# An investigation of a nonlocal entanglement of two uncoupled atoms embedded
in a coherent cavity field and the associated phase space distribution: one
quantum non-linear process

M. S. Ateto111E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Mathematics Department, Faculty of Science at Qena, South Valley University,
83523 Qena, Egypt

###### Abstract

Entanglement properties of two uncoupled atoms embedded in a coherent field
distribution through one quantum transition process is studied. A case of non-
linear Hamiltonian of the problem is considered through which the effect of a
non-linear media is illustrated. Moreover, the effect of the frequency
difference between the interatomic transition and the electromagnetic field is
also analyzed. We show that, adjusting the considered parametres of the non-
linear media and frequency difference leads to a strong control of the degree
of entanglement where excellent periodicity of entanglement evolution can be
obtained which is very important in predicting the behavior of transmitted
information through the application of various information processing schemes.
We present a detailed and comparative study of atom-atom entanglement for two
cases corresponding to different injections of the two atoms into the cavity
field. Moreover, we present an answer to the question: How does the quantum
phase space structure for a composite system relate to the entanglement
characteristics of the corresponding quantum system? We demonstrate how the
entanglement in nonlinear tripartite systems can be associated with a
delocalization in the phase space distribution.

###### pacs:

03.65.Ud, 03.67.-a, 03.67.Bg, 05.30.-d 03.67.Mn

Keywords: Coherent distribution, Nonlocal entanglement, Two qubits, one
quantum process, Concurrence, Kerr medium, Husimi $Q$-distribution

##  I Introduction

The recent rapid development of quantum information theory has largely
stimulated research on nonclassical phenomena, with the main focus on the
generation of entangled states that are required for tasks such as quantum
teleportation, dense coding, or certain types of quantum key distribution
protocols. In recent years, there are much interest to study entanglement of
two-level systems. These systems gain such importance because of the
efficiency in representing information in most quantum information processing
schemes. Particular and great interest is devoted to the generation of
entangled states in two-atom systems, since they can represent two qubits that
considered the base blocks of building the quantum gates that are essential to
implement various quantum protocols. Furthermore, there have been many
proposals for generating atomic entanglement and entanglement between cavity
modes through atom-photon interaction PWK04 ; BBPS96 ; BDSW96 ; VP96 ; V02 ;
VT99 ; VPRK95 ; W98 ; GMN06 ; CZDKAW00 ; BGASMNN07 ; PK491 ; PK691 ; KLMCK93 ;
ATETO109 ; Ateto07 . Some notable experimental demonstrations have also been
performed HMNWBRH979 .

However, the physical nature of the interacting objects and the character of
their mutual coupling control strongly the degree of quantum entanglement. A
number of studies have shown that entanglement can be created between to
objects which do not interact directly with each other but interact with
either a common field or heat bath or thermal cavity field BGASMNN07 ;
ATETO109 ; Ateto07 ; KLAK02 ; BRAUN02 ; ATETO209 .

The formation of atom-photon entanglement and the subsequent generation of
correlations between spatially separated atoms have been shown using the
micromaser MLBEHW96 ; ASMNN01 ; RSWRWR991 ; M02 ; HMNWBRH979 . The
effectiveness of micromaser FIJAME86 ; PK88 ; PB93  in generating entanglement
has been appreciated as it is strongly considered as a practical device for
processing information. It stores radiation for times significantly longer
than the duration of the interaction with any single atom PB93 . The
interaction of an atom with the intracavity field of a micromaser will,
generally, leave the atom-field system in an entangled state. The long cavity
lifetime means that the memory of this entanglement can influence the
interaction with subsequent atoms and a nonlocal correlations between these
successive atoms can be induced leading to a violation of Bell’s inequality
PB93 .

To our knowledge, the study of such a problem when the cavity field contains a
coherent distribution is still insufficient. The major difficulty may be due
to the complex calculations and long analytical mathematics, specially when
the system Hamiltonian becomes nonlinear. We are interested in the opportunity
to use the micromaser with coherent distribution as a source of nonlocal
entanglement and answer the important questions: how and to what extent can
the degree of atom-atom entanglement be cotrolled in these circumstances?
Also, can the Husimi $Q$-distribution be used as a helpful method to quantify
nonlocal entanglement dynamics? When applied to a system of two qubits, it’s
shown that the concurrence W98  may be expressed in terms of the second moment
of the Husimi $Q$-function. More strikingly, when applied to a system of three
qubits, the expression for the moment contains all three bipartite concurrence
terms. In this way this measure captures all classes of entanglement, not only
bipartite or otherwise SUGITA03 .

Our purpose here is to trace analytically and numerically the problem of a
nonlocal entanglement created between two successive atoms (two-qubits)
through surrounding the coherent cavity field with a nonlinear medium, namely,
Kerr-like medium. The Kerr medium FALI95 ; ATETO109 ; ATETO209 ; ATETO10 ;
Ateto07 ; WLG06  can be modelled as an anharmonic oscillator with frequency
$\omega$. Physically this model may be realized as if the cavity contains two
different species of atoms, one of which behaves like a two-level atom and the
other behaves like an anharmonic oscillator in the single-mode field of
frequency $\omega$ JOPU92 . Such a model is interesting by itself. The cavity
mode is coupled to the Kerr medium as well as to the two-level atoms. A Kerr-
like medium can be useful in many respects, such as detection of nonclassical
states Hillaer91 , quantum nondemolition measurement CHCOWO92 , investigation
of quantum fluctuations ZHGCLM00 , generation of entangled macroscopic quantum
statesGerry99 ; PKH03 , and quantum information processing PACH00 ; VFT00 .

An outline of paper is as follows: In Sec. II we describe in details the
physical mechanism of the interaction and introduce the modified model of the
total system that accommodates processes such as nonlocal interaction in
nonlinear media. We obtain the wave function of the first part of interaction
on which the final wave function of the total system depends. In Sec. III, we
describe the technique we are going to use in quantifying the degree of
entanglement. In addition, we calculate the the eigenvalues of the non-
Hermitian operator that used to compute the mathematical form of entanglement
measure. Different kinds of atoms injection into the cavity are introduced in
sections IV and V, where the corresponding wave functions and their associated
density matrix operators are computed. In Set. VI, we give a little historical
review about the Husimi $Q$-function, where that function is calculated for
different circumstances separately with detailed discusion included. Our
concluding remarks are shown in Set. VII.

##  II The full system and its solution

We consider two two-level atoms with the same transition frequency traverse a
high $Q$ ($Q\approx 10^{9}$) single mode cavity one after the other. The
flight time of the atoms is, however, long enough so that there is no
appreciable overlap between the atomic wave functions BGASMNN07 ; DGMN04 ;
LEW96 ; AMNN01 ; PB93 ; Ateto07 ; EGSWH96 ; MHNWBTH97 ; ELSW002 . With this
assumption we ignore the possibility of energy exchange between the atoms,
although, secondary correlations develop between them. The entanglement of
their wave functions with the cavity photons can be used to formulate local-
realist bounds on the detection probabilities for the two atoms LEW96 ; PB93 .
The generation of nonlocal correlations between the two atomic states emerging
from the cavity can in general be understood using the Horodecki theorem HHH95
. It is assumed that the atom field interaction time is shorter than the
lifetime of the cavity, so that the cavity relaxation will not be considered.
The cavity field is assumed to be filled with a nonlinear medium, namely, Kerr
medium FALI95 ; ATETO109 ; ATETO10 ; ATETO209 ; Ateto07 ; WLG06 . Also, it is
assumed that the eigenfrequency of the atomic subsystem differs from that of
cavity field subsystem. Under the rotating wave approximation (TWA), the
Hamiltonian in the interaction picture (assuming that $\hbar=1$)

$\widehat{H}_{int}=\frac{\Delta}{2}~{}\widehat{\sigma}_{z}+f(\chi,\widehat{n})+\lambda~{}\bigl{(}\widehat{\sigma}_{-}~{}\widehat{a}^{{\dagger}}+\widehat{\sigma}_{+}~{}\widehat{a}\bigr{)},$
(1)

where $f(\chi,\widehat{n})=\chi(\widehat{n}^{2}-\widehat{n})$ represents the
nonlinear term with,

