# Bogoliubov theory of the laser linewidth and application to polariton
condensates

Ivan Amelio  Institute of Quantum Electronics ETH Zurich, CH-8093 Zurich,
Switzerland  Iacopo Carusotto  INO-CNR BEC Center and Dipartimento di Fisica,
Università di Trento, 38123 Povo, Italy

###### Abstract

For a generic semi-classical laser dynamics in the complex Ginzburg-Landau
form, we develop a Bogoliubov approach for the computation of the laser
emission linewidth. Our method provides a unifying perspective of the
treatments by Henry and Petermann: both broadening mechanisms are ascribed to
the non-orthogonality of the Bogoliubov modes, which live in a space with
doubled degrees of freedom. As an example of application, the method allows to
study the interplay of driven-dissipation, interactions and spatial
inhomogeneity typical of polariton condensates. The traditional theory of the
Henry and Petermann factors is found to fail dramatically in the presence of
sizable polariton-polariton interactions. In particular, also in a strong
confining potential, the intrinsically multi-mode nature of the density
fluctuations has to be considered in order to describe quantitatively phase
diffusion à la Henry.

##  I Introduction

The emission spectrum of a lasing system reflects the complex dynamics of
spatial and temporal fluctuations. For a single mode laser, Schawlow and
Townes predicted the linewidth of the Lorentzian spectral peak to scale as the
inverse of the emitted power Schawlow and Townes (1958). However, corrections
to this simple formula have been first proposed by Petermann Petermann (1979),
who noticed that deviations of the shape of the lasing mode from the eigenmode
of a lossless cold cavity determine a broadening of the laser linewidth Henry
(1986); Siegman (1989a, b); Hamel and Woerdman (1989, 1990); Grangier, Ph. and
Poizat, J.-Ph. (1998). Then, the crucial role of the fluctuations of the
intensity, hence of the nonlinear refractive index, was first considered by
Henry close to threshold Henry (1982), but only in the last decade
nonlinearities have been fully included in the treatment of the linewidth
Chong and Stone (2012); Pick  _et al._ (2015, 2019). As a further broadening
mechanism, we have recently shown Amelio and Carusotto (2020) that in low
dimensions the lack of long-range order can determine a dramatic broadening of
the linewidth, which will not scale with the inverse of the number of photons
in the system as predicted by Schawlow and Townes. In particular, in one
dimension the scaling is determined by universal Kardar-Parisi-Zhang exponents
Amelio  _et al._ (2021). Finally, for large semiconductor arrays and random
lasers it can occur that many modes lase together and the spectra become
complex Türeci  _et al._ (2006, 2008).

In this work we develop a theoretical framework for the computation of the
linewidth of a generic semi-classical laser dynamics via the solution of a
Bogoliubov eigenproblem. The formalism allows for an efficient and transparent
study of broadening effects of the Henry and Petermann type for a laser device
operating in the single mode regime. Multimode lasing, instead, is currently
not included in our treatment and KPZ effects Amelio and Carusotto (2020) are
intrinsically beyond any linear response approach. The Bogoliubov method has
been originally developed for the computation of the collective modes of
quantum fluids Pitaevskii and Stringari (2016) and can be applied to a
remarkably broad range of physical situations, including non-equilibrium
fluids of light Carusotto and Ciuti (2013), acoustic Hawking radiation Recati
_et al._ (2009) and (disordered) topological lasers Amelio and Carusotto
(2020); Zapletal  _et al._ (2020); Loirette-Pelous  _et al._ (2021). With
respect to the laser linewidth, the method has already been used in Zhang  _et
al._ (2018) for two coupled resonators close to the exceptional point and in
Amelio and Carusotto (2020) to compute the Petermann factor of a topological
laser. Here a more general presentation of the method is given and it is shown
to provide a very convenient treatment when spatial inhomogeneities and
refractive index nonlinearities interplay. One main insight is that the
Petermann and Henry factors can be explained in a unifying perspective as both
arising from the non-orthogonality of the Bogoliubov modes, which live in a
space with doubled degrees of freedom.

This technology is then applied to the fluids of light Carusotto and Ciuti
(2013), having particularly in mind polariton condensates Kasprzak  _et al._
(2006). While the Bogoliubov method is routinely applied to study the
fluctuation modes in these systems Wouters and Carusotto (2007) and the
linewidth is typically large enough to be easily measured Love  _et al._
(2008); Whittaker and Eastham (2009), the Petermann broadening of the
linewidth has never been considered in this context. In our numerical study we
consider different trapping regimes and pump spot sizes to elucidate the role
of gain guiding and interactions in shaping the lasing mode and determining
the linewidth. In particular we point out that, while Henry’s mechanism (i.e.
phase diffusion driven by intensity and refractive index fluctuations) catches
most of the linewidth broadening, it is not sufficient to consider the density
dynamics in the single mode approximation.

The paper begins with a review of the traditional theory of the linewidth,
then we introduce our method in full generality and finally we apply it to
polariton fluids.

##  II Linewidth theory

We consider a general complex Ginzburg–Landau equation (CGLE) of the form

$i\partial_{t}\psi(\vec{x},t)=\mathcal{W}[n]\psi(\vec{x},t)+\sqrt{D(\vec{x})}\xi$
(1)

where the operator $\mathcal{W}[n]$ can depend on position and contain
differential operators, but only depends on the field through its
instantaneous density $n=|\psi|^{2}$. Here we restrict to classical white
noise satisfying
$\langle\xi^{*}(\vec{x},t)\xi(\vec{x}^{\prime},t^{\prime})\rangle=2\delta(\vec{x}-\vec{x}^{\prime})\delta(t-t^{\prime})\
,\ \ \ \langle\xi(\vec{x},t)\xi(\vec{x}^{\prime},t^{\prime})\rangle=0.$

The steady-state lasing mode $\psi_{0}(\vec{x},t)$ satisfies
$i\partial_{t}\psi_{0}(\vec{x},t)=\mathcal{W}[n_{0}]\psi_{0}(\vec{x},t)=\omega_{0}\psi_{0}(\vec{x},t)$
with $\omega_{0}$ the real-valued laser frequency.

###  II.1 Henry’s linewidth

A naive derivation of the linewidth can be performed by allowing the intensity
and phase of the lasing mode to fluctuate and neglecting other spatial modes.
Two scalar equations for the phase and density fluctuations are obtained
integrating Eq. (1) over $\int d\vec{x}\ \psi^{*}_{0}(\vec{x})$, respectively

$\partial_{t}\phi=\bar{\mu}I+\sqrt{\frac{\bar{D}}{n_{tot}}}\xi_{\phi},\ \ \
\partial_{t}I=-\bar{\Gamma}I+2\sqrt{\frac{\bar{D}}{n_{tot}}}\xi_{I}.$ (2)

Here the bar stands for spatial averaging weighted with the steady-state
density $\bar{f}\equiv\frac{1}{n_{tot}}\int d\vec{x}\
n_{0}(\vec{x})f(\vec{x})$ and the (averaged) interaction strength and density
relaxation rate are

$\bar{\mu}=\overline{{\rm Re}\frac{\partial\mathcal{W}}{\partial n}n_{0}}\ ,\
\ \ \ \bar{\Gamma}=-2\overline{{\rm Im}\frac{\partial\mathcal{W}}{\partial
n}n_{0}}.$ (3)

Integrating Eqs. (2) yields the Henry linewidth Henry (1982)

$\gamma_{H}=\frac{\bar{D}}{n_{tot}}(1+\alpha^{2})$ (4)

where the first factor $\gamma_{ST}=\frac{\bar{D}}{n_{tot}}$ is the Schawlow-
Townes result and $\alpha=2\bar{\mu}/\bar{\Gamma}$ is the Henry factor
correction. The linewidth is the diffusion rate of the phase, related to the
field correlation function via
$g^{(1)}(t)\sim\exp\left[{-\frac{\gamma_{H}}{2}t}\right]$.

###  II.2 Petermann’s factor

The main problem with this approach is that projecting through $\int d\vec{x}\
\psi^{*}_{0}(\vec{x})$ is justified only for linear hermitian problems, where
all the modes are mutually orthogonal. As first noted by Petermann Petermann
(1979), the modes of a laser device are generally not orthogonal and
spontaneous emission into these other modes will eventually result in “excess”
noise in the laser mode. The total linewidth would be given by the Schawlow-
Townes result broadened by the product of the Henry and Petermann factor
$\mathcal{K}_{0}$

$\gamma_{H\times
P}=\mathcal{K}_{0}\times(1+\alpha^{2})\times\gamma_{ST}=\mathcal{K}_{0}\times\gamma_{H}.$
(5)

The traditional treatment of the Petermann broadening requires the strong
assumption of being close to the lasing threshold, where the nonlinear device
approaches a linear amplifier. This provides us the linear operator
$\mathcal{W}[0]$ of which we can consider the right eigenmodes
$\hat{\mathcal{W}}\varphi_{k}=\omega_{k}\varphi_{k}$ and their adjoints (or
left eigenmodes)
$\hat{\mathcal{W}}^{\dagger}\tilde{\varphi}_{k}=\omega_{k}^{*}\tilde{\varphi}_{k}$.

With the extra assumption that noise has uniform strength (which is a
reasonable hypothesis in a uniformly pumped Fabry-Perot laser with
inhomogeneous losses concentrated at the mirrors), one can perform the
projection via $\int dx\ \tilde{\varphi}_{G}^{*}$ and arrive at a concise
expression for $\mathcal{K}_{0}$:

$\mathcal{K}_{0}=\int d\vec{x}\ \tilde{\varphi}_{0}^{*}\tilde{\varphi}_{0}.$
(6)

Since we assumed the normalizations $\int d\vec{x}\
\tilde{\varphi}_{k}^{*}\varphi_{l}=\delta_{kl}$ and $\int d\vec{x}\
{\varphi}_{k}^{*}\varphi_{k}=1$, the adjoint modes are not simultaneously
normalizable and in general $\mathcal{K}_{0}\geq 1$. A Petermann factor of
about 1.5 was first observed by Hamel and Woerdman Hamel and Woerdman (1990)
in a semiconductor laser with large outcoupling.

