Abstract

The Hong–Ou–Mandel interferometer is a versatile tool for analyzing the joint properties of photon pairs, relying on a truly quantum interference effect between two-photon probability amplitudes. While the theory behind this form of two-photon interferometry is well established, the development of advanced photon sources and exotic two-photon states has highlighted the importance of quantifying precisely what information can and cannot be inferred from features in a Hong–Ou–Mandel interference trace. Here we examine Hong–Ou–Mandel interference with regard to a particular class of states, so-called quantum frequency combs, and place special emphasis on the role spectral phase plays in these measurements. We find that this form of two-photon interferometry is insensitive to the relative phase between different comb line pairs. This is true even when different comb line pairs are mutually coherent at the input of a Hong–Ou–Mandel interferometer and the fringe patterns display sharp temporal features. Consequently, Hong–Ou–Mandel interference cannot speak to the presence of high-dimensional frequency-bin entanglement in two-photon quantum frequency combs.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (4)

M. Kues, C. Reimer, J. M. Lukens, W. J. Munro, A. M. Weiner, D. J. Moss, and R. Morandotti, “Quantum optical microcombs,” Nat. Photonics 13(3), 170–179 (2019).
[Crossref]

H.-H. Lu, J. M. Lukens, B. P. Williams, P. Imany, N. A. Peters, A. M. Weiner, and P. Lougovski, “A controlled-not gate for frequency-bin qubits,” npj Quantum Inf. 5(1), 24 (2019).
[Crossref]

P. Imany, J. A. Jaramillo-Villegas, M. S. Alshaykh, J. M. Lukens, O. D. Odele, A. J. Moore, D. E. Leaird, M. Qi, and A. M. Weiner, “High-dimensional optical quantum logic in large operational spaces,” npj Quantum Inf. 5(1), 59 (2019).
[Crossref]

C. Reimer, S. Sciara, P. Roztocki, M. Islam, L. R. Cortés, Y. Zhang, B. Fischer, S. Loranger, R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, W. J. Munro, J. Azaña, M. Kues, and R. Morandotti, “High-dimensional one-way quantum processing implemented on d-level cluster states,” Nat. Phys. 15(2), 148–153 (2019).
[Crossref]

2018 (5)

H.-H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-Optic Frequency Beam Splitters and Tritters for High-Fidelity Photonic Quantum Information Processing,” Phys. Rev. Lett. 120(3), 030502 (2018).
[Crossref]

H.-H. Lu, J. M. Lukens, N. A. Peters, B. P. Williams, A. M. Weiner, and P. Lougovski, “Quantum interference and correlation control of frequency-bin qubits,” Optica 5(11), 1455–1460 (2018).
[Crossref]

P. Imany, J. A. Jaramillo-Villegas, O. D. Odele, K. Han, D. E. Leaird, J. M. Lukens, P. Lougovski, M. Qi, and A. M. Weiner, “50-GHz-spaced comb of high-dimensional frequency-bin entangled photons from an on-chip silicon nitride microresonator,” Opt. Express 26(2), 1825–1840 (2018).
[Crossref]

P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, and A. M. Weiner, “Characterization of coherent quantum frequency combs using electro-optic phase modulation,” Phys. Rev. A 97(1), 013813 (2018).
[Crossref]

R.-B. Jin and R. Shimizu, “Extended wiener–khinchin theorem for quantum spectral analysis,” Optica 5(2), 93–98 (2018).
[Crossref]

2017 (4)

2016 (1)

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351(6278), 1176–1180 (2016).
[Crossref]

2015 (1)

Z. Xie, T. Zhong, S. Shrestha, X. Xu, J. Liang, Y.-X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. C. Wong, and C. Wei Wong, “Harnessing high-dimensional hyperentanglement through a biphoton frequency comb,” Nat. Photonics 9(8), 536–542 (2015).
[Crossref]

2014 (2)

2011 (1)

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

2010 (1)

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J. M. Merolla, and K. P. Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82(1), 013804 (2010).
[Crossref]

2009 (3)

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[Crossref]

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New J. Phys. 11(10), 103052 (2009).
[Crossref]

H. P. Specht, J. Bochmann, M. Mucke, B. Weber, E. Figueroa, D. L. Moehring, and G. Rempe, “Phase shaping of single-photon wave packets,” Nat. Photonics 3(8), 469–472 (2009).
[Crossref]

