# CDM and SIDM Interpretations of the Strong Gravitational Lensing Object
JWST-ER1

Demao Kong Department of Physics and Astronomy, University of California,
Riverside, California 92521, USA  Daneng Yang Department of Physics and
Astronomy, University of California, Riverside, California 92521, USA  Hai-Bo
Yu Department of Physics and Astronomy, University of California, Riverside,
California 92521, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

van Dokkum et al. (2023) reported the discovery of JWST-ER1, a strong lensing
object at redshift $z\approx 2$, using data from the James Webb Space
Telescope. The lens mass within the Einstein ring is $5.9$ times higher than
the expected stellar mass from a Chabrier initial mass function, indicating a
high dark matter density. In this work, we show that a cold dark matter halo,
influenced by gas-driven adiabatic contraction, can account for the observed
lens mass. We also derive constraints on dark matter self-interacting cross
section per particle mass $\sigma/m$. For $\sigma/m\gtrsim 0.3~{}{\rm
cm^{2}/g}$ within a $10^{13}~{}M_{\odot}$ halo, the density profile is too
shallow to be consistent with the measured lens mass of JWST-ER1, while
$\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ is well allowed. Intriguingly, self-
interacting dark matter with $\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ can
simultaneously explain the dark matter density profile of early-type galaxies
at redshift $z\approx 0.2$, where adiabatic contraction is not observed
overall.

Dark matter (353); Strong gravitational lensing (261); Early-type galaxies
(429); Galaxy dark matter halos (1880)

††software: NumPy (Harris et al., 2020), SciPy (Virtanen et al., 2020),
Matplotlib (Hunter, 2007), Colossus (Diemer, 2018)

##  1 Introduction

Recently, van Dokkum et al. (2023) used data from the James Webb Space
Telescope COSMOS-Web survey and found a strong lensing object JWST-ER1 at
redshift $z\approx 2$. The object consists of a complete Einstein ring with a
radius of $6.6~{}{\rm kpc}$ and a compact early-type galaxy (JWST-ER1g) with
an effective radius of $1.9~{}{\rm kpc}$. Within the ring, the total mass is
$6.5\times 10^{11}~{}M_{\odot}$, while the stellar mass mass is $1.1\times
10^{11}~{}M_{\odot}$ assuming a Chabrier initial mass function (Chabrier,
2003). The mass gap is large, indicating that JWST-ER1g has a dense halo.
However, for a typical Navarro-Frenk-White (NFW) halo (Navarro et al., 1997),
the dark matter density is not high enough to fully account for the mass gap,
unless the halo mass is $10^{14}~{}M_{\odot}$ (van Dokkum et al., 2023), which
would be rare at $z\sim 2$, significantly higher than expected in the stellar
mass-halo mass relation (Behroozi et al., 2013).

Mercier et al. (2023) also analyzed the JWST-ER1 system and found the total
mass is $3.7\times 10^{11}~{}M_{\odot}$ within the Einstein ring, a factor of
$2$ smaller than that in van Dokkum et al. (2023), and a $10^{13}~{}M_{\odot}$
halo is favored for JWST-ER1g. The main reason is that their redshift
measurements of the background source are different: $z\approx 2.98$ in van
Dokkum et al. (2023), whereas $z\approx 5.48$ in Mercier et al. (2023); their
other measurements are well consistent.

Another complication is that the population of low-mass stars could be higher
than expected. For example, Salpeter-like initial mass functions can give rise
to a good fit (van Dokkum et al., 2023), though the bottom-light Chabrier form
is overall favored for quiescent galaxies at $z\gtrsim 2$ (Esdaile et al.,
2021; Belli et al., 2014). While more work is needed to further improve our
understanding of the JWST-ER1 system, we explore its rich implications for
probing dark matter physics.

In this work, we interpret the observations of JWST-ER1 in collisionless cold
dark matter (CDM) and self-interacting dark matter (SIDM) scenarios. In CDM,
the halo can become denser due to adiabatic contraction induced by the infall
and condensation of baryons (Blumenthal et al., 1986; Gnedin et al., 2004). We
will show that the contraction effect on the halo is beyond the stellar
effective radius and it is important for the interpretation of JWST-ER1. After
including adiabatic contraction in modeling the halo of JWST-ER1g, the favored
halo mass is reduced to $3\times 10^{13}~{}M_{\odot}$, to be consistent with
the measurement in van Dokkum et al. (2023); or $6\times 10^{12}~{}M_{\odot}$
in Mercier et al. (2023).

We will also demonstrate that SIDM with a self-interacting cross section per
particle mass $\sigma/m\gtrsim 0.3~{}{\rm cm^{2}/g}$ in a
$10^{13}~{}M_{\odot}$ halo is disfavored, as the halo becomes too shallow to
be consistent with the measured lens mass of JWST-ER1, while $\sigma/m\approx
0.1~{}{\rm cm^{2}/g}$ is well allowed. These are the first constraints on SIDM
from strong lensing objects at the highest redshift discovered to date, and
they are largely robust to the exact value of the lens mass of JWST-ER1.

Furthermore, we will show that SIDM with $\sigma/m\approx 0.1~{}{\rm
cm^{2}/g}$ can simultaneously explain NFW-like dark matter density profiles of
early-type galaxies at $z\approx 0.2$. For these galaxies, contracted CDM
halos are overall too dense to be consistent with the measurements of strong
lensing observations, see, e.g., Shajib et al. (2021). Our analysis is based
on a concrete example J1636+4707 from Shajib et al. (2021). Additionally, we
will discuss our SIDM constraints in the context of SIDM models that have been
proposed to explain the diversity of dark matter distributions in other
galactic systems.

##  2 Initial conditions

In this section, we present initial conditions for modeling the mass
distribution of the early-type galaxy JWST-ER1g. van Dokkum et al. (2023) used
a Sérsic profile (Sérsic, 1963) to fit the light distribution of stars in
JWST-ER1g, and found that the effective radius is $R_{e}=1.9\pm 0.2~{}{\rm
kpc}$ and the Sérsic index is $n=5.0\pm 0.6$. The total stellar mass is
$1.3^{+0.3}_{-0.4}\times 10^{11}~{}M_{\odot}$, assuming a Chabrier initial
mass function (Chabrier, 2003).

In our study, we use a Hernquist profile to model the stellar distribution
(Hernquist, 1990)

$\rho_{b}(r)=\frac{M_{b}}{2\pi a^{3}}\frac{a^{4}}{r\left(r+a\right)^{3}},$ (1)

where $M_{b}$ is the total stellar mass and $a$ is the characteristic radius.
We take $M_{b}=1.3\times 10^{11}~{}{M_{\odot}}$ and $a=0.79~{}{\rm kpc}$.
Since the Hernquist profile is spherically symmetric and has a simple
analytical form, it can be easily implemented in semi-analytical methods and
N-body simulations that we will use.

We take the profile in Equation 1 to calculate the projected mass profile of
the stars as

$M_{b}(R)=2\pi\int_{0}^{R}\int_{-\infty}^{+\infty}\rho_{b}(\sqrt{R^{\prime
2}+z^{2}})dzR^{\prime}dR^{\prime},$ (2)

and compare it to the Sérsic profile with $R_{e}\approx 1.9~{}{\rm kpc}$ and
$n=5$ of JWST-ER1g (van Dokkum et al., 2023). Figure 1 shows the comparison
between the Hernquist (solid-black) and Sérsic (dashed-black) stellar
profiles. The agreement is within $15\%$ for $R\gtrsim 0.4~{}{\rm kpc}$. The
Hernquist profile can underestimate the mass by $30\%$ towards the center
$R\sim 0.1~{}{\rm kpc}$, but even with the deficiency, the contraction effect
is sufficiently strong.

For the dark matter halo of JWST-ER1g, we assume an NFW profile as the initial
condition (Navarro et al., 1997) ,

$\rho_{\rm NFW}(r)=\frac{\rho_{s}r^{3}_{s}}{r(r+r_{s})^{2}},$ (3)

where we take the scale density and radius as $\rho_{s}=7.8\times
10^{6}~{}M_{\odot}/{\rm kpc^{3}}$ and $r_{s}=70~{}\rm{kpc}$, respectively. The
corresponding halo mass is $M_{200}=3.0\times 10^{13}~{}{M_{\odot}}$ and
concentration $c_{200}=4.6$ from the concentration-mass relation at $z=2$ with
the scatter $0.11\mathrm{dex}$ (Dutton & Macciò, 2014). The concentration is
$1\sigma$ above the cosmological median at $z\approx 2$. The halo mass is
consistent with the expectation from the stellar-halo mass relation (Behroozi
et al., 2013). Note we have used the following relations

$\rho_{s}=\frac{c^{3}_{200}}{3f(c_{200})}\Delta\rho_{\rm
crit}(z),~{}r_{s}=\frac{r_{200}}{c_{200}},$ (4)

where $\Delta\approx 200$, $\rho_{\rm crit}(z)$ is the critical density of the
universe at redshift $z$, $r_{200}$ is the virial radius of the halo, and
$f(c_{200})=\ln(1+c_{200})-c_{200}/(1+c_{200})$.

