Abstract

We theoretically demonstrate an anisotropic quantum vacuum created by a judiciously designed hyperbolic metamaterial. An electric dipole located nearby shows strong orientation dependence in the decay rate. With a proper arrangement of the ellipsoid-shaped CdSe/ZnSe quantum dot relative to the Ag/TiO2 metamaterial, the anisotropies of quantum vacuum and quantum dot are harnessed to achieve an extraordinary quantum interference between radiative decay channels of orthogonal transitions. The ratio between cross damping term and spontaneous decay rate, Γijii, which never exceeds unity in previously reported works reaches 1.04 in our numerical results. The corresponding evolution of excited state population in quantum dot is also dramatically modified.

© 2016 Optical Society of America

1. Introduction

Quantum interference (QI) arising from the spontaneous emission (SE) of two nearly degenerate excited states to a common ground state leads to a number of remarkable phenomena such as coherent population trapping [1], ultranarrow spectral lines [2], and gain without inversion [3]. However, for the interference effects to occur the transition dipole moments of SE process should be nonorthogonal. This condition is rarely met in real atomic systems. In 2000, Agarwal proposed to break the anisotropic nature of the quantum vacuum (QV) to achieve QI for orthogonal transitions [4]. It paves the way for human controlling coherence and interference in light-matter interactions via QV engineering. Considerable efforts have been devoted to the realization of anisotropic vacuum-induced interference, including an atom in the vicinity of metallic surface [5, 6] or embedded in a photonic crystal [7, 8]. Unfortunately, given the atomic dipole moments assumed in these cases, the ratio between cross damping term and spontaneous decay rate is (Γ||)/(Γ + Γ||) which can approach indefinitely close to but never exceed one.

Hyperbolic metamaterials (HMMs), the iso-frequency curve of which is hyperbolic as opposed to circular as in conventional media, lie at the heart of many novel devices such as hyperlens [9], hypergratings [10], hyperbolic waveguides [11], and single-photon sources [12]. Due to the singularity in the bulk photonic density of states (PDOS) along certain directions, the SE in HMMs is dramatically modified, which provides us with a promising opportunity to attain strongly anisotropic SE rate via PDOS engineering.

In this article, we propose a system of a single anisotropic quantum dot (QD) in the proximity of a bulk HMM and theoretically demonstrate an extraordinary QI occurring in the light-matter interaction. The metamaterial is judiciously designed to tailor the QV and induce strongest anisotropy in the decay rate of an electric dipole. Meanwhile, the transition dipole moments of the QD are also exploited to maximize the ratio between cross damping term and spontaneous decay rate. As a result, the parameter is boosted in the system and reaches a high value beyond the limit of one in the numerical study. The decay of excited state population in QD is therefore considerably slowed down. Our work creates new possibilities of studying extraordinary QI for QED, solid-state quantum optics, quantum information processing, etc.

2. Basic device design

The quantum system of interest is shown in Fig. 1(a). A CdSe/ZnSe QD is suspended in air above a Ag/TiO2 multilayer stack. The naturally formed QD is usually ellipsoidal, as exaggeratedly depicted in Fig. 1(b). The single-exciton state of QD is split by anisotropy into two linearly crossed-polarized substates |y and |z [13]. As illustrated in Fig. 1(c), the substate whose polarization is along the major axis of the ellipsoid has lower energy than that along the minor axis. The energy difference between substates is about 150 μeV which is much smaller than that between single-exciton and vacuum states. Therefore the upper levels are nearly degenerate and have almost the same energy of 2.278 eV or equivalently 545 nm−1 [14]. Here we concern only with the single-exciton and vacuum states, so the QD is a perfect three-level quantum system which is commonly used in QI studies. Optical transitions |y|v and |z|v are allowed in the system while |y|z are forbidden by the selection rules [15]. The alternating layers of Ag (12.25 nm) and TiO2 (12.75 nm) constitute an HMM which breaks the symmetry of QV fluctuations and induces strong anisotropy in the decay rate of an electric dipole located nearby. More details on the design and analysis of the HMM will be given below.

 figure: Fig. 1

Fig. 1 Schematic of (a) metamaterial structure and (b) anisotropic QD. (c) Three lowest energy levels and allowed optical transitions in QD.

