i
Entanglement: Engineering and its Dynamical
Studies in Dissipative Environments
By
Rabia Tahira
CIIT/FA08-PPH-011/ISB
Ph.D Thesis
In
Physics
COMSATS Institute of Information Technology
Islamabad- Pakistan Spring, 2012
ii
COMSATS Institute of Information Technology
Entanglement: Engineering and its Dynamical
Studies in Dissipative Environments
A Thesis Presented to
COMSATS Institute of Information Technology, Islamabad
In partial fulfillment
of the requirement for the degree of
Ph.D
(Physics)
By
Rabia Tahira
FA08-PPH-011
Spring, 2012
iii
Entanglement: Engineering and its Dynamical
Studies in Dissipative Environments
A Post Graduate Thesis submitted to the Department of Physics as partial
fulfillment of the requirement for the award of the Degree of Ph.D (Physics).
Name Registration Number
Rabia Tahira CIIT/FA08-PPH-011/ISB
Supervisor
Dr. Manzoor Ikram
Ex. Professor, Centre for Quantum Physics, CIIT, Islamabad
National Institute of Lasers and Optronics, Islamabad
April, 2012
iv
Final Approval
This thesis titled
Entanglement: Engineering and its Dynamical
Studies in Dissipative Environments
By
Rabia Tahira
CIIT/FA08-PPH-011/ISB
Has been approved
For the COMSATS Institute of Information Technology, Islamabad
External examiner-1:_________________________________________
Prof. Dr. Aslam Baig
Department of Physics, Quaid-i-Azam University, Islamabad
External examiner-2:_________________________________________
Dr. Mian Mohammad Ashraf
Principal Scientific Officer,
Pakistan Institute of Laser and Optics, Rawalpindi
Supervisor:____________________________________________
Prof. Dr. Manzoor Ikram
Centre for Quantum Physics, CIIT, Islamabad
HoD: _________________________________________________
Dr. Mahnaz Haseeb
Department of Physics, CIIT, Islamabad
Dean, Faculty of Science _________________________________
Prof. Dr. Arshad Saleem Bhatti
CIIT, Islamabad
v
Declaration
I Rabia Tahira CIIT/FA08-PPH-011/ISB hereby declare that I have produced the work
presented in this thesis, during the scheduled period of study. I also declare that I have
not taken any material from any source except referred to wherever due that amount of
plagiarism is within acceptable range. If a violation of HEC rules on research has
occurred in this thesis, I shall be liable to punishable action under the plagiarism rules of
the HEC.
Date: _________________ Signature of the student:
____________________
(Rabia Tahira)
CIIT/FA08-PPH-011/ISB
vi
Certificate
It is certified that Rabia Tahira with Reg. No. CIIT/FA08-PPH-011/ISB has carried out
all the work related to this thesis under my supervision at the Centre for Quantum
Physics, Department of Physics, COMSATS Institute of Information Technology,
Islamabad and the work fulfills the requirement for the award of PhD degree.
Date: _________________
Supervisor:
______________________________
Dr. Manzoor Ikram
Ex. Professor, Centre for Quantum Physics, CIIT
National Institute of Lasers and Optronics, Islamabad
Head of Department:
_________________________
Dr. Mahnaz Haseeb
COMSATS Institute of Information
Technology, Islamabad.
vii
Dedicated
To
Allah (Glorified and Exalted He is), Hazrat Muhammad
(P.B.U.H.)
To
the memory of my parents
Safia Tahira and Khurshid Ahmed, my uncle
Anees urRehman and my Abbi.
viii
ACKNOWLEDGEMENTS
Beginning with the name of ALLAH, the most beneficent and the most merciful, who is
entire source of all knowledge and wisdom endowed to mankind and who blessed me
with enough courage to carry out this research and bless me with so many helpful people
without whom, it would have been impossible for me to carry out my work. I glorify
Almighty ALLAH and ask blessings and send salutations upon His last Prophet
MUHAMMAD (PBUH) and praise and venerate may submit to him. All praise is to
ALLAH whose favors are the sole cause for enabling me to accomplish the tedious job of
furnishing this work.
It is my privilege to express deep sense of cordial gratitude to my respected supervisor,
Prof. Dr. Manzoor Ikram, for his inestimable guidance, keens interest, constant help and
intellectual stimulation throughout the course of this research work. I really appreciate his
kind way of encouraging and directing me to improve the final draft of my research work.
I express deep appreciation to the chairman and HoD, Department of Physics, for
providing all the facilities and conductive atmosphere during my course work. I would
like to thank all the faculty members of the department of physics for having
indispensable guidance and support during my course and research work.
Very especially, I am so much indebted to Dr. M. Suhail Zubairy for his inspiring
guidance, precious suggestions and continuous help in my research work. No words can
ever cover my gratitude and appreciations to him. I, specially want to acknowledge Dr.
Hyunchul Nha for his help and guidance in my research papers. I am grateful to Dr.
Ashfaq H. Khosa, Dr. Fazal Ghafoor and Dr. Faheel Hashmi and all the staff and students
at the Centre for Quantum Physics for their help in my research work.
It would always remain incomplete without my thanks to my parents and to my family. I
am so much grateful to my brother Saad and my Abbi; their prayers always work for me.
Very especially, thanks to my uncles Ch. Saadat Ali, Ch. Khalil urRehman and Ch. Jalil
urRehman who really motivated me throughout my education. I am also thankful to all
my cousins and friends whose best wishes and help lighten the load of the work.
Rabia Tahira
CIIT/FA08-PPH-011/ISB
ix
ABSTRACT
Entanglement: Engineering and its Dynamical Studies in
Dissipative Environments
This thesis covers the two inter-related topics of recent research in the field of quantum
information theory. One topic is about the robust generation of entanglement among
Gaussian states in quantum optical passive and active devices like beam splitters and
quantum beat lasers and the other topic is about the degradation of entanglement due to
decoherence when entangled states interact with the surrounding environments. The
subject of entanglement generation is studied in continuous variable systems for two
initial separable single-mode Gaussian states in beam splitters and in quantum beat lasers.
As a general treatment, one single-mode Gaussian state is defined in terms of arbitrary
values of nonclassicality and purity while the other single-mode Gaussian state is
considered as the thermal field. The role of different parameters in the presence of
thermal noise on the entanglement generation is explored, for example, the
nonclassicality and purity of single-mode Gaussian state; and angle of beam splitter or
driving field strength in quantum beat laser are important for robust entanglement
generation. For entanglement analysis, logarithmic negativity is implied as a measure of
entanglement for two-mode Gaussian states.
Second topic investigates the dynamics of a class of initial entangled states in dissipative
environments. It is the study of decoherence mechanism in different systems and is
crucial as real systems always interact with the surrounding environments. The dynamics
of the two-qubit atomic systems and high-dimensional bipartite field states inside the two
high-Q cavities surrounded by thermal environment are investigated. The two-qubit
atomic systems are explored as both interacting (close) and non-interacting (distant)
systems. In the interacting systems, atoms are considered close together so that the atoms
may exchange energy, thus the role of collective damping and dipole-dipole interaction
becomes important. It is also noted that entanglement may be generated for initial
separable states in thermal environment. Wootters concurrence is used as a quantitative
measure of entanglement for two-qubit atomic systems.
x
For the entangled bipartite field states in thermal environment, the high dimensional
states as non-interacting systems are studied. It is concluded that sudden death of
entanglement (SDE) always occurs in non-interacting systems in thermal environments.
The increase in the temperature of the environment results in earlier disappearance of
entanglement. For entanglement analysis, negativity is used for high-dimensional
entangled field states in high-Q cavities.
xi
List of thesis Papers
This thesis is based on the following papers.
1. “Entanglement of Gaussian States using Beam Splitter”, R. Tahira, M. Ikram,
H. Nha and M. S. Zubairy, Phys. Rev. A 79, 023816 (2009).
2. “Entanglement Dynamics of High-Dimensional Bipartite Field States Inside the
Cavities in Dissipative Environments”, R. Tahira, M. Ikram, S. Bougouffa and
M. S. Zubairy, J. Phys. B: At. Mol. Opt. Phys. 43 035502 (2010).
3. “Entanglement Dynamics of Spatially Close Bipartite Atomic Systems in
Thermal Environment”, R. Tahira, M. Ikram and M. S. Zubairy, Opt. Commun.
284, 3643 (2011).
4. “Gaussian State Entanglement in Quantum Beat Laser”, R. Tahira, M. Ikram,
H. Nha and M. S. Zubairy, Phys. Rev. A 83, 054304 (2011).
xii
Table of Contents
1. Introduction………………………………………………………………. 1
1.1 Quantum Entanglement……………………………………………. 1
1.2 Quantum Information Science……………………………………... 4
1.3 Entanglement Generation………………………………………….. 6
1.4 Decoherence……………………………………………………….. 8
1.4.1 Local decay dynamics……………………………………… 9
1.4.2 Entanglement Dynamics…………………………………… 9
1.5 Thesis Layout……………………………………………….......... 11
2. Entanglement Generation via Beam Splitter…………………………. 13
2.1 Quantum Mechanics of Beam Splitters…………………………... 14
2.2 Gaussian States and CV Systems………………………………… 16
2.2.1 Gaussian States……………………………………………. 18
2.2.2 Purity……………………………………………………… 20
2.2.3 Nonclassicality……………………………………………. 20
2.3 Quantitative Measure of Entanglement…………………………... 22
2.4 Entanglement Using a Beam Splitter……………………………... 25
2.4.1 Inputs……………………………………………………… 25
2.4.2 Entanglement at the Outputs……………………………... 27
2.4.3 Optimal Beam Splitter Settings…………………………... 29
2.4.4 50:50 Beam Splitter Settings……………………………... 30
2.4.5 General Beam Splitter Settings............................................. 31
3. Entanglement Generation via Quantum Beat Laser............................. 35
3.1 Entanglement in Quantum Beat Laser............................................. 36
3.1.1 Model.................................................................................... 38
3.1.2 Entanglement Analysis......................................................... 42
4. Entanglement Dynamics of Two-Qubit Systems................................... 51
xiii
4.1 Qubit................................................................................................ 53
4.2 Entangled States.............................................................................. 53
4.3 Entanglement Measures................................................................... 55
4.3.1 Wootters Concurrence.......................................................... 55
4.3.2 Von Neumann Entropy......................................................... 56
4.3.3 Negativity............................................................................. 56
4.4 Two-Qubit Atomic Systems............................................................ 57
4.4.1 Model.................................................................................... 58
4.4.2 Sudden Death of Entanglement............................................ 62
4.4.3 Initial Pure Entangled States................................................. 63
4.4.4 Initial Unentangled States..................................................... 72
5. Entanglement Dynamics of Two-Qudit Systems................................... 74
5.1 Model............................................................................................... 75
5.2 Bipartite Field States....................................................................... 78
5.2.1 Case-I:................................................................................... 78
5.2.2 Case-II:................................................................................. 80
5.2.3 Case-III:................................................................................ 83
6. Conclusion................................................................................................. 85
A Equations of Motion of the density matrix elements (Eq. (4.24)) and
their solutions for vacuum reservoir 90
B Equations of motion of the density matrix elements (Eq. (5.5)) and
their solutions for vacuum reservoir 93
xiv
List of Figures
2-1 The quantum mechanical description of a beam splitter, a₁, a₂ and a₃, a₄
represent two single-mode inputs and outputs, respectively...................... 15
2-2 The 3D plots of logarithmic negativity versus mean thermal photon number
n and angle of beam splitter θ. The fixed parameters are purity u=1 and
nonclassical depdth τ at (a) τ=0.2, (b) τ=0.4 and (c) τ=0.45....................... 33
2-3 Logarithmic negativity as a function of purity u and beam splitter angle θ at
a fixed level of thermal noise n (a) n=1 and (b) n=4 for nonclassical depth
τ=0.45......................................................................................................... 34
2-4 The critical thermal noise n_{c} as a function of the purity and beam
splitter angle θ for a fixed nonclassical depth τ=0.4. The plot in (b) shows a
magnified view over a narrow range of u close to 1................................... 34
3-1 (a) Atomic levels in V-configuration (b) Atomic medium inside a doubly
resonant cavity. E₁, E₂ represent the two emitted modes of radiation fields
and E₃ represent the driving field, of frequencies v₁, v₂ and v₃,
respectively................................................................................................. 37
3-2 (a) Entanglement as a function of non-classicality τ and dimensionless time
Gt at Δ=0,n=0,u=1 and Ω=1000 kHz. (b) Entanglement as a function of
average photon numbers n in the thermal state and dimensionless time Gt at
Δ=0,τ=0.45,u=1 and Ω=1000 kHz. (c) Entanglement as a function of purity
u and dimensionless time Gt at n=0,τ=0.45, and Ω=400 kHz. The other
parameters are taken as γ=20 kHz, r_{a}=22 kHz, g=43 kHz and k=1.5
kHz.............................................................................................................. 45
3-3 (a) Entanglement as a function of driving field strength Ω/γ and
dimensionless time Gt at Δ=0,n=0,u=1 and τ=0.45. (b) Cross-sections of
the Fig. 3a at Ω=800 (solid line),1200 (dotted),1600 (dashed),and 2000 kHz
(dotted-dashed). Inset at smaller values of Rabi frequency Ω=200
xv
(solid),300 (dotted),400 (dashed),and 500 kHz (dotted-dashed).The other
parameters are the same as in Fig. 2-5........................................................ 48
3-4 (a) Entanglement as a function of non-classicality τ and dimensionless time
Gt at Δ=0,n=0,u=1 and Ω=400 kHz. (b) Entanglement as a function of
average photon numbers n in the thermal state and dimensionless time Gt at
Δ=0,τ=0.45,u=1 and Ω=400 kHz. (c) Entanglement as a function of purity u
and dimensionless time Gt at n=0,τ=0.45, and Ω=1000 kHz, The other
parameters are the same as in Fig. 2-5........................................................ 50
4-1 The plot shows the entanglement dynamics of initial mixed entangled state
(Eq. (4.25)). For a>(1/3) we observe SDE and exponential decay when
0≤a<(1/3).................................................................................................... 62
4-2 The distance dependence is shown for dipole-dipole interaction Ω₁₂ (solid
line) and collective damping terms γ₁₂ (dotted line).................................. 64
4-3 Entanglement dynamics of initial entangled state |Ψ> at (a) φ=0 and (b)
φ=π is shown for R=0.075λ, α₂=√(0.9) at different temperatures, n=0 (solid
line), n=0.1 (dotted line) and n=0.2 (dashed line)...................................... 67
4-4 The plot of entanglement dynamics versus (a) initial probability fixing
R=0.05λ, (b) normalized distance between the atoms R/λ fixing α₂=√(0.9)
and time for |Ψ> at n=0.1 and φ=0............................................................. 68
4-5 The schematic diagram demonstrates the |Φ> state decay dynamics. The
intermediate levels |Ψ⁽⁺⁾> and |Ψ⁽⁻⁾> are symmetric and asymmetric states,
respectively................................................................................................. 69
4-6 Entanglement dynamics of initial entangled state |Φ> for R=0.075λ at
different temperatures, n=0 (solid line), n=0.01 (dotted line) and n=0.02
(dashed line)................................................................................................ 70
4-7 The plots of entanglement dynamics for |Φ> is shown versus (a) initial
probability fixing R=0.05λ and n=0.1, (b) normalized distance between the
atoms R/λ fixing α₁=√(0.9) and n=0 and time............................................ 71
xvi
4-8 Entanglement dynamics of initial unentangled states (a) |a₁,b₂> for
R=0.075λ, n=0 (solid line), n=0.1 (dotted line) and n=0.2 (dashed line), (b)
|a₁,a₂> for R=0.075λ, n=0 (solid black), n=0.01 (dotted line), n=0.02
(dashed line)........................................................................................... 73
5-1 Two identical and independent cavities contain initial entangled states of
fields. The fields inside the cavities interact with their own
environment................................................................................................ 76
5-2 Exponential decay dynamics of initial entangled (a) NOON state
and (b) at all times is shown versus the initial probabilities of the
states C₀₂² and C₀₁², respectively............................................................... 79
5-3 The role of temperature on entanglement dynamics is shown for maximally
entangled (a) NOON state and (b) for n=0 (solid line),
n=0.1 (dotted line) and n=0.2 (dashed line)................................................ 79
5-4 Exponential decay dynamics of initial entangled (a) and (b)
at all times is shown versus the initial probabilities of the states
C₂₂² and C₁₁², respectively......................................................................... 81
5-5 The role of temperature on entanglement dynamics is shown for maximally
entangled states (a) and (b) for n=0 (solid line), n=0.1
(dotted line) and n=0.2 (dashed line).......................................................... 81
5-6 Negativity is plotted (a) versus time and initial probability of the state (Eq.
(5.16)) C₁₁² fixing C₀₂=C₂₀= √(((1-C₁₁²)/2)) and (b) for maximally
entangled state for n=0 (solid line), n=0.1 (dotted line) and n=0.2
(dashed line)................................................................................................ 84
AB
10
AB
10
AB
10 AB
10
AB
20
AB
20
AB
20 AB
20
|AB0
xvii
List of Abbreviations
EPR: Einstein, Podolsky and Rosen
Qubit: Quantum bit
CV: Continuous variable
QIP: Quantum information processing
CEL: Correlated emission laser
QBL: Quantum beat laser
SDE: Sudden death of entanglement
ℏ: Planck's constant
τ: Nonclassicality
u: Purity
PPT: Positive partial transposition
χ(x): Characteristic function
: Mean thermal photon number
: Critical thermal noise
γ: Atomic decay rate
: Annihilation operator
: Creation operator
: Density matrix
: Dipole lowering operator
: Dipole raising operator
: Hamiltonian operator
Ω: Rabi frequency
γ₁₂: Collective damping
Ω₁₂: Dipole-dipole interaction
SDT: Sudden death time
n
nc
â
â
Ĥ
Chapter 1
Introduction
1.1 Quantum Entanglement
In quantum theory, the state of a system can be described with a wavefunction. A wave-
function is de�ned in Hilbert space as superposition of all possible states with complex
probability amplitudes. It contains all the information needed to know about a given
quantum system and physical quantities are measured from the wavefunction. Quantum
states obey superposition principle and are probabilistic. According to the superposition
principle, the physical state of a quantum system can be in a linear superposition of
all possible states of the system. The measurement on a quantum system destroys the
superposition. This can be understood by the following example. Let us consider the
double-slit experiment, a well known device to demonstrate the interference of both light
and particles. The system is prepared in coherent superposition at the two slits which
results in interference fringes at the screen. So, if a detector is placed at any one of the
two slits, the interference pattern vanishes.
The quantum systems are in pure or mixed states. A quantum state is pure when
Tr (�2) = 1 and is mixed when Tr (�2) < 1. The wavefunction of a pure system can be
de�ned as
ji =Xi
�i j ii ; (1.1)
1
such thatXi
j�ij2 = 1; where �i�s are complex probability amplitudes. The mixed
quantum systems are represented by density matrix, de�ned as
� =Xi
pi j ii h ij (1.2)
whereP
i pi = 1. The state j ii may represent many body systems, such cases may
represent the correlations between two or more subsystems. Quantum entanglement is
another manifestation of quantum theory that cannot be explained classically. Quantum
entangled systems cannot be described as individual systems and contain correlations
in the composite state of the two or more subsystems . So, the quantum entangled
states cannot be written in the product form. The best examples to mathematically
demonstrate bipartite entangled systems are the maximally entangled Bell states, given
by
���� =1p2[ja1; b2i � jb1; a2i] ; (1.3)����� =
1p2[ja1; a2i � jb1; b2i] ; (1.4)
where ai and bi represent the excited and ground states of i�th (i = 1; 2) atom. These
states show maximum correlations between two entangled atoms. In general, we can�t
describe the composite systems in independent parts of it.
The term entanglement was �rst laid down by Erwin Shrodinger who called it the
characteristic trait of quantum mechanics. In 1935 and onward, there began a debate
on the legality of quantum mechanics to be a complete theory. Einstein, Podolsky and
Rosen [1] suggested a Gedanken experiment as a criticism of quantum mechanics. EPR
argued that physical quantities which can be predicted with certainty must preserve the
idea of locality and reality. For a theory to be "complete", these values must therefore
be incorporated into the theoretical description of the system. If quantum mechanics is
complete in this sense, it follows that non-commuting observables can never be simulta-
2
neously "elements of reality" as if one property can be predicted with certainty, the other
cannot, so quantummechanics is an incomplete theory. EPR objection on incompleteness
of quantum theory was countered by Bohr [2]. But the question remained unanswered
until Bell [3] introduced the inequalities. Bell�s inequalities de�ned the boundary between
classical and quantum mechanical correlations and is being experimentally veri�ed since
long [4, 5, 6, 7, 8]. The Bell�s inequalities can be satis�ed by any local variables and are
violated in the case of entanglement. Bell and Clauser et al. [3, 9] showed that quan-
tum correlations violate inequalities that must be satis�ed by any classical local hidden
variable model. Werner [10] addressed the problem of hidden variables, classical to EPR
correlations and discussed the classical states that satisfy the Bell�s inequalities. Apart
from Bell�s inequalities, there are many entanglement characterization and quanti�cation
schemes discussed in the later chapters of the thesis.
