Super-exponential scrambling of Out-of-time-ordered correlators

Introduction.— Akin to quantum butterfly effects, quantum scrambling is the process of encoded information spreading from local degrees of freedom to multiple degrees of freedom, hence comes its unattainability by measuring the local operators Swingle19 ; Lewis19 . However, through the time evolution of out-of-time-ordered correlators (OTOCs)  Xu20 ; Prakash20 ; Sekino08 ; Landsman19 ; Rozenbaum20 , this elusive process can be well quantified, which explains the enthusiastic and extensive study of OTOCs in many frontiers of physics such as quantum holography, quantum chaos and black hole physics. Recent research has proved that OTOCs act as an effective indicator of quantum phase transition Dag19 ; Sahu19 ; Wangqian19 ; Heyl2018 , many-body localization Rammensee18 ; Ray18 ; Huang16 and quantum entanglement Garttner18 . Besides, experimental progress has made it possible to observe the OTOCs in atomic-optics setups Rammensee18 ; Ray18 and nuclear spins Li2017 . Up to now, a wide range of OTOCs with logarithmic, power-law or exponential growth (EG) has been found in various systems including many-body systems and quantum chaotic systems.

As proposed in landmark studies of quantum chaos, the OTOCs can be used to describe the exponential instability in quantum dynamics Hashimoto2017 ; Mata2018B ; Fortes19 ; Dora2017 ; Rozenbaum17 ; Rozenbaum19 ; Yan19 . As for systems with the well-defined classical limit, the EG will occur within Ehrenfest time with a rate determined by the classical Lyapuonv exponent Mata2018B ; Fortes19 ; Rozenbaum17 ; Lakshmi19 . Interestingly, the mathematically verified correlation between OTOCs and Loschmidt echo provides theoretical basis for the irreversibility of scrambling dynamics Zurek20 . Note that, the boundary set by chaos dynamics for exponential scrambling of OTOCs, which means that an exponent actually marks the greatest rate of increase for OTOCs in a quantum system, is obtained by the conjecture of thermalization in quantum systems with a large number of degrees of freedom Murthy19 ; Maldacena16 . At present, massive research efforts have been focused on how many-body chaos affects the dynamics of OTOCs. For instance, a recent study has shown that the OTOCs’ EG is equal to the classical Lyapunov exponent in the presence of interatomic interaction Mata2018 . However, we noticed that the growth rate of OTOCs in previous systems fails to break the exponential limit. Therefore, is it safe to say that the EG actually tops all OTOCs’ rates of growth in quantum systems?

Model and results.— We consider a Schrödinger system with the temporally modulated nonlinear interaction, the corresponding Hamiltonian reads Zhao16 ; Zhao19

where the angular momentum operator $p=-i\hbar_{\mathrm{eff}}\partial/\partial\theta$ with $\hbar_{\mathrm{eff}}$ being the effective Planck constant, $\theta$ is angle coordinate, $g$ denotes the nonlinear interaction strength. All variables are properly scaled and thus in dimensionless units. An arbitrary quantum state is expanded in terms of the complete basis of angular momentum operator ($p|\varphi_{n}\rangle=n\hbar_{\mathrm{eff}}|\varphi_{n}\rangle$), i.e., $|\psi\rangle=\sum_{n}\psi_{n}|\varphi_{n}\rangle$, hence, it is periodical in $\theta$, i.e., $\psi(\theta)=\psi(\theta+2\pi)$. The one-period evolution operator from time $t$ to $t+1$ is given by $U(t,t+1)=\exp\left(-i{p^{2}}/{2\hbar_{\mathrm{eff}}}\right)\exp\left[-i{g|\psi(\theta,t)|^{2}}/{\hbar_{\mathrm{eff}}}\right]$. The quantum OTOCs are defined as the average of the squared commutator, i.e., $C(t)=-\langle[\hat{A}(t),\hat{B}(0)]^{2}\rangle$, where $\hat{A}(t)=e^{iHt}A(0)e^{-iHt}$ is the time-dependent operator in Heisenberg picture, with $\langle\cdot\rangle$ being the average of the initial state. In the many-body systems, the average of $C(t)$ comes from thermal states. For the quantum mapping systems, however, there is no definition of thermal states, as the temperature tends to be infinitely large after long time evolution Rigol14 . Indeed, our previous investigations have proved that the system in Eq. (1) exhibits the unbounded heating, which is quantified by the exponentially-fast growth of mean energy $\langle p^{2}\rangle$ Zhao16 ; Zhao19 .

