Venkat Pai
Bosonic Cold Atoms : ���from Continuum to Lattice
School on Physics of Cold Atoms February 10-14, 2014 @HRI
• Lecture I : Bose-Einstein Condensation (BEC), BEC of Cold Atoms in Harmonic Traps, Effect of Interactions, and Gross-Pitaevskii Equation
• Lecture II : More on Gross-Pitaevskii Equation, Optical Lattices, and Mott Insulator - Superfluid Transition
• Lecture III : More on Lattice Bosons, Long-Range Interactions, Quantum Simulators, and Strong Coupling Theory
Bosonic Cold Atoms : from Continuum to Lattice
• Slides will be available at : http://www.hri.res.in/~cmschool/coldatom/
Plan of Lectures
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap
• BEC : Effect of Interaction, Suprfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Description
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap Potential
• BEC : Effect of Interaction, Suprfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Description
• Consider non-interacting particles in thermal equilibrium
• Two characteristic length scales : Inter-particle separation (depends on density) and thermal de-Broglie wavelength (depends on temperature)
Classical versus Quantum Regimes : Density and Temperature
Classical :
Quantum :
• High temperature, low density : Classical
• Low temperature, high density : Quantum
Classical versus Quantum Particles : Uncertainty Principle and Indistinguishability
• Classical particles : The state described by positions and momenta in phase space (can be accurately measured). Particles are distinguishable
• Quantum particles : The state described by wavefunctions. Position and momentum cannot be simultaneously measured accurately. Uncertainty relation leads to indistinguishability of particles. They are identical.
• When two identical particles scatter off each other, they cannot be identified separately if uncertainty in position is larger than minimum separation during collision
• There is no observable change in the system when they are exchanged
Exchange Symmetry of Wavefunctions
• Two particle wavefunction :
• denote positions of particles 1 and 2.
• Observable is the probability density
• This should be invariant under particle exchange
• Hence,
• If we insist that two exchanges lead back to original wavefunction,
• Hence, can have only symmetric or antisymmetric wavefunctions under any two particle exchange
Exchange Symmetry of Wavefunctions : Bosons versus Fermions
• Symmetric wavefunctions : Bosons
• Anti-symmetric wavefunctions : Fermions
• Coordinates of particles :
• Quantum states ( e.g., energy eigenstates) :
• Two Fermions cannot be in the same state : Pauli Exclusion Principle
• Two or more Bosons, however, can be in the same state : leads to BEC
Bosons versus Fermions : Occupation of Quantized Energy Levels
• 2 Fermions in 3 states
• 2 Bosons in 3 states
• 2 Classical (MB) in 3 states
• Due to exchange symmetry, number of allowed states for a quantum system is different from that of classical and that makes a difference in partition function!
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap Potential
• BEC : Effect of Interaction, Suprfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Description
• Consider a grand canonical ensemble of bosons (conserved number) to be distributed among various single particle energy levels at temperature , and chemical potential
Bosons : Grand Canonical Partition Function
• Partition function involves sum over various quantum states (here, the Fock space states)
• For free particles
Bosons : Distribution Function and Average Energy
• Number of particles
• Average Energy
• Distribution function is sharply peaked at low energies
• At low temperatures, if , state with has infinite population
• States with have “negative” population
Bosons : Problem with the Chemical Potential
• Chemical potential, at best, can be as large as the lowest single particle energy; in the thermodynamic limit, it cannot be positive
• “Non-interacting” He atoms
BEC Heuristics
• What is the chemical potential so that ground state population is comparable to “almost” entire number of particles (Avogadro number)?
• Macroscopic occupation of the ground state :
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap Potential
• BEC : Effect of Interaction, Suprfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Description
• Free (non-interacting) Bosons in three dimensions : Introducing density of states
Bose-Einstein Condensation : Theory
• Number Density
• For small , chemical potential large and negative (classical limit)
Bose-Einstein Condensation : Theory
• As temperature decreases, fugacity approaches 1 and chemical potential to zero at some characteristic temperature
• Below this temperature, we cannot use
• The lowest energy state carries a macroscopic population and shields all excited states from too close an approach to the chemical potential
• Bosons start condensing to the ground state, with a finite fraction (that approaches 1 at absolute zero temperature) occupying the lowest state.
Bose-Einstein Condensation : Theory
• The rest of the bosons distribute among the excited states (at any non-zero temperature) leading to coexistnce of a “condensate” and “normal bosons”
• We must single out the ground state occupation separately, while the population of all other states can be summed up
• Condensate density increases with decreasing temperature
• Since
Bose-Einstein Condensation : Theory
Occupation of excites states
• Specific heat is continuous across the transition, but with a cusp at
• Coexistence of two phase : liquid/vapor or condensates/normal boson
• Liquid-Gas : Condensation occurs in real space
• BEC : Condensation occurs to the lowest energy states (for free particles in momentum space)
• Liquid-Gas : Transition determined by Interactions
• BEC : (possibly!) the only free particle system where a thermal phase transition takes place, determined entirely by quantum statistics. Transition temperature determined by competition between inter-particle separation and thermal de-Broglie wavelength
BEC and Liquid-Gas Transition
Classical Particles to BEC (Thermal Evolution): Towards Macroscopic Quantum Wavefunction
Ketterle (2002)
• Spin polarized Hydrogen
• Excitons, Polaritons
• Normal-Superfluid transition in
BEC Search : Possible Systems
Number density
• Ultracold Atoms, Possibly Cavity Photons!
Annett (2004)
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap Potential
• BEC : Effect of Interaction, Suprfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Description
• Atoms are trapped and confined in a harmonic potential
BEC in harmonic traps
• Alters the enrgy levels and density of states
BEC in harmonic traps
• When
• Scaling argument
Townsend et al. (1997)
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap Potential
• BEC : Effect of Interactions, Superfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Description
• Ground state Condensate Fraction depletes
Effect of Interactions
• System develops Superfluidity (flows without viscosity, upto a critical velocity)
• Energy spectrum changes dramatically; System to be described in terms of quasiparticles with modified dispersion
• Consider a particle of mass injected into a fluid with a velocity
Landau criterion for Superfluidity
• What is the nature of the quasiparticle dispersion so that the test particle does not create excitations in the fluid (that lead to dissipation)?
• Minimum value of which satisfies this equation (when )
• If quasiparticles have free-particle like dispersion, i.e.,
Landau criterion for Superfluidity (contd.)
• Observed critical velocities are smaller than
• If quasiparticles have phonon-like like dispersion, i.e.,
• A different kind of excitations at larger : rotons
Phonon Roton
Annett (2004)
Theory of Weakly Interacting Bosons
• Zero momentum condensate and small number of excitations
Theory of Weakly Interacting Bosons (contd.)
Theory of Weakly Interacting Bosons (contd.)
• New quasiparticle operator (Bogoliubov transformation)
Plan of the Talk - I
• Quantum Particles : Fermions vs. Bosons
• Bose Statistics : Partition Function, Chemical Potential
• Bose-Einstein Condensation (BEC)
• BEC in Cold Atoms : Effect of Trap Potential
• BEC : Effect of Interactions, Superfluidity
• BEC in Cold Atoms : Gross-Pitaevskii Decsription
• The ground state is a Macroscopic Condensate and has a Macroscopic Wavefunction
• What would be the effective Schrodinger equation satisfied by this state?
• This state is a Bosonic coherent state (more on that later)
Towards a description of the Condensate : Gross-Pitaevskii Equation
• Minimizing the energy in the SF ground state, gives
Towards a description of the Condensate : Gross-Pitaevskii Equation
• Chemical potential is introduced to maintain macroscopic, constant normalization of wavefunction
• Bose-Einstein Condensation in Dilute Gases : C. J. Pethick and H. Smith
• Superconductivity, Superfluids, and Condensates : J. F. Annett
• Bose-Einstein Condensation : L. P. Pitaevskii and S. Stringari
• Quantum Liquids : A. J. Leggett
References
Books
• Calculate the Partition Function for non-interacting Fermions
• Calculate the BEC transition temperature in two dimensions
• Calculate the specific heat as a function of temperature for free particle BEC in three dimensions
• If , show that, in general,
• Work out, in detail, the theory of weakly interacting Bosons
Problems
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