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Bose-Einstein Condensation

PHYS 4315R. S. Rubins, Fall 2009

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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 87Rb (NIST), 23Na (MIT) and 7Li (Rice U.). Since then, BEC has been observed around the world, and 1H (MIT) and 4He France.

• Samples typically contain of the order of 105 - 106 atoms, in which several thousand form the condensate, with transition temperatures in the range 300 – 600 nK.

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BEC: Scientific Entanglements

BEC belongs to atomicphysics, condensed matterphysics and stat. mech.

It could not have beenproduced without the toolsof optics and laser physics,the manipulation ofmagnetism and fluiddynamics, and the use ofnew techniques in lowtemperature physics.

BEC is a deep entanglementof fields, giving rise to atotally new field of physics.

See Physics Today, December 2006

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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

Ψsym = ψa(1) ψb(2) + ψa(2) ψb(1) ψa(1) ψa(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

Ψanti = ψa(1) ψb(2) – ψa(2) ψb(1) 0 if a = b.

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Boson and Fermion Gases Below 1 mK

In these Rice Universityimages of atomic clouds,those of 7Li (a boson with4 neutrons) continue tocollapse as thetemperature is lowered.

Since identical fermionscannot occupy thesame space (the Pauliexclusion principle), theatomic cloud of 6Li (afermion with 3 neutrons)shows a smaller collapse.

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BEC Photo from Rice University

• Cloud of about 70,000 7Li atoms, with about 1200 in the BEC peak at the center, at about 600 nK.

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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.

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BEC Atoms: Each in the Same Wave Function

The de Broglie

wavelength λdB = h/mv,

becomes for a quantum

gas

λdB = h/(2πmkT)1/2.

Thus λdB increases as

T is lowered, and a

phase transition to

a BEC state occurs

when λdB reaches the

atomic separation.

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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.

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Loading a Magnet Trap for Li7 (Rice U.)

• The apparatus is contained in an ultra-high vacuum at room temperature.• Hot Li7 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.

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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.

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Laser Cooling 1

• Laser cooling is achieved by using the Doppler effect to reduce vrms.

• 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.

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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 Rb87, 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 vrms on the temperature of the sample may be calculated using the equipartition theorem; i.e.

½ mvrms2 = (3/2)kT.

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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.

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Photographing the Condensate (NIST) 1

False color images show the velocity distribution just before the appearance ofBEC (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 condensatenear 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.

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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 pxx ħ.

• The temperature within the condensate may be of the order of 1 nK.