## Differential-phase-shift quantum key distribution using heralded narrow-band single photons |

Optics Express, Vol. 21, Issue 8, pp. 9505-9513 (2013)

http://dx.doi.org/10.1364/OE.21.009505

Acrobat PDF (1233 KB)

### Abstract

We demonstrate the first proof of principle differential phase shift (DPS) quantum key distribution (QKD) using narrow-band heralded single photons with amplitude-phase modulations. In the 3-pulse case, we obtain a quantum bit error rate (QBER) as low as 3.06% which meets the unconditional security requirement. As we increase the pulse number up to 15, the key creation efficiency approaches 93.4%, but with a cost of increasing the QBER. Our result suggests that narrow-band single photons maybe a promising source for the DPS-QKD protocol.

© 2013 OSA

## 1. Introduction

*μ*s have been generated from cold atoms [20

20. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. -M. Duan, and H. J. Kimble, “Generation of Nonclassical Photon Pairs for Scalable Quantum Communication with Atomic Ensembles,” Nature **423**, 731–734 (2003) [CrossRef] [PubMed] .

21. S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. **100**, 183603 (2008) [CrossRef] [PubMed] .

*et al.*[22

22. H. Yan, S. Zhu, and S. Du, “Efficient Phase-Encoding Quantum Key Generation with Narrow-Band Single Photons,” Chin. Phys. Lett. **28**, 070307 (2011) [CrossRef] .

*N*− 1)/

*N*and approaches 100% at the limit of large

*N*. Therefore, heralded narrow-band single photons with a long coherence time becomes attractive for realizing the single-photon DPS-QKD protocol.

*et al.*[22

22. H. Yan, S. Zhu, and S. Du, “Efficient Phase-Encoding Quantum Key Generation with Narrow-Band Single Photons,” Chin. Phys. Lett. **28**, 070307 (2011) [CrossRef] .

9. K. Wen, K. Tamaki, and Y. Yamamoto, “Unconditional Security of Single-Photon Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. **103**, 170503 (2009) [CrossRef] [PubMed] .

22. H. Yan, S. Zhu, and S. Du, “Efficient Phase-Encoding Quantum Key Generation with Narrow-Band Single Photons,” Chin. Phys. Lett. **28**, 070307 (2011) [CrossRef] .

*N*(>3) time slots and obtain a high key creation efficiency of 93.4% at

*N*=15. Our polarization-insensitive result implies its potential application for long-distance fiber-based QKD.

## 2. DPS-QKD Protocol

8. K. Inoue, E. Waks, and Y. Yamamoto, “Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. **89**, 037902 (2002) [CrossRef] [PubMed] .

*N*(≥ 3) time slots with equal period

*T*at Alice’s site. The keys are encoded by preparing the relative phase shift between consecutive pulses in 0 or

*π*randomly. Bob detects the incoming photon using an unbalanced M-Z interferometer setup with a path time delay difference equal to the period

*T*. Here we describe the DPS-QKD protocol using single photons by taking the example at

*N*= 3, as illustrated in Fig. 1. The detection at Bob’s site occurs in four possible time instances: (a) a photon in the first period passes through the short path; (b) a photon in the first period passes through the long path and a photon in the second period passes through the short path; (c) a photon in the second period passes through the long path and a photon in the third period passes through the short path; (d) a photon in the third period passes through the long path. Two detectors (

*D*

_{1}and

*D*

_{2}) at the output ports of Bob’s interferometer clicks for 0 or

*π*phase difference based on Alice’s modulation. Once a photon is detected, Bob records the time and which detector clicks. If the detectors click at the (b) or (c) time instances, Bob tells Alice only the information of time instances through a classical channel; otherwise, Bob discards the photon. Using the time-instance information and her phase encoding records, Alice knows which detector clicked at Bob’s site. Defining the clicks at

*D*

_{1}and

*D*

_{2}as “0” and “1” respectively, Alice and Bob can obtain a confidential bit string as a sharing key. The photon sent from Alice to Bob is one of the following four states:

_{i=1,2,3}represents the photon at time slot

*i*). As nonorthogonal with each other, the four states cannot be perfectly identified by a single measurement based on noncloning theorem [1

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

9. K. Wen, K. Tamaki, and Y. Yamamoto, “Unconditional Security of Single-Photon Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. **103**, 170503 (2009) [CrossRef] [PubMed] .

