Harmonic oscillator kicked by spin measurements: A Floquet-like system without a classical analog

We present a kicked harmonic oscillator where the impulsive driving is provided by stroboscopic measurements on an ancillary degree of freedom and not by the canonical quantization of a time-dependent Hamiltonian. The ancilla is dynamically entangled with the oscillator position, while the background Hamiltonian remains static. The dynamics of this system is determined in closed analytical form, allowing for the evaluation of a properly defined Loschmidt echo, ensemble averages, and phase-space portraits. As in the case of standard Floquet systems, we observe regimes with crystalline and quasicrystalline structures in phase space, resonances, and evidence of chaotic behavior, however, not originating from any classically chaotic system.

Figure 1

Magnetic field $\vec{B}\left(x\right)\sim x\widehat{z}$ coupling position (red) and spin (blue). Here ${\widehat{S}}_x$ measurements periodically project the system into a nonstationary state. The branch $|z\rangle |+\rangle$ ($|z\rangle |-\rangle$) experiences a harmonic potential of frequency ${\omega }_0$ displaced to the left (right).

Figure 2

Energy expectation values for two quantum trajectories determined by the outcomes of ${\widehat{S}}_x$ measurements (blue signs). The insets show the associated Husimi functions for $N=6$. Darker (lighter) regions correspond to lower (higher) probabilities. We used $v=2.0$, $z_0=(1+i)/\sqrt{2}$, and $R=0.106$.

Figure 3

(a) and (b) Averaged Loschmidt echo decay for two independent sets of 15 quantum trajectories, respectively. We used the same parameters as in Fig. 2 except for $R=\frac{1}{5}$. For the small perturbation in the Hamiltonian we set $\delta R=\frac{5}{1000}$. The inset shows logarithmic-scale plots for $N>3$. The exponential fitting is given by the green dashed lines.

Figure 4

Husimi functions for (a) $N=2$, (b) $N=5$, and (c) $N=10$ with $R=1/4$; (d) $N=5$, (e) $N=7$, and (f) $N=10$ with $R=2/5$; and (g) $N=5$, (h) $N=8$, and (i) $N=10$ with the resonant frequency ratio $R\approx 0.3710$. Lighter (Darker) colors denote higher (lower) probabilities.