## Observation of Young’s double-slit interference with the three-photon N00N state |

Optics Express, Vol. 19, Issue 25, pp. 24957-24966 (2011)

http://dx.doi.org/10.1364/OE.19.024957

Acrobat PDF (1141 KB)

### Abstract

Spatial interference of quantum mechanical particles exhibits a fundamental feature of quantum mechanics. A two-mode entangled state of N particles known as N00N state can give rise to non-classical interference. We report the first experimental observation of a three-photon N00N state exhibiting Young’s double-slit type spatial quantum interference. Compared to a single-photon state, the three-photon entangled state generates interference fringes that are three times denser. Moreover, its interference visibility of 0.49 ± 0.09 is well above the limit of 0.1 for spatial super-resolution of classical origin.

© 2011 OSA

## 1. Introduction

3. U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science **329**, 418 (2010). [CrossRef] [PubMed]

*N*is the number of quanta and the subscript refers to the spatial mode, naturally arises in generalizing the double-slit experiment to the

*N*-quantum case and was first discussed in the context of photonic de Broglie waves [4

4. J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie wave,” Phys. Rev. Lett. **74**, 4835 (1995). [CrossRef] [PubMed]

5. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature **448**, 784 (2007). [CrossRef] [PubMed]

6. J. P. Dowling, “Quantum optical metrology—the lowdown of high-N00N states,” Contemp. Phys. **49**, 125 (2008). [CrossRef]

7. A. 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 (2000). [CrossRef] [PubMed]

8. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A **63**, 063407 (2001). [CrossRef]

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

10. O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled NOON state,” Phys. Rev. A **81**, 063801 (2010). [CrossRef]

6. J. P. Dowling, “Quantum optical metrology—the lowdown of high-N00N states,” Contemp. Phys. **49**, 125 (2008). [CrossRef]

11. J. Fiurášek, “Conditional generation of N-photon entangled states of light,” Phys. Rev. A **65**, 053818 (2002). [CrossRef]

14. M. D’Angelo, A. Garuccio, and V. Tamma, “Toward real maximally path-entangled N -photon-state sources,” Phys. Rev. A **77**, 063826 (2008). [CrossRef]

15. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**, 213601 (2002). [CrossRef] [PubMed]

24. I. Afek, O. Ambar, and Y. Silberberg, “High-N00N states by mixing quantum and classical light,” Science **328**, 879 (2010). [CrossRef] [PubMed]

21. P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature **429**, 158 (2004). [CrossRef] [PubMed]

24. I. Afek, O. Ambar, and Y. Silberberg, “High-N00N states by mixing quantum and classical light,” Science **328**, 879 (2010). [CrossRef] [PubMed]

## 2. Experiment

20. H. Kim, H.-S. Park, and S.-K. Choi, “Three-photon N00N states generated by photon subtraction from double photon pairs,” Opt. Express **17**, 19720 (2009). [CrossRef] [PubMed]

*ψ*〉

_{in}is, then, transformed by passing through the PPBS to become the subscripts 1 (2) refers to the reflected (transmitted) mode of the PPBS [20

20. H. Kim, H.-S. Park, and S.-K. Choi, “Three-photon N00N states generated by photon subtraction from double photon pairs,” Opt. Express **17**, 19720 (2009). [CrossRef] [PubMed]

*N*= 3 N00N state interference as they cannot be registered at the three-photon detector.

