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Institut für Theoretische Physik und Astrophysik Universität Würzburg Bose-Einstein Condensation M.N.Kiselev Experimental observation of BEC In dilute gases of Alkali metals Annual number of published papers, which have the words “Bose” and “Einstein” in their title, abstracts or keywords (ISI database) Nobel Prize in Physics 2001 For the achievement of Bose-Einstein Condensation in dilute gases of Alkali metals… Eric A.Cornell (USA), Wolfgang Ketterle (Germany), Carl E. Wieman (USA) Outlook • Symmetry properties of many-particle wave functions: Fermions and Bosons. • Statistics of Bosons: Bose-Einstein distribution function. • Ideal Bose gas. Bose-Einstein Condensation. • Thermodynamics of the Ideal Bose gas. • Weakly interacting Bose gas. • Experiments and perspectives. *) I acknowledge the use of materials and slides from W.Ketterle Nobel Lecture 2001 and his talk given at MIT’s Teachers Program 24/06/03. Some pictures were lent to me by courtesy of W.Ketterle. 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 Identical, but classically distinguishable 3 2 1 0 3 2 1 0 3 6 9 12 10% 20% 30% 40% 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 3 2 1 0 3 2 1 0 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 3 2 1 0 3 2 1 0 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 3 2 1 0 3 2 1 0 Identical, indistinguishable 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 3 2 1 0 3 2 1 0 Bosons 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 3 2 1 0 3 2 1 0 1 1 4 3 bosons classical 11% 11% 44% 33% 10% 20% 30% 40% Bosons 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 Classical 10 % probability for triple occupancy 3 2 1 0 3 2 1 0 30 % probability for double occupancy 3 particles, total energy = 3 (Arbitrary units) 3 2 1 0 Bosons 33 % probability for triple occupancy 3 2 1 0 3 2 1 0 Bosons like to get together! 33 % probability for double occupancy Thermodynamics of 3-dimensional Ideal Bose Gas. Nε =0 ⎛ ⎛ T ⎞3/ 2 ⎞ = N ⎜1 − ⎜ ⎟ ⎟ , ⎜ ⎝ Tc ⎠ ⎟ ⎝ ⎠ ⎛T ⎞ E = 0.770 Nk BT ⎜ ⎟ ⎝ Tc ⎠ 5E ⎛ ∂E ⎞ CV = ⎜ , ⎟ = ⎝ ∂T ⎠V 2T T CV 5E S = ∫ dT = 3T T 0 ∞ k 2 dk 2 k 2 E = ∑ ε k nk = V ∫ 2 π 2 2m k 0 3/ 2 = 0.1289 m3/ 2 (k BT )5/ 2 3 1 ⎛ 2k 2 exp ⎜ ⎝ 2mk BT V, 2 F = E − TS = − E 3 m ⎛ ∂F ⎞ P = −⎜ ⎟ = 0.0851 ⎝ ∂V ⎠T 3/ 2 (k BT )5/ 2 Pressure does not depend on the volume T →0 P ⎯⎯⎯ →0 3 ⎞ ⎟ −1 ⎠ Summary: ⎛N⎞ T < Tc = 3.31 ⎜ ⎟ mk B ⎝ V ⎠ 2 2/3 particles start to collect at lowest energy until at T=0 they are all there Bose Gas undergoes a phase transition without any interaction! All the thermodynamic quantities are continuous at the transition point. 3rd - order phase transition N0 Order parameter λ CV -point in He4 Tc − T Tc T Tc T Weakly-interacting Bose Gas a d Weak interactions: Ideal Bose Gas Interacting Bose Gas ε ( p) = a d p2 ε ( p) = 2m 4π 2 U (q) ≈ a m a – scattering length, d – average distance 2 ⎧ p n0 a 3 p2 2 4π n 0 a , ⎪⎪ 4π n 0 a 2 2 ⎛ p 2 ⎞ 2m ma 2 m p +⎜ ⎟ ≈⎨ 2 2 m2 m n0 a 3 p2 p2 ⎝ ⎠ ⎪ , ⎪⎩ 2m 2m ma 2 n0 = N V 1/ 2 N 8 3/ 2 ⎛ N ⎞ ≈ 1− a ⎜ ⎟ N0 3 π ⎝V ⎠ Particles of a non-ideal Bose gas do not all have zero momentum, even in the ground state. Ordinary light divergent incoherent many small waves many modes Laser light diffraction limited (directional) coherent one big wave single mode (monochromatic) Ordinary gas atoms move around randomly divergent incoherent many small waves many modes Bose-Einstein condensate atoms in a coherent state diffraction limited (directional) coherent one big wave single mode (monochromatic) Possible candidates for the experimental observation of the Bose-Einstein Condensation U ⎛N⎞ Tc = 3.31 ⎜ ⎟ 2mk B ⎝ V ⎠ 2 2/3 Spin polarized hydrogen R Atomic Hydrogen H2 Helium molecular crystal Only 8% of particles in the condensate Strongly interacting Bose system! Excitons in semiconductors electron Strong many-body effects hole BEC @ JILA, June ‘95 (Rubidium) BEC @ MIT, Sept. ‘95 (Sodium) Distribution function Sodium, MIT Group, September 1995 ∼ 106 particles in BEC Tc ∼ 1µ K field of view Distribution function 200 µ m × 270µ m time – 1/20 s Rubidium, JILA Group, June 1995 ∼ 103 particles in BEC JILA –Joint Institute for Laboratory Astrophysics λdB = h/pT ∝ T -1/2 BEC=Tool for knowledge Condensed matter physics Many-body physics Statistical physics • Superfluidity • Quantum gases • Mesoscopic physics BEC=Tool for applications Metrology • Atomic clocks • Matter wave sensors Collisional physics • Ultracold collisions • Cold chemistry Quantum optics • Coherence of atoms • Atom laser • Entanglement Visionary long-term goals • Atom deposition – nanotechnology • Concepts for quantum computer

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