Download Phase transition of Light - Universiteit van Amsterdam

University of Amsterdam
BSc thesis
Phase transition of Light
Author:
Schelto Crone
Supervisor:
dr. Vladimir Gritsev
Report of the Bachelorproject Natuur- en Sterrenkunde 15ec
performed during 31-03-14 till 27-6-2014
July 4, 2014
Supplementary Data
Title: Phase transition of light
Author: Schelto Crone
E-mail: [email protected]
Studentnumber: 10285288
Study: Bachelor Natuur- en Sterrenkunde
Submission date: July 4, 2014
Supervisor: dr. Vladimir Gritsev
Second supervisor: dr. Diego Hofman
Institute for Theoretical Physics
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Universiteit van Amsterdam
Science Park 904, 1098 XH Amsterdam
http://iop.uva.nl/itfa/itfa.html
Abstract
In this project a study is made on Quantum-Optics models which exhibit strongly
correlated photonic behaviour. Due to this behaviour a quantum mechanical
phase transition occurs, which is investigated in this project. First a study is
made on the quantum mechanical description of the light. The quantization of
light is discussed and one of the most important models of Quantum Optics is
investigated, the Jaynes-Cummings model. This model is one of the simplest
ways to describe matter light interaction. Then two variations of the JaynesCummings model are studied which exhibit a quantum phase transition. The
first model is the Dicke-model, which contains a phase transition from a normal phase to a Superradiant phase. The phase transition is analytically derived
and presented. Then the Jaynes-Cummings lattice model is investigated. That
model exhibits a transition from a Mott-insulator phase to a Superfluid phase.
This transition is analytically investigated. The model is also numerically simulated, which also shows the phase transition. The found results are presented
in this report.
Dutch Abstract
De interactiesterkte tussen licht en materie is normaal verwaarloosbaar klein. De
interactiesterkte kan echter worden versterkt door experimentele opstellingen.
Een veelgebruikte opstelling hiervoor is een zogenaamde optische cavity. Een
optische cavity bestaat uit twee spiegels waarin een lichtbundel wordt aangebracht. Door de spiegels ontstaat er een staande lichtgolf tussen de spiegels.
Een staande golf zorgt ervoor dat de golflengte van het licht een meervoud moet
zijn van de lengte tussen de spiegels. Hierdoor zijn maar een beperkt aantal
frequenties mogelijk tussen de spiegels. Als een atoom wordt geplaatst tussen
de spiegels ontstaat er een significante koppeling tussen het atoom het licht.
Deze koppeling geeft zorgt voor interessante kwantumeffecten van het licht.
Figure 1: Een cavity met een atoom bestaande uit twee energieniveau’ s
In dit project is een theoretische studie gedaan naar deze effecten. Eerst is
er een studie gedaan naar de manier waarop licht beschreven dient te worden
in een kwantummechanische context. Daarna zijn enkele theoretische opstellingen van de kwantumoptica onderzocht waarin een zogenoemde faseovergang van
het licht is gevonden. Het licht bevindt zich in deze opstelling, afhankelijk van
enkele parameters, in verschillende toestanden. Het eerste onderzochte model
is het Jaynes-Cummings model. In dit model wordt een atoom geplaatst in
een cavity. Een grafische representatie van dit model kan worden gevonden in
figuur 1. Het atoom heeft maar twee energiebanden die toegestaan zijn, de
grondtoestand en de geëxciteerde toestand. Dit model is een zeer simpel model
van de interactie tussen materie en licht, maar wel exact oplosbaar. Daarom is
dit één van de meest bekende modellen uit de kwantumoptica. Na de JaynesCummings model is de Dicke model onderzocht. Dit model is een variatie op de
Jaynes-Cummings, in plaats van één atoom worden er meerdere atomen in de
cavity geplaatst. Er ontstaat een situatie waardoor het licht sterk met elkaar
gecorreleerd wordt. Hierdoor ontstaat er een faseovergang in dit systeem van
een ’normale’ toestand naar een zogenoemde ‘superradiant’ toestand. De ’superradiant’ toestand het systeem heeft opeens een grondtoestand met excitaties
en licht in zich. Dit komt door de sterke interactie tussen de atomen en het
licht. Deze toestand is een puur kwantummechanisch fenomeen.
Figure 2: Een voorbeeld van een Jaynes-Cummings Lattice model
Het laatste model welke is onderzocht is de zogenaamde ‘Jaynes-Cummings Lattice’ model. Dit model bestaat uit een heleboel gekoppelde Jaynes-Cummings
cavities. Een voorbeeld van dit model kan worden gevonden in figuur 2. In
de normale Jaynes-Cummings model wordt de cavity beschouwd als gesloten.
In het echt zullen de fotonen altijd uit de cavity kunnen tunnelen. Hierop is
de Jaynes-Cummings lattice model gebaseerd. Fotonen kunnen of in hun eigen
cavity bevinden, of tunnelen tussen de cavities. Vanuit de beschrijving van dit
model volgt dat er een faseovergang is tussen een ‘Mott-insulator’ toestand en
een ‘superfluid’ toestand. De Mott-insulator toestand is een toestand waar de
fotonen zich voornamelijk bevinden in hun eigen cavity. De superfluid toestand
is een fase waarin de fotonen niet meer gelocaliseerd zijn in hun respectievelijke
cavity, maar zich overal kunnen bevinden in het systeem. De overgang tussen
deze twee fase is zeer nauw, dus er is alweer een faseovergang.
3
Contents
1 Introduction
5
2 Formalism of Quantum optics
2.1 Quantization of the Electromagnetic field . . . . . . . . . . . . .
2.2 Uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Coherent state representation . . . . . . . . . . . . . . . . . . . .
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9
11
3 The
3.1
3.2
3.3
3.4
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16
18
21
Jaynes-Cummings Model
The derivation of the atom-field interaction . . . . . . .
The Rotating Wave approximation and JC hamiltionian
Solving the Jaynes-Cummings model . . . . . . . . . . .
Dynamics of JC model . . . . . . . . . . . . . . . . . . .
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4 Dicke model
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4.1 The model and its eigenstates . . . . . . . . . . . . . . . . . . . . 25
4.2 Phase transitions in the Dicke model . . . . . . . . . . . . . . . . 26
5 Jaynes-Cummings lattice model
5.1 Jaynes-Cummings Lattice Hamiltonian . . . .
5.2 Limits of the Jaynes-Cummings lattice model
5.3 Mean-field approximation . . . . . . . . . . .
5.4 Analytical Phase Boundary . . . . . . . . . .
5.5 Numerical treatment . . . . . . . . . . . . . .
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35
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6 Discussion
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7 Conclusion
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A The interaction term and the dipole approximation
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B Laplace Method
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4
1
Introduction
The interaction of matter with light is normally a weak interaction. The interaction can however be made significant by using experimental setups. A
frequently used setup to enhance the interaction is placing an atom in a cavity.
In the cavity the lightwave has a standing wave solution, therefore only a single photon mode is allowed. The setup in the cavity enhances the interaction
between the electromagnetic field and the atom. When the atom-field interaction becomes significant, strong correlated effects can occur. In this project a
theoretical study is performed to the strong correlated behaviour of these systems. The focus lies on the research of the phase transitions that can occur in
these systems. In the report two models are discussed in which a phase transition occurs due to the interaction of light with matter. To fully understand
these transitions one first need to understand the formalism of Quantum optics.
Therefore the report starts with a general introduction in the used formalism of
Quantum optics. An introduction is given to the quantization of the electromagnetic field in section 2. Then one of the most important models of theoretical
Quantum optics is investigated, the Jaynes-Cummings model. The model will
be discussed in section 3. The model describes a single two-level atom placed
in a cavity. This model is a good way to describe the interaction of matter with
a electromagnetic field. After this two systems are investigated which exhibit a
phase transition due to the strong interaction between matter and light. The
first studied model is the Dicke model, which is discussed in section 4. The
model is a variation of the Jaynes-Cummings model from a single atom to a
large amount of atoms. The model contains a phase transition between a normal phase and a Superradiant phase, which is derived and discussed. Then in
section 5 a more recently proposed model is discussed, the Jaynes-Cummings
Lattice model. This model describes an array of Jaynes-Cummings cavities.
The setup is described in a similar way as the already known Bose-Hubbard
model. The original Bose-Hubbard model contains a phase transition from a
Mott-Insulator phase to a Superfluid phase. The same phase transition can
be found in the Jaynes-Cummings Lattice model. The phase-transition is analytically investigated and numerically simulated. The results of the study are
presented in this report.
