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Tapered optical fibers are promising one-dimensional nanophotonic waveguides that can provide efficient coupling between their fundamental mode and quantum nanoemitters placed inside them. Here, we present numerical studies on the coupling of single nitrogen-vacancy (NV) centers (single point dipoles) in nanodiamonds with tapered fibers. Our results lead to two important conclusions: (1) A maximum coupling efficiency of 53.4% can be realized for the two fiber ends when the NV bare dipole is located at the center of the tapered fiber. (2) NV centers even in 100-nm-sized nanodiamonds where bulk-like optical properties were reported show a coupling efficiency of 22% at the taper surface, with the coupling efficiency monotonically decreasing as the nanodiamond size increases. These results will be helpful in guiding the development of hybrid quantum devices for applications in quantum information science.
© 2014 Optical Society of America
1. Introduction
Nitrogen-vacancy (NV) color centers in diamond have recently emerged as promising solid-state atomic systems that can be used for various quantum engineering applications [1, 2]. NV centers exhibit narrow optical transitions with lifetime-limited linewidth [3,4], robust and stable photoemission [5], and paramagnetic electron spin resonance with a very long coherence time at room temperature [6]. These properties enable broad applications in quantum information and quantum sensing, e.g., single-photon sources [7,8], quantum memories [9], and highly sensitive nanoscale magnetometers [10–12], thus rendering NV centers distinct from other solid-state quantum nanoemitters.
A promising approach for these applications is to use hybrid nanophotonic quantum devices, in which the high coupling of NV centers with photons (namely, guided modes or cavity modes) is obtained by assembling NV-embedding nanodiamonds with well-established nondiamond nanophotonic structures such as photonic crystals [13, 14], microcavities [15, 16], and tapered optical fibers [17–19]. For example, efficient coupling of single-NV centers in 20-nm-sized nanodiamonds with ultrathin tapered fibers has been demonstrated recently [17–19], being proved to work as bright and efficient single-photon sources and sensitive magnetometers.
Such efficient coupling of tapered fibers has also been known for other quantum emitters, such as atoms [20–23], colloidal quantum dots [24,25], and InAs quantum dots in GaAs waveguides [26]. In these systems, the electromagnetic environment surrounding the dipole is very different. For example, the dipole was considered to be in air for atoms and colloidal quantum dots (with a size very small compared to the optical wavelength). In contrast, the effect of the dipole of InAs quantum dots in GaAs waveguides must be considered in high-index semiconductor submicron structures. It is therefore important to properly take into account the dipolar environment to obtain efficient coupling with dipoles and tapered fibers.
In previous research on NV centers in nanodimoands, the nanodiamonds have been regarded as point dipoles, i.e., the dipole in air when sitting on the surface of tapered fibers [17–19, 27]. However, for NV centers embedded in nanodiamonds, the effect of nanodiamond size and shape may be critically important. In real experiments, nanodiamond particle size exhibits a wide distribution, ranging from 5 to 500 nm [28–30], and size selection is decisively important for developing hybrid quantum nanophotonic devices. For example, smaller nanodiamonds are preferable in view of their coupling to nanophotonic structures or magnetic sensing. Such very small nanodiamonds (e.g., 5 nm), however, exhibit fluorescence intermittency [28, 29] or short T2 coherence time of the electron spins [10, 12, 31], and hence they cannot be used for many quantum applications. In contrast, larger size nanodiamonds exhibit bulk-like optical properties but relatively low coupling and optical scattering by themselves. For these reasons, investigating the effect of size and shape of nanodiamonds on the coupling efficiency of NV centers is critically important.
In this paper, we numerically analyze the coupling of NV centers embedded in nanodiamonds with tapered optical fibers. The results show that (1) a maximum coupling efficiency of 53.4% can be realized for the two fiber ends when a NV bare dipole is located at the center of the tapered fiber; and (2) NV centers even in 100-nm-sized nanodiamonds where bulk-like optical properties were reported show a coupling efficiency of 22% at the taper surface. These results will prove helpful in guiding the development of hybrid quantum devices for applications in quantum information science.
2. Methodology
2.1. Model structure
The structure and geometry of our simulated model are shown in Fig. 1(a). The tapered fiber is a silica-glass cylinder (ns = 1.4469) with a diameter d. The dipole of the NV center is placed at the center of the tapered fiber (X = 0, Y = 0, Z = 0) or at the surface (X = d/2, Y = 0, Z = 0). Three dipolar orientations are considered: (1) radial orientation, where the dipole is directed perpendicular to the cylindrical surface (θ = 90°, ϕ = 0°), (2) azimuthal orientation, where the dipole is tangent to the cylindrical surface (θ = 90°, ϕ = 90°), and (3) axial orientation, where the dipole is parallel to the cylindrical axis (θ = 0°, ϕ = 0°). Nanodiamonds (nd = 2.43) were directly attached to the surface of the tapered fiber [see Fig. 1(b)].
Note that a NV center does not have a linear dipole but has two dipole moments of equal size in the plane perpendicular to the NV axis [13, 32, 33]. However, we regard the NV center as a linear dipole because (1) this simplification makes the subsequent discussion simpler and more intuitive and (2) these two dipoles are incoherent with each other and any resultant fluorescence polarization can be described as a superposition of the three dipole orientations (radial, azimuthal, and axial), as will be discussed in Sec. 4.3.
