⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 081101 $\mathcal{PT}$ Symmetry of a Square-Wave Modulated Two-Level System Liwei Duan (段立伟)1, Yan-Zhi Wang (汪延志)1, and Qing-Hu Chen (陈庆虎)1,2* Affiliations 1Department of Physics and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China 2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Received 11 April 2020; accepted 16 June 2020; published online 28 July 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11674285 and 11834005) and the National Key Research and Development Program of China (Grant No. 2017YFA0303002).
*Corresponding author. Email: qhchen@zju.edu.cn Citation Text: Duan L W, Wang Y Z and Chen Q H 2020 Chin. Phys. Lett. 37 081101    Abstract We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the $\mathcal{PT}$ phase diagram are captured exactly. Two kinds of $\mathcal{PT}$ symmetry broken phases are found, whose effective Hamiltonians differ by a constant $\omega / 2$. For the time-periodic dissipation, a vanishingly small dissipation strength can lead to the $\mathcal{PT}$ symmetry breaking in the $(2k-1)$-photon resonance ($\varDelta = (2k-1) \omega$), with $k=1,2,3\dots$ It is worth noting that such a phenomenon can also happen in $2k$-photon resonance ($\varDelta = 2k \omega$), as long as the dissipation strengths or the driving times are imbalanced, namely $\gamma_0 \ne - \gamma_1$ or $T_0 \ne T_1$. For the time-periodic coupling, the weak dissipation induced $\mathcal{PT}$ symmetry breaking occurs at $\varDelta_{\rm eff}=k\omega$, where $\varDelta_{\rm eff}=(\varDelta_0 T_0 + \varDelta_1 T_1)/T$. In the high frequency limit, the phase boundary is given by a simple relation $\gamma_{\rm eff}=\pm\varDelta_{\rm eff}$. DOI:10.1088/0256-307X/37/8/081101 PACS:11.30.Er, 42.82.Et, 03.65.Yz, 42.50.-p © 2020 Chinese Physics Society Article Text A non-Hermitian Hamiltonian is a natural extension of the conventional Hermitian one to describe the open quantum system. The discovery of the real spectra in non-Hermitian Hamiltonians by Bender and Boettcher[1] has stimulated enormous interest in the systems with parity-time ($\mathcal{PT}$) symmetry.[2–5] Early theoretical and experimental explorations of the non-Hermitian systems with $\mathcal{PT}$ symmetry mainly focus on the optics and photonics.[6–17] Feng et al. realized the nonreciprocal light propagation in a silicon photonic circuit which provides a way to chip-scale optical isolators for optical communications and computing.[12] Hodaei et al. stabilized single-longitudinal mode operation in a system of coupled mirroring lasers by harnessing notions from $\mathcal{PT}$ symmetry, which provides the possibilities to develop optical devices with enhanced functionality.[14] Xiao et al. achieved the first experimental characterization of critical phenomena in $\mathcal{PT}$-symmetric nonunitary quantum dynamics.[17] Recent experiments have realized the non-Hermitian magnon-polaritons systems, and higher-order exceptional points were observed, which can be used to measure the output spectrum of the cavity.[18–20] The anomalous edge state in a non-Hermitian lattice[21] has intrigued persistent attention to the combination of the non-Hermiticity and the topological phase.[22–35] The non-Bloch band theory has been developed to describe the non-Hermitian lattice systems.[22,29,32,33] Kawabata et al. established a fundamental symmetry principle in non-Hermitian physics, which paved the way towards a unified framework for non-equilibrium topological phase.[25,26] Yao et al. studied the bulk-boundary correspondence in the non-Hermitian systems and found the non-Hermitian skin effect.[29,30] Xiao et al. observed the topological edge states in $\mathcal{PT}$-symmetric quantum walks.[31] Recently, Joglekar et al. investigated a two-level system coupled to a sinusoidally varying gain-loss potential, namely, the non-Hermitian Rabi model with time-periodic dissipation.[36] They found that multiple frequency windows existed where $\mathcal{PT}$ symmetry was broken and restored. The non-Hermitian Rabi model has drawn growing attention due to its specially rich phenomena which are absent in the static counterparts.[37–46] Lee et al. found the $\mathcal{PT}$ symmetry breaking at the $(2k-1)$-photon resonance and derived the boundaries of the $\mathcal{PT}$ phase diagram by doing perturbation theory beyond rotating-wave approximation.[37] Gong et al. found that a periodic driving could stabilize the dynamics despite the loss and gain in the non-Hermitian system.[40,41] Xie et al. studied a non-Hermitian Rabi model with time-periodic coupling and found exact analytical results for certain exceptional points.[42] A synchronous modulation that combined the time-periodic dissipation and coupling was studied in Ref. [43], which provided an additional possibility for pulse manipulation and coherent control of the $\mathcal{PT}$-symmetric two-level systems. The experimental approach of the Floquet $\mathcal{PT}$-symmetric system has been proposed with two coupled high-frequency oscillators.[44] A $\mathcal{PT}$ symmetry breaking transition by engineering time-periodic dissipation and coupling has been realized through state-dependent atom loss in an optical dipole trap of ultracold $^6\mathrm{Li}$ atoms.