Fig. 1. Spatiotemporal evolutions of the deformed wavepackets described by the probability density |ψ_{±, 0 0 0}(x,t)|² of Eq. (12) with complicatedly oscillating positions which are described by the time-varying wave-peaks. Hereafter, the blue color and solid curves are associated with sign “+”, the red color and dashed curves correspond to sign “–”, and the right and left wavepackets |〈x|☺〉|² and |〈x|☹〉|² in a plot are always labeled by the live ☺ and dead ☹, respectively. When the packet |ψ_{+, 0 0 0}(x,t)|² is localized on the right or left side, we call the superposition state the kitten state or ill kitten state. All the quantities plotted in the figures of this paper are dimensionless.

Fig. 2. Spatiotemporal evolutions of the wavepackets |ψ′±,000(x,t)|2 from Eq. (13) with regular oscillations. Starting with a kitten state, the wavepacket pair approximately keeps their shapes and periodically oscillates. In an oscillating period, the wavepackets can go through many kitten states similar to the initial state in Fig. 1(b) and ill kitten states similar to the state at t = 2.6 in Fig. 2(b) with different distances between the wavepackets, which can be extracted by the ac field manipulations.

Fig. 3. Spatial distributions of the wavepackets associated with two stationary degenerate ground states for ϕ′ = 3, α = 0.1: (a) wavepackets associated with an ill kitten state with l = 0; (b) wavepackets corresponding to a kitten state with l = 1. Quantum transition between these degenerate ground states can be manipulated via ac-driven interchange of the wavepackets.

Fig. 4. (a) Time evolutions of the expected energies with the solid curve being associated with the same parameters as those of Fig. 2(a) and the dashed curve with the same parameters except for α = 1. (b) Time evolutions of the expected energies after increasing driving frequency from Ω = 2 to Ω = 5. In the insets we show that the expected energy E₀₀(t) for α = 0.1 approximately equals the stationary state energy E₀₁ indicated by the horizontal line, and the larger Ω value corresponds to smaller deviation from E₀₁. The dashed curves in both figures imply the similar conclusion for α = 1.