Introduction

Frequency combs serve as high-precision rulers for frequency and time measurement, playing a pivotal role in a wide variety of modern science and technologies1,2,3,4,5, including optical clocks, LIDAR, spectroscopy, arbitrary waveform generation, and optical neural networks. Over the past two decades, integrated combs have garnered significant research interests5,6,7,8,9,10,11,12,13,14,15,16,17,18, leading to miniaturized and chip-based photonic systems7. However, most on-chip frequency combs, such as integrated semiconductor mode-locked lasers and microresonator-based Kerr combs, mainly operate in the optical frequency domain. A fully integrated frequency comb functioning in the microwave domain remains elusive, impeding the advancement of chip-based microwave spectroscopy and integrated quantum circuits, which typically require precise microwave control. Here, we address these challenges by introducing an all-superconductor-based microcomb, featuring an elegantly simple structure, effortless operation, and ultra-low power consumption.

Our superconducting frequency comb is fully integrated, comprising two fundamental superconductor devices: a Josephson junction directly coupled to a superconducting coplanar waveguide (CPW) resonator, as illustrated in Fig. 1a (see Supplementary Fig. 1 for a photo of our device). Device fabrication is achieved through straightforward procedures utilizing standard photolithography and electron beam evaporation techniques (See Methods). The Josephson junction consists of two superconductors connected by a weak link or a thin insulating barrier. It acts as an ideal voltage-to-frequency converter, emitting photons at the Josephson frequency \({f}_{J}=2e{V}_{{{{{{{\mathrm{dc}}}}}}}}/h\), where \(2e\) is the charge of a Cooper pair, \({V}_{{{{{{{\mathrm{dc}}}}}}}}\) is the DC voltage bias across the junction, and \(h\) is Planck’s constant. Meanwhile, the superconducting CPW resonator is another crucial superconductor device widely used in high-sensitive detectors and circuits for qubit control and read-out19. We demonstrate that the combination of these two fundamental superconducting elements can generate a coherent frequency comb (Fig. 1b, c). Its on-chip generation of pulse waves and inherent compatibility with production, operation, and integration into circuit quantum electrodynamics20,21,22 offers the potential for miniaturized, low-cost, and energy-efficient quantum processors.

Fig. 1: Integrated and DC-powered superconducting microcomb.
figure 1

a Illustration of the device. Frequency comb emission is generated when a DC voltage-biased (Vdc) Josephson junction is coupled to a superconducting CPW resonator. b The power spectral density (top) and flowing supercurrent through the Josephson junction (bottom) as a function of DC bias voltage. c Frequency comb spectrum obtained at Vdc = 43 \({{{{{\rm{\mu }}}}}}{{{{{\rm{V}}}}}}\). d Extracted emission frequencies (open dots) from (c). The dashed line represents fitting to the comb formula \({f}_{m}=m{f}_{r}\).

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The Josephson junction-coupled superconducting resonator has drawn significant research interest due to its capability not only to operate as ultrasensitive sensors, such as for single photon detection23 and/or thermometry24, but also to function as a coherent photon source25,26,27,28,29,30,31,32,33,34. Particularly, in the strong coupling regime, the device generates stable microwave lasing25 and demonstrates remarkably low noise26. However, in previous investigations, the coupled devices were all operated in single mode, resulting in continuous-wave emissions. Their properties under a multimode operation have not been explored.