[45] They confirmed that a weak time-periodic dissipation could lead to $\mathcal{PT}$-symmetry breaking in $(2k-1)$-photon resonance. It should be noted that the $\mathcal{PT}$-symmetry breaking can occur in a finite non-Hermitian system, which is quite different from the quantum phase transition in the Hermitian system where the thermodynamic limit is needed.[45,47] In this letter, we study the $\mathcal{PT}$ symmetry of a two-level system with time-periodic dissipation and coupling. Instead of the widely used sinusoidal modulation,[36,37,40–42] we consider a square-wave one, which is easier to implement in the experiment of ultracold atoms[45] and has analytical exact solutions based on the Floquet theory.[48,49] The square-wave modulation has a broad range of applications in the Hermitian system. It has been used to suppress the quantum dissipation in spin chains,[50] to generate many Majorana modes in a one-dimensional p-wave superconductor system,[51] to generate large-Chern-number topological phases,[52] and so on. The square-wave modulation has also been realized in the non-Hermitian systems.[45] In this study, firstly we describe the non-Hermitian Hamiltonian of the driving two-level system. Secondly we briefly introduce the Floquet theory and derive the effective static Hamiltonian. Thirdly, we achieve the $\mathcal{PT}$ phase diagram and analyze the influence of multiphoton resonance. An equivalent Hamiltonian is obtained in the high frequency limit. Finally we give some concluding remarks. We consider a periodically driving two-level system $H(t) = H(t + T)$, with $$\begin{align} H(t) =\frac{\varDelta\left( t\right) }{2}\sigma _{x}+i \frac{\gamma \left( t\right) }{2}\sigma _{z},~~ \tag {1} \end{align} $$ where $\sigma_{x, z}$ are the Pauli matrices, $T = T_0 + T_1$ is the driving period, $\omega =2\pi /T$ is the driving frequency, $\varDelta(t)$ is the time-periodic coupling strength, and $\gamma(t)$ is the dissipation strength which leads to the periodic gain and loss. Lee et al.[37] studied the $\mathcal{PT}$ phase diagram of the non-Hermitian two-level system by doing the perturbation theory, which corresponds to $\varDelta(t)=\varDelta$ and $\gamma (t) = 4 \lambda \cos(\omega t)$. Xie et al.[42] found the exact analytical results for certain exceptional points of the two-level system with time-periodic coupling, which corresponds to $\varDelta(t)=v_0 + v_1 \cos(\omega t)$ and $\gamma (t) = \gamma$. Luo et al.[43] studied the analytical results of the non-Hermitian two-level systems with sinusoidal modulations of both $\varDelta(t)$ and $\gamma(t)$. In order to obtain the exact analytical results without using perturbation theory, we consider a synchronous square-wave modulation of both dissipation and coupling. The corresponding time-periodic parameters are $$ f (t) =\begin{cases} f _{0}, \mathrm{ ~if }~~ m T - \frac{T_0}{2} \leq t < m T + \frac{T_0}{2}, \\ f _{1}, \mathrm{ ~if }~~ m T + \frac{T_0}{2} \leq t < (m + 1)T - \frac{T_0}{2}, \end{cases}~~ \tag {2} $$ with $f=\varDelta$, $\gamma$ and $m=\dots, -1,0,1, \dots$ It is easy to confirm that the non-Hermitian Hamiltonian has a $\mathcal{PT}$ symmetry, namely $\hat{\mathcal{P}} H^†(t) \hat{\mathcal{P}} = H(t)$, where $H^†(t)$ is the Hermitian conjugate of $H(t)$ and $\hat{\mathcal{P}} = \hat{\mathcal{P}}^{-1} = \sigma_x $ is the parity operator.[2,5] This non-Hermitian system has been realized by Li et al. in the ultracold atom experiments.[45] However, they focused on a special case with only one time-periodic parameter (either dissipation or coupling), and $f_{0}=f$, $f_{1}=0$, $T_0=T_1=T/2$. We consider a more general case in which those constraints are relieved. Two time-independent Hamiltonians $H_0$ and $H_1$ appear alternately, with $$\begin{align} H_j =\frac{{\varDelta}_j }{2}\sigma _{x}+i\frac{\gamma_j }{2}\sigma _{z}, ~~ j=0, 1,~~ \tag {3} \end{align} $$ and the corresponding eigenenergies are $E_j^{\pm} = \pm h_j$, with $$\begin{align} h_{j} = \frac{\sqrt{ {\varDelta}_j ^{2}-\gamma _{j}^{2}}}{2}.~~ \tag {4} \end{align} $$ $H_j$ is one of the simplest non-Hermitian systems with $\mathcal{PT}$ symmetry.[5] When $|{\varDelta}_j| > |\gamma_j|$, the eigenenergy is real and it corresponds to the $\mathcal{PT}$-symmetric phase. When $|{\varDelta}_j| < |\gamma_j|$, the eigenenergy is imaginary and the $\mathcal{PT}$ symmetry is broken. When $|{\varDelta}_j| = |\gamma_j|$, an exceptional point (EP exists). The dynamics at each time domain is governed by the time evolution operator $$\begin{align} U_j(T_{j}) ={}& \exp (-iH_{j}T_{j}) \\ ={}&\cos (h_{j}T_{j}) I -i \mathrm{sinc} (h_{j}T_{j}) T_{j} H_{j},~~ \tag {5} \end{align} $$ where $I$ is a $2 \times 2$ identity matrix, and $\mathrm{sinc}(x)=\sin(x)/x$. According to the Floquet theory,[48,49] we can define an effective Hamiltonian $H_{\rm eff}$, which satisfies the condition $$ U_{\rm eff}(T)=\exp (-i H_{\rm eff}T) =\mathcal{T}\exp \Big[ -i\int_{-\frac{T_0}{2}}^{T-\frac{T_0}{2}}dtH(t)\Big] .~~ \tag {6} $$ The eigenenergies of the effective Hamiltonian correspond to the Floquet quasi-energies. Due to the simplicity of the square-wave modulation, the time evolution operator in a period can be written as $$ \mathcal{T}\exp \Big[ -i\int_{-\frac{T_0}{2}}^{T-\frac{T_0}{2}}dtH(t)\Big] =\exp (-i H_{1}T_{1}) \exp (-i H_{0}T_{0}).~~ \tag {7} $$ Therefore, $$ U_{\rm eff}(T)=U_{1}(T_{1})U_{0}(T_{0}).