## Two-photon interference with continuous-wave multi-mode coherent light |

Optics Express, Vol. 22, Issue 3, pp. 3611-3620 (2014)

http://dx.doi.org/10.1364/OE.22.003611

Acrobat PDF (4549 KB)

### Abstract

We report two-photon interference with continuous-wave multi-mode coherent light. We show that the two-photon interference, in terms of the detection time difference, reveals two-photon beating fringes with the visibility *V* = 0.5. While scanning the optical delay of the interferometer, Hong-Ou-Mandel dips or peaks are measured depending on the chosen detection time difference. The HOM dips/peaks are repeated when the optical delay and the first-order coherence revival period of the multi-mode coherent light are the same. These results help to understand the nature of two-photon interference and also can be useful for quantum information science.

© 2014 Optical Society of America

## 1. Introduction

3. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. **71**, S274–S282 (1999). [CrossRef]

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

*V*= 1. On the other hand, classical electromagnetic waves superposition theory provides that a HOM dip with only

*V*≤ 0.5 [5

5. J.G. Rarity, P.R. Tapster, and R. Loudon, “Non-classical interference between independent sources,” J. Opt. B: Quantum Semiclass. Opt. **7**, S171–S175 (2005). [CrossRef]

*V*= 0.5 in HOM interference is usually considered as the border between classical and quantum physics.

*quantum*nature of light, there has been a lot of research on two-photon quantum interference. These include two-photon coherence [6], quantum beating [7

7. Z.Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. **61**, 54–57 (1988). [CrossRef] [PubMed]

10. T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Characterization of single photons using two-photon interference,” Adv. Atom. Atom. Mol. Opt. Phys. **53**, 253–289 (2006). [CrossRef]

11. X.Y. Zou, L.J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. **67**, 318–321 (1991). [CrossRef] [PubMed]

12. O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled |N, 0〉+ |0, N〉 state,” Phys. Rev. A **81**, 063801 (2010). [CrossRef]

14. Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. **8**, 117–120 (2012). [CrossRef]

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

16. P. Kok, W.J. Munro, K. Nemoto, T.C. Ralph, J.P. Dowiling, and G.J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. **79**, 135–174 (2007). [CrossRef]

## 2. Two-photon interference with CW coherent light

*t*. This two-photon interference is usually configured with optical pulses and the coincidences at D1 and D2 yield a HOM interference. Since the timing is well defined for optical pulses, Δ

*t*would introduce an optical delay with respect to the reference optical pulse,

*E*

_{1}(

*t*) in Fig. 1(a). Thus, by varying Δ

*t*, one can expect to see two-photon interference with either single-photon states or coherent pulses [4

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

21. Y.-S. Kim, O. Slattery, P.S. Kuo, and X. Tang, “Conditions for two-photon interference with coherent pulses,” Phys. Rev. A **87**, 063843 (2013). [CrossRef]

*t*does not change the arbitrary nature of the inputs and therefore it would not introduce interference. However, in contrast to intuition, we will show that the two-photon interference can be observed if a certain condition is satisfied.

*T*is difficult. However, similar experiments with single-photon states can provide a hint. Legero,

*et al*investigated the two-photon interference in terms of Δ

*T*with single-photon states and observed two-photon quantum beating when the spectral frequencies of single-photon states are different [8

8. T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-resolved two-photon quantum interference,” Appl. Phys. B **77**, 797–802 (2003). [CrossRef]

10. T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Characterization of single photons using two-photon interference,” Adv. Atom. Atom. Mol. Opt. Phys. **53**, 253–289 (2006). [CrossRef]

*V*= 1. Based on this result, we can expect sinusoidal beat fringes for the two-photon classical interference with respect to Δ

*T*. As the visibility of two-photon classical interference is limited by

*V*= 0.5, we also can presume the visibility of the fringes would be

*V*≤ 0.5. We will see the two-photon beating can indeed be observed with CW coherent light.

