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Photon nonlinear mixing in subcarrier multiplexed quantum key distribution systems

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Abstract

We provide, for the first time to our knowledge, an analysis of the influence of nonlinear photon mixing on the end to end quantum bit error rate (QBER) performance of subcarrier multiplexed quantum key distribution systems. The results show that negligible impact is to be expected for modulation indexes in the range of 2%.

©2009 Optical Society of America

1. Introduction

Quantum cryptography features an unique way of sharing a random sequence of bits between users with a certifiable security not attainable with either public or secret-key classical cryptographic systems [1,2]. This is achieved by means of quantum key distribution (QKD) techniques. In essence, QKD relies on exploiting in a positive sense the laws of quantum mechanics, which are often viewed in other contexts of physics as limiting or negative [3,4]. Photonics has proved to be one of the principal enabling technologies for long-distance QKD using optical fiber links. Four main different photonic-based techniques have been reported in the literature for implementing QKD. In 1992 Bennett and co-workers [5] proposed to exploit the polarization of photons to implement the four required states by employing one circular polarization and one linear polarization basis. The main disadvantage of this method rests on the difficulty of preserving the polarization over long lengths of standard telecommunication fibers. The second approach, initially developed by Townsend and co-workers [6-8], relies on the use of optical delays and balanced interferometers at the transmitter and the receiver. The main challenge of this approach resides in keeping the interferometers free from environmental and mechanical variations and preserving the matching of the path difference at the transmitter and receiver. A third approach, based on differential phase shift quantum key distribution [9] has enabled key generation and distribution along distances over 100 Km [10] although with limited security, [11]. The fourth approach, proposed by Merolla and co-workers [12], also known as frequency coding, relies on encoding the information bits on the sidebands of either phase [13] or amplitude [14] radio-frequency (RF) modulated light. The coding principles of subcarrier multiplexing are widely employed in the field of microwave photonics [15] for a variety of applications including remote antenna beam steering and the optical processing of microwave signals [16]. Alice randomly changes the phase of the electrical signal used to drive a light modulator among four phase values, which form a pair of conjugated bases and sends the signal through a fiber link. When it arrives at Bob, he modulates the signal again using the same microwave signal frequency and thus his new sidebands will interfere with those created by Alice [12]. Over the past few years, the work done by Merolla and co-workers has led to considerable dramatic improvements in this kind of system. Originally used for implementing the Bennett 1992 B92 protocol [17],[12], it was subsequently improved by adjusting the modulator characteristics which let them demonstrate the implementation of the Bennett-Brassard 1984 BB84 protocol [2],[18], [19].

2. Theoretical fundamentals

The conceptual approach of frequency coding can be considerably improved in terms of signal speed by incorporating a multiplexing technique widely employed in microwave photonics [15] and known as subcarrier multiplexing (SCM) to provide parallel QKD. In [20] we have proposed the extension of frequency coding to multiple subcarriers, showing that it opens the possibility of parallel quantum key distribution and, therefore, of a potential substantial improvement in the bit rate of such system. The concept behind subcarrier multiplexed quantum key distribution (SCM-QKD) can be explained referring to Fig. 1.

A faint pulse laser source, emitting at frequency wo is externally modulated by N radiofrequency subcarriers by Alice. Each subcarrier, generated by an independent voltage controlled oscillator (VCO) is randomly phase modulated Φ1i, among four possible values 0, π and π/2, 3π/2, which, as mentioned above, form a pair of conjugated bases. The compound signal is then sent through an optical fiber link and upon reaching Bob’s location, are externally modulated by N identical subcarriers in a second modulator. These subcarriers are phase modulated Φ2i among two possible values 0 and π/2 which represent the choice between the two encoding bases. As a consequence, an interference single-photon signal can be generated at each of the sidebands (upper and lower) of each subcarrier. For a given subcarrier Ωi = 2πfi if Bob and Alice’s bases match, then the photon will be detected with probability 1 by either the detector placed after the filter centered at wo + Ωi or by the detector placed after the filter centered at wo − Ωi. If, on the contrary, Bob and Alice’s bases do not match there will be an equal probability of 1/2 of detecting the single photon at any of the two detectors and this detection will be discarded in a subsequent procedure of public discussion.

 figure: Fig. 1.