Entanglement is long being investigated due to its vitality in quantum information sci-
ence [11, 12, 13, 14]. There are many applications in which entanglement is a well known
key resource. These applications such as quantum teleportation, super dense coding,
quantum cryptography are being investigated in modern day research work. Quantum
computers are also based on the idea of superposition and entanglement to compute data
much faster than classical computer. Quantum computers can be used for factorization
of very large integers into prime numbers using Shor�s algorithm [15] and for searching a
particular state in the unsorted database using Grover�s algorithm [16]. For a realizable
quantum computer, there are two important requirements [17]. One is about the suc-
cessful generation of superposition or entanglement between di¤erent parts of quantum
memory registers. Second is the successful processing through quantum gates before de-
coherence destroys this superposition. These two issues at hand are the subjects of this
thesis.
3
1.2 Quantum Information Science
The recent developments in quantum science have presented many applications for faster
communication and computation. The work on quantum computers [17] and quantum
networks [18] is already being regulated. The key idea behind quantum computers is
that superposition in between di¤erent quantum memory registers may result in parallel
processing and therefore faster computation than a classical computer. There are many
analogous terms in quantum information science that come from classical information
theory. For example, a bit is the fundamental classical computation unit. A bit can be
a 0 or 1, physically, it may correspond to the charging and discharging of a capacitor.
So, a classical computer works on bits 0 and 1. The basic unit of quantum information
is a qubit. A qubit is the superposition of 0 and 1. Thus, a quantum computer works
on qubits. Physically, it may be the excited or ground state of atoms, horizontal and
vertical polarization states of photons, spin up and down of an electron and many more.
Two or more qubits may contain quantum correlations and thus can be entangled.
Quantum entanglement has been investigated as key resource in many applications of
quantum information science. Presently, widely used Rivest, Shamir and Adleman (RSA)
encryption system becomes insecure due to Shor�s algorithm [15] of prime factorization.
RSA encryption is based on the hardness or complexity of factorization of large integers
into prime numbers. It can take exponential times to factorize an integer N on a classical
computer. Shor�s algorithm can e¤ectively break RSA code using the quantum Fourier
transform and the entanglement. It runs in O�(logN)3
�times speci�cally in comparison
to the exponential times of a classical computer. Quantum algorithms like classical
algorithms use logical operations for computation. For example, Shor�s algorithm runs
until the period of a function is found using the quantum Fourier transform. Once period
is found, prime numbers can be found. Shor�s algorithm exploits the entanglement in
di¤erent parts of quantum memory registers in computation and evaluation of function.
Cryptography is being used in history for secure communication of data since long.
Since the RSA encryption system becomes insecure due to Shor�s algorithm, therefore
4
new encryption techniques are being investigated. Quantum cryptography exploits the
laws of quantum mechanics to validate secure communication. Since quantum states
collapse as a result of measurement, so when the intrusion of a third party disturbs the
data, no information is successfully processed. It involves private key distribution shared
between Alice and Bob. In this context, quantum key distribution (QKD) is one of the
most important applications of quantum science. There are many protocols suggested for
QKD for example, Bennett and Brassard [19] suggested a quantum protocol for private
key distribution in 1984 and the B92 [20] protocol suggested by C, H. Bennett. Another
protocol is the Ekert protocol [21] that is based on two-qubit entangled states shared
between Alice and Bob. Alice and Bob announce their results of measurement using
public channel and thus a random key is generated when measurement results match.
Grover�s Algorithm [16] is another example that exploits the quantum entanglement.
It helps to �nd a particular state in a large unsorted database i.e., "�nding a needle in
haystack". It makes it possible for a quantum computer to search the database with N
entries in O(pN) time which may be performed in O(N) times with a classical computer.
Entangled states shared between two distant parties have made it possible to securely
send data from one station to another station. For example, the quantum teleportation
proposed by Bennett et al [22], makes it possible to transport an unknown state from
Alice (sender) to Bob (receiver) without sending the particle itself. In this scheme, an
entangled state is shared between Alice and Bob. Alice makes a joint measurement on her
particle of the entangled state with the unknown quantum state, that is to be teleported.
The resultant two-bits of classical information is then communicated to Bob via classical
channel. Bob retrieves the original unknown quantum state by performing necessary
unitary operations on his particle of the entangled state.
Super dense coding [23] is an inverse problem to the quantum teleportation. As
in quantum teleportation, Alice sends two-bit information to Bob via classical channel
while a qubit is transmitted. In super dense coding, Alice encodes two-bits of classical
information in her qubit. She sends two-bit information using some standard two-qubit
5
unitary transformation on her qubit, to Bob. Bob can then measure and decode the
corresponding two-bits of classical information after applying the logic gates [24, 25, 26,
27] on his qubit. Thus, a two-bit classical information is transmitted by encoding it into
a single qubit, that is why it is called super dense coding.
1.3 Entanglement Generation
The generation of entanglement is one of many major areas of research in quantum
information science. Entanglement generation in bipartite systems is investigated in
qubit atomic systems [28, 29] and in �eld states inside the cavities [30, 31, 32]. Using
the nonlinear crystals as in parametric down conversion, entanglement can be generated
in horizontal and vertical polarization of photons. However, preparation of atomic state
entanglement and photon entanglement is technically di¢ cult. The subject is also studied
with light using linear and nonlinear optical setups. Light seems to be the best candidate
for the physical systems to carry out quantum information as it can be transmitted over
long distances e¢ ciently. Therefore, experiments with light in linear [33, 34] and nonlinear
optics [35] are more vigorous than with qubits. The quadratures of light are de�ned as
position and momentum variables and described as continuous variable (CV) systems.
The CV systems span the in�nite-dimensional Hilbert spaces being di¤erent to the �nite-
dimensional Hilbert spaces of the discrete systems. The CV systems are recently being
studied [36, 37] as an alternative to the discrete systems for most of quantum information
processing tasks. The CV states are demonstrated in conjugate phase space, that is with
position and momentum variables. Recently, much focus is given to the generation of
entanglement in CV systems because of its importance in quantum information processing
tasks [11, 12, 13, 14]. The possible quantum information processing schemes are being
experimentally explored with CV systems such as in teleportation [38, 39, 40, 41, 42, 43],
cryptography [44, 45, 46, 47], error correction [48] and super dense coding [49, 50, 51].
In this thesis, the generation of entanglement is studied with a speci�c class of CV
6
states, Gaussian states. The Gaussian states are a restricted class of CV states and are
important due to generation of entanglement in quantum optical systems and experi-
mental implications of many quantum information processing (QIP) tasks [52, 53, 54,
55, 56, 57, 58, 59, 60, 61]. The generation of entanglement using linear (passive) devices
like beam splitter is investigated [62, 63, 64]. The generation of entanglement in active
quantum optical systems is long being investigated with three level atoms such as in
correlated emission lasers (CEL�s) [65, 66], cascaded con�guration [67], quantum beat
lasers (QBL�s) [68, 69, 70, 71, 72] and parametric down conversion [73, 74]. In such
systems, induced coherence in atomic levels [75, 76, 77, 78] results in the generation of
entanglement.
The work on entanglement generation is particularly carried with nonclassical states
of �elds. The nonclassical states such as squeezed states [79, 80] of �eld quadratures are
being studied as bona �de measure of entanglement in CV systems. In passive devices
such as beam splitters, it is studied that one of the two inputs should be a nonclassical
�eld to generate the entanglement at the outputs. This result is extended to the active
devices such as quantum beat lasers [81].
The entanglement generated is presently characterized for Gaussian states using dif-
ferent inseparability criteria [82, 83, 84, 85, 86, 87, 88]. There are EPR inequality [82],
Simon-Duan criterion [83], Hillery-Zubairy criterion [84], and Heisenberg uncertainty.
These criteria are su¢ cient and necessary conditions for the characterization of entan-
glement in bipartite CV systems. However these fail to quantify the amount of entan-
glement present in the CV systems. Therefore, a quantitative measure of entanglement
is required. Logarithmic negativity [89] is one such measure of entanglement of two
single-mode Gaussian �elds in CV systems. In this thesis, entanglement using active and
passive devices like beam splitter and quantum beat laser is investigated. The amount
of entanglement generated is quanti�ed using logarithmic negativity.
7
1.4 Decoherence
Decoherence is the irreversible loss of information due to the interaction of the quantum
systems with the surroundings. The quantum decoherence is the quantum mechanical
interpretation of the system loss mechanism. The real systems (quantum mechanical or
even classical ones) are never isolated, they are inevitably in�uenced by their surrounding
environments. In this system-surrounding interaction, some information leaks to the
environment irreversibly. Once information is lost, we cannot tell about the original
state of the system. There are di¤erent decoherence mechanisms. One is the spontaneous
emission which causes the excited atoms to come to their lower energy (ground) states.
It originates from the de-excitation of the atom by vacuum �uctuations. Spontaneous
emission is an important property of any material system and it cannot be described
by the classical theory. Following are other decoherence mechanisms. The decoherence
in quantum optical systems occurs as the photons dissipate in the cavity walls. Ions in
traps and nuclear spins in solids decohere due to �uctuating electric and magnetic �elds,
respectively. Single electrons in quantum dots undergo decoherence due to interaction
with the surrounding electrons.
In the light of experimental investigations, the spontaneous emission leads to the
irreversible loss of information and thus is regarded as the main obstacle in practical
implementations of entanglement [90]. The decoherence mechanism and entanglement
dynamics are relative terms, because decoherence follows from the entanglement of the
system of interest with the rest of the universe. It is also responsible for the fragility of
entanglement. Because of this, understanding the basic mechanisms of decoherence has
both fundamental and practical implications. Decoherence due to individual systems has
been well established. Individual systems follow local dynamics which is always asymp-
totic. Decoherence in global systems like entangled atoms and �elds is an open issue.
In recent years, many theoretical models are presented to investigate the decoherence in
entangled systems. Following is the brief review of both local and global decay dynamics.
8
1.4.1 Local decay dynamics
The local decay dynamics is associated with how a single particle dissipates, di¤uses and
decays. In atoms, the major cause of decoherence is the spontaneous emission. Let us
consider a two-level atom interacting with the vacuum reservoir. The local dynamics
followed by atom-reservoir interaction can be described by a standard quantum map,
given by
jbiS j0iR ! jbiS j0iR ; (1.5)
jaiS j0iR !pp jbiS j1iR +
p1� p jaiS j0iR : (1.6)
where subscripts S and R represent system and reservoir, respectively. Following this
map, the lower state jbi is not a¤ected while the upper state jai either decays to ground
state jbi with probability p, creating one excitation in the environment (state j1iR), or
remains in jai, with probability 1�p. In this case, the state j1iR represents one photon in
the environment. Let us consider the atom-�eld interaction weak enough so that there is
no back reaction of the �eld to the atom which is also called Markovian approximation.
Following the Markovian approximation, p = 1 � e� t, that is, the decay probability
approaches unity exponentially in time.
As an initial pure state � jai + � jbi decays, it is coupled with the environment,
gradually losing its coherence and its purity over time. Complete decay only occurs
asymptotically in time ( p ! 1 when t ! 1), when the two-level system is again
described by the state jbi, only.
1.4.2 Entanglement Dynamics
The entangled systems are composed of two or more subsystems with individual sub-
systems following asymptotic dynamics. The question is that how composite state of
entangled systems decay or what is global dynamics. Let us consider the entangled state
9
of two two-level atoms as a special case
j�ABi = j�j jb1b2i+ ei� j�j ja1a2i ; (1.7)
which is the superposition state, such that both atoms are in either excited jaii or ground
state jbii, i = 1; 2. Here � and � are the probability amplitudes of both atoms ground
and excited components, respectively. The ei� corresponds to the relative phase factor.
It is studied that when both atoms are in their excited states, their entanglement may
�nish in �nite time [91, 92, 93, 94]. This behavior is quite in contrast to the asymptotic
decay of individual atoms. This is the so-called �nite-time disentanglement or sudden
death of entanglement (SDE). The entanglement for the above state is calculated [92] as
C (t) = max f0; 2 (1� p) j�j (j�j � p j�j)g ; (1.8)
where C represents Wootters concurrence [95]. It is used as a quantitative measure of
entanglement in two-qubit systems. From Eq. (1.8), it is obvious that when j�j � j�j
entanglement vanishes only when the individual qubits locally decay. On the other hand,
if j�j > j�j, entanglement disappears as p = j�jj�j < 1 i.e., in some �nite time, SDE. In
atomic systems, spontaneous emission is a main cause of it.
Yu and Eberly [91, 96] were the �rst to investigate the SDE in two-qubit atomic
systems in mixed state. Since then, the study on decoherence and disentanglement [97, 98]
is carried with di¤erent theoretical models [99, 100, 101, 102, 103, 104, 105, 106, 107, 108].
The entanglement dynamics, then, is extensively studied with pure [92, 94, 109, 110,
111, 112] and mixed states [91, 93, 96, 113, 114] of two-qubits in di¤erent dissipative
environments [115, 116, 117, 118]. In this regard, the necessary conditions are studied
[119] so that SDE is avoided or delayed by local operations [120, 121, 122]. All of
the above mentioned work is done with noninteracting atoms in di¤erent environments,
particularly, vacuum and thermal. It is also studied that entanglement can also be
induced in the dissipative environments [123, 124, 125, 126, 127]. In interacting atoms, it
10
is studied with initial pure [128, 129] and mixed states [130] that entanglement dynamics
undergoes SDE and revivals when the probability that both atoms are in their excited
states is large. In this thesis, we extensively studied the entanglement dynamics for both
cases of interacting and noninteracting atoms in vacuum and thermal environments.
1.5 Thesis Layout
The thesis is organized as follows. In the second and third chapter of the thesis, the
generation of entanglement in continuous variable (CV�s) systems using beam splitter
and quantum beat laser are presented, respectively. These devices provide example of
passive and active media, respectively. We de�ne the two inputs such that both are
Gaussian �elds in each case. One input is an arbitrary Gaussian state de�ned in terms
of arbitrary values of nonclassicality and purity of the state and the second input is the
thermal (classical) �eld. The analytical and numerical results are presented to show the
dependence of entanglement on di¤erent parameters.
In the fourth chapter of the thesis, we study the entanglement dynamics of two-
qubit atomic systems in dissipative environments. We consider the initial pure entangled
and unentangled two-qubit atomic systems in vacuum and thermal environments. We
study that entanglement dynamics depends upon the initial preparation of the state,
distance between the two atoms and the type of environment (thermal or vacuum). The
analytical and numerical results show that entanglement dynamics is completely di¤erent
for interacting (close atoms) and noninteracting systems (atoms distant apart). The
noninteracting systems may observe sudden death of entanglement (SDE) when initial
state contains the state with doubly excited component (both atoms are in excited state)
in vacuum environment. In contrast, we observe SDE and then revivals of entanglement in
interacting systems with the same state. We show that entanglement can be generated in
dissipative environments and asymmetric states decay slow as compared to the symmetric
state. Concurrence as a quantitative measure of entanglement is used.
11
In the �fth chapter, we study the entanglement dynamics of high dimensional qutrits
in the form of �eld states in two high-Q cavities. The cavities are surrounded by thermal
environment and �elds inside the cavities are considered non-interacting with each other.
We study that SDE always occur in thermal environment in noninteracting systems
and sudden death time (SDT) decreases with the increase in the temperature of the
environment. Since high dimensional states are studied, so negativity as a measure of
entanglement for bipartite high-dimensional states is used. The sixth and last chapter of
the thesis is about the conclusion.
12
Chapter 2
Entanglement Generation via Beam
Splitter
The generation of entanglement is long being investigated using quantum optical devices.
Experiments with such devices can only be realized in continuous variable states of the
radiation �elds. Such system are considered as continuous variables (CV�s) in compar-
ison to the discrete level systems (atoms and photons). The continuous variable (CV)
systems belong to phase space (position and momentum space) and are described using
quasi-probability distributions. The quasi-probability distributions are the coherent state
representations of the radiation �elds �rst de�ned by R. J. Glauber and E. C. G. Sudar-
shan [131, 132]. The P- and Q- representations are the quasi-probability distributions of
normal and antinormal annihilation and creation operators. Similarly, the Wigner-Weyl
distribution or W- representation is de�ned with symmetric ordered operators.
Gaussian states are restricted class of CV states and are largely explored for QIP
tasks. Gaussian states are so de�ned as their W- or P- distributions are Gaussian. An
outclass number of research contributions are made in which entanglement is character-
ized, quanti�ed and generated with active and passive devices. The passive devices such
as beam splitters and phase shifters are investigated to generate the entanglement in two
initially separable single-mode of �elds. Earlier, it is studied that to generate the entan-
13
glement at the output state of beam splitter, one of the input �elds should be nonclassical
[62, 63]. Quantum beat lasers (QBL�s), correlated emission lasers (CEL�s) and paramet-
ric down conversions are examples of active mediums. Generation of entanglement in an
active medium is investigated in the next chapter.
In this chapter, we consider the generation of entanglement in two-single mode Gaussian
�elds using beam splitter We consider the two input �elds at the beam splitter to be
Gaussian such that one is a Gaussian state de�ned in terms of arbitrary values of non-
classicality and purity and the second is the thermal �eld. The thermal �eld is de�ned
with mean thermal photon number. We study that entanglement generation depends
upon the nonclassicality, purity, mean thermal photon number and beam splitter set-
tings. Logarithmic negativity as a quantitative measure of entanglement is implied.
2.1 Quantum Mechanics of Beam Splitters
A beam splitter is a passive device which divides the incident light �eld into its re�ection
and transmission parts. This is how we classically deal the problem. If we quantum
mechanically understand the problem, we have to take into account two �elds because
of vacuum �uctuations always present at the second input port. Therefore, the beam
splitter action can be understood by the following beam splitter transformation in matrix
form 0@ a3
a4
1A =
0@ �t r
�r t
1A0@ a1
a2
1A ; (2.1)
where r (�r) and t��t�be the complex re�ection and transmission amplitude from
one(other) side. Then if beam splitter is assumed to be lossless, these parameters must
obey the reciprocity relations�with jrj = j�rj ; jtj =
���t��� jrj2 + jtj2 = 1, �rt� + r��t = 0 and
r�t + �r�t� = 0; so that incoming energy is conserved. Here a1; a2 corresponds to the two
input �elds and a3; a4 to the two output �elds correlated by the matrix as shown in Fig.
2-1. We can understand this correlation by beam splitter transformation by the following
14
examples.
Figure 2-1: The quantum mechanical description of a beam splitter, a1; a2 and a3; a4represent two single-mode inputs and outputs, respectively.
A 50:50 beam splitter re�ects half light and half transmits it. So, for a 50:50 beam
splitter, we can write [133]
a3 =1p2(a1 + ia2) and a4 =
1p2(ia1 + a2)
:Let us consider vacuum at one input and one photon at the second input. So, after beam
splitter action the state of the system j0i1 j1i2 = ay2 j0i1 j0i2 becomes
j0i1 j1i2 !1p2
�iay3 + ay4
�j0i3 j0i4 =
1p2(i j1i3 j0i4 + j0i3 j1i4) : (2.2)
Thus, the output state is an entangled state. Now, let us consider another example, a
classical coherent light is incident at on input and vacuum at the second input. The initial
state becomes j0i1 j�i2 = D2 (�) j0i1 j0i2, where the displacement operator D2 (�) =
15
exp��ay2 � ��a2
�is de�ned for the second input. After the beam splitter action, we
have
j0i1 j�i2 ! exp
��p2
�iay3 + ay4
�� ��p
2(�ia3 + a4)
�j0i3 j0i4 ;
= exp
�i�p2ay3 �
�i��p2a3
�exp
��p2ay4 �
��p2a4
�j0i3 j0i4 ;
=
���� i�p2�3
���� �p2�4
: (2.3)
So, we see that initial classical states remains uncorrelated. Kim et al has addressed the
problem with di¤erent states at the inputs of beam splitter [62]. They concluded that
nonclassicality at the any one of the inputs of beam splitter is a necessary condition for
the entanglement generation.
2.2 Gaussian States and CV Systems
Continuous variable (CV) systems are described with canonical position x and momen-
tum p = �i~ @@xvariables that satisfy canonical commutation relations [x; p] = i~ or
in phase space. Some examples of CV states are position and momentum of particles
and quadratures of electromagnetic �eld. The quadratures of electromagnetic �eld are
de�ned as vacuum, coherent, squeezed vacuum, thermal �elds and Fock states. Except
Fock states, all other �elds are Gaussian states of electromagnetic �eld [134, 135].
In CV formalism, the eigenstates of position x and momentum p = �i~ @@xoperators
satisfy the eigenvalue equations x jxi = x jxi and p jpi = p jpi : These eigenstates also sat-
isfy the orthogonality and completeness relations as hx j�xi = � (x� �x), hp j�pi = � (p� �p)
andRjxi hxj dx = 1;
Rjpi hpj dp = 1, respectively: For a single mode bosonic �eld, the
quadrature operators are de�ned as modes of quantized harmonic oscillator, given by
x =
r~2!