2008 (1)

2006 (1)

K. Wang, “Quantum theory of two-photon wavepacket interference in a beamsplitter,” J. Phys. B 39(18), R293–R324 (2006).
[Crossref]

2004 (3)

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93(7), 070503 (2004).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state,” Phys. Rev. A 70(2), 023806 (2004).
[Crossref]

M. Sagioro, C. Olindo, C. Monken, and S. Pádua, “Time control of two-photon interference,” Phys. Rev. A 69(5), 053817 (2004).
[Crossref]

2003 (1)

Y. Lu, R. Campbell, and Z.-Y. Ou, “Mode-locked two-photon states,” Phys. Rev. Lett. 91(16), 163602 (2003).
[Crossref]

2002 (1)

A. F. Abouraddy, M. B. Nasr, B. E. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65(5), 053817 (2002).
[Crossref]

2001 (1)

D. James, P. Kwiat, W. Munro, and A. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

1997 (1)

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A 56(2), 1627–1634 (1997).
[Crossref]

1996 (1)

M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, “Interferometric Bell-state analysis,” Phys. Rev. A 53(3), R1209–R1212 (1996).
[Crossref]

1992 (2)

A. Steinberg, P. G. Kwiat, and R. Chiao, “Dispersion cancellation in a measurement of the single-photon propagation velocity in glass,” Phys. Rev. Lett. 68(16), 2421–2424 (1992).
[Crossref]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45(9), 6659–6665 (1992).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, M. B. Nasr, B. E. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65(5), 053817 (2002).
[Crossref]

Alshaykh, M. S.

P. Imany, J. A. Jaramillo-Villegas, M. S. Alshaykh, J. M. Lukens, O. D. Odele, A. J. Moore, D. E. Leaird, M. Qi, and A. M. Weiner, “High-dimensional optical quantum logic in large operational spaces,” npj Quantum Inf. 5(1), 59 (2019).
[Crossref]

Aspelmeyer, M.

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New J. Phys. 11(10), 103052 (2009).
[Crossref]

Azaña, J.

C. Reimer, S. Sciara, P. Roztocki, M. Islam, L. R. Cortés, Y. Zhang, B. Fischer, S. Loranger, R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, W. J. Munro, J. Azaña, M. Kues, and R. Morandotti, “High-dimensional one-way quantum processing implemented on d-level cluster states,” Nat. Phys. 15(2), 148–153 (2019).
[Crossref]

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546(7660), 622–626 (2017).
[Crossref]

Barbieri, M.

A. Fedrizzi, T. Herbst, M. Aspelmeyer, M. Barbieri, T. Jennewein, and A. Zeilinger, “Anti-symmetrization reveals hidden entanglement,” New J. Phys. 11(10), 103052 (2009).
[Crossref]

Bellini, M.

A. Zavatta, S. Viciani, and M. Bellini, “Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state,” Phys. Rev. A 70(2), 023806 (2004).
[Crossref]

Bienfang, J. C.

Z. Xie, T. Zhong, S. Shrestha, X. Xu, J. Liang, Y.-X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. C. Wong, and C. Wei Wong, “Harnessing high-dimensional hyperentanglement through a biphoton frequency comb,” Nat. Photonics 9(8), 536–542 (2015).
[Crossref]

Bochmann, J.

H. P. Specht, J. Bochmann, M. Mucke, B. Weber, E. Figueroa, D. L. Moehring, and G. Rempe, “Phase shaping of single-photon wave packets,” Nat. Photonics 3(8), 469–472 (2009).
[Crossref]

Branczyk, A. M.

A. M. Brańczyk, “Hong–Ou–Mandel interference,” arXiv:1711.00080 (2017).

Bromberg, Y.

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351(6278), 1176–1180 (2016).
[Crossref]

Campbell, R.

Y. Lu, R. Campbell, and Z.-Y. Ou, “Mode-locked two-photon states,” Phys. Rev. Lett. 91(16), 163602 (2003).
[Crossref]

Caspani, L.

