Abstract

Quantum frequency combs from chip-scale integrated sources are promising candidates for scalable and robust quantum information processing (QIP). However, to use these quantum combs for frequency domain QIP, demonstration of entanglement in the frequency basis, showing that the entangled photons are in a coherent superposition of multiple frequency bins, is required. We present a verification of qubit and qutrit frequency-bin entanglement using an on-chip quantum frequency comb with 40 mode pairs, through a two-photon interference measurement that is based on electro-optic phase modulation. Our demonstrations provide an important contribution in establishing integrated optical microresonators as a source for high-dimensional frequency-bin encoded quantum computing, as well as dense quantum key distribution.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (5)

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: Concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017).
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[Crossref] [PubMed]

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, 622–626 (2017).
[Crossref] [PubMed]

J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scalable quantum information processing,” Optica 4, 8–16 (2017).
[Crossref]

J. A. Jaramillo-Villegas, P. Imany, O. D. Odele, D. E. Leaird, Z.-Y. Ou, M. Qi, and A. M. Weiner, “Persistent energy–time entanglement covering multiple resonances of an on-chip biphoton frequency comb,” Optica 4, 655–658 (2017).
[Crossref]

2016 (3)

2015 (2)

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

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New J. Phys. 17, 022002 (2015).
[Crossref]

2014 (2)

C. Bernhard, B. Bessire, A. Montina, M. Pfaffhauser, A. Stefanov, and S. Wolf, “Non-locality of experimental qutrit pairs,” J. Phys. A: Math. Theor. 47, 424013 (2014).
[Crossref]

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser & Photonics Rev. 8, L76–L80 (2014).
[Crossref]

2013 (2)

J. Mower, Z. Zhang, P. Desjardins, C. Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
[Crossref]

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

2010 (3)

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, 013804 (2010).
[Crossref]

L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
[Crossref]

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4, 117–122 (2010).
[Crossref]

2009 (3)

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

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

H. Takesue and Y. Noguchi, “Implementation of quantum state tomography for time-bin entangled photon pairs,” Opt. Express 17, 10976–10989 (2009).
[Crossref] [PubMed]

2008 (2)

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[Crossref] [PubMed]

2007 (2)

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[Crossref]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

2006 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

A. Agarwal, P. Toliver, R. Menendez, S. Etemad, J. Jackel, J. Young, T. Banwell, B. Little, S. Chu, W. Chen, W. Chen, J. Hryniewicz, F. Johnson, D. Gill, O. King, R. Davidson, K. Donovan, and P. J. Delfyett, “Fully programmable ring-resonator-based integrated photonic circuit for phase coherent applications,” J. Light. Technol. 24, 77–87 (2006).
[Crossref]

2005 (1)

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169 (2005).
[Crossref] [PubMed]

2004 (1)

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

2003 (2)

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

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

2002 (2)

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref] [PubMed]

D. Kaszlikowski, L. C. Kwek, J.-L. Chen, M. Żukowski, and C. H. Oh, “Clauser-horne inequality for three-state systems,” Phys. Rev. A 65, 032118 (2002).
[Crossref]

2001 (2)

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

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

2000 (2)

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441 (2000).
[Crossref] [PubMed]

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

1998 (1)

A. Steane, “Quantum computing,” Reports on Prog. Phys. 61, 117 (1998).
[Crossref]

1997 (1)

M. Żukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A 55, 2564–2579 (1997).
[Crossref]

1996 (2)

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413 (1996).
[Crossref] [PubMed]

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
[Crossref]

1989 (1)

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205 (1989).
[Crossref] [PubMed]

Acin, A.

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

Agarwal, A.

A. Agarwal, P. Toliver, R. Menendez, S. Etemad, J. Jackel, J. Young, T. Banwell, B. Little, S. Chu, W. Chen, W. Chen, J. Hryniewicz, F. Johnson, D. Gill, O. King, R. Davidson, K. Donovan, and P. J. Delfyett, “Fully programmable ring-resonator-based integrated photonic circuit for phase coherent applications,” J. Light. Technol. 24, 77–87 (2006).
[Crossref]

Aktas, D.