$f(\chi,\widehat{n})\mid n\rangle=\bigl{[}\chi(n^{2}-n)\bigr{]}\mid
n\rangle=f(\chi,n)\mid n\rangle,$ (2)

where $\widehat{a}$ ($\widehat{a}^{{\dagger}}$) is bosonic annihilation
(creation) operator for the single mode field of frequency $\omega$ and
$\lambda$ is atoms-field coupling constant, the operators
$\widehat{\sigma}_{+}=|+\rangle\langle-|$ ,
$\widehat{\sigma}_{-}=|-\rangle\langle+|$ and
$\widehat{\sigma}_{z}=|+\rangle\langle+|-|-\rangle\langle-|$ represent,
respectively, the raising, lowering and population atomic operators which obey
the commutation relations
$[\widehat{\sigma}_{+},\widehat{\sigma}_{-}]=\widehat{\sigma}_{z}$, while
$\Delta=\omega_{0}-\omega$ is the detuning parameter which represents the
difference between the atomic and cavity subsystems eigenfrequencies. We
denote by $\chi$ the dispersive part of the third order susceptibility of the
Kerr-like medium FALI95 ; ATETO109 ; ATETO10 ; ATETO209 ; Ateto07 ; WLG06 .
Since the state vector, $|\psi_{F}(0)\rangle$, of the field is represented by
a linear superposition of the number state $|n\rangle$

$|\psi_{F}(0)\rangle=\sum_{n=0}^{\infty}C^{n}|n\rangle,$ (3)

and the first atom that traverse the cavity is assumed in its upper state
$\mid+\rangle$, i. e.,

$|\psi_{A}(0)\rangle=\mid+\rangle$ (4)

the initial state vector of the interacting first-atom-field system is given
by

$|\psi_{AF}(0)\rangle=|\psi_{A}(0)\rangle\otimes|\psi_{F}(0)\rangle=\sum_{n=0}^{\infty}C^{n}|n,+\rangle,$
(5)

where $|n\rangle$ is an eigenstate of the number operator
$\widehat{a}^{\dagger}\widehat{a}=\widehat{n}$;
$\widehat{a}^{\dagger}\widehat{a}|n\rangle=n|n\rangle$, and $C^{n}$ is, in
general, complex where the square of its modulus gives the probability of the
coherent field, with mean photon $|\alpha|^{2}=\bar{n}$, to have $n$ photons
by the relation

$P(n)=|\langle
n|\psi_{F}(0)\rangle|^{2}=|C^{n}|^{2}=e^{-\bar{n}}\frac{\bar{n}^{n}}{n!}.$ (6)

Under the assumption that, first atom traverse the cavity in its upper state
$\mid+\rangle$, the joint state vector of the field and the first atom at any
instant of time $t$ can be obtained from the solution of the time-dependent
Schrödinger equation

$i\frac{d}{dt}|\psi_{AF}(t)\rangle=\widehat{H}~{}|\psi_{AF}(t)\rangle.$ (7)

The time-dependent wave function of the first-atom-field system takes the form

$|\psi_{AF}(t)\rangle=\mid U_{+}^{n}(t)\rangle+\mid U_{-}^{n+1}(t)\rangle$ (8)

which can be simply obtained by recalling the initial condition (5) and
solving the Schrödinger equation (7), where

$\mid
U_{+}^{n}(t)\rangle=\sum_{n}^{\infty}~{}C^{n}~{}\Gamma_{1}(n,t)~{}|n,+\rangle,$
(9) $\mid
U_{-}^{n+1}(t)\rangle=\sum_{n}^{\infty}~{}C^{n}~{}\Gamma_{2}(n+1,t)~{}|n+1,-\rangle,$
(10)

with the amplitudes $\Gamma_{1}(n,t)$ and $\Gamma_{2}(n+1,t)$ are,
respectively, given by

$\Gamma_{1}(n,t)=\sum_{j=1}^{2}F_{j}(n)e^{i\mu_{j}(n)t},$ (11)

and

$\Gamma_{2}(n+1,t)=-\sum_{j=1}^{2}F_{j}\frac{\mu_{j}(n)+\alpha(n)}{\lambda\sqrt{n+1}}e^{i\mu_{j}(n)t},$
(12)

with $F_{j}(n)$ is given by

$F_{j}=\frac{\mu_{k}(n)+\alpha(n)}{\mu_{kj}(n)},~{}~{}~{}~{}k\neq j=1,2,$ (13)

where the angles $\mu_{1,2}(n)$ are given by

$\mu_{1,2}(n)=-\frac{\alpha(n)+\gamma(n)}{2}\pm\sqrt{\biggl{(}\frac{\alpha(n)-\gamma(n)}{2}\biggr{)}^{2}+\lambda^{2}(n+1)}.$
(14)

with $\alpha(n)$ and $\gamma(n)$ read, respectively

$\alpha(n)=\frac{\Delta}{2}+\chi n(n-1),$ (15)
$\gamma(n)=-\frac{\Delta}{2}+\chi n(n+1).$ (16)

As already indicated above, we consider a pair of two-level atoms going
through the cavity mode one after another. Moreover, under the assumption that
time of flight through the cavity $t$ is the same for every atom BGASMNN07 ;
DGMN04 ; LEW96 ; AMNN01 ; PB93 ; Ateto07 ; EGSWH96 ; MHNWBTH97 ; ELSW002 ,
where the joint state vector of both atoms and the field may be denoted by
$|\psi_{AAF}(t)\rangle$, the corresponding atom-atom-field pure-state density
operator may be written as

$\rho_{AAF}(t)=|\psi_{AAF}(t)\rangle\langle\psi_{AAF}(t)|,$ (17)

where, the joint time-evolved wave vector, $|\psi_{AAF}(t)\rangle$, of the
tripartite system, i.e., the two atoms and the cavity field after the second
atom leaves the cavity is obtained by solving the Schrödinger equation

$i\frac{d}{dt}|\psi_{AAF}(t)\rangle=\widehat{H}_{int}|\psi_{AAF}(t)\rangle.$
(18)

It is worth to note that within the delay time between the two atoms the field
evolves towards a thermal steady state, moreover, repetition of the instant in
which the later atoms enter the cavity means the same field repeats at this
instants precisely when successive atoms exit the cavity FIJAME86 .
In order to quantify the degree of entanglement between the two atoms, the
field variables must be traced out. One may write the reduced mixed-state
density matrix of the two atoms after taking the trace over the field
variables as

$\mathbf{\rho_{AA}}(t)=\mathrm{Tr}_{F}~{}[\rho_{AAF}(t)].$ (19)

Matrix representation of the reduced density operator, Eq. (19) becomes

$\mathbf{\rho_{AA}}(t)=\left(\begin{array}[]{cccc}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{14}\\\
\rho_{21}&\rho_{22}&\rho_{23}&\rho_{24}\\\
\rho_{31}&\rho_{32}&\rho_{33}&\rho_{34}\\\
\rho_{41}&\rho_{42}&\rho_{43}&\rho_{44}\end{array}\right),$ (20)

##  III Entanglement measure

For bipartite pure states, the partial (von Neumann) entropy of the reduced
density matrices can provide a good measure of entanglement. However, for
mixed states von Neumann entropy fails, because it can not distinguish
classical and quantum mechanical correlations. For mixed states, the
entanglement can be measured as the average entanglement of its pure-state
decompositions $E_{F}(\rho)$

$E_{F}(\rho)=min\sum_{i}p_{i}E(\psi_{i}),$ (21)

with $E(\psi_{i})$ being the entanglement measure for the pure state
$\psi_{i}$ corresponding to all the possible decompositions
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$. The existence of an
infinite number of decompositions makes their minimization over this set
difficult. Wootters W98  succeeded in deriving an analytical solution to this
difficult minimization procedure in terms of the eigenvalues of the non-
Hermitian operator