In addition to the strong assumptions mentioned above, this treatment,
described by Hamel and Woerdman (1989); Grangier, Ph. and Poizat, J.-Ph.
(1998); Berry (2003), also requires to neglect the refractive index
nonlinearities. On the other hand, the treatment by Henry Henry (1986), which
includes also $\alpha\neq 0$, is still based on the Green function of the
cavity in the absence of the laser field and close to threshold, so that the
refractive index is not self-consistently reshaped by the presence of the
laser field.

In the last decade, these shortcomings have stimulated several works aimed at
considering nonlinearities Chong and Stone (2012), multimode lasing Pick  _et
al._ (2015) and quantum effects Pick  _et al._ (2019) by means of a scattering
matrix method. The review of this approach and the comparison with the
treatment sketched below goes beyond the scope of the present paper.

###  II.3 Bogoliubov method

Here we illustrate an alternative and very concise approach based on the
Bogoliubov method and anticipated in Amelio and Carusotto (2020). For a
lattice of $N$ sites labeled as $\vec{x}$ in arbitrary dimensionality (for a
continuous system one can consider a spatial grid, as typically done when
solving numerically the CGLE.), let us call $\mathcal{L}_{las}$ the $2N\times
2N$ Bogoliubov matrix of the linearized dynamics on top of the lasing steady-
state:

$i\partial_{t}\begin{pmatrix}\delta\psi\\\
\delta\psi^{*}\end{pmatrix}=\mathcal{L}\begin{pmatrix}\delta\psi\\\
\delta\psi^{*}\end{pmatrix}+\sqrt{D(\vec{x})}\begin{pmatrix}\xi\\\
-\xi^{*}\end{pmatrix},$ (7)

where, if for notational simplicity 111Since the phase freely diffuses one
cannot just integrate the noise from $t=-\infty$ but a given phase must be
arbitrarily chosen at an initial time. In practice, we can just take the
initial state unperturbed because density fluctuations will decay
exponentially fast; if one takes a perturbed initial state there will be some
extra transient in Eq. (8), which does not contribute to the linewidth. When
considering only the density fluctuations, like in Eq. (20) below, one can
integrate from $t=-\infty$ without need of an initial condition. we set the
unperturbed initial condition $\psi(\vec{x},0)=\psi_{0}(\vec{x})$, the field
fluctuation is defined from
$\psi(\vec{x},t)=(\psi_{0}(\vec{x})+\delta\psi(\vec{x},t))e^{-i\omega_{0}t}$.

Let $V=\\{V_{\vec{x}\sigma,a}\\}$ be the invertible matrix which diagonalizes
$\mathcal{L}_{las}$, where the pseudo-spin $\sigma=\uparrow,\downarrow$
indicates the particle and hole components of the Bogoliubov problem and $a$
labels the $2N$ eigenmodes. The Goldstone mode $V_{\vec{x}\sigma,G}$, that we
assume to be unique with all other excitations having a finite life-time, is
the eigenstate with zero eigenvalue. As usual, its spatial shape follows the
one of the lasing mode and we can set
$(V_{\vec{x}\uparrow,G},V_{\vec{x}\downarrow,G})=\frac{1}{\sqrt{2n_{tot}}}(\psi_{0}(\vec{x}),-\psi_{0}^{*}(\vec{x}))$
as a natural gauge choice. Notice that each Bogoliubov eigenmode, or each
column of $V$, is defined modulo a phase factor, hence the use of “gauge”
word.

The Bogoliubov problem (7) reduces to $2N$ independent equations upon
multiplication by $V^{-1}$; it is then straightforward to integrate the field

$\delta\psi(\vec{x},t)=\sum_{a}\int_{0}^{t}ds\
\pi_{a}(s)e^{-i\omega_{a}(t-s)},$ (8)

where $\omega_{a}$ are the complex Bogoliubov eigenvalues. The overlap
$\pi_{a}=-i\sum_{\vec{x}\sigma}V^{-1}_{a,\vec{x}\sigma}\sqrt{D(\vec{x})}\xi_{\vec{x}\sigma}$,
with $\xi_{\vec{x}\sigma}=(\xi_{\vec{x}},-\xi_{\vec{x}}^{*})$, represents the
projection of noise on a given mode. We also define $\pi_{ab}$ from
$\langle\pi_{a}(t)\pi_{b}(t^{\prime})\rangle=2\delta(t-t^{\prime})\sum_{\vec{x}}D(\vec{x})\left(V^{-1}_{a,\vec{x}\uparrow}V^{-1}_{b\vec{x}\downarrow}+V^{-1}_{a,\vec{x}\downarrow}V^{-1}_{b\vec{x}\uparrow}\right)\equiv\pi_{ab}\delta(t-t^{\prime})$.
One can switch to a density-phase formalism: the density and phase
contributions of the $a$-th mode are respectively

$\nu_{a}(\vec{x})=\psi_{0}^{*}(\vec{x})V_{\vec{x}\uparrow,a}+\psi_{0}(\vec{x})V_{\vec{x}\downarrow,a}$
(9)

and

$\phi_{a}(\vec{x})=\frac{i}{2n(\vec{x})}\left[\psi_{0}^{*}(\vec{x})V_{\vec{x}\uparrow,a}-\psi_{0}(\vec{x})V_{\vec{x}\downarrow,a}\right].$
(10)

The particle-hole symmetry of the Bogoliubov matrix Amelio  _et al._ (2020)
guarantees that $\nu_{a},\phi_{a}$ are real-valued. For the Goldstone mode
$a=G$ one has that $\nu_{G}=0$ and
$\phi_{G}(\vec{x})\equiv\phi_{G}=-i/\sqrt{{2n_{tot}}}$ is a constant,
consistently with the fact that the Goldstone mode corresponds to a global
phase rotation.

The crucial observation is that, since all modes $a\neq G$ decay exponentially
in time, in the large $t$ limit the phase-phase correlator gets a constant
contribution from the $a\neq G$ modes plus the linear in $t$ contribution

$\langle\delta\phi^{2}\rangle\sim\pi_{GG}\phi_{G}^{2}t$ (11)

coming from the Goldstone mode, hence

$\gamma_{Bogo}=\pi_{GG}\phi_{G}^{2}=-\frac{2}{n_{tot}}\sum_{\vec{x}}D_{\vec{x}}\
(V^{-1})_{G,\vec{x}\uparrow}(V^{-1})_{G,\vec{x}\downarrow}.$ (12)

Notice that the very last expression is not gauge invariant, but assumes the
Goldstone mode to be defined as
$(V_{\vec{x}\uparrow,G},V_{\vec{x}\downarrow,G})=\frac{1}{\sqrt{2n_{tot}}}(\psi_{0}(\vec{x}),-\psi_{0}^{*}(\vec{x}))$;
in particular, in this gauge the minus sign gets cancelled by the product
$(V^{-1})_{G,\vec{x}\uparrow}(V^{-1})_{G,\vec{x}\downarrow}$. Equations (10)
and (11) are instead in a gauge covariant and invariant form, respectively.

Notice that $\mathcal{L}$ is typically a very sparse matrix and
$(V^{-1})^{\dagger}$ are the eigenmodes of $\mathcal{L}^{\dagger}$. For very
large numerical grids, the Goldstone column, corresponding to the zero
eigenvalue, can be efficiently computed via the Lanczos algorithm. However,
the Lanczos finds the normalized column
$[(V^{-1})^{\dagger}]_{\vec{x}\sigma,G}$, while $(V^{-1})_{G,\vec{x}\sigma}$
is in general not normalized if $V_{\vec{x}\sigma,G}$ is. The correct
normalization, which is fundamental here, can be found by requiring
$\sum_{\vec{x},\sigma}(V^{-1})^{\dagger}_{\vec{x}\sigma,G}V^{*}_{\vec{x}\sigma,G}=1$
with $\sum_{\vec{x},\sigma}|V_{\vec{x}\sigma,G}|^{2}=1$ .

Figure 1:  Petermann-like broadening $\gamma_{Bogo}/\gamma_{ST}$ as a function
of the pump-spot width $\ell$ for the model of Eq. (15) with $g=0$. The full
circles correspond to $\gamma_{Bogo}/\gamma_{ST}$ computed according to Eqs.
(12) and (13), while the empty circles denote $\mathcal{K}_{0}$ obtained from
$\mathcal{W}[n_{0}]$ according to formula (6), which assumes uniform noise and
closeness to threshold. (left) The case without trapping $\Omega=0$ is shown
as a function of $\ell/\ell_{0}$ and for two different pump powers (normalized
to threshold). For small pump spots the strong gain guiding introduces a lot
of excess noise. Inset: the pumping threshold also increases for smaller
spots, since a larger outflow has to be compensated. (right) The harmonic trap
yields another length scale $\ell_{\Omega}$. For stronger trapping, the modes
of the system resemble the eigenstates of the oscillator Hamiltonian and
excess noise is reduced. For $\ell/\ell_{\Omega}\sim 3$ higher harmonics will
lase, leading to a multimode (MM) phase with no steady-state.

###  II.4 Discussion

Crucially, if the adjoint mode to the Goldstone is the Goldstone itself
$(V^{-1})^{*}_{G,\vec{x}\sigma}=V_{\vec{x}\sigma,G}$ (a sufficient condition
for this is unitarity $VV^{\dagger}=1$), the simple Schawlow-Townes expression
holds for the linewidthSchawlow and Townes (1958):

$\gamma_{ST}=\frac{\bar{D}}{n_{tot}}$ (13)

with $n_{tot}=\sum_{\vec{x}}n_{\vec{x}}$ and
$\bar{D}=\sum_{\vec{x}}D_{\vec{x}}n_{\vec{x}}/\sum_{\vec{x}}n_{\vec{x}}$.

