Bose - Einstein Condensation

1. Bose - Einstein Condensation Zemsky Danny

2. Outline of Talk • History 2) Criterion for Bose-Einstein condensation 3) How is BEC looks like? 4) Quantum description of BEC 5) GROSS- PITAEVSKII EQUATION The Mean-Field Approximation and General Solution 6) The realization of BEC - Cooling Techniques • Vortices • Interference of two condensates. • Feshbach resonance.

3. Introduction • 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 (BEC).

4. Bose and Einstein • 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 theories, except at very low temperatures.

5. 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.

6. 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, kbBoltzmann constant and  the chemical potential.

7. In the Bose-Einstein distribution, the number of particles in the energy range dE is given by n(E)dE, where z is the fugacity, defined by 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

8. The critical (or transition) temperature Tc 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): Ne 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.

9. Therefore where Below this temperature most of the atoms will be part of the BEC. For example, sodium has a critical temperature of about 2μK.

10. Transition temperature The number of excited particles at temperatures below the critical temperature can be rewritten as The number of particles at the ground state (and therefore in the condensate) N0 is given by In fact, the condensate fraction, i.e. how many of the particles are in the BEC, is represented mathematically as, where N0 is the number of atoms in the ground state.

11. The system undergoes a phase transition and forms a Bose-Einstein condensate, where a macroscopic number of particles occupy the lowest-energy quantum state. • The temperature and the density of particles n at the phase transition are related as n 3dB= 2.612. • 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.

12. The fraction of population of atoms in different state

13. dB = de Broglie wavelength = Planck’s constant m = mass T = temperature 2. Matter Waves and Atoms 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

14. 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.

15. Matter Waves and Atoms At high temperatures, a weakly interacting gas can be treated as a system of “billiard balls”. At high temperature, dBis small, and it is very improbable to find two particles within this distance. In a simplified quantum description, the atoms 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.

16. When the gas is cooled down the de Broglie wavelength increases. At the BEC transition temperature, dB becomes comparable 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.

17. 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.

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.

19. Fermions and Bosons • 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 23Na atom has 11 protons, 12 neutrons and 11 electrons.

20. Ground state properties of dilute-gas Bose–Einstein condensates • Binary collision model • Mean-field theory • Gross-Pitaevskii equation • Thomas-Fermi approximation • Vortex states and vortex dynamics • Feshbach resonance • Atom Laser • Interference

21. Binary collision model • At very low temperature the de Broglie wavelengths of the atoms are very large compared to the range of the interatomic potential. This, together with the fact that the density and energy of the atoms are so low that they rarely approach each other very closely, means that atom–atom interactions are effectively weakand dominated by (elastic) s-wave scattering . • The s-wave scattering length”a” the sign of which depends sensitively on the precise details of the interatomic potential . • a > 0 for repulsive interactions. • a < 0 for attractive interactions.

22. Mean Field theory and the GP equation In the Bose-Einstein condensation, the majority of the atoms condense into the same single particle quantum state and lose their individuality. Since any given atom is not aware of the individual behaviour of its neighbouring atoms in the condensate, the interaction of the cloud with any single atom can be approximated by the cloud's mean field, and the whole ensemble can be described by the same single particle wavefunction.

23. The Mean-Field Approximation and General Solution GROSS- PITAEVSKII EQUATION In |0> , each of the N particles occupies a definite single- particle state, so that its motion is independent of the presence of the other particles. Hence, a natural approach is to assume that each particle moves in a single-particle potential that comes from its average interaction with all the other particles. This is the definition of the self-consistent mean-field approximation.

24. Mean-field theory • Decompose wave function into two parts. • One is the condensate wave function, which is the expectation value of wave function. • The other is the non-condensate wave function, which describes quantum and thermal fluctuations around this mean value but can be ignored due to ultra-cold temperature.

25. where and are the boson field operators that create and annihilate particle at the position r, respectively. V(r-r’)is the two body interatomic potential. The Mean-Field Approximation The many-body Hamiltonian describing N interacting bosons confined by an external potential Vtrap is given, in second quantization, by

26.    is the two-body potential. This full potential is commonly approximated by a simplified binary collision pseudo-potential The Interaction Potential treating binary collisions as hard-sphere collisions. The effective interaction strength U0 is related to the s-wave scattering length a by where m is the atomic mass.