## 3. Theoretical analysis with the superposition of electromagnetic waves

*E*(

_{i}*t*) at the inputs (

*i*= 1, 2) and outputs (

*i*= 3, 4) of the BS as depicted in Fig. 1(a). Assuming that the amplitudes of

*E*

_{1}(

*t*) and

*E*

_{2}(

*t*) are the same, |

*E*

_{0}|, the two inputs of electric field are represented by where

*j*= 1, 2,

*ω*and

_{j}*ϕ*are the angular frequencies and phases of

_{j}*E*. The unitary transformation of a BS gives the output electric fields as where

_{j}*t*is the time of flight from

_{j}*E*(

_{j}*t*) to

*E*

_{3}(

*t*) and/or

*E*

_{4}(

*t*). Note that the relative time difference between

*t*

_{1}and

*t*

_{2}can be defined as the relative optical delay Δ

*t*=

*t*

_{2}−

*t*

_{1}.

*ω*for each input, there is typically non-zero spectral bandwidth. The non-zero spectral bandwidth produces a finite coherence length, which is determined by the degree of first-order coherence 0 ≤ |

_{j}*γ*(Δ

*t*)| = 1 between

*E*

_{1}(

*t*) and

*E*

_{2}(

*t*+ Δ

*t*), where |

*γ*(Δ

*t*)| = 0 for incoherent inputs while |

*γ*(Δ

*t*)| = 1 for completely coherent inputs. With the degree of first-order coherence |

*γ*(Δ

*t*)| and Eqs. (1) and (2), the output intensities at D1 and D2 are given as

*ω*and Δ

*ϕ*refer the frequency and phase difference between two inputs, i.e., Δ

*ω*=

*ω*

_{2}−

*ω*

_{1}and Δ

*ϕ*=

*ϕ*

_{2}−

*ϕ*

_{1}. Here, 〈

*x*〉 represents the average of

*x*over many events. When

*E*

_{1}and

*E*

_{2}have the same frequencies,

*ω*

_{1}=

*ω*

_{2}=

*ω*

_{0}, the intensities

*I*

_{3}and

*I*

_{4}can be represented as When the two inputs are coherent such that Δ

*ϕ*is a constant, Eq. (4) shows sinusoidal oscillations with respect to Δ

*t*with an envelope defined by |

*γ*(Δ

*t*)|. This is single-photon interference. For incoherent inputs in which Δ

*ϕ*randomly varies, the sinusoidal oscillations will be washed out since 〉sin[Δ

*ϕ*]〉 = 0.

*𝒜*=

*ω*

_{2}

*t*

_{2}−

*ω*

_{1}

*t*

_{1}− Δ

*ϕ*. For incoherent inputs, the second and third terms of Eq. (5) vanish since 〉sin

*𝒜*〉 = 0. Note that the last term of Eq. (5) does not disappear as it has a square of sin

*𝒜*term. Dropping the constant |

*E*

_{0}|

^{4}, Eq. (5) can be simplified as

*γ*(Δ

*t*)|. If two inputs of CW coherent light are completely independent and incoherent to each other from the beginning, |

*γ*(Δ

*t*)| = 0 at all times. In this case, no interference can be attained since the second term of Eq. (6) disappears. If, however, the inputs somehow have a non-zero first-order coherence, |

*γ*(Δ

*t*)| ≠ 0, and have randomized phases, we can see interference. This condition can be achieved if, for instance, the two inputs originated from a single laser and their phases are randomized after they are separated. Figure 1(b) shows a typical way to implement this condition. Note that two independent acousto-optic modulators, AOM1 and AOM2, disturb the phase coherence between two inputs,

*E*

_{1}(

*t*) and

*E*

_{2}(

*t*), and thus 〉sin[Δ

*ϕ*]〉 = 0.