$T=\rho\tilde{\rho},$ (22)

where the tilde denotes the spin-flip of the quantum state, which is defined
as

$\tilde{\rho}=(\sigma_{x}\otimes\sigma_{x})\rho^{\ast}(\sigma_{x}\otimes\sigma_{x}),$
(23)

where $\sigma_{x}$ is the Pauli matrix, and $\rho^{\ast}$ is the complex
conjugate of $\rho$ where both are expressed in a fixed basis such as
$\\{|+\rangle,|-\rangle\\}$.
In terms of the eigenvalues of $T=\rho\tilde{\rho}$, $E_{F}(\rho)$ (known as
the entanglement of formation) takes the form

$E_{F}(\rho)=h\biggl{[}\frac{1}{2}+\frac{1}{2}\sqrt{1-C^{2}(\rho)}\biggr{]},$
(24)

where $C(\rho)$ is called the concurrence and is defined as

$C(\rho)=\max\biggl{(}0,\sqrt{\mathcal{E}_{1}}-\sqrt{\mathcal{E}_{2}}-\sqrt{\mathcal{E}_{3}}-\sqrt{\mathcal{E}_{4}}\biggr{)},$
(25)

with the $\mathcal{E}$’s representing the eigenvalues of $T=\rho\tilde{\rho}$
in descending order, and,

$h(y)=-y\log y-(1-y)\log(1-y),$ (26)

is the binary entropy. The concurrence is associated with the entanglement of
formation $E_{F}(\rho)$, Eq.(24), but it is by itself a good measure for
entanglement. The range of concurrence is from 0 to 1. For unentangled atoms
$C(\rho)=0$ whereas $C(\rho)=1$ for maximally entangled atoms.
Recalling (23) and assuming that that the matrix
$T=\rho_{AA}(\sigma_{x}\otimes\sigma_{x})\rho_{AA}^{\ast}(\sigma_{x}\otimes\sigma_{x})$,
needed for calculation of the concurrence, has the form

$T=\left(\begin{array}[]{cccc}T_{11}&T_{12}&T_{13}&T_{14}\\\
T_{21}&T_{22}&T_{23}&T_{24}\\\ T_{31}&T_{32}&T_{33}&T_{34}\\\
T_{41}&T_{42}&T_{43}&T_{44}\end{array}\right),$ (27)

the next step is to find the eigenvalues of the above matrix $T$. To achieve
our goal, we are supposed to solve the characteristic equation of

$\mathrm{Det}(T-\mathcal{E}\mathbf{I})=0,$ (28)

Equation (28) is a polynomial equation of degree 4

${\mathcal{E}}^{4}+c_{3}{\mathcal{E}}^{3}+c_{2}{\mathcal{E}}^{2}+c_{1}{\mathcal{E}}+c_{0}=0,$
(29)

with

$c_{3}=-T_{11}-T_{22}-T_{33}-T_{44},$ (30)
$c_{2}=-|T_{13}|^{2}-|T_{14}|^{2}-|T_{12}|^{2}-|T_{34}|^{2}-|T_{24}|^{2}+(T_{11}+T_{22})(T_{33}+T_{44})+T_{11}T_{22}+T_{33}T_{44},$
(31)
$c_{1}=|T_{12}|^{2}(T_{33}+T_{44})+|T_{23}|^{2}(T_{11}+T_{44})|T_{24}|^{2}(T_{11}+T_{33})+|T_{34}|^{2}(T_{11}+T_{22})|T_{13}|^{2}(T_{22}+T_{44})+|T_{14}|^{2}(T_{22}+T_{33})$
$-\Re(T_{23}T_{34}T_{42})-\Re(T_{21}T_{32}T_{13})-\Re(T_{21}T_{42}T_{14})-\Re(T_{31}T_{14}T_{43})-T_{11}(T_{11}T_{33}+T_{22}T_{33}+T_{11}T_{22}),$
(32)
$c_{0}=T_{11}T_{22}T_{33}T_{44}+|T_{13}|^{2}|T_{24}|^{2}+|T_{14}|^{2}|T_{23}|^{2}-T_{11}T_{22}|T_{34}|^{2}-T_{11}T_{44}|T_{23}|^{2}-T_{33}T_{44}|T_{12}|^{2}$
$-T_{11}T_{33}|T_{24}|^{2}-T_{22}T_{33}|T_{14}|^{2}+T_{11}\Re(T_{32}T_{24}T_{43})+T_{22}\Re(T_{31}T_{14}T_{43})+T_{33}\Re(T_{21}T_{42}T_{14})+T_{44}\Re(T_{32}T_{13}T_{43})$
$-\Re(T_{21}T_{32}T_{14}T_{43})-\Re(T_{21}T_{42}T_{13}T_{34})-\Re(T_{31}T_{42}T_{14}T_{23}).$
(33)

The roots of Eq.(29) are as follows

$\mathcal{E}_{1}=-\frac{1}{4}\biggl{[}c_{3}-\sqrt{\frac{W}{3}}-4O\biggr{]}$
(34)
$\mathcal{E}_{2}=-\frac{1}{4}\biggl{[}c_{3}-\sqrt{\frac{W}{3}}+4O\biggr{]}$
(35)
$\mathcal{E}_{3}=-\frac{1}{4}\biggl{[}c_{3}+\sqrt{\frac{W}{3}}-4N\biggr{]}$
(36)
$\mathcal{E}_{4}=-\frac{1}{4}\biggl{[}c_{3}+\sqrt{\frac{W}{3}}+4N\biggr{]}$
(37)

with

$O=\frac{1}{2}\sqrt{\frac{O_{1}+O_{2}+O_{3}}{6}}$ (38)
$N=\frac{1}{2}\sqrt{\frac{O_{1}+O_{2}-O_{3}}{6}}$ (39)

where

$O_{1}=3c_{3}^{2}-8c_{2}-V^{1/3}$ (40)
$O_{2}=4V^{-1/3}[3c_{1}c_{3}-12c_{0}-c_{2}^{2}]$ (41)
$O_{3}=\sqrt{\frac{3}{w}}[4c_{3}c_{2}-c_{3}^{3}-8c_{1}]$ (42)

and

$V=V_{1}+\sqrt{\frac{V_{2}+V_{3}+V_{4}+V_{5}}{144}}$ (43)

with

$V_{1}=8c_{2}^{2}+36[3(c_{1}^{2}+c_{0}c_{3}^{2})-c_{1}c_{2}c_{3}-c_{0}c_{2}]$
(44)
$V_{2}=-54c_{2}[c_{0}c_{1}c_{3}^{3}+c_{1}^{3}c_{3}+8c_{0}(c_{1}^{2}+c_{0}c_{3}^{2})]$
(45) $V_{3}=3c_{1}c_{3}[2c_{0}(4c_{2}^{2}+3c_{1}c_{3})-c_{1}c_{2}^{2}c_{3}]$
(46)
$V_{4}=12[(c_{1}c_{3})^{3}+4c_{0}c_{2}^{2}(8c_{0}-c_{2}^{2})+c_{2}^{3}(c_{1}^{2}+c_{0}c_{3}^{2})]$
(47) $V_{5}=c_{0}^{2}[81c_{3}^{4}+576c_{1}c_{3}-768c_{c}]$ (48)

and

$W=W_{1}+W_{2}$ (49)

where

$W_{1}=3c_{3}^{2}-8c_{2}+2V^{1/3}$ (50)
$W_{2}=8V^{-1/3}[c_{2}^{2}+3(4c_{0}-c_{1}c_{3})]$ (51)

In the following we consider two different cases of injection of the second
atom into the cavity subsystem that give us clear insight into the mechanism
of how and to what extent the degree of entanglement can be controlled and
enhanced.

