4.3 High Charge Ring Flows in Condensates
4.3.3 Stabilisation of High Charge Ring Flow
s(r, z) =(z2 + (D−r)2). (4.32)
On any closed loop threaded through the single circular ring singularity implicit in Eq. (4.31), the phase η varies by 2πq, where q is the ring vortex charge. The initial state thus contains one charge q ring vortex. Its singularity is located at
r =D, z = 0 in our coordinates.
4.3.3
Stabilisation of High Charge Ring Flow
The low density region of a vortex ring in a BEC has a toroidal shape, which matches the shape of the high intensity region of a focussed optical vortex laser
beam. In the presence of the optical vortex potential, it becomes energetically favorable for the BEC ring singularity to be located at the maximum of laser light intensity. This prevents the contraction of the vortex ring. This mechanism is similar to the stabilisation of line vortices within a harmonically trapped BEC cloud by laser pinning, which we applied to Skyrmions in the second part of section 4.2.4. -200 0 200 0 1 2 -100 0 100 0 0.5 20 0 20 0 2 4 20 0 20 0 2 4 r [µm] r [µm] z [µm] z [µm] n /10 [1/m ] 19 3 Μ b) c) d) e) n /10 [1/m ] 1/2 9 3/2 n /10 [1/m ] 1/2 9 3/2
Figure 4.18: Cross-sections of density and velocity structures for different sta- tionary ring flows. In (a) the shape of the optical vortex potential is also shown. Parameter set for case (i): V0 = 48ωr,w0 = 3µm, q= 7, indicated by solid lines
in (b-e). Parameter set for case (ii): V0 = 99ωr, w0 = 2.8µm, q = 9, indicated
by dashed lines in (b-e). (a) Shows the BEC density (red highest, blue lowest) in the rz plane for case (i). The arrows indicate the BEC flow, thin white lines are isocontours of the optical vortex potential at Vv = 1×10−32J (outer lines) and
Vv = 1×10−31J (inner lines). The intersection of ring singularities with the rz
plane is marked by ×(◦) for mathematically positive (negative) circulation. (b) BEC density and (c) Mach number M atr = 0 alongz axis. (d) BEC amplitude in radial direction atz = 0. (e) BEC amplitude in radial direction atz = 170µm.
Examples of stationary states resulting from the imaginary time evolution are shown in Fig. 4.18. In a harmonic trap withωr = 7.8×2πHz andωz = 0.5×2πHz
the BEC contains 2.2×106atoms. The subplots are for two different sets of optical
vortex parameters and ring flow charges: case (i) and case (ii) as given in the caption. Case (ii) represents the more intense optical vortex. It can be seen in Fig. 4.18 (a) that the single charge 7 ring singularity that was present in the initial state has broken up into a regular stack of seven unit charge ring singularities. It is known for vortex structures in BEC, that multiple unit charge vortices are usually energetically favoured over a single high charge vortex [38]. For case (i) the seven individual ring singularities are responsible for the multi-peak structure of the Mach number shown in Fig. 4.18 (c). If the parameters are altered towards those of case (ii), the influence of the singularities on the flow in the tunnel is reduced. This corresponds to the transition from the solid to the dashed lines in Fig. 4.18 (b-c). The overall topological structure of the high light intensity region surrounded by a BEC ring flow in a harmonic trap (Fig. 4.18) is reminiscent of that of a Skyrmion and can thus be thought of as an “Atom-Light Skyrmion”.
By the same argument used in section 4.2.2 for the case of Skyrmions, the integral along the z axis of the flow velocity is directly connected to the ring vortex charge q. The quantization condition for our high charge ring flow is &
C v · dl = mhq, where C denotes any closed contour threading through all the
ring singularities. In this case Eq. (4.11) becomes: Lo/2
−Lo/2
vz(r = 0, z)dz =
h
mq, (4.33)
where Lo here denotes the length of the optical tunnel in the z direction, which
is implicitly governed by the beam waist parameter w0 of the optical vortex in
Eq. (4.25).
The method we propose for stabilising the ring singularities by pinning should be experimentally realisable with present technology. The procedure to form the ring flow consists of initially creating a BEC at rest, already in the presence of the optical vortex. Then a ring flow structure with high vorticity is seeded by means of phase imprinting, using the suggestion that exists for the ring component of a Skyrmion [139]. In the presence of dissipation the BEC is expected to evolve towards stationary states, like those in Fig. 4.18.
As can be seen in Fig. 4.18 (c) the flow velocity through the tunnel in the core of the optical vortex can be much higher than those we found in the core of a Skyrmion (compare section 4.2.6). There are two main possibilities to increase
the core flow velocity from a given state: (i) We can shorten the optical tunnel by reducing the focal width w0 in equations (4.25) and (4.26). A shorter tunnel
increases the velocity according to Eq. (4.33). The variation can be done in a continuous fashion numerically, by using a previously found stable ring flow as initial state of simulations with a shortened tunnel. (ii) We can increase the vortex charge q. However this is only possible in a discrete fashion and, in contrast to (i), numerically requires a complete new imaginary time solution of Eq. (2.18) from a suitable initial state. The highest core Mach number that we numerically found was M = 0.87.
As expected [162], supersonic flows turned out to be energetically unstable. This manifested itself in the imaginary time evolution as a decay of the total ring flow charge from q1 to q2 < q1, whenever the parameters were such that a final
state with charge q1 would breach the speed of sound.
4.3.4
Summary and Outlook
We have introduced a novel arrangement for the study of persistent ring flows in a BEC. It utilises optical vortices, laser beams with a phase singularity at the centre and hence a zero intensity hollow core. If tightly focussed, the highest intensity region of the optical vortex can fit into the low density region of the matter-wave ring vortex in the BEC and hence stabilise it against contraction. We showed numerically that energetically stable BEC states with ring flows of charges up to q= 9 exist in such a system.
If a Skyrmion is viewed as a method to stabilise the ring vortex component, the optical vortex stabilisation introduced in the second part of this chapter is more flexible. Even the stabilisation of a charge 2 ring flow in a skyrmion requires 7×106 atoms [156]. According to our discussion of the relation between
flow quantisation, core flow velocity and minimal Skyrmion size in sections 4.2.2, 4.2.6 and 4.3.3, we expect this number to increase for higher ring charges. In comparison, case (ii) of section 4.3.3 represents a stable charge 9 ring flow in a condensate of only 2.2×106 atoms.
We have also demonstrated that in the proposed trapping configuration the BEC is flowing through the optical tunnel with velocities close to the speed of sound. No supersonic energetically stable flows were found, as expected. However despite the energetic instability, it is not necessary that a supersonic flow is also dynamically unstable [64]. As we are interested in supersonic flows in a BEC for the purpose of analogue gravity, introduced in section 1.3, we investigated whether
it is possible to dynamically bring the optical tunnel flow into the supersonic regime to create an analogue black hole. This research is described in chapter 5.