Quantum many-body dynamics of driven-dissipative Rydberg polariton systems

Abstract

We present the successful solution of the notoriously hard but experimentally highly relevant many-body problem of driven-dissipative Rydberg polaritons in an optical lattice. From the continuous model, we derive a dispersion relation and construct the respective Wannier functions of the bands. We derive a Bose-Hubbard-like model for the dark-state polaritons in consideration of the interaction with the surrounding bright-state polariton bands. We then study the dynamics of this system with the Lindblad master equation to also include dissipative processes. To solve it, we use a variational approach in combination with a hard-sphere constraint to describe the Rydberg blockade. We compare these results with the solution from a Monte-Carlo wave function simulation where we find a good agreement between the two solutions for larger system sizes. We then use the variational approach to obtain a result for a system size of $N= 40$, which far exceeds the possible site number obtainable via Monte-Carlo simulations. In the end, we also study the time correlation between measurements of two photons leaving the system after they travelled through the entire lattice. The problem of the description of the infinite Hilbert space that occurs in bosonic fields is also a highly relevant topic in the context of many-body systems. Here, we show a new way to use the variational principle to describe any bosonic field by the usage of the P-representation of the density matrix in combination with the formulation of the master equation in terms of equations of motion. By investigating the bistability region of the driven-dissipative Jaynes-Cummings model, we show that our method exceeds conventional mean-field descriptions. We then extend the new approach by adding correlation functions as additional variational parameters to describe correlations between different parts of the Hilbert space. This allows us to investigate the effective three-boson model which describes Rydberg atoms inside a driven cavity with dissipation channels for the atoms and the cavity. The Rydberg-Rydberg interaction leads to correlations between the atoms which in turn cause a squeezing effect of the different bosonic modes. By a transformation of the model into the polariton picture, we show how the squeezing mostly affects the dark-state polaritons and its scaling with the pumping rate of the cavity. The last part focuses on the dissipative Ising model and its realisation through the coupling of internal states of ions and their vibrational modes. Furthermore, we present a measurement protocol that reduces unwanted detuning terms that can arrive from systematic errors in experiments.