1. 1 Bose-Einstein Condensation PHYS 4315 R. S. Rubins, Fall 2009

2. 2 About BEC In 1924, Einstein applied Satyendra Bose’s explanation of blackbody radiation to matter, predicting the phenomenon known as Bose-Einstein condensation (BEC). BEC is a quantum mechanical phase-transition, thought to be responsible for superfluidity in liquid helium. Not until 1995 was it observed in isolated atoms, in 87 Rb (NIST), 23 Na (MIT) and 7 Li (Rice U.). Since then, BEC has been observed around the world, and 1 H (MIT) and 4 He France. Samples typically contain of the order of 10 5 - 10 6 atoms, in which several thousand form the condensate, with transition temperatures in the range 300 – 600 nK.

3. 3 BEC: Scientific Entanglements BEC belongs to atomic physics, condensed matter physics and stat. mech. It could not have been produced without the tools of optics and laser physics, the manipulation of magnetism and fluid dynamics, and the use of new techniques in low temperature physics. BEC is a deep entanglement of fields, giving rise to a totally new field of physics. See Physics Today, December 2006

4. 4 Bosons and Fermions Identical particles follow either Bose-Einstein or Fermi-Dirac statistics. Bosons have integer angular momentum quantum numbers (e.g. photons, atoms with an even no. of neutrons.). They have symmetrical wavefunctions; i.e.; if two particles (1 and 2) are in the states a and b, then Ψ sym = ψ a (1) ψ b (2) + ψ a (2) ψ b (1)  ψ a (1) ψ a (2) if a = b. Fermions have half-integer angular momentum quantum nos. (e.g. electrons, nucleons, atoms with an odd no. of neutrons.). They have antisymmetrical wavefunctions; i.e.; if two particles (1 and 2) are in the states a and b, then Ψ anti = ψ a (1) ψ b (2) – ψ a (2) ψ b (1)  0 if a = b.

5. 5 Boson and Fermion Gases Below 1 mK In these Rice University images of atomic clouds, those of 7 Li (a boson with 4 neutrons) continue to collapse as the temperature is lowered. Since identical fermions cannot occupy the same space (the Pauli exclusion principle), the atomic cloud of 6 Li (a fermion with 3 neutrons) shows a smaller collapse.

6. 6 BEC Photo from Rice University Cloud of about 70,000 7 Li atoms, with about 1200 in the BEC peak at the center, at about 600 nK.

7. 7 BEC: a Phase Transition in an Ideal Gas Like the ferromagnetic transition at the Curie point of iron (1043 K), BEC is a phase transition, but unlike the ferromagnetic transition, which occurs because of the strong interaction between iron atoms, BEC occurs in an ideal gas, for which interatomic forces are negligible.

8. 8 BEC Atoms: Each in the Same Wave Function The de Broglie wavelength λ dB = h/mv, becomes for a quantum gas λ dB = h/(2πmkT) 1/2. Thus λ dB increases as T is lowered, and a phase transition to a BEC state occurs when λ dB reaches the atomic separation.

9. 9 Interference Between BEC Waves Like the interference patterns that may be produced by the coherent light from lasers, BEC waves show interference phenomena. However, unlike laser beams, which are in non- equilibrium states, a BEC wave is an equilibrium state.

10. 10 Loading a Magnet Trap for Li 7 (Rice U.) The apparatus is contained in an ultra-high vacuum at room temperature. Hot Li 7 atoms, emitted from an oven at 800 K, form an atomic beam. The atomic beam is slowed by an oppositely directed laser beam, and deflected by a second laser beam towards a magnetic and optical trap. Another laser beam collimates the deflected atomic beam, and optically pumps it, so that each atom is in the same magnetic state. Once in the trap, the atomic beam is contained by a set of six laser beams.

11. 11 Magnetic Trap (Rice U.) If the magnetic moment of an atom is parallel to the magnetic field, it will be attracted to a local minimum of the field, which occurs at the center of the magnet distribution. If the direction of the magnetic moment is reversed, the center of the distribution becomes a local maximum, which causes that atom to leave. The magnetic field at the minimum must not be zero, otherwise the atomic moments may spontaneously reverse their directions. In practice, the field at the minimum was 0.1 T. Atoms in the trap may be lost by collisions in which the moment direction is reversed.

12. 12 Laser Cooling 1 Laser cooling is achieved by using the Doppler effect to reduce v rms. Two opposing laser beams of equal intensity are each tuned to the low frequency side of an optical transition. The beam opposing the atom’s motion is blue-shifted to higher frequencies, so that the force on it is increased. The beam in the same direction as the atom’s motion is red-shifted to lower frequencies, so that the force on it is decreased.

13. 13 Laser Cooling 2 The net effect of the two opposing laser beams is to reduce the magnitude of the velocity component of each atom along the axis of the two beams. Three orthogonal pairs of lasers are used to slow the motions of atoms moving in all directions. Using laser cooling for Rb 87, the NIST group in Boulder, achieved temperatures of 10 μK, which are still ten to a hundred times too high for observing BEC. The effect of reducing v rms on the temperature of the sample may be calculated using the equipartition theorem; i.e. ½ mv rms 2 = (3/2)kT.

14. 14 Evaporative Cooling 1 This method is analogous to the cooling of a hot liquid by evaporation. The fastest moving atoms move furthest from the minimum, to a position of highest energy (see the upper atom shown in the figure). Magnetic resonance is used to reverse the moments of the most energetic atoms, causing them to leave the trap, which is now an energy maximum. Slowly reducing the radio frequency removes progressively cooler atoms. At the end, only about 1% of the atoms remain in the trap, and the temperature is reduced by a factor of about 100, giving a temperature of the order of 100 nK.

15. 15 Photographing the Condensate (NIST) 1 False color images show the velocity distribution just before the appearance of BEC (right), just after it (center), and for a nearly pure condensate (right). To increase the sample size, the magnetic trap is turned off. The excited- state (thermal) atoms move out faster, leaving the condensate near the center of the trap. These photographs were taken after the atoms had moved for about 0.05 s. The thermal cloud is almost circular, while the condensate cloud is elliptical.

16. 16 Photographing the Condensate (NIST) 2 The right frame has a horizontal dimension of 40 – 50 μm, equivalent to about 1500 atoms forming a single wave. The shape of the peak is related to the elliptical shape of the trap, giving a vivid demonstration of the uncertainty principle  p x  x  ħ. The temperature within the condensate may be of the order of 1 nK.