2.2. Simulations
Data were obtained computationally using the three-dimensional finite-difference time domain (FDTD) method (Lumerical, FDTD package). The computational region is a box of 6 × 6 × 30 μm3, as shown in Fig. 1(b). Both ends of the silica-glass cylinder (the tapered fiber) are placed outside of the simulation region. Absorbing perfectly matched layers (PMLs) are used as the end walls of this computational region. The reflectivity from the PMLs is set to <1 × 10−6. With this arrangement of PMLs, reflections from the boundary walls are checked and no significant reflection is observed. An automatic nonuniform mesh is used with a relatively high accuracy. The typical mesh size is 3 nm around the interface where a high refractive index difference exists (specifically, silica/air, silica/diamond, and diamond/air interfaces) and 24.5 nm in other areas. We did not observe any significant grid dispersion along the tapered fiber axis.
The dipole is placed 4.4 μm from the bottom of the boundary box; this is sufficiently far from the boundary walls to suppress the effect of reflection from the boundary. We apply symmetric boundary conditions to the XZ plane for radial and axial dipole orientations and antisymmetric ones for the azimuthal dipole orientation to predefine the polarization of the fundamental mode and to reduce computational time. We confirmed the convergence of the simulation by changing the number of mesh points per wavelength. Note that the dipole is not placed at the center of the simulation region. This is because we have to take enough distance between the dipole and a power monitor located at Z=22.6 μm.
The dipole is assumed to have a wavelength of 637 nm where the zero-phonon lines of NV centers are located [1]. We considered a classical electric dipole antenna for the NV point dipole as well as the previous report [34], which emits a power of in a homogeneous media of index n, where μ0, p0, and c are the vacuum permeability, the electric dipole moment, and the speed of light, respectively. We then calculated the time-averaged Poynting flux over the surface of a rectangular box enclosing the dipole to get the actual radiated power from the dipole owing to the inhomogeneous environment surrounding the dipole. The spontaneous emission rate was then given by Sbox/Pbulk.
2.3. Definition of the physical quantities
We obtain three physical quantities in the present work: (1) the spontaneous emission rate of the NV dipole, Γ, (2) the power coupled to the fundamental guided mode of the tapered fiber for both fiber ends, Pcouple, and (3) the coupling efficiency, η. These three quantities are linked through
The spontaneous emission rate (Γ) varies depending on the structures around which the dipole is located. Pcouple is obtained as the power propagating along the ±Z direction at the position sufficiently far from the dipole position where the electric field is mostly supported by the fundamental mode. Note that fluorescence from the dipole is equally divided into two directions of tapered fibers, which has been confirmed experimentally [17]. Note also that Γ and Pcouple are normalized to the emitted power of the dipole in bulk diamond throughout our discussion.This appropriate Z position can be determined by the overlap integral between the electric field profile of the dipole emission and that of the fundamental mode. The overlap integral indicates the percentage of the emission coupled to the fundamental mode. The electromagnetic field of the fundamental mode is calculated by means of a mode analysis solver within the same FDTD package. The overlap integral, W, is defined as
where the subscripts 1 and 2 designate the dipole emission and the fundamental mode, respectively [35–37].Note that the single-photon generation rate is given by Pcouple at the saturation limit of the continuous wave excitation (the limiting case of the excitation power close to infinity). The single-photon emission rate in the pulsed excitation approaches Pcouple too at the saturation limit when parameters such as excitation power or the excitation frequency are optimized. In some applications using pulsed excitation, used for two-photon quantum interference, the single-photon emission rate per excitation pulse becomes important. In such cases, the single-photon emission rate is given by the coupling efficiency η only.
3. Guided modes of tapered fibers and point dipoles coupled to tapered fibers
3.1. Mode analysis of the tapered fiber
We first analyze the guided modes of ultrathin tapered fibers by using mode analysis. Figure 2 shows the electric field distribution of the fundamental mode (quasi-linearly polarized HE11) of the tapered fiber with a diameter of 300 nm, where λ = 637 nm. Ex and Ez are the main electric field components, and the Ey is negligibly small. The electric field (Ex and Ez and hence intensity) is localized at the surface of the tapered fiber. As the diameter becomes small compared to the wavelength, the localized field at the surface increases (see [38, 39] for the details of physics of such ultrathin tapered fibers). Such a strong surface localization has been known and used for applications such as atom trapping [23, 38, 40–42].
Fig. 2 Two-dimensional cross section of the electric field of the fundamental guided mode of the tapered fiber with a diameter of 300 nm and a wavelength of 637 nm. (a) Intensity and (b) each electric field component (specifically their E2 values). The color bar shows a logarithmic scale. White circles indicate the surface of the tapered fiber.
This fundamental mode (quasi-linearly polarized HE11) of the tapered fiber is connected to the fundamental mode of the single-mode fiber with almost 100% efficiency owing to adiabatic tapering. Indeed, this near perfect transmission of such ultrathin tapered fibers has been reported in many papers [20,43–45]. We therefore think that the power coupled to the fundamental mode of tapered fibers can be extracted at the single-mode fiber ends without any loss.
3.2. Coupling of single NV bare dipoles with 300-nm-diameter tapered fibers
We simulate the coupling of single NV bare dipoles with 300-nm-diameter tapered fibers, which means that single-point dipoles are directly placed in vacuum or in the tapered fibers without diamond layers. This simulation is equivalent to using very small nanodiamonds compared to the optical wavelength, as well as colloidal quantum dots. Such very small nanodiamonds have been reported recently as “single-digit nanodiamonds” [46].
Given the three dipolar orientations as shown in Fig. 1, we can consider five geometries in our simulations: (a) radial orientation at the center of the fiber (X = 0, θ = 90°, ϕ = 0°), (b) axial orientation at the center (X = 0, θ = 0°, ϕ = 0°), (c) radial orientation at the surface (X = d/2, θ = 90°, ϕ = 0°), (d) azimuthal orientation at the surface (X = d/2, θ = 90°, ϕ = 90°), and (e) axial orientation at the surface (X = d/2, θ = 0°, ϕ = 0°). Note that azimuthal orientation becomes equivalent to radial orientation at the center of the tapered fiber.
