Quantum random walks Andre Kochanke Max-Planck-Institute of Quantum Optics 7/27/2011

Motivation 2

3 ? ? ? ? ? ? ? ?

Overview Density matrix formalism Randomness in quantum mechanics Transition from classical to quantum walks Experimental realisation 4

Density matrix approach Two state system 5 1 0

Density matrix approach Two state system Density operator 6 0 1

Density matrix approach Density operator 7 Pure stateMixed state 0 1

Galton box 8 Binomial distribution

Galton box Statistical mixture First four steps 9

Quantum analogy Used Hilbert space Specify subspaces

Quantum analogy Evolution with shift and coin operators

Quantum analogy Evolution with shift and coin operators

Quantum analogy Evolution with shift and coin operators

Quantum analogy State transformation Density matrix transformation 14

Quantum analogy 15

Quantum analogy 16 Position pcpqpcpq Variances pcpqpcpq pcpqpcpq Position 100 steps

Phase shift Transformed density matrix Average Decoherence effect Decoherence 17

Different realisations C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”, PRA 72, (2005) M. Karski et al., “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009) A. Schreiber et al., “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”, PRL 104, (2010) F. Zähringer et al., “Realization of a Quantum Walk with One and Two Trapped Ions”, PRL 104, (2010) 18

Setup 19 CCD Microwave Dipole trap laser Objective Fluorescence picture Cs Microwave M. Karski et al., Science 325, 174 (2009)

Setup 20

Setup 21

Results 22 M. Karski et al., Science 325, 174 (2009)

Results 23 M. Karski et al., Science 325, 174 (2009) Theoretical expectation 6 steps

Results 24 Theoretical expectation M. Karski et al., Science 325, 174 (2009) 6 steps

Results 25 Theoretical expectation

Results 26 Theoretical expectation M. Karski et al., Science 325, 174 (2009)

Results 27 Gaussian fit M. Karski et al., Science 325, 174 (2009)

Conclusion The density matrix formalism allows you to describe cassical and quantum behavior Karski et al. showed how to prepare a quantum walk with delocalized atoms The quantum random walk is not random at all 28 M. Karski et al., Science 325, 174 (2009)

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References C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”, PRA 72, (2005) M. Karski et al., “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009) SOM for “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009) A. Schreiber et al., “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”, PRL 104, (2010) F. Zähringer et al., “Realization of a QuantumWalk with One and Two Trapped Ions”, PRL 104, (2010) M. Karksi, „State-selective transport of single neutral atoms”, Dissertation, Bonn (2010) C. C. Gerry and P. L. Knight, „Introductory Quantum Optics“, Cambridge University Press, Cambridge (2005) M. A. Nielsen and I. A. Chuang, „Quantum Computation and Quantum Information“, Cambridge University Press, Cambridge (2000) 30