~~ \tag {8} $$ From Eqs. (5) and (8), we can achieve the effective time evolution operator $$\begin{align} &U_{\rm eff}(T) =[\cos (h_{1}T_{1}) \cos ( h_{0}T_{0})\\ & +\frac{1}{4}(\gamma _{1}\gamma _{0}-{\varDelta}_{1}{\varDelta}_{0}) T_{1}T_{0}\mathrm{sinc}(h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0})] I \\ &-i\frac{1 }{2}[{\varDelta}_{0} T_{0}\cos (h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0})\\ & + {\varDelta}_{1} T_{1}\cos (h_{0}T_{0}) \mathrm{sinc}(h_{1}T_{1})] \sigma _{x} \\ &+\frac{1}{4} ({\varDelta}_{0} \gamma _{1}- {\varDelta}_{1}\gamma _{0}) T_{1}T_{0}\mathrm{sinc}(h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0}) \sigma _{y} \\ &+\frac{1}{2}[\gamma _{0}T_{0}\cos (h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0})\\ &+\gamma _{1}T_{1}\cos (h_{0}T_{0}) \mathrm{sinc}(h_{1}T_{1})] \sigma _{z}.~~ \tag {9} \end{align} $$ Since $h_{j}$ can be either a pure real number in the $\mathcal{PT}$-symmetric phase or a pure imaginary one in the $\mathcal{PT}$ symmetry broken phase, both $\cos (h_{j}T_{j}) $ and $\mathrm{sinc}(h_{j}T_{j}) $ must be real numbers. Accordingly, the coefficients before $I$, $\sigma _{y}$ and $\sigma _{z}$ must be real, while those before $\sigma _{x}$ must be imaginary. It is easy to confirm that the effective Hamiltonian can only be in the form of $$\begin{alignat}{1} H_{\rm eff} =\frac{J }{2}\sigma _{x}+i\Big(\frac{\mathit{\varGamma} _{y}}{2}\sigma _{y}+\frac{\mathit{\varGamma} _{z}}{2}\sigma _{z}\Big) +\frac{n\omega }{2}I,~~ \tag {10} \end{alignat} $$ with $n=0,1$. The eigenenergies of $H_{\rm eff}$, or the Floquet quasi-energies of $H(t)$ would be $E^{\pm}=\pm h + \frac{n \omega}{2}$, where $$\begin{align} h =\frac{\sqrt{ J^{2}-\varGamma_y ^{2}-\varGamma_z ^{2}}}{2}.~~ \tag {11} \end{align} $$ The effective time evolution operator can be rewritten as $$\begin{alignat}{1} U_{\rm eff}(T) ={}& \exp (-iH_{\rm eff}T) \\ ={}&(-1)^n \cos (hT) I \\ &- i \frac{(-1)^n T}{2}\mathrm{sinc}(hT) J \sigma _{x} \\ &+ \frac{(-1)^n T}{2}\mathrm{sinc}(hT) (\mathit{\varGamma} _{y}\sigma _{y} +\mathit{\varGamma} _{z}\sigma _{z}).~~ \tag {12} \end{alignat} $$ By comparing the coefficients before $I$, $\sigma_x$, $\sigma_y$, and $\sigma_z$ in Eqs. (9) and (12), we can directly obtain $$\begin{alignat}{1} &(-1)^n \cos(h T) = \cos (h_{1}T_{1}) \cos \left( h_{0}T_{0}\right) \\ &+\frac{1 }{4}\left(\gamma _{1}\gamma _{0}-{\varDelta}_{1} {\varDelta}_{0}\right) T_{0}T_{1}\mathrm{sinc}(h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0}),~~ \tag {13} \end{alignat} $$ $$\begin{align} J ={}& \frac{(-1)^n }{T\mathrm{sinc}(hT)}[{\varDelta}_{0} T_{0}\cos (h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0}) \\ &+ {\varDelta}_{1} T_{1}\cos (h_{0}T_{0}) \mathrm{sinc}(h_{1}T_{1})],~~ \tag {14} \end{align} $$ $$\begin{align} \varGamma_y ={}& \frac{(-1)^n}{2\,T \mathrm{sinc}(hT)}({\varDelta}_{0} \gamma _{1}- {\varDelta}_{1} \gamma _{0}) T_{0}T_{1}\\ &\cdot\mathrm{sinc}(h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0}),~~ \tag {15} \end{align} $$ $$\begin{align} \varGamma_z ={}& \frac{(-1)^n}{T \mathrm{sinc}(hT)}[\gamma _{0}T_{0}\cos (h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0})\\ & +\gamma _{1}T_{1}\cos (h_{0}T_{0}) \mathrm{sinc}(h_{1}T_{1})].~~ \tag {16} \end{align} $$ Once we obtain $J$, $\varGamma_y$, $\varGamma_z$ and $n$, the effective Hamiltonian (10) will be finally determined. The major differences of the effective Hamiltonian and the original one are the dissipation $\varGamma_y$ in $y$-axis and the additional constant $\omega / 2$. We will show later that the additional constant is closely related with the $\mathcal{PT}$ symmetry broken phases and the exceptional points. One can easily confirm that the effective Hamiltonian has a $\mathcal{PT}$ symmetry, namely $\hat{\mathcal{P}} H^†_{\rm eff} \hat{\mathcal{P}} = H_{\rm eff}$, since $\hat{\mathcal{P}} \sigma_x \hat{\mathcal{P}} = \sigma_x$, $\hat{\mathcal{P}} \sigma_y \hat{\mathcal{P}} = -\sigma_y$ and $\hat{\mathcal{P}} \sigma_z \hat{\mathcal{P}} = -\sigma_z$. When $|\cos(h T)| < 1$, $h$ must be a real number and the $\mathcal{PT}$ symmetry is preserved. For the $\mathcal{PT}$-symmetric phase, we suppose that the eigenenergies are $E_{\pm}^{(n)} = \pm h^{(n)} + \frac{n \omega}{2}$. From Eq. (13), we can obtain $\cos(h^{(0)} T) = -\cos(h^{(1)} T)$. Then, $h^{(1)} T = h^{(0)} T + \pi$, which leads to $h^{(1)} = h^{(0)} + \frac{\omega}{2}$. Finally, $E_+^{(0)} = E_+^{(1)} + \omega$ and $E_-^{(0)} = E_-^{(1)}$. As is well known, the Floquet quasi-energies are periodic with period $\omega$, and the total quasi-energies should be $E_{\pm}^{(n)} + l\omega$ with $l = 0, \pm1, \pm2, \dots$ Therefore, $E_{\pm}^{(0)}$ and $E_{\pm}^{(1)}$ are equivalent. From now on, we only consider $n=0$ in the $\mathcal{PT}$-symmetric phase. When $h$ is an imaginary number, $\cos(h T)>1$, it corresponds to the $\mathcal{PT}$ symmetry spontaneous breaking. There are two kinds of $\mathcal{PT}$ symmetry broken phases, and their effective Hamiltonians differ by a constant. For simplicity, we assign the right-hand side of Eq. (13) to $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$, namely, $$\begin{alignat}{1} &\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1) = \cos (h_{1}T_{1}) \cos ( h_{0}T_{0})\\ &~~~~ +\frac{1 }{4}(\gamma _{1}\gamma _{0}-{\varDelta}_{1} {\varDelta}_{0}) T_{0}T_{1}\mathrm{sinc}(h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0}).