Macroscopic Zeno Effect and Stationary Flows in Nonlinear Waveguides with Localized Dissipation

We theoretically demonstrate the possibility of observing the macroscopic Zeno effect for nonlinear waveguides with localized dissipation. We show the existence of stable stationary flows, which are balanced by losses in the dissipative domain. The macroscopic Zeno effect manifests itself in the nonmonotonic dependence of the stationary flow on the strength of the dissipation. In particular, we highlight the importance of the dissipation parameters in observing the phenomenon. Our results are applicable to a large variety of systems, including the condensates of atoms or quasiparticles and optical waveguides.

Figure 1

Possible experimental scenarios to observe the MZE: nonlinear optical waveguide (upper left), a magnon waveguide (upper right), a plasmonic nanostructure (lower left), an atomic BEC in a waveguide, and two reservoirs (lower right).

Figure 2

(a) Density distributions $n(x)$ for symmetric flows for $g=1$ and $\ell =4$ and different values of ${\Gamma }_0$. Solid line: ${\Gamma }_0=0.01$; dashed line: ${\Gamma }_0=1$; dotted line: ${\Gamma }_0=10$. (b) Density distributions $n(x)$ for antisymmetric flows for $g=1$ and $\ell =1$. Solid line: ${\Gamma }_0=0.1$; dashed line: ${\Gamma }_0=1$; dotted line: ${\Gamma }_0=10$. (c) and (d) Current vs strength of the dissipation for symmetric flows (with $\ell =4$) and antisymmetric flows (with $\ell =1$) obtained for $g=0.1$ and $g=1$; stable (unstable) flows correspond to the solid (dotted) fragments of the curves. (e) Currents and instability increments vs width of the defect for symmetric [$(s)$] and antisymmetric [$(a)$] flows for $g=1$ and ${\Gamma }_0=1$. In all panels, ${\rho }_{\infty }=1$.

Figure 3

Evolution of the density $|\Psi (t,x){|}^2$ starting from the initial data $\Psi (0,x)=1$. For all the shown panels, ${\rho }_{\infty }=g={\Gamma }_0=1$. (a) The width of the defect is $\ell =4$. The generation of the symmetric stationary flow occurs. (b) and (c) The width of the defect is $\ell =2$ and $\ell =6$. The symmetric flows are unstable, and therefore no stationary flow is established.