*E*

_{1}(

*t*) and

*E*

_{2}(

*t*) are the same, Δ

*ω*= 0, the cosine term in Eq. (6) goes to 1 for any Δ

*T*. Assuming |

*γ*(Δ

*t*)| has a Gaussian distribution function, one can expect to observe a HOM dip with a visibility of

*V*= 0.5 while scanning Δ

*t*. It is remarkable that the HOM dip originated from the first-order coherence between the two inputs, even though the single-photon interference is erased by the randomized phases. Note that the HOM dip can be measured for any Δ

*T*, even when Δ

*T*is much larger than the coherence time

*t*of the light source. This result is somewhat counter-intuitive since the coincidences for

_{c}*t*≪ Δ

_{c}*T*originate from photons that are temporally separated at the BS. As these photons did not have temporal overlap at the BS, one can naively think the electric fields do not interfere, thus they should not show interference. Equation (6) shows that this intuition is incorrect and the electric fields do interfere even if they do not have temporal overlap. This shows that the electromagnetic waves rather than the photons are responsible for interference in classical physics [22

22. L. de Broglie and J.A.E. Silva, “Interpretation of a Recent Experiment on Interference of Photon Beams,” Phys. Rev. **172**, 1284–1285 (1968). [CrossRef]

21. Y.-S. Kim, O. Slattery, P.S. Kuo, and X. Tang, “Conditions for two-photon interference with coherent pulses,” Phys. Rev. A **87**, 063843 (2013). [CrossRef]

*ω*≠ 0, one can measure a sinusoidal oscillation with respect to Δ

*T*. It is notable that the coincidences always have the minimum at Δ

*T*= 0 because Eq. (6) is independent of Δ

*ϕ*. The visibility of the oscillation is determined by |

*γ*(Δ

*t*)|

^{2}. When |

*γ*(Δ

*t*)| = 1, one can measure the sinusoidal oscillation with

*V*= 0.5. As the frequency of the oscillation is determined by the difference between the frequencies of

*E*

_{1}and

*E*

_{2}, Δ

*ω*, the origin of the oscillation is a two-photon beating. As we qualitatively investigated earlier, the two-photon interference in terms of Δ

*T*indeed reveals beating fringes with limited visibility.

*T*as long as |

*γ*(Δ

*t*)| ≠ 0. In practice, we can consider a finite oscillation by considering a finite coherence time in terms of Δ

*T*. If we can somehow regulate the coherence between

*E*

_{1}(

*t*) and

*E*

_{1}(

*t*+ Δ

*T*), we can consider the degree of first-order coherence between

*E*

_{1}at time

*t*and

*t*+ Δ

*T*as, 0 ≤ |Γ(Δ

*T*)| ≤ 1, and it gives a finite interference in terms of Δ

*T*. Using |Γ(Δ

*T*)|, Eq. (6) can be modified as Note that we express Eq. (7) as a function of Δ

*t*and Δ

*T*rather than

*T*

_{1}and

*T*

_{2}since it is dependent on these variables.

*ω*= 0 and (b) for Δ

*ω*= 2

*π*× 3 MHz. The full widths at half maximum (FWHM) of the Gaussian |

*γ*(Δ

*t*)| and |Γ(Δ

*T*)| are assumed to be 0.67 ps and 1.18

*μ*s, respectively, and

*T*= 10.57 ps. When Δ

_{p}*ω*= 0, we can see repeated HOM dips while varying Δ

*t*. The visibility of the HOM dips is a maximum at Δ

*T*= 0, and as Δ

*T*increases, the visibility decreases. For Δ

*ω*≠ 0, we can see sinusoidal oscillations rather than simple dips with respect to Δ

*T*. It is remarkable that we can attain either HOM peaks or dips while scanning of optical delay Δ

*t*according to the detection time difference Δ

*T*: If Δ

*T*is chosen so as to have minimum(maximum) coincidences, HOM dips(peaks) would appear in terms of Δ

*t*. Note that the repeated two-photon interference is expected when Δ

*t*are multiples of

*T*for both the Δ

_{p}*ω*= 0 and Δ

*ω*≠ 0 cases.

