11footnotetext: These authors contributed equally to this work.22footnotetext:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Enhanced metrology at the critical point of a many-body Rydberg atomic
system

Dong-Sheng Ding1,2,3,∗,† Zong-Kai Liu1,2,3,∗ Bao-Sen Shi1,2,3,‡ Guang-Can
Guo1,2,3 Klaus Mølmer4,§ Charles S. Adams5,¶ 1CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei 230026,
China  2CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  3Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China  4Aarhus Institute of Advanced Studies, Aarhus University,
Høegh-Guldbergs Gade 6B, DK-8000 Aarhus C, Denmark
Center for Complex Quantum Systems, Department of Physics and Astronomy,
Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  5Department of
Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom

(September 3, 2024)

The spectral properties of an interacting many-body system may display
critical character and have potential applications in precision metrology.
Here, we demonstrate such many-body enhanced metrology for microwave (MW)
electric fields in a non-equilibrium Rydberg atomic gas. Near criticality the
high sensitivity of Rydberg atoms to external MW electric fields, combined
with many-body enhancement induces significant changes in the optical
transmission. We quantify this behavior using the Fisher information. For
continuous optical transmission at the critical point, the Fisher information
is three orders of magnitude larger than in independent particle systems, the
measured data provides an equivalent sensitivity of 49
nV/cm/$\textrm{Hz}^{1/2}$. The reported results constitute a milestone towards
the application of many-body effects in precision metrology.

## Introduction

Ensembles of well-controlled neutral atoms are ideal systems to explore many-
body physics [1, 2, 3, 4]. In particular, the controllable interactions among
highly-excited Rydberg atoms hold promise for studies of quantum information
and many-body physics [5, 6, 7]. Benefiting from the large interaction volume
of Rydberg atoms, a small change in the Rydberg state population can induce a
global macroscopic phase transition between non-interacting (NI) and
interacting (I) phases [8]. Laser-induced density-dependent energy shifts of
Rydberg states offer a convenient platform to directly observe non-equilibrium
phase transitions and bistability [7, 9, 10, 11, 12], and to study dynamical
analogues of forest fire [8] and epidemic spreading [13, 14]. In contrast to
other optically bistable systems [15, 16, 17, 18, 19], Rydberg ensemble
experiments can be performed without the need of optical cavity feedback and
cryogenic temperatures.

Exploring the non-equilibrium dynamics of the Rydberg system under external
fields is intriguing. The emergent thermodynamic and spectroscopic properties
of a many-body system of interacting Rydberg atoms present open questions both
in theory [20, 21, 22, 11, 23] and experiments [7, 8]. Due to the large dipole
moment, the Rydberg atoms are highly sensitive to system noise and external
electric fields [24, 25, 26, 27, 28, 29]. Most dramatically, the macroscopic
change in the optical response near a critical point [8, 12] presents a
resource for increased metrological sensitivity [30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40]. Accompanying the divergent susceptibility near the critical
point, optical probing of the system is highly sensitive to small variations
of physical parameters. Critical systems may thus display sensing errors with
a generic scaling $\sim 1/\sqrt{N^{\gamma}t^{\lambda}}$ with
$\gamma,\lambda>1$ [37, 41, 42], where $N$ is the number of atoms and $t$ is
the measurement time.

In this article, we demonstrate how Rydberg criticality provides a method for
high-sensitivity probing of external parameters. We exploit the extreme
sensitivity of the optical transmission at the critical point to probe
external MW fields. Due to the critical slowing down near phase transitions,
we need to take into account how the system dynamics do not adiabatically
follow the stationary state, but rather smooths the system response. This
leads to a non-integer power dependence of the Fisher Information (FI) on the
duration of the detuning scans. The behavior around criticality is observed to
enhance the FI by a factor of up to more than $10^{3}$ compared to a non-
interacting ensemble.

Figure 1: The principle of single-body [top row] (many-body [bottom row])
Rydberg metrology (a) Energy diagram for a two-level atom model, showing the
ground state $\left|g\right\rangle$ and Rydberg state $\left|r\right\rangle$
with spontaneous radiation rate, $\Gamma$. The atoms are driven from the
ground state to the Rydberg state by a laser with Rabi frequency, $\Omega$ and
detuning, $\Delta$. They are also exposed to a MW field with electric field
component $E_{mw}$. In the many-body case, the Rydberg resonance is modified
by the many-body interaction strength, $V={C_{6}}/{r^{6}}$ [where $C_{6}$ is
the van der Waals coefficient and $r$ is the distance between Rydberg atoms],
and the population of the Rydberg atoms $\rho_{rr}$; see more details in the
Method Sections. (b) The blue and red curves represent the spectrum with and
without external MW field, which induces a shift $\delta$. The measurement
sensitivity is highest when the derivative $\mathrm{d}\rho_{rr}/d\Delta$ is
maximal, as indicated by the points $p_{w1}$ and $p_{w2}$. The steeper slope
near the critical point in the many-body case [bottom row] results in enhanced
measurement sensitivity. (c) The many-body advantage corresponds to a
metrological ruler with thinner tick marks $\delta\omega$ than in the single-
body case. The transmission spectra are shifted by an external electrical
field forming a ruler with unfixed ticks.

## Results

Many-body metrology model. We consider a model of $N$ interacting two-level
atoms with a ground state $\left|g\right\rangle$ and a Rydberg state
$\left|r\right\rangle$ (with decay rate $\Gamma$) [Fig. 1(a)]. A laser couples
these atoms with Rabi frequency $\Omega$ and detuning $\Delta$. We derive the
Rydberg population $\rho_{rr}$ via mean-field approximation (i.e.,
$\Delta\rightarrow\Delta-V\rho_{rr}$, where $V$ is the average many-body
interaction strength from dipole interaction or ions collisions), $\delta$ is
the external field induced-frequency shift on Rydberg state
$\left|r\right\rangle$, as mentioned in Method Sections,

$\rho_{\mathrm{rr}}=\frac{\Omega^{2}}{4\left(\Delta-V\rho_{\mathrm{rr}}\right)^{2}+2\Omega^{2}+\Gamma^{2}}.$
(1)

Due to the interaction, the spectrum has a population-dependent shift
$V\rho_{rr}$, thus inducing a steep edge of $\rho_{rr}$ with a maximum
derivative

$\left.\frac{\mathrm{d}\rho_{rr}}{\mathrm{d}\Delta}\right|_{\Delta=\Delta_{c}}=\frac{1}{V+\sqrt{(\Gamma^{2}+2\Omega^{2})/3\rho_{rr}^{2}}},$
(2)

where $\Delta_{c}$ corresponds to the detuning at which the derivative gets
its maximum. We note that the $\mathrm{d}\rho_{rr}/\mathrm{d}\Delta$ is
increased due to the interaction strength $V$ (here $V<0$); more details can
be found in Method Sections. The derivative
$\mathrm{d}\rho_{rr}/\mathrm{d}\Delta$ diverges at the system’s critical point
[8], exhibiting a method of high-precision measurement [30, 31, 32, 33, 34,
35, 36]. A measurement is realized by detecting the transmission of an optical
probe field. When applying external fields (such as the electric component of
external microwave MW fields), the measurement precision is limited by the
maximum slope of the Rydberg resonance. This is indicated by the points
$p_{w1}$ ($p_{w2}$) in [Fig. 1(b)]. Compared to the non-interacting case [top
row], the slope in the vicinity of the critical point [bottom row] is
significantly enhanced. For metrology applications, the sensitivity of many-
body case is enhanced by a ratio,

$\text{{\hbox{\beta}}=}\left.\frac{\mathrm{d}\rho_{rr}}{\mathrm{d}\Delta}\right|_{V\neq
0}/\left.\frac{\mathrm{d}\rho_{rr}}{\mathrm{d}\Delta}\right|_{V=0}.$ (3)

In Fig. 1(c) we illustrate how this many-body enhancement is like having a new
ruler with much finer markings.

Figure 2: Optical transmission spectra with and without phase transition. The
transmission spectra with (red) and without (black) phase transition. The
dashed lines A and B show the maximum slopes near half transmission.
$\Delta{\omega}/2\pi\sim$ 6 MHz shows the bandwidth of the transmission
spectrum without interactions. The photon counts are given for a measurement
time $t$ = 20 $\mu$s for $2\pi\times$0.036 MHz detuning interval.

The measurement sensitivity is determined by the variation of the transmission
signal around the critical point and the photon counting noise in the
measurement record. By exploring the linear slope of transmission in a narrow
interval around the critical point the sensitivity can be expressed in terms
of the Fisher Information [43, 44],

$F(\Delta)=\frac{\overline{\mu}^{\prime}(\Delta)^{2}}{\overline{Var(\mu)}}$
(4)

where $\Delta$ is the parameter that we want to determine, $\overline{\mu}$
represents the mean value of the difference in photon numbers accumulated in
fixed time intervals by a differential detector exposed to a reference beam
and a beam passing through the atomic cloud. $Var(\mu)$ is the variance of the
differential signal, i.e the sum of the variances of the two separate and
indendent counting signals. Note that Eq.(4) expresses the usual signal-to-
noise ratio, and the Cramér-Rao bound,

$\delta\Delta\geq\frac{1}{\sqrt{F(\Delta)}}.$ (5)

yields the usual estimation error for the counting signal.

