## Manipulating spatial qudit states with programmable optical devices

Optics Express, Vol. 17, Issue 13, pp. 10688-10696 (2009)

http://dx.doi.org/10.1364/OE.17.010688

Acrobat PDF (387 KB)

### Abstract

The study of how to generate high-dimensional quantum states (qudits) is justified by the advantages that they can bring for the field of quantum information. However, to have some real practical potential for quantum communication, these states must be also of simple manipulation. Spatial qudits states, which are generated by engineering the transverse momentum of the parametric down-converted photons, have been until now considered of hard manipulation. Nevertheless, we show in this work a simple technique for modifying these states. This technique is based on the use of programmable diffractive optical devices, that can act as spatial light modulators, to define the Hilbert space of these photons instead of pre-fabricated multi-slits.

© 2009 Optical Society of America

## 1. Introduction

1. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 21–27 (2003).
[CrossRef]

2. J. Plewa, E. Tanner, D. Mueth, and D. G. Grier, “Processing carbon nanotubes with holographic optical tweezers,” Opt. Express **12**, 1978–1981 (2004).
[CrossRef] [PubMed]

3. A. Gogo, W. D. Snyder, and M. Beck, “Comparing quantum and classical correlations in a quantum eraser,” Phys. Rev. A **71**, 052103 (2005).
[CrossRef]

4. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: Application to quantum key distribution,” Appl. Opt. **47**, A32–A42 (2008).
[CrossRef] [PubMed]

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316 (2001).
[CrossRef] [PubMed]

6. E. Yao, S. Franke-Arnold, J. Courtial, and M. J. Padgett, “Observation of quantum entanglement using spatial light modulators,” Opt. Express **14**, 13089–13094 (2006).
[CrossRef] [PubMed]

7. M. Stütz, S. Gröblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. **90**, 261114 (2007).
[CrossRef]

8. A. K. Jha, B. Jack, E. Yao, J. Leach, R.W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A **78**, 043810 (2008).
[CrossRef]

7. M. Stütz, S. Gröblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. **90**, 261114 (2007).
[CrossRef]

*et al*., [21

21. I. Moreno, P. Veláquez, C. R. Fernández-Pousa, and M. M. Sánchez-López, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. **94**, 3697–3702 (2003).
[CrossRef]

## 2. Experiment

### 2.1. Experimental Setup

22. C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A **57**, 3123–3126 (1998).
[CrossRef]

20. L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. **94**, 100501 (2005).
[CrossRef] [PubMed]

23. L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich, and S. Pádua, “Characterizing entanglement in qubits created with spatially correlated twin photons,” Phys. Rev. A **76**, 032314 (2007).
[CrossRef]

24. G. Lima, F.A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, “Generating mixtures of spatial qubits,” Opt. Commun. **281**, 5058–59062 (2008).
[CrossRef]

25. X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming Spatial Entanglement Using a Domain-Engineering Technique,” Phys. Rev. Lett. **101**, 233601 (2008).
[CrossRef] [PubMed]

26. W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A **79**043817 (2009).
[CrossRef]

^{+}-ion laser operating at 351.1 nm and with an average power of 150 mw is focused into a 5-mm-long

*β*-barium borate crystal cut for type-II parametric down-conversion luminescence. Degenerated down-converted photons with the wavelength of 702 nm are selected using interference filters with small bandwidths (10 nm FWHM). The LCD is placed at the propagation path of the idler photon at a distance of 600 mm from the crystal. Our SLM is composed of two polarizers (

*P*

_{1}and

*P*

_{2}) and a twisted nematic liquid crystal display (CRL-Opto, model XGA2) with a spatial resolution of 1024×768 monochrome pixels. The polarizers are placed close to the LCD panel (2 cm away from each side). The LCD panel is connected to an interface PCB and it allows us to drive the programmable optoelectronics device from a computer. The horizontal and vertical pixel dimensions are 23

*µ*m and 16

*µ*m, respectively. Furthermore, the separations between pixels are 3

*µ*m in the horizontal and 10

*µ*m in the vertical directions. Initially, the SLM is addressed with a digital four-slit. The slit’s width is 2

*a*=101

*µ*m and the distance between two consecutive slits is

*d*=208

*µ*m (See Fig. 1(b)). After being transmitted by the SLM, the idler photon propagates through a 150 mm focal length lens (

