OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 10898–10905
« Show journal navigation

Remote state preparation of three-dimensional optical vortices

Ming Su and Lixiang Chen  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 10898-10905 (2014)
http://dx.doi.org/10.1364/OE.22.010898


View Full Text Article

Acrobat PDF (1453 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We propose a feasible scheme to remotely prepare three-dimensional (3D) optical vortex lines. Our scheme relies on the complete description of high-dimensional orbital angular momentum (OAM) entanglement in terms of the Laguerre-Gaussian modes. It is theoretically demonstrated that by simply changing the pump beam waist, we can remotely prepare the target photons in the vortex states of 3D interesting morphology, appearing as twisted vortex strands, separated vortex loops, and vortex link or knot. Furthermore, we employ the biphoton Klyshko picture to illustrate the conservation law of the OAM index and the spreading effect of the radial mode index p, where the Schmidt numbers are calculated to show the high-dimensional capacity of the quantum channels involved in the present remote state preparation.

© 2014 Optical Society of America

1. Introduction

The aim of quantum communication is to manipulate and transmit quantum information with higher efficiency and better security. The quantum teleportation, a portocol originally proposed by Bennett et al. [1

1. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef] [PubMed]

], describes how Alice helps Bob to remotely prepare a qubit state unknown to either by consuming 1 entanglement bit (ebit) and 2 classical bits (cbits). In contrast, in the protocol of remote state preparation (RSP) [2

2. C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett. 87, 077902 (2001). [CrossRef] [PubMed]

], similar task can be done by consuming only 1 ebit and 1 cbit, provided that Alice knows the state in advance but Bob does not. Both protocols rely on the entanglement to work, however, unlike the teleportation, the Bell-state measurement is not required in RSP. Besides, the communication cost is one question that concerned by many researchers [3

3. H.-K. Lo, “Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity,” Phys. Rev. A 62, 012313 (2000). [CrossRef]

5

5. S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004). [CrossRef] [PubMed]

]. Considering the trade-off between classical communication and entanglement, RSP has attracted a growing research interest in recent years, and several RSP demonstrations, such as the low-entanglement RSP [6

6. I. Devetak and T. Berger, “Low-entanglement remote state preparation,” Phys. Rev. Lett. 87, 197901 (2001). [CrossRef] [PubMed]

], the optimal RSP of mixed states [7

7. D. W. Berry and B. C. Sanders, “Optimal remote state preparation,” Phys. Rev. Lett. 90, 057901 (2003). [CrossRef] [PubMed]

], RSP in high-dimension space [8

8. B. Zeng and P. Zhang, “Remote-state preparation in higher dimension and the parallelizable manifold sn−1,” Phys. Rev. A 65, 022316 (2002). [CrossRef]

], and continuous variable RSP [9

9. Z. Kurucz, P. Adam, Z. Kis, and J. Janszky, “Continuous variable remote state preparation,” Phys. Rev. A 72, 052315 (2005). [CrossRef]

], have been reported. Besides, RSP of arbitrary photon polarization states [10

10. N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, and P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. 94, 150502 (2005). [CrossRef] [PubMed]

] and hybrid vector-polarization states [11

11. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010). [CrossRef]

] have been demonstrated, respectively.

2. RSP of 3D optical vortices

The proposed experimental scheme is depicted in Fig. 1. The OAM-entangled photon pairs are produced via the type-I collinear SPDC by a UV pump laser incident on a BBO crystal. The degenerate down-converted signal and idler beams are then separated by a non-polarizing beam splitter (BS). We start with the complete description of OAM entanglement in terms of the LG modes. The LG mode is a natural choice to describe twisted photons carrying OAM. In the cylindrical coordinates (R, ϕ, z), the normalized form of a LG mode is mathematically expressed by,
LGp(R,ϕ,z)=p!π(||+p)!R||exp(iϕ)(w2+iz/k)||+1exp(R22(w2+iz/k))×Lp||(R2w2+z2/k2w2)(w2iz/kw2+iz/k)p,
(1)
where w is the beam waist width, Lp||() is the generalized Laguerre polynomials, p and are the radial and azimuthal mode indices, respectively. With a fundamental Gaussian pump, the two-photon state can be described completely in terms of the LG modes as follows [20

20. J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003). [CrossRef]

22

22. A. M. Yao, “Angular momentum decomposition of entangled photons with an arbitrary pump,” New J. Phys. 13, 053048 (2011). [CrossRef]

],
|ΨSPDC=,ps,piCps,pi,|,ps|,pi,
(2)
where Cps,pi, denotes the coincidence amplitude for finding one signal photon in the mode of LGps and one idler photon in the mode of LGpi. Under the thin crystal approximation, the expression of Cps,pi, can be analytically given by [21

21. F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011). [CrossRef]

],
Cps,pi,Kpi,ps||(1γi2+γs2)pi(1+γi2γs2)ps(2γiγs)||(1+γi2+γs2)pi+ps+||×F12[pi,pspips||;1(γi2+γs2)21(γi2γs2)2],
(3)
where Kpi,ps||=(pi+ps+||)!pi!ps!(ps+||)!(pi+||)!, 2F1(·) is the Gauss hypergeometric function, and γs (γi) denotes the ratio of the pump beam waist wp to the signal (idler) beam waist ws (wi). Generally, the SPDC photon pairs are not produced in the maximally entangled state. For the OAM index, there is a limited spiral bandwidth, namely, the LG modes of lower OAM are generated more frequently than the higher-order ones. While for the azimuthal index, ps and pi are not perfectly correlated, namely, a single pi is generally spread to a spectrum of ps, as indicated by Eq. (3).