It is important to emphasize that the Fisher Information refers to the actual
counting signals. While a detector may output a photon rate in counts per
second, we must independently assess or estimate the variance of the signal
for the time duration of the measurement. We explore experiments with
different duration and we shall thus present the actual counts in given time
intervals and their variance to obtain the proper assessment of the
metrological sensitivity. We shall also observe that during a frequency scan
in finite time, the non-interacting (interacting) atomic system does (does
not) attain its stationary state. This leads to a linear (non-linear)
dependence of the Fisher information on the measurement time.

Figure 3: Transmission spectra and the associated Fisher information. Panels
(a)-(c) show the transmission spectra across the phase transition (shaded
regions), obtained in a total measurement time of $t=1.7$ $\mu$s, $t=10$
$\mu$s, and $t=1$ ms. The inset in (b) shows the corresponding derivative
$d\overline{\mu}/d\Delta_{s}$. Panels (d)-(f) show the trivial spectra
(without phase transition) for the same measurement times $t=1.7$ $\mu$s,
$t$=10 $\mu$s, and $t$= 1 ms. Panel (g) shows the Fisher information
associated with the determination of the steepest point on the transmission
curves for different values of the total measurement time $t$ (note that the
Fisher Information for the non-interacting case is magnified manually by a
factor 300. The red and blue curves are fitted by the function
$F=A(t/t_{0})^{\lambda}$, where the fit parameters are given in the main text.
In this process, the red data in (g) is obtained from the maximum of
$F(\Delta)|_{\Delta_{s}=\Delta_{c}}=(\mathrm{d}\overline{\mu}/\mathrm{d}\Delta_{s})^{2}/Var(\mu)$,
while for the blue data in (g), the FI at the critical point are obtained by
considering 30 data points around $\Delta_{c}$ in (d)-(f) to reduce the
fluctuations from the instability of the laser power and the cell temperature.
The error bars determined in (g) are statistics from the three repeated
experiments.

Measured derivative and Fisher information. For the experiment, we employ two-
photon excitation with a probe beam and a coupling laser beam with Rabi
frequencies (detuning) $\Omega_{p}$ ($\Delta_{s}$) and $\Omega_{c}$
($\Delta_{c}$). We measure the transmission spectra, as shown in Fig. 2(a).
The transmission depends on $\Omega_{p}$, and we can prepare the system with
and without phase transition $\Omega_{p}<\Omega_{p,c}$ and
$\Omega_{p}>\Omega_{p,c}$, where $\Omega_{p,c}\text{/}2\pi=5.6$ MHz is the
threshold Rabi frequency of the probe field. Our system displays a second
order dynamical phase transition between two stationary states with different
excitation densities [21]. The two spectra display the main distinct character
of the transmission of a non-interacting system (the black curve in Fig.
2(a)), and an interacting many-body system (the red curve in Fig. 2(a)). The
derivative $\mathrm{d}\mu/\mathrm{d}\Delta_{s}$ of the transmission is very
large at the phase transition point $\Delta_{c}$ of the red curve in Fig.
2(a), while it explores a weaker, finite maximum near half maximum of the
black black curve in Fig. 2(a).

Figure 4: Change of transmission spectra by application of MW fields. (a) The
transmission spectra under the field amplitude $E_{\mathrm{mw}}=0$
$\mathrm{mV/}\mathrm{cm}$ (black) and $E_{\mathrm{mw}}=3.8$
$\mathrm{mV/}\mathrm{cm}$ (red) with probe Rabi frequency
$\Omega_{p}=2\pi\times 5.5$ MHz below the critical value $\Omega_{p,c}=2$ for
the phase transition. (b) The transmission spectra under the field amplitude
$E_{\mathrm{mw}}=0$ $\mathrm{mV/}\mathrm{cm}$ (black) and
$E_{\mathrm{mw}}=3.8$ $\mathrm{mV/}\mathrm{cm}$ (red) with the probe Rabi
frequency $\Omega_{p}=2\pi\times 8.2$ MHz, above the critical value for the
phase transition. In these two cases, the frequency of the applied MW field is
set as $2\pi\times 16.68$ $\mathrm{GHz}$. The big circular points $p_{s}$ and
$p_{m}$, marked with red and gray in Fig. 4(a) and (b), correspond to the
position of the steepest slope. The direction of scanning $\Delta_{c}$ is from
the red-detuning towards the blue detuning. Here, $\Delta_{s}$ is swept from
-$2\pi\times$30 MHz to -$2\pi\times$24 MHz with a sweep rate $v_{s}$ =
$2\pi\times$0.0055 MHz/$\mu$s.

In the experiment, we sweep the detuning $\Delta_{s}$ and observe the
transmission at each $\Delta_{s}$. As explained above, the FI is not only
governed by the number of excited and thus interacting Rydberg atoms but also
depends on the measurement time $t$ [37], defined as the time the probe laser
explores a small interval around each detuning $\Delta_{s}$. In Fig. 3(a-c) we
consider the behavior above criticality with $\Omega_{p}\text{/}2\pi=7.9$ MHz
and in Fig. 3(d-f), we consider the behavior below criticality with
$\Omega_{p}\text{/}2\pi=5.5$ MHz. We observe that for
$\Omega_{p}\text{/}2\pi=7.9$ MHz, the transmission profile near the critical
point becomes steeper as the measurement time $t$ is increased, while, for
$\Omega_{p}\text{/}2\pi=5.5$ MHz, the transmission spectra are almost
identical. This implies that the FI is inclined to be linearly dependent on
the time $t$ for the data in Fig. 3(d-f)], while a different dependence
appears for the data in Fig. 3(a-c)].

The values of the Fisher Information are shown in Fig. 3(g) for different
measurement times $t$, we find that the FI is fitted well by the form
$F=A(t/t_{0})^{\lambda}$, where $A=2.0\times 10^{4}$ MHz-2 and $\lambda=1.28$
for $\Omega_{p}\text{/}2\pi=7.9$ MHz, while $A=10$ MHz-2 and $\lambda=1$ for
$\Omega_{p}\text{/}2\pi=5.5$ MHz. In our system, when $t=1$ ms we achieve a
large enhancement ratio up to $10^{3}$ by comparing these two cases. We
conclude that one can extract more information by the interacting many-body
system than by independent systems, and that we can extract even more
information by continuous measurements for long times. The non-integer power-
law dependent behavior of the fit to the FI is caused by the critical slowing
down and thus deviation of the atomic dynamics from the stationary state
around the critical point [45, 46, 47]. This smooths the maximum slope and
causes a more than linear suppression of the FI for the shorter measurement
times in Fig. 3(a-c). We also plot the derivative
$d\overline{\mu}/d\Delta_{s}$ against the detuning $\Delta_{s}$ in the
vicinity of the critical detuning at $t=10\mu s$, see the inset in Fig. 3(b).
We find that the derivative $d\overline{\mu}/d\Delta$ has a power law
dependence on detuning
$d\overline{\mu}/d\Delta_{s}=\chi\left|\Delta_{s}/\Delta_{0}+11.3\right|^{-\alpha}$,
where $\chi=0.02$ $\textrm{MHz}^{-1}$, $\Delta_{0}/2\pi$ = 1 MHz, and
$\alpha=2\pm 0.1$ is the fitted power-law exponent. This detuning-dependent
susceptibility is caused by the increased interaction near the critical point
[48], as the change of detuning tunes the Rydberg population $\rho_{rr}$ and
hence the interaction. For the non-interacting case, the atomic system follows
the stationary state even for the fast sweeps in our experiments, and the FI
is linearly dependent on time $t$. There is more noise in Fig. 3 (c) and (f)
than Fig. 3 (a) and (d), due to low frequency noises appearing for the longer
measurement times.

