Introduction

The concept of band structure for periodic systems plays a central role in understanding the electronic properties of solid-state systems as well as the photonic properties of photonic crystals and metamaterials1. Recently, there has been significant interest in creating analogous periodic systems not in real space but in synthetic space, allowing one to explore higher-dimensional physics with a structure of fewer physical dimensions2,3,4,5,6,7,8,9,10,11,12. 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 structures8,9,13,14,15,16,17,18. Moreover, the concept is interesting for applications such as unidirectional frequency translation, quantum information processing, nonreciprocal photon transport, and spectral shaping of light19,20,21,22,23,24,25,26,27,28,29,30. While most of the investigations of the frequency dimension have been theoretical, there have been a few recent experimental realizations24,31. Other realizations of synthetic lattices use the hyperfine spin states in cold atoms4,5,6,32,33,34,35,36 or the orbital angular momentum (OAM) of photons7,37,38,39. In general, similar to standard solid-state 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 electro-optic modulator (EOM). By periodically driving the modulator at a frequency commensurate with the mode spacing or free-spectral 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 temporally-resolved 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 long-range coupling, photonic gauge potentials and nonreciprocal bands. We anticipate that the band-structure-measurement technique introduced here can be applied to a wide variety of geometries which utilize synthetic frequency dimensions, including those that show nontrivial topological physics8,9,17,18,40,41,42.

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 group-velocity dispersion, the longitudinal modes of a ring resonator are equally spaced by the FSR, ΩR/2π = c/ngL, where ng 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 on-resonance coupling where the modulation signal’s periodicity TM = 2π/ΩM matches the roundtrip time of the ring, TR = 2π/ΩR.

Fig. 1
figure 1

Dynamically modulated ring cavity and its band structure in the synthetic frequency dimension. a A sinusoidal signal VM modulates the index of a part of the ring (left) at the mode spacing ΩR, creating coupling J between the different frequency modes (right). EOM electro-optic phase modulator. b Band structure of the nearest-neighbor coupled 1D tight-binding model, εk = 2J cos kΩ. c Theoretical time-resolved steady-state response of the ring (Eq. 16) for a cavity loss rate 2γ = J/5. The time t plays the role of the Bloch quasimomentum k

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The equations of motion for the amplitude of the m − th mode am for such a ring modulated at frequencies commensurate with its FSR can be written as (see Supplementary Note 1),

$$\dot a_m(t) = im{\it{\Omega }}\,a_m + i\mathop {\sum}\limits_n {J_{mn}} (t)\,a_n(t)$$
(1)

where \(\dot a_m \equiv {\mathrm{d}}a_m/{\mathrm{d}}t\), ΩR = ΩM = Ω and T = 2π/Ω. Jmn(t) = Jmn(t + T) is the coupling introduced by the periodic modulation signal VM(t). Througout this paper, all frequencies are measured against the resonance frequency of the 0-th order mode. In Supplementary Note 1 we justify that Jmn(t) depends only on n − m and derive the explicit relation between Jn−m(t) and VM(t) (Supplementary Eq. 30). By going to a rotating frame defined by bm = ame−imΩt, the equations of motion become,

$$i\dot b_m = - \mathop {\sum}\limits_n {J_{n - m}} (t)\,b_ne^{i(n - m){\it{\Omega }}t}$$
(2)

Defining a column vector \(\left| b \right\rangle \equiv ( \ldots ,b_{m - 1},b_m,b_{m + 1}, \ldots )^T = \mathop {\sum}\nolimits_m b_m|m\rangle\), where |m〉 is the m-th unmodulated cavity mode, Eq. (2) can be written as a matrix equation

$$i{\kern 1pt} \left| {\dot b} \right\rangle = H(t)\left| b \right\rangle .$$
(3)

Here H(t) is the Hamiltonian with the matrix elements, Hmn(t) = 〈m|H(t)|n〉 = −Jn−m(t)ei(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 Hmn 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 time-translation symmetry H(t) = H(t + T). This leads to Floquet bands43,44 with quasienergies εk that are defined in the interval [−Ω/2, Ω/2]. The relationship between the quasienergy εk 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 \(\left| b \right\rangle = ({\it{\Omega }}/2\pi ){\int}_{ - \pi /\Omega }^{\pi /\Omega } {\mathrm{d}} k\,\tilde b_k|k\rangle\). Equation (3) then reads,