Results

Coherent comb generation

We demonstrate that a Josephson junction coupled to a superconducting resonator can generate a self-phase-locked coherent frequency comb under multimode operation. The coupling between a Josephson junction and a resonator is proportional to the Josephson energy \({{{{{{\boldsymbol{E}}}}}}}_{{{{{{\boldsymbol{J}}}}}}}{{{{{\boldsymbol{\equiv }}}}}}{{{{{\boldsymbol{h}}}}}}{{{{{{\boldsymbol{I}}}}}}}_{{{{{{\boldsymbol{c}}}}}}}{{{{{\boldsymbol{/}}}}}}{{{{{\boldsymbol{2}}}}}}{{{{{\boldsymbol{e}}}}}}\), where \({{{{{{\boldsymbol{I}}}}}}}_{{{{{{\boldsymbol{c}}}}}}}\) is superconducting critical current25,28. To directly ensure strong coupling, we employ a Josephson junction with a large \({{{{{{\boldsymbol{I}}}}}}}_{{{{{{\boldsymbol{c}}}}}}}\). This is achieved by fabricating a sizable Josephson junction (approximately 4 µm2), resulting in a relatively large supercurrent of ID ≈ 2 µA in the lasing state (Fig. 1b), which surpasses by more than two orders of magnitude compared to that in the continuous-wave Josephson laser25. To achieve multimode operation within the frequency range of interest (typically 1–10 GHz), we fabricate a superconducting resonator with a low free spectrum range of 0.75 GHz. The strong coupling allows stable microwave lasing when a dc voltage bias Vdc \({{{{{\boldsymbol{\ge }}}}}}{{{{{\boldsymbol{20}}}}}}\) µV is applied to the Josephson junction (Fig. 1b).

Figure 1c shows the comb spectrum measured at Vdc = 43 μV. First, we will prove that the observed spectrum in Fig. 1c indeed represents a coherent frequency comb. The raised background between 3–10 GHz in Fig. 1c is a consequence of the limited bandwidth of amplifiers. Therefore, our subsequent investigations will focus on the frequency range in the detection bandwidth. The frequencies of a comb are given by the formula: \({f}_{m}={f}_{0}+m{f}_{r}\), where \({f}_{0}\) denotes the carrier offset frequency, \({f}_{r}\) represents the repetition frequency, and m stands for the mode number1,2,3,4,5. The extracted mode frequencies are shown in Fig. 1d, align perfectly with fitting to the comb formula. The result indicates a negligible offset frequency (\({f}_{0}\approx 0\) GHz). Consequently, our superconducting microcomb can be expressed by a simplified comb formula:

$${f}_{m}=m{f}_{r}$$
(1)

Additionally, Fig. 1c, d demonstrate that the repetition frequency \({f}_{r}\) nicely aligns with the free spectrum range (0.75 GHz) of the half-wave resonator.

The evenly spaced spectral lines alone are not sufficient for correlations among the comb teeth. To examine the coherence state of the emitted signal, we first proved that each individual mode has a stable phase using the heterodyne detection technique (Supplementary Fig. 2). One direct consequence of phase-coherent comb is the generation of pulses in the time domain1,2,3,4,5. Figure 2a displays a comb spectrum, and its corresponding time-dependent waveform is shown in Fig. 2b, c. The waveform reveals a sequence of microwave pulses, indicating that all spectrum modes are phase-locked and maintain a stable phase relationship. Moreover, every pulse displays identical features, as seen in Fig. 2d by the perfect overlap of two pulse waves. This indicates that all pulses are in-phase, and also suggests a carrier offset frequency of 0 Hz, thus confirming that all the spectral lines share a uniform phase. The coherence of the comb modes can be further validated through a unique coherent injection-locking effect, as demonstrated below.

Fig. 2: Coherent pulse-wave.
figure 2

a Comb spectrum obtained at Vdc = 83 μV. b–d Pulse waveforms corresponding to the spectrum in (a). The zoom-in waveform (b) of the region enclosed by the green-dashed box in (c) displays two highlighted pulse waves with purple and green, which overlap perfectly in (d).

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The emission power of each individual comb line, ranging up to subpicowatt (see Supplementary Fig. 3), is consistent with that of the continuous-wave (or single-mode) source25. The microcomb’s performance could be further enhanced by refining the structures and parameters of the Josephson junction-coupled resonators26. The stability of the frequency comb is determined by the linewidth of the spectral lines (see Supplementary Fig. 4). In general, the overall linewidth is influenced by the quality factor of the resonator, the stability of the DC bias voltage, and environmental electromagnetic and thermal noise. Moreover, our experiments demonstrate that the linewidth of our comb increases quadratically with the mode number (or frequency), as illustrated in Supplementary Fig. 5. This finding is consistent with the observations in quantum-limited optical combs35,36.

