Transition from order to chaos in reduced quantum dynamics

We study a damped kicked top dynamics of a large number of qubits ($N\to \infty$) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant $r\in [0,1]$, which plays the role of the single control parameter. In the parameter range for which the classical dynamics is chaotic, while varying $r$ we find the universal period-doubling behavior characteristic to one-dimensional maps: period-2 dynamics starts at $r_1\approx 0.3181$, while the next bifurcation occurs at $r_2\approx 0.5387$. In parallel with period-4 oscillations observed for $r\le r_3\approx 0.5672$, we identify a secondary bifurcation diagram around $r\approx 0.544$, responsible for a small-scale chaotic dynamics inside the attractor. The doubling of the principal bifurcation tree continues until $r\le r_{\infty }\sim 0.578$, which marks the onset of the full scale chaos interrupted by the windows of the oscillatory dynamics corresponding to the Sharkovsky order. Finally, for $r=1$ the model reduces to the standard undamped chaotic kicked top.

Figure 1

The plot of $f(z^{*},r)$ for: $r=0.6$ (blue), $r=0.9$ (orange), $r=0.972\approx r_b$ (green), $r=0.99$ (red)

Figure 2

The first 100 steps of the evolution of $z_t$ for different values of $r$ and a random initial state.

Figure 3

Bifurcation diagrams for the $Z$ coordinate of the Bloch vector with cycles identified according to the Sharkovsky order. Note secondary bifurcation diagrams located inside the four-cycle around $r\approx 0.545$ and magnified in Fig. 7.

Figure 4

Visualization of a strange attractor: 10 000 steps of a trajectory stemming from a random initial state dynamics for $r=0.75$. Black points denote subsequent positions of the Bloch vector.

Figure 5

The estimation of the correlation dimension for $r=0.75$. The slope near the inflection point is best fitted with the linear dependence given by $lnC=8.80+1.84ln\varepsilon$.

Figure 6

Bifurcation diagram for the $Z$ coordinate of the Bloch vector (top) to be compared with Lyapunov exponents ${\lambda }_j$ plotted as functions of the system parameter $r$. The onset of chaos corresponds to positivity of the largest exponent ${\lambda }_1$.

Figure 7

Magnification of Fig. 3: (a) secondary bifurcation diagrams occurring at $r\approx 0.544$ inside the four-cycle of the main bifurcation tree; (b) magnification of the rectangle from the upper panel show a ternary structure at $r\approx 0.540$.

Figure 8

Secondary bifurcation diagrams for the $X$ and $Y$ coordinates of the Bloch vector visible in the same parameter range, $r\approx 0.545$, as shown in Fig. 7.