Figure 5: Transmission under different amplitudes of the MW field for the
single-body case (a) and the many-body case (b). The amplitude $E_{mw}$ is
changed from $E_{mw}$ = 0 mV/cm to $E_{mw}$ = 4 mV/cm in steps of duration 5
$\mu$s. The black and red solid lines are the fit linear functions y=0.014 (x
+ 11.5) and y=0.89 (x - 1.81) respectively. (a1-a3) and (b1-b3) are the
critical-point histogram of the transmission distribution under the different
amplitudes of the MW field with the single-body case and the many-body case,
respectively. The dashed lines are fitted Gaussian functions. The data are
taken under equivalent experimental conditions (such as scan rate, acquisition
time and averaging) in the few- and many-body cases

Transmisssion spectra with and without MW fields. The advantages for metrology
appear due to the critical response of the system upon variation of external
perturbations. To further study this critical response we apply a MW electric
field with amplitude $E_{\mathrm{mw}}$ and detuning $\Delta_{\mathrm{mw}}$ to
continuously drive the Rydberg transition $51D_{3/2}-52P_{1/2}$. The main
effect of the microwave field here is to (1) induce a small AC Stark shift
which moves the critical point, (2) change the population of the Rydberg atoms
while it has a negligible effect on the $C_{6}$ van der Waals coefficient and
the resonant dipole-dipole interactions. As shown in Fig. 4(a) and (b), this
shifts the transmission spectra of the Rydberg system subject to the probe
Rabi frequencies $\Omega_{p}/2\pi=5.5$ MHz and $\Omega_{p}/2\pi=8.2$ MHz,
corresponding to the non-interacting and interacting many-body systems,
respectively. When we apply the MW field with $E_{\mathrm{mw}}=3.8$
$\mathrm{mV/}\mathrm{cm}$, the spectra show a small red-shift $\delta_{s}$
[Fig. 4(a)] and $\delta_{m}$ [Fig. 4 (b)], respectively. For the many-body
system, the high sensitivity of the frequency shift allows us to sense the
strength $E_{mw}$ of an applied MW field. The FI
$F(\Delta)|_{\Delta_{s}=\Delta_{c}}$ for the black data in Fig. 4 (a) is $F$ =
$1.4\times 10^{-3}\textrm{MHz}^{-2}$, and is much smaller than that for the
black data in Fig. 4 (b) [$F$ = 0.27 $\textrm{MHz}^{-2}$]. The corresponding
minimum uncertainty $\delta\Delta/2\pi\sim$ 0.3 MHz corresponds to an
uncertainty of the applied field $\Delta E_{mw}$ = 1.9 mV/cm by considering
the energy shift proportional to $E_{mw}^{2}$ when $E_{mw}$ is small [here the
stark shift $\delta_{m}\sim E_{mw}^{2}$ follows by a Taylor expansion,
$\delta_{{\rm
m}}\sim-\Delta_{mw}/2+\sqrt{\Delta^{2}_{mw}+\Omega^{2}_{mw}}/2$], see more
data in the supplementary materials. As a result, the many-body system can
sense the strength $E_{mw}$ = 3.8 mV/cm directly by measuring the spectrum
shift [$\delta_{m}$ = 2$\pi\times 1.2(0.3)$ MHz]. In comparison, the spectral
shift $\delta_{s}$ is indistinguishable for independent atoms, which are thus
not sensitive enough to sense the same MW field by monitoring of the spectral
shift.

Optical response under electric fields with different amplitudes. The many-
body metrological ruler has a thinner tick mark and thus a better precision
than the single-body ruler, because the optical response is stronger in the
many-body case when subject to a small frequency shift. We can also measure
the optical transmission at the position of the steepest slope $p_{w1}$ (Fig.
1(b)) when the atoms are subject to a MW electric field
$E_{mw}\textrm{sin}(f_{0}t)$ where $f_{0}/2\pi$ = $16.60$ GHz is near resonant
with the RF transition $51D_{3/2}$ \- $52P_{1/2}$.

We measure the transmission when increasing the amplitude of the MW with the
detuning $\Delta_{s}$ fixed near the maximum slope under the many- and single-
body conditions, as shown in Fig. 5 (a) and Fig. 5 (b), respectively. For the
single-body condition, the transmission is not sensitive to the variance of
the amplitude $E_{mw}$. For the many-body condition, the change of the
$E_{mw}$ makes the system cross the critical point and the transmission signal
is highly sensitive to the field around values of $E_{mw}=2$ mV/cm (this
position can be tuned by the coupling detuning), as shown in Fig. 5 (b). To
evaluate the sensitivity, we fit the data near criticality with a linear
function $y=k(x+x_{0})$ ($k=\overline{\mu}^{\prime}(E_{mw})$), and obtain the
ratio of the slopes $k_{2}/k_{1}=63.57$, where $k_{1}$ and $k_{2}$ represent
the linear coefficients for the single- and many-body cases. As the almost
same variance $Var(\mu)$ for these two cases, see the following figures
(a1-a3) and (b1-b3), we can obtain enhanced ratio for the FI:
($k_{2}^{2}/Var(\mu))/(k_{1}^{2}/Var(\mu))>4000$. From the $E_{mw}$-dependent
transmission, we can distinguish the standard deviation of the amplitude
$\delta E_{mw}$ = 1.4 mV/cm for the non-interacting case and $\delta E_{mw}$ =
22 $\mu$V/cm for the interacting many-body case with 5 $\mu s$ data
acquisition time per data point. By considering multiple sequential
independent measurements, we estimate the equivalent sensitivity 49
nV/cm/$\textrm{Hz}^{1/2}$.

## Discussion

Although the previous work in Ref. [12] clearly shows effects sensitive to the
electric field, that work mainly elaborated on how the presence of ionized
Rydberg atoms induce a linear shift of the critical point. In contrast, the
critical behavior of the interacting many-body system has not previously been
employed for sensing. The criticality induced by the interacting Rydberg atoms
depends on the Rydberg atom number $N$ and interaction strength $V$ [8], and
the increase of the population $\rho_{rr}$ or the interaction strength
$V\rho_{rr}$ enhances the non-linearity of the criticality. In our system, we
have a large interacting number of atoms, and the energy splitting is far from
the one of $N$ isolated systems, cf. previous theoretical works [31, 32, 33,
34, 35, 36]. Specifically, the interaction induced non-linear FI dependence on
measurement time of a single frequency scan shows an unique advantage on
sensing, which agrees with the theoretical simulations in the supplemental
materials.

In summary, we have demonstrated the critical behavior of interacting Rydberg
atoms and characterised its metrological consequences. The Fisher information
for the estimation of a weak microwave field shows an enhancement of order
$10^{3}$ by the use of interacting many-body systems. Concerning the use of
the Fisher Information and Cramér-Rao bound, we note that not only the narrow
detuning interval with the highest slope, but the entire transmission signal
contributes in an integral manner to the sensitivity of the experiments, cf. a
similar analysis of spatial image processing [49, 50]. Our analysis captures
the main contribution to that integral and thus constitutes a lower limit to
the FI. Passing from the measurement of frequencies, the experiments derive
their improved sensitivity towards MW electric fields and show that the
Rydberg non-equilibrium system can act as a versatile high-sensitivity
metrological resource.

## Method

Experiment setup. We adopt a two-photon transition scheme to excite an atomic
ground-state to a Rydberg-state, using a probe field resonantly driving the
atomic transition $5S_{1/2},F=2\rightarrow 5P_{1/2},F^{\prime}=3$, and a
coupling field, driving the transition $5P_{1/2},F^{\prime}=3\rightarrow
51D_{3/2}$. An MW electric field 1 (or 2) may be applied to drive an RF
transition between two different Rydberg states $51D_{3/2}$ and $52P_{1/2}$(or
$50F_{5/2}$). The MW electric fields used in our experiment are generated by
two RF sources and two frequency horns. A 795 nm laser is split by a beam
displacer into a probe beam and an identical reference beam, which are both
propagating in parallel through a heated Rb cell (length $10\,\mathrm{cm}$).
The temperature is set as $44.6^{\circ}\mathrm{C}$, corresponding to the
atomic density of $9.0\times 10^{10}$ $\mathrm{cm^{-3}}$. One probe beam is
overlapped with a counter-propagating coupling beam to constitute the Rydberg-
EIT process. The two transmission signals are detected on a differencing
photodetector.

Generation and calibration of MW fields. The MW fields used in our experiment
are generated by two RF sources and two frequency horns. The RF source 1 works
in the range from DC to 40 GHz, another is in the range from DC to 20 GHz. The
frequency horns are set close to the Rb cell. The RF frequency between Rydberg
D and P/F states are calculated according to the algorithm in Ref. [51]. We
use a spectrum analyzer (Ceyear 4024F, 9 kHz $\sim$ 32 GHz) and an antenna
(380 MHz $\sim$ 20 GHz) to receive the MW fields then to calibrate the
amplitude of MW fields in the centre of the Rb cell.