*L*), which is 300 mm from the LCD. The idler photon is detected with the experimental setup in two distinct configurations. The first one is used for measuring the image of the aperture mounted in the LCD, and the second one for measuring the interference pattern of the transmitted idler beam at the focal plane of lens

_{i}*L*. At the first configuration, the avalanche photodiode detector (

_{i}*D*) is placed at the LCD plane of image formation which is at 1200 mm from the crystal. At the second configuration, this detector is moved to the focal plane of

_{i}*L*lens which is at 1050 mm from the crystal. The spatial mode of the signal photon is defined by pinholes placed along its propagation path. After the transmission through these pinholes, the signal photons are then focused with the lens

_{i}*L*inside the detector

_{s}*D*.

_{s}*L*has a focal length of 300 mm and it is placed at 600 mm from the crystal.

_{s}*D*is kept fixed at the distance of 900 mm from the crystal. The photo detectors

_{s}*D*and

_{i}*D*are connected to a circuit used to record the singles and coincidences counts.

_{s}20. L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. **94**, 100501 (2005).
[CrossRef] [PubMed]

22. C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A **57**, 3123–3126 (1998).
[CrossRef]

27. P. H. S. Ribeiro, C. H. Monken, and G. A. Barbosa, “Measurement of coherence area in parametric downconversion luminescence,” Appl. Opt. **33**, 352–355 (1994).
[CrossRef] [PubMed]

*γ*〉

*is the spatial mode of the signal photon defined by the pinholes of its propagation*

_{s}*l*〉

*is a single photon state defined, up to a global phase factor, as*

_{i}### 2.2. Amplitude-only modulation of a down-converted beam

21. I. Moreno, P. Veláquez, C. R. Fernández-Pousa, and M. M. Sánchez-López, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. **94**, 3697–3702 (2003).
[CrossRef]

28. J. A. Davis, I. Moreno, and P. Tsai. “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. **37**, 937–945 (1998).
[CrossRef]

29. P. Mogensen and J. Gckstad, “Phase-only optical encryption,” Opt. Lett. **25**, 566–568 (2000).
[CrossRef]

30. J. A. Coy, M. Zaldarriaga, D. F. Grosz, and O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. **35**, 15–19 (1996).
[CrossRef]

31. J. Nicolas, J. Campos, and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid-crystal devices,” J. Opt. Soc. Am. A **19**, 1013–1020 (2002).
[CrossRef]

32. A. Márquez, C. Iemmi, and I. Moreno “Quantitative predictions of the modulation behavior of twister nematic liquid crystal displays based on a simple physical model,” Opt. Eng. **40**, 2558–2564 (2001).
[CrossRef]

*X*,

*Y*,

*Z*and

*W*[21

21. I. Moreno, P. Veláquez, C. R. Fernández-Pousa, and M. M. Sánchez-López, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. **94**, 3697–3702 (2003).
[CrossRef]

33. C. R. Fernández-Pousa, I. Moreno, N. Bennis, and C. Gómez-Reino, “Generalized formulation and symmetry properties of reciprocal non-absorbing polarization devices: application to liquid-crystal displays,” J. Opt. Soc. Am. A **17**, 2074–2080 (2000).
[CrossRef]

*ϖ*is a global phase shift. The method of Moreno

*et. al*[21

**94**, 3697–3702 (2003).
[CrossRef]

*i*1n corresponds, for example, to the optical configuration at which the initial and the second polarizers of the SLM are set to the linear horizontal polarization direction. In Fig. 2(b), we can see the values for the Jones matrix coefficients of our SLM as a function of its grey level. In Fig. 2(c), the predicted and the experimental curves for the transmission of the SLM, as a function of its grey level, are shown for a specific optical configuration, where our SLM is supposed to be modulating only the amplitude of a incident light at the wavelength of 702 nm. In this configuration, the phase being modulated should be constant regardless of the grey level that the pixels of the liquid crystal display are set for.

