Download Spontaneous Emission and Superradiance

Spontaneous Emission and
Superradiance
Bachelorarbeit
zur Erlangung des akademischen Grades
Bachelor of Science
(BSc)
eingereicht an der
Fakultät für Mathematik, Informatik und Physik
der Universität Innsbruck
von
Andreas Kruckenhauser
Betreuer:
Univ. Prof. Dr. Helmut Ritsch
Institut für Theoretische Physik
Innsbruck, am 21. Juni 2015
Abstract
An atom in an excited electronic state coupled to the electromagnetic
field, will decay via spontaneous emission to a lower energy state.
This thesis discusses spontaneous emission of several near atoms.
We treat the quantisation of the free electromagnetic field and dipole
interaction between the field and atom, the theoretical concepts
of time evolution in quantum mechanics in both, open and closed
systems. We study the Schrödinger, Heisenberg and Interaction
picture for closed systems, define the density matrix and develop the
master equation for open systems from its time evolution.
As an example we establish the master equation in Lindblad form
of an atom, where we find the spontaneous emission or Einstein A
coefficient as a result.
A central result is the superradiance, the enhanced collective emission
of light from several atoms, which results from the established master
equation for a system of N identical atoms. The emission is discussed
in the example of two atoms.
I
Contents
1 Introduction
1
2 Theoretical Concepts
2.1 Quantisation of the free Electromagnetic Field
2.1.1 Classical Field . . . . . . . . . . . . . .
2.1.2 Quantisation . . . . . . . . . . . . . .
2.2 System Dynamics . . . . . . . . . . . . . . . .
2.2.1 Schrödinger Picture . . . . . . . . . . .
2.2.2 Heisenberg Picture . . . . . . . . . . .
2.2.3 Interaction Picture . . . . . . . . . . .
2.3 Open Quantum Systems . . . . . . . . . . . .
2.4 Master Equation . . . . . . . . . . . . . . . .
2.4.1 Density Operator . . . . . . . . . . . .
2.4.2 Master Equation . . . . . . . . . . . .
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3 Spontaneous Emission of a Single Atom
3.1 Two Level Atom . . . . . . . . . . . . . . . . .
3.2 Dipole Approximation . . . . . . . . . . . . .
3.2.1 Minimal Coupling . . . . . . . . . . . .
3.2.2 Approximation . . . . . . . . . . . . .
3.3 Spontaneous Emission . . . . . . . . . . . . .
3.3.1 Representation of an Operator . . . . .
3.3.2 Master Equation for a Two-Level Atom
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in Thermal Equilibrium
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16
4 Collective Atom Dynamics
4.1 Superradiance Master Equation for N Two Level
4.1.1 Hamiltonian . . . . . . . . . . . . . . . .
4.1.2 Derivation of the Master Equation . . .
4.2 Example for Two Two-Level Atoms . . . . . . .
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Atoms
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21
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5 Conclusions
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Bibliography
29
II
Chapter 1
Introduction
The fact that matter can spontaneously radiate light has been known for a long time
in the form of Fluorescence and Luminescence. The fist theoretical description of
spontaneous emission was done by Victor Weisskopf and Eugen Wigner in 1930. They
used Dirac’s light-theory and calculated the natural linewidth of atomic crossovers.
They have shown, that an excited electronic state is not an stationary state, as we know
it from Schrödinger’s equation of the electron core system. The nature of this evolution
is due to the coupling of the electromagnetic vacuum fluctuations to the atom [10].
By studying these light sources some other effects like superradiance, stimulated emission
and absorption were discovered. At first glance superradiance and stimulated absorption
look the same. But in fact they are two independent effects. The emitted radiation
of a system with stimulated emission is proportional to the number of atoms and the
light source’s density, for superradiance only the density of atoms is relevant. This was
first discussed by the theory of Dicke. This theory describes the collective spontaneous
decays of more than one atom. These decays are not independent and lead to a higher
photon flux for short times as the same amount of independent decays. This short time
photon flux is proportional to N2 for high numbers, where N is the number of atoms.
Therefore Dicke called it an optical bomb [2].
The main reason why quantum optics was found, was a new kind of light source the laser. Quantum optics is a field theory which discusses the interaction between
electromagnetic fields and quantum mechanical systems. In quantum optics the theory
of open systems has been a theme since its birth, because sources of light are open
systems. In future superradiance may be used for lasers, which have a linewidth more
than 1000× below the current standard. This can be used for a new generation of
atomic clocks, optical lattices and laser cooling methods. But a lot of development and
work is necessary to fullfill this goal.[6].
1
Chapter 2
Theoretical Concepts
In this chapter we present the theoretical fundament, which are needed for the understanding of the next chapters. We start with a quantum model of the free electromagnetic
field. Over the time evolution of closed and open systems to the master equation of an
electron interacting with vacuum fluctuations.
2.1 Quantisation of the free Electromagnetic Field
In this section the quantisation is achieved by an heuristic approach, where aspects as
Lorentz covariance will not be discussed.
2.1.1 Classical Field
First we state the classical field, which is described by Maxwell’s equations. If the
region is free from charge (ρ(~r, t) = 0) and currents (~j(~r, t) = 0), like in the vacuum, the
~ r, t) and magnetic field B(~
~ r, t) are given by
Maxwell equations for the electric field E(~
~ r, t) = 0,
divE(~
~ r, t) = 0,
divB(~
~ r, t)
∂ B(~
,
∂t
~
~ r, t) = 1 ∂ E(~r, t) .
rotB(~
c2 ∂t
~ r, t) = −
rotE(~
(2.1a)
(2.1b)
~ r, t) and electric potential Φ(~r, t). In
A usual way is to define the vector potential A(~
~ = 0). These potentials have
vacuum it is very useful to use the Coulomb-gauge (divA
to fullfill the following equations, which can be deduced from the Maxwell equations
1 ∂2 ~
A(~r, t) − 4A(~r, t) = 0,
c2 ∂t2
1 Z ρ(~r, t) 3 0
Φ(~r, t) =
d r = 0.
4π0 |~r − ~r0 |
2
(2.2a)
(2.2b)
2 Theoretical Concepts
The first equation, is a wave equation, which has the general result
~ r, t) =
A(~


X
~


Re  A~k,λ~e~k,λ eik~r q~k,λ (t) .
(2.3)
~k,λ
Here A~k,λ are the amplitudes of the potential in direction of the polarisation vector ~e~k,λ ,
~k is the wave-vector and q~ (t) the time dependent term with the dimension of a length.
k,λ
The electric and magnetic field can be calculated from the potentials


~
X
~
~ r, t) = −grad Φ − ∂ A(~r, t) = Re 
A~k,λ~e~k,λ eik~r q̇~k,λ (t)
E(~

,
∂t
~
(2.4a)
k,λ
~ r, t) = rotA(~
~ r, t) =
B(~


X
A~
Re 
e~k,2
k,1 |k|~
~
− A~k,2 |k|~e~k,1 ieik~r q~k,λ (t) .
