2. Three-wave interactions

The homogeneous, classical three-wave equations for the decay interaction are given by

(2.1)\begin{gather} \partial_{t}A_{1} =gA_{2}A_{3}, \end{gather}

(2.2)\begin{gather}\partial_{t}A_{2} =-g^{*}A_{1}A_{3}^{*}, \end{gather}

(2.3)\begin{gather}\partial_{t}A_{3} =-g^{*}A_{1}A_{2}^{*}, \end{gather}

where $A_{j}$ is the amplitude of the $j$th wave, $A_{j}^{*}$ is its complex conjugate and $g$ is the coupling coefficient (Jurkus & Robson Reference Jurkus and Robson1960; Jaynes & Cummings Reference Jaynes and Cummings1963; Kaup et al. Reference Kaup, Reiman and Bers1979; Reiman Reference Reiman1979). In addition to the Hamiltonian

(2.4)\begin{equation} H=gA_{1}^{*}A_{2}A_{3}-g^{*}A_{1}A_{2}^{*}A_{3}^{*},\end{equation}

the interaction supports two other constants of motion

(2.5)\begin{gather} s_{2} =I_{1}+I_{3}, \end{gather}

(2.6)\begin{gather}s_{3} =I_{1}+I_{2}, \end{gather}

where $I_{j}=A_{j}^{*}A_{j}$ is the wave action of the $j$th wave. Because the wave amplitudes $A_{j}$ are not real-valued, their quantum analogues will not be observables. Thus, to directly compare the classical and quantum three-wave interactions, we will consider the second-order differential equation for the first wave action

(2.7)\begin{equation} \partial_{t}^{2}I_{1}= 2|g|^{2}(s_{2}s_{3}-2(s_{2}+s_{3})I_{1}+3I_{1}^{2}),\end{equation}

which is obtained directly by differentiating $I_{1}$ and making substitutions using (2.1)–(2.3) and (2.5) and (2.6). The dynamics for $I_{2}$ and $I_{3}$ is the same as that for $I_{1}$ thanks to the constants $s_{2}$ and $s_{3}$. Note that the second-order differential equation for $I_{1}$ is decoupled from the equations for $I_{2}$ and $I_{3}$ (which is not the case for the first-order differential equation for $I_{1}$). Because there is a symmetry between $s_{2}$ and $s_{3}$, in what follows we will assume $s_{3}\ge s_{2}$.

We can further simplify this equation by scaling it by $s_{2}$ and making the time parameter dimensionless so

(2.8)\begin{equation} \partial_{\tau}^{2}x=2(c-2(1+c)x+3x^{2}),\end{equation}

with the scaled wave amplitude $x=I_{1}/s_{2}$, the constant $c=s_{3}/s_{2}$ and the time parameter $\tau =|g|\sqrt {s_{2}}t$. In this form, the three initial conditions $A_{1}(t=0)$, $A_{2}(t=0)$ and ${A_{3}(t=0)}$ necessary to integrate (2.1)–(2.3) become initial conditions on the first wave's scaled action $x$, $x(\tau =0)$, its first derivative, $\partial _{\tau }x(\tau =0)$, and the constant $c$. Equation (2.8) is integrable, and one solution can be written as a Jacobi elliptic function

(2.9)\begin{equation} x(\tau)=c\ \text{sn}^{2}(\,p+\tau,c),\end{equation}

where $p$ is a constant determined by the initial conditions. Note that, although (2.8) is a second-order equation, the above solution only has a single free parameter. This is because the solution space for (2.8) is much larger than that of the original problem given by (2.1)–(2.3), to which (2.9) is also a solution. The space of initial conditions $A_{1}(0)$, $A_{2}(0)$ and $A_{3}(0)$ is not surjective onto the space of $c$, $x(0)$ and $\partial _{\tau }x(0)$. Indeed, because $s_{2}$, a constant determined by the initial conditions, has been scaled out of (2.8), we are not free to determine $x(0)$ and $\partial _{\tau }x(0)$ independently. There are additional requirements that $c\ge 1$ and $0\le x(0)\le 1$. It is also clear from (2.8) that the constant $c$ will act as the modulator of the nonlinearity of the interaction. Taking $c=1$, its lowest value since we have assumed $s_{3}\ge s_{2}$, maximizes the effect of the nonlinear term $x^{2}$, while taking $c\rightarrow \infty$ the system becomes linear and the Hamiltonian becomes that of the algebraic discrete quantum harmonic oscillator (May & Qin Reference May and Qin2023a).