5
2
2.1
Formalism of Quantum optics
Quantization of the Electromagnetic field
The quantum description of light requires the electromagnetic field to be quantized. The derivation follows closely the derivation found in [1] and [2]. To
perform this quantization first look at the Maxwell equations for an electromagnetic field in vacuum. These are given in the known form as
∇·E=0
(1)
∇·B=0
(2)
∂B
∇×E=−
∂t
1 ∂E
∇×B= 2
c ∂t
(3)
(4)
With these equations a electromagnetic vector potential A and a normal electric
potential φ can be associated. The fields then obey the equations
B=∇×A
∂A
E=−
− ∇φ
∂t
(5)
(6)
In the definition of the electromagnetic vector potential A a degree of freedom
remains. Due to this degree of freedom a gauge transformation is allowed. For
the following derivations the Coulomb gauge is used, which is given as
∇·A=0
(7)
Combining this with equation 4 and using the fact that for free space ∇φ = 0
gives:
1 ∂
∂A
(−
− ∇φ)
2
c ∂t
∂t
1 ∂2A
∇(∇ · A) − ∇2 A = 2 2
c ∂t
1 ∂2A
∇2 A = 2 2
c ∂t
∇ × (∇ × A) =
(8)
So in a vacuum the electromagnetic vector potential obeys a wave equation. Now
consider a cubic region space of length L. This region of space can be considered
as a box with arbitrary boundaries. For this box the vector potential can be
written as a Fourier expansion of the modes of the box. This gives
X X
(9)
A(r, t) =
ekλ Ãkλ (t)eik·r + e∗kλ Ã∗kλ (t)e−ik·r
k λ=1,2
Here λ is the polarisation of the wave, k is the wave-number and ekλ is the
polarisation vector. The Ãkλ (t) is the amplitude for each Fourier mode. The
6
wave-number k obeys the standard quantization condition
ki = 2πni /L
(10)
With ni ∈ Z. This is a summation over all possible modes of A and all possible
polarisation directions λ. But the vector potential also obeys the wave-equation
8. So for a single mode this gives:
k 2 Ãkλ (t) = −
1 ∂ 2 Ãkλ (t)
c2
∂t2
(11)
Or using ω = ck
∂ 2 Ãkλ (t)
∂t2
The solution for this equation is given by an exponential
ωk2 Ãkλ (t) = −
Ãkλ (t) = Akλ e−iωk t
Using this for equation 9 the vector potential can be written as:
X X
A(r, t) =
ekλ Akλ e−iωk t+ik·r + ekλ A∗kλ eiωk t−ik·r
(12)
(13)
(14)
k λ=1,2
From this expression the corresponding electric and magnetic field can be found
using equation 5 and 6. This gives
X X
E=i
ωk (−ekλ Akλ e−iωk t+ik·r + ekλ A∗kλ eiωk t−ik·r )
(15)
k λ=1,2
Analogue one can find the magnetic field as
B=i
X X k × ekλ
(−ekλ Akλ e−iωk t+ik·r + A∗kλ eiωk t−ik·r )
k
(16)
k λ=1,2
Now the general expressions for the free electric and magnetic field in terms
of the electromagnetic vector potential have been found. The more interesting
thing to do is to look at the energy stored inside the field. Classically this is
given by
Z
1
E=
(0 E2 + µ0 B2 )dV
(17)
2
Substituting the according electric and magnetic field gives the expression for
the energy. To solve the integral the following identities need to be used
Z
0
dV e±i(k−k )·r = V δk,k0
Z
0
dV e±i(k+k )·r = V δk,−k0
(18)
7
This gives for the energy the expression
Z
0
1X X
E=
dV (0 ωk ωk0 (−Akλ Ak0 λ0 ekλ ek0 λ0 ei((k+k )·r−2ωk ωk0 t) )+
2
0
k λ,λ =1,2
0
(−A∗kλ A∗k0 λ0 ekλ ek0 λ0 ei(−(k+k )·r+2ωk ωk0 t) )+
0
0
(A∗kλ Ak0 λ0 ekλ ek0 λ0 ei((k−k )·r ) + (Akλ A∗k0 λ0 ekλ ek0 λ0 ei((k−k )·r )) + B2 (19)
The magnetic part is very analogue to the electric part. Using the above mentioned identities one can write
1X X
V ((Akλ A∗kλ0 + A∗kλ Akλ0 )
E=
2
0
k λ,λ =1,2
1
k × ekλ · k × ekλ0 )
µ0
− (Akλ A−kλ0 e−2iωk t + A∗kλ A∗−kλ0 e2iωk t )
1
× (0 ωk2 ekλ · e−kλ0 −
k × ekλ · k × e−kλ0 )
µ0
× (0 ωk2 ekλ · ekλ0 +
(20)
This expression can be simplified by using the following identities
(k × ekλ ) · (k × e±kλ0 ) = k 2 ekλ · e±kλ0
ekλ · ekλ0 = δλ,λ0
(21)
(22)
Rewriting this with some algebra gives the final expression for the energy as
XX
E=
0 V ωk2 (Akλ A∗kλ + A∗kλ Akλ )
(23)
k
λ
This gives a general expression for the energy of the classic electromagnetic field.
i
Due to the quantization-condition ki = 2πn
L the energy contains a summation of
quantized wave-modes. To quantize the electromagnetic field each mode of the
electromagnetic field is associated by a simple quantum harmonic oscillator. So
for each mode - analogue to the quantum harmonic oscillator - the formalism of
ladder operators and number states can be applied. The electromagnetic field is
then a summation of these number state for different wave-numbers. When the
length of the box goes to infinity(L → ∞) the quantization condition does not
vanish. Therefore the electromagnetic field is now quantized. The annihilation
and a destruction operator for a specific wave-number k and polarisation λ is
given as
√
âkλ |nikλ = n |n − 1ikλ
(24)
√
â†kλ |nikλ = n + 1 |n + 1ikλ
(25)
The |nkλ i is the state which represents the number of photons in an specific
mode. For the electromagnetic field these state are commonly known as the
8
Fock state representation. The states are - similar to the quantum harmonic
oscillator - orthogonal to each other. The annihilation and destruction operators
obey the commutation relation written as:
[âkλ , â†kλ ] = δk,k0 δλ,λ0
(26)
The Hamiltonian for a single harmonic oscillator is given by the expression
1
1
(27)
H = ~ω(n̂ + ) = ~ω(↠â + )
2
2
1
(28)
= ~ω(↠â + (â↠− ↠â)
2
1
= ~ω(↠â + â↠)
(29)
2
Here we use the commutation relationship given by equation 26. This also
shows two important properties of the Fock-state representation. The Fockstate representation is simultaneous a eigenstate of the number operator and
the Hamiltonian of the system. This can easily be seen as
√
(30)
n̂ |ni = ↠â |ni = ↠n |n − 1i = n |ni
1
1
H |ni = ~ω(n̂ + ) |ni = ~ω(n + ) |ni
(31)
2
2
So the Fock-state representation gives us a known number of photons(excitation
of the electromagnetic field) and energy. If the electromagnetic field is taken as
a sum of single harmonic oscillators the expression for the Hamiltonian becomes
XX 1
H=
~ωk (âkλ â†kλ + â†kλ âkλ )
(32)
2
k
λ
So the quantized electromagnetic field is given by a sum of harmonic oscillators.
When considering the expression for the classic field (eq. 23) it can be seen that
the quantization rules for the electromagnetic field are:
~
)1/2 âkλ
(33)
20 V ωk
~
A∗kλ → (
)1/2 â†kλ
(34)
20 V ωk
The constant in front of the operator are chosen so that the Hamiltonian really
becomes a summation over just the harmonic oscillators.
Akλ → (
2.2
Uncertainty principle
The Heisenberg uncertainty principle can be generalised for photons. The position and momentum operator in terms of ladder operator can be written as
r
~
(a + a† )
(35)
x̂ =
2mω
r
~mω
p̂ = i
(a − a† )
(36)
2
9
Analogue to these operator a dimensionless position and momentum operator
can be defined as
1
(37)
X1 = (a + a† )
2
1
X2 = (a − a† )
(38)
2i
These operators are called the quadrature operators. With the known commutation relation for a and a† we can write
[X1 , X2 ] = X1 X2 − X2 X1
1
1
= (a + a† )(a − a† ) − (a − a† )(a + a† )
4i
4i
1
= ((a2 − (a† )2 − aa† + a† a − a2 + (a† )2 + a† a − aa† )
4i
i
i
= (aa† − a† a) =
(39)
2
2
Heisenberg uncertainty relationship for two hermitian operators is given as
1
|<C>|
2
So using this with the operators given by equation 37 and 38 gives
[A, B] = iC → ∆A∆B ≥
(40)
1
i
1
|< >|=
(41)
2
2
4
So there is an uncertainty relationship for the operators X1 , X2 . When considering a single mode of the electromagnetic field this can be written as
∆X1 ∆X2 ≥
E = Ce(ae−iωt + a† eiωt )
(42)
Which can be rewritten in the term of operators X1 , X2 to give
E = Ce(a(cos (ωt) − i sin (ωt)) + a† (cos (ωt) + i sin (ωt))
= Ce((a + a† ) cos (ωt) + i(a − a† ) sin (ωt))
= 2Ce(X1 cos (ωt) + X2 sin (ωt))
(43)
Also the Hamiltonian can be expressed as
H = ω(X12 + X22 )
(44)
This gives for the Fock state
1
H |ni = ω(X12 + X22 ) |ni = (n + ) |ni
(45)
2
Here it follows
q that the operators X1 , X2 only have values allowed on a circle
with radius (n + 12 ). The variance of X1 , X2 is given by
(∆X1 )2 = (∆X2 )2 =
10
1
1
(n + )
2
2
(46)
So our uncertainty relationship gives
1
1
1
(n + ) ≥
2
2
4
(47)
Here follows from that only the |0i state of the Fock state representation fulfills the minimum uncertainty. The more numbers are present the higher the
uncertainty becomes. Therefore for large n the Fock-state are not the ideal
representation of the Electromagnetic field. A solution for this is the Coherent
state representation[3] [1] [2].
2.3
Coherent state representation
A problem of the Fock-state representation is that it exhibits no phase information. The state has an exact known number representation, but due to then
the phase is fully unknown. This can also be seen by looking at the value
hn| E |ni = C(hn| (a + a† |ni) = 0
(48)
Here only one mode of the electric field is considered and the orthogonality of
the number state is used(< n|n + 1 >= 0). The Fock-state is not the only used
basis for quantization of the electromagnetic field. Another frequently used basis
of the radiation field is the coherent state representation. The coherent states
is a superposition of different number-state of photons, and therefore have a
better defined phase than the Fock states. The coherent states are generated
by applying the Glauber displacement operator on the zero-field state|0i. The
displacement operator is given by
D̂(α) = exp (α↠− α∗ â)
(49)
where α is a complex number. The operator is unitary since
D̂† (α) = exp (−α↠+ α∗ â) = D̂(−α)
(50)
The displacement operator can be rewritten using the operator theorem
e(Â+B̂) = e eB̂ e−[Â,B̂]/2
(51)
The thorough derivation of this identity can be found in [4]. Using this theory
on the displacement operator gives
2
D̂(α) = e−|α|
/2 αa† −α∗ a
11
e
e
(52)
This gives a definition of the coherent state|αi as
|αi = D̂(α) |0i
= exp(−α↠+ α∗ â) |0i
= exp (−|α|2 /2) exp (αa† ) exp (−α∗ a) |0i
= exp (−|α|2 /2) exp (αa† ) |0i
∞
X
αn
√ a† |0i
= exp (−|α|2 /2)
n!
n=0
= exp (−|α|2 /2)
∞
X
αn
√ |ni
n!
n=0
(53)
P∞ 1 n
 ) and that a |0i = 0. The new
Here we have used the fact that e = n=0 ( n!
found coherent state α is already normalised, because:
< α|α >= h0| D−α Dα |0i =< 0|0 >= 1
(54)
The importance of the coherent state representation is that |αi is the eigenstate
of the annihilation operator a. This can easily be seen by writing
a |αi = exp (−|α|2 /2)
= exp (−|α|2 /2)
= exp (−|α|2 /2)
∞
X
αn
√ a |ni
n!
n=0
∞
X
αn √
√
n |n − 1i
n!
n=0
∞
X
0
αn +1
√
a |n0 i
0 + 1!
n
n0 =0
0
∞
X
αn
√ a |n0 i = α |αi
= α exp (−|α| /2)
n0 !
0
n =0
2
(55)
With the use of the commutation relationship one can see that the coherent
state |αi is not the eigenstate of the creation operator a† . If one would follow
the same procedure as used by equation 55 one would find that the coherent
state is a so called left eigenstate of a† . It follows that
hα| a† = hα| α∗
(56)
When one would calculate the photon number probability distribution P(n) for
a coherent state |αi one would find that
2
P (n) = | hn| |αi |2 = e−|α| |α|2n /n!
(57)
This is exactly a Poisson distribution with an expected photon number < n̂ >=
|α|2 . As it turns out the coherent state presentation of the electromagnetic field
12
is the representation with the lowest possible uncertainty e.q. ∆X1 ∆X2 = 14 .
