System and definitions

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System and definitions
er
In harmonic trap (ideal)
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Dilute interacting Bosons
Single particle field operators
Macroscopic occupation assumption
Homogeneous result
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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)
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A short review of scat. theory
Eigenvalue scattering problem
Fourier Trans.
Born Approx.
Low k limit (s-wave)
Indistinguishable particles
Effective potential!
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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)
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GPE ground state properties
TDGPE
Ansatz normalization
TIGPE
Note energy is not a good quantum number
(nonlinear problem!)
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Numerical solution of TDGPE
Imaginary time evolution
Interacting ground-state
Non-interacting ground-state
(Mean-field repulsion causes increase in Size)
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Thomas-Fermi approx.
Neglect kinetic term
Relaxed
T.F.
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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)
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Homogeneous Bogoliubov spectrum
Interaction vs. Quantum
Pressure
healing length
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Bragg Spectroscopy
M. Kozuma, et. al., PRL 82, 871 (1999). J.
Stenger, et. al., PRL 82, 4569 (1999).
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The Measured Excitation Spectrum
(using Bragg spectroscopy)
Liquid Helium (scaled for comparison)
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Phonon Region
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Superfluidity!
Landau criteria
Interactions lead to superfluidity!
Superfluid velocity
A few mm/sec in experimental systems!
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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
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Many body theory (homogeneous)
Neglected
Bogoliubov Transform
Atomic commutation relations give
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Many body theory (homogeneous)
Eliminate off-diagonal third line
Convenient representation
Solution of quasi-particle amplitudes
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Diagonalized Hamiltonian
Energy spectrum (again)
Ground state is a highly non-trivial Superposition
of all momentum states
Ground state energy
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Quasi-particle physics
Inverse transformation
Particle creation
Particle Annihilation
Quasi-particle factors for repulsive condensates
Low k limit
High k limit
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Quasi-particle physics
???
Dont Forget Bosonic Enhancement!
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Quantum depletion of S.P. ground state
Evaluate the non-single-particle component of the
ground state at T0
About 1 for standard experiments
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Attractive collapse!
Complex energy unstable to excitation!
Finite size can save us (cutoff in Low ks)
Experimental values A few thousand atoms!
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Structure factor and Feynman relation
Static structure factor (Fourier transform of the
density-density correlation function)
T0
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Static Structure Factor

• Measure of
• Response at k
• Fluctuations with wave-number k

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Excitation Spectrum of Superfluid 4He
D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121,
1266 (1961).
D. G. Henshaw, Phys. Rev. 119, 9 (1960).
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Higher order Beliaev and Landau damping
Akq The many-body suppression factor
Beliaev
k-q
k
q
q
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Damping rate
Fermi golden rule
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Damping rate
Excitations
Impurities
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Points not covered
- Inhomogeneous Bogoliubov theory - Beyond T0 -
Coherent collisions of excitations (FWM) -
Hydrodynamic representation of GPE - Na3 1
theory and experiment