Universität Karlsruhe Phys. Rev. Lett. 97, (2006)

And beyond – The ‘dream’ of Quantum Computation Richard P. Feynman A brand new degree of freedom – Real-world electronics has still only used half of its potential – The future: Non-volatile, ultrafast MRAM Spin logic New detectors …

Some recent efforts to control a single electron spin Single-shot read-out of an individual electron spin in a quantum dot J. M. Elzerman, R. Hanson, et al. Nature 430, 431 (2004 ).

… or two spins! Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots J. R. Petta, A. C. Johnson, et al. Science (2005)

Spin decay mechanisms – Magnetic noise (hyperfine coupling to nuclear spins) – Electric noise (spin-orbit mediated)

Generalized Theory of Relaxation F. Bloch. Phys. Rev. 105, 1206 (1957).

Environment charge fluctuations or phonons produce noisy electric fields But electric fields do not couple to electron spin! Indirect spin decay: Spin-orbit (SO) coupling So, err… what is spin-orbit coupling? Spin-orbit mediated decay “Coupling between the orbital and the spin degrees of freedom arising from relativistic corrections to the Schrödinger equation” Uh? In semiconductor heterostructures:

A: Momentum dependent ‘magnetic field’ ( B ext =0 ) B: Quasiclassically: as the electron moves a distance dr in time dt the spin is rotated by U, which doesn’t depend on dt (‘geometric’) C: Motion induced by random external forces (phonons) induces a diffusion of the electron spin Measurement, control, and decay of quantum-dots spins W. A. Coish, V. N. Golovach, J. C. Egues, D. Loss. cond-mat/

Detailed model for an electron in a quantum dot Low energy effective dynamics Low energy effective dynamics

The two eigenstates (without noise) are actually ‘pseudospin’ states For small external magnetic fields only the ‘twist’ contribution yields a finite relaxation rate (no Van-Vleck type cancellations) This is related to the fact that time reversal doesn’t alter the first two, and therefore Kramer’s theorem prevails (at B=0 ) Spin-flip transitions between Zeeman sublevels in semiconductor quantum dots A. V. Khaetskii and Y. V. Nazarov. Phys. Rev. B. 64, (2001).

Three relaxation times T 1 (one for each process) Different environments, different spectral densities: it affects the relaxation rate Piezoelectric phonons Charge fluctuations

Spin relaxation rate resulting from each of the different processes (in a symmetrical GaAs lateral quantum dot with 1 Kelvin unperturbed level spacing) Crossover magnetic field: B ** =15 mT

At B=0 the environment dephases the two pseudospin states: each one acquires an opposite (and random) geometrical phase “A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R(t) in its Hamiltonian H(R), will acquire a geometrical phase factor exp{i  (C)} in addition to the familiar dynamical phase factor”  Pure geometric dephasing Pointer basis = normal to the heterostructure Quantal phase factors accompanying adiabatic changes M. V. Berry, Proc. R. Soc. Lond. A 392, (1984)

Even in a strongly quantum system at B=0, a noisy electric environment can cause geometric spin dephasing and relaxation through spin-orbit coupling