Simulation And Modulation In Synthetic Dimensions Based On Optical Orbital Angular Momentum

Since Feynman proposed the quantum simulation in the 1980s,it has experienced more than 40 years of development.Quantum simulation has achieved great success in theory and experiment,and has become an important branch of quantum information science.Many phenomena in condensed matter physics,particle physics,cosmology and atomic physics have been simulated in different experimental platforms,such as neutral atoms,ion traps,coupled cavity arrays,quantum dots,superconducting circuits and photonic crystals.However,the above traditional systems utilizing spatial degrees of freedom are hard to simulate physics.However,due to the limitation of the 3-dimensional physical space in the real world,the above quantum simulation platforms are hard to simulate models in high-dimensional(greater than 3-dimensional)and long-range interaction systems.The non-spatial degrees of freedom,which are called as synthetic dimensions,are utilized to overcome these difficulties.Combining synthetic dimensions and spatial degrees of freedom,in principle,people can build systems in higher dimensions.Moreover,the coupling between cites in the synthetic dimension is not restricted by the real spatial distance,so the effective long-range transition can be realized.In synthetic dimensions,great success,especially in photonic systems,has been achieved,such as high-dimensional topological systems,high-dimensional dynamics,two-dimensional solitons and non-Hermitian systems.Compared with the real spatial dimensions,synthetic dimensions have two significant advantages.Firstly,many non-spatial degrees of freedom have no upper limit in theory,such as frequency,orbital angular momentum(OAM),momentum,pulse sequence.This feature gives these degrees of freedom a unique advantage in studying physics in large-scale systems.Secondly,through the ingenious design of the coupling scheme,a system with several degrees of freedom in synthetic dimensions can be simulated by a small number of devices in the real space.By modulating a small number of physical devices in the real space,effective collaborative operations between a large number of degrees of freedom in synthetic dimensions can be realized(e.g.,the tunneling coefficients between adjacent sites in synthetic lattices can be tuned at the same time),and the control complexity is significantly reduced.These advantages make synthetic dimensions useful for scalable quantum simulation.Besides simulating physical systems,synthetic dimensions are also helpful to the design of all-optical devices.Based on the modulation of these degrees of freedom in synthetic dimensions,many schemes have been proposed,such as optical isolator,single pulse manipulation,photonic quantum storage and photonic quantum computation.In this thesis,the OAM modes in a single degenerate optical cavity constitute the synthetic dimension.Compared with other optical synthetic dimensions,the couplings between OAM modes can be introduced more freely.This feature brings two conveniences.First,when we use several devices to couple OAM modes,the influence between devices is very small,which makes the system versatile to construct lattices.Second,the system does not occupy the time dimension,in which we can study the time-varying problem.These advantages make our optical OAM system attractive for scalable quantum simulation.First,we introduce the framework of the photonic OAM lattice in degenerate cavity.Then,we give the scheme of the input and output process in the OAM lattice by the scattering theory.Finally,based on the above research methods,we propose the schemes to simulate the effective electric fields and a local Kerr nonlinearity in the optical OAM lattice,and show that those schemes can be utilized as photonic quantum storage and two-photon correlation filter,respectively.The details are as follows:1.We briefly review the idea and concept of synthetic dimension as well as three classes of synthetic dimensions.Then we introduce synthetic dimensions based on optical OAM modes in degenerate optical cavity.Next,the eigenmode of OAM in the optical cavity,Laguerre Gauss mode,is derived,and the degenerate conditions of the cavity are given by the transfer matrix and Huygens-Fresnel integral.The effective tight-binding Hamiltonian is derived by mapping the OAM modes in the cavity to the coupled-resonator optical waveguide.Finally,we introduce a series of devices to manipulate the photonic OAM,and review the schemes to realize input-output channels,two-site unit-cell and open boundary lattices in the degenerate optical cavity.2.We consider the input and output problems of the degenerate cavity.The input and output are realized by locally coupling OAM synthetic lattice to a waveguide.This process can be regarded as single impurity scattering.Therefore,the relation between the incident state and the outgoing state is given by the scattering eigenstate method and the S-matrix method respectively.As an example,we analyze the processes of the single-photon input and output,and give the transmission and reflection amplitudes of quasi-monochromatic waves.Both methods can be extended to arbitrary lattice systems.However,the former method is intuitive and easy to derive,but specifically related to the local Hamiltonian.The latter method is general but cumbersome.3.We simulate the electric field in OAM and propose a scheme of photonic quantum storage.In earlier researches of OAM-based quantum simulation,effective magnetic field and related effects have been realized.We consider the electric field simulation in OAM lattice to complete the system.Due to the novel design,we can simulate both static and time-dependent electric fields,which allows to study Bloch oscillations and a variety of related effects like Bloch translation,dynamic localization and super Bloch oscillation.We present the entire process of simulation,including initial state preparation,evolution and measurement.In the measurement of the wavepacket in the OAM lattice,we propose a non-intrusive measurement scheme to detect the system dynamics,which can be used as a general measurement scheme in the degenerate optical cavity.4.After studying the single-photon effect in the simulated electric field,we introduce photon-photon interaction into the OAM space to study the two-photon effect.The local Kerr nonlinear interaction is realized in the OAM lattice by the polariton.We solve the two-photon S-matrix of the system,and analyze the background fluorescence,wave functions in the frequency space and the second-order correlation function of the waveguide mode.In the frequency space,the two-photon wave functions show sharp multipeak distributions,which are significantly different from the spontaneous parametric down-conversion.The two-photon correlation properties are significantly dependent on the incident frequency and the properties of the OAM lattice.In addition,by tuning the coupling parameters and incident frequency,the single photon wavepacket can be almost completely transferred to the synthetic lattice,thus wavepacket in the waveguide is completely introduced by Kerr interaction.The outgoing photons are strongly correlated even with small Kerr nonlinearity.Combining with the high tunability of the degenerate cavity,our system can be used as a tunable two-photon correlation filter.