## Highly indistinguishable heralded single-photon sources using parametric down conversion |

Optics Express, Vol. 20, Issue 14, pp. 15275-15285 (2012)

http://dx.doi.org/10.1364/OE.20.015275

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### Abstract

We theoretically and experimentally investigate the conditions necessary to realize highly indistinguishable single-photon sources using parametric down conversion. The visibilities of Hong–Ou–Mandel (HOM) interference between photons in different fluorescence pairs were measured and a visibility of 95.8 ± 2% was observed using a 0.7-mm-long beta barium borate crystal and 2-nm bandpass filters, after compensating for the reflectivity of the beam splitter. A theoretical model of HOM interference visibilities is proposed that considers non-uniform down conversion process inside the nonlinear crystal. It well explains the experimental results.

© 2012 OSA

## 1. Introduction

3. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science **306**, 1330–1336 (2004). [CrossRef] [PubMed]

4. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) **409**, 46–52 (2001). [CrossRef]

5. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987). [CrossRef] [PubMed]

6. T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-NOT gate in the coincidence basis,” Phys. Rev. A **65**, 062324 (2002). [CrossRef]

7. H. F. Hofmann and S. Takeuchi, “Quantum phase gate for photonic qubits using only beam splitters and postse-lection,” Phys. Rev. A **66**, 024308 (2002). [CrossRef]

8. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature (London) **426**, 264–267 (2003). [CrossRef]

11. R. Okamoto, H. F. Hofmann, S. Takeuchi, and K. Sasaki, “Demonstration of an optical quantum controlled-NOT gate without path interference,” Phys. Rev. Lett. **95**, 210506 (2005). [CrossRef] [PubMed]

12. R. Okamoto, J. L. O’Brien, H. F. Hoffman, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science **323**, 483–485 (2009). [CrossRef] [PubMed]

13. B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. **5**, 134–140 (2009). [CrossRef]

14. R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn C-NOT gate a photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA **108**, 10067–10071 (2011). [CrossRef] [PubMed]

15. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. **49**, 2325–2338 (2002). [CrossRef]

16. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science **316**, 726–729 (2007). [CrossRef] [PubMed]

17. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. **85**, 2733–2736 (2000). [CrossRef] [PubMed]

18. Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Express **15**, 14244–14250 (2007). [CrossRef] [PubMed]

12. R. Okamoto, J. L. O’Brien, H. F. Hoffman, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science **323**, 483–485 (2009). [CrossRef] [PubMed]

14. R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn C-NOT gate a photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA **108**, 10067–10071 (2011). [CrossRef] [PubMed]

19. T. Nagata, R. Okamoto, H. F. Hofmann, and S. Takeuchi, “Analysis of experimental error sources in a linear-optics quantum gate,” New J. Phys. **12**, 043053 (2010). [CrossRef]

12. R. Okamoto, J. L. O’Brien, H. F. Hoffman, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science **323**, 483–485 (2009). [CrossRef] [PubMed]

13. B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. **5**, 134–140 (2009). [CrossRef]

16. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science **316**, 726–729 (2007). [CrossRef] [PubMed]

20. T. Jennewein, R. Ursin, M. Aspelmeyer, and A. Zeilinger, “Performing high-quality multi-photon experiments with parametric down-conversion,” J. Phys. B **42**, 114008 (2009). [CrossRef]

21. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. **45**, 595–604 (1998). [CrossRef]

**323**, 483–485 (2009). [CrossRef] [PubMed]

13. B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. **5**, 134–140 (2009). [CrossRef]

15. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. **49**, 2325–2338 (2002). [CrossRef]

22. H. R. Zhang and R. P. Wang, “Theory of fourfold interference with photon pairs from spatially separated sources,” Phys. Rev. A **75**, 053804 (2007). [CrossRef]

23. M. Barbieri, “Effects of frequency correlation in linear optical entangling gates operated with independent photons,” Phys. Rev. A **76**, 043825 (2007). [CrossRef]

24. R. Kaltenbaek, R. Prevedel, M. Aspelmeyer, and A. Zeilinger, “High-fidelity entanglement swapping with fully independent sources,” Phys. Rev. A **79**, 040302(R) (2009). [CrossRef]

## 2. Theoretical model

5. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987). [CrossRef] [PubMed]

*N*(

*δτ*) is the coincidence counts,

*δτ*is the optical delay time between the photons,

*N*is the maximum coincidence counts,

_{max}*V*is the visibility of the HOM dip, and

*τ*is the coherence time of the input photons. Here, we assume that the input two photons have the same wavelength and the same coherence time.