�a+ ay
�and p = �i
r~!2
�a� ay
�; (2.4)
16
where a; ay are quantummechanical annihilation and creation operators. These operators
always satisfy the commutation relation�a; ay
�= 1.
One way to formulate electromagnetic �eld quadratures is the quasi-probability repre-
sentations. These are coherent state representations of the radiation �eld [136, 137, 138,
139]. The coherent states are the eigenstates of annihilation operator as a j�i = � j�i
and are minimum uncertainty states. Here � is the phase space complex variable de�ned
as � = x+ip2: The given state � of a quantum system is de�ned with Glauber-Sudarshan
[131, 132] P-representation using normalization condition 1�
Rd2� j�i h�j = 1 as
� =1
�2
Zd2�
Zd2� j�i h�j � j�i h�j ; (2.5)
� =
Zd2�P (�) j�i h�j ; (2.6)
such thatRd2�P (�) = 1. The useful concept related to the Glauber-Sudarshan P -
representation is the notion of classicality and nonclassicality. Since P (�) � 0 8 classical
states and can be highly singular for nonclassical states. So, the CV states can be clas-
sical (vacuum coherent, thermal and fock states, for example) and nonclassical states
(squeezed vacuum state) of radiation �elds [140]. Nonclassical states such as squeezed
vacuum states are being studied as the resource of entanglement in quantum optical
systems [62, 63, 64, 81]. The P-distribution is associated with the normal ordering of
annihilation and creation operators. There are other distributions such as Q- and W-
representation which are associated with the antinormal and symmetric distributions,
respectively. The generator of each distribution is a characteristic function, de�ned ac-
cording to the ordering of operators as
XP (�) � Tr�� exp
��ay�exp (���a)
; (2.7)
XW (�) � Tr�� exp
��ay � ��a
�; (2.8)
XW (�) � Tr�� exp (���a)
��ay�: (2.9)
17
The Fourier transform of each gives the corresponding quasi-probability distribution,
given by
P (�) =1
�2
Zd2� exp (��� � ���)XP (�) ; (2.10)
W (�) =1
�2
Zd2� exp (��� � ���)XW (�) ; (2.11)
Q (�) =1
�2
Zd2� exp (��� � ���)XQ (�) : (2.12)
In the following, these distributions are more explicitly derived for Gaussian states in
bipartite systems. Gaussian states are described with the �rst and second moments of
�eld quadratures.
2.2.1 Gaussian States
Let�s consider a system of harmonic oscillators described with the annihilation (creation)
operator ak�ayk
�, k = 1; ::::n; satisfy commutation relation
hak; a
yl
i= �kl. For n mode
harmonic oscillators, the Hamiltonian for the system is given by
H = ~nXk=1
!k
�aka
yk +
1
2
�: (2.13)
In Cartesian decomposition of the mode operators, the position and momentum operators
ak =1p2~!k
(!kxk + ipk) are de�ned by
xk =
r~2!k
�ak + ayk
�; pk = i
r~!k2
�ak � ayk
�: (2.14)
The corresponding commutation relation is [xk; pl] = i~�kl: The commutation relation
can be generalized as
[Rk; Rl] = i~�kl; (2.15)
18
where �kl are the symplectic matrices, de�ned for n-modes as � = �nk=1wk, w =0@ 0 1
�1 0
1A and Rk (k = 1; ::::; 2n) denotes the canonical variables: The covariance ma-
trix can be de�ned from Eq.(2.15) as
Vkl =1
2hfRk; Rlgi � hRki hRli ; (2.16)
where fA;Bg = AB + BA represent the anticommutator. For a given quantum state �,
hAi = �A = Tr [A�] gives the expectation value of the observable A.
The state � of n-mode CV systems is called Gaussian if its Wigner function can be
de�ned as
W (�) =exp
n�12(�� ��)T V �1
� (�� ��)o
(2�)npjV�j
; (2.17)
with � = 1p2X, where X is de�ned as the row vector (x1; y1; :::::xn; yn)
T . The matrix V �1�
is related to the covariance matrix V Eq. (2.16) by V� = 12V: In Cartesian coordinates,
Eq. (2.17) can be written as
W (X) =exp
n�12
�X � �X
�TV �1 �X � �X
�o(2�)n k2n
pjV j
; (2.18)
where k = 1p2. Correspondingly, the characteristic function is given by
� (�) = exp
��12�TV � + i� �X
�; (2.19)
using the identity
ZdnX exp
��12XTQ�1X + i�TX
�=q(2�)n jQj exp
��12�TQ�
�; (2.20)
for some covariance matrix denoted as Q. The second term in Eq. (2.19) represents the
local displacement, so can be discarded in case of entanglement.
19
2.2.2 Purity
Pure Gaussian states can be related to the covariance matrix [141]. The purity � = Tr [�2]
of a Gaussian state can be realized as overlap between Wigner function Wk (�), given by
� (V ) = �nk2nZd2nXW 2 (X) =
1
(2k)2npjV j
: (2.21)
2.2.3 Nonclassicality
There are classical Gaussian states e.g., vacuum, thermal and coherent states of the
radiation �eld as well as non-classical Gaussian states, squeezed state. The nonclas-
sicality of the given Gaussian state is de�ned by the amount of squeezing in one of
the quadratures of the radiation �eld and its maximum value is 12[140]. There are a
number of measures to calculate the degree of nonclassicality for a single-mode state
[140, 142, 143, 144, 145, 147]. The Glauber and Sudarshan P distribution is a delta func-
tion for a coherent state and is singular for all other pure states, de�ned in terms of charac-
teristic function � (�) � Tr fD (�) �g ; where D (�) = exp�� j�j2
2
�exp
��ay�exp (���a)
is the displacement operator, by taking its inverse Fourier transform as
P (�1; ��1) =
1
�2
Zd2�1� (�1; �
�1) e
12j�1j2��1��1+��1�1 : (2.22)
The P - function for the classical states is positive and is either negative or singular
for quantum states. Thus, for nonclassical states, the integral in Eq. (2.22) cannot be
evaluated. So, an interpolation is de�ned in terms of R- function which lies in between
P and Q- function. The R- function is just the convolution transformation of the P -
function [140], de�ned as
R (� ; �1; ��1) =
1
��
Zd2�1e
� 1�j�1��1j2P (�1; �
�1) ; (2.23)
20
where � is a positive parameter de�ned such that R- function becomes positive-de�nite
for � � �m. The �m is called nonclassical depth, it is de�ned such that �m = 0 for
an arbitrary coherent state and for � = 1 we have R(z; 1) = Q(z); so the range of the
nonclassical depth can be 0 � �m � 1: For Gaussian state, R- function can be written in
terms of some arbitrary covariance matrix by using the P - function in covariance matrix
form, given by
P (z) =1
�2
Zd2x exp
��xy
�V1 �
1
2
�x+ xyE1z
�; (2.24)
where xy = (��; �) and zy = (��; �) and E1 is given by
E1 =
0@ 1 0
0 �1
1A : (2.25)
Here V1 corresponds to the single-mode covariance matrix. Similarly the R (� ; �1; ��1)
(Eq. (2.23)) can be written as
R (� ; y) =1
��
Zd2z exp
��z
yz
2�+yyz
�� yyy
2�
�P (z) ; (2.26)
where yy = (��; �). Substituting Eq. (2.24) into Eq. (2.26), we can obtain the following
form
R (� ; y) =1
�3�
Zd2z exp
��z
yz
2�+
�xyE +
yy
�
�z
� Zd2x exp
��12xy�V1 �
1
2
�x� yyy
2�
�:
(2.27)
By using Fourier-Gauss integral [146]
Zd2z exp
��12zyAz
�exp
�zyx�=exp
�12xTTA�1x
�pdetA
; (2.28)
21
valid for all matrices A = TATT > 0 and all columns x, where
T =
0@ 0 1
1 0
1A : (2.29)
Evaluating the �rst integral in Eq. (2.27), we may have
Zd2z exp
��12zy�1
�
�z + zy
�E1x+
1
�y
��= � exp
�1
2
���xyx+ 2yyE1x+
1
�yyy
��:
(2.30)
Then, Eq.( 2.27) can be simpli�ed as
R (� ; y) =1
�3
Zd2x exp
��12xy�V1 �
1
2+ �
�x
�exp
�1
2yyE1x
�: (2.31)
The positive de�nite behavior of R (� ; y) requires that
V1 �1
2+ � > 0: (2.32)
For Gaussian states, the degree of nonclassicality lies in�0; 1
2
�: Thus, � can be de�ned
as the nonclassical depth. For any arbitrary 2 � 2 covariance matrix V , the degree of
nonclassicality is given by
� = max
�0;1
2� �1
�(2.33)
where �1 is the minimum eigenvalue of the covariance matrix V1:
2.3 Quantitative Measure of Entanglement
The inseparability criteria are the entanglement characterization schemes de�ned for
bipartite states [82, 83, 84, 85, 86, 87]. All known separability criteria de�ned for two-
mode Gaussian states like EPR inequality [82], Simon- Duan criterion [82, 83], PPT
criterion [86, 88] and Hillery- Zubairy criterion for higher order moments [84], these only
22
verify that entanglement exists in regions where these criteria are violated. For applicable
QIP tasks, we need to know the amount of entanglement present.
To quantify the amount of entanglement for mixed state is one of the basic challenges
in quantum theory. Although it is well explored for qubit systems in both pure and
mixed state. However, it is di¢ cult to quantify the amount of entanglement in higher-
dimensional systems particularly in mixed state due to the complex geometry of the
states in corresponding Hilbert spaces. For continuous systems of the in�nite dimensional
Hilbert space, the positive partial transposition (PPT) criterion is the su¢ cient and
necessary for 1 � n- modes Gaussian states. According to PPT criterion, the partial
transposed matrix should have positive eigenvalues for separable systems and vice versa.
The most general de�nition of separability of initial mixed states de�ned by a density
operator � is that the state can be written as a convex combination of the product state,
given as
� =Xi
pi�iA �iB; (2.34)
where 0 6 pi 6 1 andP
i pi = 1. If the given state cannot be written in the above form,
it is entangled. To quantify the amount of entanglement in the given two-mode Gaussian
state, we consider the logarithmic negativity [89] instead of just checking the violation of
any above mentioned criteria. The logarithmic negativity is a measure of entanglement
for two-mode continuous variable systems, de�ned as
N � log2jj�PTjj; (2.35)
where jj:jj denotes the trace norm such that for any operator A, jjAjj � trpAyA and
�PT describes the partially transposed density operator. For a general n-mode Gaussian
state, there always exists a symplectic unitary transformation that maps the given state
to a tensor product of n-independent thermal states [148]. The trace norm can be cal-
culated for any real covariance matrix Vr from the so-called symplectic eigenvalues of
�(Vr�)2 where � comes from the commutation relation Eq. (2.15); for any general n-
23
mode Gaussian state. In our case, we need to calculate the symplectic eigenvalues of
the two-mode covariance matrix under partial transposition [89] that simply changes
the signs of matrix elements corresponding to the momentum variable of the transposed
mode [83].
For any two-mode output Gaussian �elds, we evaluate the output covariance matrix
Vout =
0@ A C
Cy B
1A ; (2.36)
where A, B and C are 2� 2 submatrices de�ned for the second order correlations of the
canonical operators, A and B are de�ned for individual one mode Gaussian states while
C shows the cross correlations. The sub-matrices A, B and C are de�ned as
A =
0@ n1 +12
�m1
�m�1 n1 +
12
1A ; (2.37)
B =
0@ n2 +12
�m2
�m�2 n2 +
12
1A ; (2.38)
C =
0@ ms �mc
�m�c m�
s
1A ; (2.39)
where n1 =Day1a1
E; n2 =
Day2a2
E; m1 = ha21i ; m2 = ha22i ; ms =
Da1a
y2
Eand mc =
ha1a2i : The characteristic equation becomes
�4 � (Det[A] + Det[B]� 2Det[C]) �2 +Det[Vout] = 0; (2.40)
which is the 4th order equation, let its second order positive roots be ��. The logarithmic
negativity can be written as
N = maxf0;�log2(2��)g+maxf0;�log2(2�+)g; (2.41)
24
we neglect the larger root which is 2�+ � 1 so that negativity depends upon only smaller
root 2�� to evaluate the entanglement of output two-mode Gaussian state.
2.4 Entanglement Using a Beam Splitter
The beam splitter action on the two single mode �elds with complex amplitudes �1 and
�2 can be described at the two output ports by using the beam splitter transformation
matrix MB: The complex �eld amplitudes at the two outputs can be denoted as �1 and
�2; by using beam splitter transformation MB; we have0@ �1
�2
1A = MB
0@ �1
�2
1A ; (2.42)
MB =
0@ cos � sin �ei'
� sin �e�i' cos �
1A ; (2.43)
where � is the beam splitter orientation, ' is the phase di¤erence between re�ected and
transmitted �elds and the beam splitter transmittance is cos2 �:
2.4.1 Inputs
We assume the two single mode �elds �1 and �2 at the inputs are Gaussian, so that at
one input we have an arbitrary Gaussian state and on the other input we have thermal
�eld. In general, a Gaussian state can be de�ned by the characteristic function in terms
of covariance matrix V1 as
� (x) = exp
��12xyV1x
�; (2.44)
V1 =
0@ a b
b� a
1A ; (2.45)
25
where xy = (��1; �1) denotes the row vector and the covariance matrix V1 is de�ned in
terms of arbitrary components a and b; where a is real and b = jbj ei� is complex. We can
write the covariance matrix elements in terms of nonclassicality (Eq. (2.33)) and purity
Eq. (2.21) of the single-mode Gaussian state.
The minimum eigenvalue of the covariance matrix V1 i.e., �1 = a � jbj ; so the Eq.
(2.33) becomes
� = max
�0;�a+ jbj+ 1
2
�: (2.46)
The other parameter needed to de�ne the one mode Gaussian state may be the mixedness
in a prepared quantum state described by a density matrix operator. It is de�ned in terms
of the purity u = tr (�2) of the quantum state and lies between 0 and 1. For completely
mixed state u = 0 and for pure state u = 1. For the given one mode covariance matrix
V1, it can be followed by the Eq. (2.21) as
u =1
2pdetV1
=1
2qa2 � jbj2
: (2.47)
Now, we can express the arbitrary components of the arbitrary Gaussian state in terms
of nonclassicality � and purity u from Eqns (2.46 and 2.47)as
a =1
4u2(1� 2�) +1
4(1� 2�); (2.48)
jbj = 1
4u2(1� 2�) �1
4(1� 2�): (2.49)
The state at second input of the beam splitter is assumed to be the thermal �eld. The
thermal �eld is de�ned as
�th =Xn
nn
(1 + n)n+1jni hnj ; (2.50)
26
where n is mean thermal photon number. It is given by
�n =1
exp�
}�kBT
�� 1
; (2.51)
where kB is the Boltzmann constant and T is the absolute temperature of the reser-
voir. This state can be realized in terms of the mean thermal photon number. So, the
covariance matrix for the Gaussian state de�ned at second input is
V2 =
0@ n+ 12
0
0 n+ 12
1A ; (2.52)
and purity of thermal state can be obtained as
uth =1
(2n+ 1): (2.53)
2.4.2 Entanglement at the Outputs
The input �elds of a lossless beam splitter are de�ned to be Gaussian states such that one
Gaussian state is de�ned in terms of the arbitrary values of nonclassicality � and purity u
and the second Gaussian state is the thermal noise. The arbitrary value of nonclassicality
at one input gives full treatment of the two mode output of the beam splitter from classical
states (� = 0) to highly nonclassical states�� = 1
2
�. The characteristic function for the
two-mode input �elds � (�1; �2) is followed by
� (�1; �2) = exp
��12wyVinw
�; (2.54)
where wy = (��1; �1; ��2; �2) is a row vector of the complex amplitudes of the �eld modes
at the two inputs. The total input covariance matrix Vin � V1 � V2 can be explicitly
27
written as
Vin =
0BBBBBB@a b 0 0
b� a 0 0
0 0 n+ 12
0
0 0 0 n+ 12
1CCCCCCA : (2.55)
Thus, the input state of the beam splitter is separable and is de�ned by 4� 4 matrix as
given in Eq. (2.55). The beam splitter transformation for the two inputs is also de�ned
by a 4� 4 unitary matrix UB as
UB =
0BBBBBB@cos � 0 ei' sin � 0
0 cos � 0 e�i' sin �
�e�i' sin � 0 cos � 0
0 �ei' sin � 0 cos �;
1CCCCCCA : (2.56)
Thus, the covariance matrix for the output state of the two-mode Gaussian state is
de�ned as
Vout = U yBVinUB =
0@ A C
Cy B
1A ; (2.57)
where A;B and C are 2� 2 matrices, obtained in form as
A =
0@ a cos2 � +�n+ 1
2
�sin2 � b cos2 �
b� cos2 � a cos2 � +�n+ 1
2
�sin2 �
1A ;
B =
0@ a sin2 � +�n+ 1
2
�cos2 � be�2i' sin2 �
b�e2i' sin2 � a sin2 � +�n+ 1
2
�cos2 �
1A ;
C = sin � cos �
0@ �a� n� 1
2
�ei' be�i'
b�ei'�a� n� 1
2
�e�i'
1A : (2.58)
28
The characteristic function for the two output complex amplitudes of �eld modes �1 and
�2 can be written as
� (�1; �2) = exp
��12vyVoutv
�; (2.59)
where vy � (��1; �1; ��2; �2) is the row vector de�ned for the output �eld amplitudes.
Using submatrices A, B and C to �nd root �� (Eq. (2.40)), the negativity (Eq. (2.41))
is obtained as
N = maxf0;�12log2
S �
rS2 � (2n+ 1)
2
u2
!g; (2.60)
where
S � 1
2[(n� � + 1)S+ � (n+ �)S� cos 4�] ; (2.61)
with
S� �1
u2(1� 2�) � (2n+ 1):
It is obvious that negativity does not depend on the phase shift ' introduced by the
beam splitter.
2.4.3 Optimal Beam Splitter Settings
Let us �nd the optimal choice of beam splitter settings. From Eq. (2.61), we can see that
extreme values of S can be obtained at � = 0 and �4. At � = 0, we can �nd
S = S0 �1
2u2+(2n+ 1)2
2; (2.62)
and at � = �4; we have
S = S�4� (2n+ 1)
2
�1
u2(1� 2�)) + 1� 2��: (2.63)
Since S is positive at extreme values i.e., at � = 0 and �4; therefor S is positive for the
whole range of beam splitter angles �. The logarithmic function in Eq. (2.60) monotoni-
cally decreases with S: Therefore, logarithmic negativity is maximum at the largest value
29
of S.
For the case of S� > 0, i.e., 1u2(1�2�) > (2n+1), therefore thermal photon number n is
relatively small. Then, the maximum value of S occurs at � = �4(50:50 beam splitter). If
the thermal photon number is relatively large i.e., for the case of 1u2(1�2�) < (2n+1), the
maximum occurs at � = 0. This case essentially corresponds to no beam-splitter action
and leads to no entanglement at all. Therefore, it can be concluded that the optimal
choice of beam splitter is 50:50 regardless of all other given parameters � ; u, and n. We
can also note that logarithmic negativity has no phase ' dependence.
2.4.4 50:50 Beam Splitter Settings
We obtain a very simpli�ed expression of logarithmic negativity for the optimal choice
of beam splitter settings, i.e., at � = �4; as
N = maxf0;�log2p(2n + 1)(1� 2�)g: (2.64)
We can see that entanglement only depends upon the thermal noise �n and amount of
nonclassicality in one-mode Gaussian state. It is thus independent of purity u of the
one-mode Gaussian state. From Eq. (2.64), we can �nd the critical thermal noise at
which the entanglement vanishes. It is obtained as following
nc =�
1� 2� : (2.65)
The critical thermal noise only depends upon the amount of nonclassicality in the input
Gaussian state. The critical temperature gets higher for higher amount of nonclassical-
ity and approaches to in�nity as � reaches its maximum value, i.e. 12. Therefore, for
maximum amount of nonclassicality, entanglement exists at every level of thermal noise.
On the other hand, when thermal noise exceeds the critical value (i.e., when n � nc), it
results in vanishing entanglement at the outputs. Therefore, the resistance to noise can
always be understood as an equivalent measure of single-mode nonclassicality.
