††thanks: Deceased, June 5 2020.

# Utilizing single-mode squeezing and non-Gaussianity metrics to investigate
bipartite and tripartite entanglement in three-mode ponderomotive
optomechanics simulations

Kahlil Y. Dixon1<EMAIL_ADDRESS>Lior Cohen2 Narayan Bhusal1 Jesse Frank1
Jonathan P. Dowling1 Thomas Corbitt1 1Department of Physics & Astronomy,
Louisiana State University, LA 70803, USA  2Department of Electrical, Computer
& Energy Engineering, University of Colorado Boulder, CO 80309, USA

###### Abstract

Quantum entanglement is a crucial resource for a wide variety of quantum
technologies. However, the current state-of-art methods to generate quantum
entanglement in optomechanical systems are not as efficient as all-optical
methods utilizing nonlinear crystals. This article proposes a new scheme in
which two single-mode squeezed light fields are injected into an
optomechanical cavity. We demonstrate through our numerical simulations that
the quantum entanglement can be substantially enhanced with the careful
selection of squeezing strength and squeezing angle of the two quadrature
squeezed light fields. Our results represent a significant improvement in
output bipartite photon-photon entanglement over the previously demonstrated
schemes using two coherent light fields as inputs. These simulations predict a
maximum increase in Gaussian entanglement by a factor of about 6, as well as
increases in the quantum noise and non-Gaussianity of the output light. This
non-Gaussianity is attributed to tripartite entanglement between the two
optical fields and the optomechanical oscillator (OMO). At particular
squeezing angles, the Gaussianity can be increased, thus introducing a method
of optically controlling the intracavity entanglement. These mechanics can
benefit various optical quantum technologies utilizing optomechanical
entanglement and continuous variable quantum optics.

††preprint: APS/123-QED

##  I Introduction

Quantum entanglement is one of the most important resources for a variety of
quantum technologies such as quantum metrology [1, 2], quantum communication
[3], quantum sensing [4], quantum simulation [5], and quantum imaging [6].
Producing entanglement via exploitation of the three-wave mixing process in
optomechanical cavities (OMC) is an attractive and innovative process for
generating entangled photons [7, 8]. However, the current performances of
optomechanical cavities to produce entanglement are not as robust as other
methods utilizing nonlinear crystals [9].

Our previous study has suggested that ponderomotive (optomechanical)
entanglement is highly dependent on input quantum noise [10]. Conveniently,
squeezed light—a non-classical state of light widely used in quantum optical
applications—has suppressed/enhanced quadrature noise that can be tailored
based on the squeezing strength and squeezing angle [11, 12, 13]. In order to
further investigate the relationship between quantum noise and the strength of
pondermotive entanglement, we inject such a light field, squeezed coherent
light. The primary motivation of this project is to study the effect of the
input quantum noise on the nonlinear dynamics in the optomechanical cavity
(OMC). We expected that, under certain conditions, the output bipartite
optical entanglement should also be enhanced by the input squeezing [14].
Furthermore, there are now several publications discussing non-Gaussian output
from OMCs, with only Gaussian inputs [15, 16, 17, 18]. This phenomenon is only
reported in systems that exhibit the full $\chi^{(3)}$ nonlinear interaction
or in linearized systems that exhibit strong quantum coupling [16, 15]. This
is not the case for these simulations. Thus, we utilize these Gaussianity
metrics to examine higher-order statistics that present in the two-mode
statistics of tripartite entangled states [19, 20]. The results of the
simulation (at least) strongly imply that the high-order correlations appear
from the quadruple homodyne measurement of two (of three) modes of a
tripartite entangled system.

We report (“linearized”) simulations of an optomechanical cavity in the
unresolved sideband regime that exhibits intracavity quantum noise dominance
(though in the weak quantum coupling regime) [14, 21, 22]. Moreover, the
injection of single-mode squeezed light enhances both optical bipartite
entanglement output from optomechanical cavity and tripartite entanglement of
the intracavity quantum states [8]. These simulations assume sideband
squeezing of the input coherent light into an optomechanical cavity with a
small movable end mirror optomechanical oscillator ( rest mass of $\approx
50$ng). These simulations are performed in a parameter regime that yields
bipartite optical ponderomotive entanglement without input squeezing (the
intracavity quantum noise dominance regime) (squeezing strength $r$ equal to
zero) [23, 10, 24].

Since our simulations demonstrate non-Gaussian continuous-variable output,
popular methods of entanglement quantification are obscured, specifically the
logarithmic negativity and Duan’s measure of separability [25, 26]. After
considering limited resources, these devices may be competitive to
conventional entanglement methods in certain conditions, especially those that
can exploit the non-Gaussian continuous-variable quantum states.

Figure 1: Schematic diagram of the proposed experimental setup. The two
frequency-locked pump lasers and nonlinear crystals produce two single-mode
squeezed light fields. The two quadrature squeezed light fields are injected
into the optomechanical cavity. Balanced-homodyne measurements are proposed to
quantify the output fields’ entanglement from the optomechanical cavity. The
two input fields are detuned from the same cavity resonance (carrier detuning
coefficient 0.3 and -1.5 for the subcarrier).

##  II Ponderomotive quantum coupling

Optomechanics is critical to a plethora of precision devices and instruments
[27]. Optomechanics (and Cavity Optomechanics) applications are a go-to
solution for classical optics-based technologies and an emerging technology
for Photonics and continuum Quantum Optics applications [28, 29].
Optomechanical devices are optical devices that contain a mechanical
oscillator coupled to the propagating optical modes. Optomechanics even
appears as an attractive method for cutting edge quantum computing devices
such as quantum memory and quantum teleportation for quantum computing IO
device development [30]. The utility of optomechanics will further expand into
Quantum Information Science technologies as a means of producing quantum
states and quantum measurements in various applications.

The simulations all consider the optical apparatus depicted in figure 1. Two
coherent states are squeezed by arbitrary black-box squeezers that impart the
squeezing parameters, $\xi_{1}$ and $\xi_{2}$ (where
$\xi_{j}=r_{j}e^{i\theta_{j}}$), on their respective coherent state. For the
purpose of this report, consider these squeezers to have easily tunable
squeezing strength ($r_{1}$ and $r_{2}$) and squeezing angle ($\theta_{1}$ and
$\theta_{2}$); at times swap out the squeezing strength $r_{j}$ to a tunable
nonlinearity coefficient, $\mu_{1}$ and $\mu_{2}$ (where
$r_{j}=\sqrt{P_{j}}\mu_{j}$), to better consider management (maximization) of
quantum resources. The optomechanical cavity is a cantilever shaped movable
$\mu$mirror [31, 23]. Which has higher-order vibrational modes that affect
output entanglement [10, 24]. Though there is some discussion of sideband
frequency dependence in the appendix, the sideband frequency dependence in
these simulations follows the same behavior as the unsqueezed case discussed
in a previous publication.

First, consider two unsqueezed coherent states input into the cavity that may
yield bipartite entangled optical mixed quantum states. The Hamiltonian for
dual coherent fields incident onto an optomechanical cavity with a $\mu$mirror
oscillator (movable endmirror with small mass) has the following optical-
mechanical interaction term:

$\displaystyle\
H_{int}=\hbar(g_{1}\hat{\mathbf{n}}_{1}+g_{2}\hat{\mathbf{n}}_{2})\sum_{k}(-1)^{k}\frac{(\hat{\mathbf{X}}_{\mu}/L-1)^{k}}{k!}~{},$
(1)

where $g$ is some optomechanical coupling constant, $\hat{\mathbf{X}}_{\mu}$
is the position quadrature operator corresponding to the $\mu$mirror,
$\hat{\mathbf{n}}_{j}=\hat{\mathbf{a}}^{\dagger}_{j}\hat{\mathbf{a}}_{j}$ is
the photon number of the $j^{th}$ intracavity field, and L is the cavity
length when input quantum noise is zero [32].

####  II.0.1 Linearized Hamiltonian

The linearization provides a more intuitive Hamiltonian and is valid in the
simulation’s regime. The goal is to write the interaction Hamiltonian using
the fluctuations in the optical cavity field around a steady-state average
coherent amplitude, $\alpha_{j}$. For a cavity driven by coherent fields
$\ket{\alpha_{j}}$ this involves the substitution
$\hat{\mathbf{a}}_{j}\rightarrow\alpha_{j}+\delta\hat{\mathbf{a}}_{j}$ into
the Hamiltonian.

$\displaystyle
H_{int}\approx(g^{\prime}_{1}\hat{\mathbf{n}}_{1}+g^{\prime}_{2}\hat{\mathbf{n}}_{2})\hat{\mathbf{X}}_{\mu}{}$
(2) $\displaystyle
H_{int}^{lin}=\sum_{j}g^{\prime}_{j}(\delta\hat{\mathbf{a}}^{\dagger}_{j}+\alpha_{j}^{*})(\delta\hat{\mathbf{a}}_{j}+\alpha_{j})\hat{\mathbf{X}}_{\mu}{}$
$\displaystyle
H_{int}^{lin}=\sum_{j}g^{\prime}_{j}(\delta\hat{\mathbf{a}}^{\dagger}_{j}\delta\hat{\mathbf{a}}_{j}+|\alpha_{j}|^{2}+\alpha_{j}\delta\hat{\mathbf{a}}^{\dagger}_{j}+\alpha^{*}\delta\hat{\mathbf{a}}_{j})\hat{\mathbf{X}}_{\mu}{}$

Take the new rest position of the mechanical oscillator to be displaced by the
average radiation pressure force,
$g^{\prime}_{j}|\alpha_{j}|^{2}\langle{\hat{\mathbf{X}}_{\mu}}\rangle~{}$.
Then,

$H_{int}^{lin}=\sum_{j}g^{\prime}_{j}(\delta\hat{\mathbf{a}}^{\dagger}_{j}\delta\hat{\mathbf{a}}_{j}+\alpha_{j}\delta\hat{\mathbf{a}}^{\dagger}_{j}+\alpha_{j}^{*}\delta\hat{\mathbf{a}}_{j})\hat{\mathbf{X}}_{\mu}~{}.$
(3)

