Oscillons in coupled Bose-Einstein condensates

Long-lived, spatially localized, and temporally oscillating nonlinear excitations are predicted by numerical simulation of coupled Gross-Pitaevskii equations. These oscillons closely resemble the time-periodic breather solutions of the sine-Gordon equation but decay slowly by radiating Bogoliubov phonons. Their time-dependent profile is closely matched with solutions of the sine-Gordon equation, which emerges as an effective field theory for the relative phase of two linearly coupled Bose fields in the weak-coupling limit. For strong coupling the long-lived oscillons persist and involve both relative and total phase fields. The oscillons decay via Bogoliubov phonon radiation that is increasingly suppressed for decreasing oscillon amplitude. Possibilities for creating oscillons are addressed in atomic gas experiments by collision of oppositely charged Bose-Josephson vortices and direct phase imprinting.

Figure 1

Time evolution of the oscillon with imprinted SG breather parameters, $u=0.5$ and $\nu =0.01$ [Eq. (9)]: (a) The density profile of the two superposed atomic fields, $|{\psi }_1+{\psi }_2{|}^2$. (b) The spatial distribution of the relative phase, ${\varphi }_1-{\varphi }_2$. (c) The spatial distribution of the total phase, ${\varphi }_1+{\varphi }_2$. (d) The density profile of the grand canonical GP energy $E$. (e) The measured frequency of the oscillon. (f) The computed localized energy of the oscillon: blue circle indicates the energy $E_B$ obtained according to Eq. (12); black line indicates the energy $E$ obtained by integrating Eq. (1); red line indicates the energy $H_{\mathrm{SG}}$ obtained by integrating Eq. (6).

Figure 2

Formation of GP solitons with the imprinted SG breather parameters, $u=0.5$ and $\nu =0.5$ [Eq. (9)]: (a) The density profile of the two superposed atomic fields, $|{\psi }_1+{\psi }_2{|}^2$. (b) The spatial distribution of the relative phase, ${\varphi }_1-{\varphi }_2$. (c) The spatial distribution of the total phase, ${\varphi }_1+{\varphi }_2$. (d) The density profile of the grand canonical GP energy $E$. Note that the imprinted SG breather is unstable which instantly decays into two GP solitons.

Figure 3

Time evolution of the oscillon with imprinted SG breather parameters, $u=10$ and $\nu =0.5$ [Eq. (9)]: (a) The density profile of the two superposed atomic fields, $|{\psi }_1+{\psi }_2{|}^2$. (b) The spatial distribution of the relative phase, ${\varphi }_1-{\varphi }_2$. (c) The spatial distribution of the total phase, ${\varphi }_1+{\varphi }_2$. (d) The density profile of the grand canonical GP energy $E$. (e) The measured frequency of the oscillon. (f) The computed localized energy of the oscillon: blue circle indicates the energy $E_B$ obtained according to Eq. (12); black line indicates the energy $E$ obtained by integrating Eq. (1); red line indicates the energy $H_{\mathrm{SG}}$ obtained by integrating Eq. (6). The deviation of the red and black lines in (f) indicates that the strongly coupled condensates cannot be described by the SG equation for the relative phase.

Figure 4

Oscillon stability: (a) The ratio of the remnant GP energy of the oscillon to that of the initial state at $t=15T_0$ as a function of ${\omega }_{\mathrm{imp}}$ and $\nu$ (see text). (b) The relative frequency as a function of ${\omega }_{\mathrm{imp}}$ and $\nu$. The upper green regions in (a) and (b) indicate where the SG breather initial condition cannot be imprinted. The unstable (black) region in (a) corresponds to the lower green region in (b), as ${\omega }_{\mathrm{meas}}$ is undefined.

Figure 5

Collision of two Josephson vortices ($\nu =0.005$): (a) The density profile of the two superposed atomic fields, $|{\psi }_1+{\psi }_2{|}^2$. (b) The spatial distribution of the relative phase, ${\varphi }_1-{\varphi }_2$. The two countermoving Josephson vortices with velocities $v=\pm 0.1$ collide at the instant $t=800$ forming an oscillon. As shown in (a), the quasibreather continuously radiates Bogoliubov sound waves to the background superfluid, causing an up-chirp of the oscillon's oscillation frequency.

Figure 6

Dynamics following a Gaussian phase imprint as described in the text. Panels (a) and (c) show the density profile of the superposition $|{\psi }_1+{\psi }_2{|}^2$, and (b) and (d) show the relative phase ${\varphi }_1-{\varphi }_2$ for two different sets of initial parameters: $u=2$, $\nu =0.01$, $\epsilon =0.2$ [(a) and (b)] and $u=10$, $\nu =0.5$, $\epsilon =0.2$ [(c) and (d)]. In both cases a long-lived breather is created as seen in the relative phase.