## 4. Experiment and result

*γ*(Δ

*t*)| while randomizing their phases, we built a Mach-Zehnder (MZ) interferometer with two BS and an optical delay. Two independent AOMs disturb the phase coherence between the two inputs while maintaining |

*γ*(Δ

*t*)|. After the phases are randomized, Fig. 1(b) can be considered as Fig. 1(a) by considering the second BS of Fig. 1(b) as the BS of Fig. 1(a).

21. Y.-S. Kim, O. Slattery, P.S. Kuo, and X. Tang, “Conditions for two-photon interference with coherent pulses,” Phys. Rev. A **87**, 063843 (2013). [CrossRef]

*t*, see Fig. 3(b). It shows a clear recurrence of the MZ interference envelops with a period of

*T*= 10.57 ± 0.07 ps [23

_{p}23. S.-Y. Baek, O. Kwon, and Y.-H. Kim, “High-resolution mode-spacing measurement of the blue-violet diode laser using interference of fields created with time delays greater than the coherence time,” Jpn. J. Appl. Phys. **46**, 7720–7723 (2007). [CrossRef]

*ω*≠ 0, since an AOM adds the spectral frequency to the deflected beam according to the driving RF signal frequency. The RF signal frequency difference between AOM1 and AOM2, Δ

*f*, introduces Δ

*ω*= 2

*π*Δ

*f*. Note that additional frequency modulation (FM) noise input to an AOM will disturb the phase within the same arm, thus degrading |Γ(Δ

*T*)| as Δ

*T*increases [21

**87**, 063843 (2013). [CrossRef]

*μ*〉 = 8 × 10

^{−3}for the TCSPC window, 10 ns. First, we measured the coincidence as a function of the detection time difference Δ

*T*while Δ

*t*is fixed at 0. Figure 4 presents the coincidence for (a) Δ

*ω*= 0, and (b) Δ

*ω*= 2

*π*× 3 MHz. We also present the data with different FM noise signals which degrade |Γ(Δ

*T*)| when Δ

*T*is large. The FM noise signals are quantified by the standard deviations of the RF signal frequency,

*σ*

_{FM}. Note that we present the data with coincidence counts normalized so that the coincidence without two-photon interference is 1. Since the simulation result depicted in Fig. 2 is also normalized in the same manner, this data presentation is convenient to compare theory and experiment.

*ω*= 0. In general, the coincidences show a regular HOM dip regardless of Δ

*T*. Note that each data point can be considered as a HOM dip since Δ

*t*= 0, corresponding to the optical path length difference where a HOM interference occurs. The only exception to this general statement is that when

*σ*

_{FM}is sufficiently large, the visibility of the HOM dips decrease as Δ

*T*increases and the visibility decreases faster when

*σ*

_{FM}is larger. From Fig. 3, we can calculate the coherence time of our diode laser to be about

*t*= 2.4 ps. Noting that the time scale in Fig. 4 is

_{c}*μ*s, we find that two-photon interference occurs even when

*t*≪ Δ

_{c}*T*. Interestingly, this holds even if we input FM noise in order to degrade |Γ(Δ

*T*)|.

*ω*≠ 0, the coincidences show a sinusoidal oscillation which corresponds to a two-photon beating fringe as depicted in Fig. 4(b). Similar to the Δ

*ω*= 0 case, the oscillation continues even when

*t*≪ Δ

_{c}*T*. The envelop of the oscillation is determined by the FM noise and they are identical to the Δ

*ω*= 0 case. The oscillation frequency is found to be 3.02 MHz which corresponds to the RF signal frequency difference. Note that the error bars represent the experimental standard deviations. In the following data, we omit the error bars for clear data presentation, however, they are similar with those in Fig. 4(b).