_{c}*Ṽ*for an ideal reflectivity of 50%. However, the reflectivity

*R*of a beam splitter (BS) in actual experiments is not exactly 50%. The visibility

*Ṽ*for the ideal reflectivity can be calculated from the measured visibility

*V*using

### 2.1. Visibility of HOM dip between photons in a pair

*Ṽ*of HOM dips between photons in a pair which can be calculated from an experimentally measured

_{s}*V*using Eq. (2). Spatial, temporal, or polarization mode mismatch between a pair of input photons can degrade the quality of HOM dips. However, unlike the case of two photons belonging to different pairs (which is discussed below), it is not necessary to consider undesirable timing jitters between the incident photons because the two photons are generated from a single pump photon and thus no timing jitter occurs for the type-I phase matching condition (Fig. 1(b)). In this context,

_{s}*Ṽ*is a useful parameter for evaluating the effect of mode mismatch on HOM dips.

_{s}### 2.2. Visibility of HOM dip between photons in different pairs

*Ṽ*of HOM dips between photons in different pairs, which can be extracted from experimentally obtained

_{d}*V*using Eqs. (1) and (2), can be predicted theoretically. As mentioned above, in this case it is necessary to consider both the mode mismatch and undesirable timing jitters between the input photons to estimate

_{d}*Ṽ*. There are two main causes for undesirable timing jitters: a non-zero pump pulse duration (Fig. 1(d)) and a GVM between the pump pulse and the daughter photons (Fig. 1(e)). In this paper, we introduce the ‘visibilities’

_{d}*V*and

_{pump}*V*to consider these two effects. In the first approximation,

_{GVM}*Ṽ*can be written as [20

_{d}20. T. Jennewein, R. Ursin, M. Aspelmeyer, and A. Zeilinger, “Performing high-quality multi-photon experiments with parametric down-conversion,” J. Phys. B **42**, 114008 (2009). [CrossRef]

*V*, which is given by [25

_{pump}25. M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling independent pulsed photon sources,” Ann. N.Y. Acad. Sci. **755**, 91–102 (1995). [CrossRef]

*τ*is the pump pulse duration,

_{p}*τ*is the coherence time of the trigger photons (photons 1 and 4 in Fig. 1(c)), and

_{t}*τ*is the coherence time of the input photons (photons 2 and 3 in Fig. 1(c)). Equation (4) suggests that

_{i}*V*can be maximized by increasing

_{pump}*τ*and

_{i}*τ*with respect to

_{t}*τ*. This is because the increase in

_{p}*τ*and

_{i}*τ*corresponds to the broadening of the width of the photonic wave packets of the signal and the idler photons (Fig. 1(d)) so that the effect of the timing jitter due to non-zero pump pulse duration is reduced. In experiments, the smaller the band width of the band pass filters for trigger and input photons are, the larger

_{t}*τ*and

_{i}*τ*are.

_{t}*Ṽ*describes the effect of mode mismatch between the input photons, which can be estimated from the experimentally obtained

_{s}*V*.

_{s}*V*will be discussed in detail in the following section.

_{GVM}### 2.3. Visibility degradation due to group velocity mismatch

*v*(

_{ge}*λ*) and

*v*(

_{go}*λ*) are respectively the group velocities of the extraordinary and ordinary rays for

*λ*, which is the wavelength of the daughter photons.

*V*, we calculate the HOM dip function

_{GVM}*F*(

*T*,

*t*) by accounting for the effect of timing jitters between the input photons as follows. where

*T*is the maximum timing jitter (

*T*=

*L*× Δ

*u*, where

*L*is the crystal length),

*g*(

*τ*) is the photon pair creation rate at time

*τ*after the pulse enters the crystal, and

*δτ*is the optical delay time.

