Phase engineering of Matter Waves:
Creation of Dark Solitons in Bose-Einstein condensates

Like the harmonic oscillator which archetypically describes oscillation processes, solitons are nowadays the paradigm of nonlinear waves. Dark solitons as an important class of macroscopically excited Bose condensed states are of particular interest within the new field of nonlinear atom optics to explore nonlinear properties of matter waves.
Solitonlike solutions of the Gross-Pitaevskii equation are closely related to similar solutions in nonlinear optics describing the propagation of light pulses in optical fibres. Here, bright soliton solutions correspond to short pulses where the dispersion of the pulse is compensated by the self-phase modulation, i.e. the shape of the pulse does heuristically not change. Similarly, optical dark solitons correspond to intensity minima within a broad light pulse [17].
In the case of nonlinear matter waves, bright solitons are expected only for an attractive interparticle interaction (s-wave scattering length $a<0$) [18], whereas dark solitons, also called ``kink states'', are expected to exist for repulsive interactions ($a>0$). Recent theoretical studies discuss the dynamics and stability of dark solitons [19,20,21,22], as well as concepts for their creation [16,23,24].
Conceptually, solitons as particlelike objects provide a link of BEC physics to fluid mechanics, nonlinear optics, and fundamental particle physics.

Dark solitons in matter waves are characterised by a local density minimum and a sharp phase gradient of the wave function at the position of the minimum (Fig. 1(a)).

Figure 1: (a) Density and phase distribution of a dark soliton state with $\Delta \phi _{k} \sim \pi$. The density minimum has a width of $\sim l_{0}$. (b) Phase imprinting potential, $U_{pi}$, and associated phase distribution.

Due to the balance between the repulsive interparticle interaction trying to reduce the minimum and the phase gradient trying to enhance it, the shape of the soliton does not change. The macroscopic wave function of a dark soliton in a cylindrical harmonic trap forms a plane of minimum density (DS plane) perpendicular to the symmetry axis of the confining potential. Thus, the corresponding density distribution shows a minimum at the DS plane with a width of the order of the (local) correlation length. A dark soliton in a homogeneous 1D BEC of density $n_{0}$ is described by the wave function (see [22], and references therein)

$$\Psi_k(x) = \sqrt{n_0}\left( i\cdot\frac{v_k}{c_s}+ \sqrt... ...rac{x-x_k}{l_0}\sqrt{1-\frac{v_k^2}{c_s^2}}\right]\right) ,$$ (1)

with the position $x_k$ and velocity $v_k$ of the DS plane, the correlation length $l_0=(4\pi a n_0)^{-1/2}$, and the speed of sound $c_s=\sqrt{4\pi a n_0}\hbar/m$, where $m$ is the mass of the atom. For $T=0$ in 1D, dark solitons are stable. In this case, only solitons with zero velocity in the trap center do not move, otherwise they oscillate along the trap axis [20]. However, in 3D at finite temperature, dark solitons exhibit thermodynamic and dynamical instabilities. The interaction of the soliton with the thermal cloud causes dissipation which accelerates the soliton. Ultimately, it reaches the speed of sound and disappears [22]. The dynamical instability originates from the transfer of the (axial) soliton energy to the radial degrees of freedom and leads to the undulation of the DS plane, and ultimately to the destruction of the soliton. This instability is essentially suppressed for solitons in cigar-shaped traps with a strong radial confinement [21]. As can be seen from equation (1), the local phase of the dark soliton wave function varies only in the vicinity of the DS plane, $x\approx x_{k}$, and is constant in the outer regions, with a phase difference $\Delta \phi_{k}$ between the parts left and right to the DS plane (see Fig. 1(a)). To generate dark solitons experimentally, we apply the method of phase imprinting [16], which also allows one to create vortices and other textures in Bose-Einstein condensates. A homogeneous potential, $U_{pi}$, generated again by the dipole potential of a far detuned laser beam is applied to one half of the condensate wave function (Fig. 1(b)). The potential is pulsed on for a time $t_{p}$, such that the wave function locally aquires an additional phase factor $e^{-i\Delta \phi_{pi}}$, with $\Delta \phi_{pi} = U_{pi}\cdot t_{p}/\hbar \sim \pi$. The pulse duration $t_{p}$ is chosen to be short compared to the correlation time of the condensate, $t_{c}=\hbar /\mu$, where $\mu$ is the chemical potential. This ensures that the effect of the light pulse corresponds approximately to a change of the phase of the BEC, whereas changes of the density during this time can be neglected.
Note, however, that at larger times due to the imprinted phase $\Delta \phi_{pi}$ (Fig. 1(b)) one expects an adjustment of the phase and of the density distribution in the condensate. This will lead to the formation of a dark soliton with $\Delta \phi_{k} \not= \Delta \phi_{pi}$ in general, and also to additional structures. Fig. 2 shows simulations of the 1D Gross-Pitaevskii equation for the dynamics of the condensate wavefunction after the phase imprinting at $t=0\,$ms. After $0.2\,$ms the imprinted phase step leads to the formation of a density minimum and a maximum both travelling in opposite directions. These two structures ``take away'' part of the initial phase gradient, $\Delta \phi_{pi}$. Due to matter wave dispersion and the repulsive interparticle interaction the density maximum moves with the speed of sound and broadens, whereas the density minimum travels with smaller velocity and preserves its shape. This density minimum thus corresponds to a moving dark soliton.

