# Investigation of the Effect of Quantum Measurement on Parity-Time Symmetry

Wei-Chen Wang  Department of Physics, College of Liberal Arts and Sciences,
National University of Defense Technology, Changsha 410073, China
Interdisciplinary Center for Quantum Information, National University of
Defense Technology, Changsha 410073, China  Yi Xie  Department of Physics,
College of Liberal Arts and Sciences, National University of Defense
Technology, Changsha 410073, China  Interdisciplinary Center for Quantum
Information, National University of Defense Technology, Changsha 410073, China
Man-Chao Zhang  Department of Physics, College of Liberal Arts and Sciences,
National University of Defense Technology, Changsha 410073, China
Interdisciplinary Center for Quantum Information, National University of
Defense Technology, Changsha 410073, China  Jie Zhang  Department of Physics,
College of Liberal Arts and Sciences, National University of Defense
Technology, Changsha 410073, China  Interdisciplinary Center for Quantum
Information, National University of Defense Technology, Changsha 410073, China
Chun-Wang Wu  Department of Physics, College of Liberal Arts and Sciences,
National University of Defense Technology, Changsha 410073, China
Interdisciplinary Center for Quantum Information, National University of
Defense Technology, Changsha 410073, China  Ting Chen  Department of Physics,
College of Liberal Arts and Sciences, National University of Defense
Technology, Changsha 410073, China  Interdisciplinary Center for Quantum
Information, National University of Defense Technology, Changsha 410073, China
Bao-Quan Ou  Department of Physics, College of Liberal Arts and Sciences,
National University of Defense Technology, Changsha 410073, China
Interdisciplinary Center for Quantum Information, National University of
Defense Technology, Changsha 410073, China  Wei Wu  Department of Physics,
College of Liberal Arts and Sciences, National University of Defense
Technology, Changsha 410073, China  Interdisciplinary Center for Quantum
Information, National University of Defense Technology, Changsha 410073, China
Ping-Xing Chen<EMAIL_ADDRESS>Department of Physics, College of Liberal
Arts and Sciences, National University of Defense Technology, Changsha 410073,
China  Interdisciplinary Center for Quantum Information, National University
of Defense Technology, Changsha 410073, China

###### Abstract

Symmetry, including the parity-time ($\mathcal{PT}$)-symmetry, is a striking
topic, widely discussed and employed in many fields. It is well-known that
quantum measurement can destroy or disturb quantum systems. However, can and
how does quantum measurement destroy the symmetry of the measured system? To
answer the pertinent question, we establish the correlation between the
quantum measurement and Floquet $\mathcal{PT}$-symmetry and investigate for
the first time how the measurement frequency and measurement strength affect
the $\mathcal{PT}$-symmetry of the measured system using the
${}^{40}\mathrm{Ca}^{+}$ ion. It is already shown that the measurement at high
frequencies would break the $\mathcal{PT}$ symmetry. Notably, even for an
inadequately fast measurement frequency, if the measurement strength is
sufficiently strong, the $\mathcal{PT}$ symmetry breaking can occur. The
current work can enhance our knowledge of quantum measurement and symmetry and
may inspire further research on the effect of quantum measurement on symmetry.

Key Word: quantum measurement, $\mathcal{PT}$-symmetry, ion trap

PACS number(s): 03.65.Ta, 03.65.Vf, 03.65.Yz

##  I Introduction

Symmetries determine the interactions of elementary particles [1] and classify
the different phases of complex systems. For nearly a decade, a striking
discovery revealed that the parity-time ($\mathcal{PT}$)-symmetric
Hamiltonians, despite their non-Hermitian nature, can have real eigenvalues
[2, 3, 4, 5, 6, 7]. By controlling the parameters of the
$\mathcal{PT}$-symmetric system, spontaneous $\mathcal{PT}$ symmetry breaking
can occur at an exceptional point (EP) [8, 9], where both the eigenvalues and
eigenstates of the system coalesce. Thus, there are two phases, one being the
spontaneous $\mathcal{PT}$ symmetry breaking phase (PTBP), another being the
spontaneous $\mathcal{PT}$ symmetry unbreaking phase (PTSP). Recently,
$\mathcal{PT}$-symmetric systems have been successfully implemented in
classical optical systems [10, 11, 12] and quantum systems by exploiting the
properties of open quantum systems [13, 14, 15, 16, 17]. This has laid the
experimental foundation for further research on quantum
$\mathcal{PT}$-symmetry. Compared with static time-independent
$\mathcal{PT}$-symmetric systems, time-dependent Floquet
$\mathcal{PT}$-symmetric systems [18, 19, 20] have attracted greater
attention.