~~ \tag {17} \end{alignat} $$ If $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$ is greater than $1$, then $n=0$. If $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$ is less than $-1$, then $n=1$. The exceptional points correspond to $h=0$. From Eq. (13), we can easily find that the exceptional points occur when $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1) = \pm 1$, where $+$ ($-$) corresponds to $n=0$ ($1$). Unlike the static Hamiltonians $H_j$ whose eigenenergies can only be $0$ in the exceptional points, the quasi-energies of the driven two-level system can be either $0$ for $n=0$ or $\omega/2$ for $n=1$. Once the parameters ${\varDelta}_j$, $\gamma_j$, $T_j$ of the driving two-level systems are obtained, we can calculate $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$, from which one can determine whether the $\mathcal{PT}$ symmetry is broken or not. Next, we concentrate on multiphoton resonance. For the two-level system with square-wave modulated dissipation and time-independent coupling, the multiphoton resonance refers to the case in which the coupling strength $\varDelta$ of the two-level system is an integral multiple of the driving frequency $\omega$. A vanishingly small dissipation strength can lead to the $\mathcal{PT}$ symmetry spontaneous breaking in the $(2k-1)$-photon resonance case ($k=1,2\dots$), which has been found in the two-level system with sinusoidal[37] and square-wave[45] modulated dissipations. For the two-level system with a square-wave modulated coupling, one may naively think that the necessary condition for the weak dissipation induced $\mathcal{PT}$ symmetry breaking is that both ${\varDelta}_0$ and ${\varDelta}_1$ are integral multiples of $\omega$. However, it is not just the case. The $\mathcal{PT}$ phase transition induced by the weak dissipation in the multiphoton resonance indicates that $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$ deviates from $\pm 1$ once the dissipation occurs. We expect that the necessary condition is $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0=0, \gamma_1=0, T_0, T_1)=\pm 1$. From Eq. (17), we can reach $$\begin{align} &\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0=0, \gamma_1=0, T_0, T_1) \\ =& \cos \Big(\frac{{\varDelta}_{1}T_{1}}{2}\Big) \cos \Big( \frac{{\varDelta}_{0}T_{0}}{2}\Big)\! - \!\sin \Big(\frac{{\varDelta}_{1}T_{1}}{2}\Big) \sin\Big(\frac{{\varDelta}_{0}T_{0}}{2}\Big)\\ =& \cos\Big(\frac{{\varDelta}_{0}T_{0}+{\varDelta}_{1}T_{1}}{2}\Big)\\ =& \cos \Big(\frac{{\varDelta}_{\rm eff} T}{2}\Big), \end{align} $$ where $$\begin{align} {\varDelta}_{\rm eff}=\frac{{\varDelta}_0 T_0 + {\varDelta}_1 T_1}{T}.~~ \tag {18} \end{align} $$ Therefore, the necessary condition for the $\mathcal{PT}$ phase transition induced by the weak dissipation should be ${\varDelta}_{\rm eff}= k \omega$. In other words, the driving frequency should resonate with the effective coupling strength ${\varDelta}_{\rm eff}$, rather than ${\varDelta}_0$ or ${\varDelta}_1$. When $k$ is an even number, $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0=0, \gamma_1=0, T_0, T_1)=1$. A weak dissipation can lead to $\varPi>1$ which corresponds to the $\mathcal{PT}$ symmetry broken phase with $n=0$, or $\varPi < 1$ which corresponds to the $\mathcal{PT}$-symmetric phase. Similarly, when $k$ is an odd number, a weak dissipation can lead to $\varPi < -1$ which corresponds to the $\mathcal{PT}$ symmetry broken phase with $n=1$, or $\varPi>-1$ which corresponds to the $\mathcal{PT}$-symmetric phase. We firstly consider the two-level system with only square-wave modulated dissipation. The coupling strength is time-independent, namely ${\varDelta}_0={\varDelta}_1=\varDelta$, which leads to ${\varDelta}_{\rm eff}=\varDelta$. According to the former analysis, we expect that the $\mathcal{PT}$ phase transition at weak dissipation occurs when ${\varDelta} =k \omega$. However, Li et al. only showed the $\mathcal{PT}$-symmetry breaking in $(2k-1)$-photon resonance,[45] namely ${\varDelta}=(2k-1)\omega$. In Fig. 1(a), we recover the $\mathcal{PT}$ phase diagram near the one-photon resonance in Ref. [45], by setting $T_0 = T_1 = T / 2$, $\gamma_0=\gamma$ and $\gamma_1=0$. The boundary of the phase diagram can be determined by either $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$ (Fig. 1(b)) or the imaginary part of the quasi-energies (Fig. 1(c)). Near the one-photon resonance region, $\varPi$ is less than $-1$ and the imaginary part of the quasi-energies is nonzero, which indicates that it corresponds to a $\mathcal{PT}$ symmetry broken phase with $n=1$.
cpl-37-8-081101-fig1.png
Fig. 1. (a) $\mathcal{PT}$ phase diagram near the one-photon resonance, showing $\mathcal{PT}$-symmetric phase (grey), and $\mathcal{PT}$ symmetry broken phases with $n=1$ (black). (b) $\varPi$ (Eq. (17)) as a function of $\omega/\varDelta$ at $\gamma/{\varDelta}=0.2$. The dashed line represents $\varPi=-1$, below which corresponds to $\mathcal{PT}$ symmetry broken phase with $n=1$. (c) Real (black lines) and imaginary (red lines) parts of the quasi-energies as a function of $\omega/\varDelta$ at $\gamma/{\varDelta}=0.2$. The other parameters are ${\varDelta}_0={\varDelta}_1={\varDelta}=1$, $T_0=T_1=T/2$, $\gamma_0=\gamma$ and $\gamma_1=0$.
