Extracting classical Lyapunov exponent from one-dimensional quantum mechanics

The commutator $[x(t),p]$ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. It reduces to a c-number and does not show any quantum fluctuations for arbitrary states. Related to this nature, the quantum Lyapunov exponent computed through the out-of-time-order correlator (OTOC) $\langle [x(t),p]^2 \rangle $ precisely agrees with the classical one. Hence, the OTOC may be regarded as an ideal indicator of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the commutator $[x(t),p]$ and the OTOCs might be seen in various situations, too. In order to clarify this point, as a first step, we investigate OTOCs in one-dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growth reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using a suitably localized wave packet near the peak, and the second one is taking a limit akin to the large-$N$ limit in the noncritical string theories.


Introduction
Inverted harmonic oscillators (IHOs) are ubiquitous in nature. If we drew a random potential on paper, we would see as many peaks as valleys. The valleys will be approximated by harmonic oscillators (HOs), and the peaks will be approximated by IHOs. Needless to say, HOs play indispensable roles in physics, particularly in stable systems. Correspondingly, IHOs play crucial roles in unstable systems.
These examples show some instabilities, and they may be quantified by Lyapunov exponents at the classical level. Recently, as a counterpart to this quantity in quantum mechanics, the out-of-time-order correlator (OTOC) [27] defined by has attracted attention [28,29,30]. Here, we use the Heisenberg picture, W and V are some operators in the system, and W (t) = e iHt/ W (0)e −iHt/ by using the Hamiltonian H. where λ is the Lyapunov exponent. Hence, OTOCs may quantify butterfly effects in quantum mechanics. 2 However, the relation between OTOCs and Lyapunov exponents is subtle. First, we assume the classical-quantum correspondence, and it does not work, in general. Second, even 1 Particle motion in an exact IHO (1.3) is integrable and not chaotic. However, in typical chaotic systems, the appearance of their chaotic behaviors is explained through motions near hyperbolic fixed points with broken homoclinic orbits [1], and the hyperbolic fixed points are approximately described by IHOs. Thus, IHOs capture several essential properties of chaos. This is one motivation of this work. 2 In this article, the terminology "butterfly effect" is used for the sensitive dependence of the initial condition, i.e., a finite positive Lyapunov exponent. Note that the sensitive dependence is usually related to some instabilities of the system, and it occurs even in nonchaotic systems.
if the classical-quantum correspondence is satisfied at the early stage of the time evolution, it may break down after the Ehrenfest time and the exponential development may not be observed after that. Thus, detecting exponential developments in quantum systems is harder than in the classical ones [30]. Hence, it is valuable to understand when we observe C(t) ∼ e 2λt in order to reveal properties of the OTOCs. Since IHOs also show butterfly effects, it is natural to investigate the OTOCs in IHOs in detail and study the application to the aforementioned systems.
In this article, for simplicity, we consider the IHO in one-dimensional quantum mechanics [31,32,33,34], Here, λ is the Lyapunov exponent, as we will see soon. The quantum Lyapunov exponent in the IHO has been computed by evaluating an OTOC in Ref. [31], and it exactly agrees with the classical one. Particularly, the results of Ref. [31] imply the following relation, Here, the left-hand side evaluates the OTOC in the IHO (1.3) for any normalizable quantum states, and the right-hand side is the Poisson bracket for any initial conditions in classical mechanics. 3 Since cosh(λt) ∼ e λt at a large t, this relation shows that the Lyapunov exponent of this system is λ in both classical and quantum mechanics, as Ref. [31] found. Note that this relation works for any time even after the Ehrenfest time, which is typically given by t ∼ 1 λ log 1 [30]. 4 Furthermore, this relation suggests that the commutator [x(t), p(0)] does not show any quantum fluctuations and the deviation is precisely zero. Thus, the commutator [x(t), p(0)] exhibits quite peculiar properties in the IHO, which cannot be seen in other observables, like x(t) and p(t). These results suggest that the commutator [x(t), p(0)] may be regarded as an ideal indicator of the butterfly effect in the IHO.
Then, it is natural to ask whether these remarkable properties of the OTOCs hold in more general situations. In order to understand this question, we study one-dimensional quantum mechanics with polynomial potentials [34,35]. In classical mechanics, the particle 3 In Eq. (1.4), the n = 1 case in the left-hand side may not be suitable to be called an OTOC. However, Eq. (1.4) shows the exponential development at large t and diagnoses the butterfly effect, and we loosely call it an OTOC in this article. 4 The Ehrenfest time t ∼ 1 λ log 1 is estimated as the timescale that a wave packet spreads over the curvature scale of the IHO. However, the domain of the IHO (1.3) is infinite (−∞ ≤ x ≤ ∞), and it may be reasonable that the naive Ehrenfest time does not work in our case. motions confined in the potentials are periodic and not chaotic. However, if the potential has a hill, the hill will be approximated by an IHO, and the system shows a butterfly effect near there. We discuss when the OTOCs reproduce this classical Lyapunov exponent of the hill in quantum mechanics. As is expected through the classical-quantum correspondence, we see that suitably localized wave packets correctly reproduce the Lyapunov exponent. In addition, if we take a limit similar to the large-N limit in the noncritical string theories [5,6,7,8], the correct Lyapunov exponent will be obtained through more general states such as energy eigenstates.
The organization of this article is as follows. In Sec. 2, we study the nature of the OTOCs in the IHO (1.3) in detail. In Sec. 3, we investigate the OTOCs in more general potential cases. In Sec. 4, we argue that one can understand our results in the → 0 limit in terms of classical mechanics. Section 5 contains conclusions and discussions.

