Abstract

Ultra-thin metallic nanodisks, supporting localized plasmon (LP) modes, are used as a platform to facilitate high entanglement between distant quantum emitters (QEs). High Purcell factors, with values above 103, are probed for a QE placed near to an ultra-thin metallic nanodisk, composed of the noble metals Au, Ag, Al, and Cu. The disk supports two sets of localized plasmon modes, which can be excited by QEs with different transition dipole moment orientations. The two QEs are placed on opposite sides of the nanodisk, and their concurrence is used as a measure of the entanglement. We observe that the pair of QEs remains entangled for a duration that surpasses the relaxation time of the individual QE interacting with the metallic disk. Simultaneously, the QEs reach the entangled steady state faster than in the case where the QEs are in free space. Our results reveal a high concurrence value for a QES separation distance of 60 nm, and a transition energy of 0.8 eV (λ = 1550 nm). The robustness exhibited by this system under study paves the way for future quantum applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

Y. Luan, L. McDermott, F. Hu, and Z. Fei, “Tip- and Plasmon-Enhanced Infrared Nanoscopy for Ultrasensitive Molecular Characterizations,” Phys. Rev. Appl. 13(3), 034020 (2020).
[Crossref]

2019 (3)

R. A. Maniyara, D. Rodrigo, R. Yu, J. Canet-Ferrer, D. S. Ghosh, R. Yongsunthon, D. E. Baker, A. Rezikyan, F. J. García de Abajo, and V. Pruneri, “Tunable plasmons in ultrathin metal films,” Nat. Photonics 13(5), 328–333 (2019).
[Crossref]

Z. M. Abd El-Fattah, V. Mkhitaryan, J. Brede, L. Fernández, C. Li, Q. Guo, A. Ghosh, A. R. Echarri, D. Naveh, F. Xia, J. E. Ortega, and F. J. García de Abajo, “Plasmonics in atomically thin crystalline silver films,” ACS Nano 13(7), 7771–7779 (2019).
[Crossref]

V. Karanikolas, I. Thanopulos, and E. Paspalakis, “Strong interaction of quantum emitters with a WS2 layer enhanced by a gold substrate,” Opt. Lett. 44(8), 2049–2052 (2019).
[Crossref]

2018 (2)

M. Cuevas, “Enhancement, suppression of the emission and the energy transfer by using a graphene subwavelength wire,” J. Quant. Spectrosc. Radiat. Transfer 214, 8–17 (2018).
[Crossref]

F. Zhang, J. Ren, X. Duan, Z. Chen, Q. Gong, and Y. Gu, “Evanescent-field-modulated two-qubit entanglement in an emitters-plasmon coupled system,” J. Phys.: Condens. Matter 30(30), 305302 (2018).
[Crossref]

2017 (3)

F. Zhang, D. Zhao, Y. Gu, H. Chen, X. Hu, and Q. Gong, “Detuning-determined qubit-qubit entanglement mediated by plasmons: An effective model for dissipative systems,” J. Appl. Phys. 121(20), 203105 (2017).
[Crossref]

S.-A. Biehs and G. S. Agarwal, “Qubit entanglement across ɛ-near-zero media,” Phys. Rev. A 96(2), 022308 (2017).
[Crossref]

I. Thanopulos, V. Yannopapas, and E. Paspalakis, “Non-Markovian dynamics in plasmon-induced spontaneous emission interference,” Phys. Rev. B 95(7), 075412 (2017).
[Crossref]

2016 (3)

N. Iliopoulos, A. F. Terzis, V. Yannopapas, and E. Paspalakis, “Two-qubit correlations via a periodic plasmonic nanostructure,” Ann. Phys. 365, 38–53 (2016).
[Crossref]

V. D. Karanikolas, C. A. Marocico, and A. L. Bradley, “Tunable and long-range energy transfer efficiency through a graphene nanodisk,” Phys. Rev. B 93(3), 035426 (2016).
[Crossref]