When we further decrease the driving frequency $\omega$ to the two-photon resonance region, we find that a weak dissipation can also lead to the $\mathcal{PT}$ symmetry breaking, which is not observed in Ref. [45]. As depicted in Fig. 2(a), the $\mathcal{PT}$ symmetry broken region is much narrower than that in the one-photon resonance case. In addition, the driving frequency $\omega$ at the phase boundary tends to decrease with increasing $\gamma$. Therefore, the $\mathcal{PT}$ symmetry breaking occurs at the region where $\omega$ is a bit less than $\varDelta / 2$. Near the two-photon resonance, $\varPi$ is greater than $1$ and the imaginary part of the quasi-energies is nonzero, which indicates that it corresponds to a $\mathcal{PT}$ symmetry broken phase with $n=0$. Figure 3(a) is a generalization of Figs. 1(a) and 2(a), which extends the range of $\omega$. The driving two-level system has a much richer phase diagram than the static one. Clearly, a vanishingly small dissipation strength can lead to the $\mathcal{PT}$ symmetry spontaneous breaking in both $(2k-1)$- and $2k$-photon resonances, which is consistent with our criteria ${\varDelta}= k \omega$. To explain the behavior of the $\mathcal{PT}$ symmetry breaking near the $2k$-photon resonance, we reexamine $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$ in Eq. (17) in more detail. We suppose that $\gamma_0 = \gamma \ll \varDelta$, $\gamma_1 = \lambda \gamma$, $T_0=T_1=T/2$, and $\varDelta \simeq 2k \omega$. When $\gamma$ tends to zero, $h_i T_i$ tends to $k \pi$. The first term on the right-hand side of Eq. (17) tends to one while the second term tends to zero. If the second term is greater than zero, it can lead to $\varPi>1$ and the $\mathcal{PT}$ symmetry broken phase with $n=0$. Since $\left(\gamma _{0}\gamma _{1}-\varDelta ^{2}\right) T_{0}T_{1}/4$ in the second term is less than zero, one needs that $\mathrm{sinc}(h_{1}T_{1}) \mathrm{sinc}(h_{0}T_{0}) < 0$, or $\sin(h_{1}T_{1}) \sin (h_{0}T_{0}) < 0$. Then, the condition for the occurrence of $\mathcal{PT}$ symmetry breaking is that one of $h_i T_i$ should be less than $k\pi$, while the other one should be greater than $k\pi$. If $\varDelta$ is a bit less than $2k\omega$, a finite $\gamma$ will always decrease $h_i$, which leads to both $h_i T_i < \varDelta T_i / 2 < k\pi$ and $\varPi < 1$. Therefore, no $\mathcal{PT}$ symmetry breaking occurs when $\varDelta < 2k \omega$. If $\varDelta$ is a bit larger than $2k\omega$, one can always find certain $\gamma$ which satisfies the condition for the occurrence of $\mathcal{PT}$ symmetry breaking, as long as $\lambda\ne\pm1$. Figure 3(a) corresponds to $\lambda=0$. Therefore, a finite $\gamma$ can lead to the $\mathcal{PT}$ symmetry breaking near $2k$-photon resonance.
cpl-37-8-081101-fig2.png
Fig. 2. (a) $\mathcal{PT}$ phase diagram near the two-photon resonance, showing $\mathcal{PT}$-symmetric phase (grey), and $\mathcal{PT}$ symmetry broken phase with $n=0$ (white). (b) $\varPi$ (Eq. (17)) as a function of $\omega/\varDelta$ at $\gamma/{\varDelta}=0.2$. The dashed line represents $\varPi=1$, above which corresponds to $\mathcal{PT}$ symmetry broken phase with $n=0$. (c) Real (black lines) and imaginary (red lines) parts of the quasi-energies as a function of $\omega/\varDelta$ at $\gamma/{\varDelta}=0.2$. The other parameters are ${\varDelta}_0={\varDelta}_1={\varDelta}=1$, $T_0=T_1=T/2$, $\gamma_0=\gamma$ and $\gamma_1=0$.
When $\lambda=+1$, namely $\gamma_0=\gamma_1$, the Hamiltonian (1) becomes time-independent, which is trivial. When $\lambda=-1$, namely $\gamma_0=-\gamma_1=\gamma$, $h_0$ will equal $h_1. h_1 T_1 = h_0 T_0$ if $T_0=T_1$, which leads to the $\mathcal{PT}$-symmetric phase with $\varPi < 1$ near the $2k$-photon resonance, as shown in Fig. 3(b). Following the above analysis, we can easily prove that an imbalanced driving time $T_0 \ne T_1$ can lead to the $\mathcal{PT}$ symmetry breaking when $\gamma_0=-\gamma_1$, as depicted in Fig. 3(c). The $\mathcal{PT}$ symmetry breaking near $2k$-photon resonance induced by the imbalanced driving time $T_0 \ne T_1$ is more obvious than that induced by $\gamma_0 \ne -\gamma_1$, when the dissipation strength is very weak. Therefore, the imbalanced driving time $T_0 \ne T_1$ is a more efficient method to access the $\mathcal{PT}$ symmetry breaking near $2k$-photon resonance in the experiments. Figures 3(a) and 3(c) verify our conclusion that the $\mathcal{PT}$ symmetry breaking induced by weak dissipation generally occurs at both $2k$- and $(2k-1)$-photon resonances, namely ${\varDelta}=k\omega$. The $\mathcal{PT}$ symmetry breaking at $2k$-photon resonance disappears only if $\gamma_0=-\gamma_1$ and $T_0=T_1$, as shown in Fig. 3(c).
cpl-37-8-081101-fig3.png
Fig. 3. $\mathcal{PT}$ phase diagram for time-periodic dissipation near the multiphoton resonance, showing $\mathcal{PT}$-symmetric phase (grey), and $\mathcal{PT}$ symmetry broken phases with $n=0$ (white) and $n=1$ (black): (a) $\gamma_0=\gamma$, $\gamma_1=0$, $T_0=T_1$; (b) $\gamma_0=-\gamma_1=\gamma$, $T_0=T_1$; (c) $\gamma_0=-\gamma_1=\gamma$, $T_0=0.55T$, $T_1=0.45T$.
