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14
15\title{Post Quantum Cryptography}
16\author{George Brumpton}
17\begin{document}
18\date{}
19\maketitle
20
21\newpage
22\section{Introduction}
23
24\subsection{Quantum Optics}
25\subsubsection{Concepts and Applications}
26At it's heart, quantum optics is the study of light and the way that it interacts with matter. It is an important area of physics that has a considerable influence over everyday life. A large part of quantum optics concerns lasers and their applications. These applications are numerous and include areas such as: medicine, metrology and measurement, biology and biophysics,  and in information and quantum information studies. The advances in laser technology has opened the doors to many new experiments and utilizations. Using lasers, matter can now be manipulated at the quantum level, allowing us to investigate the wave nature of atoms and have shown similarities between these waves and electromagnetic waves.   Furthermore, lasers have been used as a means to cool and trap atoms and ions, most famously in the case of creating Bose-Einstein condensates, in which lasers are used to effectively slow down atoms, causing them to form an ensemble of super cooled atoms, which are all statistically identical [1].   
27\\
28
29\subsubsection{History}
30It cannot be disputed that light is one of man's most valuable natural resources. From growing crops for food to simply illuminating the world, light has always had a pivotal role in nature and without it, there would not be human life. It has always, and continues to fascinate and baffle scientists. From it's lack of mass and the unimaginable speed that it travels. ``Descartes thought that light spreads out instantaneously''  [2]. Photons, despite not being individually visible to the human eye, give way to everything we see and allow us to observe all the colours and shapes of the world.
31\\
32For centuries the structure of light was not known. Newton believed that light was made up of individual corpuscles. On the other hand, Huygens postulated that light was a wave, much like waves on disturbed water. Eventually however, owing to the work of Einstein on the photoelectric effect, we now view light as being made up of energy quanta. These energy quanta do not behave as just a wave or a particle, but as both, depending on the situation. 
33In the last century, quantum optics has enjoyed large advances in technology. By as early as  the 1960's, it was possible to place atoms within resonators, or to place them extremely close to mirrors. Fast forward to the 1980's and scientists could study the interaction between a single mode of the electromagnetic field and a single atom [3].
34\\
35
36\subsection{Quantum Information and Computation}
37 More recently there has been a greater focus on implementing concepts from quantum optics to information studies. Developments from quantum mechanics have allowed for information to be transferred more securely, and with less noise. Quantum optics has lead to developments in quantum information and quantum computing via the qubit. The qubit is simply the quantum analogy of a classical bit. The bit is the smallest unit of classical information, whereas the qubit is the smallest unit of quantum information. Qubits have many advantages over their classical counterparts. Rather than being just discrete values of either 0 or 1, qubits can be in a mixture, or superposition of these states, thus making them continuously valued. They also have a few other properties that separate them from classical bits. For instance they are bound by the no-cloning theorem, meaning quantum information and qubits cannot be copied. Additionally, qubits can become entangled with one another, meaning a qubits state can be affected just by measuring a different qubit. Research into qubits and their properties have shown that they can be exploited to be used for computational purposes. It has been shown that quantum computers have the capability to perform tasks much more quickly than modern classical computers. Areas of research in quantum optics, such as creation of single photon states and entangled states [1], have led to the development and manufacture of basic elements used in processing quantum information and quantum computing [4,5].     \\
38
39\subsection{Cryptography}
40
41\subsubsection{History}
42The aims of cryptography are simple, send a message from one party to another whilst keeping the message secret from anyone else who may try to read it. Cryptography is not a new concept and has been used for as long as people have wished to send secret messages. It has been used in wars, espionage and even love letters. Marie Antoinette used cryptography to aid her love-life [6].  One of the earlier forms of cryptography dates back to the days of ancient Rome and ancient Greece, and the aptly named Caesar cypher. This method was used to encrypt secret messages by merely shifting each letter of the alphabet backwards or forwards some number of spaces, known as the key. For instance, say every letter was shifted 4 for places up in the alphabet, the word CRYPTOGRAPHY would become GVCTXSKVETLC. This is a very basic type of cypher and is extremely vulnerable, since there are only 25 possible keys. Any computer can simply try shifting the letters forwards, one to twenty-five places and thus easily decrypting the message. This kind of attack is known as a brute force attack and is the easiest method of decrypting the Caesar cypher.\\
