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Shaking the BEC

Lecture : Dynamics of a quantum gas 5.1 Collective modes of a trapped quantum gas 5.2 Sound 5.3 M easuring Bogoliubov excitations 5.4 Solitons 5.5 Quantized Vortices - creating and observing vortices - vortex lattice - Critical Rotation. Shaking the BEC.

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Shaking the BEC

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  1. Lecture:Dynamics of a quantum gas5.1Collective modes of a trapped quantum gas 5.2 Sound 5.3 Measuring Bogoliubovexcitations 5.4 Solitons5.5Quantized Vortices - creating and observing vortices - vortex lattice - Critical Rotation

  2. Shaking the BEC Quadrupole oscillations “Non-destructive” observation of a time-dependent wave function 5 milliseconds per frame Sloshing motion 10 msec. per frame a very sensitive measurement tool: any change in the potential will change the oscillation frequency application: Atom-Surface interaction, Van deer Waals and Casimir Polder interaction

  3. Shaking the BECtemperature and density dependence

  4. Scissors ModeO.M. Marago et al. Phys. Rev. Lett. 84, 2056 - 2059 (2000)

  5. Sound propagationM.R. Andrews PRL79, 553 (1997) Sound = propagating density perturbations

  6. Bogoliubov Excitation Spectrum high energies: particle like excitations coeff of Bogoliubov transf: cross over low energies: excitations: sound waves sppeed of sound: c coeff of Bogoliubov transf:

  7. Bragg Spectroscopy

  8. Bragg Spectroscopy

  9. Excitation Spectrum of a Bose-Einstein Condensate PRL 88, 120407 (2002) Dynamic structure factor S(k,w):response to excitation Static structure factor S(k):Fourier transform of the density correlation function

  10. Solitons Soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed GP equation: Soliton solutions for repulsive interaction: dark soliton stationary solution moving solution velocity u is related to the density ratio nmin/n0 change of phase though the soliton soliton solutions for attractive interaction: bright soliton self bound states localized in space

  11. Solitons John Scott Russell (Scottishengineer1808-1882) September 1844: ``I was observingthemotionof a boatwhich was rapidlydrawnalong a narrowchannelby a pair ofhorses, whentheboatsuddenlystopped - not so themassofwater in thechannelwhichithadput in motion; itaccumulatedroundtheprowofthevessel in a stateofviolentagitation, thensuddenlyleavingitbehind, rolledforwardwithgreatvelocity, assumingthe form of a large solitaryelevation, a rounded, smooth and well-definedheapofwater, whichcontinueditscoursealongthechannelapparentlywithoutchangeof form ordiminutionofspeed. I followedit on horseback, andovertookit still rolling on at a rate ofsomeeightorninemiles an hour, preservingits original figuresomethirtyfeetlongand a footto a footand a half in height. Itsheightgraduallydiminished, and after a chaseofoneortwomiles I lost it in thewindingsofthechannel. Such, in themonthof August 1834, was myfirstchance interview withthatsingularandbeautifulphenomenonwhich I havecalledthe Wave of Translation''. • solitonsassolutionsofnonlinear differential equationswhich • representwavesof permanent form; • arelocalised, so thattheydecayorapproach a constantatinfinity; • caninteractstronglywithothersolitons, but theyemergefromthecollisionunchanged apart from a phase shift. • Manyexactly solvable modelshavesolitonsolutions, includingtheKorteweg-de Vries equation, thenonlinear Schrödinger equation, thecouplednonlinearSchrödingerequation, andthesine-Gordon equation. • The mathematicaltheoryoftheseequationsis a broadandveryactivefieldofmathematicalresearch.

  12. Dark Solitons in a BEC

  13. Excitation Spectrum

  14. Structure of a Vortex axial symmetry: cylindrical coordinates from he velocity field we get an additional term in the energy cylindrical GP equation for f scaled variables uniformmedium approximate solution numerical solutionenergy per vortex

  15. Making Vortices

  16. Vortex Formation vortices are created by surface instabilities: Landau criterion: above a critical velocity, the flow at the surface becomes turbulent and breaks apart into vortices

  17. 3-dim Structure

  18. Large number of Vortices

  19. Crystallization of the Vortex lattice

  20. Fast RotationQuantum Hall states in rotating BEC Rotating harmonic trap: transform into the co rotating frame formally equivalent to a particle with charge q*moving in an effective magnetic field B* the Hamiltonian is there for reminiscent to the Quantum Hall effect: filling factor n To achieve these states the rotation frequency ahs to be ~ trap frequency Then the centrifugal potential exactly compensates the trapping potential -> looks like a free 2d system with a coupling to vector potential like in electro magnetism

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