Multimode Bosonic State Tomography with Single-Shot Joint Parity Measurement of a Trapped Ion

Abstract: The full characterization of a continuous-variable quantum system is a challenging problem. For the trapped-ion system, a number of methods of reconstructing the quantum states have been developed, including the measurement of the Q quasi-probability function and the density matrix elements in the Fock basis, but these approaches are often slow and difficult to scale to multi-mode states. Here, we demonstrate a novel and powerful scheme for reconstructing a continuous-variable quantum state that uses the direct single-shot measurement of the joint parity of the phonon states of a trapped ion. We drive a spin-dependent bichromatic beam-splitter interaction that coherently exchanges phonons between different harmonic oscillator modes of the ion. This interaction encodes the joint parity information into the relative phase between the two spin states, enabling measurement of the combined phonon-number parity across multiple modes in a single shot. Leveraging this capability, we directly measure multi-mode Wigner quasi-probability distributions to perform quantum state tomography of an entangled coherent state, and show that the generated state is non-positive under partial transpose, confirming its entanglement. We further show that the single-shot joint parity measurement can be used to detect parity-flip errors in real time. By post-selecting the parity measurement outcomes, we experimentally demonstrate the extension of the bosonic state lifetime, effectively implementing an error mitigation technique. Lastly, we identify the various sources of error affecting the fidelity of the spin-dependent beam-splitter operation and study the feasibility of high fidelity operations. The interaction studied in this work can be easily extended to more than two modes, and is highly relevant to continuous-variable quantum computing and quantum metrology.