Figure 1: The projected mass profile of the Hernquist profile used for
modeling the stellar distribution of JWST-ERg, compared to the Sérsic profile
from van Dokkum et al. (2023).

##  3 CDM interpretation

In response to the infall and condensation of baryons, a CDM halo becomes
denser through the process of adiabatic contraction, and this effect can be
modeled using a semi-analytical method (Blumenthal et al., 1986). In this
work, we follow the one in Gnedin et al. (2004), which was calibrated against
hydrodynamical simulations. One introduces an adiabatic invariant
$M(\bar{r})r$, where $M(\bar{r})$ is the total mass within the orbital-
averaged radial position $\bar{r}$. It follows the relation $\bar{x}=Ax^{w}$
with $x\equiv r/r_{200}$, $A\approx 0.85\pm 0.05$ and $w\approx 0.8\pm 0.02$.
Assuming that baryonic and dark matter follow the same initial radial profile,
we can use the adiabatic invariant and obtain the final radius of dark matter
particles $r_{f}$, which are initially located at $r$,

$\frac{r}{r_{f}}=1-f_{b}+\frac{M_{b}(\bar{r}_{f})}{M_{i}(\bar{r})},$ (5)

where $f_{b}\equiv M_{b}/M_{200}$ is the baryon mass fraction,
$M_{b}(\bar{r}_{f})$ is the final baryon mass profile, and $M_{i}(\bar{r})$
the initial total mass profile. In our study, the final baryon and initial
total density profiles are given by Equations 1 and 3, respectively. For given
$r$, we solve Equation 5 to obtain $r_{f}$, and obtain the density profile of
a contracted CDM halo as $M_{{\rm dm}}(r_{f})=(1-f_{b})M_{i}(r)$. In practice,
we use a public code provided by Jiang et al. (2023) to calculate $M_{\rm
dm}(r_{f})$.

Figure 2: JWST-ER1g: The projected halo mass profile of the contracted CDM
halo (solid-black), the initial NFW halo (dashed-black), the SIDM halos with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ (solid-magenta) and $0.3~{}{\rm cm^{2}/g}$
(solid-orange). The corresponding SIDM halos without including the influence
of baryons are shown for comparison (dotted). The measured value is
$5.4^{+3.7}_{-1.5}\times 10^{11}~{}{M_{\odot}}$ within the Einstein radius
$6.6~{}{\rm kpc}$ (red diamond), after subtracting the baryon mass from the
total enclosed mass (van Dokkum et al., 2023).

Figure 2 shows the projected mass profile of the contracted (solid-black) and
initial (dashed-black) CDM halos. We see that the contracted CDM halo is
sufficiently dense to be consistent with the projected halo mass within the
Einstein radius $6.6~{}{\rm kpc}$ of JWST-ER1g $5.4^{+3.7}_{-1.5}\times
10^{11}~{}M_{\odot}$ (red diamond), while the initial halo is too shallow. The
projected mass within $6.6~{}{\rm kpc}$ is increased by $30\%$ from $3.8\times
10^{11}~{}M_{\odot}$ to $5.4\times 10^{11}~{}M_{\odot}$ due to adiabatic
contraction. Thus the CDM scenario can explain the strong lensing observations
of JWST-ER1 even with a bottom-light Chabrier initial mass function.

The choice of $M_{200}$ and $c_{200}$ values is not unique, but we have
checked that the halo mass is the dominant factor. If the halo mass is
$M_{200}=10^{13}~{}{M_{\odot}}$, a factor of $3$ smaller than the fiducial
value we assume, the concentration needs to be increased to $c_{200}\approx
9.5$, which is $4\sigma$ above the cosmological median at $z\approx 2$ (Dutton
& Macciò, 2014). For a fixed halo mass, the projected mass of a contracted CDM
halo has a mild dependence on the initial concentration; see Appendix A for
details.

From Figure 2, we also see that the impact of adiabatic contraction on the
halo is beyond the central region characterized by the stellar effective
radius $1.9~{}{\rm kpc}$. This is because the initial distribution of gas is
spread out over the entire halo, and as the gas cools and condenses towards
the center, the entire halo contracts (Blumenthal et al., 1986; Gnedin et al.,
2004). We have further checked that the contraction effect is strong enough to
explain the high mass of JWST-ER1g even if the stellar potential grows from
the center; see Appendix B for the demonstration using N-body simulations.

In Appendix C, we present a similar analysis using the measurements from
Mercier et al. (2023). After the contraction effect is included, a CDM halo
with $M_{200}\approx 10^{13}~{}M_{\odot}$ is too dense to be consistent with
the measured lens mass in Mercier et al. (2023) and a $6\times
10^{12}~{}{M_{\odot}}$ halo with low concentration is favored, which is on the
lower end of the stellar mass-halo mass relation for the given stellar mass
(Behroozi et al., 2013).

##  4 SIDM interpretation

In this section, we study the interpretation of JWST-ER1 in the SIDM scenario
(Tulin & Yu, 2018; Adhikari et al., 2022). Dark matter self-interactions could
thermalize the inner halo and produce a shallow density core, which seemed
contradictory to the observations of JWST-ER1. However, in the presence of
baryons, SIDM thermalization actually leads to a small core and a high inner
halo density (Kaplinghat et al., 2014; Kamada et al., 2017; Creasey et al.,
2017; Elbert et al., 2018; Robertson et al., 2018, 2019, 2021; Sameie et al.,
2018; Despali et al., 2019; Santos-Santos et al., 2020; Feng et al., 2021;
Sameie et al., 2021; Zhong et al., 2023).

To model the SIDM density profile for JWST-ER1g, we follow a semi-analytical
method in Kaplinghat et al. (2014, 2016), which has been tested extensively
using hydrodynamical SIDM simulations, see, e.g., Robertson et al. (2018,
2021). In the inner region, where SIDM thermalization occurs over the age of
the galaxy, the halo density profile $\rho_{\rm iso}(r)$ can be obtained by
solving the isothermal Jeans equation

$\sigma^{2}_{0}\nabla^{2}\ln\rho_{\rm iso}=-4\pi G(\rho_{\rm iso}+\rho_{b}),$
(6)

where $G$ is the Newton constant, $\sigma_{0}$ is the 1D velocity dispersion
of dark matter particles, $\rho_{b}(r)$ is the final baryon density profile.
In the outer region, where the thermalization effect can be negligible, the
halo is in the collisionless limit and can be modeled with a CDM density
profile of $\rho_{\rm cdm}(r)$. The boundary is set by the characteristic
radius $r_{1}$, at which self-scattering occurs once per age of the galaxy
$t_{\rm age}$, i.e.,

$\frac{\sigma}{m}\rho_{\rm
iso}(r_{1})\frac{4}{\sqrt{\pi}}\sigma_{0}t_{\rm{age}}=1,$ (7)

where $\sigma/m$ is the self-interacting cross section per particle mass. The
full SIDM density profile is composed of two parts: $\rho_{\rm iso}$ for
$r<r_{1}$ and $\rho_{\rm cdm}$ for $r>r_{1}$; they satisfy the matching
conditions $\rho_{\rm iso}(r_{1})\approx\rho_{\rm cdm}(r_{1})$ and $M_{\rm
iso}(r_{1})\approx M_{\rm cdm}(r_{1})$.

We set $\rho_{b}(r)$ to be the Hernquist profile in Equation 1 and $\rho_{\rm
cdm}(r)$ to be the contracted CDM profile shown in Figure 2 (solid-black), and
$t_{\rm age}=3.4~{}{\rm Gyr}$, corresponding to the redshift $z\approx 2$. We
also assume a constant cross section, which should be regarded an effective
cross section for the halo (Yang & Yu, 2022; Yang et al., 2024; Outmezguine et
al., 2023; Yang et al., 2023b; Fischer et al., 2023). For a given value of
$\sigma/m$, we solve Equations 6 and 7 to find the SIDM density profile. In
our numerical study, we again use the public code by Jiang et al. (2023) to
obtain the SIDM density profile, and further test it using N-body simulations;
see Appendix B.

