## The Hong-Ou-Mandel effect in the context of few-photon scattering |

Optics Express, Vol. 20, Issue 11, pp. 12326-12340 (2012)

http://dx.doi.org/10.1364/OE.20.012326

Acrobat PDF (910 KB)

### Abstract

The Hong-Ou-Mandel effect is studied in the context of two-photon transport in a one-dimensional waveguide with a single scatterer. We numerically investigate the scattering problem within a time-dependent, wave-function-based framework. Depending on the realization of the scatterer and its properties, we calculate the joint probability of finding both photons on either side of the waveguide after scattering. We specifically point out how Hong-Ou-Mandel interferometry techniques could be exploited to identify effective photon–photon interactions which are mediated by the scatterer. The Hong-Ou-Mandel dip is discussed in detail for the case of a single two-level atom embedded in the waveguide, and dissipation and dephasing are taken into account by means of a quantum jump approach.

© 2012 OSA

## 1. Introduction

1. 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]

2. Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys. **7**, 155 (2005). [CrossRef]

3. V. Giovannetti, D. Frustaglia, F. Taddei, and R. Fazio, “Electronic Hong-Ou-Mandel interferometer for multimode entanglement detection,” Phys. Rev. B **74**, 115315 (2006). [CrossRef]

4. I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science **307**, 1733–1734 (2005). [CrossRef] [PubMed]

*per se*. To date, photon pairs can be routinely sent through optical fibers [5

5. H. Takesue, “1.5 *μ*m band Hong-Ou-Mandel experiment using photon pairs generated in two independent dispersion shifted fibers,” Appl. Phys. Lett. **90**, 204101 (2007). [CrossRef]

6. A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. **2**, 224 (2011). [CrossRef] [PubMed]

7. S. M. Wang, S. Y. Mu, C. Zhu, X. Y. Gong, P. Xu, H. Liu, T. Li, S. N. Zhu, and X. Zhang, “Hong-Ou-Mandel interference mediated by the magnetic plasmon waves in a three-dimensional optical metamaterial,” Opt. Express **20**, 5213–5218 (2012). [CrossRef] [PubMed]

8. P. Longo, P. Schmitteckert, and K. Busch, “Dynamics of photon transport through quantum impurities in dispersion-engineered one-dimensional systems,” J. Opt. A: Pure Appl. Opt. **11**, 114009 (2009). [CrossRef]

9. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems,” Phys. Rev. A **83**, 063828 (2011). [CrossRef]

*T*

_{1}-type and dephasing of

*T*

_{2}-type into account. We conclude the paper in Sec. 5 and give a short outlook on possible future work.

## 2. Fundamentals

### 2.1. The quantum-mechanical beam splitter as a four-port device and its relation to scattering problems

*π*/2 and we thus have

1. 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]

1. 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]

*k*is transformed into a transmitted and a reflected momentum state according to |out〉 =

*Ŝ*|in〉 with where

*r*and

_{k}*t*signify the reflection and transmission amplitudes, respectively. For an in-state in the left lead, we need to demand

_{k}*k*> 0, for the right lead

*k*< 0. We thus have 4 “ports” in total, namely

*k*> 0 and

*k*< 0 for the left and the right lead, respectively (cf. Fig. 1). We assume the magnitude of the momentum, |

*k*|, to be unchanged here. In other words, we assume the single-particle scattering process to be elastic and the Hamiltonian to be time-reversal symmetric.

### 2.2. Hong-Ou-Mandel interferometry as a probe for photon–photon interactions

### 2.3. Types of impurities and their single-particle scattering solution

#### 2.3.1. General form of the scattering equation

*x*

_{0}. This Hamiltonian describes a chain of identical and equally spaced coupled optical resonators which form a waveguide whose dispersion relation is centered around the resonators’ resonance frequency

*ω*. Such a system could, for instance, be realized by appropriately placed micro-disk resonators. In that case, the nearest-neighbor hopping constant

*J*is defined as the overlap integral of the electromagnetic field modes of adjacent resonators [12

12. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

*J*is chosen such that the fibers dispersion relation is well-described around a certain operating wavelength of interest. Similar to Ref. [13

13. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. **101**, 100501 (2008). [CrossRef] [PubMed]

*E*=

*ε*=

_{k}*ω*− 2

*J*cos(

*ka*) is the eigenenergy corresponding to wavenumber

*k*∈ [−

*π*/

*a*,

*π*/

*a*] (for lattice constant a). The reflection amplitude is The function

*G*(

*E*) depends on the actual realization of the impurity. Note that 1 +

*r*=

_{k}*t*and the conservation of probability, |

_{k}*r*|

_{k}^{2}+ |

*t*|

_{k}^{2}= 1, hold. We choose the zero of the energy of the free waveguide to lie in the middle of the cosine band, i.e., we set

*ω*= 0 in the following.