43A similar, but stronger type of cypher is the general substitution cypher. Instead of shifting the letters up or down the alphabet, this cypher replaces each letter in the decrypted message, or plaintext, to create the encrypted message, or ciphertext. \\
44
45
46\begin{tabular}{||c c c  c c c c c c c c c c c||} 
47 \hline
48Plain Alphabet & a & b & c & d & e & f & g & h  & i & j & k & l & m  \\ [0.5ex] 
49 \hline
50Cipher Alphabet  & W & Y & M & C & H & R & G & O & A & J & U & T & N  \\ 
51 \hline\hline
52Plain Alphabet & n  & o & p  & q & r & s & t & u & v & w & x & y & z
53 \\
54 \hline
55Cipher Alphabet  &E &B &S &L &Z &F &K &I &X &P &D &V &Q
56  \\ \hline
57  
58\end{tabular}
59  \\
60\begin{indent}
61
62\begin{tabular}{|| c c c c c c c c c c c c c c ||} 
63 \hline
64 Plaintext message: && cryptography &&&&&&&&&&&
65  \\ [0.5ex] 
66 \hline
67  Ciphertext message: &&MZVSKBGZWSOV&&&&&&&&&&& \\ [0.5ex] 
68 \hline
69\end{tabular}
70\end{indent}
71\\
72\begin{indent}
73
74Table 1: Example of using the general substitution cypher to encrypt the\\ \indent
75word `cryptography'. The letters of the alphabet have been substituted with\\ \indent
76others randomly, to create a cipher alphabet, using which one can compose \\ \indent encrypted messages.\\
77
78\end{indent}
79
80\noindent The general substitution cypher is a lot more secure than the Caesar cypher, in that it has around $4$x$10^{26}$ different keys [7]. This means that the brute force method is impractical as it would take an absurd amount of computing power and computing time, thus making the cypher secure to brute force attacks. Nonetheless, this doesn't mean the cypher is secure. Two methods were discovered that would help in cracking the cypher. The first was to guess words that could be in the message, for example `Dear', `and' and `the' could appear often, and discovering these words could help in decryption. The next method is frequency counting. The idea behind this is to count how many times each different letter appears. A high frequency letter in the cipher alphabet would almost certainly decrypt to a high frequency letter in the plain alphabet. The letter `e', for example, accounts for roughly 13\% of the letters in any given sufficiently large piece of English text [7]. Using frequency counting, and occasionally guessing words and patterns of letters, general substitution cyphers can be cracked and are therefore not considered secure [8].
81\\
82
83\subsubsection{Modern Cryptography}
84Invented in the 1880's and first heavily used in WW2, the one time pad is the only currently known unbreakable method of encryption and is `commonly acknowledged as the holy grail of cryptography' [9]. The one time pad works by giving both the sender and receiver of the secret message a sheet which contains a random key. It is vital that both the sender and receiver have identical keys. The sender encrypts their message using the key and sends it on. The receiver then uses the key to decrypt the message. As long as the key is completely random, the key is at least as long as the message, and the sender and receiver use their sheet only once and destroy it afterwards, the message cannot be read by anyone who might intercept it [10].  
85\\
86
87\subsubsection{Quantum Cryptography}
88While cryptography is still used to send secret messages in areas such as espionage, it is used more heavily in protecting data and information. For example sending bank details over the internet when purchasing something online. However, with the advent of quantum computers and the speed in which quantum information technologies are advancing, there are many threats on the horizon to current encryption methods [6]. For example, the RSA cypher, which is heavily used to transmit data securely, relies on the fact that modern classical computers cannot factorise a huge co-prime integer back into two prime numbers in any plausible amount of time. While the RSA cypher is secure against classical computers, quantum computers can solve the facorisation problem exponentially faster and thus could potentially render RSA redundant [4]. \\
89As a result of the potential vulnerabilities to classical key distribution techniques, the field of quantum key distribution (QKD) was born. The first method of QKD was invented by Charles Bennett and Gilles Brassard in 1984 and is known as BB84 [11]. The BB84 protocol involves the sender, commonly dubbed Alice, randomly polarizing photons in either a verticle/horizontal basis, or a diagonal (+45\degree/-45\degree) basis and sends them on to the receiver, commonly named Bob, who randomly chooses a basis to measure the photons in. The two parties then publicly declare which basis they used and if they used the same basis they obtain a key bit. Anytime they disagree on the basis used they simply discard the result. Due to the random nature of obtaining a key bit, Alice and Bob will have an error rate of not obtaining key bits. This protocol has safeguards against eavesdroppers. Firstly, if an eavesdropper, Eve, intercepts a photon before it reaches Bob, her presence is immediately obvious since Bob will not receive the photon. Eve however, could simply intercept a photon and then send a new one to Bob. Although this solves the problem of Bob not receiving a photon, Eve still has to guess the bases that Alice and Bob are using, thus introducing additional error. If when comparing their measurements, Alice and Bob discover an significant increase in their error rate then the presence of Eve is evident and they simply discard their key.\\