In the case of a spatially uniform system lasing in the $\vec{k}=0$ mode with
amplitude $\sqrt{n_{0}}$, the different wavevectors decouple in the Bogoliubov
problem and the $2\times 2$ sector $\mathcal{L}_{las}(\vec{k}=0)$ is
diagonalized by the matrix

$V_{\sigma,p}(\vec{k}=0)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&\frac{1-i\alpha}{\sqrt{1+\alpha^{2}}}\\\
-1&\frac{1+i\alpha}{\sqrt{1+\alpha^{2}}}\\\ \end{pmatrix}$ (14)

(the first column $p=G$ here is the Goldstone mode and
$[V^{-1}(k=0)]_{G,\sigma}=\frac{1}{\sqrt{2}}(1+i\alpha,-1+i\alpha)$ ) from
which one recovers the Henry formula Eq. 4. In particular, notice that for
$\alpha=0$ the mode matrix $V$ is unitary: the Henry factor can then be
interpreted as the non-orthogonality of the modes of (14) with respect to the
particle-hole index (as opposed to the spatial or momentum index). The same
conclusions apply to a point-like laser, which physically means that the
system dynamics is essentially described by using a single mode of a lossless
cold cavity. Alternatively, one can reason in terms of many conservative modes
that are well separated in energy, while losses and pumping couple different
modes. If the coupling is negligible one can stick to a single mode
approximation. If also $\alpha=0$, the sector corresponding to the lasing mode
features $(V^{-1})^{*}_{G,\vec{x}\sigma}=V_{\vec{x}\sigma,G}$ and there is no
broadening.

In the opposite scenario, the refractive index nonlinearity ${\rm
Re}\frac{\partial\mathcal{W}}{\partial n}$ is negligible, but gain and losses
strongly couple different conservative modes of the idealized cavity. This is
the generic case for a system with localized gain or losses and closely spaced
conservative modes (i.e a large cavity or a weak trapping potential). In this
case, the broadening of the linewidth with respect to Eq. (13) will be
interpreted as a Petermann-like factor, since the non-unitarity of $V$ comes
from the coupling of different conservative modes. However, we stress here
that in general the two broadening mechanisms will not be separable. This will
be particularly true where the refractive index and gain profiles are modified
by the intense field with respect to the cold cavity, so that both the real
and imaginary part of $\mathcal{W}$ contribute nonlinearly to the shape of the
lasing mode.

As a final remark, it is worth reminding that the Bogoliubov theory presented
here is only valid in the high spatial coherence regime, where KPZ-like
effects and the broadening related to the lack of long-range order are
negligible Amelio and Carusotto (2020).

##  III Polariton condensates

As an illustrative example of the general theory, we now move to consider the
linewidth of a condensate of light Carusotto and Ciuti (2013), with special
focus on the case of incoherently driven polariton fluids Kasprzak  _et al._
(2006). For simplicity we will restrict our analysis to the so called
adiabatic scenario where the dynamics of carriers is traced out, so that one
is left with the following CGLE Wouters and Carusotto (2007):

$i\partial_{t}\psi=\left[-\frac{1}{2m}\nabla^{2}+gn+V(x)+\right.\\\
\left.+\frac{i}{2}\left(\frac{P(x)}{1+n/n_{S}}-\gamma(x)\right)\right]\psi+\sqrt{D(x)}\xi,$
(15)

which is linearized to the Bogoliubov matrix:

$\mathcal{L}=\begin{pmatrix}-\frac{1}{2m}\nabla^{2}+2gn_{0}+V-\omega_{0}+\frac{i}{2}\left(\frac{P}{1+n/n_{S}}-\gamma\right)-\frac{i}{2}\Gamma&g\psi_{0}^{2}-\frac{i}{2}\tilde{\Gamma}\\\
-g(\psi_{0}^{*})^{2}-\frac{i}{2}\tilde{\Gamma}^{*}&+\frac{1}{2m}\nabla^{2}-2gn_{0}-V+\omega_{0}+\frac{i}{2}\left(\frac{P}{1+n/n_{S}}-\gamma\right)-\frac{i}{2}\Gamma\end{pmatrix}$
(16)

Here $\Gamma(x)=\frac{n_{0}(x)/n_{S}}{(1+n_{0}(x)/n_{S})^{2}}P(x)$ gives the
local rate for density relaxation and depends on the density itself; we also
used
$\tilde{\Gamma}(x)=\frac{\psi_{0}^{2}(x)/n_{S}}{(1+n_{0}(x)/n_{S})^{2}}P(x)$.
In the following we will assume homogeneous losses $\gamma$ and a Gaussian
pump spot $P(x)=P\exp(-x^{2}/2\ell^{2})$ of width $\ell$. Noise will be taken
with a diffusion shape $D(x)\propto P(x)$, in analogy with spontaneous
emission. For the external potential we take a harmonic trap
$V(x)=\frac{m}{2}\Omega^{2}x^{2}$.

While here for simplicity we will report results in 1D, we expect that the
physics remains qualitatively the same in 2D.

###  III.1 Non-interacting untrapped condensate

As a first step, we will neglect the refractive index nonlinearity and set
$g=0$. In the absence of trapping ($\Omega=0$), there are two typical scales
that compete: the width of the pump $\ell$ and the wavelength
$\ell_{0}=1/\sqrt{m\gamma}$, corresponding to a kinetic energy of the order of
the dissipation rate. Lasing is possible when the fluid can be localized by
the pump spot, a situation we will refer to as gain guiding. Small
$\ell/\ell_{0}$ means that the pump is trying to excite lasing in a narrow
region: the large outflowing current makes the effective loss rate of the
lasing mode much greater than $\gamma$. Correspondingly, $P_{th}/\gamma\gg 1$,
as shown in the inset of Fig. 1.a. For $\ell\to\infty$ the power threshold
approaches $\gamma$, the result for a uniform system where the flow of the
fluid is negligible.

Moreover, generating a fluid in a region small compared with $\ell_{0}$ comes
with a price in terms of coherence. This is shown in Fig. 1.a, where the ratio
$\gamma_{Bogo}/\gamma_{ST}$ is plotted in solid dots as a function of
$\ell/\ell_{0}$ for $P/P_{th}=$1.05 (cyan) and 1.4 (magenta); as expected, it
diverges for small $\ell/\ell_{0}$ and goes to 1 for large spots. In this
setting, $\gamma_{Bogo}/\gamma_{ST}$ can be safely interpreted as a Petermann
factor. The empty circles refer to $\mathcal{K}_{0}$ computed from
$\hat{\mathcal{W}}[n_{0}]$ according to Eq. (6), which provide a good
approximation in this regime. As it is apparent from the plot, the results
depend weakly on $P/P_{th}$ (actually $P$ enters in $\bar{\Gamma}$ and
provides a third length scale $\ell_{P}\sim 1/\sqrt{m\bar{\Gamma}}$; zooming
in on the tails of Fig. 1 one could see that $\gamma_{Bogo}/\gamma_{ST}$ gets
very close to 1 for $\ell\gg\ell_{P}$ (not shown)).

###  III.2 Non-interacting trapped condensate

For $\Omega>0$, also the length scale $\ell_{\Omega}=1/\sqrt{m\Omega}$ has to
be taken into account, yielding the interesting behavior of
$\gamma_{Bogo}/\gamma_{ST}$ shown in Fig. 1.b. For
$\ell,\ell_{0}>\ell_{\Omega}$ trapping is very effective and the lasing mode
approaches the ground-state of the harmonic oscillator: this is the regime of
conservative cavity confinement and the Petermann-like factor is close to 1.
The pump term couples higher harmonic modes only perturbatively, so that at
zeroth order the modes of the systems are orthogonal. On the contrary, for
weak trapping (large $\ell_{\Omega}/\ell_{0}$) and narrow pump spots (small
$\ell/\ell_{\Omega}$ i.e. small $\Omega/\gamma$) we recover the regime of gain
guiding described above for $\Omega=0$ and the corresponding divergence of the
linewidth. We can think of this regime as the very localized gain deforming
the harmonic ground state; on the other hand, when $\ell/\ell_{\Omega}\sim
1.75$ the pump spot and the ground state of the harmonic trap fit each other
in size and $\gamma_{Bogo}/\gamma_{ST}$ is very close to 1. Anyway, even for
small $\ell/\ell_{\Omega}$ the conservative limit is eventually recovered for
$\Omega\gg P$.

An increase of the Petermann-like factor occurs approaching
$\ell/\ell_{\Omega}\sim 3$ and an instability develops for larger pump spots
(for which the data are truncated). Indeed, if the pump diameter $\ell$ is
larger than the harmonic oscillator wave-function, two or more conservative
modes can feel enough gain and be enough spatially separated to lase
simultaneously. As a result, there is no steady-state and the emission
spectrum consists of different peaks. Related physics was reported in 2D
polariton systems, where a steady-state can be reached in which the rotational
symmetry is spontaneously broken by the formation of an array of vortices
Keeling and Berloff (2008). The coherence properties of this multi-mode regime
go beyond the scope of the present work.

###  III.3 Interacting trapped condensate

When $g>0$ the Henry factor can give a very large contribution to the
linewidth. In Fig. 2 we plot $\gamma_{Bogo}/\gamma_{H}$ versus $\alpha$ for
different trapping strengths. Each data set is obtained by increasing the
refractive index nonlinearity $g$, which results in the Henry factor
$\alpha=2\bar{\mu}/2\bar{\Gamma}$, computed inserting into Eq. (3) the steady-
state profile found numerically for each $g$.

Trapping, gain and interactions contribute in a complex way to the shape of
the laser mode and, as a consequence, to the linewidth. The first
consideration is that the Henry prediction $\gamma_{H}$ is at order zero a
decent approximation to the linewidth (we recall that the linewidth broadening
goes like $1+\alpha^{2}$ and can be very large, so that for $\alpha\sim 3$ one
expects a broadening by a factor of $\sim 10$ and the error of the green and
blue data by a factor 2 is still better than using the vanilla Schawlow-Townes
linewidth). However, for a given $\alpha$, the ratio
$\gamma_{Bogo}/\gamma_{H}$ does not get monotonically closer to 1 when
increasing the trapping $\Omega$, as it would be expected in virtue of an
improved single-mode approximation. The issue with this poor performance of
the $\gamma_{H}$ approximation will be elucidated in the next subsection.

