BEC collisionsQuantum dynamics simulation in a macroscopic system

Piotr Deuar(LPTMS, Orsay)

ENS-Lyon, 2 Juillet 2009

Collaborations:

Gora Shlyapnikov (LPTMS )Chris Westbrook (Institut d’Optique)Vanessa LeungAurelién PerrinDenis Boiron

Karen Kheruntsyan (Uni. Queensland )Marek Trippenbach (Warsaw University)Paweł ZińJan Chwedeńczuk

To what degree is [ BEC = coherent matter wave ] ?

• Basic “canonical” description:N ≫ 1 atoms all in one orbital ( =⇒ needs low T ).

• Mean-field description is the coherent boson field Ψ(x, t).

• Dynamics “usually” has obeyed the mean field Gröss-Pitaevskii (GP) equation, withg = 4π~2as/m

i~∂Ψ(x, t)

∂t=

{−~

2∇2

2m+V (x)+g|Ψ(x, t)|2

}Ψ(x, t)

• similar to coherent laser field with Kerr χ(3) nonlinearity.

• Well, it’s been a good description, UNLESS:1. Motion is supersonic — speed of sound c(x) =

√gn(x)/m

2. OR, Details on scales smaller than the healing lengthξ(x) = ~/mc

√2 are important

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).

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 correlations - density g(2)

g(2)(k,k′) =〈n̂(k)n̂(k′)〉〈n̂(k)〉〈n̂(k)〉

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

Back-to-Back k′ ∼−k

Collinear k′ ∼ k

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.

k*k*k*

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.

( Simulations to be described below )

What about simple analytic models?

This has been done with Bogoliubov quasiparticles + extra simplifying assumptions.Works in special cases:

Limit of very fast collision

0 30 60 90 120 150 180θ[deg]

0

0.5

1

1.5

2

g1(θ)

g2(θ)

From P. Ziń et al., PRL 94 200401 (2005)

Moderate speed (ab initio simulation)

−15 −10 −5 0 5 10 150

1

2

3

4

5

vy [mm/s]

radial

axial tangential

t=253µ s t=404µ s t=632µ s

g+(2)

|g+(1)|

a

g−(2)

|g−(1)|

PD & P. Drummond, PRL 98 120402 (2007)

Short-time evolution

0 200 400 6000

1

2

3BB

CL

t [µs]

w(B

B/C

L)x

/w(S

)x

From:

M. Ögren & K.V. Kheruntsyan,

PRA 79 021606(R) (2009)

Phase grain formation – a complicated business

k*k*k*

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

Understanding the dynamics – The plan

Ĥ =Z

dxΨ̂†(x){−~

2∇2

2m+

g2

Ψ̂†(x)ψ̂(x)}

Ψ̂(x)

• Discretize the Hamiltonian onto a lattice : produces a Bose-Hubbard Hamiltonian

• Simulate the system numerically

• Pick a regime where the complicated business happens– Bose enhancement significant– Speed not too supersonic– Spherical condensates for simplicity

• Dissect the Hamiltonian term by term to see which process produces what.

Bogoliubov Hamiltonian

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

2. Substitute into full Ĥ

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. pair production

+g[φL(t)2+φR(t)2

](ψ̂†B)

2+h.c.

}off-resonant

φ(x, t) = φL(x, t)+φR(x, t)

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)

A “technical” difficulty

• Experimentally realistic situations require 105−107 lattice points.

• Standard Bogoliubov quasiparticle evolution procedure requires diagonalization,finding of eigenstates, etc.

• For a strongly evolving condensate as here — need to diagonalize at each timestep, and project old ψ̂B(x, t) onto new ψ̂B(x, t +∆t).

SOLUTION:

Instead, the dynamics of ψ̂B can be treated stochastically using the positive-Prepresentation. . .

Positive-P method

One writes the density matrix of the system on M lattice points in coherent states|ψ j(x)〉 = eψ j(x)ψ̂

†B(x)|0〉 as

ρ̂ = |Ψ〉〈Ψ| =Z

D2Mψ1(x)D2Mψ2(x) P(ψ1(x),ψ2(x), t) |ψ1(x)〉〈ψ2(x)|

– The distribution P(. . .) can be guaranteed non-negative real

– The complete quantum evolution of the state

i~∂ρ̂∂t

=[ĤB, ρ̂

]

is equivalent to the random walk of an ensemble of 2M random variables ψ j(x, t).

– Expectation values of observables are equivalent to ensemble averages of thevariables.

– Most complexity gets shoved into the ensemble, and hopefully averages out for mostquantities of interest.