*T*≠ 0, we can see either HOM dips or peaks depending on the conditions. In particular, when Δ

*ω*= 0, we can always observe the HOM dips although the visibility of the interference decreases as

*σ*

_{FM}increases. On the other hand, we can measure either HOM dips or peaks for Δ

*ω*≠ 0. These phenomena actually come from the oscillation of two-photon interference in terms of Δ

*T*. If we choose Δ

*T*to correspond to a maximum coincidence, e.g., Δ

*T*= 0.16

*μ*s or 1.49

*μ*s, one can see the HOM peaks rather than dips while scanning Δ

*t*. For a Δ

*T*which corresponds to a minimum coincidences, e.g., Δ

*T*= 0.32

*μ*s, HOM dips appear. Note that the visibility of the HOM dips/peaks are affected by the FM noise, thus for a large

*σ*

_{FM}, the HOM dips/peaks are suppressed. It is worth noting that the repeating property of HOM dips/peaks is preserved.

## 5. Conclusion

*V*≤ 0.5. While varying the optical delay of the interferometer, HOM dips or peaks are observed depending on the chosen detection time difference. The HOM dips/peaks are repeated whenever the optical delay are multiples of the first-order coherence revival period of multi-mode coherent light. With the increasing interest of using a multi-mode CW diode laser in quantum information science due to the easy and inexpensive implementation, these results help to understand the nature of two-photon interference and also can be useful for quantum information science.

## Acknowledgments

## References and links

1. | T. Young, |

2. | E. Hecht, |

3. | L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. |

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

5. | J.G. Rarity, P.R. Tapster, and R. Loudon, “Non-classical interference between independent sources,” J. Opt. B: Quantum Semiclass. Opt. |

6. | A.K. Jha, Coherence property of the entangled two-photon field produced by parametric down-conversion, Ph.D. thesis, University of Rochester, NY, 2009. |

7. | Z.Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. |

8. | T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-resolved two-photon quantum interference,” Appl. Phys. B |

9. | T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. |

10. | T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Characterization of single photons using two-photon interference,” Adv. Atom. Atom. Mol. Opt. Phys. |

11. | X.Y. Zou, L.J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. |

12. | O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled |N, 0〉+ |0, N〉 state,” Phys. Rev. A |

13. | H.-T. Lim, Y.-S. Kim, Y.-S. Ra, J. Bae, and Y.-H. Kim, “Experimental realization of an approximate partial transpose for photonic two-qubit systems,” Phys. Rev. Lett. |

14. | Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. |

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

16. | P. Kok, W.J. Munro, K. Nemoto, T.C. Ralph, J.P. Dowiling, and G.J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. |

17. | K. Wang and D.-Z. Cao, “Subwavelength coincidence interference
with classical thermal light,” Phys. Rev.
A |

18. | J. Cheng and S.-S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. |

19. | J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical sub wavelength interference with a pseudo thermal light source,” Phys. Rev. Lett. |

20. | Y.H. Zhai, X.-H. Chen, D. Zhang, and L.-A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A |

21. | Y.-S. Kim, O. Slattery, P.S. Kuo, and X. Tang, “Conditions for two-photon interference with coherent pulses,” Phys. Rev. A |

22. | L. de Broglie and J.A.E. Silva, “Interpretation of a Recent Experiment on Interference of Photon Beams,” Phys. Rev. |

23. | S.-Y. Baek, O. Kwon, and Y.-H. Kim, “High-resolution mode-spacing measurement of the blue-violet diode laser using interference of fields created with time delays greater than the coherence time,” Jpn. J. Appl. Phys. |

24. | O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Coherence properties of spontaneous parametric down-conversion pumped by a multi-mode cw diode laser,” Opt. Express |

25. | O. Kwon, K.-K. Park, Y.-S. Ra, Y.-S. Kim, and Y.-H. Kim, “Time-bin entangled photon pairs from spontaneous parametric down-conversion pumped by a cw multi-mode diode laser,” Opt. Express |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(270.1670) Quantum optics : Coherent optical effects

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: December 13, 2013

Manuscript Accepted: January 14, 2014

Published: February 6, 2014

**Citation**

Yong-Su Kim, Oliver Slattery, Paulina S. Kuo, and Xiao Tang, "Two-photon interference with continuous-wave multi-mode coherent light," Opt. Express **22**, 3611-3620 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3611

