Abstract
When two Bose–Einstein condensates collide with high collisional energy, the celebrated matterwave interference pattern appears^{1}. For lower collisional energies, the repulsive interaction energy becomes significant, and the interference pattern evolves into an array of grey solitons^{2,3}. But the lowest collisional energies, producing a single pair of solitons, have not been probed so far. Here, we report on experiments using density engineering on the healing length scale^{3,4} to produce such a pair of solitons. We see evidence that the solitons evolve periodically between vortex rings and solitons. The stable, periodic evolution is in sharp contrast to the behaviour seen in previous experiments^{5,6} in which the solitons decay irreversibly into vortex rings through the socalled snake instability^{7,8,9,10,11,12,13}. The evolution can be understood in terms of conservation of mass and energy in a narrow condensate.
Main
We consider two condensates separated in space, and with an initial collisional energy E_{col}, which could be in the form of kinetic energy, or potential energy in a magnetic trap. The condensates are then allowed to collide in the z direction. For negligible interactions, the two colliding condensates will interfere like two noninteracting plane waves, forming a standingwave interference pattern with wavefunction ψ∝cos(k z), where the wavelength of the pattern is λ=2π/k, and k is given by ℏ^{2}k^{2}/2m=E_{col}/N, where m is the atomic mass and N is the number of atoms. The density of the pattern is therefore of the form ^{2} n=ψ^{2}∝cos^{2}(k z). The initial energy E_{col} supplies the kinetic energy of the standing wave, proportional to ∂^{2}ψ/∂ z^{2}, as well as the negligible interaction energy N μ/4 (see Supplementary Information). However, if E_{col}is decreased to become comparable to N μ/4, the form of the fringes will be modified by interactions. It will become energetically favourable to form an array of grey solitons, rather than a cosinesquared pattern^{2}. This soliton formation process is relevant for the operation of an atom interferometer^{14}. The criterion E_{col}<N μ/4 can be written as λ>4πξ, where ξ is the healing length, the approximate width of a grey soliton^{15,16}. In other words, as E_{col}decreases, the wavelength of the standingwave interference pattern increases until the wavelength reaches the size of a soliton. At this point, the fringes will evolve into solitons. A cosinesquared fringe and a soliton are actually very similar in form, as seen in Fig. 1a,b. As E_{col} decreases further, the number of solitons will decrease, maintaining conservation of energy^{3}. The lowest collisional energies will result in two grey solitons travelling in opposite directions. Such a low collisional energy is achieved in our experiment by initially separating the two condensates by a barrier with a width of the order of ξ, as shown in Fig. 1c. The energy of this initial configuration relative to the ground state is of the order of E_{col}=N μ(ξ/L), where L is the length of the condensate in the z direction. This is comparable to the energy of a grey soliton computed in one dimension, which is of the order of
where v is the subsonic soliton velocity and c is the speed of sound in the condensate^{13}. E_{col} is thus sufficient to create a pair of solitons.
Solitons can also be created by phase engineering^{17,18,19}. There is a phase step across the grey soliton, given by Δφ=2cos^{−1}(v/c). By imposing this phase step, the density profile will evolve into that of a soliton. We can understand this process by considering the current density . By imposing the phase step, this current density is significant only in the region of the step. This corresponds to a divergence in the flow field, . As the condensate obeys the continuity equation , the divergence depletes the density, forming the soliton minimum.
We use density engineering to form solitons, a complementary technique to phase engineering. We impose the density profile, and the phase evolves into that of a soliton. The process is illustrated in Fig. 1d, which shows two solitons separated by a distance d. They are moving in opposite directions, with the corresponding opposite phase steps. Our initial configuration shown in Fig. 1c can be thought of as the d=0 case of Fig. 1d. Our initial configuration thus corresponds to two overlapping, counterpropagating solitons.