Table 1 summarizes the simulation results and Fig. 3 shows the cross sections (XZ or YZ depending on the dipole orientations) of the emission from the NV dipole in these five geometries (in which 300-nm diameter tapered fibers are considered). In all cases, the electric field exhibits a typical dipolar emission pattern at around Z = 0—specifically, the well-known cos2 θ dependence [47]. The electric field far from the dipole position (Z = 22.6 μm) is mainly supported by the fundamental mode, as the overlap integral shows more than 0.99, which allows us to regard the electric field remaining at Z = 22.6 μm as Pcouple/2. It is obvious that axial and azimuthal dipole orientation [Figs. 3(b), 3(d), and 3(e)] give a prominent free-space radiation at Z = 0 compared to the radial orientation [Figs. 3(a) and 3(c)], inevitably resulting in the low coupling efficiency (Table 1).
Table 1. Coupling efficiencies and other parameters for five dipolar geometries. Γ and Pcouple are normalized to the emitted power of the dipole in homogeneous silica media. The overlap integral W is the value at Z = 22.6 μm. Note that Pcouple and η are of both fiber ends.
Fig. 3 Cross sections (X = 0 for the azimuthal dipole orientation, and Y = 0 for all other dipole configurations) of the electric field intensity of the dipolar emission and schematics of the geometries for the five cases: (a) radial and (b) axial dipoles at the center and (c) radial, (d) azimuthal, and (e) axial dipoles at the surface. The square of the electric field intensity is mapped. The color bar shows a logarithmic scale. Note that we show X=0 cross-section for the azimuthal dipole orientation because the azimuthal orientation has an asymmetric plane in X = 0. The electric field pattern of the guided mode shown in Fig. 2 should be rotated by 90° in this case.
The highest coupling efficiency is achieved when the dipole is oriented radially at the center, giving 49.2% (in case of 300-nm-diameter tapered fiber). When the radial dipole reaches the surface, the coupling efficiency decreases to 31.4%. This reduction can naturally be understood by the small overlap between the dipolar emission at Z = 0 and the fundamental mode. In more detail, when the dipole comes to the surface [(Fig. 3(c)], the electric field is concentrated only at the dipole side of the tapered fiber and is diminished on the other side. Since the fundamental mode has a field maximum symmetrically at the both taper surfaces [see Fig. 2(a)], the overlap eventually decreases when the dipole comes to the surface.
The coupling efficiencies for azimuthal and axial orientations at the taper surface are 18.4% and 12.6%, respectively. Interestingly, the coupling efficiency of the axial dipole at the surface (12.6%) is larger than that for the axial orientation at the center (where η ≈ 0). The axial dipole does not emit along the exact Z direction owing to the angular dependence of the dipole (i.e., cos2 θ), so that no coupling occurs when the dipole is placed at the center. The emission directed along the non-Z direction, however, may couple to the fundamental mode when the dipole reaches the surface, which increases the coupling efficiency.
3.3. Dependence of the coupling efficiencies on the diameter of tapered fibers
Figure 4 shows plots of the coupling efficiency with the fundamental mode of the tapered fiber as a function of the taper diameter for the above four dipolar configurations. It can be seen that up to 53.4% of the total dipole emission is coupled to the fundamental mode when the dipole is placed at the center with a radial orientation (X = 0, θ = 90°, ϕ = 0°). This maximal coupling efficiency of 53.4% is realized at a diameter of 350 nm, a value of almost half the emission wavelength (λ = 637 nm) [50, 51]. Incorporating single-digit nanodiamonds [30] into such ultrathin tapered fibers can also achieve high coupling efficiency, although it is technically challenging. We however point out that such efforts to incorporate nanodiamonds into nano- or microfibers have already been reported [52].
Fig. 4 Plots of the coupling efficiency as a function of the taper diameter for the four dipolar geometries. The inset is a comparison of our simulation (squares) for average dipolar orientations (the average of the three orientations) with the previously reported result in Ref. [25] (dotted line), which is taken from the literature. The fiber size parameter is defined as πd/λ. We used the same parameters used in [25] (specifically, λ and d) and calculated the coupling efficiency for average dipolar orientations.
When the radial dipole moves to the surface of the tapered fiber, the maximal coupling efficiency decreases to 30.8%, occurring at 300 nm and becoming slightly shifted from 350 nm for the case of a radial dipole at the center. When the dipole is azimuthally oriented and placed at the surface (X = d/2, θ = 90°, ϕ = 90°), a maximum of 19.4% is obtained at a diameter of 250 nm. When the dipole is oriented in the axial direction (X = d/2, θ = 0°, ϕ = 0°), the maximum coupling efficiency is further reduced to 13.4% at a diameter of 350 nm. Note that two small kinks appearing at 580 and 800 nm (prominently observed in the blue dots) come from the generation of other waveguide modes owing to the large taper diameter (see Ref. [27] for further details).
The inset of Fig. 4 is a comparison of our simulations with results from the previously published paper. We used the same parameters used in [25] (specifically, λ and d) and calculated the coupling efficiency for average dipolar orientations. Our results are consistent with the previous data, verifying the reliability of our simulations. Note that the data in [25] (the dotted line in the inset of Fig. 4) were calculated by using the analytical solution. The difference between the FDTD simulations and this analytical solution is sufficiently small compared to the observed uncertainty of the experimental values reported in [25] or in other similar experiments [24].
It should be noted that the coupling efficiency of tapered fibers can be high over the large wavelength range, thereby providing the high efficiency for the whole NV emission spectrum from 637 nm to 750 nm. We calculated coupling efficiencies for cases where the radial dipole is placed at the center of 300-nm-diameter taper and obtained values of 48.5 %, 47.5%, 46.2%, 44.4%, and 42.4% for 652 nm, 667 nm, 683 nm, and 700 nm, respectively. It is therefore promising to use tapered fibers for other color centers in nanodiamonds, such as silicon vacancy [48] or chromium related centers [49].