30
2.4.5 General Beam Splitter Settings
Now, let us consider the general beam splitter settings with an arbitrary beam-splitter
angle �. If the beam splitter angle deviates from its optimum value as � = �+�4, � is small
error. Then, the 12transmittance in beam splitter contains a fractional error given by
e � �2: The critical value of thermal noise nc can be found as
nc ��
1� 2�
�1� 2e2 (1� �) (1� u2)
1� u2 (1� 2�)2�; (2.66)
where nc is obtained by requiring 2S = 1+(2n+1)2
u2: The above equation shows that critical
thermal noise now depends upon nonclassicality � as well as purity u of the single-mode
Gaussian �eld : This changes our de�nition of critical thermal noise against the measure
of nonclassicality. For very small value of nonclassicality, i.e. if � � 12; the dependence on
purity vanishes and we have nc � � (1� 2e2) : On the other hand, when nonclassicality
approaches it maximum value i.e., � ! 12; the threshold level of noise depends upon the
purity since now nc � 12�4� (1� e2 (1� u2)) : We can numerically calculate the value of
thermal noise at the choice of � = �12; for example. The critical value becomes nc = 0:75
for � = 0:3 and u = 1 (that is pure state), and nc � 0:36 for � = 0:4 and u = 0:2 (that
is mixed state). It shows that the higher nonclassicality leads to the lower critical noise
level.
Pure-state Input
Let us consider the arbitrary Gaussian state to be pure i.e., u = 1. Now, entanglement
only depends upon the beam splitter angle �, mean thermal photon number n and the
amount of nonclassicality � . We obtain a plot of logarithmic negativity versus beam
splitter angle � and the thermal photon number n for di¤erent values of � in Fig. 2-2. It
is obvious that logarithmic negativity is maximum when � = �4(50:50 beam splitter) and
it decreases as beam splitter angle deviates from �4. In this case, we obtain the analytic
expression of critical thermal noise as nc = �1�2� , which is remarkably independent of
31
angle �. Therefore, although the degree of entanglement varies with the beam-splitter
parameter �, the entanglement disappears at the same level of noise nc = �1�2� regardless
of �. It is also obvious that entanglement increases with the amount of nonclassicality
irrespective of the presence of noise.
Mixed-state Input
The logarithmic negativity depends upon the degree of nonclassicality, mean thermal
photon number as well as purity of the state in general beam splitter settings. We obtain
a plot of logarithmic negativity as a function of the purity u and the beam-splitter angle
� by �xing the degree of nonclassicality � = 0:45. In Fig. 2-3, the plot is so obtained for
di¤erent values of mean thermal photon number for (a) n = 1 and (b) n = 4: We can
see that entanglement is generated at the output of beam splitter for a broader range of
beam splitter angles � by increasing purity u. The increase in the thermal noise results
in the smaller region in which entanglement is generated. In Fig. 2-4, we obtain a plot
of the critical noise nc versus purity u and the beam-splitter angle � for a �xed amount
of nonclassicality � = 0:4. As the purity u increases, the distribution of critical value nc
becomes broader with respect to the beam-splitter angle until it becomes �at at u = 1.
32
2-1
2:pdf
Figure 2-2: The 3D plots of logarithmic negativity versus mean thermal photon number�n and angle of beam splitter �. The �xed parameters are purity u = 1 and nonclassicaldepdth � at (a) � = 0:2, (b) � = 0:4 and (c) � = 0:45:
33
2-2
3:pdf
Figure 2-3: Logarithmic negativity as a function of purity u and beam splitter angle � ata �xed level of thermal noise �n (a) �n = 1 and (b) �n = 4 for nonclassical depth � = 0:45:
2-3
4:pdf
Figure 2-4: The critical thermal noise �nc as a function of the purity and beam splitterangle � for a �xed nonclassical depth � = 0:4: The plot in (b) shows a magni�ed viewover a narrow range of u close to 1:
34
Chapter 3
Entanglement Generation via
Quantum Beat Laser
The generation of entanglement in active quantum optical systems is long being inves-
tigated. These are correlated emission lasers (CEL�s) [65, 66], cascaded con�guration of
three-level atoms [67], quantum beat lasers (QBL�s) [68, 69, 70, 71, 72] and parametric
down conversion [73, 74]. The entanglement in such systems is generated as the emitted
�eld becomes correlated due to induced coherence by the external driving �eld. These
systems are thus exploited with di¤erent initial states of the systems such as coherent,
vacuum and squeezed vacuum states. Di¤erent inseparability criteria are being imple-
mented to characterize the entanglement in such systems.
In the previous chapter, we consider the entanglement generation using beam splitter
with two single-mode Gaussian �elds. In this chapter, we extend the study on entan-
glement generation in active devices like quantum beat lasers (QBL�s). We consider the
generation of entanglement in two single-mode Gaussian �elds using the similar inputs as
in beam splitter case. The two input states are supposed to be Gaussian such that one is
an arbitrary Gaussian state and the second is the thermal �eld. The arbitrary Gaussian
state is de�ned in terms of nonclassical depth and purity whereas thermal �eld is de�ned
with mean thermal photon number. We study that entanglement generation depends
35
upon the nonclassicality, purity and mean thermal photon number. Entanglement gener-
ation also depends on the driving �eld strength. In strong drive case, maximum amount
of entanglement is generated when one Gaussian state is highly nonclassical. Although
di¤erent entanglement characterization schemes are long being applied to show the en-
tanglement region. For example, Shahid et al [70] applied the Hillery-Zubairy criterion
to show the entanglement for a class of initial separable states in QBL�s. However, we
use the logarithmic negativity as a quantitative measure of entanglement.
3.1 Entanglement in Quantum Beat Laser
The generation of entanglement in active mediums such as correlated emission lasers
(CEL�s) and quantum beat lasers (QBL�s) is long being characterized for two single-mode
�elds. In this chapter, both analytical and numerical work is presented to quantify the
amount of entanglement generated in QBL�s. A QBL works on the interference phenom-
enon that can only be understood by fully quantum mechanical treatment [149]. QBL�s
can be realized in a doubly resonant cavity with active medium consisting three-level
atoms with two excited levels sharing the same ground level, in so-called V -con�guration,
as shown in Fig. 3-1. In V -con�guration, atoms have three states i.e., jai, jbi and jci,
the energy levels of jai and jbi are higher than that of jci : The fully quantized atom-�eld
state vector can be written as
j (t)i = � ja; 0i+ � jb; 0i+ �1 jc; 11; 02i+ �2 jc; 01; 12i ; (3.1)
where �; �; �1; �2 are the corresponding probability amplitudes. The �eld state j1ii =
ayi j0ii represent one photon of frequency �i, decaying from mode i = 1 or 2: Hence electric
�eld operator de�ned for each mode is given by
Ei = "i (ai exp (i (ki:r � �it)) + c:c:) : (3.2)
36
Figure 3-1: (a) Atomic levels in V-con�guration (b) Atomic medium inside a doublyresonant cavity. E1; E2 represent the two emitted modes of radiation �elds and E3represent the driving �eld, of frequencies v1, v2 and v3, respectively.
37
It is easy to see that expectation value of electric �eld operator vanishes asDE1
E=
"1���1 ha jci exp (i (k1:r � �1t)) = 0, due to orthogonal property of quantum states: In
similar fashion,DE2
Ecan be shown to be zero. But higher order terms may not be zero.
It can be followed as
DEy1E2
E= "1"2�
�1�2 hc jci exp (i (� (k1 � k2) :r + (�1 � �2) t)) ; (3.3)
where hc jci becomes equal to 1 according to the orthonormality of quantum states.
This result shows that V - type atoms exhibits quantum beats, a well known interference
phenomenon. The reason is that path information is missing giving rise to interference
just like in a double-slit experiment.
Recently, Shahid et al [70] has presented the entanglement analysis for a class of
Gaussian states in a QBL system. They use the Hillery-Zubairy [84] and Simon-Duan
criterion [83] to characterize the entanglement in QBL systems. We analyze the system
using logarithmic negativity as a quantitative measure of entanglement. We use the
similar inputs as done in the beam splitter case and investigate the role of di¤erent
parameters on the generation of entanglement.
3.1.1 Model
The Fig. 3-1 shows a three level atom in V-con�guration. The two excited levels jai and
jbi are coupled by the classical driving �eld E3 = }ab~ where }ab = e ha r bi represent the
dipole moment and is the Rabi frequency. The atoms are placed in a doubly resonant
cavity. The optical set-up for a QBL is shown in Fig. 3-1. It is a doubly resonant cavity
and a beam splitter coupling the two emission modes �1 and �2 of the cavity and �3 is
the driving �eld frequency. The free and interaction parts of the system Hamiltonian are
H0 =Xi=a;b;c
~!i jii hij+ ~�1ay1a1 + ~�2ay2a2; (3.4)
38
HI = ~g1�a1 jai hcj+ ay1 jci haj
�+ ~g2
�a2 jbi hcj+ ay2 jci hbj
��~2
�e�i(�+�3t) jai hbj+ ei(�+�3t) jbi haj
�; (3.5)
where a1�ay1
�and a2
�ay2
�are the annihilation (creation) operators for the two-mode
cavity �elds at frequencies �1 and �2, respectively. Here H0 represents the free atomic
and �eld part. The coupling strengths between the cavity mode and atoms associated
with the jai � jci and jbi � jci transitions are represented by g1 and g2, respectively.
The driving �eld frequency �3 is supposed to be resonant with the induced transition
�3 = !a�!b between levels jai and jbi. The interaction part of Hamiltonian is written so
that the transitions from jai ! jci and jbi ! jci are dipole allowed and treated quantum
mechanically but only up to second order in the corresponding coupling constants. While
the transition jai ! jbi is dipole forbidden but can be induced by the external driving
�eld : The transition from jai ! jbi has been treated semiclassically to all orders in Rabi
frequency. The master equation of motion for the �eld density matrix can be written as
[150]
�� (t) = �1
2
h��11a1a
y1�+ �11�a1a
y1 � (�11 + ��11) a
y1�a1
i� k1
�ay1a1�� 2a1�a
y1 + �ay1a1
�� 12
h��22a2a
y2�+ �22�a2a
y2 � (�22 + ��22) a
y2�a2
i� k2
�ay2a2�� 2a2�a
y2 + �ay2a2
�� 12
h��21a
y1a2�+ �12�a
y1a2 � (�12 + ��21) a
y1�a2
iei� � 1
2
h��12a1a
y2�+ �21�a1a
y2
� (�21 + ��12) ay2�a1
ie�i�; (3.6)
where k1 and k2 are the decay rates for the cavity modes �1 and �2, respectively. The
phase angle � in the above equation is given by � = �+(�1 � �2 � �3) t. In our analysis,
we assume the exact resonance �3 = �1 � �2, which leads to � = � (phase of external
�eld). The detunings are denoted as �1 � !a � !c � �1 and �2 � !b � !c � �2 and the
atomic decay rate is :
39
The coe¢ cients �11; �22, �12 and �21 are given by
�11 =g2ra
2 ( 2 + 2)
�(2 2 + 2 + i ) [ � i (�� =2)]
2 + (�� =2)2+�2 2 + 2 � i
�� [ � i (� + =2)]
2 + (� + =2)2
�; (3.7)
�12 =g2ra
2 ( 2 + 2)
� � i (�� =2) 2 + (�� =2)2
(� i )� � i (� + =2)
2 + (� + =2)2( + i )
�; (3.8)
�21 =g2ra
2 ( 2 + 2)
�(2 2 + 2 + i ) [ � i (�� =2)]
2 + (�� =2)2��2 2 + 2 � i
�� [ � i (� + =2)]
2 + (� + =2)2
�; (3.9)
�22 =g2ra
2 ( 2 + 2)
� � i (�� =2) 2 + (�� =2)2
(� i ) + � i (� + =2)
2 + (� + =2)2( + i )
�; (3.10)
where we assume g1 = g2 = g and �1 = �2 = �. Here ra is the atom injection rate.
The coe¢ cients �11and �22 characterize the gain of the cavity modes while �12 and �21
corresponds to the atomic coherence induced by the driving �eld . The time evolution
of characteristic function is given by
_� (�1; �2; t) = Tr�_�(t)e�1a
y1��
�1a1e�2a
y2��
�2a2�: (3.11)
� (�1; �2; t) = exp
�� t2
�1
2(�11 + ��11 + 2k1) �
�1�1 +
1
2(�22 + ��22 + 2k2) �
�2�2
+1
2(�12 + ��21) e
i���1�2 +1
2(�21 + ��12) e
�i��1��2 � (��11 � 2k1) �1
@
@�1
� (�11 � 2k1) ��1@
@��1� (�22 � 2k2) ��2
@
@��2� (��22 � 2k2) �2
@
@�2
���12e�i��1@
@�2� ��21e
i��2@
@�1� �12e
i���1@
@��2� �21e
�i���2@
@��1
��� � (�1; �2; 0) : (3.12)
40
where the time evolution of characteristic function for two-mode �eld can be calculated
using master equation for density matrix (Eq. (3.6)). Its solution is obtained upon
integrating Eq. (3.11) and � (�1; �2; 0) is the characteristic function de�ned for initial
�eld modes. The characteristic function for the two output modes for Gaussian �elds
can be written as
� (�1; �2; t) = exp
��12yTVty
�(3.13)
where yT = (��1; �1; ��2; �2) and Vt is the time dependent covariance matrix of the form
Vt =
0@ A C
CT B
1A : (3.14)
The QBL system is quite di¤erent from beam splitter action. The beam splitter linear
transformation just correlates the re�ection and transmission parts whereas QBL�s cor-
relate the two emitted modes. Therefore, the time evolution of QBL�s is studied. In
quantum beat laser case, we use the same input state as considered in beam splitter
case. Now, we have to calculate the time dynamics of second order moments of the �eld
modes in terms of annihilation and creation operators. The time evolution of the master
equation is included to describe this dissipative dynamics. The submatrices in Eq. (3.14)
A, B and C can be de�ned in terms of the time dependent second order moments of the
�eld modes with the corresponding annihilation and creation operators
A =
0@ n1 (t) +12
�m1 (t)
�m�1 (t) n1 (t) +
12
1A ; (3.15)
B =
0@ n2 (t) +12
�m2 (t)
�m�2 (t) n2 (t) +
12
1A ; (3.16)
C =
0@ ms (t) �mc (t)
�m�c (t) m�
s (t)
1A ; (3.17)
41
where n1 (t) =Day1a1
Et; n2 (t) =
Day2a2
Et; m1 (t) = ha21it ; m2 (t) = ha22it ; ms (t) =D
a1ay2
Etand mc (t) = ha1a2it :
3.1.2 Entanglement Analysis
We consider the two single-mode �elds to be Gaussian. One �eld is in arbitrary Gaussian
state de�ned in terms of nonclassicality � and purity u and the other is the thermal
�eld. The initial two mode characteristic function can be written as � (�1; �2; t = 0) =
exp��12yTViny
�where yT = (��1; �1; �
�2; �2). The total input is denoted by Vin = V1�V2,
the initial covariance matrix with V1 and V2; as given by Eqns. (2.45) and (2.52),
respectively. The characteristic function for the two-mode Gaussian �eld at time t is
given by Eq. (3.13). The equations of motion for the second order �eld moments can be
obtained using the master equation (3.6) as
�n1 =
�1
2(�11 + ��11)� 2k1
�n1 +
1
2
��12e
�i�m�s + ��12e
i�ms
�+1
2(�11 + ��11) ; (3.18)
�n2 =
�1
2(�22 + ��22)� 2k2
�n2 +
1
2
���21e
�i�m�s + �21e
i�ms
�+1
2(�22 + ��22) ; (3.19)
�ms =
�1
2(�11 + ��22)� (k1 + k2)
�ms +
1
2
���21e
�i�n1 + �12e�i�n2
�+1
2(�12 + ��21) e
�i�;
(3.20)
�m1 = (�11 � 2k1)m1 + �12e
�i�mc; (3.21)
�m2 = (�22 � 2k2)m2 + �21e
i�mc; (3.22)
�mc =
�1
2(�11 + �22)� (k1 + k2)
�mc +
1
2�21e
i�m1 +1
2�12e
�i�m2: (3.23)
The time-dependent elements of the covariance matrix Vt can be solved using the
given initial conditions. On obtaining the covariance matrix Vt we can analyze the en-
tanglement using Eq. (2.41). The degree of entanglement N evolves in time a function
of various parameters such as the strength of the driving �eld , nonclassicality � , pu-
rity u, and the thermal noise n of the initial Gaussian states. The major controlling
parameter which greatly a¤ect the entanglement is the strength of the driving �eld :
42
It produces the coherence between the two lasing levels and is essentially responsible for
entanglement. Before investigating the output entanglement in detail let us �rst analyze
the entanglement at the extreme values of the driving �eld strength, i.e., when <<
and >> . Afterwards, we proceed to the behavior of entanglement at intermediate
values of driving �eld.
Weak Driving Field <<
Let us consider the strength of the driving �eld much less than the spontaneous decay
rates of the atomic levels on resonance (� = 0). The terms corresponding to the atomic
coherence �12 and �21 becomes �12 = �21 � 2iG= : Similarly, the �eld gain terms �11and �22 turn out to be �11 = 2G � 2g2ra=
2, �22 = 3G2=2 2 � 0, : It is thus clear
that there is no entanglement at all times since the coherence terms �12 and �21 are very
small compared to the gain terms.
Strong Driving Field >>
It has been shown earlier that the lasing process in the quantum beat laser occurs only in
the limit that the classical driving �eld is very strong [74]. With a very large driving �eld
>> (� = 0), the coe¢ cients are given by �11 � 2G 2=2 � 0; �22 � 3G 2=22 � 0,
and �12 = �21 � 2iG = approximating master equation (Eq. (3.6)) as
�� (t) =
2iG
h�ay1a2�� �ay1a2
�ei� +
�a1a
y2�� �a1a
y2
�e�i�
i� k1
�ay1a1�� 2a1�a
y1 + �ay1a1
�� k2
�ay2a2�� 2a2�a
y2 + �ay2a2
�; (3.24)
where G = g2ra= 2: This master equation can be rewritten in the form
�� (t) = �i
hHeff ; �
i� k1
�ay1a1�� 2a1�a
y1 + �ay1a1
�� k2
�ay2a2�� 2a2�a
y2 + �ay2a2
�;
(3.25)
43
with the e¤ective Hamiltonian
Heff =2iG
�ay1a2e
i� + a1ay2e�i��: (3.26)
Therefore in the strongly driven limit, the two mode quantum beat laser behaves like an
optical beam splitter. In the previous chapter, we have shown that two output beams
of a beam splitter can become entangled if one of the input beams is nonclassical The
solutions to the time dependent elements of the covariance matrix Vt in this limit can be
obtained as
n1 =1
4e�2kt
�2a+ 2n� 1 + (2a� 2n� 1) cos
�2G t
��; (3.27)
n2 =1
4e�2kt
�2a+ 2n� 1� (2a� 2n� 1) cos
�2G t
��; (3.28)
m1 = jbj ei�e�2kt cos2�G t
�; (3.29)
m2 = � jbj ei(�+2�)e�2kt sin2�G t
�; (3.30)
ms =i
4(1� 2a+ n) e�2kt sin
�2G t
�; (3.31)
mc =i
2jbj ei(�+�)e�2kt sin
�2G t
�: (3.32)
Using these matrix elements we can evaluate the output entanglement between the two
modes of the quantum beat laser which varies with the strength of the driving �eld ,
nonclassicality � , purity u, and the thermal noise n. Since the driving �eld is kept very
large in comparison to decay rate ; we here address the e¤ect of other parameters on
the entanglement. In Fig. 3-2, we see the e¤ect of nonclassicality, purity, and thermal
noise of initial states upon entanglement. It is obvious that entanglement increases with
the nonclassicality � of the input Gaussian state. It not only increases the magnitude of
entanglement but also enlarges its temporal duration. On the other hand, the entangle-
ment decreases with the temperature or the average photon number n in the other input
44
2-5
6:pdf
Figure 3-2: (a) Entanglement as a function of non-classicality � and dimensionless timeGt at � = 0; n = 0; u = 1 and = 1000 kHz. (b) Entanglement as a function of averagephoton numbers n in the thermal state and dimensionless time Gt at� = 0; � = 0:45; u =1 and = 1000 kHz. (c) Entanglement as a function of purity u and dimensionless timeGt at n = 0; � = 0:45; and = 400 kHz. The other parameters are taken as = 20 kHz,ra = 22 kHz, g = 43 kHz and k = 1:5 kHz.
45
mode. The purity of the initial Gaussian state does not show any signi�cant e¤ect on
entanglement.
