1. Atomic BEC in microtraps: Heisenberg microscopy of Zitterbewegung
Markku Jääskeläinen

2. 1995: BEC Matter-wave analogues, Atom Chips, Atomtronics,
QPC for neutral atoms…
Also a source for experimental realization
of many other condensed matter systems.

3. 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.

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

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

6. Ultracold matter

7. 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.

8. 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, (2005)
Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark states
J. Ruseckas, et.al., PRL (2005)

9. 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

10. 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.

11. Tripod scheme: Spin-Orbit coupling

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

14. 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

15. 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

16. 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.

17. 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.

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

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

20. 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.

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

22. 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:

23. 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.

24. 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

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