††thanks: Current address: Department of Physics, Yale University, New Haven,
CT 06520, USA††thanks: These authors contributed equally to this
work.††thanks: These authors contributed equally to this work.

# Third-order exceptional point in an ion-cavity system

Jinuk Kim  Department of Electrical Engineering, Pohang University of Science
and Technology (POSTECH), 37673 Pohang, Korea  Taegyu Ha  Department of
Electrical Engineering, Pohang University of Science and Technology (POSTECH),
37673 Pohang, Korea  Donggeon Kim  Department of Electrical Engineering,
Pohang University of Science and Technology (POSTECH), 37673 Pohang, Korea
Dowon Lee  Department of Electrical Engineering, Pohang University of Science
and Technology (POSTECH), 37673 Pohang, Korea  Ki-Se Lee  Department of
Electrical Engineering, Pohang University of Science and Technology (POSTECH),
37673 Pohang, Korea  Jongcheol Won  Department of Electrical Engineering,
Pohang University of Science and Technology (POSTECH), 37673 Pohang, Korea
Youngil Moon  Department of Electrical Engineering, Pohang University of
Science and Technology (POSTECH), 37673 Pohang, Korea  Moonjoo Lee
<EMAIL_ADDRESS>Department of Electrical Engineering, Pohang
University of Science and Technology (POSTECH), 37673 Pohang, Korea

###### Abstract

We investigate a scheme for observing the third-order exceptional point (EP3)
in an ion-cavity setting. In the lambda-type level configuration, the ion is
driven by a pump field, and the resonator is probed with another weak laser
field. We exploit the highly asymmetric branching ratio of an ion’s excited
state to satisfy the weak-excitation limit, which allows us to construct the
non-Hermitian Hamiltonian $(H_{\textrm{nH}})$. Via fitting the cavity-
transmission spectrum, the eigenvalues of $H_{\textrm{nH}}$ are obtained. The
EP3 appears at a point where the Rabi frequency of the pump laser and the
atom-cavity coupling constant balance the loss rates of the system. Feasible
experimental parameters are provided.

Exceptional point (EP) is a singularity in which two, or more eigenvalues and
eigenvectors coalesce in a parameter space [1]. In the vicinity of the EP, the
non-Hermitian physical system exhibits unconventional phenomena that do not
occur in the Hermitian system. For example, strong spectral change was
measured in photonics [2], parity-time-symmetry-associated research were
performed [3], and the topology was investigated extensively [4]. While such
works were mostly done with the second-order EPs (EP2s), recent studies
focused on multi-level structures where higher-order EPs can emerge [5].

More intriguing phenomena would be explored with the higher-order EPs. The
response of the system becomes more sensitive to external perturbations [6],
and the non-trivial topology would be more complex and richer as the order
increases [7, 8]. Higher-order EPs have been observed in various experimental
settings, including acoustic cavity [9], micro-ring [10], dielectric sphere
[11], electronic circuit [12], acoustic metagrating [13], and membrane
[8]—these experiments were performed in classical domain. Realization of EP in
quantum realm, generally more difficult in practice [14], would make quantum
mechanics play in conjunction with this singularity: The EP effects would
manifest by a post-selection measurement [15], and quantum noise can give an
impact to the EP-based sensor [16].

Here, we propose a scheme for observing an EP3 in a full quantum system,
consisting of a single trapped ion coupled to an optical resonator. A lambda-
type, atomic three-level configuration is considered. While a weak transition
of the ion is pumped by a laser field from the cavity side, the other strong
transition of the ion, to which the cavity frequency is tuned, is probed with
a weak laser field along the cavity axis. We exploit the highly asymmetric
branching ratio of the ion’s excited state to satisfy the weak-excitation
limit: This allows us to neglect quantum jump operators and obtain
$H_{\textrm{nH}}$. By fitting the cavity-transmission spectrum, all three
eigenvalues of $H_{\textrm{nH}}$ are obtained. We also identify that the EP3
appears at one point of the parameter space, where both the atom-cavity
coupling constant $g$ and Rabi frequency $\Omega$ by the pump laser balance
the atomic and cavity decay rates. From the perspective of atomic
spectroscopy, the EP3 is equivalent to a crossover point from cavity
electromagnetically induced transparency (EIT) to Autler-Townes splitting
(ATS) [17, 18, 19, 20]. Experimental parameters are examined with a 40Ca+ ion
and fiber-based cavity.

We remark that our system operates with trapped ions where the gate operation,
qubit-state detection, and other controls can be done at a state-of-the-art
level [21]. Here we utilize one of the capabilities, nanoscopic position
control [22], to change one of the parameters, $g$, which constitutes a key
ingredient for the observation of the EP3. Recently EP2s were observed with
single trapped ions in free space [23, 24], and our work describes an
extension of one more order with an optical resonator. We also note that,
while this ion-cavity settings have mostly been used for investigating quantum
optics [25, 26, 27] and quantum networks [28, 29, 30] to date, our work
provides new research opportunities associated with non-Hermitian physics,
employing both coherent ion-cavity interaction and cavity dissipation.

