Cold atoms Lecture 2. 10. October 2007 BEC for independent particles Two basic models: BEC in an ideal gas vs. in a trapped atomic cloud Problems with thermodynamic limit BEC for independent particles LAST TIME Ideální kvantové plyny Ideal quantum gases at a finite temperature Ideal quantum gases at a finite temperature A gas with a fixed average number of atoms A gas with a fixed average number of atoms A gas with a fixed average number of atoms A gas with a fixed average number of atoms A gas with a fixed average number of atoms A gas with a fixed average number of atoms Condensate concentration Plane waves in a cavity Density of states Condensate concentration Condensate concentration Condensate concentration From a gas to an inhomogeneous system Physical interpretation of BEC Where are the condensate Bosons? LAST TIME Digression: simple interpretation of T[C ]LAST TIME Digression: simple interpretation of T[C ]LAST TIME What is the nature of BEC? LAST TIME What is the nature of BEC? Where are the condensate atoms? Where are the condensate atoms? Where are the condensate atoms? Continuation Where are the condensate atoms? Continuation Where are the condensate atoms? Summary Thermodynamics of BEC Capsule on thermodynamics Grand canonical ensemble Thermodynamic functions of an ideal gas BEC in an ideal gas Homogeneous one component phase: boundary conditions (environment) and state variables Homogeneous one component phase: boundary conditions (environment) and state variables Digression: which environment to choose? Digression: which environment to choose? Grand canonical ensemble Definition following Gibbs General treatment for independent particles Thermodynamic functions of an ideal gas Grand canonical ensemble - definition Grand canonical ensemble – general definition Fluctuations I. – global quantities Grand canonical statistical sum for independent bosons Grand canonical statistical sum for independent bosons Non-interacting bosons in a trap Useful digression: energy units LAST TIME Trap potential LAST TIME Trap potential LAST TIME Ground state orbital and the trap potential LAST TIME Ground state orbital and the trap potential LAST TIME Ground state orbital and the trap potential Filling the trap with particles: IDOS, DOS Filling the trap with particles Filling the trap with particles Exact expressions for critical temperature etc. How good is the thermodynamic limit How good is the thermodynamic limit How good is the thermodynamic limit How good is the thermodynamic limit How sharp is the transition The end