1. Atomic BEC in microtraps: Localisation and guiding
Markku Jääskeläinen

2. Sweden?

3. Topics centered around quantum dynamics in reduced dimensions
Quantum dynamics in guided matter waves
BEC in double well traps
Atomic gauge fields and spin-orbit coupling
Ring traps and gyroscopy

4. What are ‘Atom Chips’ ?
Micro-traps for manipulation of ultracold atoms (molecules)
Integrated optics with material particles

5. Why Matter-wave Chips ? Precision metrology & navigation Atomtronics
Molecular chips
Ultracold chemical reactions on chip
Low Dimensional condensed matter

6. How? Example: B-field + wire

7. How? Example: Optical trap
Crossed beams

8. Matter-wave integrated optics
Optical elements
Modelling:
Full simulation (expensive)
Approximations & simulations

9. Full simulations: Numerical solution of Partial Differential Equation
Finite differences, Finite elements, Pseudo spectral, Method of lines, etc
Consumes CPU-time and memory
It is nice to know that the computer understands the problem. I would like to understand it to. Eugene Wigner

10. Modelling approach: For a 2D potential energy surface
with a minimal path
V(x,y)
We can define a local Frenet frame using path length and transverse distance to bottom as coordinates.
Transfering to the new coordinates and expanding around the minima, we arrive at a system of 1D-equations to solve
where the couplings induce transitions between different longitudinal wavefunctions.

11. Mode-coupled guiding of matter-waves
We can gain understanding by studying a simpler system and using decades of knowledge in integrated optics – analogies.
Life gets easier if the A & B matrices can be nelected – adiabatic propagation like opt fibres in Hakutas talk
Note: here I dont talk about interactions, which can be taken into account, but easier if weaker than trapping energy

12. What about interactions?
…and nonlinear modecoupling unless transverse trapping is strong.

13. Beam splitter Fundamental building block, also nontrivial.
How du we split one mode into several?
Fundamental question: Coherence?
Classical scattering of atoms OR splitting of matter waves?

14. Beam splitter: Quantum optics
Two modes in, two modes out.
SU(2)
2 -> 2
OK!
1 -> 2
???

15. Explicit model: Harmonic for large and small separations
Groundstate known
Constant groundstate energy

16. After splitting we want two independent modes!
What modes?
We have:
We want:
After splitting we want two independent modes!

17. Local modes
Answer:
mix parity subset
to produce local modes

18. Local modes
Localisation at guide minima
choose mixing angle

19. Propagation of local modes
Each mode sees effects of changing potential
W(x) local tunneling rate

20. Experiment – Coherent or not?
Experiment with BEC, mode populations <n> variable 0-10.
First split that used BEC and probed ground state splitting.

21. Splitting occurs into all guides
How can this be understood?
Classical:
Scattering with sensitive dependence on position, velocity etc.

22. Quantum dynamics:
A localised mode in one arm is a superposition of n=0 & n=1 at the crossing.
The n=1 mode sees a barrier and is reflected.

23. Quantum or Classical?
IF quantum and classical dynamics give identical result, can we argue that phenomena are quantum?
Solution: uniquely quantum signature, nonclassical reflection, interference.

24. Topic switch: Double well BEC
We have seen that split quantum states can be seen as independent states.
Superposition of CM positions – surely quantum!

25. Double well BEC - Experiments
Oberthaler group (Heidelberg)
PRL 95, (2005)
Direct observation of
Tunneling in single bosonic
Josephson junction
Optical trapping, crossed beams

26. Double well BEC - Experiments
Schmiedmayer group
(Heidelberg)
Nature Physics, 1, 57 (2005)
Magnetic microtraps
above current carrying
wires
‘Atom chip’ experiment

27. Double(or few) well BEC – Exp.
Interference
after expansion
Nontrivial many-body physics occurs!
Nonlinear metrology – addition of weak tilt

28. Goal & Motivation:
Our goal is to model the dynamics and explain experimental signature – ‘contrast resonance’, and explore possibility for ultraprecise metrology.

29. Many particles – how?
For a split condensate each atom can hide in one of the two modes
Many atoms – second quantisation in Heisenberg picture

30. Quantum dynamics on sphere Schwinger representation SU(2)
# of atoms = N = 2J
Compare: polarisation, two level system as spin etc
To understand the dynamics, we use the internal state representation
Z is population diference, x and y are cosin and sine, i.e. Give the relative phase AND statistical properties i.e. coherence

31. Interference of many atoms Release of trap gives ballistic
expansion of modes
+ interference
Particle density:
Visibility:

32. Visibility of many particle interference?
We see the sum of all atoms doing interference – populations and phase distribution matters?

33. Visibility depends on time
Atoms tunnel L<->R and shift phase with time.
As a result we see different visibility if we look at different times.
We see expectation value of distribution.
If all particles are on one side, noone to intefere with!

34. Semiclassical trajectories
Initial energy = energy at NP
Condition for vanishing visibility:
8 Dec, 2005

35. Visibility dynamics Semiclassical dynamics Exact quantum dynamics
“Contrast resonance”
N = 5, 50, 500, 5000

36. Explanation:
Disappearance of visibility in time from quantum dynamics.
Sensitive dependence on parameter tuning.
Semiclassical explanation give condition – predicted and experimentally verified!