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

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.

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:

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 $H_{eff}$,

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:

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:

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].

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

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

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),

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 $\gamma t_{1}$.

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.

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).

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?

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.