Fig. 1. SCM_QKD system Layout [20]

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The advantages of SCM-QKD however come at a price. System design is expected to be more demanding and, especially, the impact of nonlinear photon mixing by harmonic distortion and intermodulation needs to be thoroughly addressed. In this letter, we provide, for the first time to our knowledge, this analysis, considering the influence of nonlinear photon mixing on the end-to-end quantum bit error rate (QBER).

We assume that the output signal from the laser, which is well above threshold can be described by a coherent state |ψ〉 and that the probability of observing a photo-count using a detector with efficiency ρ at time t is proportional to [21]:

P=ρψE(t)E+(t)ψ

Where:

E+(t)=jwξ(w)awejwt
E(t)=jwξ(w)aw+ejwt
ξ(w)=ħw2ε0V0

εo is the dielectric permittivity, Vo the mode volume, aw, aw + the annihilation and creation operators respectively. The coherent state describing the field at the output of Alice’s modulator can be expressed as:

ψA=(1+ejΨ1)αwojejΦ1NejΨ1αwoΩNjejΦ1N1ejΨ1αwoΩN1
jejΦ11ejΨ1αwoΩ1jejΦ11ejΨ1αwo+Ω1
jejΦ1N1ejΨ1αwo+ΩN1jejΦ1NejΨ1αw0+ΩN

The coherent state at the output of Bob’s modulator is given by:

ψB=(1+ejΨ1)(1+ejΨ2)αwoj{(1+ejΨ1)ejΨ2ejΦ2i+(1+ejΨ2)ejΨ1ejΦ1iejD(Ωi)}αwoΩi
j{(1+ejΨ1)ejΨ2ejΦ2i+(1+ejΨ2)ejΨ1ejΦ1iejD(Ωi)}αwo+Ωi
ej(Ψ1+Ψ2)ejD(Ωl)ej(Φ1l+Φ2k)αwoΩlΩkej(Ψ1+Ψ2)ejD(Ωl)ej(Φ1lΦ2k)αwoΩl+Ωk
ej(Ψ1+Ψ2)ejD(Ωl)ej(Φ1lΦ2k)αwo+ΩlΩkej(Ψ1+Ψ2)ejD(Ωl)ej(Φ1l+Φ2k)αwo+Ωl+Ωk

In Eq. (4) the subindices k and l run across all their possible values and:

D(Ωi)=β1LΩi+β2L2Ωi2

Represents the fiber link chromatic dispersion operator for subcarrier Ωi. Now, assuming that an optical filter centred at wo + Ωi is placed after Bob’s modulator to select the content of that sideband we can obtain the coherent state at its output by adding the contribution of the nonlinear terms |αwolk〉, |αwolk〉 such that Ωl ± Ωk = Ωi. To include them properly we must bear in mind that for these nonlinear photon beating terms:

αwo+ΩlΩk=m2αwo+Ωi

Where m represents the subcarrier index of modulation, which is assumed to be the same for all of them.

Therefore from Eq. (3)-Eq. (6) we get:

ψ+Ωi=00000cwo+Ωiαwo+Ωi

Where:

cwo+Ωi=[j{(1+ejΨ1)ejΨ2ejΦ2i++(1+ejΨ2)ejΨ1ejΦ1iejD(Ωi)}m2ej(Ψ1+Ψ2){ejD(Ωl)ej(Φ1lΦ2k)+Ωkk,lΩl=Ωi+ejD(Ωr)ej(Φ1r+Φ2s)Ωrr,s+Ωs=Ωi}]

A similar procedure for the corresponding lower sideband yields:

