# Quantum field theoretical study of an effective spin model in coupled
optical cavity arrays

Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar,
Bangalore 5600 80, India.
E-Mail<EMAIL_ADDRESS>
Phone: 091 80 23612511/23619034, Fax: 091-80-2360-0228

###### Abstract

Atoms trapped in microcavities and interacting through the exchange of virtual
photons can model an anisotropic Heisenberg spin-1/2 lattice. We do the
quantum field theoretical study of such a system using the Abelian
bosonization method followed by the renormalization group analysis. We present
interesting physics due to the presence of exchange anisotropy. An infinite
order Kosterliz-Thouless-Berezinskii transition is replaced by second order XY
transition even an infinitesimal a small anisotropy in exchange coupling is
introduced. We predict a quantum phase transition between Mott insulating and
photonic superfluid phase due to detuning between the cavity and laser
frequency, a large detuning favours the photonic superfluid phase. We also do
the analysis of Jaynes and Cumming Hamiltonian to support results of quantum
field theoretical study.

Pacs: 42.50.Dv, 42.50.Pq, 03.67.Bg, 75.10.Jm
Keywords:Quantum Many Body Models, Polariton, Cavity QED and Spin Chain Model

##  I I. Introduction

The physics of strongly correlated system is interesting in its own right and
manifests in different branches of physics. Some of the important correlated
physics appears in natural oxide materials rao  and some of them are in
engineered materials, like the correlated physics in Josephson junction array
lik1 , Bose-Einstein condensation and optical lattice dens ; zoller .
Therefore one can raise the question what is the further source of correlated
physics in the state of enginnering ?. The recent experimental success in
engineering strong interaction between the photons and atoms in high quality
micro-cavities opens up the possibility to use light matter system as quantum
simulators for many body physics green ; hart1 ; hart2 ; ji ; byrn ; caru ;
bhas ; toma ; zhao ; pipp ; rossi ; horn ; blat .
Here we would like to discuss the basic physics of micro-optical cavity very
briefly. A micro-cavity can be created in a photonic band gap material by
producing a localised defect in the structure of the crystal, in such a way
that light of a particular frequency can not propage outside the defect area.
Large arrays of such micro-cavities have been produce. Photon hopping between
neighboring cavities has been observed in the microwave and optical domains.
Many body Hamiltonians can be created and probed in coupled cavity arrays.
There atoms are used for detection and also for the generation of interaction
between photons in the same cavity. As the distance between the adjacent
cavities is considerably larger than the optical wave length of the resonant
mode, individual cavities can be addressed. These artificial system could act
as a quantum simulator. In this optical cavities system we study the different
quantum phase of polariton ( a combined excitations of atom-photon
interactions. ) by using spin models that conserve the total number of
excitations. Best of our knowledge, at first we have done the explcit quantum
field theoretical calculations of this type of system.

At first we would like to discuss the generation of the spin model for such
type of systems. It has already been discussed in the literature but we
mention this process very briefly for the sake of completeness. In this
description we will follow the Ref. hart1  and Ref. hart2 .

Micro-cavities of the photonic crystal are coupled through the exchange of
photons. Each cavity consists of one atom with three levels in the energy
spectrum, two of them are long lived and represents two spin states of the
system and the other is excited. Externally applied laser and cavity modes
couple to each atom of the cavity. It may induce the Raman transition between
these two long lived levels. Under a suitable detuning between the laser and
the cavity modes virtual photons are created in the cavity which mediate an
interaction with another atom in a neighboring cavities. One can eliminated
the excited states of atomic level photon states by choosing the appropriate
detuning between the applid laser and cavity modes. Then one can achieve only
two states per atom in the long lived state and the system can be described by
a spin-1/2 Hamiltonian hart1 ; hart2 .