General Treatment
The time dependent solutions of the di¤erential equations at the intermediate driving
�elds on resonance (� = 0) yield
n1 =A11D11
+e��t
D11
�A12 � 2
�2G� k
�4 + �2
��[B11 sin (�t) +B12 cos (�t)]
; (3.33)
n2 =A21D22
+e��t
D22
�A22 + 2
�2G� k
�4 + �2
��[B21 sin (�t) +B22 cos (�t)]
; (3.34)
ms =�ie�i��D11
�A31 � e��t
�A32 �
��2G+ k
�4 + �2
��(B31 sin (�t) +B32 cos (�t))
�;
(3.35)
m1 =1
2 (2 + 3�2 + �4)bei�e��t
��4 + �2
��2 +
�4 + 2�2 + �4
�cos (�t)� 2Y sin (�t)
;
(3.36)
m2 = �(4 + �2)2�2
2Y 2bei(�+2�)e��t [1� cos (�t)] ; (3.37)
mc = i(4 + �2)�
(2 + 3�2 + �4)Ybei(�+�)e��t
��Y +
�2 + 3�2 + �4
�sin (�t) + Y cos (�t)
;(3.38)
where
� = 2k � 4G
(4 + �2); and � =
2Gp(1 + �2) (�4 + �4)4 + 5�2 + �4
; (3.39)
� = = ;G = g2ra= 2 and Y =
p(1 + �2) (�4 + �4): (3.40)
The other coe¢ cients Aij, Bij and Dii are given below
A11 = �8G2Y�1 + �2
� �2 + �2
� h2G2�2 � 8Gk
�1 + �2
�+ k2
�4 + �2
�2i; (3.41)
A12 = 2GY �2�16k + 8 (G� 2k) (2a� n) + 2 (�2k (�2 + 4a+ n) +G (3 + 2a+ 2n))�2
+ k (1� 2a� 2n)�4�G2�2 � 4Gk
�1 + �2
�+ k2
�1 + �2
� �4 + �2
��; (3.42)
A21 = �8G2Y 3�2G2 � 6Gk + 3k2
�4 + �2
��; (3.43)
46
A22 = 2GY �2�4 + �2
� �G2�2 � 4Gk
�1 + �2
�+ k2
�1 + �2
� �4 + �2
�� �2G�3�2 + 2n�
�2 + �2���k�4 + �2
� ��4� �2 + 2n
��2 + �2
��+ 2a
�4 + �2
� �2G� k
�4 + �2
���;
(3.44)
A31 = 8G2Y k (G� 3k)
�8 + 14�2 + 7�4 + �6
�; (3.45)
A32 = 2GY�16k + 8 (G� 2k) (2a� n) + 2 (�2k (�2 + 4a+ n) +G (3 + 2a+ 2n))�2
�+ k (1� 2a� 2n)�4
�G2�2 � 4Gk
�1 + �2
�+ k2
�4 + 5�2 + �4
��; (3.46)
B11 = 2GY2�(1 + 2a)G2�2 � 4Gk (1 + 2a)
�1 + �2
�� k2 (1� 2a)
�1 + �2
� �4 + �2
��;
(3.47)
B12 = GY��G2�2
�4 + 8a+ 6�2 + 4 (a+ n)�2 + (�1 + 2a� 2n)�4
�+ 8Gk
�1 + �2
��2 + 4a+ (1 + 2a+ 2n)�2 + (�1 + a� n)�4
�+ k2
�4 + 5�2 + �4
��4� 2a
�4 + 2�2 + �4
�+�2
�2 + �2 + 2n
��2 + �2
���; (3.48)
B21 = 4GY2�G2 (1 + n)�2
��2 + �2
�� 2Gk
�1 + �2
� ��4n+ (3 + 2n)�2
�+k2n
��8� 6�2 + 3�4 + �6
��; (3.49)
B22 = GY 2nk2�4 + 5�2 + �4
� h(1� 2a)�2
�4 + �2
�2+ 2n
��8 + �2
�i�8Gk
�1 + �2
�n��8 + �6
�� �2
�2 + �2 � �4 + a
�4 + �2
�2�i� G2�2
h16� 2n
��8 + �6
�+ �2
�16 + 8�2 � �4 + 2a
�4 + �2
�2�io; (3.50)
B31 = G�2 + 3�2 + �4
� �G2�2
�4 (2 + 2a+ n) + (�1 + 2a� 2n)�2
�+ k2
�4 + 5�2 + �4
��4 (�1 + 2a+ n) + (�1 + 2a� 2n)�2
�� 8Gk
�1 + �2
� �� (1 + n)
��2 + �2
�+a�4 + �2
��; (3.51)
B32 = GY��8Gk
�1 + �2
� �2 + 4a� 2n+ (2 + a+ n)�2
�+G2�2 [8a� 4n
+(3 + 2a+ 2n)�2�+k2
�4 + 5�2 + �4
� �8a� 4 (1 + n) + (�1 + 2a+ 2n)�2
�;
(3.52)
D11 = 8GY�2 + 3�2 + �4
� �2G� k
�4 + �2
�� �G2�2 � 4Gk
�1 + �2
�+ k2
�4 + 5�2 + �4
��;
(3.53)
D22 = 8GY3�2G� k
�4 + �2
�� �G2�2 � 4Gk
�1 + �2
�+ k2
�4 + 5�2 + �4
��; (3.54)
47
D22 =Y 2
(2 + 3�2 + �4)D11: (3.55)
2-6
7:pdf
Figure 3-3: (a) Entanglement as a function of driving �eld strength = and dimension-less time Gt at � = 0; n = 0; u = 1 and � = 0:45. (b) Cross-sections of the Fig. 3a at = 800 (solid line); 1200 (dotted); 1600 (dashed);and 2000 kHz (dotted-dashed). Insetat smaller values of Rabi frequency = 200 (solid); 300 (dotted); 400 (dashed);and 500kHz (dotted-dashed).The other parameters are the same as in Fig. 2-5.
Once again the entanglement N varies with the parameters , � , u, and n. We here
discuss the e¤ect of these parameters on entanglement particularly the role of driving �eld
producing the coherence in the atomic levels. In Fig. 3-3(a), we show the e¤ect of on
the entanglement. The value of must be larger thanp2 as the lesser values makes �
in Eq. (3.39) complex resulting in no entanglement. The amount of entanglement which
is very small at low , increases with reaching the maximum value at about � 20 ,
48
and then begins to decrease with : The duration of the entanglement generally increases
with the strength of the driving �eld , which is clearly demonstrated in Fig. 3-3 (b) in
which the cross-sections of the plots of Fig. 3-3(a) are shown.
In Fig. 3-4(a), we show the e¤ect of non-classicality of the input Gaussian state
upon entanglement. The amount of entanglement along with the survival time of the
entanglement increases with the nonclassicality � and approaches to its maximum value
as � ! 1=2: In Fig. 3-4(b) we show the e¤ect of thermal noise on entanglement. Increase
in the thermal noise reduces the amount and the duration of entanglement. The purity
of the initial Gaussian state does not show any signi�cant e¤ect on entanglement.
49
2-7
8:pdf
Figure 3-4: (a) Entanglement as a function of non-classicality � and dimensionless timeGt at � = 0; n = 0; u = 1 and = 400 kHz. (b) Entanglement as a function of averagephoton numbers n in the thermal state and dimensionless time Gt at� = 0; � = 0:45; u =1 and = 400 kHz. (c) Entanglement as a function of purity u and dimensionless timeGt at n = 0; � = 0:45; and = 1000 kHz, The other parameters are the same as inFig. 2-5.
50
Chapter 4
Entanglement Dynamics of
Two-Qubit Systems
The two major issues needed to be explored extensively for a realizable quantum com-
puter are robust entanglement generation and understanding the decoherence mechanism
or entanglement dynamics. In the previous chapters, we have studied the generation of
entanglement in continuous variable systems in passive and active devices where beam
splitter is a passive and quantum beat laser is an active device. There are many earlier
explorations for the generation of entanglement in discrete systems, for example in atoms
and �eld states [28]. The key idea behind the study of entanglement dynamics is that
entanglement is needed to be preserved for successful quantum computation and quan-
tum information tasks. The study of entanglement dynamics is important because real
quantum systems always interact with the surrounding environment and information loss
is irretrievable.
There is a lot of work done in recent years on the entanglement dynamics of initial
entangled bipartite systems (pure and mixed) in dissipative environments. There are
many remarkable results in the study of entanglement dynamics in vacuum environment.
First is the Yu and Eberly [91] work that �nds the �nite-time disentanglement or sudden
death of entanglement (SDE) in a two-qubit mixed entangled state. The work shows that
51
entanglement dynamics depends upon the initial mixing of the state. Although the local
decay dynamics of qubits is asymptotic, the global dynamics may observe SDE depending
upon the initial preparation of entangled systems. This work is extended by Ikram et
al [93], studying the entanglement dynamics of di¤erent initially entangled mixed states.
They investigated that the states with doubly excited component (both atoms in the
excited state) are more fragile to the environment.
In this chapter, we consider the entanglement dynamics of initial pure states of two-
level atoms in thermal and vacuum environments. The atomic qubits are studied for both
interacting (close) and noninteracting (distant) systems. We consider the initial general
pure state of the two-qubit atoms and �nd an analytical expression of entanglement for
the full 4�4 matrix [94] in noninteracting systems. We observe that higher the proba-
bility of doubly excited component earlier is the sudden death of entanglement (SDE).
Further that entanglement vanishes in thermal environment earlier than their asymptotic
times. The interacting (close) qubit systems are earlier explored by Ficek and Tanas [128]
in vacuum environment. They suggested that entanglement dynamics also depend upon
the separation between the qubits. They observe the SDE and revivals of entanglement
in vacuum environment for initially entangled with doubly excited component. Here, we
address the same problem for a set of di¤erent initial entangled and unentangled states
of two qubits in both thermal and vacuum environments. The case of the generation of
entanglement in thermal environment for the initial unentangled states is more interest-
ing. The key idea is the coherent dipole-dipole interaction between the atoms which is
negligible at large distances.
The chapter is organized as follows. The �rst three section presents preliminary no-
tions about qubits, entangled states and entanglement measures for bipartite systems.
The last section investigates the entanglement dynamics of atoms in thermal and vac-
uum environment. We conclude that SDE always take place in thermal environment in
noninteracting systems.
52
4.1 Qubit
The classical unit of information is 0 and 1, so the classical computer�s machine lan-
guage relies on the 0�s or 1�s. The quantum unit of information is a qubit which is the
superposition of the two possible states like
j i = � j0i+ � j1i ; (4.1)
where � and � are complex probability amplitudes such that normalization condition
j�j2 + j�j2 = 1 is always satis�ed. So, the classical outcome is either 0 or 1 while quan-
tum outcome is probabilistic, the state is always in superposition of 0 and 1. Thus a
qubit is a two-level system and can be represented by vectors in the Hilbert space such
that H 2 C2. These vectors are de�ned by any orthonormal basis. So, all the assumption
of quantum mechanics are always applied to the qubit. Thus it is a superposition state
de�ned in orthonormal basis and the measurement result is probabilistic. A qubit can
also be represented with a matrix by de�ning j0i as
0@ 1
0
1A and j1i as
0@ 0
1
1A : A qubit
can be a two-level atom (one electric dipole allowed transition), the horizontal and verti-
cal polarization of electric �eld, no and one photon states of the cavity, spin-up and down
of an electron. Qubits are also realized in quantum dots and superconducting circuits.
In a quantum computer, many qubits are stored to form a quantum register, so a quan-
tum computation task requires logic operations on all the qubits in the register. Many
other applications are exploited by de�ning qubit entangled states like in teleportation,
cryptography and superdense coding.
4.2 Entangled States
Quantum systems can be in pure or mixed state, a mixed system can�t be described by
a state vector. The quantum states of a pure and mixed systems are entangled if these
can�t be described with individual systems. Let us consider a bipartite systems de�ned
53
as A and B, the composite state of the two-qubit pure systems can be written as
��(+)�AB
=1p2(j0A; 1Bi+ j1A; 0Bi) ; (4.2)��(�)�
AB=
1p2(j0A; 1Bi � j1A; 0Bi) ; (4.3)���(+)�
AB=
1p2(j0A; 0Bi+ j1A; 1Bi) ; (4.4)���(�)�
AB=
1p2(j0A; 0Bi � j1A; 1Bi) ; (4.5)
these are the 4 famous Bell or EPR states that contain maximum correlations between
two possible outcomes. In general, the quantum state of two-qubit pure systems is called
entangled if
jiAB 6=Xi;j
�ij�� i�
A
�� j�B; (4.6)
withP
i;j j�ijj2 = 1. On the other way round, the quantum state of two-qubit pure system
is separable if it can be written as the tensor product of the state for each system A and
B. Let us consider an example, the following state contains no correlations between the
two systems A and B, if the state of system A is measured, system B remains in the
superposition
jiAB =1p2(j0A; 1Bi+ j0A; 0Bi) =
1p2j0Ai (j1Bi+ j0Bi) : (4.7)
For a mixed system, a composite state �AB of two subsystems �A and �B is separable or
unentangled if written as in Eq. (2.34). So, a mixed state of quantum system is entangled
if it can�t be written as the product of the density matrices de�ned for local systems A and
B. Unlike pure states, mixed states cannot be de�ned with state vectors. Thus, mixed
entangled states cannot be written as (convex combinations) locally prepared states.
The key idea behind the entanglement is that quantum systems may have correlations
independent of the condition of locality i.e. quantum correlations.
54
4.3 Entanglement Measures
There are many schemes to characterize the entanglement in bipartite systems like one
is the Schmidt decomposition. If the composite state of the bipartite systems is
jiAB =Xi
ci juiiA jviiB ; (4.8)
where juii and jvii are the basis vectors de�ned for individual systems A and B, then
nonzero eigenvalues of the reduced density matrix �A = TrB (�AB) de�ne Schmidt number
d. So, if the Schmidt number d is 1 that corresponds to one nonzero eigenvalue of the
reduced density matrix �A; the composite state of bipartite system is separable. If d > 1
then system is entangled. There is another mostly used characterization scheme that
corresponds to the negative eigenvalues of the partial transposed density matrix, this is
the positive partial transposition or PPT criterion [86, 88]. If the initial state of the
bipartite system is de�ned by the density matrix �AB (Eq.(2.34)) ; then PPT is necessary
and su¢ cient criterion to characterize the entanglement if the partial transposed matrix
�ijkl ! �kjil gives negative eigenvalues: The entanglement characterization schemes show
the entanglement present in the system but we always need to quantify the amount of it.
There are many entanglement measures for example Wootters concurrence formula [95],
Von Neumann entropy and Negativity [89].
4.3.1 Wootters Concurrence
The concurrence formula is de�ned for two-qubit systems and is based on the bit �ip
operation �y: For a single qubit state j i, the overlap with �y-�ipped state���~ E = �y j i
de�nes the concurrence as c =���h j ~ E���. For two-qubit systems, we de�ne the matrix M
as
M (t) = � (t) (�y �y) �� (t) (�y �y) ; (4.9)
55
where � (t) is the time dependent density matrix de�ned for 2 � 2 systems. The square
roots of the eigenvalues of the 4� 4 matrix M de�ne the concurrence as
C (t) = max [0;� (t)] ; (4.10)
where � (t) =p�1 (t)�
p�2 (t)�
p�3 (t)�
p�4 (t); �i (t)�s are eigenvalues written in
descending order. Concurrence ranges from C (t) = 0 for a separable state to C (t) = 1
as for maximally entangled Bell states.
4.3.2 Von Neumann Entropy
Shannon�s entropy H = �P
i pi log2 pi represents the amount of information in classical
random variables. The Von Neumann entropy is its quantummechanical version replacing
classical probability distributions with density matrices. Let us de�ne �AB as the density
matrix of the composite system A and B. The eigenvalues (�i; i = 1; 2) of the reduced
density matrix �A = TrB (�AB) de�nes Von Neumann entropy as
E = �Xi
�i log2 �i: (4.11)
4.3.3 Negativity
The negativity is another measure of entanglement based on PPT criterion de�ned for
the density matrices. PPT criterion is a necessary and su¢ cient condition of entangle-
ment in two-qubits, high-dimensional systems [86] and for two-mode Gaussian states [89].
According to it, the negative eigenvalues of the partial transposed matrix characterize
the entanglement. For bipartite systems, the partial transposition �TA interchanges the
density matrix elements corresponding to system A only, as �ijkl ! �kjil. The negativity
is de�ned as
N (t) = �2Xi
�i; (4.12)
56
where �0is are negative eigenvalues of the partial transposed matrix �TA. Therefore,
entanglement can be de�ned as
E = max [0; N (t)] : (4.13)
4.4 Two-Qubit Atomic Systems
The initial quantum state dynamics in dissipative environments attributes a hot research
in recent years. In this regard, initial mixed and pure entangled states are exploited
mostly and most importantly for two-qubit systems in di¤erent dissipative environments.
The study of entanglement dynamics of two-qubit systems is important as they suggest
the basic system evolution and because we have all quantitative measures of entanglement
for two-qubit systems only. The multi-qubits and higher dimensional entangled systems
o¤er complicated geometry of entangled states and there is no su¢ cient and necessary
measure of entanglement.
In this chapter, we consider the interacting and noninteracting qubit systems dynam-
ics in thermal and vacuum environments. As the initial pure atomic system interacts with
the environment it dissipates its energy due to spontaneous emission, this phenomenon is
called decoherence and spontaneous emission is assumed to be the major cause of it. We
say that environment is watching and interferes with our quantum results and avoiding
this break in is impossible in real situations. Even in ideal vacuum environments, quan-
tum states lose their purity and hence coherence with time as systems evolves. Single
qubit atomic systems obey the half-life law, it means that uncorrelated atoms always
decay asymptotically.
Quantum entangled atoms show di¤erent dynamics, it depends upon the initial state
of the atoms, the surrounding environment and distance between the two atoms. For
atoms separated apart (noninteracting), it is shown that atoms come to the ground state
in some �nite time depending upon the initial mixing of the state and as a result their
entangled state becomes separable, the so-called �nite-time disentanglement or sudden
57
death of entanglement (SDE) [91, 92]. The later study has shown that SDE depends
strictly on the initial mixing of the state and surrounding environments [93]. Particu-
larly, the entangled states containing initial doubly excited components (both atoms in
excited states) show SDE even in vacuum environments. For noninteracting atoms in
their independent thermal environments, SDE always happen for any type of entangled
state. We study the entanglement dynamics of a general two-qubit (noninteracting) pure
systems and �nd that the doubly excited component is the only reason for SDE in vacuum
[94].
It is studied that if atoms are close enough so that they are in the range of their
resonant wavelengths, then system dynamics depends upon the separation between the
atoms, too. Thus interacting qubits suggest di¤erent dynamics in thermal environment.
We study the SDE and then revivals of entanglement in thermal environment for an
initial entangled state with doubly excited component. The asymmetric state dynamics
is slow as compared to the symmetric state; it also o¤ers higher amount of entanglement
during the evolution. The initial unentangled states become entangled with time due to
coherent dipole-dipole interaction between two qubits. In this chapter, we study in detail
the e¤ect of each parameter like distance between the atoms, initial type of entangled
or unentangled states and environments on system evolution with time. We present our
results both analytically and numerically and use Wootters concurrence formula as a
measure of entanglement.
4.4.1 Model
Let us start with the Hamiltonian for two identical qubits (that is two-level atoms), the
free �eld part of it is given by
HF =X~ks
~!~ks
�ay~ksa~ks +
1
2
�; (4.14)
58
where a~ks�ay~ks
�is the annihilation (creation) operator de�ned for the �eld mode ~ks
with the propagation vector ~k, index of polarization s and transition �eld frequency !~ks:
Therefore, we consider the coupling of two atoms with the three dimensional quantized
electromagnetic �eld. We�ll neglect the constant energy term in above equation that
corresponds to 12. The free atomic part of the Hamiltonian is given by
HA =2Xi=1
Ei�ii =1
2
2Xi=1
~!i�iz; (4.15)
where Ei = ~!i (i = 1; 2) is the energy (!i is the atomic transition frequency) and �ii is
the dipole raising and lowering operator of the i�th atom. It is de�ned as �+i = jaiii hbj =���i��, jaii (jbii) is the excited (ground) state for i�th atom and �
zi =
12(jaiii haj � jbiii hbj)
is the energy operator. The atomic part is written so as the large energy terms coming
from Ea�aa+Eb�bb =12~!(�aa��bb)+ 1
2(Ea + Eb) for each atoms are neglected. The total
free part of the Hamiltonian is the sum of free atomic and �eld parts, i.e., H0 = HA+HF :
The interaction part of Hamiltonian in the electric-dipole approximation is
H1 = d:E (�!r ) = �i~X~ks
2Xi=1
�~di:~g~ks (~ri)
��+i + ��i
�a~ks �H:C:
�; (4.16)
where ~di is the transition dipole moment operator for the i�th atom and H.C. denotes
Hermitian conjugate. The coupling constant ~g~ks (~ri) is given by
~g~ks (~ri) =
r!k
2"o~Ve~ks exp
�i~k � ~r
�; (4.17)
where V is the normalization volume and e~ks is the electric �eld polarization vector de-
�ned for the three dimensional �eld and evaluated at the position ~ri of the ith atom. The
interaction picture Hamiltonian can be written by using the transformation eiH0tH1e�iH0t;
59
we obtain the following form
H = �i}2Xi=1
X~ks
[~di:~g~ks (~ri) �+i a~kse
�i(!k�!i)t + ~di:~g~ks (~ri) ��i a~kse
�i(!k+!i)t �H:C:]; (4.18)
We consider general quantum reservoir theory to derive the master equation to study
the system evolution in thermal and vacuum environments [150]. Accordingly, we can
derive the master equation for two two-level atoms in thermal environment with the
Hamiltonian Eq. (4.18) as
_�SR = �i
}[H (ti) ; �S (ti) �R (ti)]�
1
}2
Z t
ti
[H (ti) ; [H (t1) ; �S (t) �R (ti)]]dt1; (4.19)
here ti is the initial time of interaction. This is the combined state of system S and
environment denoted by R and is derived from the second iteration of the Schrodinger
equation in density matrix form i} _�SR = [H(t) ; �SR (t)]. The second term in above
equation shows the �rst order system-environment correlation. The reduced state of the
system is obtained by taking trace over reservoir modes
_�S = �i
}TrR[H (ti) ; �S (ti) �R (ti)]�
1
}2TrR
Z t
ti
[H (ti) ; [H (t1) ; �S (t) �R (ti)]]dt1:
(4.20)
We assume the reservoir to be in equilibrium and de�ne a solution �SR = �S (t)�R (ti)+
�c (t) where �c (t) is the higher order correlation term and we take TrR (�c (t)) = 0: Using
the interaction picture Hamiltonian Eq. (4.18), we have the following equation of motion
for the reduced density matrix of the atoms interacting with their local environments
[151]
@�
@t= �{!o
2Xi=1
[�zi ; �]� {Xi6=j
ij[�+i �
�j ; �]�
1
2�n
2Xi;j=1
ij���i �
+j �� 2�+j ���i + ���i �
+j
��12(�n+ 1)
2Xi;j=1
ij��+i �
�j �� 2��j ��+i + ��+i �
�j
�; (4.21)
60
where �n is the mean thermal photon number. In the derivation of equation of motion,
we also use Born-Markov and rotating wave approximations. Thus environment has not
feedback and history of interaction is lost. Since we assume identical atoms so ii � :
so that spontaneous decay rate for each atom is same. Here ij and ij are the collective
damping and dipole-dipole interaction terms, respectively, de�ned as
ij =3
2
�1�
�~d:~rij
�2� sin [korij]korij
+3
2
�1� 3
�~d:~rij
�2�"cos [korij](korij)
2 � sin [korij](korij)
3
#;
(4.22)
ij = �34
�1�
�~d:~rij
�2� cos [korij]korij
+3
4
�1� 3
�~d:~rij
�2�"sin [korij](korij)
2 +cos [korij]
(korij)3
#;
(4.23)
where ko = !o=c; rij = jri � rjj is the distance between the two atoms, ~d is the unit vector
de�ned along the atomic transition dipole moments and rij is the unit vector along the
interatomic axis.
