Phase Dislocations in Bose-Einstein Condensatesshakes76/Files/presentation-Honours.pdf · 2005-12-27 · Project Overview The main objective of the project was to numerically simulate

Phase Dislocations in Bose-Einstein Condensates


Shekhar S. Chandra1

October 22, 2005

1Supervisor: Assoc. Prof. Michael MorganShekhar S. Chandra Phase Dislocations in Bose-Einstein Condensates


Project Overview

The main objective of the project was to numerically simulateBose-Einstein Condensates (BECs) and study the Aharonov-BohmEffect in BECs. We also provided an unambiguous way toexperimentally retrieve the phase of the complex scalar fieldassociated with BECs for this result.

Shekhar S. Chandra Phase Dislocations in Bose-Einstein Condensates


Outline

1 IntroductionBose-Einstein CondensationGross-Pitaevskii Theory

2 Aharonov-Bohm (AB) EffectThe SetupThe Classical AnalogueThe Quantum AB Effect

3 Phase RetrievalGeneralised Gerchberg-Saxton (GGS) AlgorithmResults

4 SolitonsGross-Pitaevskii Theory for Two-Component BECsResults

5 Conclusion



Introduction

Bose-Einstein Condensation

Bose-Einstein Condensation

The de Broglie Wavelength isgiven as

λdB =h

p, (1)

where the thermal momentump =

√2mkbT



Introduction

Gross-Pitaevskii Theory

Gross-Pitaevskii Theory

This is the mean field theory that is encompassed in the TimeDependent Gross-Pitaevskii (TDGP) Equation, independentlydeveloped by Gross & Pitaevskii in 1961.

i~∂Ψ

∂t=

[− ~2

2m∇2 +

1

2mω2|r |2 + g |Ψ|2

]Ψ, (2)

where Ψ is the complex order parameter, m is the mass of theatom involved, ω is the angular frequency of the trap and g is theself-interaction co-efficient.



Aharonov-Bohm (AB) Effect

The Setup


Although the B field is zero outside the solenoid, the chargedparticles undergo path dependent phase changes due to thenon-zero vector potential in and outside the solenoid.

Figure: Aharonov-Bohm effect setup where the two paths (ABF & ACF)undergo different phase shifts.




The Classical Analogue


Berry et al. constructed a water wave analogue to theAharonov-Bohm Effect.

Figure: Aharonov-Bohm effect in water waves.






Berry et al. constructed a water wave analogue to theAharonov-Bohm Effect.

Figure: Aharonov-Bohm effect in water waves.




The Quantum AB Effect

Quantum Construct qC++ & Closing the AB Effect Loop

qC++ is a C++ Toolkit withObject Oriented Design torapidly develop QuantumMechanical Simulations,visualize them and is fullyopen source.




The Quantum AB Effect

The Quantum AB Effect - Results

Figure: The phase dislocations observed in the numerically simulatedBEC as an analogue to the Aharonov-Bohm Effect.



Phase Retrieval

Generalised Gerchberg-Saxton (GGS) Algorithm

Generalised Gerchberg-Saxton (GGS) Algorithm

The Generalised Gerchberg-Saxton Algorithm (iterative phaseretrieval technique) developed by Tan et al. (2003) wasimplemented for the study of the phase dislocations and solitons.



Phase Retrieval

Results

Results

Figure: The results of the GGS Phase Retrieval Technique of Tan et al.(2003) to nucleated vortices. Phase retrieval is valid only to the phases ofthe inner region for the above image.



Solitons

Gross-Pitaevskii Theory for Two-Component BECs

Gross-Pitaevskii Theory for Two-Component BECs

Each component is described by a TDGP equation withself-interaction and inter-component interaction terms, whichinclude examples of a mixture of two different bosonic atoms, suchas 41K & 87Rb, as well as bosons with different internal spin states.

i~∂Ψ1

∂t=

[− ~2

2m∇2⊥ +

1

2mω2|r |2 + g11|Ψ1|2 + g12|Ψ2|2

]Ψ1, (3)

i~∂Ψ2

∂t=

[− ~2

2m∇2⊥ +

1

2mω2|r |2 + g21|Ψ1|2 + g22|Ψ2|2

]Ψ2, (4)



Solitons

Results

Results

Solitons are spatially confined (localised) solutions to non-linearsystems. In other words, they are non-dispersive solutions withoutthe superposition principle.

Figure: The observation of dark soliton rings in one of the components ofa two-component BEC.



Conclusion

Conclusion

Phase retrieval can be used to study the exotic phase structures ofphase dislocations and solitons in BECs. Future work may include:

Using the phase retrieval technique to measure of theself-interaction co-efficient in BECs.

Using the phase retrieval technique to study the diverse rangeof topological defects in BECs.

The extension of the above into multi-component BECs andinto 3-dimensions.



Conclusion

Conclusion







Conclusion

Conclusion







Conclusion

Acknowledgements

I wish to thank my Supervisor Assoc. Prof. Michael Morgan for hisinsight, guidance and critique, without him this project would nothave been possible. Dr. Rotha Yu for his experience and expertisepertaining to the numerical work of the project. Dr. DavidPaganin for his insight and input into the project.


Documents

Phase Dislocations in Bose-Einstein Condensatesshakes76/Files/presentation-Honours.pdf · 2005-12-27 · Project Overview The main objective of the project was to numerically simulate