Collisions of matter-wave solitons
Abstract
Solitons are localised wave disturbances that propagate without changing shape, a result of a nonlinear interaction which compensates for wave packet dispersion. Individual solitons may collide, but a defining feature is that they pass through one another and emerge from the collision unaltered in shape, amplitude, or velocity, but with a new trajectory reflecting a discontinuous jump. This remarkable property is mathematically a consequence of the underlying integrability of the one-dimensional (1D) equations, such as the nonlinear Schrödinger equation, that describe solitons in a variety of wave contexts, including matter-waves[1, 2]. Here we explore the nature of soliton collisions using Bose-Einstein condensates of atoms with attractive interactions confined to a quasi-one-dimensional waveguide. We show by real-time imaging that a collision between solitons is a complex event that differs markedly depending on the relative phase between the solitons. By controlling the strength of the nonlinearity we shed new light on these fundamental features of soliton collisional dynamics, and explore the implications of collisions in the proximity of the crossover between one and three dimensions where the loss of integrability may precipitate catastrophic collapse.
Department of Physics and Astronomy, Rice University, Houston, TX 77005 Centre of Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122, Australia Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
The name “soliton” is meant to convey the particle-like qualities of localised and non-dispersing wave packets, and is often reserved for those solitary waves that pass through one another without changing form. The wave packets studied here are not true solitons due to the presence of a harmonic confining potential which breaks integrability. Furthermore, as quasi-D objects, they reside near the border between D and D, where integrability is also broken. Nonetheless, the proximity to this border does not necessarily affect the soliton dynamics over experimentally relevant time scales. We will thus use the term soliton more generally to refer to non-dispersing wave packets created by a nonlinearity. While the propagation of individual solitons is now well-understood, having been extensively studied both experimentally and theoretically[3], their interactions with each other have been much less explored. While true solitons pass through one another, they nonetheless exhibit an effective interaction produced by interference of the two wave packets, with a force falling off exponentially with separation[4]. The sign and magnitude of the interaction depends on the relative phase, as was first demonstrated experimentally for optical solitons in both the time[5] and spatial[6] domains.
Studies of matter-wave solitons have mainly examined properties of single solitons[7, 8, 9, 10], with the study of soliton interactions limited to those occurring in soliton trains[8, 11] and collisions between multiple solitons resulting from a quench[12]. We provide further insight into the collisional dynamics of matter-wave solitons through the controlled formation of soliton pairs, and explicitly demonstrate that the discontinuous jump observed in soliton collisions[13] is a general property of the nonlinear interaction.
Our methods for producing a degenerate gas of atoms have been described previously[14] and are summarized in the Methods section. A Bose-Einstein condensate of atoms in the state is formed by evaporative cooling at a scattering length of and is confined in a cylindrically symmetric harmonic trap with radial and axial oscillation frequencies of and , respectively. After forming the condensate, a cylindrically-focussed blue-detuned Gaussian laser beam directed perpendicular to the long axis of the confining potential is used to cut the condensate in half, and acts as a barrier between the two condensates (Fig. 1a). The scattering length is then adiabatically ramped from to via the broadly-tunable Feshbach resonance of the state[15] to form a pair of solitons with a centre-to-centre separation of and near equal amplitude ( atoms / soliton). Once the pair is formed, the barrier is quickly () turned off. Thus, the solitons suddenly find themselves at the classical turning points of the harmonic trap and begin to accelerate towards the centre. We confirm that these wave packets are nondispersive by observing the absence of expansion when the axial confinement frequency is suddenly reduced, while a wave packet with a small, repulsive scattering length rapidly expands (see Supplementary Fig. ).
We find that the two solitons interact with a randomly distributed relative phase over different experimental runs, so we use a minimally-destructive phase contrast imaging method, polarisation phase-contrast imaging[16], to obtain multiple images of the soliton pair as they oscillate and collide in the harmonic trap. This imaging technique plays a key role for observing and interpreting the collisional dynamics since it allows us to take multiple images within a single realisation of the experiment (see Methods).
We infer the relative phase difference through comparison with numerical simulations of the D and D Gross-Pitaevskii equations (GPEs). Figures 1b and 1c show two experimental realisations in which the relative phase difference is and , respectively. Images are taken every one eighth of a trap period (). Figures 1b and 1c show trajectories over one complete period, corresponding to two collisions. For , a clear anti-node is observed during the collision at the centre, giving the appearance of an attractive interaction, while in the case, interference results in a central node and the interaction between solitons is effectively repulsive.