In a nearly onedimensional (1D) condensate, solitons are stable objects^{20,21}. In contrast, in 3D, the stable nonlinear modes are vortex rings, as well as hybrids of solitons and vortex rings^{22}. In previous studies in 3D, grey solitons were seen to decay irreversibly into vortex rings through the snake instability^{7,8,9,10,11,12}, which was originally measured in optical solitons^{13}. We find a very different, oscillatory behaviour, shown in Fig. 2 by a simulation of the 3D Gross–Pitaevskii equation^{15} (GPE). The soliton evolves into a vortex ring, which subsequently evolves back into a soliton. This process repeats itself many times with a period of 7–8 ms. We call this a ‘periodic soliton/vortex ring’, which involves the repeated revival of a soliton, which is an unstable object in our system. The simulation shows that the periodic soliton/vortex ring is stable, surviving the reflection from the end of the condensate, as well as the subsequent collision in the middle of the condensate. The periodic soliton/vortex ring has a constant phase step (0.7π for our experiment), as shown in Fig. 2j,k.
First, we will explain the evolution of the soliton into a vortex ring in terms of conservation of energy. Owing to the higher density near the centre of the condensate, the soliton moves fastest in this region^{18}. This causes the soliton to curve^{18}, as shown in Fig. 2f. Owing to the curvature, the area of the soliton has increased, corresponding to an increase in the soliton energy. To conserve the energy of the soliton, the increase in area must be compensated by an increase in v, by equation (1). This corresponds to a reduction in the depth of the soliton, as shown in Fig. 1b. The soliton thus moves with everincreasing speed and decreasing depth, until the density minimum vanishes altogether, leaving a vortex ring, as shown in Fig. 2c,g,i. The appearance of the vortex ring does not violate Kelvin’s theorem (conservation of circulation), because the vortex core can enter and exit the condensate through the nearby boundary^{23}.
The fate of vortex rings in a quantum fluid is a longstanding field of study^{23,24}. In superfluid ^{4}He at nonzero temperatures, it was suggested that the radius of a vortex ring shrinks in time owing to dissipation, until it evolves into a roton. We can understand the very different evolution of our vortex ring into a soliton in terms of conservation of mass. We consider a vortex ring in an infinite homogeneous condensate^{15,25}, as illustrated^{24} in Fig. 2l. The density is constant in time, maintained by the zero divergence of the flow (). In other words, after the flow traverses the centre of the vortex ring from right to left, it returns to the right outside the vortex ring. In contrast, our vortex ring is close to the edge of the condensate, as shown in Fig. 2c,g,i. The flow lines therefore have no return path to the right side, causing . The divergence depletes the density, forming the soliton minimum. This situation is very similar to that of phase engineering of a condensate described above. In fact, the vortex ring has a phase step, as shown in Fig. 2j. The phase step occurs on a length scale Δz of the order of the radius of the vortex ring, which is also the radius R of the condensate. Thus, for creating solitons from vortex rings, R should not be too much larger than ξ. This can also be understood energetically. The periodic soliton/vortex ring requires that the soliton and the vortex ring have the same energy. The ratio of these energies contains a factor of R/ξ, placing an upper limit on R. Using the GPE simulation, we find that the vortex ring evolves into a soliton for , corresponding to for our trap frequencies. Note that it is the radial variation of the condensate rather than the axial variation that causes the evolution.
We carry out density engineering through a narrow, bluedetuned laser barrier, which is adiabatically ramped up in τ_{ramp}=200 ms to a height U=h(3,000 Hz), creating a Bose–Einstein condensate (BEC) with a density minimum, as shown in Fig. 3a. The potential is then rapidly turned off in 1 ms, resulting in the initial condition illustrated in Fig. 1c. This sequence is indicated by the solid curve in Fig. 3b. As seen in Fig. 3d, the two counterpropagating solitons can be distinguished after 2 ms, which result from the interference between the two condensates in Fig. 3a, colliding with low energy. Figure 3d corresponds to Fig. 1d. Figure 3c shows the simulation at 2 ms with imaging effects, including integration of the density perpendicular to the image, finite resolution and finite depth of field. The experimental images in Fig. 3 agree well with the simulated images. The resolution is taken to be 2 μm, which gives the best visual agreement between the experiment and the simulation. This is larger than the measured 1.2 μm resolution of the imaging system. We attribute this difference to defocusing of the imaging system with its short depth of field.