We consider a setup in which a single trapped ion is coupled to an optical
resonator (Fig. 1). The atomic energy levels consist of two ground states
$|$g${}_{1}\rangle$, $|$g${}_{2}\rangle$ and one excited state $|$e$\rangle$.
The $|$g${}_{2}\rangle$–$|$e$\rangle$ transition is pumped resonantly with a
laser, from the cavity side (orthogonal to the cavity axis), of which we can
change $\Omega$ by adjusting the laser power. The cavity frequency
$\omega_{\textrm{c}}$ is resonantly tuned to the
$|$g${}_{1}\rangle$–$|$e$\rangle$ transition, and the cavity is weakly probed
with a laser of the frequency $\omega_{\rm{p}}$. The ion position is movable
along the cavity axis, by changing the voltages to the dc electrodes, for
varying $g$ [22]. This control can be done with a precision on the order of
nanometers, so that we can change $g$ by moving the ion from a node to an
antinode of the cavity field. Consequently, via changing the laser power and
the ion position, we steer an ion-cavity quantum state toward the EP3 in the
parameter space of $g$ and $\Omega$.

Figure 1:  (a), (b) Trapped ion in an optical cavity. Laser from cavity side
(red arrow) drives $|$g${}_{2}\rangle$–$|$e$\rangle$ transition;
$|$g${}_{1}\rangle$–$|$e$\rangle$ transition is coupled to cavity. Cavity is
probed by a weak laser field and the output transmission is measured with a
single photon counting module (SPCM). Ion position is movable along the cavity
axis (green arrow). (c) Energy levels of ion-cavity quantum states and their
interactions. $\Omega,\gamma_{1}\gg\gamma_{2}$. (d) Feasible experimental
setting with the ion and fiber-based cavity.

The $H_{\textrm{nH}}$ for our ion-cavity state is given by

$H_{\rm{}_{nH}}=\begin{pmatrix}0&\Omega/2&0\\\ \Omega/2&-i\gamma&g\\\
0&g&-i\kappa\end{pmatrix},$ (1)

where the Planck constant $\hbar=1$, $\gamma=\gamma_{1}+\gamma_{2}$ is the
total atomic decay rate, $\gamma_{i}$ is the atomic decay rate from
$|$e$\rangle$ to $|$g${}_{i}\rangle$ $(i=1,2)$, and $\kappa$ is the cavity
decay rate. Three ion-photon states of $|$g${}_{2},n=0\rangle$,
$|$e$,0\rangle$, $|$g${}_{1},1\rangle$ ($n$ indicates the number basis of
cavity photons) comprise the bases of $H_{\rm{nH}}$. We provide more details
including the derivation of $H_{\rm{nH}}$ in Ref. [31].

Figure 2:  (a) Real and (b) imaginary parts of eigenvalues as functions of $g$
and $\Omega$. Green, red, and blue sheets correspond to three eigensurfaces of
$\lambda_{i}$ $(i=1,2,3)$, respectively. $\kappa/\gamma=7.00$. (c) Slicecut of
real-part eigensurfaces at Re[$\lambda_{i}$] = 0. Three sheets overlap at
purple region. (d) Two surfaces of imaginary parts overlap at colored region.
Black lines are EP2, and open black circles correspond to EP3. Log-log plot of
real (blue circle) and imaginary (red circle) parts of eigenvalue differences
$\lambda_{1}-\lambda_{3}$, and polynomial fits (lines) as a function of
$g-g_{\rm{EP3}}$ in (e) and $\Omega-\Omega_{\rm{EP3}}$ in (f), in the vicinity
of EP3.  Figure 3:  Cavity photon number (cavity transmission) as a function
of $\Delta(=\omega_{\textrm{p}}-\omega_{\textrm{c}})$. Black points are
results of quantum Monte Carlo simulations, and blue line is fitting result
with Eq. (3). Parameters are $g/\gamma=4.60$ and $\Omega/\gamma=2.00$ in (a),
and $g_{\textrm{EP3}}/\gamma=3.41$ and $\Omega_{\textrm{EP3}}/\gamma=3.29$ in
(b), and $g/\gamma=2.00$ and $\Omega/\gamma=6.00$ in (c). Spectrum of (b) is
obtained at the EP3 condition. $\kappa/\gamma=7.00$.

We discuss two conditions for obtaining the $H_{\rm{nH}}$. Note that, since
the system needs to be operated in the weak-excitation limit for neglecting
quantum jump operators, $|\textrm{g}_{1},0\rangle$ should be populated for
most of the interaction time, i.e.,
$|\psi\rangle\simeq|$g${}_{1},0\rangle+\alpha|$g${}_{2},0\rangle+\beta|$e$,0\rangle+\delta|$g${}_{1},1\rangle$
($|\alpha|,|\beta|,|\delta|\ll 1$). The first condition consists of weak
driving of the cavity, such that the cavity mean photon number is much less
unity [32]: This would make the occupation of $|\textrm{g}_{1},1\rangle$
sufficiently small. The second condition is that $\Omega$ and $\gamma_{1}$
must be much larger than $\gamma_{2}$, leading $|\textrm{e},0\rangle$ and
$|\textrm{g}_{2},0\rangle$ to decay to $|\textrm{g}_{1},0\rangle$ immediately.
We can increase $\Omega$ by controlling the pump laser power, and satisfy
$\gamma_{1}\gg\gamma_{2}$ via choosing the ion species of 40Ca+, which has an
excited state $4^{2}$P1/2 with very asymmetric decay rates [31]. While all
simulations in the main text are done with $\gamma_{2}=0$, we compare this
result to the simulations with actual values of 40Ca+ ion, and find that the
differences of the extracted eigenvalues are negligible [31].