The two-qubit systems correspond to 4�4 density matrix �: If we de�ne the density
matrix elements using the basis j1i ! ja1; a2i ; j2i ! ja1; b2i ; j3i ! jb1; a2i ; j4i !
jb1; b2i, we can write the 4�4 density matrix � as
�full (t) =
0BBBBBB@�11 (t) �12 (t) �13 (t) �14 (t)
�21 (t) �22 (t) �23 (t) �24 (t)
�31 (t) �32 (t) �33 (t) �34 (t)
�41 (t) �42 (t) �43 (t) �44 (t)
1CCCCCCA : (4.24)
The equations of motion and the solutions to the density matrix elements are given in
the Appendix-A for the full matrix.
61
4.4.2 Sudden Death of Entanglement
Single qubit decay dynamics is always asymptotic in time. It implies that an excited atom
takes in�nite time to come to the ground state via spontaneous emission. It is studied
that entangled qubits decay dynamics may be �nite in time. The so-called sudden death
of entanglement (SDE) or �nite-time disentanglement occurs when both atoms come to
the ground state in �nite times in contrast to their local asymptotic dynamics. Yu and
Eberly �rst reported the SDE phenomenon using an initial mixed entangled state [91]
as noninteracting two-qubit systems in vacuum environment. They studied that SDE
depends upon the initial mixing of the state. Yu and Eberly had [91] used the following
mixed entangled state
� (0) =a
3ja1; a2i ha1; a2j+
1� a
3jb1; b2i hb1; b2j+
1
3(ja1; b2i+ jb1; a2i) (ha1; b2j+ hb1; a2j) ;
(4.25)
where a is the amount of mixing in the state. The concurrence can be calculated for this
state using Eq. (4.24) in Eq. (4.9) as
Figure 4-1: The plot shows the entanglement dynamics of initial mixed entangled state(Eq. (4.25)). For a > 1
3we observe SDE and exponential decay when 0 � a < 1
3:
62
C(t) =2
3maxf0; �23 (t)�
p�11 (t) �44 (t)g: (4.26)
The values of density matrix elements �11 (t) ; �44 (t) and �23 (t) are given in Appendix-
A (here setting 12, 12 = 0; as for noninteracting systems). Clearly, if �23 (t) <p�11 (t) �44 (t) we observe SDE. The plot in Fig. 4-1 shows the SDE for a > 1
3and
asymptotic decay dynamics for 0 � a < 13. It implies that entanglement dynamics de-
pends upon the initial mixing of the state. We observe two extreme values of mixing
when a = 0 and 1. When a = 0 the doubly excited component ja1; a2i vanishes and de-
cay is asymptotic. On the other hand, when a = 1 the component jb1; b2i vanishes and
we stay with the doubly excited component and Bell state. The entanglement dynamics
is then �nite in time, i.e. C(t) = 0 for all t � td (td is the disentanglement time).
4.4.3 Initial Pure Entangled States
Let us consider the initial general pure state of a two-qubit atomic system as
j (0)i = �1 j1i+ �2ei�1 j2i+ �3e
i�2 j3i+ �4ei�3 j4i ; (4.27)
where �i�s are the probability amplitudes withP
i j�ij2 = 1 and ��s are the relative
phases. The concurrence for this state requires the solution of the full 4 � 4 matrix in
Eq. (4.24). For atoms separated apart i.e., rij � � (i; j = 1; 2) so that ij, ij � 0, we
�nd the concurrence (Eq. (4.10)) for the state Eq. (4.27) as [94]
C(t) = max�0; e� tC(0)� 2�21e�2 t
�e t � 1
�; (4.28)
where c(0) = 2p�21�
24 + �22�
23 � 2�1�2�3�4 cos�; � = �1 + �2 � �3. The sudden death
time (SDT) can be calculated from c(t) = 0, obtained as
td =1
log
�2�21
2�21 � C (0)
�: (4.29)
63
From concurrence expression, we see that SDE occurs when either initial concurrence
is zero or �1 > 12
��4 �
pj�24 � 4�2�3j
�: SDT (Eq. (4.29)) also depends upon the
probability �21 of doubly excited component ja1; a2i in the state (Eq.(4.27)). Larger is
the probability �21, earlier is the �nite-time disentanglement (SDE).
For atoms close enough so that rij � � (i; j = 1; 2), collective damping ij and
dipole-dipole interaction ij dominates in atomic dynamics since the system can exchange
energy as a result of the dipole-dipole interaction. Assuming ~d:~rij = 0 for two atoms,
i.e., parallel dipoles, 12 and 12 takes the form
12 =3
2
"sin [korij]
korij+cos [korij]
(korij)2 � sin [korij]
(korij)3
#; (4.30)
12 =3
4
"�cos [korij]
korij+sin [korij]
(korij)2 +
cos [korij]
(korij)3
#: (4.31)
The distance dependence of the dipole-dipole interaction 12 and collective damping
0
10:pdf
Figure 4-2: The distance dependence is shown for dipole-dipole interaction 12 (solidline) and collective damping terms 12 (dotted line).
64
terms 12 are shown in Fig. 4-2. It shows non-zero values at small interatomic distance
(r12 < �) and nearly zero when separation between the two atoms are greater than atomic
transition wavelength. For the general state Eq. (4.27), we may not obtain a simpli�ed
expression for concurrence (as we have for separated case) because more variables are
involved. We can more elaborate the role of each probability amplitude by considering
special cases. For this purpose, we de�ne a X-shape matrix that covers the dynamics of
all Bell-type states as
�(t) =
0BBBBBB@�11(t) 0 0 �14(t)
0 �22(t) �23(t) 0
0 �32(t) �33(t) 0
�41(t) 0 0 �44(t)
1CCCCCCA : (4.32)
Using Eq. (4.9), we can �nd the eigenvalues of the matrix M as
�1(2)(t) =p�23(t)�32(t)�
p�22(t)�33(t); (4.33)
�3(4)(t) =p�14(t)�41(t)�
p�11(t)�44(t): (4.34)
Case-I:
Let us consider a superposition state that is Bell-type state with arbitrary probability
amplitudes �2, �3 with their corresponding components, given by
ji = �2 ja1; b2i+ �3ei� jb1; a2i : (4.35)
The � (t) of the concurrence expression in Eq. (4.10) can be written as
� (t) = 2�p
�22(t)�33(t)�p�11(t)�44(t)
�: (4.36)
65
For vacuum environment i.e., �n = 0, we can write the explicit expression for concurrence
as follows
� (t) = e� t��(�2�22 � 1
�cos 212t� 2�2�3 sin� sin 212t)2 (cosh 12t� 2�2�3
� cos� sinh 12t)2]: (4.37)
The analytical results with the mean thermal photon number �n are more complicated so
not produced here, only numerical results are shown in plots.
For noninteracting atoms � (t) = �2�3e� t, that is both atoms come to ground state
in in�nite time. C (t) = 0 only when either �2 or �3 are zero or when times approaches
in�nity in atomic scales.
In interacting dipole (spatially close) systems, entanglement dynamics also depends
upon the relative phase � as shown in Eq. (4.37). For � = 0 and �2 = �3, state is
symmetric and second term in Eq. (4.37) becomes zero. First term also become zero and
we have Bell state��(+)� (Eq. (4.2)) dynamics. We can obtain concurrence from the last
term as � (t) = e�( + 12)t. The symmetric state dynamics is boosted by addition of the
term 12 in spontaneous decay rate : The state decays fast and there are no oscillations
caused by the energy exchange between the two dipoles. For � = � and �2 = �3; we
have asymmetric state and second term in Eq. (4.37) again become zero. For Bell state��(�)� (Eq. (4.3)), �rst term also become zero again and only the last term retain so that� (t) = e�( � 12)t. The asymmetric state decays slowly as compared to the symmetric
state. The concurrence becomes independent of dipole-dipole interaction term 12 for
Bell states��(+)� and ��(�)�, i.e., �2 = �3 =
1p2in the Eq. (4.37).
The concurrence plots at di¤erent values of mean thermal photon numbers are shown
in Fig. 4-3 for ji at � = 0, � and �2 =p0:9. The increase in the mean thermal
photon number decreases the amount of entanglement. The asymmetric state o¤ers
higher amount of entanglement during evolution as compared to the symmetric state.
Although both symmetric and asymmetric states decay asymptotically. The plots show
66
4-1
11:pdf
Figure 4-3: Entanglement dynamics of initial entangled state ji at (a) � = 0 and (b)� = � is shown for R = 0:075�; �2 =
p0:9 at di¤erent temperatures, �n = 0 (solid line),
�n = 0:1 (dotted line) and �n = 0:2 (dashed line).
the oscillations that are due to the presence of dipole-dipole interaction 12 term in � (t) :
The oscillatory behavior exists only at initial times which is damped out due to dissipative
evolution with time. We obtain a 3D plot of concurrence in Figs. 4-4 (a) and (b)
against time and initial probability �22 and distance between the two atoms R=� = r12=�;
respectively, for state ji at � = 0. Fig. 4-4(a) shows the similar oscillatory behavior
at the extreme values of the probability �22 and smooth decay dynamics for intermediate
values particularly for Bell state, as discussed above.
The entanglement dynamics also depends upon the distance between qubits. The
distance dependence is shown in Fig. 4-4(b) by �xing the initial probability as �22 = 0:9:
In Fig. 4-4(b), the rapid oscillations show rapid energy exchange between atoms at very
small distances. As distance between the two atoms increases these oscillations die away.
For R > 0:3�, we observe the smooth asymptotic behavior.
67
4-2
12:pdf
Figure 4-4: The plot of entanglement dynamics versus (a) initial probability �xing R =0:05�; (b) normalized distance between the atoms R=� �xing �2 =
p0:9 and time for
ji at �n = 0:1 and � = 0:
Case-II:
Now consider a combination of doubly excited component and the component in which
both atoms are in their ground states, given by
j�i = �1 ja1; a2i+ �4ei� jb1; b2i : (4.38)
We obtain two � (t) of concurrence, de�ned as
�1 (t) = 2�p
�14(t)�41(t)�p�22(t)�33(t)
�; (4.39)
�2 (t) = 2�p
�22(t)�33(t)�p�11(t)�44(t)
�: (4.40)
so that concurrence is de�ned as
C (t) = max [0;�1 (t) ;�2 (t)] : (4.41)
68
For close atoms (interacting dipoles), we obtain explicit expressions for �1 (t) and �2 (t)
Figure 4-5: The schematic diagram demonstrates the j�i = �1 ja1; a2i + �4ei� jb1; b2i
state decay dynamics. The intermediate levels��(+)� and ��(�)� are symmetric and
asymmetric states, respectively.
for �n = 0 as
�1 (t) =2�21e
�2 t
( 2 � 212)
�� 212 + 2
�(1� e t cosh 12t) + 2 12e
t sinh 12t)�+ 2�1�4e
� t;
(4.42)
�2 (t) =�2�21e�2 t( 2 � 212)
�� 212 + 2
�(1� e t cosh 12t) + 2 12e
t sinh 12t)�� 2�1( 2 � 212)
e� t
��2�21e
� t �� 212 + 2�(cosh t� cosh 12t) + 2 12 ( sinh 12t� 12 sinh t)
�+ �24
� 2 � 212
� 12 : (4.43)
There is no contribution of the phase factor �:We have two � (t) of concurrence because
when the doubly excited component comes to the ground state, it follows both symmetric
and asymmetric state dynamics, as shown in Fig. 4-5. Therefore, when the population
decays from the symmetric state, the decay to the ground state is fast. Rather stable
dynamics is studied when population from asymmetric state decays to the ground state.
69
The analytical results with the mean thermal photon number �n are more complicated so
not produced here, only numerical results are shown in plots.
4-3a
14:pdf
Figure 4-6: Entanglement dynamics of initial entangled state j�i for R = 0:075� atdi¤erent temperatures, �n = 0 (solid line), �n = 0:01 (dotted line) and �n = 0:02 (dashedline).
For noninteracting qubits, we have
� (t) = max�0; 2
��1�4e
� t � �21e�2 t �e t � 1��� : (4.44)
It is obvious that for j�1j � j�4j entanglement disappears only when the individual
qubits completely decay, i.e., we never observe SDE. Anyhow, for j�1j > j�4j, we observe
SDE. Thus, we see that locally equivalent pure states with the same initial quantum
correlations behave very di¤erently as a result of interaction with the environment and
simple local operations performed on initial state can change disentanglement time from
�nite to in�nite.
Now consider the interacting qubits, we have two values �1 (t) and �2 (t) of concur-
rence. The plots in Figs. 4-6 and 4-7(a) show the collapse and revival of entanglement
as system evolve with time for j�1j > j�4j that undergoes only SDE in noninteracting
qubits case. The �rst collapse and revival occurs for very small range �21 = 0:88 to 0:93
70
of initial probability, collapse occurs as �1 (t) become negative and revival due to �2 (t)
that survives only for a very short time, the �2 (t) then becomes negative for always. The
second revival occurs as �1 (t) becomes positive. This term remains positive and decays
asymptotically for all values of �1 and follows asymmetric state dynamics. Although the
amount of entanglement is very small in this case.
4-3
15:pdf
Figure 4-7: The plots of entanglement dynamics for j�i is shown versus (a) initial prob-ability �xing R = 0:05� and �n = 0:1; (b) normalized distance between the atoms R=��xing �1 =
p0:9 and �n = 0 and time.
The collapses and revivals also depend upon the distance between the atoms. The
distance dependence is shown in Fig. 4-7(b) by �xing the initial probability �21 = 0:9;
that is for j�1j2 > j�4j2 : At small inter-atomic distances, we observe both collapses and
revivals of entanglement. As the distance increases the �rst revival dies away and we stay
with the second revival. The second revival appears due to the contribution from �1 (t)
term only, the slow decay dynamics then follows. For very small inter-atomic distances,
we see only �rst revival. It is due to the contribution from �2 (t) term only, the fast decay
dynamics is followed. As the inter-atomic distance increases, the height of �rst revival
71
decreases and second revival just appears. The entanglement dynamics is very slow after
then at relatively small distances. At large inter-atomic distances, we see �nite-time
disentanglement (SDE) and no revival. The plot in Fig.4-6 shows the e¤ect of increase in
the environment temperature. As temperature increases, we have relatively small values
of concurrences, still collapse and revivals appear. At relatively higher temperatures, we
only observe collapse and no revivals.
4.4.4 Initial Unentangled States
For an initial unentangled state ji = ja1; b2i, i.e. when either �2 or �3 is zero in Eq.
(4.35), we have
� (t) = e� t�cosh2 ( 12t)� cos2 (212t)
� 12 : (4.45)
For noninteracting (distant) qubits, the initial separable state always remain separable.
Now we can see that entanglement is generated in close atoms case. As the two qubits
exchange energy due to coherent dipole-dipole interaction, entanglement is generated.
Although it dissipates away with time. So, initially we see oscillations that are being
damped with time, as shown in Fig. 4-8(a). The initial unentangled state becomes en-
tangled even at higher values of mean thermal photon number. The e¤ect of temperature
on this initial separable state is shown in Fig. 4-8(a) for di¤erent values of mean thermal
photon number �n.
For initial unentangled state of the form ji = ja1; a2i ; that is when probability of
both atoms being in their ground state is zero. The plot in Fig. 4-8(b) shows that �rst
revival never appears as �2 (t) remains negative. The second revival appears as �1 (t)
become positive as system evolve with time. Thus, initial unentangled state become
entangled with time. The state then decays asymptotically with time . The plot in Fig.
4-8(b) also shows the e¤ect of temperature on entanglement. We see that entanglement
is always generated for non-zero but for very small values of mean photon number �n:
72
4-4
16:pdf
Figure 4-8: Entanglement dynamics of initial unentangled states (a) ja1; b2i for R =0:075�, n = 0 (solid line), n = 0:1 (dotted line) and �n = 0:2 (dashed line), (b) ja1; a2ifor R = 0:075�, n = 0 (solid black), n = 0:01 (dotted line), n = 0:02 (dashed line).
73
Chapter 5
Entanglement Dynamics of
Two-Qutrit Systems
In the previous chapter, we studied the entanglement dynamics of two-qubit atoms and
used the Wootters concurrence formula [95] as a measure of entanglement. Each 2-qubit
system comprises the two dimensional Hilbert space H = HA HB: We studied that
entanglement dynamics depends upon the initial separation between the systems and
initial preparation of the states (whether entangled or not). In this chapter, we consider
the same problem for bipartite qudit systems. Qudits are multi-level versions of qubits
and comprise d-dimensional Hilbert space. In our case, we consider d = 3, qutrit is
another name for a three-dimensional system. Although, there are many entanglement
characterization schemes for bipartite systems, as mentioned before. We imply the posi-
tive partial transposition or PPT criterion for higher dimensional systems. The state of
any higher-dimensional bipartite system is separable if it can be written as
� =Xi
pi�iA �iB; (5.1)
where �A and �B are the density matrices for two subsystems A and B. According to PPT
criterion, a new density matrix � =P
i pi (�iA)T �iB should also be a legitimate density
74
matrix for quantum states. It implies that the eigenvalues of � are also nonnegative;
this is the necessary condition for Eq. (5.1) to be valid for separable systems. Right
away, we need to quantify the amount of entanglement in higher-dimensional systems
where Wootters concurrence formula is no more valid. Henceforth, we use negativity
which is based on PPT criterion to quantify the amount of entanglement in higher-
dimensional bipartite systems. In this chapter, we study the entanglement dynamics of
higher-dimensional bipartite �eld states inside the two cavities [152]. The two cavities
are assumed to be separated and identical systems. The initial �eld states are pure
entangled states. Although the decay dynamics with one photon transitions is studied
in the previous chapter but in this chapter we proceed with the entanglement analysis
with the new measure of entanglement, i.e., negativity. We also make the comparison
between single photon entangled states and two-photon states dynamics. We also discuss
the generation of each type of �eld entangled state and the state with �xed number of
photons.
5.1 Model
Let us consider a bipartite system represented by two high-Q cavities A and B; as shown
in Fig. 5-1. The �eld states inside the cavities are assumed to be in pure form, in general,
given by
jAB (t)i =2X
P;Q=0
CPQ jPA; QBi ; (5.2)
whereCPQ�s are the probability amplitudes with the normalization condition2X
P;Q=1
jCPQj2 =
1, P and Q represent the number of photons in cavity A and B, respectively. Each cavity
is the independent system and their is no direct interaction between the cavities except
the initial entanglement assumed between the cavity �eld states. The interaction picture
75
Figure 5-1: Two identical and independent cavities contain initial entangled states of�elds. The �elds inside the cavities interact with their own environment.
