Diffusion and localization for the Chirikov typical map

We consider the classical and quantum properties of the “Chirikov typical map,” proposed by Boris Chirikov in 1969. This map is obtained from the well-known Chirikov standard map by introducing a finite-number $T$ of random phase-shift angles. These angles induce a random behavior for small time-scales $\left(t<T\right)$ and a $T$-periodic iterated map which is relevant for larger time-scales $\left(t>T\right)$. We identify the classical chaos border $k_c\sim T^{-3/2}⪡1$ for the kick parameter $k$ and two regimes with diffusive behavior on short and long time scales. The quantum dynamics is characterized by the effect of Chirikov localization (or dynamical localization). We find that the localization length depends in a subtle way on the two classical diffusion constants in the two time-scale regime.

Figure 1

Kicked rotator where the kick force rotates in time with a kick force angle $\alpha \left(t\right)$ that is a periodic function in time.

Figure 2

(Color online) Classical Poincaré sections of 50 trajectories of length $t_{\text{max}}=10000$ of the Chirikov typical map for one particular realization of the random-phase shifts with $T=10$ and three values: $k=0.05$, $k=0.078$, and $k=0.1$. The positions $\left({\theta }_t,p_t\right)∊\left[0,2\pi \right[\times \left[0,2\pi \right[$ are shown after applications of the full map, i.e., $T$-times iterated typical map with: $t\text{mod}T=0$. The value of the middle figure corresponds to the critical value $k_c\approx 0.078$ of Eq. (7) for the transition to global diffusive chaos.

Figure 3

(Color online) The ratio $D/D_0$ of the classical diffusion constant $D$ of the Chirikov typical map over the theoretical diffusion constant $D_0=k^2/2$, assuming perfectly uncorrelated phases ${\theta }_t$, as a function of the scaling parameter $x=k{T}^{3/2}$ for $10\le T\le 1000$ and $T^{-3/2}\le k\le 1$. The classical diffusion constant $D$ has been obtained from a linear fit of the average variance $⟨\delta p_t^2⟩$ in the time interval $10T\le t\le 100T$. The average variance has been calculated for 400 different realizations of the random-phase shifts and 25 different random initial conditions $\left({\theta }_0,p_0\right)∊\left[0,2\pi \right[\times \left[0,2\pi \right[$ for each realization. The vertical full (blue) line represents the classical chaos border at $k_c{T}^{3/2}={\pi }^2/4\approx 2.47$ (compare Fig. 2) and the gray rectangle at the left shadows the nondiffusive regime where the numerical procedure (incorrectly) yields small positive values of $D$ (see text). The full curved (black) line represents the scaling function $D/D_0=f\left(x\right)$ as given by Eq. (13). The inset shows $D{T}^3$ as a function of the scaling parameter $x=k{T}^{3/2}$ and the full (red) line represents the scaling function $x^{2}f\left(x\right)/2$. The full vertical (blue) line again represents the classical chaos border.

Figure 4

(Color online) Quantum evolution of the Chirikov typical map at $k=0.1$ and $T=10$ in the semiclassical limit with $\hslash =2\pi /N$ and with the Hilbert space dimension $N=2^{12}$ (left column) and $N=2^{16}$ (right column) at times $t=0$ (first row), $t=20$ (second row), $t=60$ (third row), $t=100$ (fourth row), and $t=150$ (fifth row). Shown are Husimi functions with maximum values at red (gray), intermediate values at green (light gray) and minimum values at blue (black) at the lower half of one elementary cell $\theta ∊\left[0,2\pi \right[$, $p∊\left[0,\pi \right[$. The initial condition is a coherent Gaussian state centered at ${\theta }_0=0.8\times 2\pi$ and $p_0=0.25\times 2\pi$ with a variance (in $p$ representation) of $\Delta p=2\pi /\sqrt{12N}$. The resolution corresponds to $\sqrt{N}$ (64 or 256) squares in one line. The realization of the random-phase shifts is identical to that of Fig. 2. Note that at the considered value $k=0.1$ the global classical dynamic is diffusive but requires iteration times of $t\sim 10000$ to fill one elementary cell (see Fig. 5).

Figure 5

(Color online) Quantum evolution of the Chirikov typical map as in Fig. 4 with the same coherent Gaussian state as initial condition, same values $k=0.1$ and $T=10$, $\hslash =2\pi /N$ but at the iteration time $t=20000$, and at different values of $N=2^8$, $N=2^{10}$ (first row), $N=2^{12}$, $N=2^{14}$ (second row), $N=2^{16}$, and classical simulation (third row). For the classical map $20000$ trajectories have been iterated up to the same time $t=20,000$ with random initials conditions very close to the initial position at ${\theta }_0=0.8\times 2\pi \pm 0.002$ and $p_0=0.25\times 2\pi \pm 0.002$. Colors are as in Fig. 4.