90A potentially more efficient method of QKD is to use continuous source, such as a laser. This approach is known as continuous variable quantum key distribution (CVQKD), and has advatanges over using single photon sources. For example lasers are much easier to measure than single photons and CVQKD systems only require standard telecommunication technologies [12]. Coherent states from laser pulses are commonly used as a continuous source for CVQKD.\\
91
92
93
94
95
96\subsection{Project Outline}
97The ultimate motivation of this project was to investigate to what extent incoherent or thermal sources of light can be purified to resemble coherent states of light, such as pulses from a laser. A thermal source may be, for example, a light bulb's incandescent filament [13], meaning as the filament heats up it emits light. In [14] the purification of noisy coherent states is investigated and is shown that it is experimentally feasible. However, there is no current theory on the purification of incoherent, or thermal states into coherent states. Thermal states are easy and cheap to generate, hence research like this project into purifying them could prove beneficial and valuable to CVQKD. The methods of purification of thermal states discussed in this project will rely on the concepts of displacement and attenuation, and the background theory and simulation of these effects will be discussed in detail. This projects end goal will be to simulate displaced, attenuated thermal states, to analyse their relation to coherent states. \\
98
99
100
101\section{Background Theory}
102
103\subsection{Quantum States}
104
105\subsubsection{Vacuum State}
106It is natural to first define what is meant by the vacuum state. The vacuum state, expressed as $\ket{0}$, is the state which contains no photons or physical particles and therefore has the lowest possible energy. Since there is always a lowest energy state, the vacuum state always exists. [15]\\
107It it necessary to point out that $\ket{0}$ can also refer to the ground state of a system, which is simply the lowest energy state of the system. This, however, is different to the vacuum state. The ground state can refer to the lowest energy state of a particle, or a system of many particles. This is clearly not the vacuum state since the vacuum state contains no physical particles or photons.\\
108\subsubsection{Fock States}
109The next step is to define a Fock state, also known as a photon-number state. These states are denoted by $\ket{n}$, where n is the exact number of photons in the state. For example $\ket{1}$ is the Fock state containing a single photon, and $\ket{2}$ is the multi-photon Fock state with 2 photons. All photons in the state are identical, meaning they have the same frequency [15]. \\
110\noindent Fock states are related to the vacuum states through so called annihilation and creation operators, $a$ and $a^{\dag}$ respectively. These operators satisfy the commutation relation given by $[a,a^{\dag}] = aa^{\dag} - a^{\dag}a = 1$ [15].\\
111These states are pairwise orthogonal, hence\\ $\braket{n_i}{n_j} = 0$ for $n_i \neq n_j $ and $\braket{n_i}{n_j} = 1$ for $n_i = n_j$. \\
112They are also normalised  since they obey the condition $\braket{n}{n} = 1$.
113\\
114Since Fock states contain a precise number of photons, they are very difficult to create. Currently, Fock states with only small numbers of photons can be created in the laboratory. One way that these states can be created is to place a singular atom in a cavity, which could be, for example, two highly reflective mirrors. A laser interacts with the trapped atom and excites it. The atom then emits a photon which is trapped inside the cavity once the atom has left. This can only be done for small amounts of photons and thus, only Fock states with small photon-numbers can be created using this method [16]. 
115\\
116
117\subsubsection{Number Operator}
118An additional way of viewing Fock states is by studying the  quantum harmonic oscillator. This has a Hamiltonian of the form $H = \hbar \omega a^{\dag}a$.\\
119This Hamiltonian contains the number operator, $a^{\dag}a$, which is so called because it counts the number of photons that are present in a given state. The number operator has eigenvalues $n_i$ and eigenvectors $\ket{n_i}$ . The eigenvalue $n_i$ corresponds to the number of photons in the state, hence the name "number operator", while the eigenvector $\ket{n_i}$ is the corresponding Fock state. As a result of this, it is clear that any $\ket{n}$ given Fock state contains exactly n photons.