##  IV injection of two excited atoms successfully

In this case, the time dependent state vector of the full system can be
expressed in the form

$|\psi_{AAF}(t)\rangle=\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}\mid
n\rangle+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}\mid
n+1\rangle+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}\mid
n+1\rangle+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}\mid n+2\rangle,$ (52)

which can be simply obtained by solving the Shrödinger equation (18), where

$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}[\Gamma_{1}(n,t)]^{2}\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\bigr{>},$ (53)
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n,t)\Gamma_{2}(n+1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
-\\\ \end{array}\bigr{>},$ (54)
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n+1,t)\Gamma_{2}(n+1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
+\\\ \end{array}\bigr{>},$ (55)
$\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{2}(n+1,t)\Gamma_{2}(n+2,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\bigr{>},$ (56)

Using (17) to obtain the full density operator and after the application of
(19) to obtain the reduced atomic density operator which can be put, after
using the notations $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}\equiv\mid 1\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}\equiv\mid 2\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}\equiv\mid 3\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}\equiv\mid 4\rangle$, in the form

$\rho_{AA}(t)=\sum_{i,j=1}^{4}~{}\rho_{ij}(t)|i\rangle\langle j|,$ (57)

where

$\rho_{11}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{1}(n,t)\bigr{|}^{4},$
(58)

$\rho_{12}(t)=\sum_{n}C^{n+1}C^{\ast
n}\bigl{[}\Gamma_{1}(n+1,t)\bigr{]}^{2}\Gamma_{1}^{\ast}(n,t)\Gamma_{2}^{\ast}(n+1,t),$
(59) $\rho_{13}(t)=\sum_{n}C^{n+1}C^{\ast
n}\bigl{|}\Gamma_{1}(n+1,t)\bigr{|}^{2}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$
(60) $\rho_{14}(t)=\sum_{n}C^{n+2}C^{\ast
n}\bigl{[}\Gamma_{1}(n+2,t)\bigr{]}^{2}\Gamma_{2}^{\ast}(n+1,t)\Gamma_{2}^{\ast}(n+2,t),$
(61)
$\rho_{22}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{1}(n,t)\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2},$
(62)
$\rho_{23}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n+1,t),$
(63) $\rho_{24}(t)=\sum_{n}C^{n+2}C^{\ast
n}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t)\bigl{|}\Gamma_{2}(n+2,t)\bigr{|}^{2},$
(64)
$\rho_{33}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{1}(n+1,t)\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2},$
(65) $\rho_{34}(t)=\sum_{n}C^{n+2}C^{\ast
n}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t)\bigl{|}\Gamma_{2}(n+2,t)\bigr{|}^{2},$
(66)
$\rho_{44}(t)=\sum_{n}\bigl{|}C^{n}\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+1,t)\bigr{|}^{2}\bigl{|}\Gamma_{2}(n+2,t)\bigr{|}^{2},$
(67)

For $\chi=\Delta=0.0$, the results reported in Ref. BGASMNN07  are
straightforward obtained, which analyzed in details but for low and high
average photon number $\bar{n}$.

Figure 1: The concurrence $C(\lambda t/\pi)$ (solid red curve) and the total
population $\rho_{22}(\lambda t/\pi)+\rho_{33}(\lambda t/\pi)$ (dashed blue
curve) for $\bar{n}=10$, $\delta=0.0$ where (a) $\chi/\lambda=0.0$ (b)
$\chi/\lambda=0.2$ (c) $\chi/\lambda=0.5$ (c) $\chi/\lambda=1.0$

Figure 2: The same as Fig. 1 but for $\chi/\lambda=0.0$ where (a) $\delta=0.5$
(b) $\delta=1.0$ (c) $\delta=10.0$ (d) $\delta=15.0$ (e) $\delta=20$

Figure 3: The same as Fig. 2 but when $\chi=0.5$ and (a) $\delta=1.0$ (b)
$\delta=10.0$

In order to discuss the dynamical feasibility of our scheme in the micromaser
regime, in which the lifetime of the cavity is longer than atom field
interaction time, the concurrence $C(\lambda t/\pi)$ and the corresponding
total population $\rho_{22}(\lambda t/\pi)+\rho_{33}(\lambda t/\pi)$, as a
reference state, are depicted simultaneously in figures 1, 2 and 3 for the
Rabi angle $\lambda t/\pi\in[0,16]$. The mean photon number $\bar{n}$ is taken
about $\bar{n}=10$, the detuning parameter $\delta=\Delta/\lambda$ carries
various values such as $\delta=0.0,0.5,1,10,15$ and $20$, while the Kerr
parameter $\chi/\lambda$ varies such as $\chi/\lambda=0.0,0.2,0.5$ and
$\chi/\lambda=1.0$. From Fig. 1a, where the interaction occurs resonantly in
the absence of the nonlinear media, one can see clearly that the concurrence
$C$ exhibits irregular collections of oscillations where the total population
oscillates but with amplitudes maximize (no longer exceed the value $0.25$)
when $\rho_{22}+\rho_{33}\approx 0.4$; the atoms in the mixed state
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$. Thus, the lack of the entanglement between the two atoms
can be attributed to a large population of the product states
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\ \end{array}\bigr{>}$ and
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\ \end{array}\bigr{>}$ which,
in turn, decay on the same time scale as $C$. For significantly high average
photon number $\bar{n}$, it was shown that BGASMNN07 , quantum effects which
are predominant primarily when the photon number is low, help to increase the
peak value of the concurrence $C$.

On resonance interaction between the atoms and a cavity filled with a Kerr
medium of a parameter of order $10^{-1}\lambda$, such as $\chi=0.2\lambda$,
the total population shows pseudo periodical behavior with amplitudes minima
decrease as time evolves where the revival period, $t_{R}$, elongates as time
goes on. The concurrence $C$ loses its irregular collections where instant
oscillations appear after wide time intervals accompany with amplitudes
decrease ($\approx 0.3$ in the beginning) as evolves , Fig. 1b. A significant
and interesting distinction in the case when $\chi=0.5\lambda$ is that both
$C$ and $\rho_{22}+\rho_{33}$ display a perfect $\pi$ periodicity of
oscillations. The total population $\rho_{22}+\rho_{33}$ exhibits collapses
and revivals with the revival period $t_{R}$ is shorter and each revival is a
perfect carbon copy of its predecessor. The concurrence $C$ shows oscillation
packets composed of three peaks with the strongest,$=2.5$, lies at the center
where the total population collapses. Moreover, the concurrence $C$ minimizes
where the total populations $\rho_{22}+\rho_{33}$ revives while the strong
entanglement and collapse of population occur in parallel manner, see Fig. 1c.
This periodicity is due to the fact that analytical expressions of both $C$
and $\rho_{22}+\rho_{33}$ can be summed exactly. Thus, the matching value of
the Kerr parameter plays more efficient rule than that of quantum effects that
leads to increase the peak value of the concurrence $C$ BGASMNN07 . Opposite
results are obtained when a considerable strength of the Kerr medium is
applied to the cavity field. In this case, a pseudo periodicity for $C$ is
observed where the total population preserves its periodicity accompanied with
clear decreasing in its amplitudes. In addition, one can’t build a clear
insight about the relation between curves of both $C$ and
$\rho_{22}+\rho_{33}$, Fig. 1d. Thus, from the above it is found that the Kerr
medium can be used to adjust the entanglement degree between qubits, which may
provide some reference and even basis for the quantum control of multiple
qubits.

A surprise is that the case of a non-resonance interaction does not destroy
the entanglement phenomenon, see Fig. 2. For small detuning such that
$\delta=0.5$, there is no significant change, Fig. 2a, but when $\delta$
increases to unity, the correlations time between the atoms becomes longer
with the gradual appearance of nonzero values of $C$, see Fig. 2b. Strictly
speaking, the destruction of entanglement occurs for the case when the
detuning parameter $\delta$ is absent but not the Kerr parameter
$\chi/\lambda$, Fig. 1d.

The most intersting and surprising is the case when the inter-atomic
transition is considered to be significantly high compared with the cavity
frequency such that $\delta=10.0,15.0$ and $20.0$ where the nonlinear media is
considered to be absent. This is revealed in Figs. 2c-e. We see that the total
population exhibits the known collape and revival phenomena periodically.
Also, we notice that for the increase of detuning parameter $\delta$, the
collapses become longer and the beginning strong overlap of the revivals
becomes weaker; while the amplitude of the revivals decreases as values of
$\delta$ increase. The entanglement evolution divided into three regions. The
first one is at the half of the collapse time, where the entanglement degree
oscillates with bigger amplitude for short time scale. The oscillation maximum
is related crucially to the value of $\delta$, where it occurs at time
$\lambda t=(\frac{\delta}{2}+1)\pi$. The last two regions are in the beginning
and end of the collape time. In this case, we see that the atom-atom
entanglement remain close to its maximum and show intrinsic oscillations only.
This is in fact is true for all values of the detuning parameter. The detuning
between the field and the atoms leads to an emergence of collapses and
revivals, this statement is always true only in the absence of the nonlinear
media, see Figs. 2c-e. Overall, we find that the detuning parameter acts as a
control parameter for the atom-atom entanglement, where a perfect periodicity
can be obtained for suitable high detuning. The revival of the entanglement at
time $\lambda t=(\frac{\delta}{2}+1)\pi$ is more easily understood by
reference to the density matrix of the system. As expected, there is no
entanglement before the populations of the state
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$ and the state
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}$ depopulates, but some entanglement appears for longer
times, Figs. 2b-e, and the concurrence becomes equal to the population of the
state $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$, see Fig. 1c.