### 2.3. Results

34. G. Lima, L. Neves, I F. Santos, J. G. Aguirre Gómez, C. Saavedra, and S. Pádua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A **73**, 032340 (2006).
[CrossRef]

*D*is scanned transversely to the SLM-multi-slit image should be equally weighted. One can clearly see the good agreement between our theory and the obtained experimental results. The measurements performed at the image plane, however, do not guarantee that the state of the SLM-qudit can indeed be represented by a coherent superposition of the logical spatial states as given by Eq. (4). There are other distinct types of states that would generate the same results; for example: a pure state where the amplitudes of the superposition are complex and also a completely incoherent mixed state. Therefore, it is necessary to implement another type of measurement which exclude these possibilities. This is done by measuring the coincidence interference pattern formed at the focal plane of the

_{i}*L*lens. For doing this measurement, the idler detector is now moved to a distance of 1050 mm from the crystal which corresponds to the focal plane of

_{i}*L*. For a pure state whose amplitudes have distinct phases, the maximal peak of this pattern would be shifted out of the transverse center of the SLM-multi-slit. If the state is completely incoherent, there will be no interference at all. The coincidence interference pattern recorded for the four-slit [100, 100, 100, 100] is shown in Fig. 4(a). One can clearly see the interference fringe structure without any phase shift. The solid curve represents the detection probability calculated by considering the state of Eq. (4) and a extra term which accounts for the finite size of the source. This reduces the idler-degree of transverse coherence at the LCD-plane and explain the less-than-one visibility of the patterns observed. The good fit between our experimental results and this theoretical curve means that Eq. (4) can indeed be seen as a good approximation for the experimentally generated initial state.

_{i}*ζ*

_{2}〉

*and for |*

_{i}*ζ*

_{3}〉

*are shown in Fig. 3(b) and Fig. 3(c), respectively. In order to verify that the amplitude-modulation of the slits with different grey levels was not introducing complex phases or de-coherence to the state, we measured again the interference patterns of the these new four-slits. The patterns recorded with [100, 75, 50, 25] and with [50, 100, 25, 100] are shown in Fig. 4(b) and Fig. 4(c), respectively. Again we have the coincidence interference patterns without phase shifts. These curves also fit well with the theoretical predictions for the detection probabilities calculated in the same way explained before, but now considering Eq. (5) and Eq. (6) for the qudit superpositions. These results, together, demonstrate that the amplitude-only modulation of programmable spatial light modulators can indeed be used for modifying, in a controlled way, the spatial qudit states.*

_{i}## 3. Conclusion

35. G. Lima, F. A. Torres-Ruiz, L. Neves, A Delgado, C Saavedra, and S. Pádua, “Measurement of spatial qubits,” J. Phys. B **41**, 185501 (2008).
[CrossRef]

36. G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A **78**, 012307 (2008).
[CrossRef]

37. A. B. Klimov, C. Muoz, A. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A **77**, 060303(R) (2008).
[CrossRef]

## Acknowledgments

## References and links

1. | D. G. Grier, “A revolution in optical manipulation,” Nature |

2. | J. Plewa, E. Tanner, D. Mueth, and D. G. Grier, “Processing carbon nanotubes with holographic optical tweezers,” Opt. Express |

3. | A. Gogo, W. D. Snyder, and M. Beck, “Comparing quantum and classical correlations in a quantum eraser,” Phys. Rev. A |

4. | M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: Application to quantum key distribution,” Appl. Opt. |

5. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

6. | E. Yao, S. Franke-Arnold, J. Courtial, and M. J. Padgett, “Observation of quantum entanglement using spatial light modulators,” Opt. Express |

7. | M. Stütz, S. Gröblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. |

8. | A. K. Jha, B. Jack, E. Yao, J. Leach, R.W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A |

9. | H. Bechmann-Pasquinucci and A. Peres, “Quantum Cryptography with 3-State Systems,” Phys. Rev. A |

10. | T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A |

11. | D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubit,” Phys. Rev. Lett. |

12. | J. S. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. |

13. | A. Aspect, “Bells inequality test: more ideal than ever,” Nature |

14. | Yu. I. Bogdanov, M. V. Chekhova, S. P. Kulik, G. A. Maslennikov, A. A. Zhukov, C. H. Oh, and M. K. Tey, “Qutrit State Engineering with Biphotons,” Phys. Rev. Lett. |

15. | G. Vallone, E. Pomarico, F. De Martini, and P. Mataloni, “Experimental realization of polarization qutrits from nonmaximally entangled states,” Phys. Rev. A |