For the two-level system with only square-wave modulated coupling, the dissipation strength is time-independent, namely $\gamma_0=\gamma_1=\gamma$. Figure 4 shows the $\mathcal{PT}$ phase diagram for time-periodic coupling near the multiphoton resonance. Li et al. studied the influence of the time-periodic coupling on the non-Hermitian two-level system based on a simpler model with ${\varDelta}_0=\varDelta$, ${\varDelta}_1=0$ and $T_0=T_1=T/2$,[45] which corresponds to Fig. 4(a). They concluded that the $\mathcal{PT}$ phase transition induced by the weak dissipation occurs at ${\varDelta} = 2k \omega$, which is consistent with our results ${\varDelta}_{\rm eff}=k \omega$ due to ${\varDelta}_{\rm eff}=\varDelta/2$. Figure 4(b) introduces a nonzero ${\varDelta}_1$ and imbalanced driving time $T_0 \ne T_1$, which cannot be explained by Ref. [45]. However, ${\varDelta}_{\rm eff}=k \omega$ can still provide the right condition under which the $\mathcal{PT}$ phase transitions occur. The $\mathcal{PT}$ symmetry broken phase with $n=0 $ ($1$) occurs when $k$ is even (odd), which is also consistent with the above analysis.
cpl-37-8-081101-fig4.png
Fig. 4. $\mathcal{PT}$ phase diagram for time-periodic coupling near the multiphoton resonance, showing $\mathcal{PT}$-symmetric phase (grey), and $\mathcal{PT}$ symmetry broken phases with $n=0$ (white) and $n=1$ (black). (a) ${\varDelta}_0=1$, ${\varDelta}_1=0$, $T_0=0.5T$. (b) ${\varDelta}_0=1$, ${\varDelta}_1=-0.2$, $T_0=0.55T$.
If the driving frequency is very large, namely $\omega \gg {\varDelta}_j, \gamma_j$, the period $T$ tends to zero. We suppose that $T_0$ and $T_1$ are of the same order as $T$. Expanding $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$ to the second order of $T$, we obtain $$\begin{align} &\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1) \\ \simeq& \Big(1 - \frac{h_{1}^2T_{1}^2}{2}\Big) \Big( 1 - \frac{h_{0}^2T_{0}^2}{2}\Big) +\frac{1 }{4}(\gamma _{1}\gamma _{0}-{\varDelta}_{1} {\varDelta}_{0}) T_{1}T_{0} \\ \simeq& 1 + \frac{1 }{4}[(\gamma _{1}\gamma _{0}-{\varDelta}_{1} {\varDelta}_{0}) T_{1}T_{0} - 2 h_{0}^2T_{0}^2 - 2 h_{1}^2T_{1}^2 ]\\ ={}& 1 + \frac{1}{8} [ (\gamma_0 T_0 + \gamma_1 T_1)^2 - ({\varDelta}_0 T_0 + {\varDelta}_1 T_1)^2 ]\\ ={}& 1 + \frac{1}{8} (\gamma_{\rm eff}^2 - {\varDelta}_{\rm eff}^2) T^2,~~ \tag {19} \end{align} $$ where $$\begin{align} \gamma_{\rm eff}=\frac{\gamma_0 T_0 + \gamma_1 T_1}{T}.~~ \tag {20} \end{align} $$ Therefore, the exceptional points, as well as the $\mathcal{PT}$ phase boundary, are located at $\gamma_{\rm eff}=\pm {\varDelta}_{\rm eff}$. If $|\gamma_{\rm eff}| < |{\varDelta}_{\rm eff}|$, it will correspond to the $\mathcal{PT}$-symmetric phase. Otherwise, the $\mathcal{PT}$ symmetry will be broken with $n=0$. Alternately, if we expand Eqs. (14)-(16) to the lowest order of $T$, we can find $$\begin{align} J \simeq {\varDelta}_{\rm eff},\quad \varGamma_y \simeq 0, \quad \varGamma_z \simeq \gamma_{\rm eff},~~ \tag {21} \end{align} $$ which give rise to the effective Hamiltonian as follows: $$\begin{align} H_{\rm eff} \simeq \frac{{\varDelta}_{\rm eff} }{2}\sigma _{x}+i\frac{\gamma_{\rm eff} }{2}\sigma _{z}.~~ \tag {22} \end{align} $$ It leads to the same $\mathcal{PT}$ phase boundary. In a word, we find that when the driving frequency is very large, the Floquet effective Hamiltonian is equivalent to a static one with time-averaged coupling and dissipation strength. When ${\varDelta}_0 \simeq -{\varDelta}_1$, ${\varDelta}_{\rm eff}$ tends to zero and one can easily achieve the $\mathcal{PT}$ symmetry broken phase no matter how large ${\varDelta}_j$ is. When $\gamma_0 \simeq -\gamma_1$, $\gamma_{\rm eff}$ tends to zero and one can easily preserve the $\mathcal{PT}$ symmetry no matter how large $\gamma_j$ is.
cpl-37-8-081101-fig5.png
Fig. 5. $\mathcal{PT}$ phase diagram, showing $\mathcal{PT}$-symmetric phase (grey), and $\mathcal{PT}$ symmetry broken phase with $n=0$ (white) at ${\varDelta}_0={\varDelta}_1={\varDelta}=1$, $\omega=3$, $T_0=0.4T$, $T_1=0.6T$. The red dashed lines refer to the analytical results in the high frequency limit.