##  V injection of a ground atom after the excited one

In this case, the time dependent state vector of the full system can be
expressed in the form

$|\psi_{AAF}(t)\rangle=\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}\mid
n\rangle+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}\mid
n-1\rangle+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}\mid
n+1\rangle+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}\mid n\rangle,$ (68)

where

$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n-1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
-\\\ \end{array}\bigr{>},$ (69)
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}(n,t)\Gamma_{2}(n,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\bigr{>},$ (70)
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}\Gamma_{1}^{\ast}(n+1,t)\Gamma_{2}(n+1,t)\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\bigr{>},$ (71)
$\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}}(t)\biggr{>}=\sum_{n}C^{n}~{}[\Gamma_{2}(n+1,t)]^{2}\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\
+\\\ \end{array}\bigr{>},$ (72)

Following the same procedure as in the previous section, and after using the
notations $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}\equiv\mid 1\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}\equiv\mid 2\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}\equiv\mid 3\rangle$,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}\equiv\mid 4\rangle$, the elements of the reduced atomic
density matrix (57) read

$\rho_{11}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{1}(n,t)|^{2}|\Gamma_{1}(n-1,t)|^{2},$
(73) $\rho_{12}(t)=\sum_{n}C^{n}C^{\ast
n+1}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n-1,t)\Gamma_{1}^{\ast}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$
(74) $\rho_{13}(t)=\sum_{n}C^{n+k}C^{\ast
n}|\Gamma_{1}(n,t)|^{2}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$ (75)
$\rho_{14}(t)=\sum_{n}|C^{n}|^{2}\Gamma_{1}(n,t)\Gamma_{1}^{\ast}(n-1,t)[\Gamma_{2}^{\ast}(n+1,t)]^{2},$
(76)
$\rho_{22}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{1}(n,t)|^{2}|\Gamma_{2}(n,t)|^{2},$
(77) $\rho_{23}(t)=\sum_{n}C^{n+2}C^{\ast
n}\Gamma_{1}(n+2,t)\Gamma_{2}(n+2,t)\Gamma_{1}(n,t)\Gamma_{2}^{\ast}(n+1,t),$
(78) $\rho_{24}(t)=\sum_{n}C^{n+2}C^{\ast
n}|\Gamma_{2}(n+1,t)|^{2}\Gamma_{1}(n+1,t)\Gamma_{2}^{\ast}(n+1,t),$ (79)
$\rho_{33}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{1}(n,t)|^{2}|\Gamma_{2}(n+1,t)|^{2},$
(80) $\rho_{34}(t)=\sum_{n}C^{n}C^{\ast
n+1}\Gamma_{1}^{\ast}(n,t)\Gamma_{2}(n+1,t)[\Gamma_{2}^{\ast}(n+2,t)]^{2},$
(81) $\rho_{44}(t)=\sum_{n}|C^{n}|^{2}|\Gamma_{2}(n+1,t)|^{4},$ (82)

Figure 4:  The concurrence $C(\lambda t/\pi)$ (solid red curve) and the total
population $\rho_{22}(\lambda t/\pi)+\rho_{33}(\lambda t/\pi)$ (dashed blue
curve) for $\bar{n}=10$, $\delta=0.0$ where (a) $\chi/\lambda=0.0$ (b)
$\chi/\lambda=0.2$ (c) $\chi/\lambda=0.5$ (c) $\chi/\lambda=1.0$

Figure 5: The same as Fig. 4 but for $\chi/\lambda=0.0$ where (a) $\delta=0.5$
(b) $\delta=1.0$ (c) $\delta=10.0$ (d) $\delta=15.0$ (e) $\delta=20$

Figure 6: The same as Fig. 5 but when $\chi=0.5$ and (a) $\delta=1.0$ (b)
$\delta=10.0$

To better understand the situation and gain clearer insight into the
entanglement dynamics, hereafter we shall analyze the evolution of the
concurrence $C(\lambda t/\pi)$ and the total populations $\rho_{22}(\lambda
t/\pi)+\rho_{33}(\lambda t/\pi)$ where the parametres remain as the same in
the past section.

Generally, a similar conclusion is reached for the present case but with
considerable distinctions in its details. As in the case of two excited atoms,
there is no initial entanglement between the atoms, and at early times the
entanglement builds up rapidly to a maximum appearing at short time $\lambda
t\approx 2.5\pi$, Fig. 4a. However, comparing with the case of two successive
excited atoms, shown in the previous section, we see that the entanglement
maxima that obtained in the present case are generally lower than that
obtained with two successive excited atoms, see Figs. 1-3 and 4-6. It is
interesting to note that after passing through the maximum, the entanglement
decays with two different time scales, Figs. 2 and 5. This effect is more
pronounced if we choose the interaction with atoms to be initially in
different quantum states. Thus, atom-atom entanglement depends crucially on
the coherent transfer of the population between the atoms states, specially in
the presence of a nonlinear media as well as the detuning parameter $\delta$.
In this process, the population is efficiently transferred from the more
populated state, $\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\bigr{>}$ to the state,
$\bigl{|}\begin{array}[]{l}\vspace{-2ex}+\\\ -\\\
\end{array}\bigr{>}+\bigl{|}\begin{array}[]{l}\vspace{-2ex}-\\\ +\\\
\end{array}\bigr{>}$ before it decays to the initial state leading to the
enhancement of the entanglement.

##  VI Phase spase distribution

We devote this section to concentrate on a function, which considered an
important one of the functions, $W$, (Husimi) $Q$, and (Glauber-Sudershan) $P$
functions, that is the Husimi $Q$-function which has the nice property of
being always positive and further advantage of being readily measurable by
quantum tomographic techniques BOTAWA98 ; MANTOM97 . It has been shown from
earlier studies EWIGNER32 ; ZWIGNer32 ; KAGL69 ; HICOSCWG84  that these quasi-
probability functions are important for the statistical description of a
microscopic system and provide insight into the nonclassical features of the
radiation fields. The homodyne measurement of an electromagnetic field gives
all possible linear combinations of the field quadratures. The average of the
random outcomes of the measurement is connected with the marginal distribution
of any quasi-probability used in quantum optics. In fact, Husimi $Q$-function
is not only a convenient tool to calculate the expectation values of anti-
normally ordered products of operators, but also to give some insight into the
mechanism of interaction for the model under consideration. The relation
between the phase-space measurement; Husimi $Q$-function; and the classical
information-theoretic entropy associated with quantum fields was introduced by
Wehrl WEHRL79 , which referred to as the Shannon information of the Husimi
Q-function. Thus, Husimi $Q$-function can be related to quantum entanglement
in different approaches WEHRL79 ; PEKRPELUSZ86 ; FAGU06 ; CAALCARA09 ; HUFAN09
; MIMAWA00 ; MIWAIM01 ; BERETA84 . Furthermore, separable state is represented
by a localized wave packet in phase space. Since coherent states are the most
localized states in the Husimi representation, it is argued that SUGITA03
delocalization of the Husimi distribution implies correlation-hence
entanglement- between system particles.
The Husimi $Q$-function can be given in the form as HICOSCWG84 ; HUSIMI40 ;
FuSOLO001

$Q(\alpha)=\frac{\langle\alpha\mid\rho_{F}\mid\alpha\rangle}{\pi},$ (83)

where $\rho_{F}$ is the reduced density operator of the cavity field given by
tracing over the atomic variables of the full density operator (17). The state
$\mid\alpha\rangle$ represents the well-known coherent state with amplitude
$\alpha=X+iY$. Inserting the obtained $\rho_{F}$ into Eq. (83), we can easily
obtain the Husimi $Q$-function of the cavity field

