Investigation On Photonic Gauge Potential In Synthetic Frequency Dimensions↑

The rapid development of topological properties in condensed matter physics helps people have greater understanding of electronic cognition and manipulation.Since photons can sometimes simulate electronic behavior while photonic structures have higher flexibility than that of electrons,and photonic crystals are similar to solid crystals in periodicity,people gradually expand the research to the field of topological photonics,such as unidirectional-transmitting waveguides,optical Bloch oscillations,optical topological insulators,etc.The topological properties of photons in real space are defined by photonic gauge potential and photonic AB phase which are the analogues of,in real-space light field,classic electro-magnetic field and the AB phase of electrons in electro-magnetic field,respectively.In addition to the study of topological photonics in photonic real space,it is popular to explore topological photonics in synthetic dimension as well.By using various internal degrees of freedom of photons,or by exploiting the parameters of system to form synthetic dimensions,one can create synthetic space with extra spatial dimensions.Various possibilities such as photon manipulation and topological edge states along synthetic dimension can be achieved by dynamically modulating these internal degrees of freedom.Synthetic dimension enables systems with lower dimensions to exhibit properties that only persist in systems with higher dimensions,which simplifies experimental conditions to some extent.Moreover,synthetic dimension offers great possibilities of studying and manipulating the degrees of freedom of photons.Among the degrees of freedom of photons,the photonic frequency is a competitive choice,as photonic structures naturally support modes with different frequencies which keep changing in linear time-independent systems.Synthetic frequency dimension provides a potential platform for studying unidirectional transmission in waveguides,photonic band structure,optical Bloch oscillations,etc.Considering the influence of the group velocity dispersion（GVD）of the waveguide in the synthetic dimension can provide a theoretical basis for further experimental research based on fiber loop or lithium niobate chip.In this work,based on dynamically modulating the waveguide ring resonator system by the electro-optical modulator（EOM）,simulation of the unidirectional edge states under the influence of GVD in the photonic synthetic frequency dimension is conducted.In addition,if the band structure of the synthetic frequency lattice is allowed to vary as a function of time,it will then lead to a richer set of physical properties.The method of directly measuring the dynamic band trajectory of synthetic frequency lattice using the time-resolved transmission spectroscopy can enrich the possibilities in quantum simulation and quantum information processing.Therefore,simulations and experiments of directly measuring the energy band trajectory of photonic synthetic frequency lattice are designed.This work first considers a one-dimensional array with 5 coupled ring resonators and two external waveguides.Each resonator undergoes modulation from an inserted EOM.We choose 31 sideband modes in each ring resonator so that we can configure a 5×31 lattice in the synthetic frequency space.As we set the modulation phases as₉)=9)/2,there exists an effective magnetic field in the synthetic dimension which then cultivates topological edge states.If one ignores the GVD in waveguides,the edge states propagate along the real spatial boundary and the artificial spectral boundary.However,if one takes GVD into consideration,the modulation frequency is then no longer resonant with the mode spacing between the nearest resonant modes.After the light propagating a round trip in the ring,the frequency mismatching generated by the off-resonant modes due to GVD will lead to the situation that the output filed accumulates an additional phase₈)which induces a unidirectional vague boundary at the synthetic frequency dimension.What’s more,we find it possible to control the effective area of the vague boundary in synthetic dimension by changing the magnitude of GVD.The larger the GVD,the smaller the effective area is.In particular,a relatively large GVD will break the synthetic frequency dimension.On the other hand,we again construct a synthetic lattice along the frequency dimension by dynamically modulating a ring resonator with an EOM.We at first repeat the experiments of measuring stationary band structures under on-resonant modulation,and push it forward to further studies under the influence of long-range coupling and circumstances of existing both nearest and long-range coupling.Our main work focuses on the measurement of dynamic band trajectories under near-resonant modulation.We set the modulation frequencyΩto be slightly different from the FSR,then theoretically demonstrate the equivalence between the frequency detuning and the effective force.Hence,we conclude that the off-resonant modulated one-dimensional ring resonator system is analogous to a one-dimensional solid system under constant force.We choose deferent modulation frequencies detuned from FSR to emulate different effective forces.Through theoretical calculations of band structures of 1D solid system,numerical simulations and experimental measurements by using time-resolved transmission spectroscopy of dynamic band trajectories in the1D synthetic frequency lattice,we once again demonstrate the similarity between the two systems.Afterwards,we use the same method,but 2 times or 3 times of FSR as the modulation frequency,to measure the dynamic band structure under long-range coupling.At last,we explore the effect of the modulation strength to the band trajectories and transmission spectroscopies to have a further study of the band of photonic synthetic lattices.Our work is important for further studies of photons and solid system in synthetic space.This work extends the topological photonics and band theory to the artificially constructed synthetic frequency dimension,and provides a theoretical basis for the influence of GVD of the waveguide in actual experiments.When characterizing the physics of the time-reversal manipulation of light,super Bloch oscillation and dynamic positioning of light under time-varying force,this work provides a convenient way of directly measuring the dynamic band trajectories.