Fig. 1 The proposed scheme for RSP of 3D optical vortices (see the text for details).

Owing to the OAM conservation, we always have couples of azimuthal indices for signal and idler modes, namely, i = −s. In contrast, the correlation between ps and pi breaks down in a practical experiment, as the crystal size places an effective upper limit to pump beam waist [21

21. F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011). [CrossRef]

]. A most usually encountered case is that the pump beam size approaches that of signal and idler ones, namely, wp = w = 1 mm in our case.

Fig. 2 RSP of two- and three-strand vortex lines. (a) and (d) are the desired holograms; (b) and (e) are 3D vortex link and knot [16] at Alice’s side; (c) and (f) are the simulations of desired 3D structures of two- and three-strand vortices at Bob’s side, respectively.
Fig. 3 The transverse intensity patterns of the target photons. (a), (b) and (c): RSP of two-strand vortices; (d), (e) and (f): RSP of three-strand vortices.

Fig. 4 The 3D structures of the remotely prepared optical vortices with different pump beam waists wp, where the upper and bottom lows are prepared with = 2 and 3, respectively. (a), (d): Vortex strands (wp = 1 mm); (b), (e): Vortex loops (wp = 1.5 mm); (c), (f): Vortex link and knot (wp = 3 mm).

3. Biphoton Klyshko picture and the Schmidt number

As is well known, high-dimensional entangled systems are desirable for enhancing the robustness and security of quantum communications [23

23. H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000). [CrossRef]

], and for increasing the information capacity of a photon [24

24. P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012). [CrossRef] [PubMed]

]. So it is also crucial for us to investigate the high-dimensional quantum nature of our RSP protocol. A most convenient and direct approach to quantify the amount of entanglement in a pure two-photon entangled state is the Schmidt number, which is given as κ=1/TrA(ρA2)=1/TrB(ρB2), with ρA = TrB(|Ψ〉 〈Ψ|) and ρB = TrA(|Ψ〉 〈Ψ|) being the reduced density matrices for the two subsystems, respectively. The Schmidt number also characterizes the number of effective entangled modes involved in the entangled state [25

25. C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]

] or the dimensionality of that Hilbert space [26

26. J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multimode fields,” Phys. Rev. A 77, 033845 (2008). [CrossRef]

]. According to Eqs. (4) and (5), we know that the two-photon state post-selected by our RSP scheme can be written as,
|Ψps=pi(C0,pi0,0|0,0|0,pi+C0,pi,|,0|,pi+C1,pi,|,1|,pi+C2,pi,|,2|,pi).
(6)

After a straightforward calculation based on Eq. (6), we obtain the Schmidt numbers κ different pump beam waists wp, as are listed in Table 1. One can see that in all cases the Schmidt numbers are obviously beyond the two-dimensional thinking with polarization entanglement, which therefore suggests that there may be many entangled spatial modes participating in process of our RSP protocol.

Table 1. The Schmidt numbers for different pump beam waists in our RSP scheme.

table-icon
View This Table

Fig. 5 The simplified Klyshko picture for visualization of biphoton channels sustained by the LG modes involved in our RSP scheme.

4. Conclusion

Acknowledgments

References and links

1.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef] [PubMed]

2.

C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett. 87, 077902 (2001). [CrossRef] [PubMed]

3.

H.-K. Lo, “Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity,” Phys. Rev. A 62, 012313 (2000). [CrossRef]

4.

A. K. Pati, “Minimum classical bit for remote preparation and measurement of a qubit,” Phys. Rev. A 63, 014302 (2000). [CrossRef]

5.

S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004). [CrossRef] [PubMed]

6.

I. Devetak and T. Berger, “Low-entanglement remote state preparation,” Phys. Rev. Lett. 87, 197901 (2001). [CrossRef] [PubMed]

7.

D. W. Berry and B. C. Sanders, “Optimal remote state preparation,” Phys. Rev. Lett. 90, 057901 (2003). [CrossRef] [PubMed]

8.

B. Zeng and P. Zhang, “Remote-state preparation in higher dimension and the parallelizable manifold sn−1,” Phys. Rev. A 65, 022316 (2002). [CrossRef]

9.

Z. Kurucz, P. Adam, Z. Kis, and J. Janszky, “Continuous variable remote state preparation,” Phys. Rev. A 72, 052315 (2005). [CrossRef]

10.

