## Multidimensional quantum information based on single-photon temporal wavepackets |

Optics Express, Vol. 20, Issue 28, pp. 29174-29184 (2012)

http://dx.doi.org/10.1364/OE.20.029174

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### Abstract

We propose a multidimensional quantum information encoding approach based on temporal modulation of single photons, where the Hilbert space can be spanned by an in-principle infinite set of orthonormal temporal profiles. We analyze two specific realizations of such modulation schemes, and show that error rate per symbol can be smaller than 1% for practical implementations. Temporal modulation may enable multidimensional quantum communication over the existing fiber optical infrastructure, as well as provide an avenue for probing high-dimensional entanglement approaching the continuous limit.

© 2012 OSA

## 1. Introduction

## 2. Photon temporal-profile qudits

*j*, the photon is modulated by a pattern

*j*round trips it has acquired the mode structure

_{.}Such optical field integration filters have been demonstrated in classical applications based on fiber gratings [22

22. Y. Park, T.-J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express **16**(22), 17817–17825 (2008). [CrossRef] [PubMed]

*d*times for a Hilbert space of dimension

*d*, and the photon incident in state

*d*-dimensional basis (Fig. 1(a)). Such projection measurement can be also extended to characterize the multidimensional entanglement in the time domain, as shown in Fig. 1(b).

20. F. Wolfgramm, X. Xing, A. Cerè, A. Predojević, A. M. Steinberg, and M. W. Mitchell, “Bright filter-free source of indistinguishable photon pairs,” Opt. Express **16**(22), 18145–18151 (2008). [CrossRef] [PubMed]

## 3. Phase-flip encoding scheme

*n*time intervals which have equal intensity integrals, and a phase of

*0*or

*π*radians is written on each of the

*n*time sections (Fig. 2(a) ). By assigning certain phase profile patterns to these equal intensity-integral intervals, one can have a zero overlap integral (Eq. (5) for orthogonal phase profiles. A systematic method for the selection of these orthogonal states is implemented by using columns or rows of the Walsh matrix [23

23. J. L. Walsh, “A Closed Set of Normal Orthogonal Functions,” Am. J. Math. **45**(1), 5–24 (1923). [CrossRef]

*n*-by

*-n*Walsh matrix with

*n>*2 exists if

*n*is a multiple of 4. Such a phase profile can be written as:where

*n*, depends on the desired encoding dimension,

*d*: for

*d*<10, the optimal

*n*= 32, whereas for

*d*>10 the optimal n = 64. A finite speed of EOM, results also in slightly reduced efficiency of detection because the phase flip is not instantaneous, and thus even for a detection profile designed to undo an initial profile the photon spectrum will be modified and partially reflected by the filter.

## 4. Linear phase-ramp scheme

*d*is only limited by the dimensionless quantity

*N*,

*d≤N*, as defined in Eq. (9). For a

*d*-dimensional qudit, the linear phase rate

*Δ*is chosen to befor integer numbers

_{k}*k*> 0 and

*d*≥2. Some example of phase profiles for

*d =*16 and their frequency spectra are given in Fig. 3(a) and Fig. 3(b), respectively.

*d≤N*and set

*N*= 60 for different EOM modulation bandwidth just to show the behavior of the scheme. For a constant

*N*, the ERS increases with increasing encoding dimension, because of increasing overlap of the spectra of the two encoding profiles. The rate of the ERS increase with

*d*is larger for smaller

*N*, which is a result of smaller spectral spacing between the different profiles. When the encoding dimension

*d*approaches or exceeds

*N*, the ERS saturates due to the fact that the spectra of the different profiles are already largely overlapping, and additional overlap does not affect the errors significantly. A faster EOM modulation speed leads to a smaller overlap, and thus a lower ERS. For example, with a 1ns modulation speed for a 100ns photon, the ERS can be made lower than 0.6% for encoding dimensions up to 10. The dominant factor of ERS is the spectral overlap of states with different temporal profiles, which is determined by the spacing in frequency between two consecutive phase profiles. In Fig. 3(d), we have shown the ERS as a function of the normalized dimension

*d’ = (d-1)/N*, which physically corresponds the number of symbols one encodes per available frequency bin determined by the EOM speed. For a given

*d’*, the spacing in frequency is the same. The ERS for different EOM speed converges at

*d’*~1, as expected. For small

*d’*, the ERS is slightly higher for higher dimensions. This is because for the same

*d’*, the fixed frequency spacing leads to a fixed intersymbol interference rate for a given pair of symbols, but the larger number of symbols means the total ERS is higher.