(2.4b)
~k
For the second equation we chose two orthogonal polarisation vectors and used ~e~k,λ ·~k = 0,
~
~
which is a property of the e.m. field. Hence ~e~k,1 = ~e~k,2 × |~kk| and ~e~k,2 = −~e~k,1 × |~kk| . We
now choose periodic boundaries appropriate for a cavity with side-length L, therefore
only waves with a wave-vector ~k = (kx,nx , ky,ny , kz,nz ) with ki,ni = ni 2π
, where i = x, y, z
L
and ni = ±1, ±2, ±3, . . . are allowed. The Hamiltonian H for the total field energy in
the Volume L3 is
H=
Z
V =L3
2
2 X L3 0 0 ~
2 ~
2
2
dV
E(~
r, t) + c B(~r, t) =
A~2k,λ q̇~k,λ
(t) + c2 A~2k,λ |~k|2 q~k,λ
(t)
2
2
~
k,λ
2
p~k,λ (t)
q 2 (t)
X
1X
2
2 2
2 ~k,λ
=
m~k q̇~k,λ (t) + m~k ω~k q~k,λ (t) =
+ m~k ω~k
,
2~
2m~k
2
~
k,λ
(2.5)
k,λ
with m~k,λ = L3 0 A~2k,λ (which has the dimension of a mass), the angular velocity ω~k = c|~k|
and the canonical momentum p~k,λ = q̇~k,λ (t)m~k . One sees that the energy of the system
is a sum of independent harmonic oscillator energies, i.e. each mode and polarisation is
equivalent to a harmonic oscillator.
2.1.2 Quantisation
The quantisation can be made if we identify the p~k,λ and q~k,λ in equation (2.5) as
operators which follow this commutator relations [8]
h
h
i
q~k,λ , p~k0 ,λ0 = ih̄δ~k,~k0 δλ,λ0 ,
i
h
q~k,λ , q~k0 ,λ0 = 0,
3
(2.6a)
i
p~k,λ , p~k0 ,λ0 = 0.
(2.6b)
2 Theoretical Concepts
It is common to make a canonical transformation to a~k,λ and a~†k,λ
1
a~k,λ = q
2m~k h̄ω~k
1
a~†k,λ = q
2m~k h̄ω~k
(2.7a)
(2.7b)
m~k ω~k q~k,λ + ip~k,λ ,
m~k ω~k q~k,λ − ip~k,λ .
In terms of a~k,λ and a~†k,λ , equation (2.5) becomes
X
H = h̄
ω~k
~k,λ
a~†k,λ a~k,λ
1
+
.
2
(2.8)
The operators a~k,λ and a~†k,λ follow the commutator relations derived from (2.6a) and
(2.6b)
h
i
h
i
a~k,λ , a~†k0 ,λ0 = δ~k,~k0 δλ,λ0 ,
h
a~k,λ , a~k0 ,λ0 = 0,
(2.9a)
i
a~†k,λ , a~†k0 ,λ0 = 0.
(2.9b)
These operators are often called annihilation (a~k,λ ) and creation (a~†k,λ ) operator. Now,
we can write the electric and magnetic field in terms of these operators
~ r, t) =
E(~
X
~k,λ
~
E~k,λ~e~k,λ a~k,λ + a~†k,λ eik~r ,
~
X
~ r, t) = 1
B(~
E~k,λ~e~k,λ a~k,λ − a~†k,λ eik~r ,
c~
(2.10a)
(2.10b)
k,λ
h̄ω 1
2
where E~k,λ = 20 ~Vk
has the dimenison of an electric field. The sum over the ground
state energies in equation (2.8) is infinite. This is a difficulty in quantisation of the
em. field. Practical experiments measuring the total energy of the field do not lead to
any divergence. A detailed discussion may be found in [9]. Hence, we can choose the
P
energy’s zero point to be E0 = h̄2 ~k,λ ω~k , therefore equation (2.8) becomes
H = h̄
X
~k,λ
ω~k a~†k,λ a~k,λ .
(2.11)
We define Fock or number states, which are needed in the following chapters. A number
state satisfies the eigenvalue equation
a~†k,λ a~k,λ |n~k,λ i = n~k,λ |n~k,λ i ,
4
(2.12)
2 Theoretical Concepts
where a~†k,λ a~k,λ is the number operator, which gives the number of photons n~k,λ in the
mode (~k, λ). A number state can easily be generated from its ground state |0~kλ i (which
q
solves h0~k,λ | a~†k,λ a~k,λ |0~k,λ i = 0) by using the equations a~†k,λ |n~k,λ i = n~k,λ + 1 |n~k,λ + 1i
and a~k,λ |n~k,λ i = √n~k,λ |n~k,λ − 1i as
|n~k,λ i =
a~k,λ
q
n~
k,λ
(n~k,λ !
|0~k,λ i .
(2.13)
2.2 System Dynamics
In quantum mechanics there are different ways to describe the time evolution of a closed
system. The expectation value of any observable does not depend on the chosen way.
2.2.1 Schrödinger Picture
In the Schrödinger picture the closed system-dynamics is given by a time-independent
Hamiltonian ĤS . The solutions of the stationary Schrödinger equation (2.14) are called
eigenstates |ψ(t0 )ii to the energy (eigenvalue) Ei ,
ĤS |ψ(t0 )ii = Ei |ψ(t0 )ii .
(2.14)
Every normalized solution is called wave-function or state vector. In the Schrödinger
picture the state vector evolves with time. The time evolution of the system is given by
the unitary time evolution operator Û (t, t0 ). This operator follows directly from the
time-dependent Schrödinger-equation,
ih̄
d
|ψ(t)ii = ĤS |ψ(t)ii ,
dt
(2.15)
where |ψ(t)ii = Û (t, t0 ) |ψ(t0 )ii is the formal solution, with Û (t, t0 ) = exp(− h̄i ĤS (t−t0 )).
It is common to write the time evolution operator as Û (t) with the convention t0 = 0.
Therefore, the time evolution of an observable Â(t) is A(t) = hφ(t)| Â(t) |φ(t)i whereas
the total time derivative is given by ddt = ∂∂t .
A simple example is the energy E, which is evaluated with the Hamiltonian ĤS .
hη(t)| ĤS |η(t)i = hη(0)| Û (t)† ĤS Û (t) |η(0)i
h = hη(0)|
i ĤS |η(0)i = E. We assumed that
|η(0)i is an Eigenvector of ĤS and used ĤS , Û (t) = 0.
2.2.2 Heisenberg Picture
The main difference between the Schrödinger picture and the Heisenberg picture is
that the time evolution of an Observable is incorporated in its operator. An operator
5
2 Theoretical Concepts
in the Heisenberg picture is defined by ÂH (t) = Û (t)† Â(t)Û (t), in which Â(t) is the
known operator in the Schrödinger picture. The time evolution equation or Heisenberg
equation is simply derived from the Schrödinger picture
dÂH (t)
d i
i
∂ Â(t)
=
Û (t)† Â(t)Û (t) = ĤS ÂH (t) − ÂH (t)ĤS +
dt
dt
h̄
h̄
∂t
=
i
∂ Â(t)
ĤS , ÂH (t) +
h̄
∂t
h
i
!
H
!
.
(2.16)
H
The expectation value of an operator ÂH (t) is A(t) = hφ(0)| ÂH (t) |φ(0)i =
= hφ(0)| Û (t)† Â(t)Û (t) |φ(0)i = hφ(t)| Â(t) |φ(t)i. As you can see, the result is the
same as in the Schrödinger picture. If an operator in the Schrödinger picture is time
independent and commutes with the Hamiltonian, then the expectation value is a
conserved quantity.
2.2.3 Interaction Picture
The Interaction or Dirac picture is very important in this thesis, because it is used
to derivate the master equation and calculate the system dynamics in the following
chapters. It is often used if no exact analytic solutions of the system in the Schrödinger
picture is known or the Heisenberg equation can not be solved. In the Interaction
picture both, state vector and observable are time-dependent [5].