This can be seen by calculating
(∆X1 )2 =< X12 > − < X1 >2
1
1
= hα| ((a† )2 + a2 + a† a + aa† |αi − (hα| (a† + a) |αi)2
4
2
1 2
1
∗ 2
∗ 2
= (α + (α ) + 1 + 2αα∗) − ( (α + α ))
4
2
1
1
∗ 2
∗ 2
= (1 + (α + α ) − (α + α ) ) =
4
4
(58)
Here the eigenvalues equation of the coherent state is used( a |αi = α |αi and
hα| a† = hα| α∗ ). The same procedure can be followed to calculate (∆X2 )2 = 41 .
Adding these equations together one finds that
(∆X1 )2 (∆X2 )2 =
1
1
→ ∆X1 ∆X2 =
16
4
(59)
So instead of the Fock-state representation the uncertainty does not scale with
then number of photons. Therefore the state better resemblance a classical light
state then the Fock-state representation. A representation between the Fockstate and the Coherent state representation can also be made. This is called
the squeezed state representation. In this representation the variance in X1 , X2
differ, but they still obey the uncertainty relationship. All these states are
different representation of the electromagnetic field. This representations can
be used for the most important model in Quantum optics, the Jaynes-Cummings
model [3] [1] [2].
13
3
3.1
The Jaynes-Cummings Model
The derivation of the atom-field interaction
The Jaynes-Cummings model is one of the most used model in Quantum Optics.
It describes the interaction between a single two state atom and a light field.
The light field in a cavity. Due to the cavity only a single mode of the field
needs to be considered for this model. The reason why this model is so widely
used is that it is exactly solvable. The Hamiltonian of this model contains an
atomic energy term, a electromagnetic field term and an interaction term for
the interaction between the electromagnetic field and the atom. This gives the
following Hamiltonian:
H = Hatom + Hf ield + Hint
(60)
In this model the assumption is that the atom has only two states, an excited
state |ei and a ground state |gi. This turns out to be a reasonable assumption
when dealing with photons with an ω close to the transition energy. Note that
the hat - to indicate an operator - is dropped from here for convenience. Also
from here ~ = 1. The Hf ield part is then given in general by
Hf ield =
X
k
1
ωk (a†k ak + )
2
(61)
Because we now place the atom inside a cavity only one photon mode is allowed.
Therefore the Hf ield simplifies and becomes
1
Hf ield = ω(a† a + )
2
(62)
Here the ω stands for the frequency of the allowed mode. The atomic part of the
Hamiltonian and the interaction part can be found by applying a minimal substitution. The minimal substitution is often a good way to describe a coupling
between a field and an atom. It is done by writing
p2 → (P − qA)2
(63)
here q is the charge of the atom, and A is again the electromagnetic field potential. Using this on the atomic and interaction part gives
Hminimalsubstitution =
1
[p − qA(r, t)]2 + V (x)
2m
(64)
Expanding this gives
Hminimalsubstitution =
1 2
[p − qp · A(r, t) − qA(r, t) · p + q 2 A2 (r, t)] + U (x)
2m
(65)
= Hatom + Hint
(66)
14
Here Hatom is the Hamiltonian of the atom without interaction and Hint is the
interaction Hamiltonian. The A2 term is often neglected, because the contribution scales with q 2 , so it is very small. Due to the coulomb gauge(∇ · A = 0) A
and p commute. The atom is approximated as a two-state atom. Therefore the
Hamiltonian can be written in terms of its eigenstates and vectors. This gives
Hatom =
(|ei he| − |gi hg|)
2
(67)
Where is the transition energy of the atom. The zero energy is placed between
the |gi and |ei state. Therefore the possible eigenstates are Eg = −/2 and Ee =
/2. Because this model is isomorphic to a spin- 12 system it is possible to define
the known spin operators σ + = |ei hg|, σ − = |gi he| and σz = |ei he| − |gi hg|.
Substituting these in the atomic Hamiltonian gives
σz
2
(68)
q
p · A(r, t)
m
(69)
Hatom =
The interaction term is found as
Hint = −
For the cavity we apply the dipole approximation. This says that we can assume
the field A to be slow changing and therefore approximate it as an dipole in the
cavity. Using this the interaction term becomes
Hint = −
q
p · A(r, t) ≈ −qr · E = −d · E
m
(70)
Here the part d = qr stands for the dipole term of the atom and E is the
electric field in the cavity. A more thorough justification of this approximation
can be found in appendix A. The derivation in the appendix follows closely the
derivation found in [3]. When looking at equation 15 it is clear that the electric
field is a superposition of annihilation and creation operators. Therefore the
electric field can be written as:
E = |E|a† + a)
(71)
Here |E| is some normalization constant. The normalization constant can be
found by looking at the expressions of the electric energy. This is one the one
hand still given as
1
Eem = ω(n + )
(72)
2
With n the number of excitations of the electromagnetic field. The electric
energy can also be calculated using the identity
Z
Eem = hn| |E|2 d3 r |ni
(73)
15
Using this and equation 71
2
Z
Eem = |E|
= |E|2
2
= |E|
Z
Z
d3 r hn| (a† a + aa† ) |ni
d3 r hn| (2n̂ + 1) |ni
2d3 r(2n + 1)
(74)
(75)
When defining V =
R
d3 r it reduces to
1
0 |E|2 V (2n + 1) = ω(n + )
2
(76)
This gives
r
|E| =
ω
20 V
(77)
Now an expression for the dipole field needs to be found. Due to parity considerations the dipole moment can be expressed in terms of the off-diagonal atomic
transitions. This gives
d = d(|gi he| + |ei hg|) = d(σ − + σ + )
(78)
Combining the found expressions for the electric field and the dipole moment
the interaction term becomes
r
ω
Hint = d
(a + a† )(σ + + σ − )
20 V
= g(a + a† )(σ + + σ − )
(79)
q
Where g is the atom-field coupling constant defined as g = d 2ω0 V . We have
now found the full atomic interaction Hamiltonian
H = Hatom + Hf ield + Hint
1
= σz + ω(a† a + ) + g(a + a† )(σ + + σ − )
2
2
(80)
This Hamiltonian is called the Rabi-Hamiltonian. To finally get to the JaynesCumming Hamiltonian one last approximation needs to be made.
3.2
The Rotating Wave approximation and JC hamiltionian
The Hamiltonian of equation 80 is the complete Hamiltonian to describe the
interaction of a two-state atom with a electromagnetic field. The JaynesCummings Hamiltonian however yields the Rotating Wave approximation. In
16
this approximation the fast rotating terms of the interaction term are neglected.
This can be achieved be looking at the time dependence of the Hamiltonian in
the interaction picture. First we rewrite equation 80 as
1
H = H0 + Hint = ( σz + ω(a† a + )) + g(a + a† )(σ + + σ − ))
2
2
(81)
We look at the time dependence of this Hamiltonian in the interaction picture.
Our time-evolution operator is given by
Ut = e
iH0 t
For the derivation this identity is used
X
f (λn ) |λn i hλn |
f (Â) =
(82)
(83)
n
Here  is a hermitian operator with eigenstates |λn i and eigenvalues λn . Using
this identity the unitary operator can be written as (using a† a = n̂)
X
†
ei(a a+1/2)ωt =
ei(n+1)ωt |ni hn|
(84)
n
This gives:
Ut aUt† =
X
ei(l+1/2)ωt |li hl|
n
l
×
X
(n + 1/2)1/2 |ni hn + 1|
X
e
−i(m+1/2)ωt
|mi hm|
m
=
XXX
=
X
l
m
e
ei(l−m)ωt (n + 1/2) |ni hn + 1| δln δm,n+1
n
−iωt
(n + 1)1/2 |ni hn + 1| = ae−iωt
(85)
n
Here the fact that the ei 2 σz part of H0 commutes with the annihilation and
creation operators is used. Analogue for the creation operator
U a† U † = a† eiωt
(86)
For the atomic part the identity of equation 83 can again be used. This time
the atomic part of the Hamiltonian commutes with the a and a† . Therefore we
can write
0
0 0
0
exp (−it 2 − )
(87)
U σ + U † = exp (it 2 − )
0 2
1 0
0 2
Using equation 83 we get
U σ + U † = σ + eiωt
−
†
− −iωt
Uσ U = σ e
17
(88)
(89)
Adding these equations together gives an interaction Hamiltonian in the interaction picture of:
Hint,I (t) = Ut Hint,S Ut†
(90)
= Ut g(σ + a + σ − a +
+ † i(ω+)t
= g(σ a e
+
+ σ ae
σ + a† + σ − a† )Ut†
− −i(ω+)t
+ σ ae
−i(ω−)t
− † i(ω−)t
+σ a e
)
(91)
(92)
(93)
If frequency of the field is assumed to be close to the transition frequency (ω ∼ ),
two rapidly oscillating terms are found. The σ + a† and σ − a terms are multiplied
by a rapid oscillating exponent. Due to the rapid oscillation these terms can be
approximated on a long time range as constant. Therefore these terms will not
contribute to the dynamics of the system. The Rotating wave approximation
(RWA) says to neglect these terms because they do not contribute much to the
dynamics of the system. There is also a physical argument to neglect these
terms. These terms represent either a photon absorbed and the atom falling to
its ground state(σ − a) or a photon emitted and the atom excited to its excited
state(σ + a† ). Although these are possible transitions, conservation of energy
tells us that these are unlikely phenomenon. The system needs to get energy of
the vacuum to allow this transitions. Although this is physically possible, the
likelihood is much smaller then the other terms. Therefore it is reasonable to
neglect these contributions to the Hamiltonian. Applying this approximation
to the Hamiltonian gives us the complete Jaynes-Cummings model. The complete Hamiltonian given under the rotating wave approximation and the dipole
approximation is then
H = H0 + Hint =
3.3
1
σz + ω(a† a + ) + g(aσ + + a† σ − )
2
2
(94)
Solving the Jaynes-Cummings model
A reason that the Jaynes-Cummings model is so important is that it is exactly
solvable. In this section the solution of the Jaynes-Cummings model will be
presented. The Hamiltonian can be written in the form of
H0 = ~ω(
σz
~∆
+ a† a +
σz )
2
2
(95)
Where ∆ = − ω is the detuning parameter. Also the factor 12 of the vacuum
energy is neglected because it does not contribute to the dynamics of the system
and in this section ~ 6= 1. The atom can be in found in either the |e, ni state,
where the atom is excited and there are n photons present, or the |g, n + 1i state,
where the atom is in the ground state and there are n + 1 photons. These states
are the only states the interaction term couples. Therefore to solve the JaynesCummings model only the specific subspace of {|e, ni , |g, n + 1i} needs to be
considered. A conserved factor n can be defined as n = a† a+ 21 σz . This quantity