Figure 5 shows the $\mathcal{PT}$ phase diagram at ${\varDelta}_0={\varDelta}_1=\varDelta$, $\omega/{\varDelta}=3$ and $T_0/T_1=2/3$. The phase boundary $\gamma_{\rm eff}=\pm {\varDelta}_{\rm eff}$ fits well with the exact results. In summary, we have studied a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Two time-independent Hamiltonians $H_0$ and $H_1$ appear alternately. Comparing with the formerly well-known sinusoidal modulation, the square-wave modulation has three advantages: firstly, exact analytical solutions can be achieved by employing the Floquet theory. Secondly, the $\mathcal{PT}$ phase diagram becomes richer. Thirdly, the square-wave modulation has been realized in the ultracold atoms experiment.[45] Based on the Floquet theory, we achieve an effective Hamiltonian with $\mathcal{PT}$ symmetry. We define a parameter $\varPi({\varDelta}_0, {\varDelta}_1, \gamma_0, \gamma_1, T_0, T_1)$, from which one can derive the boundaries of the $\mathcal{PT}$ phase diagram exactly. The driving two-level system has a much richer phase diagram than the static one. Two kinds of $\mathcal{PT}$ symmetry broken phases are found, whose effective Hamiltonians differ by a constant $\omega / 2$. When $\varPi>1$, the $\mathcal{PT}$ symmetry broken phase with $n=0$ occurs. When $\varPi < -1$, the $\mathcal{PT}$ symmetry broken phase with $n=1$ occurs. When $-1 < \varPi < 1$, the $\mathcal{PT}$ symmetry is preserved. With the help of $\varPi$, we firstly study the $\mathcal{PT}$ phase transition with only square-wave modulated dissipation near multiphoton resonance. The coupling strength is time-independent with ${\varDelta}_0={\varDelta}_1=\varDelta$. A weak dissipation can lead to the $\mathcal{PT}$ symmetry breaking near the $(2k-1)$-photon resonance (${\varDelta}=(2k-1)\omega$), which has been observed in the ultracold atom experiment.[45] We predict that the $\mathcal{PT}$ symmetry breaking near the $2k$-photon resonance (${\varDelta}=2k\omega$) can also happen as long as the dissipation strengths or the driving times are imbalanced, with $\gamma_0 \ne -\gamma_1$ or $T_0 \ne T_1$. Our studies pave a way to access the $\mathcal{PT}$ symmetry broken phase near the $2k$-photon resonance in the experiments. For the $\mathcal{PT}$ phase transition with square-wave modulated coupling, we define an effective coupling strength ${\varDelta}_{\rm eff}=\left({\varDelta}_0 T_0 + {\varDelta}_1 T_1\right)/T$. The weak dissipation induced $\mathcal{PT}$ symmetry breaking can occur only if ${\varDelta}_{\rm eff}=k\omega$. In the high frequency limit, we achieve a simple relation $\gamma_{\rm eff} = \pm {\varDelta}_{\rm eff}$, which gives the $\mathcal{PT}$ phase boundary. When ${\varDelta}_0 \simeq -{\varDelta}_1$, one can easily achieve the $\mathcal{PT}$ symmetry broken phase no matter how large the coupling strength $|{\varDelta}_j|$ is. When $\gamma_0 \simeq -\gamma_1$, one can easily preserve the $\mathcal{PT}$ symmetry no matter how large the dissipation strength $|\gamma_j|$ is. References Real Spectra in Non-Hermitian Hamiltonians Having P T SymmetryComplex Extension of Quantum MechanicsIntroduction to 𝒫𝒯-symmetric quantum theoryMaking sense of non-Hermitian HamiltoniansNonlinear waves in PT -symmetric systemsPhysical realization of -symmetric potential scattering in a planar slab waveguideTheory of coupled optical PT-symmetric structuresBeam Dynamics in P T Symmetric Optical LatticesOptical Solitons in P T Periodic PotentialsObservation of P T -Symmetry Breaking in Complex Optical PotentialsObservation of parity–time symmetry in opticsNonreciprocal Light Propagation in a Silicon Photonic CircuitParity–time synthetic photonic latticesParity-time-symmetric microring lasersPseudo-Parity-Time Symmetry in Optical SystemsTopologically protected bound states in photonic parity–time-symmetric crystalsObservation of Critical Phenomena in Parity-Time-Symmetric Quantum DynamicsObservation of the exceptional point in cavity magnon-polaritonsHigher-order exceptional point in a cavity magnonics systemResearch progress of hybrid cavity-magnon systemsAnomalous Edge State in a Non-Hermitian LatticeTopological Band Theory for Non-Hermitian HamiltoniansBiorthogonal Bulk-Boundary Correspondence in Non-Hermitian SystemsTopological Phases of Non-Hermitian SystemsTopological unification of time-reversal and particle-hole symmetries in non-Hermitian physicsSymmetry and Topology in Non-Hermitian PhysicsEdge Modes, Degeneracies, and Topological Numbers in Non-Hermitian SystemsSecond-Order Topological Phases in Non-Hermitian SystemsEdge States and Topological Invariants of Non-Hermitian SystemsNon-Hermitian Skin Effect and Chiral Damping in Open Quantum SystemsObservation of topological edge states in parity–time-symmetric quantum walksNon-Hermitian Chern BandsNon-Bloch Band Theory of Non-Hermitian SystemsA new way to construct topological invariants of non-Hermitian systems with the non-Hermitian skin effectTopological phases of a non-Hermitian coupled SSH