In cases of weak quantum coupling, the term
$g^{\prime}_{j}\delta\hat{\mathbf{a}}^{\dagger}_{j}\delta\hat{\mathbf{a}}_{j}$
is negligible since the other terms have factors $\alpha_{j}$. Thus the
Hamiltonian simplifies to,

$\displaystyle
H_{int}^{lin}=\sum_{j}g^{\prime}_{j}(\alpha_{j}\delta\hat{\mathbf{a}}^{\dagger}_{j}+\alpha_{j}^{*}\delta\hat{\mathbf{a}}_{j})\hat{\mathbf{X}}_{\mu}{}$
(4) $\displaystyle
H_{int}^{lin}=\sum_{j}g^{\prime}_{j}(\alpha_{j}\delta\hat{\mathbf{a}}^{\dagger}_{j}+\alpha_{j}^{*}\delta\hat{\mathbf{a}}_{j})(\hat{\mathbf{b}}^{\dagger}+\hat{\mathbf{b}})~{},$

where $x_{zpf}g^{\prime}_{j}\rightarrow g^{\prime}_{j}$. In our simulations
the two input fields are blue and red-detuned from the cavity resonance. This
means that the dominate terms in the above Hamiltonian become the following:

$g^{\prime}_{1}\alpha_{1}\delta\hat{\mathbf{a}}^{\dagger}_{1}\hat{\mathbf{b}}^{\dagger}+g^{\prime}_{2}\alpha_{2}\delta\hat{\mathbf{a}}^{\dagger}_{2}\hat{\mathbf{b}}+g^{\prime}_{1}\alpha_{1}^{*}\delta\hat{\mathbf{a}}_{1}\hat{\mathbf{b}}+g^{\prime}_{2}\alpha_{2}^{*}\delta\hat{\mathbf{a}}_{2}\hat{\mathbf{b}}^{\dagger}~{}.$
(5)

These competing photon-phonon processes (for example the input relation for
the carrier field,
$g^{\prime}_{1}\alpha_{1}\delta\hat{\mathbf{a}}^{\dagger}_{1}\hat{\mathbf{b}}^{\dagger}$,
competes with the input relation for the subcarrier field
$g^{\prime}_{2}\alpha_{2}\delta\hat{\mathbf{a}}^{\dagger}_{2}\hat{\mathbf{b}}$
and likewise for the outputs) will cause additional fluctuations in the mean
cavity coherent amplitudes, $\alpha_{j}$; though not in this linearized
simulation, previous work has shown that a single cavity field has an
intricate dependence on the change in cavity linewidth [33]. However, in this
simulation, the cavity photon fluctuations can be seen in the sideband photon
numbers ($\delta\hat{\mathbf{a}}_{j}^{\dagger}\delta\hat{\mathbf{a}}_{j}$),
though not significant enough to move the system into the quantum coupling
regime.

The optomechanical coupling modulation could also be due to two-tone
optomechanical instabilities that occur within particular choices of carrier
and subcarrier field detunings in the unresolved sideband regime [34, 35, 36].

##  III Squeezed input boosts entanglement

Following the Quantum Langevin method, the Langevin equation for squeezed
light injection is very similar to the unsqueezed OM entanglement calculations
in previous publications [37, 38]. Since the coupling matrix, $\mathbf{K}$,
does not change for lossless squeezing, we only consider alterations to the
input noise spectra $\langle\mathbf{G}\rangle$ [39, 18]. Recall that the
quantum Langevin equations calculate (in Heisenberg’s picture of quantum
mechanics) the time derivative of the intracavity quadrature operators. Let
$\mathbf{u}_{c}$ be the vector of intracavity mode operators, and the vector
$\mathbf{u}_{N}$ be the vector of input mode operators. Then, the intracavity
modes change in time according to the relation:

$\displaystyle\dot{\mathbf{u}}_{c}=\mathbf{K}\mathbf{u}_{c}+\mathbf{u}_{N}~{}.$
(6)

After Fourier transforming to the sideband frequency $\Omega$, applying cavity
input-output theory to establish the vector of quadratures of the output
optical fields $v$, and arranging the terms into a covariance matrix, we see
that the correlations in the input noises $\langle\mathbf{G}\rangle$ evolves
into the output optical correlations, covariance matrix $\mathbf{V}$:

$\mathbf{V}(\Omega)=Re(\mathbf{Q}\langle{\mathbf{G}}(\Omega)\rangle\mathbf{Q}^{\dagger}),$
(7)

where $\mathbf{Q}=\sqrt{2\gamma_{c}}\mathbf{M}(\Omega)+\mathbf{I}$, and
$\mathbf{M}(\Omega)=[\mathbf{K}+i\Omega\mathbf{I}]^{-1}$ .

To quantify the entanglement with the logarithmic negativity $E_{N}$ begin
with the direct computation of the elements of the quadrature covariance
matrix, $\mathbf{V}$:

$E_{N}=\max[~{}0,-\ln{\sqrt{2\eta-2\sqrt{\eta^{2}-4\det\mathbf{V}}}}],$ (8)

where $\eta=\det V_{11}+\det V_{22}-2\det V_{12}$. This is a popular metric
for Gaussian entanglement in continuous variable systems. One reason for its
popularity is that for quantum Gaussian states, logarithmic negativity amounts
to an exact entanglement cost [40, 41].

Note that the quadrature covariance matrix is of the form:

$\displaystyle\mathbf{V}=\left(\begin{array}[]{cccc}\left\langle{\mathbf{X}}_{1}{\mathbf{X}}_{1}\right\rangle_{+}&\left\langle{\mathbf{X}}_{1}{\mathbf{Y}}_{1}\right\rangle_{+}&\left\langle{\mathbf{X}}_{1}{\mathbf{X}}_{2}\right\rangle_{+}&\left\langle{\mathbf{X}}_{1}{\mathbf{Y}}_{2}\right\rangle_{+}\\\
\left\langle{\mathbf{Y}}_{1}{\mathbf{X}}_{1}\right\rangle_{+}&\left\langle{\mathbf{Y}}_{1}{\mathbf{Y}}_{1}\right\rangle_{+}&\left\langle{\mathbf{Y}}_{1}{\mathbf{X}}_{2}\right\rangle_{+}&\left\langle{\mathbf{Y}}_{1}{\mathbf{Y}}_{2}\right\rangle_{+}\\\
\left\langle{\mathbf{X}}_{2}{\mathbf{X}}_{1}\right\rangle_{+}&\left\langle{\mathbf{X}}_{2}{\mathbf{Y}}_{1}\right\rangle_{+}&\left\langle{\mathbf{X}}_{2}{\mathbf{X}}_{2}\right\rangle_{+}&\left\langle{\mathbf{X}}_{2}{\mathbf{Y}}_{2}\right\rangle_{+}\\\
\left\langle{\mathbf{Y}}_{2}{\mathbf{X}}_{1}\right\rangle_{+}&\left\langle{\mathbf{Y}}_{2}{\mathbf{Y}}_{1}\right\rangle_{+}&\left\langle{\mathbf{Y}}_{2}{\mathbf{X}}_{2}\right\rangle_{+}&\left\langle{\mathbf{Y}}_{2}{\mathbf{Y}}_{2}\right\rangle_{+}\\\
\end{array}\right)$ (13)

were $\left\langle u^{*}v\right\rangle_{+}~{}=~{}\frac{\left\langle
u^{*}v+v^{*}u\right\rangle}{2}$, or in block form:

$\displaystyle\centering\mathbf{V}=\left(\begin{array}[]{cc}V_{11}&V_{12}\\\
V_{21}&V_{22}\\\ \end{array}\right).\@add@centering$ (16)

Of the 16 terms in the optical output covariance matrix, the four diagonal
terms are (somewhat) trivial, and six others can be calculated directly from
the remaining six. Therefore, we only need to calculate six terms to construct
the matrix (see appendix). The six terms can be written and calculated a six
products terms of the form [42]:

$\displaystyle\hat{\mathbf{S}}_{\ell}^{\dagger}(\xi_{\ell})\hat{\mathbf{Y}}_{\ell}\hat{\mathbf{S}}_{\ell}(\zeta_{\ell})=\hat{\mathbf{S}}_{\ell}^{\dagger}(\xi_{\ell}-\zeta_{\ell})\hat{\mathbf{\gamma}}_{y}(\zeta_{\ell})~{},$
(17)

with $\zeta=r^{\prime}e^{i\theta^{\prime}}$, and
$\hat{\mathbf{\gamma}}_{y}(\zeta_{\ell})=\frac{i}{2}[(\cosh{r^{\prime}_{\ell}}+e^{i\theta^{\prime}_{\ell}}\sinh{r^{\prime}_{\ell}})\hat{\mathbf{a}}^{\dagger}-(\cosh{r^{\prime}_{\ell}}+e^{-i\theta^{\prime}_{\ell}}\sinh{r^{\prime}_{\ell}})\hat{\mathbf{a}}]$.

$\displaystyle\langle{\hat{\mathbf{G}}}\rangle_{j,j}=$ (18)
$\displaystyle\leavevmode\resizebox{182.1196pt}{}{$\left(\begin{array}[]{cc}\frac{A_{j}^{2}\alpha+B_{j}^{2}\alpha^{*2}+A_{j}B_{j}(2|\alpha|^{2}+1)}{2}&\frac{i(B_{j}^{2}\alpha^{*2}-A_{j}^{2}\alpha^{2}+A_{j}B_{j})-1}{2}\\\
-\frac{i(B_{j}^{2}\alpha^{*2}-A_{j}^{2}\alpha^{2}+A_{j}B_{j})-1}{2}&\frac{-A_{j}^{2}\alpha-
B_{j}^{2}\alpha^{*2}+A_{j}B_{j}(2|\alpha|^{2}+1)-1}{2}\end{array}\right)$}$

where $A_{j}=\cosh{r_{j}}-e^{-i\theta_{j}}\sinh{r_{j}}$ and
$B_{j}=\cosh{r_{j}}-e^{i\theta_{j}}\sinh{r_{j}}$.