Figure 2 shows the projected halo mass profile of the SIDM halos assuming
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ (solid-magenta) and $0.3~{}{\rm cm^{2}/g}$
(solid-orange). The projected halo mass decreases as the cross section
increases. At $z\approx 2$, the halo is at the stage of core formation and a
larger cross section leads to a larger core and smaller inner density. With
the measurement uncertainty, the strong lensing object JWST-ER1 disfavors SIDM
with $\sigma/m\gtrsim 0.3~{}{\rm cm^{2}/g}$ in early-type galaxies with a
$10^{13}~{}{M_{\odot}}$ halo, but $\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ is
well allowed. In Appendix C, we show that the SIDM constraints are slightly
stronger when using the measurements in Mercier et al. (2023), but overall,
they are largely robust to the exact value of the lens mass of JWST-ER1.

Figure 2 also shows the projected mass profile for the SIDM halos with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ (dotted-magenta) and $0.3~{}{\rm cm^{2}/g}$
(dotted-orange) without including the baryons, i.e., neglecting the term
$\rho_{b}(r)$ in Equation 6. These SIDM halos are too shallow to explain the
lens mass of JWST-ER1. In fact, they have a $20~{}{\rm kpc}$ density core. In
contrast, the SIDM thermalization leads to the increase of the inner density
by several orders of magnitude in the presence of the baryons. Thus JWST-ER1
provides a crucial test on the mechanism of tying dark matter and baryon
distributions in SIDM, which was first proposed in Kaplinghat et al. (2014).

It is interesting to note that for $\sigma/m=0.3~{}{\rm cm^{2}/g}$, the SIDM
density profile at $t_{\rm age}\approx 3.4~{}{\rm Gyr}$ well resembles the
initial NFW profile without adiabatic contraction. From Equation 7, we see
that $\sigma/m$ and $t_{\rm age}$ are degenerate in determining the SIDM
density profile. In other words, the density profile with $\sigma/m=0.3~{}{\rm
cm^{2}/g}$ and $t_{\rm age}\approx 3.4~{}{\rm Gyr}$ is equivalent to that with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ and $t_{\rm age}\approx 10~{}{\rm Gyr}$. Thus
in SIDM with $\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ dark matter halos of
early-type galaxies at $z\sim 0.2$ are expected to be NFW-like, i.e., their
density profiles are shallower than those predicted in contracted CDM halos,
as we demonstrate in the next section.

##  5 Early-type galaxies at $z\sim 0.2$

Figure 3: J1636+4707: The surface density profile of the SIDM halo with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ (solid-magenta), the stars (cyan) and the NFW-
like halo (gray) from observations ($1\sigma$ band) (Shajib et al., 2021). For
comparison, the CDM halo with adiabatic contraction (solid-black) and the SIDM
halo without baryons (dotted-magenta) are shown.

Shajib et al. (2021) analyzed imaging data of a sample of $23$ galaxy-galaxy
lenses, and found that the dark matter halos of the elliptical galaxies at
$z\sim 0.2$ are NFW-like and on average there are no signs of adiabatic
contraction. This result is robust to the assumption of stellar mass-to-light
ratio models. With the insights gained in the last section, we explore the
SIDM interpretation of the NFW-like halo properties of early-type galaxies at
$z\sim 0.2$.

Take J1636+4707 as a concrete example. We use the Hernquist profile in
Equation 1 to fit the observed stellar density profile in Shajib et al. (2021)
and determine the relevant parameters as $M_{b}=2.2\times 10^{11}~{}M_{\odot}$
and $a=2.7~{}{\rm kpc}$. Note that J1636+4707 has a stellar size of
$R_{e}\approx 6.5~{}{\rm kpc}$, much larger than $R_{e}=1.9~{}{\rm kpc}$ of
JWST-ER1g. The initial NFW halo parameters are found to be $\rho_{s}=5.6\times
10^{6}~{}M_{\odot}/{\rm kpc^{3}}$ and $r_{s}=67~{}{\rm kpc}$, corresponding to
a halo mass of $M_{200}=3.0\times 10^{13}~{}M_{\odot}$ at $z\approx 0.2$. We
set $\sigma/m=0.1~{}{\rm cm^{2}/g}$ and $t_{\rm age}=11.3~{}{\rm Gyr}$
($z\approx 0.2$).

Figure 3 shows the surface density profile of the SIDM halo (solid-magenta),
as well as the NFW-like halo (gray) and the baryon component (cyan) inferred
from observations within the $1\sigma$ credible range (Shajib et al., 2021).
For comparison, we also show the CDM halo with adiabatic contraction (solid-
black) and the SIDM halo without baryons (dotted-magenta). The SIDM density
profile is well consistent with the NFW-like profile from the lensing
measurements (Shajib et al., 2021). On the other hand, the contracted CDM halo
is too dense towards the inner regions, while the SIDM halo without baryons
has a $20~{}{\rm kpc}$ core and is too shallow. We have further confirmed
these results using controlled CDM and SIDM N-body simulations; see Appendix
B.

##  6 Discussion and Conclusion

We have shown that a contracted CDM halo can explain the high halo density of
JWST-ER1g at $z\approx 2$. This in turn puts strong constraints on the dark
matter self-interaction cross section in $10^{13}~{}M_{\odot}$ halos:
$\sigma/m\gtrsim 0.3~{}{\rm cm^{2}/g}$ is disfavored, while $\sigma/m\approx
0.1~{}{\rm cm^{2}/g}$ is well allowed. These constraints are largely robust to
the exact value of the lens mass of JWST-ER1. Intriguingly, SIDM with
$\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ can also explain NFW-like halo
properties of early-type galaxies at low redshift $z\approx 0.2$, as the SIDM
halo further evolves and the density profile becomes shallower.

Despali et al. (2019) performed hydrodynamical SIDM simulations of formation
of early-type galaxies assuming $\sigma/m=1~{}{\rm cm^{2}/g}$. They showed
that when baryons are included, the difference between CDM and SIDM
predictions becomes small and they are largely consistent with various
observational constraints; see Despali et al. (2022); Mastromarino et al.
(2023); McDaniel et al. (2021) for further investigations.

Overall, their findings are in good agreement with ours, i.e., the baryons
play a vital role in explaining the observations of early-type galaxies. We
have checked that the simulated SIDM halos in Despali et al. (2019) that could
potentially host JWST-ER1g are not dense enough; see their Figure 7.
Nevertheless, their most massive SIDM halos are cored at $z\approx 0.2$,
shallower than the corresponding CDM counterparts, in alignment with what we
found. More work is needed to test SIDM predictions of early-type galaxies for
$\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ using hydrodynamical simulations.

Studies show that $\sigma/m\lesssim 0.1~{}{\rm cm^{2}/g}$ in
$10^{15}~{}M_{\odot}$ cluster halos (Kaplinghat et al., 2016; Andrade et al.,
2021; Eckert et al., 2022; Adhikari et al., 2024), and $\sigma/m\lesssim
1~{}{\rm cm^{2}/g}$ in $10^{14}~{}{M_{\odot}}$ group halos (Sagunski et al.,
2021). Our constraints are consistent with those previous studies, but on a
lower mass scale $\sim 10^{13}~{}M_{\odot}$. On the other hand, for dwarf
galaxies, $\sigma/m\gtrsim 10~{}{\rm cm^{2}/g}$ is favored to explain the
diversity in the dark matter distribution, see, e.g., Valli & Yu (2018); Ren
et al. (2019); Sameie et al. (2020); Zavala et al. (2019); Kahlhoefer et al.
(2019); Correa (2021); Silverman et al. (2022); Correa et al. (2022); Yang et
al. (2023a); Nesti et al. (2023); Nadler et al. (2023); Zhang et al. (2024).
For a viable SIDM model, the cross section must be velocity-dependent,
decreasing towards massive halos (Tulin & Yu, 2018). Interestingly, in the
velocity-dependent SIDM model proposed in Nadler et al. (2023) to explain
observations of extreme halo diversity, the predicted effective cross section
is $0.1~{}{\rm cm^{2}/g}$ in $10^{13}~{}{M_{\odot}}$ halos, consistent with
our constraints from the early-type galaxies.

In the future, more objects like JWST-ER1 could be discovered with JWST and
they would further test our CDM and SIDM interpretations. It would be of great
interest to perform SIDM fits to a large sample of early-type galaxies at low
redshift, such as those in Shajib et al. (2021); Tan et al. (2023).
Cosmological hydrodynamical simulations would be needed to further understand
the formation and evolution of early-type galaxies in both CDM and SIDM
scenarios.

