Kolizje kondensatów BEC i symulacja

mikroskop owejijnej dynamiki kwantowej

Piotr Deuar(IF PAN)

FUW, 8 Października 2009

Collaborations on the topic:

Chris Westbrook (Institut d’Optique)Denis BoironValentina KrachmalnicoffJean-Christophe JaskulaMarie BonneauGuthrie Partrigde

Marek Trippenbach (Warsaw University)Paweł Ziń (IPJ)Jan Chwedeńczuk (INFM-BEC, Trento)Karen Kheruntsyan (Uni. Queensland )Aurelién PerrinVanessa Leung (Uni. Virginia)

Issues I will try to touch on

• Some similarities and differences between ultracold atoms and quantum optics

• When is a mean-field description of BECs inadequate?

• How BEC collisions are an example of this inadequacy

• What BEC collision experiments look like

• How to go beyond a mean field description in a “straightforward” way

• Four wave mixing with atoms vs. with photons

• The growth of spontaneous condensates: “phase grains” (maybe)

Common description of a “BEC” – the HamiltonianBoson field operators Ψ̂(x).

Ĥ =Z

dx Ψ̂†(x){−~

2∇2

2m+V(x)+

g2

Ψ̂†(x)Ψ̂(x)}

Ψ̂(x)

– In terms of mode occupations for a grid with mesh grid volume ∆V

Ψ̂(x) ∼ âx√∆V

; [âx, â†y] = δxy

– Get something that looks quantum-optics like

Ĥ = ∑x

{â†x H0 âx+χ

(â†x

)2â2x

}

with χ = g/2∆V, and H0 = −~2/2m∇2+V(x).

– Similar to Kerr χ(3) nonlinearity.

Common description of a “BEC”:mean field GP equation

The Boson field Ψ̂(x) is approximately a macroscopic wavefunction Ψ(x) for asingle mode occupied by all N atoms.

Ψ(x) Obeys the (superfluid) Gröss-Pitaevskii (GP) equation:

i~∂Ψ(x, t)

∂t=

{−~

2∇2

2m+V(x)+g|Ψ(x, t)|2

}Ψ(x, t)

What does the GP description miss?

1. Incoherent atoms

2. Supersonic effects— above the speed of sound c(x) =

√gn(x)/m, motion is no longer superfluid.

3. Details on scales smaller than the healing length ξ(x) = ~/mc√

2.

Example where this lack is serious: supersonic BEC collisions.

Supersonic BEC collision

Schematic in x-space:

original condensate

atoms scattered into an ≈ spherical shell

second condensate produced by Bragg optical transition

2vQ

Lasers in x-space View in k-space

From A. Perrin et al.,

PRL 99 150405 (2007)

Experimental examples

From A.P. Chikkatur et al., PRL 85, 483 (2000).

Integrated density

From J.M. Vogels et al.,

PRL 89, 020401 (2002).

(b)

z/xr

−2 0 22

From N. Katz et al., PRL 95, 220403 (2005).

Why is mean field no good here?

i~∂Ψ(x, t)

∂t=

{−~

2∇2

2m+V(x)+g|Ψ(x, t)|2

}Ψ(x, t)

In the halo, initial condensate field Ψ(x,0) is zero, and so stays that way.

(It’s a spontaneous process initially)

( Simulations to be described below )

Metastable He ∗ experiment

Uses multi-channel plate – Allows mapping of 3D atom distribution

From M. Schellekens et al.,

Science 310, 648 (2005).

