We show the utility of the weak value amplification to observe the quantum interference between two close lying atomic coherent states in a post-selected atomic cat state, produced in a system of N identical two-level atoms weakly interacting with a single photon field. Through the observation of the negative parts of the Wigner distribution of the post-selected atomic cat state, we find that the post-selected atomic cat state becomes more nonclassical when the post-selected polarization state of the single photon field tends toward becoming orthogonal to its pre-selected state. We show that the small phase shift in the post-selected atomic cat state can be amplified via measuring the peak shift of its phase distribution when the post-selected state of the single photon field is nearly orthogonal to its pre-selected state. We find that the amplification factor of 15 [5] can be obtained for a sample of 10 [100] atoms. This effectively provides us with a method to discriminate between two close lying states on the Bloch sphere. We discuss possible experimental implementation of the scheme, and conclude with a discussion of the Fisher information.
Single-photon optical nonlinearities face a fundamental trade-off: maximum nonlinearity requires both spectral resonance (narrow bandwidth) and high peak intensity (short duration), constraints that are incompatible due to the time-energy uncertainty relation. We demonstrate experimentally that this limitation does not need to exist in cases involving post-selection. We measure a cross-phase shift (XPS) produced by a resonant photon from a narrow-band source that is first transmitted through a cold atomic cloud and then localized in time through detection. The peak size of this XPS is greatly enhanced compared to that of Gaussian single-photon-level pulses without post-selection, benefiting from the narrow bandwidth of the resonant prepared state and the high intensity of the post-selected state simultaneously. We measure enhancements in the peak XPS of 6$\pm
1atanopticaldepth(OD)of2.4\pm$0.1, and our results are in qualitative agreement across a range of optical depths with the recently developed weak value theory of atomic excitation [Thompson et al., APL Quantum 2, 036108 (2025)] for such post-selected photons. This work uncovers new consequences of having simultaneous knowledge of frequency and time, raising new foundational questions about how a particle behaves, and interacts with other systems, when its preparation and post-selection are non-commuting.