The quasi-one-dimensional nature of our system coupled with the ability to form soliton pairs with a strong nonlinearity allows us to observe the rich dynamics inherent in a system at the edge of integrability. The strength of the nonlinearity is parametrised by , where is the critical atom number, , and m is the atomic mass. A soliton is unstable to collapse for [17]. Although collapse is relevant only for attractive interactions, we also use to parametrise the strength of the nonlinearity for repulsive condensates. For values of , we observe that in-phase collisions () sometimes result in annihilation (Fig. 2a), or fusion of the soliton pair (Fig. 2b), although more typically we observe partial collapses in which the atom number and the oscillation amplitude are reduced after multiple collisions. These effects can be understood as the result of density-dependent inelastic collisions in which the system becomes effectively three-dimensional[18, 19, 20]. Similar effects have been observed in nonlinear optics[21]. We find from the GPE simulations that collisions with and are unstable to collapse. The observation that collisions with do not always lead to collapse (e.g. Fig. 1b), is consistent with the shot-to-shot variation in of (see Methods). For the same nonlinearity, out-of-phase collisions () are extremely robust against collapse and survive many oscillations in the trap, as predicted theoretically[18, 20, 22]. Although on the edge of integrability, we have observed solitons with and to survive more than collisions (Fig. 2c).
The defining property of solitons passing through one another without change of shape, amplitude, or speed seems to be at odds with the observations presented in Fig. 1c, where solitons with apparently reflect from one another. This apparent paradox is resolved by noting that the effective interaction is a wave phenomenon[4], where interference gives the appearance of reflection, when in fact, the solitons do pass through one another. We experimentally demonstrate this by forming pairs of solitons with unequal atom numbers by removing atoms from one side using a short duration, near resonant pulse of light before ramping the field to form solitons. This allows us to identify, or tag, a particular soliton and to follow its trajectory before and after the collision. In Fig. 3 we show one such realisation in which a soliton pair was formed with a ratio in atom number. While a minimum does appear between the solitons during the collision, as expected for an effectively repulsive interaction, the trajectories show that they do pass through one another. The experiment does not rule out the possibility that the solitons reflect while exchanging particles during the collision. The D GPE simulations, however, demonstrate that particle exchange is a relatively small effect for the large collisional velocity in our experiment, in agreement with previous theoretical studies[19, 20].
A close inspection of the oscillations shown in Fig. 2c reveals that the solitons oscillate at a higher frequency than the usual dipole frequency, . This increased oscillation frequency is a consequence of a jump in the phase of the trajectories of the colliding, harmonically confined, solitons. Without axial confinement, the phase jump manifests as a sudden change in position relative to the original trajectory, as shown in the simulation of Fig. 4a. Although the phase of the trajectory is modified by the collision, the speed of the soliton following the collision is not. The jump is a consequence of the nonlinearity of the system[2], and was first observed with optical solitons[13].
We studied this effect by measuring the oscillation frequency of pairs of solitons for different strengths and sign of the nonlinearity (a condensate with also oscillates without dispersion in the presence of harmonic confinement[23]). At each value of , the measured axial density profiles were used to calculate the average harmonic potential energy per atom at different times during the oscillation, and subsequently fit to determine the oscillation frequency (see Methods). Oscillations for , , and are plotted in Fig. 4b for a total of three trap periods, in each case. Since the potential energy per atom is plotted, a total of six oscillations are observed. The frequency for the attractive (repulsive) interactions clearly leads (lags) when compared to the non-interacting () case. The relative frequency shifts are plotted in Figure 4c and we find them to be in reasonable agreement with numerical simulations obtained by solving the D GPE. We observe that the relative shift also provides a sensitive measurement of the zero-crossing, which is in excellent agreement with a previous determination[15]. The frequency shift is independent of , indicating that it is unrelated to the phase-dependent interactions previously discussed.
We have developed a simple analytical approximation to calculate the frequency shift. The shift arises from the mean-field interaction in which one soliton changes the potential landscape experienced by the other soliton. The phase shift is dominated by the incoherent (density-density) terms in the interaction, and we neglect all other interaction terms in the GPE in comparison. This approximation is valid for relatively weak nonlinearity and for fast moving solitons (see Methods). Although a phase shift is expected in integrable systems, the analytical approximation accounts for harmonic confinement and thus provides an estimate of the frequency shift for either sign of the nonlinearity. The analytically predicted relative frequency shift is , in which is the oscillation amplitude of a single soliton, and () is the axial (radial) harmonic oscillator length. is negative for . This approximation provides a simple, intuitive picture. For , the attraction between atoms accelerates the solitons as they approach one another, and decelerates them as they move away. The same occurs for , but with opposite sign. A similar effect has been observed for repulsive condensates in which oscillations of one condensate induced motion in the other[24, 25].
Our studies elucidate the role of integrability, relative phase, spatial dimensionality, and mean-field interactions in soliton collisions. A natural extension of this work would involve control over the relative phase between solitons, and better control of the strength of the nonlinearity. This would enable us to study collisions in a more controlled manner, providing the ability to further explore the transition between integrable and non-integrable systems, and to study the formation of soliton molecules[26]. Finally, this geometry may be applicable to atom soliton interferometry, demonstrated recently using a Bragg beamsplitter[27] rather than the tunnel barrier adopted in our geometry.
