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arXiv:1102.0035 · Physical Review B

Greenberger-Horne-Zeilinger states in a quantum dot molecule

Anand Sharma, Paweł Hawrylak

Published Mar 21, 2011• 15 citations
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Abstract

We present a microscopic theory of a lateral quantum dot molecule in a radial magnetic field with a Greenberger-Horne-Zeilinger (GHZ) maximally entangled three particle ground state. The quantum dot molecule consists of three quantum dots with one electron spin each forming a central equilateral triangle. The antiferromagnetic spin-spin interaction is changed to the ferromagnetic interaction by additional doubly occupied quantum dots, one dot near each side of a triangle. The magnetic field is provided by micromagnets. The interaction among the electrons is described within an extended Hubbard Hamiltonian which is solved by using exact diagonalization techniques. The set of parameters is established for which the ground state of the molecule in a radial magnetic field is well approximated by a GHZ state.

DOI
10.1103/physrevb.83.125311

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We present a microscopic theory of a lateral quantum dot molecule in a radial magnetic field with a Greenberger-Horne-Zeilinger (GHZ) maximally entangled three particle ground state. The quantum dot molecule consists of three quantum dots with one electron spin each forming a central equilateral triangle. The antiferromagnetic spin-spin interaction is changed to the ferromagnetic interaction by additional doubly occupied quantum dots, one dot near each side of a triangle. The magnetic field is provided by micromagnets. The interaction among the electrons is described within an extended Hubbard Hamiltonian which is solved by using exact diagonalization techniques. The set of parameters is established for which the ground state of the molecule in a radial magnetic field is well approximated by a GHZ state.
Exact-diagonalization calculations for N=3N=3N=3 electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection Sz=1∕2{S}_{z}=1∕2Sz​=1∕2 are similar to the representative class of strongly entangled states referred to as the WWW states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.
Entangled state analysis is one of the basic techniques in quantum information processing. However, it is impossible to fully distinguish the three-photon Greenberger–Horne–Zeilinger (GHZ) states with only linear optical elements. Here, we propose a deterministic scheme to complete three-photon spatial-polarization hyperentangled GHZ-state analysis (HGSA) by using the giant optical Faraday rotation induced by an excess electron spin in a quantum dot inside a one-sided optical microcavity as a result of cavity quantum electrodynamics. It is divided into two steps. The first step is used to distinguish the eight three-photon GHZ states in the spatial-mode degree of freedom (DOF) without destroying the polarization states. The second step is used to distinguish the eight GHZ states in the polarization DOF. This scheme can be generalized to N-photon spatial-polarization HGSA and has useful applications in quantum communication protocols.
We study a two-electron quantum dot molecule in a magnetic field by the direct diagonalization of the Hamiltonian matrix. The ground states of the molecule with the total spin S = 0 and S = 1 provide a possible realization for a qubit of a quantum computer. Switching between the states is best achieved by changing the magnetic field. Based on an analysis of the wave function, we show that the system consists of composite particles formed by an electron and flux quanta attached to it. This picture can also be used to explain the spin phase diagram.
In this paper, we present a proof of principle of the formation of pure maximally entangled states from the Greenberger-Horne-Zeilinger (GHZ) class, in the experimental context of charged quantum dots. Each qubit must be identified as a pair of quantum dots, sharing an excess electron, coupled by tunneling. The electron-electron interaction is accounted for and is responsible for the coupling between the qubits. The interplay between coherent tunneling events and many-body interaction gives rise to the formation of highly entangled states. We begin by treating the problem of encoding three qubits in a system with three pairs of quantum dots, and the numerical analysis of the exact quantum dynamics to find the conditions for the generation of the GHZ states. An effective two-level model sheds light on the role of a high-order tunneling process behind the dynamics. The action of the main decoherence process, the charge dephasing, is quantified in the process. We then evaluate the physical requirements for the dynamical generation of GHZ states in an NNN-qubit scenario, and its challenges.
Optically controlled exciton dynamics in coupled quantum dots is studied. We show that the maximally entangled Bell states and Greenberger-Horne-Zeilinger (GHZ) states can be robustly generated by manipulating the system parameters to be at the avoided crossings in the eigenenergy spectrum. The analysis of population transfer is systematically carried out by using a dressed-state picture. In addition to the quantum dot configuration that has been discussed by Quiroga and Johnson [Phys. Rev. Lett. 83, 2270 (1999)], we show that the GHZ states also may be produced in a ray of three quantum dots with a shorter generation time.