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As the photon wavelength of interest, i.e., 545 nm, is much larger than the unit cell of the metamaterial, the effective medium theory (EMT) is valid over a wide range of wave vectors, from 0 to 20k0 [16], where k0 = ω/c is the wave vector of light in vacuum. This covers both propagating and evanescent waves that make major contributions to the spontaneous decay of a quantum emitter [17]. Hence, the EMT is applied to analyze the HMM-induced enhancement of SE, namely the Purcell effect. The schematic illustration is shown in Fig. 2(a). The QD is modeled as an electric dipole due to its deep-subwavelength size [18]. The HMM is equivalent to an effective medium whose nonzero components of the dielectric tensor are given by [16, 19]

εxx=εyy=ε||=fεm+(1f)εd,εzz=ε=εmεdfεd+(1f)εm,
where εm (εd) is the dielectric constant of the metallic (dielectric) component, f = tm /(tm + td) is the fill fraction of metal in the unit cell and tm (td) is the thickness of metal (dielectric). With εAg = −12.63 + 0.42i, εTiO2 = 6.76 at 545nm [20] and tAg = 12.25 nm, tTiO2 = 12.75 nm, the HMM has a fill fraction of 49% and components of effective permittivity tensor ε||=2.74+0.21i, ε=27.23+0.95i. Note that the imaginary part of permittivity is negligible when compared with the real part. Thus, the iso-frequency surface for the p-polarized waves propagating in such a strongly anisotropic metamaterial can be described by [16, 17, 19]
kx2+ky2|ε|kz2|ε|||=k02,
which is schematically depicted in Fig. 2(b). Obviously, the proposed structure is a type II HMM, the iso-frequency surface of which is a single-sheeted hyperboloid. It can support bulk propagating waves which have wave vectors much larger than those allowed in vacuum. When the polarization of the dipole is perpendicular to the planar interface of the HMM, the SE will be greatly enhanced due to additional coupling of the emitted evanescent waves to the high-k metamaterial states [21]. On the other hand, the dispersion relation for s-polarized waves can be expressed as
kx2+ky2+kz2=|ε|||k02,
which prohibits the existence of propagating waves in the material. As a matter of fact, the medium exhibits metal-like behavior in this situation. So when the dipole is polarized parallel to the metamaterial interface, the SE will be suppressed at moderate distance d from the adjacent surface, as reported in [22] and [23]. The anisotropy of the SE rate is verified using finite-difference time domain (FDTD) techniques. The normalized decay rate, namely the Purcell factor, is obtained by utilizing the ratio between numerically calculated total emitted power from a dipole with and without the presence of the HMM [24, 25]. Figure 2(c) shows the normalized decay rate of an electric dipole placed above the HMM. As the distance d increases from 50 to 100 nm, the Purcell factor increases (decreases) monotonically for the y (z) dipole. The maximum anisotropy of the Purcell factor is reached at d = 50 nm with Γyy0 = 0.52 and Γzz0 = 3.21, where Γ0 is the decay rate in vacuum. It will be harnessed to achieve extraordinary QI in the following study.

 figure: Fig. 2

Fig. 2 (a) Electric dipole and effective medium model for the system. (b) Hyperbolic dispersion relation of p-polarized waves in HMM. (c) Anisotropic decay rate of a quantum emitter versus the distance d.

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Now we investigate the SE of ellipsoid-shaped QD stimulated by QV fluctuations. When using a linearly polarized pump, the single-exciton state |y (|z) can only be excited from the vacuum state |v by πyz)-polarized waves [26]. This is a manifestation of the fact that the transition dipole moments of |y|v and |z|v are orthogonal. The excitons on two upper levels have approximately equal radiative lifetimes [14], indicating the same strength of transition. Thus, the dipole moment of QD can be written as