16. | B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. OBrien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating Biphotonic Qutrits,” Phys. Rev. Lett. |

17. | S.-Y. Baek, S. S. Straupe, A. P. Shurupov, S. P. Kulik, and Y.-H. Kim, “Preparation and characterization of arbitrary states of four-dimensional qudits based on biphotons,” Phys. Rev. A |

18. | R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-Type Test of Energy-Time Entangled Qutrits,” Phys. Rev. Lett. |

19. | A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath Entanglement of Two Photons,” Phys. Rev. Lett. |

20. | L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. |

21. | I. Moreno, P. Veláquez, C. R. Fernández-Pousa, and M. M. Sánchez-López, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. |

22. | C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A |

23. | L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich, and S. Pádua, “Characterizing entanglement in qubits created with spatially correlated twin photons,” Phys. Rev. A |

24. | G. Lima, F.A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, “Generating mixtures of spatial qubits,” Opt. Commun. |

25. | X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming Spatial Entanglement Using a Domain-Engineering Technique,” Phys. Rev. Lett. |

26. | W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A |

27. | P. H. S. Ribeiro, C. H. Monken, and G. A. Barbosa, “Measurement of coherence area in parametric downconversion luminescence,” Appl. Opt. |

28. | J. A. Davis, I. Moreno, and P. Tsai. “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. |

29. | P. Mogensen and J. Gckstad, “Phase-only optical encryption,” Opt. Lett. |

30. | J. A. Coy, M. Zaldarriaga, D. F. Grosz, and O. E. Martinez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. |

31. | J. Nicolas, J. Campos, and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid-crystal devices,” J. Opt. Soc. Am. A |

32. | A. Márquez, C. Iemmi, and I. Moreno “Quantitative predictions of the modulation behavior of twister nematic liquid crystal displays based on a simple physical model,” Opt. Eng. |

33. | C. R. Fernández-Pousa, I. Moreno, N. Bennis, and C. Gómez-Reino, “Generalized formulation and symmetry properties of reciprocal non-absorbing polarization devices: application to liquid-crystal displays,” J. Opt. Soc. Am. A |

34. | G. Lima, L. Neves, I F. Santos, J. G. Aguirre Gómez, C. Saavedra, and S. Pádua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A |

35. | G. Lima, F. A. Torres-Ruiz, L. Neves, A Delgado, C Saavedra, and S. Pádua, “Measurement of spatial qubits,” J. Phys. B |

36. | G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A |

37. | A. B. Klimov, C. Muoz, A. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 11, 2009

Revised Manuscript: May 18, 2009

Manuscript Accepted: June 8, 2009

Published: June 11, 2009

**Citation**

G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, "Manipulating spatial qudit states with programmable optical devices," Opt. Express **17**, 10688-10696 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10688

