Nearly Quantum-Limited Josephson-Junction Frequency-Comb Synthesizer

While coherently driven Kerr microcavities have rapidly matured as a platform for frequency-comb formation, such microresonators generally possess weak Kerr coefficients; consequently, triggering comb generation requires millions of photons to be circulating inside the cavity. This suppresses the role of quantum fluctuations in the dynamics of the comb. In this paper, we realize a minimal version of coherently driven Kerr-mediated microwave-frequency combs in the circuit quantum electrodynamics (cQED) architecture, where the fluctuations of the quantum vacuum are the primary limitation on comb coherence. We achieve a comb phase coherence of up to 35 $\mu \mathrm{s}$, approaching the theoretical device quantum limit of 55 $\mu \mathrm{s}$ and vastly longer than the inherent lifetimes of the modes, of 13 ns. The ability within cQED to engineer stronger nonlinearities than optical microresonators, together with operation at cryogenic temperatures, and the excellent agreement of comb dynamics with quantum theory indicates a promising platform for the study of complex dynamics of quantum nonlinear systems.

Figure 1

The system schematic and flux sweep. (a) The two-mode device can be represented by a single-cavity mode (blue, ${\omega }_a$) coupled linearly to a single Kerr mode (red, ${\omega }_b$). We create a comb in this device by driving with a carefully chosen single microwave drive of strength $\eta$ at frequency ${\omega }_d$, which interacts with the device to create a series of tones at regularly spaced output frequencies (the “comb”). The linear mode plays the role of mediating a delayed self-interaction of the nonlinear mode (right panel), with kernel $F(\tau )=e^{-\tau /{\chi }_a}$. In the strong coupling regime, the non-Markovian nature of the interaction fundamentally modifies the stability of the nonlinear mode, enabling comb formation. (b) The cQED implementation of the schematic in (a). The nonlinear-mode inductance is formed by 25 Superconducting QUantum Interference Devices (SQUIDs) in series; a false-color SEM image of a component SQUID indicates the small asymmetry employed to alleviate hysteresis. The SQUID array is coupled to an antenna that both forms the capacitance of the nonlinear mode and controls its dipole coupling with the linear mode; the latter is the $\lambda /4$ mode of a coaxial three-dimensional (3D) cavity, fabricated of copper to permit the passage of an external dc flux that threads all the SQUIDs. The input signal drives the cavity through port 1, and $S_{11}(\omega )$ is monitored. (c) A color plot of the reflected signal versus the frequency from port 1 [$S_{11}(\omega )$] for a range of applied coil bias currents and/or applied SQUID fluxes. The linear- and nonlinear-mode frequencies are highlighted by blue squares and red dots, respectively. By fitting for the bare-mode frequencies, we determine $g/2\pi =87.6956$ MHz, as well as the bare-mode frequencies, represented by the dashed lines.

Figure 2

The phase diagram and the comb spectrum. (a) The theoretically predicted phase diagram in ${\mathrm{\Delta }}_{db}$-${\mathrm{\Delta }}_{da}$ space, indicating observable phases characterized by the number and stability of classical fixed points (FPs) over a range of experimentally accessible drive powers ($-132$ dBm to $-67$ dBm; see text for details). Here, the unstable regime (shaded red) exhibiting no stable fixed points (SFPs) is entered by fixing the drive frequency to ${\omega }_d=4.9085(2\pi )$ GHz, varying the nonlinear-mode frequency along the direction of the green arrow via a flux sweep (see the schematic) and observing the output power spectra. (b) Typical power spectra as a function of increasing drive power, along the indicated cross section of the experimental phase diagram in (c). At low powers, the system exhibits single-frequency output at ${\omega }_d$ (1), marked gray in the phase diagram. With increasing drive power (2), the system enters a regime with a regularly spaced multifrequency output, with spacing $\mathrm{\Delta }$, which is plotted in the phase diagram in drive-${\mathrm{\Delta }}_{db}$ space. With stronger driving power (3), the spacing $\mathrm{\Delta }$ increases, while fewer side peaks are observed above the noise background, before the system ultimately exits the unstable regime and single-frequency output resumes. The theoretical phase diagram is plotted in the top panel of (c) for comparison.

