Seeing Hofstadter's butterfly in atomic Fermi gases

We propose a way to detect the fractal energy spectrum of the Hofstadter model from the density distributions of ultracold fermions in an external trap. At low temperature, the local compressibility is proportional to the density of states of the system, which reveals the fractal energy spectrum. However, thermal broadening and noises in the real experimental situation inevitably smear out fine features in the density distribution. To overcome this difficulty, we use the maximum entropy method to extract the density of states directly from the noisy thermal density distributions. Simulations show that one is able to restore the core feature of the Hofstadter's butterfly spectrum with current experimental techniques. By further reducing the noise or the temperature, one can refine the resolution and observe fine structures of the butterfly spectrum.

Figure 1

The compressibility (in unit of $1/J$) of the Hofstadter model vs the chemical potential $\mu$ and flux $\varphi$ at different temperatures.

Figure 2

Local observables in a 3D harmonic trap. The lines with symbols show the density $\rho$, the compressibility $\kappa$, and the entropy $s$ with $N=60000,\alpha =0.006$, and $\varphi =1/3$. The entropy per particle is (a) $S/N=1.0k_B$ and (b) $S/N=0.4k_B$. The corresponding temperatures are $T/J=0.873$ and $T/J=0.329$ respectively. (c) The temperature of the cloud vs entropy per particle $S/N$ calculated using LDA with the equation of state given by the quantum transfer matrix method [25, 26].

Figure 3

Deconvolve the thermal broadening effect. (a) Density $\rho$ vs chemical potential $\mu$ at high ($T=0.8J$) and low ($T=0.05J$) temperatures, $\varphi =1/3$. (b) The corresponding compressibilities which at low temperature ($T=0.05J$) reveal the DOS of the system. The solid line (red online) shows the DOS restored from the $T=0.8J$ density distribution using the maximum entropy approach.

Figure 4

Restoring the DOS from noisy data and imperfect temperature measurement. The blue (gray) line with error bars show the density profile at $T/J=0.4$ with noises following the fluctuation-dissipation theorem. The lines with symbols show DOS restored from the noisy data using the temperature $T_{\mathrm{expt}}/J=0.3,0.4$ and $0.5$ in the Fermi-Dirac kernel. The dashed green (gray) line shows the DOS restored from the exact $\rho \left(\mu \right)$ data.

Figure 5

The deconvoluted Hofstadter spectrum using thermal density distributions. [(a), (b)] without noises (infinite samples) and [(c), (d)] obtained from 100 noisy samples.