Figure 2: The simulation shows the same two-mode periodicity due to the
dependence on two squeezing angles. This dependence is consistent with our
presented theoretical results Quantum Langevin calculations. The maximum
output $E_{N}$ varies strongly with input squeezing and angle. Temperature is
set to $4K$ (the simulation fails at room temperature), and see table 1 for
other parameters used. The maxima occur when input squeezing angles are odd
integer multiples of $\pi/2$ and minima at even integer multiples of $\pi/2$.
The last plot in the figure shows the results from applying Duan’s measure in
the same simulated conditions. Their discrepancy is due to non-Gaussianity in
the output fields. Figure 3: Above is a qualitative plot for the logarithmic
negativity from solving the Quantum Langevin equations, as a function of
squeezing angle (common to modes one and two $arg(\xi)=\arg(\zeta)$). This
result demonstrates a failure to quantify entanglement (even on an arbitrary
scale) at some input squeezing angle with $r_{1}=r_{2}\approx 0.3$. The peak
in the entanglement with the same periodicity in squeezing angle as in the
computational results.

The two measures of entanglement used here are only ideal for continuous-
variable states that are Gaussian. Notably, the two measures (logarithmic
negativity and Duan’s measure of inseparability) do not agree (figure 2). The
optical states produced by these simulations non-Gaussian. Here, applying the
logarithmic negativity yields indeterminate entanglement for moderate
squeezing in the simplified system (see figure 3). Figure 3 is a plot of a
more ideal system where the losses are zero and many noise factors are not
considered. Considering realistic conditions (losses, thermal noise, more
realistic OMO dampening, and higher-order mechanical modes in the OMO) bounds
the $E_{N}$ to a finite number as seen in figure 2, yet the entanglement
maxima and minima are at near opposite squeezing angle combinations (relative
to Duan’s measure). Although recent theoretical studies expect that the
logarithmic negativity can approach infinity for ideal cases of Gaussian
entanglement, those apply to cases of infinite squeezing [43]. Here the
Logarithmic negativity fails at very reasonable values of squeezing strength.

##  IV Gaussianity metrics

To further study the non-Gaussianity of the output optical fields we use the
Genoni’s metric for deviation from Gaussianity for ideal systems [44, 45, 46]:

$\delta(\hat{\rho})=S(\rho_{G})=S(\mathbf{V})$ (19)

where $S$ is the Von Neumann entropy, $\mathbf{V}$ is the variance matrix of
$\rho_{G}$ in standardized form [26] and $\rho_{G}$ is the Gaussian
description of the state $\rho$. The von Neumann entropy of a Gaussian system
is a function of the covariance matrix’s symplectic eigenvalues [47]. Figure 4
shows the difference in Guassianity,

$\Delta\delta=\delta(\hat{\rho}(\xi_{1},\xi_{2}))-\delta(\hat{\rho}(0,0))~{}.$
(20)

As we include an alternate measure of entanglement, so too shall we include an
alternate for Gaussianity. The more direct method of confirming non-Gaussian
statistics is to measure the higher-order cumulants. To measure the higher-
order cumulants we increase the number of parallel homodyne detection from two
(dual homodyne; see figure 1) to four (quadruple homodyne). To calculate the
fourth-order cumulants in the ponderomotive regime, one must find:

$\mathbf{\kappa}_{jk\ell
p}=\langle{Q_{j}Q_{k}Q_{\ell}Q_{p}}\rangle-3\langle{Q_{j}Q_{k}}\rangle\langle{Q_{\ell}Q_{p}}\rangle$
(21)

(where $\langle{Q}\rangle=0$) [48]. Both measures are used in figure 4. To
roughly estimate the degree of deviation from Gaussianity with the cumulant we
take the absolute sum of the fourth-order matrix,$\delta_{\kappa}=\sum_{jk\ell
p}|\kappa_{jk\ell p}|$; to roughly measure the change in Gaussianity,
$\Delta\delta_{\kappa}$, after applying squeezing to the injected field,

$\Delta\delta_{\kappa}=\sum_{jk\ell p}|\kappa_{jk\ell
p}(\xi_{1},\xi_{2})|-|\kappa_{jk\ell p}(0,0)|~{}.$ (22)

Figure 4: All four plots are at zero loses and $10$nK and equal squeezing
strengths ($r_{1}=r_{2}=0.3$) to simplify the measures of Gaussianity and show
changes with squeezing angle. A: The top left plot is the logarithmic
negativity ($E_{N}$). The logarithmic negativity is calculated from the
Gaussian statistics of the state only. Therefore, if only the Gaussian fields
are considered, the entanglement persists and is still boosted by the
squeezing. B: The top right plot depicts results from Duan’s measure of
separability $R$ rewritten here as $1-R$ to be positive for entanglement and
zero or less for any separable state. Duan’s measure of separability is more
reliable than the logarithmic negativity when involving non-Gaussian
statistics. The peaks in this plot occur at different angles versus the
$E_{N}$. Since the $E_{N}$ does not consider the higher-order statistics,
these new maxima must be due to the non-Gaussian entanglement adding to the
measured value. If this is the case, the non-Gaussianity should be higher at
these points, and the following plots are in agreement. C: The bottom left
plot is the measured deviation from Gaussianity ($\delta$) in the output
optical fields. The higher the value, the more non-Gaussian the state. This
implementation of this metric only holds for pure states, so we have set
temperature and loss to their minimums for this figure. The angles for higher
Gaussianity (lower $\delta$) directly coincide with the regions of maximum
logarithmic negativity. This suggests that when these systems are tuned to
produce more Gaussian states, the output entanglement is at least somewhat
conserved. D: The last plot in the figure is of the fourth-order cumulant. The
fourth-order cumulant measure is somewhat a more direct yet unrefined measure
of non-Gaussianity. For the most part, higher values for this measure mean
that there is likely more non-Gaussianity, especially where the two measures
agree. Interestingly, the injection of squeezed light can both increase and
decrease the Gaussianity of the output light. The strength of this effect, of
course, varies with squeezing strength (see figure 8).

In Figure 4, entanglement measure results are also included. All four plots
are at zero losses and $10$nK and equal squeezing strengths
($r_{1}=r_{2}=0.3$) to simplify Gaussianity measures and show changes with
squeezing angle. The logarithmic negativity is calculated from the Gaussian
statistics of the state only. Therefore, if only the Gaussian fields are
considered, the entanglement persists and is still boosted by the squeezing.
This happens at the angles where Duan’s measure is minimum. Since the $E_{N}$
does not consider the higher-order statistics, the maxima in Duan’s measure
must be due to the non-Gaussian entanglement in the optical output adding to
the measured value. The results from the Gaussianity metrics further confirm
this. Where there is non-Gaussian entanglement, the fourth-order cumulant
measure should be higher since the correlations for non-Gaussian entanglement
persist throughout the higher-order cumulants (keeping them nonzero). Using
Genoni’s measure, however, a different depiction of the changes in the overall
Gaussianity. This metric will be positive not only when there is tripartite
entanglement but when there is bipartite entanglement between one optical
field and the OMO. This is because it measures how the pure three-mode state
was poorly approximated by the two-mode Gaussian state created from the
measurement data. Consequentially, this minima in this measure highlight where
the squeezing boosts the Gaussianity, which are the same regions where the
squeezing boosts the $E_{N}$. These regions must not pertain to tripartite
entangled states and thus correspond to the production of Gaussian photon-
photon entanglement by the OMC. The $E_{N}$ is maximum where the squeezing
increases the Gaussianity. However, where the squeezing reduces the
Gaussianity, the $E_{N}$ is not zero, implying higher overall entanglement at
points of low Gaussianity. Duan’s measure here shows that the non-Gaussian
regions have the highest relative entanglement. The fourth-order cumulant
better reflects regions of high non-Gaussian entanglement, confirmed by the
Duan measure results since it depends on some of the higher-order correlations
in the cumulant matrix. The cumulant metric is maximum where tripartite
entanglement is maximum. These results are consistent with the quantum noise
results, when input quantum noise is maximum entanglement (including
tripartite entanglement)is maximum.

##  V OMC entanglement device efficacy

Optomechanical cavity-based entanglement generating devices can yield
entanglement without squeezing the coherent input fields. However, this may
not be efficient enough to rival two-mode squeezed vacuum entanglement
generation methods. The figures above and below show much higher (factor of 6
for $E_{N}$) output entanglement for squeezed input fields, yet the $E_{N}$
yields are not competitive for intense squeezing (about $r>0.5$ or larger than
a decibel).

$\begin{array}[]{|c|c|c|}\text{Parameter}&\text{Notation}&\text{Stable and
$E_{N}\neq 0$}\\\ \hline\cr\text{Temperature}&T&4\text{K}\\\ \text{Circulating
carrier }\text{power}&P_{1}&0.2816\text{W}\\\ \text{Circulating subcarrier
power}&P_{2}&0.2238\text{W}\\\ \text{Loss}&L_{s}&25ppm\\\ \text{Carrier
}\text{detuning}&d_{1}&0.3\\\ \text{Subcarrier detuning}&d_{2}&-1.5\\\
\text{Quality factor}&Q&17000\\\ \text{Cavity
}\text{Length}&L_{n}&0.01\text{m}\\\ \end{array}$

Table 1: Set of variable simulation parameters, and a stable configuration
that yield non-zero entanglement.

###  V.1 Quantum Noise

Two coherent states input into an optomechanical cavity has been shown
theoretically and experimentally to generate entanglement [9, 7, 49, 14].
Moreover, recent work has provided more insight into what drives this quantum
process. Specifically, the relationships between output entanglement and input
quantum noise; not only does the output entanglement thrive when the quantum
noise is dominant in the optomechanical cavity, but that, at low thermal noise
(lower than room temperature for this system), the quantum noise itself is
more indicative of the output entanglement than the ratio. Thus, to increase
the entanglement output from the optomechanical cavity, we need to increase
the input quantum noise while keeping the mechanical oscillator’s temperature
reasonably low. For the oscillator to maintain the low relative classical
noise, the input light must remain at or near minimum uncertainty at the
input; increasing the fluctuations in the input light’s amplitude
consequentially decreases the fluctuations in that light’s phase to maintain
this quantum only input noise [13]. Thus, we intentionally amplify the
intracavity quantum noise by injecting displaced squeezed states (or squeezed
coherent light).