*ψ*〉

*is heralded in mode 2 (at the entrance of the single mode fiber SMF) whenever there is a single-photon detection at SPC1 [20*

_{p}20. H. Kim, H.-S. Park, and S.-K. Choi, “Three-photon N00N states generated by photon subtraction from double photon pairs,” Opt. Express **17**, 19720 (2009). [CrossRef] [PubMed]

*ψ*〉

*is transformed to the spatial two-mode three-photon N00N state |*

_{p}*ψ*〉

*, by using the mode converter shown in Fig. 1(b) and 1(c). The mode converter, which consists of half-wave plates (HWP), birefringent prisms (BP), quartz plates, and a polarizer (Pol), is based on the following operation principle. The first and second BPs are aligned so that their optic axes are oriented respectively at +*

_{s}*θ*and −

*θ*with respect to the vertical polarization. The first two HWPs are used for rotating the horizontal-vertical polarization basis of the photons so that they overlap with the rotated optic axes of BPs. The third HWP and the quartz plates are used to match the polarization states of photons in spatial modes

*a*and

*b*. An additional horizontal polarizer (Pol) is used to clean up the polarization state so that all three photons in Eq. (4) are guaranteed to have the same polarization.

*d*= 2

*L*sin

*θ*between the modes

*a*and

*b*is determined by the angle

*θ*and the beam walk-off (i.e., e- and o-ray separation)

*L*of a single BP (see Fig. 1(c)). In the experiment, considering the numerical aperture (0.12) and mode field diameter (5.6

*μ*m) of SMF, the 1/

*e*

^{2}beam diameter of each spatial mode is estimated to be 1.4 mm. The beam spacing

*d*was chosen as

*d*= 2.2 mm, which leads to a mode overlap of 0.8 %. Our mode converter design provides excellent interferometric phase stability between the spatial modes

*a*and

*b*.

*ψ*〉

*, a single-mode fiber (identical to SMF) tip was scanned at the focus of a lens (15 mm focal length) by using a piezo-controlled translation stage, see Fig. 1(b). The group delay between the spatial modes*

_{s}*a*and

*b*was compensated by a set of mirrors (not shown in the figure) in front of the focusing lens. The other end of the fiber tip was connected to three single-photon detectors (SPC2∼4) via a set of 3 dB fiber beamsplitters. The three-fold coincidence SPC2–SPC3–SPC4 triggered by SPC1 constitutes the proper measurement for the heralded three-photon N00N states, |

*ψ*〉

*and |*

_{p}*ψ*〉

*.*

_{s}## 3. Temporal interference of three-photon N00N state

*ψ*〉

*is directly responsible for the quality of the spatial N00N state |*

_{p}*ψ*〉

*, which in turn affects the double-slit interference visibility with |*

_{s}*ψ*〉

*. Hence, it is of utmost importance to ensure that the three-photon polarization N00N state is prepared with high purity. Thus, we have first measured the temporal interference fringes due to |*

_{s}*ψ*〉

*as in other N00N state experiments [10*

_{p}10. O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled NOON state,” Phys. Rev. A **81**, 063801 (2010). [CrossRef]

15. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**, 213601 (2002). [CrossRef] [PubMed]

19. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature **429**, 161 (2004). [CrossRef] [PubMed]

24. I. Afek, O. Ambar, and Y. Silberberg, “High-N00N states by mixing quantum and classical light,” Science **328**, 879 (2010). [CrossRef] [PubMed]

**17**, 19720 (2009). [CrossRef] [PubMed]

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

*ψ*〉

*, we replace QWP2 with a HWP2 and a horizontal polarizer (Pol.) as shown inset of Fig. 1(a) and connect the output end of SMF to the fiber-coupled three-photon detector (SPC2–SPC3–SPC4) depicted in Fig. 1(b). It is then possible to introduce a phase difference*

_{p}*χ*between the left-circular and right-circular polarization modes of |

*ψ*〉

*by rotating HWP2 by*

_{p}*χ*/4 and the temporal N00N state interference can be observed in four-fold coincidences between the trigger detector and the three-photon detector. The experimental data for the temporal interference measurements are shown in Fig. 2. The error bars denote statistical uncertainty calculated as square root of the measured counts. We first measured, as a reference, the heralded single-photon interference shown in Fig. 2(a). In Fig. 2(b), the response of the three-photon detector, three-fold coincidences among SPC2–SPC3–SPC4, is shown. In this case, the first two terms of Eq. (2) affect the outcome and the three-fold coincidence probability

*P*is calculated to be where it is assumed that the three-photon detector consists of three single-photon detectors connected with 3 dB fiber beamsplitters as shown in Fig. 1(b) and

*η*is the detection efficiency at each detector. The first term is due to the heralded three-photon N00N state term, i.e., the first term in Eq. (2), and the second/third terms are from the second term in Eq. (2). The amplitudes expressed as |etc〉 in Eq. (2) do not contribute to the outcome of the three-photon detector. The experimental data in Fig. 2(b) is in good agreement with the above theoretical calculation.

