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 • g has finite range for  > 1 • Box:  =D/2 no BEC for D≤2 • H.O.  =D no BEC for D≤1…..

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…

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

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

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

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

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

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!....

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

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

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

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

BONUS PROBLEM Derive the momentum distribution

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

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

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

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

Results

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

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

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

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

Questions?(Quinten?)

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