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Almeida, M. P.

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

Aspelmeyer, M.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169 (2005).
[Crossref] [PubMed]

Azaña, J.

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, 622–626 (2017).
[Crossref] [PubMed]

Babazadeh, A.

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: Concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017).
[Crossref] [PubMed]

Banwell, T.

A. Agarwal, P. Toliver, R. Menendez, S. Etemad, J. Jackel, J. Young, T. Banwell, B. Little, S. Chu, W. Chen, W. Chen, J. Hryniewicz, F. Johnson, D. Gill, O. King, R. Davidson, K. Donovan, and P. J. Delfyett, “Fully programmable ring-resonator-based integrated photonic circuit for phase coherent applications,” J. Light. Technol. 24, 77–87 (2006).
[Crossref]

Barbieri, M.

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

Barreiro, J. T.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[Crossref]

Belabas-Plougonven, N.

Bentivegna, M.

Bernhard, C.

C. Bernhard, B. Bessire, A. Montina, M. Pfaffhauser, A. Stefanov, and S. Wolf, “Non-locality of experimental qutrit pairs,” J. Phys. A: Math. Theor. 47, 424013 (2014).
[Crossref]

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

Bessire, B.

C. Bernhard, B. Bessire, A. Montina, M. Pfaffhauser, A. Stefanov, and S. Wolf, “Non-locality of experimental qutrit pairs,” J. Phys. A: Math. Theor. 47, 424013 (2014).
[Crossref]

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

Bienfang, J. C.

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Figures (6)

Fig. 1
Fig. 1 (a) Microscope picture of the microring and U-grooves to support fiber coupling. (b) Joint spectral intensity for comb line pairs from 3 to 40. The background accidentals are not subtracted in this measurement and the coincidence to accidental ratio is about 10:1. (c) Illustration of biphoton spectrum after phase modulation. (d) Experimental setup.
Fig. 2
Fig. 2 Analogy between a 1-bit delay interferometer for forming projections of time-bin qubits and a frequency splitter for forming projections of a frequency-bin qubit. The green frequency bins after the frequency splitter are phase modulation sidebands from |1〉 and |2〉.
Fig. 3
Fig. 3 (a) Coincidence dip as a function of sideband frequency to maximize the indistinguishability. (b) Coincidences of the S6I6 and S7I7 superposition versus phase applied on S7I7. (c) Coincidences of the S5I5 and S6I6 superposition versus phase applied on S6I6. The coincidences reported are in (a) 20 minutes. and (b), (c) 10 minutes. and after background subtraction. Each data point was measured three times to obtain the standard deviation indicated by the error bars.
Fig. 4
Fig. 4 Phase modulation scheme for quantum state tomography. Red peaks represent the input signal and idler, each of which is in one of two frequency bins. Blue curves represent projections of signal and idler after the phase modulator (frequency splitter) into three new frequency positions. Solid blue is a projection of the superposition state; dashed blue peaks represent a projection from a single signal or idler frequency bin.
Fig. 5
Fig. 5 (a) Real and (b) imaginary parts of the estimated density matrix for comb line pairs S6I6 and S7I7.
Fig. 6
Fig. 6 Illustration of overlapped phase modulation sidebands for comb line pairs S5I5, S6I6 and S7I7.

Tables (2)

Tables Icon

Table 1 Projection measurements for frequency-bin density matrix estimation. For each measurement coincidences were acquired over a 10-minute period. A dash (-) indicates that the phase setting indicated by the respective column is not involved in the projection measurement indicated by the respective row; hence coincidence counts were not obtained.

Tables Icon

Table 2 Parameters for evaluations of the CGLMP inequality. The coincidences were measured in 10-minute spans; measurements were done three times to obtain standard deviations. To achieve the maximum and minimum number of coincidences, the phases of ϕ S x ( a ) = ϕ I y ( b ) = 0 and ϕ S x ( a ) = ϕ I y ( b ) = π / 3 were put on the biphotons, respectively. To calculate each of the probabilities that appear in Eq. (8), the corresponding coincidence counts have to be divided by the maximum number of coincidences Pmax(0, 0).