This work was supported by the John Templeton Foundation under grant ID #61884
and the U.S. Department of Energy under grant No. de-sc0008541. The opinions
expressed in this publication are those of the authors and do not necessarily
reflect the views of the John Templeton Foundation.

##  Appendix A Varying model parameters

Figure 4: The projected halo mass profiles for six different sets of the model
parameters for JWST-ER1g: $M_{200}$, $c_{200}$, and $M_{b}$. The line style is
the same as in Figure 2.

Figure 4 shows the projected halo mass profiles for six different sets of the
model parameters in the case of JWST-ER1g. For both CDM and SIDM halos, the
projected halo mass profiles have a mild dependence on the initial halo
concentration and the total baryon mass. If the halo mass is
$10^{13}~{}{M_{\odot}}$ or lower, the projected mass within the Einstein
radius will be too low to explain the high dark matter density of JWST-ER1g,
even if we increase the total baryon mass to $2.0\times 10^{11}~{}M_{\odot}$,
$2\sigma$ higher than the mean reported in van Dokkum et al. (2023).

##  Appendix B Controlled N-body simulations

Figure 5: Left panel: The projected halo mass profile of the simulated SIDM
halo at $t=0~{}{\rm Gyr}$ (solid-green), $3.4~{}{\rm Gyr}$ (dashed-green), and
$11.3~{}{\rm Gyr}$ (dashed-dotted-green) for JWST-ER1g. Right panel: The
surface density profile of the simulated SIDM halo at $t=0~{}{\rm Gyr}$
(solid-green) and $11.3~{}{\rm Gyr}$ (dashed-green) for J1636+4707. For both
simulated halos, $\sigma/m=0.1~{}{\rm cm^{2}/g}$. Other curves are the same as
those in Figures 2 and 3.

We perform controlled N-body simulations to model the CDM and SIDM halos of
JWST-ER1g and J1636+4707. For JWST-ER1g, we take the initial NFW halo
parameters as $\rho_{s}=7.8\times 10^{6}~{}{M_{\odot}}/{\rm kpc^{3}}$ and
$r_{s}=70~{}{\rm kpc}$. At the center of the halo, we grow a baryonic
potential following a Hernquist profile. The scale radius is fixed to
$a=0.79~{}{\rm kpc}$ initially and remains a constant, while the mass grows
linearly to $M_{b}=1.3\times 10^{11}~{}M_{\odot}$ within $5~{}{\rm Gyr}$.
Similarly, for J1636+4707, the initial halo parameters are $\rho_{s}=5.6\times
10^{6}~{}{M_{\odot}}/{\rm kpc^{3}}$ and $r_{s}=67~{}{\rm kpc}$, and a growing
Hernquist potential with $a=2.7~{}{\rm kpc}$ that reaches $M_{b}=2.2\times
10^{11}~{}M_{\odot}$. For both cases, we let their CDM halos fully relax in
the presence of the baryonic potential and obtain their contracted CDM density
profiles. Then, we turn on dark matter self-interactions with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ and follow gravothermal evolution of the
halos. We use the public code SpherIC (Garrison-Kimmel et al., 2013) to
generate initial conditions and the code GADGET-2 (Springel et al., 2001;
Springel, 2005), implemented with an SIDM module (Yang & Yu, 2022), to perform
N-body simulations. The total number of simulation particles is $10^{6}$
($2\times 10^{6}$) for JWST-ER1g (J1636+4707), the particle mass is $3.9\times
10^{7}~{}M_{\odot}$ ($1.9\times 10^{7}~{}M_{\odot}$). For both cases, the
softening length is set to $1~{}{\rm kpc}$. Note when reporting evolution time
of the simulated SIDM halos, we do not account for the time for growing the
potential and relaxing the CDM halos.

Figure 5 (left panel) shows the projected mass profile of the contracted CDM
halo ($t=0~{}{\rm Gyr}$, solid-green), the SIDM halo at $t=3.4~{}{\rm Gyr}$
(dashed-green) and $11.3~{}{\rm Gyr}$ (dashed-dotted-green) from our N-body
simulations, in the case of JWST-ER1g. Compared to the contracted CDM halo
induced by baryon infall and condensation (solid-black), our simulated CDM
halo is less dense and the baryons have less impact over large radii. This is
because we purposely grow the potential from the center of the halo, while the
semi-analytical method in Gnedin et al. (2004) is calibrated against
cosmological hydrodynamical simulations that take into account the contraction
effect on the entire halo. Even with the conservative approach to model
adiabatic contraction, our contracted CDM halo is sufficiently dense to be
consistent with the measurement within the uncertainty (red diamond), further
confirming our result that a CDM halo with adiabatic contraction can explain
the high halo density of JWST-ER1g.

Furthermore, the projected mass of the simulated SIDM halo at $t=3.4~{}{\rm
Gyr}$ (dashed-green) agrees with that of the SIDM halo with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ from the semi-analytical method (solid-
magenta) within $10\%$. The latter is slightly denser, because it was produced
by matching to the contracted CDM halo based on Gnedin et al. (2004) (solid-
black); see Equation 7 and we have set $t_{\rm age}=3.4~{}{\rm Gyr}$. We also
show the projected mass profile of the simulated SIDM halo at $t=11.3~{}{\rm
Gyr}$, corresponding to $z\approx 0.2$, and it is close to the initial NFW
profile (dashed-black). Thus SIDM predicts that the halo of JWST-ER1 would be
NFW-like if it further evolved from $z\approx 2$ to $0.2$. It also well agrees
with the SIDM halo with $\sigma/m=0.3~{}{\rm cm^{2}/g}$ at $t_{\rm
age}=3.4~{}{\rm Gyr}$ from the semi-analytical method, confirming our
expectation from the scaling relation $t_{\rm age}\propto(\sigma/m)^{-1}$ as
indicated in Equation 7.

Figure 5 (right panel) shows the surface density of the SIDM halo at
$t=11.3~{}{\rm Gyr}$ (dashed-dotted-green), as well as the contracted CDM halo
($t=0~{}{\rm Gyr}$, solid-green) from our controlled N-body simulations, in
the case of J1636+4707. We again see that the simulated SIDM halo becomes NFW-
like after gravothermal evolution, and its surface density is well within the
range from the measurement Shajib et al. (2021) (gray band). Since the
baryonic potential grows from the center of the halo in our N-body
simulations, its impact on the halo has a smaller range compared to that using
the method in Gnedin et al. (2004) (solid-black).

##  Appendix C Alternative fits

Figure 6: Left panel: The projected mass profile of the contracted CDM halo
(solid-black), the initial NFW halo (dashed-black), the SIDM halos with
$\sigma/m=0.1~{}{\rm cm^{2}/g}$ (solid-magenta) and $0.3~{}{\rm cm^{2}/g}$
(solid-orange) for JWST-ER1g based on the measurement in Mercier et al. (2023)
(blue diamond). Right panel: The projected mass profile of the contracted CDM
halo (solid-black) and the initial NFW halo (dashed-black), where the halo
parameters are chosen such that the NFW halo without adiabatic contraction
could explain the measurement in Mercier et al. (2023) (blue diamond).

We perform a fit to the projected halo mass of JWST-ER1g reported in Mercier
et al. (2023), i.e., $(2.46\pm 0.30)\times 10^{11}~{}M_{\odot}$. We use the
Hernquist profile to model the stellar distribution and fix the parameters as
$M_{b}=1.4\times 10^{11}~{}M_{\odot}$ and $a=0.62~{}{\rm kpc}$. The halo mass
is $M_{200}=6.0\times 10^{12}~{}M_{\odot}$ and concentration is
$c_{200}=2.30$, which is $2\sigma$ below the median at $z\approx 2$. The
corresponding initial NFW parameters are $\rho_{s}=1.9\times
10^{6}~{}M_{\odot}/{\rm kpc^{3}}$ and $r_{s}=80~{}{\rm kpc}$.