Figure 3

Comb coherence. (a) The coherence time $T_{\mathrm{coh}}$ extracted by measuring $G^{(1)}(\tau )$ [Eq. (2)] for the same operating parameters as Fig. 2. The inset shows the approximate $T_{\mathrm{coh}}$ calculated using Floquet analysis of linearized SDEs (see the text). (b) A cross section of the phase diagram along the dashed line in (a), in black. The blue curve and shaded region indicate the theoretically calculated coherence time due to nonlinearity alone (pure dephasing ${\gamma }_{\phi }=0$). The orange curve shows $T_{\mathrm{coh}}$ for ${\gamma }_{\varphi }=2.0(2\pi )$ kHz, demonstrating good agreement with the experimental result. (c) The experimental (top panel) and theoretical (lower panel) $G^{(1)}(\tau )$ for ${\gamma }_{\phi }=2.0(2\pi )$ kHz at positions (1) (stable regime), (2) (threshold of comb formation), and (3) (higher drive power in comb regime). (d) The numerical results for the variation of the coherence time with nonlinearity. The purple and green points correspond to parameters for devices A (25 SQUIDs) and B (5 SQUIDs), respectively, obtained by varying the nonlinearity alone. The experimental results for the devices are shown by the orange square and diamond; corresponding $G^{(1)}(\tau )$ are marked by the same symbols as in (c). The curves are fits to $T_{\mathrm{coh}}=a({\gamma }_{\phi }+b\mathrm{\Lambda }{)}^{-1}$, with $(a,b)=(1.19,0.55)$ for device A and $(0.94,0.40)$ for device B. (e) Top panel: the $I$-$Q$ trace at positions (1), (2), and (3), showing the two-dimensional (2D) projection of the limit-cycle orbit, which decreases in radius with increasing power. Lower panel: the theoretical effective radius of the limit cycle $r_{\mathrm{eff}}$ (solid blue) and the standard deviation of the noise projected tangential to limit cycle, $\delta n$ (solid red, right-hand axis), scaled by their values at threshold. The relative decrease in $r_{\mathrm{eff}}$ combined with the increase in $\delta n$ point toward a reduction in coherence time with increasing drive power (see the text).

Figure 4

Temporal instabilities. (a) By fixing ${\omega }_a$ and ${\omega }_b$ and varying ${\omega }_d$ (top panel), the system can be driven to the regime with no stable fixed points along the purple arrow in Fig. 2, while leaving the underlying mode structure unchanged. The resulting phase diagram plotting the observed comb spacing $\mathrm{\Delta }$ is shown, with the theoretical prediction in the inset. For ${\mathrm{\Delta }}_{db}/2\pi \lesssim -30$ MHz and strong enough driving, a distinct regime emerges (dark gray) where the output spectrum broadens significantly. In (b), we show the typical evolution of the spectrum across the white dashed line, chosen to show a large variation in the comb spacing. In dashed red are the underlying polariton resonances, indicating that the emergent comb peaks do not exactly coincide with these resonances. (c) The time dynamics as observed via $I$-$Q$ traces, with both axes scaled by $⟨A⟩=\sqrt{⟨I^2⟩+⟨Q^2⟩}$ at position (1), for ease of direct comparison. In the stable comb regime (1), the cavity response settles into an obvious orbit as before; the inset shows a 500-ns trace after $t=40\mu \mathrm{s}$, demonstrating the stable orbit. In the unstable regime (2), the response shows large deviations over time and no periodic phase-space trajectory is observed.

Figure 5

The calculated maximal Lyapunov exponent ${\lambda }_M$ in drive-${\mathrm{\Delta }}_{db}$ space. The panel on the right framed in blue indicates the detuning range explored in Figs. 2 and 3 of the main text. The solid orange curve indicates the linear stability boundary. In the white regions, ${\lambda }_M<0$ is negative and the system is therefore stable. In the light-gray regions, ${\lambda }_M\approx 0$, indicating a stable limit cycle (LC). The dark regions are where ${\lambda }_M>0$ and the system exhibits chaotic dynamics. Typical long-time dynamics projected in the nonlinear-mode phase space are plotted in (a)–(c) corresponding to dynamics in the stable fixed point, the stable limit cycle, and the chaotic regimes, respectively.

Figure 6

The coherence times as a function of ${\gamma }_{\phi }$. The colored lines and points show numerically obtained $T_{\mathrm{coh}}$ values from the simulation of Eqs. (3) as a function of the drive power, while the dashed diamonds indicate experimental values.