Fourier analysis of many-body transition amplitudes and states

Abstract Particle-exchange symmetry shapes the counting statistics of many-body systems by giving rise to multiple alternative evolutions whose transition amplitudes interfere. Here, we apply a Fourier transform over the symmetric group S N to the collection of N ! many-body transition amplitudes connecting two states of an N -particle system. This provides a decomposition of the counting statistics of many-body interference experiments into contributions associated with distinct irreducible exchange symmetries. As a first application of our formalism, we consider bosons or fermions which are rendered partially distinguishable from one another through unobserved degrees of freedom, and whose state is also subjected to a Fourier analysis. We thereby obtain a systematic description of partial distinguishability, which is both an important experimental noise source and a possible handle for the control of many-body interference. As a second application, we identify mechanisms responsible for completely destructive interference in many-body systems with other exchange symmetries than those of bosons or fermions. For an interferometer implementing the discrete Fourier transform unitary, we predict many instances of such suppressed transitions.