Dark-Soliton Molecules in an Exciton-Polariton Superfluid

The general theory of dark solitons relies on repulsive interactions and, therefore, predicts the impossibility to form dark-soliton bound states. One important exception to this prediction is the observation of bound solitons in nonlocal nonlinear media. Here, we report that exciton-polariton superfluids can also sustain dark-soliton molecules, although the interactions are fully local. With a novel all-optical technique, we create two dark solitons that bind together to form an unconventional dark-soliton molecule. We demonstrate that the stability of this structure and the separation distance between two dark solitons is tightly connected to the driven-dissipative nature of the polariton fluid.

Popular Summary

Solitons are localized waves propagating with unchanged shape in a nonlinear medium. Since their discovery in 1834 by Scottish civil engineer Russell in the shallow waters of the Union Canal, solitons have been observed in a variety of systems such as plasmas, molecules, and quantum fluids. Solitons can be bright, with a localized bump, or dark, with a dip on a homogeneous background. In this work, we use a novel all-optical technique to imprint dark solitons in a type of quantum fluid and find that two dark solitons can bind together to form a “dark-soliton molecule.”

Bright solitons are known to attract or repel each other, but previous work has shown that dark solitons experience only mutual repulsive forces, except in media with nonlocal long-range interactions. Our work shows this is not always the case: Dark solitons can attract one another in media with local interactions as well.

In our experiments, the medium is a quantum fluid of polaritons, quasiparticles composed of a photon coupled to an electric dipole. We show that two dark solitons can be imprinted parallel to each other and bound to form a stable “molecule,” which propagates on the fluid over macroscopic distances. Remarkably, the distance between the solitons is independent of the fluid properties and is determined only by the intrinsic dissipation of the polariton system. This behavior is in sharp contrast with observations in other quantum fluids such as atomic Bose-Einstein condensates.

Our results are a significant advance in the understanding of nonlinear phenomena in out-of-equilibrium quantum fluids, and our all-optical technique opens a way to the systematic study of quantum turbulence.

Figure 1

Experimental setup. A Ti:sapphire laser beam with engineered shape and phase is focused on the microcavity sample with a small incidence angle to create a polariton fluid that propagates in the $y$ direction. (a) shows the theoretical bistability curve resulting from the quasiresonant pumping. The gray region corresponds to the low-density regime (LD), while the red one has much higher polariton density (HD). The white part delimits the bistable hysteresis cycle. The phase profile of the excitation beam is customized with the SLM, as plotted in (b). The beam is then spatially filtered to smooth the phase jump: (c) illustrates how the opening of the filtering slit oriented along the $x$ direction influences the soliton pair propagation in the polariton fluid. (d) shows the shape of the pump: The white dashed line corresponds to the upper bistability frontier, while the black rectangle indicates the detection field of view. The blue area inside the high-intensity region indicates where the phase is shifted by $\pi$. The smoothing of the phase jump in the $y$ direction due to the spatial filtering is indicated by the color scale. The real space image gives access to the density and phase maps (interfering with a reference beam), whereas the conditions of the driving field are extracted from the momentum space detection.

Figure 2

Impression of parallel dark solitons. (a) Propagation of two parallel dark solitons. (i) Phase pattern created by the SLM, with the same field of view as the detection. The phase modulation is sent only on the bottom part of the image. (ii) Density (logarithmic scale) and phase at high power, where almost all the pumped area is above the bistability range and, therefore, gets the phase profile corresponding to the driving one. The top of the picture corresponds to the limit of the excitation: One can see a thin bistable region just before the density of polariton abruptly drops to the low-density regime. (ii) At lower intensity: The white area corresponds to the bistable part of the fluid where the solitons propagate freely, until they reach the lower part of the bistability range. The energy detuning $\mathrm{\Delta }E$ is 0.13 meV, the fluid velocity is $1.35\mu \mathrm{m}/\mathrm{ps}$, $c_s=0.31\mu \mathrm{m}/\mathrm{ps}$, and $m^{*}=2.20\times {10}^{-34}\mathrm{kg}$. (b) Impression of two pairs of dark solitons. (i) Phase pattern designed by the SLM, with the same field of view as the detection. (ii) Density (logarithmic scale) and (iii) phase of the four solitons propagating through the bistable part of the fluid.

Figure 3

Set of solitons impression with different spatial filtering. In (a), the phase jump is not filtered: On the fluid, it is blurred by the flow, but the solitons do not propagate on the bistable region. From (b) to (d), the filtering of the spatial components is gradually increased. The solitons start to propagate, but the angles between them remain too high and they bounce on each other, vanishing along their propagation. In (e), the spatial components are filtered enough so that they can align and propagate parallel, remaining dark solitons. For this set, the detuning between the laser and the lower polariton branch $\mathrm{\Delta }E$ is 0.11 meV, the flow speed is $v_f=0.66\mathrm{\mu m}/\mathrm{ps}$, the speed of sound $c_s=0.42\mathrm{\mu m}/\mathrm{ps}$, and the effective polariton mass $m^{*}=1.04\times {10}^{-34}\mathrm{kg}$.

Figure 4

Set of solitons aligning to each other for different imprinting distances. The red area indicates the imprinting region of the fluid. The fluid flow is from bottom to top. (a) Experimental data, with the density map on top and the corresponding phase map below. Solitons get closer together along their propagation, until reaching a minimal distance from which they align parallel. For this set, the energy detuning $\mathrm{\Delta }E$ is 0.34 meV, for a fluid velocity of $1.00\mathrm{\mu m}/\mathrm{ps}$, a speed of sound of $0.67\mathrm{\mu m}/\mathrm{ps}$, and an effective polariton mass of $m^{*}=1.24\times {10}^{-34}\mathrm{kg}$. (b) Numerical simulations which show the same behavior: As soon as the driving intensity is reduced enough to reach the bistable regime, imprinted solitons enter a transition region where they get closer, until a minimal distance from which they align.

Figure 5

(a) Evolution of the distance between the solitons along their propagation for different initial imprinting distances. Each color line corresponds to one picture with a different imprinted separation distance. Solitons tend to align at an equilibrium distance from each other. The energy shift $\mathrm{\Delta }E=0.34\mathrm{meV}$, the fluid velocity is $1.00\mu \mathrm{m}/\mathrm{ps}$, $c_s=0.67\mu \mathrm{m}/\mathrm{ps}$, and $m^{*}=1.24\times {10}^{-34}\mathrm{kg}$. The black dashed line illustrates the equilibrium separation distance at $4.8\mu \mathrm{m}$. The distances are extracted from the fit of the transverse soliton profile, shown in the inset. (b) Summary of the experimental equilibrium separation distances for different hydrodynamic conditions, as a function of the Mach number. It stays constant around $4.75\mu \mathrm{m}$. The parameters of each set are listed in Table 1. (c) Numerical simulations done by tuning only the energy shift $\mathrm{\Delta }E$ and keeping constant the fluid velocity and the effective mass as the experimental ones.

Figure 6

Numerical simulations of the equilibrium separation distance as a function of the decay rate of the polariton (red dots). The blue dot corresponds to the experimental value for our sample. The error bars result from fluctuation of the solitons in the region where the distance is averaged.