Propagation of collective modes in non-overlapping dipolar Bose-Einstein Condensates

Recent experiments on dipolar Bose-Einstein Condensates (dBECs) of chromium [1], dysprosium [2], and erbium atoms [3], have stimulated a growing interest in ultracold dipolar quantum gases [4] due to the long-range and anisotropic nature of such interaction. The research on this kind of interaction is expecting to give rise to important new features not only at microscopic level, but at macroscopic level as well. The development of new experimental techniques focused on the realization of molecular gases with large electric polar moment, where the dipolar effects are particularly strong, has led to the hope that challenging new frontiers in this field can be opened.

The theoretical research in dipolar gases has tended to focus mostly on the study of the anisotropic effects of the dipolar interaction [4, 5], rather than on the long-range ones, where there are only few proposals: cat states in triple-well potentials [6, 7] and propagation of center-of-mass modes studied by means of a variational method [8, 9].

The effect of the dipole-dipole interaction on the frequency of a collective mode has been experimentally measured in a single dBEC [10]. However, a central issue that remains opened is to understand the mechanism of the expansion and propagation of collective modes between non-overlapping condensates.

The aim of the present work is to study the propagation of dipolar, monopolar and quadrupolar modes induced by the long-range nature of the dipolar interaction by solving the full 3D time-dependent Gross-Pitaevskii equation (TDGPE). We consider two dipolar Bose-Einstein condensates harmonically trapped in a double-well configuration such that the overlap between the two clouds is negligible as well as the corresponding tunneling effect. The only effective force acting between the two dBECs is the one produced by the long-range behaviour of the dipolar interaction.

First, we study the propagation of center-of-mass (dipolar) modes between two non-overlapping pancake-shaped condensates. Our time-dependent numerical results are in agreement with previous ones obtained within the variational method [8, 9]. Then, we investigate the propagation of other excitations (monopolar and quadrupolar modes), in the bilayer system. That is, the appearance and shift of new characteristic frequencies. Finally, as an extension, we test our model in a triple-well configuration, comparing the results with the case of the double-well configuration, and finding out that the classical model gives a qualitative description of the dipolar coupling. The coupled-pendulum model is used to understand qualitatively the results obtained before.

Bose-Einstein condensates at low temperatures can be described in the mean-field framework provided a large number of particles and a weakly interacting system. The TDGPE in presence of the dipolar interaction reads:

$$i\hbar\frac{\partial\Psi(\vec{r},t)}{\partial t}=\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{ext}(\vec{r})+g|\Psi(\vec{r},t)|^{2}+V_{D}(\vec{r})\right]\Psi(\vec{r},t)\,,$$

(1)

where $\Psi(\vec{r},t)$ is the wavefunction of the condensate, which is normalized to the number of particles $N=\int|\Psi(\vec{r},t)|^{2}d\vec{r}$, $m$ is the mass of the bosons, and $V_{ext}(\vec{r})$ is the trapping potential. The contact interaction potential is characterized by the coupling constant $g=4\pi\hbar^{2}a/m$, where $a$ is the $s$-wave scattering length. The mean-field dipolar interaction is given by

$$V_{D}(|\vec{r}-\vec{r}\,^{\prime}|,\theta)=d^{2}\,\frac{1-3\cos^{2}\theta}{|\vec{r}-\vec{r}\,^{\prime}|^{3}}\,.$$

(2)

Here $\vec{d}=\sqrt{\mu_{0}/4\pi}\,\vec{\mu}_{m}$, where $\mu_{0}$ is the magnetic permeability of vacuum and $\vec{\mu}_{m}$ is the magnetic dipole moment of the atoms. All the atoms are assumed to have the same dipole moment, both in magnitude and orientation given a polarization direction. The angle $\theta$ is the angle between the magnetization axis and the relative position between two dipols. We can see from Eq. (2) that in contrast to contact interaction, the dipolar interaction is anisotropic, due to the term $\cos^{2}\theta$, and long-range, due to the $1/|\vec{r}-\vec{r}\,^{\prime}|^{3}$ dependence.