Quantum measurement is a lasting research topic whose history dates back to
the birth of quantum mechanics. However, the physical mechanisms of quantum
measurement still contain some unknowns. The mathematical formalism of the
quantum measurement was introduced by von Neumann [21]. From a physical point
of view, measurement is the interaction between the measured system and an
external system, thus playing the role of a measuring apparatus. Measurement
frequency and strength are two important features of the measurement. The
evolution of the measured system is hindered (even stopped) when the
measurement frequency increases from small to large, a well-known quantum Zeno
effect [22, 23, 24, 25]. Moreover, based on the strength of quantum
measurement, two complementary measurement schemes have been widely
investigated in various quantum systems: von Neumann’s projective ’strong’
measurement and Aharonov’s weak measurement [26, 27]. The quantum weak
measurement can be used to achieve an ultrasensitive measurement [28, 29]. The
recent discovery of weak–to–strong transition of quantum measurement in a
trapped-ion system [30] has stimulated further research into the mechanisms of
quantum measurements.

How quantum measurement affects the symmetry of a quantum system is a
fundamental question. Some previous studies Ref.[31] are related to this
topic, but their conclusion remains far from the actual answer. Here, we
design a quantum Floquet $\mathcal{PT}$-symmetric system adapted to the
existing quantum measurement model by using the trapped-ion system, where in
the frequency and strength of the measurement can be modulated conveniently.
We analyze the effect of measurement frequency and strength on the
$\mathcal{PT}$ symmetry and observe that the measurement at high frequencies
would break the $\mathcal{PT}$ symmetry; as long as the measurement strength
is strong enough, the $\mathcal{PT}$ symmetry breaking can occur even if the
measurement frequency is not fast enough. This work may improve the existing
knowledge of quantum measurement and symmetry and inspire further research in
this area.

Figure 1: (Color online) The experimental and theoretical models of quantum
measurement (a) The energy levels of the ${}^{40}\mathrm{Ca}^{+}$ ion with the
internal states $|0\rangle$, $|1\rangle$, and $|\omega\rangle$ corresponding
to the energy levels ${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$,
${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$, and ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$
with a flat continuum photon field, respectively. (b) An experimental scheme
for the pulse measurement is proposed, based on the internal states of
${}^{40}\mathrm{Ca}^{+}$. The 854 nm laser controls the measurement process,
while the 729 nm laser controls the evolution between the internal states of
${}^{40}\mathrm{Ca}^{+}$. (c) The pulse time sequence of laser control. The
coupling strength $\Omega$ between the internal states is controlled by the
intensity of the 729 nm laser, and the evolution time $t_{0}$ is controlled by
the duration time of the 729 nm laser. The interaction strength $\gamma$ is
controlled by the intensity of the 854 nm laser, and the measurement time
$t_{1}$ is controlled by the duration of the 854 nm laser.

##  II Theoretical and experimental schemes

Let us now consider a simple scheme, as shown in FIG. 1(a). In the
${}^{40}\mathrm{Ca}^{+}$ system, the level
${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$ is set to $|0\rangle$, and the metastable
level ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$ is set to $|1\rangle$.
${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$ and ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$
can be coupled by the 729 nm laser, and the coupling strength is determined by
the laser intensity. Electrons can be excited from
${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$ to level
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ by the 854 nm laser. Since the natural
linewidth of ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ is large, i.e., the lifetime
of ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ is very short, photons drain the
population away from ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ rapidly (toward
another level not shown in the figure). When the 854 nm laser is turned on, if
the electron is in ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$, it gets excited to
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ and spontaneously emits photons.
Theoretically, the process can be regarded as a measurement of
${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$, whether or not the photon emitted
spontaneously from ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$, is detected.
Therefore, according to the scheme given in [32], the Hamiltonian is composed
of the internal states $|0\rangle$ and $|1\rangle$, and the photon field can
be written as:

$\displaystyle H_{total}$ $\displaystyle=$
$\displaystyle\frac{\Omega}{2}(|0\rangle\langle 1|+|1\rangle\langle 0|)$ (1)
$\displaystyle+$
$\displaystyle\int\mathrm{d}\omega\omega|\omega\rangle\langle\omega|+\sqrt{\frac{\gamma}{\pi}}\int\mathrm{d}\omega(|\omega\rangle\langle
1|+|1\rangle\langle\omega|).$

The photon field and ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ constitute the
measuring apparatus, described by ${|\omega\rangle\langle\omega|}$, and the
measured system is composed of the internal states $|0\rangle$ and
$|1\rangle$, wherein $\Omega$ is the coupling strength between $|0\rangle$ and
$|1\rangle$. Moreover, $\gamma$ is the interaction strength between the
measuring apparatus and the measured system [32], determined by the intensity
of the 854 nm laser and the natural linewidth of
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$. Solely focusing on the dynamic evolution
of the measured system, the Hamiltonian $H_{total}$ can be reduced to an
effective Hamiltonian $H_{eff}$,

$\displaystyle H_{eff}=-i\gamma|1\rangle\langle
1|+\frac{\Omega}{2}(|0\rangle\langle 1|+|1\rangle\langle 0|).$ (2)

This relation yields the Rabi oscillations of frequency $\Omega/2$, but at the
same time absorbs the population of $|1\rangle$, thereby performing a
”measurement.” Thus, using the ${}^{40}\mathrm{Ca}^{+}$ system, we design a
continuous measurement model [33, 34, 35, 36] wherein the 729 nm laser enables
the evolution of the quantum states, while the 854 nm laser performs the
measurements.