$$i{\dot{\tilde{b}}}_k = \left\langle {k|H(t)|b} \right\rangle = H_k(t)\,\tilde b_k$$
(4)
$$H_k(t) = - \mathop {\sum}\limits_s {J_s} (t)\,e^{is{\mathrm{\Omega }}t - is{\mathrm{\Omega }}k};s \in {\Bbb Z}$$
(5)

where we have used 〈k|H(t)|k′〉 = δ(k − k′)Hk(t). H(t) is already diagonal in k-space at each instant t due to its modal translational symmetry. Since Hk(t) is also time-periodic, the Floquet quasienergies εk,n and eigenfunctions ψkn(t) are well-defined and satisfy

$$(H_k(t) - i\partial _t){\kern 1pt} \psi _{kn}(t) = \varepsilon _{k,n}{\kern 1pt} \psi _{kn}(t),$$
(6)

with ψkn(t) = ψkn(t + T), and εk,n = εk + nΩ.

The above discussion was for a closed system. Next, we turn to an open system, where the ring is coupled to through- and drop-port waveguides (Fig. 2a), and show how its band structure can be read-out directly by time-resolved 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 input-output equations are,

$$\dot b_m = - \gamma b_m + i\mathop {\sum}\limits_s {J_s} (t)\,e^{is\Omega t}b_{m + s} + i\sqrt \gamma e^{ - i(\omega + m\Omega )t}s_{{\mathrm{in}}}$$
(7)
$${\hskip-6pc}s_{{\mathrm{out}}}(t) = i\sqrt \gamma \mathop {\sum}\limits_m {b_m} (t)\,e^{im\Omega t} = i\sqrt \gamma \left. {\tilde b_k(t)} \right|_{k = t}$$
(8)

where sin 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 sout. 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:

$$i\partial _t\left| b \right\rangle = ( - i\gamma + H(t))|b\rangle - \sqrt \gamma e^{ - i\omega t}|s_{{\mathrm{in}}}\rangle$$
(9)
Fig. 2
figure 2

Experimental setup. a Schematic. b Detailed setup. Cavity length ~13.5 m, Cavity mode spacing or free-spectral range ΩR = 2π ⋅ 15.04 MHz. EOM electro-optic phase modulator, PD photodiode, SOA semiconductor optical amplifier, amp RF amplifier

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At steady-state, we can write,

$$\left| {b(t)} \right\rangle = e^{ - i\omega t}|b{\prime} (t)\rangle ;\,|b{\prime} (t)\rangle = |b{\prime} (t + T)\rangle$$
(10)

From Eqs. (9) and (10) we have,

$$\left\langle k \right|\left[ {\omega - (H(t) - i\partial _t) + i\gamma } \right]\left| {b{\prime} } \right\rangle = - \sqrt \gamma \langle k|s_{{\mathrm{in}}}\rangle$$
(11)

or,

$$(\omega - (H_k(t) - i\partial _t) + i\gamma )\tilde b{\prime}_k = - \sqrt \gamma s_{{\mathrm{in}}}T\delta (k - t)$$
(12)

Since |b′〉 is time periodic, the eigenstates ψkn(t) of the Floquet Hamiltonian Hk(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 \(\left\langle {f(t)|g(t)} \right\rangle _T \, = (1/T){\int}_0^T {\mathrm{d}}t\,f^ \ast (t) \cdot g(t)\):

$$\begin{array}{l}\frac{1}{T}{\int}_0^T {\mathrm{d}}t\,\psi _{kn}^ \ast (t)(\omega - (H_k(t) - i\partial _t) + i\gamma )\tilde b{\prime}_k(t)\\ \quad \quad \quad \quad \quad \quad = - \sqrt \gamma s_{{\mathrm{in}}}{\int}_0^T {\mathrm{d}} t\,\,\psi _{kn}^ \ast (t){\kern 1pt} \delta (k - t)\end{array}$$
(13)

Using Eq. (6) in Eq. (13), the inner product is,

$$\left\langle {\psi _{kn}|\tilde b{\prime}_k} \right\rangle _T = - \frac{{\sqrt \gamma s_{{\mathrm{in}}}{\kern 1pt} \psi _{kn}^ \ast (t = k)}}{{\omega - \varepsilon _k - n\Omega + i\gamma }}$$
(14)

