Abstract
Synthetic dimensions have garnered widespread interest for implementing high dimensional classical and quantum dynamics on lowdimensional geometries. Synthetic frequency dimensions, in particular, have been used to experimentally realize a plethora of bulk physics effects. However, in synthetic frequency dimension there has not been a demonstration of a boundary which is of paramount importance in topological physics due to the bulkedge correspondence. Here we construct boundaries in the frequency dimension of dynamically modulated ring resonators by strongly coupling an auxiliary ring. We explore various effects associated with such boundaries, including confinement of the spectrum of light, discretization of the band structure, and the interaction of boundaries with oneway chiral modes in a quantum Hall ladder, which exhibits topologically robust spectral transport. Our demonstration of sharp boundaries fundamentally expands the capability of exploring topological physics, and has applications in classical and quantum information processing in synthetic frequency dimensions.
Introduction
The concept of synthetic dimensions^{1,2,3}, whereby various degrees of freedom of atoms or photons are used to mimic spatial dimensions, is of significant recent interest for simulating highdimensional phenomena on systems with fewer geometric dimensions. Synthetic dimensions have been formed by coupling states labeled by degrees of freedom such as spin^{1,4}, frequency^{5,6}, orbital angular momentum (OAM)^{7}, time bins^{8,9,10} or transverse spatial supermodes^{11}. Many interesting physical effects, including nontrivial topological phenomena and effective gauge fields for neutral ultracold atoms or photons, have been realized in synthetic dimensions.
Specifically for topological phenomena, constructing a sharp boundary in the synthetic dimension is of central importance. An essential concept in topological physics is the bulkedge correspondence, which relates the existence and properties of edge modes in a finite lattice to the quantized topological invariant of the corresponding bulk (infinite) lattice. For Hermitian systems, examples of bulkedge correspondence include the oneway chiral edge states at the boundary of a Chern insulator^{12}, the zeroenergy edge modes of a SuSchriefferHeeger model^{13}, and the recently discovered corner modes of a higherorder topological insulator^{14,15,16}. Moreover, the bulkedge correspondence has also been generalized to nonHermitian systems, leading to intriguing phenomena such as the nonHermitian skin effect^{17,18,19,20}. Creating a boundary in the synthetic dimension is essential for further exploration of such phenomena in synthetic space. In addition, the creation of boundaries in synthetic dimensions is important for applications such as implementing arbitrary linear transformations for frequency conversion, quantum circuits, and photonic neural networks^{21}.
A prominent approach to create synthetic dimensions is to use the frequency modes of a ring resonator. Synthetic frequency dimensions have enabled experimental demonstrations of a plethora of bulk physical effects. For Hermitian systems, examples of these effects include Bloch oscillations^{22,23,24,25}, effective electric and magnetic gauge fields^{26,27,28,29,30}, spinorbit coupling and consequent spinmomentum locking^{27}, complex longrange coupling^{31,32}, and chiral currents originating from the nontrivial topology of the quantum Hall effect^{27}. For nonHermitian systems, nontrivial eigenvalue topology such as topological winding or braiding of the energy bands have also been recently observed in frequency dimensions^{33,34}. However, experimentally probing the edge implications of these bulk topological phenomena has remained an open challenge in synthetic frequency dimensions. Unlike systems in real space, synthetic lattices created using frequency modes typically do not have a welldefined boundary. In the absence of boundaries or defects, the robustness of light transport^{35}, which is one of the hallmarks of topological phenomena, has not been observed along the frequency axis.
In this paper, we provide an experimental demonstration for constructing boundaries in synthetic frequency dimensions. Previous theoretical works have investigated syntheticspace boundary effects by assuming sharp^{5} or gradual^{36} changes in the groupvelocity dispersion of the waveguide forming the ring resonator, by strongly coupling an auxiliary ring^{21}, or by including memory elements^{37}. Here we experimentally realize the approach of coupling to auxiliary ring resonators. We observe that an excitation within the finite lattice stays confined between the boundaries in synthetic space, resulting in the discretization of the band structure in reciprocal space. We also implement boundaries in a synthetic quantum Hall ladder geometry and demonstrate oneway propagation of topological chiral edge states that are immune to back reflection despite the presence of a boundary, thus constituting an observation of topologically robust transport of light along the frequency axis. With the added functionality of creating sharp edges, we anticipate the observation of higherdimensional boundary phenomena that have been beyond the purview of realspace or syntheticspace topological photonics.