#### 2.3.2. Local on-site potential

*H*=

*H*

_{leads}+

*H*

_{pot}, where is the contribution due to the impurity, which is part of the tight-binding chain itself and

*g*is the strength of the local on-site potential (cf. Fig. 2).

*G*(

*E*) then simply becomes (cf. Eq. (7)) Consequently, the reflection amplitude takes the form Thus, the balanced beam splitter with

#### 2.3.3. Local two-level system

8. P. Longo, P. Schmitteckert, and K. Busch, “Dynamics of photon transport through quantum impurities in dispersion-engineered one-dimensional systems,” J. Opt. A: Pure Appl. Opt. **11**, 114009 (2009). [CrossRef]

9. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems,” Phys. Rev. A **83**, 063828 (2011). [CrossRef]

13. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. **101**, 100501 (2008). [CrossRef] [PubMed]

21. D. Witthaut and A.S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. **12**, 043052 (2009). [CrossRef]

*H*=

*H*

_{leads}+

*H*

_{TLS}, where The transition energy of the two-level atom is denoted by Ω and

*V*is the atom–cavity coupling strength (cf. Fig. 2).

### 2.4. Influencing variables for the Hong-Ou-Mandel dip

*R*̂ has the form

*ρ*denotes the density matrix and the Lindbladian reads

*T*

_{1}-type is described by the relaxation operator

*T*

_{2}-type requires

22. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. **70**, 101–144 (1998) [CrossRef]

23. K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B **10**, 524–538 (1993) [CrossRef]

## 3. Time evolution, initial states, and physical quantities

*t*)〉 stands for the state vector of radiation field and scatterer [8

8. P. Longo, P. Schmitteckert, and K. Busch, “Dynamics of photon transport through quantum impurities in dispersion-engineered one-dimensional systems,” J. Opt. A: Pure Appl. Opt. **11**, 114009 (2009). [CrossRef]

9. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems,” Phys. Rev. A **83**, 063828 (2011). [CrossRef]

14. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. **104**, 023602 (2010). [CrossRef] [PubMed]

22. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. **70**, 101–144 (1998) [CrossRef]

23. K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B **10**, 524–538 (1993) [CrossRef]

24. Y. Saad, “Analysis of some Krylov subspace approximations to the matrix exponential operator,” SIAM Journal on Numerical Analysis **29**, 209–228 (1992). [CrossRef]

26. M. Pototschnig, J. Niegemann, L. Tkeshelashvili, and K. Busch, “Time-domain simulation of the nonlinear Maxwell equations using operator-exponential techniques,” IEEE Trans. Ant. Propagat. **57**, 475–483 (2009). [CrossRef]

_{x1x2}is a boson-symmetric product of single-particle pulses launched from different ends of the waveguide (cf. Fig. 3). To be precise, the wave function is of the form where

*k*

_{0}, center

*x*, and width

_{c}*s*. Unless stated otherwise, we choose

*s*

^{(1)}=

*s*

^{(2)}≡

*s*= 7

*a*,

*Na*is the total length of the waveguide. The relative displacement Δ

*is varied from −30*

_{x}*a*to +30

*a*in order to record the Hong-Ou-Mandel dip. When Δ

*= 0, both pulses initially have the same distance to the scatterer. The waveguide consists of*

_{x}*N*= 199 sites and the scatterer couples to site

*x*

_{0}= (

*N*+1)/2 = 100. For the remainder, we take the nearest-neighbor hopping strength

*J*> 0 as our fundamental energy scale. Consequently, time is measured in units of

*h*̄/

*J*. Moreover, lengths are given in units of the lattice constant

*a*so that carrier wave numbers have the unit 1/

*a*. We choose Δ

*t*= 0.1

*h*̄/

*J*, which is smaller than any other time scale in the system, as the fundamental time step in the stochastic time evolution. For more information on the details of the simulation technique itself, we refer to Ref. [8