20. T. Jennewein, R. Ursin, M. Aspelmeyer, and A. Zeilinger, “Performing high-quality multi-photon experiments with parametric down-conversion,” J. Phys. B **42**, 114008 (2009). [CrossRef]

*g*(

*τ*) is given by:

21. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. **45**, 595–604 (1998). [CrossRef]

*g*(

*τ*) with

*F*[

*T*,

*δτ*] is no longer a simple Gaussian function. Therefore, in this study, we numerically calculated the HOM dip function

*F*[

*T*,

*δτ*] and searched for

*δτ*which gives the minimum value to determine

*V*.

_{GVM}## 3. Experimental setup

28. S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B **21**, 591–598 (2004). [CrossRef]

*f*= 600 mm) so that the Rayleigh length is about 100 mm and the beam waist size at the focused spot is about 300

*μ*m. As shown in Fig. 2, the pump beam is reflected by a dichroic mirror that is located 5 mm behind the crystal so that two pairs of parametric fluorescence photons are simultaneously generated in opposite directions. The angle between the pump and generated photons inside the crystal is set to 3.2 degrees. For this phase matching condition, the GVM between the pump (extraordinary ray) and daughter (ordinary ray) photons is 185 fs/mm. Note that for these thin crystals, the walk-off effect (about 50 and 100

*μ*m for thicknesses of 0.7 and 1.5 mm, respectively) is much smaller than the pump beam waist size (300

*μ*m) and can thus be neglected.

_{max}=99%, CW=780 nm) (FWHM 4-nm: specially ordered, Barr Associates Inc, T

_{max}= 92%, CW=780 nm). Note that we carefully selected filters that have the same center wavelength. We inserted an optical delay for Signal 1 to obtain a HOM dip. The half-wave plate (HWP) and the quarter-wave plate (QWP) inside the optical delay are adjusted to maximize the visibility of HOM dips. To obtain HOM dips for different pairs, we measured four-fold coincidence counts between four detectors D1 to D4 (SPCM-AQR13FC, Perkin Elmer) using a homemade coincidence circuit by varying the optical delay. To obtain HOM dips for a pair of photons, we measured two-fold coincidence counts between detectors D1 and D2 using similar experimental setups. For example, to obtain a HOM dip between Signal 2 and Idler 2, we coupled the PMF in the Idler 2 output to the connector immediately before the optical delay.

*R*of the fiber beam splitter differs slightly from 50%; it depends on the bandwidth of the incident light.

*R*was measured to be 47.2 and 46.2% for bandwidths of 2 and 4 nm, respectively. Each photon pair is guided into polarization maintaining fibers after being transmitted through BPFs of same bandwidth. The HOM dip is measured by varying the optical delay.

29. Y. Kawabe, H. Fujiwara, S. Takeuchi, and K. Sasaki, “Investigation of the spatial propagation properties of type-I parametric fluorescence by use of tuning curve filtering method,” Jpn. J. Appl. Phys. **46**, 5802–5808 (2007). [CrossRef]

## 4. Experimental results

### 4.1. HOM interference between a pair of daughter photons

*N*are plotted as dots and the error bars are

*τ*and

*V*as free parameters.

_{s}*V*and

_{s}*τ*for the four conditions are the same for two different pairs (Signal 1, Idler 1 and Signal 2, Idler 2) within the errors. This implies that these two photon pair sources are almost identical. For

*V*, we obtained average visibility 97.5 ± 0.2% with 2-nm bandpass filters and 96.9 ± 0.4% with 4-nm bandpass filters. After compensating for the non-ideal beam splitting ratio mentioned in the previous section using Eq. (3),

_{s}*Ṽ*= 98.2 ± 0.2% on average for a 2-nm bandpass filter, and

_{s}*Ṽ*= 98.0±0.3% on average for 4-nm bandpass filters. These values are very close to unity; the slight deviations from unity may be caused by imperfect frequency or polarization mode matching.

_{s}### 4.2. HOM interference between photons in different pairs

*τ*and

*V*as free parameters.

