Bosonic many-body topological phase with synthetic time dimensions

Optical networks composed of interconnected waveguides are a versatile platform to simulate bosonic physical phenomena. Significant work in the non-interacting regime has demonstrated the capabilities of this platform to simulate many exotic effects such as photon transport in the presence of gauge fields, dynamics of quantum walks, and topological transition and dissipation phenomena. However, the extension of these concepts to simulating interacting quantum many-body phenomena such as the Bose-Hubbard and the fractional quantum Hall (FQH) physics has remained elusive. In this work, we address this problem and demonstrate a framework for quantum many-body simulation as well as drive and dissipation in photonic waveguides. Specifically, we show that for waveguide photons, a tunable on-site interaction can be simulated using a photon-number-selective phase gate. Furthermore, we show that such a phase gate can be physically implemented using three-level-atom-mediated photon subtraction and addition, and can achieve a gate fidelity arbitrarily close to one for any input photon number. We apply this approach to bosonic lattice models and propose circuits that can accurately simulate the Bose-Hubbard and FQH Hamiltonian as benchmarking examples. Moreover, we show how to simulate the Lindbladian evolution with engineered drive and dissipation such that the steady state of the Lindbadlian corresponds to the ground state of desired Hamiltonians. Our scheme extends the time-multiplexed waveguide photonic simulation platform to the strongly interacting quantum many-body regime while retaining all of its crucial advantages, such as single-site addressability, Hamiltonian parameter controllability, and hardware efficiency. Moreover, we quantitatively analyze the effect of dominant sources of experimental imperfections on our proposals, and show that many of the quantum many body effects associated with our simulation protocols can be readily observed under realistic conditions.