Spatial correlations in quantum walks with two particles

Spatial correlations in quantum walks with two particles M. Štefaňák (1), S. M. Barnett 2, I. Jex (1) and T. Kiss (3) (1) Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic (2) Department of Physics, University of Strathclyde, Glasgow, Scotland (3) Department of Quantum Optics and Quantum Information, Research Institute for Solid State Physics and Optics, Budapest, Hungary M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 1 / 31

Outline 1 Introduction 2 Quantum Walk of two Particles on a Line 3 Spatial Correlations 4 Meeting Problem 5 Conclusions M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 2 / 31

Motivation Generalization of a classical random walk for a quantum particle (Aharonov et al. 93) Unitary time evolution, randomness due to measurement Application Database search (Shenvi et al. 03) Universal quantum computation (Childs 09, Lovett et al. 10) Random Walk Probabilities Trajectories add up Diffusion σ t Quantum Walk Probability amplitudes Trajectories interfere Wave propagation σ t M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 4 / 31

Quantum Walk on a Line Particle lives on 1-D lattice position space H P = l 2 (Z) = Span { m m Z} Moves in a discrete time steps on a lattice RW : m m 1, m + 1 = QW : m m 1 + m + 1 Does not preserve orthogonality { 0 1 + 1 orthogonal 2 1 + 3 } nonorthogonal To make the time evolution unitary we need an additional degree of freedom coin space H C = Span { L, R } M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 5 / 31

Quantum Walk on a Line Time evolution equation ψ(t) = U t ψ(0), U = S (I P C) Initial state ψ(0) position + coin ( e.g. 0 (ψ L L + ψ R R ) ) Displacement operator S = m ( ) m 1 m L L + m + 1 m R R Coin flip C - rotates the coin state before the step itself e.g. Hadamard matrix H = 1 2 ( 1 1 1 1 ) M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 6 / 31

Quantum Walk on a Line M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 7 / 31

Asymptotic Methods Fourier analysis, stationary phase approximation (Nayak et al. 00) ψ(k, t) = Ũt (k)ψ, Ũ(k)v j (k) = e iω j (k) v j (k) ψ(m, t) = e i(ω j (k)t mk) ( v j (k), ψ ) v j (k)dk j Weak limit theorems (Konno 02, Grimmett et al. 04) ( ) X n = t j Hadamard walk ( ω j (k) ) n ( vj (k), ψ ) 2 dk + O(t 1 ) P(y = x/t) = 1 y ( (ψ L + ψ R )ψ L + (ψ L ψ R )ψ R ) πt 1 2y 2 (1 y 2 ) M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 8 / 31

Asymptotic distribution of the Hadamard walk M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 9 / 31

Asymptotic distribution of the Hadamard walk M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 10 / 31

Asymptotic distribution of the Hadamard walk M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 11 / 31

Experiments Quantum walk on a line Atom in an optical trap (Karski et al. 09) Trapped ion (Schmitz et al. 09) Linear quantum optics (Schreiber et al. 10) M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 12 / 31

Motivation Non-classical features Particles can be entangled Particles can be indistinguishable fermions or bosons Experiments Current experiments 1-dimensional quantum walk n-particles on a line n-dimensional walk Our study Restrict to two non-interacting particles on a line Identify the effects of entanglement and statistics M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 14 / 31

Distinguishable particles Non-interacting particles performing a Hadamard walk on a line H TOT = (H P H C ) 1 (H P H C ) 2 Their time evolution is independent Ψ(t) = UTOT t Ψ(0), U TOT = U 1 U 2 Both start the walk from the origin Ψ(t = 0) = 00 Ψ C Initial coin state Ψ C separable x entangled M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 15 / 31

Distinguishable particles Initial coin state Separable coin states Ψ C = ψ 1 ψ 2 Two-particle state remains separable Probability distribution factorizes P(m, n, t) = P 1 (m, t) P 2 (n, t) Entangled coin states Probability distribution does not factor Can be expressed in terms of single-particle amplitudes P(m, n, t) {P 1 (m, t), P 2 (n, t)} + interference terms M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 16 / 31

Distinguishable particles mapping to a 2-D walk Two particles on a line single particle on a plane Coin operator tensor product of two Hadamard matrices C = H H We can use the methods developed for single-particle walks M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 17 / 31

Indistinguishable particles Motivated by linear optical networks (beam-splitters etc.) Creation operators â (m,l) particle at m, coin state L Dynamics is defined on a one-particle level Single step - transformation of creation operators â (m,l) 1 ) (â 2 (m 1,L) + â (m+1,r) â (m,r) 1 ) (â 2 (m 1,L) â (m+1,r) Commutator for bosonic operators, anti-commutator for fermionic Two-particle state in terms of single-particle amplitudes Indistinguishability entanglement of coin states M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 18 / 31

Spatial correlations Two particles start the walk from the origin with coin state Ψ C Probability to be on the same side of the line w.r.t. origin 0 0 t t P s (t) = P(m, n, t) + P(m, n, t) m= t n= t m=1 n=1 Problem Dependence of P s on the initial coin state Ψ C M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 20 / 31

Separable coin states P(m, n, t) factorizes into single-particle distributions Asymptotic probability distribution replace sums by integrals Within this approximation P s is time-independent limit Simple expression in the Hadamard basis H χ ± = ± χ ±, χ ± 2 ± 2 2 2 = L ± R 2 2 Initial coin state Ψ C = ψ 1 ψ 2, ψ i = x (i) + χ+ + x (i) χ P s = 1 4 ( ) (2 + (2 x (1) + 2 1)(2 x (2) + 2 1) M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 21 / 31

Separable coin states Bounds 3 4 P s 1 4 Max same eigenstate, min opposite eigenstate M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 22 / 31

Entangled coin states Map to 2-D walk with coin C = H H, find asymptotic distribution Simplifies in the basis of the coin C Initial coin state ( ) C χ i χ j = i j χ i χ j, i, j = ±1 Ψ C = x ++ χ + χ + + x + χ + χ + x + χ χ + + x χ χ P s = 1 (2 + x ++ 2 + x 2 x + 2 x + 2) 4 Same bounds as for separable states Max (min) Ψ C is the eigenstate of C with ev. +1 (-1) M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 23 / 31

Meeting problem Two particles start the walk from the origin with coin state Ψ C Probability that the particles meet anywhere on the lattice t P m (t) = P(n, n, t) n= t Problem Asymptotic behavior of P m on the initial coin state Ψ C M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 25 / 31

Meeting problem Quantum walk covers almost uniformly interval of length t In this interval P(m, t) 1 t Rough estimate P m (t) area covered (probability) 2 = t 1 t 2 = 1 t More detailed analysis contribution from the peaks P m (t) ln t t Holds when the initial coin state Ψ C is not eigenstate of C = H H with eigenvalue 1 For C Ψ C = Ψ C the peaks are canceled no ln t correction M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 26 / 31

Asymptotic behavior of the meeting probability M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 27 / 31

Asymptotic behavior of the meeting probability M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 28 / 31

Asymptotic behavior of the meeting probability M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 29 / 31

Conclusions Summary Introduction of quantum walk with more particles Analysis of spatial correlations and meeting In the present model the non-classical features are not significant MŠ, T. Kiss, I. Jex and B. Mohring, J. Phys. A 39, 14965 (2006) MŠ, S. M. Barnett, T. Kiss, B. Kollár and I. Jex, in preparation More particles More complicated graphs Outlook Interactions distance-dependent coin etc. M. Štefaňák (FNSPE CTU in Prague) Correlations in Quantum Walks 26. 8. 2010 31 / 31