Cold atoms Lecture 5. 14^th November, 2007 Preliminary plan/reality in the fall term Recapitulation Offering many new details and alternative angles of view Operators Operators Hamiltonian Hamiltonian Hamiltonian On symmetries and conservation laws Hamiltonian of a homogeneous gas Hamiltonian of a 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 Bogolyubov transformation Bogolyubov transformation Bogolyubov transformation Bogolyubov transformation Bogolyubov transformation 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 What next ??? • Off-diagonal long range order and the Bogolyubov ground state • Coherent state as the GP vacuum • Spontaneous symmetry breaking Off-Diagonal Long Range Order Analysis of BEC on the one-particle level ODLRO as a measure of coherence in the system OPDM for homogeneous systems OPDM for homogeneous systems: ODLRO OPDM for homogeneous systems: ODLRO OPDM for homogeneous systems: ODLRO From OPDM towards the macroscopic wave function From OPDM towards the macroscopic wave function From OPDM towards the macroscopic wave function ODLRO for interacting bosons ODLRO for interacting bosons ODLRO for interacting bosons ODLRO for interacting bosons ODLRO for interacting bosons ODLRO for interacting bosons 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 New vacuum and the shifted field operators 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 More about the approximate vacuum 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