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From 1D “BEC” to the Tonks-Girardeau gas

From 1D “BEC” to the Tonks-Girardeau gas. Mark Awadalla Idse Heemskerk Gerben Schooneveldt Laura van der Noort. Overview. Is BEC possible in 1D How to go from 3D to 1D The exact solution to the 1D bose gas Experiments. To BEC or not to BEC?. N constant g has finite range BEC

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From 1D “BEC” to the Tonks-Girardeau gas

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  1. From 1D “BEC” to the Tonks-Girardeau gas Mark Awadalla Idse Heemskerk Gerben Schooneveldt Laura van der Noort

  2. Overview • Is BEC possible in 1D • How to go from 3D to 1D • The exact solution to the 1D bose gas • Experiments

  3. To BEC or not to BEC? • N constant • g has finite range BEC • g has finite range for  > 1 • Box:  =D/2 no BEC for D≤2 • H.O.  =D no BEC for D≤1…..

  4. To BEC or not to BEContinued • No BEC in D=1 in systems with finite compressibility • Hohenberg inequality: No BEC in D≤2 • And in general: no phase transitions in 1D So one might conclude that BEC is impossible in 1D…

  5. But…. • The exact solution shows that GP/MF results are correct in some regime • D=1 is a critical case anyway • Experiments confirm the results of the MF approach

  6. Becoming 1D step 1: putting them on a line • Take a very anisotropic harmonic trap: <<1 • Make • Define the 1D density • Now

  7. Becoming 1Dstep 2: kill the radial excitations • Thomas-Fermi: an1>>1 • In terms of the non-interacting states, many excited states are occupied: radial degree of freedom not frozen out • This looks 1D but isn’t: 3D cigar • So we must make an1<<1 for real 1D

  8. MF 3D vs. 1D: How to see the difference • From the 1D GP eq. you get for the cigar • And for the 1D MF

  9. The applicability of MF • MF/GP works when • 3D: • 1D: • So in 3D you need n small and in 1D big

  10. Beyond MF: an exact solution • 1D hardcore boson gas • This was first solved by E.H Lieb and W. Liniger in 1963 by means of a so-called Bethe ansatz. • In the limit we will recover our MF results! • In the limit something special will happen!....

  11. The Bethe ansatz:2 particle bose gas • A solution to this Hamiltonian is given by (check for yourself!;)) • And the ratio between the amplitudes can be written

  12. The Bethe ansatz:The Bethe equations • Now we quantize the momenta by periodic BC’s • This generalizes for N particles to the Bethe equations

  13. Bethe becomes Lieb-Liniger • Thermodynamic limit • Now you pretty much know everything. (Once you solve it. )

  14. Solutions • Solving for : MF results. (Identifying ) • Solving for : Fermionic behaviour! • Momentum density for will look like • And the normal density

  15. BONUS PROBLEM Derive the momentum distribution

  16. To summarize.. • 3D cigar (Thomas-Fermi) • 1D Mean Field • Tonks-Girardeau

  17. Experimental realization of a TG-gas: step 1 Nature, May 2004; Paredes, Widera, Murg, Mandel, Fölling, Cirac, Shlyapnikov, Hänsch, Bloch • 2D optical lattice • Harmonic potential in z-direction

  18. Experimental realization of a TG-gas: step 2 • Adding an optical potential in the z-direction • Varying the potential:  = Eint / Ekin ≈ 5-200

  19. The actual measurement • Removing all potentials • “Making a picture” after 16 ms • Averaging over the array of 1D gases • This gives you the momentum distribution

  20. Results

  21. Another way to reach TG-limit • Increasing interaction strength by decreasing ar

  22. Another optical trap Science, August 2004; Kinoshita, Wenger, Weiss • 2D lattice of 1D-condensates • Harmonic z-potential • Just a few particles per condensate. • No extra optical latice • =5

  23. Magnetic trap Next summer? Jan-Joris, et al. • Just one condensate, containing 103-104 particles • Box potential; potentially any potential • =5?

  24. Conclusion • A 3D BEC can effectively be made 1D • MF ceases to apply at a certain point • Luckily the 1D Bose gas is exactly solvable and we get predictions for that situation • Both cases and the transition between them have been measured. • Not all measurements are equally convincing

  25. Questions?(Quinten?)

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