Interaction between atoms

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1 Interaction between atoms MICHA SCHILLING HAUPTSEMINAR: PHYSIK DER KALTEN GASE INSTITUT FÜR THEORETISCHE PHYSIK III UNIVERSITÄT STUTTGART

2 Outline 2 Scattering theory slow particles / s-wave scattering Interaction in a dilute gas Gross-Pitaevskii Equation Bogoliubov Theory Higher order approximation quantum depletion

3 Scattering 3 Ansatz for scattering: : scattering amplitude Schrödinger s equation (spherical coordinates): : angular momentum : interaction potential

4 Asymptotic solution 4 stationary wave : phase of scattering amplitude: partial cross-section:

5 Slow particles scattering starting with (eq. from slide 3): 5 derive: expression for phase to calculate in the limit of slow particles low energies low temperatures ultra cold gases result:, and is small neglect everything but ( s-wave scattering)

6 Assumption 1 6 slow particles: large wavelength small : radius of action of the field : neglect - & -term

7 Assumption 1 7 slow particles: large wavelength small : radius of action of the field : neglect - & -term with the general solution:

8 Assumption 2 8 : neglect -term equation of free motion with the solution:

9 In the limit of slow particles, S-wave scattering 9 scattering amplitude: total cross section: replace full potential with effective potential, having the same s-wave scattering length

10 Effective potential 10 Pseudo potential: figure: schematic representation of the interatomic potential and the effective potential; from L.Pitaevskii, S. Stringari: Bose-Einstein Condensation, p.28

11 Scattering length of the effective potential pseudo potential gives scattering length : 11 first order (Born approximation): higher order:, because

12 Microscopic Lennard-Jones microscopic: L.J. potential has positive scattering length 12 for most gases: figure: Lennard-Jones potential from: University of Cambridge webside

13 Interaction in a dilute gas slow particles low energy particles low temperatures dilute gas: in BEC: replace interaction potential by pseudo potential valid for all scattering lengths here:, weak interactions use perturbation theory for interaction potential 13

14 Equation of motion Hamiltonian for interaction: (2 nd quantization) 14 Heisenberg representation timedependence of the field operators equation of motion:

15 Gross-Pitaevskii equation macroscopic occupation of one specific mode (BEC) replace with order parameter replace potential by pseudo potential: varies slowly in the range of interaction in the arguments of 15 Gross-Pitaevskii equation: with density of the gas

16 Application of G.P.equation 16 non-uniform gases in external fields BEC trapped in magnetic field describing vortices similar to theory of superfluidity (Ginzburg& Pitaevskii)

17 again Hamiltonian for interaction: Bogoliubov theory 17 transformation:

18 Bogoliubov Hamiltonian 18 with interaction potential:

19 Lowest order approximation replace:, neglect: & with and therefore: lowest order Born approximation: 19 ground state energy

20 Sound velocity of the BEC ground state energy:, with 20 pressure: compressibility: sound velocity:

21 Higher order approximation 21 lowest order neglect higher order just neglect quadratic terms of replace replace new Hamiltonian:

22 linear transformation: Bogoliubov transformation 22 with:

23 Dispersion law & ground state excited states of interacting Bose gas gas of non-interacting quasi-particles 23 ground state with higher order approximation: replace with in momentum space: (Lee and Young, 1957)

24 Elementary excitations Bogoliubov dispersion law for elementary excitations: 24 large momenta( ): small momenta( ): consistence: figure: Bogoliubov excitation spectrum: dispersion law for elementary excitations; from L.Pitaevskii, S. Stringari: Bose- Einstein Condensation, p.34

25 setting: at Healing length 25 characteristic interaction length: : thickness of the layer from a density of to

26 Occupation numbers occupation number of the quasi-particles: 26 occupation number of the particles:

27 Quantum depletion number of atoms in the condensate: 27 particle occupation number at K: integration at K: same parameter as corrections to the ground state energy

28 Conclusion 28 scattering with low energy particles: neglect everything but the s-wave scattering use an effective soft potential instead of the real interaction-potential Gross-Pitaevskii equation: describe a dilute gas in an external field at K lowest order approximation Bogoliubov Theory: lowest order approximation coincides with G.P.equation without external field higher order approximation: excitations of the condensate describes Quantum depletions

29 Thank you for your attention! questions?

30 30

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