Fig. 1 shows the schematic phase diagram of our description to generate the xy
spin interaction of the system. The Hamiltonian of the system consists of
three parts:

$H~{}=~{}{H_{A}}~{}+~{}{H_{B}}~{}+~{}{H_{C}}$ (1)

Hamiltonians are following

${H_{A}}~{}=~{}\sum_{j=1}^{N}{{\omega}_{e}}|e_{j}><e_{j}|~{}+~{}{\omega}_{ab}|b_{j}><b_{j}|$
(2)

$j$ is the cavity index. ${\omega}_{ab}$ and ${\omega}_{e}$ is the energy of
the state $|b>$ and the excited state respectively. The energy level of state
$|a>$ is set zero. The following Hamiltonian describes photons in the cavity,

${H_{C}}~{}=~{}{{\omega}_{C}}\sum_{j=1}^{N}{{a_{j}}}^{\dagger}{a_{j}}~{}+~{}{J_{C}}\sum_{j=1}^{N}({{a_{j}}}^{\dagger}{a_{j+1}}+h.c),$
(3)

where ${a_{j}}^{\dagger}$ creates a photon in cavity $j$, ${\omega}_{C}$ is
the energy of photons and $J_{C}$ is the tunneling rate of photons between
neighboring cavities. Interaction between the atoms and photons and also by
the driving lasers are described

${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+{g_{a}}{a_{j}})|e_{j}><a_{j}|+h.c]+[a\leftrightarrow
b].$ (4)

Here ${g_{a}}$ and ${g_{b}}$ are the couplings of the cavity mode for the
transition from energy states $|a>$ and $|b>$ to the excited state.
${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of lasers with
frequencies ${\omega}_{a}$ and ${\omega}_{b}$ respectively.

They have derived an effective spin model by considering the following
physical processes: A virtual process regarding emission and absorption of
photons between the two stable states of neghiboring cavities. The resulting
effective Hamiltonian is

${H_{xy}}=\sum_{j=1}^{N}B{{\sigma}_{j}}^{z}~{}+~{}\sum_{j=1}^{N}({J_{1}}{{\sigma}_{j}}^{\dagger}{{\sigma}_{j+1}}^{-}~{}+~{}{J_{2}}{{\sigma}_{j}}^{-}{{\sigma}_{j+1}}^{-}+h.c)$
(5)

When $J_{2}$ is real then this Hamiltonian reduces to the XY model. Where
${{\sigma}_{j}}^{z}=|b_{j}><b_{j}|~{}-~{}|a_{j}><a_{j}|$,
${{\sigma}_{j}}^{+}=|b_{j}><a_{j}|$, ${{\sigma}_{j}}^{-}=|a_{j}><b_{j}|$

$H_{xy}~{}=~{}\sum_{i=1}^{N}B({{\sigma}_{i}}^{z}~{}+~{}{J_{x}}{{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}~{}+~{}{J_{y}}{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}).$
(6)

With ${J_{x}}=(J_{1}+J_{2})/2$ and ${J_{y}}=(J_{1}-J_{2})/2$.

Here we discuss very briefly about an effective
${{\sigma}_{i}}^{z}{{\sigma}_{i}}^{z}$ in such a system. Authors of Ref.hart1
; hart2  have proposed the same atomic level configuration but having only one
laser of frequency ${\omega}$ that mediates atom-atom coupling through virtual
photons. Another laser field with frequency $\nu$ is used to tune the
effective magnetic field. They described the one-dimensional case. In this
case the Hamiltonian ${H_{AC}}$ will change but the Hamiltonians $H_{A}$ and
$H_{C}$ will not.