No Quantum Fluctuation of the OTOC in the IHO
We prove the relation (1.4). We start from classical mechanics. The classical solution of the Hamiltonian (1.3) is given by where x(0) and p(0) are the initial conditions of the position x(t) and momentum p(t). Then, we can compute the Poisson bracket as and the second equality in the relation (1.4) is satisfied.
Next, we consider quantum mechanics. As Refs. [31,33] pointed out, we obtain through the Hadamard lemma, and it leads to Since this quantity is a c-number, the relation (1.4) is satisfied for any normalizable states. . We take λ = 1 and put the infinite potential walls at x = ±2. We prepare the Gaussian wave packets with (∆x) 2 = (∆p) 2 centered at (x, p) = (−1, 1.3) at t = 0 and evaluate their time evolutions in the quantum mechanics with = 1/10, 1/25, 1/50, and 1/100. We also compute the corresponding quantities for a single classical particle; they are depicted by the black dashed lines. Note that x(t) shows that the wave packets hit the potential wall at x = 2 around t ∼ 2.5. All the OTOCs agree very well until the hits, and they are independent of . Thus, the relation (1.4) works as long as we ignore the effect of the potential walls. Note that if we replace λ with iω, (ω ∈ R) in the Hamiltonian (1.3), we obtain a similar relation for the harmonic oscillator (HO), Actually, we can derive this relation directly from the relation m|[x(t), p(0)]|n = i δ mn cos ωt, which we can easily obtain by solving the HO through the standard method [36].

OTOC in General Potentials
So far, we have seen that the quantum fluctuations of the OTOCs in the IHO and HO are exactly zero. This is because x(t) in Eq. (2.1) is linear in x(0) and p(0), and it will not be true in the general potential V (x). On the other hand, if the potential V (x) has a hill (valley), the region near the hill (valley) will be approximated by the IHO (HO), and the quantum fluctuations of the OTOCs will be suppressed. Particularly, a classical particle near the hill will show a butterfly effect, and the Lyapunov exponent is computed from the curvature of the potential as Then, one question is whether one can obtain λ saddle through the OTOCs without using the localized wave packets. Particularly, energy eigenstates are a useful basis of the Hilbert space, and it is natural to try to evaluate the OTOCs for these states. However, energy eigenstates generally do not represent a localized particle in the position space, and obtaining λ saddle from them seems nontrivial. In order to test it, we numerically evaluate the OTOCs Although the exponent in the V (x) = −ax 2 + bx 8 case is close to 2λ saddle , it is significantly smaller in the V (x) = −ax 2 + bx 4 case. 7 In addition, the results must depend on the energy level. For example, if the energy is close to the ground state, the particles are localized near 6 We have attempted to evaluate the OTOCs of the energy eigenstates in the IHO with the infinite potential walls, which we have used in the numerical study of the wave packets in Fig. 1. However, we found that the convergence of numerical computations was not good, and we could not obtain reliable results. We presume that the infinite potential walls are problematic. Actually, if we evaluate {x(t), p(0)} 2 of a classical particle in an infinite potential well (without the IHO potential) and take an average over the initial position so that it corresponds to the semiclassical energy eigenstate, we can easily see that {x(t), p(0)} 2 diverges. This divergence will be resolved in quantum mechanics, but it may cause larger numerical errors. 7 In the V (x) = −ax 2 + bx 4 case, the exponential development of [x(t), p(0)] 2 is roughly [x(t), p(0)] 2 ∼ exp (λ saddle t) = exp (2λ saddle t). Similar behaviors have been observed in other models too [32], and Ref. [32] argued that the OTOCs may be suppressed by exp (−λ saddle t) in thermal ensembles. the bottom of the potential, and we would observe the cos-type behaviors as in (2.5) rather than exponential developments.
Actually, if we tune the energy level and such that the OTOCs for the energy eigenstate show the exponential developments (see Fig. 3). We observe that both the OTOCs [x(t), p(0)] and [x(t), p(0)] 2 show the exponential growth with the Lyapunov exponent λ saddle as we take → 0. This limit is related to the doublescaling limit in the noncritical string theories [5,6,7,8], and we call this energy E cr the . We employ the same data in Fig. 2 except the energies. In quantum mechanics, since we cannot take E = E cr exactly, we choose the closest one with E − E cr > 0, and we take E = 0.0001 in classical mechanics, correspondingly. We observe that all OTOCs show the exponential growth with the Lyapunov exponent λ saddle as → 0, and the relation (1.4) is approximately satisfied.
critical energy.
This result can be explained as follows. In the Wentzel-Kramers-Brillouin (WKB) approximation, the probability density of the energy eigenfunction ρ is proportional to the inverse of the classical momentum, ρ ∝ 1/|p|. At E = E cr = V (x saddle ), the momentum near and the density ρ shows a divergence Therefore, when we evaluate observables, the contribution of the saddle points would domi-nate. 8 However, quantum corrections make this divergence milder. By combining these two effects, the OTOCs exhibit behaviors similar to the IHO near the saddle point as E → E cr and → 0. Note that, if we prepare some states that are constructed from energy eigenstates whose energies are close to E cl , their OTOCs may also show the exponential growth. 9