In this work, we consider a bilayer system confined in the following double well potential:

$$V_{ext}(x,y,z)=\frac{1}{2}m\left(\omega_{\perp}^{2}\,(y^{2}+z^{2})+\omega_{x}^{2}\,min[(x+x_{c})^{2},(x-x_{c})^{2}]\right)\,.$$

(3)

The shape of each single dBEC depends on the trapping frequencies $\omega_{\perp}$ and $\omega_{x}$. The distance between the two dBECs is given by $L=2\,x_{c}$. In the numerical calculation we have used the following parameters: $2\,x_{c}=3\mu$m, $m=52\;\mbox{uma}$ (⁵²Cr atoms) , total number of particles $N=5000$ ($2500$ bosons in each dBEC), and trapping frequencies $\omega_{x}=1500\times 2\pi\,\,\mbox{Hz}\gg\omega_{\perp}=50\times 2\pi\,\,\mbox{Hz}$. Thus, each dBEC has a pancake-shaped geometry (Fig. 1). The oscillator length in the $x$ direction that gives a longitudinal length scale is: $l_{x}=\sqrt{\hbar/m\omega_{x}^{2}}=23.21\mbox{nm}$. The scattering length choosed for the particles is $a=0.001\;a_{B}$, where $a_{B}$ is the Bohr radius.

The two condensates are non-overlapping, that is, the barrier height is large enough to avoid the tunneling of particles. But they must be close enough in order to analyze the long-range properties of the interaction since it decays as $1/|\vec{r}-\vec{r}\,^{\prime}|^{3}$. Finally, the dipole moments are aligned in the $x$-direction, parallel to the symmetry axis of the pancakes. This corresponds to the attractive configuration (the repulsive configuration would imply the dipoles to be oriented in the plane of the pancake shaped condensate). The purpose of this choice is to ensure, first, the stability of the condensate [4] (cigar-shaped geometry brings earlier to the collapse than the pancake-shaped), and second, other dipole moment orientations would decrease dipolar effects.

Before presenting the numerical simulations, a qualitative description of the propagation of a collective mode between two non-overlapping dBECs can be found in the frame of classical mechanics [8]. We consider the classical analogy of two-coupled pendulums, where the natural oscillation frequency of each pendulum ($\omega_{0}$) corresponds to the frequency of the excited mode, and the coupling string $\alpha$ corresponds to the dipolar coupling. The two natural modes of oscillation are:

	$\displaystyle\omega_{in}$	$\displaystyle=$	$\displaystyle\omega_{0}$		(4)
	$\displaystyle\omega_{out}$	$\displaystyle=$	$\displaystyle\omega_{0}\sqrt{1+2\alpha/\omega_{0}^{2}}\,,$		(5)

which are the in-phase mode and the out-of-phase mode, respectively. In the dipolar drag effect when the center-of-mass mode is excited, $\omega_{in}$ is the frequency of the trap in the direction of the initial displacement (Kohn’s theorem).

The parameter $\alpha$, depends on the dipole moment $\vec{\mu}_{m}$, the distance between the two condensates $2\,x_{c}$ and their density gradient. Whereas the first two parameters are easily tunable, the latter is more difficult to control (in spite of the fact that it can be changed by varying all the parameters). Therefore, these two parameters and the frequency of the excited mode will detune the shift of the out-of-phase mode frequency $\omega_{out}$ [8].