is conserved because the number of photons and atomic excitations is conserved
18
in the system. This can be seen because in the commutator relationship [H, n] =
0. Because n is conserved we can subtract this quantity from the Hamiltonian.
Using this the Hamiltonian can be rewritten as:
H̃0 = H0 − ~ωn =
~∆
σz
2
(96)
The full Hamiltonian of the system then becomes
H̃ =
~∆
σz + ~g(aσ + + a† σ − )
2
(97)
The interaction term only couples the subspace {|e, ni , |g, n + 1i}n . Therefore
only this subspace needs to be considered for solving the Hamiltonian. The
Hamiltonian can then be written in matrix form as
he, n| H̃ |e, ni
he, n| H̃ |g, n + 1i
H̃ =
(98)
hg, n + 1| H̃ |e, ni hg, n + 1| H̃ |g, n + 1i
he, n| H̃0 |e, ni
he, n| Hint |g, n + 1i
=
(99)
hg, n + 1| Hint |e, ni hg, n + 1| H̃0 |g, n + 1i
The H̃int term only works on the off-diagonal elements, while the H̃0 term
applies only on the diagonal terms. Applying the operators on the states gives
for the Hamiltonian
√
+∆/2
g n+1
√
(100)
H̃ = ~
g n+1
−∆/2
~ ∆ Ωn
=
(101)
2 Ωn −∆
√
Where Ωn is the Rabi frequency defined as Ωn := 2g n + 1. The energy eigenstates can be found by finding the eigenvalues of matrix 101. This gives
Ẽ± = ±~
Where Rn :=
energy
p
Rn
2
(102)
Ω2n + ∆2 . Adding the conserved number term gives for the full
Rn
)
(103)
2
And using the same procedure the eigenstates can be found by solving for the
eigenvector of matrix 101. This gives for the eigenstates
E± = ~((n + 1)ω ±
θn
θn
) |e, ni + cos ( ) |g, n + 1i
2
2
θn
θn
|n−i = cos ( ) |e, ni − sin ( ) |g, n + 1i
2
2
|n+i = sin (
19
(104)
(105)
Where
sin θn = p
cos θn = p
Ωn
(Rn − ∆)2 + Ω2n )
Rn − ∆
(Rn − ∆)2 + Ω2n )
(106)
(107)
Or more compact
tan (2θn ) =
Ωn
∆
(108)
The states of equations 104 and 105 are called dressed states of the JaynesCummings Hamiltonian. The states are an entanglement of the two basis states
|e, ni and |g, n + 1i with a simple rotation θn . The state |e, ni and |g, n + 1i are
the eigenstates of the Hamiltonian without the interaction term. The interaction
term couples the two possible states, and the system becomes entangled. The
system is maximally entangled when sin θn = cos θn . This is when θn = π4 .
From equation 108 it follows that this is the case when ∆ → 0. So when
the system is on resonance( = ω) the entanglement is maximum. This is
physically reasonable, because when the system is on resonance the interaction
term allows the greatest amount of mixing of the states. When the system is
not on resonance (|∆| 0) the states behave like the non-interaction states
|e, ni and |g, n + 1i. This can clearly be seen in figure 3. This figure shows a
graph where the energy of the states is plotted versus the detuning parameter
∆. The dotted lines represent the energy terms for the Hamiltonian without
interaction terms. It is clear that when the detuning is almost zero, the system
has a great coupling. But when the detuning is large, the energy behaves like
there is no coupling. Only when the detuning parameter is close to zero does
the interaction term contribute significantly.
20
Figure 3: A Plot of the Dressed states versus the detuning parameter [5]
3.4
Dynamics of JC model
In this section the time evolution of the Hamiltonian given by equation 95 will
be derived. This gives great insight in the dynamics of the Jaynes-Cummings
model. In the Schrödinger picture the time evolution of any state can be written
- when the Hamiltonian does not explicitly depend on time - as
i
|Ψ(t)i = e− ~ Ht |Ψ(0)i
(109)
Here |Ψ(0)i is the state of the system at t = 0. |Ψ(0)i can be expanded in the
basis of the eigenstates of the Hamiltonian. This gives
|Ψ(0)i =
∞
X
(|n+i hn + |Ψ(0)i + |n−i hn − |Ψ(0)i)
(110)
n=0
Now using equation 109 and defining cn± (0) = hn ± |Ψ(0)i the wave-function
can be written as
|Ψ(t)i =
=
∞
X
i
i
(e− ~ Ht cn+ (0) |n+i + e− ~ Ht cn− (0) |n−i)
n=0
∞
X
i
i
(e− ~ E+ t cn+ (0) |n+i + e− ~ E− t cn− (0) |n−i)
(111)
(112)
n=0
Where E± are the eigenvalues of the Hamiltonian given by equation 103. This is
allowed because the Hamiltonian does not explicitly depend on time. Plugging
21
in the expression for the energy then gives
|Ψ(t)i =
∞
X
e−i(n+1)ωt (e−i
Rn
2
t
cn+ (0) |n+i + ei
Rn
2
t
cn− (0) |n−i)
(113)
n=0
When changing again to the interacting picture one can remove the ei(n+1)ωt
term. In the interaction picture this state can be written as
|Ψ(t)iint = ei(n+1)ωt |Ψ(t)ischrd
(114)
Looking at a given n the expression for the state becomes
|Ψ(t)iint,n = cn+ (0)e−i
Rn
2
t
|n+i + cn− (0)ei
Rn
2
t
|n−i
(115)
We can define the coefficients for the kets as time coefficients so that cn± (t) =
Rn
cn± (0)e∓i 2 t . Writing this in matrix form gives
!
Rn
cn− (0)
cn− (t)
ei 2 t
0
(116)
=
Rn
cn+ (0)
cn+ (t)
0
e−i 2 t
For the dynamics it would be nice to look at the coefficients in the basis of
[|e, ni , |g, n + 1i]. The unitary transformation given by equations 104 and 105
can be used to transform the basis. This gives in matrix form
cos θn − sin θn
U=
(117)
sin θn
cos θn
Using this on the matrix given by equation 116 we can write
!
Rn
0
ce,n (0)
ce,n (t)
ei 2 t
†
U
(118)
=U
Rn
cg,n+1 (0)
cg,n+1 (t)
0
e−i 2 t
!
Rn
Ωn
sin
(
cos ( R2n t) − i Rδn sin ( R2n t)
−i R
t)
ce,n (0)
2
n
=
Ωn
cg,n+1 (0)
sin ( R2n t)
−i R
cos ( R2n t) + i Rδn sin ( R2n t)
n
(119)
This gives a description of the time evolution of the complex coefficients of the
begin state. Two interesting cases can be found. The first case is when the
cavity is on resonant frequency. When the system is on resonance(∆ = 0) the
parameters Rn and
Ωn are the
same.
If the system starts in the initial state
ce,n (0)
1
|Ψ(0)n in = |e, ni (
=
) the complex coefficients become
cg,n+1 (0)
0
Ωn
t)
2
Ωn
cg,n+1 (t) = −i sin (
t)
2
ce,n (t) = cos (
22
(120)
(121)
This gives the dynamics of the state as
|Ψ(t)in = cos (
Ωn
Ωn
t) |e, ni − i sin (
t) |g, n + 1i
2
2
(122)
It is more interesting to look at the probabilities to find the system in a certain
state. These probabilities can be derived as
1
Ωn
t) = (1 + cos (Ωn t))
2
2
Ωn
1
= sin2 (
t) = (1 − cos (Ωn t))
2
2
Pe,n = cos2 (
Pg,n+1
(123)
(124)
√
This clearly shows the system oscillating with frequency Ωn = 2g n + 1. These
process is known as Rabi-oscillations or Rabi flopping. The changes oscillate
opposite of each other with a frequency Ωn . Another interesting setup is when
the field is prepared in a coherent state instead of a Fock state (Still on resonance
condition ∆ = 0). Suppose the atom - starting out in state |ei - interacts with
a coherent field |αi. The average number of photons is then given by n̄ = |α|2 .
The chance to find the atom in the excited state is now given by(using equation
123)
Pe (t) =
∞
X
n=0
Pn Pe,n (t) =
∞
∞
X
X
Pn
1
(1 + cos (Ωn t)) = (1 +
Pn cos (Ωn t)) (125)
2
2
n=0
n=0
The field is prepared for a coherent state |αi. The probability for a coherent
state is given by
2n
2 |α|
Pn = e−|α|
(126)
n!
Combining all the expressions the probability to find the system in the excited
state is given as
Pe (t) =
∞
2n
2 X |α|
1
(1 + e−|α|
cos (Ωn t))
2
n!
n=0
(127)
These function does not look very special, but when plotting this function in a
graph the function shows some interesting behaviour. The chance to find the
atom in the excited state first collapses. After a certain time however the system
has a revival and the chance to find the atom in the excited state is nonzero.
23
Figure 4: Collapse and revival for |α| = 5
A graph of function 127 is given in figure 4 for |α| = 5. When looking at
equation 127 it becomes clear that the dynamics is only dependent on cos (Ωn t).
The cosine is at its maximum when Ωn t is a multiple of 2π. This gives us the
condition that the revival time
Ωn tr = 2πj → tr =
2π|α|
g
(128)
The revival is strong the first time. The further the time progresses the collapse
and revival effects become more blurred, but are still present. This effect is
a pure quantum-mechanical effect due to the quantization of light. Therefore
there is no classic analogue to this effect. This shows some interesting features
fabricated by the Jaynes-Cummings model.
24
4
Dicke model
Another model used in quantum optics is the Dicke model. This model is an
extension of the Jaynes-Cummings model. In this model the cavity is composed
of a large amount of atoms instead of the single atom found in the JaynesCummings model. Due to the large amount the system shows collective behaviour of the photons interacting with the atoms. One of the most prominent
collective behaviour is a phase transition of the model. The phase transition
is a transition from a normal phase to a Superradiance phase. In the normal
phase the ground-state of the system contains no excitations. This means that
the expected number of photons and atomic excitations is zero. This means
quantitative
< a† a > = 0
< Jz > = −
(129)
N
2
(130)
In the Superradiant phase however these quantities increased dependending on
the coupling strength. Therefore the ground-state of the system does contains
radiation, therefore the name Superradiant phase. The transition between the
normal phase and the Superradiant phase is instant. The phase transition is
derived in part 4.2. But first discuss the model in a little more detail( with
again ~ = 1).
4.1
The model and its eigenstates
The Dicke model has an Hamiltonian similar to the Jaynes-Cummings Hamiltonian, but instead of a single atom in is expanded to N atoms. It is written
as
N
X
( σiz + g(a† σi− + aσi+ ))
HDicke = a† aω +
(131)
2
i=1
Where σiz,± stands for the respective spin operator for the respective atom.