After long time of flight,n(x, t) → n(k,0)

From A. Perrin et al., PRL 99 150405 (2007)

Allows measurement of HBT correlations - density g(2)

g(2)(x,y) = 〈â†xâ

†yâyâx〉

〈n̂(x)〉〈n̂(y)〉

Hanbury Brown-Twiss short-rangecorrelations g(2)(z,z+ ∆z) in a singlecloudSchellekens et al., Science 310, 648 (2005)

in a Boson He4 and a Fermion He3

cloud. Note how healing lengthξ(x) = ~/mc

√2 ∝ 1/m

Jeltes et al., Nature 445, 402 (2007)

Density correlations in the scattered halo

g(2)(k,k′) =〈n̂(k) [n̂(k′)−δkk′]〉

〈n̂(k)〉〈n̂(k′)〉

From A. Perrin et al., PRL 99 150405 (2007)

Back-to-Back k′ ∼−k(pairing)

Collinear k′ ∼ k(HBT)

Another interesting issue – Phase grains

Coherence: g(1)(k,k+δk) =〈Ψ̂†(k)Ψ̂(k+δk)〉√〈n̂(k)〉〈n̂(k+δk)〉

– Locally coherent regions. |g(1)| ≫ 0 Norrie et al., PRL 94, 040401 (2005)– Scattering rate into such a coherent region with n atoms is ∝ (1+n)→ Bose stimulation if n & 1 leads to rapid coherent growth of occupation of the phasegrain→ Mini condensates formed.

0 0.2 0.4 0.6 0.8 1 1.20

1

2radialradial

along collision

in halo

phase grainsize

0 0.2 0.4 0.6 0.8 1 1.20

100

200

phase grainoccupation

0 0.5 1 1.50

0.5

1

1.5

BEC overlap

Simulation beyond mean field: Bogoliubov Hamiltonian

1. Write Ψ̂(x, t) = φ(x, t)+ ψ̂B(x, t)

2. Substitute into full Ĥ =R

dx Ψ̂†(x){−~2∇22m +

g2 Ψ̂

†(x)Ψ̂(x)}

Ψ̂(x)

3. Assume ψ̂B(x, t) is orthogonal to φ(x, t).

4. Assume δN the number of particles contained in ψ̂B is ≪ N, the total number.

5. Remove terms of high order in δN/N (quantum depletion) from Ĥ to obtain ĤB

6. For later convenience, separate right- and left-moving condensates (velocities ≈±kC)into φ(x, t) = φL(x, t)+φR(x, t).

time-dependent Bogoliubov Hamiltonian

ĤB =Z

dx

{ψ̂†B

(−~

2∇2

2m

)ψ̂B K.E.

+2g|φ(t)|2ψ̂†Bψ̂B collective potential+2gφL(t)φR(t)(ψ̂†B)

2+h.c. halo pair production

+g[φL(t)2+φR(t)2

](ψ̂†B)

2+h.c.

}pair production near BECs

GP equations for condensates:

i~dφR(x, t)

dt=

{− ~

2

2m∇2+g

[|φR(x, t)|2+2|φL(x, t)|2+φ∗L(x, t)φR(x, t)

]}φR(x, t)

i~dφL(x, t)

dt=

{− ~

2

2m∇2+g

[|φL(x, t)|2+2|φR(x, t)|2+φ∗R(x, t)φL(x, t)

]}φL(x, t)

Cute and useful feature: can remove terms to see what process affects whatobservation.

A “TECHNICAL” DIFFICULTY:

• Experimentally realistic situations require 105−107 lattice points.

• Standard Bogoliubov quasiparticle evolution procedure requires diagonalization ofĤB, finding of eigenstates, etc. This is unlikely given the size of the space!

SOLUTION: Instead, the dynamics of ψ̂B can be treated stochastically using a methodstraight from quantum optics – the positive-P representation. . .

i~dψ1(x, t)

dt=

[− ~

2

2m∇2+2g|φ(x, t)|2

]ψ1(x, t)

= +gφ(x, t)2ψ2(x, t)∗+ i√

igψ1(x, t) ξ1(x, t)

i~dψ2(x, t)

dt=

[− ~

2

2m∇2+2g|φ(x, t)|2

]ψ2(x, t)

= +gφ(x, t)2ψ1(x, t)∗+ i√

igψ2(x, t) ξ2(x, t)

Here, ξ j(x, t) are independent Gaussian random variable fields with mean zero andvariances 〈ξi(x, t)ξ j(x′, t ′)〉 = δi j δ(x−x′)δ(t − t ′). And e.g. 〈Ψ̂B†Ψ̂B〉 = 〈ψ∗2ψ1〉stoch

Four wave mixing with ATOMS vs with PHOTONS

Z

XY

XY-plane

• Spontaneous scattering into empty “halo” modes as with light

• ATOMS: Halo radius NOT at expected |v| = v0 collision velocity, but smaller

• ATOMS: Halo ellipsoidal if condensates non-spherical

• ATOMS: Superradiance-like effect, but small.