Hamiltonian can be obtained after electric-dipole approximation as
H (t) = ~Xj=A;B
Xk
g(j)k
��b(j)k
�yaje
�i(���k)t +H:C:
�; (5.3)
where H.C. denotes the Hermitian conjugate, b(j)k�bjk
�yare the k�th reservoir mode an-
nihilation (creation) operator with frequency �k for cavity j = A;B, g(j)k is the coupling
constant of the reservoir-�eld system and aj�ayj
�is annihilation(creation) operator for
the particular two-cavities �eld modes. We consider the general reservoir theory again
to derive the master equation for cavity �elds to study the entanglement dynamics. The
environment surrounding the cavities is supposed to be in thermal state with mean ther-
mal photon number �n1 for cavity-A and �n2 for cavity-B, as shown in Fig. 5-1. Using the
Eq. (5.3) and Eq. (4.20), we arrive at the following reduced equation of motion for �eld
states [150] after rotating wave and Born-Markov approximations as
76
_� = �2Xi=1
(�ni + 1)
2ki
�ayi ai�� 2ai�a
yi + �ayi ai
��
2Xi=1
�ni2ki
�aia
yi�� 2a
yi�ai + �aia
yi
�;
(5.4)
where ki is the cavity decay rate and �ni is the mean thermal photon number for the i�th
cavity. In deriving this master equation, we assume that there is no feedback from the
environment and history of interaction is lost. Thus, we assume that the correlation time
between the �elds and the reservoirs is much shorter than the characteristic time required
for the dynamical evolution of �eld states such as spontaneous emission time and SDT
so that the Markov approximation is valid. We can write the density matrix elements
by de�ning the basis as j1i ! j0A; 0Bi ; j2i ! j0A; 1Bi ; j3i ! j0A; 2Bi ; j4i ! j1A; 0Bi ;
j5i ! j1A; 1Bi ; j6i ! j1A; 2Bi ; j7i ! j2A; 0Bi ; j8i ! j2A; 1Bi ; j9i ! j2A; 2Bi for a 9�9
density matrix �. Since we need the partial transposed state to measure the entanglement,
so the composite matrix can be written under partial transposition with respect to the
cavity-A as
�TA (t) =
0BBBBBBBBBBBBBBBBBBBBB@
�11 (t) �12 (t) �13 (t) �41 (t) �42 (t) �43 (t) �71 (t) �72 (t) �73 (t)
�21 (t) �22 (t) �23 (t) �51 (t) �52 (t) �53 (t) �81 (t) �82 (t) �83 (t)
�31 (t) �32 (t) �33 (t) �61 (t) �62 (t) �63 (t) �91 (t) �92 (t) �93 (t)
�14 (t) �15 (t) �16 (t) �44 (t) �45 (t) �46 (t) �74 (t) �75 (t) �76 (t)
�24 (t) �25 (t) �26 (t) �54 (t) �55 (t) �56 (t) �84 (t) �85 (t) �86 (t)
�34 (t) �35 (t) �36 (t) �64 (t) �65 (t) �66 (t) �94 (t) �95 (t) �96 (t)
�17 (t) �18 (t) �19 (t) �47 (t) �48 (t) �49 (t) �77 (t) �78 (t) �79 (t)
�27 (t) �28 (t) �29 (t) �57 (t) �58 (t) �59 (t) �87 (t) �88 (t) �89 (t)
�37 (t) �38 (t) �39 (t) �67 (t) �68 (t) �69 (t) �97 (t) �98 (t) �99 (t)
1CCCCCCCCCCCCCCCCCCCCCA
:
(5.5)
The equations of motion in terms of the density matrix elements are derived and given
along with their solutions in Appendix-B. Since we assume identical cavities, so we can
write cavity �eld decay rates k1 = k2 = k:
77
We use the negativity to calculate the amount of entanglement in two-photon and
one-photon �eld initial entangled states. The negativity stands on the PPT criterion and
is formulated from negative eigenvalues �i of the partial transposed matrix �TA, given by
N (t) = max[0;�2Xi
�i (t)]: (5.6)
Like all other entanglement measures, N = 0 denotes the separable or unentangled states
and N = 1 for maximally entangled states.
5.2 Bipartite Field States
In this section, we study some special cases of entangled �eld states and compare the
entangled states dynamics of two-photons with one photon entangled states numerically
and analytically.
5.2.1 Case-I:
Let us consider the two-photons NOON state as the initial �eld entangled state in the
two cavities, followed by Eq. (5.2)
���(1)AB (0)E = C20 j2A; 0Bi+ C02 j0A; 2Bi : (5.7)
Assuming vacuum surroundings for both cavities, �ni = 0 (i = 1; 2), we obtain negativity
as following
N (t) = max
�0; e�2kt �
�1� ekt
�2+
q(1� ekt)4 + 4C202C
220
�: (5.8)
Setting C02 = C20 =1p2, we have maximally entangled state with two photons. This
kind of initial entangled �eld state can be generated as suggested schematically in [153],
has its potential applications in quantum lithography and Heisenberg-limited metrology.
78
5-1
18:pdf
Figure 5-2: Exponential decay dynamics of initial entangled (a) NOON state���(1)AB (0)E =p
1� C202 j2A; 0Bi+C02 j0A; 2Bi and (b)����(1)AB (0)E =p1� C201 j1A; 0Bi+C01 j0A; 1Bi at
all times is shown versus the initial probabilities of the states C202 and C201, respectively.
5-2
19:pdf
Figure 5-3: The role of temperature on entanglement dynamics is shown for maximallyentangled (a) NOON state jAB (0)i = 1p
2[j2A; 0Bi+ j0A; 2Bi] and (b) jAB (0)i =
1p2[j1A; 0Bi+ j0A; 1Bi] for �n = 0 (solid line), �n = 0:1 (dotted line) and �n = 0:2 (dashed
line).
79
In Fig. 5-2, there is a 3D comparison in plots of negativity versus probability and time
for the given NOON state (Eq. (5.7)) and the state, given by
����(1)AB (0)E = C01 j0A; 1Bi+ C10 j1A; 0Bi : (5.9)
It is obvious that decay is asymptotic at all times and for all values of initial probabilities
C202 and C201 for the two states. Although the given NOON state (Eq. (5.7)) decays fast
due to presence of more photons in the �elds but it always decays asymptotically (no
SDE). The Fig. 5-3 show the decay dynamics of maximally entangled NOON state (i.e.,
C02 = C20 =1p2) and maximally entangled Bell state
����(1)AB (0)E (i.e., C01 = C10 =1p2) for
di¤erent values of mean thermal photon number �n1 = �n2 = �n. We can see that���(1)AB (0)E
decays fast in comparison to����(1)AB (0)E and its sudden death time (SDT) also decreases.
It can also be seen that sudden death time decreases as mean thermal photon number
�n increases for the two states. Certainly, SDE always occurs in thermal environment as
reported earlier [93, 94].
5.2.2 Case-II:
Let us consider another initially entangled state, given by
���(2)AB (0)E = C00 j0A; 0Bi+ C22 j2A; 2Bi : (5.10)
Assuming �ni = 0 (i = 1; 2), we obtain the explicit negativity expression as
N (t) = maxh0; 2C22
��C22e�4kt
��1 + ekt
�2+ C00e
�2kt�i: (5.11)
Setting C00 = C22 =1p2; we obtain another maximally entangled state. This kind of
entangled �eld state can be generated by swapping the state of zero and two photons in
Eq. (5.7), as discussed above [154, 155]. In Fig. 5-4, there is a 3D comparison in plots of
negativity versus initial probability for the state and time for the state���(2)AB (0)E (Eq.
80
5-3
20:pdf
Figure 5-4: Exponential decay dynamics of initial entangled (a)���(2)AB (0)E =p
1� C222 j0A; 0Bi + C22 j2A; 2Bi and (b)����(2)AB (0)E = p
1� C211 j0A; 0Bi + C11 j1A; 1Biat all times is shown versus the initial probabilities of the states C222 and C
211, respec-
tively.
5-4
21:pdf
Figure 5-5: The role of temperature on entanglement dynamics is shown for maxi-mally entangled states (a) jAB (0)i = 1p
2[j0A; 0Bi+ j2A; 2Bi] and (b) jAB (0)i =
1p2[j0A; 0Bi+ j1A; 1Bi] for �n = 0 (solid line), �n = 0:1 (dotted line) and �n = 0:2 (dashed
line).
81
(5.10)) and the state , given by
����(2)AB (0)E = C00 j0A; 0Bi+ C11 j1A; 1Bi : (5.12)
The state����(2)AB (0)E is supposed to show SDE when C11 > C00 [94]. The Fig. 5-4(b) shows
that SDE occurs when the initial probability of the doubly excited component C211 for
the state����(2)AB (0)E exceeds 0:5. Larger is this probability shorter is the SDT. Relatively,
decay dynamics is shown to be asymptotic in Fig. 5-4(a) for the state���(2)AB (0)E at
larger values (greater than 0.5) of initial probability C222: The plots in Fig. 5-5 show the
decay dynamics of maximally entangled state���(2)AB (0)E (i.e., C00 = C22 =
1p2) and the
Bell state����(2)AB (0)E (i.e., C00 = C11 =
1p2) for di¤erent values of mean thermal photon
number �n1 = �n2 = �n. The (SDT) increases for���(2)AB (0)E in comparison. For the initial
state given in Eq. (5.10), the sudden death time is calculated as
td =1
kln
� pC22p
C22 �pC00
�: (5.13)
It also shows that SDE for this two-photon entangled state in each cavity also happens
when C22 > C00: Above equation also explains that sudden death time increases when
two-photon transition probability increases. This delay is because of the presence of
intermediate transition states before the state comes completely to ground state. Again
SDT decreases as the probability C222 increases. Let us consider the initial state����(2)AB (0)E
and calculate its negativity as [94]: Now, we can write the negativity for two-photon
entangled state���(2)AB (0)E from Eq.(5.11) as
N (t) = N1 (t) e�kt �2� e�kt
�; (5.14)
where
N1 (t) = C (t) = 2�C00C11e
�2kt � C211e�2kt �ekt � 1�� : (5.15)
82
It is so written as initially maximally entangled state is assumed, that is by setting all
initial probability amplitudes in entangled states���(2)AB (0)E and ����(2)AB (0)E are equal to
1p2: The e¤ect of temperature is shown in Fig. 5-5. We can see that sudden death time
decreases with increase in the mean thermal photon number and SDE always takes place
for �n > 0:
5.2.3 Case-III:
Let us consider the special case of an initial entangled state with �xed number of photons
in the two cavities as
jAB (0)i = C02 j0A; 2Bi+ C11 j1A; 1Bi+ C20 j2A; 0Bi : (5.16)
This state becomes maximally entangled by setting C02 = C11 = C20 = 1=p3. The
generation of this kind of state is discussed in [31]. According to this scheme, two ex-
cited two-level atoms pass by two empty high-Q cavities for pre-determined interaction
times gt1A = 0:7708, gt1B = 1:5708, gt2A = 0:6667 and gt2B = 1:1107: In the end, the
probability of detection of two atoms in the ground state will be 0:9535: This kind of
maximally entangled state can be an important quantum resource for the quantum tele-
portation of �eld state of the kindXk
Ck jki : [154]. The e¤ect of probabilities on the
entanglement dynamics is shown in the Fig. 5-6(a). The 3D plot is obtained by initially
assuming C02 = C20 =
q1�C2112
; that is the probabilities of single excited components
of each �eld modes are supposed to be equal. The Fig. 5-6(a) also explains that en-
tanglement reaches 1 and then declines linearly until C11 = 1 when the state becomes
unentangled. SDE also supposed to happen as the doubly excited component j1A; 1Bi is
present in this kind of entangled state. In Fig. 5-6(b), there is a plot of entanglement
dynamics for �n � 0 for initial maximally entangled state with �xed number of photons1p3(j0A; 2Bi+ j1A; 1Bi+ j2A; 0Bi) : We can see that SDE always happen for �n > 0 and
SDT decreases with increase in mean thermal photon number �n.
83
5-5
22:pdf
Figure 5-6: Negativity is plotted (a) versus time and initial probability of the state
(Eq. (5.16)) C211 �xing C02 = C20 =
q1�C2112
and (b) for maximally entangled statejAB (0)i = 1p
3(j0A; 2Bi+ j1A; 1Bi+ j2A; 0Bi) for �n = 0 (solid line), �n = 0:1 (dotted
line) and �n = 0:2 (dashed line).
84
Chapter 6
Conclusion
Quantum theory o¤ers many concepts that can�t be demonstrated classically. One such
phenomenon is entanglement that is a coherent superposition in di¤erent parts of com-
posite systems. Entanglement is being investigated for its vital applications in many
QIP tasks. Superposition in di¤erent parts of quantum memory registers o¤ers parallel
processing and faster computation. There are two crucial points largely being investigated
for a realizable quantum computer [17]. One is the generation of coherent superposition
or entanglement in di¤erent quantum registers and second is the control on quantum
logical operations before decoherence occurs.
Generation of entanglement and the study of entanglement loss due to decoherence
are the subjects of this thesis. The generation of entanglement is explored in quantum
optical devices (with both passive and active ones). The generation of entanglement is
investigated with special class of continuous variable (CV) states, Gaussian states. The
problem is studied with initial separable single-mode Gaussian light beams at the inputs
of beam splitter and quantum beat laser. The inputs are de�ned so that one Gaussian
beam is de�ned with arbitrary value of nonclassicality and purity, an arbitrary Gaussian
state. The second input is a classical Gaussian state i.e., a thermal �eld. The logarithmic
negativity as a quantitative measure of entanglement is implied to investigate the depen-
dence of entanglement on given parameters. The given parameters are nonclassicality �
85
and purity u of the arbitrary Gaussian state and mean thermal photon number n repre-
senting classical thermal �eld. It also depends upon the beam splitter settings (angle �)
in beam splitter case and classical driving �eld in quantum beat laser system.
The beam splitter is initially assumed to be lossless. For a half transmitting mirror,
a 50:50 beam splitter, we �nd that entanglement only depends upon the amount of
nonclassicality � of the input Gaussian state and mean thermal photon number n of the
thermal noise at the second input. A threshold for thermal �eld is observed, entanglement
at the output disappears above a critical value of thermal noise. The critical thermal
noise is also independent of purity of input Gaussian state for 50:50 beam splitter. On
the other hand, for a maximum value of nonclassicality i.e., � ! 12entanglement persists
at every level of thermal noise.
For general beam splitter settings with a certain deviation in transmittance from half,
entanglement then depends upon this deviation, nonclassicality and purity of arbitrary
Gaussian �eld and mean thermal noise. The dependence of entanglement and critical
thermal noise on purity of arbitrary Gaussian state is investigated both analytically and
numerically. It is observed that entanglement increases with the increase in the amount
of nonclassicality � and decreases with the mean thermal photon number n: A 50:50 beam
splitter setting is the optimal choice, since we have maximum amount of entanglement
for each value of thermal noise n. For general beam splitter settings, entanglement at the
outputs depends upon angle of beam splitter, purity and nonclassicality of single-mode
Gaussian state, mean thermal noise and deviation in transmittance.
Quantum beat laser (QBL) is another device which can generate entanglement be-
tween two modes of the radiation �eld. It works on the interference phenomenon to
generate the entanglement at the outputs. It consists of a doubly resonant cavity with
active medium of three-level atoms in V-con�guration. These three-level atoms in V-
con�guration are prepared in coherent superposition by a classical driving �eld so that
the two excited levels share the same ground level. Thus, the two modes in QBL may
get entangled depending upon the classical driving �eld that drives the upper two levels.
86
The same inputs are considered as we studied in beam splitter case, i.e. two separable
Gaussian states as the initial states of two modes in a doubly resonant cavity. We study
the problem with di¤erent cases of driving �eld strength with respect to the atomic de-
cay rates. The weak driving �eld results in no entanglement due to weak coherence.
The strong driving �eld results in the robust generation of entanglement. It comes out
to be a special case of beam splitter with losses. In this case, generation of entangle-
ment depends upon nonclassicality � and mean thermal photon number n of the initial
Gaussian states. Purity u of the arbitrary Gaussian state shows no signi�cant e¤ect on
the amount of entanglement generated. The amount of entanglement generated increases
with the increase in the amount of nonclassicality and decreases with the increase in the
temperature (or mean thermal photon number) of the initial thermal �eld.
The generation of entanglement in QBL�s is also investigated for intermediate range
of driving �eld strengths. The role of each parameter such as nonclassicality; purity of
the arbitrary Gaussian state, mean thermal photon number n and the strength of the
driving �eld on the generation of entanglement is studied. It is observed that amount of
entanglement generated increases with the increase in the driving �eld strength, but at a
certain value of driving �eld strength that is around 20 , it starts deceasing, however
its time duration increases. Entanglement also increases by increasing the amount of
nonclassicality and decreases by increasing the thermal noise at the intermediate values
of driving �eld strengths. Purity of the single-mode Gaussian state has no signi�cant
e¤ect on the entanglement generation.
Decoherence is the main obstacle in QIP tasks since it ceases the coherent superposi-
tion of quantum states. The major cause of decoherence in atomic systems is spontaneous
emission of atoms. The uncorrelated atoms obey half-time law and always decay asymp-
totically. It is observed with two two-level (i.e., two qubits) entangled atoms that entan-
glement may �nish as atoms decay to ground state in �nite time [91, 92]. It is studied
that entanglement time dynamics during interaction with the dissipative environment de-
pends upon the initial preparation of pure [92, 94] or mixed entangled states [91, 96, 93].
87
Finite-time disentanglement or so-called sudden death of entanglement (SDE) occurs if
the initial probability of both excited atoms is large as compared to other probability
amplitudes.
Ficek and Tanas [128] studied that entanglement dynamics also depends upon the
distance between the atoms. This problem is explored more extensively in this thesis for
thermal environments with interacting and noninteracting atomic qubits. SDE depends
upon the initial entanglement, distance between qubits and the nature of dissipative en-
vironment. It is studied that SDE always occurs in thermal environment for all initial
entangled noninteracting qubits [94]. Sudden death time (SDT) decreases with the in-
crease in the temperature of the surrounding environment. In interacting qubits where
interatomic spacings are in the range of their resonant wavelength and can exchange
energy, entanglement dynamics is asymptotic for symmetric and asymmetric states. The
symmetric states decay faster than asymmetric states and amount of entanglement de-
creases with the increase in the temperature of surroundings. The initial entangled state
with doubly excited component with higher probability may observe SDE and then re-
vivals in interacting qubits. This occurs because the doubly excited component decays
to the intermediate levels (symmetric and asymmetric states) before atoms come to their
respective ground states. SDE and revivals also depends upon the distance between
the atoms. At large distances we only observe SDE and no revival. It is studied that
initial unentangled states become entangled in dissipative environments due to dipole-
dipole interaction. Wootters concurrence formula [95] is used as a quantitative measure
of entanglement for both interacting and noninteracting atomic qubits.
The problem of decoherence mechanism or entanglement dynamics in dissipative en-
vironments is further extended for high dimensional qudits bipartite �eld states. A set
of di¤erent initial entangled pure state of a bipartite system are considered in two inde-
pendent and identical cavities. The entanglement dynamics is studied with the initial
entangled states of two and one photons inside the two cavities. Entanglement dynamics
of two photon NOON state is faster as compared to the one photon symmetric state.
88
Then, the initial entangled state with either two photons in each cavity or no photons
are considered and the dynamics with the initial entangled state with either one or no
photon in both cavities is compared. In this case, entanglement dynamics is although
fast but SDT is delayed as compared to doubly excited entangled state with one photon
in both cavities. This occurs because of intermediate levels present between the doubly
excited component with two photons in each cavity and zero photon state of each cavity.
The initial entangled state with �xed number of photons in each cavity is considered.
It is investigated that as the probability of doubly excited component with one pho-
ton in both cavities increases, SDT decreases. The decay is always asymptotic if this
component becomes zero. In each case, the amount of entanglement decreases with the
increase in the temperature of surrounding environment. SDE always occurs for non-zero
temperature environments in noninteracting systems.
89
Appendix A
Equations of motion of the density
matrix elements (Eq. (4.24)) and
their solutions for vacuum reservoir
The equantions of motion of the density matrix elements (Eq. (4.24)) for the thermal
reservoir can be obtained from the master equation (Eq. (4.21)) as
_�11 = �2 (n+ 1) �11 + n (�22 + �33) + n 12 (�23 + �32) ; (A.1)
_�12 = ���2n+
3
2
� + {!o
��12 �
��n+
1
2
� 12 � {12
��13 + n 12�24 + n �34; (A.2)
_�13 = ���n+
1
2
� 12 � i12
��12 �
��2n+
3
2
� + {!o
��13 + n �24 + n 12�34; (A.3)
_�14 = � [(2n+ 1) + 2{!o] �14; (A.4)
_�22 = (n+ 1) �11 � (2n+ 1) �22 ��n+
1
2
� 12 (�23 + �32) + n �44 + {12 (�23 � �32)(A.5)
_�23 = (n+ 1) 12�11 ��n+
1
2
� 12 (�22 + �33)� (2n+ 1) �23 + n 12�44 + {12 (�22 � �33) ;
(A.6)
_�24 = (n+ 1) 12�12 + (n+ 1) �13 ��2n+
1
2
� �24 �
�n+
1
2
� 12�34 � {!o�24 � {12�34;
(A.7)
90
_�33 = (n+ 1) �11 ��n+
1
2
� 12 (�23 + �32)� (2n+ 1) �33 + n �44 � {12 (�23 � �32) ;
(A.8)
_�34 = (n+ 1) �12 + (n+ 1) 12�13 �n
2 12�22 � (n+ 1)
122�24 �
�2n+
1
2
� 12�34
�{12�24 � {!o�34; (A.9)
_�44 = (n+ 1) [ 12 (�23 + �32) + (�22 + �33)]� 2n �44; (A.10)
�21 = ��21; �31 = ��13; �41 = ��14; �32 = ��23; �42 = ��24; �43 = ��34: (A.11)
The solutions of these equations for vacumm reservior (n = 0) are
�11 = �21e�2 t; (A.12)
�12 =�12e�
t2(3 +2{!)