4. Effect of nanodiamond structures on the coupling of NV centers with tapered fibers
4.1. Enhancement and inhibition of spontaneous emission of NV centers in nanodiamonds
The spontaneous emission rate of the dipole strongly depends on the local density of photonic states around it. In particular, the difference of refractive index (nd/nair = 2.43) at the air–diamond interface causes large reflection, so that this enhancement or inhibition becomes prominent. We therefore need to consider how the variety of nanodiamond shapes and sizes affects our present studies on the NV coupling to the tapered fibers.
We first consider the simplest case where the NV dipole is placed at the center of a spherical nanodiamond. The analytical solution for the spontaneous emission rate is known in this case [54–56, 67] and is , where k0 = 2π/λ and a is the radius of the sphere (see [54, 56] for the exact expression for U(k0a)). This analytical solution allows us to verify the validity of our FDTD simulations.
Figure 5(a) shows a plot of the spontaneous emission rate of the FDTD simulations as a function of the diameter of the spherical nanodiamond together with the analytical solutions. The FDTD simulations are in good agreement with the analytical solution over the whole diameter range. The spontaneous emission rate is strongly inhibited in the small-size region from 0 to 200 nm. It shows a periodical enhancement and inhibition as the nanodiamond size increases. The first enhancement peak appears at a size of 370 nm with 2.0 times enhancement more than that in bulk. Second- or higher order enhancements appear almost periodically as the size increases.
Fig. 5 Plots of the spontaneous emission rate of NV centers as a function of nanodiamond size. We considered three nanodiamond shapes: (a) spherical, (b) cubical, and (c) pyramidal. The red line in (a) is the analytical solution. The dashed lines connecting the dots in (b) and (c) are drawn to guide the eye. The insets are the close-up of the region L ≤ 200 nm. The spontaneous emission rate is normalized to that in bulk diamond. The length of the nanodiamond strucutre and the position of the dipole are shown on the left side of each plot.
The analytical solutions give a limit of the spontaneous emission rate when the nanodiamond size approaches zero: , which is 0.059 using the diamond index of nd = 2.43. This limiting value should not depend on the shape of the nanodiamond. We further calculate the spontaneous emission rate in other nanodiamond shapes (cubes and pyramids) using FDTD. The results are shown in Figs. 5(b) and 5(c) for cubes and pyramids, respectively. They, indeed, asymptotically approach this value as the nanodiamond size goes to zero, showing good consistency of the FDTD simulations with the analytical solutions.
The shape of the nanodiamonds does not seem to affect the spontaneous emission rate in the region size smaller than 400 nm; the spontaneous emission is strongly inhibited in the small-size region, with inhibition first appearing between 300 and 400 nm. Since the spherical nanodiamonds exhibited the simplest periodic modulations of the spontaneous emission rate, we consider only spherical nanodiamond structure in the following sections.
It should be mentioned that the emission rate calculated here (both computationally and analytically) does not agree quantitatively with the experimental data for the excited-state lifetime of NV centers in nanodiamonds. Reported NV lifetimes in nanodiamonds (typically having a mean particle size of 25 nm) range from 20 to 40 ns [1, 57, 58], which is much shorter than the theoretical limit of the lifetime (0.059−1 × τbulk ∼ 190 ns, where τbulk = 11 ns [2]). The reason for this discrepancy has been presumably due to the presence of non-radiative decay in diamond. The nod-radiative decay shortens the excited-state lifetime and decreases the fluorescence quantum efficiency of NV centers. It therefore can alleviate the suppression of the spontaneous emission rate in the very small nanodiamond region and can diminish the emission enhancement. There are various non-radiative decay channels in diamond, such as NV− transitions to neutral NV state or quenching due to some other defects, and it has not been fully clarified yet (interested readers are referred to the literature [34,57,58]). Despite of this discrepancy, the theoretical simulations of the spontaneous emission rate performed here, at least, can give a crude explanation for how the emission rate changes depending on the size, shape, and dipole position. We therefore think that our calculations can help demonstrate how our subsequent calculations of the coupling efficiency and the emission rate will vary depending on the size, shape, and dipole position.
4.2. Dependence of the coupling efficiency on nanodiamond size
We next analyze the coupling efficiency of the NV dipole embedded in spherical nanodiamonds with tapered fibers. Figures 6(a) and (b), respectively, show a schematic of the model of the simulation and the coupling efficiency and the spontaneous emission rate as a function of the size of spherical nanodiamonds, Φ. The dipole position and the dipole orientation are assumed to be the center of the nanodiamond and the radial dipole orientation (X = 150 nm + Φ/2, θ = 90°, ϕ = 0°), respectively. The diameter of the tapered fiber is 300 nm. The coupling efficiency monotonically decreases as the nanodiamond size increases. This monotonic reduction can be simply explained by the fact that the dipole moves away from the taper surface.
Fig. 6 Dependence of the coupling efficiency on nanocrystal size. (a) Schematic of the geometries. (b) Plots of the coupling efficiency and the emission rate as a function of the nanocrystal size Φ. The dotted line indicates the emission rate of the NV center without tapered fibers, which is exactly the same as Fig. 5(a). (c) Plot of the guided power (Pcouple) as a function of the nanocrystal size. Pcouple is normalized to the spontaneous emission rate of the dipole in bulk diamond. The dipole has radial orientation.