Here, we consider the case of a pure state, i.e., a Gaussian wavepacket $\psi(\theta,0)=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$. As in ref. Rozenbaum17 , $\hat{A}=\hat{B}=p$, namely

Our main result is the analytical prediction of the Super-EG scrambling of the quantum OTOCs

where

$\alpha$, $\beta$ and $\eta$ are prefactors and $\mathcal{N}_{0}$ is the normalization constant of the initial state (usually $\mathcal{N}_{0}=1$). Numerical results of OTOCs are in good agreement with the analytical expression (see Fig. 1). It is worth noting that EG is usually believed to be the boundary of the scrambling of OTOCs in chaotic systems Maldacena16 , therefore, our finding of the super-EG scrambling sheds new light in the field of quantum information Dressel18 ; Alonso19 . We have also investigated, both numerically and analytically, the OTOCs defined as $C(t)=-\langle[\theta(t),p]^{2}\rangle$. Interestingly, we found the scaling

where $\mu$ is the time-dependent coefficient, $N$ is number of basis, namely the dimension of the system, $\nu$ is a constant, and the growth rate reads

with $\tilde{\mathcal{N}}(t)=\langle\psi(t)|\theta^{2}|\psi(t)\rangle$ Supple . It is obvious that the $C(t)$ approaches to infinite with the increase of the dimension of the system $N$, which demonstrates that there is no bound on the OTOCs in our system.

The bound of the exponential growth of OTOCs applies to many-body systems with finite temperature where the thermal states are well defined. For such systems, the two-point correlator in OTOCs saturates rapidly. The exponential growth of OTOCs mainly results from the four-point correlator Maldacena16 . It has proved that the periodically-driven systems are equivalent to the systems with infinite temperature Rigol14 , for which the thermal states cannot be well defined without an effective Hamiltonian. These essential differences lead to the inconsistency between the super-EG of OTOCs in our system and the bound of chaos in ref. Maldacena16 . Remarkably, for periodically-driven systems, the two-point correlator contributes mainly to the OTOCs, and the four-point correlator saturates rapidly Rozenbaum17 ; Ueda2018 . In the following, we will show that the two-point correlator part exhibits the super-exponential growth with time. Here, the super-EG of OTOCs is rooted in the exponentially-fast diffusion of mean energy. It is a clear evidence that the scrambling dynamics is closely related to quantum thermalization, which is highlighted in such interdisciplinary topics as quantum information scrambling and quantum chaos. On the other hand, as a variant of kicked rotor model, our system is an ideal platform for investigating the wavepacet dynamics, such as quantum walk Dadras18 and topologically-protected transport Ho12 in momentum-space lattice. Therefore, our finding opens a new prospective in the field of the scrambling dynamics in momentum-space lattice.

We proceed to evaluate the scrambling in the semiclassical limit. While approaching to the semiclassical limit, the quantum commutator reduces to Possion bracket $[p(t),p(0)]=\hbar_{\mathrm{eff}}\{p(t),p(0)\}=\hbar_{\mathrm{eff}}{\partial p(t)}/{\partial x(0)}$. Then, a natural definition of the classical OTOCs are

where $\langle\cdot\rangle$ denotes the average of the ensemble of classical trajectories Rozenbaum17 . In numerical calculations, the classical OTOCs are approximated as $C_{cl}(t)\approx\langle(\delta p(t)/\delta x(0))^{2}\rangle$, where $\delta p$ and $\delta x$ denote the difference of the two nearest neighboring trajectories Rozenbaum17 .

We have proved that the system in Eq. (1) is mathematically equivalent to a generalized kicked rotor (GKR) model Zhao16 ; Zhao19 ; Supple , whose Hamiltonian takes the form

The kicking strength dependent on the Fourier components of the quantum state reads

The classical mapping equation of this GKR model can be formulated as

where $p(t)$ and $\theta(t)$ indicate the classical momentum and angle variables after the $t$-th kick.