Fisher information and Cramér-Rao bound. In parameter estimation, the Cramér-
Rao bound sets a lower limit to the statistical estimation error by $\nu$
independent experiments, $(\delta\theta)_{min}=1/\sqrt{\nu\cdot F(\theta)}$.
Here $F(\theta)$ is the Fisher Information (FI) which has the value

$F\text{({\hbox{\theta}})}=\sum_{\mu}\frac{1}{L(\mu,\theta)}\left(\frac{\partial
L(\mu,\theta)}{\partial\theta}\right)^{2},$ (6)

where $L(\mu,\theta)$ is the likelihood function for the possible measurement
outcome $\mu$, conditioned on the parameter $\theta$ [52]. In our experiment
we subtract two counting signals which may both be well described by Poisson
distributions with mean values $\overline{\mu}_{1}$ and $\overline{\mu}_{2}$,
and the same values for their variances. Both count numbers are large, and in
our experiments the observed noise is dominated by electronic noise, and hence
the distributions are both well approximated by Gaussian distributions. The
difference signal is thus described by a Gaussian distribution with mean value
$\overline{\mu}=\overline{\mu}_{1}-\overline{\mu}_{2}$ and variance
$\sigma^{2}=\overline{\mu}_{1}+\overline{\mu}_{2}$. For a Gaussian
distribution with the likelihood [53, 54]

$L(\mu,\theta)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1}{2}(\frac{\mu-\overline{\mu}(\theta)}{\sigma})^{2}}$
(7)

one can see that

$\frac{\partial
L(\mu,\theta)}{\partial\theta}=\left(\frac{\mu-\overline{\mu}}{\sigma^{2}}\right)L(\mu,\theta)\overline{\mu}^{\prime}(\theta).$
(8)

We hence obtain

$\displaystyle F\text{($\theta$)}$
$\displaystyle=\sum_{\mu}\frac{1}{L(\mu,\theta)}\left(\left(\frac{\mu-\overline{\mu}}{\sigma^{2}}\right)L(\mu,\theta)\overline{\mu}^{\prime}(\theta)\right)^{2}$
$\displaystyle=\sum_{\mu}L(\mu,\theta)\frac{(\mu-\overline{\mu})^{2}}{\sigma^{4}}\left(\overline{\mu}^{\prime}(\theta)\right)^{2}$
$\displaystyle=\frac{Var(\mu)}{\sigma^{4}}\left(\overline{\mu}^{\prime}(\theta)\right)^{2}=\frac{\left(\overline{\mu}^{\prime}(\theta)\right)^{2}}{\sigma^{2}}$
(9)

where $\overline{\mu}^{\prime}(\theta)$ denotes the derivative of the mean
$\overline{\mu}(\theta)$ w.r.t $\theta$. There is a further contribution to
the FI due to the dependence of the variance $\sigma^{2}$ on $\theta$. Its
value is $\frac{1}{2\sigma^{4}}(\frac{d(\sigma^{2})}{d\theta})^{2}$, and for
our system it plays a less important role.

Signal to ratio analysis. For a given Gaussian incident probe and reference
beams with mean photon signal $\mu_{0}$ in one second, the on-resonance
absorption coefficient of the atoms and the loss of the optical path is
$1-\zeta$ (corresponding to the transmission ratio $\zeta$), a differencing
photodetector with efficiency $\eta$, records a Poisson distributed number of
clicks with mean value $\eta\mu_{0}$ and variance $\eta\mu_{0}$ per second,
which outputs a voltage signal. The difference between two beams is much
weaker and has the mean value

$\mu=\zeta\eta\mu_{0}[1+\varepsilon(\Delta_{s})]t-\zeta\eta\mu_{0}t=\zeta\eta\mu_{0}\varepsilon(\Delta_{s})t$
(10)

where $\varepsilon(\Delta_{s})$ is the transmission probability induced by
Rydberg-EIT effect, $t$ is the considered time interval. As the input photon
number of each beam is very large $\mu_{0}\sim 10^{14}$ photons per second for
$\Omega_{p}/2\pi$ = 7.9 MHz, the variance of the difference of the two beams
per second is the sum of the means because the difference of two Gaussian-
distributed variables is also a Gaussian distributed variable: $Var(\mu)$ =
$2\zeta\eta\mu_{0}+\zeta\eta\mu_{0}\varepsilon(\Delta_{s})$ $\sim$
2$\zeta\eta\mu_{0}$. $\zeta=20.6\%$ for $\Omega_{p}/2\pi$ = 7.9 MHz,
$\zeta=14.6\%$ for $\Omega_{p}/2\pi$ = 6.5 MHz, $\zeta=8.7\%$ for
$\Omega_{p}/2\pi$ = 5.5 MHz. The output voltage signal of the differencing
photodetector could be converted into the photon number by a voltage
conversion ratio $G=5.3\times 10^{7}$ V/W. The 0.5 voltage output signal
corresponds to $\sim 1\times 10^{10}$ photon numbers per second. The coupling
detuning $\Delta_{s}$ is swept with a rate of $v_{s}$ and with M sampling rate
[means that there are average M data points in the swept detuning
$\Delta_{s}$], the transmission spectrum could be measured by accumulating the
photon numbers in each detuning interval. The fast scan accumulate small
photon numbers for each interval, while the slow scan get large photon
numbers. If we scan the detuning of coupling laser $\Delta_{s}$ from red- to
blue-detuning and vice versa, we could observe the bistability, see also in
Refs. [7, 55, 8]. The bistability shifted by the MW fields could be found in
the supplementary materials.

Figure 6: Theoretical simulations of interacting two-level atoms. (a) Rydberg
state population $\rho_{rr}$ as a function of laser detuning $\Delta$ for
different interaction strengths $V$.(b) The derivative of Rydberg population
with respect to $\Delta$.

Non-linearity of interacting atoms. To elucidate the sensitivity of the
Rydberg atoms to the frequency for different interaction strength, we consider
a two-level atoms model with ground state $\left|g\right\rangle$ and Rydberg
state $\left|r\right\rangle$ (with spontaneous radiation rate $\Gamma$), which
are coupled by a laser with Rabi frequency $\Omega$ and detuning from
resonance $\Delta$. After mean-field approximation (i.e.,
$\Delta\rightarrow\Delta-V\rho_{rr}$, where $V$ is the many-body interaction
term from dipole interaction or ions collisions, and $\rho_{rr}$ is the
population of the Rydberg state), the steady-state solution for a two-level
optical Bloch equation is [20]

$\begin{array}[]{cc}\dot{\rho}_{\mathrm{gr}}=i\frac{\Omega}{2}\left(\rho_{\mathrm{rr}}-\rho_{\mathrm{gg}}\right)+i\Delta_{\mathsf{\mathrm{eff}}}\rho_{\mathrm{gr}}-\frac{\Gamma}{2}\rho_{\mathrm{gr}}\\\
\dot{\rho}_{\mathrm{rr}}=-i\Omega\left(\rho_{\mathrm{gr}}-\rho_{\mathrm{rg}}\right)-\Gamma\rho_{\mathrm{rr}}\end{array}$
(11)

where $\Delta_{\mathrm{eff}}$ is the effective detuning
$\Delta_{\mathrm{eff}}=\Delta-V\rho_{rr}$ by considering an interaction
strength $V$. We obtain an equation about $\rho_{rr}$ for bistability as
follows

$V^{2}\rho_{\mathrm{rr}}^{3}-2V\Delta\rho_{\mathrm{rr}}^{2}+\left(\Delta^{2}+\Omega^{2}/2+\Gamma^{2}/4\right)\rho_{\mathrm{rr}}-\Omega^{2}/4=0$
(12)

By taking the derivative of both sides, we finally get the relationship
between the slope of the steep edge of the hysteresis loop
$\max(d\rho_{rr}/d\Delta)$ and the interaction of Rydberg atoms $V$.

$\frac{\mathrm{d}\rho_{rr}}{\mathrm{d}\Delta}=-\frac{8\left(\Delta\rho_{rr}-\rho_{rr}^{2}V\right)}{\Gamma^{2}+4\Delta^{2}+12\rho_{rr}^{2}V^{2}-16\Delta\rho_{rr}V+2\Omega^{2}}$
(13)

The critical point is defined when this derivative reaches infinity
$\mathrm{d}\rho_{rr}/\mathrm{d}\Delta$ $\Rightarrow$ $\infty$, from which the
threshold of the Rydberg population is obtained, i.e.,

$\rho_{th}=\frac{\sqrt{-3\Gamma^{2}V^{2}+4\Delta^{2}V^{2}-6V^{2}\Omega^{2}}+4\Delta
V}{6V^{2}}.$ (14)

The relation between $\rho_{rr}$, $\mathrm{d}\rho_{rr}/\mathrm{d}\Delta$, and
$\Delta$ is demonstrated in Fig. 6. And by letting the derivative equals to 0,
i.e., $\mathrm{d}\rho_{rr}/\mathrm{d}\Delta=0$, we obtain the analytical
expression of maximum derivative
$\left.\mathrm{d}\rho_{rr}/\mathrm{d}\Delta\right|_{\Delta=\Delta_{c}}$ and
the corresponding detuning $\Delta_{c}$ as follows

$\Delta_{c}=\frac{1}{6}\left(6\rho_{rr}V-\sqrt{3}\sqrt{\Gamma^{2}+2\Omega^{2}}\right)$
(15)
$\frac{\mathrm{d}\rho_{rr}}{\mathrm{d}\Delta}(\Delta_{c})=\frac{1}{V+\sqrt{(\Gamma^{2}+2\Omega^{2})/3\rho_{rr}^{2}}}$
(16)

Data availability. The data that support this study are available at Github
[56] (many-body-enhanced-metrology).