Next, we obtain the eigenvalues of $H_{\rm{nH}}$ as functions of $g$ and
$\Omega$ numerically. Figs. 2(a) and (b) show three Riemann sheets with real
and imaginary parts of the eigenvalues, respectively. The EP3 appears at the
position where the three eigensurfaces coalesce, with
$\lambda_{\rm{EP3}}=-i(\gamma+\kappa)/3$ at $g_{\rm{EP3}}$ and
$\Omega_{\rm{EP3}}$ [31]. We denote $g_{\rm{EP3}}$ and $\Omega_{\rm{EP3}}$ as
the atom-cavity coupling constant and the Rabi frequency of the pump laser at
the EP3, respectively. We also investigate the scaling behavior of the
eigenvalues in the vicinity of the EP3 (Figs. 2(e) and (f)). The real and
imaginary parts, Re$[\lambda_{1}-\lambda_{3}]$ and
Im$[\lambda_{1}-\lambda_{3}]$ are fit with polynomial functions
$a+b\cdot(g-g_{\rm{EP3}})^{1/3}$ and
$a^{\prime}+b^{\prime}\cdot(\Omega-\Omega_{\rm{EP3}})^{1/3}$ with the
coefficients $a,b,a^{\prime},$ and $b^{\prime}$, showing good agreement with
the calculation results. This agreement presents that our singular point
exhibits a representative feature of the EP3.

We proceed to describe our scheme for extracting eigenvalues from actual
experiments. We “simulate” the experiment with a quantum Monte Carlo (qMC)
calculation [31]. The cavity is probed at the bare cavity mean photon number
0.014 such that the weak-excitation condition is satisfied. For three sets of
$g$ and $\Omega$ the simulation results are presented in Fig. 3. When
$g>\Omega$, cavity EIT gives rise to the central transparency peak with
vacuum-Rabi spectrum (Fig. 3(a)); when $g<\Omega$, in a broad cavity spectrum,
we point out two “dips” that are caused by the pump-induced dressed states,
which reflects the pronounced feature of ATS (Fig. 3(c)); the EP3 appears in-
between (Fig. 3(b)).

We then extract the eigenvalues $\lambda_{i}$ $(i=1,2,3)$ of $H_{\rm{nH}}$, in
a given $g$ and $\Omega$, from the qMC simulated spectrum. We formulate the
equation of motion with $H_{\rm{nH}}$ as

$i\frac{d}{dt}|\psi\rangle=\left(H_{\rm{nH}}-\Delta
I\right)|\psi\rangle+\begin{pmatrix}0\\\ 0\\\ \epsilon\\\
\end{pmatrix},\quad|\psi\rangle=\begin{pmatrix}\alpha\\\ \beta\\\ \delta\\\
\end{pmatrix}$ (2)

where $\Delta=\omega_{\rm{p}}-\omega_{\rm{c}}$, and $\epsilon$ denotes the
weak amplitude of the probe laser [31]. Correspondingly we find an analytic
expression of the cavity transmission

$T=\left|\frac{\Delta(i\kappa-\Delta+\lambda_{2}+\lambda_{3})\kappa+\lambda_{1}(\Delta\kappa-i\lambda_{2}\lambda_{3})}{(\Delta-\lambda_{1})(\Delta-\lambda_{2})(\Delta-\lambda_{3})}\right|^{2},$
(3)

and through fitting the measured spectrum with Eq. (3), three eigenvalues of
$\lambda_{i}$ are obtained. More details are described in Ref. [31]. The
fitting results are shown in Fig. 3, agreeing with the qMC simulation spectrum
very well.

Figure 4:  Real and imaginary parts of the numerically obtained eigenvalues of
$H_{\rm{nH}}$ are shown in lines. Green line is $\lambda_{1}$, red is
$\lambda_{2}$, and blue is $\lambda_{3}$. Points are obtained from fitting the
transmission spectrum, results of quantum Monte Carlo simulation, with Eq.
(3). (a) $\Omega/\gamma=2.50$, (b) $\Omega/\gamma=3.29$, (c)
$\Omega/\gamma=3.50$, (d) $g/\gamma=2.50$, (e) $g/\gamma=3.41$ and (f)
$g/\gamma=3.50$. EP3 appears in (b) and (e).

Fig. 4 shows the extraction of the eigenvalues with this model for several $g$
and $\Omega$. The solid lines are real and imaginary parts of the numerically
obtained eigenvalues of $H_{\rm{nH}}$. The points correspond to real and
imaginary parts, extracted from fitting the transmission spectrum with Eq.
(3). Numerically obtained eigenvalues agree well with the extraction from the
fitting results, also confirming that our model is correct.

Finally, we seek a feasible experimental setting to observe the EP3. We
consider that a 40Ca+ ion [28, 26, 27] is coupled to a fiber-based optical
resonator. We choose the strong 42S1/2 to 42P1/2 transition
($\gamma_{1}/(2\pi)=10.8$ MHz) at $397$ nm, coupled to the cavity. The
specification of the cavity includes radii of curvature of 350 $\mu$m for both
fiber mirrors, a cavity length of 500 $\mu$m, a maximum atom-cavity coupling
constant $g_{0}/(2\pi)=49.7$ MHz, and $\kappa/{(2\pi)}=80.0$ MHz (finesse of
$\sim 1880$) [33, 34, 35]. The 32D3/2 to 42P1/2 transition
($\gamma_{2}/(2\pi)=0.74$ MHz) at 866 nm corresponds to the weak transition,
pumped by a power-tunable laser from the cavity side. In this configuration,
the 42S1/2 state is mostly occupied for satisfying the weak-excitation
condition. We expect that $g_{\rm{EP3}}/(2\pi)=38.9$ MHz, which would be
reached by moving the ion position between an antinode and a node of the
cavity field; $\Omega_{\textrm{EP3}}/(2\pi)=37.7$ MHz. Asymmetric mirror
coating would make the transmission measurement more effective [36].