Coherent injection-locking effect

Injection-locking is a widely used technique for effectively narrowing the linewidth of a laser source25,26,37,38. In optical frequency combs, it has been demonstrated only in a few cases in quantum cascade lasers by injecting a radio-frequency (RF) modulation at the cavity roundtrip frequency39,40,41,42. In our superconducting microcomb, we showcase a distinct coherent injection-locking effect that can be achieved by injecting an external microwave at any arbitrary comb mode. For instance, we select an arbitrary mode, such as minj = 7, as the injection tone (Fig. 3a). Then, an injection signal finj is applied and swept around f7 = 5.26 GHz. We measure the spectra at frequencies fsen around various sensing modes msen (Fig. 3a). The resulting spectra maps are illustrated in Fig. 3b–d (additional results in Supplementary Fig. 6). We find that all modes are simultaneously locked within the same injection frequency range Δfinj = 8.08 MHz (see Fig. 3b, d for definition and Fig. 3k for extracted values). This reaffirms that all spectral lines are coherently phase-locked.

Fig. 3: Coherent injection-locking effect.
figure 3

a A comb spectrum showing five observed modes. b–d Spectrum maps obtained by sweeping the injection frequency finj around f7 at a fixed power (Pinj = −82 dBm), and at various sensing modes msen = 6 (b), 7 (c), and 9 (d), respectively. The emission signals are locked within the range Δfinj and Δfsen. e–g Comparison of spectra between free-running (black) and injection-locking (red) states. Insets show the linewidth, which is ≤1 Hz for all locked modes. h–j Spectra maps obtained by varying injection power (Pinj) with an off-resonance tone (finj = f7-Δfoff; f7 = 5.26 GHz, Δfoff = 8 MHz). The shifts of the locked tone from the free-running tones are indicated by Δflock. The long-dashed lines in (b–d, h–j) are fittings to extended Alder’s Eq. (2). k–m Plots of Δfinj and Δfoff (k), Δfsen (l), and Δflock (m) as functions of mode number m.

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This unique coherent injection-locking effect results in an ultra-high-resolution comb, with the linewidth of all comb teeth significantly narrowed down to ≤1 Hz, as shown in Fig. 3e–g and their insets. It’s worth noting that the observed linewidth of 1 Hz is constrained by the resolution bandwidth of the spectrum analyzer used in the experiments26, implying the potential for even narrower linewidths.

The injection-locking range Δfinj of our microcomb widens with the injection power Pinj (Supplementary Fig. 7). Specifically, when msen=minj (Fig. 3c, f, i), the injection-locking phenomena replicate those observed in the continuous-wave source25,26, and can therefore be explained by the Adler’s theory43. However, the results for the comb modes with msen ≠ minj cannot be described by Adler’s equation (see Supplementary Fig. 8a,b). We find that the locked sensing frequency range Δfsen (see Fig. 3b–d for definition) is proportional to m (Fig. 3l). This implies that the radiation frequencies of the injection-locked comb satisfy the comb formula \((1)\) over the entire locked frequency range. Consequently, this introduces an in-situ tunable superconducting microcomb, allowing for adjustable \({f}_{r}\) within the range of \(\Delta {f}_{{{{{{{\mathrm{inj}}}}}}}}/{m}_{{{{{{{\mathrm{inj}}}}}}}}\).

The coherent injection-locking leads to a unique frequency-pulling effect, particularly noticeable when msen ≠ minj. Figure 3h–j illustrate these effects when finj is fixed at an off-resonance tone (the corresponding on-resonance injection results are shown in Supplementary Fig. 9), shifted from the free-running emission tone by Δfoff = 8 MHz. When msen = minj, the emission line is gradually pulled toward finj with increasing injection power Pinj, ultimately locking at finj (Fig. 3i). This behavior mirrors that observed in a continuous-wave source25,26. However, the remarkable outcome arises when msen ≠ minj. In this case, although no injection tone is applied around each sensing mode, notable emission lines are induced at the corresponding tones, termed “induced-injection tone” and labeled as fid in Fig. 3b, d, h, 3j. The linewidth of the fid lines is considerably narrower than those of the free-running emissions and the side-band harmonics (see Supplementary Fig. 10). Furthermore, neither fid emission nor Kerr comb generation are observed with RF injection under zero DC bias voltage (Supplementary Fig. 11). These indicate that the fid emissions originate from the mutual interactions among Josephson photons, injection photons, and the resonator’s multi-modes, setting the fid signals apart from the Kerr combs generated directly from RF pumping44.

