Bosonic Cold Atoms : from Continuum to Latticecmschool/coldatom/talks/venkat_1.pdf · •...

Venkat Pai

Bosonic Cold Atoms : ��from Continuum to Lattice

School on Physics of Cold Atoms February 10-14, 2014 @HRI

[email protected]

•  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





•  BEC in Cold Atoms : Effect of Trap Potential



•  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!

•  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 :

•  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)


•  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.


•  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


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)

•  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)






•  BEC : Effect of Interactions, Superfluidity


•  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)






•  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

Bosonic Cold Atoms : from Continuum to Latticecmschool/coldatom/talks/venkat_1.pdf · •...

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