Atomic BEC in microtraps: Heisenberg microscopy of Zitterbewegung Markku Jääskeläinen

1995: BEC Matter-wave analogues, Atom Chips, Atomtronics, QPC for neutral atoms… Also a source for experimental realization of many other condensed matter systems.

2D Electron Gas Ballistic transport: conductance quantisation Figure courtesy: C. Beenakker, Leiden University Figure courtesy: Quantum Electronic Devices group, UNSW, Sydney Ballistic transport: conductance quantisation Diffusive transport: Ohmic conductance etc.

Transport in a quantum wire A parabolic quantum wire in a 2DEG Separation ansatz: y x

Modes/Sub-bands We find the solutions Giving plane waves, and HyperPhysics, Copyright C.R. Nave (2005) Energy spectrum

Ultracold matter

Gauge Fields in Low Dimensional Quantum Dynamics Appearance of Gauge Structure in Simple Dynamical Systems F. Wilczek & A. Zee, PRL 52, 2111 (1984) Only U(1), i.e. Abelian gauge structure.

Nonabelian Gauge Fields for Ultracold matter Cold Atoms in Non-Abelian Gauge Potentials: From the Hofstadter “Moth” to Lattice Gauge Theory K. Osterloh et.al. PRL 95, 010403 (2005) Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark states J. Ruseckas, et.al., PRL 95 010404 (2005)

Simple example: Lambda system Interaction Hamiltonian in RWA Dark state: Adiabatically- dressed state: Schrödinger equation for a particle coupled to a gauge field! Gauge field given by spatial variation of generalized Rabi frequencies

From Spin–orbit coupling in quantum gases Victor Galitski & Ian B. Spielman Nature 494, 49–54 (07 February 2013) a, Typical level diagram. In our experiments, a pair of lasers—often counter-propagating—couple together a selected pair of atomic states labelled by and that together comprise the atomic ‘spin’. These lasers are arranged in a two-photon Raman configuration that uses an off-resonant intermediate state (grey). These lasers link atomic motion along the x direction to the atom’s spin creating a characteristic spin–orbit coupled energy-momentum dispersion relation. b, Minima location.

Tripod scheme: Spin-Orbit coupling

Spin-Orbit coupling: Band structure of GaAs Inversion asymmetries: Structural (SIA) – Rashba Bulk/Lattice (BIA) - Dresselhaus

Spin-Orbit coupled quantum wire We study a parabolic quantum wire in a 2DEG with spin-orbit coupling, both Rashba, and linear Dressel- haus. Similar analytical expressions possible for cylindrical symmetry etc y x

Semiclassical dynamics Classical dynamics takes place in phase-space, (x,p). We can investigate classical-like dynamics in the Heisenberg picture Dynamical factoring of expectation values of operator products

Semiclassical equations of motion Velocity field: Note: only Rashba here, i.e. Force from potential: Spin precession: Precession vector: Set of ordinary differential equations, living in a classical phase-space.

Semiclassical Zitterbewegung Contribution from SO is perpendicular to both spin and z-axis Zig-zag trembling motion, Zitterbewegung. First discovered by Schrödinger in 1930, originally in the Dirac equation for a free electron.

Zitterbewegung in phase space. Rashba Orbit in phase-space: transverse cut is harmonic with spin precessing perpendicular to z-axis and momentum.

Zitterbewegung in phase space. Rashba Dresselhaus Difference between Rashba and Dresselhaus is in orientation of spin precession.

Quantum transport: fixed energy state Standard separation ansatz, solve for sub-bands Kramers degeneracy We use the k-states corresponding to a fixed energy, and form two orthogonal states.

Quantum Zitterbewegung Charge density exhibits the zig-zag pattern of Zitterbewegung Spin polarisation varies in synchronisation with charge oscillation

Heisenberg microscopy Heisenbergs famous gedanken-experiment showed that if we measure the coordinate With some uncertainty, the momentum can only be measured with an uncertainty given by:

Heisenberg microscopy Husimi distribution in phase space is a coherent state, a Gaussian displaced to with average momentum is a probability distribution, gives the probability of finding the electron at x with momentum p when measuring under minimal un- certainty conditions.

Bloch-vector: charge & spin Any two-level system can be described using a Bloch-vector representation. We introduce the phase-space charge density: together with the phase-space spin density, three components

Along the wire, the charge density exhibits Zitterbewegung in the transverse phase-space The spin density exhibits the associated precession.