μ=μ0(|yv|ey+|zv|ez+H.c.)
where μ0 is the magnitude, and ey (ez) is the unit vector corresponding to the transition |y|v (|z|v). As shown in Fig. 3, we place the anisotropic QD in the vicinity of the HMM. The y' axis which is the short axis of ellipsoid-shaped QD deviates by an angle of θ from the standard y axis. The equation of motion for the reduced density matrix elements (of |y'and|z'states in the QD) under rotating wave and Wigner-Weisskopf approximations can be written as [5, 25, 27, 28]
ρ˙y'y'=Γy'y'ρy'y'12(Γz'y'ρy'z'+Γy'z'ρz'y'),ρ˙z'z'=Γz'z'ρz'z'12(Γy'z'ρz'y'+Γz'y'ρy'z'),ρ˙y'z'=Γy'y'+Γz'z'2ρy'z'Γy'z'2ρz'z'Γy'z'2ρy'y',ρy'y'+ρz'z'+ρvv=1,
where energy splitting between the excited states is neglected for the sake of simplicity. The spontaneous decay rates Γii (i = y’, z’) and cross damping terms Γij (i, j = y’, z’, i ≠ j) are given by [4, 23, 29]
Γ=2ω02ε0c2μIm[G(r0,r0,ω)]μ,
where G(r0,r0,ω) is the dyadic Green function in the anisotropic QV, and μ is the dipole moment formulated in Eq. (4) with y and z replaced by y’ and z’. The orientation dependence of Γ can be expressed as
(Γy'y'Γy'z'Γz'y'Γz'z')=(cosθsinθsinθcosθ)(Γyy00Γzz)(cosθsinθsinθcosθ),
where Γyy and Γzz are the anisotropic SE rates at θ = 0 and can be derived from Fig. 2(c). The ratio between cross damping term and spontaneous decay rate of |y' state is defined as
Γy'z'Γy'y'=(ΓzzΓyy)sinθcosθΓyycos2θ+Γzzsin2θ,
which reaches its maximum of (ΓzzΓyy)/2ΓyyΓzzat tan2 θ = Γyyzz. Substituting in the anisotropic decay rates obtained at d = 50 nm, i.e., Γyy = 0.52Γ0 and Γzz = 3.21Γ0, the parameter maximizes at 1.04 when θ = 22°. It breaks the limit of one suggested in previously reported works [4–8]. As a matter of fact, the ratio can exceed unity whenever the anisotropy of the SE rate is strong enough, i. e.,Γzz/Γyy>3+22. For example, a ratio of 11.16 can be achieved with Γyy0 = 0.01 and Γzz0 = 5 in the monolayer plasmonic nanoshell system considered in Fig. 3 of [5]. So the extraordinary QI near HMM is only a proof of concept. More nanophotonic systems can be designed to make this phenomenon more evident. Here we choose HMM because it is easy to be fabricated and there has already been several experimental works on it [12, 21]. It seems to be more feasible to perform the experimental demonstration using HMM in the near future. The influence of this breakthrough on the time evolvement of the quantum system will be discussed in detail later.

 figure: Fig. 3

Fig. 3 Cross section of the structure in the YOZ plane. The angle between y' and y axes is denoted by θ.

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3. Demonstration of QI

To conduct the demonstration, the QD is initially prepared in three typical states, |y',|z'and(|y'|z')/2. The evolution of the system is calculated via Eq. (5) with the position-specific parameters that maximize Γy'z'y'y'. When the initial state |Ψ0=|y' (ρy'y' = 1, ρz'z' = 0, ρy'z' = 0), the evolution of the excited state populations at d = 50 nm and the transient coherence (real part of ρy'z') versus the distance d are plotted in Figs. 4(a) and 4(b), respectively. Nonzero coherence, along with nonzero population in the state|z', is a clear signature of anisotropic QV-induced QI between two orthogonal transitions. The coherence becomes larger as the QD gets closer to the HMM, resulting from stronger anisotropy of the Purcell factor at shorter distance. Similar phenomena can be observed in Figs. 4(c) and 4(d) with |Ψ0=|z' (ρy'y' = 0, ρz'z' = 1, ρy'z' = 0). However, the spontaneous decay rate Γz'z' is larger than Γy'y', which leads to greater degradation of the QI effect. So the coherence in Fig. 4(d) is weaker than that in Fig. 4(b). The influence of extraordinary Γy'z'y'y' on the system becomes especially prominent in the case |Ψ0=(|y'|z')/2(ρy'y' = 1/2, ρz'z' = 1/2, ρy'z' = −1/2). In Figs. 4(e) and 4(f), we have plotted the evolution of the excited state population ρy'y' at d = 50 nm with θ = 0° (minimum Γy'y'), 22° (maximum Γy'z'y'y'), and 45° (maximum Γy'z'). Obviously, the decay of the excited state |y' is much slower under the maximum Γy'z'y'y' condition, allowing sufficient time for observation. In order to provide an insightful view, we derive the analytical expression of ρy'y' with Eq. (5).