Now, let us rewrite Eq. (2) as
$H_{eff}=H_{\mathcal{PT}}-\frac{i\gamma}{2}\mathbf{I}$, where
$H_{\mathcal{PT}}=\frac{\Omega}{2}\sigma_{x}-i\frac{\gamma}{2}\sigma_{z}$ is
the $\mathcal{PT}$-symmetric Hamiltonian with balanced gain and loss,
$\sigma_{x(z)}$ is the Pauli matrix, and $\mathbf{I}$ is the identity
operator. This continuous measurement model corresponds to the static passive
$\mathcal{PT}$-symmetric system [16], which is time-independent. By solving
the eigenvalue of the Hamiltonian $H_{\mathcal{PT}}$, the expression of the
discriminant of the $\mathcal{PT}$ symmetry is obtained as follows:

$\displaystyle\left\\{\begin{aligned}
&\frac{\gamma}{\Omega}<1,&&\textit{PTSP}\ ,\\\
&\frac{\gamma}{\Omega}>1,&&\textit{PTBP}\ .\end{aligned}\right.$ (3)

However, the continuous measurement model cannot define the measurement
frequency and strength well. Hence, we must consider a pulse measurement model
[37, 38, 39, 40]. The experimental scheme suitable for
${}^{40}\mathrm{Ca}^{+}$ system is shown in FIG. 1(b). In this scheme, instead
of constantly interacting with ions, the laser is divided into pulses, and the
pulse time sequence is shown in FIG. 1(c). The system is first evolved under
the 729 nm laser drive for $t_{0}$. Then, the 729 nm laser is switched off,
and the 854 nm laser is switched on to measure the quantum state, with the
duration time of $t_{1}$, and so on. In addition to controlling the pulse
duration, one can also control the pulse intensity. Under the action of 729 nm
and 854 nm laser pulses, the quantum state alternately performs the
”evolution-measurement,” which is the pulse measurement model of
${}^{40}\mathrm{Ca}^{+}$ system.

Referring to the Hamiltonian $H_{eff}$ of the continuous measurement model,
the Hamiltonian $H_{eff}(t)$ of the pulse measurement model can be written as:

$\displaystyle H_{eff}$ $\displaystyle(t)=-i\gamma(t)|1\rangle\langle
1|+\frac{\Omega(t)}{2}(|0\rangle\langle 1|+|1\rangle\langle 0|)$
$\displaystyle H_{eff}$
$\displaystyle(t)=H_{\mathcal{PT}}(t)-\frac{i\gamma(t)}{2}\mathbf{I}$
$\displaystyle\left\\{\begin{aligned} &\Omega(t)=\Omega,\gamma(t)=0,&&0<t\leq
t_{0},\\\ &\Omega(t)=0,\gamma(t)=\gamma,&&t_{0}<t\leq t_{0}+t_{1},\\\
\end{aligned}\right.$

Where
$H_{\mathcal{PT}}(t)=\frac{\Omega(t)}{2}\sigma_{x}-i\frac{\gamma(t)}{2}\sigma_{z}$.
$H_{eff}(t)$ is a periodic time-dependent Hamiltonian, and
$H_{eff}(t+T)=H_{eff}(t)$, where $T=t_{0}+t_{1}$ is the period of the
Hamiltonian $H_{eff}(t)$. Notably, $H_{\mathcal{PT}}(t)$ is no longer a static
$\mathcal{PT}$-symmetric system. According to Eq. (3), the $\mathcal{PT}$
symmetry of $H_{\mathcal{PT}}(t)$ alternates with time. The $\mathcal{PT}$
symmetry of $H_{\mathcal{PT}}(t)$ becomes difficult to describe, and the
evolution characteristics become ambiguous. Therefore, we will describe the
$\mathcal{PT}$ symmetries of such time-dependent systems using the Floquet
theorem [18, 19].