Figure 6 (left panel) shows the projected halo mass profile of the contracted
CDM halo (solid-black), the initial NFW halo (dashed-black), as well as the
SIDM halos with $\sigma/m=0.1~{}{\rm cm^{2}/g}$ (solid-magenta) and
$0.3~{}{\rm cm^{2}/g}$ (solid-orange). The contracted halo is sufficiently
dense to be consistent with the measurement in Mercier et al. (2023) (blue
diamond). In contrast, the NFW halo is too shallow. The constraints on SIDM
are slightly stronger than those based on van Dokkum et al. (2023), as Mercier
et al. (2023) reported smaller uncertainties in the measurement of the
projected halo mass. Overall, our SIDM constraints from the JWST-ER1 object
are largely robust to the exact value of the lens mass, and it is clear that
$\sigma/m\approx 0.1~{}{\rm cm^{2}/g}$ is allowed.

Additionally, as shown in Figure 6 (right panel), a $10^{13}~{}M_{\odot}$ NFW
halo could be dense enough to explain the measurement in Mercier et al.
(2023). However, the contracted CDM halo is too dense.

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Geometry of q-bic Hypersurfaces

Raymond Cheng

Submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

under the Executive Committee

of the Graduate School of Arts and Sciences

Columbia University

Raymond Cheng

All Rights Reserved

to 0pt plus 0.7filAbstract2pc
Geometry of \(q\)-bic Hypersurfaces

Raymond Cheng

Traditional algebraic geometric invariants lose some of their potency in
positive characteristic. For instance, smooth projective hypersurfaces may be
covered by lines despite being of arbitrarily high degree. The purpose of this
dissertation is to define a class of hypersurfaces that exhibits such
classically unexpected properties, and to offer a perspective with which to
conceptualize such phenomena.

Specifically, this dissertation proposes an analogy between the eponymous
\(q\)-bic hypersurfaces—special hypersurfaces of degree \(q+1\), with \(q\)
any power of the ground field characteristic, a familiar example given by
the corresponding Fermat hypersurface—and low degree hypersurfaces, especially quadrics and
cubics. This analogy is substantiated by concrete results such as: \(q\)-bic
hypersurfaces are moduli spaces of isotropic vectors for a bilinear form; the
Fano schemes of linear spaces contained in a smooth \(q\)-bic hypersurface are
smooth, irreducible, and carry structures similar to orthogonal Grassmannian;
and the intermediate Jacobian of a \(q\)-bic threefold is purely inseparably
isogenous to the Albanese variety of its smooth Fano surface of lines.

Johan, what a pleasure and privilege it has been. You bring such
clarity, and imbue such spirit to everything that one cannot help but
brave the darkness and indifference of this world in attempts to steal a
bit of truth and beauty. I hope that, some day, I might return with
something that truly delights.

Chao Li, Giulia Saccà, Will Sawin, and Jason Starr: thank you for
agreeing to read a demanding draft of this manuscript, serving on my
defense committee, and sharing helpful comments and questions. Alongside
David Hansen, Eric Katz, Daniel Litt, and Alex Perry: you have taught me
so much and have immensely shaped my outlook on mathematics. I am
grateful for all your guidance.

My fellow travellers Remy van Dobben de Bruyn,
Shizhang Li, Qixiao Ma,
Carl Lian, Monica Marinescu, Dmitrii Pirozhkov,
Noah Olander,
Roy Magen, and
Morena Porzio:
I so cherish our fellowship, and hope that,
in spite of the mountains and oceans that may separate us,
that we might share time and space again.

My dear friends: you have sustained me with your companionship and have brought
me such joy through the years. Among many, I would like to
specially mention:
Daniel Cizin for our late night doorway conversations;
Clara Dolfen for fostering such a warm community;
Sam Mundy for coffee, beer, and much more;
Lauren Tang for sharing language and friendship;
Renatta Picciotto for Moldus III, casatiello, and more;
and finally, to you for everything—thank you.

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# Experimental observation of real spectra in Parity-Time symmetric ZRC dimers
with positive and negative frequencies

Stéphane Boris Tabeu 1,2,3<EMAIL_ADDRESS>Fernande Fotsa-Ngaffo4
Senghor Tagouegni 1,2 Kazuhiro Shouno 5 Aurélien Kenfack-Jiotsa2,6 1
Laboratory of Mechanics, Materials and Structures, Department of Physics,
Faculty of Science,University of Yaounde I, P.O. Box 812,Yaounde, Cameroon  2
Nonlinear Physics and Complex Systems Group, Department of Physics, The Higher
Teacher’s Training College, University of Yaounde I, P.O. Box 47, Yaounde,
Cameroon  3 Laboratory of Electronics and Signal Processing, Department of
Electrical and Telecommunication Engineering, National Advanced School of
Engineering,University of Yaounde I, P.O. Box 8390, Yaounde,Cameroon  4
Institute of Wood technologies, University of Yaounde I, P.O. Box 306,
Mbalmayo, Cameroon  5 Graduate School of Systems and Information Engineering,
University of Tsukuba, Ibaraki 305-8573, Japan  6 High-Tech and Slow
Technology (HITASTEC), P.O. Box 105, Yaounde, Cameroon

(August 27, 2024)

###### Abstract

We present in this work the first experimental observation of oscillations in
Parity-Time symmetric ZRC dimers. The system obtained is of first order
ordinary differential equation due to the use of imaginary resistors. The
coupled cells must share the same type of frequency: positive or negative. We
observed the real and imaginary parts of the voltage across the components of
a ZRC cell. Exceptional points are well identified. This work may be very
useful in the generation of new type of oscillators. It can also be used in
the design of new optoelectronic devices for major applications in the
transport of information and the mimics of two-level systems for quantum
computing.

Non-Hermitian systems, Parity-time symmetry, Imaginary resistor, ZRC-Dimer,
Negative frequency

##  I  Introduction

Non-Hermitian systems are today a major fields of investigations in different
specialities. In 1998, Carl Bender demonstrated that a non-Hermitian systems
can have a real spectra without being Hermitian opening the road to Parity-
Time (PT) Symmetry a1 ; a2 ; a3 . In Quantum Mechanics, this reality is
related to the potential which satisfied
$V\left(x\right)={V^{*}}\left({-x}\right)$ with an even real part and odd
imaginary part a1 ; a2 ; a3 ; a4 ; a5 ; a6 . In Optics, the refractive index
of the medium must satisfied a relation of the same nature
$n\left(x\right)={n^{*}}\left({-x}\right)$ a7 ; a8 ; a9 ; a10 ; a11 ; a12 ;
a13  and $\varepsilon\left(x\right)={\varepsilon^{*}}\left({-x}\right)$ for
dielectric permittivity in Metamaterials a14 ; a15 ; a16 ; a17 . The notion of
Parity-Time Symmetry is also extended to many order fields such as Photonics
a7 ; a8 ; a9 ; a10 ; a11 ; a12 ; a13 , Mechanics a18 ; a19 ; a20 ,
Optomechanics a20 ; a21 ; a22 ; a23 , Acoustics a24 ; a25 ; a26  and
Electronics a27 ; a28 ; a29 ; a30 ; a31  just to name few. In 2010, Reuter
realized the first experimental demonstration of PT-symmetry in Optics by
coupling two waveguides having equal amount of gain and loss a7 . In 2011,
Schindler et al used two active RLC circuits to introduce the notion in
Electronics a27 ; a28 . The loss cell was represented by a natural positive
resistor and the gain cell by a negative resistor using Negative impedance
converter. In 2013, the team of Carl Bender experimentally demonstrate the
real spectra in coupled mechanic oscillators a18 . The Rabi oscillations
occurred near the the exceptional point in the response of the system. More
after it was presented the counterpart of Parity-Time symmetry: The Anti-
Parity Time symmetry. In this new Physics, the loss and the gain are
exclusively in the dimers a30 ; a32 ; a33 ; a34 ; a35 ; a36 ; a37 ; a38 . In
Quantum Mechanics, the potential follows the relation
$V(x)=-{V^{*}}\left(-x\right)$ , the refractive index
$n(x)=-{n^{*}}\left(-x\right)$ in Optics and
$\varepsilon\left(x\right)=-{\varepsilon^{*}}\left({-x}\right)$ for the
dielectric permittivity in Metamaterials. Now the real parts of the potential,
the refractive index of the medium and the permittivity are odd and their
imaginary parts even giving rise to the coupled of Gain- Gain cells or Loss-
Loss cells to made Anti-Parity Time symmetry. In others fields, the rotating
frame is used to create indirectly positive and negative frequencies in the
cells of the dimer such as in lasers gyroscopes a36 . It also used the Coupled
Mode Theory (CMT) and adiabatic elimination in optical waveguides , slowly
varying envelope approximation in Nanophotonics to achieve the same reality.
Many others works are also presented in these directions a30 ; a31 ; a32 ; a33
; a34 ; a35 ; a36 ; a37 . Tabeu et al presented in a30  how to achieve Parity-
Time symmetry and Anti-Parity time symmetry in electronics using imaginary
resistors a31 ; a39 ; r1  to have directly the system of first order ordinary
differential equation without the coupling mode theory or the rotating frame.
The configurations presented were with Gain-Loss, Gain-Gain and Loss-Loss in
the dimer. All these (Anti)-Parity-time systems have many applications such as
sensing and telemetry, non-reciprocal transport of information, generation of
qubits for quantum computing, wireless power transfer, switches, single mode
power Lasers a9 ; a10 ; a11 ; a12 ; a17 ; a30 ; a31 ; a34 ; a38 ; r1 ; a40 ;
a41 ; a42 ; a43 ; a44 ; a45  and others .

