# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
###### Abstract
The quantum simulator, on the other hand, starts in the state \(|\psi(0)\rangle\) and evolves to \(|\psi(\zeta)\rangle\) under the action of the operator \(\hat{U}^{\prime}=\exp\left(i\hat{O}^{\prime}\zeta/\hbar\right)\). Since we have a system and a simulator, there exists a correspondence relating these elements, that is a correspondence between \(|\phi(0)\rangle\leftrightarrow|\psi(0)\rangle\), \(|\phi(\zeta)\rangle\leftrightarrow|\psi(\zeta)\rangle\), \(\hat{U}\leftrightarrow\hat{U}^{\prime}\), and thus, \(\hat{O}\leftrightarrow\hat{O}^{\prime}\). This is revealed via measurements on both systems. The greater the accuracy of this correspondence, the more one can trust the simulator [3]. In this work, we present an integrated quantum photonic simulator for atom-field quantum interacting systems. It is based on optical gratings and can be regarded as a basic brick for constructing more complex photonic simulators. First of all, we would like to stress the high interest in quantum photonic simulators; for instance, integrated photonic structures allow simulating a number of condensed matter effects such as Anderson localization, Mott transition, etc. [6], and more recently, anyonic interaction has been simulated by using an array of channel optical waveguides with a helically bent axis [7]. In our case, we will simulate in an integrated photonic device the dynamical and topological properties of two very well-known quantum interacting systems. This is relevant since a way to check the fidelity of a simulation is testing the same physical model of the dynamics with different systems and then comparing the results [5]. The two systems to be simulated are the well-known interaction of a spin-1/2 particle under a time-dependent external classical magnetic field (see for instance [8]) and the interaction of a two-level atom with an external classical electric field (see for instance [9]) which are used, at present, for implementing quantum information devices for quantum processing, quantum computation, and so on [8]. It is also well known that these mechanical systems require a very complex technology. We will take a standard photonic device, that is a two-mode planar guide modulated by an integrated grating (see for instance [10]) in which a single-photon quantum state is propagated. We will show that it can emulate both the dynamical and geometrical properties of the two aforementioned systems. As mentioned, this photonic simulator based on integrated optical gratings can be regarded as a basic brick for more complex photonic simulators; thus, the results obtained can be extended to multi-level systems or to several concatenated two-mode systems simulating concatenated temporal operations. Addressing practical issues, this simulator will also benefit from some advantages of classical and quantum integrated photonics [6,11,12], such as the fast and energetically efficient operation and miniaturization capabilities of integrated photonics, which favor scalability. Moreover, integrated photonic devices are becoming some of the most powerful and appealing technologies for classical and quantum information [13; 14]. Finally, we must stress that although we present a photonic device for quantum simulation, it can be also used to implement quantum operations and/or effects with the photonic device itself such as logic gates, geometric phases, and so on. The plan of the paper is as follows: In Section 2, we briefly present the Hamiltonian of the quantum interacting systems, in a suitable form for their photonic simulation, along with the physical parameters and some useful solutions. In Section 3, starting from quantum states as single-photon states, an integrated photonic device for the quantum simulation of both a spin-magnetic field coupling and light-matter interaction is presented, along with the limits of this simulation. In Section 4, quantum simulation of geometric phases is studied. Finally, a summary is presented in Section 5.