Before that, we discuss the behavior of the traditional Petermann factor
$\mathcal{K}_{0}$ from Eq. (6). Looking at the fast blowing up of the red
empty circles, it is apparent that the standard expression $\gamma_{H\times
P}$ in Eq. (5) completely misses the exact result, overestimating the
linewidth, more and more dramatically for larger $g$. Therefore, Eq. (6) seems
to be useless for the interacting system.

Figure 2:  Ratio $\gamma_{Bogo}/\gamma_{H}$ (solid line and full circles)
versus $\alpha$, which quantifies the relevance of refractive index
fluctuations and corresponds to increasing the nonlinearity $g$. Different
trapping frequencies $\Omega$ are reported; for larger $\Omega$, one can see
that the single mode approximation holds for small enough $\alpha$, so that
$\gamma_{Bogo}/\gamma_{H\times P}$ gets close to one, but this argument fails
at finite $\alpha$, signalling a limitation of the Henry’s approach. Empty
symbols and dashed lines correspond to $\mathcal{K}_{0}$ from the standard
treatment of Eq. (6): it is evident that the standard Petermann times Henry
factor treatment of Eq. (5) does not work at all in the presence of
interactions, calling for the full Bogoliubov treatment. Here we have fixed
$P=2\gamma$ and pump diameter $\ell/\ell_{0}\simeq 2.8$.

###  III.4 Improved Henry’s scheme

Figure 3:  Ratio $\gamma_{Bogo}/\gamma_{H,nn}$ for increasing interactions.
The regime is exactly the same of Fig. 2, but now $\gamma_{H,nn}$ accounts for
the density driven drift of the phase with the density fluctuations computed
beyond the single mode approximation. This shows that Henry’s intuition is
physically correct, but that density dynamics is microscopically complex.
Importantly, $\gamma_{Bogo}/\gamma_{H,nn}$ goes to 1 for increasing trapping.

As we show below, the quantitative disagreement of $\gamma_{H}$ and
$\gamma_{Bogo}$ highlighted in Fig. 2 should not undermine Henry’s intuition
that density fluctuations have the greatest impact on the drift of the phase.
The issue with Fig. 2 is that $\gamma_{H}$ in the Henry’s scheme is computed
in the single mode approximation, while density fluctuations are notoriously
short ranged and short lived Chiocchetta and Carusotto (2013). Here we show
that Henry’s mechanism accounts quantitatively for the exact linewidth
provided the phase mode is driven by the full density dynamics. This means
that the density-density correlator should be computed retaining all the
Bogoliubov modes.

For this purpose, we make use of the ansatz

$\psi(x,t)=\psi_{0}(x,t)\left[1+\frac{\delta
n(x,t)}{2n_{0}(x)}-i\phi(t)+...\right].$ (17)

Crucially only the global phase fluctuations are considered, while the full
spatial dependence of the density dynamics is taken into account. The spatial
uniformity of $\phi(t)$ is the only approximation we make in this subsection
and which cuts off Petermann-like correlations.

After integration by $\int dx\ \psi^{*}_{0}(x)$ the equation for the phase
becomes

$\partial_{t}\phi=g\overline{\delta
n}+\sqrt{\frac{\bar{D}}{n_{tot}}}\xi_{\phi},$ (18)

where we remind that according to the bar notation established above
$\overline{\delta n}$ is the spatially averaged density fluctuation with
weight $n_{0}(x)$. Once again we remark that this expression is not exact, but
some non-orthogonality between the modes has been neglected. However, it
allows for a more refined expression for the linewidth with respect to Eqs.
(2), since it doesn’t assume that density fluctuates in a perfectly correlated
way across the system. After integration, one gets at the leading order in $t$

$\langle(\phi(t)-\phi(0))^{2}\rangle\simeq\frac{\bar{D}}{n_{tot}}t+2g^{2}t\int_{0}^{\infty}d\tau\
\langle\overline{\delta n}(\tau)\ \overline{\delta n}(0)\rangle.$ (19)

For an inhomogeneous system the dynamics of $\overline{\delta n}$ is rather
complex, since it contains contributions from all the modes. Nonetheless, the
density-density correlator can be computed exactly using the full Bogoliubov
theory or numerically via relatively cheap simulations of the duration of a
few $\Gamma^{-1}$.

We now sketch the computation of $\langle\overline{\delta n}(\tau)\
\overline{\delta n}(0)\rangle$ using Bogoliubov’s method. Since the undamped
Goldstone mode does not contribute to the density fluctuations, one has

$\delta n(x,t)=\sum_{a\neq G}\int_{-\infty}^{t}ds\
\nu_{a}(x)\pi_{a}(s)e^{-i\omega_{a}(t-s)},$ (20)

where the integral can start from $-\infty$ since the initial condition for
$\delta n$ becomes irrelevant after a time of order $\Gamma^{-1}$. Upon
integration, the correlator reads

$\langle\overline{\delta n(t)}\ \overline{\delta
n(t^{\prime})}\rangle=-i\sum_{a,b\neq
G}\frac{\bar{\nu}_{a}\bar{\nu}_{b}\pi_{ab}}{\omega_{a}+\omega_{b}}e^{-i\omega_{a}|t-t^{\prime}|}.$
(21)

Finally, the linewidth reads

$\gamma_{H,nn}=\frac{\bar{D}}{n_{tot}}+g^{2}\sum_{a,b\neq
G}\frac{\bar{\nu}_{a}\bar{\nu}_{b}\pi_{ab}}{-\omega_{a}\omega_{b}}.$ (22)

In the case of a point-like or uniform laser there are only two $k=0$ modes,
denoted as $a=G,A$ ($A$ stands for “amplitude”), and one has
$\langle\overline{\delta n}(\tau)\ \overline{\delta
n}(0)\rangle=2\frac{n_{tot}\bar{D}}{\Gamma}e^{-\Gamma|\tau|},\pi_{AA}=2D(1+\alpha^{2}),\bar{\nu}_{A}=\sqrt{\frac{2n_{0}}{1+\alpha^{2}}}$
and $\omega_{A}=-i\Gamma$, which leads to the standard Henry linewidth.

The ratio $\gamma_{Bogo}/\gamma_{H,nn}$ is plotted in Fig. 3 for increasing
$\alpha$ for exactly the same regime of Fig. 2 (for comparison $\alpha$ is
also defined in the same way, even though it is a definition motivated by the
single-mode approximation). From the approximate flatness of the curves, it is
clear that for any $g$ the linewidth is well captured by the Henry mechanism
but for a factor, which, as it is more evident from the red points, is of the
order of $\gamma_{Bogo}/\gamma_{H,nn}(g=0)$. Therefore, one would be tempted
to interpret the discrepancy between $\gamma_{Bogo}$ and $\gamma_{H,nn}$ as a
Petermann-like effect. However, notice that $\gamma_{Bogo}/\gamma_{H,nn}$ can
be less than one (blue points), which is impossible in the standard theory of
the Petermann factor. This supports our claim that in general there is no
natural way of disentangling the non-orthogonality of the Bogoliubov modes due
to spatial and particle-hole indices.

We have then shown that density fluctuations, if properly calculated beyond
the single-mode approximation, account for most of the linewidth broadening.
Crucially, notice that for any $\alpha$ the ratio
$\gamma_{Bogo}/\gamma_{H,nn}$ tends systematically to 1 for stronger trapping,
in contrast to Fig. 2.

##  IV Conclusions

To summarize, we have presented a general linear response treatment of the
linewidth of laser devices operating in a single mode regime with good spatial
coherence. Our Bogoliubov scheme allows for a unified treatment of the
Petermann and of the Henry broadening mechanisms in terms of the non-
orthogonality of modes which live in a space with doubled degrees of freedom.

As a most relevant example, we have applied this technique to calculate the
linewidth broadening due to gain guiding and density fluctuations for a
polariton condensate described by a generalized Gross-Pitaevskii equations.
The complex interplay of the harmonic trapping, the pump spot size and the
interactions has been discussed in detail. In particular, we have shown that
one needs to include density-density correlations beyond the single mode
approximation in order to quantitatively account for the interaction-induced
Henry-like broadening.

Since the linewidth of a polariton condensate is a easily measurable quantity
Love  _et al._ (2008); Whittaker and Eastham (2009), the main difficulty to
experimentally probe the different broadening mechanisms appears to be the
calibration of the experiment; indeed, it is not straightforward to measure
the number of polaritons in the cavity and the noise strength, so any physical
conclusion should rely on ratios of linewidths in different regimes.

Finally, the Bogoliubov method presented here can be straightforwardly
generalized to deal with more realistic microscopic models, accounting for the
dynamics of the carriers by rate equations Wouters and Carusotto (2007) or by
the full Maxwell-Bloch equations Zhang  _et al._ (2018). In future works, it
would be also interesting to study quantum effects to the linewidth Pick  _et
al._ (2019) within a Wigner representation formalism Berg  _et al._ (2009).

## Acknowledgements

We are grateful to Stefan Rotter for a few tips on the Petermann factor. I. C.
acknowledges financial support from the H2020-FETFLAG-2018-2020 project
"PhoQuS" (n.820392), and from the Provincia Autonoma di Trento.