_{d}*Ṽ*, we obtained 95.8 ± 2% for the case with a 0.7-mm-long crystal and a 2-nm bandpass filter. To the best of our knowledge, this visibility equals the highest visibility ever reported [24

_{d}24. R. Kaltenbaek, R. Prevedel, M. Aspelmeyer, and A. Zeilinger, “High-fidelity entanglement swapping with fully independent sources,” Phys. Rev. A **79**, 040302(R) (2009). [CrossRef]

*Ṽ*decreased to 91.9 ± 2% due to the larger GVM mismatch. With wider bandpass filters,

_{d}*Ṽ*for 0.7 and 1.5-mm-long crystals are 90.9 ± 2% and 83.7 ± 2%, respectively.

_{d}### 4.3. Comparison of theoretical model and experimental results

*Ṽ*of HOM dips between photons in different pairs. The horizontal and vertical axes represent the BBO crystal length and

_{d}*Ṽ*, respectively. The blue and red dots represent data obtained using the 2 and 4-nm bandpass filters, respectively. The blue and red solid lines are plots of Eq. (3) for the cases with 2 and 4-nm bandpass filters, respectively. Since the GVM for the phase matching condition used is 185 fs/mm,

_{d}*T*in Eq. (9) is 130 and 278 fs for 0.7 and 1.5-mm long crystals, respectively. We used the average

*Ṽ*of 98.2 % and 98.0 % (Sec. 4.1) for 2 and 4-nm bandpass filters, respectively. To calculate

_{s}*V*in Eq. (5), we used the estimated coherence times of the signal and idler photons in Table 1 (402 fs and 255 fs for 2 and 4-nm bandpass filters, respectively) for both

_{pump}*τ*and

_{t}*τ*, and we set

_{i}*τ*= 200 fs.

_{p}*Ṽ*, which reflects the effect of mode mismatch between the input photons, and

_{s}*V*, which is determined by the effect of the timing jitter due to non-zero pump pulse duration (Fig. 1(d)). The difference of the intersection points between the two theoretical curves is due to

_{pump}*V*. As we discussed in Sec. 2.2,

_{pump}*V*is larger when the band widths of the band pass filters for trigger and input photons are smaller. Because of the broadening of the width of the photonic wave packets of the signal and the idler photons, the different wave packets can still overlap in spite of the timing jitter (Fig. 1(d)).

_{pump}*Ṽ*as the crystal length increase is determined by

_{d}*V*, the effect of GVM inside the crystal (Fig. 1(e)). When the crystal is longer, the timing jitter caused by GVM becomes significant, resulting in the reduction in the visibility. However, the deterioration is somewhat moderate for the blue solid line (BPF = 2nm) when compared to the red solid line (BPF = 4nm). This is because the photonic wave packets of the signal and the idler photons with the broadened width (BPF = 2nm) can overlap better.

_{GVM}**42**, 114008 (2009). [CrossRef]

*Ṽ*will be between the solid and dotted lines.

_{d}## 5. Conclusion

## Acknowledgments

## References and links

1. | C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proceedings of IEEE International Conference on Computers Systems and Signal Processing175–179 (1984). |

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

3. | V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science |

4. | E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) |

5. | C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

6. | T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-NOT gate in the coincidence basis,” Phys. Rev. A |

7. | H. F. Hofmann and S. Takeuchi, “Quantum phase gate for photonic qubits using only beam splitters and postse-lection,” Phys. Rev. A |

8. | J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature (London) |

9. | N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch, A. Gilchrist, J. L. O’Brien, G. J. Pryde, and A. G. White, “Demonstration of a simple entangling optical gate and its use in Bell-state analysis,” Phys. Rev. Lett. |

10. | N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, “Linear optics controlled-phase gate made simple,” Phys. Rev. Lett. |

11. | R. Okamoto, H. F. Hofmann, S. Takeuchi, and K. Sasaki, “Demonstration of an optical quantum controlled-NOT gate without path interference,” Phys. Rev. Lett. |

12. | R. Okamoto, J. L. O’Brien, H. F. Hoffman, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science |

13. | B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. |

14. | R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn C-NOT gate a photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA |

15. | H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. |

16. | T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science |

17. | N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. |

18. | Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Express |

19. | T. Nagata, R. Okamoto, H. F. Hofmann, and S. Takeuchi, “Analysis of experimental error sources in a linear-optics quantum gate,” New J. Phys. |

