Quantum dynamics of dissipative Kerr solitons

Dissipative Kerr solitons arising from parametric gain in ring microresonators are usually described within a classical mean-field framework. Here, we develop a quantum-mechanical model of dissipative Kerr solitons in terms of the Lindblad master equation and study the model via the truncated Wigner method, which accounts for quantum effects to leading order. We show that, within this open quantum system framework, the soliton experiences a finite coherence time due to quantum fluctuations originating from losses. Reading the results in terms of the theory of open quantum systems allows us to estimate the Liouvillian spectrum of the system. It is characterized by a set of eigenvalues with a finite imaginary part and a vanishing real part in the limit of vanishing quantum fluctuations. This feature shows that dissipative Kerr solitons are a specific class of dissipative time crystals.

Figure 1

Schematic representation of the generation of a Kerr optical frequency comb using a high-$Q$ Kerr optical ring microresonator. A continuous-wave source drives the ring, which induces the propagation of a soliton (depicted in red) along the ring. The output signal shows the optical frequency comb. When all the resonator modes participate in the parametric process, the nonlinear dynamics give rise to a DKS.

Figure 2

Integrated dispersion relation $D_{\mathrm{int}}\left(l\right)$ versus relative mode number $l$. The red markers show the 101 spectral modes that are considered in the numerical simulations. The parameters are given in the text and correspond to typical experimental situations [18].

Figure 3

(a) Time evolution of the total number of photons in the ring for different values of $\tilde{N}$. (b) Total number of photons in the ring and intracavity power versus the scaling parameter $\tilde{N}$. The values are taken at ${\tau }^{*}=60$, where the photon number reached a stationary distribution.

Figure 4

Snapshot of the mode occupation at ${\tau }^{*}$ for (a) $\tilde{N}=1$, (b) ${10}^{-2}$, (c) ${10}^{-4}$, (d) $2.5\times {10}^{-5}$, and (e) $6.3\times {10}^{-6}$. The red dashed lines show the GPE prediction for the mode occupation. Parameter values are ${\sigma }_0=-1.024\kappa , D_1=1.8587\times {10}^3\kappa , D_2=2.02\times {10}^{-2}\kappa , g/2\pi =0.49\times {10}^{-9}\kappa$, and $F=1.8\times {10}^4$.

Figure 5

Snapshot of the field density in real space at (a), (e), (i), and (m) $\tau =0.5\times {10}^4\kappa /2T\approx 8.5$, (b), (f), (j), and (n) $\tau =1.5\times {10}^4\kappa /2T\approx 25.4$, (c), (g), (k), and (o) $\tau =2.5\times {10}^4\kappa /2T\approx 42.3$, and (d), (h), (l), and (p) $\tau =3.5\times {10}^4\kappa /2T\approx 59.2$, for (a)–(d) $\tilde{N}={10}^{-2}$, (e)–(h) ${10}^{-4}$, (i)–(l) $2.5\times {10}^{-5}$, and (m)–(p) $6.3\times {10}^{-6}$. The density is plotted as a function of the coordinates $x=Rcos\left(\theta \right)$ and $y=Rsin\left(\theta \right)$, with $R$ being the radius of the ring resonator. The soliton is depicted at a time multiple of $T=2\pi /D_1\approx 4.2\times {10}^{-12}$ s; the rotation period of the soliton is along the ring. For this choice of time, the peak of the soliton always occupies the same position in the ring, allowing an easier comparison between the different plots.

Figure 6

Occupation density in real space for (a) and (d) $\tilde{N}={10}^{-2}$, (b) and (e) ${10}^{-4}$, and (c) and (f) $4.4\times {10}^{-5}$ at (a)–(c) $\tau =6$ and (d)–(f) $\tau =15$. Solid lines are averages over ${10}^4$ trajectories, while three single trajectories are plotted as crosses, stars, and diamonds.

Figure 7

(a) Liouvillian gap $\mathrm{\Lambda }$ versus $\tilde{N}$. The error bars of the fit (in red) show the standard error (the 95% confidence interval) of each point. The power-law fit of the Liouvillian gap for $\tilde{N}\ge 6.3\times {10}^{-4}$ is shown by the solid line. The coefficients of the fit are $\mathrm{\Lambda }/\kappa ={\tilde{N}}^a\times b$, with $a=-0.99\pm 0.01$ and $b=(6.3\pm 0.3)\times {10}^{-7}$. Inset: the Liouvillian gap versus the total number of photons inside the ring microresonator at long times (see Fig. 3). The dashed line represents a power-law fit.

Figure 8

Schematic representation of the spectrum of the Liouvillian. The spectrum always has one zero eigenvalue, corresponding to the steady state (red cross). A set of eigenvalues with a vanishing real part and equally spaced imaginary parts emerges in the classical limit of large $\tilde{N}$ (blue circles). The Liouvillian gap is the distance between the complex eigenvalues with the largest real part and the imaginary axis.

Figure 9

Fourier spectrum obtained by considering 51 modes for (a) $\tilde{N}=1$, (b) $\tilde{N}=4\times {10}^{-4}$, (c) $\tilde{N}={10}^{-4}$, (d) $\tilde{N}=2.5\times {10}^{-5}$, and (e) $\tilde{N}=1.1\times {10}^{-5}$. The gray region shows the $95\%$ standard deviation. The red dashed lines represent the envelope of the Fourier spectrum obtained from the GPE.