The spontaneous emission rate in Fig. 6(b) exhibits inhibition in the smaller nanodiamond and enhancement at 370 nm, as does that without the tapered fiber (i.e., for nanodiamond only, the dotted line in Fig. 6(b) or 5(a)). The spontaneous emission rate is almost the same as that without the tapered fiber where the nanodiamond size is larger than 250 nm; the resonance enhancement of the spontaneous emission rate is prominent under the presence of tapered fibers. In contrast, it is enhanced in the region of Φ = 0–200 nm owing to the presence of the tapered fiber because the local photonic density of states is increased by the existence of the tapered fiber. Specifically, this enhancement is a factor of 2.16 for 50-nm nanodiamonds. Figure 6(c) is a plot of Pcouple as a function of nanodiamond size, which gives the maximum of the fiber-coupled single-photon emission rate in the CW excitation in real applications. Because Pcouple is a product of the coupling efficiency and the spontaneous emission rate, this has a maximum at around 370 nm. This maximal Pcouple at Φ = 370 nm is increased by a factor of ∼3 compared to that of 50-nm-sized nanodiamond. (Pcouple at 50 nm and at 370 nm are respectively 0.042 and 0.1328).
Importantly, the coupling efficiency for the radial dipole is 22% even in 100-nm-sized nanodiamonds [see Fig. 6(b)]. NV centers in 100-nm-sized nanodiamonds can exhibit bulk-like properties, such as very narrow zero-phonon-line (ZPL) optical transitions [59] and long T2 electron spin coherence time [30]. Indeed, we have recently reported a 1.2-GHz ZPL in the direct photoluminescence spectrum [59], which is comparable to the values obtained in pure bulk diamonds [7, 60]. By assembling such an excellent nanodiamond with tapered fibers and using it in cryogenic environment (see Ref. [61–64] for the detail), it may be possible to realize hybrid nanophotonic devices that can be employed for narrow-band single-photon sources, with high coupling of the NV–photon interaction. The present results therefore provide a useful guideline for the development of hybrid NV–taper systems for indistinguishable single-photon sources, quantum phase gates, and quantum memory.
4.3. Dependence of the coupling efficiency on dipole orientation
We next analyze the dependence of the coupling efficiency and the spontaneous emission rate on the dipole orientations in nanodiamonds. We considered again cases in which the taper diameter was 300 nm and the NV dipole was located at the center of the nanodiamonds. Figures 7(a) and 7(b), respectively, show a schematic of the geometry and the dependence of the coupling efficiency on the nanodiamond size for the three dipole orientations. The dipole-orientation dependence of the coupling efficiency exhibits the largest value of 23.8% for radial orientations followed in order by 11% for azimuthal and 7% for axial orientations in 100-nm-sized nanodiamonds. This order is the same as that for the case without nanodiamonds (see Fig. 4).
Fig. 7 Dependence of the coupling efficiency on the three orientations. (a) Schematic of the geometries. (b) Plots of the coupling efficiencies as a function of the nanocrystal size Φ for radial, azimuthal, and axial orientations. (c) Plots of the coupling efficiencies as a function of the dipole–taper surface distance Φ/2, where the dipole is in vacuum.
In any dipolar orientation, the coupling efficiency decreases as the nanocrystal size increases. This monotonic reduction can again be explained by the decrease of the overlap between dipolar emission and the guided mode by which the dipole moves away from the tapered fiber. In fact, the coupling efficiency follows the same trend as the case of a dipole located in vacuum (with no nanodiamond structure). In Fig. 7(c), we show the coupling efficiency of the free-space dipole (in vacuum) for the three orientations at the corresponding taper–dipole distances [65]. This fact clearly indicates the importance of the taper–dipole distance in obtaining a high coupling efficiency.
4.4. Dipole positional dependence of the coupling efficiency in 370-nm-sized nanodiamonds
As shown in Fig. 6(b), Pcouple was enhanced in 370-nm-sized nanodiamonds. Such an enhancement may be important when using this system as single-photon sources in CW excitation [66]. To further investigate this enhancement, the dependence of the coupling efficiency on the dipole position in this nanodiamond was investigated. The coupling efficiency is calculated by changing the position of the NV center in the nanodiamond.
Figures 8(a) and (b) show a schematic of the model and the coupling efficiency and the total emission rate as a function of dipole position δ, respectively. The coupling efficiency, as expected from the other results, monotonically decreases as the dipole moves away from the tapered fiber. The spontaneous emission rate exhibits strong enhancement at 185 nm from the dipole position, which is the exact center of the nanodiamond.
Fig. 8 Dependence of the coupling efficiency on the NV dipole position in the nanocrystals. (a) Schematic of the geometries. (b) Plots of the coupling efficiency and the spontaneous emission rate as a function of the dipole position in 370-nm-sized nanocrystal, δ. (c) Plot of the guided power (Pcouple) as a function of the dipole position. The dipole has radial orientation.
Figure 8(c) shows the single-photon emission rate (Pcouple) from the fiber end. One can extract a high emission rate when the dipole is at the center of the 370-nm-sized nanocrystal. The photon emission rate coupled with the fiber is enhanced by a factor of up to ∼3 compared with that for a nanodiamond size of 50 nm [see Fig. 6(c)]. This enhancement of the photon emission rate may be used for the development of bright single-photon sources.
Note that very sharp resonance peaks of the spontaneous emission rate may appear when the dipole comes in close proximity to the edge of the nanodiamond sphere. Such enhancement in the spontaneous emission rate of the dipole in dielectric spheres was predicted from analytical solutions [55, 56] and was subsequently demonstrated in experiment [67]. We however did not simulate such cases because placing a dipole very near the dielectric boundary requires more careful treatment in FDTD, and more importantly, the spherical nanodiamond used here is only an example of various nanodiamond shapes. The real nanodiamonds should have different shapes from the ideal spheres, thereby showing different cavity enhancement of NV centers. Although the coupling between tapered fibers and ideal spherical nanodiamonds is important, such simulations are far beyond the scope of the present paper.
