Cold atoms Lecture 7. 21^st February, 2007 Preliminary plan/reality in the fall term Recapitulation Offering many new details and alternative angles of view BEC in atomic clouds Nobelists I. Laser cooling and trapping of atoms Doppler cooling in the Chu lab Doppler cooling in the Chu lab Nobelists II. BEC in atomic clouds Trap potential Ground state orbital and the trap potential BEC observed by TOF in the velocity distribution Ketterle explains BEC to the King of Sweden Simple estimate of T[C] (following the Ketterle slide) Interference of atoms Today, we will be mostly concerned with the extended (" infinite" ) BE gas/liquid Microscopic theory well developed over nearly 60 past years Interacting atoms Importance of the interaction – synopsis Many-body Hamiltonian Mean-field treatment of interacting atoms Many-body Hamiltonian and the Hartree approximation Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation – homogeneous gas Field theoretic reformulation (second quantization) Purpose: go beyond the GP approximation, treat also the excitations Field operator for spin-less bosons Action of the field operators in the Fock space Action of the field operators in the Fock space Field operator for spin-less bosons – cont'd Operators Hamiltonian Hamiltonian Hamiltonian On symmetries and conservation laws Hamiltonian is conserving the particle number Gauge invariance of the 1st kind Hamiltonian of a homogeneous gas Hamiltonian of the homogeneous gas Hamiltonian of the homogeneous gas Hamiltonian of the homogeneous gas Hamiltonian of the homogeneous gas Hamiltonian of the homogeneous gas Summary: two symmetries compared Hamiltonian of the homogeneous gas Hamiltonian of the homogeneous gas Bogolyubov method Originally, intended and conceived for extended (rather infinite) homogeneous system. Reflects the 'Paradoxien der Unendlichen' Basic idea Basic idea Approximate Hamiltonian Approximate Hamiltonian Approximate Hamiltonian Approximate Hamiltonian Approximate Hamiltonian Approximate Hamiltonian Approximate Hamiltonian Approximate Hamiltonian Bogolyubov transformation Bogolyubov transformation Bogolyubov transformation – result Bogolyubov transformation – result More about the sound part of the dispersion law Particles and quasi-particles What is the Bogolyubov approximation about Trying to understand the Bogolyubov method Notes to the contents of the Bogolyubov theory ODLRO in the Bogolyubov theory ODLRO in the Bogolyubov theory ODLRO in the Bogolyubov theory ODLRO in the Bogolyubov theory ODLRO in the Bogolyubov theory Three methods of reformulating the Bogolyubov theory Three methods of reformulating the Bogolyubov theory I. explicit construction of the classical part of the field operators Quotation from Landau-Lifshitz IX Quotation from Landau-Lifshitz IX Quotation from Landau-Lifshitz IX Quotation from Landau-Lifshitz IX II. the condensate represented by a coherent state Reformulation of the Bogolyubov requirements Reformulation of the Bogolyubov requirements About the coherent states New vacuum and the shifted field operators L1: Thermodynamics: which environment to choose? L1: Thermodynamics: which environment to choose? L1: Thermodynamics: which environment to choose? L1: Homogeneous one component phase: boundary conditions (environment) and state variables L1: Homogeneous one component phase: boundary conditions (environment) and state variables New vacuum and the shifted field operators New vacuum and the shifted field operators New vacuum and the shifted field operators General case: the approximate vacuum General case: the approximate vacuum General case: the approximate vacuum General case: perturbative expansion General case: perturbative expansion General case: perturbative expansion General case: perturbative expansion General case: perturbative expansion General case: perturbative expansion General case: the Bogolyubov transformation General case: the Bogolyubov transformation General case: the Bogolyubov transformation Detail: the mean-field for a homogeneous system Detail: the mean-field for a homogeneous system Detail: the mean-field for a homogeneous system Detail: the mean-field for a homogeneous system III. broken symmetry and quasi-averages Zero temperature limit of the grand canonical ensemble Degenerate ground state Degenerate ground state Rovnovážná struktura molekul AB[3 ]Equilibrium structure of the AB[3] molekules Equilibrium structure of the AB[3] molekules Equilibrium structure of the AB[3] molekules Broken continouous symmetries in extended systems Bose condensate - degeneracy of the ground state Symmetry breaking – removal of the degeneracy Symmetry breaking – removal of the degeneracy How the symmetry breaking works – ideal BE gas How the symmetry breaking works – ideal BE gas How the symmetry breaking works – ideal BE gas The end