Dark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate

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1 Dark & Bright Solitons in Strongly Repulsive Bose-Einstein Condensate Indu Satija, George Mason Univ & National Institute of Standard and Tech ( NIST) collaborator:radha Balakrishnan, Institute of Mathematical Sciences, Chennai, India Physical Review Letters: 103, , 2009

2 outline Soliton/solitary waves BEC Weakly interacting BEC and GPE ( NLSE) Dark solitons of GPE Strongly Interacting BEC: HCB ( in contrast to ferminization approach, use spin-coherent states to obtain equation for the order-parameter: non-gpe-type New exotic soliton for particle-hole Imbalance Soliton Dispersion Summary & Open Questions

3 Solitary Waves & Solitons A certain class of nonlinear equations support solitary waves/soliton solution Solitary waves: travelling wave, move in spendid isolation, holding their shape, size & velocity as it moves Solitons: after a collision of two solitary waves, if each retains its shape, size and velocity, such a solitary wave is called soliton. Therefore, soliton is a solitary wave with special collision property ( only their phase can change) In physics literature, a solitary wave is often referred as soliton, due to its particle-like behavior. Such solutions arise in a variety of fields such as hydrodynamics, plasma physics, magnetism... and also in BOSE EINSTEIN CONDENSATES

4 Discovered by John Scott Russel in 1834: he called them wave of translation:i was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation"

6 For dilute gas where range of interatomic forces << particle separation a: s-wave scattering length

9 Dark Soliton : absence of atoms that can propagate without changing shape in the condensate Solitons flatten out at sound velocity

11 Strongly nteracting BEC: Hard-Core Bosons

12 Hamiltonian: Extended Bose-Hubbard Order Parameter

13 Discrete-non-GPE: (HGPE)

14 Comparison with DNLSE (GPE)

15 Canonical Formulation

16 Hard Core Bosons With infinite onsite replusion ( HCB limit), Ψ S + = S x + is y η =< S + > Spin-Coherent States i τ i > = τ i > = τ > Define: τ i tan(θ i /2)e iφ i 1 1+ τ i 2exp[ τ is i 0 > τ i >= cos(θ i /2) + > +sin(θ i /2)e iφ i > η i = 1 2 sin(θ i)e iφ i ρ s = 1 4 sin2 (θ i ) ρ i = sin 2 (θ i /2) ρ s = ρ(1 ρ) ρ p ρ h

17 Glauber coherent states: ˆΨ(r) = 1 V k a k e ik.r a k α k > = α k α k > α k > = exp[ α k a k 0 > ψ(r) = 1/ V k α k e ik.r k α k > = α >= {Ψ} > ˆΨ(r) {Ψ} > = Ψ(r) {Ψ} > < ˆΨ ˆΨ > = < ˆΨ >< ˆΨ > The coherent states are normalized, are not orthogonal but form a complete set, 1/π d 2 α k α k >< α k = I

18 Two species of solitons

19 Analytic solution vs Numerical solution

20 Soliton profile for total density GPE-type non-gpe-type

21 Quarter-filling non-gpe-type non-gpe-type GPE-type Gpe-type Stationary soliton Soliton with velocity close to sound velocity

$fraction Two Species of$

22 Brightening of the soliton in condensate fraction Two Species of Solitons GPE-type Exotic: Immortal

23 As velocity changes from zero to sound velocity, kink transforms to a vortex

24 3/4-filling

25 half-filling Condensate soliton is identical to GPE

26 GPE-solitons in the condensate are dark and have Zero amplitude at sound velocity non-gpe-type in the condensate are some-what brightened and survive all the way upto sound velocity: Immortal Soliton non-gpe-type soliton exists provided the background density is different from half-filling

27 Above Sound Velocity

29 Solitary Waves or Solitons?? Bill Reinhardt has studied the collision properties of the dark solitary waves and shown that they are SOLITONS ( unpublished) Collision properties of bright solitons are under investigation

30 Summary New species of solitons has been predicted in strongly repulsive BEC Unlike GPE soliton, they brighten-up away from halffilling Unlike GPE solitons, they survive all the way upto sound velocity and then brekup into periodic train for velocities bigger than sound velocity. We refer them as immortal/ persistent soliton Immortal solitons are transformed into periodic trains above sound velocity The immortal solitons could have exotic collision properties

31 Open Questions?? Does HGPE provide useful insight into the dynamics of strongly interacting quantum many body BEC? Is the brightening of dark soliton a robust phenomena for very strongly interacting BEC? Relationship between particle-hole asymmetry and Immortal Soliton Immortal soliton travel with highest possible speed and retain their shape and hence can be a good candidate for quantum information propagation for fast information travel. Experimental realization

32 I hope it will not shock experimental physicists too much if I say that we donot accept their observations unless they are confirmed by theory,.. Sir Arthur Eddinton, 11 September 1933

Solitons in optical fibers. by: Khanh Kieu

Solitons in optical fibers. by: Khanh Kieu Solitons in optical fibers by: Khanh Kieu Project 9: Observation of soliton PD (or autocorrelator) ML laser splitter Er-doped Fiber spool 980nm pump PD (or autocorrelator) P 0 is the free parameter What