Dynamical Instability

Train creation, scenario A

An example of a scenario resulting in a train of grey solitons is presented in Fig. 5.5. In this simulation we linearly ramp up a hump laser beam, centred at zf,1 = −20µm over a time of 1s, with parameters as given in table 5.3. The

obstacle accelerates the BEC flow, shown in Fig. 5.5 (b).

It can be seen that thezcoordinate at which the flow first becomes supersonic does not coincide with the maximum of our hump potential Vh,1, as we might

expect considering the potential at r = 0 alone. The reason is the interplay between the varying width of the narrow channel in the core of the optical vortex with the variations in the combined trapping and hump potential on the z axis:

It follows from the regularity condition at the horizon, Eq. (5.8), that A and V

need to have the same sign there. Due to the symmetry of the optical vortex potential underz → −z, we haveA <0 forz <0 andA >0 forz >0. The sign of the potential slope V at r = 0 can be found from Fig. 5.5 (d). Taking into account that at point X in Fig. 5.5 (d) the flow is not yet sufficiently accelerated, it can only become supersonic near point Y, where the sign condition can again be fulfilled. This is indeed what is found in the dynamics.

Once the local speed of sound is reached, localised density dips develop and propagate downstream (towards positive z, in the direction of BEC flow). We observe that the flow through the narrow optical tunnel remains quasi-one di- mensional at all times, i.e. the flow velocity there depends only on z. When the first soliton reaches the outer region of the trap, it decays. The decay excites oscillations of the whole cloud. In Fig. 5.5 (c) we compare the solitons emitted in our simulations with the analytical expression for grey solitons [215, 216] given by nsol =nbg(z)|u(z, z0)|

2

, where

u(z, z0) =Btanh [B(z−z0)/ξ] +iD, (5.29)

with D and B related by D2 _+B2 _{= 1. The wavefunction n}

sol represents a dip

to a fraction D2 _{of the inhomogeneous background density n}

bg at z = z0. B

describes the soliton’s width. The background healing length ξ is evaluated at the location of the soliton (z0): ξ = 1/

4πnbg(z0)as. After adjusting nbg, z0

and D, the width of the soliton is fixed, and the excellent agreement shown in Fig. 5.5 (c) thus demonstrates the applicability of 1D theory.

Transverse effects of soliton propagation, scenario B

Despite the success of one-dimensional soliton theory in describing the grey soli- tons in our optical tunnel, the core of the optical tunnel is merely a part of our three-dimensional atom-light skyrmion. We will hence take a quick look at the BECs behaviour in the dimensions transverse to the tunnel axis. Fig. 5.6 shows density color maps in therz plane and locations of ring singularities at the initial time t= 0 and at 0.555 s after adding a hump potential. In scenario B, two grey solitons are emitted. Following the evolution in the rz plane, we find that the solitons propagate without connection to the initial ring singularities. Those sin- gularities slowly shift their position in time, to reflect changes of the overall flow structure along the z-axis. However we also see the temporary creation of addi- tional ring vortex-antivortex pairs, not present in the initial flow. These always

-2000 0 200 5 46 48 50 52 54 0 0.5 1_{c) d)} z [µm] z [µm] z [µm] z [µm] 0 50 100 0 1 2 3 4 5 6 7 0 20 40 60 0 2 4 6 8 10 12 14 a) b) t=0 t=0.21s t=0.42s t=0.48s t=0.55s t=0.62s t=0.68s t=0 t=0.21s t=0.42s t=0.48s t=0.55s t=0.62s t=0.68s ρ /10 [1/m ] 19 3 ρ /10 [1/m ] 19 3 t=0.68s Μ V / h ωx _X Y

Figure 5.5: Generation of a grey soliton train. The initial state is that of case (ii) in table 5.3. A potential hump centred at z = −20 µm is ramped up over 1 s. (a) Time evolution of condensate density along the z-axis. The graphs for subsequent time samples are shifted upwards by 1 unit. The zero of density is indicated with an elongated tick mark for the last three samples. (b) Evolution of the Mach number. Successive samples are shifted by 2. Thin horizontal lines indicate M = 1 for each time sample. The peaks are cut off at M = 2; their amplitude is as high as 600 for almost dark solitons. (c) Fit of the functional shape described in the text (dashed grey) onto a soliton in the train (solid black). The fit parameters are nbg = 7.6×1018/m3 and D = 0.33. (d) Monotonically

increasing hump potential for the same time samples as in panel (a). The symbols X and Y indicate locations in the flow relevant to the discussion in this section.

r[µm] r[µm] z[ µ m] z[ µ m]

a)

b)

Figure 5.6: BEC density in therz plane (red highest, blue lowest). The intersec- tion of ring singularities with therz plane is marked by×(◦) for mathematically positive (negative) circulation. (a) Initial state for case (ii) in table 5.3. (b) Situ- ation 0.555 s after the sudden addition of the obstacle potential for scenario B in table 5.3. The density dips belonging to the two emitted grey solitons are visible in the core. For this snapshot, each is surrounded by a ring vortex-antivortex pair. While the 9 ring singularities of the initial state have slightly shifted po- sition to reflect changes in the overall flow structure, they are not found to be connected to soliton movement.

appear at the same z-coordinate as the solitons and co-propagate with them for short times. After that they are annihilated again. Radially, the singularity pairs appear within the low BEC density region where the optical vortex potential is strong. The whole evolution for this case, including creation and destruction of singularities, is shown in the moviesoliton ring hybrids? gpe.mov on the CD. Following the ring pair creation up lay outside the scope of this thesis. Previ- ous work on the 3D structure of BEC excitations in tight, quasi-one dimensional confinement, however, suggests that our solitons evolve into soliton vortex-ring hybrids [217]. These have already been observed experimentally [144]. The opti- cal tunnel would present a clean system for their study.