${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+\frac{{\Lambda}_{a}}{2}e^{-i{{\nu}_{a}}t}{g_{a}}{a_{j}})|e_{j}><a_{j}|+h.c]+[a\leftrightarrow
b].$ (7)

Here, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the
driving laser with frequency ${\omega}$ on transition $|a>\rightarrow|e>$
$|b>\rightarrow|e>$, whereas ${\Lambda}_{a}$ and ${\Lambda}_{b}$ driving laser
with frequency ${\nu}$ on transition $|a>\rightarrow|e>$ $|b>\rightarrow|e>$.
They have eliminated adiabatically the excited atomic levels and photons by
considering the interaction picture with respect to $H_{0}=H_{A}~{}+~{}H_{C}$.
They have considered the detuning parameter in such a way that the Raman
transitions between two level supressed. They have also chosen the parameter
in such a way that the dominant two-photon processes are those that involve
one laser photon and one cavity photon each but the atom does no transition
between levels a and b. Whenever two atoms exchange a virtual photon then both
of them experience a Stark shift plays the role of an efective
${{\sigma}^{z}}{{\sigma}^{z}}$ interaction. Then the effective Hamiltonian
reduce to

${H_{zz}}~{}=~{}\sum_{j=1}^{N}(\tilde{B}{{\sigma}_{j}}^{z}~{}+~{}{J_{z}}{{\sigma}_{j}}^{z}{{\sigma}_{j+1}}^{z})$
(8)

Analytical expressions for $\tilde{B}$, $J_{1}$, $J_{2}$ and $J_{z}$ has given
in Ref. hart2 . These two parameters can be tuned independently by varying the
laser frequencies. They have obtained an effective model by combining
Hamiltonians $H_{xy}$ and $H_{zz}$ by using Suzuki-Trotter formalism. The
effective Hamiltonian simulated by this procedure is

$H_{spin}~{}=~{}\sum_{j=1}^{N}(B_{tot}{{\sigma}_{j}}^{z}~{}+~{}\sum_{{\alpha}=x,y,z}J_{\alpha}{{\sigma}_{j}}^{\alpha}{{\sigma}_{j+1}}^{\alpha})$
(9)

where $B_{tot}=B+\tilde{B}$. It has been shown in Ref. hart2  that $J_{y}$ is
less than than $J_{x}$. It is clear from analytical expressions for $J_{x}$
and $J_{y}$ that the magnitudes of ${J_{1}}$ and $J_{2}$ are different. This
result of numerical simulations trigger us to define a model, which has given
below to study the quantum phases of this system and also the transition among
them and at the same time this subject is in the state of art of engineering.

##  II II. Renormalization Group study of model Hamiltonian

We consider the anisotropic Heisenberg spin-1/2 Hamiltonian on a one
dimensional lattice. The XYZ Heisenberg Hamiltonians is defined as:

$\displaystyle H_{XYZ}~{}=~{}\sum_{n}~{}[$
$\displaystyle(1+a)~{}S_{n}^{x}S_{n+1}^{x}~{}+~{}(1-a)~{}S_{n}^{y}S_{n+1}^{y}$
(10)
$\displaystyle+~{}\Delta~{}S_{n}^{z}S_{n+1}^{z}~{}+~{}h~{}S_{n}^{z}~{}]~{},$

where $S_{n}^{\alpha}$ are the spin-1/2 operators. We assume that the $XY$
anisotropy $a$ and the $zz$ coupling $\Delta$ satisfy $-1\leq\Delta\leq 1$,
and $0<a\leq 1$ and magnetic field strength $h\geq 0$. The Hamiltonian
$H_{XYZ}$ is invariant under the transformation $S_{n}^{x}\rightarrow-
S_{n}^{x}$, $S_{n}^{y}\rightarrow-S_{n}^{y}$, $S_{n}^{z}\rightarrow-
S_{n}^{z}$, actually it is a $Z_{2}$ symmetry. For finite $h$, $Z_{2}$
symmetry is absent when $S_{n}^{z}\rightarrow-S_{n}^{z}$. Here $h\sim
B_{tot}$, ${\Delta=J_{z}}$, ${J_{1}=1}$ and ${J_{2}}=a$.
Spin operators can be recasted in terms of spinless fermions through Jordan-
Wigner transformation and then finally one can express the spinless fermions
in terms of bosonic fields gia . We recast the spinless fermions operators in
terms of field operators by this relation gia .