Based on the classical mapping equations, we numerically investigate the classical OTOCs for different interaction $g$. In numerical simulations, we set the initial values of $p$ and $\theta$ as random variables with the probability distribution given by Gaussian function in phase space. The difference of initial trajectories is $\delta\theta(0)=10^{-5}$. Interestingly, the classical OTOCs increase in the super-EG way, i.e.,

which is in good agreement with our theoretical prediction (see Fig. 2(a)). In addition, we numerically investigate the maximal Lyapunov exponent $\lambda=\lim_{t\rightarrow\infty}\lim_{\delta\theta\rightarrow 0}\langle\log[\delta p(t)/\delta\theta(0)]\rangle/t$. Remarkably, the maximal Lyapunov exponent $\lambda$ linearly increases with time $\lambda(t)\propto\gamma t$ (see Fig. 2(a)), which is in consistence with the theoretical prediction in Eq. (18). This clearly demonstrates the existence of the hyper-chaotic dynamics Andreev19 . Different from previous research, in which the classical OTOCs depend on the $\lambda$ in an exponential way of $C_{cl}(t)\propto e^{\lambda t}$ Mata2018B ; Fortes19 ; Rozenbaum17 ; Lakshmi19 , here we get the super-EG with the form $C_{cl}(t)\propto e^{\gamma t^{2}}$. Compared with traditional quantum systems Guarneri17 ; Gisin95 , richer physics will be exhibited in the periodically-modulated nonlinear Schrödinger system from the perspective of either quantum dynamics or semi-classical dynamics.

Theoretical analysis.— It is straightforward to decompose the OTOCs in Eq. (2) as

where we define the terms in the right of the above equation as

Our numerical investigation shows that the contribution of the first term $C_{1}(t)$ is larger than the other two about several orders of magnitude, which means $C(t)\approx C_{1}(t)$ Ueda2018 . The two-point correlation function quantifies the quantum irreversibility measured by the expectation value of $p^{2}$ under the perturbation of $p$ at the time $t$. Hereinafter, we will show how the exponentially-fast diffusion of mean energy ($\langle p^{2}\rangle\propto e^{\gamma t}$) induces the super-EG of $C_{1}(t)$.

The definition of $C_{1}(t)$ in Eq. (13) means that there are four steps in the calculation of this part. The first step is the forward time evolution of an initial state from $t_{0}$($=0$) to $t_{+}=t_{0}+t$, which yields the quantum state $|\psi(t_{+})\rangle={U}(t)|\psi(0)\rangle$. In the second step, the operator $p$ is exerted on the state $|\psi(t_{+})\rangle$, i.e., $|\tilde{\psi}(t_{+})\rangle\equiv p|\psi(t_{+})\rangle={p}{U}(t)|\psi(0)\rangle$. The norm of this state has the expression $\mathcal{N}_{t_{+}}=\langle\tilde{\psi}(t_{+})|\tilde{\psi}(t_{+})\rangle=\langle\psi(0)|{U}^{\dagger}(t){p}^{2}{U}(t)|\psi(0)\rangle$, which is just the mean energy at the time $t_{+}$. Our previous investigation has shown that the mean energy exponentially increases Zhao16 ; Zhao19 , i.e.,

where we adopted the condition $\mathcal{N}_{0}=\langle\psi(0)|\psi(0)\rangle=1$. The third step is the time reversal from $t_{+}$ to $t_{0}$ ($t$ steps), which results in a state $|\varphi(t_{0})\rangle\equiv{U}^{\dagger}(t)|\tilde{\psi}(t_{+})\rangle={U}^{\dagger}(t)\hat{p}\hat{U}(t)|\psi(0)\rangle$. Finally, in the fourth step, one can obtain the expectation value of $p^{2}$ with the state $|\varphi(t_{0})\rangle$, i.e., $C_{1}(t)\approx\langle\varphi(t_{0})|p^{2}|\varphi(t_{0})\rangle$ $=\langle\tilde{\psi}(t_{+})|{U}(t)p^{2}{U}^{\dagger}(t)|\tilde{\psi}(t_{+})\rangle$.