120Further more, it can be deduced that\\
121\\
122$a\ket{n} = \sqrt{n}\ket{n-1}$ \hspace{1cm} (2.1) \\
123$a^{\dag}\ket{n} = \sqrt{n+1}\ket{n+1}$ \hspace{0.325cm} (2.2)\\
124\\
125Evidently, the $a$ operator creates a state with one less photon than the original. It is henceforth called the annihilation operator for photons. Similarly, $a^{\dag}$ creates a state that contains one more photon than the original, and is therefore the creation operator for photons.\\
126These relations clearly link the vacuum state, or ground state to other Fock states, as the creation operator applied to the vacuum state will give a new Fock state. Since
127$a^{\dag}\ket{0} = \ket{1}$, it is apparent that a Fock state containing n photons be generated by applying the creation operator $n$ times [15,16].\\
128
129
130
131\subsubsection{Coherent States}
132In contrast to the Fock states, which are difficult to generate and are referred to as the most quantum states, coherent states most closely resemble classical fields of light, such as laser pulses [15] and are therefore reffered to as the most classical states. In Fock space, coherent states are defined as \\
133\\
134$\ket{\alpha}$ = exp$(-\frac{1}{2}|\alpha^2|)      \sum\limits_{n=0}^\infty \cfrac{\alpha^n}{\sqrt{n!}} \ket{n}  $ \hspace{1cm} (2.3)\\
135\\
136where $\alpha$ is a complex number. Coherent states are normalised since $\braket{\alpha}{\alpha} = 1$ and the state $\ket{\alpha}$ contains an average number of photons of $|\alpha^2|$. \\
137Coherent states are also defined as the eigenstates of the annihilation operator [17] and have the eigenvalue $\alpha$, hence \\
138\\
139$a\ket{\alpha} = \alpha \ket{\alpha}$\\
140\\
141where $a$ is the annihilation operator for photons. \\Coherent states will be looked at in greater detail at a later stage in this project, when considering photon-number statistics and the displacement operator. \\
142
143\subsubsection{Thermal States}
144The final state that is relevant to this project is a thermal state. Hot bodies, such as candle flames or filaments in an incandescent light bulb, emit electromagnetic radiation. This electromagnetic radiation is a source of thermal photons and hence a source of thermal states. Properties of thermal light are found by using the laws of statistical mechanics to the radiation that is inside an enclosed cavity that is at a certain temperature T [18]. This radiation has a spectrum of modes which each oscillate with a different angular frequency $\omega_i$. Looking at a singular mode of the radiation, the probability of finding n photons in the mode is given by\\
145\\
146$P_\omega(n) = \cfrac{\exp(-E_n/k_B T)}{\sum\limits_{n=0}^\infty   \exp(-E_n/k_B T)}$ \hspace{1cm} (2.4)\\
147\\
148
149\noindent
150where $k_B$ is the Boltzmann constant and $E_n$ is the energy of the radiation.
151Thermal states are central in this project and thus will be studied to greater detail in the coming sections.
