Observation of Two-Dimensional Localized Jones-Roberts Solitons in Bose-Einstein Condensates

Jones-Roberts solitons are the only known class of stable dark solitonic solutions of the nonlinear Schrödinger equation in two and three dimensions. They feature a distinctive elongated elliptical shape that allows them to travel without change of form. By imprinting a triangular phase pattern, we experimentally generate two-dimensional Jones-Roberts solitons in a three-dimensional atomic Bose-Einstein condensate. We monitor their dynamics, observing that this kind of soliton is indeed not affected by dynamic (snaking) or thermodynamic instabilities, that instead make other classes of dark solitons unstable in dimensions higher than one. Our results confirm the prediction that Jones-Roberts solitons are stable solutions of the nonlinear Schrödinger equation and promote them for applications beyond matter wave physics, like energy and information transport in noisy and inhomogeneous environments.

Figure 1

(a) Experimental setup for phase imprinting. An imprint pattern displayed on a digital micromirror device (DMD) is imaged by a microscope objective onto the atoms (see inset). The detection light is superimposed on a polarizing beam splitting cube and transverses the atomic cloud, whose shadow is imaged with a second microscope objective onto the chip of a CCD camera. (b) Gross-Pitaevskii simulations of the creation of JRS via phase engineering. A triangular phase pattern whose lower vertex is approximately at the center of the BEC generates two JRSs that travel across the BEC. The time between frames is 10 ms.

Figure 2

(a) Experimental column density profiles of the BEC taken at different times after the initial imprinting. The time of flight is 10 ms. (b) Corresponding numerical simulations (in trap) using the Gross-Pitaevskii equation. In both (a) and (b), two elongated but spatially localized rarefaction pulses (density dips), known as Jones-Roberts solitons, are formed close to the center of the BEC after 5 ms from the initial imprinting. After formation, the two rarefaction pulses maintain an almost constant finite size until they reach the border of the BEC. The discrepancies between (a) and (b) are mainly due to the switching off of the trapping potential, that is not simulated in (b). The density defects that are visible at the border of the BEC are due to unavoidable spurious sound waves stemming from the phase imprint and are unrelated to the physics of the JRS.

Figure 3

(a) Points: measured relative depth over time for the JRS (filled blue) and DPS (open orange) as a function of time. Error bars are one $\sigma$ statistical error. The blue solid line is the depth predicted by the numerical simulations rescaled to take into account the finite resolution of our imaging system ($\simeq 1\mu \mathrm{m}$). The orange dashed line is an exponential fit to the data. The inset shows an absorption image of our JRSs reaching the border of the BEC and breaking into VA pairs. The experimental observations are in good agreement with the simulations. The phase structure illustrates that the phase winds in opposite directions around the two winding points. (b) Experimental column density profiles of DPS at different evolution times showing the decay due to snaking instability. (c) Points: measured orientation of the JRSs. Error bars are one $\sigma$ statistical error. Filled black and open red points correspond to solitons traveling towards left and right, respectively. The solid lines are the prediction from the numerical simulations. The orientation shows small variations over time due to sound waves traveling through the BEC and a substantial discrepancy with respect to the direction of travel $\varphi$, indicated by the dashed lines. The inset shows the definition of the angles $\theta$ and $\varphi$.

Figure 4

(a) Evolution of the long axis $\alpha$ of the JRSs as a function of time. Points are the experimental data while the solid line is the result of the numerical simulations. The small wiggles are due to sound waves traveling across the BEC. The scale on the left (right) relates to the experimental data (simulations). The scaling factor of 3 comes from the 10 ms time of flight and is in accordance with the numerical simulations. The dashed line (related to the scale on the right) is the mean value of the KPC for times $>5\mathrm{ms}$, and the shaded band takes into account the fluctuations of the speed of the solitons. (b) The same as (a) but for the short axis $\beta$. The different scaling factor takes into account also the limited resolution of our imaging ($\simeq 1\mu \mathrm{m}$). Error bars for (a) and (b) are one $\sigma$ statistical error.