Spontaneous topological defects in theformation of a Bose-Einstein condensate

Matthew Davis1, Ashton Bradley1,∗, Geoff Lee1, Brian Anderson2

1ARC Centre of Excellence for Quantum-Atom Optics, University of Queensland, Brisbane, Australia.2College of Optical Sciences, University of Arizona, Tuscon, USA.

∗Now at Jack-Dodd Centre for Quantum Technology, University of Otago, Dunedin, New Zealand.

Funding: Australian Research Council, National Science Foundation, University of Queensland.. – p.1

UQ theory 2 talks:Andrew Sykes: Force on a slow moving impurity due to quantum fluctuations in a 1D BEC (Wed 1250).

Joel Corney: Quantum dynamics of ultracold Fermions (Thu 830).

Ashton Bradley∗: Scale invariant thermodynamics of a toroidally Bose gas (Fri 1720).

Simon Haine: Generating number squeezing in a BEC through nonlinear interaction (Fri 1740).

UQ theory 2 posters:Andy Ferris: Detection of continuous variable entanglement without a coherent local oscillator

Geoff Lee: Coherence properties of a continuous-wave atom laser at finite temperature

Sarah Midgley: A comparative study of simulation methods for the dissociation of molecular BECs

Tania Haigh: Macroscopic superpositions in small double well condensates

Jacopo Sabbatini: Topological defect formation in 87Rb Bose Ferromagnet with quantum noise

Michael Garrett: Bose-Einstein Condensation in a Dimple Trap

Chao Feng: Mean-field study of superfluid critical velocity in a trapped Bose-Einstein condensate

UQ experiment:Erik van Ooijen: Macroscopic superpositions in small double well condensates

Leif Humbert: Towards an all-optical BEC in optical toroidal traps

Sebastian Schnelle: Ultra-cold atoms in a time averaged optical potential

. – p.2

Overview

1. Bose-Einstein condensation “phase transition” in a trap.

2. Finite temperature Bose gases.

3. Simulating condensate formation.

4. Observations of spontaneous vortices in BEC.

5. What causes spontaneous vortices in BEC?

6. Condensate formation in flat / elongated systems.

. – p.3

What is a Bose-Einstein condensate (BEC)?

It is a mesoscopic quantum many-body system:

• Typically 105 – 107 atoms in single translational quantum state.

• Matter equivalent of a single mode laser.

• Confined by lasers / magnetic fields in a vacuum.

• 100,000 times less dense than atmosphere.

• Form at temperatures around 100 nK.

• Size ≈ 100 µm.

To a good approximation:

• All atoms in a BEC share the same wave function.

How does it arise from a boring old incoherent thermal gas?. – p.4

The BEC phase transition

. – p.5

The BEC phase transition

0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

1.2

T / Tc

N0 /

N

. – p.6

Non-equilibrium finite temperature BEC

Challenge: is it possible to develop a practical non-equilibriumformalism for finite temperature Bose gases?

Desirable features:

• Can deal with inhomogeneous potentials.

• Can treat interactions non-perturbatively.

• Calculations can be performed on a reasonable time scale (sayunder one week).

• Application to condensate formation.

Possibilities:Positive-P method, ZGN formalism, quantum kinetic theory, 2PI . . .

. – p.7

Stochastic Gross-Pitaevskii methodProcedure: Split field operator into low- and high-energy pieces.

Ψ̂(x) = ψ̂(x) + φ̂(x).

Treat high-energy region usingquantum kinetic theory.

Treat low-energy region in trun-cated Wigner approximation —highly occupied, classical fieldmodes.

Essentially GPE approximation:ψ(x) = 〈ψ̂(x)〉.

ER

RNC

RC

{

{

Condensate Band

Non-Condensate Band

Beyond S-Wave

Position

Energy

ES

. – p.8

Stochastic Gross-Pitaevskii equation (SGPE)

dψ(x) = − i

~LGPψ(x) dt+ P

{

G(x)

kBT(µ− LGP )ψ(x) dt+ dWG(x, t)

}

.

First term — standard GPE (but with energy cutoff).

Second term — growth: coupling to thermal cloud described bychemical potential µ and temperature T .

• G(x) ≈ γkBT : collision rate (from quantum Boltzmann integral).

• dWG(x, t) ∝√γkBT : driving noise associated with growth.

Could solve a quantum Boltzmann equation for thermal cloud. . .. . . but instead we model bath dynamics with time dependent µ, T .

P : Project dynamics onto the low-energy basis (e.g. SHO states.)

. – p.9

Thermal equilibrium

Growth term causes SGPE to evolve any initial condition to thermalequilibrium for given µ, T .

