Bose einstein condensation

In the quantum level, there are profound differences between fermions (follows Fermi-Dirac statistic) and bosons (follows Bose-Einstein statistic).

As a gas of bosonic atoms is cooled very close to absolute zero temperature, their characteristic will change dramatically.
More accurately when its temperature below a critical temperature Tc, a large fraction of the atoms condenses in the lowest quantum states .

This dramatic phenomenon is known as Bose-Einstein condensation

1. BOSE-EINSTEIN CONDENSATION The institute of science AKSHAY APPA KHORATE DHIRAJKUMAR HIRALAL HIRALAL

2. 1. Introduction Condensation (BEC). In the quantum level, there are profound differences between fermions (follows Fermi-Dirac statistic) and bosons (follows Bose-Einstein statistic). As a gas of bosonic atoms is cooled very close to absolute zero temperature, their characteristic will change dramatically. More accurately when its temperature below a critical temperature Tc , a large fraction of the atoms condenses in the lowest quantum states . This dramatic phenomenon is known as Bose-Einstein

3. Bose and Einstein theories, except at very low temperatures. In 1924 an Indian physicist named Bose derived the Planck law for black-body radiation by treating the photons as a gas of identical particles. Einstein generalized Bose's theory to an ideal gas of identical atoms or molecules for which the number of particles is conserved. The equations, which were derived by Einstein didn't predict the behavior of the atoms to be any different from previous

4. Einstein found that when the temperature is high, they behave like ordinary gases. However, at very low temperatures Einstein's theory predicted that a significant proportion of the atom in the gas would collapse into their lowest energy level. This is called Bose-Einstein condensation . The BEC is essentially a new state of matter where it is no longer possible to distinguish between the atoms.

5. Criterion for Bose-Einstein condensation 1. Ideal Bose gas The Pauli principle does not apply in this case, and the low- temperature properties of such a gas are very different from those of a fermion gas. The properties of BE gas follow from Bose-Einstein distribution. Here T represents the temperature, k b Boltzmann constant and  the chemical potential. k B T k k k n   1 ,  n  N ,   1 e  (  k   )  1

6. In the Bose-Einstein distribution, the number of particles in the energy range dE is given by n(E)dE, where g ( E ) z  1 e E / k B T n(E)   1 z is the fugacity, defined by z  e  / k B T where μ is the chemical potential of the gas, and the density of states g(E) (which gives the number of states between E and E+dE) is given (in three dimensions) for volume V by  2m  3/ 2 V E g(E)  4  2  3

7. The critical (or transition) temperature T c is defined as the highest temperature at which there exists macroscopic occupation of the ground state. The number of particles in excited states can be calculated by integrating n(E)d(E): B E d E   2m  3/ 2 V 2 3  1 E / k T  N e   n(E)dE   4   z e  1 N e is maximal when z=1 (and thus μ=0), and for a condensate to exist we require the number of particles in the excited state to be smaller than the total number of particles N.

8. B  2m  3/ 2 V  2mk T  3/ 2 V E dE   2m  3/ 2 V  3 / 2 x dx z  1 e E / k B T  3   3  N e      k T    B    N  2   2   1       e x 4  2  3  1 4  2  3 4  2  3 Therefore   3    3   2.314  2   2      w he r e  4  2 N  2 / 3  T c T    2mk 2.315V B    2 Below this temperature most of the atoms will be part of the BEC. For example, sodium has a critical temperature of about 2μK.

9. Transition temperature In fact, the condensate fraction, i.e. how many of the particles are in the BEC, is represented mathematically as, 2 3   1    T C   T  N N where N is the number of atoms in the ground state. The number of excited particles at temperatures below the critical temperature can be rewritten as  T  3/ 2 N e  N    T c  The number of particles at the ground state (and therefore in the condensate) N is given by e T   T  3/ 2    N  N  N  N 1       c   

10.  The system undergoes a phase transition and forms a Bose-Einstein condensate, where a macroscopic number of particles occupy the lowest-energy quantum state. 3  The temperature and the density of particles n at the phase transition are related as n  3 dB = 2.612 . 2 3 T 2     N( T )  N  2  2    V   3  mk B   BEC is a phase-transition solely caused by quantum statistics, in contrast to other phase-transitions (like melting or crystallization) which depend on the inter- particle interactions.

