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

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

Goal & Motivation: Our goal is to model the dynamics and explain certain experimental signatures – visibility, ‘contrast resonance’ – squeezing Also, 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 – field operator

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

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

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.

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

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

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

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

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

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

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.

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

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

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

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

Tilted potential Small gradient added Experiment shows narrow disappearance of visibility

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

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!

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

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

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.