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# Statistical Representation of Spacetime

Hamidreza Simchi<EMAIL_ADDRESS>Department of Physics, Iran
University of Science and Technology, Narmak, Tehran 16844, Iran
Semiconductor Technoloy Center, P.O.Box 19575-199, Tehran, Iran

###### Abstract

It is assumed that the spacetime is composed by events and can be explained by
partially ordered set (causal set). The parent events born two kinds of
children. Some children have a causal relation with their parents and other
kinds have not. It is assumed that evolution of the population is only happen
by the causal children. The assumed population can be modeled by finite
(infinite) dimension Leslie matrix. In both finite and infinite cases, it is
shown that the stationary state of the population always exists and the matrix
has positive eigenvalues. By finding the relation between the statistical
information of the population and the stationary state, a probability matrix
and a Shannon-like entropy is defined. It is shown that the change in entropy
is always quantized and positive and in consequence, the world is inflating.
We show that the vacuum energy can be attributed to the necessary done work
for preserving the causal relation between the parents and the children
(cohesive energy). By assuming that the sum of cohesive energy and kinetic
energy of the denumerable causal spacetime is equal to the heat, which flows
across a causal horizon, we find the relation between energy-momentum tensor
and discrete Ricci tensor which can be called the Einstein state equation.
Finally, it is shown that the constant of proportionality $\eta$ between the
entropy and the area is proportional to $\frac{k_{B}}{l_{p}^{2}}$ at Planck
scale which is in good agreement with the Hawking’s result.

††preprint: APS/123-QED

##  I Introduction

After the discovery of quantum physics, it became possible to describe the
electronic and structural properties of materials based on their constituent
elements, namely atoms and electrons. Using the concepts related to atomic
orbitals and their hybridization, the cause of the stability of the material
structure was described [1], and based on the energy band theory, and many-
body theory, the electronic properties and the quantum transport properties of
the majority carriers were well described [2-5] so that the computational
results were consistent with what was seen in the laboratory. This process
continued with the discovery of more elementary particles, and scientists
tried to describe strong nuclear forces [6] and weak forces [7] based on the
properties of related elementary particles. Hence, it can be said that quantum
physics, many-body theory, and quantum field theory are attempts to express
and interpret the macroscopic properties of a physical system based on its
atomic and sub-atomic microscopic components. Of course, it should be noted
that many efforts were made to unifying the known forces in nature so that
finally, by presenting a standard model [8], these efforts were completed,
without considering the force of gravity.

In contrast, the theory of relativity is not an attempt to describe the
macroscopic properties of a system based on its microscopic properties. This
theory can be considered as a description of how energy and matter interact
with the gravitational field (spacetime field) [9-10]. It is based on how
spacetime field interacts with matter that the planets move in a certain
orbit, light is deflected when it passes near the sun, the existence of black
holes in the galaxy is predicted, and a scientific description is given for
some other cosmic phenomena. Of course, the efforts made to determine the
cause of black hole radiation [11] and the thermodynamic description of
Einstein’s relativistic equation [12], which is made by using a combination of
quantum physics and Einstein’s theory of relativity, must be taken into
account.

If we take quantum gravity theory as a theory for describing the observable
macroscopic phenomena based on the atomic and sub-atomic microscopic
components that make up the spacetime field, then when the problem scale
approaches to the Planck scale, the question arises as to what these
microscopic components are and what are their characteristics? It is well
known that one of the important features of worldline is its causal property.
This means that at any point of spacetime (present) one set of events can be
considered as past and another set as future event related to the present. Due
to the constant and maximum speed of light in the vacuum and this causal
relationship between the present, the future and the past, some events will
not have any physical connection with the present. In order to mathematically
represent this causal property of events, the partially ordered causal set,
which has been studied by mathematicians before [13], may be used.

Attempts to use partially ordered sets as the microscopic elements that make
up spacetime go back many years. By defining the null, parallel lines and
planes and proving numerous theorems involving them, Robb has described the
relativity using the discrete spacetime (i.e., casual structure) [14,15]. It
has been shown that the casual structure of a spacetime, together with a
conformal factor, determine the metric of a Lorentzian spacetime, uniquely
[16,17]. Therefore, one can recover the conformal metric by using the before
and after relations amongst all events [18]. Now, if one has a measure for the
conformal factor, he/she can recover the entire metric and spacetime [18]. Of
course, ,t Hooft [19] and Myrheim [20] have independently found the causal set
theory too. Extensive research is being done today on the theory of causal
sets and its application in the formulation of the theory of relativity at the
Planck scale [21]. In one group of this research, by focusing on the causal
events, an attempt is made to study the static and dynamic state
characteristics of the causal set [22, 23], while a bunch of research focuses
on the causal relationship between events [24]. Of course, the third group, by
assuming the irreversibility of time and explaining and using the principles
of energy and momentum conservation and the absence of red shift in the
momentum relation at Planck scale, have introduced energetic causal set theory
that includes both classical and quantum cases [25, 26]. Ref. 21 is a fine
review on the causal set theory and its outlook.

Now, the question that can be asked is whether spacetime can be considered as
a population consisting of discrete members (parents) between whom there is a
causal relationship (such as a partially ordered set) that has evolved with
the birth of new members (children) of this set and $i$-th step goes to
$(i+1)$-th step and define entropy for this set and study it, statistically?
In this article, we try to answer this question in the affirmative. We assume
that spacetime is composed by events which is shown by the partially ordered
set. At each $i$-th step, the ordered set is called the parent set, and at
each $(i+1)$-th step, the ordered set is called children set. Some children
have causal relation with their parents and some have not. It is assumed that
the evolution of the system is only done by causal children. By using finite
and infinite Leslie matrix, the system is modeled and it is shown that the
stationary state always exists. We show that, since for preserving the
causality relation between parents and children some works should be done, the
entropy of the system is quantized and increases by the birth of new children.
It means that we encounter the universe inflation. Finally, by using the
Jacobson’s procedure [12], we calculate the energy-momentum tensor and in
consequence the Einstein state equation including discrete Ricci tensor.

The structure of the article is as follows. The basic assumptions are
introduced in section II and the Leslie matrix model is introduced in section
III. The quantized entropy and Einstein state equation are studied in section
IV and V, respectively. The summary is provided in section VI.

##  II Assumptions

Let us to consider a causal set $C$ which is a locally finite, partially
ordered set. The order relation is shown by ‘$\prec$ ‘ and the set has the
below properties [13, 22]:

  1. 1.

For all $x\in C$ , $x\nless x$ (Irreflexive)

  2. 2.

For all $x,y,z\in C$ if $x\prec y$ and $y\prec z$ then $x\prec z$ (Transitive)

  3. 3.

For all $x,y\in C$ if $x\prec y$ and $y\prec x$ then $x=y$ (Antisymmetric)

  4. 4.

For all $x,y\in C$ we have $\left|\left(z\in C\middle|x\prec z\prec
y\right)\right|<N_{0}$ (Locally finite)

The causal set can be considered as age-structured populations if adding
elements to the causal set are considered as individuals born. At each step of
growth, two kinds of babies may be born. In the first kind, $m_{i}$ number of
individuals born have the order relation with their parents and in the second
kind, some individuals born have not. For the second kind, $b_{i}$ stands for
the proportion of individuals born. The individuals born with order relation
are placed on the causal worldline and the individuals born without order
relation are not [27]. It is assumed that the evolution of the population is
only done by causal children. As Fig.1 shows, one can find the number of
causal individuals born (non-causal individual born) by counting the red
(blue) cycles from the beginning up to the $i$-th step. All of the red cycles
may be considered as parents in the $i$-th step.

Figure 1:  (Color online) An example of a causal set as a population growth.
The events in red cycles are placed on the causal worldlines. The non-dashed
red color cycles show an example of the casual worldline. The blue cycles show
the non-causal events. One can find the number of causal individuals born
(non-causal individuals born) by counting the red (blue) cycles from the
beginning up to the i-th step.

##  III Matrix Representation

Let us to define the below Leslie matrix [28]

$M_{\text{ij}}=\left\\{\begin{matrix}m_{j}\ >0\ \ \ \ \ \ for\ i=1\\\
0<b_{i}\leq 1\ \ \ \ \ for\ i=j+1\\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Otherwise\\\ \end{matrix}\right.\ $ (1)

i.e.,

$M=\begin{pmatrix}\begin{matrix}m_{1}&m_{2}&m_{3}\\\ b_{1}&0&0\\\ 0&b_{2}&0\\\
\end{matrix}&\begin{matrix}m_{4}&m_{5}&\cdots\\\ 0&0&\cdots\\\ 0&0&\cdots\\\
\end{matrix}&\begin{matrix}\cdots&\cdots&m_{n}\\\ \cdots&\cdots&\cdots\\\
\cdots&\cdots&\cdots\\\ \end{matrix}\\\ \begin{matrix}0\ \ \ &0&\text{\ \ \ \
}b_{3}\\\ 0&0&\ \ 0\\\ 0&0&\ \ 0\\\ \end{matrix}&\begin{matrix}0&0&\cdots\\\
b_{4}&0&\cdots\\\ 0&b_{5}&\cdots\\\
\end{matrix}&\begin{matrix}\cdots&\cdots&\cdots\\\ \cdots&\cdots&\cdots\\\
\cdots&\cdots&\cdots\\\ \end{matrix}\\\ \begin{matrix}0\ \ &0&\ \ \ 0\\\
\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots\\\
\end{matrix}&\begin{matrix}0&0&b_{6}\\\ \vdots&\vdots&\vdots\\\
\vdots&\vdots&\vdots\\\ \end{matrix}&\begin{matrix}\cdots&\cdots&\cdots\\\
\vdots&\vdots&\vdots\\\ \vdots&b_{n-1}&0\\\ \end{matrix}\\\ \end{pmatrix}$ (2)

Generally, based on the Perron–Frobenius theorem [29, 30], The eigenvalues of
$M$ are real and it has eigenvalues with all positive elements such that
$M\vec{u}=\lambda\vec{u}$ and $\vec{v}M=\lambda\vec{v}$.