First, the Hamiltonian in the Schrödinger picture is split into two parts H = H0 + V (t),
where H0 is the part where exact analytic results are known and V (t) is the rest, which
can depend on time. In our case it is the interaction term between the system and
the reservoir (thermal bath). The idea is to make the observables time-dependent in
terms of H0 , while the explicit time-dependency induced by the interaction of V (t)
is included in the state vectors. This can be done by defining the state vector as
|ψ(t)iD = ÛH† 0 (t) |ψi, where ÛH0 (t) = exp(− h̄i Ĥ0 (t)) and |ψi = |ψ(t)i is the state vector
in the Schrödinger-picture. The transformation of an observable into the interaction
picture is given by the claim that the expectation value does not change,
!
hψ(t)|D ÂD (t) |ψ(t)iD = hψ(t)| Â(t) |ψ(t)i
= hψ(t)|D ÛH† 0 (t)Â(t)ÛH0 (t) |ψ(t)iD .
(2.17)
Therefore, an operator in the interaction picture is given by ÂD (t) = ÛH† 0 (t)Â(t)ÛH0 (t).
The dynamic of an observable can be derived similarly to the Heisenberg equation. The
6
2 Theoretical Concepts
motion of the state vector can be calculated from its total time derivation,
d
|ψ(t)iD =
dt
!
d †
d
ÛH0 (t) |ψ(t)i + ÛH† 0 (t) |ψ(t)i
dt
dt
i
i
= Ĥ0 ÛH† 0 (t) |ψ(t)i − ÛH† 0 (t)Ĥ |ψ(t)i
h̄
h̄
i †
= ÛH0 (t) Ĥ0 − Ĥ ÛH0 (t) |ψ(t)iD
h̄
i
= − V̂D (t) |ψ(t)iD ,
h̄
†
with V̂D (t) = ÛH0 (t)V̂ (t)ÛH0 (t).
(2.18a)
(2.18b)
It looks like the Schrödinger equation with the interaction term transformed into the
interaction picture.
2.3 Open Quantum Systems
In reality it is impossible to isolate the system of interest from its surroundings. When
solving the system of interest we always have to consider that there is an interaction
between the system and its surroundings, which can affect the systems solution. Usually,
we are not able to track the system and environment (or reservoir), neither theoretically
nor experimentally. Hence we seek a description which gives the main influence of the
surroundings, without considering the full description of the environment. This is given
by the master equation, which is discussed in section 2.4. An open system consists of
two parts, the system of interest S and the reservoir R. When these two systems are
combined, they are a closed system and its time evolutions is given in section 2.2. The
Hamiltonian of the full system is given by
Ĥ = ĤS ⊗ 11R + V + 11 ⊗ ĤR ,
(2.19)
where the index R or S means that these is an operator in the Hilbert space of R or
S. Therefore, the Hilbert space of the complete system is a product space of R and S.
The operator V has parts in both spaces and gives the interaction between S and R.
From now on we drop the indices and tensor products.
2.4 Master Equation
The master equation which is formally a Lindblad equation in quantum mechanics
describes a non-unitary time evolution of a density operator.
7
2 Theoretical Concepts
2.4.1 Density Operator
The density operator is the most universal way to describe the state of a quantum
mechanical system and is usually defined in the Schrödinger-picture.
The description or measurement of a state is unique, if a complete set of commuting
observables (CSCO) is given or measured or given at time t. If the set of observables is
incomplete, there is missing information about the system’s state. This is also called
a mixed state. A mixed state is a statistical ensemble with probabilities pi , that the
system is in the pure state |ψi i, with hψi |ψj i = δi,j . A pure state is reached, when a
CSCO is measured at time t. Some examples for a mixed state are the spin-polarisation
in the Stern-Gerlach experiment or the state of a macroscopic solid (it’s not possible
to measure the state of 1023 electrons at the same time). The density operator in a
discrete basis is defined as
ρ=
X
pi |ψi i hψi | ,
(2.20a)
i
with 0 ≤ pi ≤ 1,
X
pi = 1.
(2.20b)
i
Hence, the expectation value of an observable  is given by
A=
X
X
pi hψi | Â |ψi i =
i
=
pi hψi | 11Â11 |ψi i
i
XX
pi hψi |ηn i hηn | Â |ηm i hηm |ψi i
i m,n
=
X
hηn | Â |ηm i
X
m,n
=
X
pi hψi |ηn i hηm |ψi i
i
Am,n ρm,n =
m,n
X
(Aρ)m
m
= T r[Âρ] = T r[ρÂ]
(2.21)
Tr[.] is the trace and we assume that the wave function |ηi i is an Eigenvector to Â. The
expectation value is a statistical average over the different properties pi , which results
from the incomplete information. The time evolution of the density operator or Von
Neumann equation can be derivated from the Schrödinger equation,
∂ρ(t) X
=
pi
∂t
i
"
!
#
∂
∂
|ψi i hψi | + |ψi i hψi |
∂t
∂t
X
i
i
=
pi − Ĥ |ψi i hψi | + |ψi i hψi | Ĥ
h̄
h̄
i
i
ih
= − Ĥ, ρ(t) .
h̄
8
(2.22)
2 Theoretical Concepts
2.4.2 Master Equation
In this section, the master equation for a system coupled to a reservoir is derived. First
we begin with a complete Hamiltonian H = HS + HR + V , where HS , HR are the
Hamiltonians for the System S and Reservoir R and V is the interaction term in the
Schrödinger picture. To evaluate the time evolution of the density operator χ(t) of the
whole system (S ⊗ R), we have to transform the Von Neumann equation (2.22) into
the interaction picture. For this we need the unitary operator transformation used in
(2.17) with H0 = HS + HR . From now on we use the ∼ symbol for an operator in the
interaction picture
∂
i
χ(t) = − [H(t), χ(t)]
∂t
h̄
i
∂
ih
ÛH0 (t)χ̃(t)ÛH† 0 (t) = − ÛH0 (t)H̃(t)ÛH† 0 (t), ÛH0 (t)χ̃(t)ÛH† 0 (t)
∂t
h̄
!
h
i
∂
i
i
UH0 (t) − [H0 , χ̃(t)] + χ̃(t) UH† 0 (t) = − ÛH0 (t) H̃(t), χ̃(t) ÛH† 0 (t)
h̄
∂t
h̄
i
∂
i
ih
i
− [H0 , χ̃(t)] + χ̃(t) = − [H0 , χ̃(t)] −
Ṽ (t), χ̃(t)
h̄
∂t
h̄
h̄
i
∂
ih
χ̃(t) = − Ṽ (t), χ̃(t) .
(2.23)
∂t
h̄
The formal integration of (2.23) gives
χ̃(t) = χ(0) +
i
1 Z t 0h
dt Ṽ (t0 ), χ̃(t0 ) .
ih̄ 0
(2.24)
As you can see this equation can be solved by iteration. After one more iteration
equation (2.24) can be written as
χ̃(t) = χ(0) +
Z t
0
0
dt
!
i
h
ii
1 h
1 Z t0 00 h
0
0
00
00
Ṽ (t ), χ̃(0) − 2
dt Ṽ (t ), Ṽ (t ), χ̃(t )
,
ih̄
h̄ 0
(2.25)
and after a differentiation of the whole equation, the time evolution is
i
h
ii
1 h
1 Z t 0h
˙
Ṽ (t), χ̃(0) − 2
dt Ṽ (t), Ṽ (t0 ), χ̃(t0 ) .
