Scaling laws of the out-of-time-order correlators at the transition to the spontaneous $\cal{PT}$-symmetry breaking in a Floquet system

The Hamiltonian of the PTKR model in dimensionless unites reads

where the kicking potential $V(\theta)=K\left[\cos(\theta)+i\lambda\sin(\theta)\right]$ satisfies the $\cal{PT}$-symmetric condition $V(\theta)=V^{*}(-\theta)$ Longhi2017 ; West2010 ; Zhao19 . The parameters $K$ and $\lambda$ indicate the strength of the real and imaginary parts of the kick potential, respectively. The $p=-i\hbar_{\text{eff}}{\partial}/{\partial\theta}$ is angular momentum operator, $\theta$ is the angle coordinate, and $\hbar_{\text{eff}}$ denotes the effective planck constant. The time $t_{n}(=0,1,2\ldots)$ is integer, indicating kicking numbers. The eigenequation of angular momentum operator is $p|\varphi_{n}\rangle=n\hbar_{\text{eff}}|\varphi_{n}\rangle$ with eigenstate $\langle\theta|\varphi_{n}\rangle=e^{in\theta}/\sqrt{2\pi}$ and eigenvalue $p_{n}=n\hbar_{\text{eff}}$. On the basis of $|\varphi_{n}\rangle$, an arbitrary quantum state can be expanded as $|\psi\rangle=\sum_{n}\psi_{n}|\varphi_{n}\rangle$.

For time-periodic systems, i.e., $\textrm{H}(t+T)=\textrm{H}(t)$, the Floquet theory predicts the eigenequation of the evolution operator $U|\psi_{\varepsilon}\rangle=e^{-i\varepsilon}|\psi_{\varepsilon}\rangle$, where the eigenphase $\varepsilon$ is referred to as quasienergy. One-period time evolution of a quantum state of the PTKR system is given by $|\psi(t_{j+1}\rangle=U|\psi(t_{j}\rangle$ with the Floquet operator

This demonstrates that in numerical simulations, one period evolution is split into two steps, namely the kicking evolution $U_{K}$ and the free evolution $U_{f}$. The kick evolution is realized in angle coordinate space, i.e., $\psi^{\prime}(\theta)=U_{K}(\theta)\psi(\theta,t_{j})$. Then, one can utilize the fast Fourier transform to change the state $\psi^{\prime}(\theta)$ to angular momentum space, thereby obtaining its component $\psi^{\prime}_{n}$ on the eigenstate $|n\rangle$. Finally, the free evolution is conducted in angular momentum space, i.e., $\psi_{n}(t_{j+1})=U_{f}(p_{n})\psi^{\prime}_{n}$. By repeating the same procedure, one can get the quantum state at arbitrary time Casati79 .

It is straightforward to prove that the Floquet operator of the PTKR satisfies the $\cal{PT}$ symmetry $U=(\mathcal{P}\mathcal{T})^{\dagger}U\mathcal{P}\mathcal{T}$, where $\cal{P}$ and $\cal{T}$ are the parity and time reversal operators, respectively. Based on conventional understanding of quantum mechanics, one asserts that the two operators, i.e., $U$ and $\cal{PT}$ have simultaneous eigenstates, that is to say, the quasieigenstate $|\psi_{\varepsilon}\rangle$ is also the eigenstate of the $\cal{PT}$ operator, i.e., $\cal{PT}|\psi_{\varepsilon}\rangle=\pm|\psi_{\varepsilon}\rangle$. This conclusion is indeed valid for positive quasienergies $\varepsilon>0$. However, a notable feature of the PTKR system is that complex quasienergies $\varepsilon=\varepsilon_{r}\pm\varepsilon_{i}$ emerge when the strength of the imaginary part of the complex potential exceeds a threshold value, i.e., $\lambda>\lambda_{c}$ Longhi2017 ; West2010 ; Zhao19 . The threshold value $\lambda_{c}$ is just the exceptional points of the system. It can be proven that the quasieigenstate $|\psi_{\varepsilon}\rangle$ is no longer an eigenstate of the $\cal{PT}$ operator due to the complex quasienergies, thus demonstrating the spontaneous $\mathcal{PT}$-symmetry breaking. An intrinsic quality of the PTKR system is that $\cal{PT}$ symmetry is helpful in protecting the real spectrum of the Floquet operator.