##  III Results and discussion

According to the Hamiltonian in Eq. (II), the operator $U(T,0)$ of an
”evolution-measurement” can be written as:

$\displaystyle U(T,0)=\begin{bmatrix}e^{-\gamma t_{1}}cos(\Omega
t_{0}/2)&-ie^{-\gamma t_{1}}sin(\Omega t_{0}/2)\\\ -isin(\Omega
t_{0}/2)&cos(\Omega t_{0}/2)\end{bmatrix}.$ (5)

Here, $\Omega t_{0}/2$ is the measurement interval related to the measurement
frequency. We set the initial state $\rho(0)$ of the measured system as
$|0\rangle\langle 0|$. After applying the $U(T,0)$ $n$ times, the probability
that the measured system is still in the initial state becomes

$\displaystyle P_{0}$ $\displaystyle=$ $\displaystyle|\langle
0|U^{n}(T,0)\rho(0)U^{\dagger n}(T,0)|0\rangle|^{2}.$ (6)

where $P_{0}$ is the survival probability. The expression of $U(T,0)$ is
similar to Eq.(47) in the literature [37]. When $e^{-\gamma t_{1}}\rightarrow
1$, the evolution of the measured system is consistent with the general
unperturbed Rabi oscillation, the measurement is fuzzy, and the wave packet of
the measuring apparatus is not separated [37]. When $e^{-\gamma
t_{1}}\rightarrow 0$, literature [37] describes that the measurement is very
strong, and the wave packet of the measuring apparatus becomes sharp. As
discussed above, $\gamma t_{1}$ is related to the separation distance of the
wave packet of the measuring apparatus. Therefore, $\gamma t_{1}$ can be
defined as the measurement strength, and an increase in $\gamma t_{1}$ implies
that the measurement strength changes from weak to strong. Finally, from a
physical point of view, the operator $U(T,0)$ can be described as follows: the
measured system evolves $t_{0}$with the driving strength $\Omega/2$ and is
subsequently measured with the strength $\gamma t_{1}$.

To discuss the $\mathcal{PT}$ symmetry of the pulse measurement model, we
consider the Floquet Hamiltonian $\mathcal{H}_{F}(t)=H_{eff}(t)-i\partial_{t}$
[18, 19]. By calculating the eigenvalue of
$\mathcal{H}_{\mathcal{PT}}=H_{\mathcal{PT}}(t)-i\partial_{t}$ [41], we can
obtain the expression for the discriminant of $\mathcal{PT}$ symmetry of the
measured system (detailed derivation is in the appendix),

$\displaystyle\left\\{\begin{aligned} &cos^{2}(\frac{\Omega
t_{0}}{2})cosh^{2}(\frac{\gamma t_{1}}{2})<1,&&\textit{PTSP},\\\
&cos^{2}(\frac{\Omega t_{0}}{2})cosh^{2}(\frac{\gamma
t_{1}}{2})>1,&&\textit{PTBP}.\end{aligned}\right.$ (7)

When $cos^{2}(\frac{\Omega t_{0}}{2})cosh^{2}(\frac{\gamma t_{1}}{2})=1$, we
obtain the EP of the measured system. Eq. (7) indicates that the
$\mathcal{PT}$ symmetry of the measured system is determined by both the
measurement interval $\Omega t_{0}$ and strength $\gamma t_{1}$.

Figure 2: (Color online) (a) PT phase diagram of the measured system in the
Floquet representation. (b) The survival probability $P_{0}$ when the
measurement strength is fixed, $\gamma t_{1}=0.3$, and the measurement
interval $\Omega t_{0}=0.05,0.1,0.5,1.0$. (c) The survival probability $P_{0}$
when the measurement interval is fixed, $\Omega t_{0}=0.1$, and the
measurement strength $\gamma t_{1}=0.01,0.05,0.2,0.5$.

Thus, we establish a correlation between the general quantum measurement model
and the Floquet $\mathcal{PT}$-symmetry system. Next, we investigate the
effects of measurement interval and measurement strength on the $\mathcal{PT}$
symmetry. According to Eq. (7), FIG. 2(a) shows the $\mathcal{PT}$ symmetric
phase diagram of the measured system with respect to the measurement interval
$\Omega t_{0}$ and measurement strength $\gamma t_{1}$. By choosing different
$\Omega t_{0}$ and $\gamma t_{1}$, we can obtain different symmetry phases of
the measured system.

We set the initial state of the measured system as $|0\rangle$. The survival
probability of the measured system is shown in FIG. 2(b), when fixing the
measurement strength ($\gamma t_{1}=0.3$). The measured system is in the PTSP,
when $\Omega t_{0}=0.5,1$. When $\Omega t_{0}=0.05,0.1$, the measured system
is in PTBP. This implies that when the measurement frequency is fast enough,
the coherent evolution between the internal states disappears, and the
measured system changes from PTSP to PTBP. When $\Omega t_{0}$ is further
reduced, the evolution of the measured system is significantly slower in the
PTBP. Therefore, the measurement at a high frequency breaks the $\mathcal{PT}$
symmetry of the measured system. The dynamic characteristic of the unbroken
$\mathcal{PT}$ symmetry system is the oscillation and that of the broken
$\mathcal{PT}$ symmetry system is an exponential decrease in the survival
probability.

