Counter-factual carving exponentially improves many-body carved state fidelity

We propose a new, "counter-factual" method for carving a broad class of entangled states of many atoms coupled to a cavity mode, resulting in an exponentially better scaling of the carved state infidelity with the cavity cooperativity compared to previous methods. For many atoms initialized in a classical spin coherent state, different non-classical spin components, or Dicke states, shift the Lorentzian cavity lineshape by an amount dependent on the atom-cavity coupling strength. For a finite energy shift between different Dicke states, the polynomial tail of the lineshape means that directly probing the shifted spectrum can only herald a nonclassical state with an infidelity scaling polynomially with the cavity quality. Instead, we propose addressing a single photon "source atom" within the cavity with a spectrum of tones tuned to the shifted cavity resonances to engineer photon emission rates which depend on the atomic ensemble state. A heralding measurement of the internal state of the source atom consistent with no photon emission projects the ensemble onto a state where the faster decaying spin components have been exponentially suppressed. Carving "counter-factually," by post-selecting on events where a photon was never emitted from the source atom, then enables an exponentially better scaling of the carved state fidelity with the cavity cooperativity. Applying many tones to the source atom allows arbitrary amplitude and phase control of the Dicke state components. Applications of the method include producing high-fidelity many-atom Dicke states and GHZ states useful for quantum metrology, and graph states useful for measurement based quantum computing.