Transfer of Bose-Einstein condensates through discrete breathers in an optical lattice

We study the effect of discrete breathers (DBs) on the transfer of a Bose-Einstein condensate (BEC) in an optical lattice using the discrete nonlinear Schrödinger equation. In previous theoretical (primarily numerical) investigations of the dynamics of BECs in leaking optical lattices, collisions between a DB and a lattice excitation, e.g., a moving breather (MB) or phonon, were studied. These collisions led to the transmission of a fraction of the incident (atomic) norm of the MB through the DB, while the DB can be shifted in the direction of the incident lattice excitation. Here we develop an analytic understanding of this phenomenon, based on the study of a highly localized system—namely, a nonlinear trimer—which predicts that there exists a total energy threshold of the trimer, above which the lattice excitation can trigger the destabilization of the DB and that this is the mechanism leading to the movement of the DB. Furthermore, we give an analytic estimate of upper bound to the norm that is transmitted through the DB. We then show numerically that a qualitatively similar threshold exists in extended lattices. Our analysis explains the results of the earlier numerical studies and may help to clarify functional operations with BECs in optical lattices such as blocking and filtering coherent (atomic) beams.

Figure 1

DB solutions for the symmetric case ${\psi }_1={\psi }_3$ [Eq. (7)]. For a nonlinearity $\lambda >5.04$, four symmetric solutions exist. The solution for $N_2>1/2$ is termed a bright breather, while the solution for $N_2\to 0$ corresponds to a dark breather [see Eq. (12)]. The dashed vertical line at $N_2=1/3$ marks the asymptote for the phase-wise and antiphase-wise time-periodic solutions for $\lambda \to \infty$.

Figure 2

(a) The lower part of the PN energy landscape exhibits three minima separated by saddle points. (b) The phase space of the trimer is restricted to lie between the two parts of the PN shell, which consists of the lower and upper parts of the PN landscape, which are shown in this panel. (c) Contour plot of (a), the lower part of the PN energy landscape. The three minima and saddle points are clearly visible. The minimum at $A_1=A_3=0.17$ in (c) corresponds to the bright breather. The figure is plotted for $\lambda =3$.

Figure 3

Dynamics on the PN landscape for increasing total energy of the trimer. (a) (left) A contour plot of the lower PN energy landscape $H_{\mathrm{PN}}^l$ is shown for total energy below the rim ($E_t=-1.32<E_{\mathrm{thrs}}=-1.311$). A projection of the orbit onto the $A_1$–$A_3$ plane is over-plotted (black curve). (a) (middle) the corresponding amplitudes $A_i(t)$ indicate that the maximum amplitude remains at the central site. The dashed vertical line marks the time interval $[0,25]$ for which the orbits in the left picture are plotted. (a) (right) A sketch of the initial condition shows that the excitation at site $1$ is slightly below threshold. (b) Destabilization of the DB for $E_t=-1.310>E_{\mathrm{thrs}}$ just above the rim. We see that the rim of the PN landscape clearly restricts the dynamics and governs the destabilization process of the DB (left panel). The norm migrates from site $2$ to site $1$, see the crossing of amplitudes $A_2(t)$ (green/light-gray line) and $A_1(t)$ (blue thick line) for short times $t<10$ (middle panel). (c) For higher total energy $E_t=-1.28$ the bottleneck at the rim widens and the maximum norm transmitted to site $3$ is increased. (d) For even higher total energy $E_t=-1.04$ the orbit explores large parts of the phase space and visits all three sites. The grey shaded area (left panel) is forbidden by the upper PN landscape $H_{\mathrm{PN}}^u$. In all cases $\lambda =3,$ ${\delta }_{\varphi }=\pi$. For other values of ${\delta }_{\varphi }$ the same qualitative behavior is found.

Figure 4

Norm transfer through a bright breather. The maximum (atomic) norm at site $3$ detected after the collision of the DB with a lattice excitation is shown as a function of the total energy of the trimer. The discrete symbols indicate three different values of the initial phases. For $E_t<E_{\mathrm{thrs}}$ the DB is stable and practically no transfer of norm takes place on short time scales. For $E_t>E_{\mathrm{thrs}}$ we observe instability of the DB centered at site $2$: the breather migrates to site $1$ and norm is transferred to site $3$. An upper bound to $max[N_3(t)]$ is calculated from the PN landscape, both analytically [dotted dashed line, cf. Eq. (21)] and umerically (solid line). The analytical calculation is performed in the limit for large $\lambda$ and therefore deviates slightly from the exact numerical result. We used $\delta {\varphi }_{23}=\pi$ which is the typical case observed in [28] for DBs in an extended leaking optical lattice and $\lambda =3$.

Figure 5

The destabilization of the DB in the trimer (solid lines) is compared to the trimer with a third linear site (dashed lines). On the time scale, where the destabilization takes place, the dynamics is qualitatively very similar—see enlargements in upper pictures showing a crossing of $A_2(t)$ (green/light-gray line) and $A_1(t)$ (blue thick line) at $t\approx 2.5$. Total energy is $E_t=-1.31>E_{\mathrm{thrs}}$ and $\lambda =3$. (a) The initial condition (18) is given by $\delta A=0.016$, ${\delta }_{\varphi }=\pi$, which is the same as in Fig. 3b. (b) The initial condition is determined by the parameters $\delta A=0.152$, ${\delta }_{\varphi }=\pi /2$.

Figure 6

Demonstration that the proposed mechanism for the destabilization and migration of a DB works in an extended lattice ($M=101$ wells). (a) Unperturbed breather solution from the trimer centered in the middle of the lattice at site $51$. The color code represents the (atomic) norm $|{\psi }_n(t)|{}^2$ (left panel). In the right panel the amplitudes of the central site of the initial breather, $A_{51}$ (green/light-gray line), and the two neighbors to the left, $A_{50}$ (blue thick line) and $A_{49}$ (red line), are monitored. (b)–(d) Increasing perturbation at the left neighboring site $n=50$. (b) The total energy of the local trimer $({\psi }_{c-1},{\psi }_c,{\psi }_{c+1})$ is just above $E_{\mathrm{thrs}}$. The breather is barely stable but no migration takes place. (c) The breather migrates by one site toward the perturbation (that is, to the left) after $t\approx 5$ time steps. (d) During the migration process, a moving breather is initiated to the right. For $t>5$ the right panel of (c) and (d) monitors a local trimer, with the breather having moved into its center. Initial conditions are given by Eq. (23) with (a) ${\delta }_A={\delta }_{\varphi }=0$, (b) ${\delta }_A=0.016$, ${\delta }_{\varphi }=\pi$, (c) ${\delta }_A=0.03$, ${\delta }_{\varphi }=\pi$, and (d) ${\delta }_A=0.14$, ${\delta }_{\varphi }=\pi$.