Results
Creation of boundaries in one dimension
Consider a single ring resonator of length L_{0} made of a waveguide with group velocity v_{g} (Fig. 1a). In the absence of group velocity dispersion, the ring supports cavity modes equispaced in frequency by the freespectral range (FSR) Ω_{R} = 2πv_{g}/L_{0}. To excite these modes we couple the ring with an external waveguide at an amplitude coupling ratio γ_{0}. The resulting transmission spectrum, assuming that all the ring modes are critically coupled with an internal loss rate equal to the external coupling loss rate, is shown in Fig. 1c. The spectrum features a periodic array of resonant dips equally spaced by the FSR. These modes can be coupled to form a onedimensional (1D) synthetic frequency lattice by electrooptically modulating the refractive index of a small portion of the ring at a modulation frequency Ω_{M} = Ω_{R}^{5,6,32}. The Hamiltonian for such a system is^{32,38},
where \({b}_{m}\,({b}_{m}^{{{{\dagger}}} })\) is the annihilation (creation) operator for a mode at frequency ω_{m} = ω_{0} + mΩ_{R}. For a single ring with ω_{0} ≫ Ω_{R}, a very large number of modes (M > 100) can be coupled along the synthetic frequency dimension, as demonstrated experimentally in refs. ^{25,32}. Thus a single modulated ring closely approximates the bulk behavior (M → ∞) of a lattice.
To truncate such a lattice and create boundaries, we couple an auxiliary ring resonator of a smaller length L_{a} < L_{0}, corresponding to a larger FSR Ω_{R,a} = 2πv_{g}/L_{a} (Fig. 1b). Here we have assumed that the auxiliary ring is made of a waveguide with the same group velocity as the main ring, and is coupled to the main ring via a directional coupler with an amplitude coupling coefficient γ_{a}. Note that similar geometries have previously been used for optical communications, flatband lattices, reconfigurable frequency conversion, and demonstrating coupledresonator induced transparency^{39,40,41,42,43,44,45}.
As an illustration, Fig. 1d shows the spectral positions of the main cavity and auxiliary ring modes for N = L_{0}/L_{a} = 6 in the absence of modulation. The corresponding transmission spectrum is plotted in Fig. 1e. Near the frequencies where the resonances from the two rings align, if \({\gamma }_{a} \, > \, {\gamma }_{0}^{2}/2\), a splitting is induced (Fig. 1f). Here \({\gamma }_{0}^{2}\) is the power splitting ratio of the directional coupler between the inputoutput waveguide and the main ring. Unlike the spectrum in Fig. 1c, the spectrum here in Fig. 1e is no longer periodic with respect to translation by Ω_{R} along the frequency axis.
When the modulation is again introduced in the main ring with a modulation frequency Ω_{M} = Ω_{R}, the modulation can induce the transition between some of the modes. Specifically in Fig. 1e, the green arrows represent the allowed modulationinduced couplings along the frequency dimension, whereas the red crosses represent inhibition of the coupling to modes that are perturbed by the auxiliary ring. A series of several finite lattices are formed, which are separated by the split resonances induced by the auxiliary ring. The presence of the auxiliary ring thus can introduce a sharp boundary in the synthetic dimension.
Characterization of the unmodulated resonators
To experimentally characterize the resonator in the absence of modulation, we measure the transmission spectra (Fig. 2) in an experimental realization of the setup shown in Fig. 1a. The details of the experiments, which are implemented using fiber rings, are provided in Supplementary Information Section I. Without the auxiliary ring, the transmission features a set of resonant dips, with minimum transmission \({T}_{\min }\approx 0.7\) that are similar for all the dips. These dips correspond to the resonances of the main ring. The frequency spacing of the nearest resonances as a function of the order of resonances is plotted as the blue line in Fig. 2c. We see that the frequency spacing is nearly a constant. In the presence of coupling to the auxiliary ring, there is a marked increase in \({T}_{\min }\) near the main cavity modes that are aligned to the auxiliary ring modes (Fig. 2b). The increase in \({T}_{\min }\) is in accordance with scattering matrix simulations including a loss in the auxiliary ring (inset of Fig. 2b), and this loss was ignored in Fig. 1d, e for simplicity. Around the resonant frequencies of the auxiliary ring, we see that the resonances of the coupled system are no longer equally spaced (orange line in Fig. 2c). In addition, for the coupled system, the frequency spacings between modes far away from the resonances of the auxiliary ring, which we define as the FSR of our coupled ring system, is smaller as compared to the FSR of the main ring by itself (Fig. 2c, see Supplementary Section II for an analytical derivation of this effect).
Measurement of boundary effects in 1D lattice space
For the remainder of the paper, we will consider a modulated resonator. We first demonstrate the effect of a boundary created by the auxiliary ring by measuring the steadystate intensity distribution in the synthetic frequency dimension (Fig. 3) in the presence of modulation. We excite the system at a frequency ω_{in} near one of the resonances of the main ring, the order of which is denoted by m_{0}. ω_{in} is gradually swept, and the detuning \({{\Delta }}\omega ={\omega }_{{{{{{{{\rm{in}}}}}}}}}{\omega }_{{m}_{0}}\) forms the vertical axis in Fig. 3a, b, d–f. At each input frequency, the frequencylattice distribution of the steadystate cavity field is obtained from a heterodyne measurement of the transmitted field^{46}. This frequency sideband number is denoted by m − m_{0} along the horizontal axis in Fig. 3.
In the absence of the auxiliary ring, the transmitted field contains a large number of sidebands (Fig. 3a). This experimental data matches well with the simulated spectrum in Fig. 3b which was calculated using a Floquet scattering matrix analysis. The steadystate field intensity of the mth sideband away from the input falls off exponentially as \({\sim} {\exp}( m{m}_{0} /{\tau }_{p}J)\) (see Fig. 3c bottom), for large m − m_{0}^{25}, where τ_{p} and J are the ring photon lifetime and the modulation strength respectively.
On the other hand, when the auxiliary ring is coupled to the main ring, the output field contains a far smaller number of sidebands. This indicates that within the ring, the only modes excited are those that lie between the two boundaries along the frequency axis (experiment: Fig. 3d, simulations: Fig. 3e). We also observe interference fringes created by reflections from the boundaries. Note that the strengths of the fringes increase with an increase in the modulationinduced coupling strength, since light is able to traverse along the frequency axis for longer distances before getting dissipated. However, the strong confinement of light to within the boundaries is preserved as long as the splitting induced by the auxiliary ring resonator is larger than 2J. Figure 3f illustrates the spectra upon exciting various lattice sites within the two boundaries. This result was obtained by sweeping the input laser detuning over a large range Δω ≫ Ω_{R}. Since the measured heterodyne spectrum is always referenced to the input laser frequency mode m_{0}, we observe a shift of the output spectrum towards lower frequency sidebands as m_{0} increases.
Measurement of 1D boundary effects in reciprocal space
An infinite lattice that obeys discrete translational symmetry can be characterized by a conserved continuous quantum number, the Bloch quasimomentum k ∈ [0, 2π), which labels the bulk properties in reciprocal space. For each k, one or more continuous bands are formed which correspond to the eigenenergy spectrum of the infinite lattice. In the frequency synthetic dimension, the wavevector along the frequency axis corresponds to a time variable. We have previously demonstrated a syntheticspace band structure spectroscopy technique^{32}. In this technique, we scan the input frequency of a continuouswave laser. For each frequency, after the transient dissipates, we measure the transmission intensity as a function of time. Since the time corresponds to the wavevector k along the synthetic frequency dimension, the resulting twodimensional plot of transmission as a function of frequency and wavevector then provides a measure of the bandstructure. An example of such a measurement, for our system in the absence of the auxiliary ring, is shown in Fig. 4a. The locations of the peaks in the frequencywavevector space closely match the band structure of a onedimensional tightbinding model with nearestneighbor coupling.
We repeat the same measurement in the presence of the auxiliary ring (Fig. 4b). We see strong excitation of the system only at a discrete set of frequencies, as expected since the presence of the two boundaries results in a discrete set of eigenstates. For each of these eigenstates, the wavevector components spread over a range, centered at approximately where the wavevector would be at the same frequency for the infinite system. The experimental results in Fig. 4b agree excellently with numerical simulation results shown in Fig. 4c based on a Floquet scattering matrix analysis of the coupled ring system. Moreover, the numerical results indicate that the discrete eigenfrequencies that we observe in Fig. 4b agree with tightbinding simulations (Fig. 4d) where open boundaries are imposed on the two ends of a finite lattice, providing further evidence of a sharp boundary that we create.
Demonstration of boundary effects in a quantum Hall ladder
We now demonstrate the effect of boundary on a topologically nontrivial system, the twoleg quantum Hall ladder^{47}, and show how it enables us to observe topologically robust transport of light along the frequency axis. To construct a twoleg quantum Hall ladder, we use a setup schematically shown in Fig. 5a, where we couple a pair of main ring resonators. The main ring on the left is in addition coupled to an auxiliary ring. We ensure that the FSR of the main ring on the right matches the FSR of the coupled system consisting of the main ring on the left together with the auxiliary ring. We modulate both of the main rings at a frequency Ω_{M} = 2π ⋅ 5.28 MHz, which matches the FSR, with a relative phase difference ϕ in the modulations on the two rings^{5}. The resulting Hamiltonian then describes a twoleg quantum Hall ladder^{4,27,47} (Fig. 5b):
where N_{L} and N_{R} represent the number of frequency modes in the left and right legs of the ladder respectively, and N_{L} < N_{R} due to the presence of the auxiliary ring that couples to the main ring on the left. J is the modulationinduced hopping along the synthetic frequency dimension. K represents the coupling between the two legs of the ladder, determined by the splitting ratio of the directional coupler that couples the two main rings together. The model in Eq. (2) exhibits a uniform effective magnetic flux ϕ permeating each square plaquette of the lattice. For ϕ ≠ 0, π, timereversal symmetry is broken; such a model then supports oneway chiral states on each leg which are immune to back reflections from the boundary or corner (Fig. 5g, h). This oneway nature derives from a parent 2D quantum Hall insulator which manifests strong topological protection^{47,48}. Thus, the setup allows us to study the interaction of boundaries with the topologically protected oneway chiral modes in a quantum Hall ladder.
To demonstrate the effect of the boundary as induced by the auxiliary ring, we excite the left main ring in the setup as shown in Fig. 5a. We choose the excitation frequency to match one of the lattice sites away from the boundary (Fig. 5c and g). In the case of ϕ = 0, the band structure for an infinite twoleg system is shown in Fig. 5d. Since the system has timereversal symmetry, the eigenstates equally occupy the left and the right legs and the system does not exhibit any chiral behavior. Consequently, with the excitation as shown in Fig. 5c, we expect that the generated field will propagate to both sides of the excitation site. Also, we expect to see interference fringes between the site of excitation and the boundaries. In Fig. 5e, we show the experimental results for this case where we measure the spectrum of the transmitted light via heterodyne detection (see Methods). We indeed observe that the output field contains strong components on both sides of the excitation site m = m_{0}. In Fig. 5f, we plot the amplitude at various lattice sites for Δω/Ω_{R} = 0.11. We observe interference fringes due to the presence of the boundaries (indicated by green vertical lines), as exemplified by the dips at m − m_{0} = ±2.
In the case of ϕ = π/2, the band structure for the infinite system is shown in Fig. 5h. Since the system breaks timereversal symmetry, the eigenstates show asymmetry in occupation between the left leg and the right leg, as illustrated in Fig. 5h where the color gradient shows the projection of the eigenstate on the left leg. Hence, with the excitation shown in Fig. 5g where the left leg is excited, we expect that the generated field will propagate to higher frequencies for the lower band, and to lower frequencies for the upper band, as determined by the sign of the group velocities of the chiral modes in Fig. 5h. Also, we do not expect to see interference fringes between the site of excitation and the boundaries, since the oneway nature of the chiral modes should suppress back reflection from the boundaries (schematics in Fig. 5g). In Fig. 5i, we show the experimental results for this case where we measure the spectrum of the transmitted light via heterodyne detection. Strikingly different from Fig. 5e, we indeed observe that the output field contains frequency components almost exclusively for modes to the left of the excitation (m − m_{0 }≤ 0) for the upper band, in the oneway detuning range shaded in pink in Fig. 5h. The direction of frequency conversion switches for the lower band. In Fig. 5f, we plot the experimentally measured amplitude at various lattice sites as the orange curve, which agrees well with Floquet scattering matrix simulations (red dots). The oneway nature, as well as the absence of interference fringes, are borne out in this amplitude distribution in frequency lattice space. Fig. 5j plots the amplitude distribution for a wide range of detuning Δω, corresponding to the excitation of different lattice sites m_{0} along the frequency dimension. We observe that the topological robustness of light transport, as evidenced by the oneway nature and the lack of fringes, persists as we excite modes with different distances from the boundary. Note that the persistence of oneway propagation in the twoleg ladder limit attests to the topological robustness of the full 2D quantum Hall lattice independent of the boundary along the frequency axis. This is because the ladder preserves the modal structure of the edge states of the full 2D lattice in spite of the removal of all the bulk sites from the full 2D lattice, as predicted theoretically in ref. ^{47}.
Discussion
We have demonstrated the construction of sharp boundaries in synthetic dimensions by coupling an auxiliary ring resonator to a dynamically modulated ring, using a platform based on optical fibers. Recent progress in nanophotonic electrooptic modulators^{49,50} incorporated into lowloss microring resonators provide opportunities for scalable onchip integration of such concepts. This approach can be generalized to higher dimensions for exploring nontrivial topological boundary phenomena^{25,51,52}, both in conventional topological insulators as well as in higherorder topological insulators. While our demonstrations were limited to the simplest case of nearestneighbor coupling, there are several ways to create boundaries in the presence of longrange coupling^{28,32}, a feature that is readily accessible in synthetic frequency dimensions. Examples include: (i) using multiple incommensurate rings, (ii) using perturbations to the cross section of the ring (as recently demonstrated in ref. ^{53,54}), and (iii) using dispersion engineering of the waveguide that comprises the ring. Our results also show that the energy of a synthetic lattice can be confined to a finite number of sites by coupling to additional auxiliary resonators, which is critical in efficient implementations of linear transformations or matrixvector multiplications^{21}. Our work should significantly advance the capabilities of synthetic dimensions in both topological photonics and for quantum^{55} and classical signal processing.