The previous result can be generalized to $n$ coupled pendulums. The eigenvalues of the problem are:

$$\omega_{ev}=\omega_{0}\sqrt{1+(2+\gamma)\alpha/\omega_{0}^{2}}\,.$$

(6)

And the parameter $\gamma$ can be found by solving the following equation [11]:

$$(1+\gamma)^{2}\xi(\gamma,n-2)-2(1+\gamma)\xi(\gamma,n-3)+\xi(\gamma,n-4)=0\,,$$

(7)

where

$$\xi(\gamma,n)=(-1)^{n}\,\frac{\sinh((n+1)\Omega(\gamma))}{\sinh{\Omega(\gamma)}}\,,\;\mbox{with}\;\;\;\Omega(\gamma)=\mbox{arccosh}(-\frac{\gamma}{2})\,.$$

(8)

A nice long range effect that can be realized in a bilayer configuration is the dipolar drag. It is well known, that when the center of mass mode (dipole mode) is excited in a single BEC confined in a harmonic trap (by displacing the condensate out of its equilibrium position and releasing it, or giving some velocity to the center of mass of the condensate), the condensate oscillates harmonically with the same frequency as the characteristic trap frequency in the direction of the initial perturbation. This is still true in presence of long-range interactions that is for a dipolar Bose-Einstein condensate in a harmonic potential. However, when the dipole mode is excited in one dBEC of a bilayer system, although there is no tunneling between both condensates, the center of mass of the other dBEC starts to oscillate, as a result of the long-range character of the dipolar interaction, and both the in-phase and the out-of-phase dipole modes of the bilayer system are excited. Whereas the in-phase mode (center of mass) is independent of the dipolar interaction, the frequency of the out-of-phase mode depends on it. Dipolar drag is one of the so called propagation of collective modes, produced by the long-range nature of the dipolar interaction. It has been already studied in electron gases in uniform bilayer systems, under the name of Coulomb drag [12, 13].

Let us extend the dipolar drag effect to the case, of a triple-well trap. Some studies have been carried out in triple-well configurations related with dipolar effects [18], phase diagrams [6] or symmetry breaking [7]. In the classical analogy of three coupled-pendulums (6-8), the eigenvalues of the problem are: $\omega_{in}=\omega_{\perp}$, $\omega_{out,1}=\omega_{\perp}\sqrt{1+\alpha/\omega_{\perp}^{2}}$ and $\omega_{out,2}=\omega_{\perp}\sqrt{1+3\alpha/\omega_{\perp}^{2}}$.

In Fig. 6, we show the displacements of the centers of mass when only the left condensate is perturbed. All three frequencies appear in the movement, and, as a consequence, the displacements of the centers of mass have a more complex behaviour.

Although there are three eigenmodes, it is possible to excite only two frequencies by choosing the appropiate initial conditions. If we perturb only the condensate of the center, looking to the eigenvectors of the problem, only the two frequencies that preserve the symmetry of the system, $\omega_{in}$ and $\omega_{out,2}$, will appear in the movement of the center of mass (the other frequency corresponds to a non-symmetric mode).

In Fig. 7, we present the numerical results obtained by solving the TDGPE exciting only the center of mass mode of the centered dBEC. As expected, only two frequencies are involved in the movement of the three non-overlapping condensates. We have obtained the corresponding frequencies by performing a Fourier analysis and we have compared them with the case of only two dBECs:

$$\frac{(\omega_{out,2}^{2}-\omega_{\perp}^{2})_{3well}}{(\omega_{out}^{2}-\omega_{\perp}^{2})_{2well}}\approx 1.6\,.$$

(17)

It is very close to the value of the ratio $1.5$ obtained in the classical analogy. This means that the classical coupled-pendulum model yields a qualitative description of the dipolar coupling between few non-overlapping dBECs. Another remarkable fact is that, indeed, the magnitude of the shift is increased for three condensates compared with the case of two.

In the present work, the full TDGPE has been numerically solved to study the long-range character of the dipolar interaction. We have investigated the propagation of collective modes, such as dipolar and quadrupolar excitations, between non-overlapping pancake-shaped dBECs. A classical coupled-pendulum model has been proposed.

We have analyzed the effect of different parameters in the propagation of the excitations. The dipole moment enhances the dipolar effects, whereas the enlargement of the scattering length and the distance between condensates leads the system to be dominated by the contact interaction which hides the desired long-range effects.