χ̃(t)
=
ih̄
h̄ 0
(2.26)
Note, that this equation is exact, equation (2.22) is only cast into a form, where
reasonable approximations can be made. Now, we define the reduced density operator
ρ(t) of the system as
ρ(t) = T rR [χ(t)] ,
(2.27)
where the trace is only taken over the reservoir. We assume that the interaction is
turned on at time t=0, therefore the density operator factorizes as χ(0) = ρ(0)R0 , where
9
2 Theoretical Concepts
R0 is the initial reservoir density operator. Performing the partial trace in equation
(2.27) we get
Z t
nh
h
iio
nh
io
˙ρ̃(t) = 1 T rR Ṽ (t), χ̃(0) − 12
dt0 T rR Ṽ (t), Ṽ (t0 ), χ̃(t0 )
,
ih̄
h̄ 0
(2.28)
with T rR (χ̃) = e(i/h̄)HS t ρe(−i/h̄)HS t = ρ̃. Note, that this time evolution is non-unitary
from this point on. An example is the spontaneous decay (see chapter 3) of an atom
(system) in the vacuum field (reservoir). When an atomic state decays, a photon is sent
out into the reservoir. Therefore the energy is not obtained. It is also not a pure state
any more, because tracing over degrees of freedom means, that reservoir observables
can not be evaluated. Hence, a CSCO cannot be found.
If the interaction Hamiltonian V has the specific form
V = h̄
X
βi Γi ,
(2.29)
i
where Γi and βi are proportional to lowering or rising operators of the reservoir and
system, respectively, then the first term in equation (2.28) can be evaluated,
T rR
nh
io
Ṽ , χ(0)
n
o
= T rR e(i/h̄)H0 t V (t)e−(i/h̄)H0 t χ(0) + . . .
=
X
χn hun | V |un i + . . .
n
=
X
hun | Γi |un i βi + · · · = 0
(2.30)
i,n
where |un i is the orthonormal basis of the reservoir.
The density operator factorizes at t = 0 and we assume that the coupling V between
S and R is weak. However, χ(t) should only show deviations in the order of V from
an uncorrelated state. The reservoir is a very large system and should be therefore
unaffected by the interaction. Then the density operator can be written as χ̃(t) =
ρ̃(t)R0 + O(V ).
The first major approximation is a the so-called Born approximation. We neglect terms
higher than second order in V , so equation (2.28) reads
Z t
nh
h
iio
˙ = − 12
ρ̃(t)
dt0 T rR Ṽ (t), Ṽ (t0 ), ρ̃(t0 )R0
.
(2.31)
h̄ 0
The detailed discussion of this approximation can be found in the work of Haake
[3]. An important property of this equation is, that it is not Markovian, the density
operator depends on its past history. The second major approximation is therefore the
replacing of ρ(t0 ) by ρ(t). This can be done if the correlation functions in eq. (2.34a)
and (2.34b) are proportional to δ(t − t0 ). After these two approximations (Born-Markov
approximation) it looks like:
Z t
h
iio
nh
˙ρ̃(t) = − 12
dt0 T rR Ṽ (t), Ṽ (t0 ), ρ̃(t)R0
.
h̄ 0
10
(2.32)
2 Theoretical Concepts
A detailed discussion of the Markov-approximation can be found in [1] .
Therefore it becomes
˙ =−
ρ̃(t)
XZ t
i,j
=−
i,j
+
=−
0
XZ t
h
0
nh
h
dt0 T rR
nh
β̃i (t)Γ̃i (t), β̃j (t0 )Γ̃j (t0 ), ρ̃(t)R0
iio
β̃i (t)β̃j (t0 )ρ̃(t) − β̃j (t0 )ρ̃(t)β̃i (t) Γ̃i (t)Γ̃j (t0 )R0
ρ̃(t)β̃j (t0 )β̃i (t) − β̃i (t)ρ̃(t)β̃j (t0 ) Γ̃j (t0 )Γ̃i (t)R0
XZ t
i,j
dt0 T rR
0
dt0
nh
i
io
i
β̃i (t)β̃j (t0 )ρ̃(t) − β̃j (t0 )ρ̃(t)β̃i (t) hΓ̃i (t)Γ̃j (t0 )iR
h
i
o
+ ρ̃(t)β̃j (t0 )β̃i (t) − β̃i (t)ρ̃(t)β̃j (t0 ) hΓ̃j (t0 )Γ̃i (t)iR ,
(2.33)
where we used the correlation functions
h
i
(2.34a)
h
i
(2.34b)
hΓ̃i (t)Γ̃j (t0 )iR = T rR R0 Γ̃i (t)Γ̃j (t0 ) ≈ δ(t − t0 ),
hΓ̃j (t0 )Γ̃i (t)iR = T rR R0 Γ̃j (t0 )Γ̃i (t) ≈ δ(t − t0 ).
This is the master equation which is needed for the next two chapters.
If the master equation is given in the interaction picture, the back transformation to
the Schrödinger picture is
ρ̇ =
1
˙ (i/h̄)H0 t .
[H0 , ρ] + e−(i/h̄)H0 t ρ̃e
ih̄
11
(2.35)
Chapter 3
Spontaneous Emission of a Single
Atom
In the following chapter we want to derive a model for spontaneous emission of an
atom, which is in a thermal bath at zero temperature. That means that the electric and
magnetic fields are zero on average, but not their variances. Hence, vacuum fluctuations
are interacting with the atom and causing it to decay.
3.1 Two Level Atom
A two level system is a very important
model for an atom, which is often used
in quantum mechanics. It consists of the
corresponding two states, the ground (|1i)
and excited (|2i) state. These states energies E1 and E2 , with E1 ≤ E2 . We
consider the special case where the two
levels are the two deepest bound electronic
states of the valence electron in an atom
(see figure 3.1).
These approximations are valid as long as
the atom is prepared in state |1i or |2i (or
a superposition) and no higher states can
be reached. This can be achieved if the
electric field is in thermal equilibrium. If
it is not in thermal equilibrium, there is
a chance that state |3i or higher can be
reached, which is not part of this description.
The Hamiltonian of a two level system is Figure 3.1: Schematic representation of the
hydrogen term diagram, with a two level system
12
3 Spontaneous Emission of a Single Atom
given by
ĤA = |1i h1| E1 + |2i h2| E2 .
(3.1)
Every two level system can mathematically be described by the Pauli spin matrices.
These matrices are defined as
!
0 1
σx =
= |1i h2| + |2i h1| ,
1 0
(3.2a)
!
0 −i
σy =
= i |1i h2| − i |2i h1| ,
i 0
(3.2b)
!
1 0
σz =
= |2i h2| − |1i h1| ,
0 −1
!
(3.2c)
!
0
1
if we set |1i =
and |2i =
. Therefore the Hamiltonian can be written as
1
0
1
1
(3.3)
ĤA = (E1 + E2 )11 + (E2 − E1 )σz .
2
2
The first term in eq. (3.3) is a constant energy shift and can eliminated if we move the
atomic energies to the mean of E1 and E2 . We identify the atomic transition frequency
ωA = (E2 − E1 )/h̄ and write equation (3.3) as
1
ĤA = h̄ωA σz .
2
(3.4)
3.2 Dipole Approximation
In this section we discuss the derivation of the dipole interaction between a charged
particle and the electromagnetic field.
3.2.1 Minimal Coupling
The force on a classical particle in an electromagnetic field is the Lorentz force. Hence
the equation of motion is given by
i
h
~ r, t) + ~r˙ × B(~
~ r, t) .
m~r¨ = q E(~
(3.5)
Here q is the charge of the particle and m its mass. It can be shown that the Hamilton
function
1 ~ r, t) 2 + qΦ(~r, t)
H=
p~ − q A(~
(3.6)
2m
leads to (3.5). This Hamilton function can be won by replacing the kinematic momentum
~ r, t) in the Hamilton function of a free particle
p~ by the canonical momentum p~ − q A(~
and add qΦ(~r, t). This is called principle of minimal coupling.