When msen ≠ minj, as depicted in Fig. 3h, j, while the emissions of all the modes are pulled toward fid tones with increasing Pinj, the fid tones themselves also shift with Pinj. Consequently, this results in the locked frequency range Δflock≠Δfoff (see Fig. 3h, j). Figure 3m demonstrates that Δflock is also proportional to m with \(\Delta {f}_{{{{{{{\mathrm{lock}}}}}}}}=\frac{m}{{m}_{{{{{{{\mathrm{inj}}}}}}}}}\Delta {f}_{{{{{{{\mathrm{off}}}}}}}}\), perfectly aligned with the requirements of the comb formula \((1)\) for the emissions within the locking range.

There is no existing theory that can describe our observed coherent injection-locking effect. We have derived an extended Adler’s equation to quantitatively describe these unique phenomena. As analyzed above, in the locking range Δfinj, all comb emissions are described by the comb formula \((1)\). In the unlocking range outside Δfinj, the emission signals are given by (please refer to Method for detailed derivation):

$${f}_{m,n}={\frac{m}{{m}_{{{{{{{\mathrm{inj}}}}}}}}}}({f}_{{{{{{{\mathrm{inj}}}}}}}}+{f}_{h})+n{f}_{h}$$
(2)

Here, fh denotes the repetition frequency of the harmonic emissions induced by off-resonance injection, and \(n{f}_{h}\) represents the nth harmonic shift. The fh is given by:

$${f}_{h}=({f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}}}-{f}_{{{{{{{\mathrm{inj}}}}}}}})\sqrt{1-(\frac{\Delta {f}_{{{{{{{\mathrm{inj}}}}}}}}/2}{{f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}}}-{f}_{{{{{{{\mathrm{inj}}}}}}}}})}$$
(3)

where \(\Delta {f}_{{{{{{{\mathrm{inj}}}}}}}}=\alpha \sqrt{{P}_{{{{{{{\mathrm{inj}}}}}}}}}\) (with \(\alpha\) as a constant related to cavity losses), and \({f}_{{m}_{{inj}}}\) is the mode frequency at \({m}_{{inj}}\). Notably, when m=minj, the extended Alder’s Eq. (2) reduces to the standard Alder’s equation. The perfect fittings in Fig. 3b–d, 3h–j indicate that our extended Alder’s Eq. (2) accurately describes all the coherent injection-locking effects in our superconducting microcomb.

In previous studies of continuous-wave superconducting lasers, the downconversion of higher-order Josephson frequencies to the resonator’s fundamental mode has been demonstrated25. However, the upconversion of Josephson photons to higher mode has not been reported. In Fig. 4a, we present a comb spectrum (red) within our typical experimental bandwidth of 3–10 GHz. The applied DC bias voltage across the Josephson junction is Vdc = 36.387 µV, corresponding to a Josephson frequency fJ = 17.60 GHz. As illustrated in Fig. 4a, all the comb modes with fm < fJ are generated through downconversion, and any modes with fm > fJ (if existing) would result from upconversion. However, emissions with fm > 17.60 GHz lie beyond the bandwidth of our spectrum measurements, preventing their direct observation.

Fig. 4: Up- and down-conversion of Josephson photons into multiple modes.
figure 4

a Illustration of the up- and down-conversion. A comb spectrum (red) is obtained at Vdc = 36.39 μV, corresponding to a Josephson frequency of 17.6 GHz. b–d Coherent injection-locking effects with msen = 7 and for injections at minj = 1 (b), 6 (c), and 54 (d), respectively. The dashed lines are fittings to extend Alder’s Eq. (2).