Figure 5: Above plots correspond to the dual single-mode squeezed input
simulations (table 1) with $r_{1}=r_{2}=0.8$. Without squeezed input light,
the quantum to thermal noise ratio greatly determined the cavity’s
entanglement output. When $r>0$, this somewhat holds, but the overall increase
in entanglement is much less than when unsqueezed coherent light is input. In
the two plots, the quantum to thermal noise ratio is maximum at the input
squeezing angles where output entanglement is maximum (at about $(n\pi,m\pi)$
where $n$ and $m$ are some arbitrary integers); these findings are in full
agreement with the entanglement simulation results. The boosted quantum noise
in the injected field still corresponds to a boosted output entanglement even
when working with non-Gaussian states.

While entanglement generation methods’ advantages will vary with system
parameters, the maximal overall benefit provided by OM methods to entanglement
production only seems slight at first glance. This middling benefit to
entanglement is seemingly in contrast to the significant increase in the
quantum noise (see figure 5). The expectation is that the quantum noise drives
(or is an indicator of) the optomechanical entanglement. Figure 5 shows
notable quantum variances that are at most two orders of magnitude higher than
the unsqueezed input variances [10], yet the output entanglement seems only
weakly boosted.

The output entanglement could be increasing proportionally to the increase in
input quantum noise, but the increase in non-Gaussianity obscures the evidence
in our metrics, effectively hiding the entanglement in the non-Gaussian bulk
of the optical state. Concurrently, the temperature dependence is greatly
lessened. This effect was also found to be the case in other publications with
squeezed light injection into OMCs [37]. This is likely due to the squeezing
changing the OMO’s zero-point fluctuations or temperature, and this effect
should be investigated further in the ponderomotive regime. Thus, we observe
an increase in output entanglement is about six times the entanglement output
instead of about 30 times the output (as is the case for the quantum to
thermal noise ratio).

Figure 6: The above plots demonstrate the utility of optomechanically enhanced
entanglement generation methods concerning access to high-quality nonlinear
crystals. The value of Duan’s measure of the output cavity field is higher
(for Gaussian states, this implies more entanglement) than squeezed light with
beamsplitter methods at all values of crystal nonlinearity. The Duan Measure
of bipartite entanglement, $1-R$, in the two plots is devoid of singularities
that appear in the logarithmic negativity. The input power for the carrier and
subcarrier is kept constant at 46$\mu$W and 1.1 mW, respectively; thus, the
squeezing strength is $r$ where $r_{j}=\mu_{j}\sqrt{P_{j}}$. In regions of
higher Gaussian entanglement we can compare the OMC boosted entanglement rates
with those from conventional entanglement methods that also utilize squeezing.
For this comparison assume the conventional methods (SPDC) produce sideband
entanglement. We see the OMC-enhanced circuit could be preferable at low
squeezing when input coupling loss is zero; this result is more akin to
intracavity squeezing. The horizontal line represents the OMC performance when
input coupling is considered. In this case there is no advantage offered by
squeezing the input fields due to poor coupling in the circuit $\approx
50ppm$.

###  V.2 Squeezing Resources Usage Efficacy

The simulations show OMC-based circuits have the potential for applications as
entangling devices that rival conventional means of entanglement production.
Consider situations where losses are low and nonlinear crystals of high
nonlinearity and length, are scarce. Then in a setup with two lasers of set
output powers $p_{1}$ and $p_{2}$, the maximum entanglement produced with the
nonlinear crystal is proportional to the amount of squeezing the can be
obtained from the crystal and light sources. The squeezing strength relates to
the pump power and the effective non-linearity $\mu$ as
$r=\mu\sqrt{p_{1}+p_{2}}$ [50]. Our laboratory simulations take
$p_{1}=46.24\mu$W and $p_{2}=1.1$mW (note that if $\mu_{1}=\mu_{2}$ then
$r_{1}\neq r_{2}$ and if $r_{1}=r_{2}$ then $\mu_{1}\neq\mu_{2}$). Using
these, we can plot (figure 6) the entanglement from conventional methods and
compare it to entanglement from methods that inject squeezed light into the
optomechanical cavities (crystal + optomechanical entanglement method); to
better consider means of producing entanglement are already available and
demonstrates the advantage of OMC integration. Duan’s measure is sufficient
but not necessary for non-Gaussian states. It is not reliable in comparing
entanglement output between entangling devices if one or both produce non-
Gaussian entanglement.

##  VI Conclusions

Entanglement and squeezing are valuable quantum resources in quantum
technologies. Quantum Cavity Optomechanics contains new means of producing
squeezed and entangled light. Generating entanglement via squeezed light
injection into sideband unresolved OMCs may well be a competitive method for
producing, enhancing, or processing entanglement from single-mode squeezing
resources for continuous-variable quantum devices. To this end, an
understanding of the output fields’ Gaussianity is necessary. These
simulations show a non-trivial relationship between input squeezing,
intracavity entanglement, two-mode Gaussianity, and intracavity quantum noise.
At the right squeezing angles, implementing an optomechanical cavity in a
conventional entanglement circuit could increase output entanglement,
quantified by Duan’s measure of separability, by a factor of 2 or greater due
to the increase in the injected quantum noise. These results are consistent in
our theoretical results (Quantum Langevin) and our simulations (two-photon
formalism quadrature propagation). The increase in quantum noise due to
squeezing helps mitigate unwanted effects due to thermal noises, as was the
case in previous studies [37]. When the OMO is entangled with one or both
optical fields, the output fields from these OMCs are non-Gaussian CV fields
in these cases and are not well quantified by the logarithmic negativity or
Duan’s measure. Lastly, the intracavity entanglement can be controlled by
controlling the squeezing angle of the input light, and in the region of
increased Gaussianity, there is increased bipartite optical entanglement
(Logarithmic negativity).

## Acknowledgement

KD, LC, and NB dedicate this paper to the memory of their former supervisor,
late Professor Jonathan P. Dowling. KD, LC, NB, and JPD would like to
acknowledge the Air Force Office of Scientific Research, the Army Research
Office, the Defense Advanced Research Projects Agency, and the National
Science Foundation. We would also like to thank Professor Mark Wilde for
important discussions. This material is based upon work supported by the
National Science Foundation under Grant No. PHY-1806634.

##  VII Appendix

###  VII.1 Computational methods

Much of the computational framework from our previous project was preserved
for this work. Here the matrix $\mathscr{S}$ describes how the sideband
operators transform due to single-mode squeezing

$\displaystyle\mathscr{S}(r,\theta)=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(23)
$\displaystyle\left(\begin{array}[]{cc}\cos{2\theta}\sinh{r}+\cosh{r}&\sin{2\theta}\sinh{r}\\\
\sin{2\theta}\sinh{r}&\cosh{r}-\cos{2\theta}\sinh(r)\\\
\end{array}\right)~{}.$ (26)

Since the program assumes squeezing of the sideband fields prior to injection,
none of the squeezing is lost due to the cavity input coupling.

The two-photon formalism’s flexibility allows for the thorough simulation of
the sideband quadratures’ homodyne measurement [51, 24, 10, 52, 31, 23]. Such
flexibility creates a robust platform for understanding how the output
entanglement and quantum noise are affected by laboratory complications and
loss parameters.

###  VII.2 Squeezing angle dependence

In this section we note the steps to calculate the expected value of
$\hat{\mathbf{S}}^{\dagger}(\xi)\hat{\mathbf{S}}(\zeta)$. Before we begin we
note that
$\hat{\mathbf{S}}^{\dagger}(\xi)\hat{\mathbf{S}}(\zeta)=\hat{\mathbf{1}}$ when
$\xi=\zeta$ and
$\hat{\mathbf{S}}^{\dagger}(\zeta)\hat{\mathbf{S}}(\xi)=\hat{\mathbf{S}}(\zeta)$
when $\xi=2\zeta$ (will be shown in the following notes). Let’s begin with the
following:

$\displaystyle\hat{\mathbf{S}}^{\dagger}(\xi)\hat{\mathbf{S}}(\zeta)=e^{-\xi^{*}\hat{\mathbf{a}}^{2}+\xi\hat{\mathbf{a}}^{\dagger
2}}~{}e^{\zeta^{*}\hat{\mathbf{a}}^{2}-\zeta\hat{\mathbf{a}}^{\dagger 2}}$
(27)

Using the Baker-Campbell-Hausdroff (BCH) formula we can write this product as
follows (note:
$\mathbf{A}=-\xi^{*}\hat{\mathbf{a}}^{2}+\xi\hat{\mathbf{a}}^{\dagger 2}$ and
$\mathbf{B}=\zeta^{*}\hat{\mathbf{a}}^{2}-\zeta\hat{\mathbf{a}}^{\dagger 2}$
):

$\begin{array}[]{r}\hat{\mathbf{S}}^{\dagger}(\xi)\hat{\mathbf{S}}(\zeta)=e^{-\xi^{*}\hat{\mathbf{a}}^{2}+\xi\hat{\mathbf{a}}^{\dagger
2}}e^{\zeta^{*}\hat{\mathbf{a}}^{2}-\zeta\hat{\mathbf{a}}^{\dagger 2}}\\\
=\exp\left[\left(-\xi^{*}+\zeta^{*}\right)\hat{\mathbf{a}}^{2}+(\xi-\zeta)\hat{\mathbf{a}}^{\dagger
2}+\hat{\mathbf{Z}}\right]\end{array}$ (28)

where
$\hat{\mathbf{Z}}=\frac{1}{2}\hat{\mathbf{k}}_{0}+\frac{1}{12}([\hat{\mathbf{f}}_{2}(-\xi),\hat{\mathbf{k}}_{0}]-\frac{1}{12}[\hat{\mathbf{f}}_{2}(\zeta),\hat{\mathbf{k}}_{0}])+...$
with
$\hat{\mathbf{k}}_{0}=\frac{1}{2}[-\xi^{*}\hat{\mathbf{a}}^{2}+\xi\hat{\mathbf{a}}^{\dagger
2},\zeta^{*}\hat{\mathbf{a}}^{2}-\zeta\hat{\mathbf{a}}^{\dagger 2}]$.

