Superposed quantum evolutions across chaotic and regular regimes

Abstract

While the superposition of quantum evolutions is known to produce interference effects, the interference between evolutions with regular and chaotic classical limits remains largely unexplored. Here, we use a Mach-Zehnder interferometer to investigate the superposition of two quantum evolutions, implemented via post-selection, and to compare it with the corresponding classical mixture. The quantum kicked top provides a natural platform for this study, as its classical dynamics ranges from regular to mixed to fully chaotic depending on the Hamiltonian parameters. We show that when a regular evolution is superposed with a chaotic one, the resulting subsystem entropy can exceed that of the classical mixture, provided the contribution of the chaotic branch dominates in the superposed quantum evolution. We further demonstrate that entropy production in such superpositions is strongly influenced by the structure of the underlying classical phase space. We further show that increased entropy generation can occur for purely regular dynamics at small values of the chaos parameter, given an appropriate choice of post-selection. These results reveal a nontrivial interplay between classical chaos and quantum interference in superposed quantum dynamics

Figures (8)

• Figure 1: Stroboscopic map showing the classical time evolution over 200 kicks for 289 initial states. Each initial state is evolved using the classical map in Eq. \ref{['eq:classical_eom']} for different values of the chaos parameter $\kappa$ and the amount of $y$-precession per period $\alpha=\pi/2$. The black dotted line indicates $\theta = 2.25$. As $\kappa$ increases, the dynamics of the initial states along the black dotted line evolve from regular motion to a mixed regular–chaotic regime, and ultimately to fully chaotic behaviour.
• Figure 2: A Mach-Zehnder interferometer setup. The input system consists of a composite system of spin and ancilla qubit. The parameters $\sigma_1$ and $\sigma_2$ represent the beam splitter angles for $BS_1$ and $BS_2$, respectively. Path 1 consists of the $n$-fold application of the quantum kicked top unitary with chaos parameter $\kappa_1$ ($U^n_{\kappa_1}$) while path 2 consists of the $n$-fold quantum kicked top unitary with chaos parameter $\kappa_2$ ($U^n_{\kappa_2}$). The two detectors at the output ports are labeled $D_1$ and $D_2$.
• Figure 3: Time-averaged subsystem entropy difference $\Delta S$, averaged over 200 periods of the kicked top unitary. The spin coherent state is initialized at $(\theta,\phi) = (2.25, 1.1)$, while the ancilla qubit is initialized in the state $\ket1$. (a) $\Delta S$ as a function of the beam splitter angles, with $\sigma_1=\sigma_2$, with 31 points in the interval $(0,\pi)$. Different curves correspond to different pairs of chaos parameters $\kappa$, with $\kappa_1 = 0.5$ (regular dynamics) fixed and $\kappa_2$ varied from $1$ to $5$. (b) $\Delta S$ as a function of the chaos parameter difference $\kappa_2-\kappa_1$, sampled with 20 points in the interval $(0.1,9.5)$. Different curves correspond to different beam splitter angles $\sigma_1 = \sigma_2$. (c) Contour plot showing the dependence of $\Delta S$ on the chaos parameter difference $\kappa_2-\kappa_1$ and the beam splitter angle $\sigma_{1,2}$, sampled with 20 and 31 points, respectively. The black dotted line indicates $\Delta S=0$ in panels (a) and (b).
• Figure 4: Time-averaged subsystem entropy difference $\Delta S$, averaged over 200 periods of the kicked top unitary. The spin coherent state is initialized at $(\theta,\phi) = (2.25, 1.1)$, and the ancilla qubit is initialized in the state $\ket1$. (a) $\Delta S$ as a function of the beam splitter angles, with $\sigma_1=\sigma_2$, sampled with 31 points in the interval $(0,\pi)$. Different curves correspond to different pairs of chaos parameters $\kappa$, with $\kappa_1 = 10$ (chaotic dynamics) fixed and $\kappa_2$ varied from $7.5$ to $1.5$. (b) $\Delta S$ as a function of the chaos parameters difference $\kappa_1-\kappa_2$, sampled with 20 points in the interval $(0.1,9.5)$. Different curves correspond to different beam splitter angles $\sigma_1 = \sigma_2$. (c) Contour plot showing the dependence of $\Delta S$ on the chaos parameters difference $\kappa_1-\kappa_2$ and the beam splitter angle $\sigma_{1,2}$, sampled with 20 and 31 points, respectively. The black dotted line indicates $\Delta S=0$ in panels (a) and (b).
• Figure 5: Time-averaged subsystem entropy difference $\Delta S$, defined in Eq. \ref{['eq:single qubit entropy difference']}, averaged over 200 periods of the kicked top unitary for three pairs of chaos parameters $(\kappa_1,\kappa_2)$. Each curve corresponds to a different beam splitter mixing angle with $\sigma_1=\sigma_2$. For each curve, we sample 21 spin coherent initial states at fixed polar angle $\theta=2.25$ and uniformly spaced azimuthal angles $\phi\in[-\pi,\pi]$. These initial conditions correspond to the states lying on the black dotted line in the classical stroboscopic maps of Fig. \ref{['fig:classical_map']}. The black dotted horizontal line in these figures indicates $\Delta S=0$. The spin size is $j=25$.