There is a summation over all N atoms, this is frequently simplified written as
H = a† aω + Jz + g(a† J− + aJ+ )
(132)
Where
Jz,± =
N
X
σiz,±
(133)
i=1
The eigenstates is a combination of all the single parts of individual Hilbert
space of single dressed states. This is written as
|Ψi = |±i1 ⊗ |±i2 ⊗ |±i3 ⊗ ... ⊗ |±iN |ni
25
(134)
Another representation frequently used is the angular momentum analogy. The
states are then written down as
|Ψi = |ni |J, mi
(135)
This is still a eigenstate of the Hamiltonian, but the states are highly degenerate. With this representation an exact solution can be found for the energyspectrum. This is not easily done and only the solution is presented here. The
full derivation can be found in [6] and [7]. The energy eigenstates are found by
pertubation theory and can be represented as:
p
(136)
E ≈ nω + j 4g 2 n + ∆2 + E 1
Here j is the degeneracy number, and E 1 is the first order correction. The more
interesting feature of the Dicke-model is the phase transition which is presented
in the next section[8]
4.2
Phase transitions in the Dicke model
[9] The Dicke model in a generalisation of the Jaynes Cummings model to N
atoms. This model show an interesting phase transition. The phase transition
was first presented by Hepp and Lieb[10]. In this section this phase transition
will be derived using a the coherent state representation of the Dicke-model.
The section follows closely the derivation given by Wang and Hioe[11]. This
derivation shows in an elegant way the phase transition in the Dicke model.
The derivation starts by placing N-atoms in a cavity. The kinetic energy of the
atoms is for this derivation neglected. The Hamiltonian of this model in the
rotating-wave approximation is then given by:
H = H0 + HI =
X
s
N
N
X
1
1 Z X 1 X 0
√
ωs (a†s as + ) +
σj +
(λs as σj+ + λ0s a†s σj− )
2
2
2
2
s
j=1
j=1
(137)
Here the subscript s is for the s-th photon mode, subscript j stands for the j-th
atom. The coupling constant is now written as λs for a single photon mode.
The Hamiltonian is essentially the Jaynes-Cummings Hamiltonian summed for
N atoms and s modes. The V stands for the volume of the cavity. If we assume
the atoms to be in a cavity there is only one mode. Then the Hamiltonian can
be written as:
H = a† a +
N
X
1
λ
( 0 σjZ + √ (aσj+ + a† σj− ))
2
2
N
j=1
(138)
Here 0 = /ω, so the √transition
is measured in terms of the ω of
√ strength
√
the cavity and λ = λ0 N /ω V = λ0 ρ/ω. If we assume the atoms to be
in equilibrium we can calculate the canonical partition function Z(N, T ). In
general the partition function is defined as:
Z(N, T ) = T r(e−βH )
26
(139)
With β = kB1T . The trace of a matrix in invariant in every basis. The used basis
in this derivation is the basis of coherent state representation |αi. This ket is
the eigenstate of the annihilation operator, so a |αi = α |αi and hα| a† = hα| α∗ .
The trace’; of the partition function can then be written as the sum over all
possible states. This gives
X
X Z d2 α
hs1 ..sn | hα| e−βH |αi |s1 ...sn i
(140)
Z(N, T ) =
...
π
s =±1
s =±1
1
N
So here the summation is over the |gi , |ei possible states of all N atoms ( represented as ±). The double integral integrates over all possible coherent state
representation in the cavity. The factor of π1 is due to the normalisation of
the coherent states.
√ Now the altered√annihilation and destruction operators are
defined as b = a/ N and b† = a† / N . Using this allows the Hamiltonian to
be written as:
N
X
0
λ
(b† b + σJz + (b† σj− + bσj+ ))
H=
(141)
2
2
j=1
Due to the definition of b† b the a† a part can be moved inside the summation.
This gives a nice expression for the Hamiltonian. The next step is to simplify the
integrand of equation 140. This can be achieved by calculating the expectation
value hα| e−βH |αi. The exponent of this equation can be written as:
X
hα| e−βH |αi = hα|
(−βH)n /n! |αi
(142)
n
Where
H=
N
X
λ
(b† b + σJz + (b† σj− + bσj+ ))
2
2
j=1
(143)
Lets look at the commutation relationship between b and b† . The commutation
relationship is given by:
[b, b† ] = 1/N [a, a† ] = 1/N
(144)
In the Dicke model we can approximate the commutation relationship in the
limit of N → ∞ as zero. There is a slight
√ problem with
√ this approximation. The
field operators are defined as b† = a† / N and b = a/ N . Therefore in the limit
N → ∞ the commutator vanishes, but the field operators them-self must not
become zero. Now lets look at a single power l of equation 142(hα| (−βH)l /l! |αi.
Lets imagine the power l is expanded and look at the b† b terms. In the expansion
every term has at least a b† b, and at the most l b† b terms. The field operators
are the only operators in the Hamiltonian that operate on |αi. Imagine a term
written in the form of:
b† bb† bb† b...b† b
(145)
These field operators commute in the N → ∞ limit, so every field operators
term we can write in the so-called anti-normal order(b† )l (b)k . In this order the
27
expectation value for the coherent state will be calculated by the definition of
a coherent state
hα| (b† )l (b)k |αi = (α∗ )l (α)k
(146)
After having calculated this for all terms, the expansion can be undone and the
power of l can be written in the form of
hα| (−βH)l /l! |αi =
N
X
λ
1
(−β(α∗ α +
( σJZ + √ (α∗ σj− + ασj+ ))))
l!
2
2
N
j=1
(147)
This procedure can be performed for each term of equation 142. Then the summation can be rewritten in the original exponent. This gives for the expectation
value
hα| e−βH |αi = exp(−β(α∗ α +
N
X
λ
( σJZ + √ (α∗ σj− + ασj+ ))))
2
2
N
j=1
(148)
Now to simplify equation 140 the following identity is defined:
hj =
0 Z
λ
σJ + √ (α∗ σj− + ασj+ )
2
2 N
(149)
This and equation 148 allows the integrand of equation 140 to be reduced to
2
hα| e−βH |αi = e−β|α| e−β
PN
j=1
hj
(150)
Now the integral can be rewritten in a solvable form as
Z 2 X
PN
X
2
d α
Z(N, T ) =
...
hs1 ...sn | e−β|α| e−β j=1 hj |s1 ...sn i
π s =±1 s =±1
1
N
Z 2 X
X
2
d α
−βhj
=
...
e−β|α| hs1 ...sn | ΠN
|s1 ...sn i
j=1 e
π s =±1 s =±1
1
N
Z 2 X
X
2
d α
−βhj
...
e−β|α| ΠN
|sj i
(151)
=
j=1 hsj | e
π s =±1 s =±1
1
N
Every summation is only over the possible states ±. Therefore the integrand
can be rewritten as a multiplication of the possible states of the atoms. This
gives
Z 2
d α −β|α|2
Z(N, T ) =
e
(h+1| e−βh |+1i + h−1| e−βh |−1i)N
π
Z 2
d α −β|α|2
=
e
(T re−βh )N
(152)
π
Here h is the operator earlier defined. The operator h can be written in matrix
form as
!
1 0
λα
√
2
N
h = λα∗
(153)
√
− 12 0
N
28
This matrix can be solved in for its eigenvalues and eigenstates. The matrix is
diagonal in the basis of its eigenvectors with its eigenvalues on the diagonal. The
trace is a summation over the diagonal, and the trace of a matrix is independent
of the basis of the matrix. Therefore the trace over the matrix can expressed as
the sum of the eigenvalues of the matrix. The eigenvalues of h are given as:
0
4λ2 |α|2 1
x = ±( )(1 + 02
)2
2
N
Using this the integral of equation 152 can be written as
Z 2
d α −β|α|2 +β|x|
Z(N, T ) =
e
(e
+ e−β|x| )N
π
Z 2
d α −β|α|2
=
e
(2 cosh β|x|)N
π
(154)
(155)
The integral is over the entire space of α. To solve the integral we first note that
Rthe2 entire integral is real and converges when2 |α| → ∞. ∗ Because the integral
d α is an integral over the complex plane(d α = dαdα ) the integral can be
rewritten in polar coordinates. This gives
Z 2
Z ∞
Z 2π
Z ∞
d α
dθ
=
rdr
=2
rdr
(156)
π
π
0
0
0
Equation 155 then becomes
Z ∞
2
Z(N, T ) =
2rdre−βr (2 cosh β|x|)N
(157)
0
r2
N
and dy = 2r
dr gives
Z ∞ N
N e−βyN (2 cosh β|x|)N
Z(N, T ) =
0
Z ∞
=N
dy exp[N (−βy + ln(2 cosh β|x|))]
Substituting y =
(158)
0
It is good to note that the eigenvalue equation also changes under the substitution. The —x— becomes
0
|x| = ( )(1 +
2
0
= ( )(1 +
2
4λ2 |α|2 1
0
4λ2 r2 1
) 2 = ( )(1 + 02 ) 2
02
N
2
N
4λ2 y 1
)2
02
(159)
The integral 158 can be solved by the use of Laplace method. More information
about this method can be found in appendix B. Using Laplace method gives:
C
β0
4λ2 y 1
Z(N, T ) = N √
max exp (N (−βy + ln (2 cosh (
(1 + 02 ) 2 )))) (160)
2
N 0≤y≤∞
29
Here the second derivative of the exponent and constants are placed in C. To
find the maximum we differentiate the exponent. Define
φ(y) = −βy + ln (2 cosh (
β0
4λ2 y 1
(1 + 02 ) 2 ))
2
(161)
Then
β0
4λ2 y 1
4λ2 y 1
β0
(1 + 02 ) 2 ))−1 × (2 sinh (
(1 + 02 ) 2 ))
2
2
β0
4λ2 y − 1 4λ2
×
(1 + 02 ) 2 · ( 02 )
4
β0
4λ2 1
4λ2 y 1
= −β + tanh (
(162)
(1 + 02 ) 2 )(1 + 02 )− 2 (βλ2 /0 )
2
φ0 (y) = −β + (2 cosh (
Now we can set φ0 = 0 to find the maximum. This gives
4λ2 y
β0
4λ2 y 1
0
(1 + 02 ) = tanh (
(1 + 02 ) 2 )
2
λ
2
4λ2 y
To see the behaviour better we define η = (1 + 02 ), This gives
β0
0
η
=
tanh
(
η)
λ2
2
(163)
(164)
Now the maximum needs to be found in the region of 0 ≤ y ≤ ∞ or in terms
of η the region is given as 1 ≤ η ≤ ∞. When looking at equation 164 three
0
different regions can be defined. If λ2 < 0 the fraction λ2 > 1. Then equation
164 has no solution in the region 1 ≤ η ≤ ∞. A tanh function can only take
values in the region tanh x ∈ [−1, 1]. When the function 164 does not have a
solution the maximum value of the integral lies on the place y = 0.