5. Conclusions
In conclusion, we have simulated coupling of NV centers in nanodiamonds with tapered optical fibers by using the three-dimensional FDTD method to investigate how the nanodiamond structures affect the coupling of the NV dipole with tapered fibers. The results lead to the following important conclusions: (1) A maximum coupling efficiency of 53.4% for the two fiber ends can be realized when the NV bare dipole is located at the center of the tapered fiber. Note that one may place a fiber-Bragg-grating mirror in front of either of the fiber outputs so that all the photons are output from one end. (2) NV centers even in 100-nm-sized nanodiamonds where bulk-like optical properties were reported show a coupling efficiency of 22% at the taper surface. These results will be helpful in guiding the development of hybrid quantum devices for applications in quantum information science.
The most important indication from these conclusions may be that we are able to get a coupling efficiency of 22% for the radial dipole even in 100-nm-sized nanodiamonds [see Fig. 6(b)]. The NV centers in 100-nm-sized nanodiamonds can show bulk-like properties, such as very narrow ZPL optical transitions [7, 59, 60]. The present results therefore should encourage assembling nanodiamonds possessing bulk-like NV centers with tapered fibers.
The recent realization of deterministic coupling of single-NV nanodiamonds with tapered fibers could be a promising step toward this assembling [18]. One can first characterize ZPL linewidths of NV centers in nanodiamonds at cryogenic temperatures to find those having ”excellent” NV centers. With some markers to identify the same nanodiamonds, they can be deterministically pick-& -placed to tapered fibers. By cooling such assembled systems to cryogenic temperatures, narrow-band bright single photon sources using NV centers may be realized in fiber-integrated quantum circuits. Thus further researches on the deterministic coupling of nanodiamonds to tapered fibers and on the preparation of nanodiamonds incorporating bulk-like NV centers would be necessary in the future.
Acknowledgments
The authors acknowledge helpful discussions with Hideki Fujiwara and Ryo Okamoto. MA is a recipient of a fellowship from JSPS (“The Invitation Fellowship for Research in Japan”). We gratefully acknowledge financial support from MEXT-KAKENHI Quantum Cybernetics (No. 21101007), JSPS-KAKENHI (Nos. 26220712, 23244079, 25620001, 23740228, 26706007, and 26610077), JST-CREST, JSPS-FIRST, the Project for Developing Innovation Systems of MEXT, the G-COE Program, and the Research Foundation for Opto-Science and Technology.
References and links
1. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, “The nitrogen-vacancy colour centre in diamond,” Phys. Rep. 528, 1–45 (2013). [CrossRef]
2. F. Jelezko and J. Wrachtrup, “Single defect centres in diamond: a review,” Phys. Stat. Solidi (A) 203, 3207–3225 (2006). [CrossRef]
3. P. Tamarat, T. Gaebel, J. Rabeau, M. Khan, A. Greentree, H. Wilson, L. Hollenberg, S. Prawer, P. Hemmer, F. Jelezko, and J. Wrachtrup, “Stark shift control of single optical centers in diamond,” Phys. Rev. Lett. 97, 083002 (2006). [CrossRef] [PubMed]
4. Y. Shen, T. M. Sweeney, and H. Wang, “Zero-phonon linewidth of single nitrogen vacancy centers in diamond nanocrystals,” Phys. Rev. B 77, 33201 (2008). [CrossRef]
5. C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85, 290–293 (2000). [CrossRef] [PubMed]
6. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nature Mat. 8, 383–387 (2009). [CrossRef]
7. H. Bernien, L. Childress, L. Robledo, M. Markham, D. Twitchen, and R. Hanson, “Two-photon quantum interference from separate nitrogen vacancy centers in diamond,” Phys. Rev. Lett. 108, 043604 (2012). [CrossRef] [PubMed]
8. A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89, 187901 (2002). [CrossRef] [PubMed]
9. E. Togan, Y. Chu, A. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. Sørensen, P. Hemmer, A. Zibrov, and M. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010). [CrossRef] [PubMed]
10. J. Maze, P. Stanwix, J. Hodges, S. Hong, J. Taylor, P. Cappellaro, L. Jiang, M. G. Dutt, E. Togan, A. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin, “Nanoscale magnetic sensing with an individual electronic spin in diamond,” Nature 455, 644–647 (2008). [CrossRef] [PubMed]
11. G. Balasubramanian, I. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, H. Tobias, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, “Nanoscale imaging magnetometry with diamond spins under ambient conditions,” Nature 455, 648–651 (2008). [CrossRef] [PubMed]
12. C. Degen, “Scanning magnetic field microscope with a diamond single-spin sensor,” Appl. Phys. Lett. 92, 243111 (2008). [CrossRef]
13. T. Schröder, A. W. Schell, G. Kewes, T. Aichele, and O. Benson, “Fiber-integrated diamond-based single photon source,” Nano Lett. 11, 198–202 (2011). [CrossRef]
14. O. Benson, “Assembly of hybrid photonic architectures from nanophotonredic constituents,” Nature 480, 193–199 (2011). [CrossRef] [PubMed]