As shown in FIG. 2(c), when the measurement frequency is fixed ($\Omega
t_{0}=0.1$), and the measurement strength $\gamma t_{1}$ is changed from weak
to strong, the evolution of the measured system will also change
significantly. When $\gamma t_{1}=0.01,0.05$, the measured system is in the
PTSP. The evolution of the measured system is oscillation, and there are still
coherent transitions between the internal states. When $\gamma t_{1}=0.2,0.5$,
the strong measurement strength breaks the $\mathcal{PT}$-symmetry of the
measured system. The evolution of the measured system becomes slower as the
measurement strength becomes stronger in the PTBP. Therefore, we find that
even if the measurement frequency is not fast enough, as long as the
measurement strength is strong enough, the $\mathcal{PT}$ symmetry of the
measured system will be broken.

Figure 3: (Color online) (a) The survival probability $P_{0}$ when the
measurement interval is fixed, $\Omega t_{0}=0.050(2)$, and the measured
system is in PTSP. (b) The survival probability $P_{0}$ when the measurement
interval is fixed, $\Omega t_{0}=0.050(2)$, and the measured system is in the
PTBP. The error bars represent the standard deviation of the measurements.

Next, we experimentally observe the effect of the measured strength on the
$\mathcal{PT}$ symmetry. We set the measurement interval as $\Omega
t_{0}=0.050(2)$ and the initial state as $|0\rangle$. The dynamic evolution of
the measured system is shown in FIG. 3(a). For the measurement strength
$\gamma t_{1}=0.0044(2),0.0132(5),0.026(1)$, the measured system is in PTSP.
In FIG. 3(b), $\gamma t_{1}=0.34(1),0.68(3),1.36(6)$; the measured system is
in PTBP. The dynamic evolution of the measured system is no longer an
oscillation but an exponential decay. When the measured system is in PTBP,
with the increase in the measurement strength $\gamma t_{1}$, the transition
rate from $|0\rangle$ to $|1\rangle$ becomes slower.

Interestingly, although both static $\mathcal{PT}$-symmetry system and Floquet
$\mathcal{PT}$-symmetry system satisfy the $\mathcal{PT}$-symmetry, the
expression of the discriminant Eq. (7), obtained by solving the eigenvalues of
$\mathcal{H}_{\mathcal{PT}}(t)$, is quite different from Eq. (3). Since
$\mathcal{H}_{\mathcal{PT}}(t)$ is time-dependent, Eq. (7) depends not only on
$\gamma$ and $\Omega$, but also on time $t_{0}$ and $t_{1}$. Hence, we
experimentally investigated the difference between the Floquet
$\mathcal{PT}$-symmetric system and static passive $\mathcal{PT}$-symmetric
system. According to Eq. (3), when $\gamma/\Omega=2$, the static
$\mathcal{PT}$-symmetric system is in PTBP and the dynamic evolution of the
system should no longer oscillate. For the Floquet $\mathcal{PT}$-symmetric
system, the experimental results are demonstrated in FIG. 4. We set the
initial state as $|0\rangle$. When $\Omega t_{0}=0.050(2)$, $\gamma
t_{1}=0.0040(2),0.0140(7)$, the evolution of the measured system is still
under oscillation. Therefore, the $\mathcal{PT}$ symmetry is not broken. When
$\gamma t_{1}=0.40(2),0.80(4)$, the evolution of the measured system is
exponential, and the $\mathcal{PT}$ symmetry is broken. The experimental
results do not agree with the conclusion given by Eq. (3) but agree with the
conclusion given by Eq. (7).

Figure 4: (Color online) The survival probability $P_{0}$ when $\Omega
t_{0}=0.050(2)$ and $\gamma/\Omega=2.0(1)$. When $\gamma/\Omega=2$, the pulse
measurement model can still be in PTSP.

##  IV Conclusion

We probe the effect of quantum measurement on the symmetry of the measured
system and establish the correlation between the quantum measurement and the
Floquet $\mathcal{PT}$-symmetry. Additionally, an experimental scheme to
investigate the $\mathcal{PT}$ symmetry of the measured system was designed by
the trapped-ion system. Using Floquet theorem, we obtain the expression of the
discriminant of $\mathcal{PT}$ symmetry of the measured system, which is very
different from that of the static $\mathcal{PT}$-symmetric system. We observe
that the measurement interval $\Omega t_{0}$ and strength $\gamma t_{1}$
affect the $\mathcal{PT}$ symmetry of the measured system. When the
measurement strength $\gamma t_{1}$ is fixed and the measurement frequency is
fast enough, the $\mathcal{PT}$ symmetry of the measured system is broken, and
the evolution of the system is hindered by the measurement. Conversely, when
the measurement interval $\Omega t_{0}$ is fixed, the sufficiently strong
measurement will break the $\mathcal{PT}$ symmetry of the measured system.

There are some open questions to be investigated further. What is the physical
mechanism of $\mathcal{PT}$ symmetry breaking caused by quantum measurement?
Can the measurement affect the other symmetry?

##  V Acknowledgments

This work is supported by the National Basic Research Program of China under
Grant No.2016YFA0301903, the National Natural Science Foundation of China
under Grant.12074433, 12004430, 12174447, 12174448, 11904402.