In conclusion, we have theoretically explored the emergence of EP3 and a
measurement method in an ion-cavity system. In the lambda-type level
configuration the EP3 appears in the parameter space of $g$ and $\Omega$. The
eigenvalues of $H_{\rm{nH}}$ are obtained through fitting the cavity-
transmission spectrum. Our work provides interesting possibilities to
investigate the non-Hermitian physics in the ion-cavity setting.

We thank Kyungwon An for insightful comments. This work has been supported by
BK21 FOUR program and Educational Institute for Intelligent Information
Integration, National Research Foundation (Grant No. 2019R1A5A1027055 and
Grant NO. 2020R1I1A2066622), Institute for Information & communication
Technology Planning & evaluation (IITP, Grant No. 2022-0-01040), Samsung
Science and Technology Foundation (SSTF-BA2101-07, SRFC-TC2103-01) and Samsung
Electronics Co., Ltd. (IO201211-08121-01).

Our data are available at https://doi.org/10.5281/zenodo.7821387.

## References

  * Wiersig [2020] J. Wiersig, Photon. Res. 8, 1457 (2020).
  * Zhang  _et al._ [2018] J. Zhang, B. Peng, c. K. Özdemir, K. Pichler, D. O. Krimer, G. Zhao, F. Nori, Y.-x. Liu, S. Rotter, and L. Yang, Nat. Photon. 12, 479 (2018).
  * Feng  _et al._ [2017] L. Feng, R. El-Ganainy, and L. Ge, Nat. Photon. 11, 752 (2017).
  * Bergholtz  _et al._ [2021] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021).
  * Mandal and Bergholtz [2021] I. Mandal and E. J. Bergholtz, Phys. Rev. Lett. 127, 186601 (2021).
  * Wu  _et al._ [2021] Y. Wu, P. Zhou, T. Li, W. Wan, and Y. Zou, Opt. Express 29, 6080 (2021).
  * Zhong  _et al._ [2018] Q. Zhong, M. Khajavikhan, D. N. Christodoulides, and R. El-Ganainy, Nat. Commun. 9, 4808 (2018).
  * Patil  _et al._ [2022] Y. S. S. Patil, J. Höller, P. A. Henry, C. Guria, Y. Zhang, L. Jiang, N. Kralj, N. Read, and J. G. E. Harris, Nature 607, 271 (2022).
  * Ding  _et al._ [2016] K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Phys. Rev. X 6, 021007 (2016).
  * Hodaei  _et al._ [2017] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017).
  * Wang  _et al._ [2019] S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, Nat. Commun. 10, 832 (2019).
  * Xiao  _et al._ [2019] Z. Xiao, H. Li, T. Kottos, and A. Alù, Phys. Rev. Lett. 123, 213901 (2019).
  * Fang  _et al._ [2021] X. Fang, N. J. R. K. Gerard, Z. Zhou, H. Ding, N. Wang, B. Jia, Y. Deng, X. Wang, Y. Jing, and Y. Li, Commun. Phys. 4, 271 (2021).
  * Wu  _et al._ [2019] Y. Wu, W. Liu, J. Geng, X. Song, X. Ye, C.-K. Duan, X. Rong, and J. Du, Science 364, 878 (2019).
  * Naghiloo  _et al._ [2019] M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Nat. Phys. 15, 1232 (2019).
  * Lau and Clerk [2018] H.-K. Lau and A. A. Clerk, Nat. Commun. 9, 4320 (2018).
  * Anisimov  _et al._ [2011] P. M. Anisimov, J. P. Dowling, and B. C. Sanders, Phys. Rev. Lett. 107, 163604 (2011).
  * Zhu  _et al._ [2013] C. Zhu, C. Tan, and G. Huang, Phys. Rev. A 87, 043813 (2013).
  * Tan and Huang [2014] C. Tan and G. Huang, J. Opt. Soc. Am. B 31, 704 (2014).
  * Peng  _et al._ [2014] B. Peng, c. K. Özdemir, W. Chen, F. Nori, and L. Yang, Nat. Commun. 5, 5082 (2014).
  * Bruzewicz  _et al._ [2019] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Appl. Phys. Rev. 6, 021314 (2019).
  * Guthöhrlein  _et al._ [2001] G. R. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
  * Wang  _et al._ [2021] W.-C. Wang, Y.-L. Zhou, H.-L. Zhang, J. Zhang, M.-C. Zhang, Y. Xie, C.-W. Wu, T. Chen, B.-Q. Ou, W. Wu, H. Jing, and P.-X. Chen, Phys. Rev. A 103, L020201 (2021).
  * Ding  _et al._ [2021] L. Ding, K. Shi, Q. Zhang, D. Shen, X. Zhang, and W. Zhang, Phys. Rev. Lett. 126, 083604 (2021).
  * Dubin  _et al._ [2010] F. Dubin, C. Russo, H. Barros, A. Stute, C. Becher, P. Schmidt, and R. Blatt, Nat. Phys. 6, 350 (2010).
  * Lee  _et al._ [2019a] M. Lee, K. Friebe, D. A. Fioretto, K. Schüppert, F. R. Ong, D. Plankensteiner, V. Torggler, H. Ritsch, R. Blatt, and T. E. Northup, Phys. Rev. Lett. 122, 153603 (2019a).
  * Takahashi  _et al._ [2020] H. Takahashi, E. Kassa, C. Christoforou, and M. Keller, Phys. Rev. Lett. 124, 013602 (2020).
  * Stute  _et al._ [2012] A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, Nature 485, 482 (2012).
  * Walker  _et al._ [2018] T. Walker, K. Miyanishi, R. Ikuta, H. Takahashi, S. Vartabi Kashanian, Y. Tsujimoto, K. Hayasaka, T. Yamamoto, N. Imoto, and M. Keller, Phys. Rev. Lett. 120, 203601 (2018).
  * Krutyanskiy  _et al._ [2023] V. Krutyanskiy, M. Galli, V. Krcmarsky, S. Baier, D. A. Fioretto, Y. Pu, A. Mazloom, P. Sekatski, M. Canteri, M. Teller, J. Schupp, J. Bate, M. Meraner, N. Sangouard, B. P. Lanyon, and T. E. Northup, Phys. Rev. Lett. 130, 050803 (2023).
  * [31] Supplemental Document .
  * Choi  _et al._ [2010] Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, Phys. Rev. Lett. 104, 153601 (2010).
  * Ballance  _et al._ [2017] T. G. Ballance, H. M. Meyer, P. Kobel, K. Ott, J. Reichel, and M. Köhl, Phys. Rev. A 95, 033812 (2017).
  * Lee  _et al._ [2019b] M. Lee, M. Lee, S. Hong, K. Schüppert, Y.-D. Kwon, T. Kim, Y. Colombe, T. E. Northup, D.-I. D. Cho, and R. Blatt, Phys. Rev. Applied 12, 044052 (2019b).
  * Ong  _et al._ [2020] F. R. Ong, K. Schüppert, P. Jobez, M. Teller, B. Ames, D. A. Fioretto, K. Friebe, M. Lee, Y. Colombe, R. Blatt, and T. E. Northup, New J. Phys. 22, 063018 (2020).
  * Lee  _et al._ [2022] D. Lee, M. Kim, J. Hong, T. Ha, J. Kim, S. Kang, Y. Choi, K. An, and M. Lee, Opt. Continuum 1, 603 (2022).
  * Carmichael [2008] H. J. Carmichael,  _Statistical Methods in Quantum Optics 2: Non-Clasical Fields_ (Springer-Verlag Berlin Heidelberg, 2008).