Finally, we can write the output field from Eq. (8) by using Eq. (10) and then expanding \(\tilde b{\prime}_k(t)\) in the ψkn basis,

$$\begin{array}{*{20}{l}} {s_{{\mathrm{out}}}(t;\omega )} \hfill & = \hfill & {i\sqrt \gamma e^{ - i\omega t}\mathop {\sum}\limits_n \left. {\psi _{kn}(t)\langle \psi _{kn}|\tilde b{\prime}_k\rangle _T} \right|_{k = t}} \hfill \\ {} \hfill & = \hfill & { - e^{ - i\omega t}s_{{\mathrm{in}}}\,\left. {\mathop {\sum}\limits_n \frac{i\gamma\, |\psi _{kn}(t)|^2}{{\omega - \varepsilon _k - n\Omega + i\gamma }}} \right|_{k = t}} \hfill \end{array}$$
(15)

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 |Jn−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 Tout = |sout/sin|2 as,

$$T_{{\mathrm{out}}}(t = k;{\mathrm{\Delta }}\omega ) = \frac{{\gamma ^2}}{{({\mathrm{\Delta }}\omega - \varepsilon _k)^2 + \gamma ^2}}|\psi _{kn}(t)|^4$$
(16)

Equation (16) shows that for a fixed input detuning Δω that is within a band of the system, the temporally-resolved transmission exhibits peaks at those times t for which the system has an eigenstate with εk = 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 real-space Floquet systems45,46. These proposals do not directly reveal the k-dependence 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 Jn−m(t) and keeping only the terms on the right-hand side of Eq. (2) that are time-independent. This allows us to define an effective time-independent 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 Js(t) = −2J1 cos Ωt, then the system has the band structure of a 1D nearest-neighbor-coupled tight-binding model, εk = 2J1 cos kΩ (Fig. 1b). In Fig. 1c we plot the numerically calculated time-resolved 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 electro-optic lithium niobate phase modulator, as shown in Fig. 2b. The ring has a roundtrip length of ~3.5 m, corresponding to a mode spacing ΩR = 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 ΩR 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 frequency-dimension coupling precluded the mapping out of the entire band structure48.

To measure the time-resolved transmission that is necessary to read out the band structure, we monitor the through- and drop-port 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 line-by-line raster scan fashion.

Experimental results

We plot the experimentally measured band structure in Fig. 3a–c, where the modulation voltage has a form VM(t) = V1 cos Ωt, and observe excellent agreement with the theoretically calculated band structure for a nearest-neighbor 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/ΩM-periodic along the time axis. The response is also periodic along the Δω-axis. Both of these periodic responses are expected due to the modal translational symmetry and time periodicity as discussed earlier.

Fig. 3
figure 3

Band structure for on-resonance modulation ΩM = ΩR = 2π ⋅ 15.04 MHz. a–c Experimentally measured time-resolved transmission. d–f Theoretical transmission based on Eq. (15). The bottom x-axis represents time (the quasimomentum) in units of π/ΩR, while the top x-axis of (a–c) presents the time in real units. The y-axis represents the input laser’s detuning from the ring’s resonance frequency, allowing us access to different quasienergies εk (left: real units, right: normalized to FSR). On increasing the coupling J (theory) or the amplitude of modulation Vpp (experiment), cosinusoidal bands with increasing extent in quasienergy are observed, in agreement with εk = 2J cos kΩ. See also Supplementary Movie 1. g–i Corresponding time-averaged transmission through the modulated cavity. In (g), typical Lorentzian resonances of the unmodulated cavity are seen. In (h), the photonic density of states (DOS) in synthetic space is observed. The experimental data (blue) matches the expected density of states (red dashed) of a 1D tight-binding model with nearest neighbor coupling. In the bottom panels (c), (f, i), the modulation-induced coupling is strong enough to cause overlap of Floquet bands

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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 ΩM, the photodiode provides a time-averaged 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 synthetic-space 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 waveguide-cavity coupling (γ → 0), and assuming |ψkn(t)| to be independent of k. As a demonstration, the red dashed lines in Fig. 3h denote the DOS of a 1D lattice with nearest-neighbor 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 band49,50,51.

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 tight-binding lattice52. Such a flexibility is unique to synthetic space and is unmatched in either solid-state materials or photonic crystals. As an illustration, long-range coupling can be achieved by using a modulation with a frequency that is a multiple of the FSR31,52. Figure 4a shows the measured band structure of the system when ΩM = 2ΩR = 2π ⋅ 30.08 MHz, which creates a lattice with only next-nearest neighbor coupling. This system has a response that is periodic at a frequency of 2ΩR. Thus, the first Brillouin zone extends from k = −π/2ΩR to π/2ΩR, which is half the extent shown in Fig. 4a. The resulting measurement shown in Fig. 4a agrees with the band structure for a tight-binding model with only next-nearest neighbor coupling.