Figure 6

(Color online) Illustration of the Chirikov localization for the quantum Chirikov typical map. Shown is the (averaged) absolute squared wave function in $p$ representation as a function of the integer quantum number $n$ associated to $p=\hslash n$. The initial state at $t=0$ is localized at $n=0$ and the quantum typical map has been applied up to times $t=t_{\text{max}}$ with $t_{\text{max}}$ chosen such that the initial classical diffusion is well-saturated at $t_{\text{max}}/4$ due to quantum effects (see Fig. 7). In order to determine numerically the localization length, first ${\left|{\psi }_n\right|}^2$ has been time-averaged for the time interval $t_{\text{max}}/4\le t\le t_{\text{max}}$ and then a further ensemble average (over 100 realizations of the random-phase shifts) of ${\left|{\psi }_n\right|}^2$ (upper red/gray curve) or of $\text{ln}\left({\left|{\psi }_n\right|}^2\right)$ (lower blue/black curve) has been applied. The localization lengths are obtained as the (double) inverse slopes of a linear fit (with relative weights $w_n\sim {\left|{\psi }_n\right|}^2$) of $\text{ln}\left({\left|{\psi }_n\right|}^2\right)$ versus $n$. Here the parameters are $N=2^{12}$, $k=0.5$, $T=100$, $t_{\text{max}}=546132$, and $\hslash =2\pi /\left(17+\gamma \right)\approx 0.357$ with the golden number $\gamma =\left(\sqrt{5}-1\right)/2\approx 0.618$. The fits are given by the thick straight lines. The theoretical localization length is ${\ell }_0=k^{2}T/\left(2{\hslash }^2\right)\approx 98.3$ while the numerical fits provide ${\ell }_{\psi }\approx 231.7$ (upper curve) and ${\ell }_{\text{ln}\psi }\approx 187.5$ (lower curve).

Figure 7

(Color online) The ensemble averaged variance of the expectation value of $n^2$ as a function of $t$ for the same parameters (and the same realizations of the random-phase shifts) as in Fig. 6. The upper horizontal (red) line corresponds to the saturation value ${\ell }_{\text{diff}}^2$ (obtained from a time average of $⟨n^2⟩$ in the interval $t_{\text{max}}/4\le t\le t_{\text{max}}$) with ${\ell }_{\text{diff}}\approx 146.8$ being the localization due to saturation of diffusion. The short lower horizontal (green) line corresponds to ${\ell }_0^2$ with the theoretical localization length ${\ell }_0\approx 98.3$ (see also Fig. 6). The inset shows the initial diffusive regime at shorter time scales. The straight (red/gray) line shows the value $D_0/{\hslash }^2\approx 0.983$ associated to the initial diffusion: ${\hslash }^2⟨n^2⟩=⟨p^2⟩\approx D_{0}t$ and with the theoretical diffusion constant $D_0=k^2/2$.

Figure 8

(Color online) Localization length determined by three different numerical methods as a function of the theoretical value ${\ell }_0=k^{2}T/\left(2{\hslash }^2\right)$ in a double-logarithmic representation. The data points correspond to ${\ell }_{\psi }$ (red crosses) obtained from a linear fit of $\text{ln}⟨{\left|{\psi }_n\right|}^2⟩$ versus $n$ (see Fig. 6), ${\ell }_{\text{ln}\psi }$ (green squares) obtained from a linear fit of $⟨\text{ln}{\left|{\psi }_n\right|}^2⟩$ versus $n$, and ${\ell }_{\text{diff}}$ (blue triangles) obtained from the saturation of quantum diffusion (see Fig. 7). The straight full (blue) line represents $\ell =2{\ell }_0$. For clarity the data points for ${\ell }_{\text{ln}\psi }$ (or ${\ell }_{\text{diff}}$) have been shifted down by a factor of 4 (or 16). The value $\hslash =2\pi /\left(17+\gamma \right)\approx 0.357$ is exactly as in Fig. 6 while $k∊\left\{0.1,0.15,0.2,0.3,0.5,0.7,1.0\right\}$ and $T∊\left\{10,15,20,30,50,70,100\right\}$. The Hilbert space dimension is mostly $N=2^{12}$ except for a few data points with the largest values of ${\ell }_0$ where we have chosen $N=2^{14}$. The ensemble averages have been performed over 100 different realizations of the random-phase shifts.

Figure 9

(Color online) Same as in Fig. 8 but with fixed values for the classical parameters $k=0.2$ and $T=50$ while $\hslash =2\pi /\left(\tilde{N}+\gamma \right)$ varies with the integer variable $\tilde{N}$ in the interval $8\le \tilde{N}\le 200$ (i.e., $0.03132\le \hslash \le 0.7291$). The number of realizations of the random-phase shifts is 20.

Figure 10

(Color online) The localization length $\ell ={\ell }_{\psi }$ versus the theoretical value ${\ell }_0=k^{2}T/\left(2{\hslash }^2\right)$ for $k=0.2$, $T=10$ and the same values of $\hslash$ as in Fig. 9. The straight full (red) upper line corresponds to $\ell =2{\ell }_0$. The lower full (blue) curve corresponds to the corrected expression $\ell =2{\ell }_{\text{cl}}\left({\ell }_0\right)=2{\ell }_0/\left[h\left({\ell }_0\right)+\sqrt{h{\left({\ell }_0\right)}^2+2B{\ell }_0}\right]$, $h\left({\ell }_0\right)=\left(1-2A{\ell }_0\right)/2$, $A\approx 0.03121$, and $B\approx 0.1177$ due to a modified classical dynamics (see text for explanation). The number of random-phase realizations is 50. The inset shows the classical diffusion for the same classical parameters $k=0.2$, $T=10$. The data points (red crosses) are (selected) numerical data and the full (blue/black) curve represents the numerical fit: $⟨p^2⟩=D_{0}t\left(T+At\right)/\left(T+Bt\right)$ providing the two parameters $A$ and $B$ for the corrected localization length ${\ell }_{\text{cl}}$ in the main figure. The two straight lines correspond to the initial diffusion at short-time scales with diffusion constant $D_0=k^2/2$ and the final diffusion at long time scales with $D=D_0\left(A/B\right)\approx 0.2652D_0$. This value of $D$ coincides quite well with the value of the scaling curve in Fig. 3 at $x=k{T}^{3/2}\approx 6.32$.