AB INITIO SIMULATIONS OF COLLISIONS AB INITIO SIMULATIONS OF COLLISIONS AB INITIO SIMULATIONS OF COLLISIONS AB INITIO SIMULATIONS OF COLLISIONS AB INITIO SIMULATIONS OF COLLISIONS AB INITIO SIMULATIONS OF COLLISIONS OF HELIUM CONDENSATESOF HELIUM CONDENSATESOF HELIUM CONDENSATESOF HELIUM CONDENSATESOF HELIUM CONDENSATESOF HELIUM CONDENSATESOF HELIUM CONDENSATESOF HELIUM CONDENSATES

K é Kh t 1 P D 2 V K h l i ff3 J C J k l 3 G P t id 3 M B 3 D B i 3 C W tb k3Karén Kheruntsyan1, P. Deuar2, V. Krachmalnicoff3, J‐C. Jaksula3, G. Partridge3, M. Bonneau3, D. Boiron3, C. Westbrook3y , , , , g , , ,1ARC Centre of Excellence for Quantum Atom Optics University of Queensland Brisbane QLD 4072 Australia1ARC Centre of Excellence for Quantum‐Atom Optics, University of Queensland, Brisbane, QLD 4072, Australia2Institute of Physics Polish Academy of Sciences Al Lotników 32/46 02 668 Warsaw Poland2Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02‐668 Warsaw, Poland3Laboratoire Charles Fabry de l’Institut d’Optique Univ Paris‐Sud Campus Polytechnique 91127 Palaiseau Cedex FranceLaboratoire Charles Fabry de l Institut d Optique, Univ. Paris‐Sud, Campus Polytechnique, 91127 Palaiseau Cedex, France

2 Hamiltonian and positive P equations 3 Simulation parameters1 Motivation 2. Hamiltonian and positive‐P equations 3. Simulation parameters1. Motivation p q p2 • Total No of 4He* atoms in the initial BEC: 105• Interested in mechanisms of pair production of  )1(ˆˆˆˆˆˆ 2

3 02

UdH x Total No. of  He  atoms in the initial BEC: 10

I i i l di i BEC i h

p pparticles for quantum atom optics

)1(22

mdH x

P iti P ( h ) th d th t b d• Initial condition: pure BEC in a coherent state, particles for quantum‐atom optics

Positive‐P (phase‐space) method: the quantum many‐body  split into two counter‐propagating halves(pair correlated photons have been pivotal to dynamics, governed by the Hamiltonian (1), is simulated exactly

split into two counter propagating halves(p p pquantum optics; e g demonstrations of EPR )()(2/)()0( kk

dynamics, governed by the Hamiltonian (1), is simulated exactly via stochastic differential equations for c fields [2]:

quantum optics; e.g., demonstrations of EPR d d i l ti f B ll’ i liti )

)exp()exp(2/)()0,( 00 zikzikn xxvia stochastic differential equations for c‐fields [2]:

• Harmonic trap: 47/1150/1150 Hzparadox and violations of Bell’s inequalities) Harmonic trap: 47/1150/1150 Hz

R l ti lli i l it 2 2 7 36 /• Pair‐correlated atoms in BEC collisions have been ~),( 22itx • Relative collision velocity: 2v0=2 x 7.36 cm/sPair correlated atoms in BEC collisions have been detected in [1]

),()/(~)/(2

),(1

200

2 tUiUiit

xx

• Numerical lattice: 1024 x 48 x 112 lattice pointsdetected in [1]

)()()(2 100mt

Numerical lattice: 1024 x 48 x 112 lattice points

N b f d i l t d 5505024• Modified experiments are underway; the new ~~~),(~

22itx • Number of modes simulated: 5505024Modified experiments are underway; the new geometry gives better detection access

),(~)/(~)/(~2

),(2

200

2 tUiUiit

xx

• Positive‐P simulations run for 70 μsgeometry gives better  detection access

)()()(2 200mt

ξ ( t) i t ith

Positive P simulations run for 70 μs

N b f t h ti t j t 2000• First‐principles simulations are now possible for 

)()()()( )3( • ξi(x,t) – noise terms, with • Number of stochastic trajectory averages: 2000p p p

the entire collision duration (in contrast to [2]))()(),(),( )3( tttt ijji xxxx

• s wave scattering interaction • Computing time: 6‐7 days on 10 CPUSthe entire collision duration (in contrast to [2]) jj

maU /4 2• – s-wave scattering interaction Computing time: 6 7 days on 10 CPUS maU /40

Figure 3.Figure 3. Real‐space densityFigure 1Figure 1 Schematic diagram of

Figure 3.Figure 3. Real space density of the atomic cloud after

ii i di ib i ( P l f h )