## 3. Experimental setup and photon source characterization

23. S. Du, J. Wen, and M. H. Rubin, “Narrowband Biphoton Generation Near Atomic Resonance,” J. Opt. Soc. Am. B **25**, C98–C108 (2008) [CrossRef] .

^{85}Rb magneto-optical trap (MOT) [24

24. S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A Dark-Line Two-Dimensional Magneto-Optical Trap of 85Rb Atoms with High Optical Depth,” Rev. Sci. Instrum. **83**, 073102 (2012) [CrossRef] [PubMed] .

*μ*K. In presence of counter-propagating pump (

*ω*, 780 nm) and coupling (

_{p}*ω*, 795 nm) laser beams, the phase-matched Stokes (

_{c}*ω*, 780 nm) and anti-Stokes (

_{s}*ω*, 795 nm) photon pairs are generated [21

_{as}21. S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. **100**, 183603 (2008) [CrossRef] [PubMed] .

21. S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. **100**, 183603 (2008) [CrossRef] [PubMed] .

*D*

_{0}(PerkinElmer SPCM-AQ4C) heralds the generation of its paired anti-Stokes photon and synchronizes the timing for the entire experimental system. The anti-Stokes photons are successively sent through an electro-optical amplitude modulator (EOAM, fiber-based,10 GHz, EOspace) and an electro-optical phase modulator (EOPM, fiber-based, 10 GHz, EOspace), which are driven by a two-channel arbitrary waveform generator (Tektronix AFG 3252). In this way, we produce heralded single anti-Stokes photons with the waveform consisting of

*N*time slots [25

25. P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-Optic Modulation of Single Photons,” Phys. Rev. Lett. **101**, 103601 (2008) [CrossRef] [PubMed] .

*π*between two adjacent sequential pulses.

*D*

_{3}during the MOT loading stage. With a PZT-mounted prism inserted into the Path-L, we can actively lock the M-Z interferometer for a complete constructive or destructive interference. Coincidence counts between

*D*

_{0}and the two single-photon detectors (

*D*

_{1}and

*D*

_{2}, PerkinElmer SPCM-AQRH-16-FC) at the output ports of the interferometer are recorded by a time-to-digital converter (Fast Comtec P7888) with 2 ns bin width. The experiment runs at a repetition rate of 600 Hz with a 30% time window for the DPS-QKD experiment.

*μ*W and 1.6 mW respectively. The optical depth at the anti-Stokes transition is about 45. With the EOAM operating at its maximum transmission, the unmodulated Stokes-anti-Stokes coincidence counts for 300 s run time is shown as the plot(1) (red curve) in the Fig. 2(b) and (c). The heralded single photon has a temporal length of about 350 ns. The experimentally detected photon pair rate is about 375 pair/s. After taking into account the two-photon detection efficiency of 2.27% [including the photon detector quantum efficiencies (50% each), fiber-fiber coupling efficiencies at MOT (70%), EOM transmissions (50% each), fiber connection efficiency (81%), and filter transmissions (80% each)], it corresponds to 16520 pair/s produced from the source. Using the EOMs, we modulate the single photon into 3 time slots, which is illustrated as plot(2) (blue curve) in the Fig. 2(b). The full width at half maximum of the pulse is 5 ns, and the time interval

*T*is 12 ns. In this 3-pulse modulation case, the utilization efficiency, defined as the ratio of modulated photon rate to the unmodulated photon rate, is only about 7.1%. We notice, as the interferometer in Bob’s configuration is 1-bit delay, the interference of probability amplitudes only occurs between two adjacent time slots. As long as the difference between adjacent pulses is sufficiently small, it is not necessary to generate identical probability amplitudes across all the pulses for