We assume that the initial state is expanded as $|\psi(t_{0})\rangle=\sum_{\varepsilon}\rho_{\varepsilon}|\psi_{\varepsilon}\rangle$. Then, after the $n$th kick, the quantum state has the expression $|\psi(t_{n})\rangle=U^{t_{n}}|\psi(t_{0})\rangle=\sum_{\varepsilon}\rho_{\varepsilon}e^{-i\varepsilon_{r}t_{n}}e^{\varepsilon_{i}t_{n}}|\psi_{\varepsilon}\rangle$, whose norm $\mathcal{N}=\langle\psi(t_{n})|\psi(t_{n})\rangle$ exponentially increases with time due to positive $\varepsilon_{i}$. We numerically investigate the time evolution of $\cal{N}$ for different $\lambda$. Without loss of generality, we choose a Gaussian wavepacket, i.e., $\psi{(\theta,t_{0})}=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$ with $\sigma=10$ as the initial state in numerical simulations. Figure 1(a) shows that for very small $\lambda$ (e.g., $\lambda=0.01$ and 0.05), the value of $\mathcal{N}$ equals almost to unity with time evolution, which implies that quasienergies are all real. Interestingly, for sufficiently large $\lambda$ (e.g., $\lambda=0.15$), $\mathcal{N}$ increases exponentially with time, i.e., $\mathcal{N}=e^{\mu t}$, and the growth rate $\mu$ increases with the increase of $\lambda$. The non-unitary feature of the Floquet operator, $U_{K}=\exp[K\lambda\sin(\theta)/\hbar]$, leads to the growth of the norm. A rough estimation of the norm yields a time dependence of the form $\mathcal{N}\propto\exp(K\lambda t/\hbar)$, indicating the relation of the growth rate $\mu\propto\lambda$, which is confirmed by our numerical results [see inset in Figure 1(a)]. We further investigate the long-time average value of the norm, $\bar{\mathcal{N}}=\sum_{n=1}^{N}\mathcal{N}(t_{n})/N$, for a wide range of $\lambda$. Figure 1(b) shows that, for a specific $\hbar_{\text{eff}}$ (e.g., $\hbar_{\text{eff}}=0.1$), $\bar{\mathcal{N}}$ remains at unity for $\lambda$ smaller than a threshold value $\lambda_{c}$, beyond which it monotonically increases with $\lambda$. It is reasonable to believe that the threshold value $\lambda_{c}$ corresponds to the emergence of spontaneous $\cal{PT}$-symmetry breaking.

Recently, the long short-term memory (LSTM) network has been exploited to extract the character of time series and thus to predict the phase diagram of quantum diffusion Mano21 . Based on the character of the time evolution of $\mathcal{N}$, we conducted supervised training on the LSTM network and used it to evaluate the feature of $\mathcal{N}(t)$, namely, whether $\mathcal{N}(t)=e^{\mu t}$ or not, for different system parameters. This highly effective machine learning method outputs the probability $\rho$ of the time series $\mathcal{N}(t)$ to be exponentially increasing or not, which can predict the phase diagram of spontaneous $\mathcal{PT}$-symmetry breaking. Interestingly, our results show that the $\rho$ increases with the increase of both $K$ and $\lambda$ [see Fig. 1(c)]. We identify two phases in the parameters space $(K,\lambda)$, the boundary of which is clearly visible in Fig. 1(c). We further investigate the $\lambda_{c}$ for different $K$ and $\hbar_{\text{eff}}$. Our results demonstrate that the critical parameter $\lambda_{c}$ increases with the increase of $\hbar_{\text{eff}}$ and decreases with the increase of $K$ [see Fig. 1(d)]. This behavior is rooted in the fact that the mean spacing level $\Delta$ of the quasienergies of the quantum kicked rotor (QKR) model is proportional to $\hbar/K$ West2010 . The smaller the $\Delta$ is, the easier it is for the non-Hermitian parameter $\lambda$ to cause the coalescence of two quasienergies, implying the relation $\lambda_{c}\propto\hbar/K$.

