We present a mechanism to protect quantum information stored in an ensemble of nuclear spins in a semiconductor quantum dot. When the dot is charged the nuclei interact with the spin of the excess electron through the hyperfine coupling. If this coupling is made off-resonant, it leads to an energy gap between the collective storage states and all other states. We show that the energy gap protects the quantum memory from local spin-flip and spin-dephasing noise. Effects of nonperfect initial spin polarization and inhomogeneous hyperfine coupling are discussed. DOI: 10.1103/PhysRevLett.103.010502 PACS numbers: 03.67.Pp, 03.65.Yz, 73.21.La, 76.70.Àr An essential ingredient for quantum computation and long-distance quantum communication is a reliable quantum memory. Nuclear spins in semiconductor nanostructures are excellent candidates for this task. With a magneton 3 orders of magnitude weaker than electron spins, they are largely decoupled from their environment, and the hyperfine interaction with electron spins allows one to access ensembles of nuclear spins in a controlled way [1][2][3][4][5][6][7][8][9][10]. In particular, the quantum state of an electron spin can be mapped onto the nuclear spins, giving rise to a longterm memory [3][4][5][6][7]. Nevertheless, memory lifetimes are limited, e.g., by dipole-dipole interactions among the nuclei. In this Letter, we demonstrate that the presence of the electron spin in the quantum dot substantially reduces the decoherence of this collective memory associated with surrounding nuclear spins. The virtual transitions between electronic and nuclear states can be used to produce an energy shift proportional to the number of excitations in the storage spin-wave mode. This isolates the storage states energetically and protects them against nuclear-spin flips and spin diffusion.Consider a quantum dot charged with a single excess electron as indicated in Fig. 1. The electron spinŜ is coupled to the ensemble of underlying nuclear spinsÎ j by the Fermi contact interaction,where A is the average hyperfine interaction constant, A % 90 eV for GaAs, and % j is proportional to the electron density at the position of the jth nucleus, P j % j ¼ 1. For convenience, we introduce the collective operatorŝ A P j % jÎ j . The first term in Eq. (1) yields the Overhauser field, an effective magnetic field for the electron, and also the Knight shift for each nuclei. The flip-flop terms in Eq. (1) As will be shown here, the same can be used to provide a protective energy gap. Fully polarized nuclei.-We start by reconsidering the storage of a qubit in collective nuclear states [3]. In the case when all the nuclear spins are initially polarized in the Àz direction (zero temperature limit), the j#i e and j"i e spin states of the electron are mapped onto the nuclear-spin states j0i j À I; ÀI; . . . ; ÀIi and j1i A  þ j0i / X j % j j ÀI;...;ðÀI þ 1Þ j ;...;ÀIi; (2) respectively.Ĥ JC couples the state j0ij "i e to j1ij #i e with an angular frequency ¼ Að P j % 2 j 2IÞ 1=2 . The detuning between ...

We present a full quantum mechanical three dimensional theory describing an electromagnetic field interacting with an ensemble of identical atoms. The theory is constructed such that it describes recent experiments on light-matter quantum interfaces, where the quantum fluctuations of light are mapped onto the atoms and back onto light. We show that the interaction of the light with the atoms may be separated into a mean effect of the ensemble and a deviation from the mean. The mean effect of the interaction effectively give rise to an index of refraction of the gas. We formally change to a dressed state picture, where the light modes are solutions to the diffraction problem, and develop a perturbative expansion in the fluctuations. The fluctuations are due to quantum fluctuations as well as the random positions of the atoms. In this perturbative expansion we show how the quantum fluctuations are mapped between atoms and light while the random positioning of the atoms give rise to decay due to spontaneous emission. Furthermore we identify limits, where the full three dimensional theory reduce to the one dimensional theory typically used to describe the interaction.

We present a three-dimensional theory of stimulated Raman scattering (SRS) or superradiance. In particular we address how the spatial and temporal properties of the generated SRS beam, or Stokes beam, of radiation depends on the spatial properties of the gain medium. Maxwell equations for the Stokes field operators and of the atomic operators are solved analytically and a correlation function for the Stokes field is derived. In the analysis we identify a superradiating part of the Stokes radiation that exhibit beam characteristics. We show how the intensity in this beam builds up in time and at some point largely dominates the total Stokes radiation of the gain medium. We show how the SRS depends on geometric factors such as the Fresnel number and the optical depth, and that in fact these two factors are the only factors describing the coherent radiation.Comment: 21 pages 14 figure