**11**, 114009 (2009). [CrossRef]

*t*)〉 at all times, the calculation of arbitrary physical quantities becomes possible. According to Refs. [8

**11**, 114009 (2009). [CrossRef]

**83**, 063828 (2011). [CrossRef]

14. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. **104**, 023602 (2010). [CrossRef] [PubMed]

**= {1,2,...,

*x*

_{0}− 1}) and the other on the right-hand side (

**= {

*x*

_{0}+ 1,...,

*N*}) of the scatterer. Here, the site

*x*

_{0}to which the scatterer is coupled is explicitly excluded from the summation since any excitation which might be trapped [14

14. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. **104**, 023602 (2010). [CrossRef] [PubMed]

**11**, 114009 (2009). [CrossRef]

*, i.e., as a function of the spatial separation of both incoming wave packets (cf. Fig. 3), Eq. (20) reproduces the famous Hong-Ou-Mandel dip.*

_{x}## 4. Results and discussion

**59**, 2044–2046 (1987). [CrossRef] [PubMed]

### 4.1. Local on-site potential in a tight-binding waveguide

*g*= 2 |

*J*||sin(

*k*

_{0}

*a*)| and operate at a carrier wavenumber of

*= 0. For large separations, both wave packets pass the scatterer individually. Thus, one can read off the single-photon transmission probability at Δ*

_{x}*∼ ±30*

_{x}*a*which is (nearly) perfect 50%. The reason why the local on-site potential works that well as a beam splitter for non-monochromatic excitations is because the reflectivity does not change for small deviations around the central carrier wavenumber of

*∂*|

_{k}*r*|

_{k}^{2}|

_{k=π/2a}= 0 (not shown). This special property is due to a combination of the scatterer being part of the chain, i.e., it is not side-coupled, and the cosine dispersion of the tight-binding waveguide. Furthermore, effects due to non-linear dispersion are reduced since the group-velocity dispersion is zero at

### 4.2. Local two-level system in a tight-binding waveguide—the Hong-Ou-Mandel effect as a probe for photon–photon interactions

**83**, 063828 (2011). [CrossRef]

**104**, 023602 (2010). [CrossRef] [PubMed]

#### 4.2.1. Influence of the atomic transition energy on the Hong-Ou-Mandel dip

*J*has several consequences. First, the atom–photon detuning

*V*is decreased. From Fig. 5 we can read off the tendency that the deviation from a perfect Hong-Ou-Mandel dip becomes more pronounced as the detuning is reduced. This is in line with Refs. [9

**83**, 063828 (2011). [CrossRef]

**104**, 023602 (2010). [CrossRef] [PubMed]

*V*∼

*J*and the resonance condition is fulfilled.

*→ ±∞) is nearly immune to changes in the atom–photon detuning. Since this limit represents individual particles passing the device, the fading of the Hong-Ou-Mandel dip must be due to effective photon–photon interactions whose effects can—at least in theory—be separated from signatures which are only induced by beam splitter imperfections.*

_{x}*P*

_{LR}= 0.5. The two-level atom acts as a dispersive beam splitter and—in contrast to the on-site potential—the atomic degree of freedom is side-coupled. Additionally, we operate in the non-linear regime of the dispersion relation (

*∂*|

_{k}*r*|

_{k}^{2}|

_{k=3π/4a}≠0 (not shown). As an example, we chose Ω =

*J*in all subsequent considerations.

#### 4.2.2. From the harmonic oscillator to the two-level system

**83**, 063828 (2011). [CrossRef]

**104**, 023602 (2010). [CrossRef] [PubMed]

*b*and

*b*

^{†}. Specifically, the formulation

*U*= 0 and a two-level system in the limit

*U*→ ∞.

*U*. In the absence of interaction (

*U*= 0), the Hong-Ou-Mandel dip becomes “perfect” besides the beam splitter imperfections due to the single-excitation transport characteristics. For

*U*> 0, i.e, in the interacting system, the fading of the Hong-Ou-Mandel dip depends non-monotonically on the value of the anharmonicity until it saturates in the limit

*U*→ ∞.