15. J. Wolters, A. W. Schell, G. Kewes, N. Nusse, M. Schoengen, H. Doscher, T. Hannappel, B. Lochel, M. Barth, and O. Benson, “Enhancement of the zero phonon line emission from a single nitrogen vacancy center in a nanodiamond via coupling to a photonic crystal cavity,” Appl. Phys. Lett. 97, 141108 (2010). [CrossRef]
16. P. E. Barclay, C. Santori, K.-M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled optical cavity-qed system,” Opt. Express 17, 8081–8097 (2009). [CrossRef] [PubMed]
17. T. Schröder, M. Fujiwara, T. Noda, H.-Q. Zhao, O. Benson, and S. Takeuchi, “A nanodiamond-tapered fiber system with high single-mode coupling efficiency,” Opt. Express 20, 10490–10497 (2012). [CrossRef] [PubMed]
18. L. Liebermeister, F. Petersen, A. v. Münchow, D. Burchardt, J. Hermelbracht, T. Tashima, A. W. Schell, O. Benson, T. Meinhardt, A. Krueger, A. Stiebeiner, A. Rauschenbeutel, H. Weinfurter, and M. Weber, “Tapered fiber coupling of single photons emitted by a deterministically positioned single nitrogen vacancy center,” Appl. Phys. Lett. 104, 031101 (2014). [CrossRef]
19. X. Liu, J. Cui, F. Sun, X. Song, F. Feng, J. Wang, W. Zhu, L. Lou, and G. Wang, “Fiber-integrated diamond-based magnetometer,” Appl. Phys. Lett. 103, 143105 (2013). [CrossRef]
20. K. Nayak, P. Melentiev, M. Morinaga, F. L. Kien, V. Balykin, and K. Hakuta, “Optical nanofiber as an efficient tool for manipulating and probing atomic fluorescence,” Opt. Express 15, 5431–5438 (2007). [CrossRef] [PubMed]
21. M. Das, A. Shirasaki, K. Nayak, M. Morinaga, F. Le Kien, and K. Hakuta, “Measurement of fluorescence emission spectrum of few strongly driven atoms using an optical nanofiber,” Opt. Express 18, 17154–17164 (2010). [CrossRef] [PubMed]
22. K. Nayak, F. Le Kien, M. Morinaga, and K. Hakuta, “Antibunching and bunching of photons in resonance fluorescence from a few atoms into guided modes of an optical nanofiber,” Phys. Rev. A 79, 021801 (2009). [CrossRef]
23. A. Goban, K. Choi, D. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. Stern, and H. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012). [CrossRef] [PubMed]
24. M. Fujiwara, K. Toubaru, T. Noda, H. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11, 4362–4365 (2011). [CrossRef] [PubMed]
25. R. Yalla, F. Le Kien, M. Morinaga, and K. Hakuta, “Efficient channeling of fluorescence photons from single quantum dots into guided modes of optical nanofiber,” Phys. Rev. Lett. 109, 063602 (2012). [CrossRef] [PubMed]
26. M. Davanço, M. Rakher, W. Wegscheider, D. Schuh, A. Badolato, and K. Srinivasan, “Efficient quantum dot single photon extraction into an optical fiber using a nanophotonic directional coupler,” Appl. Phys. Lett. 99, 121101 (2011). [CrossRef]
27. M. R. Henderson, S. Afshar V, A. D. Greentree, and T. M. Monro, “Dipole emitters in fiber: interface effects, collection efficiency and optimization,” Opt. Express 19, 16182–16194 (2011). [CrossRef] [PubMed]
28. C. Bradac, T. Gaebel, N. Naidoo, M. Sellars, J. Twamley, L. Brown, A. Barnard, T. Plakhotnik, A. Zvyagin, and J. Rabeau, “Observation and control of blinking nitrogen-vacancy centres in discrete nanodiamonds,” Nat. Nanotech. 5, 345–349 (2010). [CrossRef]
29. V. N. Mochalin, O. Shenderova, D. Ho, and Y. Gogotsi, “The properties and applications of nanodiamonds,” Nat. Nanotech. 7, 11–23 (2011). [CrossRef]
30. G. Kucsko, P. Maurer, N. Yao, M. Kubo, H. Noh, P. Lo, H. Park, and M. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–58 (2013). [CrossRef] [PubMed]
31. J. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. Hemmer, A. Yacoby, R. Walsworth, and M. Lukin, “High-sensitivity diamond magnetometer with nanoscale resolution,” Nat. Phys. 4, 810–816 (2008). [CrossRef]
32. T. P. M. Alegre, C. Santori, G. Medeiros-Ribeiro, and R. G. Beausoleil, “Polarization-selective excitation of nitrogen vacancy centers in diamond,” Phys. Rev. B 76, 165205 (2007). [CrossRef]
33. A. Batalov, V. Jacques, F. Kaiser, P. Siyushev, P. Neumann, L. Rogers, R. McMurtrie, N. Manson, F. Jelezko, and J. Wrachtrup, “Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond,” Phys. Rev. Lett 102, 195506 (2009). [CrossRef] [PubMed]
34. F. Inam, T. Gaebel, C. Bradac, L. Stewart, M. Withford, J. Dawes, J. Rabeau, and M. Steel, “Modification of spontaneous emission from nanodiamond colour centres on a structured surface,” New J. Phys. 13, 073012 (2011). [CrossRef]
35. M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]
36. M. Davanço and K. Srinivasan, “Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole,” Opt. Lett. 34, 2542–2544 (2009). [CrossRef] [PubMed]
37. M. Davanço and K. Srinivasan, “Hybrid gap modes induced by fiber taper waveguides: Application in spectroscopy of single solid-state emitters deposited on thin films,” Opt. Express 18, 10995–11007 (2010). [CrossRef] [PubMed]
38. F. L. Kien, J. Liang, K. Hakuta, and V. Balykin, “Field intensity distributions and polarization orientations in a vacuum-clad subwavelength-diameter optical fiber,” Opt. Commun. 242, 445–455 (2004). [CrossRef]
39. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]
40. F. Le Kien, V. Balykin, and K. Hakuta, “Atom trap and waveguide using a two-color evanescent light field around a subwavelength-diameter optical fiber,” Phys. Rev. A 70, 063403 (2004). [CrossRef]
41. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010). [CrossRef] [PubMed]
