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Home page > Research Teams > Bose-Einstein Condensates

Rotating condensates and vortex lattices

Here we describe several studies that we performed between 1999 and 2005 with rotating condensates, containing one or several quantized vortices.

Composition of the team

Permanent member:
- Jean Dalibard, CNRS
Postdocs:
- Kirk Madison (1999-2001)
- Peter Rosenbusch (2001-2002)
- Zoran Hadzibabic (2003-2007)
PhD students:
- Frédéric Chevy (1998-2002)
- Vincent Bretin (2000-2004)
- Sabine Stock (2002-2006)

Contents of this page

Why vortices ?
Nucleation and vizualisation of vortices
Mesurement of angular momentum
Phase of a one-vortex state
Shape and excitation of a vortex line
Abrikosov lattice
Rapidly rotating condensate

Why vortices?

The rotation of a quantum fluid illustrates in a striking way the constraints set by Quantum Mechanics on the velocity field of a quantum macroscopic object. Consider a fluid formed by particles of mass $m$ with a local density $\rho(\vec r)$ . If the fluid is described by the macroscopic wave function

$\psi(\vec r)=\sqrt{\rho(\vec r)}\;e^{i\theta(\vec r)}$

then the velocity field in a point where the density is non-zero is given by:

$\vec v(\vec r)=\frac{\hbar}{m}\vec \nabla \theta$

The curl of this velocity field is always zero. Therefore it cannot coincide with the velocity field of a classical object in uniform rotation at angular frequency $\Omega$ :

$\vec v_{\rm clas.}(\vec r)=\vec \Omega\times \vec r$

The rotation of a quantum fluid generally involves the nucleation of vortices. A vortex is a line along which the density is zero and around which the circulation of velocity is quantized:

$\oint \vec v(\vec r)\cdot d\vec r=n\frac{h}{m}$

where $n$ in integer. In the experiments described below, the observed vortices correspond to $n = 1$ . The existence of these vortices has been predicted by Onsager and Feynman. The quantization of the circulation of the velocity was proven experimentally by J. Vinen and co-workers, and the vortices were observed for the first time in liquid helium by R. Packard and his team. Similar objects appear in type II supraconductors placed in a magnetic field.

After the discovery of gaseous Bose-Einstein condensates in 1995, the quest for vortices in these systems has been very active. The first vortex was observed in Boulder in 1999, in a mixture of two condensates, the first condensate rotating around the other one. This vortex was nucleated by optically printing the required phase \exp(i\thetaA few months later, our team succeeded in observing several vortices, by setting the condensate into rotation using a laser stirrer. The study of vortices is an important element of this field of research. They appear in all domains where quantum macroscopic physics enters into play, and gaseous condensates constitute very well suited systems to study the nucleation and the dynamics of these "universal" objects. In the context of gaseous BEC, they have also been observed at MIT and Oxford University.

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Nucleation and visualisation of vortices

We first prepare a condensate with typically 300 000 atoms. It is confined in a magnetic trap which is axi-symmetric with respect to the z axis (horizontal). The oscillation frequency along z is approximately 10 Hz. In the transverse plane, the oscillation frequency is notably larger, and it can be adjusted between 100 and 200 Hz. The condensate is cigar-shaped, with a length along z of the order of 100 micrometers, and a diameter in the xy plane of the order of 6 micrometers.

We set the condensate in rotation using a laser beam, which plays the same role as the spoon that stirs a cup of tea. The position of the laser stirrer is controlled by two acousto-optic modulators.

The direct detection of vortices inside the condensate is not possible. Indeed the size of the vortex core, i.e., the distance over which the condensate density is notably reduced because of rotation, is only a fraction of a micrometer. This is too small to be measured by standard optical methods. We use a destructive imaging technique, using a time-of-flight sequence. At a given time we switch off the magnetic trap confining the condensate, which rapidly expands because of the repulsive interactions between atoms. After a time of the order of 25 milliseconds, all distances in the transverse plane have been scaled by a factor between 20 and 40. The vortex core, whose size is now of the order of 10 micrometers, can then be detected.