Positive-P : evolution equations

GP + appropriate noise

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 jδ(x− x′)δ(t − t ′).

The condensate field φ(x, t) = φL(x, t) + φR(x, t) itself evolves according to theunchanged GP evolution from before

Positive-P : caveats• Noisy and limited precision ∝ √number of trajectories

• Amplification of noise can produce a signal-to-noise catastrophe after someevolution time.

Positive-P : benefits• Contains all two-body processes in the Hamiltonian

With some small print

• Complexity of the description grows very slowly — linear in lattice size/atom number!107 atoms or lattice points is fine to simulate on a PC.With some small print

• Uncertainty in results can be reliably computed from the ensemble

• Equations and their integration in t are only of similar complexity to the usual meanfield GP equations.

• Lucky bonus: for our Bogoliubov Hamiltonian ĤB, there is no noise catastrophe.

What does one see?

k*k*k*

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

t

rela

tive

g(

2)(k

*,−

k *)

g(2)(k*,−k

*) with 1σ error

quantum depletion(relative)

BEC overlap(relative)

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

QUESTIONS

– The dominant, resonant process is pair production. kC& − kC =⇒ k& − kSuch models give g(2)(k,−k) →≥ 2 (perfect pairs) at long times.– How can the other “weak” processes destroy the pairing – and by so much?

– Why do the phase grains become elongated radially?

– Why do they degrade after the collision is finished?

– Other fun issues: halo mean radius < kC, anisotropy in radius and width, evolutionafter apparent end of collision,. . .

Pairing loss mechanism 1Consider only rudimentary condensate evolution (stiff wavepackets)

idφL,R

dt = − ~2

2m∇2φL,R, but various ĤB

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

t

rela

tive

g(

2)(k

*,−

k *)

quantum depletion(relative)

BEC overlap(relative)

pairs only

pairs + φj2full H

Bpairs + φ

j2+mf potential

pairs + mf potential

ĤB = ψ̂†B(−~2∇22m

)ψ̂B +2gφLφR(ψ̂†B)2+h.c.

ĤB = ψ̂†B(−~2∇22m

)ψ̂B +gφ2(ψ̂†B)2+h.c.

ĤB = ψ̂†B(−~2∇22m

)ψ̂B +2g|φ|2ψ̂†Bψ̂B +2gφLφR(ψ̂†B)2+h.c.

ĤB = ψ̂†B(−~2∇22m

)ψ̂B +2g|φ|2ψ̂†Bψ̂B +gφ2(ψ̂†B)2+h.c.

k*k*k* k*k*k*

=⇒ After production, pairs can veer away from back-to-back alignment by rollingaround on the anisotropic collective mean field of the condensates

Pairing loss mechanism 2Consider only rudimentary Bogoliubov process (pairs+K.E.)

ĤB = ψ̂†B(−~2∇22m

)ψ̂B +2gφLφR(ψ̂†B)2+h.c., but various GP evolutions

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

t

rela

tive

g(

2)(k

*,−

k *)

quantum depletion(relative)BEC overlap

(relative)

full GPKE + shift + broadening + interf

KE + broadening + interference

KE + interferenceKE only

condensate velocity shift

i~dφRdt = − ~2m∇2φRi~dφRdt =

[− ~2m∇2+gφ∗LφR

]φR

i~dφRdt =[− ~2m∇2+g

(|φR|2+φ∗LφR

)]φR

i~dφRdt =[− ~2m∇2+g

(2|φL|2+ |φR|2+φ∗LφR

)]φR

k*k*k*k*k*k*

Not fully understood – clearly due to nonlinear evolution of BECs during the collision,and primarily due to mutual repulsion between wavepackets (Is the resulting anisotropyor the extra velocity kick more important?) Why is g(2) non-monotonic?

Pairing loss mechanism 3

There is some degradation of pairing even in the simplest pairing Hamiltonian withstiff wavepackets:

k*k*k*

Probable cause: finite wavepacket size → variable mean-field energy cost toproduce a pair depending on position→relaxation of requirements for pair to have zero total momentum → some atoms withk not necessarily paired with exactly −k.

Finally

• BECs display rich non-mean-field dynamics when beyond superfluidity

• Numerics can be used to gain physical understanding via a term-by-term dissectionof the dynamics.

• Bogoliubov dynamics can be simulated without the need to find any eigenstates (oreigenvalues).

• Related topics that are amenable to this approach:– Scattering around an impurity or sharp potential– Fast flow over disordered potential– Shocks introduced by rapidly varying potentials.