*N*>3. Therefore, to utilize single photons efficiently, we produce

*N*pulses following the slowly varying envelope of the unmodulated single-photon waveform, as shown in Fig. 2(c). Although it leads to a certain cost of increasing QBER, the utilization efficiency is significantly improved when

*N*reaches a large value, such as

*N*=15 which is shown as plot(2) (blue color) in Fig. 2(c). The utilization efficiency for

*N*=15 reaches about 20.2%. We further implement two passive polarization-independent beam splitters (BS1, BS2) to reduce the polarization sensitivity of the M-Z interferometer on the receiver. The measured visibility of interference fringes as a function of incident polarization angle is displayed in Fig. 2(d). The red dots represent the experiment data, while the blue line is the average visibility of 97.6%. Thus our optical detection setup at Bob’s side is insensitive to the polarization change which is crucial for long-distance fiber-based QKD.

## 4. DPS-QKD Experimental demonstration

*N*= 3, we test all four possible phase modulation patterns: (0, 0, 0), (

*π*, 0, 0), (0,

*π*, 0) and (0, 0,

*π*). For each fixed encoding pattern, we record the coincidence counts for 300 s run time between the detector

*D*

_{0}and the detectors

*D*

_{1}and

*D*

_{2}. After the measurement, Bob discards the photons detected during the first and last time instances [(a) and (d) in Fig. 1], and compares his detection events with Alice’s encoded pattern through a classical communication channel. The measured coincidence counts are displayed in Fig. 3, where the corresponding fixed phase modulation pattern is shown above in the overhead table. The QBERs for pattern (0, 0, 0), (

*π*, 0, 0), (0,

*π*, 0) and (0, 0,

*π*) are 2.98%, 4.48%, 10.67% and 6.18% respectively. Comparing the error rate of the four patterns, we find that a phase change always results in a higher error rate. This is mainly caused by the limited 240 MHz bandwidth of our arbitrary waveform generator. As a result, the step waveforms sent to the EOPM have finite rise and fall times of 2.5 ns. During this rise (or fall) time, phase shift is neither 0 nor

*π*, resulting in the imperfect destructive interference at the outputs of M-Z interferometer. The error rates are expected to be reduced significantly if we use a faster waveform generator to control the phase shift more precisely. An alternative way to reduce QBER is to exclude these error events from the detection time window. We can set the single-photon detection at the middle of each interference time slot, only within a small data window which does not include the rise and fall times of phase modulation. We confirm this by reducing the detection time window to 2 ns, and obtain a QBER as low as 3.06%, which is below the threshold for unconditional security in the DPS-QKD scheme. In the experiment, the measured photon-counting rate of shaped

*N*=3 pulses is 12 count/s and the key creation efficiency is 66.6%. With the 5 dB transmission line loss, the final sifted secure key rates for 12-ns data window and 2-ns data window are 3.8 bit/s and 1.7 bit/s.

*N*(>3) time slots, we also perform the operations of QKD in different

*N*(>3) cases. The number of possible phase modulation patterns increases exponentially with

*N*. As an example, Fig. 4 shows the measured results for the following four phase patterns at

*N*=15: (a) (0, 0, 0,

*π*,

*π*, 0,

*π*,

*π*, 0, 0, 0,

*π*,

*π*,

*π*, 0), (b) (0, 0, 0,

*π*,

*π*, 0,

*π*,

*π*, 0,

*π*,

*π*, 0, 0,

*π*, 0), (c) (

*π*, 0, 0, 0, 0,

*π*,

*π*, 0, 0,

*π*,

*π*, 0, 0,

*π*,

*π*), and (d) (

*π*,

*π*, 0, 0,

*π*, 0, 0, 0, 0,

*π*,

*π*, 0, 0, 0, 0). These patterns are generated based on pseudo-random process in a computer. In this case, with the full detection window (12 ns), the key generation rate is 18.4 bit/s with a QBER of 9.41%. As we reduce the detection window to 2 ns, the key generation rate becomes 4.4 bit/s and the QBER is 6.69%. As compared with

*N*=3, the key creation efficiency reaches 93.4%.

