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Quantum weak measurements, wavepacket shifts and optical vortices are universal wave phenomena, which originate from fine interference of multiple plane waves. These effects have attracted considerable attention in both classical and quantum wave systems. Here we report on a phenomenon that brings together all the above topics in a simple one-dimensional scalar wave system. We consider inelastic scattering of Gaussian wave packets with parameters close to a zero of the complex scattering coefficient. We demonstrate that the scattered wave packets experience anomalously large time and frequency shifts in such near-zero scattering. These shifts reveal close analogies with the Goos-Hänchen beam shifts and quantum weak measurements of the momentum in a vortex wavefunction. We verify our general theory by an optical experiment using the near-zero transmission (near-critical coupling) of Gaussian pulses propagating through a nano-fibre with a side-coupled toroidal micro-resonator. Measurements demonstrate the amplification of the time delays from the typical inverse-resonator-linewidth scale to the pulse-duration scale.

Number-phase minimum uncertainty states and number states are shown to be realized by quantum nondemolition measurements of the photon number. Interphase modulation in a Kerr medium establishes a quantum-mechanical correlation between the signal and probe waves initially in coherent states. State reduction is realized by homodyne detection of a probe quadrature-phase amplitude. The number uncertainty can be arbitrarily reduced by increasing the probe intensity. The minimum uncertainty relation between the number and sine operators approximately holds until a near-number state is realized.

Number-phase minimum uncertainty states and number states are shown to be realized by quantum nondemolition measurements of the photon number. Interphase modulation in a Kerr medium establishes a quantum-mechanical correlation between the signal and probe waves initially in coherent states. State reduction is realized by homodyne detection of a probe quadrature-phase amplitude. The number uncertainty can be arbitrarily reduced by increasing the probe intensity. The minimum uncertainty relation between the number and sine operators approximately holds until a near-number state is realized.

Number-phase minimum uncertainty states and number states are shown to be realized by quantum nondemolition measurements of the photon number. Interphase modulation in a Kerr medium establishes a quantum-mechanical correlation between the signal and probe waves initially in coherent states. State reduction is realized by homodyne detection of a probe quadrature-phase amplitude. The number uncertainty can be arbitrarily reduced by increasing the probe intensity. The minimum uncertainty relation between the number and sine operators approximately holds until a near-number state is realized.