Dynamical Fractal in Quantum Gases with Discrete Scaling Symmetry

Inspired by the similarity between the fractal Weierstrass function and quantum systems with discrete scaling symmetry, we establish general conditions under which the dynamics of a quantum system will exhibit fractal structure in the time domain. As an example, we discuss the dynamics of the Loschmidt amplitude and the zero-momentum occupation of a single particle moving in a scale invariant $1/r^2$ potential. In order to show these conditions can be realized in ultracold atomic gases, we perform numerical simulations with practical experimental parameters, which shows that the dynamical fractal can be observed in realistic timescales. The predication can be directly verified in current cold atom experiments.

Figure 1

The Weierstrass function $W(x)$. Top: $b=1$ and $ab<1$, regular curve; middle: $b=2$ and $ab=1$, the transition point; bottom: $b=3$ and $ab>1$, fractal curve. $a=1/2$ is fixed for all three cases. The inset is a zoom in of the detailed structure around the red point, which shows the self-similarity behavior of $W(x)$.

Figure 2

The binding energies $E_n$, overlap coefficients ${\alpha }_n$ and ${\tilde{\alpha }}_n$ calculated using a real potential $V(x)$ given by Eq. (16) with realistic parameters given in the main text.

Figure 3

Real part of the Loschmidt echo amplitude $\mathcal{L}(t)$ calculated using the same experimental parameters as in Fig. 2. The bottom plot is an enlargement of the top plot by a scale of ${\lambda }^2=3$, which shows a clear self-similar pattern. The inset in the lower panel shows that the $\mathcal{L}(t)$ is indeed a smooth function in very short timescale. The time interval of the inset is around $300\mu \mathrm{s}$.

Figure 4

Normalized zero-momentum occupation ${\rho }_0(t)=\{[n_0(t)]/[n_0(0)]\}$ calculated using the same experimental parameters as Figs. 2 and 3. The bottom plot is an enlargement of the top plot by a scale of ${\lambda }^2=3$, which indicates self-similarity. The inset in the lower panel shows ${\rho }_0(t)$ is a smooth function in short timescale. The time interval of the inset is around $200\mu \mathrm{s}$.