$Q=\frac{1}{\pi}(\langle\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\mid\rho_{F}\mid\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\rangle+\langle\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\mid\rho_{F}\mid\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\rangle),$ (84)

where

$\mid\alpha\rangle=e^{-\mid\alpha_{0}\mid^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha_{0}^{n}}{\sqrt{n!}},$
(85)

###  VI.1 Injection of two excited atoms one by one

In this case, the $Q$-function is given by

$Q=\frac{1}{\pi}\biggl{(}\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}},n\biggr{>}\biggr{|}^{2}+\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}},n+1\biggr{>}\biggr{|}^{2}\biggr{)},$ (86)

where for the states $\mid\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\rangle$ and $\mid\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\rangle$, $\rho_{F}$ reads

$\rho_{F}=\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n\biggr{>}\biggl{<}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n\biggr{|}+\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\
\end{array}}(t),n+1\biggr{>}\biggl{<}U_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}}(t),n+1\biggr{|},$ (87)

in this case

$\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}}{n!}[\Gamma_{1}(n,t)]^{2},$
(88) $\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\biggl{|}U_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t),n+1\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}\alpha^{\ast
2}}{\sqrt{n!(n+2)!}}\Gamma_{2}(n+1,t)\Gamma_{2}(n+2,t),$ (89)

$\alpha=X+iY.$ (90)

Figure 7: Conour plots for Husimi $Q$-function for $\bar{n}=10$, $\delta=0.0$,
$\lambda t=5\pi/3$ where (a) $\chi/\lambda=0.0$ (b) $\chi/\lambda=0.2$ (c)
$\chi/\lambda=0.5$ (d) $\chi/\lambda=1.0$

Figure 8: The same as Fig. 7 but when $\delta=10.0$ and (a) $\chi/\lambda=0.0$
(b) $\chi/\lambda=1.0$

Figure 9: The same as Fig. 7 but for $\chi/\lambda=0.5$ where (a) $\lambda
t=\pi/6$ (b) $\lambda t=\pi/4$ (c) $\lambda t=\pi/3$ (d ) $\lambda t=\pi/2$

Here, we present a detailed analytical discussion about how the Husimi
$Q$-function plays an important role in quantifying atom-atom entanglement.
For this reason, we have pictured the numerical results obtained as contour
plots in figures 7-9. The parameters was considered as follows: the average
photon number $\bar{n}$ is kept fixed for all plots such as $\bar{n}=10$, the
detuning parameter carries values such as $\delta=0.0,10.0$, and the Kerr
parameter $\chi/\lambda$ varies as $\chi/\lambda=0.0,0.2,0.5,1.0$, where the
Rabi angle $\lambda t=5\pi/3$ is kept fixed for figures 7 and 8, while for
figure 9 it changes as $\lambda t=\pi/6,\pi/4,\pi/3,\pi/2$, where the Kerr
parameter $\chi/\lambda$ is half integer.
Form figure 7, by which we examine the effect of the Kerr parameter
$\chi/\lambda$ at a fixed Rabi angle $\lambda t=5\pi/3$, we observe that, in
the absence of the Kerr parameter, Fig. 7a, the Husimi $Q$-function composed
of three components, two blobs centered at the points (0, $\pm 4$) and a
symmetric circular peak (Poisson band) centered at the point (4, 0) and
strongly localized in the right half of the available phase space, i.e., the
interaction region THMAMAPO07 , once more, the squeezing effect on the number
of photons can be observed from the figure. Experimental and theoretical
studies showed that squeezing of matter wave fields (squeezed states) is
closely related to entanglement RSCNB03 ; RECL97 ; DSCZ00 ; SDCZ01 ; ATETO010
; PMBLMH04 ; FIUNIO03 . Moreover, we observe that the Husimi distribution
blobs delocalized almost around the right half of the phase space, i. e., the
interaction region of phase space and have structures with almost positive
density throughout the available phase space. It is worth to note that the
bifurcating of the blobs corresponds to the collapse. This can be clearly seen
if we compare the numerical calculations for total populations and
$Q$-function. It was shown that the bifurcation of the Husimi distribution,
which is the signature of the formation of Schrödinger cat states, implies
correlation, and hence entanglement VAOR95 ; ORPAKA95 ; JEOR94 ; MIMAWA00 ;
MIWAIM01 ; SUGITA03 ; HIMCMI05 , see also Fig. 1a. As soon as the effect of
the Kerr medium is considered, a clear difference in the figures shape
appears, where the sensibility of the Husimi $Q$-function with the
nonlinearity parameter $\chi/\lambda$ is very intersting. We observe that the
Husimi distribution is taken gradually towards the non-interaction region,
namely, left half of the available phase space, where we observe that the two
blobs rotating in the complex coherent state parameter plane in the
counterclockwise directions with the same speed where two overlapping regions
appear with both terms are appreciably vanishing. But note also that the
intersection areas with this prescription are not in the same position. Upon
the increase of the Kerr parameter then one can observe that the overlapping
becomes dominant and the Husime distribution are taken completely into the
interaction region. As a consequence, strong entanglement degree between the
two atoms can be built up at that time, see Figs. 1c and 7c. The conclusion
will be extremely interesting when we look at the behaviors of both the
concurrence $C$ and the Husimi distribution once the Kerr medium is taken into
account, specially at the moment when the Kerr medium becomes strong at which
entanglement diminishes and the Husimi distribution is strongly localized and
taken into non-interaction region again, Figs. 1d and 7d. It is important to
note that, localization of the Husimi distribution around a fixed point
implies low degree of entanglement SUGITA03 ; HIMCMI05 .

Our observations will become clear when we look at the behaviors of both the
concurrence $C$ and the Husimi distribution at the moment when detuning
parameter possesses a value. The results showed in Figs. 2c, 3b and 8. A clear
delocalization of the Husimi distribution but into the right half of phase
space, the interaction region, corresponds to longer time interval of
entanglement, Figs. 2c and 8a, while the opposite is true, Figs. 3b and 8b. An
extremely interesting observations can be obtained when we consider various
values of the Rabi angle $\lambda t$, while the nonlinear media is switched
on. The results show strongly the role that the Hisimi distribution play in
quantifying entanglement, where at all moments but not $\lambda t=\pi/2$, once
the suitable Kerr parameter is chosen, stronger entanglement can be built up
since the Husimi distribution is delocalized but completely spreads between
two or three points into the right half of phase space, the interaction
region, see also Fig. 1c.

###  VI.2 Injection of excited and ground atoms one by one

In this case, for the states $\mid\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\rangle$ and $\mid\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\rangle$, $\rho_{F}$ reads

$\rho_{F}=\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n-1\biggr{>}\biggl{<}S_{\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\
\end{array}}(t),n-1\biggr{|}+\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\
\end{array}}(t),n+1\biggr{>}\biggl{<}S_{\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}}(t),n+1\biggr{|}.$ (91)

The $Q$-function is given by

$Q=\frac{1}{\pi}\biggl{(}\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\
+\\\ \end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}},n-1\biggr{>}\biggr{|}^{2}+\biggl{|}\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\
-\\\ \end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}},n+1\biggr{>}\biggr{|}^{2}\biggr{)},$ (92)

where

$\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}+\\\ +\\\
\end{array}}(t),n-1\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}\alpha_{0}}{\sqrt{n!(n+1)!}}\Gamma_{1}(n+1,t)\Gamma_{2}(n+1,t),$
(93) $\biggl{<}\alpha,\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}\biggl{|}S_{\begin{array}[]{l}\vspace{-2ex}-\\\ -\\\
\end{array}}(t),n+1\biggr{>}=e^{-(\mid\alpha\mid^{2}+\mid\alpha_{0}\mid^{2})/2}\sum_{n}\frac{(\alpha_{0}\alpha^{\ast})^{n}\alpha^{\ast}}{\sqrt{n!(n+1)!}}\Gamma_{1}^{\ast}(n,t)\Gamma_{2}(n+1,t),$
(94)