The two images below have been taken by measuring the absorption of a laser beam whose frequency is resonant with the rubidium atoms. The left image has been obtained after a rotation at a relatively low frequency. One does not observe any difference with a condensate that has not been stirred. The right image corresponds to a higher rotation frequency. A hole is clearly visible at the center of the condensate. This is a vortex. The critical rotation for this vortex to appear is approximately $0.7\, \omega_\perp$ , where $\omega_\perp$ is the oscillation frequency in the xy plane, orthogonal to the cigar axis. This frequency is notably larger than that predicted by a reasoning based upon thermodynamics. It corresponds to the resonance frequency of the rotating quadrupole mode.

K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000) : Vortex formation in a stirred Bose-Einstein condensate.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, cond-mat/0004037, Jour. Mod. Optics 47, 2715 (2000) : Vortices in a stirred Bose-Einstein condensate.

K.W. Madison, F. Chevy, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 86, 4443 (2001) : Stationary states of a rotating Bose-Einstein condensate: Routes to vortex nucleation.

M. Cozzini, S. Stringari, V. Bretin, P. Rosenbusch, and J. Dalibard, Phys. Rev. A 67, 021602 (2003): Scissors mode of a rotating Bose-Einstein condensate

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Mesure of angular momentum

To measure the angular momentum of the rotating condensate, we have used a technique which has been suggested theoretically by Sandro Stringari and Francesca Zambelli (Phys. Rev. Lett. 81, 1754 (1998)). We excite the two quadrupole modes of the condensate, $m = +2$ and $m = -2$ , and we measure their frequency. If the condensate does not rotate, these frequencies are equal. On the contrary, if the condensate has a non-zero angular momentum $L_z$ per atom, the difference between the two frequencies reads:

$\omega_+-\omega_-=\frac{2L_z}{Mr_\perp^2}$

where $M$ is the atom mass and $r_\perp$ the radius of the condensate. The measure of the difference between the two frequencies then allows us to find the value of the angular momentum, as soon as we know the size of the condensate (which is easy). This measure is quite analogous to that of the precession of the oscillation plane of the Foucault pendulum, which reveals the rotation of the Earth. The result of the measurement of the angular momentum as a function of the rotation frequency $\Omega$ is shown in the figure below. When the condensate is stirred at a frequency below the critical frequency, the angular momentum is zero. Just at the critical frequency, which corresponds to the apparition of a centered vortex, we measure an average angular momentum per atom equal to $\hbar$ ( $h/(2\pi)$ ). The angular momentum then increases regularly as new vortices appear and get closer to the condensate center. For very large stirring frequencies, the condensate does not rotate anymore and the angular momentum is again 0.

F. Chevy, K. Madison, and J. Dalibard, Phys. Rev. Lett. 85, 2223 (2000) : Measurement of the angular momentum of a rotating condensate.

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Phase of a one-vortex state

Using a "Young’s double-slit" type experiment, we have observed the interference of our condensate with itself. We have checked that the phase pattern of a one-vortex state is indeed $\exp(i\theta)$ , where $\theta$ is the azimuthal angle around the z axis. The interference pattern which characterizes this phase distribution presents a "dislocation" in its center, whereas the interference fringes obtained with a non rotating condensate are straight. Depending on the relative phase between the two interfering copies of the condensate, the dislocation appears as a bright or dark fringe.


Franges d’interférence en l’absence de vortex (gauche) et en présence de vortex (2 images de droite)

F. Chevy, K.W. Madison, V. Bretin, and J. Dalibard, Phys. Rev. A 64 031601R (2001) : Interferometric detection of a single vortex in a dilute Bose-Einstein condensate.

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Shape and excitation of the vortex line

The pictures showed above have been obtained by observing the condensate along the vortex axis. To access the shape of the vortex line, we have installed a transverse observation system. We have found that the vortex line is not always straight and that it often has the shape of a U or a N. We have measured the evolution of this shape as a function of time. We have found that the line, initially straight, becomes curved and localized on the side of the condensate after a time of the order of 10 seconds. This is probably a consequence of a slight static anisotropy of our trap in the xy plane, which implies that the angular momentum along the z axis is not strictly conserved.

We have also observed the Kelvin mode of the vortex line. We first excited the transverse quadrupole mode $m=-2$ of condensate with a single positively charged vortex, and we observed its desexcitation in two kelvons (quanta of the Kelvin mode). The two kelvons propagate in opposite direction (conservation of linear momentum) and have the same energy. Each kelvon has an angular momentum $m=-1$ so that only the quadrupole mode $m=-2$ can decay through this channel (not the $m=+2$ ).