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Acknowledgment. D-S.D thanks for discussions with professor Jun Ye from JILA.
Z-K.L. appreciates instructive discussions with Dr. Tian-Yu Xie. We
acknowledge funding from National Key Research and Development Program of
China (2017YFA0304800), the National Natural Science Foundation of China
(Grant Nos. U20A20218, 61525504, 61722510, 61435011), the Innovation Program
for Quantum Science and Technology (2021ZD0301100), Anhui Initiative in
Quantum Information Technologies (AHY020200), the Youth Innovation Promotion
Association of Chinese Academy of Sciences under Grant No. 2018490, EPSRC
through grant agreements EP/M014398/1 (“Rydberg soft matter”), EP/R002061/1
(“Atom-based Quantum Photonics”), EP/L023024/1 (“Cooperative quantum optics in
dense thermal vapours”), EP/P012000/1 (“Solid State Superatoms”), EP/R035482/1
(“Optical Clock Arrays for Quantum Metrology”), EP/S015973/1 (“Microwave and
Terahertz Field Sensing and Imaging using Rydberg Atoms”), the Danish National
Research Foundation through the Center of Excellence for Complex Quantum
Systems (Grant agreement No. DNRF156), as well as, DSTL, and Durham
University.).

Author contributions. D-S.D. conceived the idea and implemented the physical
experiments with Z-K.L. Z-K.L., D-S.D., and K.M. employ the Fisher
information. D-S.D., Z-K.L., and K.M. derived the equations, plotted figures,
and wrote the manuscript. All authors contributed to discussions regarding the
results and analysis contained in the manuscript. D-S.D., B-S.S.,G-C.G., K.M.,
and C-S.A. support this project.

Competing interests. The authors declare no competing interests.


# Observational properties of Coherent Quantum Black Holes

Abdybek Urmanov<EMAIL_ADDRESS>Department of Physics, Nazarbayev
University, Kabanbay Batyr 53, 010000 Astana, Kazakhstan  Hrishikesh
Chakrabarty<EMAIL_ADDRESS>Department of Physics,
Nazarbayev University, Kabanbay Batyr 53, 010000 Astana, Kazakhstan  Daniele
Malafarina<EMAIL_ADDRESS>Department of Physics, Nazarbayev
University, Kabanbay Batyr 53, 010000 Astana, Kazakhstan

###### Abstract

We consider null and time-like geodesics around a spherically symmetric, non-
rotating Coherent Quantum Black Hole (CQBH). The classical limit of the
geometry of CQBH departs from that of the Schwarzschild spacetime at short
scales and depends on one parameter $R_{s}$ which can be interpreted as the
physical radius of the ‘quantum’ core. We study circular orbits, photon rings,
and lensing effects and compare them with the Schwarzschild metric. Using the
relativistic ray-tracing code GYOTO, we produce a simulation of the shadow and
show that thin accretion disks around a CQBH possess unique ring structures
that distinguish them from other theoretical models.

##  I Introduction

The term ‘black hole mimickers’ refers to theoretical astrophysical objects
that closely resemble black holes in observational appearance, exhibiting
similar features such as photon rings and accretion disks while at the same
time being described by a theoretical model distinct from classical black
holes Cardoso and Pani (2019).

Various types of black hole mimickers have been proposed over the years as
candidates for the extreme compact objects we observe in the universe. These
include gravastars Cardoso et al. (2016a); Visser and Wiltshire (2004);
Chirenti and Rezzolla (2007); Harko et al. (2009); Broderick and Narayan
(2007), boson stars Vincent et al. (2016); Olivares et al. (2020); Marunović
and Murković (2014); Rosa et al. (2023), wormholes Einstein and Rosen (1935);
Berry et al. (2020a, b); Bambhaniya et al. (2022); Tsukamoto (2020), regular
black holes Casadio et al. (2024a); Hayward (2006); Bardeen (2014, 2019);
Bambi (2013); Simpson and Visser (2022); Mazza et al. (2021); Abdujabbarov et
al. (2016); Lemos and Zaslavskii (2008); Bonanno et al. (2024) and deformed
compact objects Carballo-Rubio et al. (2018a); Shaikh et al. (2022);
Narzilloev et al. (2020); Abdikamalov et al. (2019); Chowdhury et al. (2012).
In recent times, significant progress in experimental observations, such as
the imaging of the shadow of the supermassive black hole at the core of the
M87 galaxy Akiyama et al. (2019a, b) and the detection of gravitational waves
from binary mergers Abbott et al. (2016) has presented the opportunity to test
the validity of these alternative models and in turn potentially constrain
proposed modification to General Relativity (GR) Li et al. (2022); Shaikh
(2023); Johnson-Mcdaniel et al. (2020); Cardoso et al. (2016b).

In the present article we consider an alternative black hole model, called
Coherent Quantum Black Hole (CQBH), which is obtained from a simple
prescription for quantization that was proposed by Casadio Casadio (2022a,
2021, b). The CQBH, despite being a simple toy model, presents several
features that make it appealing as a hypothetical candidate for an
astrophysical compact object. First of all it is built with a minimal approach
for the quantization of gravity in the vicinity of the classical black hole
singularity which produces a compact massive core of finite size. Secondly,
the core of the resulting object may be located below the classical event
horizon. This results in the CQBH having an outer horizon which mimics the
black hole event horizon while lacking an inner Cauchy horizon, thus avoiding
the problems related to causality violations and mass inflation Price (1972);
Simpson and Penrose (1973); Poisson and Israel (1989, 1990); Carballo-Rubio et
al. (2018b). Finally, the CQBH produces a spacetime geometry which bears the
unique signature of its core. This means that quantum corrections in the
strong curvature region produce macroscopic effects which are potentially
measurable. From the point of view of observations this may allow for
observational tests to constraint the size of the object’s core and
consequently the scale of the modifications to GR. In Casadio et al. (2024b)
it was shown that the CQBH is consistent with the Bekenstein-Hawking
thermodynamic entropy for black holes while in Casadio et al. (2022) a charged
CQBH was obtained. In this article we study the motion of massive test
particles and photons in the geometry of the CQBH and investigate how future
observations may be used to test the physical viability of this hypothetical
compact object.

The paper is organized as follows: In section II we briefly review the line
element and its properties. Section III and IV are dedicated to modelling the
motion of massive and massless particles respectively. In particular in
section III we model thin accretion disks in the geometry of the CQBH while in
section IV we model gravitational lensing. In section V we use GYOTO, a
general relativistic raytracing code Vincent et al. (2011); Aimar et al.
(2024); Paumard et al. (2024) to simulate the image of the CQBH’s shadow.
Finally in section VI we summarize the results and their implications.
Throughout the article we use geometrized units setting $c=1$ with the metric
signature $\\{-,+,+,+\\}$.

##  II The spacetime

The geometry of the static and spherically symmetric CQBH in Schwarzschild
coordinates is given by

$ds^{2}=-(1+2V_{\rm QN})dt^{2}+\frac{dr^{2}}{1+2V_{\rm QN}}+r^{2}d\Omega^{2},$
(1)

where $d\Omega^{2}$ is the line element on the unit sphere and

$V_{\rm QN}=\frac{2V_{\rm N}}{\pi}{\rm Si}\left(\frac{r}{R_{s}}\right),$ (2)

represents the effective quantum analog of the Newtonian potential $V_{\rm
N}=-G_{\rm N}M/r$. Here $R_{s}$ is a parameter describing the size of the
quantum core and

${\rm Si}(x)=\int_{0}^{x}\frac{\sin{z}}{z}dz,$ (3)

is the Sine integral function. The other constants are $G_{\rm N}$, i.e.
Newton’s constant and $M$, which describes the ADM mass of the CQBH. It is
important to notice that the parameter $R_{s}$ in the CQBH model is obtained
by introducing an Ultra-Violet (UV) cutoff scale to regularize the UV
divergence due to the vanishing volume and infinite density of the classical
source of the Schwarzschild metric. Therefore the metric must be understood as
a simple toy model that captures the essence of the classical configuration
emerging from the coherent quantum states. The resulting core is ‘classical’
in the sense that it is larger than the Planck length and of the order of the
Schwarzschild radius Casadio (2021). In fact $R_{s}$ may be larger, smaller or
equal than the size of the horizon radius (see Fig. 1). Due to this additional
parameter in the metric, the solution violates the no-hair theorem Calmet et
al. (2022).