N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, and P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. 94, 150502 (2005). [CrossRef] [PubMed]

11.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010). [CrossRef]

12.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293 (2009). [CrossRef]

13.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

14.

M. Berry and M. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Roy. Soc. A 457, 2251–2263 (2001). [CrossRef]

15.

M. Berry and M. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+ 1 spacetime,” J. Phys. A: Math. Gen. 34, 8877 (2001). [CrossRef]

16.

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). [CrossRef]

17.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004). [CrossRef] [PubMed]

18.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118–121 (2010). [CrossRef]

19.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106, 100407 (2011). [CrossRef] [PubMed]

20.

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003). [CrossRef]

21.

F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011). [CrossRef]

22.

A. M. Yao, “Angular momentum decomposition of entangled photons with an arbitrary pump,” New J. Phys. 13, 053048 (2011). [CrossRef]

23.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000). [CrossRef]

24.

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012). [CrossRef] [PubMed]

25.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]

26.

J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multimode fields,” Phys. Rev. A 77, 033845 (2008). [CrossRef]

27.

M. D’Angelo and Y. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005). [CrossRef]

28.

L. Chen, J. Leach, B. Jack, M. J. Padgett, S. Franke-Arnold, and W. She, “High-dimensional quantum nature of ghost angular Young’s diffraction,” Phys. Rev. A 82, 033822 (2010). [CrossRef]

29.

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153 (2014). [CrossRef]

OCIS Codes
(260.1960) Physical optics : Diffraction theory
(050.4865) Diffraction and gratings : Optical vortices
(270.5585) Quantum optics : Quantum information and processing
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Quantum Optics

History
Original Manuscript: February 19, 2014
Revised Manuscript: April 8, 2014
Manuscript Accepted: April 23, 2014
Published: April 29, 2014

Citation
Ming Su and Lixiang Chen, "Remote state preparation of three-dimensional optical vortices," Opt. Express 22, 10898-10905 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10898


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef] [PubMed]
  2. C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett. 87, 077902 (2001). [CrossRef] [PubMed]
  3. H.-K. Lo, “Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity,” Phys. Rev. A 62, 012313 (2000). [CrossRef]
  4. A. K. Pati, “Minimum classical bit for remote preparation and measurement of a qubit,” Phys. Rev. A 63, 014302 (2000). [CrossRef]
  5. S. A. Babichev, B. Brezger, A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004). [CrossRef] [PubMed]
  6. I. Devetak, T. Berger, “Low-entanglement remote state preparation,” Phys. Rev. Lett. 87, 197901 (2001). [CrossRef] [PubMed]
  7. D. W. Berry, B. C. Sanders, “Optimal remote state preparation,” Phys. Rev. Lett. 90, 057901 (2003). [CrossRef] [PubMed]
  8. B. Zeng, P. Zhang, “Remote-state preparation in higher dimension and the parallelizable manifold sn−1,” Phys. Rev. A 65, 022316 (2002). [CrossRef]
  9. Z. Kurucz, P. Adam, Z. Kis, J. Janszky, “Continuous variable remote state preparation,” Phys. Rev. A 72, 052315 (2005). [CrossRef]
  10. N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. 94, 150502 (2005). [CrossRef] [PubMed]
  11. J. T. Barreiro, T.-C. Wei, P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010). [CrossRef]
  12. M. R. Dennis, K. O’Holleran, M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293 (2009). [CrossRef]
  13. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]
  14. M. Berry, M. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Roy. Soc. A 457, 2251–2263 (2001). [CrossRef]
  15. M. Berry, M. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+ 1 spacetime,” J. Phys. A: Math. Gen. 34, 8877 (2001). [CrossRef]
  16. J. Leach, M. Dennis, J. Courtial, M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). [CrossRef]
  17. J. Leach, M. R. Dennis, J. Courtial, M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004). [CrossRef] [PubMed]
  18. M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118–121 (2010). [CrossRef]
  19. J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106, 100407 (2011). [CrossRef] [PubMed]
  20. J. P. Torres, Y. Deyanova, L. Torner, G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003). [CrossRef]
  21. F. M. Miatto, A. M. Yao, S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011). [CrossRef]
  22. A. M. Yao, “Angular momentum decomposition of entangled photons with an arbitrary pump,” New J. Phys. 13, 053048 (2011). [CrossRef]
  23. H. Bechmann-Pasquinucci, W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000). [CrossRef]
  24. P. B. Dixon, G. A. Howland, J. Schneeloch, J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012). [CrossRef] [PubMed]
  25. C. K. Law, J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]
  26. J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multimode fields,” Phys. Rev. A 77, 033845 (2008). [CrossRef]
  27. M. D’Angelo, Y. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005). [CrossRef]
  28. L. Chen, J. Leach, B. Jack, M. J. Padgett, S. Franke-Arnold, W. She, “High-dimensional quantum nature of ghost angular Young’s diffraction,” Phys. Rev. A 82, 033822 (2010). [CrossRef]
  29. L. Chen, J. Lei, J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153 (2014). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited