In the center of mass frame of a binary collision, the scattered particles
come out "back to back" because of momentum conservation. In the language of
deBroglie waves, the same process is called spontaneous, four wave mixing
and the oppositely directed wave vectors of the outgoing waves is fixed by a
phase matching condition.
We have observed such a process by producing two colliding Bose-Einstein
condensates using stimulated Raman transitions. A 3D reconstruction of the
collision, obtained with our position sensitive detector, is shown in the
figure. One sees a spherical shell represented by circles of varying diameter. In the mid plane
of the sphere one can see two unscattered pancake-shaped condensates I and
II. One can also see a third condensate III, produced by imperfect
polarization of the Raman beams, and a fourth condensate IV which results
from stimulated four wave mixing of condensates I, II and III.
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Each frame represents a 2.4 ms time slice of the atomic cloud as
it passes the plane of the detector.
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3D reconstruction of the scattering halo generated by the collision
of two BECs after time-of-flight. The two colliding condensates are
on the equatorial plane of the sphere. The two condensates generated
by four-wave-miging are visibles on the top and the bottom of the
sphere. Click on the image to download a 3D animation (.avi) of the scattering halo
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Using the atom positions, we can study correlation functions for back to
back pairs. We can also observe pairs of atoms emitted in the same
direction. This is another manifestation of the Hanbury Brown Twiss effect. Although
the back to back correlation is easily understood in terms of classical
particles, the HBT peak is necessarily an interference phenomenon, and
therefore the process is quantum mechanical. The HBT effect here gives us a measure of the size of the pair production
region and therefore allows us to confirm that the momentum spread of a back
to back pair, is limited chiefly by the uncertainty principle.
In the summer of 2006, we transported our detector to the VU Amsterdam to collaborate with the metastable He group there. The group in Amsterdam, had recently produced a degenerate gas of the fermionic isotope 3He* using sympathetic cooling by 4He*. Much like in our experiment of 2005 in Orsay, we released the cloud of atoms onto the detector which was placed below the trap. The detector gives the arrival times and positions of individual atoms with a quantum efficiency estimated to be 10%. With this information we can plot a histogram of separations in 3D for all the pairs in a cloud. In Amsterdam, it was possible to use the same apparatus to make measurements on both fermions and bosons and clearly show their contrasting behavior. The figure below shows normalized pair separation histograms taken at the same temperature (about 500 nK), for fermions and bosons.
The fermions show "anti-bunching"
(see "dégroupement" for a French version),
i.e. a tendency to avoid each other, due purely to quantum statistical effects. Interactions between the atoms are entirely negligible. This antibunching effect is reminiscent of antibunching of photons, but it is different in that the Pauli exclusion principle (or the exchange anti-symmetry of wavefunctions) forbids more than one atom to occupy the same phase space cell, and thus antibunching is unavoidable.
We have also demonstrated that a diverging atomic lens in the form of a blue-detuned, focussed laser beam, can be used to change the size of the atom source as viewed from the detector. Decreasing the source size and increases the correlation length at the detector. Since the antibunching contrast is limited by the detector resolution, which is not small compared to the correlation length, the defocussing technique allows us to increase the anti-bunching contrast.
In 1956, two Astronomers, Hanbury Brown and Twiss, showed that photons emitted by a thermal light source, such as a sodium lamp or a star, behaved in a surprising way. They showed that those photons tended to arrive in groups despite the chaotic nature of the source. This bunching
(see "dégroupement" for a French version)
effect was especially surprising since there is no physical interaction between the photons. Later on, this effect was shown to be related to the quantum mechanical nature of those photons. Quantum mechanics separates all particles in two populations: bosons and fermions. These two classes,obey different statistics compared to classical particles. Bose statistics tend to favor configurations in which individual particles end up in a same quantum state (a Bose-Einstein Condensate is one dramatic example), whereas Fermi statistics exclude those configurations.
