  Nanoscale phase coexistence and inhomogeneity are ubiquitous in correlated
electron materials, existing in doped Mott insulators, manganites, and
high-temperature superconductors. The small length scales and lack of contrast
mechanisms make it extremely challenging to measure real-space images of the
phase coexistence with high resolution. Yet, images aid our understanding of
how phase coexistence and domain boundaries dictate the exotic material
properties. Here we show that resonant soft-X-ray holography, previously
employed to image magnetic domains, can be used to image phase separation on
the nanoscale. We observe nucleation and growth of nanometre-sized metallic
domains out of the insulating phase of the prototypical correlated material
VO2, using linearly polarized coherent synchrotron radiation. By spectrally
resolving the holograms, we extract differential soft X-ray absorption spectra
with 50 nm spatial resolution. Furthermore, when combined with ultra-bright and
ultra-short X-ray sources, X-ray holography could capture both nanoscale
spatial variations and temporal fluctuations that occur close to the transition
temperatures or are induced by femtosecond light pulses that cannot be observed
with alternative imaging methods.