Example: above Tc in an oblate harmonic trap.

Density slice (z=0) of classical region

x

y

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

Column density of classical region

x

y

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

Anderson lab parameters: (ν⊥, νz) = (7.8, 17.3) Hz, µ ≈ 0, T ≈ 60 nK.. – p.10

Simulating BEC formation

1. Begin with µi, Ti above critical point for BEC.

2. Sample initial state using ergodic evolution of SGPE.

3. Model evaporative cooling:⇒ Ramp µi → µf (up) and Ti → Tf (down).

4. Watch the condensate band rethermalise to new equilibrium.

5. Repeat a number of times (200).

6. Analysis: e.g. average over many trajectories to determinecondensate number N0 via Penrose-Onsager criterion.

. – p.11

Results

2 3 4 5 60

0.2

0.4

0.6

0.8

Time (s)

Vor

tex

prob

abili

ty

Expt result rangeExpt observation timesSGPE

0

1

2

3

4

5

6

N0 (

105 a

tom

s)

Expt measurementsSGPE

. – p.12

Experimental and numerical column densities

a

b

c

. – p.13

. – p.14

Effect of ramp of µ and T

Previous results for an instantchange in thermal cloudparameters:µ : 0 → 25 ~ω⊥

T : 54 nK → 38 nK

What happens for a finitetime ramp?

Dotted lines indicate the endof the ramp.

Vortex prob. fairly insensitivefor ramps up to ∆t = 4T⊥.

0

1

2

3

4

5

N0 (

105 a

tom

s)

a

0 1 2 3 40

0.2

0.4

0.6b

Time (s)

Vor

tex

prob

abili

ty

. – p.15

What is special about these simulations/experiments?

Nothing in particular!

The evaporative cooling procedure is a standard rf ramp. However:

• Final cooling is in a very weakly trapped “pancake” Bose gas —many classical modes ⇒ shorter correlation length at Tc.

• Extra-long time-of-flight imaging (59 ms) with magnetic levitation.

• Imaging axis is parallel to symmetry axis of the TOP trap.

David Hall also reports seeing spontaneous vortices in BECs.

Jean Dalibard said that occasionally they saw holes in condensates ina TOP trap.

What happens in a spherical trap?

. – p.16

What causes vortices to appear?

Initial noise? (Fluctuations in the initial condition.)

Dynamical noise? (Atoms entering/leaving the low-energy region.)

SGPE:

dψ(x) = − i

~LGPψ(x) dt+ P {γ(µ− LGP )ψ(x) dt+ dWG(x, t)} .

Noise correlations: 〈dWG(x, t)∗dWG(x′, t′)〉 = 2γkBTδ(x − x′)δ(t− t′)〉.

Numerical experiment:

1. Initial state noise only — set final temperature Tf = 0.

2. Dynamical noise only — start with 132 atoms in SHO ground state.

. – p.17

Results

2 3 4 5 60

0.2

0.4

0.6

0.8

Time (s)

Vor

tex

prob

abili

ty

Expt result rangeExpt observation timesFull SGPESGPE init. noiseSGPE growth noise

0

1

2

3

4

5

6

N0 (

105 a

tom

s)

Expt measurementsFull SGPESGPE init. noiseSGPE growth noise

. – p.18

Testing Kibble-Zurek mechanism

Kibble-Zurek mechanism predicts the scaling of the vortex densitywith the “speed” of the phase transition.

Experimentally: we need to make the BEC form faster.

How? Squash the pancake — increases collision rate.

Early results suggest a factor of 20 increase is possible.

Example movie.

. – p.19

What about cigar-shaped systems?

Regions merge to formdark solitons.

These are stable in aquasi-1D system.

Images from Peter Engels(Washington State Univ.)

(ν⊥, νz) = (400, 7) Hz.

. – p.20

Summary [C. N. Weiler et al., Nature, 455, 938 (2008).]

Spontaneous vortices have been observed in condensate formation,and have been modelled with a stochastic GPE.

We find vortex statistics in broad agreement with experiment.

Simulations suggest defects occur due to initial thermal fluctuations.

Outlook

More vortices in flat “pancake” systems.

Solitons occur in elongated “cigar” systems.

Test of the Kibble-Zurek mechanism in a Bose gas.

. – p.21

Matt’s arXiv selection

Weekly email that filters arXiv.org for you!

Includes all abstracts with relevance to the physics of ultra-cold atoms.

Typically 20–30 abstracts / week (from about 300).

Currently about 450 subscribers.

See http://www.physics.uq.edu.au/people/mdavis/ to sign up.

. – p.22