*V*= 0.86 ± 0.08) heralded three-photon N00N state interference with three-times greater phase resolution than the single-photon case shown in Fig. 2(a). Even at a much higher pump power of 300 mW, similar results are observed, see Fig. 2(e) and Fig. 2(f), albeit with somewhat reduced visibility (

*V*= 0.74 ± 0.07). Note that the background noise by triple-pairs at 300 mW pump power is much larger than it of at 70 mW pump power since as we increase the pump power, the SPDC efficiency amplitude

*γ*also increases. The observed three-photon N00N state visibilities after the background noise subtracted, however, are well above the classical limit of 0.1 either in 70 mW or 300 mW of pump power [26

26. S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express **12**, 5735 (2004). [CrossRef] [PubMed]

27. I. Afek, O. Ambar, and Y. Silberberg, “Classical bound for Mach-Zehnder superresolution,” Phys. Rev. Lett. **104**, 123602 (2010). [CrossRef] [PubMed]

## 4. Spatial interference of three-photon N00N state

*ψ*〉

*, is prepared with sufficiently high purity, we now proceed to demonstrate the Young’s double-slit interference with the N00N state |*

_{p}*ψ*〉

*. For this measurement, QWP2 is now restored at its original location and the |*

_{s}*ψ*〉

*is transformed to the |*

_{p}*ψ*〉

*state with the mode converter shown in Fig. 1(b). Spatial interference fringes are measured with the detection scheme also shown in Fig. 1(b) and the pump power is increased to 400 mW.*

_{s}*ψ*〉

*are shown in Fig. 3. As was in the temporal interference data, the error bars are the standard deviations which correspond to the square roots of the measured counts. We first measured the spatial profile of the beam at the focus of the lens by blocking mode*

_{s}*b*. The single-photon and the three-photon spatial profiles are shown in Fig. 3(a) and Fig. 3(b), respectively. Note that both the single-photon and the three-photon states are heralded by single-photon detection at SPC1. Fitting the data with a Gaussian function exp[−(

*x*/

*w*

_{0})

^{2}], we find that the 2

*w*

_{0}widths are 10.8 ±0.4

*μ*m for the single-photon case and 6.2 ±0.6

*μ*m for the three-photon case. This is in good agreement with the theoretical estimation that the three-photon probability distribution is proportional to the cube of the single-photon one.

*a*and

*b*are now open and interference fringes are measured in two-fold and four-fold coincidences at the focus of the lens as a function of the scanning fiber tip position, see Fig. 3(c) and Fig. 3(d). The fitting curves are Gaussian envelops multiplied by a raised sine curve. The size of Gaussian envelop is obtained from the spatial profile without interference shown in Fig. 3(a) and Fig. 3(b). Here the visibility is calculated from the offset and amplitude of the sinusoidal modulation. In theory, the fringe patterns are expected to show sinusoidal modulations within the respective spatial profiles of the single- and three-photon states and the modulation frequency for the three-photon N00N state should be three times more than that of the single-photon state. The experimental data, which show fringe spacing of 6.0

*μ*m for the single-photon state and 2.0

*μ*m for the three-photon N00N state, are thus in good agreement with the theory. We point out that, although the mode field diameter of the scanning fiber tip is larger than the fringe spacing due to the three-photon N00N state, the fact that the fiber is single-mode at 780 nm allows us to measure a spatial fringe spacing smaller than the mode field diameter without sacrificing the visibility. See the Appendix for details.