13
3 Spontaneous Emission of a Single Atom
3.2.2 Approximation
~ therefore the
The quantisation of equation (3.6) is made by replacing p~ → −ih̄5,
Schrödinger equation has the form
1 ∂
~ − q A(r̂,
~ t) 2 + qΦ(r̂, t) + V (r̂) |ψi .
−ih̄5
ih̄ |ψi =
∂t
2m
(3.7)
We add another potential V (r̂), which in our case is the coulomb potential of the core
Figure 3.2: Representation of the vectors in an core-electron system.
at the position of the electron.
The main approximation is to replace the position operator r̂, which is the location of
~ and Φ (see
the electron by the position of the core ~rA , which is fixed in the terms of A
figure 3.2). This approximation is justified if the electromagnetic field force is nearly
constant over the expansion of the atom. Therefore the wavelength λ must be greater
than the expansion of the atom. For example, the wavelength of visible light is about
λ = 500nm and the Bohr radius is a0 = 0.53Å, hence there is a difference of about 4
orders of magnitude. Therefore, equation (3.7) becomes
∂
1 ~ − q A(~
~ rA , t) 2 + qΦ(~rA , t) + V (r̂) |ψi .
ih̄ |ψi =
−ih̄5
∂t
2m
(3.8)
The gauge freedom of classical electrodynamics can be used to
~ 0 (~r, t) = A(~r, t) + grad Λ(~r, t),
A
∂
Φ0 (~r, t) = Φ(~r, t) − Λ(~r, t),
∂t
14
(3.9a)
(3.9b)
3 Spontaneous Emission of a Single Atom
where Λ(~r, t) is an arbitrary scalar function. Starting from the Coulomb gauge, then
~ rA , t) = 0 and Φ(~rA , t) = 0. If we choose Λ(~rA , t) = −(~r − ~rA )A(~
~ rA , t), the vector
divA(~
and scalar potential becomes
~ 0 (~rA , t) = A(~
~ rA , t) − grad ~rA(~
~ rA , t) = 0,
A
(3.10a)
∂
~ rA , t) = (~r − ~rA ) ∂ A(~
~ rA , t).
(~r − ~rA )A(~
∂t
∂t
~ rA , t) = −grad Φ0 (~rA , t) −
The classical field can be calculated as E(~
∂ ~
∂ ~
−grad (~r − ~rA ) ∂t
A(~rA , t) = − ∂t
A(~rA , t). Therefore equation (3.8) reads
Φ0 (~rA , t) = Φ(~rA , t) +
∂
−h̄2
~ rA , t) + V (r̂) |ψi .
ih̄ |ψi =
4 −q(r̂ − ~rA )E(~
∂t
2m
"
(3.10b)
∂
A0 (~rA , t)
∂t
=
#
(3.11)
The second term on the right side looks like the energy of a dipole in a electric field.
Hence we identify q(r̂ − ~rA ) as the dipole operator, which couples to the electric field
[7].
3.3 Spontaneous Emission
Now, we want to deduce the master equation for an atom surrounded by a thermal bath
at temperature T=0 (i.e. the vacuum), which acts as a reservoir of harmonic oscillators
(see section 2.1.2). The first step for this has already been done in section 2.4.2.
3.3.1 Representation of an Operator
If a complete set of orthonormal states as used in (3.1), |1i and |2i, is given, every
Operator  can be rewritten as
 = 11Â11 =
X
|ni hn| Â |mi hm| .
(3.12)
m,n
As an example we expand the dipole operator q(r̂ − ~rA ) = qr̂0 , where r̂0 is the distance
between the electron and core in states of the two level atom
qr̂0 = q
2
X
hn| r̂0 |mi |ni hm|
m,n=1
= q (h2| r̂0 |2i |2i h2| + h1| r̂0 |1i |1i h1| + h1| r̂0 |2i |1i h2| + h2| r̂0 |1i |2i h1|)
= q (h1| r̂0 |2i |1i h2| + h2| r̂0 |1i |2i h1|) = ~d1,2 σ− + ~d2,1 σ+ .
(3.13)
Here, we used h1| r̂0 |1i = h2| r̂ |2i = 0 under the assumption that those states are
symmetric, which guarantees a zero dipole moment. We also defined the dipole matrix
elements, which are
∗
h1| r̂0 |2i = ~d1,2 = ~d2,1 ,
(3.14)
the lowering σ− = |1i h2| and rising σ+ = |2i h1| operator.
15
3 Spontaneous Emission of a Single Atom
3.3.2 Master Equation for a Two-Level Atom in Thermal
Equilibrium
The Hamiltonian which has the form of H = HS + HR + V , with
1
HS = h̄ωA σz ,
2
X
HR =
h̄ω~k a~†k,λ a~k,λ ,
(3.15a)
(3.15b)
~k,λ
∗
~ rA , t).
V = − ~d1,2 σ− + ~d1,2 σ+ E(~
(3.15c)
The atomic Hamiltonian is given from (3.4), the Hamiltonian of the reservoir is from
(2.11) and the interaction term is from equation (3.11).
The interaction term has to be rewritten
∗
~ rA , t) =
V = − ~d1,2 σ− + ~d1,2 σ+ E(~
=−
X ~
ikrA
e
∗
~
E~k,λ~e~k,λ a~k,λ~d2,1 σ− + eikrA E~k,λ~e~k,λ a~k,λ~d2,1 σ+
~k,λ
∗
~
~
+ e−ikrA E~k,λ~e~k,λ a †~k,λ ~d2,1 σ− + e−ikrA E~k,λ~e~k,λ a~†k,λ~d2,1 σ+ .
(3.16)
This has to be transformed into the interaction picture of HS + HR and changes to
Ṽ = −
X ~
ikrA
e
∗
~
E~k,λ~e~k,λ a~k,λ~d2,1 σ− e−i(ωA +ω~k ) + eikrA E~k,λ~e~k,λ a~k,λ~d2,1 σ+ e−i(ωA −ω~k )
~k,λ
∗
~
~
+ e−ikrA E~k,λ~e~k,λ a~†k,λ~d2,1 σ− e−i(−ωA +ω~k ) + e−ikrA E~k,λ~e~k,λ a~†k,λ~d2,1 σ+ ei(ωA +ω~k ) . (3.17)
Now the rotating wave approximation has to be done, therefore all fast oscillating terms
like e−i(ωA +ω~k ) and ei(ωA +ω~k ) are eliminated [4]. After the transformation back into the
Schrödinger picture the interaction term becomes
V =
X
~k,λ
h̄ κ~∗k,λ a~†k,λ σ− + κ~k,λ a~k,λ σ+ ,
i~krA
s
with κ~k,λ = e
ω~k ~e~k,λ , ~d2,1 .
2h̄0 V
(3.18a)
(3.18b)
We identify some terms from equation (2.29)
β1 = σ− ,
Γ1 = Γ† =
β2 = σ+
X
~k,λ
(3.19a)
κ~∗k,λ a~†k,λ ,
Γ2 = Γ =
X
~k,λ
16
κ~k,λ a~k,λ .
(3.19b)
3 Spontaneous Emission of a Single Atom
These operators must be transformed into the interaction picture:
Γ̃1 (t) = Γ̃† (t) =
κ~∗k,λ a~†k,λ eiω~k t ,
(3.20a)
κ~k,λ a~k,λ e−iω~k t
(3.20b)
X
~k,λ
Γ̃2 (t) = Γ̃(t) =
X
~k,λ
and
β̃1 (t) = σ− e−iωA t ,
β̃2 (t) = σ+ eiωA t .
(3.21)
By inserting these operators into equation (2.33), where the Markov approximation is
not done, we get
˙ =−
ρ̃(t)
Z t
0
dt0
0
n
[σ− σ− ρ̃(t0 ) − σ− ρ̃(t0 )σ− ] e−iωA (t+t ) hΓ̃† (t)Γ̃† (t0 )iR + h.c.