**83**, 063828 (2011). [CrossRef]

**104**, 023602 (2010). [CrossRef] [PubMed]

*U*-term become most pronounced if the atom–photon detuning is zero. However, in the Hong-Ou-Mandel setup, the resonance condition is not fulfilled since the scatterer acts as a beam splitter. To further understand this non-monotonicity, it is helpful to consider the detuning between two impinging photons and the energy they had in case they double-occupied the atomic site, i.e.,

*δ*=

*n*Ω+

*Un*(

*n*−1) −

*nε*

_{k0}, where

*n*= 2 and

*ε*

_{k0}is the single-photon energy. If the single-particle resonance condition, i.e., Ω =

*ε*

_{k0}, was fulfilled,

*δ*would grow monotonically as

*U*is increased. In the Hong-Ou-Mandel dip in Fig. 6,

*δ*changes its sign as

*U*grows. This eventually leads to the non-monotonic dependence of the depth of the Hong-Ou-Mandel dip.

#### 4.2.3. Influence of dissipation and dephasing

*T*

_{1}) as well as from pure dephasing, i.e., the randomization of the phase relation between the atom’s ground and excited state, (subsumed in time constant

*T*

_{2}).

*T*

_{1}-times on the shape of the Hong-Ou-Mandel dip. Once a photon is lost, i.e., the

*T*

_{1}-relaxation operator was applied to the two-particle state (cf. Sec. 2.4), the wave function collapses to a single-particle state and two-particle coincidences become impossible, which leads to a less pronounced Hong-Ou-Mandel dip. However, since the definition of the joint probability

*P*

_{LR}in Eq. (20) is normalized to the total probability, the single-photon limit is independent of the value of

*T*

_{1}. Note that in the quantum jump approach as described in Refs. [22

22. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. **70**, 101–144 (1998) [CrossRef]

23. K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B **10**, 524–538 (1993) [CrossRef]

*T*

_{2}-times the single-photon limit seems to be practically unaffected. Only very short dephasing times lead to a significant change in the single-photon transport which results in the beam splitter being unbalanced and thus the single-photon limit changes.

**11**, 114009 (2009). [CrossRef]

*T*

_{2}-times comparable to the temporal overlap

*τ*of the wave packet at the position of the atom lead to an enhanced transmission. As a crude estimate,

*τ*∼

*s*/

*v*

_{g}, where

*v*

_{g}= 2

*aJh*̄

^{−1}sin(

*k*

_{0}

*a*) is the group velocity.

*T*

_{1}and

*T*

_{2}) and the actual interaction between two excitations is impossible and one still would have to speculate to which degree an imperfect Hong-Ou-Mandel dip really is the signature for effective photon–photon interactions. We would like to emphasize that the previous studies were driven by the explicit knowledge of the stationary, i.e., the monochromatic, single-particle solution yielding a condition for the balanced beam splitter. This condition is, as was shown, not perfectly met for pulses of finite width. However, given a fixed width of the wave packets and fixed values of

*T*

_{1}and

*T*

_{2}, the condition for the balanced beam splitter can be recovered in a trial-and-error fashion by tuning the two-level system’s transition energy or coupling strength such that in a single-photon setup reflection and transmission occur with equal probability. In addition to that, carefully designed and/or tunable dispersion relations such as those available in photonic crystal waveguides [27

27. K. Busch, G. v. Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. **444**, 101–202 (2007). [CrossRef]

*T*

_{1}and/or

*T*

_{2}times. In that case, however, one still would need sufficient knowledge about possible sources of either

*T*

_{1}-dissipation or

*T*

_{2}-dephasing, since these two quantities cannot be clearly separated from one another in the Hong-Ou-Mandel dip.

## 5. Conclusion and outlook

*T*

_{1}-type as well as pure dephasing of

*T*

_{2}-type. Due to the normalization of the joint probability to the total probability in the system,

*T*

_{1}-relaxation only affects the depth of the Hong-Ou-Mandel dip.

*T*

_{2}-dephasing can also change the offset since the single-photon transmittance is modified. Knowing these properties, Hong-Ou-Mandel interferometry techniques can—at least in principle—also be exploited to probe the environment.

28. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. **2**, 849–855 (2006). [CrossRef]

33. M. T. C. Wong and C. K. Law, “Two-polariton bound states in the Jaynes-Cummings-Hubbard model,” Phys. Rev. A **83**, 055802 (2011). [CrossRef]

34. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. **3**, 807–812 (2007). [CrossRef]

36. R. Yan, P. Pausauskie, J. Huang, and P. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. USA **106**, 21045–21050 (2009). [CrossRef] [PubMed]

**11**, 114009 (2009). [CrossRef]

**83**, 063828 (2011). [CrossRef]

## Acknowledgments

## References and links

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

2. | Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys. |

3. | V. Giovannetti, D. Frustaglia, F. Taddei, and R. Fazio, “Electronic Hong-Ou-Mandel interferometer for multimode entanglement detection,” Phys. Rev. B |

4. | I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science |

5. | H. Takesue, “1.5 |

6. | A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. |

7. | S. M. Wang, S. Y. Mu, C. Zhu, X. Y. Gong, P. Xu, H. Liu, T. Li, S. N. Zhu, and X. Zhang, “Hong-Ou-Mandel interference mediated by the magnetic plasmon waves in a three-dimensional optical metamaterial,” Opt. Express |

8. | P. Longo, P. Schmitteckert, and K. Busch, “Dynamics of photon transport through quantum impurities in dispersion-engineered one-dimensional systems,” J. Opt. A: Pure Appl. Opt. |

9. | P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems,” Phys. Rev. A |

10. | C. G. Gerry and P. L. Knight, |

11. | Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. |

12. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

13. | L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. |

14. | P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. |

15. | J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. |

16. | E. Rephaeli, J. T. Shen, and S. Fan, “Full inversion of a two-level atom with a single-photon pulse in one-dimensional geometries,” Phys. Rev. A |

17. | J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A |

18. | J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. |

19. | J. T. Shen and S. Fan, “Strongly correlated multi-particle transport in one dimension through a quantum impurity,” Phys. Rev. A |

20. | T. Shi and C. P. Sun, “Lehmann-Symanzik-Zimmermann reduction approach to multiphoton scattering in coupled resonator arrays,” Phys. Rev. B |

21. | D. Witthaut and A.S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. |

22. | M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. |

23. | K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B |

24. | Y. Saad, “Analysis of some Krylov subspace approximations to the matrix exponential operator,” SIAM Journal on Numerical Analysis |

25. | J. Niegemann, L. Tkeshelashvili, and K. Busch, “Higher-order time-domain simulations of Maxwell’s equations using Krylov-subspace methods,” J. Comput. Theor. Nanosci. |

26. | M. Pototschnig, J. Niegemann, L. Tkeshelashvili, and K. Busch, “Time-domain simulation of the nonlinear Maxwell equations using operator-exponential techniques,” IEEE Trans. Ant. Propagat. |

27. | K. Busch, G. v. Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. |

28. | M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. |

29. | A. Greentree, C. Tahan, J. Cole, and L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. |

30. | D. Angelakis, M. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A |

31. | M. I. Makin, J. H. Cole, C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Time evolution of the one-dimensional Jaynes-Cummings-Hubbard Hamiltonian,” Phys. Rev. A |

32. | J. Q. Quach, C.-H. Su, A. M. Martin, A. D. Greentree, and L. C. L. Hollenberg, “Reconfigurable quantum metamaterials,” Opt. Express |

33. | M. T. C. Wong and C. K. Law, “Two-polariton bound states in the Jaynes-Cummings-Hubbard model,” Phys. Rev. A |

34. | D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. |

35. | M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature |

36. | R. Yan, P. Pausauskie, J. Huang, and P. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. USA |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(290.0290) Scattering : Scattering

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 16, 2012

Revised Manuscript: May 3, 2012

Manuscript Accepted: May 5, 2012

Published: May 16, 2012

**Citation**

Paolo Longo, Jared H. Cole, and Kurt Busch, "The Hong-Ou-Mandel effect in the context of few-photon scattering," Opt. Express **20**, 12326-12340 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12326