A. Y. Nikitin, P. Alonso-González, S. Vélez, S. Mastel, A. Centeno, A. Pesquera, A. Zurutuza, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Real-space mapping of tailored sheet and edge plasmons in graphene nanoresonators,” Nat. Photonics 10(4), 239–243 (2016).
[Crossref]

2015 (8)

F. J. García de Abajo and A. Manjavacas, “Plasmonics in atomically thin materials,” Faraday Discuss. 178, 87–107 (2015).
[Crossref]

K. J. Tielrooij, L. Orona, A. Ferrier, M. Badioli, G. Navickaite, S. Coop, S. Nanot, B. Kalinic, T. Cesca, L. Gaudreau, Q. Ma, A. Centeno, A. Pesquera, A. Zurutuza, H. de Riedmatten, P. Goldner, F. J. García de Abajo, P. Jarillo-Herrero, and F. H. L. Koppens, “Electrical control of optical emitter relaxation pathways enabled by graphene,” Nat. Phys. 11(3), 281–287 (2015).
[Crossref]

Y. N. Gartstein, X. Li, and C. Zhang, “Exciton polaritons in transition-metal dichalcogenides and their direct excitation via energy transfer,” Phys. Rev. B 92(7), 075445 (2015).
[Crossref]

A. González-Tudela, C.-L Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9(5), 320–325 (2015).
[Crossref]

S. A. H. Gangaraj, A. Nemilentsau, G. W. Hanson, and S. Hughes, “Transient and steady-state entanglement mediated by three-dimensional plasmonic waveguides,” Opt. Express 23(17), 22330–22346 (2015).
[Crossref]

H. R. Haakh and D. Martín-Cano, “Squeezed Light from Entangled Nonidentical Emitters via Nanophotonic Environments,” ACS Photonics 2(12), 1686–1691 (2015).
[Crossref]

K. V. Nerkararyan and S. I. Bozhevolnyi, “Entanglement of two qubits mediated by a localized surface plasmon,” Phys. Rev. B 92(4), 045410 (2015).
[Crossref]

J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotechnol. 10(1), 2–6 (2015).
[Crossref]

2014 (3)

I. Silveiro and F. Javier García de Abajo, “Plasmons in inhomogeneously doped neutral and charged graphene nanodisks,” Appl. Phys. Lett. 104(13), 131103 (2014).
[Crossref]

A. Manjavacas and F. J. García de Abajo, “Tunable plasmons in atomically thin gold nanodisks,” Nat. Commun. 5(1), 3548 (2014).
[Crossref]

F.-P. Schmidt, H. Ditlbacher, U. Hohenester, A. Hohenau, F. Hofer, and J. R. Krenn, “Universal dispersion of surface plasmons in flat nanostructures,” Nat. Commun. 5(1), 3604 (2014).
[Crossref]

2013 (1)

L. Gaudreau, K. J. Tielrooij, G. E. D. K. Prawiroatmodjo, J. Osmond, F. J. G. de Abajo, and F. H. L. Koppens, “Universal distance-scaling of nonradiative energy transfer to graphene,” Nano Lett. 13(5), 2030–2035 (2013).
[Crossref]

2012 (2)

F.-P. Schmidt, H. Ditlbacher, U. Hohenester, A. Hohenau, F. Hofer, and J. R. Krenn, “Dark Plasmonic Breathing Modes in Silver Nanodisks,” Nano Lett. 12(11), 5780–5783 (2012).
[Crossref]

U. Hohenester and A. Trügler, “MNPBEM – A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183(2), 370–381 (2012).
[Crossref]

2011 (3)

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

A. Gonzalez-Tudela, D. Martín-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of Two Qubits Mediated by One-Dimensional Plasmonic Waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
[Crossref]

D. Martín-Cano, A. González-Tudela, L. Martín-Moreno, F. J. García-Vidal, C. Tejedor, and E. Moreno, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B 84(23), 235306 (2011).
[Crossref]

2010 (2)

S. Savasta, R. Saija, R. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: Vacuum Rabi Splitting with a Single Quantum Dot in the Center of a Dimer Nanoantenna,” ACS Nano 4(11), 6369–6376 (2010).
[Crossref]