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##  Appendix A Appendix

###  A.1 A1. Discrimination of PT symmetry

First, according to the properties of $\mathcal{PT}$-symmetry, several methods
can be used to determine whether the system is in the $\mathcal{PT}$ symmetry
phase (PTSP) or $\mathcal{PT}$ symmetry breaking phase (PTBP). For example,
the Hamiltonian $H_{\mathcal{PT}}$ and $\mathcal{PT}$ operator have common
eigenstates in PTSP, while no common eigenstates in PTBP. The second method is
a widely accepted method, which involves the calculation of the eigenvalue of
the Hamiltonian $H_{\mathcal{PT}}$.

Next, we discuss how to determine the symmetry of Floquet
$\mathcal{PT}$-symmetry system. First, we introduce the Floquet method.
Floquet method can be used to solve the periodic time-dependent Hamiltonian;
for example, the Schrodinger equation of the time-dependent Hamiltonian $H(t)$
is:

$\displaystyle
H\left(t\right)\Psi\left(t\right)=i\hbar\frac{\partial}{\partial
t}\Psi\left(t\right).$ (A1)

If $H(t)=H(t+T)$, where T is the period. Then, according to the Floquet’s
theory, there are solutions to Eq. (A1)

$\displaystyle\Psi_{\alpha}\left(t\right)=\exp{\left(-\frac{i\epsilon_{\alpha}t}{\hbar}\right)}\Phi_{\alpha}\left(t\right),$
(A2)

where $\Psi_{\alpha}\left(t\right)$ is called the Floquet state;
$\Phi_{\alpha}\left(t\right)=\Phi_{\alpha}(t+T)$ satisfies periodicity and is
known as the Floquet mode; $\epsilon_{\alpha}$ is a constant number that does
not vary with time and is defined as a quasi-energy level. Hence, given
$\Phi_{\alpha}\left(t\right)$ and $\epsilon_{\alpha}$ for a particular
$H\left(t\right)$, the solution of the wave function $\Psi\left(t\right)$ at
any time $t$ can be obtained as:

$\displaystyle\Psi\left(t\right)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}c_{\alpha}\Psi_{\alpha}\left(t\right)$ (A3)
$\displaystyle=$
$\displaystyle\sum_{\alpha}c_{\alpha}\exp{\left(-\frac{i\epsilon_{\alpha}t}{\hbar}\right)}\Phi_{\alpha}\left(t\right),$

where the coefficient $c_{\alpha}$ can be obtained from the initial wave
function
$\Psi\left(0\right)=\sum_{\alpha}c_{\alpha}\Psi_{\alpha}\left(0\right)$. By
substituting Eq. (A2) into Eq. (A1), the eigen-equations about Floquet mode
and quasi-energy can be obtained as:

$\displaystyle\mathcal{H}\left(t\right)\Phi_{\alpha}\left(t\right)=\epsilon_{\alpha}\Phi_{\alpha}\left(t\right),$
(A4)

where
$\mathcal{H}\left(t\right)=H\left(t\right)-i\hbar\frac{\partial}{\partial t}$
is the Floquet Hamiltonian.

The evolution of the wave function $\Psi\left(t\right)$ satisfies the
following form:

$\displaystyle U\left(T+t,t\right)\Psi\left(t\right)=\Psi\left(T+t\right).$
(A5)

Substitute Eq. (A3) into Eq. (A5) to obtain

$\displaystyle U\left(T+t,t\right)$
$\displaystyle\exp{\left(-\frac{i\epsilon_{\alpha}t}{\hbar}\right)}\Phi_{\alpha}\left(t\right)$
(A6) $\displaystyle=$
$\displaystyle\exp{\left(-\frac{i\epsilon_{\alpha}(T+t)}{\hbar}\right)}\Phi_{\alpha}\left(T+t\right).$

Since $\Phi_{\alpha}\left(t\right)=\Phi_{\alpha}(t+T)$, the formula above can
be obtained

$\displaystyle U\left(T+t,t\right)\Phi_{\alpha}\left(t\right)$
$\displaystyle=$
$\displaystyle\exp{\left(-\frac{i\epsilon_{\alpha}T}{\hbar}\right)}\Phi_{\alpha}\left(t\right)$
(A7) $\displaystyle=$ $\displaystyle\eta_{\alpha}\Phi_{\alpha}\left(t\right),$

which shows that the Floquet modes are the eigenstates of the one-period
propagator. Therefore, we can find the Floquet modes and quasi-energies
$\epsilon_{\alpha}=-\hbar\ln(\eta_{\alpha})/T$ by numerically calculating
$U\left(T+t,t\right)$ and diagonalizing it.