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### References

- D. G. Grier, "A revolution in optical manipulation," Nature 424, 21-27 (2003). [CrossRef]
- J. Plewa, E. Tanner, D. Mueth and D. G. Grier, "Processing carbon nanotubes with holographic optical tweezers," Opt. Express 12, 1978-1981 (2004). [CrossRef] [PubMed]
- A. Gogo,W. D. Snyder and M. Beck, "Comparing quantum and classical correlations in a quantum eraser," Phys. Rev. A 71, 052103 (2005). [CrossRef]
- M. T. Gruneisen, W. A. Miller, R. C. Dymale and A. M. Sweiti, "Holographic generation of complex fields with spatial light modulators: Application to quantum key distribution," Appl. Opt. 47, A32-A42 (2008). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001). [CrossRef] [PubMed]
- E. Yao, S. Franke-Arnold, J. Courtial and M. J. Padgett, "Observation of quantum entanglement using spatial light modulators," Opt. Express 14, 13089-13094 (2006). [CrossRef] [PubMed]
- M. Stutz, S. Groblacher, T. Jennewein and A. Zeilinger, "How to create and detect N-dimensional entangled photons with an active phase hologram," Appl. Phys. Lett. 90, 261114 (2007). [CrossRef]
- A. K. Jha, B. Jack, E. Yao, J. Leach, R.W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Fourier relationship between the angle and angular momentum of entangled photons," Phys. Rev. A 78, 043810 (2008). [CrossRef]
- H. Bechmann-Pasquinucci and A. Peres, "Quantum Cryptography with 3-State Systems," Phys. Rev. A 85, 3313-3316 (2000).
- T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, "Security of quantum key distributions with entangled qudits," Phys. Rev. A 69, 032313 (2004). [CrossRef]
- D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski and A. Zeilinger, "Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubit," Phys. Rev. Lett. 85, 4418-4421 (2000). [CrossRef] [PubMed]
- J. S. Bell, "On the problem of hidden variables in quantum mechanics," Rev. Mod. Phys. 38, 447-452 (1966). [CrossRef]
- A. Aspect, "Bells inequality test: more ideal than ever," Nature 398, 189-190 (1999). [CrossRef]
- Yu. I. Bogdanov, M. V. Chekhova, S. P. Kulik, G. A. Maslennikov, A. A. Zhukov, C. H. Oh and M. K. Tey, "Qutrit State Engineering with Biphotons," Phys. Rev. Lett. 93, 230503 (2004). [CrossRef] [PubMed]
- G. Vallone, E. Pomarico, F. De Martini and P. Mataloni, "Experimental realization of polarization qutrits from nonmaximally entangled states," Phys. Rev. A 76, 012319 (2007). [CrossRef]
- B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. OBrien, K. J. Resch, A. Gilchrist and A. G. White, "Manipulating Biphotonic Qutrits," Phys. Rev. Lett. 100, 060504 (2008). [CrossRef] [PubMed]
- S.-Y. Baek, S. S. Straupe, A. P. Shurupov, S. P. Kulik and Y.-H. Kim, "Preparation and characterization of arbitrary states of four-dimensional qudits based on biphotons," Phys. Rev. A 78, 042321 (2008). [CrossRef]
- R. T. Thew, A. Acın, H. Zbinden and N. Gisin, "Bell-Type Test of Energy-Time Entangled Qutrits," Phys. Rev. Lett. 93, 010503 (2004). [CrossRef]
- A. Rossi, G. Vallone, A. Chiuri, F. De Martini and P. Mataloni, "Multipath Entanglement of Two Photons," Phys. Rev. Lett. 102, 153902 (2009). [CrossRef] [PubMed]
- L. Neves, G. Lima, J. G. A. Gomez, C. H. Monken, C. Saavedra and S. Padua, "Generation of Entangled States of Qudits using Twin Photons," Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]
- I. Moreno, P. Velaquez, C. R. Fernandez-Pousa and M. M. Sanchez-Lopez, "Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display," J. Appl. Phys. 94, 3697-3702 (2003). [CrossRef]
- C. H. Monken, P. H. S. Ribeiro and S. Padua, "Transfer of angular spectrum and image formation in spontaneous parametric down-conversion," Phys. Rev. A 57, 3123-3126 (1998). [CrossRef]
- L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich and S. Padua, "Characterizing entanglement in qubits created with spatially correlated twin photons," Phys. Rev. A 76, 032314 (2007). [CrossRef]
- G. Lima, F.A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, S. Padua, "Generating mixtures of spatial qubits," Opt. Commun. 281, 5058-5906 (2008). [CrossRef]
- X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu and N. B. Ming, "Transforming Spatial Entanglement Using a Domain-Engineering Technique," Phys. Rev. Lett. 101, 233601 (2008). [CrossRef] [PubMed]
- W. H. Peeters, J. J. Renema and M. P. van Exter, "Engineering of two-photon spatial quantum correlations behind a double slit," Phys. Rev. A 79043817 (2009). [CrossRef]
- P. H. S. Ribeiro, C. H. Monken and G. A. Barbosa, "Measurement of coherence area in parametric downconversion luminescence," Appl. Opt. 33, 352-355 (1994). [CrossRef] [PubMed]
- J. A. Davis., I. Moreno, and P. Tsai. "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998). [CrossRef]
- P. Mogensen and J. Gckstad, "Phase-only optical encryption," Opt. Lett. 25, 566-568 (2000). [CrossRef]
- J. A. Coy, M. Zaldarriaga, D. F. Grosz, and O. E. Martinez, "Characterization of a liquid crystal television as a programmable spatial light modulator," Opt. Eng. 35, 15-19 (1996). [CrossRef]
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