Abstract

Spatial qudits are D-dimensional (D2) quantum systems carrying information encoded in the discretized transverse momentum and position of single photons. We present a proof-of-principle demonstration of a method for preparing arbitrary pure states of such systems by using a single phase-only spatial light modulator (SLM). The method relies on the encoding of the complex transmission function corresponding to a given spatial qudit state onto a preset diffraction order of a phase-only grating function addressed at the SLM. Fidelities of preparation above 94% were obtained with this method, which is simpler, less costly, and more efficient than those that require two SLMs for the same purpose.

© 2013 Optical Society of America

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References

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2013 (1)

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

2012 (1)

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

2011 (2)

2010 (1)

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 82, 055801 (2010).
[CrossRef]

2009 (1)

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

2008 (3)

2005 (2)

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

2004 (2)

2001 (3)

J. Fiurášek, Phys. Rev. A 64, 024102 (2001).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, Opt. Eng. 40, 2558 (2001).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

1999 (1)

1994 (1)

Z. Zhang, G. Lu, and F. T. S. Yu, Opt. Eng. 33, 3018 (1994).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Aguirre-Gómez, J. G.

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

Ali-Khan, I.

M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Bagnoud, V.

Barnett, S. M.

S. M. Barnett, Quantum Information (Oxford University, 2009).

Boyd, R. W.

M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Cabello, A.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Campos, J.

Cañas, G.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

Chiuri, A.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

Cottrell, D. M.

Davis, J.

A. Márquez, C. Iemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, Opt. Eng. 40, 2558 (2001).
[CrossRef]

Davis, J. A.

De Martini, F.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

Delgado, A.

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

Etcheverry, S.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

Fernández, E.

Fiurášek, J.

J. Fiurášek, Phys. Rev. A 64, 024102 (2001).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Gómez, E. S.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

Guzmán, R.

Howell, J. C.

M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Iemmi, C.

Jennewein, T.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Kolenderski, P.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Laflamme, R.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Lima, G.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

Lizana, A.

Lu, G.

Z. Zhang, G. Lu, and F. T. S. Yu, Opt. Eng. 33, 3018 (1994).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Márquez, A.

Mataloni, P.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

Monken, C. H.

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

Moreno, I.

Mosk, A. P.

Neves, L.

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011).
[CrossRef]

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 84, 012330 (2011).
[CrossRef]

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 82, 055801 (2010).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

Nogueira, W. A. T.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011).
[CrossRef]

O’Sullivan-Hale, M. N.

M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Pádua, S.

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

Rossi, A.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

Saavedra, C.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Sinha, U.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Solís-Prosser, M. A.

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 84, 012330 (2011).
[CrossRef]

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 82, 055801 (2010).
[CrossRef]

Torres-Ruiz, F. A.

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

Vallone, G.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

van Putten, E. G.

Vargas, A.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Vellekoop, I. M.

Volpini, M.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Xavier, G. B.

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

Youning, L.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Yu, F. T. S.

Z. Zhang, G. Lu, and F. T. S. Yu, Opt. Eng. 33, 3018 (1994).
[CrossRef]

Yzuel, M.

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. Yzuel, Appl. Opt. 43, 6278 (2004).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, Opt. Eng. 40, 2558 (2001).
[CrossRef]

Yzuel, M. J.

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Zhang, Z.

Z. Zhang, G. Lu, and F. T. S. Yu, Opt. Eng. 33, 3018 (1994).
[CrossRef]

Zhao, T.

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

Zuegel, J. D.

Appl. Opt. (3)

J. Phys. B (1)

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, J. Phys. B 41, 185501 (2008).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Opt. Eng. (2)

Z. Zhang, G. Lu, and F. T. S. Yu, Opt. Eng. 33, 3018 (1994).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, Opt. Eng. 40, 2558 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Phys. Rev. A (4)

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 84, 012330 (2011).
[CrossRef]

P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein, Phys. Rev. A 86, 012321 (2012).
[CrossRef]

M. A. Solís-Prosser and L. Neves, Phys. Rev. A 82, 055801 (2010).
[CrossRef]

J. Fiurášek, Phys. Rev. A 64, 024102 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef]

M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Sci. Rep. (1)

S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, Sci. Rep. 3, 2316 (2013).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

S. M. Barnett, Quantum Information (Oxford University, 2009).

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Figures (4)

Fig. 1.
Fig. 1.

Ten-pixels period gratings: (a) binary and (b) blazed. (c) Upper panels: amplitude and phase components of the aperture T(x) for preparing the state 0.56|0+0.83ei0.63π|1. Lower panels: corresponding phase masks for preparing this state at the first diffraction order. (d) Normalized intensity distributions and theoretical prediction for measurements of this state at near (upper panel) and far field (lower panel). (e) Real and imaginary parts of the experimentally reconstructed density matrix using the blazed grating. The ideal density matrix is shown as a blue wire grid, and the obtained fidelity is F=0.996.

Fig. 2.
Fig. 2.

Experimental setup (details in text).

Fig. 3.
Fig. 3.

Bloch spheres showing the fidelities of preparation of spatial qubit states using (a) binary and (b) blazed gratings. The latitude θ[0,π] and the longitude ϕ(π,π] parametrizes an arbitrary pure state |ψ=cos(θ/2)|0+eiϕsin(θ/2)|1 on the sphere surface.

Fig. 4.
Fig. 4.

(a) Three- and (b) seven-dimensional qudit states with their corresponding normalized intensity distributions for measurements at near and far field, and the experimentally reconstructed density matrices using the blazed grating. The ideal density matrices are shown as blue wire grids and the obtained fidelities are shown in the insets.

Tables (1)

Tables Icon

Table 1. Average Fidelities of Preparation for D-level Qudits

Equations (3)

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

|ψ==0D1β˜|,
e1bin=2π2(1cosφ0),
e1bla=sinc2(1φ02π),

Metrics