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82(7), 075427 (2010).
[Crossref]

2009 (1)

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

2008 (1)

S. Scheel and S. Y. Buhmann, “Macroscopic Quantum Electrodynamics - Concepts and Applications,” Acta Phys. Slovaca 58(5), 675–809 (2008).
[Crossref]

2007 (2)

I. V. Bondarev and B. Vlahovic, “Entanglement of a pair of atomic qubits near a carbon nanotube,” Phys. Rev. B 75(3), 033402 (2007).
[Crossref]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76(3), 035420 (2007).
[Crossref]

2006 (2)

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum Optics with Surface Plasmons,” Phys. Rev. Lett. 97(5), 053002 (2006).
[Crossref]

W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8(4), S87–S93 (2006).
[Crossref]

2004 (1)

P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science 306(5698), 1002–1005 (2004).
[Crossref]

2002 (3)

I. V. Bondarev, G. Y. Slepyan, and S. A. Maksimenko, “Spontaneous decay of excited atomic states near a carbon nanotube,” Phys. Rev. Lett. 89(11), 115504 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66(6), 063810 (2002).
[Crossref]

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65(11), 115418 (2002).
[Crossref]

2001 (1)

W. K. Wootters, “Entanglement of formation and concurrence,” Quantum Information and Computation 1(1), 27–44 (2001).

1999 (1)

S. Scheel, L. Knöll, and D.-G. Welsch, “Spontaneous decay of an excited atom in an absorbing dielectric,” Phys. Rev. A 60(5), 4094–4104 (1999).
[Crossref]

1998 (2)

H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57(5), 3931–3942 (1998).
[Crossref]

W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998).
[Crossref]

1986 (1)

A. L. Fetter, “Magnetoplasmons in a two-dimensional electron fluid: Disk geometry,” Phys. Rev. B 33(8), 5221–5227 (1986).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

1969 (1)

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

Abd El-Fattah, Z. M.

Z. M. Abd El-Fattah, V. Mkhitaryan, J. Brede, L. Fernández, C. Li, Q. Guo, A. Ghosh, A. R. Echarri, D. Naveh, F. Xia, J. E. Ortega, and F. J. García de Abajo, “Plasmonics in atomically thin crystalline silver films,” ACS Nano 13(7), 7771–7779 (2019).
[Crossref]

Agarwal, G. S.

S.-A. Biehs and G. S. Agarwal, “Qubit entanglement across ɛ-near-zero media,” Phys. Rev. A 96(2), 022308 (2017).
[Crossref]

Alonso-González, P.

A. Y. Nikitin, P. Alonso-González, S. Vélez, S. Mastel, A. Centeno, A. Pesquera, A. Zurutuza, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Real-space mapping of tailored sheet and edge plasmons in graphene nanoresonators,” Nat. Photonics 10(4), 239–243 (2016).
[Crossref]

Andrew, P.

P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science 306(5698), 1002–1005 (2004).
[Crossref]

Badioli, M.

K. J. Tielrooij, L. Orona, A. Ferrier, M. Badioli, G. Navickaite, S. Coop, S. Nanot, B. Kalinic, T. Cesca, L. Gaudreau, Q. Ma, A. Centeno, A. Pesquera, A. Zurutuza, H. de Riedmatten, P. Goldner, F. J. García de Abajo, P. Jarillo-Herrero, and F. H. L. Koppens, “Electrical control of optical emitter relaxation pathways enabled by graphene,” Nat. Phys. 11(3), 281–287 (2015).
[Crossref]

Baker, D. E.

R. A. Maniyara, D. Rodrigo, R. Yu, J. Canet-Ferrer, D. S. Ghosh, R. Yongsunthon, D. E. Baker, A. Rezikyan, F. J. García de Abajo, and V. Pruneri, “Tunable plasmons in ultrathin metal films,” Nat. Photonics 13(5), 328–333 (2019).
[Crossref]

Barnes, W. L.