## 5. Discussion and conclusion

### 5.1. Discussion

9. K. Wen, K. Tamaki, and Y. Yamamoto, “Unconditional Security of Single-Photon Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. **103**, 170503 (2009) [CrossRef] [PubMed] .

26. P. Grangier, G. Roger, and A. Aspect, “Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences,” Europhys. Lett. **1**, 173–179 (1986) [CrossRef] .

*N*

_{0}is the Stokes counts at

*D*

_{0},

*N*

_{01}and

*N*

_{02}are the twofold coincidence counts, and

*N*

_{012}is the threefold coincidence counts. An ideal coherent light source gives

*N*time slots, we obtain the

*N*cases shown in Fig. 5(a). The measured

8. K. Inoue, E. Waks, and Y. Yamamoto, “Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. **89**, 037902 (2002) [CrossRef] [PubMed] .

*N*in sending a photon successfully. At Bob’s side, photons at the first time slot in the short path and the last time slot in the long path do not contribute to the key and thus the maximum key detection efficiency of a single photon is (

*N*− 1)/

*N*. Therefore the total key creation efficiency of the conventional scheme scales as (

*N*− 1)/

*N*

^{2}which decreases to zero at the limit of large

*N*[the blue dashed line in Fig. 5(b)]. In our experimental setup, the sending efficiency at Alice’s site is always 1 and thus the key creation efficiency scales as (

*N*−1)/

*N*which approaches 1 at the limit of large

*N*. Figure 5(b) shows the difference between our experiment scheme and conventional DPS-QKD scheme in the key creation efficiency as a function of

*N*. The experimental data (black dot) agrees well with the theory (red solid line).

*N*in Fig. 5(d). It is clear that the key generation rate increases with the increase of

*N*. A larger

*N*offers a higher utilization efficiency of a single photon and a higher final key creation rate. Under the security condition, the product of the QBER and the key generation rate maybe an appropriate figure of merit for the QKD system and it can be used to optimize the value of

*N*.

### 5.2. Conclusion

*N*= 3, we obtain a QBER of 3.06% with a 2-ns photon counting window [Fig. 5(c)], which meets the requirement of unconditional security. We also conduct the experiment with

*N*(>3) time slots, and the measurement of key creation efficiency agrees well with the theory, showing a significant improvement compared with conventional DPS-QKD scheme. The dependence of conditional autocorrelation

*N*are studied systematically. Note that even though the QBER values for the cases with

*N*>3 are higher than the known security threshold of 4.12% [9

**103**, 170503 (2009) [CrossRef] [PubMed] .

*N*=9. Meanwhile, the use of more sophisticated classical postprocessing techniques such as two-way classical communications [27

27. D. Gottesman and H. -K. Lo, “Proof of Security of quantum key distribution with two-way classical communications,” IEEE Trans. Inf. Theor. **49**, 457–475 (2003) [CrossRef] .

*N*>3 already demonstrated in this experiment may become secure, further showing the advantage of this scheme. However, this improvement has yet to be proven. Our results suggest the potential application of narrow-band single-photon source in quantum key generation and distribution. Our polarization insensitive setup is suitable for fiber-based long distance QKD systems.

## Acknowledgments

## References and links

1. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys. |

2. | C. H. Bennett and G. Brassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in |

3. | A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. |

4. | C. H. Bennett, “Quantum Cryptography Using any Two Nonorthogonal States,” Phys. Rev. Lett. |

5. | C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum Cryptography without Bell’s Theorem,” Phys. Rev. Lett. |

6. | A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. |

7. | H. -K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. |

8. | K. Inoue, E. Waks, and Y. Yamamoto, “Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. |

9. | K. Wen, K. Tamaki, and Y. Yamamoto, “Unconditional Security of Single-Photon Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett. |

10. | E. Waks, H. Takesue, and Y. Yamamoto, “Security of Differential-Phase-Shift Quantum Key Distribution against Individual Attacks,” Phys. Rev. A |

