1. One-Dimensional Bose Gases with N-Body Attractive Interactions
International Workshop – Nonlinear Physics: Theory and Experiment V
Gallipoli June , 2008
One-Dimensional Bose Gases
with N-Body Attractive Interactions
(Phys. Rev. A 77, (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: Bosonsh
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. A 75, (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. A 11, 265 (1975)‏

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

12. 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:

13. the normalization gives
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. Vakhitov-Kolokolov criterion:
Stability analysis
Vakhitov-Kolokolov criterion:
However, for
it is
for each