Abstract
There has been significant recent interest in synthetic dimensions, where internal degrees of freedom of a particle are coupled to form higherdimensional lattices in lowerdimensional physical structures. For these systems, the concept of band structure along the synthetic dimension plays a central role in their theoretical description. Here we provide a direct experimental measurement of the band structure along the synthetic dimension. By dynamically modulating a resonator at frequencies commensurate with its mode spacing, we create a periodically driven lattice of coupled modes in the frequency dimension. The strength and range of couplings can be dynamically reconfigured by changing the modulation amplitude and frequency. We show theoretically and demonstrate experimentally that timeresolved transmission measurements of this system provide a direct readout of its band structure. We also realize longrange coupling, gauge potentials and nonreciprocal bands by simply incorporating additional frequency drives, enabling great flexibility in band structure engineering.
Introduction
The concept of band structure for periodic systems plays a central role in understanding the electronic properties of solidstate systems as well as the photonic properties of photonic crystals and metamaterials^{1}. Recently, there has been significant interest in creating analogous periodic systems not in real space but in synthetic space, allowing one to explore higherdimensional physics with a structure of fewer physical dimensions^{2,3,4,5,6,7,8,9,10,11,12}. Synthetic dimensions are internal degrees of freedom of a system that can be configured into a lattice. For example, a synthetic dimension can be constructed from the equidistant frequency modes of a ring resonator. This synthetic frequency dimension enables one to study fundamental physics such as the effective gauge field and magnetic field for photons, 2D topological photonics in a 1D array, and 3D topological photonics in planar structures^{8,9,13,14,15,16,17,18}. Moreover, the concept is interesting for applications such as unidirectional frequency translation, quantum information processing, nonreciprocal photon transport, and spectral shaping of light^{19,20,21,22,23,24,25,26,27,28,29,30}. While most of the investigations of the frequency dimension have been theoretical, there have been a few recent experimental realizations^{24,31}. Other realizations of synthetic lattices use the hyperfine spin states in cold atoms^{4,5,6,32,33,34,35,36} or the orbital angular momentum (OAM) of photons^{7,37,38,39}. In general, similar to standard solidstate and photonic structures, all these synthetic lattices are again characterized by a band structure in synthetic space, but a direct experimental measurement of this band structure in synthetic space is lacking.
In this work we provide a direct experimental demonstration of a band structure in the synthetic dimension. Specifically, we realize the synthetic dimension in a ring resonator containing an electrooptic modulator (EOM). By periodically driving the modulator at a frequency commensurate with the mode spacing or freespectral range (FSR) of the ring, we introduce coupling between the modes and realize a synthetic frequency dimension lattice. The equidistance of the modes enables the realization of a long synthetic dimension with more than ten modes, all with uniform hopping implemented by a single modulation signal. Since the wavevector k of the reciprocal space of such a frequency lattice is the time axis, we theoretically prove that temporallyresolved measurements of the transmission through the ring reveal its band structure, and demonstrate this method in experiments. Furthermore, we show that additional frequency drives enable us to engineer the band structure and to realize complex longrange coupling, photonic gauge potentials and nonreciprocal bands. We anticipate that the bandstructuremeasurement technique introduced here can be applied to a wide variety of geometries which utilize synthetic frequency dimensions, including those that show nontrivial topological physics^{8,9,17,18,40,41,42}.
Results
Theory
We illustrate the concept that underlies the measurement of band structure in synthetic space using a simple model of a ring resonator. In the absence of groupvelocity dispersion, the longitudinal modes of a ring resonator are equally spaced by the FSR, Ω_{R}/2π = c/n_{g}L, where n_{g} and L are the group index and length of the ring respectively, and c is the speed of light. In a static ring, these modes are uncoupled from each other. One can introduce coupling between the modes by incorporating a phase modulator in the ring (Fig. 1a). Here we consider onresonance coupling where the modulation signal’s periodicity T_{M }= 2π/Ω_{M} matches the roundtrip time of the ring, T_{R} = 2π/Ω_{R}.