20. | T. Jennewein, R. Ursin, M. Aspelmeyer, and A. Zeilinger, “Performing high-quality multi-photon experiments with parametric down-conversion,” J. Phys. B |

21. | P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. |

22. | H. R. Zhang and R. P. Wang, “Theory of fourfold interference with photon pairs from spatially separated sources,” Phys. Rev. A |

23. | M. Barbieri, “Effects of frequency correlation in linear optical entangling gates operated with independent photons,” Phys. Rev. A |

24. | R. Kaltenbaek, R. Prevedel, M. Aspelmeyer, and A. Zeilinger, “High-fidelity entanglement swapping with fully independent sources,” Phys. Rev. A |

25. | M. Zukowski, A. Zeilinger, and H. Weinfurter, “Entangling independent pulsed photon sources,” Ann. N.Y. Acad. Sci. |

26. | R. Kaltenbaek, B. Blauensteiner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, “Experimental interference of independent photons,” Phys. Rev. Lett. |

27. | R. Kaltenbaek, 2008 PhD Thesis. |

28. | S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B |

29. | Y. Kawabe, H. Fujiwara, S. Takeuchi, and K. Sasaki, “Investigation of the spatial propagation properties of type-I parametric fluorescence by use of tuning curve filtering method,” Jpn. J. Appl. Phys. |

**OCIS Codes**

(270.5290) Quantum optics : Photon statistics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: April 18, 2012

Revised Manuscript: May 25, 2012

Manuscript Accepted: May 29, 2012

Published: June 22, 2012

**Citation**

Masato Tanida, Ryo Okamoto, and Shigeki Takeuchi, "Highly indistinguishable heralded single-photon sources using parametric down conversion," Opt. Express **20**, 15275-15285 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15275

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### References

- C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proceedings of IEEE International Conference on Computers Systems and Signal Processing175–179 (1984).
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, England, 2000).
- V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science306, 1330–1336 (2004). [CrossRef] [PubMed]
- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London)409, 46–52 (2001). [CrossRef]
- C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987). [CrossRef] [PubMed]
- T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, “Linear optical controlled-NOT gate in the coincidence basis,” Phys. Rev. A65, 062324 (2002). [CrossRef]
- H. F. Hofmann and S. Takeuchi, “Quantum phase gate for photonic qubits using only beam splitters and postse-lection,” Phys. Rev. A66, 024308 (2002). [CrossRef]
- J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature (London)426, 264–267 (2003). [CrossRef]
- N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch, A. Gilchrist, J. L. O’Brien, G. J. Pryde, and A. G. White, “Demonstration of a simple entangling optical gate and its use in Bell-state analysis,” Phys. Rev. Lett.95, 210504 (2005). [CrossRef] [PubMed]
- N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, “Linear optics controlled-phase gate made simple,” Phys. Rev. Lett.95, 210505 (2005). [CrossRef] [PubMed]
- R. Okamoto, H. F. Hofmann, S. Takeuchi, and K. Sasaki, “Demonstration of an optical quantum controlled-NOT gate without path interference,” Phys. Rev. Lett.95, 210506 (2005). [CrossRef] [PubMed]
- R. Okamoto, J. L. O’Brien, H. F. Hoffman, T. Nagata, K. Sasaki, and S. Takeuchi, “An entanglement filter,” Science323, 483–485 (2009). [CrossRef] [PubMed]
- B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys.5, 134–140 (2009). [CrossRef]
- R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn C-NOT gate a photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA108, 10067–10071 (2011). [CrossRef] [PubMed]
- H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt.49, 2325–2338 (2002). [CrossRef]
- T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316, 726–729 (2007). [CrossRef] [PubMed]
- N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000). [CrossRef] [PubMed]
- Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Express15, 14244–14250 (2007). [CrossRef] [PubMed]
- T. Nagata, R. Okamoto, H. F. Hofmann, and S. Takeuchi, “Analysis of experimental error sources in a linear-optics quantum gate,” New J. Phys.12, 043053 (2010). [CrossRef]
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