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, Γij/Γii, 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
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 , ultranarrow spectral lines , and gain without inversion . 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 . 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⊥ + 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 , hypergratings , hyperbolic waveguides , and single-photon sources . 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 and . 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 . 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 and are allowed in the system while are forbidden by the selection rules . 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.
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 , 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 . 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 . The HMM is equivalent to an effective medium whose nonzero components of the dielectric tensor are given by [16, 19]20] and tAg = 12.25 nm, tTiO2 = 12.75 nm, the HMM has a fill fraction of 49% and components of effective permittivity tensor , . 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]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 . On the other hand, the dispersion relation for s-polarized waves can be expressed as22] and . 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 Γyy/Γ0 = 0.52 and Γzz/Γ0 = 3.21, where Γ0 is the decay rate in vacuum. It will be harnessed to achieve extraordinary QI in the following study.
Now we investigate the SE of ellipsoid-shaped QD stimulated by QV fluctuations. When using a linearly polarized pump, the single-exciton state () can only be excited from the vacuum state by πy (πz)-polarized waves . This is a manifestation of the fact that the transition dipole moments of and are orthogonal. The excitons on two upper levels have approximately equal radiative lifetimes , indicating the same strength of transition. Thus, the dipole moment of QD can be written asFig. 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 andstates in the QD) under rotating wave and Wigner-Weisskopf approximations can be written as [5, 25, 27, 28]4, 23, 29]Eq. (4) with y and z replaced by y’ and z’. The orientation dependence of Γ can be expressed asFig. 2(c). The ratio between cross damping term and spontaneous decay rate of state is defined as4–8]. As a matter of fact, the ratio can exceed unity whenever the anisotropy of the SE rate is strong enough, i. e.,. For example, a ratio of 11.16 can be achieved with Γyy/Γ0 = 0.01 and Γzz/Γ0 = 5 in the monolayer plasmonic nanoshell system considered in Fig. 3 of . 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.
3. Demonstration of QI
To conduct the demonstration, the QD is initially prepared in three typical states, ,and. The evolution of the system is calculated via Eq. (5) with the position-specific parameters that maximize Γy'z'/Γy'y'. When the initial state (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, 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 (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'/Γy'y' on the system becomes especially prominent in the case (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'/Γy'y'), and 45° (maximum Γy'z'). Obviously, the decay of the excited state is much slower under the maximum Γy'z'/Γy'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).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.
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  to the form, where are the unit vectors corresponding to the transitionsand. 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 = Γyy/Γzz and. Note that Γy'z'/Γy'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.
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.
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).
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