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### References

- 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]
- Y. L. Lim and A. Beige, “Generalized Hong-Ou-Mandel experiments with bosons and fermions,” New J. Phys.7, 155 (2005). [CrossRef]
- V. Giovannetti, D. Frustaglia, F. Taddei, and R. Fazio, “Electronic Hong-Ou-Mandel interferometer for multimode entanglement detection,” Phys. Rev. B74, 115315 (2006). [CrossRef]
- I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science307, 1733–1734 (2005). [CrossRef] [PubMed]
- H. Takesue, “1.5 μm band Hong-Ou-Mandel experiment using photon pairs generated in two independent dispersion shifted fibers,” Appl. Phys. Lett.90, 204101 (2007). [CrossRef]
- A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011). [CrossRef] [PubMed]
- S. M. Wang, S. Y. Mu, C. Zhu, X. Y. Gong, P. Xu, H. Liu, T. Li, S. N. Zhu, and X. Zhang, “Hong-Ou-Mandel interference mediated by the magnetic plasmon waves in a three-dimensional optical metamaterial,” Opt. Express20, 5213–5218 (2012). [CrossRef] [PubMed]
- P. Longo, P. Schmitteckert, and K. Busch, “Dynamics of photon transport through quantum impurities in dispersion-engineered one-dimensional systems,” J. Opt. A: Pure Appl. Opt.11, 114009 (2009). [CrossRef]
- P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems,” Phys. Rev. A83, 063828 (2011). [CrossRef]
- C. G. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).
- Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun.63, 118–122 (1987). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett.24, 711–713 (1999). [CrossRef]
- L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett.101, 100501 (2008). [CrossRef] [PubMed]
- P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett.104, 023602 (2010). [CrossRef] [PubMed]
- J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett.30, 2001–2003 (2005). [CrossRef] [PubMed]
- E. Rephaeli, J. T. Shen, and S. Fan, “Full inversion of a two-level atom with a single-photon pulse in one-dimensional geometries,” Phys. Rev. A82, 033804 (2010). [CrossRef]
- J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A79, 023837 (2009). [CrossRef]
- J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett.98, 153003 (2007). [CrossRef] [PubMed]
- J. T. Shen and S. Fan, “Strongly correlated multi-particle transport in one dimension through a quantum impurity,” Phys. Rev. A76, 062709 (2007). [CrossRef]
- T. Shi and C. P. Sun, “Lehmann-Symanzik-Zimmermann reduction approach to multiphoton scattering in coupled resonator arrays,” Phys. Rev. B79, 205111 (2009). [CrossRef]
- D. Witthaut and A.S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys.12, 043052 (2009). [CrossRef]
- M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys.70, 101–144 (1998) [CrossRef]
- K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B10, 524–538 (1993) [CrossRef]
- Y. Saad, “Analysis of some Krylov subspace approximations to the matrix exponential operator,” SIAM Journal on Numerical Analysis29, 209–228 (1992). [CrossRef]
- J. Niegemann, L. Tkeshelashvili, and K. Busch, “Higher-order time-domain simulations of Maxwell’s equations using Krylov-subspace methods,” J. Comput. Theor. Nanosci.4, 627–634 (2007).
- M. Pototschnig, J. Niegemann, L. Tkeshelashvili, and K. Busch, “Time-domain simulation of the nonlinear Maxwell equations using operator-exponential techniques,” IEEE Trans. Ant. Propagat.57, 475–483 (2009). [CrossRef]
- K. Busch, G. v. Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101–202 (2007). [CrossRef]
- M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys.2, 849–855 (2006). [CrossRef]
- A. Greentree, C. Tahan, J. Cole, and L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys.2, 856–861 (2006). [CrossRef]
- D. Angelakis, M. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A76, 31805 (2007). [CrossRef]
- M. I. Makin, J. H. Cole, C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Time evolution of the one-dimensional Jaynes-Cummings-Hubbard Hamiltonian,” Phys. Rev. A80, 043842 (2009). [CrossRef]
- J. Q. Quach, C.-H. Su, A. M. Martin, A. D. Greentree, and L. C. L. Hollenberg, “Reconfigurable quantum metamaterials,” Opt. Express19, 11018–11033 (2011). [CrossRef] [PubMed]
- M. T. C. Wong and C. K. Law, “Two-polariton bound states in the Jaynes-Cummings-Hubbard model,” Phys. Rev. A83, 055802 (2011). [CrossRef]
- D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys.3, 807–812 (2007). [CrossRef]
- M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460, 1110–1112 (2009). [CrossRef] [PubMed]
- R. Yan, P. Pausauskie, J. Huang, and P. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. USA106, 21045–21050 (2009). [CrossRef] [PubMed]

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