152\\
153\subsection{Density Matrices}
154\subsubsection{Definition and Properties}
155An often more practical way of representing a quantum system is to use a density matrix. The density matrix is defined as the outer product of a general wavefunction and its complex conjugate\\
156\\
157$\rho = \ket{\Psi}\bra{\Psi}$ \hspace{1cm} (2.5)\\
158\\
159and the elements of the density matrix can be obtained from\\
160\\
161$\rho_{nm} = \sum\limits_{n,m} P_{nm}  \ket{n}\bra{m} $ \hspace{1cm} (2.6)\\
162\\
163where $\rho_{nm}$ are the matrix elements and $P_{nm}$ are respective probabilities.\\
164The density matrix has a few important properties. Firstly, it is Hermitian meaning that it is equal to its conjugate transpose. It is also always normalised meaning that its trace, which is the sum of the diagonal elements, is always equal to one. This property is a consequence of the diagonal elements, $\rho_{nm}$ with $n=m$, which represent the probability of occupying the state $\ket{n}$ [19]. Clearly summing over the probabilities of every state in the system will give one, hence the normalisation property.\\
165
166\subsubsection{Fock State Density Matrix}
167 The natural first density matrix to study is the density matrix of a Fock state. Given a Fock state $\ket{i}$, it's density matrix and matrix elements are simply [20] \\
168\\
169$\rho = \ket{i}\bra{i}$ \hspace{0.2cm} and   \hspace{0.2cm} 
170$\rho_{nm} = \delta_{ni}\delta_{im} $ \hspace{1cm} (2.7)  \\
171\\
172where
173$\delta_{ni} = 1$ if $n=i$ \\and \hspace{0.18cm} $\delta_{ni} = 0$ if $n\neq i$.\\
174\\
175Clearly, the density matrix for a Fock state will just contain a "1" in the diagonal of the row and column of the number of photons in the state, and 0's everywhere else. For example the density matrix for the state $\ket{2}$ would be\\
176\\ $\rho = \ket{2}\bra{2 } = $
177$\begin{pmatrix}
1780&0\\
1790&1\\
180\end{pmatrix}$.\\
181
182\subsubsection{Coherent State Density Matrix}
183The second density matrix that is crucial to this project is the coherent state density matrix. This can be derived 
184using the definition of a coherent state, given in equation (2.3),\\
185\\
186$\ket{\alpha}$ = exp$(-\frac{1}{2}|\alpha^2|)      \sum\limits_{n=0}^\infty \cfrac{\alpha^n}{\sqrt{n!}} \ket{n}  $ \\
187\\
188and using the definition of a density matrix from equation (2.6)\\
189\\
190$\rho_{nm} = \sum\limits_{n,m} P_{nm}  \ket{n}\bra{m} $\\
191\\
192we obtain the coherent state density matrix\\
193\\
194$\rho =$ exp$(-|\alpha^2|)      \sum\limits_{n=0}^\infty \cfrac{\alpha^{2n}}{n!} \ket{n}\bra{n} $. \hspace{1cm} (2.8)\\
195\\
196
197\subsubsection{Thermal Density Matrix}
198The final density matrix that is relevant to this project is the thermal density matrix. This has the form\\
199\\
200$\rho = \sum\limits_{n} \cfrac{\langle n \rangle^{n}}{(1+\langle n \rangle)^{n+1}} \ket{n} \bra{n} $ \hspace{1cm} (2.9)\\
201\\
202where\\
203\\
204$\langle n \rangle = [\exp(\hbar\omega /k_B T) - 1]^{-1} $. \hspace{0.3cm} [20]\\
205\\
206\noindent
207Density matrices play an important role in this project, as they provide a means to easily find photon-number distributions, which will be studied in the next section. The simulation and manipulation of density matrices is a large part of this project and will feature heavily in results sections.\\
208
209\subsection{Photon-number Statistics}
210The reason that density matrices are so heavily used in this project is that they provide a convenient way to view the photon-number statistics of different states. Each of the three different states that are being used in this project follow different photon-number statistics and have contrasting photon-number distributions. These distributions illustrate the probabilities of finding certain numbers of photons in a state.
211\\
212\\
213\[\includegraphics[width=7.5cm]{coherent1.png}\]
214\\
215Figure 1: Example of a photon-number distribution for the coherent state $\ket{\alpha}$ with $|\alpha| = 10$. The x-axis denotes the number of photons and the y-axis denotes probabilities. The figure thus illustrates the probabilities (Pk) of finding certain amounts of photons (k) in the state. [21, p.5]
216\\
217\\
218The distribution shown in figure 1 is an example of a coherent state, which follows a Poisson distribution [18]. However, the photon-number distributions for thermal states and Fock states do not adhere to a Poisson distribution. A Fock state, for example will simply have a peak to a maximum probability of one at it's exact photon-number. A thermal state however, will have a broader range of probabilities than a coherent state. \\
219\\
220\[\includegraphics[width=7.5cm]{thermal1.png}\]
221\\
222Figure 2: A comparison of photon-number distributions between a thermal state and a coherent state. Both states have a  mean photon-number of 10. [18, p.82].