Note that, the process of time reversal (the third step) can be viewed as a process starting from a new initial state $|\tilde{\psi}(t_{+})\rangle$ normalized to $\mathcal{N}_{t_{+}}$, and then evolving into a quantum state $|\varphi(t_{0})\rangle$ in a time interval $t$. The two-point correlator $C_{1}(t)$ is just the “mean energy” of the state $|\varphi(t_{0})\rangle$, which follows the exponentially-fast way

with the rate $\gamma_{t_{+}}=\ln\left[1+\left({g{{\cal{N}}_{t_{+}}}}/{\pi\hbar_{\mathrm{eff}}}\right)^{2}\right]$ Zhao16 ; Zhao19 . Taking Eq. (14) into account, the ${\gamma}_{t_{+}}$ has the expression

where we use the condition $e^{2\gamma t}\gg 1$. As a consequence, the OTOCs take the form

where the prefactors $\alpha$, $\beta$ and $\eta$ can not be exactly obtained, since in the process of derivation we have used approximations in Eqs. (14), (15) and (16). According to the above analysis, the super-EG of OTOCs is mainly caused by the positive feedback mechanism of the temporally-modulated interaction, which is the key point in this letter. The exponential increase of the kicking strength, which is absent in the traditional kicked rotor model, is responsible for the appearance of super-EG in the classical limits of the system.

Next, we analyze the time evolution of the classical OTOCs. For the kicked rotor model, the maximum Lyapunov depends on the kicking strength by the way of $\lambda\propto\ln(K)$ Chirikov79 ; Rozenbaum17 . In the GKR model, the definition of one of the components of the kicking strength $K_{n}$ in Eq. (9) indicates that the $K_{n}$ is the quantum correlation in momentum space. A significant feature of this system is the exponential localization of quantum states, i.e., $|\psi(p)|^{2}\sim\exp(-|p|/\xi)$ with $\xi$ being the localization length. Through simple calculation, one can obtain that the quantum correlation also has the exponential decay, i.e., $K_{n}\propto\exp(-|p|/\xi)$ Zhao16 ; Zhao19 . Then, a rough estimation of the kicking strength in the GKR model (see Eq. (10)) is $K(t)=\sum_{n}nK_{n}(t)\propto\sum_{n}ne^{-|n|\hbar_{\mathrm{eff}}/\xi}\propto\xi$. Previously, we have predicted the exponentially-fast increase of the localization length $\xi\propto e^{\gamma t}$ by means of the hybrid quantum-classical theory Zhao16 ; Zhao19 . As a consequence, the kick strength exponentially grows with time, i.e., $K\propto e^{\gamma t}$. Accordingly, the Lyapunov exponent linearly increases with time

For strong chaotic case, the classical OTOCs exponentially increase with the growth rate proportional to Lyapunov exponent Mata2018B ; Fortes19 ; Rozenbaum17 ; Lakshmi19 , i.e., $C_{cl}(t)\propto e^{\lambda t}$. Taking Eq. (18) into account, one can get $C_{cl}(t)\propto\exp(\gamma t^{2})$, which is confirmed by our numerical results (see Fig. 2).

Conclusion and prospects—We have proved the existence of OTOCs’ super-EG. The results not only give evidence to verify the association between quantum hyper-chaos and quantum scrambling, but also confirm the super-exponential sensitivity of quantum dynamics to initial conditions. Since the conventional theory has it that a EG boundary is imposed on the chaotic dynamics, our finding, by breaking traditional restraints, bears great significance in the research of information scrambling and quantum chaos Ray18 . More importantly, the super-EG is the fastest growth rate of OTOC today, for which, the underlying mechanism is the positive feedback of the periodic modulated interaction. Our finding helps review the issue of the chaotic systems’ boundary.

Experimentally, due to the generality of the nonlinear Schrödiner system, our findings will serve as a universal theory in broad fields including the cold atomic gases Dalfovo99 , nonlinear optics Szameit10 ; Longhi11 ; Blomer06 and complex Ginzburg-Landau equation in condensed-matter physics Anderson2002 . More remarkably, the interaction in the above-mentioned systems features high controllability Inouye98 ; Kevrekidis03 ; Theis04 ; Anderson2002 ; Chin10 ; Szameit10 ; Longhi11 ; Blomer06 , which will facilitate the experimental realization and observation of the predictions.