##  Appendix A non-Hermitian Hamiltonian

We derive the non-Hermitian Hamiltonian, Eq. (1) of the main text. We begin
with the total Hamiltonian describing the system of the ion, cavity field, and
pump laser:

$H_{\textrm{tot}}=\omega_{1}\sigma_{\textrm{e}}+(\omega_{1}-\omega_{2})\sigma_{\textrm{g}_{2}}+\omega_{\textrm{c}}a^{\dagger}a+g(a^{\dagger}\sigma_{\textrm{g}_{1}\textrm{e}}+\textrm{h.~{}c.})+\frac{\Omega}{2}(\sigma_{\textrm{g}_{2}\textrm{e}}e^{i\omega_{\textrm{p}\rm{{}^{\prime}}}t}+\textrm{h.~{}c.}),$
(4)

where the Planck constant $\hbar=1$, $\omega_{i}$ is the transition frequency
between $|$g${}_{i}\rangle$ and $|$e$\rangle$ $(i=1,2)$,
$\sigma_{\textrm{e(g}_{i})}$ is the projection operator onto the state
$|\textrm{e}\rangle$($|\textrm{g}_{i}\rangle$), $\omega_{\textrm{c}}$ is the
cavity frequency, $\omega_{\rm{p^{\prime}}}$ is the frequency of the pump
laser, $a^{\dagger}(a)$ is the cavity photon creation (annihilation) operator,
$g$ is the atom-cavity coupling constant,
$\sigma_{\textrm{g}_{i}\textrm{e}}=\sigma_{\textrm{eg}_{i}}^{\dagger}$ is the
transition operator between $|$e$\rangle$ and $|$g${}_{i}\rangle$, and
$\Omega$ is the Rabi frequency by the pump laser. The interaction Hamiltonian
is obtained by using the relation
$H_{\rm{int}}=i\dot{U}U^{\dagger}+UH_{\rm{tot}}U^{\dagger}$ where
$U=\textrm{exp}\left[i\left(\omega_{1}\sigma_{\textrm{e}}+(\omega_{1}-\omega_{\textrm{p}\rm{{}^{\prime}}})\sigma_{\textrm{g}_{2}}+\omega_{\textrm{c}}a^{\dagger}a\right)t\right]$,

$H_{\textrm{int}}=\Delta_{\textrm{p}\rm{{}^{\prime}}}\sigma_{\textrm{g}_{2}}+g\left(a^{\dagger}\sigma_{\rm{g}_{1}\rm{e}}+\textrm{h.~{}c.}\right)+\frac{\Omega}{2}\left(\sigma_{\textrm{g}_{2}\textrm{e}}+\textrm{h.~{}c.}\right),$
(5)

with
$\Delta_{\textrm{p}\rm{{}^{\prime}}}=\omega_{\textrm{p}\rm{{}^{\prime}}}-\omega_{2}=0$
and $\omega_{1}=\omega_{\textrm{c}}$ in our work.