We numerically investigate the time evolution of the three parts of the OTOCs, i.e., $C_{1}$, $C_{2}$, and $C_{3}$ for $\lambda<\lambda_{c}$. Figure 3(a) shows that the time dependence of $C_{1}$ and $C$ almost overlap, displaying rapid growth up to saturation. Since the real part of $C_{3}$, i.e., $\textrm{Re}(C_{3})$ fluctuates between positive and negative values, we plot the absolute value $|\textrm{Re}(C_{3})|$ in Fig. 3(b). Both $C_{2}$ and $|\textrm{Re}[C_{3}]|$ saturate after a very short time evolution. Importantly, $C_{1}$ is at least 4 orders of magnitude larger than both $C_{2}$ and $|\textrm{Re}[C_{3}]|$, leading to a perfect consistency between $C_{1}$ and $C$. Consequently, based on Eq. (3), we can safely use the approximation

where $\langle p^{2}(t_{0})\rangle_{R}=\langle\psi_{R}(t_{0})|{p}^{2}|\psi_{R}(t_{0})\rangle/\mathcal{N}_{\psi_{R}}(t_{0})$ denotes the exceptional value of energy of the state $|\psi_{R}(t_{0})\rangle$ divided by its norm $\mathcal{N}_{\psi_{R}}(t_{0})=\langle\psi_{R}(t_{0})|\psi_{R}(t_{0})\rangle$.

The normalization procedure for time reversal yields the equivalence $\mathcal{N}_{\psi_{R}}(t_{0})=\mathcal{N}_{\tilde{\psi}}(t_{n})=\langle\psi(t_{n})|p^{2}|\psi(t_{n})\rangle$, which shows that the value of $\mathcal{N}_{\psi_{R}}(t_{0})$ is just the mean energy of the state $|\psi(t_{n})\rangle$ at the time $t=t_{n}$. For $\lambda<\lambda_{c}$, the quasienergies are all real, thus the dynamics of the PTKR is the same as that of the Hermitian QKR. A noteworthy characteristic of the QKR’s energy diffusion is the phenomenon of DL, i.e. the mean energy $\langle p^{2}\rangle$ gradually approaches to saturation level with increasing time due to quantum coherence. It is reasonable to believe that the mechanism of DL suppresses the growth of both $\mathcal{N}_{\psi_{R}}(t_{0})$ and $\langle p^{2}(t_{0})\rangle_{R}$, and therefore leads to the saturation of $C(t_{n})$.

To confirm this conjecture, we consider a specific time, i.e., $t=t_{n}$, and numerically trace the evolution of $\langle p^{2}\rangle$ for both the forward ($t<t_{n}$) and backward ($t>t_{n}$) evolution. Figure 4(a) shows that for $t_{n}=2500$, $\langle p^{2}\rangle$ increases rapidly to saturation during forward time evolution from $t_{0}$ to $t_{2500}$, then jumps to a specific value at the start of the time reversal (i.e., at $t=t_{2500}$) before finally saturating for the backward evolution from $t_{2500}$ to $t_{0}$. This clearly demonstrates the emergence of the DL, which is also reflected by the probability density distribution in momentum space. We compare the momentum distributions at the end of the forward evolution (i.e., $t=t_{2500}$) and the end of time reversal (i.e., $t=t_{0}$) in Fig. 4(b). One can see that the two quantum states almost overlap with each other, both of which are exponentially localized in momentum space, i.e., $|\psi(t_{n})|^{2}\sim\exp(-|p|/L)$ [see Fig. 4(b)]. A rough estimation yields $\mathcal{N}_{\psi_{R}}(t_{0})=\langle\psi(t_{n})|p^{2}|\psi(t_{n})\rangle\sim L^{2}$ and $\langle p^{2}(t_{0})\rangle_{R}\sim\int_{-\infty}^{\infty}p^{2}\exp(-|p|/L)dp\sim L^{2}$. Plugging the two relations into Eq. (7), we can immediately get the estimation of the OTOCs, i.e., $C(t_{n})\sim L^{4}$. It is known that the localization length is in a quadratic function of $K$, i.e., $L\propto K^{2}$ Izrailev90 , which results in the relation

This clearly demonstrates that the $C$ is time-independent after the long term evolution, verifying our numerical results in Fig. 2(a) and Fig. 3(a).