ladder PT spectroscopy of the Rabi problem PT -symmetric Rabi model: Perturbation theoryNon-Hermitian Hamiltonian for a modulated Jaynes-Cummings model with PT symmetryCrossing rule for a PT -symmetric two-level time-periodic systemStabilizing non-Hermitian systems by periodic drivingPiecewise adiabatic following: General analysis and exactly solvable modelsExceptional points in a time-periodic parity-time-symmetric Rabi modelAnalytical results for a parity-time-symmetric two-level system under synchronous combined modulationsExperimental Realization of Floquet P T -Symmetric SystemsObservation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atomsDark Soliton of Polariton Condensates under Nonresonant ${ \mathcal P }{ \mathcal T }$-Symmetric PumpingEntanglement and Spin Squeezing in Non-Hermitian Phase TransitionsSolution of the Schrödinger Equation with a Hamiltonian Periodic in TimeSteady States and Quasienergies of a Quantum-Mechanical System in an Oscillating FieldFloquet control of quantum dissipation in spin chainsGenerating many Majorana modes via periodic driving: A superconductor modelTowards large-Chern-number topological phases by periodic quenching
[1] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243
[2] Bender C M, Brody D C and Jones H F 2002 Phys. Rev. Lett. 89 270401
[3] Bender C M 2005 Contemp. Phys. 46 277
[4] Bender C M 2007 Rep. Prog. Phys. 70 947
[5] Konotop V V, Yang J and Zezyulin D A 2016 Rev. Mod. Phys. 88 035002
[6] Ruschhaupt A, Delgado F and Muga J G 2005 J. Phys. A 38 L171
[7] El-Ganainy R, Makris K G, Christodoulides D N and Musslimani Z H 2007 Opt. Lett. 32 2632
[8] Makris K G, El-Ganainy R, Christodoulides D N and Musslimani Z H 2008 Phys. Rev. Lett. 100 103904
[9] Musslimani Z H, Makris K G, El-Ganainy R and Christodoulides D N 2008 Phys. Rev. Lett. 100 030402
[10] Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G A and Christodoulides D N 2009 Phys. Rev. Lett. 103 093902
[11] Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D 2010 Nat. Phys. 6 192
[12] Feng L, Ayache M, Huang J, Xu Y L, Lu M H, Chen Y F, Fainman Y and Scherer A 2011 Science 333 729
[13] Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167
[14] Hodaei H, Miri M A, Heinrich M, Christodoulides D N and Khajavikhan M 2014 Science 346 975
[15] Luo X, Huang J, Zhong H, Qin X, Xie Q, Kivshar Y S and Lee C 2013 Phys. Rev. Lett. 110 243902
[16] Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C and Szameit A 2017 Nat. Mater. 16 433
[17] Xiao L, Wang K, Zhan X, Bian Z, Kawabata K, Ueda M, Yi W and Xue P 2019 Phys. Rev. Lett. 123 230401
[18] Zhang D, Luo X Q, Wang Y P, Li T F and You J 2017 Nat. Commun. 8 1368
[19] Zhang G Q and You J Q 2019 Phys. Rev. B 99 054404
[20] Shen R C, Zhang G Q, Wang Y P and You J Q 2019 Acta Phys. Sin. 68 230305
[21] Lee T E 2016 Phys. Rev. Lett. 116 133903
[22] Shen H, Zhen B and Fu L 2018 Phys. Rev. Lett. 120 146402
[23] Kunst F K, Edvardsson E, Budich J C and Bergholtz E J 2018 Phys. Rev. Lett. 121 026808
[24] Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S and Ueda M 2018 Phys. Rev. X 8 031079
[25] Kawabata K, Higashikawa S, Gong Z, Ashida Y and Ueda M 2019 Nat. Commun. 10 297
[26] Kawabata K, Shiozaki K, Ueda M and Sato M 2019 Phys. Rev. X 9 041015
[27] Leykam D, Bliokh K Y, Huang C, Chong Y D and Nori F 2017 Phys. Rev. Lett. 118 040401
[28] Liu T, Zhang Y R, Ai Q, Gong Z, Kawabata K, Ueda M and Nori F 2019 Phys. Rev. Lett. 122 076801
[29] Yao S and Wang Z 2018 Phys. Rev. Lett. 121 086803
[30] Song F, Yao S and Wang Z 2019 Phys. Rev. Lett. 123 170401
[31] Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N et al. 2017 Nat. Phys. 13 1117
[32] Yao S, Song F and Wang Z 2018 Phys. Rev. Lett. 121 136802
[33] Yokomizo K and Murakami S 2019 Phys. Rev. Lett. 123 066404
[34] Liu J S, Han Y Z and Liu C S 2020 Chin. Phys. B 29 010302
[35] Liu J S, Han Y Z and Liu C S 2019 Chin. Phys. B 28 100304
[36] Joglekar Y N, Marathe R, Durganandini P and Pathak R K 2014 Phys. Rev. A 90 040101
[37] Lee T E and Joglekar Y N 2015 Phys. Rev. A 92 042103
[38] Bagarello F, Lattuca M, Passante R, Rizzuto L and Spagnolo S 2015 Phys. Rev. A 91 042134
[39] Moiseyev N 2011 Phys. Rev. A 83 052125
[40] Gong J and Wang Q H 2015 Phys. Rev. A 91 042135
[41] Gong J and Wang Q H 2019 Phys. Rev. A 99 012107
[42] Xie Q, Rong S and Liu X 2018 Phys. Rev. A 98 052122
[43] Luo X, Yang B, Zhang X, Li L and Yu X 2017 Phys. Rev. A 95 052128
[44] Chitsazi M, Li H, Ellis F M and Kottos T 2017 Phys. Rev. Lett. 119 093901
[45] Li J, Harter A K, Liu J, de Melo L, Joglekar Y N and Luo L 2019 Nat. Commun. 10 855
[46] Jia C Y and Liang Z X 2020 Chin. Phys. Lett. 37 040502
[47] Lee T E, Reiter F and Moiseyev N 2014 Phys. Rev. Lett. 113 250401
[48] Shirley J H 1965 Phys. Rev. 138 B979
[49] Sambe H 1973 Phys. Rev. A 7 2203
[50] Chen C, An J H, Luo H G, Sun C P and Oh C H 2015 Phys. Rev. A 91 052122
[51] Tong Q J, An J H, Gong J, Luo H G and Oh C H 2013 Phys. Rev. B 87 201109
[52] Xiong T S, Gong J and An J H 2016 Phys. Rev. B 93 184306