Dynamical entanglement and chaos: The case of Rydberg molecules

A Rydberg molecule is composed of an outer electron that collides on the residual ionic core. Typical states of Rydberg molecules display entanglement between the outer electron and the core. In this work, we quantify the average entanglement of molecular eigenstates and further investigate the time evolution of entanglement production from initially unentangled states. The results are contrasted with the underlying classical dynamics, obtained from the semiclassical limit of the core-electron collision. Our findings indicate that entanglement is not simply correlated with the degree of classical chaos, but rather depends on the specific phase-space features that give rise to inelastic scattering. Hence mixed phase-space or even regular classical dynamics can be associated with high entanglement generation.

Figure 1

Molecular reference frame. The core axis is along $OZ$ and its angular momentum $\mathbf{N}$ is along $OX$.

Figure 2

Two limiting Poincaré surfaces of sections. (a) No coupling $k=0$, (b) large coupling $k=10$ (in the generic case, see Fig. 4 below). The shaded areas correspond to quantum states with minimal $(J-L=40)$ and mean $(J=50)$ values of $N$ (see Sec. 3). Their widths correspond to $\mathrm{\Delta }N=1$. The mean $N=J$ case is not on the equatorial $OYZ$ plane, as can be understood by squaring $\mathbf{J}=\mathbf{L}+\mathbf{N}$ (where $\mathbf{N}$ is “horizontal,” parallel to $OX$) when the lengths of $N$ and $J$ are equal.

Figure 3

(Color online) Average linear entropy for generic (red boxes) and resonant $T_e=T_0$ states (black boxes), shown for different values of $k$. The rms is also shown as red dashed (generic) and solid (resonant) error bars.

Figure 4

Poincaré surfaces of section for different values of the coupling constant $k$. Top, generic; bottom, resonant cases. (a,d) $k=0.25$. (b,e) $k=1$. (c,f) $k=10$.

Figure 5

(Color online) Classical phase-space and linear entropy in the $k=0.5$ resonant situation. (Top) Poincaré surfaces of sections at the left, middle, and right ends of the energy interval shown at the bottom. (Bottom) Linear entropy of the individual eigenstates. The black dots between the dashed lines correspond to the states that entered the statistics shown in Fig. 3. Each eigenstate of energy $E$ is labeled by $\nu \left(E\right)$, the principal quantum number in the $N=J$ channel.

Figure 6

Husimi plots (top) and $B_N\left(E\right)$ coefficients (bottom) for two particular wave functions in the $k=0.5$ resonant situation near the center of the interval plotted in Fig. 5. Left level, with $\nu \left(E\right)=315.4233$, is nearly quantized on the minimum value of $N=40$ as shown by the $B_N$ distribution. Compare its Husimi plot with shaded areas in Fig. 2. It has $S_2\left(E\right)=0.03658$. The right level, with ${\nu }_E=315.6118$, quantized nearby the $+OZ$ axis, spans a greater number of $B_N$ values. Its Husimi plot overlaps with a greater number of $N$ circles indicated in Fig. 2. It has $S_2\left(E\right)=0.6881$.

Figure 7

(Color online) Short time variation of the linear entropy generated from the initial product state $F_{\mathrm{loc}}(r\approx r_{tp})\otimes \mid N_0=40⟩$ for different collision matrices labeled by the value of the coupling strength $k$: from top to bottom, $k=10$ (topmost curve, red), 1 (middle curve, black), and 0.25 (bottom curve, blue). $t$ is given in units of the Kepler period $T_e$. (a) The parameters are chosen so that the corresponding classical dynamics falls in the generic phase-space case. (b) Same as (a) for the resonant case $T_e=T_0$. In both (a) and (b), the $k=0.25$ curve is multiplied by a factor 5.

Figure 8

(Color online) Same as Fig. 7 but with the initial product state given by $F_{\mathrm{loc}}(r\approx r_{tp})\otimes \mid N_0=50⟩$ (the $k=0.25$ curve is not multiplied by a factor 5).

Figure 9

(Color online) Long-time variation of the linear entropy for the case considered in Fig. 7 [$N_0=40$, (a) generic and (b) resonant cases]. The inset in (b) blows up the vertical scale for the $k=0.25$ curve to help visualize the oscillations, also visible for $k=1$.

Figure 10

(Color online) Long-time variation of the linear entropy for the case considered in Fig. 8 [$N_0=50$, (a) generic and (b) resonant cases]. The inset in (b) details $S_2\left(t\right)$ in the region $t=$ $80$ to $110$ $T_e$.

Figure 11

The partial autocorrelation $C\left(t\right)$ defined by Eq. (26) (parameters corresponding to the $k=0.25$ resonant case, arbitrary units) is plotted in an interval centered around the revival time $T_e^{\mathrm{rev}}\simeq 105$ $T_e$.