Coherent walk and lock in driven fast-gain frequency-combs

. Locking multiple modes into a frequency comb is key for multiple metrological applications, and a great effort has been therefore invested in this challenge over the last decade. The most common techniques are based on either nonlinearities or modulation of the cavity, while the latter is considered the more controllable method to produce frequency combs. The modulation couples cavity modes and creates a lattice in a synthetic dimension with coherent walk dynamics, but typically these dynamics are overthrown by the dissipative processes, leading to a spectrum that is narrow relatively to the full frequency ladder potential. Here we propose and demonstrate that by using fast-gain we preserve the full potential of the coherent walk and lock the frequency comb at its maximum possible bandwidth. Moreover, we find in our system a unique regime of dissipative fast-gain Bloch oscillations. We demonstrate these dynamics in RF-modulated quantum cascade laser ring devices.

Quantum walk dynamics is naturally telling us the story of the lattice and of its band structure.For example, a fast walk, i.e. a wavefunction that diverges very quickly, shows that the coupling is efficient.The walk can reveal lattice disorder if the transport exponentially drops, or if there is a revival of the initial conditions, there might be a gauge field involved, e.g.Bloch oscillations.For this exact reason, in recent years, the quantum walk has become one of the main tools for quantum algorithms [1].But this type of dynamics is not reserved exclusively for quantum particles in real space lattices [2,3].Synthetic lattices, which are crystals in a dimension created by changing a property such as frequency or position in time [4][5][6], can also support quantum walk dynamics [6].
The most practical synthetic dimension used today is the frequency ladder, which is created by modulating the system with the modes frequency difference.Effectively, it also creates a synchronization of coherence and typically leads to frequency-combs, which are extremely important for applications of metrology and communication.The coherent walk is observed for example if a single frequency light is injected into an EO modulated cavity -with time, the bandwidth will expand in a pattern that follows the quantum walk dynamics.However, in most such systems, when dissipation kicks in, it breaks the dynamics, choosing one favorable supermode which is dictated by dispersion and gain curvature and influencing the coherence of the comb.In principle, nonlinearities can break this link [7].
Here we demonstrate a new type of nonlinear quantum walk and lock dynamics based on fast gain-recovery media.This type of gain, through a dissipative four-wave mixing process, uncovers the full potential of the synthetic space.The ultrafast response of the saturation allows the wavefunction in the synthetic lattice to evolve and lock on its maximum potential defined by the dispersion and mode coupling [8].We observe these dynamics in RF modulated ring quantum cascade lasers (QCLs), which then benefits from an extremely controllable wide frequency-comb spectrum at its output.Finally, we show that by detuning the modulation frequency from the ring-cavity resonance of the QCL we are able to observe nonlinear Bloch oscillations.The unique nature of the gain mechanisms in quantum cascade lasers (QCL) generally facilitates frequency comb generation without external cavities, enabling simple on-chip spectroscopy devices in the mid-IR and THz frequencies [9].QCLs are extremely important in the molecular fingerprint region of the mid-infrared, where they allow a flexible platform for developing frequency-combs with small device footprints with sufficient output power and wide spectral coverage [10].
Due to the fast gain recovery time of QCLs, these types of devices can lase in a frequency comb regime, while their action is often frequency-modulated (FM) and not pulsed [11] due to almost instantaneous saturation that pushes for uniform intensity in the cavity and thus for lower peak power [12].For example, for homodyne spectroscopy, lower peak power is advantageous, but the most critical features are the linewidth and, more importantly, the bandwidth of the laser output.
The bandwidth of QCL frequency comb is typically governed by carrier lifetimes, dispersion, and waveguide losses, and the relations between these quantities were studied in previous work [13,14].However, in ring QCLs (Fig. 1a), where in principle unidirectional propagation is allowed, the comb formation is negligible, and can even vanish.
In our current work, we inject ring QCLs with current modulated at radio-frequencies (RF) and study the dynamics of the spectrum (fig. 1b).When the modulation matches the cavity length resonance, it creates coupling between adjacent modes and induces quantum walk dynamics.Typically, in a linear system, the quantum walk would reach the lattice edges, which are here defined by the depth of the modulation and dispersion, and then obtain an aperiodic pattern.Adding linear selective dissipation can stabilize the, but usually, the resulting spectrum is relatively narrow, not utilizing the full potential of the synthetic lattice (fig.1c).However, the fast-gain saturation in our QCL devices, which acts instantaneously on the intensity, locks the quantum walk at its widest part when it reaches the edges, unlocking the full spectral potential of the modulated system (fig.1d).
We also study the dynamics when the RF modulation is slightly off resonance, which leads to an effective constant gauge field and therefore Bloch oscillations.
Typically, Bloch oscillations serve as a probe for the band structure, or in our case, of the synthetic lattice [15].It can also help to find the quantities related to topological systems, such as Zak phase in 1D chiral systems.Bloch oscillation sampling can be also extended to 2D systems, probing the Berry curvature or winding numbers [16].It is therefore critical to study Bloch oscillations also in dissipative and nonlinear systems, to understand Bloch oscillations sampling for the whole variety of systems that exist in nature.We measure nonlinear Bloch oscillations in QCL rings (fig.2).The dynamics starts with bloch oscillations, but then the state slowly stabilizes on narrower state.The top plot in Fig. 2 shows the measure signal and the bottom the simulation that reproduces our results with realistic parameters., 07021 (2023)

Fig. 1 .
Fig. 1.Coherent walk and lock in fast-gain lasers.(a) ring mid-IR QCL fast-gain device.(b) Experimental demonstration of a fast-gain quantum walk and lock in a QCL.The RF modulation takes the single mode to a quantum walk.(c) Typical active mode-locking mechanism with regular gain saturates the average power, but the dissipation at frequencies far from the gain peak leads to a narrow Gaussian spectrum.(d) Active mode-locking dynamics with fast-gain.The RF modulation takes the single mode to a quantum walk which ends at the edges of the lattice, that are defined by the ratio between the modulation depth and the dispersion.At the end of the process, the state is locked at its highest spectral potential.

Fig. 2 .
Fig. 2. Fast-gain Bloch oscillations.The detuning from the resonant cavity frequencies leads to an effective electric gauge field acting on the synthetic lattice sites.This field drives Bloch oscillations, that slowly (on a  scale) saturate due to the fastgain dynamics of the system.The left plot shows the measured phenomenon in a quantum cascade laser with fast-gain and the right is a simulation of the dynamics.