Figure 10: Conour plots for Husimi $Q$-function for $\bar{n}=10$,
$\delta=0.0$, $\lambda t=5\pi/3$ where (a) $\chi/\lambda=0.0$ (b)
$\chi/\lambda=0.2$ (c) $\chi/\lambda=0.5$ (d) $\chi/\lambda=1.0$

Figure 11: The same as Fig. 10 but when $\delta=10.0$ and (a)
$\chi/\lambda=0.0$ (b) $\chi/\lambda=1.0$

Figure 12: The same as Fig. 10 but for $\chi/\lambda=0.5$ where (a) $\lambda
t=\pi/6$ (b) $\lambda t=\pi/4$ (c) $\lambda t=\pi/3$ (d ) $\lambda t=\pi/2$

To obtain clearer insight about the efficiancy of the Husimi $Q$-function in
quantifying entanglement, a comparative discusion will be important. Here we
examine the behavior of the Husimi function under the same circumstances of
parameters as in the previous section. This is illustrated in figures 10-12.
Generally, the Husimi $Q$-function developed in manner similar to that
observed previousley with slight change such that the localized Poisson band
barely appear which means that nonclassical effect is not pronounced. In this
case the entanglement between the two atoms doesn’t live much more in opposite
to the case of two successive excited atoms, where entanglement does appear at
almost the whole time scale, see Figs. 1a, 4a, 7a, 10a. The case when we turn
on the nonlinear media is also intersting. The Husimi distribution moves
quickly towards the non-interaction region once the a weak Kerr medium is
switched on, while a strong localized Poisson band appear clearly in the
interaction region which reflects small probability of interaction and so
lesser times of entanglement, see Figs. 4b, 10b. The suitable choice of the
Kerr parameter push the Husimi distribution towards the interaction region
again, which means stronger entanglement appears at longer times, Figs. 4c,
10c. There is no something interesting in case when the nonlinear media is
strong, where almost the same behavior was seen before, for comparison see
Figs. 1d, 4d, 7d, 10d.

However, the effect of the detuning parameter $\delta$ is quite differente,
specially in the presence of strong nonlinear media. In such a case, the
Husimi distribution displays a strong delocalization and spreads between two
points in the right half of the phase space in case of zero-value nonlinear
media accompanied with a significant density throughout the available phase
space. Moreover, overlapp takes place and the squeezing effect on the number
of photons occurs can be observed. The strong Kerr media have intersting
effect where the Husimi distribution shows ring-like shape around the center
of the phase space but nevertheless have a significantly different densities
throughout the available phase space, i.e., interaction and non-interaction
regions. The effect of the matching choice the nonlinear media parameter
$\chi/\lambda$ on the Husimi distribution at different values Rabi angle
$\lambda t$ points out that enhancement of entanglement depends crucially on
this suitable choice, see Figs. 4a and 12a-c.

##  VII Conclusions

We have considered the nature of the entanglement of output two successive
atoms from a micromaser cavity for pure states input of the atoms when the
cavity distribution is coherent. We have considered two different successive
injections of the initial states of the atoms that traverse the cavity,
namely, initially excited atoms and initially excited atom followed ground
one. We have examined the effect of the frequency difference between the
inter-atomic frequency and cavity frequency individually and in coexistence of
nonlinear media on nonlocal atomic entanglement. We have also investigate the
role of the Husimi $Q$-distribution that plays to give clearer insight about
entanglement dynamics. Our conclusions are Summarized in the following:
(i) The physical nature of the interacting objects and the character of their
mutual coupling control strongly the degree of quantum atomic entanglement, in
other words, the interaction of a cavity field with two successive excited
atoms plays more efficient role in producing atomic entanglement than that of
interaction with two successive atoms traverse the cavity in dissimilar
initial states.
(ii) The nonlinear medium plays an important role in producing atomic
entanglement depending on the appropriate choice of its parameter. In this
case an excellent periodical entanglement can be produced with increased
maximum value comparing with the case when the nonlinear medium parameter is
not appropriate or zero.
(iii) The extremely interesting is the role of the detuning parameter on
producing nonlocal atomic entanglement. When the detuning parameter is
considered to be considerably high, periodical, long lived entanglement can be
obtained.
(iv) Husimi $Q$-distribution plays a very clear and efficient role in
quantifying nonlocal atomic entanglement, where bifurcating of Husimi
$Q$-distribution in whole phase space and delocalization in the interaction
region of the phase space corresponding to correlations between particles and
as a consequence nolocal atomic entanglement.

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11institutetext: University of Bedfordshire, Luton, UK
11email<EMAIL_ADDRESS>

# Parsec: A State Channel for the Internet of Value

Amit Kumar Jaiswal  0000-0001-8848-7041

###### Abstract

We propose Parsec, a web-scale State channel for the Internet of Value to
exterminate the consensus bottleneck in Blockchain by leveraging a network of
state channels which enable to robustly transfer value off-chain. It acts as
an infrastructure layer developed on top of Ethereum Blockchain, as a network
protocol which allows coherent routing and interlocking channel transfers for
trade-off between parties. A web-scale solution for state channels is
implemented to enable a layer of value transfer to the internet. Existing
network protocol on State Channels include Raiden for Ethereum and Lightning
Network for Bitcoin. However, we intend to leverage existing web-scale
technologies used by large Internet companies such as Uber, LinkedIn or
Netflix. We use Apache Kafka to scale the global payment operation to
trillions of operations per day enabling near-instant, low-fee, scalable, and
privacy-sustainable payments. Our architecture follows Event Sourcing pattern
which solves current issues of payment solutions such as scaling, transfer,
interoperability, low-fees, micropayments and to name a few. To the best of
knowledge, our proposed model achieve better performance than state-of-the-art
lightning network on the Ethereum based (fork) cryptocoins.

###### Keywords:

Blockchain Payment Channels Distributed Ledger Event Sourcing Ethereum Smart
Contracts.

##  1 Introduction

The influence of data systems in finance has been so far focused on cutting
administrative costs. However, technology could transform business models in
finance. With the help of cryptography on distributed data systems, we can
design infrastructure that not only keeps track of transactions and balances
but also allows transfer of assets and enforcement of contracts without the
need of clearinghouses, middle men, escrows and even some tasks performed by
lawyers. On these systems settlements are final and there is no possibility of
repudiation while smart contracts execute deterministically according to
predefined inputs. Institutions that enclasp this vision will have a
competitive advantage. Investing in distributed cryptographic technology and
regulatory oriented technology will pay off in the form of less
intermediation, less legal fees or exordium and ultimately an accelerated
innovation cycle where new financial services can be provided with much less
friction and where these services are much better tailored to smaller market
segments. These institutions will be able to compete in the long tail of
financial services and offer a much more competitive service for the
mainstream.

However, for the Internet of Value vision to happen, there’s need for a whole
redesign of the architectural Information Technology (IT) principles used to
build new services. In this paper, our proposed models can achieve better
performances than state-of-the-art existing models on the Ethereum Blockchain
[13] network.

##  2 Architectural Details

Figure 1: Architecture to create new services

In search of new services such as global payment solution, we created an
architecture that makes possible to quickly innovate and create new services.
In order to achieve this at web scale, we attempt to employ the Event Sourcing
[7] pattern, an approach to handle large operations on data driven by a series
of events, in which each event is recorded in an append-only datastore. In
this kind of architecture, instead of having a centralized state for all the
applications in a centralized database or a set of disconnected state holding
databases, we rely on a centralized log of state changes, which allows to
create new applications very quickly because it is only necessary to replay
the log of changes in order to build new services with new states that will be
consistent with the state being held by other services in the same company.

Event Sourcing is very powerful but has a coupling problem. In big
organizations a central team needs to own the event log.

Figure 2: Architecture based on Event Sourcing Pattern

Usually, many different teams want to access the log and they all need to
agree on the technology and permissions to write and read from it. Also teams
reading or writing intensively from or to the central infrastructure risk
causes performance problem to the whole.

Also, this kind of architecture is not suitable for collaborating among
different organizations or group of companies. For that reason, an
architecture where each business unit or application would hold a copy of the
parts the log relevant to it would be better. If the stream of data is
composed of different topics (for example, premium payments or claims), then
each business unit would be able to hold a copy of the topics relevant to it.
The above architecture would be feasible in a trusted environment but in a
collaborative environment, all transactions would have to be signed by the
different parties and the order of the transactions should be guaranteed.

