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8/23/2019 Keeling Notes
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LightMatter Interactions
and Quantum Optics.
Jonathan Keeling
http://www.standrews.ac.uk/~jmjk/teaching/qo/
http://www.standrews.ac.uk/~jmjk/teaching/qo/http://www.standrews.ac.uk/~jmjk/teaching/qo/http://www.standrews.ac.uk/~jmjk/teaching/qo/http://www.standrews.ac.uk/~jmjk/teaching/qo/8/23/2019 Keeling Notes
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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 JaynesCummings model . . . . . . . . . . . . 92.2 Semiclassical limit . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Quantum behaviour . . . . . . . . . . . . . . . . . . . . . . 11
3 Decay of a twolevel system 15
3.1 Many mode quantum model irreversible decay . . . . . . 15
3.2 Density matrix equation for relaxation of twolevel 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 Quantumdot 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 HeisenbergLangevin 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 Nogo 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 discuss 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 behaviour will be discussed in some sections, both because an understanding
of semiclassical behaviour (i.e. classical radiation coupled to quantum mechanical 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 fewlevel 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 discrete level spectra which couple to the electromagnetic field. Such conceptstherefore apply to a wide variety of systems, and a variety of characteristic energies of electromagnetic radiation. Systems currently studied ex
perimentally include: real atomic transitions coupled to optical cavities[1];
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viii INTRODUCTION
Josephson junctions in microwave cavities (waveguides terminated by reflecting boundaries)[2, 3]; Rydberg atoms (atoms with very high principle 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, including 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 general interested in calculating a response function, relating the expectedoutcome to the applied input. However, to understand the predicted behaviour, 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 pictures frequently according to which is most appropriate. When consideringopen quantum systems, a variety of different approaches; density matrixequations, HeisenbergLangevin equations and their semiclassical approximations, again corresponding to both Schrodinger and Heisenberg pictures.
The main part of this course will start with the simplest case of a singletwolevel 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 twolevel atoms, leading to collective effects. Thetechniques of open quantum systems will also be applied to describing lasing, focussing on the more quantum examples of micromasers and single
atom lasers. The end of the course will consider atoms beyond the twolevelapproximation, 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 quantised matter fields. Such a theory, e.g. the JaynesCummings 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 twolevel 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 lightmatter system, this lecture will show the relationbetween such models and the classical electromagnetism of Maxwells equations.
To reach this destination, we will follow the route of canonical quantisation; 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) = k2A(k)2. (1.19)
Thus, the field part of the Lagrangian becomes:
02
dV
E2 c2B2 = 1
0
d3k1
k2(k)2
+ 0
d3kA(k)2 c2k2A(k)2
. (1.20)
The notation
has been introduced to mean integration over reciprocalhalfspace; 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 reciprocalhalfspace. 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 longitudinal 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 c2k2A(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 nonretarded)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 =
,
qq80r r . (1.26)
Note that since the Coulomb interaction is nonretarded, both the Coulomband transverse parts of interaction must be included to have retarded interactions between separated charges.
1.3 Canonical quantisation; photon modes
We have in Eq. (1.25) a Lagrangian which can now be treated via canonical 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 proceed, 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
+ c2k2A(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 interactions 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 approximation, 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 Hamiltonian H0 obeying H0n = nn one may then rewrite the above in theform:
HAp = n,m
gk,nnm2
nmak,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 = qnrm 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 offdiagonal terms in
the twolevel 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
dnm2e2
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 coupling 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 calculating:
P(t) =
dP()eit = 
mn
dmnmn (1.41)
in linear response theory, with H0 = m
mm
m
and the perturbation
H1 = i
nm
A(t)nmdnmnm, 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 CohenTannoudji et al. [6].
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Lecture 2
The JaynesCummingsmodel
The aim of this lecture is to discuss the interaction between a single twolevel system and various different configurations of photon fields. We startby introducing the JaynesCummings 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 JaynesCummings 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 JaynesCummingsmodel, of a twolevel 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 totwolevel 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
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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 fri. Then, restricting to only the two lowest atomic levels and toa single radiation mode, one has a model of twolevel 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 introduced 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 approximation, 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 corotating 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 JaynesCummingsmodel:
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 twolevel system, we can adaptthe JaynesCummings model by neglecting the dynamics of the photon
field, and replacing the photon creation and annihilation operators withthe amplitude of the timedependent 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 raising/lowering operators for the twolevel system. To solve this problem, andfind the timedependent state of the twolevel 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 probability 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 twolevel system begins in its ground state, then the timedependentstate 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 JaynesCummings model in the rotating wave approximation.