ψΩi=00000cwoΩiαwoΩi
cwoΩi=[j{(1+ejΨ1)ejΨ2ejΦ2i++(1+ejΨ2)ejΨ1ejΦ1iejD(Ωi)}m2ej(Ψ1+Ψ2){ejD(Ωl)ej(Φ1lΦ2k)+Ωkk,l+Ωl=Ωi+ejD(Ωr)ej(Φ1r+Φ2s)Ωrr,sΩs=Ωi}]

Direct application of Eq. (1) to Eq.(7)-Eq.(10) yields the photo-count when an upper or lower sideband subcarrier is selected:

P±Ωi=cwo±Ωi2ρξ2(w0±Ωi)nwo±Ωi

Where [21]:

nwo±Ωi=αwo±Ωia+aαwo±Ωi

Is the average number of photons in the mode of frequency wo ± Ωi.

3. Quantum bit error rate derivation and discussion

From Eq. (11) we can derive a suitable end-to-end QBER expression for the SC-QKD system subject to photon nonlinear mixing following the approach of [21].

QBER(Ωi)={(1V)+(1QCNRCSOi)}ρTLμ¯i+dB2[{1+(1QCNRCSOi)}ρTLμ¯i+dB]

In Eq. (13) V represents the optical visibility achieved in the filtering process, ρ the detector efficiency, μ¯i is the average photon number in mode Ωi, dB is the dark count rate, TL the overall fiber link losses TL = TB10-αL/10 where TB represents the losses in Bob’s modulator and QCNRiCSO is the Quantum Carrier to Noise Ratio, given by:

QCNRCSOi=16m2NCSO(Ωi)

The number of composite second order terms NCSOi) depends on the number of subcarriers employed in the system and also on their frequency spacing. It is composed of the sum of second harmonic distortion terms and frequency sum and difference terms. Figure 2 shows, for example the frequency spectrum, number of second harmonic distortion terms NCSOHDi), number of frequency sum and difference terms NCSOUi)+NCSODi) and overall number NCSOi) for a low count (i.e 15 channel frequency plan) system.

 figure: Fig. 2.

Fig. 2. Frequency spectrum, number of second harmonic distortion, frequency sum and difference intermodulation terms, and overall CSO number NCSOi) for a frequency plan composed of 15 evenly spaced channels spanning from 2 to 30 GHz.

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We now make use of Eq. (13) to analyze the effect of photon nonlinear mixing on the performance of the SCM-QKD systems. We consider three different frequency plans representative of low, middle and high channel count systems. In the first case N=15 and channels are evenly spaced channels spanning from 2 to 30 GHz, in the second N=30 and channels are evenly spaced channels spanning from 1 to 30 GHz, in the third N=50 and channels are evenly spaced channels spanning from 1 to 50 GHz.

In Fig. 3.a and Fig. 3.b we present the QBER values versus the optical fiber link length in Km, obtained for the three frequency plans (N=15,N=30 and N=50) system previously presented and for the case of a frequency coded (N=1) system.

For the computation of the QBER we have taken standard values for several parameters. For instance the visibility is taken as V=98%, the detector efficiency is ρ = 0.13, α = 0.2dB/Km and TB = 9.6dB. We have considered the worst case scenario, that is, the QBER has been computed for the channel in the frequency plan exhibiting the highest NCSO value. The average number of photons per sideband at the laser output is, unless otherwise stated, μ¯ = 0.05.

In Fig. 3.a we represent the QBER values when the modulation index for the SCM channels is 2% (i.e m=0.02). Note that the impact of nonlinear photon mixing in this case is negigible for low, middle and high count channel systems. This situation changes for higher modulation indexes. For instance, in Fig. 3.b we plot the QBER results for the same systems when the modulation index of the SCM channels is a 8%. The impact of the degrading effects of nonlinear photon mixing is now apparent with a greater impact the higher the channel count of the frequency plan. It can be appreciated that, for instance, the QBER can increase by a factor of almost 50% for N=50 as compared to the case where N=1. In practice these results suggest that modulation indexes close to 2% must be employed if the SCM-QKD system is to be considered inmune to photon nolinear mixing effects.

 figure: Fig. 3.