28. E. Yablonovitch and R. B. Vrijen, “Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure,” Opt. Eng. **38**, 334 (1999). [CrossRef]

*V*is an important feature that distinguishes between quantum and classical cases. For three-photon Young’s double-slit interference, the classical limit of the fringe visibility can be calculated by considering detection of the intensity cubed (i.e., three-photon detection) rather than detection linear in intensity (i.e., single-photon detection) and has been shown to be 0.1 [26

26. S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express **12**, 5735 (2004). [CrossRef] [PubMed]

27. I. Afek, O. Ambar, and Y. Silberberg, “Classical bound for Mach-Zehnder superresolution,” Phys. Rev. Lett. **104**, 123602 (2010). [CrossRef] [PubMed]

*V*= 0.49 ±0.09 which is well above the classical limit of 0.1.

## 5. Conclusion

## Appendix : Spatial interference measurement with a single mode fiber

*l*and the envelope profile of 2

*σ*, see the inset in Fig. 4(a).

_{x}*η*of an incident beam to a resonator or an optical fiber is where

_{k}*E*

_{in}(

*x*,

*y*) and

*E*(

_{k}*x*,

*y*) refer to the electric field of the incident beam at the input plane and the wave function of the resonator (or fiber) mode, respectively [29]. The electric field

*E*

_{in}(

*x*,

*y*) at the location of the SMF tip shows interference and it can be written as where 2

*σ*(2

_{x}*σ*) denotes the horizontal (vertical) envelope. Furthermore, it is well known that SMF guides only one spatial mode that can be approximately described as a Gaussian (known as LP01 or HE11 mode), where 2

_{y}*σ*is the mode field diameter of the SMF and Δ is a position of the scanning SMF.

*l*is smaller than the mode field diameter 2

*σ*. To demonstrate that such a measurement is indeed possible, we set up the Young’s interferometer with a He-Ne laser as shown in Fig. 4(a). By varying the beam size

*d*and the beam spacing

*L*, we are able to change the interference envelope 2

*σ*as well as the fringe spacing

_{x}*l*. The mode field diameter of SMF used in our experiment is 2

*σ*= 5.4

*μm*.

*l*= 39.8

*μm*> 2

*σ*= 5.4

*μm*and the result is shown in Fig. 4(b). The fringe is clearly resolved with the visibility of 0.82. We now vary

*L*such that the expected fringe spacing

*l*= 1.9

*μm*, smaller than the mode field diameter of SMF. The experimental data are shown in Fig. 4(c) and the visibility is better than 0.89. It is clear that, due to the overlap integral of Eq. (6), we are able to measure fringe spacing smaller than the mode field diameter of SMF.

_{in}denotes the incident field. Thus, the three-photon envelope will be reduced by the factor of

## Acknowledgments

## References and links

1. | R. P. Feynman, R. B. Leighton, and M. Sands, |

2. | Y.-H. Kim, R. Yu, S. P. Kulik, and Y. Shih, “Delayed “choice” quantum eraser,” Phys. Rev. Lett. |

3. | U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science |

4. | J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie wave,” Phys. Rev. Lett. |

5. | A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature |

6. | J. P. Dowling, “Quantum optical metrology—the lowdown of high-N00N states,” Contemp. Phys. |

7. | A. 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. |

8. | P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A |

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

10. | O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled NOON state,” Phys. Rev. A |

11. | J. Fiurášek, “Conditional generation of N-photon entangled states of light,” Phys. Rev. A |

12. | H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett. |

13. | K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. |

14. | M. D’Angelo, A. Garuccio, and V. Tamma, “Toward real maximally path-entangled N -photon-state sources,” Phys. Rev. A |

15. | K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. |

16. | E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multi photon wave packet,” Phys. Rev. Lett. |

17. | M. D’Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and quantum lithography,” 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. | M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature |

20. | H. Kim, H.-S. Park, and S.-K. Choi, “Three-photon N00N states generated by photon subtraction from double photon pairs,” Opt. Express |