It has been shown that, if [30]

$l_{j}=\left\\{\begin{matrix}1\ \ \ \ \ \ \ \ \ \ \ \ \ \ for\ \ \ \ \ \ \
j=1\\\ \Pi_{r=1}^{j-1}\text{\ \ }b_{r}\ \ \ \ \ \ for\ j\geq 2\\\
\end{matrix}\right.\ $ (3)

Then

$u_{i}=\frac{l_{i}}{\lambda^{i}}\ \ \ \ ,\ \ \
v_{i}=\frac{\left(\sum_{j=i}^{n}{m_{j}u_{j}}\right)}{u_{i}}$ (4)

For example, if $M=\begin{bmatrix}m_{1}&m_{2}\\\ b_{1}&0\end{bmatrix}$ then
$u_{1}=1/\lambda_{i}$, $u_{2}=b_{1}/\lambda_{i}^{2}$, $v_{1}=\lambda_{i}$ and
$v_{2}=m_{2}$ (Appendix A).

Now, a question can be asked: what can be the physical interpretation of $M,$
$\vec{u}$ , and $\lambda_{i}$?

Fig.2 shows the new causal and non-causal children of $i$-th step when the
n-type causal parents present at $i$-th step. The parents belong to the own
$i$-th step and the all of the previous steps from the beginning. By
substituting Eq. (4) in the eigenvalue equation, one can find

$\begin{pmatrix}m_{1}&m_{2}&\begin{matrix}\cdots&\text{\ \ }m_{n}\\\
\end{matrix}\\\ b_{1}&0&\begin{matrix}0\ &\ \ 0\\\ \end{matrix}\\\
\begin{matrix}0\\\ 0\\\ \end{matrix}&\begin{matrix}\cdots\\\ \cdots\\\
\end{matrix}&\begin{matrix}\begin{matrix}0\ \ \ &0\\\ \end{matrix}\\\
\begin{matrix}b_{n-1}&0\\\ \end{matrix}\\\ \end{matrix}\\\
\end{pmatrix}\begin{pmatrix}1/\lambda\\\ b_{1}/\lambda^{2}\\\
\begin{matrix}\vdots\\\ b_{1}\cdots b_{n-1}/\lambda^{n}\\\ \end{matrix}\\\
\end{pmatrix}=\lambda\begin{pmatrix}1/\lambda\\\ b_{1}/\lambda^{2}\\\
\begin{matrix}\vdots\\\ b_{1}\cdots b_{n-1}/\lambda^{n}\\\ \end{matrix}\\\
\end{pmatrix}$ (5)

or

$\begin{pmatrix}\frac{m_{1}}{\lambda}+\frac{m_{2}b_{1}}{\lambda^{2}}+\cdots+\frac{m_{n}b_{1}b_{2}\cdots
b_{n-1}}{\lambda^{n}}\\\ b_{1}/\lambda\\\ \begin{matrix}\vdots\\\ b_{1}\cdots
b_{n-1}/\lambda^{n-1}\\\ \end{matrix}\\\ \end{pmatrix}=\begin{pmatrix}1\\\
b_{1}/\lambda\\\ \begin{matrix}\vdots\\\ b_{1}\cdots b_{n-1}/\lambda^{n-1}\\\
\end{matrix}\\\ \end{pmatrix}$ (6)

Therefore, he/she can write the below eigenvalue equation

$\begin{pmatrix}\frac{m_{1}}{\lambda}&\frac{m_{2}b_{1}}{\lambda^{2}}&\text{\ \
\ }\begin{matrix}\cdots&\frac{m_{n}b_{1}b_{2}\cdots b_{n-1}}{\lambda^{n}}\\\
\end{matrix}\\\ 1&0&\begin{matrix}\cdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &0\\\
\end{matrix}\\\ \begin{matrix}0\\\ 0\\\ \end{matrix}&\begin{matrix}\cdots\\\
\cdots\\\ \end{matrix}&\begin{matrix}\begin{matrix}\cdots\\\ 1\\\
\end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }&\begin{matrix}0\\\ 0\\\
\end{matrix}\\\ \end{matrix}\\\ \end{pmatrix}\begin{pmatrix}1\\\ 1\\\
\begin{matrix}\vdots\\\ 1\\\ \end{matrix}\\\ \end{pmatrix}=\begin{pmatrix}1\\\
1\\\ \begin{matrix}\vdots\\\ 1\\\ \end{matrix}\\\ \end{pmatrix}$ (7)

It means that he/she can define the probability element and the probability
matrix as below [30]

$p_{i}=m_{i}l_{i}/\lambda^{i}$ (8) $P_{\text{ij}}=\left\\{\begin{matrix}p_{i}\
\ for\ i=1\\\ 1\ \ \ for\ i=J=1\\\ 0\ \ \ \ \ \ \ \ \ \ Otherwise\\\
\end{matrix}\right.\ $ (9)

Figure 2:  (Color online) The relation between i-th and (i+1)-th step.

Therefore, $m_{i}u_{i}=m_{i}l_{i}/\lambda^{i}$ can be considered as the born
probability of $m_{i}$ new causal children related to the parent $u_{i}$ and
in consequence $m_{1}u_{1}+m_{2}u_{2}+\cdots+m_{n}u_{n}$ is the total born
probability of new casual children at ($i$ +1)-step.

By attention to the above descriptions, the matrix $M$ is the evolution matrix
such that $\vec{u}^{\prime}(i+1)=M\vec{u}(i)$ where
$\vec{u}(\vec{u}^{\prime})$ is the distribution of parents from the beginning
up to the $i$ ($i$ +1)-th step. It should be noted that if the term
$M_{\text{nn}}=b_{n}\neq 0$ it means that the n-th parent born non-causal
child with the probability $b_{n}$ [28]. The obtained general results satisfy
again and one should only find the new eigenvalues and eigenvectors. For
providing the physical interpretation of $\lambda$ we define the entropy in
the next section.

##  IV Entropy

Entropy is the total number of ways to rearrange the internal microstates of a
system while keeping its external macrostates unaltered [31]. Here, we
encounter the ensemble of parents with the boron probability $p_{\text{ij}}$
as internal microstates and the stationary population as the macrostate. For
studying the effect of the variation of microstates on the macrostate, we
should define the entropy.

By using the Eq. (4) we can define the new population vector $\vec{Z}$ as
below [30]

$Z_{i}=\frac{1}{\sum_{i=1}^{n}{ip_{i}}}u_{i}v_{i}=\frac{\sum_{i=1}^{n}p_{i}}{\sum_{i=1}^{n}{ip_{i}}}=\frac{1}{\sum_{i=1}^{n}{ip_{i}}}$
(10)

Therefore, $\vec{Z}P=\vec{Z}$. Then, $\vec{Z}$ stands for the stationary
distribution and in consequence we define the population entropy for the
stationary distribution as below [30]

$S=-\frac{\sum_{i=1}^{n}{p_{i}\ log(p_{i})}}{\sum_{i=1}^{n}{ip_{i}}}$ (11)

where, $p_{i}=m_{i}l_{i}/\lambda^{i}$. It should be noted that the numerator
of the Eq. (11) is similar to the Shannon entropy [32]. For providing the
physical interpretation of $\lambda$ we write

$log\lambda=\frac{\sum_{i=1}^{n}{p_{i}log(\frac{\lambda^{i}p_{i}}{p_{i}})}}{\sum_{i=1}^{n}{ip_{i}}}=\frac{1}{\sum_{i=1}^{n}{ip_{i}}}\sum_{i=1}^{n}{p_{i}(log\lambda^{i}p_{i}-logp_{i})}$
(12)

or

$log\lambda=\frac{\sum_{i=1}^{n}{p_{i}\log\lambda^{i}p_{i}}}{\sum_{i=1}^{n}{ip_{i}}}-\frac{\sum_{i=1}^{n}{p_{i}\log\left(p_{i}\right)}}{\sum_{i=1}^{n}{ip_{i}}}=\Phi+S$
(13)

or

$\delta=\Phi+S=\Phi+\frac{S^{{}^{\prime}}}{T}=\Phi+T^{{}^{\prime}}S^{{}^{\prime}}$
(14)

where
$\Phi=\frac{\sum_{i=1}^{n}{p_{i}\log\lambda^{i}p_{i}}}{\sum_{i=1}^{n}{ip_{i}}}$,
$S^{{}^{\prime}}=\sum_{i=1}^{n}{p_{i}\log\left(p_{i}\right)}$,
$T^{{}^{\prime}}=\left(T=\sum_{i=1}^{n}{ip_{i}}\right)^{-1}$, and
$\delta=log\lambda$. From thermodynamic, we know, $F=U-TS$ where $F$ and $U$
are Helmholtz free energy and internal energy, respectively. Also, $T$ and $S$
are absolute temperature and entropy, respectively. Since,
$\delta=\Phi+T^{{}^{\prime}}S^{{}^{\prime}}$ then [30]

$0=\mathrm{\Delta}\Phi+T^{{}^{\prime}}\mathrm{\Delta}S^{{}^{\prime}}$ (15)

or

$\frac{1}{T^{{}^{\prime}}}=T=\frac{\mathrm{\Delta}S^{{}^{\prime}}}{-\mathrm{\Delta}\Phi}\rightarrow\mathrm{\Delta}S^{{}^{\prime}}=-T\mathrm{\Delta}\Phi$
(16)

In thermodynamic $\frac{1}{T}=\frac{\mathrm{\Delta}S}{\mathrm{\Delta}U}$ and
in consequence the temperature of any object can be described by the amount of
heat that must be added to it to increase its entropy by one unit. Similarly,
by attention to the Eq. (16), $T$ can be described by the amount of
$-\mathrm{\Delta}\Phi$ that must be added to the population to increase its
entropy by one unit. By comparison between Eq. (14) and the equation $F=U-TS$,
one can conclude the comparison table 1.