Fig. 3. QBER values versus the optical fiber link length in Km, obtained for the three frequency plan (N=15,N=30 and N=50) systems and for the case of a frequency coded (N=1) system. 3.a) m=2%, 3.b) m=8%

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4. Conclusion

In this paper, we have analysed the influence of nonlinear photon mixing on the end to end quantum bit error rate QBER performance of subcarrier multiplexed quantum key distribution Systems. The results show that negigible impact is to be expected for modulation indexes in the range of 2%. These values are perfectly attainable with available radiofrequency and optoelectronic technologies.

Acknowledgements

The authors wish to acknowledge the financial support of the Spanish Government through Quantum Optical Information Technology (QOIT), a CONSOLIDER-INGENIO 2010 Project and the Generalitat Valenciana through the PROMETEO research excellency award programme GVA PROMETEO 2008/092.

References and links

1. S. Wiesner, “Conjugate coding”, SIGACT News 15, 77–88 (1983).

2. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984 IEEE, New York, 175–179 (1984).

3. N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002). [CrossRef]  

4. W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature London, 299, 802–803 (1982). [CrossRef]  

5. C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992). [CrossRef]  

6. P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993). [CrossRef]  

7. P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994). [CrossRef]  

8. P. D. Townsend, “Quantum Cryptography on Optical fiber networks”, Opt. Fiber Technol. 4, 345–370 (1998). [CrossRef]  

9. K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002). [CrossRef]   [PubMed]  

10. H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005). [CrossRef]  

11. M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008). [CrossRef]  

12. J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999). [CrossRef]  

13. J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999). [CrossRef]  

14. O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003). [CrossRef]  

15. J. Capmany and D. Novak, “Microwave Photonics combines two worlds”, Nature Photon. 1, 319–330 (2007). [CrossRef]  

16. J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical processing of microwave signals”, J. Lightwave Technol. 23, 702–721 (2005). [CrossRef]  

17. C. H. Bennett, “Quantum cryptography using any two non-orthogonal states”, Phys. Rev. Lett. 68, (3121) (1992). [CrossRef]   [PubMed]  

18. J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002). [CrossRef]  

19. M. Bloch, S. McLaughlin, J.M. Merolla, and F. Patois, “Frequency-coded quantum key distribution”, Opt. Lett. 32, 301–303 (2007). [CrossRef]   [PubMed]  

20. C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK, (2005).

21. N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution”, Phys. Rev. A 61 (052304) (2000). [CrossRef]  

References

  • View by:

  1. S. Wiesner, “Conjugate coding”, SIGACT News 15, 77–88 (1983).
  2. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984 IEEE, New York, 175–179 (1984).
  3. N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
    [Crossref]
  4. W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature London, 299, 802–803 (1982).
    [Crossref]
  5. C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
    [Crossref]
  6. P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993).
    [Crossref]
  7. P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
    [Crossref]
  8. P. D. Townsend, “Quantum Cryptography on Optical fiber networks”, Opt. Fiber Technol. 4, 345–370 (1998).
    [Crossref]
  9. K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002).
    [Crossref] [PubMed]
  10. H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
    [Crossref]
  11. M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008).
    [Crossref]
  12. J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
    [Crossref]
  13. J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999).
    [Crossref]
  14. O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
    [Crossref]
  15. J. Capmany and D. Novak, “Microwave Photonics combines two worlds”, Nature Photon. 1, 319–330 (2007).
    [Crossref]
  16. J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical processing of microwave signals”, J. Lightwave Technol. 23, 702–721 (2005).
    [Crossref]
  17. C. H. Bennett, “Quantum cryptography using any two non-orthogonal states”, Phys. Rev. Lett. 68, (3121) (1992).
    [Crossref] [PubMed]
  18. J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
    [Crossref]
  19. M. Bloch, S. McLaughlin, J.M. Merolla, and F. Patois, “Frequency-coded quantum key distribution”, Opt. Lett. 32, 301–303 (2007).
    [Crossref] [PubMed]
  20. C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK, (2005).
  21. N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution”, Phys. Rev. A 61 (052304) (2000).
    [Crossref]

