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System and definitions

System and definitions. er. In harmonic trap (ideal):. Dilute interacting Bosons. Single particle field operators:. Macroscopic occupation assumption:. Homogeneous result:. Dilute interacting Bosons. an operator!. Not an operator!. Inhomogeneous (time and space):.

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System and definitions

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  1. System and definitions er In harmonic trap (ideal):

  2. Dilute interacting Bosons Single particle field operators: Macroscopic occupation assumption: Homogeneous result:

  3. Dilute interacting Bosons an operator! Not an operator! Inhomogeneous (time and space): Single particle density matrix formalism: Time evolution of operator in Heisenberg Rep. Scattering theory (see ahead): Mean-field assumption – discard fluctuating part Time-Dependent Gross- Pitaevskii equation (TDGPE)

  4. A short review of scat. theory Eigenvalue scattering problem: Fourier Trans. Born Approx. Low k limit (“s-wave”) Indistinguishable particles… Effective potential!

  5. GPE – ground state properties Variational derivation + Energy functional Smallness parameter: Interaction energy: Kinetic energy: Weak interactions ≠ ideal gas behavior! (still small depletion, but strongly non-ideal)

  6. GPE – ground state properties TDGPE: Ansatz + normalization: TIGPE: Note: energy is not a good quantum number (nonlinear problem!)

  7. Numerical solution of TDGPE Imaginary time evolution: Interacting ground-state Non-interacting ground-state (Mean-field repulsion causes increase in Size)

  8. Thomas-Fermi approx. Neglect kinetic term: Relaxed T.F.

  9. Excitations – Bogoliubov equations Ansatz (plug Into TDGPE): Neglect terms of order u2, v2 and uv Bogoliubov equations (“linearized GPE”): Homogeneous system (u(r) and v(r) are plane waves):

  10. E(m) k(x-1) Homogeneous Bogoliubov spectrum Interaction vs. Quantum Pressure “healing length”

  11. Bragg Spectroscopy M. Kozuma, et. al., PRL 82, 871 (1999). J. Stenger, et. al., PRL 82, 4569 (1999).

  12. The Measured Excitation Spectrum (using Bragg spectroscopy) Liquid Helium (scaled for comparison)

  13. Phonon Region

  14. Superfluidity! Landau criteria: Interactions – lead to superfluidity! Superfluid velocity A few mm/sec in experimental systems!

  15. Many body theory (homogeneous) Assume macroscopic occupation of S.P. Ground state: Put in assumption + keep terms of order and The number operator is conserved – can be placed in

  16. Many body theory (homogeneous) Neglected: Bogoliubov Transform: Atomic commutation relations give:

  17. Many body theory (homogeneous) Eliminate off-diagonal third line: Convenient representation: Solution of quasi-particle amplitudes:

  18. Diagonalized Hamiltonian Energy spectrum: (again) Ground state is a highly non-trivial Superposition of all momentum states: Ground state energy:

  19. Quasi-particle physics Inverse transformation: Particle creation Particle Annihilation Low k limit Quasi-particle factors for repulsive condensates High k limit

  20. Quasi-particle physics ??? Don’t Forget Bosonic Enhancement!

  21. Quantum depletion of S.P. ground state Evaluate the non-single-particle component of the ground state at T=0 About 1% for “standard” experiments

  22. Attractive collapse! Complex energy – unstable to excitation! Finite size can save us (cutoff in Low k’s) Experimental values: A few thousand atoms!

  23. Structure factor and Feynman relation Static structure factor (Fourier transform of the density-density correlation function) T=0

  24. Feynman Relation Static Structure Factor • Measure of: • Response at k • Fluctuations with wave-number k

  25. Feynman Relation Excitation Spectrum of Superfluid 4He D. G. Henshaw, Phys. Rev. 119, 9 (1960). D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121, 1266 (1961).

  26. Landau k-q k Higher order – Beliaev and Landau damping AkqThe many-body suppression factor: Beliaev k-q k q q

  27. Thed function can be turned into a geometrical condition: Damping rate Fermi golden rule:

  28. Damping rate Excitations Impurities

  29. Points not covered - Inhomogeneous Bogoliubov theory - Beyond T=0 - Coherent collisions of excitations (FWM) - Hydrodynamic representation of GPE - Na3 ~ 1 – theory and experiment

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