Atomic BEC in microtraps: Localisation and guiding Markku Jääskeläinen

Sweden?

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

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

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

How? Example: B-field + wire

How? Example: Optical trap Crossed beams

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

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

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.

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

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

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?

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

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

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

Local modes Answer: mix parity subset to produce local modes

Local modes Localisation at guide minima choose mixing angle

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

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

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

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.

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.

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

Double well BEC - Experiments Oberthaler group (Heidelberg) PRL 95, 010402 (2005) Direct observation of Tunneling in single bosonic Josephson junction Optical trapping, crossed beams

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

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

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

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

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

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

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

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!

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

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

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