ρy'y'=0.73e0.52Γ0t0.25e1.86Γ0t+0.02e3.21Γ0t
for θ = 22° and
ρy'y'=0.5e0.52Γ0t
for θ = 0°, 45°. The second term on the right hand side of Eq. (9) increases with time, which slows down the decay of ρy'y'. Different from that in Eq. (10), the decay of ρy'y' in Eq. (9) is no longer quasi-exponential and has a non-negative slope at the initial instant t = 0. This is the unique characteristic of extraordinary QI whose ratio between cross damping term and spontaneous decay rate is no less than one. More interesting phenomena could be expected as the exploration of extraordinary QI in SE goes on in the future.

 figure: Fig. 4

Fig. 4 Evolution of (a) excited state populations and (b) transient coherence between the excited states in a QD placed adjacent to the HMM, initially prepared in|y'. The populations are derived at d = 50 nm while the coherence is plotted as a function of distance d. Counterparts of (a) and (b) for a QD initially prepared in|z' are depicted in (c) and (d), respectively. Decay of the excited state population ρy'y' with (e) θ = 0°, 22° and (f) θ = 22°, 45°. The QD is placed at a distance of 50 nm from the HMM and initially prepared in (|y'|z')/2.

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Last but not least, extraordinary QI is the consequence of both the anisotropy of QD and the anisotropy of QV. When the QD is spherically symmetric, the transition dipole moments are restricted by the selection rules [15] to the formμ=μ0(|21|e+|31|e++H.c.), where e±=1/2(ey±iez)are the unit vectors corresponding to the transitions|2|1and|3|1. Substituting μ into Eq. (6), we get the ratio between cross damping term and spontaneous decay rate which is (Γzz - Γyy) / (Γzz + Γyy). The expression has been presented in a number of literatures [4–8, 25, 27]. It can never exceed unity, regardless of the anisotropic QV. However, when the quantum emitter becomes ellipsoidal, transition dipole moments would take the form of Eq. (4). Substituting it into Eq. (6), the ratio is modified as Eq. (8) which can be greater than one with tan2 θ = Γyyzz andΓzz/Γyy>3+22. Note that Γy'z'y'y' will be zero if Γyy = Γzz, i. e., there will be no QI if QV is isotropic. So the anisotropy of QV is also indispensable, which is the result of the orthogonality between two decay channels. Therefore, extraordinary QI cannot occur in the absence of either anisotropy.

4. Conclusions

In conclusion, we have theoretically demonstrated an extraordinary QI between the decay of closely lying states in an anisotropic quantum emitter, implemented by engineering both the HMM-induced QV and the location and orientation of the ellipsoid-shaped QD. The proposed quantum system is studied using a density-matrix approach. Numerical results prove that the ratio between cross damping term and spontaneous decay rate breaks the limit of one and reaches an unprecedented high value of 1.04 after optimization. The slowdown of excited state population decay in QD has also been addressed and explained in this article. Our work opens a door for exploring extraordinary QI in light-matter interactions which could find applications in a variety of areas, including solid-state quantum optics, quantum information processing, and novel single-photon devices.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (Grant No. 61177056), and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (Grant No. 708038).

References and links

1. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999). [CrossRef]  

2. M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003). [CrossRef]   [PubMed]  

3. J. H. Wu, H. F. Zhang, and J. Y. Gao, “Probe gain with population inversion in a four-level atomic system with vacuum-induced coherence,” Opt. Lett. 28(8), 654–656 (2003). [CrossRef]   [PubMed]  

4. G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000). [CrossRef]   [PubMed]  

5. V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009). [CrossRef]   [PubMed]  

6. Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012). [CrossRef]   [PubMed]  

7. G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001). [CrossRef]  

8. J.-P. Xu, L.-G. Wang, Y.-P. Yang, Q. Lin, and S.-Y. Zhu, “Quantum interference between two orthogonal transitions of an atom in one-dimensional photonic crystals,” Opt. Lett. 33(17), 2005–2007 (2008). [CrossRef]   [PubMed]  

9. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]  

10. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009). [CrossRef]   [PubMed]  

11. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef]   [PubMed]  

12. M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013). [CrossRef]  

13. M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999). [CrossRef]  

14. I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006). [CrossRef]   [PubMed]  

15. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

16. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012). [CrossRef]  

17. Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012). [CrossRef]  

18. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover Publications, 1998).

19. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013). [CrossRef]  

20. M. J. Weber, Handbook of Optical Materials (CRC, 2002).

21. H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef]   [PubMed]  

22. R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995). [CrossRef]   [PubMed]  

23. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, New York, 2012).

24. L. Sun, B. Tang, and C. Jiang, “Enhanced spontaneous emission of mid-infrared dipole emitter in double-layer graphene waveguide,” Opt. Express 22(22), 26487–26497 (2014). [CrossRef]   [PubMed]  

25. P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015). [CrossRef]   [PubMed]  

26. V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999). [CrossRef]  

27. Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008). [CrossRef]   [PubMed]  

28. G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics Vol. 70 (Springer, 1974).

29. H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003). [CrossRef]  

References

  • View by:

  1. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
    [Crossref]
  2. M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
    [Crossref] [PubMed]
  3. J. H. Wu, H. F. Zhang, and J. Y. Gao, “Probe gain with population inversion in a four-level atomic system with vacuum-induced coherence,” Opt. Lett. 28(8), 654–656 (2003).
    [Crossref] [PubMed]
  4. G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
    [Crossref] [PubMed]
  5. V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
    [Crossref] [PubMed]
  6. Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
    [Crossref] [PubMed]
  7. G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
    [Crossref]
  8. J.-P. Xu, L.-G. Wang, Y.-P. Yang, Q. Lin, and S.-Y. Zhu, “Quantum interference between two orthogonal transitions of an atom in one-dimensional photonic crystals,” Opt. Lett. 33(17), 2005–2007 (2008).
    [Crossref] [PubMed]
  9. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
    [Crossref]
  10. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
    [Crossref] [PubMed]
  11. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
    [Crossref] [PubMed]
  12. M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
    [Crossref]
  13. M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
    [Crossref]
  14. I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
    [Crossref] [PubMed]
  15. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  16. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
    [Crossref]
  17. Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
    [Crossref]
  18. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover Publications, 1998).
  19. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
    [Crossref]
  20. M. J. Weber, Handbook of Optical Materials (CRC, 2002).
  21. H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
    [Crossref] [PubMed]
  22. R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
    [Crossref] [PubMed]
  23. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, New York, 2012).
  24. L. Sun, B. Tang, and C. Jiang, “Enhanced spontaneous emission of mid-infrared dipole emitter in double-layer graphene waveguide,” Opt. Express 22(22), 26487–26497 (2014).
    [Crossref] [PubMed]
  25. P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
    [Crossref] [PubMed]
  26. V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
    [Crossref]
  27. Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
    [Crossref] [PubMed]
  28. G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics Vol. 70 (Springer, 1974).
  29. H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
    [Crossref]

2015 (1)

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (2)

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

2012 (4)

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

2009 (2)

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[Crossref] [PubMed]

2008 (2)

2007 (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

2006 (2)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

2003 (3)

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

J. H. Wu, H. F. Zhang, and J. Y. Gao, “Probe gain with population inversion in a four-level atomic system with vacuum-induced coherence,” Opt. Lett. 28(8), 654–656 (2003).
[Crossref] [PubMed]

2001 (1)

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

2000 (1)

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
[Crossref] [PubMed]

1999 (3)

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

1995 (1)

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
[Crossref] [PubMed]

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

Akimov, I. A.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Andrews, J. T.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

Bacher, G.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Bayer, M.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Bian, R. X.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Borovitskaya, E.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Buhmann, S. Y.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Chen, H.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Dung, H. T.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Dunn, R. C.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

Forchel, A.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Gao, J. Y.

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Gong, Q.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Gorbunov, A.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Gu, Y.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Henneberger, F.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Hommel, D.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Irudayaraj, J.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Ishii, S.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Jacob, Z.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Jha, P. K.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Jiang, C.

Kästel, J.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Keitel, C. H.

