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# Tunneling Measurement of a Single Quantum Spin

M. Hru\u{s}ka, L.N. Bulaevskii and G. Ortiz
Los Alamos National Laboratory, US

Keywords: tunneling, spin, quantum measurement,qubit

Abstract:
We consider tunneling between electrodes via a microscopic system which can be modeled by the two-level Hamiltonian (a localized spin 1/2 probed by STM or a quantum dot). Measurements of the tunneling current $I(t)$ in such a system provide information on the orientation and dynamics of the spin and they constitute an example of indirect-continuous quantum measurements. We assume that a) coupling of the spin with electrodes is much stronger than that with environment, b) the DC magnetic field ${\bf B}$ acts on the spin, c) electrons in the electrodes are polarized, and d) distribution function of electrons in the electrodes corresponds to the thermal equilibrium at the temperature $T$. By using the non-equilibrium Keldysh method and Majorana representation for spin [1] we find conditions under which the tunneling current leads to the steady state with spin precession seen as a peak at the Larmor frequency in the spectral density of the current-current correlation function $\langle I(\omega)I(-\omega)\rangle$. This occurs, for example, when electrons in the electrodes are fully polarized in the direction ${\bf P}\perp {\bf B}$, but does not occur if electrons are weakly polarized or are polarized with ${\bf P}\parallel {\bf B}$. The height and the width of the peak in the spectral density $\langle I(\omega)I(-\omega)\rangle$ at the Larmor frequency depend on the electron temperature $T$, on the strength of the magnetic field $B$ and on the voltage applied to the electrodes. The width of the peak increases with the tunneling current and in the limit of high current the spin dynamics (spin precession) is suppressed because the width of the peak becomes much larger than the Larmor frequency (quantum Zeno effect). We describe how the tunneling current may be used to read a qubit represented by a single quantum spin 1/2. We discuss also the experimental results obtained by STM dynamic probes of spins [2,3]. \\ 1. O. Parcollet and C. Hooley, cond-mat/0202425. \\ 2. Y. Manassen, {\it et al.}, Phys. Rev. Lett. {\bf 62}, 2531 (1989); J. Magn. Reson. {\bf 126}, 133 (1997); Phys. Rev. B {\bf 61}, 16223 (2000).\\ 3. C. Durkan and M.E. Welland, Appl. Phys. Lett. {\bf 80}, 458 (2002).

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