11. | K. Inoue, E. Waks, and Y. Yamamoto, “Differential-Phase-Shift Quantum Key Distribution Using Coherent Light,” Phys. Rev. A |

12. | H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum Key Distribution over a 40-dB Channel Loss Using Superconducting Single-Photon Detectors,” Nat. Photonics |

13. | A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J. -P. Poizat, and P. Grangier, “Single Photon Quantum Cryptography,” Phys. Rev. Lett. |

14. | R. Alléaume, F. Treussart, G. Messin, Y. Dumeige, J.-F. Roch, A. Beveratos, R. Brouri-Tualle, J. -P. Poizat, and P Grangier, “Experimental Open-Air Quantum Key Distribution with a Single-Photon Source,” New J. Phys. |

15. | E. Waks, K. Inoue, C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Secure Communication: Quantum Cryptography with a Photon Turnstile,” Nature |

16. | P. M. Intallura, M. B. Ward, O. Z. Karimov, Z. L. Yuan, P. See, A. J. Shields, P. Atkinson, and D. A. Ritchie, “Quantum Key Distribution Using a Triggered Quantum Dot Source Emitting near 1.3 |

17. | A. Trifonov and A. Zavriyev, “Secure Communication with a Heralded Single-Photon Source,” J. Opt. B |

18. | A. Soujaeff, T. Nishioka, T. Hasegawa, S. Takeuchi, T. Tsurumaru, K. Sasaki, and M. Matsui, “Quantum Key Distribution at 1550 nm Using a Pulse Heralded Single Photon Source,” Opt. Express |

19. | C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental Quantum Cryptography,” J. Cryptology |

20. | A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. -M. Duan, and H. J. Kimble, “Generation of Nonclassical Photon Pairs for Scalable Quantum Communication with Atomic Ensembles,” Nature |

21. | S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. |

22. | H. Yan, S. Zhu, and S. Du, “Efficient Phase-Encoding Quantum Key Generation with Narrow-Band Single Photons,” Chin. Phys. Lett. |

23. | S. Du, J. Wen, and M. H. Rubin, “Narrowband Biphoton Generation Near Atomic Resonance,” J. Opt. Soc. Am. B |

24. | S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A Dark-Line Two-Dimensional Magneto-Optical Trap of 85Rb Atoms with High Optical Depth,” Rev. Sci. Instrum. |

25. | P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-Optic Modulation of Single Photons,” Phys. Rev. Lett. |

26. | P. Grangier, G. Roger, and A. Aspect, “Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences,” Europhys. Lett. |

27. | D. Gottesman and H. -K. Lo, “Proof of Security of quantum key distribution with two-way classical communications,” IEEE Trans. Inf. Theor. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5568) Quantum optics : Quantum cryptography

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 11, 2013

Revised Manuscript: April 5, 2013

Manuscript Accepted: April 5, 2013

Published: April 10, 2013

**Citation**

Chang Liu, Shanchao Zhang, Luwei Zhao, Peng Chen, C. -H. F. Fung, H. F. Chau, M. M. T. Loy, and Shengwang Du, "Differential-phase-shift quantum key distribution using heralded narrow-band single photons," Opt. Express **21**, 9505-9513 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9505

