The nucleation, growth and stabilization of vortex lattices

We give a simple unified theory of vortex nucleation and vortex lattice formation which is valid from the initiation process up to the final stabilization of the lattice. We treat the growth of vortex lattices from a rotating thermal cloud, and their production using a rotating trap. We find results consistent with previous work on the critical velocity or critical angular velocity for vortex formation, and predict the initial number of vortices expected before their self assembly into a lattice. We show that the thermal cloud plays a crucial role in the process of vortex lattice nucleation.

This web page should be read in conjuction with our paper Phys. Rev. Lett. 89, 260402 (2002) available from the PRL Online server or as preprint cond-mat/0205037.

The purpose of this page is to elaborate on the results presented in our paper and, in particular, to make movies of our simulations publicly available.

Equation

For numerical reasons we have simulated Eq. (3) of the paper in the stationary laboratory frame. It takes then the form

$\displaystyle (i-\gamma )\dot{\psi }=\left( -\nabla ^{2}+V_{T}(\mathbf{x},t)+u\... ...ert^{2}+i\gamma \boldsymbol {\alpha \cdot \mathrm{L}}+i\gamma \mu \right) \psi$

in our computational units of time $t_{0}=1/\omega _{r}$ , distance $r_{0}=\sqrt{\hbar /2m\omega }$ and energy $E_{0}=\hbar \omega _{r}$ .
Unless explicitly mentioned the common parameters for the simulations presented on this page are

, $\gamma =0.1$ and $\mu =12$ . For typical experimental conditions $\gamma$ would be of the order of 0.01. We have done simulations with $\gamma$ as small as 0.01 to confirm that they show the same principal behaviour though the final number of vortices in the lattice may vary slightly with different $\gamma$ .

Rotating vapour cloud

Unless mentioned otherwise the simulations start with a ground state of the trap (normalized to unity), to which we add a uniform superposition of angular momentum components l = 1 to 30 on a Gaussian radial profile centred at the Thomas-Fermi radius and with maximum amplitude of ~2 x 10^-7. This simulates the non-stimulated collisions which start the process. An unseeded simulation will proceed from numerical noise and produce similar results, but take longer.
In the movies, the angular velocity of the cloud is indicated by a rotating line through the origin. Vortices are marked as + and - according to their sense of orientation.

Rotating well above threshold for vortex nucleation: $\alpha =0.65$

MPEG movies of density profile (3.2 MB) and local chemical potential (4.0 MB) for case of Fig. 1 of the paper.
Note the large gradient of the local chemical potential accross the position of the vortices. Vortices are moving in direction of larger chemical potential until eventually the chemical potential becomes flat accross the whole condensate when the lattice is formed.

Rotating just above threshold for vortex nucleation: $\alpha =0.45$

Because the gain is very small in this case, we chose $\gamma =1$ .
MPEG movie of density profile (2.7 MB).
The critical angular velocity for our parameters is $\alpha =0.444$ for which only the l = 9 component has positive gain. For $\alpha =0.45$ , however, the l = 9 and 10 components experience gain. Thus, in the simulation we see a very irregular ring of 10 vortices approaching the condensate. Note that the final configuration is a lattice of three vortices, not just one, even though the angular velocity is just above the threshold for vortex nucleation. We also verified that no vortices form for $\alpha =0.44$ .

Only dominant l = 16 component seeded: $\alpha =0.65$

MPEG movie of density profile (3.3 MB).
Here, we only seeded the l = 16 component. The ring of vortices is very regular and remains perfect for a long time even after penetrating the condensate. Note that the angular velocity of the vortex ring is initially smaller than that of the thermal cloud, and the ring breaks up when its angular velocity finally equals that of the thermal cloud.

Rotating trap

In this case we did not seed any of the angular components. The initial state is a pure ground state of the trap. The trap geometry determines which l components will be occupied initially due to coherent mixing facilitated by the stirring. We use an elliptical trap with $\omega _{x}=1.05\,\omega$ and $\omega _{y}=1.15\,\omega$ .

Without a thermal cloud ( $\gamma =0$ ): $\Omega =0.65$

MPEG movie of density profile (3.1 MB) for case of Fig. 4 (a) of the paper.
This corresponds to the ordinary Gross-Pitaevskii equation (GPE) without any dissipational terms. The vortices do not penetrate the condensate, but stay in the low-density region. No vortex lattice is formed.
See also simulations of the pure GPE by B. M. Caradoc-Davies for the case of a single stirrer.

With a thermal cloud ( $\gamma =0.1$ ): $\alpha =\Omega =0.65$

MPEG movie of density profile (3.1 MB) for case of Fig. 4 (b) of the paper.
In this case the thermal cloud is co-rotating with the rotating trap potential. The exchange terms between condensate and thermal cloud allow the vortices to penetrate the condensate and finally settle down into a vortex lattice.

Non-equilibrium lattice ( $\alpha \neq \Omega$ ): $\alpha =0.65,\Omega =0.5$

MPEG movies of density profile (1 MB) and local chemical potential (1.4 MB). In these movies, the white line through the origin represents the long axis of the elliptical potential while the angular velocity of the cloud is indicated by two shorter green lines in the outer region.
If the thermal cloud and the trap are not rotating at the same velocity, no true equilibrium state is possible. Nevertheless, a lattice is formed that rotates with the angular velocity of the thermal cloud (not the trap!), but the exact positions of the vortices are constantly re-adjusted as the the shape of the condensate rotates with the trap.

Vortex lattice decay ( $\alpha =0$ )

MPEG movie of density profile (1.9 MB).
Vortex lattice decay can be understood as the reverse process when the angular velocity of the thermal cloud is smaller than the rotation of the lattice. If $\alpha =0$ the vortex lattice decays completely. In this case the initial state was a vortex lattice with 22 vortices, which was created using a rotating trap with $\Omega =0.75$ .

Movie site associated with the paper:

The nucleation, growth and stabilization of vortex lattices

Abstract

Equation

Rotating vapour cloud

Rotating well above threshold for vortex nucleation: $\alpha =0.65$

Rotating just above threshold for vortex nucleation: $\alpha =0.45$

Only dominant l = 16 component seeded: $\alpha =0.65$

Rotating trap

Without a thermal cloud ( $\gamma =0$ ): $\Omega =0.65$

With a thermal cloud ( $\gamma =0.1$ ): $\alpha =\Omega =0.65$

Non-equilibrium lattice ( $\alpha \neq \Omega$ ): $\alpha =0.65,\Omega =0.5$

Vortex lattice decay ( $\alpha =0$ )


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Movie site associated with the paper:

The nucleation, growth and stabilization of vortex lattices

Abstract

Equation

Rotating vapour cloud

Rotating well above threshold for vortex nucleation:

Rotating just above threshold for vortex nucleation:

Only dominant l = 16 component seeded:

Rotating trap

Without a thermal cloud ( ):

With a thermal cloud ( ):

Non-equilibrium lattice ( ):

Vortex lattice decay ()

Rotating well above threshold for vortex nucleation: $\alpha =0.65$

Rotating just above threshold for vortex nucleation: $\alpha =0.45$

Only dominant l = 16 component seeded: $\alpha =0.65$

Without a thermal cloud ( $\gamma =0$ ): $\Omega =0.65$

With a thermal cloud ( $\gamma =0.1$ ): $\alpha =\Omega =0.65$

Non-equilibrium lattice ( $\alpha \neq \Omega$ ): $\alpha =0.65,\Omega =0.5$

Vortex lattice decay ( $\alpha =0$ )