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### References

- T. Young, Lectures on Natural Philosophy, Vol. I, p. 464 (Johnson, London, 1807).
- E. Hecht, Optics (Addision-Wesely, San Francisco, 2002).
- L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71, S274–S282 (1999). [CrossRef]
- C.K. Hong, Z.Y. Ou, L. Mandel, “Measurement of sub picosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987). [CrossRef] [PubMed]
- J.G. Rarity, P.R. Tapster, R. Loudon, “Non-classical interference between independent sources,” J. Opt. B: Quantum Semiclass. Opt. 7, S171–S175 (2005). [CrossRef]
- A.K. Jha, Coherence property of the entangled two-photon field produced by parametric down-conversion, Ph.D. thesis, University of Rochester, NY, 2009.
- Z.Y. Ou, L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54–57 (1988). [CrossRef] [PubMed]
- T. Legero, T. Wilk, A. Kuhn, G. Rempe, “Time-resolved two-photon quantum interference,” Appl. Phys. B 77, 797–802 (2003). [CrossRef]
- T. Legero, T. Wilk, M. Hennrich, G. Rempe, A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004). [CrossRef] [PubMed]
- T. Legero, T. Wilk, A. Kuhn, G. Rempe, “Characterization of single photons using two-photon interference,” Adv. Atom. Atom. Mol. Opt. Phys. 53, 253–289 (2006). [CrossRef]
- X.Y. Zou, L.J. Wang, L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett. 67, 318–321 (1991). [CrossRef] [PubMed]
- O. Kwon, Y.-S. Ra, Y.-H. Kim, “Observing photonic de Broglie waves without the maximally-path-entangled |N, 0〉+ |0, N〉 state,” Phys. Rev. A 81, 063801 (2010). [CrossRef]
- H.-T. Lim, Y.-S. Kim, Y.-S. Ra, J. Bae, Y.-H. Kim, “Experimental realization of an approximate partial transpose for photonic two-qubit systems,” Phys. Rev. Lett. 107, 160401 (2011). [CrossRef] [PubMed]
- Y.-S. Kim, J.-C. Lee, O. Kwon, Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. 8, 117–120 (2012). [CrossRef]
- E. Knill, R. Laflamme, G.J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]
- P. Kok, W.J. Munro, K. Nemoto, T.C. Ralph, J.P. Dowiling, G.J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]
- K. Wang, D.-Z. Cao, “Subwavelength coincidence interference with classical thermal light,” Phys. Rev. A 70, 041801 (2004). [CrossRef]
- J. Cheng, S.-S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004). [CrossRef] [PubMed]
- J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, K. Wang, “Experimental observation of classical sub wavelength interference with a pseudo thermal light source,” Phys. Rev. Lett. 94, 173601 (2005). [CrossRef]
- Y.H. Zhai, X.-H. Chen, D. Zhang, L.-A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72, 043805 (2005). [CrossRef]
- Y.-S. Kim, O. Slattery, P.S. Kuo, X. Tang, “Conditions for two-photon interference with coherent pulses,” Phys. Rev. A 87, 063843 (2013). [CrossRef]
- L. de Broglie, J.A.E. Silva, “Interpretation of a Recent Experiment on Interference of Photon Beams,” Phys. Rev. 172, 1284–1285 (1968). [CrossRef]
- S.-Y. Baek, O. Kwon, Y.-H. Kim, “High-resolution mode-spacing measurement of the blue-violet diode laser using interference of fields created with time delays greater than the coherence time,” Jpn. J. Appl. Phys. 46, 7720–7723 (2007). [CrossRef]
- O. Kwon, Y.-S. Ra, Y.-H. Kim, “Coherence properties of spontaneous parametric down-conversion pumped by a multi-mode cw diode laser,” Opt. Express 17, 13059 (2009). [CrossRef] [PubMed]
- O. Kwon, K.-K. Park, Y.-S. Ra, Y.-S. Kim, Y.-H. Kim, “Time-bin entangled photon pairs from spontaneous parametric down-conversion pumped by a cw multi-mode diode laser,” Opt. Express 21, 25492 (2013). [CrossRef] [PubMed]

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