Table 1: Comparsion between statistical representation and thermodynamic Statiatical Representation | Thermodynamic
---|---
$\delta$ | Helmholtz Free energy
$T$ | Inverse of absolute temperature
$-T^{{}^{\prime}}S^{{}^{\prime}}$ | Entropy
$\Delta\Phi$ | Internal (mean) energy

Now, let us to assume that the causal relation between children and parents is
energetic relation. That is, after establishing a causal relationship between
a parent and a child if we assume that the amount of $-\mathrm{\Delta}\Phi$
increases by $V_{i}$ then
$\mathrm{\Delta}S=\left(\sum_{i=1}^{n}{ip_{i}}\right)^{-1}\sum_{i}{V_{i}>0}$.
Therefore, the entropy increases by growing the population and in consequence
the population will be more stable than before. Since, the causal events are
denumerable then $V_{i}$ is discrete and in consequence $\mathrm{\Delta}S$ is
too i.e., $\mathrm{\Delta}S$ is quantized. It can be assumed that, the
cohesive energy of the denumerable causal spacetime is the energy of vacuum
which is appeared in quantum filed theory as $\sum_{i}{\hslash\omega/2}$. In
consequence, if it is assumed that $V_{i}=\hslash\omega/2$, since
$\sum_{i}{V_{i}=}m\hslash\omega/2$. It means that the entropy is quantized and
equal to

$S=+\frac{m\hbar}{2}T=+\frac{m\hbar}{2}\left(\sum_{i=1}^{n}{ip_{i}}\right)^{-1}$
(17)

where, $\left(T=\sum_{i=1}^{n}{ip_{i}}\right)^{-1}=cte$. For comparison, we
can take into account the formation of a crystal. By bringing an element of
infinity closer to one element in the crystal, the two elements are bounded to
each other and the internal (cohesive) energy will be increased and stored in
the crystal [33]. It means that the cohesive energy is always negative [33].
Similarly, by adding a new child to the population and establishing the causal
relation the amount of $-\mathrm{\Delta}\Phi$ increases by the quanta of
energy and stored in the population. Therefore, it is expected that the energy
$-\mathrm{\Delta}\Phi=\sum_{i}{V_{i}(=}m\hslash\omega/2)$ will be equal to the
energy of vacuum. Of course, the causal spacetime is a disordered system and
Bloch’s theorem does not satisfy here. Increment of entropy, due to the
increasing the number of parents and causal children, cases the expansion of
the universe (universe inflation). Why we see universe inflation because its
entropy increases due to the increment of the spacetime events.

##  V Einstein State equation

It has been shown that, when $M$ is an infinite-dimensional population matrix,
under special conditions, a stationary distribution exists [34] (Appendix B).
Also, when, the index $i,j\rightarrow\infty$, we encounter an infinite
denumerable Leslie matrix, $M$. It has been shown that for
$i,j\rightarrow\infty$, an essentially unique stationary distribution exists
[35] and there is a generating function $\mathbf{G}(z)$ for calculating
$M^{n}$, where $z$ is a complex indeterminate. Now, if $u_{0}$ is the first
parent (initial condition) then $u_{n}=M^{n}\ u_{0}$[35] (Appendix B).
Therefore, the stationary distribution exists when $i,j\rightarrow\infty$ [34,
35].

Also, in spacetime dynamic, it can be assumed that the heat is energy which
flows across a causal horizon and the heat flux is given by [12]

$\delta Q=\int{T_{\text{ab}}\chi^{a}d\Sigma^{b}}$ (18)

where, all integrands are defined in Appendix C. It should be noted that the
past horizon of a local Rindler horizon (system) is instantaneously stationary
(in local equilibrium) at spacetime point $p$ [12]. We assume that each unit
area of spacetime is composed by the denumerable causal events. Each causal
event has kinetic energy $K_{i}$ and is bounded to another causal event by the
causal correspondence energy $V_{i}$. It can be assumed that the heat which
flows across a causal horizon is equal to
$\sum_{i}\left(V_{i}+K_{i}\right)\delta{A}$ where $\delta{A}$ is defined in
Appendix C. The assumed phenomenon is similar to the relaxed state of a
crystal. In the crystal, the elements vibrate around their equilibrium with
preserving the bounding between elements (preserving the cohesive energy).
Here, although the unit area of the causal spacetime is a disorder system and
the Bloch’s theorem does not satisfy we assume that

$\delta Q=-\sum_{i}\left(V_{i}+K_{i}\right)\delta{A}$ (19)

Since, the area $\delta{A}$ is composed by the denumerable causal events by
using Eq. (C2) and Eq. (19), we can write

$-\kappa\int\lambda\ R_{\text{ab}}k^{a}k^{b}{d\lambda
dA}=\\{-\sum_{i}{\left(V_{i}+K_{i}\right)\\}}\delta{A}$ (20)

where, all integrands, $\delta A$ and $\kappa$ are defined in Appendix C.

Using Eq. (C5)

$\kappa\int{\text{\ T}_{\text{ab}}k^{a}k^{b}{d\lambda
dA}}=-\\{\sum_{i}{\left(V_{i}+K_{i}\right)\\}}\int{R_{\text{ab}}k^{a}k^{b}{d\lambda
dA}}$ (21)

Therefore,

$\kappa
T_{\text{ab}}k^{a}k^{b}=-\\{\sum_{i}{\left(V_{i}+K_{i}\right)\\}}R_{\text{ab}}k^{a}k^{b}$
(22)

which is valid for all null $k^{a}$. It implies that [12]

$T_{\text{ab}}=-\frac{1}{\kappa}\sum_{i}\left(V_{i}+K_{i}\right)\\{R_{\text{ab}}+\left(-\frac{R}{2}+\Lambda\right)\\}g_{\text{ab}}$
(23)

where, $R_{\text{ab}}$ and $R$ are Ricci tensor and Ricci scalar, respectively
and $\Lambda$ is some constant. Therefore, we can imagine the unit area is
composed by the energetic spacetime events with discrete geometry. As Eq. (23)
shows, the energy-momentum tensor can be calculated by the product of the
energy stored within the discrete geometry and the discrete curvature of the
unit area. It means that energy-momentum and the stationary state as two
macroscopic parameters can be studied by the behavior of the energetic
spacetime events which are the microscopic constitutes. Eq. (23) can be called
Einstein state equation. Th Eq. (23) can be written in matrix form as follows

$T=Trace\left(\Xi\right)\left(R+\Upsilon g\right)$ (24)

where, $\Upsilon=-\frac{R}{2}+\Lambda$ and

$\Xi=-\frac{1}{\kappa}\begin{pmatrix}V_{1}+K_{1}&0&\cdots\\\
0&V_{2}+K_{2}&0\\\ \vdots&\vdots&\ddots\\\ \end{pmatrix}$ (25)

Each $V_{i}+K_{i}$ is attributed to a causal relation between two events,
after coarse-graining. Of course, if

$\sum_{i}\left(V_{i}+K_{i}\right)=-\frac{\hbar\kappa\eta}{2\pi}$ (26)

where, the minus sign stands for showing the cohesive energy behavior of the
causal relation between the events of the denumerable causal spacetime, then

$\left(\frac{2\pi}{\hbar\kappa\eta}\right)T_{\text{ab}}=\\{R_{\text{ab}}+\left(-\frac{R}{2}+\Lambda\right)\\}g_{\text{ab}}$
(27)

which is the Einstein state equation [12]. Using Eq. (26), it is possible to
obtain a limit that the discrete state (Eq. (23)) becomes a continuous state
(Eq. (27)).

Since, $k_{B}T$ has energy dimension, therefore
$k_{B}(\frac{\hbar\kappa}{2\pi})$ has energy dimension. But,
$\sum_{i}\left(V_{i}+K_{i}\right)$ is energy per unit area therefore the
coefficient $\eta/k_{B}$ has the inverse of the dimension of area and
$k_{B}\left(\frac{\hbar\kappa}{2\pi}\right)\left(\frac{\eta}{k_{B}}\right)$
has energy per area dimension. Now, if
$\frac{\eta}{k_{B}}\sim\frac{1}{l_{p}^{2}}$ where $l_{p}=$
$\sqrt{G\hslash/c^{3}}$ is the Planck length then
$\eta\sim\frac{k_{B}}{l_{p}^{2}}$. The result is in good agreement with the
Hawking’s result which is $\eta=\frac{k_{B}}{4l_{p}^{2}}$ [36].

Also, since the spacetime area $\delta A$ is composed by the causal events, it
may be possible one calculates the discrete Ricci curvature in terms of the
graph which is made by the causal events when the background-independent
coarse grain technique is applied [37, 38] and then solve the matrix Eq. (24).
It can be the subject of the future researches.

##  VI Summary

It was assumed that spacetime is composed by events (called parents). The
parents born children which some children have causal relation with their
parents and some have not. It was assumed that the only causal children take
part in the evolution of the system in each step of growth. The system has
been modeled by the Leslie matrix with finite (infinite) dimensions. In both
finite and infinite dimension, it has been shown that the stationary
population of causal events exist and for the population the relation $\Delta
S=-\Delta Q$ is satisfied where $S$ is the entropy of system and $Q$ is its
internal energy. For preserving the casual relation between parents and
children, some works should be done which is stored in the system as internal
energy $\Delta Q<0$. Since, there are denumerable casual relations, by
assuming that the stored energy is equal to $\hslash\omega/2$ for each causal
relation, it has been shown that the energy of vacuum is equal to
$m\hslash\omega/2$ where $m$ is the number of casual parents which took part
in the evolution of the system and in consequence the entropy is quantized.
Also, we assumed that $\delta A$ area of spacetime is composed by the
disordered causal events which have the causal bounding energy $V_{i}$ and the
kinetic energy $K_{i}$. By considering the past horizon of a local Rindler
horizon which is instantaneously stationary (in local equilibrium) at
spacetime point $p$, the relation between energy-momentum tensor
$T_{\text{ab}}$ and Ricci tensor has been found. The relation can be called
the Einstein state equation. Using the discrete curvature theory and the
coarse grain technique, one may solve the discrete state Einstein equation for
specific shape of denumerable causal spacetime (further future research
subjects). Finally, it was shown that the coefficient between entropy and area
i.e., $\eta\sim\frac{k_{B}}{l_{p}^{2}}$ which is in good agreement with the
Hawking’s result.