Figure 2: Phase distribution (a) and density distribution (b) of the condensate wave function after the phase imprinting obtained numerically from 1D-simulations of the Gross-Piteavskii equation for different evolution times in the magnetic trap. The dark soliton is indicated by an arrow.

In these experiments we produce Bose-Einstein condensates of $^{87}$Rb every $20\,$s containing typically $1.5\times 10^5$ atoms in the ($F=2, m_{F}=+2$) state, with less than 10% of the atoms being in the thermal cloud. The fundamental frequencies of our static magnetic trap are $\omega_{x} = 2\pi\times 14\,$Hz and $\omega_{\perp}=2\pi\times 425\,$Hz along the axial and radial directions, respectively. The condensates are cigar-shaped with the long axis ($x$-axis) oriented horizontally. For the phase imprinting potential, $U_{pi}$, we use a blue detuned, far off-resonant laser field ( $\lambda =532\,$nm) of intensity $I\approx 50\, {\rm W}/{\rm mm}^2$ pulsed for a time $t_{p}=20\, \mu$s resulting in a phase shift $\Delta \phi_{pi}$ on the order of $\pi$. Spontaneous processes can be totally neglected.
A high quality optical system is used to image an intensity profile onto the BEC, nearly corresponding to a step function with a width of the edge, $l_{e}$, smaller than $3\, \mu$m. The corresponding potential gradient leads to a force transferring momentum locally to the wave function and supporting the creation of a density minimum at the position of the DS plane for the dark soliton. Note that the velocity of the soliton also depends on $l_{e}$.
After applying the dipole potential we let the atoms evolve within the magnetic trap for a variable time $t_{ev}$. We then release the BEC from the trap (switched off within $200\, \mu$s) and take an absorption image of the density distribution after a time of flight $t_{TOF}=4\,$ms (reducing the density in order to get a good signal-to-noise ratio in the images). In a series of measurements we have studied the creation and successive dynamics of dark solitons as a function of the evolution time and the imprinted phase. Figures 3(a) and (b) show density profiles of the atomic clouds for two different evolution times in the magnetic trap, $t_{ev}$. The phase imprinting potential, $U_{pi}$, has been applied to the lower part of the condensate with $x<0$ and the potential strength in this measurement was estimated to correspond to a phase shift of $\sim \pi$.

Figure 3: Absorption images of Bose-Einstein condensates with kink-wise structures propagating in the direction of the long condensate axis for two different evolution times, $t_{ev}$, in the magnetic trap: (a) $t_{ev}=200\, \mu$s and (b) $t_{ev}=3\,$ms. The dark soliton is marked by an arrow. ( $\Delta \phi \sim \pi , N\sim 1.5\times 10^5$, and $t_{TOF}=4\,$ms).

For short evolution times (Fig. 3(a)) the density profile of the condensate shows a pronounced minimum (contrast about 40%). After a time of typically $t_{ev}\sim 1.5\,$ms, a second minimum appears. Both minima (contrast about 20% each) travel in opposite directions and in general with different velocities along the long condensate axis (Fig. 3(b)). One of the most important results is that both structures move with velocities which are smaller than the speed of sound ( $c_{s}\approx 3.7\,$mm/s for our experimental parameters) and depend on the applied phase shift $\Delta \phi_{pi}$. Therefore, the observed structures are different from sound waves in a Bose-Einstein condensate as first observed at MIT [25]. We identify the minimum moving slowly in the negative $x$ direction to be the DS plane of a dark soliton. Figure 4 shows the experimentally observed evolution of this dark soliton.

Figure 4: Position of the experimentally observed density minimum corresponding to the dark soliton versus evolution time in the magnetic trap.

With different parameter sets for the imprinted phase, $\Delta \phi_{pi}$, which is determined by the product of laser intensity and imprinting time, and the width of the imprinted phase step, $l_{e}$, the velocity of the dark soliton could be varied experimentally between $v_{k} = 2.0\,$ mm/s (Fig. 4) and $v_{k} = 3.0\,$ mm/s. In addition to the dark soliton, the dipole potential creates a density wave which ``consumes'' part of the imprinted phase $\Delta \phi_{pi}$ and which travels in the positive $x$ direction with a velocity close to $c_{s}$. After opening the trap, a complex dynamics results in the appearence of a second minimum behind the density wave (see Fig. 2). All our experimental observations agree very well with theoretical investigations and numerical simulations of the 3D Gross-Pitaevskii equation [26]. The experimental results also show clear signature of the presence of dissipation in the dynamics originating from the interaction of the dark soliton with the thermal cloud. We observe a decrease of the contrast of the dark soliton by $\sim 50\%$ on a time scale of $10\,$ms. This is in contradiction with a nondissipative dynamics, where the contrast should even increase for a dark soliton moving away from the trap center [22]. The decrease of the contrast can therefore only be explained by the presence of dissipation decreasing the energy of the dark soliton. As the lifetime of the dark soliton is sensitive to the ensemble temperature, the studies of dissipative dynamics of dark solitons will offer a possibility for thermometry of Bose-Einstein condensates in the condition where the thermal cloud is not discernible. Parallel to the work reviewed here [26], dark solitons in nearly spherical Bose-Einstein condensates were observed at NIST [27].