To provide evidence of our analytical prediction, we investigate the time-averaged value of OTOCs, i.e., $\bar{C}=\sum_{j=1}^{N}C(t_{j})/N$, numerically for different $K$. In the numerical simulations, we ensure that $N$ is large enough for the long-term saturation of $C(t)$ to be well quantified by $\bar{C}$. Our numerical results show that for a specific $\hbar$, $\bar{C}$ increases in a power-law of $K$ [see Fig. 3(c)], which is well described by our theoretical prediction in Eq. (8). This is a strong indication of the validity of our analytical analysis. Our findings of the dependence of the OTOCs on the kick strength provide an opportunity to control the operator growth with an external driven potential.

The discontinuous jump in the mean square momentum, $\langle p^{2}\rangle$, at $t=t_{2500}$, the beginning of time reversal, is due to the action of the operator $p$ on the quantum state $|\psi(t_{2500})\rangle$. This action generates the quantum state $|\tilde{\psi}(t_{2500})\rangle=p|\psi(t_{2500})\rangle$, for which the mean value is given by $\langle\tilde{\psi}(t_{2500})|p^{2}|\tilde{\psi}(t_{2500})\rangle=\langle\psi(t_{2500})|p^{4}|\psi(t_{2500})\rangle$. The exponentially-localized shape of the quantum state, $|\psi(t_{2500})|^{2}\sim\exp(-|p|/L)$ [see Fig. 4(b)], allows us to obtain the expectation values $\langle\psi(t_{2500})|p^{2}|\psi(t_{2500})\rangle\sim L^{2}\hbar_{\text{eff}}^{2}$ and $\langle\tilde{\psi}(t_{2500})|p^{2}|\tilde{\psi}(t_{2500})\rangle=\langle\psi(t_{2500})|p^{4}|\psi(t_{2500})\rangle\sim L^{4}\hbar_{\text{eff}}^{4}$, which quantitatively explains the discontinuous increase in the mean energy from $L^{2}\hbar_{\text{eff}}^{2}$ to $L^{4}\hbar_{\text{eff}}^{4}$.

Figure 5(a) shows the time evolution of $C_{1}$, $C_{2}$, $|\textrm{Re}(C_{3})|$, and $C$ for $\lambda$ just larger than the $\cal{PT}$-symmetry phase transition point $\lambda\gtrsim\lambda_{c}$. It is clear that the time dependence of $C_{1}$ corresponds perfectly to that of $C$, both of which increase following the SQG $C\propto t^{3.4}$. The time evolution of $C_{2}$ displays the QG, i.e., $C_{2}(t)\propto t^{2}$, while $|\textrm{Re}(C_{3})|$ follows the SQG $|\textrm{Re}(C_{3})|\sim t^{2.9}$. In addition, one can see that $C_{1}$ is larger than both $C_{2}$ and $|\textrm{Re}(C_{3})|$ by approximately four orders of magnitude. Therefore, it is sufficient to analyze the time evolution of the term $C_{1}$ to uncover the mechanism of the SQG of $C$.

Since the value of $C(t)$ at a specific time $t=t_{n}$ is dependent on both the mean energy $\langle p^{2}(t_{0})\rangle_{R}$ and the norm $\mathcal{N}_{\psi_{R}}(t_{0})$ of the state $|\psi_{R}(t_{0})\rangle$ at the end of time reversal [see Eq. (7)], we numerically calculate the forward and backward time evolution of $\langle p^{2}\rangle$, $\langle p\rangle$, and $\mathcal{N}$ with a fixed $t_{n}$ (e.g., $t_{n}=2500$ in Fig. 6). Figure 6(a) demonstrates that the mean energy diffuses ballistically with time $\langle p^{2}\rangle\approx\gamma^{2}t^{2}$ during the forward evolution $t_{0}\rightarrow t_{2500}$, and it displays the intrinsic time reversal during $t_{2500}\rightarrow t_{0}$. Meanwhile, the mean momentum $\langle p\rangle$ linearly increases for $t_{0}<t<t_{2500}$, and linearly decays for $t_{2500}\rightarrow t_{0}$ [see Fig. 6(b)]. Moreover, Fig. 6(a) reveals that the norm remains unity, i.e., $\mathcal{N}(t)=1$, during the forward evolution and equals the mean energy at the time $t=t_{2500}$, i.e., $\mathcal{N}_{\psi_{R}}(t_{j})=\langle p^{2}(t_{2500})\rangle$ during the time reversal. Taking the ballistic diffusion of energy into account, the following equivalence can be derived