We proceed to include the dissipation operators in the Lindblad form. The
equation of motion for the ion-cavity density matrix $\rho$ is expressed as

$\frac{d\rho}{dt}=-i\left[H_{\textrm{int}},\rho\right]+\sum_{i=1,2}{\gamma_{i}\left(2\sigma_{\textrm{g}_{i}\textrm{e}}\rho\sigma_{\textrm{eg}_{i}}-\sigma_{\textrm{eg}_{i}}\sigma_{\textrm{g}_{i}\rm{e}}\rho-\rho\sigma_{\textrm{eg}_{i}}\sigma_{\textrm{g}_{i}\textrm{e}}\right)}+\kappa\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right).$ (6)

where $\gamma_{i}$ is the atomic decay rate from $|\textrm{e}\rangle$ to
$|\textrm{g}_{i}\rangle$, and $\kappa$ is the cavity decay rate.

Given Eq. (6), the terms are rearranged for obtaining

$\displaystyle\frac{d\rho}{dt}=$
$\displaystyle-i\left[g\left(a^{\dagger}\sigma_{\textrm{g}_{1}\textrm{e}}+\textrm{h.~{}c.}\right)+\frac{\Omega}{2}\left(\sigma_{\textrm{g}_{2}\textrm{e}}+\textrm{h.~{}c.}\right),\rho\right]-i\sum_{i=1,2}\gamma_{i}\left[-i\sigma_{\textrm{eg}_{i}}\sigma_{\textrm{g}_{i}\textrm{e}},\rho\right]-i\kappa\left[-ia^{\dagger}a,\rho\right]$
(7)
$\displaystyle+2\sum_{i=1,2}{\gamma_{i}\sigma_{\textrm{g}_{i}\rm{e}}\rho\sigma_{\textrm{eg}_{i}}}+2\kappa
a\rho a^{\dagger}$ $\displaystyle\simeq$
$\displaystyle-i\left[g\left(a^{\dagger}\sigma_{\textrm{g}_{1}\textrm{e}}+\textrm{h.~{}c.}\right)+\frac{\Omega}{2}\left(\sigma_{\textrm{g}_{2}\rm{e}}+\textrm{h.~{}c.}\right)+\sum_{i=1,2}-i\gamma_{i}\sigma_{\textrm{e}}-i\kappa
a^{\dagger}a,\rho\right]$ $\displaystyle=$
$\displaystyle-i\left[H_{\textrm{nH}}^{\prime},\rho\right].$

Since our system operates in the weak-excitation limit, the state is mostly
populated in the ground state $|\textrm{g}_{1},0\rangle$, and thus the quantum
jump operators (the second line of Eq. (7)) are neglected [37]. Then we find
the equation of motion with the non-Hermitian Hamiltonian
$H_{\textrm{nH}}^{\prime}$.

Now we obtain $H_{\rm{nH}}$ in a $3\times 3$ matrix form. We define the four
involved states as the following basis

$|\textrm{g}_{1},n=0\rangle=\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix}\quad|\textrm{g}_{2},0\rangle=\begin{pmatrix}0\\\ 1\\\ 0\\\
0\end{pmatrix}\quad|\textrm{e},0\rangle=\begin{pmatrix}0\\\ 0\\\ 1\\\
0\end{pmatrix},\quad|\textrm{g}_{1},1\rangle=\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix}$ (8)

where $n$ is the basis of cavity-photon numbers. The operators are represented
by

$a=\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}\quad\sigma_{\rm{g}_{2}\rm{e}}=\begin{pmatrix}0&0&0&0\\\
0&0&1&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}\quad\sigma_{\rm{g}_{1}\rm{e}}=\begin{pmatrix}0&0&1&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}.$ (9)

With these bases and operators, it is straightforward to show that
$H_{\rm{nH}}^{\prime}$ is given by

$H^{\prime}_{\rm{nH}}=\begin{pmatrix}0&0&0&0\\\ 0&0&\Omega/2&0\\\
0&\Omega/2&-i\gamma&g\\\ 0&0&g&-i\kappa\end{pmatrix},$ (10)

with $\gamma=\gamma_{1}+\gamma_{2}$ $(\gamma_{1}\gg\gamma_{2})$. The $3\times
3$ sub-matrix, excluding the first row and column of $H^{\prime}_{\rm{nH}}$,
corresponds to $H_{\rm{nH}}$ of the main text:

$H_{\rm{nH}}=\begin{pmatrix}0&\Omega/2&0\\\ \Omega/2&-i\gamma&g\\\
0&g&-i\kappa\end{pmatrix}.$ (11)

##  Appendix B Eigenvalue and eigenvector of $H_{\textrm{nH}}$ at the third-
order exceptional point

###  B.1 Eigenvalue

We describe an analytic form of the eigenvalue of $H_{\rm{nH}}$ at the third-
order exceptional point (EP3). We also find the Rabi frequency
$\Omega_{\textrm{EP3}}$ and the atom-cavity coupling constant
$g_{\textrm{EP3}}$, which induce the emergence of the EP3. Considering the
weak-excitation limit and
$|\psi_{\textrm{tot}}\rangle\simeq|\textrm{g}_{1},0\rangle+\alpha|\textrm{g}_{2},0\rangle+\beta|\textrm{e},0\rangle+\delta|\textrm{g}_{1},1\rangle$,
the bases that contribute to the dynamics of the system are

$|1\rangle=|\textrm{g}_{2},0\rangle\quad|2\rangle=|\textrm{e},0\rangle\quad|3\rangle=|\textrm{g}_{1},1\rangle.$
(12)

The characteristic polynomial of $H_{\rm{nH}}$ is given by

$\lambda^{3}+i\left(\gamma+\kappa\right)\lambda^{2}-\left(\frac{\Omega^{2}}{4}+g^{2}+\gamma\kappa\right)\lambda-\frac{i\Omega^{2}\kappa}{4}=0$
(13)

