vortices withm=±4, but this cannot be reached in practice.
We have also used a version of this algorithm with only four intervals tracing out a closed path around a grid gap. This version will detect only singly charged vortices, which is sufficient for many applications and faster.
The vortex detection algorithm needs to take into account the fact that the low density areas of our discretised wavefunction, areas where the wavefunc-tion vanishes for all practical purposes, appear to be full of vortices. The low density regions abound with extremely weak high frequency excitations with wild phase fluctuations– extremely small floating point complex numbers with almost random phases. This can be called an “Oort cloud” in reference to a cloud of rocky or comet-like small objects in the outer regions of our solar sys-tem. Unlike these, the Oort vortices do not have any physical importance and we can avoid counting and detecting them by specifying a minimum density
|ψi|2 for the detection algorithm.
5.4 Collective cloud excitations in a BEC
As we have outlined in section 5.3.7, the most important feature of the BEC trap ground state is that it rests motionless in its trap for arbitrary lengths of simulation time. Meaningful experiments and simulations, however, will in-vestigate dynamics of the trapped BEC cloud, which can be excited in many different ways. For the time being we will look into low frequency and low energy collective excitations of the whole cloud. Low excitation energy in this context means that the energies are much smaller than the mean-field interac-tion between the atoms (the chemical potential). Excitainterac-tions of this scale are of collective nature. Angular excitations in form of vortices have been glanced in section 5.1.2 and high frequency excitations up to finite temperature effects cannot be treated reliably within the mean field approximation of the GP equa-tion.
The most simple excitations of a trapped BEC cloud are the dipole oscilla-tions performed by the cloud, when the initial state is not centered within the harmonic potential. In a TOP trap (with harmonic potential), thexandy oscil-lations are energetically degenerate and have the oscillation frequencyωx,y. The z dipole oscillation has a frequency ofωz =√
8 ωx,y due to the TOP-averaged double axial magnetic field gradient as described in section 2.2.2.
In addition to the dipole excitations, quadrupole excitations exist in several different modes. These modes do not necessarily correspond to the harmonic oscillator frequencies directly because of the atomic interactions. In a an axially symmetric trap like the TOP trap, angular momentum about the symmetry axis is conserved and m is a good quantum number, which can be used to denote the excitation states. The lowest lying quadrupole mode is the m=2 mode, which involves deformation of the cloud in thexy-plane with no motion along thez axis. This mode is doubly degenerate (m =±2) and involves two
“tidal waves” oscillating around the cloud in thexy-plane in clockwise or anti-clockwise orientation with a frequency of √
2 ωx,y. A superposition of both
ω/ωx
Figure 5.4: Energy spectrum of collective excitations of a BEC cloud in a TOP trap. Presence of a central vortex breaks the degeneracy of the |m|=2 and
|m|=1 modes.
equally excited modes will not exhibit a “rotation” when looking down onto the cloud along the z-axis.
In the low lying m=0 quadrupole mode, the radial and axial cloud sizes oscillate in anti-phase. In the high lyingm=0 mode, the axial and radial direc-tions oscillate in phase. This is often called the “breathing” mode of the BEC cloud. The frequencies for the m=0 quadrupole modes have been derived by Stringari in [128] using the GP equation and hydrodynamic theory. With the trap frequency ratio λ=ωz/ωx,y these are where the + sign stands for the “breathing” mode and the−sign stands for the low-lying mode. Further quadrupole modes are the so-called “scissors” modes, which we will look at more closely in the following section. The spectrum of the low frequency excitations of a BEC cloud in a TOP trap is shown in figure 5.4.
5.4.1 Scissors mode
The scissors mode is an odd-parity collective excitation of a BEC cloud in an anisotropic trap. It is known as an excitation in atomic nuclei, predicted theoretically by geometric models and found experimentally in 1984 [139]. In a BEC the scissors mode allows the direct observation of the superfluid nature of the condensate. The scissors mode can be excited by a small angle trap tilt about theyorxaxis. This causes an oscillatory response of the cloud about the
5.4. COLLECTIVE CLOUD EXCITATIONS IN A BEC 125 respective axis, which distinctly depends on the nature of the cloud. Marag`o et. al. have used the response for experimental investigations of a (partially) condensed cloud at finite condensate temperatures [140, 51, 141].
A theoretical investigation of the scissors mode and a calculation of its ex-citation frequency has been published by Gu´ere-Odelin in [142]. The derivation starts from the condensate irrotational flow field
v(r, t) = (~/m)∇S(r, t),
whereS is the wavefunction phase, and the superfluid hydrodynamic equations (5.9) and (5.10), as discussed above in section 5.1.2. Considering the Thomas-Fermi density distribution and a small angle scissors tilt, one additionally finds that the scissors motion is independent of compressional modes. Thus the flow is steady:
∇ ·v(r, t) = 0. (5.54)
The above constraints lead to the following expression for the condensate phase S(r, t) = (m/~)β(t)xz. (5.55) Consequently the hydrodynamic equations yield for the trap angle θ and the phase parameter [51]
θ(t) =˙ −β(t)/, β(t) = 2¯˙ ω2θ(t), (5.56) where the trap deformation parameterand the frequency ¯ω are defined as
= ωz2−ωx2
ωz2+ωx2, ω¯ =
rω2x+ω2z
2 . (5.57)
The correct initial conditions in equation (5.56) then leads to the solution θ(t) =θ0cos(ωsct), β(t) =θ0ωscsin(ωsct), (5.58) where the scissors frequency for the TOP trap case withωz =√
8 ωx becomes ωsc= 3ωx. In general, for axial trap anisotropiesλ=ωz/ω⊥, the scissors mode frequency is
ωsc=p
1 +λ2 ω⊥. (5.59)
Thus at zero temperature, the two energetically degenerate scissors modes, which are individually characterised by functions of the formf(r)xzandf(r)yz [143], represent an undamped single frequency oscillation, provided that the trap rotation angle used to excite the scissors mode is small.
The superfluid scissors mode oscillation is the result of a strongly quenched moment of inertia of the condensed cloud. For finite temperatures, the moment of inertia approaches the (larger) rigid body value of Θ =mNhx2+y2i(average overN ensemble particles) as the condensate fraction decreases. The different oscillation frequencies and amplitude damping at finite temperature can be used as a detector for BEC.