As discussed above, Floquet $\mathcal{PT}$-symmetric Hamiltonian can be
written as
$\mathcal{H}_{\mathcal{PT}}\left(t\right)=H_{\mathcal{PT}}\left(t\right)-i\hbar\frac{\partial}{\partial
t}$, where $H_{\mathcal{PT}}\left(t\right)$ is the $\mathcal{PT}$-symmetric
Hamiltonian that varies periodically with time. To evaluate the symmetry of
the Floquet $\mathcal{PT}$-symmetric system, the eigenvalue $\epsilon_{i}$ of
$\mathcal{H}_{\mathcal{PT}}\left(t\right)$ can be calculated. Since
$\epsilon_{i}$ is related to the eigenvalue $\eta_{i}$ of the time evolution
operator $U\left(T+t,t\right)$, the expression of discriminant of the symmetry
of Floquet $\mathcal{PT}$-symmetric system can be obtained. When $\eta_{i}$ is
imaginary, $\epsilon_{i}$ is real and the system is in PTSP; when $\eta_{i}$
is real, $\epsilon_{i}$ is imaginary and the system is in PTBP.

In this paper, the Hamiltonian of Floquet $\mathcal{PT}$-symmetric system is
as follows:

$\displaystyle H_{eff}$ $\displaystyle(t)$
$\displaystyle=-i\gamma(t)|1\rangle\langle
1|+\frac{\Omega(t)}{2}(|0\rangle\langle 1|+|1\rangle\langle 0|)$
$\displaystyle\left\\{\begin{aligned} &\Omega(t)\neq 0,\gamma(t)=0,&&0<t\leq
t_{0},\\\ &\Omega(t)=0,\gamma(t)\neq 0,&&t_{0}<t\leq t_{0}+t_{1},\\\
\end{aligned}\right.$

The time evolution operator within a period $T=t_{0}+t_{1}$ is given by:

$\displaystyle U(T,0)=e^{-\gamma|1\rangle\langle
1|t_{1}}e^{-i\frac{\mathrm{\Omega}}{2}(|1\rangle\langle 0|+|0\rangle\langle
1|)t_{0}}$ (A9) $\displaystyle=$ $\displaystyle\left[\begin{matrix}exp(-\gamma
t_{1})cos(\frac{\mathrm{\Omega}t_{0}}{2})&-iexp(-\gamma
t_{1})sin(\frac{\mathrm{\Omega}t_{0}}{2})\\\
-isin(\frac{\mathrm{\Omega}t_{0}}{2})&cos(\frac{\mathrm{\Omega}t_{0}}{2})\\\
\end{matrix}\right].$

The eigenvalues of $U(T,0)$ are solved as follows:

$\displaystyle\eta_{\pm}$ $\displaystyle=$ $\displaystyle\frac{1}{2}e^{-\gamma
t_{1}}[cos(\frac{\mathrm{\Omega}t_{0}}{2})+e^{\gamma t_{1}}cos(\frac{\Omega
t_{0}}{2})]$ (A10) $\displaystyle\pm$ $\displaystyle 2e^{-\frac{1}{2}\gamma
t_{1}}[\sqrt{cosh^{2}(\frac{\gamma t_{1}}{2})cos^{2}(\frac{\Omega
t_{0}}{2})-1}],$

From the above formula, $\sqrt{cosh^{2}(\frac{\gamma
t_{1}}{2})cos^{2}(\frac{\Omega t_{0}}{2})-1}$ determines whether $\eta_{\pm}$
is complex. Then, the discriminant of symmetry in the text is obtained as
follows:

$\displaystyle\left\\{\begin{aligned} &cos^{2}(\frac{\Omega
t_{0}}{2})cosh^{2}(\frac{\gamma t_{1}}{2})<1,&&\textit{PTSP}\,\\\
&cos^{2}(\frac{\Omega t_{0}}{2})cosh^{2}(\frac{\gamma
t_{1}}{2})>1,&&\textit{PTBP}\ .\end{aligned}\right.$ (A11)

In fact, it is more rigorous to solve the eigenvalue of
$\mathcal{H}_{PT}\left(t\right)=H_{PT}\left(t\right)-i\partial_{t}$. However,
the expression of the discriminant of $\mathcal{PT}$ symmetry obtained by
solving $\mathcal{H}_{F}(t)$ or $\mathcal{H}_{PT}$ is the same.

###  A.2 A2. Floquet PT symmetric systems

For different forms of Floquet $\mathcal{PT}$-symmetric systems, the dynamic
evolution of the system are quite different. For the Hamiltonian
$H_{eff}=-i\gamma(t)|1\rangle\langle 1|+\frac{\Omega(t)}{2}(|0\rangle\langle
1|+|1\rangle\langle 0|)$, we can choose different $\gamma(t)$ and
$\mathrm{\Omega}(t)$. For example, $\gamma\left(t\right)$ is a square wave
oscillating between 0 and $\gamma$ with oscillation frequency
$\mathrm{\omega}$, and $\Omega\left(t\right)$ is a square wave oscillating
between 0 and $\Omega$ with the oscillation frequency $\mathrm{-\omega}$. When
$\mathrm{\Omega}=0.1*2\pi(Mhz),\frac{\gamma}{\mathrm{\Omega}}=2$,
$\frac{2\pi\omega}{\mathrm{\Omega}}=0.5$, and initial state is $|0\rangle$, we
can obtain the dynamical evolution of $\rho_{00}(t)$, as shown in FIG. A1.
Evidently, this is completely different from the evolution of the measured
system in the text. Therefore, by controlling the system parameters, a Floquet
$\mathcal{PT}$-symmetric system can be constructed to meet the research
requirements.

