Quantum mechanics of Hyperion

This paper is motivated by the suggestion [W. Zurek, Phys. Scri. T, 76, 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become nonclassical within $20\text{years}$, but for the effects of environmental decoherence. The dynamics of quantum and classical probability distributions are compared for a satellite rotating perpendicular to its orbital plane, driven by the gravitational gradient. The model is studied with and without environmental decoherence. Without decoherence, the maximum quantum-classical (QC) differences in its average angular momentum scale as ${\hslash }^{2∕3}$ for chaotic states, and as ${\hslash }^2$ for nonchaotic states, leading to negligible QC differences for a macroscopic object like Hyperion. The quantum probability distributions do not approach their classical limit smoothly, having an extremely fine oscillatory structure superimposed on the smooth classical background. For a macroscopic object, this oscillatory structure is too fine to be resolved by any realistic measurement. Either a small amount of smoothing (due to the finite resolution of the apparatus) or a very small amount of environmental decoherence is sufficient to ensure the classical limit. Under decoherence, the QC differences in the probability distributions scale as ${({\hslash }^2∕D)}^{1∕6}$, where $D$ is the momentum diffusion parameter. We conclude that decoherence is not essential to explain the classical behavior of macroscopic bodies.

Figure 1

Orbit of satellite spinning about the $z$ axis perpendicular to the orbital plane. $\theta$ denotes the position of the satellite on the orbit, and $\varphi$ is the orentation of the satellite with respect to the semimajor axis of the orbit.

Figure 2

Poincare section for $e=0$, $\alpha =0.5$. Black circle denotes a typical initial state.

Figure 3

QC differences in $⟨J_z⟩$ vs $\tau$, for several $\beta$, with $e=0$, $\alpha =0.5$. For $\beta =\{0.0125,0.05,0.5\}$, the statistical errors are ${\sigma }_m=\{0.0002,0.0007,0.002\}$.

Figure 4

QC differences in $⟨J_z⟩$ for $\beta =0.0125$, with $e=0$, $\alpha =0.5$, ${\sigma }_m=0.7\times {10}^{-3}$.

Figure 5

Scaling of the early time QC differences in $⟨J_z⟩$ (denoted $\Delta ⟨J_z⟩$) with $\beta$, for $e=0$, $\alpha =0.5$, showing a ${\beta }^2$ dependence.

Figure 6

Variation of ${\mid \mathit{qm}-\mathit{cl}\mid }_1$ [Eq. (19)] with time and $\beta$ for the nonchaotic state $(e=0,\alpha =0.5)$. Each classical ensemble has 1 000 000 members.

Figure 7

Poincare section for a chaotic state, $\alpha =0.5$, $e=0.1$. Black circle denotes a typical initial state, with $J_z=10$ and $\delta =0.5$.

Figure 8

QC differences in $⟨J_z⟩$ vs $\tau$ for a chaotic state, $\alpha =0.5$, $e=0.1$. For $\beta =\{1\times {10}^{-3},3\times {10}^{-3},8\times {10}^{-3}\}$ the statistical errors are ${\sigma }_m=\{1.4\times {10}^{-4},1.4\times {10}^{-4},2.2\times {10}^{-4}\}$.

Figure 9

QC differences in $⟨J_z⟩$ vs $\beta$, for a chaotic state before saturation is reached, showing a ${\beta }^2$ dependence.

Figure 10

QC differences in $⟨J_z⟩$, averaged over $\tau$ from 20 to 100. Initial state is in the chaotic sea. $e=0.1$, $\alpha =0.5$. This suggests that $\Delta ⟨J_z⟩\propto {\beta }^{2∕3}$ in the saturation regime.

Figure 11

Maximum QC differences in $⟨J_z⟩$ vs $\beta$ for a chaotic state. $e=0.1$, $\alpha =0.5$. This suggests that this maximum difference scales as ${\beta }^{2∕3}$.

Figure 12

Quantum probability density for $\tau =40.0$, $\beta =0.002$.

Figure 13

${\mid \mathit{qm}-\mathit{cl}\mid }_1$ vs time for different values of $\beta$, for a chaotic state, $e=0.1$.

Figure 14

Quantum and classical probability densities for $\tau =40.0$, $\beta =0.002$. Both quantum and classical densities are convolved with a triangular filter of width 0.25 in $J_z$.

Figure 15

Variation of $\Delta ⟨J_z⟩$ with $\tau$ with and without the random potential with $\sigma ∕V_{\mathrm{ch}}=0.024$, ${\tau }_c=0.01$ (chaotic state, $\beta =0.05$).

Figure 16

Variation of ${\mid \mathit{qm}-\mathit{cl}\mid }_1$ with $\tau$ for different numbers $n$ of realizations of the random potential with $\sigma ∕V_{\mathrm{ch}}=0.012$, ${\tau }_c=0.01$ (chaotic state).

Figure 17

${\mid \mathit{qm}-\mathit{cl}\mid }_1$ vs $\tau$ for $\beta =0.05$, for varying $D={\sigma }^2{\tau }_c∕6$.

Figure 18

${\mid \mathit{qm}-\mathit{cl}\mid }_1$ vs $\tau$ for varying $\beta$, with ${\tau }_c=0.01$ and $\sigma ∕V_{\mathrm{ch}}=0.012$, for a chaotic state.

Figure 19

Quantum and classical probability distributions at $\tau =40$, with $\beta =0.0125$, $\sigma ∕V_{\mathrm{ch}}=0.012$, and ${\tau }_c=0.01$ for the chaotic state. In (b) the solid lines denote the results with the environment, and squares without the environment, showing that the classical probability distribution is not significantly affected by the environment.

Figure 20

Decay times ${\tau }_d$ of ${\mid \mathit{qm}-\mathit{cl}\mid }_1$ vs the correlation time ${\tau }_c$ of the perturbing environment, with $\beta =0.05$, $\sigma ∕V_{\mathrm{ch}}=0.012$, in the saturation regime $(\alpha =0.5,e=0.1)$.

Figure 21

1-norm of the difference between two classical ensembles, one with $⟨J_z⟩=10$, the other $⟨J_z⟩=11$. $\beta =0.05$, $\sigma ∕V_{\mathrm{ch}}=0.012$, ${\tau }_c=0.01$.

Figure 22

${\mid \mathit{qm}-\mathit{cl}\mid }_1$ vs $n^{-1∕2}$ at $\tau =40$. $n$ is the number of realizations of the random potential ($\sigma ∕V_{\mathrm{ch}}=0.012$, ${\tau }_c=0.01$ for the chaotic state).

Figure 23

Maximum values of ${\mid \mathit{qm}-\mathit{cl}\mid }_1$ vs ${\beta }^2∕D$. The points labeled $\beta$, $\sigma$, and ${\tau }_c$ represent data sets where $\beta$, $\sigma$, and ${\tau }_c$ were varied with the other two parameters held constant.

Figure 24

${\mid \mathit{qm}-\mathit{cl}\mid }_1$ for smoothed probability distributions for varying ${\sigma }_m$ and $\beta$ in the saturation regime $(\tau =20)$.