1. Introduction
The correspondence principle proposed by the Copenhagen School of Quantum Mechanics states that the dynamics of quantum systems should converge towards classical dynamics when the quantum number of the system approaches infinity. The correspondence between classical and quantum Lyapunov exponents in many-body quantum chaotic systems is one of the important bases for testing the correspondence principle. The Ehrenfest time $\tau =\tfrac{1}{\tilde{\lambda }}\mathrm{ln}N$, where $\tilde{\lambda }$ is the exponential growth rate (EGR) of the out-of-time order correlator (OTOC) and N is the particle number of the system [1], gives the time scale for which the center of the wave packet will follow classical dynamics when this wave packet moves in a smoothly varying potential [2]. In other words, the correspondence principle remains valid in this time scale.
Recently, the exponential growth behavior of the OTOC has been regarded as an indicator of chaos or quantum instability in many quantum systems [3–17]. In particular, its growth rate before the Ehrenfest time is closely associated with the classic Lyapunov exponent (CLE) λ in the large quantum number limit [13–19]. Therefore, OTOCs are not only used to diagnose quantum chaos but also to test the correspondence principle. In the context of black hole physics, an OTOC can be considered to as an indicator of chaos in gravity dual theory [20] and the corresponding upper bound of the maximum quantum Lyapunov exponent is obtained in black hole geometry [1, 3, 21]. Experimentally, the OTOC has been measured in ion trap systems [22–25] and nuclear magnetic resonance platforms [26–28].
However, an important discovery is that the OTOC in some non-chaotic regular systems also exhibits exponential growth behavior in early times, and fast scrambling emerges not only in the chaotic case but also in the regular one [29, 30]. In integrable systems it is shown that OTOCs in the vicinity of saddle points could grow exponentially since quantum instability exists at saddle points [31]. This means that quantum instability is not only possible but is the rule for generic states in the vicinity of the saddle point. These results are not surprising, because the instability of the saddle point causes its CLE to be positive. Similar behavior of the OTOC also appears in an inverted harmonic oscillator (IHO) system with a Higgs potential [32]. This further indicates that quantum instability in regular systems exists at the saddle points and their surrounding regions. For orbits which are far from the saddle point, quantum instability disappears. Then, it is natural to ask whether unstable regular orbits cause the OTOCs to exhibit exponential growth behavior before the Ehrenfest time.
To study the early behavior of OTOCs with initial states located on unstable regular orbits, we consider the simplest unstable but regular system—the IHO system—described by the Hamiltonian $\hat{H}={p}^{2}/2m-m{\omega }^{2}{q}^{2}/2$ [33]. Various aspects of this special system have been widely studied. In [34], the authors, based on their description of a complete generalized Airy function-type quantum wave solution for an IHO, noted that the dynamics of a quantum wave packet is dependent upon its initial position. Moreover, the saddle point of this regular system has an exponential sensitivity to initial conditions, as in chaotic systems [35]. The phase-space volume of a classical IHO system is unbounded; however, its corresponding volume for quantum inverted oscillators can be bounded by the system photons [36, 37]. The IHO system is not just a pure theoretical model—it has been realized experimentally [38]. In mathematics, it even challenges the Riemann hypothesis [39]. Moreover, it also shows important significance in general relativity, quantum mechanics and the chaotic domain [40–46].
Fidelity OTOCs (FOTOCs) are related to the quantum variance of Hermitian operators and provide a method for us to visualize the chaotic or scrambling dynamics of quantum systems in semi-classical phase space. Recently, their growth rate before the Ehrenfest time was shown to be associated with the CLE of chaotic systems [14, 16] and the quantum instability of saddle points in non-chaotic systems [31]. In the Schrödinger picture, the evolution of quantum wave packets in phase space is directly related to the quantum variance of momentum or the coordinate operator. On the other hand, the Husimi Q function is a quasi-probability distribution function that has been used to visualize the evolution of quantum wave packets in phase space [47–51]. In this paper, we use OTOCs to study the quantum instability of the IHO system and to see how Husimi quasi-probability wave packets behave as OTOCs grow exponentially. In addition, we also discuss the differences in Ehrenfest time between stable and unstable manifolds in an IHO system.