In order to measure the degree of time reversal for a fixed $t_{n}$, we define the ratio of mean energy between forward ($t<t_{n}$) and backward ($t>t_{n}$) time evolution as

where $\langle p^{2}(2t_{n}-t_{j})\rangle_{R}$ and $\langle p^{2}(t_{j})\rangle$ ($0\leq j\leq n$) denote the mean square of momentum for the forward evolution and time reversal, respectively. The inset in Fig. 6(b) shows that $\mathcal{R}$ is very large (i.e., $\mathcal{R}\gtrsim 10^{4}$ for $t=t_{0}$) and approaches almost one with time evolution. This reveals that the mean energy at the end of time reversal is much greater than that at the initial time, i.e., $\langle p^{2}(t_{0})\rangle_{R}\gg\langle p^{2}(t_{0})\rangle$. We further investigate the time evolution of $\langle p^{2}(t_{0})\rangle_{R}$, and find [see Fig. 6(c)] that the $\langle p^{2}(t_{0})\rangle_{R}$ increases in the power-law of time

Substituting Eqs. (9) and (11) into Eq. (7) yields the SQG of OTOCs

Figure 7 shows the probability density distribution of the state at forward $|\psi(t_{j})\rangle$ and backward $|\psi_{R}(t_{j})\rangle$ evolution in both the real and momentum space. The initial state is a Gaussian wavepacket $\psi{(\theta,t_{0})}=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$ centered at $\theta=0$ and $p=0$ [see Figs. 7(a) and (b)]. Interestingly, one can observe that the quantum state is mainly distributed in the region $0<\theta<\pi$ for both the forward and backward evolution. This is due to the fact that the action of the Floquet operator of the kicking term $U_{K}(\theta)=\exp[K\lambda\sin(\theta)/\hbar_{\text{eff}}]\exp[-iK\cos(\theta)/\hbar_{\text{eff}}]$ on a quantum state, i.e., $U_{K}(\theta)\psi(\theta)$ helps to amplify the state within the region $0<\theta<\pi$ as $K\lambda\sin(\theta)>0$. Assuming that the real part of the kicking potential provides the driven force $F=K\sin(\theta)$, the PTKR experiences a positive magnitude force $F>0$ during the forward evolution, thus the momentum grows with time, as shown in Fig. 6(b). For the time reversal, the sign of kick strength $K$ flips, i.e., $K\rightarrow-K$, so the mean momentum decreases with time evolution. Our conjecture is supported by the numerical results of momentum distributions. Figures 7(b), (d) and (f) show that the wavepacket, like a soliton, moves to the positive direction in momentum space for $t_{0}\rightarrow t_{2500}$, resulting in $\langle p\rangle=\gamma t$ [in Fig. 6(b)], and moves back to the opposite direction for $t_{2500}\rightarrow t_{0}$. In addition, the width of the wavepacket in momentum space is so narrow that one can safely use the approximation $\langle p^{2}\rangle\sim(\langle p\rangle)^{2}$, which is verified by our numerical results in Figs. 6(a).

We numerically investigate the time evolution of $C_{1}$, $C_{2}$, $C_{3}$, and $C$ for $\lambda\gg\lambda_{c}$. As shown in Fig. 5(b), all of them increase in the way of QG (i.e., $\propto t^{2}$). We use the ratios $\Delta_{12}=C_{1}/C_{2}$ and $\Delta_{13}=C_{1}/|\textrm{Re}[C_{3}]|$ to quantify the differences among $C_{1}$, $C_{2}$, and $|\textrm{Re}[C_{3}]|$. Our investigation show that both of them are larger than one, specifically $\Delta_{12}\approx 2.5$ and $\Delta_{13}\approx 22$ [see the inset in Fig. 5]. This suggests that $C_{1}$ contributes mainly to the $C$, which is verified by the good agreement between $C_{1}$ and $C$ [see Fig. 5(b)]. To reveal the mechanism of the QG of $C$, we proceed to analyze the time evolution of $C_{1}$ by thoroughly investigating both the forward and backward evolution of the mean values $\langle p^{2}\rangle$, $\langle p\rangle$, and the norm $\cal{N}$ for a given $t_{n}$.