W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8(4), S87–S93 (2006).
[Crossref]

P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science 306(5698), 1002–1005 (2004).
[Crossref]

Biehs, S.-A.

S.-A. Biehs and G. S. Agarwal, “Qubit entanglement across ɛ-near-zero media,” Phys. Rev. A 96(2), 022308 (2017).
[Crossref]

Bondarev, I. V.

I. V. Bondarev and B. Vlahovic, “Entanglement of a pair of atomic qubits near a carbon nanotube,” Phys. Rev. B 75(3), 033402 (2007).
[Crossref]

I. V. Bondarev, G. Y. Slepyan, and S. A. Maksimenko, “Spontaneous decay of excited atomic states near a carbon nanotube,” Phys. Rev. Lett. 89(11), 115504 (2002).
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ACS Nano (2)

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Figures (8)

Fig. 1.
Fig. 1. Structure under investigation. A pair of quantum emitters is placed at the center and opposite sides of a metallic nanodisk at distance $z_{1}=-z_{2}$. The nanodisk of $R$ radius composed of the Au, Ag, Al, and Cu noble metals is placed on the $x-y$ plane.
Fig. 2.
Fig. 2. Quality factor of an infinite ultra-thin metallic layer, composed of Au, Ag, Al, and Cu materials, and defined as $L_{\textrm {SP}}/\delta _{\textrm {SP}}$. The different material parameters are considered from experimentally tabulated data. The thickness of the metallic layer is $0.75\,$nm.
Fig. 3.
Fig. 3. Surface charge $\rho _{n}^{l}(\mathbf {r})$ distribution on $x-y$ plane of an ultra-thin metallic nanodisk, for different angular and radial eigenmodes. a) $l=1$ and $n=0$ dipole mode, excited by a $x-$oriented QE. b) $l=0$ and $n=1$ breathing mode, excited by a $z-$oriented QE. The position of the QE is at the center of the nanodisk. The same charge distribution is observed for all noble metal materials considered.
Fig. 4.
Fig. 4. Purcell factor of a quantum emitter, placed $30\,$nm above a metallic disk of radius of $30\,$nm, varying its emission energy. The transition dipole moment of the quantum emitter is along (a) $x$ and (b) $z$. Au, Ag, Al, and Cu materials are considered from experimentally tabulated data. The thickness of the metallic disks is $0.75\,$nm.
Fig. 5.
Fig. 5. Relaxation rate $\Gamma (\mathbf {r},\omega )$ of the individual QE and the coupling rates $\Gamma _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$, and $\Omega _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$ between a pair of quantum emitters, varying their energy $\hbar \omega$, interacting with a metallic disk. The position of the QEs is $\mathbf {r}_{1}=(0,0,z_{1})$ and $\mathbf {r}_{2}=(0,0,-z_{1})$, with $z_{1}=30\,$nm. The metallic disk has a radius of $30\,$nm, and the materials are considered Au and Ag, described from tabulated data. The transition dipole moment of the quantum emitter is along (a) $x$ and (b) $z$. The thickness of the metallic disks is $0.75\,$nm.
Fig. 6.
Fig. 6. Contour plot of coupling rates (a) $\Omega _{Cx}$ and (b) $\Gamma _{Cx}$ between a pair of QEs, varying the energy $\hbar \omega$ and the radius $R$ of a metallic disk that is placed in between them. The position of the QEs is $\mathbf {r}_{1}=(0,0,z_{1})$ and $\mathbf {r}_{2}=(0,0,-z_{1})$, $z_{1}=30\,$nm. We consider an Au nanodisk. The transition dipole moment of the quantum emitter is along $x$. The thickness of the metallic disks is $0.75\,$nm.
Fig. 7.
Fig. 7. Contour plot of concurrence $C(t)$ between a pair of QEs, where different transition energies, $\hbar \omega$ are considered and the time evolution is recorded in $\Gamma (\mathbf {r}_{1},\omega )$ units, when they interact in the presence of a metallic disk. The position of the QEs is $\mathbf {r}_{1}=(0,0,z_{1})$ and $\mathbf {r}_{2}=(0,0,-z_{1})$, $z_{1}=30\,$nm, and their transition dipole moment is along $x$. The metallic disk has a radius of $40\,$nm, and the materials considered are (a) Au CRC and (b) Ag JC, where experimentally measured data are considered. The thickness of the metallic disks is $0.75\,$nm.
Fig. 8.
Fig. 8. We consider a pair of QEs placed at $\mathbf {r}_{1}=(0,0,z_{1})$ and $\mathbf {r}_{2}=(0,0,-z_{1})$, $z_{1}=30\,$nm, and their transition dipole moment is along $x$. (a) Contour plot of the concurrence $C(t)$ between a pair of QEs, considering a transition energy of $\hbar \omega =0.8\,$eV, when they interact in the presence of a Au JC disk, where a different disk radius, $R$, is considered, and the time evolution is recorded in $\Gamma (\mathbf {r}_{1},\omega )$ units. (b) Concurrence of a pair of QEs in the presence of a pair of QEs in the presence of a Au JC nanodisk and in vacuum. The thickness of the Au disk is $0.75\,$nm.

