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Bose-Einstein Condensation M.N.Kiselev 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 Identical, 1 but classically distinguishable 0 3 2 1 0 3 310% 2 620% 1 930% 0 12 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 Identical, 2 indistinguishable 1 0 3 2 1 0 3 2 1 0 3 particles, total energy = 3 (Arbitrary units) Bosons 3 2 1 0 3 2 1 0 3 2 1 0 3 particles, total energy = 3 (Arbitrary units) Bosons 3 2 1 0 3 2 1 0 bosons classical 3 1 11% 10% 2 1 11% 20% 1 4 44% 30% 0 3 33% 40% 3 particles, total energy = 3 (Arbitrary units) Classical 3 2 10 % probability for triple occupancy 1 0 3 2 1 0 3 2 30 % probability 1 for double occupancy 0 3 particles, total energy = 3 (Arbitrary units) Bosons 3 2 33 % probability for triple occupancy 1 0 3 2 Bosons like to get together! 1 0 3 2 33 % probability 1 for double occupancy 0 Thermodynamics of 3-dimensional Ideal Bose Gas. 3/2 ⎛⎞⎛⎞T ∞ kdk222 k 1 EnV==ε NNε =0 =−⎜⎟1,⎜⎟ ∑ kk 2 22 ⎜⎟ ∫ 22π m ⎛⎞ k ⎝⎠Tc k 0 ⎝⎠ exp⎜⎟− 1 ⎝⎠2mkB T 3/2 3/2 5/2 ⎛⎞T mkT()B E ==0.770NkB T⎜⎟ 0.12893 V , ⎝⎠Tc 2 ⎛⎞∂E 5E =− E CV ==⎜⎟ , FT= E − S ⎝⎠∂T V 2T 3 T ∂F mkT3/2() 5/2 CV 5E ⎛⎞ B SdT== P =−⎜⎟ =0.0851 3 ∫ ⎝⎠∂V T 0 T 3T Pressure does not depend on the volume P ⎯⎯⎯T →0→0 Summary: 2/3 2 ⎛⎞N TT<=c 3.31 ⎜⎟particles start to collect at lowest energy until at T=0 they are all there mkB ⎝⎠ V 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 CV λ -point in He4 TTc − Tc T Tc T Weakly-interacting Bose Gas a 4π 2 d Uq()≈ a m Weak interactions: ad a – scattering length, d – average distance p2 Ideal Bose Gas ε ()p = 2m 2 2 3 ⎧ p p na0 2 2 ⎪ 4π na0 , 2 Interacting 4π na0 22 ⎛⎞p ⎪ m 2mma N ε ()pp=+≈2 ⎜⎟⎨ n0 = Bose Gas mm⎝⎠2 p 2 p 2 2 na3 V ⎪ , 0 ⎩⎪ 2m 2mma2 1/2 NN8 3/2 ⎛⎞ Particles of a non-ideal Bose gas do not ≈−1 a ⎜⎟ all have zero momentum, NV0 3 π ⎝⎠ even in the ground state. Ordinary light Laser light divergent diffraction limited (directional) incoherent coherent many small waves one big wave many modes single mode (monochromatic) Ordinary gas Bose-Einstein condensate atoms move around randomly atoms in a coherent state divergent diffraction limited (directional) incoherent coherent many small waves one big wave many modes single mode (monochromatic) Possible candidates for the experimental observation of the Bose-Einstein Condensation U Spin polarized hydrogen 2/3 2 ⎛⎞N Tc = 3.31 ⎜⎟ 2mkVB ⎝⎠ R Atomic Hydrogen molecular H 2 crystal Helium Only 8% of particles in the condensate Strongly interacting Bose system! electron Excitons in semiconductors Strong many-body effects hole BEC @ JILA, June ‘95 (Rubidium) BEC @ MIT, Sept. ‘95 (Sodium) Sodium, MIT Group, September 1995 6 ∼ 10 particles in BEC Distribution function TKc ∼ 1µ field of view 200µmm× 270µ time – 1/20 s Rubidium, JILA Group, June 1995 3 ∼ particles in BEC Distribution function 10 JILA –Joint Institute for Laboratory Astrophysics -1/2 λdB=h/p T ∝ T BEC=Tool for knowledge BEC=Tool for applications Condensed matter physics Metrology Many-body physics • Atomic clocks Statistical physics • Matter wave sensors • Superfluidity • Quantum gases • Mesoscopic physics Collisional physics • Ultracold collisions • Cold chemistry Quantum optics • Coherence of atoms Visionary long-term goals •Atom laser • Atom deposition – nanotechnology • Entanglement • Concepts for quantum computer.
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