2008 (1)

M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008).
[Crossref]

2007 (2)

J. Capmany and D. Novak, “Microwave Photonics combines two worlds”, Nature Photon. 1, 319–330 (2007).
[Crossref]

M. Bloch, S. McLaughlin, J.M. Merolla, and F. Patois, “Frequency-coded quantum key distribution”, Opt. Lett. 32, 301–303 (2007).
[Crossref] [PubMed]

2005 (2)

J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical processing of microwave signals”, J. Lightwave Technol. 23, 702–721 (2005).
[Crossref]

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

2003 (1)

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

2002 (3)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002).
[Crossref] [PubMed]

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

2000 (1)

N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution”, Phys. Rev. A 61 (052304) (2000).
[Crossref]

1999 (2)

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999).
[Crossref]

1998 (1)

P. D. Townsend, “Quantum Cryptography on Optical fiber networks”, Opt. Fiber Technol. 4, 345–370 (1998).
[Crossref]

1994 (1)

P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
[Crossref]

1993 (1)

P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993).
[Crossref]

1992 (2)

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

C. H. Bennett, “Quantum cryptography using any two non-orthogonal states”, Phys. Rev. Lett. 68, (3121) (1992).
[Crossref] [PubMed]

1983 (1)

S. Wiesner, “Conjugate coding”, SIGACT News 15, 77–88 (1983).

1982 (1)

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature London, 299, 802–803 (1982).
[Crossref]

Bennett, C. H.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

C. H. Bennett, “Quantum cryptography using any two non-orthogonal states”, Phys. Rev. Lett. 68, (3121) (1992).
[Crossref] [PubMed]

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984 IEEE, New York, 175–179 (1984).

Bessette, F.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

Bloch, M.

Blow, K. J.

P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
[Crossref]

Brassard, G.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984 IEEE, New York, 175–179 (1984).

Capmany, J.

Cova, S.

P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
[Crossref]

Curty, M.

M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008).
[Crossref]

Diamanti, E.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

Duraffourg, L.

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

Fejer, M. M.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK, (2005).

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Goedgebuer, J. P.

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

Guerreau, O.

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

Honjo, T.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

Inoue, K.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002).
[Crossref] [PubMed]

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK, (2005).

Langrock, C.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

Lütkenhaus, N.

N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution”, Phys. Rev. A 61 (052304) (2000).
[Crossref]

Malassenet, F. J.

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

Mazurenko, Y.

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

McLaughlin, S.

Merolla, J.M.

Mérolla, J-M.

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999).
[Crossref]

Moroder, T.

M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008).
[Crossref]

Novak, D.

J. Capmany and D. Novak, “Microwave Photonics combines two worlds”, Nature Photon. 1, 319–330 (2007).
[Crossref]

Ortega, B.

Pastor, D.

Patois, F.

M. Bloch, S. McLaughlin, J.M. Merolla, and F. Patois, “Frequency-coded quantum key distribution”, Opt. Lett. 32, 301–303 (2007).
[Crossref] [PubMed]

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

Phoenix, D. J. D.

P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
[Crossref]

Porte, H.

Rarity, J. G.

P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993).
[Crossref]

Rhodes, W. T.

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography”, Opt. Lett. 24, 104–106 (1999).
[Crossref]

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Sales, S.

Salvail, L.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

Smolin, J.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

Soujaeff, A.

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

Takesue, H.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

Tamaki, K.

M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008).
[Crossref]

Tapster, P. R.

P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993).
[Crossref]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Townsend, P. D.

P. D. Townsend, “Quantum Cryptography on Optical fiber networks”, Opt. Fiber Technol. 4, 345–370 (1998).
[Crossref]

P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
[Crossref]

P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993).
[Crossref]

Waks, E.