Using the quantum field-theoretic method of quantization, we can promote the wave amplitudes of (2.1)–(2.3) to operators and replace the complex conjugation with Hermitian conjugation to obtain a set of quantum three-wave equations

(2.10)\begin{equation} \begin{gathered} \partial_{t}\hat{A}_{1} =g\hat{A}_{2}\hat{A}_{3}, \\ \partial_{t}\hat{A}_{2} =-g^{*}\hat{A}_{1}\hat{A}_{3}^{{{\dagger}}}, \\ \partial_{t}\hat{A}_{3} =-g^{*}\hat{A}_{1}\hat{A}_{2}^{{{\dagger}}}. \end{gathered}\end{equation}

The operators have the canonical commutation relations $[\hat {A}_{j},\hat {A}_{k}^{{\dagger} }]=\delta _{j,k}$ for $j,k\in \{1,2,3\}$ and with $\delta _{j,k}$ the Kronecker delta function. The quantum Hamiltonian

(2.11)\begin{equation} \hat{H}={\rm i}g\hat{A}_{1}^{{{\dagger}}}\hat{A}_{2}\hat{A}_{3}-{\rm i}g^{*} \hat{A}_{1}\hat{A}_{2}^{{{\dagger}}}\hat{A}_{3}^{{{\dagger}}},\end{equation}

and the mutually commuting operators

(2.12)\begin{gather} \hat{s}_{2} =\hat{n}_{1}+\hat{n}_{3}, \end{gather}

(2.13)\begin{gather}\hat{s}_{3} =\hat{n}_{1}+\hat{n}_{2}, \end{gather}

where the number operators are defined in the usual way, $\hat {n}_{j}=\hat {A}_{j}^{{\dagger} }\hat {A}_{j}$. We will denote the eigenvalues of these Hermitian operators with the same symbols as the classical operators, e.g. $\langle \hat {s}_{2}\rangle =s_{2}$. As found by Yuan Shi (Shi et al. Reference Shi2021a), eigenvectors of the operators $\hat {s}_{2}$ and $\hat {s}_{3}$, and the eigenvectors of the number operators $\hat {n}_{1}$, $\hat {n}_{2}$ and $\hat {n}_{3}$ form a finite $d=s_{2}+1$-dimensional invariant subspace when acted on by the Hamiltonian. We can write such subspace elements as

(2.14)\begin{equation} \varPsi(t)=\sum_{j=0}^{s_{2}}\alpha_{j}(t)\psi_{j},\end{equation}

where

(2.15)\begin{equation} \psi_{j}=|s_{2}-j,s_{3}-s_{2}+j,j\rangle ,\end{equation}

has eigenvalues $\{\hat {n}_{1},\hat {n}_{2},\hat {n}_{3}\}\psi _{j}=\{s_{2}-j,s_{3}-s_{2}+j,j\}\psi _{j}$. The finiteness of the subspace may be directly seen through the action of the Hamiltonian on a subspace element $\psi _{j}$:

(2.16)\begin{align} H\psi_{j} & = {\rm i}g(s_{2}-j+1)^{1/2}(s_{3}-s_{2}+j)j^{1/2}\psi_{j-1} \nonumber\\ & \quad -{\rm i}g^{*}(s_{2}-j)^{1/2}(s_{3}-s_{2}+1+j)(\,j+1)^{1/2}\psi_{j+1}. \end{align}

If $j=s_{2}$ in the above equation, then the coefficient of $\psi _{s_{2}+1}$ will be zero and similarly for $j=0$ and $\psi _{-1}$. The action of the Hamiltonian on these subspace elements can be calculated directly from the Schrödinger equation

(2.17)\begin{equation} {\rm i}\partial_{\tau}\varPsi=H\varPsi,\end{equation}

where we have taken the constant $\hbar =1$. Writing $\varPsi (t)$ as a column vector of weights $(\alpha _{0}(t),\alpha _{1}(t),\ldots \alpha _{s_{2}}(t))$, $H$ becomes a $d\times d$ tridiagonal matrix,

(2.18)\begin{equation} H=\left(\begin{array}{@{}cccccc@{}} 0 & h_{0} & 0 & 0 & 0 & \dots\\ h_{0} & 0 & h_{1} & 0 & 0 & \dots\\ 0 & h_{1} & 0 & h_{2} & 0 & \dots\\ 0 & 0 & h_{2} & 0 & h_{3} & \dots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right), \end{equation}

with

(2.19)\begin{equation} h_{j}=\sqrt{(s_{2}-j)(s_{3}-s_{2}+1+j)(\,j+1)}. \end{equation}