H =
1
2 ga
ga + 0aa. (2.13)Rabi oscillations in the JaynesCummings 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 oscillation 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
 = e2/2 exp(a) 0, = e2/2n
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 = e2n
2nn!
sin2
g
nt
2
=1
2 1
2
e2 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 coefficients2n/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
22 (2.23)
Using this in Eq. (2.21) gives the approximate result:
Pex 12
122
2m
exp
m
2
22 + 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. Concentrating on the peak, near m = 0, there is a fast oscillation frequency g, describing Rabi oscillations. Then, considering whether the terms in the sumadd in phase or out of phase, there are two time scales: collapse of oscillations occurs when there is a phase difference of 2 across the range of termsm < 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 successive 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
22
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) = eigt22
dx exp
1
22
1 +igt
4
x2 + i
2r +
gt
2
x
=
eigt22 exp
2
2
(2r + gt/2

)2
1 + igt/4
dx exp
1
22
1 +igt
4
x i2 2r + gt/21 + 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 frequency 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 JaynesCummings oscillations are discussed by GeaBanacloche [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 twolevel systemcoupled to many photon modes, such that irreversible decay now occurs.We will use the density matrix description of the twolevel 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 difference 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, hybridises 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)
15
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16 LECTURE 3. DECAY OF A TWOLEVEL SYSTEM
3.2 Density matrix equation for relaxation oftwolevel 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 systemreservoir 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 matrix 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 straightforwardly 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 nonzero. Such correlations will be assumed to be small, but must be nonzero 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
TWOLEVEL 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 expression 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 TWOLEVEL 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 absorption (which exists only if n > 0), and leads to excitation of the twolevelsystem, 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 BoseEinstein occupation function for the photons, n = [exp()1]1, the excitation probabilityof the twolevel 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 )
2dab220ckV
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:
() =2dab2
(2)220
d
d sin3
ckcdk
3c3(ck ) = 1
40
4dab2333c3
(3.16)This formula therefore gives the decay rate of an atom in free space, associated 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 TWOLEVEL 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 twolevel 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 decoherence 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 twolevel 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 timeevolution 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 explicitly, 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 density 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 dephasing 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 twolevel 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 twolevelsystem vary, and so offdiagonal matrix elements, involving time dependent coherences, are washed out in . This is particularly an issuein solid state experiments, where the twolevel system energy may involve the dimensions of a fabricated or selfassembled 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 timedependent 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 diagonal 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 systemdependent variation of energies, and should have no time dependence. Thesame behaviour is also relevant if timedependent 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) becomes:
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(1t). Alternatively, 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 inhomogeneous 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 sufficientlynonlinear 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 photonecho 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 twolevel 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 reconverge. 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 reexamination of the question we started an earlier lecture with what is the effect of shining classical light on a twolevel 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 rearranging, 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 nonlinearity implicit in a twolevel system. Such dependence is not so surprising
given that in the Hamiltonian dynamics, the resonance width already depended on the field strength. One may note that if one considered an harmonic atomic spectrum, rather than a twolevel 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 atomphoton detuning for increasing field strength.
This problem, of a single atom coherently pumped, and incoherently decaying 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
Twolevel atom in a cavity:Purcell effect and Cavity
QED
In the previous few lectures we have discussed a single twolevel atom coupled 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 pseudomode (which itself decays),and also coupled to noncavity 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 twolevel 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 pseudomode 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 twolevel system, coupled to onedimensional radiation modes.Following the earlier discussion of systemreservoir 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)
29
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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 solutions (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
kx 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 enhancement 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 twolevel 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 anonnegligible 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 JaynesCummings model, along with relaxation of both the twolevel 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 apseudomode 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 pseudomode overlapswith a range of true eigenmodes, the probability of remaining in the pseudomode 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 pseudomode, 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 dimensions, if the cavity is not spherical, then decay into noncavity 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 generally 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 subsequent 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 PAPBCAB2 = 0is conserved, and thus writing PA = 2, PB = 2, CAB = . Substituting 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 twolevel 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 cavityenhanced 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:
=30dab230c3
,g
2
2 = 0dab220V
(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 enhancement depends both on the quality factor, and also on the mode volume. 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 offresonant reduction is not as relevant as the onresonant 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 discussion below explains the origin and significance of these numbers. In eachcase there are several stages to consider: the nature of the twolevel system, 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 twolevel 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.
37
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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 simplest 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 spinorbit 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 levelsF = 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 twolevel 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 individual 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 enhancement. 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 volume 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=
r2
/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 determined 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