The equations of motion for the amplitude of the m − th mode a_{m} for such a ring modulated at frequencies commensurate with its FSR can be written as (see Supplementary Note 1),
where \(\dot a_m \equiv {\mathrm{d}}a_m/{\mathrm{d}}t\), Ω_{R }= Ω_{M} = Ω and T = 2π/Ω. J_{mn}(t) = J_{mn}(t + T) is the coupling introduced by the periodic modulation signal V_{M}(t). Througout this paper, all frequencies are measured against the resonance frequency of the 0th order mode. In Supplementary Note 1 we justify that J_{mn}(t) depends only on n − m and derive the explicit relation between J_{n−m}(t) and V_{M}(t) (Supplementary Eq. 30). By going to a rotating frame defined by b_{m} = a_{m}e^{−im}^{Ω}^{t}, the equations of motion become,
Defining a column vector \(\left b \right\rangle \equiv ( \ldots ,b_{m  1},b_m,b_{m + 1}, \ldots )^T = \mathop {\sum}\nolimits_m b_mm\rangle\), where m〉 is the mth unmodulated cavity mode, Eq. (2) can be written as a matrix equation
Here H(t) is the Hamiltonian with the matrix elements, H_{mn}(t) = 〈mH(t)n〉 = −J_{n−m}(t)e^{i(n−m)}^{Ωt}.
This Hamiltonian H(t) has two symmetries. The first is the modal translational symmetry along the frequency axis between the equally spaced modes, since the matrix element H_{mn} depends only on m−n. This symmetry permits the definition of a conserved Bloch quasimomentum k in the associated reciprocal space. Since the reciprocal space here is conjugate to the frequency dimension, we expect it to be identified with time. We will formally show that this is indeed the case below. The second symmetry is the timetranslation symmetry H(t) = H(t + T). This leads to Floquet bands^{43,44} with quasienergies ε_{k} that are defined in the interval [−Ω/2, Ω/2]. The relationship between the quasienergy ε_{k} and the quasimomemtum k is the band structure.
Define the Bloch modes \(\left k \right\rangle = \mathop {\sum}\limits_m {e^{  im\Omega k}} m\rangle\). The state vector b〉 can be written as \(\left b \right\rangle = ({\it{\Omega }}/2\pi ){\int}_{  \pi /\Omega }^{\pi /\Omega } {\mathrm{d}} k\,\tilde b_kk\rangle\). Equation (3) then reads,
where we have used 〈kH(t)k′〉 = δ(k − k′)H_{k}(t). H(t) is already diagonal in kspace at each instant t due to its modal translational symmetry. Since H_{k}(t) is also timeperiodic, the Floquet quasienergies ε_{k,n} and eigenfunctions ψ_{kn}(t) are welldefined and satisfy
with ψ_{kn}(t) = ψ_{kn}(t + T), and ε_{k,n} = ε_{k} + nΩ.
The above discussion was for a closed system. Next, we turn to an open system, where the ring is coupled to through and dropport waveguides (Fig. 2a), and show how its band structure can be readout directly by timeresolved transmission measurements. Starting from Eq. (1), assuming all modes couple to both waveguides with equal rates γ, and by going to the rotating frame, the inputoutput equations are,
where s_{in} is the amplitude of the monochromatic input wave at frequency ω (Fig. 2a). The last step in Eq. (8). follows from the definition of \(\tilde b_k = \mathop {\sum}\nolimits_m {b_m} e^{im\Omega k}\). It explicitly shows that the quasimomentum k is mapped to the time t in the cavity output field s_{out}. By defining a column vector \(\left {s_{{\mathrm{in}}}} \right\rangle = s_{{\mathrm{in}}}\mathop {\sum}\nolimits_m {e^{  im\Omega t}} m\rangle\), we can write Eq. (7) more compactly as:
At steadystate, we can write,
From Eqs. (9) and (10) we have,
or,
Since b′〉 is time periodic, the eigenstates ψ_{kn}(t) of the Floquet Hamiltonian H_{k}(t) − i∂_{t} form a complete basis for expanding \(\tilde b{\prime}_k\). These expansion coefficients can be obtained by taking the inner product of Eq. (12) with \(\psi _{kn}^ \ast (t)\), defined as \(\left\langle {f(t)g(t)} \right\rangle _T \, = (1/T){\int}_0^T {\mathrm{d}}t\,f^ \ast (t) \cdot g(t)\):
Using Eq. (6) in Eq. (13), the inner product is,
Finally, we can write the output field from Eq. (8) by using Eq. (10) and then expanding \(\tilde b{\prime}_k(t)\) in the ψ_{kn} basis,
Equation (15) shows that the transmission at time t is exclusively determined by the quasienergies and eigenstates at k = t. For \(\gamma \ll \Omega\) and J_{n−m} < Ω/2, only the term for which nΩ is closest to the input frequency ω contributes significantly to the sum in Eq. (15). Using this n, and denoting the input detuning by Δω ≡ ω − nΩ, we can write the normalized transmission T_{out} = s_{out}/s_{in}^{2} as,
Equation (16) shows that for a fixed input detuning Δω that is within a band of the system, the temporallyresolved transmission exhibits peaks at those times t for which the system has an eigenstate with ε_{k} = Δω, k = t. Thus, measuring the times at which the transmission peaks appear in each modulation period 2π/Ω, as a function of Δω, yields the Floquet band structure of the system. This is in contrast to previous proposals of detecting density of states in realspace Floquet systems^{45,46}. These proposals do not directly reveal the kdependence of eigenenergies, and hence do not provide a direct detection of the band structure.