In this works, we proposed the experimental realization of ZRC cells and a PT-
dimer based imaginary resistors. Since the Hamiltonian of the resulted system
is the same which the one encounter after some transformations in Optics and
Photonics, it can mimics easily such these systems and gather the bridge for
new devices in Optolectronics and Photoelectronics with the avenue of optical
and photonic quantum computing. The paper is organized as follows: In Sect. II
the RLC and ZRC are presented with their frequencies. In Sect. III, the ZRC
PT-dimer is studied. Its eigenvalues are calculated and some simulations of
its dynamic behavior are driven. In Sect. IV, the experimental realization of
the dimer is presented and the oscillations across the gain and loss cells.
The paper end with a conclusion.

##  II RLC and ZRC Cells

###  II.1 Real volatge in RLC circuit

(

(a)

(

(b)  Figure 1: (a) The RLC cell is constituted of a real resistor $R$, an
inductor $L$ and a capacitor $C$. All the components are mounted in parallel.
The ZRC cell is constituted of an imaginary resistor $Z=jr$, a real resistor
$R$ and a capacitor $C$ all mounted in parallel.

a) b)

The RLC cell contains a real resistor, a capacitor and an inductor Fig.1a(a)
The Kirchhoff’s law applied at the node gives us:

${I^{R}}+{I^{L}}+{I^{C}}=0$ (1)

By applying the Ohm’s law at the borders of the components, we have a second
order form of ordinary differential equation depicted as :

$\frac{{{d^{2}}V}}{{d{t^{2}}}}+\gamma\frac{{dV}}{{dt}}+\omega_{0}^{2}V=0{\rm{}}$
(2)

With $\gamma=\frac{1}{{RC}}$ representing the damping rate and
${\omega_{0}}=\sqrt{\frac{1}{{LC}}}$ the natural frequency of the cell. We
look a solution in the form $V\propto{e^{j\alpha t}}$ and it derives from Eq.
( 2) a second order equation in $\alpha$ :

${\alpha^{2}}-j\gamma-\omega_{0}^{2}=0$ (3)

The final voltage with an appropriate choice of initial conditions is :

$V(t)=\frac{1}{2}{V_{0}}{e^{-\left({\frac{\gamma}{2}}\right)t}}\left({{e^{j\omega
t}}+{e^{-j\omega t}}}\right)$ (4)

${\alpha_{1,2}}=j\left({\frac{\gamma}{2}}\right)\pm\sqrt{\omega_{0}^{2}-{{\left({\frac{\gamma}{2}}\right)}^{2}}}=0$
(5)

with $\omega=\sqrt{\omega_{0}^{2}-{{\left({\frac{\gamma}{2}}\right)}^{2}}}$ .
The system is under-damped if $\omega>0$ . Then, the system is pseudo-
oscillatory with exactly the combination of two voltages with positive and
negative frequencies:

$V(t)={\bf{V}}\left(\omega\right)+\,{\bf{V}}\left({-\omega}\right)\ $ (6)

The result is exactly a real value.

$V(t)={V_{0}}{e^{-\left({\frac{\gamma}{2}}\right)t}}\cos\left({\omega
t}\right)$ (7)

###  II.2 Complex voltage in ZRC circuit

Figure 2: Experimental realization of ZRC circuits with positive and negative
frequencies. Pseudo-oscillations of the real
$\real\left({{\bf{V}}\left(\omega\right)}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\cos\left({\omega t}\right)$ (in blue) and the imaginary
$\imaginary\left({{\bf{V}}\left(\omega\right)}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\sin\left({\omega t}\right)$ (in cyan) parts of the voltage . The
sign of the frequency is the one of the capacitors. the experiments run a
$f\approx\pm{\rm{}}7.96{\rm{}}kHz$ . The others values of components are :
$Z=j2{\rm{}}k\Omega\,$ for the imaginary resistor ;$R=\pm 30\,k\Omega$ for the
real resistors and $C=\pm 10{\rm{}}nF$ for capacitors
Figure 3: The circuit of the ZRC Dimer. Each cell contains in parallel an
imaginary resistor ${Z_{n}}=j{r_{n}}$, a real resistor ${R_{n}}$, a capacitor
${C_{n}}$. The ZRC-cells are coupled by an imaginary resistor $z=jr^{\prime}$
or a capacitor $C^{\prime}$ or by all of them.

(

(a)

(

(b)  Figure 4:  The eigenvalues of the setup corresponding to three cases in
which we have a positive and negative capacitive couplings and at last a case
with both the capacitive and the imaginary couplings. (a) Real parts of the
eigenvalues (b) Imaginary parts of the eigenvalues

a) b)

Figure 5: Relative values of the modules of complex voltages across the gain
and loss cells ${I_{n}}={\left|{{V_{n}}\left(\tau\right)}\right|^{2}}$ with
different initial inputs. In the exact phase, there are normal oscillations
with an amplification of the initial input due the presence of gain and loss
in the setup. In the broken phase, there is an exponential growth and also a
deep difference between the behavior of gain and loss cells

(

(a)

(

(b)

(

(d)

(

(f)

a) b)
c) d)
e) f)

(

(c)

(

(e)

Figure 6:  Experimental observation of real part of voltages in the ZRC PT-
dimer at $\gamma=0.2\,{\gamma_{PT}}$ . The first case with positive frequency
and capacitive
coupling$C={\rm{}}10{\rm{}}nF\,;\,\,C^{\prime}=\,\,2\,nF\,;{\rm{}}Z=j2{\rm{}}k\Omega;\,\,c={\rm{}}0.2\,\,;\,\,\nu={\rm{}}0$:
(a) The real voltage dynamic across the loss and the gain cells
$\left({f\approx 2.23\,kHz\,\,;\,\,R\approx 59.16\,k\Omega}\right)$ (b) The
Lissajous’s curve. The second case with negative frequency and capacitive
coupling
$C={\rm{}}-10{\rm{}}nF\,;\,\,C^{\prime}=\,\,2\,nF\,;{\rm{}}Z=j2{\rm{}}k\Omega;\,\,c={\rm{}}-0.2\,\,;\,\,\nu={\rm{}}0$:
(c) The real voltage dynamic across the loss and the gain cells
$\left({f\approx-5.2\,kHz\,\,;\,\,R\approx 38.73\,k\Omega}\right)$ (d) The
Lissajous’s curve. The third case with positive frequency , capacitive and
imaginary couplings
$C={\rm{}}10{\rm{}}nF\,;\,\,C^{\prime}=\,\,2\,nF\,;{\rm{}}Z=j2{\rm{}}k\Omega;\,\,c={\rm{}}0.2\,\,;\,\,\nu={\rm{}}0.5$:
(e) The real voltage dynamic across the loss and the gain cells
$\left({f\approx 3.34\,kHz\,\,;\,\,R\approx 39.44\,k\Omega}\right)$ (f) The
Lissajous’s curve. Figure 7: View of the experiment environment

The imaginary resistor is a resistor which resistance is a pure imaginary
number. It is built indirectly by gyrators a39 . If it is associated to a
resistor which resistance is a real number, we can have a complex resistor
which impedance is in the form
${R_{C}}=R+jr=\left|{{R_{C}}}\right|{e^{j\Phi}}$ . $R$ represents the real
part and $r$ the imaginary part
$\left({\left|{{R_{C}}}\right|=\sqrt{{R^{2}}+{r^{2}}}\,\,;\tan\left(\Phi\right)=\frac{r}{R}}\right)$
. The ZRC cell is constituted by a capacitor $C$ , a real resistor $R$ and at
least an imaginary resistor $Z=jr$ a30 ; a31 . The value of components may be
positive or negative without any restrictions. The cell is depicted in Fig.
1a(b) . The Kirchhoff’s law applied at the node gives us:

${I^{Z}}+{I^{R}}+{I^{C}}=0$ (8)