0.1 Apparatus
The primary difference between the apparatus used here and that which was described previously[14] is that a pair of perpendicularly oriented laser beams, derived from a single fiber laser operating at , provide cylindrically-symmetric harmonic confinement. The beam is divided into two separate paths, directed parallel and perpendicular to the magnetic field axis, and focused at the atoms to a radius of and , respectively.
The magnetic field is controlled using a pair of coils in Helmholtz configuration, and allows us to vary the scattering length across a broad region. Initially, the BEC is formed at a field of , corresponding to a scattering length of [28]. Once the BEC is formed, a blue-detuned Gaussian beam is turned on to cut the condensate in half and act as a high barrier between the two halves. The field is adiabatically ramped down () to a final scattering length of . We find experimentally that this procedure produces two solitons that interact with a randomly distributed relative phase.
To measure the oscillation frequencies for different nonlinearities, the final scattering length is varied between and , corresponding to nonlinear strengths of and , respectively.
The barrier is a cylindrically focused beam, blue-detuned by from the S-P resonance, with a radius of perpendicular to the condensate, and a radius of along the condensate axis. A barrier height of approximately was used to split the condensate, and to maintain a centre-to-centre separation of between solitons.
Polarisation phase-contrast imaging (PPCI) was used in order to minimally perturb the atoms, allowing us to take multiple images during a single experimental run. Since the relative phase of solitons varies between experimental runs, the use of this technique was crucial for interpreting the collisional dynamics. PPCI exploits the birefringence of the scattered light from atoms in a strong magnetic field. The scattered light is interfered with the probe light using a linear polariser. The resulting image is simply related to the column density distribution[16]. With this technique the laser may be far detuned from resonance ( in this case, where ), minimizing the number of photons scattered during the imaging process. Furthermore, the beam radius of approximately provides a uniform intensity profile across the soliton pair so that any phase-shift imprinted on the solitons from the imaging beam is common for the pair.
0.2 Oscillation frequency
The axial density, , was calculated for each image and used to determine the potential energy per atom from:
(1) |
0.3 Analytical approximation of frequency shift
The quasi-1D GPE is:
(2) |
in which . Here, , as above, is the atomic scattering length, and is the axial (radial) harmonic oscillator length[29]. In the absence of the nonlinear interaction, the two-soliton state is modeled by , and is:
(3) |
We have introduced the position coordinate such that , which defines a pair of symmetric Gaussians in the harmonic trap. In the limit of large impact speed (i.e. ) the interaction Hamiltonian becomes:
(4) |
in which the coherent interaction terms are neglected due to the fast spatial-phase oscillations between the rapidly moving solitons. We treat the interaction-induced shift as a small perturbation, and write the soliton motion as . The equation of motion for the perturbation is[30]:
(5) |
in which serves as the effective mass of the soliton and the factor is due to the identity . By substituting Eqns. (3) and (4) in Eqn. (5), we find that the total spatial jump, due to the density-density interaction, is:
(6) |
and the corresponding shift in the oscillation frequency is:
(7) |
The approximate analytical approach presented here applies to a broad class of pulses in generic models, matching the known results for integrable ones.
0.4 Uncertainties
The uncertainty in the strength of the nonlinearity is due to the uncertainty in the atom number, , the determination of the scattering length, , and the radial trap frequency, . The uncertainty in arises from shot-to-shot variation in and a systematic uncertainty of due to our ability to discern atoms from the background. To measure , the trap intensity was modulated near the radial trap frequency and the resultant loss in atom number, from heating, was measured. The uncertainty in determined from a Lorentzian fit to the data is . The mapping of vs. has been previously determined, with our region of interest being near the zero-crossing[15]. A linear fit to the data near the zero crossing gives a slope of and a zero crossing crossing at , with the uncertainties derived from a systematic uncertainty in the field calibration of . This gives a systematic uncertainty in of for . Thus, the statistical and systematic uncertainties in the strength of the nonlinearity are and , respectively, for .
We thank Rodrigo Shiozaki for his help with the data acquisition system, and Maxim Olshanii for helpful discussions. Support for this work was provided by the NSF, ONR, the Welch Foundation (Grant C-1133), the Binational (US-Israel) Science Foundation (Grant No. 2010239), and an ARO-MURI Non-equilibrium many-body dynamics grant (W911NF-14-1-0003). Author contributions J.N., P.D., and D.L. performed the experiment and analysed the data. D.L. performed the numerical simulations. B.M. developed the analytical model. R.H. was involved in all aspects of the experiment. All authors discussed the results and implications and took part in preparing the manuscript. Competing Interests The authors declare that they have no competing financial interests. Correspondence Correspondence and requests for materials should be addressed to R.G.H. (email: randy@rice.edu).
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