11. The fraction of population of atoms in different state

12. 2. Matter Waves and Atoms m k T B dB 2   2   d B  = de Broglie wavelength m = mass T = temperature  = Planck’s constant Bose-Einstein condensation is based on the wave nature of particles. De Broglie proposed that all matter is composed of waves. Their wavelengths are given by

13. Matter Waves and Atoms BEC also can be explained as follows, as the atoms are cooled to these very low temperatures their de Broglie wavelengths get very large compared to the atomic separation. Hence, the atoms can no longer be thought of as particles but rather must be treated as waves. At everyday temperatures, the de Broglie wavelength is so small, that we do not see any wave properties of matter, and the particle description of the atom works just fine.

14. At high temperature,  dB is small, and it is very improbable to find two particles within this distance. I n a s i m p l i f ied quantum de s cri p tion, the ato m s can be regarded as wavepackets with an extension  x, approximately given by Heisenberg’s uncertainty relation  x= h/  p, where  p denotes the width of the thermal momentum distribution. Matter Waves and Atoms At high temperatures, a weakly interacting gas can be treated as a system of “billiard balls”.

15. When t he g a s i s c oo led do w n t he d e B r og l i e w a v elen gth increases. A t t he BE C tr an s itio n t e m p e r a tur e,  dB be c om es c omp a r a ble to the distance between atoms, the wavelengths of neighboring atoms are beginning to overlap and the Bose condensates forms which is characterized by a macroscopic population of the ground state of the system. As the temperature approaches absolute zero, the thermal cloud disappears leaving a pure Bose condensate.

16. Phase Diagram The green line is a phase boundary. The exact location of that green line can move around a little, but it will be present for just about any substance. Underneath the green line there is a huge area that we cannot get to in conditions of thermal equilibrium. It is called the forbidden region.

17. 18  Finally, if the atomic gas is cooled enough, what results is a kind of fuzzy blob where the atoms have the same wave function.

18. Not all particles can have BEC. This is related to the spin of the particles. Single protons, neutrons and electrons have a spin of ½. They cannot appear in the same quantum state. BEC cannot take place. Some atoms contain an even number of fermions. They have a total spin of whole number. They are called bosons . Example: A 23 Na atom has 11 protons, 12 neutrons and 11 electrons. Fermions and Bosons

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20. The Strange State of BEC When all the atoms stay in the condensate, all the atoms are absolutely identical. There is no possible measurement that can tell them apart. Before condensation, the atoms look like fuzzy balls. After condensation, the atoms lie exactly on top of each other (a superatom ).

21. W h at D oes a B o s e - E i n s t e i n C o n d e n s a t e Look Like? There is a drop of condensate at the center. The condensate is surrounded by uncondensed gas atoms.

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24. 64 Interference Pattern  When two Bose-Einstein condensates spread out, the interference pattern reveals their wave nature.

25. Interference between two condensates

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27. Molecular BEC from Fermions

28. This is evidence for condensation of pairs of 6 Li atoms on the BCS side of the Feshbach resonance. The condensate fractions were extracted from images like these, using a Gaussian fit function for the ‘‘thermal’’ part and a Molecular BEC from Fermions

29. 77 Vortices When the condensate is rotated, vortices appear. The angular momentum of each of them has a discrete value.

30. What Is Bose-Einstein Condensation Good For? This is a completely new area. Applications are too early to predict. The atom laser can be used in: atom optics (studying the optical properties of atoms) atom lithography (fabricating extremely small circuits) precision atomic clocks other measurements of fundamental standards hologram communications and computation. Fundamental understanding of quantum mechanics. Model of black holes .

31. References           Homepage of the Nobel e-Museum ( http://www.nobel.se/ ). BEC Homepage at the University of Colorado ( http://www.colorado.edu /physics/2000/bec/ ). Ketterle Group Homepage ( http://www.cua.mit/ketterle_group/ ). The Coolest Gas in the Universe (Scientific American, December 2000, 92-99). Atom Lasers (Physics World, August 1999, 31-35). http://cua.mit.edu/ketterle_group/Animation_folder/TOFsplit.htm http://www.colorado.edu/physics/2000/bec/what_it_looks_like.html . http://www.colorado.edu/physics/2000/bec/lascool4.html . http://www.colorado.edu/physics/2000/bec/mag_trap.html Pierre Meystre Atom Optics.