${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$
(11)

where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized fields
of right- and the left-moving fermions respectively. $k_{F}$ is Fermi wave
vector. Therefore, one can study the effect of gate voltage through arbitrary
$k_{F}$. We would like to express the fermionic fields in terms of bosonic
field by the relation

${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))},$
(12)

$r$ is denoting the chirality of the fermionic fields, right (1) or left
movers (-1). The operators $U_{r}$ commutes with the bosonic field. $U_{r}$ of
different species commute and $U_{r}$ of the same species anti-commute. $\phi$
field corresponds to the quantum fluctuations (bosonic) of spin and $\theta$
is the dual field of $\phi$. They are related by the relations
${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$.

Hamiltonian $H_{0}$ is non-interacting part of $H_{XYZ}$.

$H_{0}~{}=~{}\frac{v}{2}~{}\int~{}dx~{}[~{}(\partial_{x}\theta)^{2}~{}+~{}(\partial_{x}\phi)^{2}~{}]~{},$
(13)

where $v$ is the velocity of the low-energy excitations. It is one of the
Luttinger liquid parameters and the other is $K$, which is related to $\Delta$
by gia

$K~{}=~{}\frac{\pi}{\pi+2\sin^{-1}(\Delta)}~{}.$ (14)

$K$ takes the values 1 and 1/2 for $\Delta=0$ (free field), and $\Delta=1$
(isotropic anti-ferromagnet), respectively. The relation between $K$ and
$\Delta$ is not preserved under the renormalization, so this relation is only
correct for initial Hamiltonian. The analytical form of the spin operators in
terms of the bosonic fields are

$\displaystyle S_{n}^{x}~{}$ $\displaystyle=$
$\displaystyle~{}[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\cos({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{y}~{}$ $\displaystyle=$
$\displaystyle~{}-[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\sin({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{z}~{}$ $\displaystyle=$
$\displaystyle~{}{\sqrt{\frac{\pi}{K}}}~{}\partial_{x}\phi~{}+~{}(-1)^{n}c_{1}\cos(2{\sqrt{\pi
K}}\phi)~{},$ (15)

where $c_{i}$s are constants as given in Ref. zamo . The Hamiltonian $H_{XYZ}$
in terms of bosonic fields is the following,

$\displaystyle H_{XYZ}$ $\displaystyle=$ $\displaystyle
H_{0}+a\int\cos(2{\sqrt{\frac{\pi}{K}}}\theta(x))dx+\Delta\int\cos(4{\sqrt{\pi
K}}\phi(x))dx$ (16) $\displaystyle-h\int{{\partial}_{x}}{\phi(x)}dx$

One can get the $H_{XY}$ Hamiltonian by simply putting $\Delta=0$ in the above
Hamiltonian. In this derivation, different powers of coefficients $c_{i}$ have
been absorbed in the definition of $a,h$ and $\Delta$. The integration of the
oscillatory terms in the Hamiltonian yield negligible small ontributions, the
origin of the oscillatory terms are the spin operators. So it’s a resonably
good approximation to keep only the non-oscillatory terms in the Hamiltonian .
The Gaussian scaling dimension of these coupling terms, $a,\Delta$ are $1/K$,
$4K$ respectively. The third term ($\Delta$) of the Hamiltonian tends to order
the system into density wave phase, whereas the second term ($a$) of the
Hamiltonian favours the staggered order in the xy plane. Two sine-Gordon
couplings terms are from two dual fields. Therefore the model Hamiltonian
consists of two competing interactions between the ordered phase and the $XY$
order. This Hamiltonian contains two strongly relevant and mutually nonlocal
perturbation over the Gaussian (critical) theory. In such situation strong
coupling fixed point is usually determined by the most relevant perturbation
whose amplitude grows up according to its Gaussian scaling dimensions and it
is not much affected by the less relevant coupling terms. However this is not
the general rule, if the two operators exclude each other, i.e., if the field
configurations which minimize one perturbation term do not minimize the other.
In this case interplay between the two competing relevant operators can
produce a novel quantum phase transition through a critical point or a
critical line. Therefore we would like to study the RG euation to interpret
the quantum phases of the system.
We will now study how the parameters $a$, $\Delta$ and $K$ flow under RG. The
operators in Eq. 17 are related to each other through the operator product
expansion; the RG equations for their coefficients will therefore be coupled
to each other. Here we derive the RG equations by using perturbative
renormalization group approach scheme. We use operator product expansion to
derive these RG equations which is independent of boundary condition cardy .
RG equations themselves have been established in a perturbative expansion in
coupling constant ($g(l)$), they cease to be valid beyond the certain length
scale, where $g(l)\sim 1$ gia . The RG equations for the coefficients of
Hamiltonian $H_{XYZ}$ are