0
+ [σ+ σ+ ρ̃(t0 ) − σ+ ρ̃(t0 )σ+ ] eiωA (t+t ) hΓ̃(t)Γ̃(t0 )iR + h.c.
0
+ [σ− σ+ ρ̃(t0 ) − σ+ ρ̃(t0 )σ− ] e−iωA (t−t ) hΓ̃† (t)Γ̃(t0 )iR + h.c.
0
o
+ [σ+ σ− ρ̃(t0 ) − σ− ρ̃(t0 )σ+ ] eiωA (t−t ) hΓ̃(t)Γ̃† (t0 )iR + h.c. .
(3.22)
If we set the reservoir
in thermal
equilibrium, then the density operator R0 =
Q h̄ω a† a/kB T −h̄ω~k /kB T
~
k
e
1
−
e
,
where
kB is the Boltzmann constant. Therefore, the
j
reservoir correlation function can explicitly be calculated
hΓ̃† (t)Γ̃(t0 )iR =
X
0
|κ~k,λ |2 eiω~k (t−t ) n̄(ω~k , T ),
(3.23a)
~k,λ
hΓ̃(t)Γ̃† (t0 )iR =
X
0
h
i
|κ~k,λ |2 e−iω~k (t−t ) n̄(ω~k , T ) + 1 ,
(3.23b)
~k,λ
hΓ̃† (t)Γ̃† (t0 )iR = 0,
(3.23c)
0
hΓ̃(t)Γ̃(t )iR = 0,
(3.23d)
where
n̄(ω~k , T ) =
e−h̄ω~k /kB T
1 − e−h̄ω~k /kB T
(3.24)
is the mean number of photons for an oscillator with frequency ω~k at temperature T.
If we consider, that we discuss this phenomenon at T=0, then only hΓ̃(t)Γ̃† (t0 )iR is
drifferent from zero. This can be done since n̄(ωvisible , 300K) ≈ 10−35 , i.e. no photons
are in any visible mode at ambient temperature.
The summation in these non-zero correlation functions is changed into an integral,
where the number of modes in the wave vector volume of (k+dk)3 is still the same. For
this we have to define the mode density
V
g(~k) = 3 .
8π
17
(3.25)
3 Spontaneous Emission of a Single Atom
Making a variable change to τ = t − t0 in the integral of equation (3.22), this equation
becomes
˙ =−
ρ̃(t)
XZ t
×
dτ
0
λ
Z
[σ+ σ− ρ̃(t − τ ) − σ− ρ̃(t − τ )σ+ ]
~
d k g(~k)|κ(~k)|2 e−i(|k|c−ωA )τ + h.c. .
3
(3.26)
If we
evaluate as an example the dkx integral, we find an integral which is proportional
R∞
to 0 dkx e−ikx cτ . We can expand this integral to −∞, because the oscillating term will
average this part to zero. Now we have a Fourier transformation of a constant function,
which is proportional to δ(τ ), for a detailed discussion see [1]. Then ρ̃(t − τ ) can be
replaced by ρ̃(t), which is called the Markov approximation. Therefore, we can deduce
that
˙ =
ρ̃(t)
X
[σ− ρ̃(t)σ+ − σ+ σ− ρ̃(t)]
Z
3
dk
Z t
~
dτ g(~k)|κ(~k)|2 e−i(|k|c−ωA )τ + h.c. .
(3.27)
0
λ
We know that the last integral behaves like δ(τ ), therefore we can extend the τ integration
to infinity and get
lim
Z t
t→∞ 0
~
dτ e−i(|k|c−ωA )τ = πδ(|~k|c − ωA ) + i
P
ωA − |~k|c
,
(3.28)
where P is the Cauchy principal value. Now we find
˙ = γ + i∆ [σ− ρ̃(t)σ+ − σ+ σ− ρ̃(t)] + h.c.
ρ̃(t)
2
(3.29)
and
γ = 2π
XZ
d3 k g(~k)|κ(~k)|2 δ(|~k|c − ωA ),
(3.30a)
λ
∆=
X
λ
P
Z
d3 k
g(~k)|κ(~k)|2
.
ωA − |~k|c
(3.30b)
After some rearrangements we get the following equation,
˙ = − i ∆ [σz , ρ̃] + γ (2σ− ρ̃(t)σ+ − σ+ σ− ρ̃(t) − ρ̃(t)σ+ σ− ) .
ρ̃(t)
2
2
(3.31)
We finally transform equation (2.35) back to the Schrödinger picture and get
i
γ
ρ̇(t) = − ωA0 [σz , ρ] + (2σ− ρ(t)σ+ − σ+ σ− ρ(t) − ρ(t)σ+ σ− ) ,
2
2
18
(3.32)
3 Spontaneous Emission of a Single Atom
with ωA0 = ωA + ∆. Note that ∆ is the Lamb-Shift at temperature T=0.
This is the master equation in Lindblad form, which can be written as
i
ih
ρ̇(t) = − ĤA0 , ρ + L[ρ],
h̄
(3.33)
with ĤA0 = h̄2 ωA0 σz and the Lindblad superoperator L[ρ] = γ2 (2σ− ρ(t)σ+ − σ+ σ− ρ(t)−
ρ(t)σ+ σ− ).
We identify γ as the damping constant. We can solve this integration if we change to
spherical coordinates in ~k-space
γ = 2π
=
XZ ∞
0
λ
3
ωA
2
8π 0 h̄c3
dω
Z π
sin(θ)dθ
0
XZ π
λ
0
Z 2π
0
sin(θ)dθ
Z 2π
0
2
ω2V
ω dφ 2 3
~e~k,λ · d~1,2 δ(ω − ωA )
8π c 2h̄0 V
dφ ~e~k,λ · d~1,2
2
(3.34)
.
For each ~k we choose the orthogonal polarisation state λ1 and λ2 , such that ~e~k,λ1 · d~1,2 =
0, then
Z 2π
Z π
ωA3
~1,2 2
dφ
~
e
sin(θ)dθ
·
d
γ= 2
~
k,λ2
8π 0 h̄c3 0
0
Z
Z
π
2π
ω3
= d21,2 2 A 3
sin(θ)dθ
dφ 1 − cos2 (θ)
8π 0 h̄c 0
0
3 2
1 4ωA d1,2
=
.
(3.35)
4π0 3h̄c3
Now we can identify γ as the Einstein A coefficient, as obtained from the WignerWeisskopf theory of natural linewidth [10].
As an example we can solve the master equation from (3.32) with the denisty operator
ρ = ρ1 |1i h1| + ρ2 |2i h2|, where |2i and |1i are the excited and ground state from section
3.1.
γ
ρ̇1 |1i h1| + ρ̇2 |2i h2| = {2 |1i h1| ρ2 − 2 |2i h2| ρ2 }
(3.36)
2
If we multiply this equation from the left and right with h1| and |1i or h2| and |2i we
get two coupled differential equations
ρ̇2 = −γρ2 ,
ρ̇1 = γρ2 ,
(3.37a)
(3.37b)
ρ2 (t) = e−γt ,
ρ1 (t) = 1 − e−γt .
(3.38a)
(3.38b)
with the solutions
Where we assumed ρ2 (t = 0) = 1 and used the normalisation ρ1 + ρ2 = 1. The result is
an exponential decay of the probabilities of states.
19
3 Spontaneous Emission of a Single Atom
Figure 3.3: Plot of probability as a function of time, where the time is given in units of
1/Γ.