## 5. Superpositions and mutual information

*d*with a corresponding set of temporal profiles

*d*= 4. To generate a superposition basis, we start with a

*d × d*matrix whose entries are random numbers such that the rows or columns of the matrix form a linearly independent set of vectors. With the Gram–Schmidt process [24] we orthonormalize the set to get a superposition basis

*k*=

*j*), whereas the off-diagonal entries represent error probabilities. The superposition basis

*a*and

*b*are the amplitude and phase parameters, respectively. In order to project the photon state onto

*d*= 16, the optimal filter bandwidth Δ

_{filter}is found to be approximately 1.5Δ

_{photon;}for this value, the mutual information is 3.3 bits/symbol, while for equal-amplitude superpositions it drops considerably, but is still well above 1.

*d*the loss has a stronger effect on ERS. For very high loss levels, the ERS reaches a maximum value of 1-1/

*d*. Nevertheless, at loss levels below 60%, the loss-induced ERS is comparable to that caused by the inter-symbol interference, so that the overall ERS remains low.

27. R. T. Thew, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A **66**(1), 012303 (2002). [CrossRef]

11. M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nat. Phys. **3**(10), 692–695 (2007). [CrossRef]

12. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication,” Phys. Rev. Lett. **82**(12), 2594–2597 (1999). [CrossRef]

*et. al.*have shown, based on the Schmidt decomposition, that the effective Hilbert space of the PDC biphoton state can be finite-dimensional and therefore, it is possible to reconstruct the biphoton state with high fidelity using a finite-dimensional projection [28

28. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. **84**(23), 5304–5307 (2000). [CrossRef] [PubMed]

*M*is given by the ratio of the cavity linewidth to that of the pump. In this case superpositions of

*M*low-frequency modes are sufficient for tomography and reconstruction of the wavefunction with high fidelity. High error rates could prevent the violation of Bell’s inequalities by reducing the two-photon interference visibility below the classical field theory limit [4

4. J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. **64**(21), 2495–2498 (1990). [CrossRef] [PubMed]

6. Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Observation of nonlocal interference in separated photon channels,” Phys. Rev. Lett. **65**(3), 321–324 (1990). [CrossRef] [PubMed]

_{EOM}/Δ

_{photon}~100, even for Hilbert space dimensions as high as 10, the error rates are well below 1%, which is smaller than the error rates in previous experiments demonstrating violations of Bell’s inequalities for qubits [4

4. J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. **64**(21), 2495–2498 (1990). [CrossRef] [PubMed]

10. A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. **7**(9), 677–680 (2011). [CrossRef]

## 6. Conclusions

## Acknowledgments

## References and links:

1. | J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. |

2. | J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. |

3. | T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in bell experiments using qudits,” Phys. Rev. Lett. |

4. | J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. |

5. | J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. |

6. | Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Observation of nonlocal interference in separated photon channels,” Phys. Rev. Lett. |

7. | P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A |

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

9. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

10. | A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. |

11. | M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nat. Phys. |

12. | J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication,” Phys. Rev. Lett. |

13. | R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. |

14. | O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. |

15. | A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express |

16. | B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. |

17. | C. Polycarpou, K. N. Cassemiro, G. Venturi, A. Zavatta, and M. Bellini, “Adaptive detection of arbitrarily shaped ultrashort quantum light states,” Phys. Rev. Lett. |

18. | Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. |

19. | C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Time-bin-modulated biphotons from cavity-enhanced down-conversion,” Phys. Rev. Lett. |