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The coherent injection-locking effect provides an advanced methodology for sensing comb emissions beyond the detection bandwidth. To demonstrate this, we perform injection-locking measurements using various injection tones with minj = 1, 6, 54, as illustrated in Fig. 4a, while selecting a fixed sensing tone within the detection bandwidth, such as msen = 7. Figures 4b–d display the coherent injection-locking effects. The injection-locking phenomenon arises from the nonlinear interaction between emission tone and external injection signal43. Since no injection signal is applied at m = 7, the coherence injection-locking effect displayed in Fig. 4d suggests the presence of a comb emission tone at f54 = 39.28 GHz, which is significantly higher than the Josephson frequency fJ = 17.6 GHz. This observation provides evidence for the upconversion process of Josephson photons to higher modes. The presence of comb emissions at m = 1 (Fig. 4b) and 54 (Fig. 4d) indicates that our Josephson microcomb spans a frequency range exceeding five octaves, a span typically challenging to achieve in semiconductor-based frequency combs. It is essential to note that this octave value is limited by the upper-frequency limit of the microwave generator used in our experiments. Hence, it is highly likely that even higher modes could exist beyond this range.

Discussion

We have demonstrated a fully integrable superconducting microcomb. Unlike previously reported superconductor-based frequency combs44,45,46,47,48,49,50,51,52, which all operated under microwave pumping using externally supplied, expensive, and energy-consuming microwave sources, our superconducting microcomb is driven solely by a DC bias voltage. This makes it highly desirable for scalable and on-chip integration. The initiation of our microcomb requires simply tuning on a low-power DC bias source, with a minimal input power as low as 40 pW (Fig. 1b)—approximately nine orders of magnitude lower than the semiconductor-based frequency combs. This presents significant advancement for ultrasensitive and energy-efficient applications.

The exceptional coherence of our comb enables a unique coherent injection-locking effect previously inaccessible in frequency combs, leading to an ultra-high-resolution and tunable comb. This innovation introduces unique functionalities, such as the generation of coherent subcombs through four-wave mixing, as showcased in Supplementary Fig. 12. The simultaneous up- and down-conversion of Josephson photons significantly expands the comb’s frequency range over multiple octaves. These capabilities offer added flexibility and tunability for applications of superconducting microcomb technology.

This DC-biased superconducting microcomb with ultra-low power consumption (down to a few tens of picowatts) can work at ultra-low temperatures, aligning seamlessly with the operating conditions of superconducting quantum circuits. For a typical dilution refrigerator, which has a cooling power of hundreds of microwatts at 20 mK, in principle, it is feasible to integrate up to millions of the combs at the base temperature based on their energy efficiency. Our comb can serve as a multifrequency microwave source for multiplexed quantum measurement53. An important potential application of our superconducting microcomb is the development of a low-cost, on-chip arbitrary waveform generator. This could be achieved by controlling the intensities of the comb teeth54, which is possible by integrating our superconducting comb with a series of frequency-tunable superconducting resonators serving as adjustable filters55. This advancement promises substantial benefits for future quantum technologies. Moreover, implementing superconducting frequency combs in the terahertz domain could be feasible by using intrinsic Josephson junctions in high-temperature superconductors56. This approach could also extend the technology’s working temperature range.

Methods

Device fabrication

We fabricated four superconducting microcombs (#1-#4), and detailed parameters are listed in Supplementary Tables 1. The superconducting resonators are fabricated using a superconducting Nb film for device #1 and a Ta (\({{{{{\rm{\alpha }}}}}}\)-phase) film57 for devices #2, #3, and #4. The film was sputtered on a 10 \({{{{{\rm{mm}}}}}}\) by 10 \({{{{{\rm{mm}}}}}}\) sapphire substrate (C-plane, thickness 650 \(\,{{{{{\rm{\mu }}}}}}{{{{{\rm{m}}}}}}\)). Standard photolithography followed by reactive ion etching in a CF4 was then used to define the resonators. The Al/AlOx/Al (Al thicknesses 40 nm/80 nm) Josephson junction was fabricated using standard double-angle evaporation and lift-off techniques. To ensure good electrical contact between the Josephson junction and the resonator layers, the sample was ion-beam milled to remove residual oxides and resist residue from the surface of the resonator before the double-angle evaporation.

