Abstract

Two-dimensional continuous time quantum random walks (CTQRW) are physical processes where quantum particles simultaneously evolve in different permissible directions within discrete graphs. In order to force the quantum walkers (QWs) to evolve in such a fashion, one generally requires periodic potentials. Here, we demonstrate that two-dimensional CTQRW can be generated in free space by properly tailoring the initial wave functions. We analytically show that within a certain spatial region the arising probability distribution quantitatively resembles the probability pattern exhibited by a QW traversing a periodic lattice potential. These theoretical predictions were experimentally verified using classical laser light, appropriately shaped by a spatial light modulator. Expanding the presented results to the case of multiple walkers may open new possibilities in quantum information technology using bulk optics.

© 2014 Optical Society of America

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References

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2014 (1)

2013 (2)

2012 (1)

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

2011 (2)

O. Mulken, A. Blumen, Phys. Rep. 502, 37 (2011).
[Crossref]

J. A. Jones, Prog. Nucl. Magn. Reson. Spectrosc. 59, 91 (2011).
[Crossref]

2010 (1)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

2009 (2)

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009).
[Crossref]

A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009).
[Crossref]

2008 (1)

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

2006 (1)

V. M. Kendon, Phil. Trans. R. Soc. A 364, 3407 (2006).
[Crossref]

2003 (2)

P. L. Knight, E. Roldán, J. E. Sipe, Phys. Rev. A 68, 020301 (2003).
[Crossref]

J. Kempe, Contemp. Phys. 44, 307 (2003).
[Crossref]

1999 (2)

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, Appl. Opt. 38, 5004 (1999).
[Crossref]

1997 (1)

P. W. Shor, SIAM J. Comput. 26, 1484 (1997).
[Crossref]

1995 (1)

J. E. Sipe, Phys. Rev. A 52, 1875 (1995).
[Crossref]

1993 (1)

Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993).
[Crossref]

Aharonov, Y.

Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993).
[Crossref]

Arie, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, Nature 494, 331 (2013).
[Crossref]

Aspuru-Guzik, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Berg, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Blumen, A.

O. Mulken, A. Blumen, Phys. Rep. 502, 37 (2011).
[Crossref]

Bouwmeester, D.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009).
[Crossref]

Broome, M. A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Campos, J.

Chavez-Cerda, S.

Childs, A. M.

A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009).
[Crossref]

Christodoulides, D. N.

Cottrell, D. M.

Dattoli, G.

Davidovich, L.

Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993).
[Crossref]

Davis, J. A.

Demler, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Fedrizzi, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, Nature 494, 331 (2013).
[Crossref]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Jones, J. A.

J. A. Jones, Prog. Nucl. Magn. Reson. Spectrosc. 59, 91 (2011).
[Crossref]

Karman, G. P.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

Kassal, I.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Kempe, J.

J. Kempe, Contemp. Phys. 44, 307 (2003).
[Crossref]

Kendon, V. M.

V. M. Kendon, Phil. Trans. R. Soc. A 364, 3407 (2006).
[Crossref]

Kitagawa, T.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Knight, P. L.

P. L. Knight, E. Roldán, J. E. Sipe, Phys. Rev. A 68, 020301 (2003).
[Crossref]

Lahini, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, Nature 494, 331 (2013).
[Crossref]

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, Nature 494, 331 (2013).
[Crossref]

Lobino, M.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Marzoli, I.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

Matsuda, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Matthews, J. C. F.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Morandotti, R.

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Moreno, I.

Moya-Cessa, H.

Mulken, O.

O. Mulken, A. Blumen, Phys. Rep. 502, 37 (2011).
[Crossref]

O’Brien, J. L.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Perets, H. B.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Perez-Leija, A.

Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Politi, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Pozzi, F.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Roldán, E.

P. L. Knight, E. Roldán, J. E. Sipe, Phys. Rev. A 68, 020301 (2003).
[Crossref]

Rudner, M. S.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Schleich, W.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

Shor, P. W.

P. W. Shor, SIAM J. Comput. 26, 1484 (1997).
[Crossref]

Silberberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Sipe, J. E.

P. L. Knight, E. Roldán, J. E. Sipe, Phys. Rev. A 68, 020301 (2003).
[Crossref]

J. E. Sipe, Phys. Rev. A 52, 1875 (1995).
[Crossref]

Sorel, M.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Soto-Eguibar, F.

Szameit, A.

Thompson, M. G.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Torre, A.

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, Nature 494, 331 (2013).
[Crossref]

White, A. G.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Woerdman, J. P.

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

Woerhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Yzuel, M. J.

Zagury, N.

Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993).
[Crossref]

Zhou, X.-Q.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

Appl. Opt. (1)

Contemp. Phys. (1)

J. Kempe, Contemp. Phys. 44, 307 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, A. G. White, Nat. Commun. 3, 882 (2012).
[Crossref]

Nature (1)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, A. Arie, Nature 494, 331 (2013).
[Crossref]

Opt. Express (1)

Phil. Trans. R. Soc. A (1)

V. M. Kendon, Phil. Trans. R. Soc. A 364, 3407 (2006).
[Crossref]

Phys. Rep. (1)

O. Mulken, A. Blumen, Phys. Rep. 502, 37 (2011).
[Crossref]

Phys. Rev. A (4)

Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993).
[Crossref]

P. L. Knight, E. Roldán, J. E. Sipe, Phys. Rev. A 68, 020301 (2003).
[Crossref]

D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich, J. P. Woerdman, Phys. Rev. A 61, 013410 (1999).
[Crossref]

J. E. Sipe, Phys. Rev. A 52, 1875 (1995).
[Crossref]

Phys. Rev. Lett. (3)

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009).
[Crossref]

A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009).
[Crossref]

Prog. Nucl. Magn. Reson. Spectrosc. (1)

J. A. Jones, Prog. Nucl. Magn. Reson. Spectrosc. 59, 91 (2011).
[Crossref]

Science (1)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Woerhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, Science 329, 1500 (2010).
[Crossref]

SIAM J. Comput. (1)

P. W. Shor, SIAM J. Comput. 26, 1484 (1997).
[Crossref]

Other (1)

See Supplement 1 for a complete rigorous analysis.

Supplementary Material (1)

» Supplement 1: PDF (595 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Theoretical probability evolution. (a) Probability distribution |ψlattice(n,z)|2 expected for a quantum walker traversing a uniform lattice, where the lattice sites are labeled by n. (b) Calculated free space probability density, |ψfs(xn,z)|2, plotted for the different transverse positions xn=nπ/α. (c) The difference |ψfs(xn,z)ψlattice(n,z)|2.

Fig. 2.
Fig. 2.

Experimental setup consisting of a Helium Neon laser (λ=633nm), telescope for beam expansion, spatial light modulator (Holoeye Pluto VIS) for amplitude and phase modulation, spherical lens (f=300mm) for Fourier transformation, microscope objective (10× Olympus Plan Achromat) for imaging, and movable CCD camera (Basler Ace 1600-20gm).

Fig. 3.
Fig. 3.

Theoretical and experimental comparison for a 1D CRW in free space. (a) Theoretical free-space probability evolution corresponding to the initial wave profile given by Eq. (4) using the parameters α=125mm1 and σ=0.13mm. (b) Bottom, experimental intensity envelope as produced in our setup at z=0cm. Center, recorded intensity evolution of the light beam at the plane y=0. Top, final intensity pattern after a propagation distance of z=2.5cm. (c) Comparison between theoretical and experimental intensity distribution in the initial plane at y=0. (d) Comparison between theoretical and experimental intensity evolution at two different transverse positions. x0=0 and x1=π/α25μm.

Fig. 4.
Fig. 4.

Theoretical and experimental comparison for a 1D CRW in a coherent superposition of states. (a) Theoretical free-space probability evolution of the initial wave function ψ(x,0)=exp[x2/2σ2](J0(αx)+J0(αx2π)) using the parameters α=145mm1 and σ=1mm. (b) Bottom, experimental initial intensity profile. Center, top view of the intensity evolution. Top, intensity pattern recorded after a propagation distance of z=2.5cm.

Fig. 5.
Fig. 5.

Experimental results for a 2D CRW. Top row, experimental images taken with a CCD camera. Central row, theoretical intensity evolution. Bottom row, stitched image from all camera images, showing the 2D CRW of a “single excitation.”

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

(iz+12k2)ψ=0,
ψ(x,y,0)=Ne(x2+y2)/2σ2Jn(αx)Jn(αy),
ψ(x,y,z)=Ns{x,y}(1izkσ2)1/2exp[2ks2+iα2σ2z4(kσ2iz)]×l=(i)lJn+2l(A(s,z))Jl(B(z)).
ψ(x,0)=Nexp[x2/2σ2]J0(αx).
ψ(x,z)=l=(i)lJ2l(αx)Jl(α2z4k).
ψ(xm,z)=(i)mJm(α2z/4k),
χ(ξ,η)=exp[iM(ξ,η)[ϕ(ξ,η)+ϕG(ξ,η)]].
ψ(x,y,0)=exp[(x2+y2)/2σ2]J0(αx)J0(αy).

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