When the magnitude of J is much smaller than Ω, one can use the rotating wave approximation in Eq. (2), by Fourier expanding J_{n−m}(t) and keeping only the terms on the righthand side of Eq. (2) that are timeindependent. This allows us to define an effective timeindependent Hamiltonian, \(H_k^{{\mathrm{eff}}} =  \mathop {\sum}\nolimits_s {\tilde J_{s;{\kern 1pt} q =  s}} \,e^{  is\Omega k}\), where \(\tilde J_{s;{\kern 1pt} q} \equiv (1/T){\int}_0^T {\mathrm{d}}t\,J_s(t)e^{  iq\Omega t}\). As an example, suppose J_{s}(t) = −2J_{1} cos Ωt, then the system has the band structure of a 1D nearestneighborcoupled tightbinding model, ε_{k} = 2J_{1} cos kΩ (Fig. 1b). In Fig. 1c we plot the numerically calculated timeresolved transmission of Eq. (16) obtained by diagonalizing the full Floquet Hamiltonian without making the rotating wave approximation, which agrees well with the band structure in Fig. 1b. For details of the numerical diagonalization, see “Methods”.
Experimental setup
We implement the synthetic frequency dimension using a fiber ring resonator incorporating an electrooptic lithium niobate phase modulator, as shown in Fig. 2b. The ring has a roundtrip length of ~3.5 m, corresponding to a mode spacing Ω_{R} = 2π ⋅ 15.04 MHz (see “Supplementary Methods”)^{47}. We use a narrow linewidth continuous wave (cw) laser as the input. Its frequency could be scanned by a range much larger than Ω_{R} to observe multiple Floquet bands beyond the first Floquet Brillouin zone. The setup also includes a semiconductor optical amplifier (SOA) to partially compensate various losses, including the loss from the modulator. The residual loss and the input coupling leads to a cavity photon decay rate of 2γ = 2π ⋅ 300 kHz. The setup is stable for more than 1 ms, which is sufficient for obtaining the entire band structure. Thus there is no need for active feedback stabilization. Note that Spreeuw et al. have reported band gaps in a Sagnac fiber ring using Faraday elements and counterpropagating modes; however, the absence of frequencydimension coupling precluded the mapping out of the entire band structure^{48}.
To measure the timeresolved transmission that is necessary to read out the band structure, we monitor the through and dropport outputs on a fast photodiode (bandwidth > 5 GHz), connected to a 1 GHz oscilloscope. We scan the laser frequency slowly at 100–500 Hz such that the system reaches steady state at each frequency. This enables us to map out the band structure in a linebyline raster scan fashion.
Experimental results
We plot the experimentally measured band structure in Fig. 3a–c, where the modulation voltage has a form V_{M}(t) = V_{1} cos Ωt, and observe excellent agreement with the theoretically calculated band structure for a nearestneighbor coupled 1D lattice based on Eq. (15) (Fig. 3d–f). Both the cosine dependence of the band on the quasimomentum k(=t), and the increase of the width of the band with increasing modulation amplitude are observed. At a fixed detuning Δω, the transmission response of the system is 2π/Ω_{M}periodic along the time axis. The response is also periodic along the Δωaxis. Both of these periodic responses are expected due to the modal translational symmetry and time periodicity as discussed earlier.
The results in Fig. 3a–c were obtained using a fast photodiode with a bandwidth greater than 5 GHz. If we instead use a slower photodiode with a bandwidth less than the modulation frequency Ω_{M}, the photodiode provides a timeaveraged response, and we observe transmission spectra as shown in Fig. 3g–i. Such transmission spectra represent a direct observation of the photonic density of states (DOS) of the syntheticspace lattice, as can be seen by integrating Eq. (15) over k, which yields the imaginary part of the Green’s function for the band, and hence the DOS, in the limit of weak waveguidecavity coupling (γ → 0), and assuming ψ_{kn}(t) to be independent of k. As a demonstration, the red dashed lines in Fig. 3h denote the DOS of a 1D lattice with nearestneighbor coupling, and match the experimentally measured data well after accounting for the smearing due to a finite γ. The van Hove singularities associated with the DOS of periodic systems are also clearly visible at the edges of the band^{49,50,51}.
In the synthetic space, it is straightforward to create a wide variety of band structures by simply changing the modulation pattern. Different modulation patterns correspond to different coupling configurations in the tightbinding lattice^{52}. Such a flexibility is unique to synthetic space and is unmatched in either solidstate materials or photonic crystals. As an illustration, longrange coupling can be achieved by using a modulation with a frequency that is a multiple of the FSR^{31,52}. Figure 4a shows the measured band structure of the system when Ω_{M} = 2Ω_{R} = 2π ⋅ 30.08 MHz, which creates a lattice with only nextnearest neighbor coupling. This system has a response that is periodic at a frequency of 2Ω_{R}. Thus, the first Brillouin zone extends from k = −π/2Ω_{R} to π/2Ω_{R}, which is half the extent shown in Fig. 4a. The resulting measurement shown in Fig. 4a agrees with the band structure for a tightbinding model with only nextnearest neighbor coupling.
Moreover, the inclusion of both nearestneighbor and longrange hopping leads to a photonic gauge potential whose effects can be observed in the band structure^{52}. As a demonstration, we apply a modulation signal of the form V_{M} = V_{1} cos Ω_{R}t + V_{2} cos (2Ω_{R}t + ϕ). In this case, ϕ is the photonic gauge potential, as can be seen by representing the corresponding tightbinding lattice in terms of a collection of plaquettes, and by noticing that ϕ corresponds to a magnetic flux that threads each plaquette (Fig. 4b)^{53,54,55}. Figure 4c shows the experimentally obtained band structure for ϕ = π/2. Note that this band is asymmetric around k = 0, and hence nonreciprocal. This indicates the breaking of timereversal symmetry in the structure due to the presence of the gauge potential ϕ. In Fig. 4d we show the band structure for an even longer range hopping, obtained by applying a modulation signal V_{M}(t) = V_{1} cos Ωt + V_{2} cos 3Ωt. The range of coupling that we can achieve is limited to thirdnearest neighbor coupling by the 50MHz analog bandwidth of the arbitrary waveform generator (AWG) used in our setup (see “Supplementary Methods”). With the use of AWGs with much higher analog bandwidths exceeding 1 GHz, which are commercially available, it should be straightforward to significantly extend the range of coupling. Alternatively, one could use larger rings with a smaller FSR ~5 MHz, which would permit up to tenthnearest neighbor coupling within a 50 MHz bandwidth of the AWG.
Discussion
We have theoretically proposed and experimentally demonstrated a technique to directly measure the band structure of a system with a synthetic dimension. The fiber ring resonator with a modulator allows for independent tuning of the strength and range of the coupling along this synthetic lattice, making it dynamically tunable. By combining multiple frequency drives and incorporating longrange hopping, we have demonstrated a photonic gauge potential and its effect on the band structure.
The synthetic frequency dimension platform that we have experimentally demonstrated here, along with the band structure measurement technique, is ripe for probing systems beyond 1D^{10,11,12,18,56,57,58,60}. For example, 2D quantum Hall phenomena such as oneway edge states^{8,9,61} and synthetic Hall ribbons^{5,34,62,63,64,65} could be observed in extensions of our system, with the added benefit of frequency conversion from transport along the synthetic dimension. The dimensionality can be increased beyond 1D by using realspace dimensions^{8,9}, by using additional frequency dimensions^{26,52}, by using the Floquet dimension^{14,59,60} or by using other synthetic dimensions such as OAM in conjunction with frequency^{66} (see Supplementary Discussion). Even within 1D, there have been proposals to realize unique photon transport phenomena using dynamically modulated cavities, which could be implemented in a reconfigurable fashion in our platform^{16,67}. Longer fiber ring resonators in the pulsed regime have been previously used for realizing paritytime symmetry^{68}, optical Ising machines^{69} and soliton interactions^{70}, in a synthetic temporal dimension^{71,72,73,74} that is complementary to our cwpumped synthetic frequency dimension. In these systems, the band structure has been indirectly inferred from transport measurements in the synthetic temporal dimension, using pulses in fiber loops to simulate photonic lattices^{75}. Lastly, the advent of onchip silicon^{76,77} and lithium niobate ring resonators^{78} with modulation bandwidths higher than the FSR of onchip ring resonators can enable synthetic dimensions and topological photonics in a monolithically integrated platform.