K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002).
[Crossref] [PubMed]

Wiesner, S.

S. Wiesner, “Conjugate coding”, SIGACT News 15, 77–88 (1983).

Wootters, W. K.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature London, 299, 802–803 (1982).
[Crossref]

Yamamoto, Y.

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002).
[Crossref] [PubMed]

Zbiden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Zurek, W. H.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature London, 299, 802–803 (1982).
[Crossref]

Electron. Lett. (2)

P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Single-photon interference in a 10 Km long optical fiber interferometer”, Electron. Lett. 29, 634–635 (1993).
[Crossref]

P. D. Townsend, D. J. D. Phoenix, K. J. Blow, and S. Cova, “Design of quantum cryptography systems for passive optical Networks”, Electron. Lett. 30, 1875–1876 (1994).
[Crossref]

Eur. Phys. J. D (1)

J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois, and W. T. Rhodes, “Integrated quantum key distribution system using single sideband detection”, Eur. Phys. J. D 18, 141–146 (2002).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single-sideband detection scheme with WDM synchronization”, IEEE J. Sel. Top. Quantum Electron. 9, 1533–1540 (2003).
[Crossref]

J. Cryptology (1)

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography”, J. Cryptology 5, 3 (1992).
[Crossref]

J. Lightwave Technol. (1)

Nature London, (1)

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned”, Nature London, 299, 802–803 (1982).
[Crossref]

Nature Photon. (1)

J. Capmany and D. Novak, “Microwave Photonics combines two worlds”, Nature Photon. 1, 319–330 (2007).
[Crossref]

New J. Phys. (1)

H. Takesue, E. Diamanti, T. Honjo, C. Langrock, M. M. Fejer, K. Inoue, and Y. Yamamoto, “Differential phase shift quantum key distribution over 105 km fibre”, New J. Phys. 7, 1–12 (2005).
[Crossref]

Opt. Fiber Technol. (1)

P. D. Townsend, “Quantum Cryptography on Optical fiber networks”, Opt. Fiber Technol. 4, 345–370 (1998).
[Crossref]

Opt. Lett. (2)

Phys Rev. A, (1)

M. Curty, K. Tamaki, and T. Moroder, “Effect of detector dead times on the security evaluation of differential-phase-shift quantum key distribution against sequential attacks”, Phys Rev. A, 77 (052321) (2008).
[Crossref]

Phys. Rev. A (1)

N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution”, Phys. Rev. A 61 (052304) (2000).
[Crossref]

Phys. Rev. Lett. (3)

J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in Sidebands of Phase-Modulated Light for Quantum Cryptography”, Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

C. H. Bennett, “Quantum cryptography using any two non-orthogonal states”, Phys. Rev. Lett. 68, (3121) (1992).
[Crossref] [PubMed]

K. Inoue, E. Waks, and Y. Yamamoto, “Differential phase shift quantum key distribution”, Phys. Rev. Lett. 89, (037902) (2002).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbiden, “Quantum Cryptography”, Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

SIGACT News (1)

S. Wiesner, “Conjugate coding”, SIGACT News 15, 77–88 (1983).

Other (2)

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984 IEEE, New York, 175–179 (1984).

C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK, (2005).

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Figures (3)

Fig. 1.
Fig. 1. SCM_QKD system Layout [20]
Fig. 2.
Fig. 2. Frequency spectrum, number of second harmonic distortion, frequency sum and difference intermodulation terms, and overall CSO number NCSOi) for a frequency plan composed of 15 evenly spaced channels spanning from 2 to 30 GHz.
Fig. 3.
Fig. 3. QBER values versus the optical fiber link length in Km, obtained for the three frequency plan (N=15,N=30 and N=50) systems and for the case of a frequency coded (N=1) system. 3.a) m=2%, 3.b) m=8%

Equations (21)