223\\
224\\
225 The Poisson distribution and thus the coherent state photon distribution must obey the condition $\Delta n = \sqrt{\Bar{n}}$, where $\Delta n$ is the standard deviation and $\Bar{n}$ is the mean photon-number. A thermal state, on the other hand, will have $\Delta n > \sqrt{\Bar{n}}$, which means they follow a Super-Poissonian distribution. Conversely, a Fock state distribution will have a standard deviation of 0 and thus not follow any Poisson distribution. [18]\\
226\noindent Since this projects main goal is to investigate how closely thermal states can be purified to resemble coherent states, photon-number statistics and distributions will be used a great deal as it allows an easy way to compare the behaviour of each state.\\
227
228\subsection{Displacement Operator}
229It is essential to review the displacement operator and it's properties for this project. It's uses will become clear when discussing the method of purification of thermal states. The displacement operator is defined as\\
230\\
231$D(\alpha)= \exp(\alpha a^{\dag} - \alpha^{*} a)$ \hspace{1cm} (2.10)\\
232\\
233where $a$ and $a^{\dag}$ are the photon annihilation and creation operators respectively, and $\alpha$ is a complex number [22].
234The displacement operator displaces the system in phase space by an amount dependant on the complex parameter  $\alpha$ that is chosen [23].\\
235The displacement operator is frequently discussed when studying coherent states, since the displacement operator provides a new way to define the coherent state. This can be shown by applying the displacement operator to a coherent state, giving\\
236\\
237$D(\alpha)\ket{\beta} = \ket{\alpha + \beta} $ \hspace{1cm} (2.11)\\
238\\
239which is simply another coherent state that is shifted depending on the parameter of the displacement. Hence if the original state is set equal to the vacuum state $\ket{\beta} = \ket{0}$ and then applying the displacement operator\\
240\\
241$D(\alpha)\ket{0} = \ket{\alpha + 0} = \ket{\alpha}$ \hspace{1cm} (2.12)\\
242\\
243gives a coherent state. As a consequence of this, it is clear that any coherent state can be generated by simply displacing the vacuum state. Coherent states are thus just vacuum states that have been displaced in phase space. This is why coherent states are called the most classical of quantum states, since they are just displaced ground states instead of proper quantum excitations [24, p.122].\\
244
245\subsection{Effects of Displacement on Photon-number Distributions}
246The displacement operator can dramatically change the photon-number distribution of a quantum system. For this project it is necessary to review the  effects of displacement on Fock states and thermal states, as attempts will be made to simulate these effects numerically.\\
247It is shown in [25] that displaced Fock states are eigenstates of the displaced Hamiltonian for the quantum harmonic oscillator, and the photon-number distribution for displaced Fock states is shown to be\\
248\\
249$P_{df}(l) = \cfrac{\exp(-|\alpha|^2 ) |\alpha|^{2(l-n)}}{n! l!} \hspace{0.2cm} \times \hspace{0.2cm} \abs{\sum\limits_{k=0}^{n} \cfrac{n!l!(-1)^k |\alpha|^{2(n-k)} }{k!(n-k)!(l-k)!}^}^2 $ \hspace{1cm} (2.13)\\
250\\
251where $\alpha$ is the complex parameter of the displacement and $P_{df}(l)$ are the probabilities to find l photons in the displaced Fock state.\\
252Displaced Fock states are written as $\ket{\alpha, n} $, where $\alpha$ describes the amount of displacement and $n$ is the original Fock state. If the value of the displacement is set to zero, $\ket{0, n}$, then the state is no longer displaced and returns to a regular Fock state with photon number-distribution given by $P_f = \delta_{ln}$.\\
253Considering the photon-number distribution for a displaced number state but letting n=1, equation (2.13) reduces to\\
254\\
255$P_{df}(l) = (l!)^{-1} \exp(-|\alpha|^2) |\alpha|^{2(l-1)} (l-|\alpha|^2)^2$. \hspace{1cm} (2.14)\\
256
257
258\[\includegraphics[width=7.5cm]{Df2.png}\]
259Figure 3: Photon-number distribution of the displaced Fock state $\ket{7,1}$. Taken from [25, p.2649], however this figure is wrongly labeled as the state $\ket{7,2}$ in the literature.\\
260\\
261Clearly from equation (2.14) there should be a zero in the distribution when $|\alpha|^2 = l$, which is confirmed by figure 3.\\
262The displacement of Fock states can be used to study the effects of displacement on thermal states. Since the photon-number distribution of a thermal state shows the probabilities to find certain amounts of photons, the distribution can be expressed as weighted Fock states. For example, any photon-number distribution can be expressed as\\
263\\
264$P1\ket{1}, P2,\ket{2}, P3\ket{3},..., PN\ket{N}$ \hspace{1cm} (2.15)\\
265\\