$\begin{array}[]{l}\hat{\mathbf{k}}_{0}=\left[\xi\hat{\mathbf{a}}^{\dagger
2}-\xi^{*}\hat{\mathbf{a}}^{2},\zeta^{*}\hat{\mathbf{a}}^{2}-\zeta\hat{\mathbf{a}}^{\dagger
2}\right]\\\ =\left[\xi\hat{\mathbf{a}}^{\dagger
2},\zeta^{*}\hat{\mathbf{a}}^{2}\right]+\left[\xi^{*}\hat{\mathbf{a}}^{2},\zeta\hat{\mathbf{a}}^{\dagger
2}\right]\end{array}$ (29)

since, $[\hat{\mathbf{a}}^{\dagger
2},\hat{\mathbf{a}}^{2}]=-2(2\hat{\mathbf{n}}+1)$ and
$[\hat{\mathbf{a}}^{2},\hat{\mathbf{a}}^{\dagger 2}]=2(2\hat{\mathbf{n}}+1)$
we can write eqn 29 as

$\displaystyle 2(\xi\zeta^{*}-\xi^{*}\zeta)(2\hat{\mathbf{n}}+1)~{}.$ (30)

Figure 7: These plots examine changes in Gaussianity and entanglement changes
with small squeezing ($\mu=0.1$ relative to unsqueezed input) versus sideband
frequency, $\Omega$, at 77K. Most notable here is that while the squeezing
effects the logarithmic negativity and when it increases the Gaussianity,
Duan’s measure does not change.

Thus, the resulting operator is highly dependent on the input squeezing angle
difference and is not very sensitive to differences in squeezing strength
between the two inputs. To make this more clear we can write
$\mathbf{\hat{k}}_{0}$ in a different form than above if we assume
$\xi=r_{1}e^{i\theta}$ and $\zeta=r_{2}e^{i\phi}$:

$\displaystyle\hat{\mathbf{k}}_{0}~{}=~{}4ir_{1}r_{2}\sin{(\theta-\phi)}(2\hat{\mathbf{n}}+1)~{}.$
(31)

If the squeezing angles are the same, the term $\hat{\mathbf{k}}_{0}$
vanishes; the real parts of the squeezing parameters do not change the form of
the operator unless they are zero. Furthermore, at higher orders there are
terms with dependence on $\sin(\theta+\phi)$ and $\cos(\theta+\phi)$; there is
still some non-trivial complications to the squeezing operator $around$ the
line $\theta=\phi$. These complications may account for the peaks in the dual
single-mode squeezed input entanglement results which appear to show that the
entanglement is maximum when $\theta\approx\phi$; it is likely that when
$\theta$ exactly equals $\phi$ that the $E_{N}$ falls (or rises) sharply.

###  VII.3 Frequency dependence

Considering the results in figure 4 with those depicted in figure 7, the input
squeezing, at the angles that yield increased Gaussianity, does not change the
separability in Duan’s measure; meaning there is no net change in output
entanglement at these squeezing angles. In this configuration, the input
squeezing transfers non-Gaussian entanglement to Gaussian entanglement.

Figure 8: The above plots are of the changes in non-Gaussianity versus
squeezing strength. These ultimately show that increasing Gaussianity with
squeezing only works best for low squeezing strengths (here $\mu>0.5$) or
unbalanced squeezing.

###  VII.4 Increasing Gaussianity with input squeezing

Continuous-variable non-Gaussian systems are cumbersome analytically. Thus, a
valuable feature of these systems is that injecting squeezed light can
increase the Gaussianity of the output fields. However, in our simulations,
intense squeezing consistently decreases the output Gaussianity, see figure 8.

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# Dual Cheeger constant for weighted graphs over ordered fields

Anna Muranova  Anna Muranova: Faculty of Mathematics and Computer Science,
University of Warmia and Mazury, ul. Sloneczna 54, 10-710 Olsztyn, Poland
<EMAIL_ADDRESS>
###### Abstract.

We consider a dual Cheeger constant $\overline{h}$ for finite graphs with edge
weights from an arbitrary real-closed ordered field. We obtain estimates of
$\overline{h}$ in terms of number of vertices in graph. Further, we estimate
the largest eigenvalue for the discrete Laplace operator in terms of
$\overline{h}$ and show the sharpness of estimates. As an example we consider
graphs over non-Archimedean field of the Levi-Civita numbers.

Keywords: weighted graphs, Laplace operator, non-Archimedean field, ordered
field, Levi-Civita field, dual Cheeger constant.
Mathematics Subject Classification 2010: 05C50, 05C22, 47A75, 12J15, 39A12

## Introduction

The Cheeger isoperimetric constant $h$ was firstly introduced for compact
Riemannian manifold by Jeff Cheeger [4] in 1969 and then considered for graphs
(e.g. [5, 11]). The dual Cheeger constant $\overline{h}$ was introduced
directly on weighted graphs in [1]. The dual Cheeger constant shows how close
the graph is to bipartite and it is used in estimation of the largest
eigenvalue of discrete Laplacian. Classical approach to weighted graph assume,
that edge weights are positive real numbers. In this paper we consider the
model, which was introduced in previous works by author (see [12, 13, 14]),
where edge weights are positive elements of an arbitrary real-closed ordered
field $(\mathbb{K},\succ)$. We introduce the notion of dual Cheeger constant
$\overline{h}$ for finite graphs analogously to the classical case of field
$\mathbb{R}$ and prove its estimates in terms of number $N$ of vertices in
graph. We show, that for even $N$ holds

$\overline{h}\succeq\dfrac{N}{2(N-1)}$

and for odd $N$ holds

$\overline{h}\succeq\dfrac{N+1}{2N}.$

These estimates improve the result in [1], which states that
$\overline{h}\geq\dfrac{1}{2}$ for the field $\mathbb{R}$. The sharpness of
our result follows from examples in [1], since the equality holds for a
complete graph with all the weights equal $1$ over $\mathbb{R}$(and over any
real-closed field, due to the same calculations).

Further, we show that for any $\mathbb{K}$ the largest eigenvalue of the
Laplace operator can be estimated as

(1) $2\overline{h}\preceq\lambda_{N-1}\preceq
1+\sqrt{1-(1-\overline{h})^{2}},$

where the both equalities hold for any bipartite graph.

In Section 4, we consider graphs over a non-Archimedean field of the Levi-
Civita numbers $\mathcal{R}$ and show, that left inequality in can hold also
for complete graphs, while the right one is precise for them in a sense, that
there is a sequence of graphs, for which the difference
$|\lambda_{N-1}-1+\sqrt{1-(1-\overline{h})^{2}}|$ converges in an order
topology.

##  1\. Preliminaries

###  1.1. Real-closed ordered field

Firstly we remind a concept of real-closed ordered field. More detailed
description can be found in textbooks on Algebra see e. g. [9, 18].

######  Definition 1.

A field $\mathbb{K}$ is called ordered, if there exist a subset
$\mathbb{K}^{+}~{}\subset\mathbb{K}$ with the following properties:

  1. (1)

$0\not\in\mathbb{K}^{+}$,

  2. (2)

for any non-zero element $k\in\mathbb{K}$ either $k\in\mathbb{K}^{+}$ or
$-k\in\mathbb{K}^{+}$,

  3. (3)

if $k_{1},k_{2}\in\mathbb{K}^{+}$, then $k_{1}+k_{2}\in\mathbb{K}^{+}$ and
$k_{1}k_{2}\in\mathbb{K}^{+}$.

The total order in the ordered field is then defined as follows: for
$k_{1},k_{2}\in\mathbb{K}$ we write $k_{1}\succ k_{2}$ (or, equivalently,
$k_{2}\prec k_{1}$) if $(k_{1}-k_{2})\in\mathbb{K}^{+}$. Further $k_{1}\succeq
k_{2}$ (equiv. $k_{1}\preceq k_{2}$) if
$(k_{1}-k_{2})\in\mathbb{K}^{+}\cup\\{0\\}$.

An ordered field $\mathbb{K}$ is non-Archimedean if there exist an
infinitesimal, i.e. $\epsilon\in\mathbb{K}$ such that

${\epsilon}\prec\dfrac{1}{n}=\dfrac{1}{\underbrace{1+\dots+1}_{n\mbox{
times}}}$

for any $n\in\mathbb{N}\subset\mathbb{K}$. Otherwise, the field is called
Archimedean. All Archimedean fields are isomorphic to some (not necessary
proper) subfield of real numbers. Examples of non-Archimedean fields are
ordered field of rational functions [3, p. A.VI.21], Levi-Civita field [2, 8],
super-real field [6].

By _absolute value of $k\in\mathbb{K}$_ we mean
$|k|\in\mathbb{K}^{+}\cup\\{0\\}$ defined as

$|k|=\begin{cases}k,\mbox{ if }k\succeq 0\\\ -k,\mbox{otherwise.}\end{cases}$

The convergence of sequence $\\{k_{n}\\}_{n\in\mathbb{N}}$ to $k$ in ordered
topology of the ordered field means by definition, that for any element
$a\in\mathbb{K}$ exists $N_{0}\in\mathbb{N}$ such that for any $n\geq N_{0}$
we have

$|k_{n}-k|\preceq a.$

There are several equivalent definitions of a real-closed field [6, p. 38]. We
remind some of them.

An ordered field $\mathbb{K}$ is real-closed if one of the following holds:

  1. (1)

$\mathbb{K}$ has no proper algebraic extension to an ordered field,

  2. (2)

the complexification of $\mathbb{K}$ is algebraically closed,

  3. (3)

every positive element in $\mathbb{K}$ has a square root and every polynomial
over $\mathbb{K}$ of odd degree has a root in $\mathbb{K}$.

It is well known, that order can be introduced in any real-closed field.
Moreover, any ordered field has an algebraic real-closed extension closure,
which is unique up to isomorphism, preserving the order (Artin-Schreier
Theorem, see [6, p. 38]). Therefore, further in this note we consider
exclusively real-closed ordered fields, e.g. algebraic numbers, real numbers,
Levi-Civita field.

###  1.2. Graphs over ordered field

The concept of graph over an ordered field was introduced in previous works by
author (see [12, 13]). Graphs over the field $\mathbb{R}$ of real numbers are
classical weighted graphs. Here we shortly remind the definition. field.