Fig. 4
figure 4

Band-structure engineering with long-range hopping and synthetic gauge potentials. a Time-resolved transmission for a modulation frequency ΩM = 2ΩR, coupling exclusively next-nearest neighbors. While the band structure is similar to Fig. 3, the first Brillouin zone is halved, and two periods of the cosine band structure are seen in ΩRt∈[−π, π]. b Schematic of lattice for VM = V1 cos ΩRt  + V2 cos (nΩRt + ϕ), where V1 (solid lines) determines the strength of nearest-neighbor coupling and V2 (dashed lines) determines the long-range hopping, here shown for n = 2. The phase of the V2 links is +ϕ when going up in frequency and −ϕ when going down in frequency, as imposed by the modulation53,54. Bottom: Equivalent lattice representation by a triangular chain threaded by a phase ϕ per plaquette. c Measured nonreciprocal band structure for V2/V1 = 0.40, ϕ = π/2, n = 2, which shows strong asymmetry about k = 0. See also Supplementary Movie 2. d Same as c, but for ϕ = 0, n = 3. Black overlays in (c) and (d) indicate expected band structures

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Moreover, the inclusion of both nearest-neighbor and long-range hopping leads to a photonic gauge potential whose effects can be observed in the band structure52. As a demonstration, we apply a modulation signal of the form VM = V1 cos ΩRt + V2 cos (2ΩRt + ϕ). In this case, ϕ is the photonic gauge potential, as can be seen by representing the corresponding tight-binding 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 time-reversal 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 VM(t) = V1 cos Ωt + V2 cos 3Ωt. The range of coupling that we can achieve is limited to third-nearest neighbor coupling by the 50-MHz 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 tenth-nearest 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 long-range 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 1D10,11,12,18,56,57,58,60. For example, 2D quantum Hall phenomena such as one-way edge states8,9,61 and synthetic Hall ribbons5,34,62,63,64,65 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 real-space dimensions8,9, by using additional frequency dimensions26,52, by using the Floquet dimension14,59,60 or by using other synthetic dimensions such as OAM in conjunction with frequency66 (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 platform16,67. Longer fiber ring resonators in the pulsed regime have been previously used for realizing parity-time symmetry68, optical Ising machines69 and soliton interactions70, in a synthetic temporal dimension71,72,73,74 that is complementary to our cw-pumped 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 lattices75. Lastly, the advent of on-chip silicon76,77 and lithium niobate ring resonators78 with modulation bandwidths higher than the FSR of on-chip 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,

$$\psi _{kn}(t) = \mathop {\sum}\limits_p {e^{ip{{\Omega }}t}} \,\tilde \psi _{kn}(p) = \mathop {\sum}\limits_p {|p\rangle } \langle p|\psi _{kn}(t)\rangle _T$$
(17)

with the basis vectors |p〉 = eipΩt and the inner product

$$\left\langle {f(t)|g(t)} \right\rangle _T \ =\ \frac{1}{T}{\int}_{0}^T {\mathrm{d}} t\,f^ \ast (t) \cdot g(t).$$
(18)

In this basis, the matrix elements of the Floquet Hamiltonian are,

$$\left\langle {p{\prime}|H(t) - i\partial _t|p} \right\rangle _T \ = \ \frac{1}{T}{\int}_{0}^T {\mathrm{d}} t\,e^{i(p - p^{\prime})\Omega t}(H_k(t) + p\Omega )$$
(19)
$${\hskip4.7pc}= p\Omega \delta _{pp{\prime}} - \mathop {\sum}\limits_s {e^{ - is\Omega k}} \tilde J_{s;p{\prime} - p - s}$$
(20)

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 q-th Fourier component of the periodic hopping term Js(t) between modes m and m + s \((q \in {\Bbb Z})\). To calculate ψkn(t) and εk,n, we truncate the matrix corresponding to the Floquet Hamiltonian in the |p〉 basis to a large enough order pmax and numerically determine the eigenvalues and eigenvectors, respectively67,79.

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 ΩM 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 ΩM 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 1-ms acquisition of the drop-port transmission through the ring cavity. The drop-port output was sent to an erbium-doped fiber amplifier (EDFA) to boost the signal-to-noise 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 1-ms-long 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.