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Transcript
Optimal Gate Control of Quantum-dot Cellular Automata Yousof Mardoukhi, Esa Räsänen [email protected], [email protected] Introduction System and Methodology Quantum dots positioned on a certain lattice structure, usually square lattices, have a physical realization of cellular automata. This system is called Quantum-dot Cellular Automata (QCA). QCAs can be used for next generation of computer circutes as their time switching is fast and their energy consumption is considerabily low. The aim of this project[1] was to achieve controlability over QCAs by applying a local voltage gate and manipulate the charge distribution of a single electron QCA as a very first step for further complicated studies. System The Hamiltonian of the system is defined as follow, p2 H(x, y, t) = + Vc (x, y) + U (x, y, t) 2 (2) where Vc (x, y) is the model confining potential for the QDs. It is given by a square lattice of cavities (M × N ) having a shape of a Fermi function given by, ( ) 1 h√ i : 1 ≤ i ≤ N, 1 ≤ j ≤ M − max (x−i a)2 +(y−j a)2 −R /ξ +1 e where ξ = 0.25 is the softness of the potential and a = 6 a.u. is the lattice constant and R = 2 a.u. is the radius of the quantum dot. Voltage Gate U (x, y, t) in the Hamiltonian is the pontential of the voltage gate which can be splited up into spatial and time varying parts Control Scheme The optimisation is done by optimal control thery (OCT). the time varying part of the field is optimised iteratively by solving Schrödinger equation. By introducing the Lagrange multiplier |χ(t)i the control equations are derived. i∂t |Ψ(t)i = H(r, t)|Ψ(t)i, |Ψ(0)i = |ΦI i (1a) i∂t |χ(t)i = H(r, t)|χ(t)i (1b) |χ(T )i = Θ(|r − r0 |)|rihr|Ψ(T )i 1 f (t) = − hχ(t)|g(r)|Ψ(t)i. α (1c) (1d) The target operatior is a heaviside distribution function wich implies the localisation of the particle within a certain region of space. α is the fluence factor which is kept constant during the opitmisation. Conclusion We have elaborated a scheme within quantum optimal control theory to manipulate electric charge in quantum-dot lattices that could be applied as quantum. We have demonstrated that high-delity local control of single-electron charge in quantum dot lattices is possible up to N' 5 for both (1×N) and (2×N) lattices. Our study sheds light for the future controllability of a quantum-dot cellular automata in the case of elementary majority gates like AND and OR. We were also successful in identification of two bit digits 0 and 1 thanks to the symmetry properties of the lattice sturcture. U (x, y, t) = g(x, y) f (t), where f (t) = A sin(ωt) and g(x, y) is modeled as a Gaussian pulse shape: β −(x−x0 )2 −(y−y0 )2 ]/(2σ 2 ) [ √ e (σ 2π)2 Results 1×N: We start from the delocalised ground state. The voltage gate acts on the very left most dot and the target is aimed to localize the charge density on the very right most QD. The optimisation is done on the time-depentent part f (t), while keeping the spatial part g(x, y) and the fluence of the initial field constant.The yield at the U T end of pulse duration is found to be 91.4%. The initial pulse without optimisation has a yield of only 10%. Even after applying a frequency filter at ωth = 5 a.u. the yield becomes 84.7%. The threshold frequency as an instance for GaAs is 8.6THz which can be expected to be reached [2]. 2×N: This case study is more realistic since the lattice size of 2×2 is the premitive cell of QCA. As there are two symmetry axes present, two voltage gates are considered. In the special case of 2×2 since the cell has C and σ symmetries, it is controlable even with one voltage gate for realisation of digit 0 and 1. For a lattice size of 2×5 with the same parameters for the voltage gates as previous case of 1×6, the initial pulse yield is only 2.5% which after the optimisation the yield is increased to 94.2%. By imposing a fre- Figure 1: (a) GS charge density distribution. The target dot is marked by T. (b) Final state quency filter ωth = 5 a.u. the yield is reduced to charge distribution after optimisation. (c) Pulse 87.6%. References U T [2] Wirth et al. Synthesized light transients. Science, 334(6053):195–200, 2011. U T U T Acknowledgements The work was supported by the European Communitys FP7 through the CRONOS project, grant agreement no. 280879, the Academy of Finland, and COST Action CM1204 XLIC. CSC Scientic Computing Ltd. is acknowledged for computational resources. (4) where β = 16.04 a.u. and σ = 1.6 a.u. and the tuple (x0 , y0 ) is the position of the center of the voltage gate. shapes and (d) corresponding Fourier spectra [1] Y. Mardoukhi E. Räsänen. Towards a quantum dot cellular automata with optimal gate control. submitted, 2014. (3) 2×2: This cell is premitive cell for the purpose of computation. The diagonal charge distribution corresponds to bit 0 and anti-diagonal one to bit 1. The essence of this representation of bits lies within the symmetry of the configuration. In Fig. 2(b) and (c) since the target is not distinguishable the charge will be distributed equally between the two QDs positioned on anti-diagonal lattice points. The yield of transition from bit 0 to 1 and vice versa is > 98%. The field parameters is the same as above. Figure 2: Middle row: Field-driven (U) localization of the electron charge from the ground state (b1) to the diagonal configuration defined as bit 1 (b2). Lower row: Transformation of bit 0, corresponding to a diagonal configuration (c1) to an off-diagonal configuration defined as bit 1 (c2).