Understanding OTOCs from Classical Mechanics
As we can see in Figs First, we consider the derivation of the Poisson bracket {x(t), p(0)} n , which is the counterpart of the OTOC [x(t), p(0)] n . In classical mechanics, the energy eigenstate in quantum mechanics can be approximated by using the particles uniformly distributed on the constant energy curve in phase space (see Fig. 4). Then, physical quantities for the energy eigenstate can be computed by taking the averages of the quantities for each particle. Hence, to obtain the OTOC, we need to compute {x(t), p(0)} n for single particles and take their average.
We can compute the Poisson bracket {x(t), p(0)} for a single particle as follows. Suppose that a particle starts from (x, p) = (x(0), p(0)) at t = 0, and we define the position of this particle at time t as x(t, x(0), p(0)). Then, we can compute x(t, x(0) + ∆x, p(0)) − x(t, x(0), p(0)) ∆x . (4.1) To evaluate the right-hand side of this equation, we need to compute the positions of the two particles x(t, x(0) + ∆x, p(0)) and x(t, x(0), p(0)) (see Fig. 4). Generally, we cannot obtain the particle positions explicitly, and we evaluate them numerically and extrapolate the limit ∆x → 0. Let us consider the motions of the two particles in (4.1) in the potential V (x) = −ax 2 + bx 4 . Each particle periodically moves in phase space. The period depends on the energy of the particle, and it becomes longer as E → E cr . (Actually, it diverges at E = E cr , and its motion is not periodic anymore.) Thus, for example, in the case of the two particles plotted in Fig. 4, the period of the left particle is longer, and, when the left particle returns to its original position, the right particle moves slightly ahead of its original position. Hence, {x(t), p(0)} is almost periodic but progressively increases for each period (see Fig. 5). In addition, since the two particle motions are almost periodic, {x(t), p(0)} has to be zero at least twice for each period, and the sign of {x(t), p(0)} changes when it crosses these zero points. (Otherwise, the two particles could not return to their original positions.) In the case of the HO (2.5), t = π/2ω and 3π/2ω correspond to the zero points.
Besides, when the particles pass near the hill of the potential, they show exponential growth similar to (2.1), and {x(t), p(0)} also develops exponentially. On the other hand, when the particles move in the potential valleys, their motions are like oscillators and {x(t), p(0)} will show a cos-type behavior similar to (2.5). In this way, we can roughly explain the time evolution of {x(t), p(0)} in Fig. 5.
So far, we have discussed the properties of {x(t), p(0)} for a single particle. Now, we argue {x(t), p(0)} for the energy eigenstate by taking an average of these single particle results, and we explain the behaviors shown in Fig. 2 in quantum mechanics. We first discuss why the Lyapunov exponents are smaller than λ saddle . When we take the average, the maximum of {x(t), p(0)} dominates. As we can see in Fig. 5, the maximum appears after the exponential developments terminate, and there, the growth is much slow. Hence, the Lyapunov exponents of the energy eigenstates are always smaller than λ saddle . 10 Next, we discuss why we do not observe clear exponential developments in [x(t), p(0)] . This is because {x(t), p(0)}, for single particles, take positive and negative values, which may cancel each other out when we take the average. This is very different from the OTOC [x(t), p(0)] for the wave packets, where we have seen clear exponential developments as in Fig. 1. On the other hand, at the critical energy E = E cr , the particles near the saddle points dominate, and such cancellations are suppressed. Hence, we observe the exponential growth as in Fig. 3.

Discussions
We have studied the OTOC [x(t), p(0)] n in the IHO. They have a peculiar property of not showing any quantum fluctuations independent of the quantum states. This suggests that the OTOCs can be regarded as ideal indicators of the butterfly effect in the IHO. Hence, we expect that the IHO may work as a starting point of perturbative calculations in unstable systems and chaos, and the Lyapunov exponents of the systems may be extracted through the OTOCs. (The situation may be analogous to HOs in stable systems.) Indeed, we have seen that we can derive the Lyapunov exponents of the saddle points by preparing sufficiently localized wave packets or by employing an energy eigenstate and taking the limit E → E cr and → 0 in one-dimensional quantum mechanics. We have also shown that the properties of the OTOCs of the energy eigenstates can be understood through classical mechanics.
In addition, as we mentioned in the Introduction, IHOs have a wide range of applications from condensed matter systems to quantum gravity and string theories. Thus, it is interesting to study the implications that the OTOCs do not receive any quantum corrections in these systems.