$\displaystyle\frac{da}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}(2-\frac{1}{K})a,$ $\displaystyle\frac{d{\Delta}}{dl}~{}$
$\displaystyle=$ $\displaystyle~{}(2-4K){\Delta}$
$\displaystyle\frac{dK}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}\frac{a^{2}}{4}~{}-~{}K^{2}{O}^{2}~{},$ (17)

These RG equations have trivial (${a^{*}}=0={{\Delta}^{*}}$) fixed points for
any arbitrary $K$. Apart from that these RG equations have also two non-
trivial fixed lines, $a=\Delta$ and $a=-{{\Delta}}$ for $K=1/2$. In our study,
there are critical surfaces on which the system flows onto the non-trivial
fixed lines (${a}=\pm{{\Delta}}$). A density wave states can be characterized
when $K\rightarrow 0$ or the staggered ordered when ($K\rightarrow\infty$).
Note that the transition occuring on them are second order. Infinitesimal
amount of anisotropy will change the situation drastically, a gapless phase in
absence of anisotropy will change to the gapped phase in presence of
anisotropy. Since this gapped excitation is not directly related to
magnetization, therefore it will not favour to create the plateau phase. When
the system is in the plateau phase the transition driven by the magnetic field
is always of $z=2$ ( z is the dynamical critical exponent) and thus the
plateau shows a square-root behaviour of magnetization. When $a$ is
increasing, a second order transition drives system to the in plane XY ordered
phase , whose exponents depend on initial couplings and hence are
nonuniversal. In absence of planar anisotropy the transition to plateau state
is Kosterliz- Thouless-Brezinskii (KTB). When the inplane anisotropy is
present, then $z=1$ phase . Please see Refs. subir ; sondhi  for detail
understanding of this subject. A magnetic field larger than the relevant gap
of the system drives the system to a gapless phase. This transition is from
commensurate phase to incommensurate phase transition.

We have seen the analytical expression of $B_{tot}$ from Ref. hart2  that the
total magnetic field increase for the larger values of detuning, therefore
larger detuning drives the system from gapped (Mott-insulating) state to
gapless superfluid state.

We now discuss, how the effective repulsion will decrease as we increase the
detuning between the atomic and laser frequency. It can be explained starting
from the Jaynes-Cummings Hamiltonian jay ; horo . Janes-Cummings Hamiltonian
for a single atom is

$H_{JC}~{}=~{}{{\omega}_{C}}{{a}^{\dagger}a}+{{\omega}_{0}}|e><e|+{\lambda}({a}^{\dagger}|g><e|~{}+~{}a|e><g|)$
(18)

${{\omega}_{C}}$ and ${{\omega}_{0}}$ are the frequency of the resonant mode
of the cavity and of the atomic transition, respectively. ${\lambda}$ is the
Jaynes-Cumming coupling between the cavity mode and the two level system.
${a}^{\dagger}$ (${a}$) is the creation (annihilation) operator of a photon
inside a cavity. $|g>$ and $|e>$ is respectively the ground state and excited
states of the two level system respectively. When we consider large values of
photon and atomic frequencies compare to atom-photon coupling $\lambda$, the
number of excitations is conserved for this Hamiltonian. Suppose we consider
fixed numbers of excitation, $n$. The energy eigenvalues for $n$ excitations
is