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

Kildishev, A. V.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Knöll, L.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Kretzschmar, I.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Krishnamoorthy, H. N.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Kulakovskii, V. D.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Kümmell, T.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Kuther, A.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Lagutchev, A.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Leonardi, K.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Leung, P. T.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Li, F.-L.

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

Li, G.-X.

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

Lin, Q.

Liu, J.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Macovei, M.

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

Martin, O. J.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Menon, S.

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

Menon, V. M.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Molesky, S.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Narimanov, E.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Narimanov, E. E.

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Newman, W.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Ni, X.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Paspalakis, E.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Podolskiy, V. A.

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[Crossref] [PubMed]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Reinecke, T. L.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Reithmaier, J. P.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Ren, P.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

Schäfer, F.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Scheel, S.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Shalaev, V. M.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Shalaginov, M. Y.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Sivco, D. L.

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Smolyaninov, I. I.

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Sun, L.

Tang, B.

Thongrattanasiri, S.

Timofeev, V. B.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
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Vitanov, N. V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
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Walck, S. N.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Wang, L.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Wang, L.-G.

Wang, Y.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Weigand, R.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

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H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
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P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
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Xie, X. S.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
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Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
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Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
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V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

Zhang, H. F.

Zhang, J.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Zhang, T.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Zhang, X.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Zhu, S.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
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Appl. Phys. Lett. (2)

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
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Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
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Nano Lett. (1)

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

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A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
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Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
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Opt. Express (1)

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G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
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Phys. Rev. B (1)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
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M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
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G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
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V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
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I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
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R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Science (1)

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
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Figures (4)

Fig. 1
Fig. 1 Schematic of (a) metamaterial structure and (b) anisotropic QD. (c) Three lowest energy levels and allowed optical transitions in QD.
Fig. 2
Fig. 2 (a) Electric dipole and effective medium model for the system. (b) Hyperbolic dispersion relation of p-polarized waves in HMM. (c) Anisotropic decay rate of a quantum emitter versus the distance d.
Fig. 3
Fig. 3 Cross section of the structure in the YOZ plane. The angle between y' and y axes is denoted by θ.
Fig. 4
Fig. 4 Evolution of (a) excited state populations and (b) transient coherence between the excited states in a QD placed adjacent to the HMM, initially prepared in | y' . The populations are derived at d = 50 nm while the coherence is plotted as a function of distance d. Counterparts of (a) and (b) for a QD initially prepared in | z' are depicted in (c) and (d), respectively. Decay of the excited state population ρy'y' with (e) θ = 0°, 22° and (f) θ = 22°, 45°. The QD is placed at a distance of 50 nm from the HMM and initially prepared in (| y' | z' )/ 2 .

Equations (10)

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ε xx = ε yy = ε || =f ε m +(1f) ε d , ε zz = ε = ε m ε d f ε d +(1f) ε m ,
k x 2 + k y 2 | ε | k z 2 | ε || | = k 0 2 ,
k x 2 + k y 2 + k z 2 =| ε || | k 0 2 ,
μ= μ 0 (|yv| e y +|zv| e z +H.c.)
ρ ˙ y'y' = Γ y'y' ρ y'y' 1 2 ( Γ z'y' ρ y'z' + Γ y'z' ρ z'y' ), ρ ˙ z'z' = Γ z'z' ρ z'z' 1 2 ( Γ y'z' ρ z'y' + Γ z'y' ρ y'z' ), ρ ˙ y'z' = Γ y'y' + Γ z'z' 2 ρ y'z' Γ y'z' 2 ρ z'z' Γ y'z' 2 ρ y'y' , ρ y'y' + ρ z'z' + ρ vv =1,
Γ= 2 ω 0 2 ε 0 c 2 μIm[ G ( r 0 , r 0 ,ω)]μ,
( Γ y'y' Γ y'z' Γ z'y' Γ z'z' )=( cosθ sinθ sinθ cosθ )( Γ yy 0 0 Γ zz )( cosθ sinθ sinθ cosθ ),
Γ y'z' Γ y'y' = ( Γ zz Γ yy )sinθcosθ Γ yy cos 2 θ+ Γ zz sin 2 θ ,
ρ y'y' =0.73 e 0.52 Γ 0 t 0.25 e 1.86 Γ 0 t +0.02 e 3.21 Γ 0 t
ρ y'y' =0.5 e 0.52 Γ 0 t

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