Quantum Optics Tensor Networks in Time for the Design of Nonlinear Photonic Devices

Matrix product states (MPS) provide a memory efficient way to store high dimensional many body quantum systems [1]. Each degree of freedom is assigned a tensor and connecting each are bond matrices which capture correlations between different degrees of freedom as depicted in Fig 1. (a). Next efficient time evolution is then achieved by time evolving block decimation (TEBD) [2]. The essence of TEBD is that given a local Hamiltonian and a small time step the resulting unitary transformation can be decomposed into a collection of commuting two-site and single site gates acting on our MPS as shown in Fig 1. (b). With this formalism we care able to describe 1D waveguide systems with dispersion profiles up to arbitrary order and evolve states under a Hamiltonian with a $\chi^{(3)}$ non linearity [3]. Linear loss and two photon absorption can be included in the model through the materials imaginary dispersion data. Arbitrary spatial, temporal and spectral pump profiles may also be included in the driving of parametric processes. Given an initial quantum field and classical drives we can simulate the temporal dynamics of the quantum field. From this we extract the familiar measurable quantities including spectral content, temporal profile, correlators or any well defined observable of the field. In Fig 2. (c) We show the spectrum of a weak coherent state at $2050\text{nm}\ (\vert\alpha\vert =1)\ (\text{red})$ , which undergoes stimulated 4 wave mixing via a classical pump at 2070nm (purple). After the interaction the resulting spectra (blue) contains stimulated photon generation at the phase matched idler wavelength of 2090nm and parametric amplification of the stimulating field. Fig 2. (d) shows an example of the field expectation value and variance for the underlying quantum state before (red) and after (blue) the interaction.