######  Definition 2.

Let $\mathbb{K}$ be a real-closed ordered. _A graph oven an ordered field $K$
(OF-graph)_ is a couple $(V,b)$, where $V$ is a set of vertices (i.e.
arbitrary set) and $b:V\times V\rightarrow\mathbb{K}^{+}\cup\\{0\\}$ satisfies
the following properties:

  1. (1)

$b(x,y)=b(y,x)$ for any $x,y\in V$,

  2. (2)

$b(x,x)=0$ for any $x\in V$.

The last property mean that we consider graphs without loops. We write $x\sim
y$ whenever $b(x,y)\neq 0$ and say that there is an edge between $x$ and $y$.

A subgraph of a graph $(V,b)$ is any graph $(U,b\left|{}_{U\times U}\right.)$,
where $U\subset V$.

A path between any vertices $x,y\in V$ is any sequence
$\\{x_{i}\\}_{i=1}^{n},n\in\mathbb{N}$ such that

$x=x_{0}\sim x_{1}\sim x_{2}\sim\dots\sim x_{n}=y.$

A graph is called connected, if there exists a path between any two vertices
of it.

A connected component of a graph is any its connected subgraph that is not
part of any larger connected subgraph. Therefore, set of vertices of graph is
a disjoint union of vertices of all its connected components.

Further in this note we consider finite ($\\#V<\infty$) graphs. For the sake
of brevity futher we refer to a graph by its set of vertices (e.g. $V$).

A graph is called bipartite, if there exist a partition $V=V_{1}\cup V_{2}$ of
vertices, such that from $x\sim y$ follows that $x\in V_{1},y\in V_{2}$ or
$x\in V_{2},y\in V_{1}$.

A graph is called complete, if $x\sim y$ for any two $x,y\in V$ with $x\neq
y$.

The _normalized_ weight on edges is defined by

(2) $p(x,y)=\dfrac{b(x,y)}{b(x)},\mbox{ for any }x,y\in V,$

where $b(x)=\sum_{y}b(x,y)$. If $b(x)=0$ then the vertex $x$ is called
isolated vertex. Further we consider graphs without isolated vertices,
therefore $p(x,y)$ is well defined.

For any $V_{1}\subset V$ we denote

$b(V_{1})=\sum_{x\in V_{1}}b(x)$

and for any $V_{1},V_{2}\subset V$ with $V_{1}\cap~{}V_{2}=\emptyset$ we
denote

$b(V_{1},V_{2})=\sum_{\begin{subarray}{c}x\in V_{1},\\\ y\in
V_{2}\end{subarray}}b(x,y).$

###  1.3. Laplace operator

Our main interest is normalized Laplace operator (Laplacian) defined on the
set of functions

$\mathfrak{F}_{V}=\\{f\mid f:V\to\mathbb{K}\\}$

by

(3) $\mathcal{L}f(x)=\sum_{y\in V}(f(x)-f(y))\dfrac{b(x,y)}{b(x)}=\sum_{y\in
V}(f(x)-f(y))p(x,y).$

An eigenvalue of $\mathcal{L}$ is defined as $\lambda$ such that
$\mathcal{L}f=\lambda v$, where $v$ is called eigenfunction. It is proven in
[14] that all $N=\\#V$ eigenvalues of the Laplacian for a connected graph
belong to the same field $\mathbb{K}$ as edge weights and all the
eigenfunctions are in $\mathfrak{F}_{V}$. Moreover, it is shown there that the
largest eigenvalue of the connected graph satisfies

$\dfrac{N}{N-1}\preceq\lambda_{N-1}\preceq 2$

and $\lambda_{N-1}=2$ if and only if the graph is bipartite.

For non-connected graph the eigenvalues are union of eigenvalues of its
connected components (with multiplicities) and eigenfunctions are extension of
eigenfunctions of connected components to the whole graph, which are $0$ on
the other components.

Here and further we order eigenvalues of Laplacian as

$0=\lambda_{0}\preceq\lambda_{1}\preceq\lambda_{2}\preceq\dots\preceq\lambda_{N-1}\preceq
2.$

The multiplicity of the eigenvalue $0$ is equal to the number of connected
components of the graph.

For any function $f\in\mathfrak{F}_{V}$ we denote

$V_{f}^{+}=\\{x\in V\mid f(x)\succ 0\\}$

and

$V_{f}^{-}=\\{x\in V\mid f(x)\prec 0\\}.$

We introduce an analogue of scalar product on $\mathfrak{F}_{V}$ by

$\langle f,g\rangle=\sum_{x\in V}f(x)g(x)b(x).$

for any $f,g\in\mathfrak{F}_{V}$. The Green formula (see e.g. [12]) states,
that

$\langle\mathcal{L}f,g\rangle=\dfrac{1}{2}\sum_{x,y\in
V}(f(x)-f(y))(g(x)-g(y))b(x,y).$

##  2\. Dual Cheeger constant

The dual Cheeger constant for weighted graphs was introduced in [1]. Here we
introduce it in the same way for OF-graphs.

######  Definition 3.

Let $V$ be an OF-graph with $N=\\#V$ vertices. Then its dual Cheeger constant
is defined by

(4) $\overline{h}=\max_{\begin{subarray}{c}\emptyset\neq V_{1}\subset V,\\\
\emptyset\neq V_{2}\subset V,\\\ V_{1}\cap
V_{2}=\emptyset\end{subarray}}\dfrac{2b(V_{1},V_{2})}{b(V_{1})+b(V_{2})},$

i.e. the maximum is taken over all pairs of non-empty disjoint subsets of $V$.

######  Lemma 1.

For any connected graph the dual Cheeger constant satisfies

(5) $\overline{h}\preceq 1.$

Moreover, $\overline{h}=1$ if and only if the graph is bipartite.

The proof follows the same outline as in [1].

###### Proof.

The inequality (5) follows immediately, since $b(V_{1},V_{2})\preceq b(V_{l})$
for $l=1,2.$

Further, let $V$ is bipartite with the partition $V=V_{1}\cup V_{2}$. Then
equality in (5) is attained on this partition. Let now $\overline{h}=1$. Then

$2b(V_{1},V_{2})=b(V_{1})+b(V_{2}),$

i.e. there is no edges between the vertices from the same $V_{l}$, for
$l=1,2$. Moreover, since the graph is connected, $V_{3}=\emptyset$. Therefore,
the graph is bipartite and $V=V_{1}\cup V_{2}$ is its partition. ∎

######  Theorem 1.

(a) For a graph $V$ with $N=2K\geq 2$ vertices and at least one edge, we have

$\overline{h}\succeq\frac{N}{2(N-1)}.$

(b) For a graph $V$ with $N=2K+1>2$ vertices and at least one edge, we have

$\overline{h}\succeq\dfrac{N+1}{2N}.$

###### Proof.

(a) Let us consider only such $V_{1},V_{2}\subset V$ that $V=V_{1}\cup V_{2}$
is a partition of the graph with $\\#V_{1}=\\#V_{2}=K$. There are ${N\choose
K}=\frac{N!}{K!(N-K)!}$ such partitions. For each such $V_{1},V_{2}$ we define

$\overline{h}(V_{1},V_{2}):=\frac{2b(V_{1},V_{2})}{b(V_{1})+b(V_{2})}.$

therefore $\overline{h}\succeq\overline{h}(V_{1},V_{2})$.

For every such a partition $V=V_{1}\cup V_{2}$ we have

$\overline{h}(V_{1},V_{2})=\frac{2\sum_{x\in V_{1},y\in
V_{2}}b(x,y)}{b(V)}=\frac{2\sum_{x\in V_{1},y\in V_{2}}b(x,y)}{\sum_{x,y\in
V}b(x,y)},$

and thus

(6) $\frac{2\sum_{x\in V_{1},y\in V_{2}}b(x,y)}{\sum_{x,y\in
V}b(x,y)}\preceq\overline{h}$

Note that $b({x_{0}},y_{0})$ for some $x_{0},y_{0}\in V$ is in numerator if
and only if ${x_{0}}\in V_{1}$ and ${y_{0}}\in V_{2}$. It’s easy to count that
among all the considered partitions there are ${{N-2}\choose{K-1}}$ such
partitions for any given ordered pair $(x_{0},y_{0})$ . Now we sum up all
inequalities (6), rewrite the numerator using the last observation and obtain

${{N-2}\choose{K-1}}\frac{2\sum_{x_{0},y_{0}\in V}b(x_{0},y_{0})}{\sum_{x,y\in
V}b(x,y)}\preceq\overline{h}{N\choose K}$

from where follows

$2{{N-2}\choose{K-1}}\preceq\overline{h}{N\choose K}$

and, further,

$\overline{h}\succeq\dfrac{2(N-K)K}{N(N-1)}=\dfrac{K}{(N-1)}=\dfrac{N}{2(N-1)},$

since $N=2K$.

(b) The idea of the proof is the same, as in the part (a), but the
corresponding partitions are $\\#V_{1}=K,\\#V_{2}=K+1$. Then, following the
same outline as in (a), we get

$\overline{h}\succeq\dfrac{2(N-K)K}{N(N-1)}=\dfrac{N-K}{N}=\dfrac{N+1}{2N},$

since $N=2K+1$.

∎

Therefore, the following Corollary, which improves the result
$\overline{h}\geq\frac{1}{2}$ in [1], holds:

######  Corollary 1.

For any finite graph with at least one edge

$\overline{h}\succ\dfrac{1}{2}.$

##  3\. Largest eigenvalue of the graph Laplacian

In this section we show the relation between dual Cheeger constant and the
largest eigenvalue $\lambda_{N-1}$ of the discrete Laplacian for the finite
graph over an ordered field. For the field $\mathbb{R}$ it is done in [1] and
the result is similar to the result, which relate Cheeger constant and
smallest non-zero eigenvalue (see e. g. [5], [7] for $\mathbb{R}$ and [14] for
ordered fields).