This paper is organized as follows. In section 2 , we study the time evolution of mean photon number for different initial states in a quantum IHO system. In section 3 , we analyze the early behavior of OTOCs for unstable orbits in an IHO system and visualize the OTOCs using the Husimi Q function. Finally, we present our results and a brief summary.
2. Mean photon number in an inverted harmonic oscillator
In this section, we study a classical one-dimensional IHO system with unstable regular orbits, namely,1 ) there is a saddle point O(q = 0, p = 0) in phase space. The particles on the asymptote (p = − q) form a so-called stable manifold, since these points move to the saddle point under the action of Hamiltonian (1 ). The particles that start from other positions move to infinity, forming an unstable manifold. It should be noted that all orbits in the IHO system are unstable, and the Lyapunov exponents are the same as the saddle point, i.e. λO = 1. In this paper we set m = ω = 1.
$\begin{eqnarray}H=\displaystyle \frac{{p}^{2}}{2m}+V,\qquad V=-\displaystyle \frac{1}{2}m{\omega }^{2}{q}^{2},\end{eqnarray}$
where ω and m are the frequency and mass of the IHO, respectively. In contrast to harmonic oscillators, the classical potential V tends to minus infinity in the limit q → ∞ . In figure 1 we present the classical dynamics phase diagram of the IHO system. For the classical Hamiltonian (
Figure 1. The classical dynamics phase diagram of an IHO. The coordinate (q, p) of points O, A, B, C, D and E are (0, 0), (5, − 5), (3, 3), (–4.267, 5.643), (–3, 3) and (–4.267, –5.643), respectively. Point O is the saddle point. |
To study the quantum dynamics of an IHO, we introduce the quadratic quantization form of the position and momentum operators and set ℏ = 1,
$\begin{eqnarray}\hat{q}=({a}^{\dagger }+a)/\sqrt{2},\qquad \hat{p}={\rm{i}}({a}^{\dagger }-a)/\sqrt{2},\end{eqnarray}$
then the quantum Hamiltonian of the IHO becomes $\begin{eqnarray}\hat{H}=-\displaystyle \frac{1}{2}({a}^{2}+{a}^{\dagger 2}).\end{eqnarray}$
Note that Hamiltonian (3 ) is equivalent to Hamiltonian (1 ) in the classical limit when the system photon number Np → ∞ . As in [48, 52], we take the photon coherent state as the initial state of the system6 ) implies that the mean photon number is related to the central position of the coherent state in phase space. In other words, if the photon number of the system Np is given, the coordinate and momentum parameters of coherent states in phase space are bounded in the region (q and2 + p2)/2 ≤ Np. This shows that there is a natural boundary in phase space of a quantum IHO system with finite photon number, while the corresponding classical IHO system is unbounded.
$\begin{eqnarray}| \psi \rangle ={{\rm{e}}}^{-\beta {\beta }^{* }/2}{{\rm{e}}}^{\beta {a}^{\dagger }}| 0\rangle ,\end{eqnarray}$
with $\begin{eqnarray}\beta =(q+{\rm{i}}p)/\sqrt{2},\end{eqnarray}$
where ∣0⟩ is the ground state of the light field q and p are generalized coordinates and momentum. According to quantum theory, the mean photon number on the initial coherent state is $\begin{eqnarray}\langle {a}^{\dagger }a\rangle =\langle \psi | {a}^{\dagger }a| \psi \rangle =\beta {\beta }^{\dagger }=({q}^{2}+{p}^{2})/2.\end{eqnarray}$
Equation (In figure 2 we show the time evolutions of mean photon number ⟨a†a⟩ for initial coherent states centered at points A and B with fixed system photon number. It is seen that for a starting point located in the stable manifold of the IHO, such as point A, the mean photon number ⟨a†a⟩ first decreases and then increases to the maximum value, as shown in figure 2(a). For points far away from the saddle point under the action of a Hamiltonian, such as point B, the mean photon number grows to the maximum value directly, as shown in figure 2(b). Wherever the center of the initial wave packets is located, the mean photon number ⟨a†a⟩ increases to infinity in the classical limit case Np → ∞ . For the case with finite initial photon number Np, the curve of mean photon number changing with time first overlaps that in the classical limit. After a time tp, the the evolutionary behavior of mean photon number is no longer consistent with that in the classical limit case. This means that during the period 0 < t ≤ tp, there exists so-called classical–quantum correspondence for the system (1 ) because as t > tp the quantum behavior in the system differs from that in the classical system. We also note that the time for the mean photon number to remain consistent with the classical case increases with increase in the system photon number. This indicates that the time tp increases with the initial photon number Np, which is understandable because the classical–quantum correspondence becomes clearer in systems with larger photon numbers. Moreover, the time tp for maintaining classical–quantum correspondence is less than the time in which the mean number of photon increases to its maximum value. Strictly speaking, the time to maintain the classical–quantum correspondence in an IHO system is not determined by the time for the mean photon number to evolve to its maximum.
