If one has access to information about a quantum system both before and after a measurement, the usual remit of Heisenberg uncertainty relation (HUR) does not apply. Here, we theoretically and experimentally demonstrate that, in such a scenario, one can retrodict position and momentum measurements without being limited by HUR. The experiment uses a large ensemble of 10^11 atoms in a macroscopic vapor cell, and the results may be applicable for quantum state estimation and sensing, especially for sensitive magnetometers equally capable of measuring the amplitude and phase of an RF magnetic field both below the standard quantum limit.
Quantum interface between photons is a long- standing goal of fundamental significance, and also serves as powerful tools for quantum technologies. Remarkable advances in quantum optics have recently developed in several platforms to demonstrate the generation of optical nonlinearities at the level of individual photons, which enable a number of unique applications such as light-by-light control at quantum level. Here, we present two examples in the realm of continuous variables based on the platform of flying atoms where thermal motion of atoms with long-lived coherence of ground state mediate the coupling between the spatially separated optical channels.
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