The CQBH metric (1) reduces to the Schwarzschild metric in the weak field
limit, as can be seen by taking the limit, as $r/R_{s}\rightarrow\infty$, for
which the sine integral function ${\rm
Si}\left(r/R_{s}\right)\rightarrow\pi/2$ and $V_{\rm QN}\simeq V_{\rm N}$.
Hence, at large distances the metric is asymptotically flat. However at
distances close to the core, the properties of the CQBH differ significantly
from those of a Schwarzschild black hole. For instance the classical black
hole singularity is replaced by a finite size core with an integrable
singularity at the center and finite tidal forces Casadio (2022a); Lukash and
Strokov (2013). This suggests that observations of orbits in the vicinity of
the compact object may allow to distinguish it from a Schwarzschild black
hole.

The radius of the event horizon $R_{H}$ of a CQBH is obtained by solving the
equation $g^{rr}=1+2V_{\rm QN}=0$ and is shown in Fig. 1 as a function of the
core radius $R_{s}$. Notice that due to the presence of the Sine integral
function, the value of the horizon radius as a function of $R_{s}$ oscillates
about the Schwarzschild radius ($R_{\rm Sch}=2M$). In the following, for the
sake of simplicity, we take the ADM mass of the compact object as $M=1$, or
equivalently we rescale the time and radial coordinates $t$, $r$ and the core
radius $R_{s}$ in units of $M$. We can see that there exist values of $R_{s}$
for which the horizon of the CQBH is the same as Schwarzschild. The largest
value of $R_{s}$ for which the two horizons coincide is $R_{s}\simeq 1.06$ and
for $R_{s}\gtrsim 1.2$ the horizon radius is smaller than the core radius,
thus making the CQBH a horizonless compact object. A detailed analysis of the
CQBH can be found in Casadio and Micu (2024). In the following, our focus lies
in understanding the properties of massive and massless particle motion in the
CQBH exterior and how specific features of lensing, shadow and accretion disks
depend on the parameter $R_{s}$.

Figure 1:  The solid line represents the horizon radius $R_{H}$ of the CQBH as
a function of $R_{s}$. For comparison we show the Schwarzschild radius $R_{\rm
Sch}$ given by the horizontal dashed line. The shaded area represents the size
of the matter core showing that for $R_{s}\gtrsim 1.2$ the core is larger than
the horizon.

##  III Motion of massive particles

Black holes are not directly observable by definition. However, their
surroundings have properties that depend directly on the nature and properties
of the central object. For example, the closest stable orbit to a Kerr black
hole depends on the value of the central object’s angular momentum. Therefore,
at least in principle, it should be possible to test theoretical models of
extreme compact objects through observations of their surrounding environment,
such as accretion disks. We can model the motion of particles in the accretion
disk by considering geodesic motion for test particles (assumed to have
negligible mass as compared to the central object’s) on circular orbits. To
this aim in the following we shall write the equations of motion for test
particles.

Due to the static and spherically symmetric nature of the CQBH metric, there
exist two Killing vectors corresponding to two constants of motion for
particles on geodesics, namely the particle’s energy $p_{t}=\tilde{E}$ and
angular momentum $p_{\phi}=\tilde{L}$. Also, due to spherical symmetry, we can
restrict the analyses to the equatorial plane. Now, the conserved quantities
per unit mass $E=\tilde{E}/m$ and $L=\tilde{L}/m$ are obtained from

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{p^{t}}{m}=(1+2V_{\rm QN})^{-1}E,$ (4)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{p^{\phi}}{m}=r^{-2}L,$ (5)

where $\dot{x}^{\mu}=dx^{\mu}/d\tau$ is the tangent to the geodesic
$\\{x^{\mu}(\tau)\\}$ with $\tau$ the proper time. Using the normalization
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$ for massive particles, we find the
radial equation of motion as

$\dot{r}^{2}=E^{2}-V_{\rm eff}(r),$ (6)

where $V_{\rm eff}$ is the effective potential which for the metric (1) is
given by

$V_{\rm eff}=\left(\frac{L^{2}}{r^{2}}+1\right)\left(1+2V_{\rm QN}\right).$
(7)

In Fig. 2, we plot the effective potential with respect to the radial
coordinate $r$ and compare its behavior with that of a Schwarzschild black
hole. It is interesting to note that the effective potential for massive
particles around a CQBH exhibits an oscillatory behavior with several local
extrema for any given values of $E$ and $L$, which is in sharp contrast with
the Schwarzschild case. In Fig. 2, we chose the value of the angular momentum
$L=12$ which corresponds to particles on circular orbit rotating at the
innermost stable circular orbit (ISCO), $R_{\rm ISCO}=6$, in the Schwarzschild
spacetime. The most notable feature is that there exist multiple radii
allowing stable orbits even beyond the ISCO radius for Schwarzschild. These
numerous stable and unstable orbits could substantially affect the appearance
of the accretion disk around the CQBH. For larger values of $r$, i.e. far from
the black hole, the CQBH effective potential tends toward Schwarzschild and
the potential barrier for stable orbits becomes smaller, thus making it more
difficult for particle to circularize in a realistic accretion disk.

Figure 2:  The effective potential $V_{\rm eff}$ as a function of the radial
coordinate for massive particles in the equatorial plane of CQBH spacetime as
compared to the Schwarzschild effective potential (solid line). All cases are
shown for angular momentum $L=12$. The vertical grey line highlights the
innermost stable circular orbit for the Schwarzschild case.

For timelike geodesics, the effective potential $V_{\rm eff}$ is given by Eq.
(7) and the energy and angular momentum of a particle on a circular orbit at a
radius $r$ can be found from the conditions $V_{\rm eff}=E$ and
$V^{\prime}_{\rm eff}=0$, where the prime denotes derivative with respect to
the radial coordinate $r$. More explicitly, the second equality gives

${V^{\prime}_{\rm eff}}=\frac{2L^{2}}{r^{3}}+\frac{4L^{2}V_{\rm
QN}}{r^{3}}-\frac{2L^{2}V^{\prime}_{\rm QN}}{r^{2}}-2V^{\prime}_{\rm QN}=0,$
(8)

and using Eqs. (6) and (8), we can obtain the energy and angular momentum
$E(r)$ and $L(r)$ of particles on a given circular orbit with radius $r$ as

$\displaystyle E^{2}$ $\displaystyle=$ $\displaystyle\frac{(1+2V_{\rm
QN})^{2}}{1+2V_{\rm QN}-V^{\prime}_{\rm QN}r},$ (9) $\displaystyle L^{2}$
$\displaystyle=$ $\displaystyle\frac{r^{3}V^{\prime}_{\rm QN}}{1+2V_{\rm
QN}-V^{\prime}_{\rm QN}r}.$ (10)

Similarly, the particle’s angular velocity is given by

$\omega=\frac{d\phi}{dt}=\sqrt{\frac{V^{\prime}_{\rm QN}}{r}}.$ (11)

At distances far from the compact object, $V_{\rm QN}\simeq V_{\rm N}$, and
these quantities reduce to those of the Schwarzschild metric. Now, in
Schwarzschild we can find the radius of the ISCO from the condition $V_{\rm
eff}^{\prime\prime}=0$, which gives the radius at which the circular orbit is
marginally stable. In the case of the CQBH the condition $V_{\rm
eff}^{\prime\prime}=0$ identifies the radii at which the local extrema become
marginally stable. We get

$\displaystyle{V_{\rm eff}}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{6L^{2}}{r^{4}}-\frac{8L^{2}V^{\prime}_{\rm
QN}}{r^{3}}+\frac{12L^{2}V_{\rm QN}}{r^{4}}+$ (12)
$\displaystyle+\frac{2L^{2}V^{\prime\prime}_{\rm
QN}}{r^{2}}+2V^{\prime\prime}_{\rm QN}=0,$

where

$\displaystyle V_{\rm QN}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\frac{GM}{r^{2}}\left[{\rm
Si}\left(\frac{r}{R_{s}}\right)-\sin{\frac{r}{R_{s}}}\right],$ (13)
$\displaystyle V_{\rm QN}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\frac{GM}{r^{2}}\left[-\frac{2}{r}{\rm
Si}\left(\frac{r}{R_{s}}\right)+\frac{3}{r}\sin{\frac{r}{R_{s}}}-\frac{1}{R_{s}}\cos{\frac{r}{R_{s}}}\right].$

Figure 3:  Boundary contours for regions of stable circular orbits for the
CQBH depending on the value of the parameter $R_{s}$. The black horizontal
dashed line represents $R_{\rm ISCO}$ for the Schwarzschild case. The grey
solid and dashed lines denote the radius of the horizons for the CQBH and
Schwarzschild, respectively.