42. E. Vetsch, S. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Quantum Electron. 18, 1763 (2012).
43. M. Fujiwara, K. Toubaru, and S. Takeuchi, “Optical transmittance degradation in tapered fibers,” Opt. Express 19, 8596–8601 (2011). [CrossRef] [PubMed]
44. T. Aoki, “Fabrication of ultralow-loss tapered optical fibers and microtoroidal resonators,” Jpn. J. Appl. Phys. 49, 8001 (2010). [CrossRef]
45. R. Garcia-Fernandez, W. Alt, F. Bruse, C. Dan, K. Karapetyan, O. Rehband, A. Stiebeiner, U. Wiedemann, D. Meschede, and A. Rauschenbeutel, “Optical nanofibers and spectroscopy,” Appl. Phys. B 105, 3–15 (2011). [CrossRef]
46. J. Tisler, G. Balasubramanian, B. Naydenov, R. Kolesov, B. Grotz, R. Reuter, J.-P. Boudou, P. A. Curmi, M. Sennour, A. Thorel, M. Börsch, K. Aulenbacher, R. Erdmann, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Fluorescence and spin properties of defects in single digit nanodiamonds,” ACS Nano 3, 1959–1965 (2009). [CrossRef] [PubMed]
47. J. J. David, Classical electrodynamics (Wiley Eastern Limited, 1975).
48. E. Neu, C. Arend, E. Gross, F. Guldner, C. Hepp, D. Steinmetz, E. Zscherpel, S. Ghodbane, H. Sternschulte, D. Steinmüller-Nethl, Y. Liang, A. Krueger, and C. Becher, “Narrowband fluorescent nanodiamonds produced from chemical vapor deposition films,” Appl. Phys. Lett. 98, 243107 (2011). [CrossRef]
49. I. Aharonovich, S. Castelletto, D. A. Simpson, J. C. McCallum, A. D. Greentree, and S. Prawer, “Two-level ultrabright single photon emission from diamond nanocrystals,” Nano Lett. 9, 3191–3195 (2009). [CrossRef] [PubMed]
50. F. Le Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, “Spontaneous emission of a cesium atom near a nanofiber: Efficient coupling of light to guided modes,” Phys. Rev. A 72, 032509 (2005). [CrossRef]
51. M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Fiber-based cryogenic and time-resolved spectroscopy of pbs quantum dots,” Opt. Express 19, 1786–1793 (2011). [CrossRef] [PubMed]
52. M. Naqshbandi, J. Canning, B. C. Gibson, M. M. Nash, and M. J. Crossley, “Room temperature self-assembly of mixed nanoparticles into photonic structures,” Nature Commun. 3, 1188 (2012). [CrossRef]
53. I. Friedler, C. Sauvan, J.-P. Hugonin, P. Lalanne, J. Claudon, and J. Gèrard, “Solid-state single photon sources: the nanowire antenna,” Opt. Express 17, 2095–2110 (2009). [CrossRef] [PubMed]
54. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
55. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410 (1988). [CrossRef] [PubMed]
56. F. L. Kien, N. H. Quang, and K. Hakuta, “Spontaneous emission from an atom inside a dielectric sphere,” Opt. Commun. 178, 151–164 (2000). [CrossRef]
57. F. A. Inam, M. D. Grogan, M. Rollings, T. Gaebel, J. M. Say, C. Bradac, T. A. Birks, W. J. Wadsworth, S. Castelletto, J. R. Rabeau, and M. J. Steel, “Emission and nonradiative decay of nanodiamond nv centers in a low refractive index environment,” ACS Nano 7, 3833–3843 (2013). [CrossRef] [PubMed]
58. A. Mohtashami and A. F. Koenderink, “Suitability of nanodiamond nitrogen–vacancy centers for spontaneous emission control experiments,” New J. Phys. 15, 043017 (2013). [CrossRef]
59. H.-Q. Zhao, M. Fujiwara, M. Okano, and S. Takeuchi, “Observation of 1.2-ghz linewidth of zero-phonon-line in photoluminescence spectra of nitrogen vacancy centers in nanodiamonds using a fabry-perot interferometer,” Opt. Express 21, 29679–29686 (2013). [CrossRef]
60. A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond,” Phys. Rev. Lett. 109, 033604 (2012). [CrossRef] [PubMed]
61. M. Fujiwara, T. Noda, A. Tanaka, K. Toubaru, H.-Q. Zhao, and S. Takeuchi, “Coupling of ultrathin tapered fibers with high-q microsphere resonators at cryogenic temperatures and observation of phase-shift transition from undercoupling to overcoupling,” Opt. Express 20, 19545–19553 (2012). [CrossRef] [PubMed]
62. H. Takashima, T. Asai, K. Toubaru, M. Fujiwara, K. Sasaki, and S. Takeuchi, “Fiber-microsphere system at cryogenic temperatures toward cavity qed using diamond nv centers,” Opt. Express 18, 15169–15173 (2010). [CrossRef] [PubMed]
63. A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008). [CrossRef]
64. A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nature Phys. 5, 509–514 (2009). [CrossRef]
65. The exact distance in vacuum that corresponds to those in nanodiamond should be written in d/ndiamond. These values can be easily interpolated from the calculated results in Fig. 7(c).
66. M. E. Reimer, G. Bulgarini, N. Akopian, M. Hocevar, M. B. Bavinck, M. A. Verheijen, E. P. Bakkers, L. P. Kouwenhoven, and V. Zwiller, “Bright single-photon sources in bottom-up tailored nanowires,” Nat. Commun. 3, 737 (2012). [CrossRef] [PubMed]
67. H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. 89, 257403 (2002). [CrossRef] [PubMed]