20. | F. Wolfgramm, X. Xing, A. Cerè, A. Predojević, A. M. Steinberg, and M. W. Mitchell, “Bright filter-free source of indistinguishable photon pairs,” Opt. Express |

21. | D. Kielpinski, J. F. Corney, and H. M. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. |

22. | Y. Park, T.-J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express |

23. | J. L. Walsh, “A Closed Set of Normal Orthogonal Functions,” Am. J. Math. |

24. | G. Golub and C. VanLoan, |

25. | M. A. Nielsen and I. L. Chuang, |

26. | G. Smith, “Quantum Channel Capacities,” arXiv:1007.2855 (2010). |

27. | R. T. Thew, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A |

28. | C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. |

**OCIS Codes**

(270.5565) Quantum optics : Quantum communications

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: September 28, 2012

Revised Manuscript: November 14, 2012

Manuscript Accepted: December 3, 2012

Published: December 17, 2012

**Citation**

Alex Hayat, Xingxing Xing, Amir Feizpour, and Aephraim M. Steinberg, "Multidimensional quantum information based on single-photon temporal wavepackets," Opt. Express **20**, 29174-29184 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29174

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### References

- J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008). [CrossRef]
- J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95(26), 260501 (2005). [CrossRef] [PubMed]
- T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in bell experiments using qudits,” Phys. Rev. Lett. 104(6), 060401 (2010). [CrossRef] [PubMed]
- J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64(21), 2495–2498 (1990). [CrossRef] [PubMed]
- J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62(19), 2205–2208 (1989). [CrossRef] [PubMed]
- Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Observation of nonlocal interference in separated photon channels,” Phys. Rev. Lett. 65(3), 321–324 (1990). [CrossRef] [PubMed]
- P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A 41(5), 2910–2913 (1990). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]
- G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]
- A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011). [CrossRef]
- M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, “Entangling independent photons by time measurement,” Nat. Phys. 3(10), 692–695 (2007). [CrossRef]
- J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication,” Phys. Rev. Lett. 82(12), 2594–2597 (1999). [CrossRef]
- R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93(1), 010503 (2004). [CrossRef]
- O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008). [CrossRef] [PubMed]
- A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19(15), 13770–13778 (2011). [CrossRef] [PubMed]
- B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13(6), 065029 (2011). [CrossRef]
- C. Polycarpou, K. N. Cassemiro, G. Venturi, A. Zavatta, and M. Bellini, “Adaptive detection of arbitrarily shaped ultrashort quantum light states,” Phys. Rev. Lett. 109(5), 053602 (2012). [CrossRef] [PubMed]
- Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83(13), 2556–2559 (1999). [CrossRef]
- C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Time-bin-modulated biphotons from cavity-enhanced down-conversion,” Phys. Rev. Lett. 97(22), 223601 (2006). [CrossRef] [PubMed]
- F. Wolfgramm, X. Xing, A. Cerè, A. Predojevi?, A. M. Steinberg, and M. W. Mitchell, “Bright filter-free source of indistinguishable photon pairs,” Opt. Express 16(22), 18145–18151 (2008). [CrossRef] [PubMed]
- D. Kielpinski, J. F. Corney, and H. M. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106(13), 130501 (2011). [CrossRef] [PubMed]
- Y. Park, T.-J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008). [CrossRef] [PubMed]
- J. L. Walsh, “A Closed Set of Normal Orthogonal Functions,” Am. J. Math. 45(1), 5–24 (1923). [CrossRef]
- G. Golub and C. VanLoan, Matrix Computations (Johns Hopkins University Press, 3rd ed., 1996).
- M. A. Nielsen and I. L. Chuang, Quantum Computation And Quantum Information (Cambridge, 2000).
- G. Smith, “Quantum Channel Capacities,” arXiv:1007.2855 (2010).
- R. T. Thew, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66(1), 012303 (2002). [CrossRef]
- C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000). [CrossRef] [PubMed]

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