Thus, explicitly, we may write the Schrödinger equation as a system of $d$ coupled first-order linear differential equations:

(2.20)\begin{gather} {\rm i}\dot{\alpha}_{0} = h_{0}\alpha_{1}, \end{gather}

(2.21)\begin{gather}{\rm i}\dot{\alpha}_{1} = h_{0}\alpha_{0}+h_{1}\alpha_{2},\\ \ldots \nonumber \end{gather}

(2.22)\begin{gather}{\rm i}\dot{\alpha}_{j} =h_{j-1}\alpha_{j-1}+h_{j}\alpha_{j+1}. \end{gather}

We have taken $g=-i$ in the above equations for simplicity; however, it will be shown that the phase and magnitude of $g$ will not affect the quantum dynamics, as they did not affect the classical scaled dynamics of (2.8), below. Note that many efficient techniques have been developed for simulating sparse Hamiltonian dynamics, such as that of (2.20)–(2.22), on quantum computers. See, for example, Berry et al. (Reference Berry, Childs, Cleve, Kothari and Somma2015) and Low & Chuang (Reference Low and Chuang2017).

To compare this finite linear system with the nonlinear classical system, we need only take the expectation value of $\hat {n}_{1}$

(2.23)\begin{equation} \langle\hat{n}_{1}\rangle=\sum_{j=0}^{s_{2}}|\alpha_{j}|^{2}(s_{2}-j),\end{equation}

and compare it with the classical wave action $I_{1}$. Of course, the quantum linear dynamical equations and the classical nonlinear equations refer to different systems, so their dynamics will diverge except in the classical limit ($s_{2}\rightarrow \infty )$. Let us find the quantum equivalent of the classical (2.8) to see how the quantum dynamics might be written as a nonlinear differential equation. First, we find a second-order equation for $\hat {n}_{1}$ by using the quantum three-wave equations and making substitutions with the definitions of $\hat {s}_{2}$ and $\hat {s}_{3}$:

(2.24)\begin{equation} \partial_{t}^{2}\hat{n}_{1}= 2|g|^{2}(\hat{s}_{2}\hat{s}_{3}- (2\hat{s}_{2}+2\hat{s}_{3}+1)\hat{n}_{1}+3\hat{n}_{1}^{2}). \end{equation}

This is similar to that for $I_{1}$ in (2.7). Next, we take the expectation of this equation

(2.25)\begin{equation} \partial_{t}^{2}\langle n_{1}\rangle= 2|g|^{2}(s_{2}s_{3}- (2s_{2}+2s_{3}+1)\langle n_{1}\rangle+3\langle n_{1}\rangle^{2}+3\delta), \end{equation}

where we have defined the variance

(2.26)\begin{equation} \delta=\langle n_{1}^{2}\rangle-\langle n_{1}\rangle^{2}. \end{equation}

Finally, we scale the equation for $\langle \hat {n}_{1}\rangle$ by $s_{2}^{2}$, make the equation dimensionless by using the time parameter to absorb the coupling coefficient $g$ and define $x_{Q}=\langle n_{1}\rangle /s_{2},$ $\tau =|g|\sqrt {s_{2}}t$, and $\delta ^{\prime }=\delta /s_{2}^{2}$ to arrive at

(2.27)\begin{equation} \partial_{\tau}^{2}x_{Q}=2\left(c-2\left(1+c+\frac{1}{2s_{2}}\right)x_{Q}+3x_{Q}^{2}+3\delta^{\prime}\right),\end{equation}

which may be directly compared with (2.8). Both the quantum equation for $x_{Q}$ and the classical (2.8) for $x$ are the same except for the additional linear factor of $2x_{Q}/s_{2}$ and the inclusion of the scaled variance in the quantum system. As the dimension of the quantum system increases, so will $s_{2}$, diminishing the effect of the $2x_{Q}/s_{2}$ term; however, the effect of the variance in (2.27) will depend on the initial conditions of the quantum system, not just the total dimension. Also, note the variance is not a function of $x_{Q}.$ It must be calculated directly from the Schrödinger equation.

3. Quantum–classical correspondence

The initial conditions of the quantum system must be chosen carefully to correspond to those in the classical system. Consider an arbitrary initial condition where we write the weights of (2.23) in polar form

(3.1)\begin{equation} \alpha_{j}=r_{j}\ {\rm e}^{{\rm i}\phi_{j}},\end{equation}

with a real amplitude $r_{j}$ and argument $\phi _{j}$. Using (2.20)–(2.22), we may directly differentiate (2.23) to find

(3.2)\begin{equation} \partial_{t}\langle\hat{n}_{1}(0)\rangle=2\sum_{j=0}^{s_{2}}h_{j}r_{j}r_{j+1} \sin(\phi_{j+1}-\phi_{j}).\end{equation}

In choosing the initial condition for the classical system, for exact correspondence we would have $x(0)=x_{Q}(0)$, $\partial _{\tau }x(0)=\partial _{\tau }x_{Q}(0)$ and the nonlinearity parameters equal. This system of equations is underdetermined, however, because the quantum system has many more degrees of freedom, $f\sim O(s_{2})$, than the classical system. To deal with this, we will restrict ourselves to considering quantum initial conditions for which the real amplitude $r_{j}$ is taken to be a Gaussian over the index of $\alpha _{j}$ with a mean $\mu$ and standard deviation $\sigma$ which will depend on the dimension of the quantum system

(3.3)\begin{equation} r_{j}=\mathcal{N}\exp\left({-\frac{(\,j-\mu s_{2})^{2}}{2\sigma^{2}}}\right). \end{equation}

The normalization $\mathcal {N}$ must account for the initial condition being clipped since $r_{-1}=r_{s_{2}+1}=0$. Note that the initial scaled variance $\delta _{0}^{\prime }$ of the quantum system is only qualitatively related to the standard deviation $\sigma$ since

(3.4)\begin{equation} \delta_{0}^{\prime}=\left(\sum_{j=0}^{s_{2}}r_{j}^{2}(1-j/s_{2})^{2}\right)- \left(\sum_{j=0}^{s_{2}}r_{j}^{2}(1-j/s_{2})\right)^{2}, \end{equation}

while

(3.5)\begin{equation} \sigma=\left(\sum_{j=0}^{s_{2}}r_{j}^{2}(\,j-\mu)^{2}- \left(\sum_{j=0}^{s_{2}}r_{j}^{2}(\,j-\mu)\right)^{2}\right)^{1/2}. \end{equation}

Let us further restrict our quantum initial conditions to those which are velocity maximizing, which amounts to taking $\phi _{j+1}-\phi _{j}={\rm \pi} /2$ for all $j$. This is a prescription of the initial phase of the quantum nonlinear orbit. Since we are principally interested in correspondence over the course of many nonlinear orbits, the initial phase should not matter for our analysis. We thus need to match $\mu$ and $\sigma$ for the point on the classical orbit where the velocity is maximized. As noted in § 2, $x(0)$ and $\partial _{\tau }x(0)$ are not independent, so choosing to maximize the classical velocity also specifies a point on the classical trajectory. The solution to (2.8) at the classical inflection point (when $\partial _{\tau }^{2}x=0$) is

(3.6)\begin{equation} x_{0}=\tfrac{1}{3}(1+c-\sqrt{1+c^{2}-c}). \end{equation}

The negative root is chosen because the velocity at the inflection point

(3.7)\begin{equation} \dot{x}_{0}=2\sqrt{x_{0}}\sqrt{1-x_{0}}\sqrt{c-x_{0}} \end{equation}

becomes imaginary for the positive root. Finally, we set $\mu =x_{0}$. The choice of initial standard deviation $\sigma$ will be discussed below.

We compare the integrated classical system with initial conditions $x(0)=x_{0}$, $\partial _{\tau }x(0)=\dot {x}_{0}$ with the quantum system in figures 1, 2 and 8. As discussed above, the initial condition is Gaussian in $r$, and we have also taken $\mu =x_{0}$, $\sigma =d/5$ and the normalization $\mathcal {N}$ chosen such that the total probability is 1. In what remains of this section, we will consider the quantum–classical correspondence and various effects on that correspondence due to different choices of nonlinearity parameter $c$, the initial standard deviation $\sigma$ and the quantum dimension $d$.

Figure 1. Comparison between nonlinear classical and quantum dynamics. The initial standard deviation chosen for the quantum systems is set at $d/5$, and the nonlinearity parameter $c=1.05$.

Figure 2. Phase-space plots of the classical $(x,\dot {x})$ and quantum $(x_{Q},\dot {x}_{Q})$ systems with initial condition $(x_{0},\dot {x}_{0})$, initial quantum dimension $d=100$ and initial quantum standard deviation $\sigma =20$ for three values of the nonlinearity parameter, $c=1.1$, $1.05$ and $1.01$. The classical trajectories are shown in solid blue lines. The quantum trajectories are shown via the dashed red lines. The blue dot indicates the initial condition. Each system is evolved for three classical orbits.

In figure 1, we compare three systems, two quantum with $d=100$ and $d=400$ and the classical system, for a nonlinearity parameter $c=1.05$. Recall that $c\ge 1$, and as $c\rightarrow 1$, the nonlinearity increases. Despite the quantum and classical velocities being independently maximized, there is excellent initial correspondence between all three systems. The nonlinearity is pronounced enough that the $d=100$ system diverges from the classical solution within a couple of periods; however, the correspondence for the $d=400$ system lasts much longer, with the $d=400$ quantum system accurately capturing both the nonlinearity of the classical orbit as well as its period.

We may more carefully explore the effect of the nonlinearity parameter $c$ on the quantum–classical dynamics via phase-space diagrams of the quantum $d=100$ and classical systems in figure 2. Three classical nonlinear orbits are shown for each of the nonlinearity parameters $c=1.1$, $1.05$ and $1.01$. Decreasing the nonlinearity parameter causes the classical orbits to become more pinched. This increasingly linear relationship between $\langle \hat {n}_{1}\rangle$ and $\partial _{\tau }\langle \hat {n}_{1}\rangle$ indicates exponential-like growth, the onset of the classical instability and its quantum counterpart (see May & Qin Reference May and Qin2023b). From the quantum orbits in figure 2, it is obvious that, as the nonlinearity increases, the correspondence between the systems decreases. Investigations into exponential growth of quantum correlators in non-chaotic systems indicate that proximity to the classical fixed point leads to quantum scrambling (Xu, Scaffidi & Cao Reference Xu, Scaffidi and Cao2020). For instance, after a single pass near the classical fixed point, the $d=100$ system sharply diverges from the classical solution for $c=1.01$. On the other hand, the $d=100$ quantum system approximates the classical solution for a couple of classical periods for $c=1.05$ (see also figure 1), and the correspondence lasts for much longer for $c=1.1$. The proximate cause of the sharp divergence of the quantum system from the classical system near the fixed point is the increased classical period as $c\rightarrow 1$. Since the quantum system may only approximate the classical system for finite times, when $c\rightarrow 1$ and the classical period tends to infinity, the quantum solution will inevitably diverge from the classical solution within a single period.

Shown in figure 3 is the error as a function of the dimension of the quantum system for a nonlinearity parameter $c=1.05$. The error is calculated by averaging the absolute value of the difference between the classical and quantum systems over a single classical period. The log of this mean absolute error (MAE) is shown for each of the first four classical periods. Note that, for two uncorrelated oscillators with amplitude 1, the expected value of the MAE will be 0.25. A MAE higher than this must be due to anti-correlated phases. This occurs for low dimension in figure 3. As the dimension of the quantum simulation is increased, the error decays exponentially. There is a minimum error, however, which may be attributed to our choice of initial conditions, particularly the initial standard deviation.

Figure 3. Log–log plot of the MAE between the classical and quantum systems with $c=1.05$, averaged over each of the first four classical periods. The dimension $d$ ranges from 25 to 400.

For a target value of the MAE, how does the dimension scale with increasing nonlinearity? Recall that the nonlinearity parameter ranges from $1$, when the nonlinearity is maximized, to $\infty$, where the nonlinearity disappears. For clarity, we make the transformation $c^{\prime } = (c-1)^{-1}$, so that $c^{\prime } \in (0,\infty )$, and increasing $c^{\prime }$ increases the nonlinearity. In figure 4, we show how the dimension depends on $c^\prime$ for various target values of the MAE between the quantum and classical solutions for either the first or second classical period. For each value of the threshold error, the slope is remarkably consistent. This consistency is not limited to the short time scales either: looking at how the dimension scales with $c^{\prime }$ for a threshold error of $0.15$ during the second period, we see the same slope. We can approximate this exponential relationship

(3.8)\begin{equation} d \propto (c - 1)^\gamma \end{equation}

with $\gamma \sim -3/4$.

Figure 4. Log–log plot of the dimension versus the transformed nonlinearity parameter $c^{\prime } = (c-1)^{-1}$ for various values of the MAE between the quantum and classical solutions during the first and second classical periods. The threshold error values for the first classical period are $0.15$, $0.1$ and $0.05$. Also shown, in the red, large dashes, is the dimension necessary to keep the MAE below 0.15 during the second classical period. The original nonlinearity parameter $c$ ranges from 2 (highly linear, left side of the plot) to 1.01 (highly nonlinear, right side of the plot).

So far, the initial standard deviation of the quantum system has been taken to be $\sigma =d/5$ for simplicity; however, this does not necessarily lead to the best correspondence with the classical system. There is a trade off between long-term fidelity and the size of the initial variance. Shown in figure 5 is the growth of the variance over time given different initial standard deviations. Only the maximum variance is plotted because the magnitude of the variance oscillates with the amplitude of the wave. This can be seen in the spurts of growth of the maximum variance occurring at multiples of the period of the nonlinear oscillation. For a higher initial standard deviation, the variance grows more slowly with time.

Figure 5. Maximum variance over time for the $d=200$, $c=1.05$ quantum three-wave interaction. Results for various initial standard deviations are shown.

The slow growth of the variance with larger initial standard deviations is attributable to the spectrum of the Hamiltonian, shown in figure 6. When $c\rightarrow \infty$, the $d$ eigenvalues will be exactly linearly distributed, and for finite $c>1$, they will lie between the two lines of the figure. Importantly, even for the maximally nonlinear $c=1$, the eigenvalues with small absolute value (those with scale eigenvalue indices of around $0.5$) will still be approximately linearly distributed. As the three-wave interaction acts similarly to a perturbed quantum harmonic oscillator, we may by analogy understand that states with a large initial standard deviation are represented by the lowest energy eigenmodes—those eigenmodes which reside in the central, linear area of figure 6. With higher initial variance states having their dominant eigenfrequencies linearly distributed, they remain in correspondence with the classical dynamics for longer periods of time. Thus, despite a high initial variance coming at the cost of the quantum (2.27) beginning with a dynamics farther from that of the classical (2.8) with no variance, the effect of quantum interference is suppressed. Of course, if the initial variance is taken to be too large, (i) the initial condition may no longer be taken to be Gaussian, and (ii) the initial condition ceases to be well-approximated by lower-frequency eigenstates. An initial standard deviation of $\sigma =d/5$ strikes a balance between the need for a small growth rate of the variance with the need to prevent clipping of the initial Gaussian in the finite domain with $\alpha _{-1}=\alpha _{d}=0$ enforced.

Figure 6. Eigenvalues of the Hamiltonian scaled by the highest eigenvalue. The indices of the eigenvalues linearly increase from that of the lowest eigenvalue 0, to the highest index 1. For finite-dimensional systems, the eigenvalues will lie between the $c=1$ and $c\rightarrow \infty$ curves.

The variance may be used as a tool to determine where quantum revivals will occur. Shown in figure 7 is the variance for the $d=100$ quantum system for the first approximately 880 classical orbits. The initial standard deviation is $\sigma =20$, which corresponds to an initial variance of $\delta ^{\prime }=0.011$. Beginning around $\tau =3075$, the variance sharply decreases to a minimum of $\delta ^{\prime }=0.022$, indicating a partial quantum revival, shown in figure 8. Although the phase and amplitude of the quantum system differ from those of the classical, during the revival, the quantum system accurately captures the period of the classical orbit. When the variance is higher, for example, beginning at $\tau =1750$, it is difficult to characterize the correlation between the quantum and classical systems.

Figure 7. Variance of $d=100$ quantum system with nonlinearity parameter $c=1.05$ and initial standard deviation set to $\sigma =d/5=20$. The three labelled vertical gridlines at $\tau =0$, $1750$ and $3075$ indicate the starting times of the three elements of figure 8. The horizontal gridline shows the starting variance $\delta ^{\prime }=0.011$.

Figure 8. Comparison between classical and quantum dynamics for nonlinearity parameter $c=1.05$ beginning at three times $\tau =0$, $1750$ and $3075$. The plot beginning at $\tau =1750$ typifies high variance behaviour, and the plot beginning at time $\tau =3075$ shows a partial quantum revival, which was found by looking for relative minima in the variance of figure 7. The $d=100$ quantum system begins with a standard deviation of $\sigma =d/5=20$.