  We propose a novel 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.