The Ohm’s law at the borders of resistors involves:

$V=R{I^{R}}=Z{I^{Z}}$ (9)

From the combination of Eq. 8 and Eq. 9, we deduce the following:

$C\frac{{dV}}{{dt}}+\frac{V}{R}-j\frac{V}{r}=0$ (10)

$\frac{{dV}}{{dt}}+\left({\frac{1}{{RC}}-j\frac{1}{{rC}}}\right)V=0$ (11)

The final voltage with an appropriate initial conditions is:

$V\left(t\right)={V_{0}}{{\mathop{\rm
e}\nolimits}^{\left({-\frac{1}{{RC}}t}\right)}}{{\mathop{\rm
e}\nolimits}^{\left({j\frac{1}{{rC}}t}\right)}}$ (12)

We obtained a complex voltage across the components of the circuit given as:

$V\left(t\right)={V_{0}}{e^{\left({-\gamma t}\right)}}{e^{\left({j\omega
t}\right)}}{\rm{=}}{V_{0}}{e^{\left({-\gamma
t}\right)}}\left({\cos\left({\omega t}\right)+j\sin\left({\omega
t}\right)}\right)$ (13)

with $\gamma=\frac{1}{{RC}}$ and $\omega=\frac{1}{{rC}}$ which may be positive
or negative.
The system with positive frequency does not describe the same reality with the
one with negative frequency because $\cos\left({\omega t}\right)$ is even and
$\sin\left({\omega t}\right)$ is odd.

$\left\\{\begin{array}[]{l}{\bf{V}}\left(\omega\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\left({\cos\left({\omega t}\right)+j\sin\left({\omega
t}\right)}\right)\\\ {\bf{V}}\left({-\omega}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\left({\cos\left({\omega t}\right)-j\sin\left({\omega
t}\right)}\right)\end{array}\right.$ (14)

$\left\\{\begin{array}[]{l}\real\left({{\bf{V}}\left(\omega\right)}\right)=\real\left({{\bf{V}}\left({-\omega}\right)}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\cos\left({\omega t}\right)\\\
\imaginary\left({{\bf{V}}\left(\omega\right)}\right)=-\imaginary\left({{\bf{V}}\left({-\omega}\right)}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\sin\left({\omega t}\right)\end{array}\right.$ (15)

Since this response of the system is complex, based on the model of the
imaginary resistor proposed in Ref.a39  , one can probe the real part
$\real\left({{\bf{V}}\left(\omega\right)}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\cos\left({\omega t}\right)$ and the imaginary part
$\imaginary\left({{\bf{V}}\left(\omega\right)}\right)={V_{0}}{e^{\left({-\gamma
t}\right)}}\sin\left({\omega t}\right)$ of the system simultaneously. The sign
of the damping rate define the nature of the cell. Depending on the signs of
the components, When $\gamma<0$, we have a gain cell and when $\gamma>0$, we
have a loss cell. The experimental verification of the behavior of the cell
with positive and negative frequencies is given in Fig. 2. The real and the
imaginary parts of the voltage are presented simultaneously for each case. The
sign of the frequency is the one of the capacitor in the cell. This gives the
possibilities to achieve experimentally PT-symmetric systems with cells of
same frequency and Anti-Parity-Time (APT) symmetry with cells of opposite
frequencies without the rotating frame condition as in existing systems in
others fields of Physics. These results open new avenues for experimental non-
Hermitian Quantum Mechanics and Quaternionic Quantum Electronics.

##  III ZRC PARITY-TIME SYMMETRIC DIMER

###  III.1 The ZRC model and equations of dynamics

The ZRC PT- symmetric dimer is made by two active ZRC-cells. The cells contain
in parallel an imaginary resistor, a capacitor and a real resistor. They are
coupled by a capacitor or an imaginary resistor or by all of them. The values
of components may be positive or not without any restriction. In Fig. 3 is
presented the setup of the ZRC PT-symmetric dimer. By applying the Kirchoff’s
laws at nodes 1 and 2, we have:

$\left\\{\begin{array}[]{l}I_{1}^{R}+I_{1}^{C}+I_{1}^{Z}+I_{1}^{G}+{I^{C^{\prime}}}+{I^{r^{\prime}}}=0{\rm{}}\\\
{\rm{}}I_{2}^{R}+I_{2}^{C}+I_{2}^{Z}+I_{2}^{G}-{I^{C^{\prime}}}-{I^{r^{\prime}}}=0{\rm{}}\end{array}\right.$
(16)

with $I_{n}^{C}={C_{n}}\frac{d}{{dt}}{V_{n}}$ ;
${I^{C^{\prime}}}=C^{\prime}\frac{d}{{dt}}\left({{V_{0}}-{V_{1}}}\right)$ ;
$I_{n}^{R}=\frac{{{V_{n}}}}{{{R_{n}}}}\,\,$;
$I_{n}^{Z}=-j\frac{{{V_{n}}}}{{{r_{n}}}}\,\,$;
${I^{R^{\prime}}}=\frac{1}{{R^{\prime}}}\left({{V_{1}}-{V_{2}}}\right)$ ;
${I^{r^{\prime}}}=-j\frac{1}{{r^{\prime}}}\left({{V_{1}}-{V_{2}}}\right)$ for
all the components. These considerations lead on the general form of first
order ordinary differential equations:

$\left\\{\begin{array}[]{l}\frac{{d{V_{1}}}}{{d\tau}}=\frac{1}{\Delta}\left[{\left({1+{c_{2}}}\right)\left({j\left({1+{\nu_{1}}}\right)-{\gamma_{1}}}\right)-j{c_{1}}\Gamma{\nu_{2}}}\right]{V_{1}{\rm{}}}\\\
\,+{\rm{}}\frac{1}{\Delta}\left[{-j\left({1+{c_{2}}}\right){\nu_{1}}+{c_{1}}\Gamma\left({j\left({1+{\nu_{2}}}\right)-{\gamma_{2}}}\right)}\right]{V_{2}}\\\
\\\
\frac{{d{V_{2}}}}{{d\tau}}=\frac{1}{\Delta}\left[{-j\left({1+{c_{1}}}\right)\Gamma{\nu_{2}}+{c_{2}}\left({j\left({1+{\nu_{1}}}\right)-{\gamma_{1}}}\right)}\right]{V_{1}{\rm{}}}\\\
\,+{\rm{}}\frac{1}{\Delta}\left[{\left({1+{c_{1}}}\right)\Gamma\left({j\left({1+{\nu_{2}}}\right)-{\gamma_{2}}}\right)-j{c_{2}}{\nu_{1}}}\right]{V_{2}}\end{array}\right.$
(17)

where $\tau={\omega_{1}}t$; ${\omega_{n}}=\frac{1}{{{r_{n}}{C_{n}}}}$;
$\Gamma=\frac{{{\omega_{2}}}}{{{\omega_{1}}}}$;
${c_{n}}=\frac{{C^{\prime}}}{{{C_{n}}}}$;
${\nu_{n}}=\frac{{{r_{n}}}}{{r^{\prime}}}$;
${\gamma_{n}}=\frac{{{r_{n}}}}{{{R_{n}}}}$ and
$\Delta=1+\sum\limits_{n=1}^{2}{{c_{n}}}$.
The system of first order ordinary differential equation is PT- symmetric when
the several conditions are satisfied:

$\left\\{\begin{array}[]{l}\Gamma=1{\rm{}}\,;{\rm{}}\,{c_{1}}={c_{2}}=c{\rm{}}\\\
\,{\nu_{1}}={\nu_{2}}=\nu\,\,\,;\,\,{\gamma_{1}}=-{\gamma_{2}}=\gamma\end{array}\right.$
(18)

We can associate to our system a non-Hermitian effective Hamiltonian such that
:

$j\frac{d}{{d\tau}}\left|{\Psi\left(\tau\right)}\right\rangle={H_{eff}}\left|{\Psi\left(\tau\right)}\right\rangle$
(19)

where: $j$ is the imaginary unit $\left({{j^{2}}=-1}\right)$ ;
$\left|{\Psi\left(\tau\right)}\right\rangle={\left({{V_{1}}\left(\tau\right),{V_{2}}\left(\tau\right)}\right)^{T}}$;
${H_{eff}}={k_{0}}I+{k_{x}}{\sigma_{x}}+{k_{y}}{\sigma_{y}}+{k_{z}}{\sigma_{z}}$
. $I$ is the $2\times 2$ matrix unit and ${\sigma_{x}}$ , ${\sigma_{y}}$ and
${\sigma_{z}}$ are Pauli’s matrices.