C. Reimer, S. Sciara, P. Roztocki, M. Islam, L. R. Cortés, Y. Zhang, B. Fischer, S. Loranger, R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, W. J. Munro, J. Azaña, M. Kues, and R. Morandotti, “High-dimensional one-way quantum processing implemented on d-level cluster states,” Nat. Phys. 15(2), 148–153 (2019).
[Crossref]

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546(7660), 622–626 (2017).
[Crossref]

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351(6278), 1176–1180 (2016).
[Crossref]

C. Reimer, L. Caspani, M. Clerici, M. Ferrera, M. Kues, M. Peccianti, A. Pasquazi, L. Razzari, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Integrated frequency comb source of heralded single photons,” Opt. Express 22(6), 6535–6546 (2014).
[Crossref]

Chang, K.-C.

K.-C. Chang, X. Cheng, M. C. Sarihan, D.-D. Mendinueto, Y. S. Lee, T. Zhong, Y.-X. Gong, Z. Xie, J. H. Shapiro, F. N. C. Wong, and C. W. Wong, “High-dimensional energy-time entanglement up to 6 qubits per photon through biphoton frequency comb,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2019), p. JTu3A.6.

Cheng, X.

K.-C. Chang, X. Cheng, M. C. Sarihan, D.-D. Mendinueto, Y. S. Lee, T. Zhong, Y.-X. Gong, Z. Xie, J. H. Shapiro, F. N. C. Wong, and C. W. Wong, “High-dimensional energy-time entanglement up to 6 qubits per photon through biphoton frequency comb,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2019), p. JTu3A.6.

Chiao, R.

A. Steinberg, P. G. Kwiat, and R. Chiao, “Dispersion cancellation in a measurement of the single-photon propagation velocity in glass,” Phys. Rev. Lett. 68(16), 2421–2424 (1992).
[Crossref]

Chiao, R. Y.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45(9), 6659–6665 (1992).
[Crossref]

Chu, S. T.

C. Reimer, S. Sciara, P. Roztocki, M. Islam, L. R. Cortés, Y. Zhang, B. Fischer, S. Loranger, R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, W. J. Munro, J. Azaña, M. Kues, and R. Morandotti, “High-dimensional one-way quantum processing implemented on d-level cluster states,” Nat. Phys. 15(2), 148–153 (2019).
[Crossref]

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546(7660), 622–626 (2017).
[Crossref]

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351(6278), 1176–1180 (2016).
[Crossref]

C. Reimer, L. Caspani, M. Clerici, M. Ferrera, M. Kues, M. Peccianti, A. Pasquazi, L. Razzari, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Integrated frequency comb source of heralded single photons,” Opt. Express 22(6), 6535–6546 (2014).
[Crossref]

Cino, A.

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Figures (7)