266where $Pn_i$ are the probabilities of finding $n_i$ photons in the state, and $\ket{n_i}$ are the Fock states corresponding to those photon-numbers [26]. Since a thermal state can be expressed in this way, it is also possible to represent a displaced thermal state, however instead of regular Fock states, displaced Fock states must be used instead. Thus, to simulate a displaced thermal state, a weighted sum of all displaced Fock states in the relevent Hilbert space must be performed. For example, retrieving $Pn's$ from the diagonals of a thermal density matrix, the photon-number distribution of the displaced thermal state can be expressed in the form\\
267\\
268$P1\ket{\alpha, 1} + P2\ket{\alpha, 2} + P3\ket{\alpha, 3} + ... + PN\ket{\alpha, N} $ \hspace{1cm} (2.16) \\
269\\
270where, again, $\alpha$ denotes the parameter of the displacement.\\
271
272
273
274\subsection{Attenuation}
275The final piece of theory that is needed for this project regards attenuation. Attenuation happens due to loss within the system, thus to model a real life system this loss needs to be taken into account. As with the last section, attenuation only needs to be considered in the case of Fock states, since we can treat other states as being made up of weighted Fock states. The general formula for attenuation of a number state is given by [25] \\
276\\
277$P(l,\mu) = \begin{cases}
278{n\choose l} \mu^l (1-\mu)^{n-l} \hspace{0.2cm} (l\leq n) \\
2790 \hspace{0.2cm} (l > n)
280\end{cases}
281$ \hspace{1cm} (2.17)\\
282\\
283In this formula, $\mu = \exp(-\gamma t)$ where $\gamma$ is the damping constant and $t$ is elapsed time. Therefore, the lower the value of $\mu$, the greater the amount of attenuation.\\
284 This formula can be used to study the effects of attenuation on displaced thermal states, using similar logic from the previous section. Taking a displaced thermal state and treating it as being made up of individual weighted Fock states, formula (2.6) can be applied to each one of the individual Fock state, giving attenuated Fock states. These can them be summed together, weighted by the probabilities given by the displaced thermal distribution, to simulate an attenuated, displaced thermal state [26]. Simulating these attenuated, displaced thermal states is the main focus of this project and they will be investigated to observe their resemblance to coherent states.  
285
286
287
288
289
290
291
292
293
294
295\section{Methods and Set-up}
296\subsection{Outline}
297This project heavily relied on numerical simulation using computer code to investigate the properties of the three quantum states that have been discussed earlier, and to simulate a potential method to manipulate thermal states to make them closely resemble coherent states. Information and figures from [25] has been of great use and results from the paper will feature alongside some results of this project for comparison between the numerical simulation and current theory. \\
298
299\subsection{Set-up}
300All of the numerical simulations in this project have been coded in the programming language python. Heavy use of a python library known as QuTip [27] was used to aid in programming. QuTip enabled coding directly in a quantum mechanical framework and has been an invaluable resource to this project, as it has saved a large amount of time that would have otherwise been spent coding this framework from the ground up. This time saved allowed deeper investigation of the background theory, and granted more time to perform simulations and investigate the results.\\
301Installing QuTip was quite a challenge in of itself, and required booting the Linux operating system from a USB flash drive to enable full use of QuTips features.
302
303\subsection{Methods}
304
305\subsubsection{Density Matrices and photon-number Statistics}
306Once QuTip had been installed the first step in this project was to simulate a number of Fock, coherent, and thermal states to see if they agreed with the current theory regarding their photon-number statistics. This was achieved by first using QuTip to create density matrices for each one of the three states. For example, simulating a Fock state density matrix in a 2-level Hilbert space, with photon-number equal to two, generated the matrix \\
307\\
308INSERT MATRIX HERE.\\
309\\
310Clearly, this agrees with the theory discussed in section 2.2.2, as the matrices are identical. After generating these density matrices it was the simple case of taking the diagonal elements and plotting them against their corresponding photon numbers, to create photon-number distributions for each of the states. These distributions will be examined and discussed in detail in the results section.
311\subsubsection{Effects of Displacement on Photon-number Statistics}
312\subsubsection{Attenuation}
313\section{Results: Density Matrices and photon-number Statistics}
314\section{Results: Effects of Displacement on Photon-number Statistics}
315\section{Results: Attenuation}
316The main approach, that will be discussed in greater detail later in this section, was to simulate displacement and then attenuation of thermal states, to investigate how closely this resembled a coherent state
317\newpage
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417\end{document}