Shortcuts to Adiabatic Soliton Compression in Active Nonlinear Kerr Media

TL;DR

This study addresses fast, high-fidelity soliton self-compression in active nonlinear Kerr media by marrying a variational approximation with shortcuts to adiabaticity (STA) through inverse engineering. By deriving an Ermakov-like equation for the soliton width $a(z)$ and designing tailored profiles for $G(z)$, $\beta_2(z)$, and $\gamma(z)$, the authors achieve substantial compression over much shorter propagation lengths than conventional adiabatic methods, with fidelities up to $\sim 0.99$. The framework is shown to be robust to third-order dispersion and is extendable to decompression, while outlining practical constraints such as positivity of the control fields and boundary-condition requirements. The approach is general and transferable to other dissipative nonlinear systems, and it opens routes for integrating STA with optimal control to further refine soliton dynamics in photonic, cold-atom, and magnetic contexts.

Abstract

We implement variational shortcuts to adiabaticity for optical pulse compression in an active nonlinear Kerr medium with distributed amplification and spatially varying dispersion and nonlinearity. Starting with the hyperbolic secant ansatz, we employ a variational approximation to systematically derive dynamical equations, establishing analytical relationships linking the amplitude, width, and chirp of the pulse. Through the inverse engineering approach, we manipulate the distributed gain/loss, nonlinearity and dispersion profiles to efficiently compress the optical pulse over a reduced distance with high fidelity. In addition, we explore the dynamical stability of the system to illustrate the advantage of our protocol over conventional adiabatic approaches. Finally, we analyze the impact of tailored higher-order dispersion on soliton self-compression and derive physical constraints on the final soliton width for the complementary case of soliton expansion. The broader implications of our findings extend beyond optical systems, encompassing areas such as cold-atom and magnetic systems highlighting the versatility and relevance of our approach in various physical contexts.

Figures (6)

• Figure 1: (a) A comparison between $a(z)$ (red-solid line) obtained using inverse engineering with a final fiber length of $z_f=6$ and the adiabatic reference (blue-dashed line) with $z_f=60$. (b) The corresponding distributed gain $g(z)$ designed via inverse engineering (red-solid line) compared with the adiabatic protocol with constant $g_0$ (blue-dashed line). (c) and (d) illustrate the spatio-temporal evolution of soliton wave packets, respectively, in both the adiabatic and STA protocols. Other parameters are: $g_0=0.01$, $\beta_2=1$ and $\gamma=2$.
• Figure 2: (a) A comparison between $a(z)$ (red-solid line) obtained using inverse engineering with a final fiber length of $z_{f}=6$ and the adiabatic reference (blue-dashed line) with $z_{f}=60$. (b) The corresponding GVD $\beta_2(z)$ is designed through inverse engineering with the STA protocol shown in red-solid line, while the blue-dashed line represents the GVD profile with adiabatic protocol. (c) and (d) illustrate the spatio-temporal evolution of soliton wave packets in the adiabatic and STA protocols. Here the constant Kerr nonlinearity is set to $\gamma=2$, $\beta_2 (0)=1$ and $\beta_2 (z_f)=0.3$.
• Figure 3: (a) A comparison between the corresponding nonlinear function $\gamma(z)$ designed via inverse engineering (red-solid line) with $z_f=6$ and the adiabatic protocol $\gamma(z) = 2e^{\gamma_0 z}$ with constant $\gamma_0 =0.02$ (blue-dashed line) with $z_f=60$. (b) illustrates the spatio-temporal evolution of soliton wave packets in the STA protocol. Other parameters are: $\gamma(0)=2$, $\gamma(z_f)=6.6$, $\beta_2=1$, and $g=0$.
• Figure 4: The fidelity versus propagation distance $z$ for STA (solid line) and adiabatic protocol (dashed line) soliton compression in three sections: (a) distributed gain, (b) decreasing GVD, and (c) variable nonlinearity strength. The corresponding parameters in different cases are the same as those in Fig. \ref{['evolutionone']} to \ref{['nonlinear']}.
• Figure 5: Comparing the output intensity profile via STA protocol ($z_f =6$, red-solid line) and adiabatic protocol evolution ($z_f = 60$, blue-dashed line). Other parameters are: $\beta_3 = 0.006$, $\gamma=2$, $\beta_2=1$, and $g_0=0.01$ for adiabatic reference.