Many-Body Correlations in Phase-Space Microscopes

The recently proposed "phase-space microscope" (PSM) extends quantum gas microscopes by allowing joint, single-shot measurements of position and momentum. We show that the $n$-body correlation functions constructed from such phase-space snapshots determine, in principle, all $n$-body observables of a many-body quantum state. This establishes phase-space imaging as a complete probe of strongly correlated matter, beyond what snapshots of position and momentum can provide---a limitation that goes back to a famous question posed by Pauli. Even without the full reconstruction of the density matrices, the PSM provides valuable insight into the quantum states. We demonstrate this power through several applications: witnessing entanglement hidden from both position and momentum, including between spatially separated collective degrees of freedom; resolving pairing structure across the BEC-BCS crossover, including the phase of the pair wavefunction and the reconstruction of its two-body Wigner function; probing inhomogeneous systems and preformed pairs without long-range order, relevant to the pseudogap phase; extracting the interaction energy in a magnetic field; and measurements of the local Chern marker. Furthermore, we discuss phase imprinting in the matter-wave Fourier plane as an alternative method for probing time-reversal-symmetry-broken phases such as chiral $p$-wave superconductors. Together, these results establish the PSM as a versatile new tool for exploring fundamental features of quantum many-body phases in strongly correlated quantum matter.