�(�2 � �3) e
� t2(� 12+2{12) + (�2 + �3) e
� t2( 12�2{12)
�; (A.13)
�13 =�12e�
t2(3 +2{!)
�(��2 + �3) e
� t2(� 12+2{12) + (�2 + �3) e
� t2( 12�2{12)
�; (A.14)
�14 = �1�4e�2t( 2+{!); (A.15)
�22 =1
4
�2��22 � �23
�e� t cos (212t) +
e�t( + 12)
� 12
�(�2 + �3)
2 ( � 12) + 2�21 ( + 12)
�+et(� + 12)
+ 12
�(�2 � �3)
2 ( + 12) + 2�21 ( � 12)
��� 4�
21 (
2 + 212)
2 � 212e�2 t; (A.16)
�23 =1
4
����22 + �23
�e�t( +{12) +
e�t( + 12)
� 12
�(�2 + �3)
2 ( � 12) + 2�21 ( + 12)
�+��22 � �23
�e�t( �{12) � et(� + 12)
+ 12
�(�2 � �3)
2 ( + 12) + 2�21 ( � 12)
��+8 12�
21
212 � 2e�2 t; (A.17)
�24 =1
2
���1 (�2 + �3) ( + 12)
� 2{12e�
t2(3 +2{!+ 12�2{12) � �1 (�2 � �3) ( � 12)
+ 2{12
e�t2(3 +2{!� 12+2{12) +
e�t2( +2{!+ 12+2{12)
� 2{12(�2 + �3) (�1 ( + 12) + �4 ( � 2{12))
�e� t2( +2{!� 12�2{12)
+ 2{12(�2 � �3) (�1 ( � 12)� �4 ( + 2{12))
); (A.18)
91
�33 =1
4
��2��22 � �23
�e� t cos (212t) +
e�t( + 12)
� 12
�(�2 + �3)
2 ( � 12) + 2�21 ( + 12)
�� e�2 t
2 � 2124�21
� 2 + 212
�+et(� + 12)
+ 12
�(�2 � �3)
2 ( + 12) + 2�21 ( � 12)
��;
(A.19)
�34 =1
2
(e�
t2( +2{!� 12�2{12)
+ 2{12(�2 � �3) (�1 ( � 12)� �4 ( + 2{12)) � �1 (�2 � �3)
� ( � 12)e�
t2(3 +2{!� 12+2{12)
+ 2{12+e�
t2( +2{!+ 12+2{12)
� 2{12(�2 + �3) (�1 ( + 12)
+�4 ( � 2{12)) �e�
t2(3 +2{!+ 12�2{12)
� 2{12�1 (�2 + �3) ( + 12)
); (A.20)
�44 = �21 + �22 + �23 + �24 +( 2 + 3 212)�
21
2 � 212e�2 t � e�t( + 12)
2 ( � 12)((�2 + �3)
2 ( � 12)
+2�21 ( + 12)�� et(� + 12)
2 ( + 12)
�(�2 � �3)
2 ( + 12) + 2�21 ( � 12)
�: (A.21)
92
Appendix B
Equations of motion of the density
matrix elements (Eq. (5.5)) and
their solutions for vacuum reservoir
The equations of motion in terms of density matrix elements for the general state (Eq.
(5.5)) are given by
_�11 = �n1k1�11 � n2k2�11 + (n2 + 1) k2�22 + (n1 + 1) k1�44; (B.1)
_�12 = � (n2 + 1)k22�12 � n1k1�12 �
3
2n2k2�12 +
p2 (n2 + 1) k2�23 + (n1 + 1) k1�45;
(B.2)
_�13 = � (n2 + 1) k2�13 � n1k1�13 � 2n2k2�13 + (n1 + 1) k1�46; (B.3)
_�14 = � (n1 + 1)k12�14 �
3
2n1k1�14 � n2k2�14 + (n2 + 1) k2�25 +
p2 (n1 + 1) k1�47;
(B.4)
_�15 = � (n1 + 1)k12�15 �
3
2n1k1�15 �
3
2n2k2�15 � (n2 + 1)
k22�15
+p2 (n2 + 1) k2�26 +
p2 (n1 + 1) k1�48; (B.5)
_�16 = � (n1 + 1)k12�16 � (n2 + 1) k2�16 �
3
2n1k1�16 � 2n2k2�16 +
p2 (n1 + 1) k1�49;
(B.6)
93
_�17 = � (n1 + 1) k1�17 � 2n1k1�17 � n2k2�17 + (n2 + 1) k2�28; (B.7)
_�18 = � (n1 + 1) k1�18 � 2n1k1�18 �3
2n2k2�18 � (n2 + 1)
k22�18 +
p2 (n2 + 1) k2�29; (B.8)
_�19 = � (n1 + 1) k1�19 � (n2 + 1) k2�19 � 2n1k1�19 � 2n2k2�19; (B.9)
_�22 = � (n2 + 1) k2�22 � n1k1�22 + n2k2�11 � 2n2k2�22 + 2 (n2 + 1) k2�33 + (n1 + 1) k1�55;
(B.10)
_�23 =p2n2k2�12 �
3
2(n2 + 1) k2�23 � n1k1�23 �
5
2n2k2�23 + (n1 + 1) k1�56; (B.11)
_�24 = � (n1 + 1)k12�24 �
3
2n1k1�24 �
3
2n2k2�24 � (n2 + 1)
k22�24 +
p2 (n2 + 1) k2�35
+p2 (n1 + 1) k1�57; (B.12)
_�25 = n2k2�14 � (n1 + 1)k12�25 � (n2 + 1) k2�25 �
3
2n1k1�25 � 2n2k2�25
+2 (n2 + 1) k2�36 +p2 (n1 + 1) k1�58; (B.13)
_�26 =p2n2k2�15 � (n1 + 1)
k12�26 �
3
2(n2 + 1) k2�26 �
3
2n1k1�26 �
5
2n2k2�26
+p2 (n1 + 1) k1�59; (B.14)
_�27 = � (n1 + 1) k1�27 � (n2 + 1)k22�27 � 2n1k1�27 �
3
2n2k2�27 +
p2 (n2 + 1) k2�38;(B.15)
_�28 = n2k2�17 � (n1 + 1) k1�28 � (n2 + 1) k2�28 � 2n1k1�28 � 2n2k2�28 + 2 (n2 + 1) k2�39;
(B.16)
_�29 =p2n2k2�18 � (n1 + 1) k1�29 �
3
2(n2 + 1) k2�29 � 2n1k1�29 �
5
2n2k2�29; (B.17)
_�33 = 2n2k2�22 � n1k1�33 � 2 (n2 + 1) k2�33 � 3n2k2�33 + (n1 + 1) k1�66; (B.18)
_�34 = � (n1 + 1)k12�34 � (n2 + 1) k2�34 �
3
2n1k1�34 � 2n2k2�34 +
p2 (n1 + 1) k1�67;(B.19)
_�35 =p2n2k2�24 � (n1 + 1)
k12�35 �
3
2(n2 + 1) k2�35 �
3
2n1k1�35 �
5
2n2k2�35
+p2 (n1 + 1) k1�68; (B.20)
_�36 = 2n2k2�25 � (n1 + 1)k12�36 � 2 (n2 + 1) k2�36 �
3
2n1k1�36 � 3n2k2�36
+p2 (n1 + 1) k1�69; (B.21)
_�37 = � (n1 + 1) k1�37 � (n2 + 1) k2�37 � 2n1k1�37 � 2n2k2�37; (B.22)
_�38 =p2n2k2�27 � (n1 + 1) k1�38 �
3
2(n2 + 1) k2�38 � 2n1k1�38 �
5
2n2k2�38; (B.23)
_�39 = 2n2k2�28 � (n1 + 1) k1�39 � 2 (n2 + 1) k2�39 � 2n1k1�39 � 3n2k2�39; (B.24)
94
_�44 = n1k1�11 � (n1 + 1) k1�44 � 2n1k1�44 � n2k2�44 + (n2 + 1) k2�55 + 2 (n1 + 1) k1�77;
(B.25)
_�45 = n1k1�12 � (n1 + 1) k1�45 � (n2 + 1)k22�45 � 2n1k1�45 + 2 (n1 + 1) k1�78 (B.26)
�32n2k2�45 +
p2 (n2 + 1) k2�56; (B.27)
_�46 = n1k1�13 � (n1 + 1) k1�46 � (n2 + 1) k2�46 � 2n1k1�46 � n2k2�46 + 2 (n1 + 1) k1�79;
(B.28)
_�47 =p2n1k1�14 �
3
2(n1 + 1) k1�47 �
5
2n1k1�47 � n2k2�47 + (n2 + 1) k2�58; (B.29)
_�48 =p2n1k1�15 �
3
2(n1 + 1) k1�48 � (n2 + 1)
k22�48 �
5
2n1k1�48 �
3
2n2k2�48
+p2 (n2 + 1) k2�59; (B.30)
_�49 =p2n1k1�16 �
3
2(n1 + 1) k1�49 � (n2 + 1) k2�49 �
5
2n1k1�49 � 2n2k2�49; (B.31)
_�55 = n1k1�22 + n2k2�44 � (n1 + 1) k1�55 � (n2 + 1) k2�55 � 2n1k1�55 � 2n1k1�55
�2n2k2�55 + 2 (n2 + 1) k2�66 + 2 (n1 + 1) k1�88; (B.32)
_�56 = n1k1�23 +p2n2k2�45 � (n1 + 1) k1�56 �
3
2(n2 + 1) k2�56 � 2n1k1�56 �
5
2n2k2�56
+2 (n1 + 1) k1�89; (B.33)
_�57 =p2n1k1�24 �
3
2(n1 + 1) k1�57 � (n2 + 1)
k22�57 �
5
2n1k1�57 �
3
2n2k2�57
+p2 (n2 + 1) k2�68; (B.34)
_�58 =p2n1k1�25 + n2k2�47 �
3
2(n1 + 1) k1�58 � (n2 + 1) k2�58 �
5
2n1k1�58
�2n2k2�58 + 2 (n2 + 1) k2�69; (B.35)
_�59 =p2n1k1�26 �
3
2(n1 + 1) k1�59 �
3
2(n2 + 1) k2�59 �
5
2n1k1�59 �
5
2n2k2�59
+p2n2k2�48; (B.36)
_�66 = n1k1�33 + 2n2k2�55 � (n1 + 1) k1�66 � 2 (n2 + 1) k2�66 � 2n1k1�66
�3n2k2�66 + 2 (n1 + 1) k1�99; (B.37)
_�67 =p2n1k1�34 �
3
2(n1 + 1) k1�67 � (n2 + 1) k2�67 �
5
2n1k1�67 � 2n2k2�67; (B.38)
95
_�68 =p2n1k1�35 +
p2n2k2�57 �
3
2(n1 + 1) k1�68 �
3
2(n2 + 1) k2�68 �
5
2n1k1�68
�52n2k2�68; (B.39)
_�69 =p2n1k1�36 + 2n2k2�58 �
3
2(n1 + 1) k1�69 � 2 (n2 + 1) k2�69 �
5
2n1k1�69 � 3n2k2�69;
(B.40)
_�77 = 2n1k1�44 � 2 (n1 + 1) k1�77 � 3n1k1�77 � n2k2�77 + (n2 + 1) k2�88; (B.41)
_�78 = 2n1k1�45 � 2 (n1 + 1) k1�78 � 3n1k1�78 � (n2 + 1)k22�78 �
3
2n2k2�78 (B.42)
+p2 (n2 + 1) k2�89; (B.43)
_�79 = 2n1k1�46 � 2 (n1 + 1) k1�79 � 3n1k1�79 � (n2 + 1) k2�79 � 2n2k2�79; (B.44)
_�88 = 2n1k1�55 + n2k2�77 � 2 (n1 + 1) k1�88 � 3n1k1�88 � (n2 + 1) k2�88
�2n2k2�88 + 2(n2 + 1)k2�99; (B.45)
_�89 = 2n1k1�56 +p2n2k2�78 � 2 (n1 + 1) k1�89 � 3n1k1�89 �
3
2(n2 + 1) k2�89 �
5
2n2k2�89;
(B.46)
_�99 = 2n1k1�66 + 2n2k2�88 � 2 (n1 + 1) k1�99 � 3n1k1�99 � 2 (n2 + 1) k2�99 � 3n2k2�99;
(B.47)
where ni�s are mean photon numbers in the thermal reservoirs attached to the cavity 1(2)
and �ij = �ji. For both the reservoirs in vacuum (n = 0) and k1 = k2 = k , the solutions
of the above mentioned matrix elements are
�11 (t) = C222e�4kt + e�2kt
�C202 + C212 + C222
�+ e�2kt
�C220 + C221 + C222
�+ C201 + C210
� e�3kt�C212 + 2C
222
�+ e�2kt
�C211 + 2C
212 + 2C
221 + 4C
222
�� e�3kt
�C221 + 2C
222
�+ C202
+ C211 + C220 + C212 + C221 + C222 + C00C01 ��C201 + C210 + 2C
2022C
211
� +2C220 + 3C212 + 3C221 + 4C222�e�kt;
+C21C22) +p2e�
52kt
�C11C12 + 2C21C22
�1� 1
3e�kt
��� 1615
p2C20C22e
�3kt;
(B.48)
96
�12 (t) = e�12kt
�C00C01 + C10C11 + C20C21 +
p2C01C02 +
p2C11C12 +
2
5
p2C20C22
+2
3
p2C21C22
�+ e�
52kt�C20C21 + 2
p2C20C22
�� e�
32kt (C10C11 + 2C20C21
+p2C11C12 +
4
3
p2C20C22 +
p2C21C22
��p2e�
32kt (C01C02 + C11C12
�13 (t) = e�kt (C00C02 + C10C12 + C20C22)� e�2kt (C10C12 + 2C20C22) + C20C22e�3kt;(B.49)
�14 (t) = e�52kt�C02C12 +
p2C12C22
�� e�
32kt�C01C11 + 2C02C12 +
p2C10C20
+2p2C11C21 + 3
p2C12C22
�+ e�
12kt�C00C10 + C01C11 + C02C12 +
p2C10C20
+p2C11C21 +
p2C12C22
�+p2e�
52kt (C11C21 + 2C12C22)�
p2C12C22e
� 72kt;
(B.50)
�15 (t) = e�kt�C00C11 + 2C11C22 +
p2C01C12 +
p2C10C21
�+ 2C11C22e
�3kt
�e�2kt�4C11C22 +
p2C01C12 +
p2C10C21
�; (B.51)
�16 (t) = e�32kt�C00C12 +
p2C10C22
��p2C10C22e
� 52kt; (B.52)
�17 (t) = e�kt (C00C20 + C01C21 + C02C22)� e�2kt (C01C21 + 2C02C22) + C02C22e�3kt;
(B.53)
�18 (t) = e�32kt�C00C21 +
p2C01C22
��p2C01C22e
� 52kt; (B.54)
�19 (t) = C00C22e�4kt; (B.55)
�22 (t) = e�kt�C201 + C210 + 2C
202 + 2C
211 + 2C
220 + 3C
212 + 3C
221 + 4C
222
��2C222e�4kt � 2e�2kt
�C202 + C212 + C222
�� e�2kt
�C211 + 2C
212 + 2C
221 + 4C
222
��e�kt
�C210 + C211 + 2C
220 + C212 + 2C
221 + 2C
222
�+ e�3kt
�2C212 + 4C
222 + C221
+2C222�+ 2e�3kt
�C212 + 2C
222
�; (B.56)
�23 (t) = e�32kt (C01C02 + C11C12 + C21C22)� e�
52kt�C11C12 + 2C21C22
�2 + e�kt
��; (B.57)
�24 (t) = e�kt�C01C10 + 2C12C21 +
p2C02C11 +
p2C11C20
�+ 2C12C21e
�3kt
�e�2kt�4C12C21 +
p2C02C11 +
p2C11C20
�; (B.58)
�25 (t) = e�32kt�C01C11 + 2C02C12 +
p2C10C20 + 2
p2C11C21 + 3
p2C12C22
��2e� 5
2kt�C02C12 +
p2C12C22
��p2e�
52kt (C11C21 + 2C12C22) + 2
p2C12C22e
� 72kt
�p2e�
32kt (C10C20 + C11C21 + C12C22) ; (B.59)
97
�26 (t) =1
2
p2e�2kt
�4C11C22 +
p2C01C12 +
p2C10C21
��p2C11C22e
�3kt
�e�2kt�C10C21 +
p2C11C22
�; (B.60)
�27 (t) = e�32kt�C01C20 +
p2C02C21
��p2C02C21e
� 52kt; (B.61)
�28 (t) = e�2kt (C01C21 + 2C02C22)� 2C02C22e�3kt; (B.62)
�29 (t) = C01C22e� 52kt; (B.63)
�33 (t) = C222e�4kt + e�2kt
�C202 + C212 + C222
�� e�3kt
�C212 + 2C
222
�; (B.64)
�34 (t) = e�32kt�C10C02 +
p2C20C12
��p2C20C12e
� 52kt; (B.65)
�35 (t) =1
2
p2e�2kt
�4C12C21 +
p2C02C11 +
p2C11C20
�� e�2kt
�C11C20 +
p2C12C21
��p2C12C21e
�3kt; (B.66)
�36 (t) = e�52kt�C02C12 +
p2C12C22
��p2C12C22e
� 72kt; (B.67)
�37 (t) = C02C20e�2kt; (B.68)
�38 (t) = C02C21e� 52kt; (B.69)
�39 (t) = C02C22e�3kt; (B.70)
�44 (t) = e�kt�C210 + C211 + 2C
220 + C212 + 2C
221 + 2C
222
�� 2e�2kt
�C220 + C221 + C222
��e�2kt
�C211 + 2C
212 + 2C
221 + 4C
222
�� 2C222e�4kt + e�3kt
�C212 + 2C
222
�+2e�3kt
�C221 + 2C
222
�; (B.71)
�45 (t) = e�32kt
�C10C11 + 2C20C21 +
p2C11C12 +
4
3
p2C20C22 +
p2C21C22
��2e� 5
2kt�C20C21 + 2
p2C20C22
��p2e�
52kt (C11C12 + 2C21C22)
+8
3
p2C20C22e
�3kt +p2C21C22e
� 72kt; (B.72)
�46 (t) = e�2kt (C10C12 + 2C20C22)� 2C20C22e�3kt; (B.73)
�47 (t) = e�32kt (C10C20 + C11C21 + C12C22)� e�
52kt (C11C21 + 2C12C22) + C12C22e
� 72kt;
(B.74)
�48 (t) = e�2kt�C10C21 +
p2C11C22
��p2C11C22e
�3kt; (B.75)
�49 (t) = C10C22e� 52kt; (B.76)
98
�55 (t) = 4C222e�4kt + e�2kt
�C211 + 2C
212 + 2C
221 + 4C
222
�� 2e�3kt
�C212 + 2C
222
��2e�3kt
�C221 + 2C
222
�; (B.77)
�56 (t) = e�52kt (C11C12 + 2C21C22)� 2C21C22e�
72kt; (B.78)
�57 (t) = e�2kt�C11C20 +
p2C12C21
��p2C12C21e
�3kt; (B.79)
�58 (t) = e�52kt (C11C21 + 2C12C22)� 2C12C22e�
72kt; (B.80)
�59 (t) = C11C22e�3kt; (B.81)
�66 (t) = e�3kt�C212 + 2C
222
�� 2C222e�4kt; (B.82)
�67 (t) = C20C12e� 52kt; (B.83)
�68 (t) = C12C21e�3kt; (B.84)
�69 (t) = C12C22e� 72kt; (B.85)
�77 (t) = C222e�4kt + e�2kt
�C220 + C221 + C222
�� e�3kt
�C221 + 2C
222
�; (B.86)
�78 (t) = e�52kt�C20C21 + 2
p2C20C22
�� 2p2C20C22e
�3kt; (B.87)
�79 (t) = C20C22e�3kt; (B.88)
�88 (t) = e�3kt�C221 + 2C
222
�� 2C222e�4kt; (B.89)
�89 (t) = C21C22e� 72kt; (B.90)
�99 (t) = C222e�4kt: (B.91)
99
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