Complex chaos in the conditional dynamics of qubits

We analyze the consequences of iterative measurement-induced nonlinearity on the dynamical behavior of qubits. We present a one-qubit scheme where the equation governing the time evolution is a complex-valued nonlinear map with one complex parameter. In contrast to the usual notion of quantum chaos, exponential sensitivity to the initial state occurs here. We calculate analytically the Lyapunov exponent based on the overlap of quantum states, and find that it is positive. We present a few illustrative examples of the emerging dynamics.

Figure 1

The Julia set for the nonlinear map (6). The parameter is set to $p=1$. Gray scale indicates how fast the map converges to the stable cycle $\{-1,\infty \}$ (dark, fast; gray, slow convergence; white, no convergence).

Figure 2

(Color online) The complex parameter space $p$ of the nonlinear map (6), with the initial state being the critical point $z_0=0$. Colors indicate the length of attractive cycles. White corresponds to no convergence.

Figure 3

Iterations of the nonlinear map (6) for purely imaginary $p$ with the initial state being the critical point $z_0=0$. After ${10}^4$ iterations the absolute values of $z$ for the next 50 steps are shown.