21. | P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature |

22. | F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A |

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

24. | I. Afek, O. Ambar, and Y. Silberberg, “High-N00N states by mixing quantum and classical light,” Science |

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

26. | S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express |

27. | I. Afek, O. Ambar, and Y. Silberberg, “Classical bound for Mach-Zehnder superresolution,” Phys. Rev. Lett. |

28. | E. Yablonovitch and R. B. Vrijen, “Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure,” Opt. Eng. |

29. | A. Yariv and P. Yeh, |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.4180) Quantum optics : Multiphoton processes

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: September 16, 2011

Revised Manuscript: October 21, 2011

Manuscript Accepted: November 14, 2011

Published: November 22, 2011

**Citation**

Yong-Su Kim, Osung Kwon, Sang Min Lee, Jong-Chan Lee, Heonoh Kim, Sang-Kyung Choi, Hee Su Park, and Yoon-Ho Kim, "Observation of Young’s double-slit interference with the three-photon N00N state," Opt. Express **19**, 24957-24966 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-24957

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### References

- R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison Wesley, 1965), Vol. III.
- Y.-H. Kim, R. Yu, S. P. Kulik, and Y. Shih, “Delayed “choice” quantum eraser,” Phys. Rev. Lett.84, 1 (2000). [CrossRef] [PubMed]
- U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science329, 418 (2010). [CrossRef] [PubMed]
- J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie wave,” Phys. Rev. Lett.74, 4835 (1995). [CrossRef] [PubMed]
- A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature448, 784 (2007). [CrossRef] [PubMed]
- J. P. Dowling, “Quantum optical metrology—the lowdown of high-N00N states,” Contemp. Phys.49, 125 (2008). [CrossRef]
- A. 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 (2000). [CrossRef] [PubMed]
- P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001). [CrossRef]
- V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science306, 1330 (2004). [CrossRef] [PubMed]
- O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled NOON state,” Phys. Rev. A81, 063801 (2010). [CrossRef]
- J. Fiurášek, “Conditional generation of N-photon entangled states of light,” Phys. Rev. A65, 053818 (2002). [CrossRef]
- H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett.99, 163604 (2007). [CrossRef] [PubMed]
- K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett.99, 053602 (2007). [CrossRef] [PubMed]
- M. D’Angelo, A. Garuccio, and V. Tamma, “Toward real maximally path-entangled N -photon-state sources,” Phys. Rev. A77, 063826 (2008). [CrossRef]
- K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett.89, 213601 (2002). [CrossRef] [PubMed]
- E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multi photon wave packet,” Phys. Rev. Lett.82, 2868 (1999) [CrossRef]
- M. D’Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett.87, 013602 (2001). [CrossRef]
- Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Express15, 14244 (2007). [CrossRef] [PubMed]
- M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429, 161 (2004). [CrossRef] [PubMed]
- H. Kim, H.-S. Park, and S.-K. Choi, “Three-photon N00N states generated by photon subtraction from double photon pairs,” Opt. Express17, 19720 (2009). [CrossRef] [PubMed]
- P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature429, 158 (2004). [CrossRef] [PubMed]
- F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A74, 033812 (2006). [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 (2007). [CrossRef] [PubMed]
- I. Afek, O. Ambar, and Y. Silberberg, “High-N00N states by mixing quantum and classical light,” Science328, 879 (2010). [CrossRef] [PubMed]
- C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044 (1987). [CrossRef] [PubMed]
- S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express12, 5735 (2004). [CrossRef] [PubMed]
- I. Afek, O. Ambar, and Y. Silberberg, “Classical bound for Mach-Zehnder superresolution,” Phys. Rev. Lett.104, 123602 (2010). [CrossRef] [PubMed]
- E. Yablonovitch and R. B. Vrijen, “Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure,” Opt. Eng.38, 334 (1999). [CrossRef]
- A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed., (Oxford University Press, 2006).

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