## Appendix A

For example, if $M=\begin{bmatrix}m_{1}&m_{2}\\\ b_{1}&0\end{bmatrix}$ it can
be easily shown that, the eingenvalues are

$E=\frac{m_{1}}{2}\pm\sqrt{\frac{m_{1}^{2}}{4}+m_{2}b_{1}}:=\lambda_{1},\lambda_{2}$
(28)

and if $\vec{u}=\begin{pmatrix}u_{1}\\\ u_{2}\\\ \end{pmatrix}$ is eigenvector
then

$m_{1}u_{1}+m_{2}u_{2}=\lambda_{i}u_{1}$ (29) $b_{1}u_{1}=\lambda_{i}u_{2}$
(30)

Therefore

$\vec{u}=\begin{pmatrix}u_{1}\\\ u_{2}\\\
\end{pmatrix}=\frac{1}{\sqrt{\lambda_{i}^{2}+b_{1}^{2}}}\begin{pmatrix}\lambda_{i}\\\
b_{1}\\\
\end{pmatrix}=\frac{1}{\lambda_{i}^{2}}\frac{1}{\sqrt{\lambda_{i}^{2}+b_{1}^{2}}}\begin{pmatrix}1/\lambda_{i}\\\
b_{1}/\lambda_{i}^{2}\\\ \end{pmatrix}$ (31)

and

$\begin{pmatrix}m_{1}&m_{2}\\\ b_{1}&0\\\
\end{pmatrix}\begin{pmatrix}1/\lambda_{i}\\\ b_{1}/\lambda_{i}^{2}\\\
\end{pmatrix}=\begin{pmatrix}\left(m_{1}\lambda_{i}+b_{1}m_{2}\right)/\lambda_{i}^{2}\\\
b_{1}/\lambda_{i}\\\ \end{pmatrix}=\begin{pmatrix}1\\\ b_{1}/\lambda_{i}\\\
\end{pmatrix}=\lambda_{i}\begin{pmatrix}1/\lambda_{i}\\\
b_{1}/\lambda_{i}^{2}\\\ \end{pmatrix}$ (32)

However, if $\vec{v}M=\lambda\vec{v}$ and
$\vec{v}=\begin{pmatrix}v_{1}&v_{2}\\\ \end{pmatrix}$ then

$m_{1}v_{1}+b_{1}v_{2}=\lambda v_{1}$ (33) $m_{2}v_{1}=\lambda v_{2}$ (34)

Therefore

$\vec{v}=\begin{pmatrix}v_{1}&v_{2}\\\
\end{pmatrix}=\frac{1}{\sqrt{\lambda_{i}^{2}+m_{2}^{2}}}\begin{pmatrix}\lambda_{i}&m_{2}\\\
\end{pmatrix}$ (35)

and

$\begin{pmatrix}\lambda_{i}&m_{2}\\\
\end{pmatrix}\begin{pmatrix}m_{1}&m_{2}\\\ b_{1}&0\\\
\end{pmatrix}=\begin{pmatrix}\lambda_{i}m_{1}+m_{2}b_{1}&\lambda_{i}m_{2}\\\
\end{pmatrix}=\begin{pmatrix}\lambda_{i}^{2}&\lambda_{i}m_{2}\\\
\end{pmatrix}=\lambda_{i}\begin{pmatrix}\lambda_{i}&m_{2}\\\ \end{pmatrix}$
(36)

Now, if $M^{*}=\begin{pmatrix}0&-m_{2}\\\ -b_{1}&m_{1}\\\ \end{pmatrix}$ is
the adjoint of $M$ then

$\begin{pmatrix}0&-m_{2}\\\ -b_{1}&m_{1}\\\
\end{pmatrix}\begin{pmatrix}-m_{2}\\\ \lambda_{i}\\\
\end{pmatrix}=\begin{pmatrix}-m_{2}\lambda_{i}\\\
b_{1}m_{2}+m_{1}\lambda_{i}\\\
\end{pmatrix}=\lambda_{i}\begin{pmatrix}-m_{2}\\\ \lambda_{i}\\\
\end{pmatrix}$ (37)

Therefore, ${\vec{v}}^{*}=\begin{pmatrix}-m_{2}\\\ \lambda_{i}\\\
\end{pmatrix}$ is the eigenvector of $M^{*}$.

## Appendix B

It has been shown that [34]

Theorem#1: Let $M$ be an infinite-dimensional population matrix. If (i)
$m_{i}>\ 0$ for infinitely many $i$ and $m_{i}$, bounded, (ii) the greatest
common divisor of the subscripts $i$ of the positive $m_{i}$ is unity, and
(iii) $\text{Limit}_{i\rightarrow\infty}b_{i}=0$, then the population matrix
$M$ has essentially unique stationary distribution $\overline{x}=(x_{1},\
x_{2},\cdots)$, $x_{i}>0$.

And also [34]

Theorem#2: If (i) $m_{i}>\ 0$ for infinitely many n, and $m_{i}$ bounded; (ii)
the matrix $M$ defines a compact operator $T$ on $l_{2}$, that is the Hilbert
space of vectors $\overline{x}=(x_{1},\ x_{2},\cdots)$ for which
$\sum{\left|x_{i}\right|^{2}<\infty}$, then the equation
$M\overline{x}=\lambda\overline{x}$ has a solution $\lambda>0$ and
$\overline{x}=(x_{1},\ x_{2},\cdots)$, $x_{i}>0$.

Therefore, for $i\rightarrow\infty$ an essentially unique stationary
distribution exists [34].

However, we can write the entries of an infinite denumerable Leslie matrix as
[35]

$M_{\text{ij}}=\delta_{1j}m_{j}+\delta_{i,j+1}b_{i}$ (38)

Therefore, the corresponding discrete dynamical equation is

$u_{n}=M\ u_{n-1}$ (39)

Now, if $u_{0}$ is the first parent (initial condition) then

$u_{n}=M^{n}\ u_{0}$ (40)

Lemma: It has been shown that, there is the generating matrix $G(z)$ such that
[35]

$G\left(z\right)=\sum_{n\geq 0}{M^{n}\ z^{n}}$

where, $z$ is a complex indeterminate and the entries of $G$ is

$G_{\text{ij}}=C_{j}^{i-1}z^{i-j}+\frac{C_{1}^{i-1}\sum_{n\geq
0}{m_{j+n-1}C_{j}^{j+n-1}z^{n+i}}}{\mathrm{\Delta}(z)}$ (41)

Here,

$C_{k_{i}}^{k_{f}}=\left\\{\begin{matrix}\Pi_{{k=k}_{i}}^{k_{f}}b_{k}\text{\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for\ }k_{f}>k_{i}\\\ 1\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ for\ \ \ \ \ k_{f}=k_{i}-1\\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Otherwise\\\ \end{matrix}\right.\ $ (42)

which is similar to Eq. (2) and

$\mathrm{\Delta}\left(z\right)=1-\sum_{n\geq 0}{m_{n}C_{1}^{n}z^{n+1}}$ (43)

It should be noted that, the real root of $\mathrm{\Delta}\left(z\right)\ $is
associated with the solution of Euler–Lotka equation which is [35]

$1-\sum_{n\geq 0}{m_{n}C_{1}^{n}\left(\frac{1}{\rho}\right)^{n+1}=0}$ (44)

where, $\rho$ is the leading eigenvalue ($\rho>0)$.

## Appendix C

Let us to consider a black hole event horizon. The system is the degrees of
freedom beyond the horizon and the outside world is separated from the system
by a causality barrier. Such a system is not in equilibrium because the
horizon is expanding, contracting, or shearing. A local Rindler horizon of a
small spacelike 2-surface element$P$ whose past directed null normal
congruence to one side (which we call the inside) and has vanishing expansion
and shear at a first order neighborhood of each spacetime point$P$ is
considered [12]. The part of spacetime beyond the Rindler horizon that is
instantaneously stationary (in local equilibrium) at $P$ is considered as
system. The heat flux to the past is given by [12]

$\delta Q=\int{T_{\text{ab}}\chi^{a}d\Sigma^{b}}$ (45)

where, $T_{\text{ab}}$ is the matter energy-momentum tensor and $\chi^{a}$ is
the killing field generating boost orthogonal to $P$ and vanishing at $P$. The
integral is over a pencil of generators of the inside past horizon of $P$. Of
course, it is assumed that the temperature of the system is equal to the Unruh
temperature which is observed by observer hovering just inside the horizon. It
can be shown that [12]

$\delta Q=-\kappa\int{\lambda T_{\text{ab}}k^{a}k^{b}{d\lambda dA}}$ (46)

where, $k^{a}$ is the tangent vector to the horizon generators for an affine
parameter $\lambda$ that vanishes at $P$ and is negative to the past of $P$,
$\chi^{a}=-\kappa\lambda k^{a}$ and $d\Sigma^{a}=k^{a}{d\lambda dA}$ , where
$dA$ is the area element on a cross section of the horizon. Here, $\kappa$ is
the acceleration of the Killing orbit on which the norm of $\chi^{a}$ is unity
and it is assumed that the speed of light is equal to unity. However,
according to the Unruh effect, the Minkowski vacuum state of quantum fields is
a thermal state with respect to the boost Hamiltonian at temperature
$T=\hslash\kappa/2\pi$ and it can be shown that [12]

$\delta Q=\left(\frac{\hbar\kappa}{2\pi}\right)\eta\int\lambda\
R_{\text{ab}}k^{a}k^{b}{\lambda dA}$ (47)

Therefore, for all null $k^{a}$ [12]

$\frac{2\pi}{\hbar\eta}T_{\text{ab}}=R_{\text{ab}}+(-\frac{R}{2}+\Lambda)g_{\text{ab}}$
(48)

where, $R_{\text{ab}}$ and $R$ are Ricci tensor and Ricci scalar,
respectively. $\Lambda$ is some constant. Here, it is assumed that [12]

$\delta A=-\int{\lambda R_{\text{ab}}k^{a}k^{b}{\lambda dA}}$ (49)

which is the area variation of a cross section of a pencil of generators of
the inside past horizon of $P$.

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