Why is Halo radius smaller

It costs µ= 32gn(x) to remove a particle from the condensate (the mean-field energyfrom the repluslion of the remaining particles), but 2gn(x) to place one in a non-condensate mode. The energy balance is

~2k20

2m+

32

gn=~

2k2

2m+2gn

When the mean-field energy is removed from the Hamiltonian ĤB, the radius reverts tov0 (and ellipticity disappears)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

2

4

6

x 10−12

kx,y

/ k0s

lices

of p

uck

dens

ity n

(kx,

0) a

nd n

(0,k

y) [m

2 ]k 0

= 0

.707

1 v r

eck

0 = 0.707 k

recalong k

x: position = 0.939 s.d = 0.0925 peak = 6.67e−12

along ky: position = 0.982 s.d = 0.122 peak = 5.12e−12

Full calculation

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

2

4

6

8

x 10−12

kx,y

/ k0s

lices

of p

uck

dens

ity n

(kx,

0) a

nd n

(0,k

y) [m

2 ]k 0

= 0

.707

1 v r

ec

k0 = 0.707 k

recalong k

x: position = 0.985 s.d = 0.0884 peak = 7.9e−12

along ky: position = 0.985 s.d = 0.105 peak = 6.37e−12

No MF energy g|φ|2 terms

Why is the Halo an ellipse?

REASON 1: Particles can roll off the condensate to recover some or all of the lostmean-field energy ∝ gn(x).

BUT -

in the long condensate direction, this does nothappen because the density “falls out” from underthe particles as the condensates move away beforethe particle can roll far.

−100 −50 0 50 1000

1

2

3

4

5

x 1019

x [µm]

2ρ

0(x

,0,0

; t)

[m

−3]

t0=0

t1= 17.5 µs

t2= 35 µs

t3= 52.5 µs

t4= 70 µs

WHILE -in the short directions, a halo particle moves fast androlls off before the condensate can change much.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

19

y [µm]

2ρ

0(0

,y,0

; t)

[m

−3]

t0=0

t1=17.5 µs

t2=35 µs

t3= 52.5 µs

t4= 70 µs

4

0

1

2

3

0.97

0.5

0.36

kinetic energy gain = U0 x (0.36+0.97+0.5) x 10

19 m

−3

= U0 x 1.83 x 10

19 m

−3

= U0 x 0.76 x ρ

0(0)

δ ka/k

0=0.076

ks,y

=(0.95+0.076)k0=1.026k

0

Why is the Halo an ellipse?

REASON 2: Along the collision direction, halo particles become bogged in thepotential valley forming between the condensates (because they are slightly slowerdue to the halo radius shift mentioned before), and become deccelerated in this valley.

Why is the superradiance along the condensate small?

In standard photon-atom superradiance the atom can take on a lot of momentumbut little energy→ photon can be scattered at almost a right angle, and along long axis of BEC.

PHOTONATOM

in atom-atom superradiance all particles have the same mass, changing theallowed geometry → high-speed scattering along long axis of BEC not possible

Why do phase-grains elongate radially?

0 0.2 0.4 0.6 0.8 1 1.20

1

2radialradial

along collision

in halo

phase grainsize

0 0.2 0.4 0.6 0.8 1 1.20

100

200

phase grainoccupation

0 0.5 1 1.50

0.5

1

1.5

BEC overlap

• At late times only lagging tails of BEC wavepackets remain overlapping

• These produce scattered atoms with lower velocity

• Because significant phase space density has already built up in the largest phasegrains, they stimulate strong, coherent scattering out of the lagging BEC tailspreferentially next to the existing phase grain.

Thank you : )