We can achieve that in a manner similar to blockchain by means of a linked
stream of events using hash pointers to the previous events and by embedding
signatures of both sides of a transaction in the message itself.

Figure 3: Sub-model based on Linked stream of events

This would imply that some business units would have to act as signing
authorities to their counter-parties in external organizations. By acting like
that we would achieve the benefits of a distributed ledger but without being
able to transfer assets.

Asserting that assets can’t be transferred with only a distributed ledger can
be contentious. Indeed if both parties have proof that the transaction has
been agreed.

However, even in this case where the transaction implied a transfer of assets,
for example, in the case where the transaction implied a transfer of assets,
for example, in the case of signing an insurance policy, this would not imply
that any of the parties would pay the stipulated amounts. We would have no
guarantee that policy takers would pay the claims. For that purpose we would
need to resort to a third party escrow or even better to a Smart Contract
running in a blockchain with distributed consensus. It is arguable whether
this blockchain would have to be public or if a permissioned distributed
ledger would be able to achieve this purpose.

Figure 4: Model for State Channel and Smart Contract operation on Public
Blockchain

Here we assume it as a public blockchain capable of running Smart Contracts.
In fact the structure we described here is following an existing approach, so
called a State Channel [1] and it allows untrusted parties to exchange
transactions with the possibility to resort to an independent escrow held in a
Smart Contract operating on a public blockchain in case that one of the
parties does not fulfill its commitment.

Figure 5: Architecture of Parsec Infrastructure Layer

Finally get the big picture on how a state of the art architecture for a
financial system should look like. It would have to: (1) Be based on
transaction logs, (2) It would have to include a replication mechanism of such
logs between business units, (3) Logs would have to be enhanced to become
shared ledgers when dealing with external untrusted parties, and (4) State
channel support would have to be added for asset transfer.

##  3 Implementation

We implemented a infrastructure layer named Parsec111Code Available at:
https://github.com/quanonblocks/Parsec using Apache Kafka, a distributed data
system that can be used as an event sourcing unified log tool. It can be seen
both as a queue or as a data store. We intend to implement only the very
minimum additions to the already existing functionality based on [4, 5, 6] so
that it can work as a State Channel. For that purpose a fundamental tool of
the Apache Kafka [3] ecosystem will be Kafka Streams [8] as it already offers
many of the functionalities we need to implement.

We integrated few major functions on top of state channel to work which are
(1) Scalability and replicability is provided by Apache Kafka and Kafka mirror
maker, (2) Ability to maintain state at scale and interaction with Smart
Contract or chaincode [14] is extended by Parsec on top of Kafka Streams, and
(3) Guarantees an exact sequence of hash pointers and transaction verification
with its cryptographic signatures is developed in Parsec.

For above given functionality, we intend to provide a library running on top
of Kafka Streams under fast data solution system Landoop. Streams that
guarantees the last three points.

###  3.1 Parsec Nodes

Parsec nodes are the job instances of the Parsec Library implementing the
Parsec Software Development Kit (SDK). The Parsec library is developed on top
of the Kafka streams library with the several addition of functionality such
as support for a catalogue of various kinds of transactions and to keep an
update state for all unique Ids of the transaction topic (kafka topic). Also,
we have integrated the Hashed Timelock [9] to interact with Ethereum smart
contracts [10] for the kafka topic partitions under the nodes’ supervision for
scheduled settlements or the request of control topic.

All functionality for each Parsec job relates only to the partitions under
it’s control.

###  3.2 Parsec Transactions

A basic catalogue of predefined transactions is implemented which can be used
to add additional custom transactions.

Micropayment transaction [11] performs the following actions:

  * •

Keep a sliding array of the latest n number of transactions in order to
reconstruct the exact order of the transactions from the transaction hash
pointers. This is performed in order to cope with our order of messages or
repeated messages.

  * •

Calculate the balance after each transaction.

  * •

Commit the balance to the Smart Contract in the Ethereum network at predefined
intervals. Intervals are defined by the modulo m of the incremental
transaction sequence number.

  * •

Commit the balance to the smart contract in the Ethereum network at predefined
timeouts. The timeout duration can be specified by the checkpoint_timeout
parameter.

  * •

Under the request of the control topic, perform an unscheduled settlement
request to the smart contract.

  * •

Under the request of the control topic, perform a disputed settlement request
to the smart contract by means of transferring the signed transactions since
the last settlement.

##  4 Conclusion & Future Works

This paper attempts to address the challenge in scaling, fast transaction
processing, verification process issues simplifies with our proposed model
contains Parsec Simulator which generates a special protocol messages under
Kafka cluster with fully functional API support for existing cryptocurrencies
like Bitcoin [15] and Ethereum [2] accessible via Avro User Interface222Avro
UI: https://github.com/kaaproject/avro-ui. We also have supports to check
Schema’s [12] and protocol messages with Coyote test suite333Coyote Tester:
https://github.com/Landoop/coyote. For future work, we are looking forward to
provide additional ways to support State Management functionality.

## ACKNOWLEDGMENT

This work is supported by Quanonblocks LLP.

## References

  * [1] Dziembowski, S., Faust, S., & Hostakova, K. (2018). Foundations of state channel networks. IACR Cryptology ePrint Archive, 2018: 320.
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  * [11] Bitcoinj. Working with micropayment channels. https://bitcoinj.github.io/working-with-micropayments.
  * [12] Schema registry UI. https://github.com/Landoop/schema-registry-ui.
  * [13] Vitalik Buterin. 2014. A next-generation smart contract and decentralized application platform. https://github.com/ethereum/wiki/wiki/White-Paper.
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## Appendix

### Parsec Protocol

The Parsec protocol introduces a layer for cryptocurrency transactions to take
place, without necessarily being registered into the global blockchain.

Figure 6: Customer Journey via Parsec Protocol

Parsec Protocol

    package org.parsec.protocol

      val customerPaymentJourney = .stripMargin
      val DEFAULT_CHANNEL = "default"
      val ALLOWED_CURRENCIES = Set("ETH", "BTC")

      /**
        * The seller needs to produce an INVOICE (through Parsec channel)
        *
        * @param invoiceAddress the sellect address for the currency to be transferred to
        * @param price          the amount of currency to exchange
        * @param currency       ETH or BTC supported
        * @param invoiceType    (optional) use to pass-in other metadata about the invoice
        */
      case class Invoice(invoiceAddress: String, price: Double,
      currency: String, invoiceType: String = "") {
        assert(ALLOWED_CURRENCIES.contains(currency), s"ParsecProtocol: Currency
        $currency is not an allowed currency. Use: " + ALLOWED_CURRENCIES.mkString(" or "))

        def produceInvoiceHash = "######-####-####"
      }

      /**
        * Here the Parsec Protocol lies here!
        *
        * We need to ensure that the order of Signed Invoices is guaranteed
        * No cluster of distributed systems can guarantee that
        (due to varying network latencies etc.)
        *
        */
      case class HashPointer(transactionID: String,
      // i.e. invoiceID
                             transactionHash: String)

      case class SignedInvoice(invoiceID: String = "auto-generate-random-UUID+channel",
      // import java.util.UUID
                               invoiceSignature: String,
                               supplierAddress: String, // no need to know Private Key
                               buyerPublicKey: String,
                               buyerAddress: String,
                               currency: String,
                               price: Double,
                               channel: String = DEFAULT_CHANNEL,
                               hashPointerToPrevious: HashPointer) {
        assert(ALLOWED_CURRENCIES.contains(currency), s"ParsecProtocol:
        Currency $currency is not an allowed currency. Use: " +
        ALLOWED_CURRENCIES.mkString(" or "))
        assert(isValidBuyerAddress)

        def isValidBuyerAddress: Boolean = {
          if (currency == "ETH")
            return buyerAddress equals ethereumAddressFromPublicKey(buyerPublicKey)
          else
            return buyerAddress equals bitcoinAddressFromPublicKey(buyerPublicKey)
        }
      }

    }