Methods
Experimental details
In this section, we provide a detailed description of the experimental setup corresponding to Fig. 5 of the main text. The setups for Figs. 2–4 can be obtained by disconnecting the second main ring on the right in Fig. 5a. We use a fiber ring resonator^{32,46}, with a lithium niobate phase modulator in each ring as the electrooptic modulator (EOM). The rings are excited by a lownoise continuouswave laser (RIO Orion)^{56}, with a narrow linewidth <3 kHz. The main rings have a length of L_{0} ≈ 38.6 m, corresponding to a freespectral range (FSR) of 5.35 MHz. Both the main fiber rings are coupled to through and drop ports to enable an independent calibration of the FSR of each ring when the coupling between the two rings is absent. The FSRs of the two rings are passively equalized by measuring the FSR of each ring and adding extra lengths of fiber or freespace sections in the second ring to compensate for the difference. The auxiliary ring consists of a loop of fiber containing a fiber polarization controller. The electrical signals used to drive the two modulators are derived from the same fieldprogrammable gate array (FPGA), to ensure phase synchronization over long timescales. The phase difference between them was precisely controlled in software, and could be varied across the entire range [0, 2π). By contrast, when independent function generators were used to drive the two modulators, we observed a continuous drift in the phase offset. Hence it was important to use two modulation signals derived from the same FPGA clock. The modulation signals were amplified by RF amplifiers before driving the EOMs. Each of the main rings also had an erbiumdoped fiber amplifier (EDFA) to compensate for the losses from the EOM and other components. The auxiliary ring resonator had no EDFA. The amplified spontaneous emission noise from the EDFA was filtered using a densewavelength division multiplexing (DWDM) filter with a passband of 26.5 GHz centered at 1542.12 nm.
To enable strong coupling between the two rings (Kτ_{p} > 1), where τ_{p} is the photon lifetime in the main ring resonators and K is the coupling rate between the main rings, we use polarizationmaintaining (PM) fiber components at the junction between them. This is because the polarization axes of the two rings set by the EOMs are otherwise independent. The PM sections are illustrated by the blue fibers in Supplementary Fig. S1. The splitting ratio of the intercavity coupler is 75:25, that of the input coupler is 95:5, and that of the auxiliary ring coupler is 60:40. The through port signal was monitored on a slow photodiode (e.g., Fig. 2), whereas the drop port signal was sent to a fast photodiode with a bandwidth of 5 GHz after optical amplification with a semiconductor optical amplifier (not shown). For band structure measurement (Fig. 4), the drop port signal was directly sent to the photodiode in this manner. For frequency latticespace measurements (Figs. 3 and 5), the drop port output was first mixed with the output of an acoustooptic modulator (AOM) before sending it to the fast photodiode. The AOM shifted a part of the input laser by the RF drive frequency of 500 MHz to enable heterodyne detection of the cavity output.
Comparison with active modelocking: In our setup, the presence of both an EDFA and a modulator is similar to that of an actively modelocked laser. However, a few important differences exist: (i) The setup is operated completely below the lasing threshold; (ii) The input is around the same wavelength as the output in the 1550nm band, as opposed to lasers where the pump is at a significantly shorter wavelength than the lasing output. In our experiments, the EDFA only plays the role of mitigating roundtrip losses to achieve a high effective finesse for the cavity. If one operates the EDFA at gain higher than the roundtrip loss, especially with an amplitude modulator, actively modelocked pulses can be produced^{57,58}.
Data availability
The data generated in this study have been deposited in the Zenodo database under accession code 10.5281/zenodo.6516650 (https://doi.org/10.5281/zenodo.6516650).
Code availability
The codes used to process the data generated in this study have been deposited in the Zenodo database under accession code 10.5281/zenodo.6516650 (https://doi.org/10.5281/zenodo.6516650).
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Acknowledgements
We acknowledge David A. B. Miller for providing lab space and equipment, and Meir Orenstein and Momchil Minkov for useful discussions. This work is supported by a Vannevar Bush Faculty Fellowship from the U.S. Department of Defense (Grant No. N000141713030) and a MURI project from the U.S. Air Force Office of Scientific Research (Grant No. FA95501810379). L.Y. also acknowledges the support of the National Natural Science Foundation of China (11974245, 12122407).
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A.D., L.Y., and S.F. conceived the project. A.D. carried out the experiments with assistance from K.Y.Y. and K.W. S.B. and A.D. developed the Floquet simulations. A.D., L.Y., K.Y.Y., K.W., J.V., and S.F. discussed the results and contributed to writing the manuscript. S.F. supervised the project.
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Dutt, A., Yuan, L., Yang, K.Y. et al. Creating boundaries along a synthetic frequency dimension. Nat Commun 13, 3377 (2022). https://doi.org/10.1038/s41467022311407
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DOI: https://doi.org/10.1038/s41467022311407
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