Quantum-to-classical transition in a system with mixed classical dynamics

We study how decoherence rules the quantum-classical transition of the kicked harmonic oscillator. The system presents classical dynamics that ranges from regular to strong chaotic behavior depending on the amplitude of the kicks. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and quantum phase space distributions is proportional to a single parameter $\chi \equiv K{\hslash }_{\mathrm{eff}}^2∕4D^{3∕2}$, which relates the effective Planck constant, ${\hslash }_{\mathrm{eff}}$, to the kicking strength, $K$, and the diffusion constant, $D$. This relation between classical and quantum distributions is valid when $\chi <1$, a case that is always attainable in the semiclassical regime, independent of the value of the strength of noise given by $D$. Our results extend a recent study performed in the chaotic regime.

Figure 1

Stroboscopic phase space of the KHO for different values of the kick amplitude $K$. In (a) $K=0.5$, (b) $K=1.0$, (c) $K=1.5$, and in (d) $K=2.0$. The phase space points $(q_0,p_0)=(0,0)$, (0,1.1), and (0.7,0), marked with symbols (●), are the centers of the initial coherent states used in our numerical simulations.

Figure 2

Separation ${\mathcal{D}}_n$ between quantum and classical distribution as a function of the number of kicks for $K=0.5$. The full lines join all the values of ${\mathcal{D}}_n$ calculated for the same value of the Lamb-Dicke parameter $\eta$. In (a) the initial distribution is a coherent state centered at the phase space point $(q0,p0)=(0,1.1)$ and the curves are for $\eta =0.5$, 0.25, 0.125, 0.0625, and 0.03125. In (b) the initial coherent state is centered at the origin and the curves are for $\eta =0.5$, 0.25, 0.125, and 0.0625. The number of kicks, $n$, are rescaled by a function $n^{*}\left(\eta \right)={\eta }^{-\alpha }$, where we optimize the values of $\alpha$ in order to obtain the best collapsing of all the curves. The insets show the curves without the rescaling over the time $n$. The curves in both insets corresponds to decreasing values of $\eta$ from left to right.

Figure 3

Same as Fig. 2 but for $K=1.0$. In (a) the initial coherent state centered at $(q0,p0)=(0,1.1)$ and in (b) $(q0,p0)=(0.7,0)$. The different curves in each plot correspond to $\eta =0.5$, 0.25, 0.125, 0.0625, and 0.03125. The number of kicks, $n$, is rescaled by a function $n^{*}\left(\eta \right)={\eta }^{-\alpha }$ where we optimize the values of $\alpha$ in order to obtain the best collapsing of all the curves.

Figure 4

(Color online) Density plots of the Wigner (right) and classical distributions (left) that result from the evolution with the KHO Hamiltonian with the kick amplitude $K=0.5$ from an initial coherent state centered in the phase space point $(q_0,p_0)=(0.0,1.1)$ and with a width $\Delta q\left(0\right)=\Delta p\left(0\right)=\eta =0.25$. In (a) and (b) immediately before the kick $n=8$ and in (c) and (d) immediately before the kick $n=18$.

Figure 5

(Color online) Density plots of the Wigner (right) and classical distributions (left) that result from the evolution with the KHO Hamiltonian with the kick amplitude $K=0.5$ from an initial coherent state centered at the origin of phase space and with a width $\Delta q\left(0\right)=\Delta p\left(0\right)=\eta =0.25$. In (a) and (b) immediately before the kick $n=60$ and in (c) and (d) immediately before the kick $n=300$.

Figure 6

Renormalized distances between quantum and classical distributions, with diffusion. All the curves were generated with the initial coherent state centered in the phase-space point $(q_0,p_0)=(0,1.1)$ (see Fig. 1) with the kick amplitude $K=0.5$ in (a) and $K=1.5$ in (b). The eleven different curves correspond to $\chi \equiv K{\eta }^4∕D^{3∕2}=6.2\times {10}^{-2},4.3\times {10}^{-2},1.5\times {10}^{-2},1.1\times {10}^{-2},7.6\times {10}^{-3},3.9\times {10}^{-3},1.4\times {10}^{-3},6.8\times {10}^{-4},4.8\times {10}^{-4},2.4\times {10}^{-4},4.3\times {10}^{-5}$ in (a) and $\chi =6.5\times {10}^{-2},4.5\times {10}^{-2},3.3\times {10}^{-2},2.3\times {10}^{-2},1.2\times {10}^{-2},4.1\times {10}^{-3},2.1\times {10}^{-3},1.4\times {10}^{-3},7.2\times {10}^{-4},1.3\times {10}^{-4},4.5\times {10}^{-5}$ in (b), where we used the values of diffusion constant $D=0.1$, 0.05, 0.01, 0.005, 0.001, 0.0005 and the Lamb-Dicke parameter $\eta =0.25$, 0.125, 0.0625, 0.03125.

Figure 7

The maximum distance between quantum and classical distributions given by the value of the first peak of ${\mathcal{D}}_n$ at $n=n_{\text{peak}}$ as a function of $\chi$. The initial distribution is centered at ${\mathbf{x}}_0=(0,1.1)$ in (a) and ${\mathbf{x}}_0=(0.7,0)$ in (b). The marks (엯) are for $K=0.5$, the (∎) for $K=1.0$, and (▵) for $K=1.5$. The region of linear behavior of ${\mathcal{D}}_{n_{\text{peak}}}$ as a function of $\chi$ is highlighted by the dashed line with a unit value of the slope. The inset is borrowed from Ref. 15 when the initial distribution explores a chaotic region in phase space of the classic KHO. We use the value of the Lamb-Dicke parameters $\eta =0.5,0.25,0.125,0.0625,0.03125$, and the diffusion constant $D=5\times {10}^{-6},1\times {10}^{-5},5\times {10}^{-5},1\times {10}^{-4},5\times {10}^{-4},1\times {10}^{-3},5\times {10}^{-3},1\times {10}^{-2},5\times {10}^{-2},{10}^{-1}$.