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

P = ρ ψ E ( t ) E + ( t ) ψ
E + ( t ) = j w ξ ( w ) a w e jwt
E ( t ) = j w ξ ( w ) a w + e jwt
ξ ( w ) = ħ w 2 ε 0 V 0
ψ A = ( 1 + e j Ψ 1 ) α w o j e j Φ 1 N e j Ψ 1 α w o Ω N j e j Φ 1 N 1 e j Ψ 1 α w o Ω N 1
j e j Φ 11 e j Ψ 1 α w o Ω 1 j e j Φ 11 e j Ψ 1 α w o + Ω 1
j e j Φ 1 N 1 e j Ψ 1 α w o + Ω N 1 j e j Φ 1 N e j Ψ 1 α w 0 + Ω N
ψ B = ( 1 + e j Ψ 1 ) ( 1 + e j Ψ 2 ) α w o j { ( 1 + e j Ψ 1 ) e j Ψ 2 e j Φ 2 i + ( 1 + e j Ψ 2 ) e j Ψ 1 e j Φ 1 i e jD ( Ω i ) } α w o Ω i
j { ( 1 + e j Ψ 1 ) e j Ψ 2 e j Φ 2 i + ( 1 + e j Ψ 2 ) e j Ψ 1 e j Φ 1 i e jD ( Ω i ) } α w o + Ω i
e j ( Ψ 1 + Ψ 2 ) e jD ( Ω l ) e j ( Φ 1 l + Φ 2 k ) α w o Ω l Ω k e j ( Ψ 1 + Ψ 2 ) e jD ( Ω l ) e j ( Φ 1 l Φ 2 k ) α w o Ω l + Ω k
e j ( Ψ 1 + Ψ 2 ) e jD ( Ω l ) e j ( Φ 1 l Φ 2 k ) α w o + Ω l Ω k e j ( Ψ 1 + Ψ 2 ) e jD ( Ω l ) e j ( Φ 1 l + Φ 2 k ) α w o + Ω l + Ω k
D ( Ω i ) = β 1 L Ω i + β 2 L 2 Ω i 2
α w o + Ω l Ω k = m 2 α w o + Ω i
ψ + Ω i = 0 0 0 0 0 c w o + Ω i α w o + Ω i
c w o + Ω i = [ j { ( 1 + e j Ψ 1 ) e j Ψ 2 e j Φ 2 i + + ( 1 + e j Ψ 2 ) e j Ψ 1 e j Φ 1 i e jD ( Ω i ) } m 2 e j ( Ψ 1 + Ψ 2 ) { e jD ( Ω l ) e j ( Φ 1 l Φ 2 k ) + Ω k k , l Ω l = Ω i + e jD ( Ω r ) e j ( Φ 1 r + Φ 2 s ) Ω r r , s + Ω s = Ω i } ]
ψ Ω i = 0 0 0 0 0 c w o Ω i α w o Ω i
c w o Ω i = [ j { ( 1 + e j Ψ 1 ) e j Ψ 2 e j Φ 2 i + + ( 1 + e j Ψ 2 ) e j Ψ 1 e j Φ 1 i e jD ( Ω i ) } m 2 e j ( Ψ 1 + Ψ 2 ) { e jD ( Ω l ) e j ( Φ 1 l Φ 2 k ) + Ω k k , l + Ω l = Ω i + e jD ( Ω r ) e j ( Φ 1 r + Φ 2 s ) Ω r r , s Ω s = Ω i } ]
P ± Ω i = c w o ± Ω i 2 ρ ξ 2 ( w 0 ± Ω i ) n w o ± Ω i
n w o ± Ω i = α w o ± Ω i a + a α w o ± Ω i
QBER ( Ω i ) = { ( 1 V ) + ( 1 QCNR CSO i ) } ρ T L μ ¯ i + d B 2 [ { 1 + ( 1 QCNR CSO i ) } ρ T L μ ¯ i + d B ]
QCNR CSO i = 16 m 2 N CSO ( Ω i )

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