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One-Dimensional Bose Gases with N-Body Attractive Interactions

International Workshop – Nonlinear Physics: Theory and Experiment V Gallipoli June 12 -21, 2008 . One-Dimensional Bose Gases with N-Body Attractive Interactions. (Phys. Rev. A 77, 053608 (2008)). E. Fersino (SISSA, Trieste) ‏. In collaboration with Giuseppe Mussardo & Andrea Trombettoni.

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One-Dimensional Bose Gases with N-Body Attractive Interactions

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  1. International Workshop – Nonlinear Physics: Theory and Experiment V Gallipoli June 12 -21, 2008 One-Dimensional Bose Gases with N-Body Attractive Interactions (Phys. Rev. A 77, 053608 (2008)) E. Fersino (SISSA, Trieste)‏ In collaboration with Giuseppe Mussardo & Andrea Trombettoni Gallipoli, 17 June 2008

  2. Outlook • Brief reminder on the physics of ultracold atoms • N-Body Interactions • Mean-field equations for 3-body (and N-body) attractive interactions: bright soliton solutions (ground-state for )‏ • Infinite degeneracy of the ground-state for the 3-body interaction at a critical value of the interaction • Effect of an harmonic trap and an optical lattice: stabilization of the 3-body (localized) ground-states • Soliton solutions for the repulsive 3-body interaction

  3. Quantum statistics and temperature scales kB TC =ħ ω (0.83 N)1/3 kB TF =ħ ω (6 N)1/3

  4. Interactions are tunable via Feshbach resonances • Atoms collide in open channel at small energy • Same atoms in different hyperfine states form a bound state in closed channel • Coupling through hyperfine interactions between open and closed channel • If two channels have different magnetic moment then magnetically tunable ΔΕ=Δμ B Resonance when bound-state and continuum become degenerate

  5. Trapped ultracold atoms: Bosons h Bose-Einstein condensation of a dilute bosonic gas Probe of superfluidity: vortices

  6. Ultracold bosons in an optical lattice Another scheme to change the strength of the interaction is to add an optical lattice Vext(r)=Vr (Sin2(kx)+Sin2(ky)+Sin2(kz)) Increasing V, one passes from a superfluid to a Mott insulator

  7. Trapped ultracold atoms Ultracold bosons and/or fermions in trapping potentials provide new experimentally realizable interacting systems on which to test well-known paradigms of the statistical mechanics: -) in a periodic potential -> strongly interacting lattice systems -) interaction can be enhanced/tuned through Feshbach resonances (BEC-BCS crossover – unitary limit) -) inhomogeneity can be tailored – defects/impurities can be added -) effects of the nonlinear interactions on the dynamics -) strong analogies with superconducting and superfluid systems -) quantum coherence / superfluidity on a mesoscopic scale -) quantum vs finite temperature physics …

  8. 3-body interaction can be induced and controlled ! Different schemes have been recently proposed to realize effective 3-body interactions [1,2] [1] H.P. Buchler, A. Micheli, and P. Zoller, Nature Phys.3, 726 (2007)‏ [2] B. Paredes, T. Keilmann, and J.I. Cirac, Phys. Rev. A75, 053611 (2007)‏

  9. N-Body Attractive Contact Interactions We consider an effective attractive 3-body contact interaction and, more generally, an N-body contact interaction: contact interaction N-body attractive (c>0)‏ With

  10. 2-Body Contact Interactions it is integrable and the ground-state energy E can be determined by Bethe ansatz: N=2 Lieb-Liniger model [3]: Mean-field works for [4]: is the ground-state of the nonlinear Schrodinger equation in order to have a finite energy per particle with energy [3] J. B. McGuire, J. Math. Phys. 5, 622 (1964)‏ [4] F. Calogero and A. Degasperis, Phys. Rev. A11, 265 (1975)‏

  11. N-Body Contact Interactions N>2no Bethe solution is available – in mean-field: in order to have a finite energy per particle

  12. Bright solitons (I)‏ To determine the brigth solitons we use a mechanical analogy: By quadrature:

  13. Bright solitons (II)‏ One obtains: the normalization gives

  14. Bright solitons (III)‏ is undetermined and then E=0: an infinite degeneracy parametrized by the chemical potential occurs!

  15. Comparison with the numerical ground-state

  16. External trap: Variational analysis (I)‏ Using a Gaussian as a variational wavefunction (having width ) for the ground-state one gets: kinetic interaction external potential instability for is on the critical line for

  17. External trap: Variational analysis (II)‏ ω≠ 0 In presence of an external trap Dα < 4 : stable Dα > 4 : stable iff c < c*(ω) c< c*(ω) D=1: α = 4 : stable iff c < c* Optical lattice at critical value: Variational analysis (III)‏ With an optical lattice VOL= ε cos (x) Dα < 4 : stable Dα > 4 : stable iff c*< c < c*(ε) D=1: α = 4 : stable iff c*< c < c** c**< c*(ε)

  18. Repulsive case: dark solitons No fine tuning of the interaction required for the 3-body repulsive interaction

  19. Conclusions & Perspectives • =4 (i.e., N=3!!) is critical in 1D • Infinite degeneracy of the ground-state at the critical interaction value • Role of the external trap and optical lattice in stabilizing localized ground-states for the 3-body interaction • In perspective, the possibility to induce and tune effective 3-body interaction could become an important tool to control the nonlinear dynamical properties of matter wavepackets and induce new strongly correlated phases in ultracold atoms

  20. Dimensionless variables

  21. Stability analysis Vakhitov-Kolokolov criterion: However, for it is for each

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