27. This allows one to replace V(r’-r) with an effective interaction The Interaction Potential In a dilute and cold gas, one can nevertheless obtain a proper expression for the interaction term by observing that, in this case, only binary collisions at low energy are relevant and these collisions are characterized by a single parameter, the s-wave scattering length, independently of the details of the two-body potential.

28. The boson field operators satisfy the following commutation relations: From these relations, the Heisenberg equation of motion for         can be calculated and one obtains The Mean-Field Approximation

29. The field operators can in general be written as a sum over all participating single-particle wave functions and the corresponding boson creation and annihilation operators. where are single-particle wave functions and ai are the annihilation operators. The Mean-Field Approximation The basic idea for a mean-field description of a dilute Bose gas was formulated by Bogoliubov (1947). In general, this can be written as

30. The Mean-Field Approximation The boson creation and annihilation operators obey the commutation rules The bosonic creation and annihilation operators a+ and a are defined in Fock space through the relations : where the ni denote the bosonic populations of the particle states. Gives the number of atoms is the single-particle i state.

31. Since the main characteristic of BEC is that most participating particles condense into the lowest single particle quantum state, it is possible to separate out the condensate part            of the generalised mean field operator. In this case (with N0 n0 ), there is no significant physical difference between states with N0 and N0+1 so that operators    and    in the can be replaced by. Bogoliubov approximation With a total number of particles N, the population n0 of the lowest state is macroscopic such that n0 N0 >> 1 . This is well known as the Bogoliubov approximation.

32. Using the Bogoliubov approximation, the field operator is written as a sum of its expectation value and an operator representing the remaining populations in thermal states, which can be considered vanishingly small in the zero temperature BEC regime. This decomposition leads to the following expression for the         term in Hamiltonian. Bogoliubov approximation

33. By substituting the decomposition, within the approximation, and normalising the condensate wavefunction to As indicated above, all terms containing the perturbation operator         have been neglected Gross– Pitaevskii (GP) equation

34. is the condensate wave function • describes quantum and thermal fluctuations around this mean value. • The expectation value of is zero and in the mean- field theory its effects are assumed to be small, amounting to the assumption of the thermodynamic limit (Lifshitz and Pitaevskii, 1980). • The effects of is negligibly smallin the equation because of zero temperature (i.e., pure condensate). Meaning of the Decomposition

35. The Mean-Field Approximation Its modulus fixes the condensate density through The function (r,t) is a classical field having the meaning of an order parameter and is often called the ‘‘wave function of the condensate.’’

36. In certain cases, i.e. for eigenstates of a harmonic trap, the wavefunction can be separated into parts of spatial and time dependence The time-independent GP equation with eigenvalue representing the chemical potential of the system at zero temperature. Substituting into the time-dependent GP equation leads to the time independent GP equation

37. Numerical results

38. In BEC, the kinetic energy term       becomes small compared to the high self-energy and can be neglected Thomas-Fermi Approximation The time independent GP equation, with nonlinearity C , and for a harmonic trapping potential can be simplified in the so-called ``Thomas-Fermi Approximation"

39. M1 M2 s-Wave Scattering New coordinate system -> scattering of a particle of mass  in a potential U(r)

40. s-Wave Scattering incident wave simply a plane wave outgoing wave The differential cross section; k – the momentum of the incident wave.

41. s-Wave Scattering In the case of a central potential we expand the wave function as: where uk,l are the solutions of the radial Schrödinger equation: with for large r

42. s-Wave Scattering(cont.) and the solution is After expansion of the plane wave exp(ikz) in terms os spherical harmonics, we get and the total cross section is

43. S-scattering (cont.) From now on we discuss on the case of particles so slow that r0 is the range of the potential U (r). For cold enough collisions, only the l=0, or s-wave, partial wave will contribute to the scattering cross section. In the low-energy regime, one has approximately:

44. with we will get, S-scattering (cont.) where a is the scattering length

46. 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).

47. How Is BEC Made? Laser beam

48. Experimental realizations of BEC 1.First 108- 1011atoms are collected and precooled to 10-100K at densities around 1010 -1011cm-3using laser cooling techniques. 2.This point is typically reached with 104 -107atoms at 100 nK-1K and densities around 1014cm-3 3.The whole experimental cycle typically takes between 10 and 100 s.

49. Cooling Techniques • Laser cooling in a magneto-optical trap. • Evaporative cooling process. • Subsequent rethermalisation.

50. Laser cooling in a magneto-optical trap • The gas sample first optically trapped and cooled using laser light .