Interference in phase space and phase-only reconstruction

We demonstrate that the Fourier transform of a function leads to interference effects in phase space which allow the reconstruction of the essential properties of the original function, even when we restrict the inverse Fourier transform to the phase factor rather than the complete Fourier transform. We analyze this phase-only reconstruction, known from signal processing, in quantum mechanics for the elementary example of a superposition of two real-valued Gaussians. We employ the Wigner function as well as techniques of semiclassical quantum mechanics to show that their key features, such as their separation and their width, are indeed encoded in the phase of the Fourier transform. For this example, the amplitude-only reconstruction performing the inverse Fourier transform of the absolute value of the Fourier transform fails.