Methods
Numerical diagonalization of the Floquet Hamiltonian
In this section we outline the procedure we used to diagonalize the Floquet Hamiltonian from Eq. (5) and numerically calculate the quasienergies and eigenfunctions.
The Floquet eigenstates at each k are time periodic and can be expanded in terms of their Fourier components,
with the basis vectors p〉 = e^{ip}^{Ω}^{t} and the inner product
In this basis, the matrix elements of the Floquet Hamiltonian are,
where \(\tilde J_{s;{\kern 1pt} q} \equiv (1/T){\int}_0^T \mathrm{d} t\,J_s(t)e^{  iq\Omega t}\) is the qth Fourier component of the periodic hopping term J_{s}(t) between modes m and m + s \((q \in {\Bbb Z})\). To calculate ψ_{kn}(t) and ε_{k,n}, we truncate the matrix corresponding to the Floquet Hamiltonian in the p〉 basis to a large enough order p_{max} and numerically determine the eigenvalues and eigenvectors, respectively^{67,79}.
Experimental calibration of the frequency axis
The laser’s optical frequency can be scanned by applying a voltage to the frequency sweep input. To calibrate this relationship, we could apply a sinusoidal modulation to the intracavtiy EOM while simultaneously scanning the laser’s frequency with a linear voltage ramp. The sidebands created by the modulation provide a calibration of the frequency change with voltage. A more accurate calibration can be obtained by varying the modulation frequency Ω_{M} till the slow transmission is maximally flattened for moderate modulation amplitudes, J ≈ 0.1 Ω, as in the observation of the DOS in Fig. 3h. Multiple resonances of the cavity were simultaneously flattened for Ω_{M} close to 2π × 15.04 MHz, indicating that the resonances are equally spaced by this FSR. Since we applied a linear voltage ramp signal, we could equivalently calibrate the frequency axis with time. The relationship between voltage and frequency was found to deviate from linearity for large frequency sweeps, as shown in Supplementary Fig. 1. We observed that a quadratic fit was of sufficient accuracy to relate the frequency with the scan time over an optical frequency scanning range of 60 MHz. This fit was then used to convert the slow time axis from the raw data into the frequency axis in Figs. 3 and 4.
Data acquisition and time slicing
Each band structure measurement consisted of a 1ms acquisition of the dropport transmission through the ring cavity. The dropport output was sent to an erbiumdoped fiber amplifier (EDFA) to boost the signaltonoise ratio. The EDFA output was detected on a 5 GHz photodiode, whose output was further electrically amplified by an RF amplifier (passband 50 kHz to 14 GHz) before being sent to the 1 GHz oscilloscope. A finite lower cutoff frequency of the RF amplifier mitigates the added spontaneous emission noise from the EDFA and the intracavity SOA. We used a sampling rate of 2 GSa/s to get sufficient resolution along the (fast) time axis corresponding to the Bloch quasimomentum k in Figs. 3 and 4.
The 1mslong trace was then broken into time slices, each of duration equal to the roundtrip time of the ring (=1/(15.04 MHz) ≈66.49 ns). The time within each of these slices (the “fast” time) corresponds to k. The change of the starting time between successive slices was converted to a change in optical frequency of the laser using the calibration in Supplementary Fig. 1. Each such slice shows zero, one or more peaks depending on whether the input laser frequency is outside any band, at the edge of a band, or within a band, respectively [see Supplementary Movies 1 and 2]. Stacking up the slices vertically yields the band structure in Figs. 3a–c and 4a, c and d.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
This work is supported by a Vannevar Bush Faculty Fellowship (Grant No. N000141713030) from the U. S. Department of Defense, and by a MURI grant from the U. S. Air Force Office of Scientific Research (Grant No. FA95501710002). M.M. acknowledges support from the Swiss National Science Foundation (Grant No. P300P2_177721).
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Q.L. conceived the band structure measurement technique. A.D. designed, built, and characterized the setup, in consultation with L.Y., Q.L., D.A.B.M., and S.F. M.M. developed the coupledamplitude equations and the Floquet analysis relating the transmission to the band structure. A.D. collected and analyzed the experimental data. All authors contributed to discussion of the results and writing the manuscript. S.F. supervised the project.
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Dutt, A., Minkov, M., Lin, Q. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat Commun 10, 3122 (2019). https://doi.org/10.1038/s41467019111179
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DOI: https://doi.org/10.1038/s41467019111179
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