enhance-ment. The minimum size of the cavity is constrained by the
need to be able
to controllably insert atoms, and a typical cavity length[1, 11]
is L 200m,
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40 LECTURE 6. CAVITY QUANTUM ELECTRODYNAMICS
and beam waist w 20m. This gives a ratio of wavelength to mode
vol-ume of 3/V 105. Recalling the condition that controls the
Purcelleffect, this means again that a very high Q factor, Q
108 is needed to
reach strong coupling.In the following we consider two possible
approaches; metallic and di-
electric mirrors, and see which is capable of producing such
high qualityfactors. In relating the quality factor the
reflectivity, it is important to notethat they depend differently
on the mode index involved. In particular, ifone introduces the
finesse F, then:
Q =
=
n
, F=
R
1 R (6.4)
where is the mode spacing (which depends on the length of the
cav-
ity), and R is the intensity reflection coefficient of the
mirrors. The aboveapproximation holds for good mirrors. For very
good mirrors, i.e. T =1 R 1, one has Q = nF n/(1 R), and the cases
discussed belowwill be in this range. For a cavity with length L
200m, the mode indexwill be n 250, and so F 4 105 would be
sufficient.
Metallic mirrors
As a brief reminder, the reflectivity of a metallic mirror can
be found byconsidering a plane wave incident on its surface, and
finding the reflectionand transmission coefficients associated with
the effective (frequency de-
pendent) dielectric constant arising from its conductivity. For
a plane wavewith frequency , one has:
B = 0( + ir0)E (6.5)
Thus the refractive index n =
r i/0 and for a good conductor atlow enough frequencies n = (1
i)/20. Then, matching electric andmagnetic fields at the
interface:
Ei + Er = Et 1 + r = tBi + Br = Bt
1
r = nt
(6.6)
so r = (n 1)/(n + 1). Thus, for a good conductor one finds:
R = 1 2
20
(6.7)
For a typical metal, with 1071m1, and optical frequencies
=400THz, this gives R = 1 0.05, hence F= 60. In order that the
qualityfactor reach the required value, one would have to use a
very high index(i.e. a very large cavity), with n 107. For such a
cavity, the mode volumewould be huge, and so strong coupling
unattainable. Clearly metal cavities
are not relevant.
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6.1. OPTICAL TRANSITIONS OF ATOMS 41
Dielectric mirrors: Bragg reflectors
Much higher quality mirrors for optical frequencies can be made
using di-
electric Bragg mirrors. These consist of alternating layers of
materials withdifferent dielectric constant, with spatial period
/4, so that even if thedielectric contrast between adjacent layers
is not sufficient to cause strongreflection, the interference of
multiple reflections will lead to strong overallreflection. This is
illustrated in Fig. 6.2.
Cavity
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
R=
|r|2
/0
na=1.5, nb=2.5
26 layers
Figure 6.2: Left: Cartoon of Bragg mirror cavity. The twocolours
indicate material of two differing refractive indices.
Right:reflectivity spectrum of a Bragg mirror
The reflectivity of a Bragg mirror can be found using the
transfer matrixapproach, discussed in the appendix to this lecture.
At resonance (i.e. ata frequency such that wana = wbnb = (c/)(/2),
where wa,b are layerthicknesses and na,b the refractive indices),
the reflection coefficient is givenby:
r =1 (nb/na)N1 + (nb/na)N
. (6.8)
where N is the number of pairs of layers. For the case
illustrated in Fig. 6.2,with a dielectric contrast nb/na = 5/3, N =
13 layers is sufficient to reach afinesses ofF= 4105. Because the
overall reflection depends on destructiveinterference between the
partial reflections at each interface, the reflectanceis high only
over a range of frequencies, known as the stop band. It is
therefore important that the frequency of all normal modes of
the coupledcavity QED system lie within this stop band if one wants
to ignore thefrequency dependence of the mirrors.
Other considerations The background atomic decay rate is
deter-mined by the intrinsic properties of the atomic transition,
reduced by thegeometry of the cavity, and so this sets the scale
that must be overcomefor strong coupling. Typical values of the
corresponding g, , are givenin Table 6.1. In addition to the
timescales included within Eq. (5.18), insome experimental systems,
there is another timescale, that of the time foran atom to leave
the cavity, however this is typically larger than all other
timescales.
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42 LECTURE 6. CAVITY QUANTUM ELECTRODYNAMICS
6.2 Microwave transiti