${E_{n}}^{\pm}=n{{\omega}_{C}}+\frac{\Delta}{2}\pm\sqrt{n{\lambda}^{2}~{}+~{}\frac{{\Delta}^{2}}{4}}$
(19)

Here $\Delta~{}=~{}({\omega}_{0}-{\omega}_{c})$ and $n\geq 1$. Now we consider
an array of cavities, the basic Hamiltonian for each cavity is the same as
that of Eq.19 . Here we consider the system with one excitation of energy
$E_{1}$ in each cavity and the lowest energy of two excitation in each cavity
is $E_{2}$. Therefore to create one additional excitation in each cavity
requires energy

${E_{2}-2E_{1}}=2\sqrt{{\lambda}^{2}~{}+~{}\frac{{\Delta}^{2}}{4}}-\sqrt{2{\lambda}^{2}~{}+~{}\frac{{\Delta}^{2}}{4}}-\frac{\Delta}{2}.$

Which one may consider as an effective on-site repulsion because it measures
the difference between the energy of two and single excitation (polariton in
each cavity). This effective repulsion decreases as we increase the detune
factor. Therefore we can conclude that for $\Delta=0$, double occupation never
occurs, indicating a Mott insulating behaviour. When $\Delta$ is much larger
than the coupling $\lambda$ then the occupation number larger than one occurs
as to be expected for a photonic superfluid regime. In our quantum field
theoretical calculations, we have also predicted that the large detuning
drives the system from gapped Mott insulating phase to the gapless superfluid
phase.

##  III Conclusions

At first we have done the quantum field theoretical analysis of an effective
spin model in coupled optical cavity arrays. We have predicted two quantum
phases, Mott insulator and photonic superfludity. Anisotropy in the exchange
interaction has also created a gapped phase. An infinite order KTB transition
has been replaced by the second order XY transition. The rigorous quantum
field theoretical derivation of this manuscript is absent in all previous
studies and also we provide physical explanation of the transition process
based on Jaynes-Cummings Hamiltonian.

Acknowledgments

The author would like to acknowledge The Center for Condensed Matter Theory of
the Physics Department of IISc for providing working space and also Dr. R. C.
Sarasij for reading the manuscript very critically.

Figure 1: Level structure, driving laserrs, and relevant couplings to the
cavity mode to generate effective ${{\sigma}^{x}}{{\sigma}^{x}}$ and
${{\sigma}^{y}}{{\sigma}^{y}}$ couplings for one atom. The cavity mode couples
with strength ${g_{a}}$ and ${g_{b}}$ to transition $|a>\leftrightarrow|e>$
and $|b>\leftrightarrow|e>$ respectively. One laser frequency ${\omega}_{a}$
couples to the transition $|a>\leftrightarrow|e>$ with Rabi frequency
${\Omega}_{a}$ and another with frequency ${\omega}_{b}$ to
$|b>\leftrightarrow|e>$ with ${\Omega}_{b}$. The dominant two-photon processes
are indicated in faint arrows. Reprinted with permission from American
Physical Society. Analytical expressions and physical meaning of different
symbols have given in Ref.hart2 .  Figure 2: Level structure, driving laserrs,
and relevant couplings to the cavity mode to generate effective
${{\sigma}^{z}}{{\sigma}^{z}}$ couplings for one atom. The cavity mode couples
with strength ${g_{a}}$ and ${g_{b}}$ to transition $|a>\leftrightarrow|e>$
and $|b>\leftrightarrow|e>$ respectively. Two lasers with frequencies $\omega$
and $\nu$ couple with Rabi frequencies ${\Omega}_{a}$, respectively,
${\Lambda}_{a}$ to transition $|a>\leftrightarrow|e>$ and ${\Omega}_{b}$,
respectively, ${\Lambda}_{b}$ to transition $|b>\leftrightarrow|e>$ . The
dominant two-photon processes are indicated in faint arrows. Reprinted with
permission from American Physical Society. Analytical expressions and physical
meaning of different symbols have given in Ref.hart2 .

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