Figure A1: (Color online) the dynamical evolution of $\rho_{00}(t)$. There is
no attenuation in the system.

###  A.3 A3: Experimental Setup

In our experimental scheme, the continuous measurement model is constructed by
the internal state of ${}^{40}\mathrm{Ca}^{+}$, and the energy level structure
is shown in FIG. 1(a). The Zeeman sublevels
${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$ and ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$
of the ${}^{40}\mathrm{Ca}^{+}$ in a 5.2 G-magnetic field are chosen as the
quantum states $|0\rangle$ and $|1\rangle$. The lifetime of the excited state
${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$ is $1.168\pm 0.007s$.
${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$ is excited to
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ by the circularly polarized 854 nm laser
beam. Experimentally, because the polarization of the 854 nm laser is not
perfectly circularly polarized, electrons may be excited from
${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$ to other Zeeman sublevels of
${}^{2}\mathrm{P}_{3/2}$. Since the lifetime of
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ is $6.924\pm 0.019ns$, the electron
cannot exist stably and will emit spontaneously. According to the selection
rule and branching ratio of the spontaneous emission, the electron has about
94% chance to transition to ${}^{2}\mathrm{S}_{1/2}(m_{J}=+1/2)$ and 6% chance
to transition to ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$. The spontaneous
emission of ${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ ensures that most of the
population flows into ${}^{2}\mathrm{S}_{1/2}(m_{J}=+1/2)$ as the environment
and not back into ${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$ as the system. After
considering the polarization of 854 nm laser and branch ratio of the
spontaneous emission, the probability of this emission from
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ state to
${}^{2}\mathrm{S}_{1/2}(m_{J}=+1/2)$ is about 90%. These experimental setup
and optimizations ensure the rationality of the theoretical approximation in
this paper.

${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$ and ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$
are coupled by the 729 nm laser, and ${}^{2}\mathrm{D}_{5/2}(m_{J}=+1/2)$ and
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$ are coupled by the 854 nm laser. The 729
nm and 854 nm lasers resonate with the corresponding transition energy level,
respectively. Thus, in the interaction picture, the Hamiltonian of the system
can be written as:

$\displaystyle H_{eff}=-i\Gamma|P\rangle\langle P|$ $\displaystyle+$
$\displaystyle\frac{\Omega}{2}(|0\rangle\langle 1|+|1\rangle\langle 0|)$ (A12)
$\displaystyle+$ $\displaystyle\frac{\Omega^{\prime}}{2}(|P\rangle\langle
1|+|1\rangle\langle P|).$

where $\Omega$ is the coupling strength between $|0\rangle$ and $|1\rangle$,
and $\Omega^{\prime}$ is the coupling strength between $|1\rangle$ and
$|P\rangle$. The coupling strength can be controlled by adjusting the laser
intensity. There is an electric quadrupole transition between $|0\rangle$ and
$|1\rangle$, and an electric dipole transition between $|1\rangle$ and
$|P\rangle$; thus, in general $\Omega^{\prime}>>\Omega$ (In the experiments,
$\Omega\simeq 0.1(MHz)$, and $\Omega^{\prime}$ is in the order of $MHz$).
$\Gamma\simeq 22(MHz)$ is the natural linewidth of
${}^{2}\mathrm{P}_{3/2}(m_{J}=+3/2)$. According to the method given in
literature [32, 36], when $\Omega^{\prime},\Gamma>>\Omega$, Eq. (A12) can be
written as:

$\displaystyle H_{eff}=-i\gamma|1\rangle\langle
1|+\frac{\Omega}{2}(|0\rangle\langle 1|+|1\rangle\langle 0|).$ (A13)

Where $\gamma=\Omega^{\prime 2}/(2\Gamma)$. So we have the desired Hamiltonian
for the ${}^{40}\mathrm{Ca}^{+}$ system.

In the experimental scheme, the initial state of the measured system is
prepared to ${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$, i.e., state $|0\rangle$. The
motion state of the ${}^{40}\mathrm{Ca}^{+}$ is cooled to the ground state by
a series of laser cooling techniques, including the Doppler cooling, EIT
cooling, and sideband cooling. The average phonon number of an ion is about
0.02. After the cooling completion, the electrons are manipulated by a laser
into an ${}^{2}\mathrm{S}_{1/2}(m_{J}=-1/2)$. In the experiment, the fidelity
of the initial state is about 99.8%.