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 Gallipoli, 17 June 2008

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

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

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

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

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

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 …

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. A 75, 053611 (2007)‏

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

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. A 11, 265 (1975)‏

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

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

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

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

Comparison with the numerical ground-state

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

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)

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

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

Dimensionless variables

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