Fig. 1.
Fig. 1. General form of BFC frequency bins. The frequency offset ($\Omega$) is defined with respect to $\omega _0$ ($2\omega _0$ = pump laser frequency). Bins centered at $\Omega _{p}$ are carved from a broadband biphoton spectrum $\Phi (\Omega )$, and adjacent bins are separated from one another by $\Delta \omega$. Although the weight of each frequency bin is set by the profile of the broadband biphoton spectrum, all bins are carved from spectral filters with identical lineshapes [$f_{2}(\Omega )$ shown as an example]. Grey lines highlight pairwise entanglement across symmetric (energy-matched) bins.
Fig. 2.
Fig. 2. (a) Experimental arrangement (see text for details). PPLN, periodically-poled lithium niobate waveguide (HCPhotonics). Pulse shaper (Finisar). PBS, fiber-based polarizing beam splitter. PC, polarization controller. 50:50, fiber-based 50:50 beam splitter. SPD, single-photon detectors (Quantum Opus). TIA, time interval analyzer (PicoQuant). (b-c) Joint phase accumulated by a 6-bin BFC when phases of $\frac {\pi }{2}$ and $\pi$ are applied to frequency bins $-1$ and $2$, respectively. (b) For the arrangement with the pulse shaper placed before the PBS, any applied phase is common to both photons. (c) This is not the case when the pulse shaper is placed within the HOM interferometer as now one of the photons can accumulate phase distinct from the other.
Fig. 3.
Fig. 3. (a) Spectrum of the photon in path $B$, acquired by scanning an 18 GHz filter with the pulse shaper. (b) HOM interference trace of the unfiltered SPDC spectrum.
Fig. 4.
Fig. 4. HOM interference traces for $ |{\Psi (0)}\rangle$, $ |{\Psi (\frac {\pi }{2})}\rangle$, and $|{\Psi (\pi )}\rangle$, i.e., three states with different values of $\beta$ – the relative phase between frequency bin pairs $| {\psi _1(0)}\rangle$ and $| {\psi _2(0)}\rangle$. In this and subsequent figures, the axes of the photon spectra in path $A$ and path $B$ are reversed with respect to one another. This has been done to align frequency bins that contribute to the same two-photon basis state. For clarity, high and low frequencies are colored blue and red, respectively, and are defined with respect to the center of the photon spectrum. To facilitate an easy comparison between overlaid HOM interference traces, experimental data (solid circles) for consecutive delay steps are connected by straight lines.
Fig. 5.
Fig. 5. HOM interference traces for (a) a coherent superposition state – $ |{\Psi (0)}\rangle$ = $|{\psi _1(0)}\rangle$ + $|{\psi _2(0)}\rangle$, (b) comb line pair $|{\psi _1(0)}\rangle$ and (c) comb line pair $|{\psi _2(0)}\rangle$. (d) A comparison between HOM interference for a coherent superposition state (trace (a)) and a mixture of the constituent comb line pairs (trace (b) + trace (c)).
Fig. 6.
Fig. 6. HOM interference traces of $ |{\Psi (0)}\rangle = |{\psi _1(\alpha _1)}\rangle + |{\psi _2(0)}\rangle$ are shown for two different values of $\alpha _1$, the phase between two-photon basis states $| {-1,1}\rangle$ and $|{1,-1}\rangle$. The interferograms in (a) correspond to traces for $|{\Psi (0)}\rangle = |{\psi _1(0)}\rangle + |{\psi _2(0)}\rangle$ (green) and $|{\Psi (0)}\rangle = |{\psi _1(\frac {\pi }{2})}\rangle + |{\psi _2(0)}\rangle$ (purple). HOM interference traces for individual comb line pairs $|{\psi _1(\alpha _1)}\rangle$ and $|{\psi _2(0)}\rangle$ are shown in (b) and (c), respectively. Experimental data (solid circles) for consecutive delay steps in each trace connected by straight lines.
Fig. 7.
Fig. 7. HOM interference traces recorded for different biphoton states using the arrangement corresponding to Fig. 2(b), where the pulse shaper is placed before the HOM interferometer. (a) $ |{\Psi _a}\rangle = \sum _{p=1}^{2} |{\psi _p(0)}\rangle$, a coherent superposition of four frequency bins. (c) A coherent superposition of nine frequency bins, including a central bin at $p = 0$ (see inset for the photon spectrum). The states in (b) and (d) have joint spectral intensities identical to those in (a) and (c), respectively. However, in (b) and (d) the phase relationship between comb line pairs fluctuates within the integration time for each delay step.

Equations (10)

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Ω p = { ( p 1 2 ) Δ ω p = 1 , 2 , ( p + 1 2 ) Δ ω p = 1 , 2 , ,
| p , p = d Ω Φ ( Ω ) f p ( Ω ) | + Ω , Ω A B
| ψ p ( α p ) | p , p A B + e i α p | p , p A B ,
E ^ C ( + ) ( t ) = 1 2 d ω 1 [ e i ω 1 τ a ^ ( ω 1 ) + i b ^ ( ω 1 ) ] e i ω 1 t E ^ D ( + ) ( t ) = 1 2 d ω 2 [ i e i ω 2 τ a ^ ( ω 2 ) + b ^ ( ω 2 ) ] e i ω 2 t .
C p ( τ ; α p ) Δ t d t T R d T | v a c | E ^ C ( + ) ( t + T ) E ^ D ( + ) ( t ) | ψ p ( α p ) | 2 ,
C p ( τ ; α p ) = K p { 1 R e [ e i α p e i ( 2 p 1 ) Δ ω τ d Ω | f ( Ω ) | 2 e 2 i Ω τ ] }
| Ψ = p N c p | ψ p ( α p ) ,
C | Ψ ( τ ) = p N | c p | 2 C p ( τ ; α p ) ,
ρ ^ = p N | c p | 2 | ψ p ( α p ) ψ p ( α p ) |
| Ψ ( β ) e i β | ψ 1 ( 0 ) + | ψ 2 ( 0 ) ,

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