Stable circular orbits for massive particles around the CQBH are shown in Fig.
3, depending on the radius of the core $R_{s}$. For any given value of
$R_{s}$, there exist several regions of stable circular orbits in the CQBH
spacetime and the radius of the ISCO $R_{\rm ISCO}$ for each region
corresponds to the smallest stable circular orbit. This means that there exist
gaps between separate regions of stable orbits. This feature persists also at
large distances making the CQBH disk’s structure unique. Of course, at large
distances the gaps would become less prominent as the potential barriers
become smaller. Note that the smallest $R_{\rm ISCO}$ for the CQBH that
resembles Schwarzschild is obtained for $R_{s}\approx 0.8$. Notice also that
for values of $R_{s}\gtrsim 0.8$ the smallest ISCO radius is larger than
Schwarzschild’s.

On the other hand, the regions of stable circular orbits get closer for
smaller values of $R_{s}$ and it can be expected that for sufficiently small
values there would be stable orbits all the way to the CQBH horizon. This is
in sharp contrast to the Schwarzschild case where stable orbits can no longer
exist beyond $r=6$.

##  IV Motion of massless particles

Following the same procedure as before for massless particles, we still have
the conserved quantities from Eqs. (4) and (5) and we must now use
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$ to write the radial equation of
motion

$\dot{r}^{2}=E^{2}-\frac{L^{2}}{r^{2}}(1+2V_{\rm QN}),$ (15)

so that the effective potential for photons is given by

$V_{\rm eff}=(1+2V_{\rm QN})\frac{L^{2}}{r^{2}}.$ (16)

The photon ring is defined as the circular orbit of massless particles and it
is given by the condition $V^{\prime}_{\rm eff}=0$, explicitly

$1-V_{\rm QN}^{\prime}r+2V_{\rm QN}=0.$ (17)

This equation reduces to

$\frac{\pi r}{2G_{N}M}=3{\rm
Si}\left(\frac{r}{R_{s}}\right)-\sin\left(\frac{r}{R_{s}}\right),$

which can be solved numerically to obtain the radius of photon sphere. The
most interesting aspect is that for $R_{s}\lesssim 0.36$ there exist values of
$R_{s}$ for which the above equation has multiple solutions, indicating the
existence of unstable as well as stable orbit for photons. In Fig. 4, we
compare the effective potentials for massless particles in Schwarzschild and
CQBH for different values of $R_{s}$. Again the effective potential of the
CQBH shows an oscillatory behavior about the Schwarzschild’s effective
potential with the most interesting feature being the existence of one stable
circular orbit for photons for $R_{s}\simeq 0.2$.

Figure 4: The effective potential for massless particles in the equatorial
plane of the CQBH as a function of the radial coordinate in units of $M$. The
effective potential is shown for three values of $R_{s}$. Notice that for
$R_{s}=0.2$ (dotted line) there exist three locations for circular orbits for
photons, with one of them being stable. For comparison the solid line shows
the corresponding effective potential for Schwarzschild and the vertical grey
line highlights the Schwarzschild photon ring radius.

In Fig. 5, we show the radius of the CQBH photon ring as a function of
parameter $R_{s}$. For $R_{s}=1.06$, the horizon of CQBH is approximately
equal to the Schwarzschild one. However, at this value of the parameter, the
photon ring of the CQBH is larger than the Schwarzschild photon ring. Also,
for smaller $R_{s}$ there are regions where more than one photon ring may
appear, for example, there are three rings for $R_{s}\simeq 0.2$, and for even
smaller values of the parameter the photon rings get closer. At sufficiently
small values of $R_{s}$ we expect the existence of a photon disk which spreads
from $r\approx 2.35$ to $r\approx 3.6$.

Figure 5: The CQBH’s photon ring radius ($R_{\rm ph}$) as a function of the
parameter $R_{s}$ is represented by the solid black line in comparison with
Schwarzschild’s photon ring as the dashed black line. Also, the event horizon
for the CQBH and Schwarzschild are shown by the grey solid and dashed lines,
respectively. The vertical line shows the location of the three circular
photon orbits obtained for $R_{s}=0.2$ as can be seen by the extrema of the
effective potential in Fig. 4.

###  IV.1 Lensing

The first application of General Relativity was gravitational lensing Einstein
(1936) and since then it has been widely used to study several astrophysical
phenomena (see, for example Schneider et al. (1992); Luminet (1979); Clowe et
al. (2006); Virbhadra and Ellis (2000); Abdujabbarov et al. (2017)). In the
following, we analyze the deflection angle of light in the CQBH geometry and
compare it with that by a Schwarzschild black hole.

Using Eq. (5) and (15), we obtain

$\frac{d\phi}{dr}=\pm\frac{1}{r^{2}}\Bigg{[}\frac{1}{b^{2}}-\frac{1}{r^{2}}(1+2V_{\rm
QN})\Bigg{]}^{-1/2},$ (18)

where $b=L/E$ is the impact parameter for a massless particle. Then the
magnitude of the total angle swept by a light-ray as it travels from infinity
towards the closest approach to the CQBH and to infinity again can be obtained
by simply integrating the previous equation

$\Delta\phi=2\int_{r_{1}}^{\infty}\frac{dr}{r^{2}}\Bigg{[}\frac{1}{b^{2}}-\frac{1}{r^{2}}(1+2V_{\rm
QN})\Bigg{]}^{-1/2},$ (19)

where due to symmetry we can integrate from infinity to the turning point
$r_{1}$, defined by the condition $dr/dt\big{|}_{r_{1}}=0$ and for which
$V_{\rm eff}(r_{1})=1/b^{2}$. Introducing a new variable $w=b/r$ we rewrite
$\Delta\phi$ as

$\Delta\phi=2\int_{0}^{w_{1}}dw\left[1-w^{2}\left(1-\frac{4Mw}{\pi b}{\rm
Si}\left(b/wR_{s}\right)\right)\right]^{-1/2},$ (20)

where $w_{1}=b/r_{1}$ and we have set $G_{\rm N}=1$.

Of course, it is not possible to find an analytical solution for the integral
above, because of the presence of the sine integral function. Therefore, first
we look at the weak-field limit, i.e when $M/b\ll 1$. In the weak-field limit,
as $b/wR_{s}\rightarrow\infty$ and for large values of the argument the Sine
integral function ${\rm Si}(x)$ can be rewritten as an asymptotic series as

${\rm
Si}(x)\simeq\frac{\pi}{2}-\frac{\cos{x}}{x}-\frac{\sin{x}}{x^{2}}+\cdots$ (21)

Keeping the next to leading order term from Eq. (21) $\Delta\phi$ becomes

$\Delta\phi=2\int_{0}^{w_{1}}dw\left[1-w^{2}\left(1-\frac{2M}{b}w+H(w)\right)\right]^{-1/2},$
(22)

with

$H(w)=\frac{4MR_{s}}{\pi b^{2}}w^{2}\cos\left(\frac{b}{wR_{s}}\right).$ (23)

Notice that $H(w)$ has a term in $MR_{s}/b^{2}$ which in the weak field limit
is negligible. Then for $H=0$ Eq. (22) is same as in the Schwarzschild case
and can be integrated to find $\Delta\phi=\pi+4M/b$. Considering $H\neq 0$ and
making an expansion of the integrand in Eq. (22) for $M/b\ll 1$ we get

$\Delta\phi=\pi+\frac{4M}{b}-\frac{4MR_{s}}{\pi
b^{2}}\int_{0}^{w_{1}}dw\frac{w^{4}\cos(b/wR_{s})}{(1-w^{2})^{3/2}},$ (24)

which is related to the deflection angle for a light ray via
$\delta\phi=\Delta\phi-\pi$. The deflection angle as a function of the inverse
of the impact parameter $1/b$ for Schwarzschild (with $M=1$) and for the CQBH
with $R_{s}=0.5$ is shown in Fig. 6.

We can also calculate the critical value of the impact parameter $b_{\rm
crit}$ at which photons get captured into the photon ring. The critical impact
parameter can be calculated numerically by substituting $R_{\rm ph}$ into Eq.
(15) and solving the following equation

$\frac{1}{b_{\rm crit}^{2}}-\frac{1+2V_{QN}(R_{\rm ph})}{R^{2}_{\rm ph}}=0.$
(25)

The value of $b_{\rm crit}$ is important because it determines the radius of
the shadow of the black hole which in turn relates to the image seen by a
distant observer. Then for Schwarzschild we know that $b_{\rm
crit}=3\sqrt{3}\approx 5.196$ and from Fig. 6 for a CQBH with $R_{s}=0.5$ we
find $b_{\rm crit}\approx 4.764$. Fig. 6 also shows that lensing for CQBH and
Schwarzschild coincides in the weak-field regime. However, for any given value
of $R_{s}$ the value of the deflection angle of the CQBH as a function of
$1/b$ oscillates about the Schwarzschild value in a manner that may affect the
appearance of the accretion disk and the shadow.

Figure 6: The deflection angle $\delta\phi$ as a function of the inverse of
the impact parameter $1/b$. The solid line shows the Schwarzschild case while
the dash-dotted line shows the CQBH with $R_{s}=0.5$. The dashed line
represents the weak lensing limit. The two dotted vertical lines denote the
critical value of impact parameter $b_{\rm crit}$ at which the photons get
captured.