Figure 1. Figure 1. Schematic diagram of th lli i t Th t

of the atomic cloud after Figure 2.Figure 2. Atomic momentum distribution (positive‐P simulation, average of 2000 stochastic trajectories):the collision geometry: The two  70 μs, showing the spatial Three orthogonal cuts through the origin after 70 μs collision duration by which time the collision hasdisks represent the colliding con‐

μ , g pseparation of the twoThree orthogonal cuts through the origin after 70 μs collision duration, by which time the collision has 

essentially ceased The (nearly) spherical shell of scattered atoms is clearly seen with the darker regions

disks represent the colliding condensates; the sphere represents

separation of the two llidi d tessentially ceased. The (nearly) spherical shell of scattered atoms is clearly seen, with the darker regions densates; the sphere represents  colliding condensates.

corresponding to the colliding condensates.  A more careful quantitative analysis reveals a surprise: the the scattered atoms. The cigar  p g g f q y pscattering shell renders itself as an ellipsoid (see Fig 4); the experiment also shows this!

gshaped initial condensate is scattering shell renders itself as an ellipsoid (see Fig. 4); the experiment also shows this!shaped initial condensate is h i th iddlshown in the middle.

St h ti H t F k B li b hStochastic Hartree‐Fock‐Bogoliubov approachg pp

• Apply the HFB scheme: )(ˆ)()(ˆ ttt xxx • Apply the HFB scheme:                                            ),(),(),( 0 ttt xxx ˆ• Re‐formulate the Heisenberg equations of motion for ),(ˆ txg q

φ),(

)(ˆ 2 φ)(ˆ)(2)(ˆ)(2

),( 2222

ttUttUt

i xxxxx

),(),(2),(),(22 0000 ttUttU

mti xxxx

according to the positive P method2mt

according to the positive‐P method

‐ easier to solve than the standard Bogoliubov method in 3Deasier to solve than the standard Bogoliubov method in 3D

can incorporate the mean field dynamics of the colliding‐ can incorporate the mean field dynamics of the colliding condensates [ψ0(x,t)] as the solution to the GP equation[ψ0( , )] q

agreement with the full positive P simulation!‐ agreement with the full positive‐P simulation!

• Scattered atoms move in a complicated mean‐field potential p plandscape of the separating and expanding condensatesFigure 4Figure 4 Slice of the atomic momentum distribution through the origin in the k k plane in polar co landscape of the separating and expanding condensates

δk k kFigure 4.Figure 4. Slice of the atomic momentum distribution through the origin in the kx-ky plane in polar co‐

• Shift δk=k0–ks in the radius of the sphere can be estimated asordinates (left panel); the respective radial densities along kx and ky axis (right panel) show the  0 s p( p ) p g x y ( g p )anisotropy of the distribution and its ellipticity (ε=1 03) The axis are in units of 22222222 nmUkkkkk /mvk anisotropy of the distribution and its ellipticity (ε=1.03). The axis are in units of                     . 

2200

000

00000 2

nmUknU

kknU

knU

k ss

/00 mvk ComparisonComparison Peak position  Peak width Peak density Nscattered

20

20

000000 2222 kknU

mmnU

mnU

m

Elli ti it t i l lid d t hill f

pptable:table:

p(units of k0) (units of k0)

y(arb. units)

scattered 00

• Ellipticity:  atoms moving along y slide down a steeper hill of table:table: (units of k0) (units of k0) (arb. units)along k k k k k k the mean‐field potential and regain the energy shift U0n0 as  along kx ky kx ky kx ky p g gy U0 0

kinetic energy (radius is back to k ) unlike the atoms along xTheory 0.95±0.01 0.98±0.01 0.08±0.005 0.10±0.005 1.40±0.1 1 1400±50kinetic energy (radius is back to k0), unlike the atoms along  x

Experiment 0.88±0.02 0.94±0.02 0.09±0.01 0.11±0.01 1.48±0.16 1 1600±200Experiment  0.88±0.02 0.94±0.02 0.09±0.01 0.11±0.01 1.48±0.16 1 1600±200

REFERENCESREFERENCES[1] Ob ti f At P i i S t F W Mi i f T C llidi B Ei t i C d t A P i[1] Observation of Atom Pairs in Spontaneous Four‐Wave Mixing of Two Colliding Bose‐Einstein Condensates. A. Perrin, 

H. Chang, V. Krachmalnicoff, M. Schellekens, D. Boiron, A. Aspect, and C. I. Westbrook , Phys. Rev. Lett. 99, 150405 (2007).C a g, ac a co , Sc e e e s, o o , spect, a d C estb oo , ys e ett 99, 50 05 ( 00 )[2] Atomic four wave mixing via condensate collisions A Perrin C M Savage D Boiron V Krachmalnicoff C I Westbrook[2] Atomic four‐wave mixing via condensate collisions. A. Perrin, C. M. Savage, D. Boiron, V. Krachmalnicoff, C. I. Westbrook, 

h h ( )K. V. Kheruntsyan, New J. Physics 10, 045021 (2008).