Experiments

The devices are mounted in a dilution refrigerator with a base temperature of 20 mK. A complete circuit is shown in Supplementary Fig. 13. We adopt a similar approach to refs. 25,30,31 for the low noise biasing scheme to measure the tunneling current and provide a stable voltage bias to the device. The bias voltage is supplied by an on-chip voltage divider circuit consisting of a 10 Ω shunt resistor and a 10 Ω reference resistor. The current through the device is then measured via the voltage drop Vr across the 10 Ω reference resistor. The current \({I}_{D}\) and voltage \({V}_{{{{{{{\mathrm{dc}}}}}}}}\) of the device is given by \({I}_{D}=\,{V}_{r}/10\) and \({V}_{{{{{{{\mathrm{dc}}}}}}}}=10{I}_{{{{{{{\mathrm{bias}}}}}}}}-20{I}_{D}\). Here,\(\,{V}_{r}\) is obtained from a nano voltmeter (Keithley 2182 A) and \({I}_{{{{{{{\mathrm{bias}}}}}}}}\) is the output from a current source (Keithley 6221). Additional filtering for the circuit is provided by two 100 \({{{{{\rm{\mu }}}}}}{{{{{\rm{F}}}}}}\) chip capacitors. Additionally, all low-frequency lines are heavily filtered outside the PCB with multi-pole RC and PI low-pass filters located in the mixing chamber of the dilution refrigerator.

The device’s output signal is amplified by an amplifier chain consisting of a cryogenic amplifier (+42 dB) and two room-temperature amplifiers (+32 dB). The spectra are acquired by an Agilent N9010A spectrum analyzer (S/A), and the waveforms are recorded using a Keysight MSOV334A digital oscilloscope (D/O). The injection signal to the resonator is generated by an Agilent N5183A analog signal generator (S/G) and is attenuated by a low-temperature attenuation chain to assure that the thermal contribution of photons to the cavity is negligible.

Extended Adler’s equation

In the unlocking range outside Δfinj, the emission signals for msen = minj, which can be described by Adler’s theory43, are given by:

$${f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}},n}=({f}_{{{{{{{\mathrm{inj}}}}}}}}+{f}_{h})+n{f}_{h}$$
(3)

where fh is the repetition frequency of the harmonic emissions. The first term, finj + fh, accounts for the comb’s emission at minj, while \(n{f}_{h}\) represents the nth harmonic shift. The fh is given by:

$${f}_{h}=({f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}},0}-{f}_{{{{{{{\mathrm{inj}}}}}}}})\sqrt{1-(\frac{\Delta {f}_{{{{{{{\mathrm{inj}}}}}}}}/2}{{f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}},0}-{f}_{{{{{{{\mathrm{inj}}}}}}}}})}$$

where \(\Delta {f}_{{{{{{{\mathrm{inj}}}}}}}}=\alpha \sqrt{{P}_{{{{{{{\mathrm{inj}}}}}}}}}\) (with \(\alpha\) as a constant related to cavity losses). The comb’s emission at minj corresponds to n = 0, therefore we obtain \({f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}},0}={f}_{{{{{{{\mathrm{inj}}}}}}}}+{f}_{h}\) from Eq. (3). Utilizing the superconducting microcomb formula (1) (see the main text), we can express the comb’s emissions for any arbitrary m as \({f}_{m,0}=\frac{m}{{m}_{{{{{{{\mathrm{inj}}}}}}}}}{f}_{{m}_{{{{{{{\mathrm{inj}}}}}}}},0}={\frac{m}{{m}_{{{{{{{\mathrm{inj}}}}}}}}}}({f}_{{{{{{{\mathrm{inj}}}}}}}}+{f}_{h})\). By adding the nth harmonic shift nfh, we derive an extended Adler’s Eq. (2), as shown in the main text, for the coherent injection-locking in our superconducting combs.