##  V Shadow

$\theta=5^{\circ}$ |  |  |
---|---|---|---
$\theta=72^{\circ}$ |  |  |
$\theta=86^{\circ}$ |  |  |
| Schwarzschild | $R_{s}=1.06$ | $R_{s}=0.6$
Figure 7: Images of geometrically thin and optically thick accretion disks
around Schwarzschild and CQBH with $R_{s}=1.06$ and $R_{s}=0.6$ for
inclination angles $\theta=5^{\circ},$ $72^{\circ}$, $86^{\circ}$. It is
important to notice that despite of the fact that for $R_{s}\simeq 1.06$ the
horizon of the CQBH has value close to the Schwarzschild case the images of
the two shadows are considerably different.

We now turn our attention to the shadow image of a CQBH. The first computation
of the appearance of an accretion disk surrounding a Schwarzschild black hole
using the ray-tracing method was done by Luminet in the 1970s Luminet (1979).
To obtain the image of the shadow for the CQBH we use an open-source general
relativistic ray-tracing code called GYOTO Vincent et al. (2011); Aimar et al.
(2024); Paumard et al. (2024), which we modify to simulate the image of a thin
infinite accretion disk around the CQBH and produce images of the
gravitationally lensed disk by ray-tracing the photons emitted from the
accreting matter towards the observer. The GYOTO code integrates geodesics
backwards in time, so the initial conditions are given on the observer’s
screen in terms of position and the incidence angle of the photons. Since most
photons emitted from the accreting matter do not arrive at the observer,
backward ray-tracing deals only with photons that hit the screen thus
substantially reducing the number of calculations. To integrate the geodesic
equations, the code uses the Runge-Kutta algorithm of fourth order RK4 with an
adaptive step Press et al. (2007). The GYOTO code is also capable of producing
shadow images with other more realistic models of accretion disks such as
Novikov-Thorne Novikov and Thorne (1973), Polish doughnut Abramowicz and
Fragile (2013), etc.. However for our purposes simulating the shadow of a
geometrically thin infinite accretion disk around a CQBH is sufficient to
outline the key characteristics that distinguish it from Schwarzschild.

We modified GYOTO to include the CQBH spacetime and for computational
complexity reduction, we used the simplest default model of accretion disk
called ‘ThinDisk’, which describes the geometrically thin accretion disk Page
and Thorne (1974); Marck (1996). Geometrically thin refers to the condition
$h\ll r$, where $r$ is the radial coordinate on the disk and $h$ is the semi-
thickness of the disk. The interior radius of a disk corresponds to $R_{\rm
ISCO}$ and the outer radius of the disk is taken at infinity. Also, for
simplicity, we consider an optically thick disk, for which only photons
emitted from the disk’s surface can be observed, i.e. we take $\lambda\ll h$
where $\lambda$ is the mean free path of the photon.

For the optically thick and geometrically thin disk, according to Fanton et
al. (1997), the emitted radiation can be modelled via monochromatic radiation
with isotropic specific intensity in the rest frame of the disk. This means
that the intensity of emitted radiation $I_{\nu_{em}}$ as a function of the
emitted frequency $\nu$ behaves as:

$I_{\nu_{em}}\propto\delta(\nu_{\rm em}-\nu_{\rm line})\epsilon(r),$ (26)

where $\delta$ is a Dirac distribution, $\nu_{\rm line}$ is the frequency of
the monochromatic emission line and $\epsilon(r)$ is the surface emissivity.
Here for simplicity we follow the GYOTO’s default model ‘ThinDisk’ thus taking
$\epsilon(r)=1$. The specific intensity for distant observer is given as:

$I_{\nu_{\rm obs}}=g^{3}I_{\nu_{\rm em}},$ (27)

where $\nu_{\rm obs}$ is observed frequency and $g=\nu_{\rm obs}/\nu_{\rm em}$
accounts for the gravitational redshift. The observed flux is then connected
to the observed specific intensity via

$dF_{\nu_{\rm obs}}=I_{\nu_{\rm obs}}\cos{\alpha}d\Omega,$ (28)

where $\alpha$ is the angle between the direction of the incoming photon and
the normal to the screen and $d\Omega$ (not to be confused with the line
element on the unit sphere) is the infinitesimal solid angle around the
direction of incidence of the photon. In the code, to each pixel on the screen
corresponds a direction in the sky from which photons can be received. The
whole screen is then defined as a collection of pixels covering a given field
of view in the sky and the whole screen then covers a finite solid angle
$\Omega$. Then we can define the total observed flux as:

$F_{\nu_{\rm obs}}=\sum_{i}I_{\nu_{\rm
obs},i}\cos{\alpha_{i}}\delta\Omega_{i}$ (29)

where the subscript ‘${i}$’ corresponds to one pixel, the summation is over
the total number of pixels, $\alpha_{i}$ is the incidence angle for each
pixel, $\delta\Omega_{i}$ is the infinitesimal solid angle for each pixel, and
$I_{\nu_{\rm obs},i}$ is the observed specific intensity for each pixel.

The simulations of a thin accretion disk around a CQBH are shown in Fig. 7.
The figure shows shadow images at three different inclination angles of the
line of sight between the screen and the source, i.e. $\theta=5^{\circ},\
72^{\circ}\ \text{and}\ 86^{\circ}$. Notice that $\theta=0^{\circ}$
corresponds to the line of sight being perpendicular to the disk and
$\theta=90^{\circ}$ corresponds to the line of sight on the disk’s plane. The
CQBH shadow images are obtained for two values of the parameter $R_{s}=1.06$
and $R_{s}=0.6$. The first column shows the shadow in the Schwarzschild case
for comparison (i.e. $R_{s}=0$). It is immediately noticeable that the CQBH
exhibits periodic bright and dim patterns in the accretion disk for both
values of $R_{s}$. This alternating bright and dark regions in the accretion
disk are a consequence of multiple stable and unstable regions in the
effective potential. This behavior is in sharp contrast to the Schwarzschild
case.

The second interesting feature is that the inner edge of the accretion disk is
much closer to the horizon for the CQBH with $R_{s}=0.6$ than the
Schwarzschild case. On the other hand, for $R_{s}=1.06$, the inner edge is
almost at the same distance as the Schwarzschild case. This feature is evident
from the variable radius of the ISCO of CQBH which has a sharp dependence on
$R_{s}$ as seen from the Fig. 3. Therefore, even though for some values of
$R_{s}$ the CQBH and Schwarzschild may have similar values for the location of
the horizon or the ISCO radius, the shadow image always looks different from
Schwarzschild. This suggest that, depending on the observational capabilities
(basically on the power to resolve the bright and dim regions) of the current
and future VLBI telescope, it could be possible to distinguish a CQBH from a
Schwarzschild black hole via observations.

##  VI Discussion

Spacetime singularities are one of the most fascinating predictions of GR
while, at the same time, being an indicator of the limits of the theory. The
main problem with the singularity at the center of black holes is the
divergence of curvature and tidal forces that signals a breakdown of the
theory. There exist many attempts to resolve the black hole singularity by
introducing suitable corrections to GR in the strong field (see Malafarina
(2017) for a recent review). However, a large number of these attempts predict
regular black holes (see for example Malafarina (2023)). Such regular black
holes, while being non-singular, typically posses a Cauchy horizon, which
cause other problems such as mass inflation. Some authors claim that
singularities are not necessarily a problem since it may be impossible to test
the true nature of black hole candidates Nakao (2024). The CQBH considered
here replaces the singular center with a finite quantum core which exhibits a
classical behavior and has an integrable, i.e. tame, singularity, without a
Cauchy horizon. This makes it an appealing hypothetical candidate for
astrophysical compact objects. Moreover, we have shown that the
‘classicalization’ of the quantum corrections which produce the object’s core
has macroscopic effects that produce distinct features in the observable
properties of the CQBH’s accretion disk. Such effects may be in principle
observed thus allowing to test and eventually rule out the model.

For example, the bright and dark ring signatures in the CQBH’s shadow lead to
the possibility of distinguishing it from the Schwarzschild metric by
astrophysical observations. The periodicity of $2\pi R_{s}$ in the function
${\rm Si}(r/R_{s})$ would induce a corresponding periodicity in the separation
$d$ between nearby rings in the accretion disk. In principle this value can be
related to the distance between bright and dark rings in the image of the
accretion disk as obtained by a telescope thus allowing to put constraints on
the allowed values of $R_{s}$ from shadow and lensing observations.

## Acknowledgement

The authors would like to thank Roberto Casadio and Andrea Giusti for useful
comments and discussion. The authors acknowledge support from Nazarbayev
University Faculty Development Competitive Research Grant No. 11022021FD2926.

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