Unidirectional transport of wave packets through tilted discrete breathers in nonlinear lattices with asymmetric defects

We consider the transfer of lattice wave packets through a tilted discrete breather (TDB) in opposite directions in the discrete nonlinear Schrödinger model with asymmetric defects, which may be realized as a Bose-Einstein condensate trapped in a deep optical lattice, or as optical beams in a waveguide array. A unidirectional transport mode is found, in which the incident wave packets, whose energy belongs to a certain interval between full reflection and full passage regions, pass the TDB only in one direction, while in the absence of the TDB, the same lattice admits bidirectional propagation. The operation of this mode is accurately explained by an analytical consideration of the respective energy barriers. The results suggest that the TDB may emulate the unidirectional propagation of atomic and optical beams in various settings. In the case of the passage of the incident wave packet, the scattering TDB typically shifts by one lattice unit in the direction from which the wave packet arrives, which is an example of the tractor-beam effect, provided by the same system, in addition to the rectification of incident waves.

Figure 1

The transport scheme under consideration. Large green dots denote identical incident wave packets coming from the left in (a) and the right in (b). The middle part schematically shows the TDB, where ${\varepsilon }_n=0$, except for ${\varepsilon }_{n_c-1}<0$.

Figure 2

(a)–(c) Contour plots of energy given by Eq. (8) for different ${\varepsilon }_1$ with fixed $\lambda =3$ and ${\varepsilon }_3=0$. (d)–(f) The structure of the DB corresponding to panels (a)–(c), respectively. SP1 and SP2 denote the saddle points.

Figure 3

Dynamics of the trimer initiated by the perturbations applied at site 3 [(a),(b)] or at site 1 [(c),(d)]. Here $E_{\mathrm{thr}1}=-1.01787>E_{\mathrm{thr}2}=-1.29859$ correspond to SP1 and SP2, respectively. (a) $H=-1.319<E_{\mathrm{thr}2}$, and the orbit cannot climb over SP2. (b), (c) $E_{\mathrm{thr}2}<H=-1.263<E_{\mathrm{thr}1}$, hence the orbit can pass SP2, but cannot pass SP1. (d) $H=-1.011>E_{\mathrm{thr}1}$, and the orbit can climb over the SP1. Perturbation parameters in Eq. (12) are $\delta A_3=0.004528$ in (a) and $\delta A_3=0.032866$ in (b). Similarly, perturbations applied at site 1 are $\delta A_1=0.055125$ in (c) and $\delta A_1=0.158584$ in (d). In all the cases, $\lambda =3$, ${\varepsilon }_1=-0.5$, ${\varepsilon }_3=0$, $A_1^{\mathrm{TDB}}=0.1412,A_3^{\mathrm{TDB}}=0.1664$, and $\delta \theta =\pi$.

Figure 4

The transfer of incident waves through the stationary TDB in the long lattice ($M=101$ sites), for identical excitations arriving from the left in (a) and from the right in (b), respectively. Colors represent norms at the sites. In (c) and (d), the small and red circles depict the TDBs located, essentially, at sites $50,51$, and 52, with $A_{50}=0.1502$, $A_{51}=0.974553$, $A_{52}=0.166384$. Circles of different size and color represent the incident, reflected, and transmitted excitations. The amplitudes of the incident excitations are $A_1=A_4=0.2$ and $A_2=A_3=0.4$, and its energy is $\delta E=0.39743$. The energy thresholds are $E_{\mathrm{thr}1}=-1.14325$ in (a) and $E_{\mathrm{thr}2}=-1.30276$ in (b) for the excitations arriving from the left and right, respectively. In all the cases, $\lambda =3$, ${\varepsilon }_n=0$, except for ${\varepsilon }_{50}=-0.3$, and $E_{\mathrm{TDB}}=-1.6573$.

Figure 5

The transfer of incident waves through the stationary TDB in the long lattice ($M=101$ sites) with weaker intrinsic nonlinearity of identical excitations arriving from the left in (a) and the right in (b), respectively (cf. Fig. 4 where the effective nonlinearity of the incident pulses is stronger by a factor $\approx 3$). The TDB is located, essentially, at sites $50,51$, and 52. The amplitudes of the incident excitations are $A_1=A_4=0.16$ and $A_2=A_3=0.32$. In all the cases, $\lambda =2.5$, ${\varepsilon }_n=0$, except for ${\varepsilon }_{50}=-0.8$, ${\varepsilon }_{52}=0.5$.

Figure 6

The transfer of incident waves through the stationary TDB in the long lattice ($M=101$ sites) with different energies. Colors represent norms at the sites. The amplitudes of the incident excitations are $A_2=A_3=0.32$, and its energy is $\delta E=0.2594<0.271$ in (a) and (b); $A_2=A_3=0.35$ and $0.271<\delta E=0.2966<0.315$ in (c) and (d); $A_2=A_3=0.4$ and $\delta E=0.38395>0.315$ in (e) and (f). In all the cases, $A_1=A_4=0.18$, $\lambda =3$, ${\varepsilon }_n=0$, except for ${\varepsilon }_{50}=-0.05$.

Figure 7

The transmission, reflection, and absorption coefficients for the passage through the stationary TDB in the long lattice ($M=101$ sites) of the same incident wave packet arriving from the left and from the right, as functions of its energy, $\delta E$. The energy thresholds for the excitation coming from the left and right satisfy $E_{\mathrm{thr}1}>E_{\mathrm{thr}2}$, and $\to$ ($\nrightarrow$) implies that the incident wave can (cannot) pass the TDB. Regions I, II, and III are defined by, respectively, $E_t<E_{\mathrm{thr}2}$, $E_{\mathrm{thr}2}<E_t<E_{\mathrm{thr}1}$, and $E_t>E_{\mathrm{thr}1}$. The TDB is located, essentially, at sites $50,51$, and 52, with $A_{50}=0.163359$, $A_{51}=0.972447$, and $A_{52}=0.166316$. In all the cases, $\lambda =3$ and ${\varepsilon }_n=0$, except for ${\varepsilon }_{50}=-0.05$.

Figure 8

Contour plots of the energy given by Eq. (8) from the main text when the absolute value of ${\varepsilon }_1$ is large enough, with fixed $\lambda =3$ and ${\varepsilon }_3=0$. SP1 and SP2 denote the saddle points.