In the EP3 condition, Eq. (13) is identical with
$(\lambda-\lambda_{\rm{EP3}})^{3}=0$. Comparing the coefficients of the terms
$\lambda^{2}$ and last terms, we obtain

$\lambda_{\rm{EP3}}=-i\frac{\gamma+\kappa}{3}\quad\textrm{and}\quad\Omega_{\rm{EP3}}=\frac{2(\gamma+\kappa)}{3}\sqrt{\frac{\gamma+\kappa}{3\kappa}}.$
(14)

Inserting Eq. (14) to Eq. (13) and comparing the coefficient of the term
$\lambda^{2}$ to that of $(\lambda-\lambda_{\rm{EP3}})^{3}=0$, we also find

$g_{\textrm{EP3}}=\sqrt{\frac{(\gamma+\kappa)^{2}}{3}-\frac{(\gamma+\kappa)^{3}}{27\kappa}-\gamma\kappa}.$
(15)

###  B.2 Eigenvector

Given $\lambda_{i}$, we represent the eigenvector
$|\psi_{i}\rangle=C_{i}\left(c_{1,i}|1\rangle+c_{2,i}|2\rangle+c_{3,i}|3\rangle\right)$
with the normalization coefficient $C_{i}$. The analytic form of
$c_{1,i},c_{2,i},c_{3,i},$ and the decomposition matrix $\Xi$ are expressed as

$\begin{pmatrix}c_{1,i}\\\ c_{2,i}\\\
c_{3,i}\end{pmatrix}=\begin{pmatrix}\frac{\Omega}{2}\frac{\lambda_{i}+i\kappa}{\lambda_{i}}\\\
\lambda_{i}+i\kappa\\\
g\end{pmatrix}\quad\textrm{and}\quad\Xi=\begin{pmatrix}\frac{\Omega}{2}\frac{\lambda_{1}+i\kappa}{\lambda_{1}}&\frac{\Omega}{2}\frac{\lambda_{2}+i\kappa}{\lambda_{2}}&\frac{\Omega}{2}\frac{\lambda_{3}+i\kappa}{\lambda_{3}}\\\
\lambda_{1}+i\kappa&\lambda_{2}+i\kappa&\lambda_{3}+i\kappa\\\ g&g&g\\\
\end{pmatrix},$ (16)

allowing us to construct the non-Hermitian Hamiltonian through
$\Xi\cdot\Lambda\cdot\Xi^{-1}=H_{\rm{nH}}$, where

$\Lambda=\begin{pmatrix}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\\\ \end{pmatrix}.$ (17)

Thus the coalesced eigenvector at the EP3 is

$\begin{pmatrix}c_{1,\textrm{EP3}}\\\ c_{2,\textrm{EP3}}\\\
c_{3,\textrm{EP3}}\end{pmatrix}=\begin{pmatrix}\frac{\Omega_{\textrm{EP3}}}{2}\frac{\lambda_{\textrm{EP3}}+i\kappa}{\lambda_{\textrm{EP3}}}\\\
\lambda_{\textrm{EP3}}+i\kappa\\\ g_{\textrm{EP3}}\end{pmatrix},$ (18)

with the omission of $C_{i}$.

##  Appendix C Quantum Monte Carlo Simulation

In order to obtain the data in Fig. 3 of the main text, we perform quantum
Monte Carlo (qMC) simulation with python QuTip. The employed Hamiltonian
includes the weak cavity probing term in addition to Eq. (4),

$H_{\textrm{tot,
qMC}}=\omega_{1}\sigma_{\textrm{e}}+(\omega_{1}-\omega_{2})\sigma_{\textrm{g}_{2}}+\omega_{\textrm{c}}a^{\dagger}a+g(a^{\dagger}\sigma_{\textrm{g}_{1}\textrm{e}}+\textrm{h.~{}c.})+\frac{\Omega}{2}(\sigma_{\textrm{g}_{2}\textrm{e}}e^{i\omega_{\textrm{p}\rm{{}^{\prime}}}t}+\textrm{h.~{}c.})+\epsilon(ae^{i\omega_{\textrm{p}}t}+\textrm{h.~{}c.}),$
(19)

where $\epsilon$ is the amplitude of the probing field, $\omega_{\textrm{p}}$
is the frequency of the probe laser. Defining
$U=\textrm{exp}[i(\omega_{\textrm{p}}\sigma_{\textrm{e}}$
\+
$(\omega_{\textrm{p}}-\omega_{\rm{p^{\prime}}})\sigma_{\textrm{g}_{2}}+\omega_{\textrm{p}}a^{\dagger}a)t]$,
we obtain the interaction Hamiltonian

$H_{\textrm{int,
qMC}}=-\Delta\left(\sigma_{\textrm{e}}+\sigma_{\textrm{g}_{2}}+a^{\dagger}a\right)+g\left(a^{\dagger}\sigma_{\rm{g}_{1}\rm{e}}+\textrm{h.~{}c.}\right)+\frac{\Omega}{2}\left(\sigma_{\textrm{g}_{2}\textrm{e}}+\textrm{h.~{}c.}\right)+\epsilon(a+\textrm{h.~{}c.}),$
(20)

with $\Delta=\omega_{\textrm{p}}-\omega_{\textrm{c}}$. Eq. (20) is actually
used in the qMC simulation. The atomic and cavity decay operators are
identical with those of Eq. (6).