The second region is when λ2 > 0 . In this region equation 164 has a solution
depending on the value of β. We can define a βc as:
βc =
2
0
arctanh
0
λ2
(165)
This shows that the function have an extra constraint. The arctanh (x) is only
defined in the area x ∈ [−1, 1]. This is only the case in equation 165 when
λ2 > 0 . If β < βc a solution of the equation 164 can still not be found.
The maximum is therefore still at y = 0. But the phase transition occurs when
β > βc . In this region the equation 165 has one(and only one) solution. Imagine
the solution is given by η0 , then
β0
0
η0 = tanh
η0
2
λ
2
Writing
0
λ2 η0
(166)
= 2σ we can use the definition of η to write
y0 = λ 2 σ 2 −
30
02
4λ2
(167)
A good way to see the phase transition is by looking at the free energy of the
atoms. The free energy for a single atom is then defined as:
βf (T ) = lim (
N →∞
1
ln Z(N, T ))
N
(168)
The free energy is then given in terms of kT . In the regions λ2 < 0 or λ2 > 0
and β < βc the fre energy is given as
1
βf (T ) = ln (2 cosh ( β0 ))
2
(169)
But in the region λ2 > 0 and β > βc the expression for the free energy can be
written as
1
1
(170)
f (T ) = − (ln (2 cosh ( β0 )) − y0 )
β
2
y0 can be rewritten as:
y0 = λ2 σ 2 −
= λ2 (
=
02
4λ2
02 η0
02
)
−
4λ4
4λ2
02
(η0 − 1)
4λ2
(171)
From the fact that η0 ∈ [1, ∞] it follows that y0 ≥ 0. Here it is clear that the
free energy suddenly get a jump in value. when the parameters change towards
the critical values. This is the phase transition in the Dicke model. When the
parameters λ < λc and β < βc the system is in a normal phase. But when the
λ > λc and β > βc the system is in a so called superradiant phase. To explain
this phase lets first calculate the quantity
†
h(
T r( aNa )e−βH
a† a
)i =
N
T re−βH
(172)
This expression can be solved the same way as the partition function was solved.
The only difference is found by calculating the integrand of equation 140. The
integrand becomes
(aa† ) −βH
)e
|αi |s1 ...sn i
N
2
−βhj
=e−β|α| |α|2 ΠN
|sj i
j=1 hsj | e
hs1 ...sn | hα| (
Using this the expectation value becomes
Z ∞
a† a
h(
)i =
yrdy exp[N (−βy + ln(2 cosh β|x|))]
N
Z0 ∞
×(
dy exp[N (−βy + ln(2 cosh β|x|))])−1
0
31
(173)
(174)
Where x is still given by equation 154. Again using Laplace method to solve
the equation an expression can be found for the expectation value. This gives
for different values of λ and β:
a† a
)i = 0
N
02
a† a
)i = (λ2 σ 2 − 2 )
h(
N
4λ
02
= ( 2 (η0 − 1))
4λ
h(
when λ2 < 0 or λ2 > 0 , β < βc
(175)
when λ2 > 0 , β > βc
(176)
The same way we can find an expression for the value h( JNz )i. This gives(in T=0
limit)
1
Jz
)i = −
N
2
Jz
02
h( )i = − 2
N
2λ
h(
whenλ2 < 0 or λ2 > 0 , β < βc
(177)
Here we again find the jump in expectation value due to the change of state. The
Superradiant phase is a phase when the ground-state expectation value of the
number of photons and excitations is not zero and − 21 - what you would expect
normally. But the values are given by function which are always greater then
the original state. So due to the strongly correlated photons the ground state is
radiating. This result shows a Superradiant phase-transition behaviour in the
Dicke model. This effect is a pure quantum mechanical effect of the quantization
of light. It is good to note that in this derivation several approximations where
used. In another paper of Hioe[12] there are several generalisations added to the
model. It is shown that without the Rotating wave approximation or the dipole
approximation you still would expect a phase transition. There is also shown
that when the atoms are not kept fixed you’d also expect a phase transition.
These generalisations show that the phase transition is a phenomenon that is
deeply embedded in the Dicke model[5][9][8].
32
5
5.1
Jaynes-Cummings lattice model
Jaynes-Cummings Lattice Hamiltonian
Recently another physical system has been proposed that would exhibit collective photonic behaviour [13, 14, 15]. The system takes close resemblance to the
Bose-Hubbard model widely used in condensed matter physics. The proposed
system contains an array of optical cavities with each containing a single twolevel atom. This model behaves then like an array of Jaynes-Cummings models.
However, due to the fact that cavities are never a fully closed system, the photons can tunnel between cavities. The photons can therefore hop between each
cavity. A string of coupled cavities can be seen in figure 5. The cavities can
also be coupled in a lattice shape.
Figure 5: A string of N coupled Jaynes-Cumming cavities
The Jaynes-Cummings lattice model can be described the same way as a BoseHubbard model. The Bose-Hubbard model is a model that describes bosons
on a lattice. The bosons are only allowed to be on the lattice sites, but they
can tunnel to other lattice sites. This can as example describe bosons placed
in a high periodic potential. The bosons can only move between through the
potential by tunneling towards another lattice site. The Hamiltonian describing
this system therefore consists of two terms. The first term is a potential term
which describes the on-site repulsion of the bosons on the lattice. The second
term, the kinetic term, describes the hopping of the bosons on the lattice. The
Bose-Hubbard Hamiltonian is therefore given as
X
Hhub = Hsite + Hhop − µ
n̂i
(178)
i
Hsite
Hhop
UX
=
n̂i (n̂i − 1)
2 i
X †
= −t
bi bj
(179)
(180)
<i,j>
Here n̂i are the number of bosons on lattice site i, U is the on-site repulsion
term for the bosons and t is the hopping constant. The b† and bi term are the
respective annihilation and creation operator for bosons on a lattice. In the
hopping term the summation is over all the neighbours of site i. Their is also a
chemical potential µ , because the Bose-Hubbard model describes the system in
the Grand-Canonical-ensemble. It is good to note that the chemical potential µ
and the hopping constant t are approximated in this model to be the same for all
33
lattice site. An adaption of this model can be preformed to adapt it to describe
the Jaynes-Cumming lattice model. In the Jaynes-Cummings lattice model an
array of Jaynes-Cummings cavities is used. The cavities can be considered as the
lattice sites of the Bose-Hubbard model. Because the cavities are not perfect
the photons can tunnel to its nearest neighbours. So the photons are either
trapped inside their respective cavity, or they can hop to other cavities. The
Hamiltonian describing this model contains an energy term of the photon in
its respective cavity and again the hopping term. In the case of photons in
a cavity the energy term is given by the earlier described Jaynes-Cummings
Hamiltonian. This is given for a cavity i as
HiJC = σi+ σi− + ωa†i ai + g(σi+ ai + σi− a†i )
(181)
Here the is again the transition energy of the atom, the σi± are the atomic raiser
and lowering operator for lattice site i, and g is again the atom-field coupling
strength. All cavities are assumed to be the same, so the coupling strength is
the same for each cavity. The kinetic part of the Hamiltonian is given by the
hopping term. This is given - in analogue for the Bose-Hubbard model - by
X
HiHopping =
κ(a†j ai + a†i aj )
(182)
j
Here the κ stands for the hopping strength, again assumed to be constant for
all cavities. The a†j ai represents the photon hopping from the i-th cavity to a
neighbour. The term a†i aj stands for the photon hopping from a neighbour to the
i-th cavity. The full Hamiltonian in the grand Canonical ensemble also contains
a chemical potential µ - again approximated as the same for all cavities. Adding
all this part together we get the full Hamiltonian for our Jaynes-Cummings
lattice model
X
X
X
H=
σi+ σi− +ωa†i ai +g(σi+ ai +σi− a†i )+
κ(a†j ai +a†i aj )−
µNi (183)
i
<i,j>
i
Where here Ni stands for the number of excitations (e.g. Ni = adagger
ai + 21 σiz ).
i
So now there is a full Hamiltonian for the Jaynes-Cummings lattice model.
The ground state of the model can be in two possible phases, a Mott-insulator
phase and a Superfluid phase. In the Mott-insulator the atomic part of the
Hamiltonian dominates. Therefore the photons are trapped in their respective
cavity. The Superfluid phase is the phase when the hopping term dominates.
The excitations of the system are then not located to their respective lattice site,
but the excitations are delocalised and present in the entire array. The phases
have a narrow transition between them. Therefore there is a phase transition
between the two stages. This phase transition can be shown analytically with a
mean field treatment, and can be clearly seen when numerically simulating the
system.
34
5.2
Limits of the Jaynes-Cummings lattice model
The Jaynes-Cummings Lattice model cannot be exactly solved. There are however two interesting limits which are interesting to discuss. The first limit is
the atomic limit κg << 1. This limit means that the hopping term must be
very small, and can therefore be treated as a perturbation of the Hamiltonian.
The eigenstates of the unperturbed Hamiltonian are a product of the single
Jaynes-Cummings eigenstates |Ψi = |Ψi1 ⊗ |Ψi2 ⊗ ... where the |Ψii stands for a
Jaynes-Cummings state of lattice site i. The energy-states are the known dressed
eigenstates of the Jaynes-Cummings Hamiltonian, with correction for the chemical potential. The energy are the known eigenvalues of the Jaynes-Cummings
Hamiltonian, so therefore the energy for a single cavity can be written as:
En± = nω + ∆/2 ± [(∆/2)2 + ng 2 ]1/2 − µn
(184)
With ∆ again the detuning and n the number of excitations. The ground state
can be found by looking at the lowest eigenstate. This can either be the |0i state,
or the |n−i state( The |n+i state has by definition always a higher energy then
the |n−i state). When starting at the limit ω − µ g it becomes clear that when looking at equation 184- the number of excitations dominates the energy
value. Therefore the lowest energy state is the |0i state. When lowering the
quantity ω − µ the system reaches a point when it is energetically favourable to
add an excitation to its system. This point is reached when E0 = E1− . This
repeats for all the points where En− = E(n+1)− . Using this condition one can
find the expression
p
p
n(ω − µ) − (∆/2)2 + ng 2 = (n + 1)(ω − µ) − (∆/2)2 + (n + 1)g 2
p
p
(µ − ω) = g (∆/2g)2 + n − ∆/2g)2 + n + 1
p
p
(185)
(µ − ω)/g = (∆/2g)2 + n − ∆/2g)2 + n + 1
When
the system is at resonance this gives the simple expression (µ − ω)/g =
√
√
n − n + 1 for the ground state in the atomic limit [16]. From this it also
follows that in the atomic limit the number of excitation of the ground state
depends on the value of ω − µ. The other interesting limit is the hoppingdominating limit κg 1. In this limit the atomic part of the Hamiltonian
becomes treated as a perturbation. This model is not exact solvable, but when
the photon-atom coupling is neglected the Hamiltonian can be written in the
way of a tight-binding model
X †
X †
Htb = (ω − µ)
ai ai − κ
(ai aj + a†j ai )
(186)
i
<i,j>
This model is already well understood and independent of the shape of the
lattice the ground-state energy eigenvalues is given by
E0 = N (ω − µ) − N zκ
35
(187)
Where z is the number of neighbours. This results shows an instability for large
photon hopping strength. When zκ > (ω − µ) the ground state energy becomes
negative. The ground state does not converge for a large number of excitation
N . Therefore in this limit there is a certain instability in the model[16].