######  Theorem 2.

The largest eigenvalue $\lambda_{N-1}$ of the graph Laplacian satisfies

$2\overline{h}\preceq\lambda_{N-1}\preceq 1+\sqrt{1-(1-\overline{h})^{2}}.$

For the proof of this theorem we need the following important lemma, which is
proved in [1] for $\mathbb{K}=\mathbb{R}$ , but there the proof uses
integrals. Here we present the proof of the lemma, which is analogous to the
prove of Theorem 2 in [14] and uses the technique, used in [5] for the proof
of the relation between the smallest non-zero eigenvalue of Laplacian and
Cheeger constant.

######  Lemma 2.

Let $(V,b)$ be a graph without isolated vertices and $f\in\mathfrak{F}_{V}$ be
such that $V^{+}_{f}\neq\emptyset$ and $V\setminus V^{+}_{f}\neq\emptyset$.
Let

$h(f)=\min_{\emptyset\neq S\subset V^{+}_{f}}\dfrac{b(S,V\setminus S)}{b(S)}$

and

$f_{+}(x)=\begin{cases}f(x),x\in V^{+}_{f},\\\ 0,\mbox{
otherwise},\end{cases}$

then

$1-\sqrt{1-h^{2}(f)}\preceq\dfrac{\langle\mathcal{L}f_{+},f_{+}\rangle}{\langle
f_{+},f_{+}\rangle}\preceq 1+\sqrt{1-h^{2}(f)}.$

###### Proof.

Let us denote

$W:=\dfrac{\langle\mathcal{L}f_{+},f_{+}\rangle}{\langle f_{+},f_{+}\rangle}.$

By Green formula we can rewrite $W$ as

$W=\dfrac{\frac{1}{2}\sum_{x,y\in V}(f_{+}(x)-f_{+}(y))^{2}b(x,y)}{\sum_{x\in
V}f_{+}^{2}(x)b(x)}.$

Then multiplying the numerator and the denominator by $\sum_{x,y\in
V}(f_{+}(x)+f_{+}(y))^{2}b(x,y)$ and using Cauchy-Schwarz inequality we get

(7) $W\succeq\dfrac{\frac{1}{2}\left(\sum_{x,y\in
V}b(x,y)\left|f_{+}^{2}(x)-f_{+}^{2}(y)\right|\right)^{2}}{\left(\sum_{x\in
V}f_{+}^{2}(x)b(x)\right)\left(\sum_{x,y\in
V}(f_{+}(x)+f_{+}(y))^{2}b(x,y)\right)}.$

From the other hand,

$\displaystyle W\cdot{\sum_{x\in
V}f_{+}^{2}(x)b(x)}=\langle\mathcal{L}f_{+},f_{+}\rangle$
$\displaystyle=2\sum_{x\in V}f_{+}^{2}(x)b(x)-\left(\sum_{x\in
V}f_{+}^{2}(x)b(x)+\sum_{x,y\in V}f_{+}(x)f_{+}(y)b(x,y)\right)$
$\displaystyle=2\sum_{x\in V}f_{+}^{2}(x)b(x)-\left(\sum_{x,y\in
V}f_{+}(x)(f_{+}(x)+f_{+}(y))b(x,y)\right)$ $\displaystyle=2\sum_{x\in
V}f_{+}^{2}(x)b(x)-\left(\sum_{x,y\in
V}f_{+}(y)(f_{+}(x)+f_{+}(y))b(x,y)\right)$ $\displaystyle=2\sum_{x\in
V}f_{+}^{2}(x)b(x)-\frac{1}{2}\left(\sum_{x,y\in
V}(f_{+}(x)+f_{+}(y))^{2}b(x,y)\right),$

where in the last two lines we have switched notations for $x,y$ in the second
sum and sum up two lines and divided by two in the last line. Using this in
denominator of (7) we get

(8) $W\succeq\dfrac{\frac{1}{2}\left(\sum_{x,y\in
V}b(x,y)\left|f_{+}^{2}(x)-f_{+}^{2}(y)\right|\right)^{2}}{\left(\sum_{x\in
V}f_{+}^{2}(x)b(x)\right)^{2}\left(4-2W\right)}.$

Now our goal is to estimate the numerator of (8)

$A:=\frac{1}{2}\left(\sum_{x,y\in
V}b(x,y)\left|f_{+}^{2}(x)-f_{+}^{2}(y)\right|\right)^{2}$

in terms of $h(f)$. Firstly, let us order all the vertices of the graph as
$x_{1},x_{2}\dots,x_{N}$ in the way that $f(x_{i})\preceq f(x_{i+1})$.
Further, let us denote

$S_{k}=\\{x_{k+1},\dots,x_{N}\\}\subset~{}V\;\;\;\mbox{ and
}\;\;\;n_{0}=\max_{i}\\{i\mid x_{i}\not\in V^{+}_{f},0\leq i<N\\},$

i.e. for any $k\geq n_{0}$ we have $S_{k}\subset~{}V^{+}_{f}$.

Now we can rewrite $A$ as

$\displaystyle A$
$\displaystyle=2\left(\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}b(x_{i},x_{j})\left(f_{+}^{2}(x_{j})-f_{+}^{2}(x_{i})\right)\right)^{2}$
$\displaystyle=2\left(\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}b(x_{i},x_{j})\sum_{k=i}^{j-1}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=2\left(\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\sum_{k=i}^{j-1}b(x_{i},x_{j})\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=[\mbox{change limits of summation for $k$ and $j$}]$
$\displaystyle=2\left(\sum_{i=1}^{N-1}\sum_{k=i}^{N-1}\sum_{j=k+1}^{N}b(x_{i},x_{j})\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=[\mbox{change limits of summation for $i$ and $k$}]$
$\displaystyle=2\left(\sum_{k=1}^{N-1}\underbrace{\sum_{i=1}^{k}\sum_{j=k+1}^{N}b(x_{i},x_{j})}_{b(S_{k},V\setminus
S_{k})}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=2\left(\sum_{k=1}^{N-1}{b(S_{k},V\setminus
S_{k})}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=2\left(\sum_{k=n_{0}}^{N-1}{b(S_{k},V\setminus
S_{k})}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle\succeq
2\left(\sum_{k=n_{0}}^{N-1}{h(f)b(S_{k})}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=2h^{2}(f)\left(\sum_{k=n_{0}}^{N-1}{\sum_{i=k+1}^{N}b(x_{i})}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=[\mbox{change limits of summation for $i$ and $k$}]$
$\displaystyle=2h^{2}(f)\left(\sum_{i=n_{0}+1}^{N}{\sum_{k=n_{0}}^{i-1}b(x_{i})}\left(f_{+}^{2}(x_{k+1})-f_{+}^{2}(x_{k})\right)\right)^{2}$
$\displaystyle=2h^{2}(f)\left(\sum_{i=n_{0}+1}^{N}{b(x_{i})}\left(f_{+}^{2}(x_{i})-f_{+}^{2}(x_{n_{0}})\right)\right)^{2}$
$\displaystyle=2h^{2}(f)\left(\sum_{i=n_{0}+1}^{N}{b(x_{i})}f_{+}^{2}(x_{i})\right)^{2}=2h^{2}(f)\left(\sum_{x\in
V}{b(x)}f_{+}^{2}(x)\right)^{2},$

where we have used several times that $f_{+}(x_{k})=0$ for any $k\leq n_{0}$.
Applying the last estimate to (8) we obtain

$W\succeq\dfrac{2h^{2}(f)}{(4-2W)},$

from where immediately follows the statement of the lemma. ∎

###### Proof of Theorem 2.

The proof follows exactly the same outline as proof of the Theorem 3.2 in [1]
and we do not repeat it here. Note that the facts that

$\lambda_{N-1}=\sup_{f\neq 0}\dfrac{\langle\mathcal{L}f,f\rangle}{\langle
f,f\rangle}=\max_{f\neq 0}\dfrac{\langle\mathcal{L}f,f\rangle}{\langle
f,f\rangle},$

where maximum is attained on the corresponding eigenfunction and

$2-\lambda_{N-1}=\inf_{f\neq
0}\dfrac{\langle(2\mathcal{I}-\mathcal{L})f,f\rangle}{\langle f,f\rangle},$

where $\mathcal{I}:\mathfrak{F}_{V}\rightarrow\mathfrak{F}_{V}$ is the
identical operator, follow from decomposition of $f$ by orthonormal basis,
consisting of eigenfunctions of $\mathcal{L}$ (for more details see [14]).

∎

##  4\. Examples

###  4.1. Levi-Civita field

The Levi-Civita field was introduced by Tullio Levi-Civita in 1862 [10]. The
Levi-Civita field is the smallest non-Archimedean real-closed ordered field,
which is Cauchy complete in the order topology. Moreover, it has a subfield,
isomorphic to the field of rational functions, whose elements naturally appear
as edge weights in the theory of electrical networks (see e.g. [12, 13]).

Firstly, we recall the definition and main notations for the Levi-Civita field
$\mathcal{R}$ (see e.g. [2, 8, 15, 16, 17]).

######  Definition 4.

A subset $M$ of the rational numbers $\mathbb{Q}$ is called _left-finite_ if
for every number $r\in\mathbb{Q}$ there are only finitely many elements of $M$
that are smaller than $r$.

######  Definition 5.

[10] We define the Levi-Civita field $\mathcal{R}$ as a field of formal power
series

(9) $a=\sum_{i=0}^{\infty}a_{i}\epsilon^{q_{i}},$

with $a_{i}\in\mathbb{R}$ and $Q=\\{q_{i}\\}_{i=0}^{\infty}\subset\mathbb{Q}$
being left-finite.

Addition and multiplication are defined naturally as for formal power series.
The order is defined as follows: $a\succ 0$ if $a_{0}>0$.

Note that $\epsilon$ here is considered as a ‘fixed’ variable. It is
infinitesimal, as well as e.g. $\epsilon^{2}$,
$5\epsilon^{\frac{1}{2}}+2\epsilon$, etc.

From the definition of elements of the Levi-Civita field follows that to prove
the convergence $r_{n}\to r\in\mathcal{R}$ it is enough to prove, that for any
$m\in\mathbb{N}$ exists $N_{0}\in\mathbb{N}$ such that
$|r_{n}-r|<\epsilon^{m}$ for any $n>N_{0}$.

###  4.2. Examples of graphs over $\mathcal{R}$

######  Example 1.

Let us consider a family of complete graphs $V(n)$ with $3$ vertices and
weights as at the Figure 1 (i.e. $b(x,y)=\epsilon^{n}$ in $V(n)$).