After a time τ=5 ms, we obtain in situ images of the pair of counterpropagating vortex rings of the periodic soliton/vortex ring, as shown in Fig. 3g. In the image, the two vortex rings are seen as weak density minima. The weak density minima are also seen in Fig. 3h, which shows the profile along the central axis in Fig. 3g. Owing to the shape of a vortex ring, the density minima should be weak along this central axis. After each of the two vortex rings has evolved back into a soliton, we image the soliton stage at τ=9 ms, as shown in Fig. 3j. Two strong, slitlike solitons are seen. Figure 3k shows the greatly increased contrast along the central axis in the soliton stage, relative to the vortex ring stage in Fig. 3h. This increase in contrast is evidence for the evolution of the vortex ring into a soliton. We find that the solitons are more visible for the small atom number shown in the figure.
To measure the speed of the periodic soliton/vortex ring, we observe its position as a function of time, as shown in the integrated profiles in Fig. 4a, and the filled circles in Fig. 4b. We see that the speed (the slope) is approximately constant in time, and is therefore the same for the soliton stage and the vortex ring stage. The speed is observed to be v=0.66±0.08 mm s^{−1}, for N=3×10^{4}. This speed is of the same order of magnitude as the speed v≈(ℏ/2m R)ln(1.59R/ξ)≈0.3 mm s^{−1} predicted for a vortex ring in an infinite, homogeneous condensate^{15,23,24}. Excellent agreement between experiment and simulation is seen throughout Fig. 4.
The speed of a grey soliton is less than c, so it is interesting to compare the speed of the periodic soliton/vortex ring with c. We measure c using the technique of ref. 26. Specifically, we rapidly turn on the barrier in 100 μs and leave it on as indicated by the dashed curve in Fig. 3b, creating two counterpropagating sound pulses, as shown after 5 ms in Fig. 4g. The profiles in Fig. 4c and the open circles in Fig. 4b show the position of the sound pulses as a function of time, giving c=1.24±0.07 mm s^{−1}, which agrees with the theoretical value^{27} of . We thus find that the periodic soliton/vortex ring moves slower than the speed of sound, at v=0.53c. This subsonic motion is evident in Fig. 4e,g, in that the two arrows are closer together for the vortex rings than for the sound pulses.
In conclusion, we have achieved density engineering of a pair of counterpropagating grey solitons, which are the result of a lowenergy collision between two BECs. The close relationship between solitons and matterwave interference fringes illustrates the quantum mechanical nature of solitons in a BEC. Owing to the inhomogeneous nature of the narrow condensate, each soliton evolves into a periodic soliton/vortex ring. As vortex rings and solitons can be considered to be quasiparticles^{8,15,22,23}, perhaps the periodic soliton/vortex ring could also be described as a Rabi oscillation between these quasiparticle states. Such a Rabi oscillation is seen in optics, between two soliton states^{28}.
After submission of our manuscript, we learned of ref. 29, reporting density engineering of pairs of solitons. As the condensate was nearly onedimensional, the solitons were stable.
Methods
Our experimental apparatus is basically described in ref. 30. Potentials are focused onto the condensate through an aspheric lens with numerical aperture 0.5. Imaging is carried out through this same lens. After reaching the BEC phase, the trap frequencies are adiabatically reduced by half in 350 ms to 117 and 13 Hz in the radial and axial directions, respectively. The resulting condensate contains 1×10^{4} or 3×10^{4} ^{87}Rb atoms, as specified in the captions of Figs 3 and 4. The condensate with 3×10^{4} atoms has μ/h=580 Hz (h is Planck’s constant), corresponding to a minimum healing length of ξ=0.32 μm, which is found at the centre of the condensate. The healing length is larger at the lower densities found away from the centre.
The laser barrier is bluedetuned by 6 nm from the 780 nm resonance. The condensate is split in the radial direction by this highly elongated laser beam with an axial diameter of 1.4 μm, which is 4.4ξ (3.4ξ) for the larger (smaller) condensate. Using the 1D GPE simulations of ref. 3, as well as our 3D GPE simulations, this diameter of roughly 4ξ is the maximum that will create a single pair of solitons.
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Acknowledgements
We thank A. Soffer, M. Segev, W. Ketterle, A. Minguzzi and R. Ozeri for helpful discussions. This work was supported by the Israel Science Foundation.
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Shomroni, I., Lahoud, E., Levy, S. et al. Evidence for an oscillating soliton/vortex ring by density engineering of a Bose–Einstein condensate. Nature Phys 5, 193–197 (2009). https://doi.org/10.1038/nphys1177
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DOI: https://doi.org/10.1038/nphys1177
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