Theory of Electrons in Solids Lu J. Sham, University of California San Diego, DMR 0403465

1. time Theory of Electrons in SolidsLu J. Sham, University of California San Diego,DMR 0403465 Fig. 1 Nuclear bath trajectories How the electron spin coherence is lost and restored An electron is confined in a semiconductor quantum dot. The directions of its spin pointing up or down along a magnetic field constitute 0 or 1 of a bit of information. The quantum state of the electron spin can be in a superposition of its up and down states. This possibility of being in two states at the same time is the root of the power of quantum information or computer. Coherence is a measure of the capacity of such superposition. Maintaining coherence for a long enough time to do a useful series of operations is a fundamental element in the quantum computer. The most stubborn cause of disturbance of the electron coherence is the bath of millions of nuclear spins in the dot. Decoherence arises out of the quantum schizophrenia of the nuclear spins to follow the electron spin up state with one collective trajectory and the electron spin down state with another (blue and red curves of Fig. 1). By switching the roles of the electron up and down states, the nuclear schizophrenic paths follow the new master states. By a concatenated sequence of electron flips (Fig. 2), the nuclear multitude may be coaxed back into a single collective state, restoring the electron coherence. Fig. 3(a) shows the increase of coherence time with flips. Note the magnifying factor of the true time for the l-th level. spin flip at time  flip at 3 Fig. 2 Flip sequences coherence decoherence Fig. 3 (a) Coherence (b) Deviation from ideal

2. Theory of Electrons in SolidsLu J. Sham, University of California San Diego,DMR 0403465 Supplementary Notes The theory of decoherence of an electron spin in contact with a bath of nuclear spins is given by Yao, Liu & Sham (cond-mat/0508441, Phys. Rev. B to be published). The technical term for the schizophrenia of the nuclear bath is entanglement. If the superposition of the electron spin up (u) and down (d) states is represented by u+d, the state including the nuclear spins is initially (u+d)N, N for the collective nuclear state. The contact between the electron and the nuclear path causes the nuclear spin states to split, thus, (uB+dR). This schizophrenia disrupts the coherence between the u and d states. The nuclear state is termed collective, because of the mutual interaction, as in mass hysteria. Note that the herding of the millions of nuclear spins is done by flipping a single electron spin, a very economical method. The concatenated sequences are built by induction. (Yao, Liu & Sham, cond-mat/0604634, submitted for publication.) The first line in Fig. 2 represents a simple flip of the electron spin from up to down and vice versa; the second line is a two flip sequence made up of the first one and then its reverse; the third one is the concatenation of the second sequence and its reverse; and so on. The coherence scale is set from 1 to 0, 1 being perfect and 0 being totally incoherent, like a classical coin. To compare the coherence with one and two electron spin flips, the red and dotted blue curves in Fig. 3 (a), remember that the coherence of the blue curve is at twice the flip time, i.e. one needs to stretch out the blue curve horizontally by a factor of two. To compare the fifth concatenated sequence with the first, one needs to stretch out its curve (green dots) a factor of 32 horizontally. The high coherence plateau lasts a very long time indeed compared with the optical operation time of about 10 ps (or about 100 million operations during high coherence).

3. Theory of Electrons in SolidsLu J. Sham, University of California San Diego,DMR 0403465 • Education and Human Resource: • Wang Yao, graduated Ph.D. 6/25/06, solid state cavity QED and single electron spin decoherence and recovery for quantum computing. Now postdoctoral at U. of Texas, Austin. • Parin Dalel, a new graduate student with industrial experience and several patents, educates the group on classical computers and scalability. His research applies the information theory approach to issues in spintronics and quantum computation. His education program may be an experimental paradigm for interdisciplinary research to further quantum technology. • Semion Saikin, postdoctoral, quantum information in solids and decoherence of electron spin due to a mesoscopic ensemble of nuclear spins. • Education: • Applied Quantum Mechanics for undergraduates and graduates entering into quantum technology • I have been building and teaching a course suited to the future needs of a quantum engineer, with an online self-test on mathematics preparation and remedial action and a selection of topics more immediate to the needs than a standard quantum course. http://physicscourses.ucsd.edu/lsham Outreach: Contact with industry to explore possible technological applications of our ideas in spintronics and quantum computing, through UCSD Office of Technology Transfer and California Institute of Telecommunication and Information Technology.