P. Rosenbusch, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 89, 200403 (2002): Dynamics of a single vortex line in a Bose-Einstein condensate

V. Bretin, P. Rosenbusch, F. Chevy, G.V. Shlyapnikov, and J. Dalibard, Phys. Rev. Lett. 90, 100403 (2003): Quadrupole Oscillation of a Single-Vortex Condensate: Evidence for Kelvin Modes

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Abrikosov lattice

When the condensate is stirred at a frequency larger than the critical frequency $0.7\,\omega_\perp$ , we observe the nucleation of several vortices. When many vortices are present, they form a triangular lattice. This type of lattice is well known for type II supraconductors placed in a magnetic field, and it is called an Abrikosov lattice. The pictures below show lattices with up to 14 vortices. Using much bigger condensates, the MIT and Boulder groups have recently observed lattices with more than 100 vortices.

As shown by Feynman, if one takes the coarse grain average of the velocity field (i.e. if one averages over a distance larger than the distance between adjacent vortices), one recovers the classical velocity field:

$\vec v(\vec r)=\Omega\times \vec r$

F. Chevy, K. Madison, V. Bretin, J. Dalibard, cond-mat/0104218, Proceedings of the workshop Trapped particles and fundamental physics (Les Houches 2001), organized by S. Atutov, K. Kalabrese and L. Moi : Formation of quantized vortices in a gaseous Bose-Einstein condensate.

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Rapidly rotating condensate

The rapid rotation of a condensate corresponds to the case where the rotation frequency $\Omega$ becomes similar to the transverse trap frequency $\omega$ . The centrifugal and trapping forces nearly compensate each other and the spatial extent of the condensate becomes very large. This is an interesting regime, since it is formally equivalent to the situation leading to Quantum Hall effect in bi-dimensional gases of electrons. The description of the rotating system can be done within the Lowest Landau Level; if the number of vortices is small compared to the number of particles, a mean field description is valid. When the number of vortices reaches the number of particles, the state of the system becomes highly correlated, analogous to those appearing in fractional Quantum Hall Effect. This regime has not yet been observed experimentally.

From a theoretical point of view, we have studied the arrangement of the vortices for an interacting gas in the LLL regime, using a mean-field approach (Amandine Aftalion, Xavier Blanc, Jean Dalibard, Phys. Rev. A 71, 023611 (2005) : Vortex patterns in a fast rotating Bose-Einstein condensate). We have also studied the case of an ideal gas in fast rotation. We have shown that vortices were also present in this case, and we have related their position to the root of a random polynomial (Y. Castin , Z. Hadzibabic, S. Stock, J. Dalibard, and S. Stringari, Phys. Rev. Lett. 96, 040405 (2006): Seeing zeros of random polynomials: quantized vortices in the ideal Bose gas)

To study experimentally the fast rotation regime while avoiding the explosion of the condensate, we superimposed a quartic potential to the confining harmonic potential. The variations in the vortex lattice for increasing $\Omega$ is presented in the figure below, obtained for $\omega/(2\pi)$ =65 Hz. When \Omega\omega, the vortex lattice is well ordered. Above this frequency, the vortices are more disordered and their contrast strongly drops. This effect is probably due to a longitudinal instability of the vortex line when $\Omega$ is above $\omega$ .

Finally we studied the monopole mode of the condensate in fast rotation. The structure of this mode can indeed bring new information on the nature of the gas in this regime. The monopole oscillation shows a "entering wave" phenomenon, as shown below. This figure was obtained for $\Omega/2\pi=67$ Hz, by taking one picture every millisecond during the monopole oscillation.

P. Rosenbusch, D.S. Petrov, S. Sinha, F. Chevy, V. Bretin, Y. Castin, G. Shlyapnikov, and J. Dalibard, Phys. Rev. Lett. 88, 250403 (2002) : Critical rotation of a harmonically trapped Bose gas.

V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004): Fast Rotation of a Bose-Einstein Condensate

S. Stock, V. Bretin, F. Chevy and J. Dalibard, Europhys. Lett. 65, 594 (2004): Shape oscillation of a rotating Bose-Einstein condensate

For the monopole mode of a condensate at rest, see also: F. Chevy, V. Bretin, P. Rosenbusch, K. W. Madison, and J. Dalibard, Phys. Rev. Lett. 88, 250402 (2002) : The transverse breathing mode of an elongated Bose-Einstein condensate.

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