5.3
Mean-field approximation
A way to solve the Hamiltonian given by equation 183 is by using a mean-field
approximation. In this approximation the hopping term is rewritten with the
use of the decoupling approximation. Imagine two operators A and B. The
mean field approximation then says that these operators can be written as a
variation of a mean field parameter. Defining A = m + δA and B = m + δB
where m =< A >=< B >. Using this a decoupling approximation can be
written down as
AB = (m + δA)(m + δB)
= m2 + m(δA + δB) + δAδB
= m2 + m(A − m + B − m) + δAδB
= m(A + B) − m2 + δAδB
(188)
Now we approximate the variation of δA and δB as small to write[17]
AB =< A > B + A < B > − < A >< B >
(189)
Using this identity for the hopping term of the Jaynes-Cummings Hamiltonian
gives
X
HiHopping =
κ(a†j ai + a†i aj )
(190)
j
= zκ(< a†j > ai + a†j < ai > − < a†j >< ai > +h.c.)
(191)
Here z stands for the number of neighbours of cavity i. We can now define the
mean field parameter ψ = zκ < ai > to write
HiHopping = (ψ ∗ ai + a†j ψ −
1
|ψ|2 )
zκ
(192)
So the mean field Hamiltonian then becomes
X
1
HMF =
( − µ)σi+ σi− + (ω − µ)a†j aj + g(a†j σj− + σj+ aj ) − (ψ ∗ ai + a†j ψ − |ψ|2 )
zκ
i
(193)
The mean field Hamiltonian can still not be exactly solved, but a good analytical
approximation can be made. Further the numerical simulation becomes much
easier because the number of parameters is greatly decreased.
36
5.4
Analytical Phase Boundary
First the analytical approximation will be made. The mean field approximation
can also be used to determine the phase transition boundary between the Mottinsulating phase and the Superfluid phase. In the critical region the change of
the ground state energy is small. Therefore the expression for the energy can be
expanded in terms of the Superfluid parameter Ψ. The terms in the Hamiltonian
involving Ψ can be treated as a pertubation. This leads to an expression of the
ground state in the form of
EG = E0mf + r|Ψ|2 + O(Ψ4 ).
(194)
The odd-terms can be neglected because the Hamiltonian is invariant under
parity transformations. An expression for r can be found by using second order perturbation theory. The mean-field hopping part for a single cavity is
considered as a perturbation of the Jaynes-Cummings part of the single cavity
Hamiltonian. The Hamiltonian for a singe cavity can be written as
Hjmf = H0,j + Hj0
(195)
µ)a†j aj
H0,j = ( − µ)σi+ σi− + (ω −
1
|ψ|2
Hj0 = −(ψ ∗ ai + a†j ψ) +
zκ
+
g(a†j σj−
+ σj+ aj )
(196)
(197)
The general expression for the second order correction of the energy from perturbation theory is given as
E (2) =
X | hk (0) | H 0 |n(0) i |2
(0)
(0)
k6=n
En − Ek
(198)
The whole idea is to get an expression for the factor r. The states used for perturbation theory are the dressed states of the Jaynes-Cummings Hamiltonian,
1
and the H 0 is the Hamiltonian given by equation 197 minus the zκ
|ψ|2 part.
This is because that part does not operate on the dressed state, so it can be
omitted from the calculation. Writing for r
r = Rn +
1
zκ
(199)
Allows us to calculate the factor Rn can by the above mentioned perturbation
theory. The resonant case is still assumed(∆ = 0). This gives for Rn=0
Rn=0 =
X | hk (0) | H 0 |0i |2
(0)
k6=n
=−
X | h1α| − (ψ ∗ ai + a†j ψ) |0i |2
(0)
X | − ψ < 1α|1 > |2
(0)
α=±
(201)
E1α
α=±
=−
(200)
(0)
E0 − Ek
E1α
37
(202)
Using h1±| = √12 (h1, e| + h2, g|) and En,± = nω + ∆
2 ±
be written that(∆ = 0)
Rn=0 =
q
2
2
(∆
2 ) + ng − µn it can
X 1
1
2 µ − ω + αg
α=±
(203)
If n > 0 the same procedure can be followed
Rn>0 =
X | hk| H 0 |n−i |2
(0)
k6=n
=
(204)
(0)
En− − Ek
X | hk| − (ψ ∗ ai + a†j ψ) √12 (|n, gi + |n − 1, ei)|2
(0)
k6=n
(0)
En− − Ek
(205)
√
√
|ψ|2 X
1
| hk| (( n |n − 1, gi + n − 1 |n − 2, ei)
(0)
(0)
2
k6=n En− − Ek
√
√
+( n + 1 |n + 1, gi + n |n, ei))|2
(206)
=
The |ki which gives a nonzero product are |n + 1, ±i , |n − 1, ±i. This gives as
expression
√
√
|ψ|2 X | hn + 1, α| ( n + 1 |n + 1, gi + α n |n, ei)|2
Rn>0 =
(
+
2 α=±
En− − E(n+1)α
√
√
| hn − 1, α| n |n − 1, gi + α n − 1 |n − 2, ei |2
)
(207)
En− − E(n−1)α
√
√
|ψ|2 X
( n + 1 + α n)2
√
=
(
+
√
4 α=± n(ω − µ) − g n − ((n + 1)(ω − µ) + αg n + 1)
√
√
( n + α n − 1)2
√
(208)
√
n(ω − µ) − g n − ((n − 1)(ω − µ) + αg n − 1)
√
√
|ψ|2 X
( n + 1 + α n)2
√
=
(
√
4 α=± (µ − ω) − g( n − α n + 1)
√
√
( n + α n − 1)2
√
+
)
(209)
√
−(µ − ω) − g( n − α n − 1)
According to landau theory the phase transition happens when the parameter
r chances sign. Then the symmetry gets broken and a phase transition occurs.
One can see this by looking at the function of the energy(E = E0 +r|Ψ|2 ). When
r changes sign the function changes from a hyperbolic function to a Mexican-hat
function, and therefore there is a phase transition. This means that our phase
boundary can be calculated by imposing the condition that r = 0. Using this
gives the equation
1
1
=0→κ=
(210)
Rn +
zκ
zRn
38
Where for R0 equation 206 is used and for Rn>0 equation 209 is used.
Figure 6: Analytical phase boundary
Equation 210 is plotted in figure 6 for n = 0 to n = 3. The analytical treatment
shows that certain lobes are formed. In these lobes the system is in a Mott
insulator phase where the excitation of the system stays in their respective
cavity. Outside the lobes the photons are in a Superfluid phase. In this phase
the excitations are not limited to their cavity[16] [15] [17].
5.5
Numerical treatment
Another approach of the model is a numerical simulation. This is done on
the mean-field Hamiltonian (equation 193). This Hamiltonian is numerically
diagonaliseable for a known value of κ and µ − ω. The first step is to write the
Hamiltonian in matrix form in the basis (|g, 0i , |e, 0i , |g, 1i , |e, 1i |g, 2i . . . ). In
these basis the Hamiltonian for a single cavity can be written down as:


0
0
−zκψ
0
0
...
 0
−µ
g
−zκψ
. . .
√0


−zκψ
2zκψ
. . .
g
ω
−
µ
0
−


√
mf
Hi =  0
(211)
−zκψ
+√
ω − 2µ
2g
. . .


√0
 0

0
−
2zκψ
2g
2ω
−
2µ
.
.
.


..
..
..
..
..
..
.
.
.
.
.
.
39
For a exact treatment the matrix extends to infinity, but in the numerical treatment the Hamiltonian needs to be truncated for a certain n. The aim for the
numerical treatment is to find a convergent value of ψ =< ai >. The approach
used in this report for the diagonalisation is described below. First a trail value
of ψ is guessed, which is somewhere between 0 and 10. The diagonalisation is
done with a known parameter µ − ω and κ on the resonance condition(∆ = 0).
With this values and the guessed value for ψ the matrix is numerically diagonalised. This is done by solving for the eigenvalues and eigenstates of the
matrix. Then the lowest eigenvalue(Eg ) is determined with its respective eigenstate. This eigenstate is used to find a new ψ = hEg | ai |Eg i. With this ψ the
procedure can be repeated. This procedure is iterated until the value of ψ converges to a certain value. This value is then the order parameter for the certain
parameter κ and µ − ω. This procedure is done for multiple values of κ and
µ − ω and the found values are placed on the grid.
Figure 7: Numerical simulation of the Jaynes-Cummings Lattice model. The
grid is a 30x30 grid and the matrix is truncated at n = 40
A plot for the borders 10− 4 < κ < 1 and −2 < µ < 0 is presented in figure 7.
The phase boundaries and lobes are clearly visible in the numerical simulation.
At the boundaries when κ ≈ 1 or when µ − ω ≈ 0 the numerical error becomes
to great. Therefore the simulation at these boundaries can be neglected. The
boundaries are almost similar to the boundaries found in the analytical treatment as seen in figure 6. The founded simulation matches the one found in [16]
[18] [9].
40
6
Discussion
In this report several theoretical Quantum optics models have been studied,
and strongly correlated photon behaviour have been investigated. The question
arise in which limits the described models are still a good description. The
Jaynes-Cummings model have been derived under two important approximations, the Rotating wave approximation and the Dipole approximation. The
Dipole approximation is only applicable when the atom can be approximated
as a dipole. The approximation can be extended to by adding a quadrupole
term. It turns out that this does not change the dynamics of the system by
much. Therefore the Dipole approximation is valid in most cases. The Rotating wave approximation has much stronger limits. In [19] the limits of the
Rotating wave approximation is discussed. They state that the Rotating wave
approximation is only applicable for low values for the coupling (e.g. g/ω)
and for low detuning ∆. Outside this limits the Rotating wave approximation
is not applicable , and the physics of the system changes significantly. For
the most direct experimental setup of the Jaynes-Cummings model the Rotating wave approximation is however applicable. But for experiments which map
the Jaynes-Cummings Hamiltonian the Rotating wave approximation cannot be
made. Another good question is how these models can be experimentally tested.