Equations (22)

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ρ t = i [ H , ρ ] + j , i = 1 , 2 Γ i j 2 ( S i + S j ρ 2 S j ρ S i + + ρ S i + S j ) ,
H = ω 0 i , j = A , B S i + S i + Ω i j S i + S j .
Γ i j / Γ 0 = 6 π c ω 0 Im ( μ ^ i G ( r i , r j , ω 0 ) μ ^ j ) ,
Ω i j / Γ 0 = 6 π c ω 0 Re ( μ ^ i G ( r i , r j , ω 0 ) μ ^ j ) ,
ρ ˙ e e = 4 Γ ( r , ω ) ρ e e ,
ρ ˙ e g = ρ e g ( Γ + i ω 0 ) ,
ρ ˙ s s = 2 ( ρ s s ρ e e ) ( Γ + Γ C ) ,
ρ ˙ a a = 2 ( ρ a a ρ e e ) ( Γ Γ C ) ,
ρ ˙ a s = 2 ρ a s ( Γ Ω C ) ,
ρ e e ( t ) = ρ e e ( 0 ) e 4 Γ t ,
ρ e g ( t ) = ρ e g ( 0 ) e 2 ( Γ + ω 0 ) t ,
ρ a s ( t ) = ρ a s ( 0 ) e 2 ( Γ Ω C ) t ,
ρ a a ( t ) = ρ a a ( 0 ) e 2 ( Γ Γ C ) t Γ Γ C Γ + Γ C ρ e e ( 0 ) ( e 4 Γ t e 2 ( Γ Ω C ) t ) ,
ρ s s ( t ) = ρ s s ( 0 ) e 2 ( Γ + Γ C ) t Γ + Γ C Γ Γ C ρ e e ( 0 ) ( e 4 Γ t e 2 ( Γ + Ω C ) t ) .
C ( t ) = e Γ t sinh 2 ( 2 Γ C t ) + sin ( 2 Ω C t ) ,
G x x 0 ( r 1 , r 2 , ω ) = c 2 2 π ω 2 1 ( z 1 z 2 ) 3 ,
G z z 0 ( r 1 , r 2 , ω ) = c 2 4 π ω 2 1 ( z 1 z 2 ) 3 .
G x x ind ( r 1 , r 2 , ω ) = c 2 2 ω 2 n = 0 c n 1 ( z 1 , ω ) ( ( z 2 / R ) 2 + 1 z 2 / R ) 2 n + 2 ( z 2 / R ) 2 + 1 ,
G z z ind ( r 1 , r 2 , ω ) = ± c 2 2 ω 2 n = 1 c n 0 ( z 1 , ω ) ( ( z 2 / R ) 2 + 1 z 2 / R ) 2 n + 1 ( z 2 / R ) 2 + 1 ,
F = L SP δ SP ,
σ ( ω n l ) ω n l = 2 i ε 0 R ζ n l ,
ω n l ω p i 2 d ζ n l 8 π ε 0 R ,