Figure 2. Time evolution of the mean photon number for points A (a) and B (b) in figure 1 with different system photon numbers. |
Comparing figures 2(a) and 2(b), for fixed initial photon number Np, we find that the time to maintain the classical–quantum correspondence for the initial coherent state centered at point A is longer than that at point B. This implies that the time for which the classical–quantum correspondence is maintained in the quantum IHO depends on the central positions of the initial coherent states of system in phase space.
3. Exponential growth of the OTOC in an inverted harmonic oscillator
In this section, we study the quantum variance derived from the OTOC in an IHO system and analyze the Husimi quasi-probability wave packet during exponential OTOC growth. The OTOC is defined as [53]
$\begin{eqnarray}C(t)=\langle {[\hat{W}(t),\hat{V}(0)]}^{\dagger }[\hat{W}(t),\hat{V}(0)]\rangle ,\end{eqnarray}$
where ⟨...⟩ denotes the expectation values and $\hat{W}(t)\,={{\rm{e}}}^{{\rm{i}}\hat{H}t}\hat{W}{{\rm{e}}}^{-{\rm{i}}\hat{H}t}$. $\hat{H}$ is a quantum Hamiltonian and $\hat{W}$ and $\hat{V}$ are two arbitrary local operators.Here, we choose $\hat{W}=\hat{P}={\rm{i}}({a}^{\dagger }-a)/\sqrt{2}$ and $\hat{V}$ as a projection operator onto the initial state ∣ψ⟩, i.e. $\hat{V}=| \psi \rangle \langle \psi | $. Substituting the operators $\hat{V}=| \psi \rangle \langle \psi | $ and $\hat{W}=\hat{P}$ into equation (7 ), one has
$\begin{eqnarray}C(t)=\langle \psi (t)| {\hat{P}}^{2}| \psi (t)\rangle -\langle \psi (t)| \hat{P}| \psi (t){\rangle }^{2}\end{eqnarray}$
$\begin{eqnarray}\equiv {\rm{Var}}[\hat{P}(t)],\end{eqnarray}$
where ${\rm{Var}}[\hat{P}(t)]$ is the quantum variance of the momentum operator $\hat{P}$. This relation is similar to the definition of FOTOC [14] and enables us to visualize the OTOC in semi-classical phase space. Whether the initial states are centered at points A or B, the corresponding OTOCs grow exponentially and the duration of the exponential behavior becomes longer with increase in the system photon number, as is clearly seen in figures 3(a) and 3(b). Moreover, we find that when the system photon number is fixed, the time for OTOCs to maintain exponential behavior for initial states centered at point A is longer than that for initial states centered at point B. Figure 3 further indicates that the time to maintain classical–quantum correspondence in a quantum IHO depends not only on the initial system photon number but also on the central positions of the initial states in phase space.
Figure 3. Time evolution of the OTOC for point A (a) and B (b) in figure 1 with different system photon numbers. |
For initial states centered at points in the stable manifold, the evolution of OTOCs is completely consistent before the Ehrenfest time, and the EGRs are twice the CLE of the saddle point, i.e. $\tilde{\lambda }=2{\lambda }_{O}$, as shown in figure 4(a). Therefore, for initial states located on the stable manifold, the Ehrenfest time is $\tau =\tfrac{1}{\tilde{\lambda }}\mathrm{ln}{N}_{p}\approx 2.8$ when Np = 300. In figure 4(b), we find that OTOCs with initial states located on an unstable manifold have the same EGRs as the saddle point, yet the time for OTOCs to maintain exponential growth is related to the position of the initial states. Moreover, comparing figures 4(a) and 4(b), it is evident that the Ehrenfest time for initial states centered in the stable manifold are longest. These results illustrate that the Ehrenfest time in the IHO system depends not only on the initial system photon number but also on the central positions of the initial states in phase space.