20
Chapter 4
Collective Atom Dynamics
For want of a better term, a gas which is radiating strongly because of coherence will be
called ’superradiant’.
— Robert H. Dicke, 1954 [2]
Spontaneous emission for multiple atoms can be treated independently only if their
distance is large compared to the emitted wavelength. For small distances the coupling
between the atoms and lightfield leads to collective effects like superradiance.
4.1 Superradiance Master Equation for N Two Level
Atoms
This section contains the derivation of the master equation for superradiance effects
of N identical atoms, which are surrounded by a thermal bath at temperature T=0.
This derivation is quite similar to the derivation of the master equation for spontaneous
emission in section 3.3.
4.1.1 Hamiltonian
The Hamiltonian for the whole system is given by H = HS + HR + V , where HS and
HR are the system and reservoir Hamiltonian respectively, with the interaction term V .
We treat these N atoms like N independent two level systems. Therefore, the system
Hilbert space is a tensor product of N two level systems. The reservoir Hamilton is the
same as in section 3.3 and the interaction term is the dipole coupling between each
21
4 Collective Atom Dynamics
single atom and the lightfield. Hence those operators are
HA =
1
h̄ωA σz,α ,
2
(4.1a)
h̄ω~k a~†k,λ a~k,λ ,
(4.1b)
X
α
HR =
X
~k,λ
V =
X
α,~k,λ
h̄ κ∗α,~k,λ a~†k,λ σ−,α + κα,~k,λ a~k,λ σ+,α .
(4.1c)
Here, σz,α , σ−,α and σ+,α are the Pauli spin operators for the αth atom. The interaction
Hamiltonian is already in the rotating wave approximation
which is done in the same
q ω
~
i~krA,α
k
~
way as in section (3.17), with κα,~k,λ = e
~e d , where ~rA,α is the location
2h̄0 V ~k,λ 2,1
th
of the α atom. Now we have to transform the interaction term into the interaction
picture of H0 = HA + HR and it becomes
Ṽ =
X
α,~k,λ
h̄ κ∗α,~k,λ a~†k,λ σ−,α e−i(ωA −ω~k )t + κα,~k,λ a~k,λ σ+,α ei(ωA −ω~k )t .
(4.2)
4.1.2 Derivation of the Master Equation
We begin with the master equation (2.31) from Chapter 2. By inserting Ṽ from equation
(4.2), we find
ρ̃˙ = −
Z t
0
dt0
X X n
κβ,~l,θ κ∗α,~k,λ [σ+,β σ−,α ρ̃(t0 ) − σ−,α ρ̃(t0 )σ+,β ]
α,~k,λ β,~l,θ
0
× e−i(ωA −ω~k )t ei(ωA −ω~l)t ha~l,θ a~†k,λ i
(4.3)
R
0
+ κβ,~l,θ κ∗α,~k,λ [ρ̃(t0 )σ−,α σ+,β − σ+,β ρ̃(t0 )σ−,α ] e−i(ωA −ω~k )t ei(ωA −ω~l)t ha~†k,λ a~l,θ i
R
+
+
κ∗β,~l,θ κα,~k,λ
κ∗β,~l,θ κα,~k,λ
0
0
i(ωA −ω~k )t0 −i(ωA −ω~l)t
[σ−,β σ+,α ρ̃(t ) − σ+,α ρ̃(t )σ−,β ] e
0
0
e
i(ωA −ω~k
[ρ̃(t )σ+,α σ−,β − σ−,β ρ̃(t )σ+,α ] e
)t0
ha~†l,θ
a~k,λ i
R
e−i(ωA −ω~l)t ha~l,θ a~†k,λ i
R
,
which we already used in (3.23c) and (3.23d), such that the correlation functions
ha~†l,θ a~†k,λ i
R
= ha~l,θ a~k,λ iR = 0. In equation (2.30) we have shown that the correlation functions
ha~ a† i = ha† a~ i = 0 for ~k 6= ~l and λ 6= θ. The remaining correlation functions
l,θ
~k,λ
R
~l,θ
can be evaluated as
k,λ R
ha~k,λ a~†k,λ i = n̄(ω~k , T ) + 1,
(4.4a)
ha~†k,λ a~k,λ i = n̄(ω~k , T ).
(4.4b)
R
R
22
4 Collective Atom Dynamics
Here n̄(ω~k , T ) is the same as used in (3.24) and is the mean photon number in the k th
mode. Now we consider, that the temperature is T = 0, then n̄(ω~k , T ) = 0 and only
ha~k,λ a~†k,λ i is not zero. Therefore ρ̃˙ becomes
R
Z t
ρ̇˜ = −
dt0
0
0
n
κβ,~k,λ κ∗α,~k,λ [σ+,β σ−,α ρ̃(t0 ) − σ−,α ρ̃(t0 )σ+,β ] ei(ωA −ω~k )(t−t )
X
α,β,~k,λ
0
o
+ [ρ̃(t0 )σ+,α σ−,β − σ−,β ρ̃(t0 )σ+,α ] e−i(ωA −ω~k )(t−t ) .
(4.5)
Now, the summation over ~k is changed into an integral, with the mode density g(~k) =
from equation (3.25)
ρ̇˜ = −
Z t
0
dt
Z
d3 k g(~k)
0
V
8π 3
~
n
0
κβ,~k,λ κ∗α,~k,λ [σ+,β σ−,α ρ̃(t0 ) − σ−,α ρ̃(t0 )σ+,β ] ei(ωA −|k|c)(t−t )
X
α,β,λ
~
0
0
o
+ [ρ̃(t )σ+,α σ−,β − σ−,β ρ̃(t0 )σ+,α ] e−i(ωA −|k|c)(t−t ) .
=−
Z t
dτ
Z ∞
0
0
dω
ω2V X Z
dΩ κβ,~k,λ κ∗α,~k,λ
8π 3 c3 α,β,λ
n
× [σ+,β σ−,α ρ̃(t − τ ) − σ−,α ρ̃(t − τ )σ+,β ] ei(ωA −ω)τ
o
+ [ρ̃(t − τ )σ+,α σ−,β − σ−,β ρ̃(t − τ )σ+,α ] e−i(ωA −ω)τ .
(4.6)
In the last step we choose spherical coordinates for the integral over ~k space and a
variable change to τ = t − t0 . We have shown in section 3.3, that the correlation
functions are proportional to δ(τ ). Hence, the Markov approximation can be made by
replacing ρ̃(t − τ ) with ρ(t). Now (4.6) can be restated as
ρ̇˜ = −
Z ∞
0
ω2V X Z
dΩ κβ,~k,λ κ∗α,~k,λ
dω
8π 3 c3 α,β,λ
× [σ+,β σ−,α ρ̃(t) − σ−,α ρ̃(t)σ+,β ]
+ [ρ̃(t)σ+,α σ−,β − σ−,β ρ̃(t)σ+,α ]
Z t
dτ ei(ωA −ω)τ
0
Z t
dτ e
−i(ωA −ω)τ
.
(4.7)
0
The spherical surface integral can be evaluated by choosing the polarisation vector ~eλ,1
orthogonal to d~2,1 :
2 ~
V XZ
1
ω Z dΩ ~ 2 ∗
~
dΩ κβ,~k,λ κα,~k,λ =
|d2,1 | − ~eλ,2 · d2,1
eik(~rA,β −~rA,α )
8π 3 c3 λ
4π0 πc3 h̄ 4π
sin ξ
1
ω
ˆ
=
1 − d~2,1 · ~rˆβ,α
3
4π0 πc h̄
ξ
1
ω
=
F (ξ),
4π0 πc3 h̄
2 !