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### References

- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys.74, 145–195 (2002). [CrossRef]
- C. H. Bennett and G. Brassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, 1984 (IEEE, New York, 1984), 175.
- A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett.67, 661–663 (1991). [CrossRef] [PubMed]
- C. H. Bennett, “Quantum Cryptography Using any Two Nonorthogonal States,” Phys. Rev. Lett.68, 3121–3124 (1992). [CrossRef] [PubMed]
- C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum Cryptography without Bell’s Theorem,” Phys. Rev. Lett.68, 557–559 (2000). [CrossRef]
- A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett.98, 230501 (2007). [CrossRef] [PubMed]
- H. -K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett.108, 130503 (2012). [CrossRef] [PubMed]
- K. Inoue, E. Waks, and Y. Yamamoto, “Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett.89, 037902 (2002). [CrossRef] [PubMed]
- K. Wen, K. Tamaki, and Y. Yamamoto, “Unconditional Security of Single-Photon Differential Phase Shift Quantum Key Distribution,” Phys. Rev. Lett.103, 170503 (2009). [CrossRef] [PubMed]
- E. Waks, H. Takesue, and Y. Yamamoto, “Security of Differential-Phase-Shift Quantum Key Distribution against Individual Attacks,” Phys. Rev. A73, 012344 (2006). [CrossRef]
- K. Inoue, E. Waks, and Y. Yamamoto, “Differential-Phase-Shift Quantum Key Distribution Using Coherent Light,” Phys. Rev. A68, 022317 (2003). [CrossRef]
- H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum Key Distribution over a 40-dB Channel Loss Using Superconducting Single-Photon Detectors,” Nat. Photonics1, 343–348 (2007). [CrossRef]
- A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J. -P. Poizat, and P. Grangier, “Single Photon Quantum Cryptography,” Phys. Rev. Lett.89, 187901 (2002). [CrossRef] [PubMed]
- R. Alléaume, F. Treussart, G. Messin, Y. Dumeige, J.-F. Roch, A. Beveratos, R. Brouri-Tualle, J. -P. Poizat, and P Grangier, “Experimental Open-Air Quantum Key Distribution with a Single-Photon Source,” New J. Phys.6, 92 (2004). [CrossRef]
- E. Waks, K. Inoue, C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Secure Communication: Quantum Cryptography with a Photon Turnstile,” Nature420, 762 (2002). [CrossRef] [PubMed]
- P. M. Intallura, M. B. Ward, O. Z. Karimov, Z. L. Yuan, P. See, A. J. Shields, P. Atkinson, and D. A. Ritchie, “Quantum Key Distribution Using a Triggered Quantum Dot Source Emitting near 1.3 μm,” Appl. Phys. Lett.91, 161103 (2007). [CrossRef]
- A. Trifonov and A. Zavriyev, “Secure Communication with a Heralded Single-Photon Source,” J. Opt. B7, S772–S777 (2005). [CrossRef]
- A. Soujaeff, T. Nishioka, T. Hasegawa, S. Takeuchi, T. Tsurumaru, K. Sasaki, and M. Matsui, “Quantum Key Distribution at 1550 nm Using a Pulse Heralded Single Photon Source,” Opt. Express15, 726–734 (2007). [CrossRef] [PubMed]
- C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental Quantum Cryptography,” J. Cryptology5, 3–28 (1992). [CrossRef]
- A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. -M. Duan, and H. J. Kimble, “Generation of Nonclassical Photon Pairs for Scalable Quantum Communication with Atomic Ensembles,” Nature423, 731–734 (2003). [CrossRef] [PubMed]
- S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett.100, 183603 (2008). [CrossRef] [PubMed]
- H. Yan, S. Zhu, and S. Du, “Efficient Phase-Encoding Quantum Key Generation with Narrow-Band Single Photons,” Chin. Phys. Lett.28, 070307 (2011). [CrossRef]
- S. Du, J. Wen, and M. H. Rubin, “Narrowband Biphoton Generation Near Atomic Resonance,” J. Opt. Soc. Am. B25, C98–C108 (2008). [CrossRef]
- S. Zhang, J. F. Chen, C. Liu, S. Zhou, M. M. T. Loy, G. K. L. Wong, and S. Du, “A Dark-Line Two-Dimensional Magneto-Optical Trap of 85Rb Atoms with High Optical Depth,” Rev. Sci. Instrum.83, 073102 (2012). [CrossRef] [PubMed]
- P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-Optic Modulation of Single Photons,” Phys. Rev. Lett.101, 103601 (2008). [CrossRef] [PubMed]
- P. Grangier, G. Roger, and A. Aspect, “Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences,” Europhys. Lett.1, 173–179 (1986). [CrossRef]
- D. Gottesman and H. -K. Lo, “Proof of Security of quantum key distribution with two-way classical communications,” IEEE Trans. Inf. Theor.49, 457–475 (2003). [CrossRef]

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