Figure 4. Time evolution of the OTOC for initial coherent states centered at different points. The gray dashed line in the left panel corresponds to the exponential growth rate given by twice the classical Lyapunov exponent. Here we set the system photon number Np = 300. |
The Husimi Q function is a quasi-probability distribution function that has been used to distinguish chaotic and periodic orbits in quantum systems [47–52]. The Husimi Q function is defined as
$\begin{eqnarray}Q({q}_{1},{p}_{1})=\displaystyle \frac{1}{\pi }\langle {q}_{1},{p}_{1}| \rho | {q}_{1},{p}_{1}\rangle ,\end{eqnarray}$
where ∣q1, p1⟩ is the photon coherent state and ρ is the density matrix. In figure 5 we present the time evolution of the Husimi Q function with different initial states in the phase space of an IHO system. Figures 5(a)–(c) and 5(e)–(g) show that the evolution of quasi-probability wave packets for the initial coherent states centered at points in the stable manifold has similar behaviors during exponential growth of the OTOCs. This means that the evolution of OTOCs with initial states centered at points on the stable manifold are highly consistent before the Ehrenfest time, as shown in figure 4(a). In figures 5(i)–(l) we show the evolution of a Husimi quasi-probabilistic wave packet with initial state centered at point B, which is located on an unstable manifold. It is found that the time for the wave packet reduces due to behaviors similar to that on a stable manifold. During exponential growth of the OTOCs it is easy to see that in the IHO system the corresponding quantum wave packets spread along the phase trajectory in classical phase space. In particular we note that the evolution of quasi-probability wave packets is identical except for the position of exponential growth of the OTOCs, as shown in figures 5(b), 5(f) and 5(k). This means that whether the initial states are centered at points located on stable or unstable manifolds, the evolutionary behaviors of OTOCs before the Ehrenfest time are consistent, as shown in figure 4. In addition, we present the Husimi quasi-probabilistic wave packets after exponential growth of the OTOC in figures 5(d), 5(h) and 5(l) for points O, A and B, respectively. For initial states located on a stable manifold, when the time exceeds the corresponding Ehrenfest time some discrete wave packets appear in the upper left and lower right corners of the phase space, as seen in figures 5(d) and 5(h). These evolutionary behaviors of the Husimi Q function are significantly different from that before the Ehrenfest time. This indicates that the classical–quantum correspondence has been broken. For initial states located on the unstable manifold, discrete wave packets appear earlier in the upper left and lower right corners of the phase space, as seen in figure 5(l). This means that the initial wave packets located in the unstable manifold reach the boundary determined by the initial system photon number more quickly. Therefore, the classical–quantum correspondence in the unstable manifold breaks earlier and the Ehrenfest time is shorter than in a stable manifold. Figure 5 provides an intuitive explanation of the behaviors of the OTOCs in figure 4 and why the Ehrenfest time in quantum IHO systems is related to their initial positions.
Figure 5. Change of Husimi quasi-probabilistic wave packets with time for initial system photon number Np = 300. The top, middle and bottom panels denote, respectively, the case in which the initial coherent state is centered at points O, A and B in figure 1. |
4. Conclusion
In this paper we show that both the system photon number and the central position of the initial coherent states affect the Ehrenfest time, and the Ehrenfest times for initial states located on the stable manifold of an IHO system are longest. Moreover, the EGRs of OTOCs at any position in the IHO system are twice the CLE of the saddle point. This illustrates that quantum instability exists at arbitrarily orbits in an IHO system. For starting points located in the stable manifold of an IHO, the mean photon number first decreases and then increases to a maximum value, and the evolution of OTOCs is completely consistent before the Ehrenfest time. For points far away from the saddle point under the action of an IHO Hamiltonian, the mean photon number grows to the maximum value directly and the time for which the OTOCs maintain exponential growth depends on the central positions of the initial wave packets. Moreover, we analyze the Husimi quasi-probability wave packets of different initial states, and find that during the exponential growth of the OTOCs the evolution of quasi-probability wave packets is identical except for position. Our results could help us to understand in depth the correspondence principle and OTOCs in unstable systems.