${\sigma_{x}}=\left({\begin{array}[]{*{20}{c}}0&1\\\
1&0\end{array}}\right){\kern 1.0pt}{\kern
1.0pt};\,\,{\sigma_{y}}=\left({\begin{array}[]{*{20}{c}}0&{-j}\\\
j&0\end{array}}\right){\kern 1.0pt}{\kern
1.0pt};\,\,{\sigma_{z}}=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&{-1}\end{array}}\right){\kern 1.0pt}$ (20)

The complex coefficients ${k_{0}}$ ,${k_{x}}$,${k_{y}}$, and ${k_{z}}$ are
function of the independent real parameters of the system deduced as :

$\left\\{\begin{array}[]{l}{k_{0}}=-\left({1+c+\nu}\right)/\Delta\,\,\,\,;\,\,\,\,{k_{x}}=-\left({c-\nu}\right)/\Delta\\\
{k_{y}}=-c\gamma/\Delta\,\,\,;\,\,\,\,{k_{z}}=-j\left({1+c}\right)\gamma/\Delta\end{array}\right.$
(21)

Since the system is PT-symmetric, it satisfied the following relation:

$\left[{{\bf{PT}}\,,{H_{eff}}}\right]=0$ (22)

where ${\bf{P}}={\sigma_{x}}$ is the Parity operator and ${\bf{T}}=K$ is the
time reversal operator with $K$ being the complex conjugation operation. That
is ${k_{0}}$, ${k_{x}}$ and ${k_{y}}$ must be real whereas ${k_{z}}$ must be
purely imaginary. The eigenvalues of the Hamiltonian resulted from
$\det\left({{H_{eff}}-\omega I}\right)=0$ are given as follow:

${\omega_{\pm}}={k_{0}}\pm\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$ (23)

###  III.2 The eigenvalues of the ZRC PT-dimer and oscillations

Function of the real parameters of the system, the eigenvalues of the
effective Hamiltonian are :

${\omega_{\pm}}=\frac{{-(1+c+\nu)\pm\sqrt{\left({1+2c}\right)\left({\gamma_{PT}^{2}-{\gamma^{2}})}\right)}}}{{1+2c}}$
(24)

with ${\gamma_{PT}}=\pm\frac{{\left|{c-\nu}\right|}}{{\sqrt{1+2c}}}$.
The system is PT-unbreakable if
$c<-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$ and PT-breakable
$c>-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$. When $c=\nu$, one
have a non-Hermitian diabolic point in the unbreakable Parity-Time symmetry
and a thresholdless point in the breakable Parity-Time symmetry.

The exceptional points are where both eigenvalues and eigenvecors coalesce.
There are well identified in Fig. 4a. Three cases are presented. In the first
case only the positive capacitive coupling is activated
$\left({c={\rm{}}0.2\,\,;\,\,\nu={\rm{}}0}\right)$. When
$\gamma<{\gamma_{PT}}$, all the eigenfrequencies are real
$\left({\real\left({{\omega_{\pm}}}\right)\neq
0\,;\,\imaginary\left({{\omega_{\pm}}}\right)=0}\right)$ and we are in the
exact phase of Parity-Time Symmetry. When $\gamma={\gamma_{PT}}$, the
eigenvalues coalesce $\left({{\omega_{+}}={\omega_{-}}}\right)$ : this is the
exceptional point. Its presence in a system calls for enhancement of sensing
than the one with diabolic point, for chiral properties which are revealed
when the point is encircled with time varying components in the setup, opening
the road to explain the asymmetric transport of information in Optics and in
Photonics. It is at this point that happens unidirectional invisibility in the
scattering properties of waveguides with Parity-Time symmetry. At last, when
$\gamma>{\gamma_{PT}}$, the real parts of eigenvalues are identical
$\left({\real\left({{\omega_{+}}}\right)=\real\left({{\omega_{-}}}\right)}\right)$
and the imaginary parts emerge opposite in sign
$\left({\imaginary\left({{\omega_{+}}}\right)=-\imaginary\left({{\omega_{-}}}\right)}\right)$
: we are in the broken phase Parity-Time symmetry. In this phase the system is
nonreciprocal and there is a deep difference between gain cell and loss cell
when the initial input is set to one or another cell. The Coherent Perfect
Absorber-Laser (CPA-L) occurs in the this phase when the scattering properties
are investigated in PT-waveguides. We have the same description with the
second case with negative capacitive coupling
$\left({c=-\,0.2\,;\,\,\nu=0}\right)$ but more than $-1/2$ to be in the
breakable PT-symmetry and the third case with both positive capacitive
coupling and imaginary coupling $\left({c=\,0.2\,;\,\,\nu=0.5}\right)$. For
the third case, analytical solutions and numerical simulations with the
algorithm of Runge-Kutta order 4 are presented in Fig. 5. The behaviors of
oscillations when the input is set from the loss is equal to the one of the
initial input from gain after a beat of time. This confirms the character of
non-reciprocity which is a signature of non-Hermitian systems.

##  IV Experimental realization of oscillations in ZRC PT-Dimer

In this section is presented the experimental realization of the ZRC Parity-
Time symmetric dimer with positive and negative frequencies. The imaginary
resistor used is in the model of Ref.a39 . One can easily probe the real and
imaginary parts of the voltages across the components. We have use the
Negative Immitiance Converter to achieve negative values of components with
resistors, capacitors and operational amplifiers. The realizations are those
of the three configurations listed in the previous section at
$\gamma=0.2\,{\gamma_{PT}}$. Only the real parts of voltages are presented in
Fig. 6. In the first case, the positive natural frequency of independent cells
is $f=7.96\,kHz$. The values of different components
are:${Z_{0}}={Z_{1}}=Z=j2\,k\Omega\,;\,\,\,{R_{0}}=59.16\,k\Omega\,\,{R_{1}}=-59.16\,k\Omega;\,{C_{0}}={C_{1}}=\,C=10\,nF\,;\,\,C^{\prime}=\,2\,nF\,;\,\,jR^{\prime}=0$.
The frequency of oscillations obtained is ${f\approx 2.23\,kHz}$ (Fig.
6(a-b)). In the second case, the negative natural frequency of independent
cells is $f=-7.96\,kHz$. The values of different components
are:${Z_{0}}={Z_{1}}=Z=j2\,k\Omega\,;\,\,\,{R_{0}}=38.73\,k\Omega\,\,{R_{1}}=-38.73\,k\Omega;\,{C_{0}}={C_{1}}=\,C=-10\,nF\,;\,\,C^{\prime}=\,2\,nF\,;\,\,jR^{\prime}=0$.
The frequency of oscillations obtained is ${f\approx-5.2\,kHz}$ (Fig. 6(c-d)).
In the third case, the positive natural frequency of independent cells is
$f=7.96\,kHz$. The values of different components
are:${Z_{0}}={Z_{1}}=Z=j2\,k\Omega\,;\,\,\,{R_{0}}=39.44\,k\Omega\,\,{R_{1}}=-39.44\,k\Omega;\,{C_{0}}={C_{1}}=\,C=10\,nF\,;\,\,C^{\prime}=\,2\,nF\,;\,\,jR^{\prime}=j4\,k\Omega\,$.
The frequency of oscillations obtained is ${f\approx 3.34\,kHz}$ (Fig.
6(e-f)).

Operational amplifiers (LF356), metal-film resistors and polystyrene
capacitors are used in the experimental circuits. All the elements are tuned
to be within $0.1\%$ with respect to the desired values. For injecting the
initial conditions, we have used DG200ABA analog switches and and an external
standard voltage source. All the waveforms are acquired by Tektronix TDS3014B
digital oscilloscope. Agilent 33120A function generator is used to generate
the trigger signal for the experimental circuits (See Fig. 7)

##  V Conclusion

We have reported in this work the experimental realization of the ZRC PT-
symmetric Dimer. A comparative study is made between RLC and ZRC cells.
Contrary to the RLC cell which dynamic is of second order ordinary
differential equation with a real output, the ZRC is of first order ordinary
differential equation with a complex output. We succeed the measurements of
the real and imaginary parts of the voltage in positive and negative
frequencies. Oscillations of the PT-dimers are also probed in the exact phase
of the Parity-Time symmetry without any restriction or approximation. This
work paves the way for the design of new electronic and optolectronic devices.
It opens the investigation on the exploitation of the new band of negative
frequencies in Telecommunications and the generation of new oscillators for
simulations of qubits in quantum computing.

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