Figure 8(a) shows that the mean energy exhibits ballistic diffusion $\langle p^{2}\rangle\approx\gamma^{2}t^{2}$, during the forward evolution from $t_{0}$ to $t_{n}$, and decays as the inverse of a quadratic function, with $\langle p^{2}\rangle\propto t^{-2}$, during the backward evolution from $t_{n}$ to $t_{0}$. This decay is symmetric with respect to the $\langle p^{2}\rangle$ of $t<t_{n}$. The dynamics of the mean momentum also exhibits perfect time reversal, namely it linearly increases as $\langle p\rangle=\gamma t$ during $t_{0}\rightarrow t_{2500}$ and decreases linearly during $t_{2500}\rightarrow t_{0}$. The ratio $\mathcal{R}$ remains close to one throughout the time evolution, except at the end, i.e., $\mathcal{R}(t=2500)\approx 2.5$ [see the inset in Fig. 8(b)], providing a clear evidence of time reversal. For the forward evolution from $t_{0}$ to $t_{2500}$, the norm $\mathcal{N}(t_{j})$ is equal to unity, while for the interval $t_{2500}\rightarrow t_{0}$, it is equal to the value of $\langle p^{2}(t_{2500})\rangle$, i.e., $\mathcal{N}_{\psi_{R}}(t_{j})=\langle p^{2}(t_{2500})\rangle$ [see Fig. 8(a)]. By utilizing the ballistic diffusion of mean energy, we establish the relationship $\mathcal{N}_{\psi_{R}}(t_{0})\approx\mathcal{N}_{\psi_{R}}(t_{n})\approx\gamma^{2}t_{n}^{2}$, where $t_{n}$ is an arbitrary time. We further evaluate the behavior of $\langle p^{2}(t_{0})\rangle_{R}$ for different $t_{n}$. As shown in Fig. 8(c), the $\langle p^{2}(t_{0})\rangle_{R}$ remains almost constant at a value of one, indicating that it is independent of time. Plugging in the values of $\mathcal{N}_{\psi_{R}}(t_{0})$ and $\langle p^{2}(t_{0})\rangle_{R}$ into Eq. (7), we obtain the QG of OTOCs

The time reversal of a wavepacket’s dynamics is clearly seen in the evolution of its momentum distributions. For the forward time evolution, the quantum state is localized at the point $\theta_{c}=\pi/2$ [seen in Figs. 9(a), (c) and (e)], which is the result of the localization-effect of the imaginary part of the Floquet operator $U_{K}(\theta)$. With the wavepacket mimicking a classical particle, it experiences a kicking force of magnitude $F=K\sin(\theta_{c})=K$, resulting in a constant acceleration of momentum $\Delta p=K$, which is reflected in the linear growth of momentum. This phenomenon of the directed current is also seen in the propagation of momentum distributions in Figs. 9(b), (d) and (f), where a soliton can be observed moving unidirectionally towards the positive direction in momentum space.

During the backward evolution, the wavepacket of the real space remains centered at $\theta_{c}=\pi/2$, with a width much smaller than the corresponding state at the time of forward evolution [see Figs. 9(a), (c) and (e)]. As the particle is exposed to the kicking force with $F=-K$ during time reversal, its momentum decreases linearly in time, which is also reflected in the propagation of the wavepackets in momentum space [Figs. 9(b), (d) and (f)]. It is evident that the $|\psi(t_{n})|^{2}$ is in perfect overlap with the $|\psi_{R}(t_{n})|^{2}$, apart from the initial state [see Figs. 9(b)]. The width of $|\psi_{R}(t_{0})|^{2}$ is considerably larger than that of $|\psi(t_{0})|^{2}$, leading to the ratio of energy being larger than one, i.e., $\mathcal{R}=\langle p^{2}(t_{5000})\rangle_{R}/\langle p^{2}(t_{0})\rangle\approx 2.5$ [see the inset in Fig. 8(b)].