1. Atomic BEC in microtraps: Squeezing & visibility in interferometry
Markku Jääskeläinen

2. Double(or few) well BEC – Exp.
Manipulation of BEC, and observation of interference
after ballistic expansion
Nontrivial many-body dynamics occurs!
Nonlinear metrology – addition of weak tilt

3. Goal & Motivation:
Our goal is to model the dynamics and explain certain experimental signatures
– visibility, ‘contrast resonance’
– squeezing
Also, explore possibility for ultraprecise metrology.

4. Many particles – how?
For a split condensate each atom can hide in one of the two modes
Many atoms – second quantisation – field operator

5. Optical lattice, Bose-Hubbard model, Mott Insulator transition etc
Double well – Bose Hubbard dimer
Why do dimer instead of lattice ? Smaller Hilbert space, can solve numerically, do dynamics etc.
Lattice is a series of bosonic Josephson junctions, if we understand how one behaves, we have a good clue to many.
Single junction

6. Quantum dynamics on sphere +exponentially small correction
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
+exponentially small correction

7. Parameters:
In experiments the lattice depth is controllable parameter, i.e. barrier height and ground state width.
Effective interaction g decreases slower than tunnel split
Both F & G increase with depth.

8. Hamiltonian: Angular momentum operators – generate rotations
Nonlinear rotation – ‘twist’

9. Husimi distribution
Joint probability distribution
for number difference and
relative phase
Angular coherent state, F.T. Arechi et.al.,
PRA 6, 2211 (1972)

10. Number squeezed Phase squeezed Superfluid, Coherent Mott insulator,
Fock
Superposition
Uncertainties
Number
We see that quantum groundstates can have phase distribution with low visibility
Phase

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

12. Visibility of many particle interference?
What do experiments
measure?
We see the sum of all atoms doing interference – populations and phase distribution matters?

13. Number squeezed Phase squeezed Superfluid, Coherent Mott insulator,
Fock
Superposition
Uncertainties
Number
We see that quantum groundstates can have phase distribution with low visibility
Phase

14. Ground state visibility
SF -> MI crossover
Sensitive dependence on particle number: odd/even effect (filling).
Even at large numbers adding/removing can give V =0.5.

15. Dynamical visibility Quantum dynamics:
Atoms tunnel L<->R and shift phase with time.
As a result we see different visibility if we look at different times.
If all particles are on one side, noone to intefere with!
Example: Atoms start with equal populations
of L & R in minimal uncertainty state.
All tunnel over to L + phase squeezing

16. Quenching
Slow change of lattice depth (up or down) followed by rapid release.
The rat of changing depth matters, adiabatic following of many body state
Slow means adiabatic w.r.t. many-body groundstate

17. Slow squeezing, rapid release to lower depth
Number
squeezed
Slow squeezing, rapid release to lower depth
Superfluid,
Coherent
Visibility oscillates with decaying amplitude

18. Nonstationary state, highly squeezed:
Twisting, swirling around Jx
Oscillations of V speed up due to interactions.

19. Tilted potential Small gradient added
Experiment shows narrow disappearance of visibility

20. Tilted potential
We know that if all particles are in one well, Jx =Jy = 0.
No interference!
Example:

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

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

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

24. Metrology? Smallest gradient depends on other parameters. We remember:
Present experiments have probed acceleration

25. Number
squeezed
Mott insulator,
Fock
Visibility limits largest G we an work with; initial state should not be too squeezed.
Uncertainties

26. Understanding a single junction explains lattice dynamics – up to certain times.
Metrology in principle possible, further improvements possible.
Modelling & theory go hand in hand with experiments in Ultracold physics.