ˆ
+ 1 − 3 d~2,1 · ~rˆβ,α
2 !
cos ξ sin ξ
− 3
ξ2
ξ
!!
(4.8)
23
4 Collective Atom Dynamics
~
ˆ
~
r
with ξ = |~k||~rβ,α |, ~rA,β − ~rA,α , d~2,1 = |dd~2,1 | and ~rˆβ,α = |~rβ,α
. The time integral was
β,α |
2,1
already evaluated in (3.28) . By using the spontaneous emission constant γ from (3.35),
equation (4.7) becomes
ρ̇˜ =
Z ∞
dω
0
X
α,β
3ω 3 γ
F
4ωA3 π
ω
|~rβ,α |
c
× {[2σ−,α ρ̃(t)σ+,β − σ+,β σ−,α ρ̃(t) − ρ̃(t)σ+,β σ−,α ] πδ(ω − ωA )
P
+ [ρ̃(t)σ+,α σ−,β − σ+,α σ−,β ρ̃(t)] i
ωA − ω
For the second term we use
Z ∞
3ω 3 γ
ω
P
±
F
|~rβ,α |
.
∆α,β =
dω
3
4ωA π
c
ωA ± ω
0
(4.9)
(4.10)
The integral of the first term can be evaluated and equation (4.9) changes to
X 3γ ωA
˜
ρ̇ =
F
|~rβ,α | [2σ−,α ρ̃(t)σ+,β − σ+,β σ−,α ρ̃(t) − ρ̃(t)σ+,β σ−,α ]
4
c
α,β
o
+
+ i ∆−
α,β + ∆α,β [ρ̃(t), σ+,α σ−,β ] .
(4.11)
Note, that the rotating wave approximation does not give the correct result for the
Lamb shift, therefore ∆+
α,β must be included [1]. After some conversions, (4.11) is given
by
X Γα,β
ρ̇˜ =
[2σ−,α ρ̃(t)σ+,β − σ+,β σ−,α ρ̃(t) − ρ̃σ+,β σ−,α ]
α,β 2
+i
X
α,β,α6=β
X
∆α,β
[ρ̃(t), σ+,α σ−,β ] + i
∆α [ρ(t), σz,α ] .
2
α
(4.12)
−
F ωc |~rβ,α | , where Γα,α = γ and ∆α,β = ∆+
We defined Γα,β = 3γ
α,β + ∆α,β . Now, we
2
have to transform the master equation back into the Schrödinger picture with equation
(2.35) and get
i
ρ̇(t) = − [HS0 , ρ(t)] + L[ρ],
(4.13)
h̄
with
X1
X
HS0 =
h̄ (ωA + ∆α,α ) σz,α +
h̄∆α,β σ+,β σ−,α
(4.14)
α 2
α,β,α6=β
and the Lindblad Superoperator
X Γα,β
L[ρ] =
[2σ−,α ρ(t)σ+,β − σ+,β σ−,α ρ(t) − ρ(t)σ+,β σ−,α ] .
α,β 2
(4.15)
As an example we plot Γα,β , for α 6= β and ~rα,β · d~1,2 = 0 as an function of |~rβ,α | in
figure 4.1.
24
4 Collective Atom Dynamics
Figure 4.1: Γα,β , for α 6= β as an function of |~rβ,α |, with ~rα,β · d~1,2 = 0.
4.2 Example for Two Two-Level Atoms
In this section we solve the master equation from (4.13) for two identical atoms and
discuss its photon flux.
In the product Hilbert space of two atoms a complete orthonormal basis is given by
|22i = |2i1 ⊗ |2i2 , |21i = |2i1 ⊗ |1i2 , |12i = . . . , |11i = . . . , where the indices denote
the first or second atom and |1i , |2i are the state vectors of the two level system from
section 3.1. It can be shown, that these vectors are no eigenvectors to HS0 from (4.14).
Therefore we define one triplet
|1, 1i = |22i ,
1
|1, 0i = √ (|21i + |12i) ,
2
|1, −1i = |11i
(4.16b)
1
|0, 0i = √ (|21i − |12i) ,
2
(4.16d)
(4.16a)
(4.16c)
and a singlet state as
which are eigenvectors of HS0 . A schematic representation, of the eigenvalues can be
seen in figure 4.2.
Now, we define the projecting operator as
Prm,r0 m0 = |rmi hr0 m0 | ,
25
(4.17)
4 Collective Atom Dynamics
Figure 4.2: Schematic representation of the state’s energies and their decay rates.
which satisfy
hPrm,r0 m0 i = ρrm,r0 m0 (t).
(4.18)
These operators fulfil the following identities,
σ+,1 σ−,1 + σ+,2 σ−,2 = P10,10 + P00,00 + 2P11,11 ,
1
σ+,1 σ−,2 = (P10,10 − P00,00 − P00,10 + P10,00 ) .
2
(4.19a)
(4.19b)
Hence, we can rewrite equation (4.13) where we use (4.18) and therefore get four coupled
differential equations for the diagonal elements of the reduced density operator
ρ̇11 = −2Γρ11 ,
ρ̇10 = (Γ + Γ1,2 ) ρ11 − (Γ + Γ1,2 ) ρ10 ,
ρ̇00 = (Γ − Γ1,2 ) ρ11 − (Γ − Γ1,2 ) ρ00 ,
ρ̇1−1 = (Γ + Γ1,2 ) ρ10 + (Γ − Γ1,2 ) ρ00 .
(4.20a)
(4.20b)
(4.20c)
(4.20d)
The emission rates are shown in figure 4.2 and
solution for the coefficients
the numeric
~
in figure 4.3. For this example we assumed ~r1,2 · d1,2 = 0 ,ρ22 (t = 0) = 1, which can
be achieved by a short π-pulse and |~r1,2 | ωcA = 0.6 π.
If Γ + Γ1,2 > Γ or Γ − Γ1,2 > Γ, then this decay channel it is called superradiant, if its
below Γ it is subradiant, see figure 4.1, where α = 1 and β = 2. In the case, that Γ1,2 is
nearly Γ, then the state |00i is extreme stable and is therefore subradiant.
26
4 Collective Atom Dynamics
Figure 4.3: Numerical plot of probabilities for superradiance effects as a function of
time, where the time is given in units of 1/Γ.
Now, we can calculate the number of emitted photons. We know, that every time
an electron decays, a photon is emitted. Hence, the photon flux I is proportional
d P
to dt
α hσ+,α σ−,α i. Therefore we get I ∝ 2Γρ2 for the photon flux for spontaneous
emission of two independent atoms (see (3.38a)). For the supperradiant case we get
I ∝ 2Γρ22 + (Γ + Γ1,2 )ρ10 + (Γ − Γ1,2 )ρ10 . As an example, where |~r1,2 |ωA /c = 2π 10−1 ,
we get a higher photon flux due to superradiant than with spontaneous emission of
independent atoms (see figure 4.4).
Figure 4.4: Decay rate (which is proportional to the photon flux) for superradiance and
spontaneous emission as a function of time, where the time is given in units of 1/Γ.
27
Chapter 5
Conclusions
As a result of the quantisation of the free electromagnetic field, it was shown, that each
classical mode is equivalent to an harmonic oscillator. With the result of the dipole
approximation it was possible to describe the interaction of the atoms and field.
We specially discussed the time evolution of the density operator of open systems,
where we made a Born Markov approximation. This master equation was solved for
the spontaneous emission of one atom. As a result we got the exponential decay of the
excited state.
Further we derived the master equation for an ensemble of N identical atoms and solved
it for the case of two atoms.From the discussion of the decay rates,we learned that
the superradiant system’s photon flux is higher for short times,then the system with
spontaneous decay.
28
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29