##  Appendix D Fitting function of cavity-transmission spectrum

We derive the fitting function for the cavity-transmission spectrum, Eq. (3)
of the main text, which is needed for extracting the eigenvalues. When the
amplitude of the probe field is very weak such that the cavity mean photon
number is much less than unity, the equation of motion for
$|\psi\rangle=\alpha|\textrm{g}_{2},0\rangle+\beta|\textrm{e},0\rangle+\delta|\textrm{g}_{1},1\rangle$
can be expressed as

$\frac{d}{dt}|\psi\rangle=-i(H_{\rm{nH}}-\Delta
I)|\psi\rangle-i\begin{pmatrix}0\\\ 0\\\ \epsilon\\\
\end{pmatrix},\quad|\psi\rangle=\begin{pmatrix}\alpha\\\ \beta\\\ \delta\\\
\end{pmatrix},$ (21)

with the identity matrix $I$, and $\epsilon\ll\kappa$.

In the steady state, we find the state vector

$\begin{pmatrix}\alpha_{\textrm{ss}}\\\ \beta_{\textrm{ss}}\\\
\delta_{\textrm{ss}}\\\ \end{pmatrix}=(\Xi\Lambda\Xi^{-1}-\Delta
I)^{-1}\begin{pmatrix}0\\\ 0\\\ -\epsilon\\\ \end{pmatrix}=\Xi(\Lambda-\Delta
I)^{-1}\Xi^{-1}\begin{pmatrix}0\\\ 0\\\ -\epsilon\\\ \end{pmatrix},$ (22)

where $\alpha_{\textrm{ss}}$, $\beta_{\textrm{ss}}$, and
$\gamma_{\textrm{ss}}$ are the steady-state coefficients. From the analytic
solution of Eq. (22), we find

$\delta_{\textrm{ss}}=-\frac{\Delta(i\kappa-\Delta+\lambda_{2}+\lambda_{3})\kappa+\lambda_{1}(\Delta\kappa-i\lambda_{2}\lambda_{3})}{(\Delta-\lambda_{1})(\Delta-\lambda_{2})(\Delta-\lambda_{3})\kappa}\epsilon.$
(23)

This leads to the cavity-transmission spectrum

$T=\left|\frac{\delta_{\textrm{ss}}}{\epsilon/\kappa}\right|^{2}=\left|\frac{\Delta(i\kappa-\Delta+\lambda_{2}+\lambda_{3})\kappa+\lambda_{1}(\Delta\kappa-i\lambda_{2}\lambda_{3})}{(\Delta-\lambda_{1})(\Delta-\lambda_{2})(\Delta-\lambda_{3})}\right|^{2},$
(24)

with the eigenvalues of $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$. Via
fitting the qMC simulation data with Eq. (24), identical with Eq. (3) of the
main text, we obtain the three eigenvalues.

##  Appendix E Extracting eigenvalues with calcium ion

In the qMC simulations of the main text, the decay rate from
$|\textrm{e}\rangle$ to $|\textrm{g}_{2}\rangle$ $(\gamma_{2})$ is neglected.
In order to examine the influence of $\gamma_{2}$ to the cavity transmission
spectrum, correspondingly to the extraction of the eigenvalues, we herein
perform numerical simulations including the actual value of $\gamma_{2}$. All
other parameters are identical with those in the main text. In Figs. 5(a)–(c),
we obtain the cavity transmission spectrum with and without $\gamma_{2}$,
compare those two results, and find that the difference is negligibly small.
Therefore, this comparison confirms that the actual experiment with 40Ca+ ion
is suitable for extracting the eigenvalues of $H_{\textrm{nH}}$. We fit the
transmission spectrum of actual values of 40Ca+ ion with Eq. (24), showing
excellent agreement (Fig. 5(d)–(f)). The extraction results of all eigenvalues
for several $g$ and $\Omega$ are presented in Fig. 6.

Figure 5:  (a)–(c) Cavity photon number (cavity transmission) as a function of
$\Delta(=\omega_{\textrm{p}}-\omega_{\textrm{c}})$. Red points are results of
numerical simulations with actual value of $\gamma_{2}$ of 40Ca+, and blue
points are with $\gamma_{2}=0$. (d)–(f) Quantum Monte Carlo simulation results
with $\gamma_{2}$ are black points, blue line is fitting result with Eq. (24).
Parameters are $g/\gamma=4.60$ and $\Omega/\gamma=2.00$ in (a), (d), and
$g_{\rm{EP3}}/\gamma=3.38$ and $\Omega_{\rm{EP3}}/\gamma=3.27$ in (b), (e),
and $g/\gamma=2.00$ and $\Omega/\gamma=6.00$ in (c), (f). Spectrum of (b) and
(e) are obtained at the EP3 condition. $\kappa/\gamma=6.94$.  Figure 6:  Real
and imaginary parts of the numerically obtained eigenvalues of $H_{\rm{nH}}$
are shown in lines. Green line is $\lambda_{1}$, red is $\lambda_{2}$, and
blue is $\lambda_{3}$. Points are obtained from fitting the transmission
spectrum, the result of quantum Monte Carlo simulation with actual values of
40Ca+. (a) $\Omega/\gamma=2.50$, (b) $\Omega/\gamma=3.27$, (c)
$\Omega/\gamma=3.50$, (d) $g/\gamma=2.50$, (e) $g/\gamma=3.38$, and (f)
$g/\gamma=3.50$. EP3 appears in (b) and (e). $\kappa/\gamma=6.94$.