$x$$y$$z$$\epsilon^{n}$$1$$1$ Figure 1.

One can check, that non-constant eigenvectors of the Laplacian are

$v_{1}(n)=(-1,1,0)\mbox{ and
}v_{2}(n)=\left(-\dfrac{1}{1+\epsilon^{n}},-\dfrac{1}{1+\epsilon^{n}},1\right)$

with the corresponding eigenvalues

$\lambda_{1}(n)=\dfrac{1+2\epsilon^{n}}{1+\epsilon^{n}}\mbox{ and
}\lambda_{2}(n)=\dfrac{2+\epsilon^{n}}{1+\epsilon^{n}}=2-\epsilon^{n}+\epsilon^{2n}-\epsilon^{3n}\dots,$

where for $\lambda_{2}(n)$ we have represented ratio as formal power series.

Considering all the posibilities, one can see that the dual Cheeger constant

$\overline{h}(n)=\dfrac{2}{2+\epsilon^{n}}=1-\dfrac{\epsilon^{n}}{2}+\dfrac{\epsilon^{2n}}{4}-\dfrac{\epsilon^{3n}}{8}\dots$

is attained on the partition of vertices $\\{x,y\\}\cup\\{z\\}$. Then

$1+\sqrt{1-(1-\overline{h}(n))^{2}}=2-\dfrac{\epsilon^{2n}}{8}+\dfrac{\epsilon^{3n}}{8}\dots.$

Note that the difference

$1+\sqrt{1-(1-\overline{h}(n))^{2}}-2\overline{h}(n)\to 0,\mbox{ as
}n\to\infty$

in a sence of the ordered topology. Indeed, the graph $V(n)$ differ from a
bipartite graph by the edge $x\sim y$, whose weight $b(x,y)=\epsilon^{n}$ is
infinitesimal and the weights converge to $0$ as $n\to\infty$.

######  Example 2.

Let us consider a complete graph $V$ with $N=2K$ vertices and the following
edge weights: let fix a partition of vertices $V=V_{1}\cup V_{2}$ with
$\\#V_{1}=\\#V_{2}=K$ and let $b(x,y)=\epsilon^{n}$ (where $n\in\mathbb{N}$ is
fixed) if $x,y$ belong to the same $V_{i}$ and $b(x,y)=1$ otherwise. Firstly
we calculate the dual Cheeger constant for this graph. For any $x\in V$ we
have $b(x)={K+(K-1)\epsilon^{n}}$. Let $U_{1},U_{2}$ be disjoint subsets of
$V$.

1 case. Let $\\#U_{1}=K_{1}>K$ and then $\\#U_{2}=K_{2}<K$. In this case

$\overline{h}(U_{1},U_{2}):=\dfrac{2\sum_{x\in U_{1},y\in
U_{2}}b(x,y)}{(K_{1}+K_{2})(K+(K-1)\epsilon^{n})}$

is maximazed for fixed $K_{1},K_{2}$ as

$\overline{h}(K_{1},K_{2}):=\dfrac{KK_{2}+(K_{1}-K_{2})\epsilon^{n}}{(K_{1}+K_{2})(K+(K-1)\epsilon^{n})},$

i.e. when $U_{1}\supset V_{1}$ (or $U_{1}\supset V_{2}$). Further,

(10) $\displaystyle\overline{h}(K_{1},K_{2})=$
$\displaystyle\dfrac{2(KK_{2}+(K_{1}-K)\epsilon^{n})}{(K_{1}+K_{2})(K+(K-1)\epsilon^{n})}$
(11) $\displaystyle\preceq$
$\displaystyle\dfrac{2K(K_{2}+1)}{(K+1+K_{2})(K+(K-1)\epsilon^{n})},$

since $K_{1}\succeq K+1$ and $(K_{1}-K)\epsilon^{n}$ is infinitesimal.
Therefore,

$\overline{h}(K_{1},K_{2})\leq\overline{h}(V_{1},U),$

where $\\#U=K_{2}+1,U\subset V_{2}$ and it is enough to consider the following
case.

2 case. Let $\\#U_{1}=K_{1}\leq K$ and $\\#U_{2}=K_{2}\leq K$. If

$\overline{h}(U_{1},U_{2}):=\dfrac{2\sum_{x\in U_{1},y\in
U_{2}}b(x,y)}{(K_{1}+K_{2})(K+(K-1)\epsilon^{n})}$

then the maximum of $\overline{h}(U_{1},U_{2})$ for fixed $K_{1},K_{2}$ is
attained when all the edges between $U_{1}$ and $U_{2}$ have weight $1$ (which
is possible for $U_{1}\subset V_{1}$ and $U_{2}\subset V_{2}$). This maximum
is

$\overline{h}(K_{1},K_{2})=\dfrac{2K_{1}K_{2}}{(K_{1}+K_{2})(K+(K-1)\epsilon^{n})}$

and further, maximazing the ratio ${K_{1}K_{2}}/(K_{1}+K_{2})$ gives

$\max_{K_{1}\leq K,K_{2}\leq
K}\overline{h}(K_{1},K_{2})=\dfrac{2K^{2}}{2K(K+(K-1)\epsilon^{n})}=\dfrac{K}{K+(K-1)\epsilon^{n}},$

where the maximum is attained on $K_{1}=K_{2}=K$.

From all the above, the dual Cheeger constant for the given graph is

$\displaystyle\overline{h}$ $\displaystyle=\dfrac{K}{K+(K-1)\epsilon^{n}}$
$\displaystyle=1-\left(\frac{K-1}{K}\right)\epsilon^{n}+\left(\frac{K-1}{K}\right)^{2}\epsilon^{2n}+\left(\frac{K-1}{K}\right)^{3}\epsilon^{3n}\dots,$

where we have represented ratio as formal power series.

Therefore, taking $n$ large enough we can get $|1-\overline{h}|\prec r$ for
any given $r\in\mathcal{R}$. It is an intuitively clear result, since such a
graph is close to bipartite (all edges in the same part of partition have
weight $\epsilon^{n}$, while all edges between $V_{1}$ and $V_{2}$ have weight
$1$).

The matrix of Laplacian in this case is

$\frac{1}{B}\left(\begin{array}[]{ccccc|ccccc}B&-\epsilon^{n}&\dots&-\epsilon^{n}&-\epsilon^{n}&-1&-1&\dots&-1&-1\\\
-\epsilon^{n}&B&\dots&-\epsilon^{n}&-\epsilon^{n}&-1&-1&\dots&-1&-1\\\
\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\\
-\epsilon^{n}&-\epsilon^{n}&\dots&B&-\epsilon^{n}&-1&-1&\dots&-1&-1\\\
-\epsilon^{n}&-\epsilon^{n}&\dots&-\epsilon^{n}&B&-1&-1&\dots&-1&-1\\\
\hline\cr-1&-1&\dots&-1&-1&B&-\epsilon^{n}&\dots&-\epsilon^{n}&-\epsilon^{n}\\\
-1&-1&\dots&-1&-1&-\epsilon^{n}&B&\dots&-\epsilon^{n}&-\epsilon^{n}\\\
\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\\
-1&-1&\dots&-1&-1&-\epsilon^{n}&-\epsilon^{n}&\dots&B&-\epsilon^{n}\\\
\lx@intercol\hfil\underbrace{-1\;\;\;\;-1\;\;\;\dots\;\;\;-1\;\;-1}_{K}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\underbrace{-\epsilon^{n}\;\;-\epsilon^{n}\;\;\dots\;\;\;-\epsilon^{n}\;\;\;\;B}_{K}\hfil\lx@intercol\\\
\end{array}\right),$

where $B=b(x)={K+(K-1)\epsilon^{n}}$. Then the eigenvectors are:

  * •

$v_{0}=(1,1,\dots,1,1)$ with an eigenvalue $\lambda_{0}=0$;

  * •

$v_{m}=(-1,0,\dots,0,1,0,\dots,0),$ where $1$ stays at the place $(m+~{}1)$
for $m=1,\dots,K-1$ with eigenvalues

$\lambda_{m}=\dfrac{K+K\epsilon^{n}}{K+(K-1)\epsilon^{n}};$

  * •

$v_{m}=(\underbrace{0,0,0\dots 0}_{K},-1,0,\dots,0,1,0,\dots,0),$ where $1$
stays at the place $(m+2)$ for $m=K,\dots,N-2$ with the same eigenvalues as in
the previous case

$\lambda_{m}=\dfrac{K+K\epsilon^{n}}{K+(K-1)\epsilon^{n}};$

  * •

$v_{N-1}=({\underbrace{1,1,\dots,1,1}_{K}},{\underbrace{-1,-1,\dots,-1,-1}_{K}}),$
with the eigenvalue

$\lambda_{N-1}=\dfrac{2K}{K+(K-1)\epsilon^{n}}.$

Note that $\lambda_{N-1}=2\overline{h}$ and can be written as formal power
series as

$\displaystyle\lambda_{N-1}=2-2\frac{K-1}{K}\epsilon^{n}+2\left(\frac{K-1}{K}\right)^{2}\epsilon^{2n}-2\left(\frac{K-1}{K}\right)^{3}\epsilon^{3n}\dots.$

From the other hand,

$\displaystyle 1+\sqrt{1-(1-\overline{h})^{2}}$
$\displaystyle=1+\sqrt{\dfrac{K^{2}+2K(K-1)\epsilon^{n}}{(K+(K-1)\epsilon^{n})^{2}}}$
$\displaystyle=2-\frac{(K-1)^{2}}{2K^{2}}\epsilon^{2n}+\frac{(K-1)^{3}}{K^{3}}\epsilon^{3n}\dots.$

Therefore, the estimate

$\lambda_{N-1}\preceq 1+\sqrt{1-(1-\overline{h})^{2}}$

is precise up to the order $\epsilon^{n}$ for the considered graph and the
difference

$1+\sqrt{1-(1-\overline{h})^{2}}-\lambda_{N-1}$

converge to $0$ in a sence of convergence in order topology in $\mathcal{R}$
if we consider a family of graphs with $n\to\infty$.

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