Equations (18)

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| Ψ = k = 1 N α k | k , k SI
| k , k SI = d Ω Φ ( Ω k Δ ω ) | ω P + Ω , ω P Ω SI
| S = 1 2 ( | 1 + e i ϕ | 2 )
| Ψ 8 = 1 2 ( | 1 S + e i ϕ S | 2 S ) 1 2 ( | 1 I + e i ϕ I | 2 I ) = 1 2 | 1 , 1 SI + 1 2 | 1 , 2 SI + i 2 | 2 , 1 SI + i 2 | 2 , 2 SI = ( 1 2 , 1 2 , i 2 , i 2 )
= ν = 1 16 ( C Ψ ν | ρ ^ | Ψ ν n ν ) 2 2 C Ψ ν | ρ ^ | Ψ ν
C = ν = 1 4 n ν
ρ ^ = [ 0.4388 + 0.0000 i 0.0115 0.0699 i 0.0721 0.0193 i 0.3745 + 0.0166 i 0.0115 + 0.0699 i 0.0574 + 0.0000 i 0.0279 0.0244 i 0.0084 0.0227 i 0.0721 + 0.0193 i 0.0279 + 0.244 i 0.0281 + 0.0000 i 0.0280 0.0211 i 0.3745 0.0166 i 0.0084 + 0.227 i 0.0280 + 0.0211 i 0.4757 + 0.000 i ]
I 3 = 3 [ P 11 ( 0 , 0 ) + P 21 ( 0 , 1 ) + P 22 ( 0 , 0 ) + P 12 ( 0 , 0 ) ] 3 [ P 11 ( 0 , 1 ) + P 21 ( 0 , 0 ) + P 22 ( 0 , 1 ) + P 12 ( 1 , 0 ) ] 2
Φ S k x ( a ) = ( k 5 ) ϕ S x ( a ) = 2 π 3 ( k 5 ) ( a + α x )
Φ I k y ( b ) = ( k 5 ) ϕ I y ( b ) = 2 π 3 ( k 5 ) ( b + β y )
ρ ^ = λ | ψ ψ | + ( 1 λ ) ρ ^ N
| ψ = 1 3 [ | 5 , 5 SI + | 6 , 6 SI + | 7 , 7 SI ]
ρ ^ N = 1 9 [ | 5 , 5 5 , 5 | SI + | 5 , 6 5 , 6 | SI + | 5 , 7 5 , 7 | SI + | 6 , 5 6 , 5 | SI + | 6 , 6 6 , 6 | SI + | 6 , 7 6 , 7 | SI + | 7 , 5 7 , 5 | SI + | 7 , 6 7 , 6 | SI + | 7 , 7 7 , 7 | SI ]
Π ^ S x ( a ) = 1 3 [ | 5 S + e i ϕ S x ( a ) | 6 S + e i 2 ϕ S x ( a ) 7 | S ] [ 5 | S + e i ϕ S x ( a ) 6 | S + e i 2 ϕ S x ( a ) 7 | S ]
Π ^ I y ( a ) = 1 3 [ | 5 I + e i ϕ I y ( b ) | 6 I + e i 2 ϕ I y ( b ) 7 | I ] [ 5 | I + e i ϕ I y ( b ) 6 | I + e i 2 ϕ I y ( b ) 7 | I ]
P x y ( a , b ) = Tr { ρ ^ Π ^ S x ( a ) Π ^ I y ( b ) } = λ ψ | Π ^ S x ( a ) Π ^ I y ( b ) | ψ + 1 λ 9 m = 5 7 n = 5 7 m n | Π ^ S x ( a ) Π ^ I y ( b ) | m n SI
m n | Π ^ S x ( a ) Π ^ I y ( b ) | m n SI = 1 9
ψ | Π ^ S x ( a ) Π ^ I y ( b ) | ψ = 1 27 | 1 + e i [ ϕ S x ( a ) + ϕ I y ( b ) ] + e i 2 [ ϕ S x ( a ) + ϕ I y ( b ) ] | 2

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