The combination of our metastable Helium BEC with a position sensitive micro-channel plate detector allowed us to observe this bunching behavior in 3 dimensions. To perform the experiment, we simply released a cloud of ultra-cold atoms from a magnetic trap onto the detector. After their 308ms time of flight, the arrival times and positions of each atom were recorded and the separation of all pairs was computed and histogrammed.
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The Hanbury Brown and Twiss bunching effect for He* atoms. The left graph shows the arrival time correlation (converted to position z), the right graph the detector plane correlation.
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The atom bunching signal corresponds to the bump in the 1st figure at separations less than 1mm in the run at 0.55 microK. The 2nd two dimensional figure shows the correlation function in the plane of the detector. The asymmetry is due to the asymmetry of the spatial distribution of the source.
In order explore atom correlations with the metastable Helium BEC apparatus, the group acquired a new position sensitive micro-channel plate detector with a delay line anode (from Roentdek). Though very common in particle and nuclear physics, this is the first use in a cold-atom physics experiment. The detector can hand high mean particle rates (up to 10MHz for the delay-line) while measuring position and arrival times of individual particles with good resolutions (230&mu m and 1ns respectively for the moment). This makes the detector particularly suitable for our experimental conditions especially because of our MHz particle rates (for a duration of 20 ms).
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Rough sketch of the He* experiment. (Not to scale) In the center in red, the atomic cloud. On each side, the coils of the "cloverleaf" creating the trapping field; 47cm underneath : the micro-channel plate with the delay-line anode that allows for 3 dimensional imaging.
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The detector allows us to reconstruct the atomic cloud on an atom by atom basis. The figure shows a 3D reconstruction of a BEC hitting the detector. The well known "pancake" shape is clearly evident.
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An atom cloud after time of flight. The pancake shape is characterisitc of a BEC. (click on the image to download the 3D animation)
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The observation of BEC of He* hinged, among other things, on the elastic and inelastic collision properties of the atom at micro-K temperatures. Encouraging theoretical predictions existed, but no experimental information was available. The crucial question was whether elastic collisions, characterized by their scattering length a, were sufficiently rapid compared to inelastic processes in a spin polarized sample, involving presumably both two-body and three-body collisions. Conventionally, the two-body collision rate constant is denoted by, and the three-body rate constant by L.
Because of the large internal energy of He*, a significant fraction if not all of the inelastic collisions result in ions which can be detected by a microchannel plate. Thus monitoring of the ion products in a BEC permits a qualitatively new observation of BEC. The figure below shows the detected ion rate during evaporation through BEC. The sudden increase in the ion production rate is due to the increase in density associated with the BEC transition. This type of data also allows one to reproducibly place a cloud near the BEC transition point Tc.
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Observed ion rate during evaproration. The red curve, showing an abrupt increase, corresponds to evaporation through BEC. The blue curve corresponds to an evaporation ramp which was ended before achieving BEC.
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Using clouds Tc, it is possible to extract values for the inelastic rate constants. Surprisingly, it is also possible to get an accurate estimate of the elastic scattering length from the ionization measurements. The key idea in these measurements is to use the fact that at Tc the density of the sample is well known using the theory for a weakly interacting Bose gas. A second important ingredient is that in a BEC, the chemical potential &mu is simply related to the density and the scattering length. Accurate measurements of Tc are possible by observing the expansion of clouds of atoms, either at Tc or &mu in a BEC. The known density estimates Tc allow one to use the ion rate to get the ionization rate constants and knowing the ionization rate constants, we can get infer the density in a BEC from the ion rate. The BEC density together with the a measurement of the chemical potential finally gives a value for the scattering length, which is independent of the absolute calibration of the MCP.
Our final results are [2,3,4]:
a = 11.3 (+2.5,) nm
&beta = 0.9 (+1.7,-0.8) 10-14 cm3/s
L = 2.5 (+4.5,-2.7) 10-27 cm6/s
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