Although the physics of the Jaynes-Cummings model can be well described theoretically, it is not a easy model to realise. Evidence for the quantization of the
electromagnetic field and the predicted results of the Jaynes-Cummings model
have been found by [20]. The phase transition in the Dicke model is recently being tested [9], but results have yet to be confirmed. Recently another system has
been used which maps the Dicke model [21]. In this Dicke-like system the phase
transition has been confirmed. This result is promising, but more research is
needed. For the Jaynes-Cummings lattice model experimental verification has
not been performed. The model is a recently proposed model, therefore not
much research is done on the model. It is also not an easily realised model
and experimental verification will take some time. The phase transition from a
Mott-Insulator phase to a Superfluid phase is already experimentally found in
the Bose-Hubbard model[22]. Therefore the phase transition is likely to also occur in the Jaynes-Cummings lattice model. An alternative way to test the above
mentioned models is found in circuit-QED[16]. In this method the Hamiltonian
of the Jaynes-Cummings model is mapped by a circuit setup. he Hamiltonian
is the same of the of an Jaynes-Cummings model, but the parameters are more
easily adjustable. Therefore this shows promising testing of the proposed models. The photon is not in a cavity, but on a placed on a chip. The atom is
not a real atom, but an artificial atom. This is a quantum object that exhibits
the same behaviour as an atom, but is not a real atom. With circuitQED the
Jaynes-Cummings model has been realised. The research on the Dicke model is
already done with a few atoms, but not for large N. This approach is promising
to study the Dicke model and the Jaynes-Cummings model. The physics found
should then confirm the predicted physics of the ’normal’ cavity model.
41
7
Conclusion
A theoretical study has been performed on strong photonic behaviour in two
quantum optic models. The first researched model is the Dicke model. A
phase transition has been found between a normal phase and the Superradiant
phase. The Superradiant phase is a quantum mechanical phase in which the
ground state of the system contains radiation. This phase transition depends on
the value of the coupling constant. When the value of the coupling constant is
greater then the critical value, a the system changes phase for a normal phase to
a Mott-Insulator phase and the phase transition occurs. The phase transition
has been derived from the Dicke Hamiltonian. The other studied model is
the Jaynes-Cummings Lattice model. This recently proposed model contains a
phase transition from a Mott-Insulator phase to a Superfluid phase. The phase
transition depends of the hopping constant κ and the parameter µ − ω. The
phase transition has been analytically investigated and numerically simulated.
42
Acknowledgement
I would like to thank my supervisor dr. Vladimir Gritsev for his support and help
during the project. The project would not be the same without his supervision.
Also I would like to thank the Institute of Theoretical Physics and University
of Amsterdam for granting me the opportunity to perform this project.
References
[1] R. Loudon. ”The Quantum Theory of Light”. Oxford Science Publications,
third edition, 2000.
[2] D.F.Walls and G.J.Milburn. ”Quantum Optics”. Springer, first edition
edition, 1994.
[3] M. Orszag. ”Quantum Optics. Including Noise Reduction, Trapped Ions,
Quantum Trajectories, and Decoherence”. Springer, Second edition, 2008.
[4] S.M. Barnett and P.M. Radmore. ”Methods in Theoretical Quantum Optics”. Oxford Science Publications, first edition, 1997.
[5] M. Haroche and J-M Raimond. Exploring the Quantum Atoms, Cavities,
and Photons. Oxford Graduate texts, first edition, 2013.
[6] Michael Tavis and Frederick W. Cummings. Exact solution for an nmolecule¯radiation-field hamiltonian. Phys. Rev., 170:379–384, Jun 1968.
[7] L. M. Narducci, M. Orszag, and R. A. Tuft. Energy spectrum of the dicke
hamiltonian. Phys. Rev. A, 8:1892–1906, Oct 1973.
[8] Barry M. Garraway. The dicke model in quantum optics: Dicke model
revisited. Phil. Trans. R. Soc. A, 369:1137–1155, March 2011.
[9] Kristian Gotthold Baumann. Experimental realization of the dicke quantum phase transition, 2011.
[10] K. Hepp and E. H. Lieb. On the superradiant phase transition for molecules
in a quantized radiation field: the dicke maser model. Annals of Physics,
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[11] Y. K. Wang and F. T. Hioe. Phase transition in the dicke model of superradiance. Phys. Rev. A, 7:831–836, Mar 1973.
[12] F. T. Hioe. Phase transitions in some generalized dicke models of superradiance. Phys. Rev. A, 8:1440–1445, Sep 1973.
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phase transitions of light. Nature, 2:856–861, Nov 2006.
43
[14] Dimitris G. Angelakis, Marcelo Franca Santos, and Sougato Bose. Photonblockade-induced mott transitions and xy spin models in coupled cavity
arrays. Phys. Rev. A, 76:031805, Sep 2007.
[15] Michael J. Hartmann, Fernando G. S. L. Brandao, and Martin B. Plenio.
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– 855, Nov 2006.
[16] Jens Koch and Karyn Le Hur. Superfluid˘mott-insulator transition of light
in the jaynes-cummings lattice. Phys. Rev. A, 80:023811, Aug 2009.
[17] S. Sachdev. ”Quantum phase transitions”. Cambridge University Press,
second edition, 2011.
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[19] Michael Tomak, Omar E. Araby, Mikhail Pletyukhov, and Vladimir Gritsev. Strongly interacting polaritons in coupled arrays of cavities. Juli 2013.
[20] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche. Quantum rabi oscillation: A direct test of field
quantization in a cavity. Phys. Rev. Lett., 76:1800–1803, Mar 1996.
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Chuanwei Zhang, and Peter Engels. Dicke-type phase transition in a
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44
A
The interaction term and the dipole approximation
For the derivation of the Jaynes-Cummings model a dipole approximation is
made on the p · A term. In this section the approximation is justified in a more
thorough way. The justification closely follows reference [3]. The derivation
starts by writing the wave function as a unitary transformation:
iq
|ψ(t)i = e ~ r·A |χ(t)i = U |χ(t)i
(212)
Using this expression in the schrödinge equation gives
∂
(U |χ(t)i) = HU |χ(t)i
∂t
∂U
∂ |χ(t)i
+ i~
|χ(t)i = HU |χ(t)i
i~U
∂t
∂t
The the expression is multiplied by U † = U −1 from the left to write
i~
i~
∂ |χ(t)i
= H 0 |χ(t)i
∂t
(213)
(214)
(215)
∂U
∂t
Applying the definition of the unitary transformation gives:
H 0 = U † HU − i~U †
(216)
∂U
∂A
= qr
= −qr · E
(217)
∂t
∂t
Now we need to calculate the U † HU . First the commutation relations of p and
U is calculated. This gives
i~U †
[pi , U ] |ψi = (pi U − U pi ) |ψi = −i~(
∂U
∂ |ψi
∂ |ψi
|ψi + U
−U
)
∂xi
∂xi
∂xi
(218)
This gives
[pi , U ] = −i~
∂r · A
∂U
= i~U
∂xi
∂xi
(219)
And in general
[p, U ] = qU ∇(r · A)
(220)
2
Analogue we calculate [p , U ]
[p2 , U ] = U (−iq~∇2 (r · A) + q 2 (∇(r · A))2 + 2e∇(r · A) · p)
(221)
†
Using this and the fact that only p does not commute with U , U HU can be
calculated writing
1 † 2
q
q2 2
U p U − A · U † pU +
A +V
2m
m
2m
q2 2
p2
q~ 2
e2
=
A +V +
+
∇ (r · A) +
(∇(r · A))2
2m
2m 2im
2m
q
e
q2
+ ∇(r · A) · p − A · p − A · (∇(r · A))
m
m
m
U † HU =
45
(222)
Now the dipole approximation or Long Wave approximation is applied. The
Length of the electromagnetic wave is assumed to be much larger then the
length of the dipole field A. Therefore the approximation can be made that
A(r, t) = eik·(r0 +r)−iωt
= eik·r0 (1+ik·r+...)−iωt
≈ eik·r0
(223)
This implies that all the derivatives of ∂i A = 0. The chain rule allows us to
finally write
X ∂r
∂A
(
·A+r·
)
∂xi
∂xi
i
X
∂A
=
(Ai + r ·
))
∂x
i
i
X
=
Ai
∇(r · A) =
(224)
i
Using the same procedure gives(using the coulomb gauge(∇ · A = 0):
∇2 (r · A) =
X
i,j
xi
∂2A
=0
∂x2j
(225)
This gives a final expression of:
p2
q2 2
q
q2 2
A +V +
+
A +
2m
2m 2m
m
q
q2
+ A · p − A · p − A2
m
m
p2
+ V = H 0 + qr · E
=
2m
U † HU =
(226)
(227)
Adding this all together gives
H0 =
p2
+ V − qr · E
2m
(228)
And finally the interaction term with dipole approximation can be written as
Hint = −qr · E
(229)
And this is the form used for the Jaynes-Cummings Hamiltonian. There are
certain regimes when this approximation is not applicable, but for the setups
used in this report this approximation is justifiable.
46
B
Laplace Method
The Laplace Method is an technique of approximating certain integrals. The
method in this report is used to approximate the partition function of the DickeHamiltonian. The method is used for integrals in the form of
Z b
I=
e−λp(t) q(t)dt
(230)
a
The main idea behind the approximation is that the function p(t) has a maximum value at t = t0 . This maximum value will probably, due to the exponent,
also be the maximum value of the entire function in the integral. The values
that contribute the most to the final value of the integral will be in the neighbourhood of p(t0 ). The values close to p(t0 ) are probably close to its maximum
value. These values will also exponentially increase. Therefore the approximation is that only the values close to the maximum t0 will significantly contribute
to the integral. For the Laplace method the function p(t) is first expanded
around its maximum value p(t0 ). This gives
1
p(t) = p(t0 ) + p0 (t0 )(t − t0 ) + p00 (t0 )(t − t0 )2 + O(t3 )
2
(231)
Because t0 is the maximum for p(t), the first derivative must be equal to zero.
Neglecting higher then quadratic terms gives us an integral of
Z b
2
1 00
I≈
(232)
e−λ(p(t0 )+ 2 p (t0 )(t−t0 ) ) q(t)dt
a
Because only values close to the maximum contributes to the integral, the exact
boundaries are not significant. The only demand set on a and b for this approximation is that a < t0 < b and that a and b are not close to t0 . If this condition
is fulfilled, the exact value of a and b are not important because only values
close to t0 will contribute to the integral. Therefore the boundaries can be set
at infinity. The benefit is that now the integral changes to a known Gaussian
integral. This allows us to write
Z ∞
2
1 00
−λp(t0 )
I≈e
q(t0 )
e−λ 2 p (t0 )(t−t0 ) dt
(233)
−∞
Here the function q(t) is approximated as the value at q(t0 ). Now - using the
R∞
p
2
Gaussian integral −∞ e−a(x+b) dx = πa - the integral becomes
s
I≈
2π
q(t0 )e−λp(t0 )
λp00 (t0 )
(234)
And this is the Laplace method. It changes an integral solving problem into a
searching for maximum problem and can therefore approximate hard to solve
integrals.
47