1. Introduction

Coupling quantum emitters to a common reservoir may result in an effective dipole-dipole interaction (DDI) between them. DDIs mediated by photons, either in the range of microwaves for coupling superconducting qubits [1] or in the visible range for quantum dots [2, 3], molecules, or nitrogen vacancy centers in diamond [4], are a key element in fundamental research on light-matter interactions [5–7] and quantum information processing [8, 9]. For example, entanglement generation between quantum emitters due to DDIs has been proved in a number of environments, including left-handed materials [10], plasmonic waveguides [11], surface plasmons supported by nanowire [12, 13] or graphene [14], etc., even if emitters were initially prepared in an unentangled state. It opens the possibility of manipulating entanglement between qubits via waveguide/cavity QED setups. However, up to now only the dynamics of entanglement between two qubits induced by DDIs has been investigated in literature [10–14] while tens of entangled qubits are usually required for advanced applications such as quantum cryptography, quantum teleportation, and quantum computing [15–18]. Therefore, extending the discussion to more qubits and developing corresponding waveguide/cavity QED system is an intriguing and promising task in both quantum optics and nanophotonics.

In this paper, we study the dynamics of the single-exciton states of three and four quantum emitters coupled via DDIs and how nanophotonic structures can be exploited to control the DDIs between them. With a judicious design, long-range and efficient DDIs can be facilitated in waveguides as well as cavities, which make the three- or four-qubit W state subradiant and all the other states superradiant. In this way, all the other states decay to a negligible amount in a relatively short time and high-purity W state is left in the system. Our work proves that efficient generation and control of long-range DDIs between multiple emitters can be realized by designing proper nanophotonic structures, which will find applications in modern quantum communication and computing systems.

2. Cooperative behavior of three and four quantum emitters

The quantum systems of interest are shown in Fig. 1. Three or four identical quantum emitters such as quantum dots [19] or rare-earth ions [20] are embedded in a nanophotonic structure, as depicted in Figs. 1(a) and 1(b). Each emitter has two levels and serves as a qubit by denoting ground and excited states as$|0〉$and$|1〉$, respectively. Assuming that only one of the emitters is initially excited, all the states of interest and transitions between them are illustrated in Figs. 1(c) and 1(d).

 

Fig. 1 Coupling of (a) three and (b) four quantum emitters through the vacuum field. Each emitter has two energy levels and can be viewed as a qubit. Scheme of the quantum states involved and transitions between them (red dashed line with double arrows) in the case of (c) three and (d) four emitters.

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The dynamics of the density matrix ρ for three or four qubits is described by a master equation [11, 12, 21]

The requirement of identical DDIs between quantum emitters can be met by using highly symmetric configurations. As depicted in Figs. 2(a) and 2(b), three emitters are located at the vertices of an equilateral triangle while their dipole orientations are either along the medians of the triangle [Fig. 2(a)] or perpendicular to the plane of the triangle [Fig. 2(b)]. In the latter case, when the distance between emitters decreases to zero,${\gamma }_i={\gamma }_0$and Eq. (5) becomes the description of Dicke superradiance [24]. The configuration of four emitters is unique as illustrated in Fig. 2(c). Four emitters are located at the vertices of a regular tetrahedron with their dipole orientations along the medians of the tetrahedron. In the following section, we will put these configurations into various nanophotonic structures to manipulate the DDIs between qubits and the collective spontaneous emissions of entangled states. But before we proceed, we would like to study the scenario of free space first to get an intuitive view of DDI. The dyadic Green’s function in free space or homogeneous environments has an analytical expression [25]

 

Fig. 2 Configurations of three quantum emitters with their dipole orientations (a) in and (b) out of the plane of equilateral triangle. (c) Configuration of four quantum emitters with equal DDIs. (d) Noncoherent mutual interaction strength in free space (normalized to individual spontaneous emission rate) as a function of dipole-dipole distance (normalized to free space wavelength). The red, blue and green solid lines correspond to the configurations in Figs. 2(a), 2(b) and 2(c), respectively.

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3. Parallel plate waveguide for three-dipole interaction

We propose a parallel plate waveguide (PPW) consisting of an air gap inside perfect electric conductor (PEC) or silver, as illustrated in Fig. 3(a). Three quantum emitters are placed in the middle of the gap. For the three-dipole configuration in Fig. 2(a), they couple to the TE mode supported by PPW which exhibits epsilon-near-zero (ENZ) behavior at the cut-off wavelength. The zero propagation constant in PPW eliminates the oscillation effect in γ_i caused by the term e^ikr in Eq. (7) while the confinement in the z direction slows down the speed at which the envelope of γ_i approaches zero. In this manner, strong DDI can be achieved in a range comparable to λ₀. Hereafter, we use finite-difference time domain (FDTD) techniques to calculate the noncoherent mutual interaction term γ_i in various nanophotonic structures. The red solid line in Fig. 3(b) shows the DDI in a PPW composed of PEC with the air gap width d = λ₀/2. Obviously, the PPW works in the ENZ regime where γ_i has no oscillation and a long interaction distance. The ENZ nature can also be acquired in a PPW composed of silver at visible or near-infrared frequencies. Due to the penetration depth into silver, the ENZ wavelength is longer than 2d, e. g., λ₀ = 760 nm for d = 340 nm. The blue dashed and green dotted lines in Fig. 3(b) show the DDIs in silver PPWs operating in the ENZ and propagating regime, respectively. As a result of silver losses, γ_i decays faster in the ENZ waveguide than its PEC counterpart. However, γ_i/γ₀ = −0.4 can still be obtained at r = λ₀ in the ENZ waveguide, much better than in the propagating waveguide where the DDI is further weakened by the oscillation. In Fig. 3(c), we plot the phase progression in the ENZ waveguide made of silver. As expected, only a small phase advance is observed over several wavelengths. For the three-dipole configuration in Fig. 2(b), the quantum emitters couple to the TM mode supported by PPW which exhibits no cut-off no matter how narrow the air gap is. The DDIs between z-oriented dipoles are displayed in Fig. 3(d) using the same waveguide parameters as in Fig. 3(b). Strong oscillations emerge as a consequence of the propagation effect while slight variations in phase exist between different waveguides. In summary, PPW can be used to extend the interaction distance between the embedded emitters when operating in the ENZ regime. Unfortunately, the scheme only works for the three-dipole configuration drawn in Fig. 2(a).

 

Fig. 3 (a) Schematic of three dipoles placing in a PPW composed of PEC or silver. The width of the air gap is denoted as d. The cross section of the system in the YOZ plane is shown in the inset. (b) Noncoherent mutual interaction strength in PPW (normalized to individual spontaneous emission rate) as a function of dipole-dipole distance (normalized to free space wavelength). The dipole configuration is illustrated in Fig. 2(a). The red solid, blue dashed and green dotted lines represent the PEC ENZ, silver ENZ and silver propagating operation, respectively. (c) Phase progression of the in-plane electric field in the XOY plane of a silver PPW operating at its ENZ wavelength. The orientation of the dipole is indicated in the figure. (d) Counterpart of Fig. 3(b) with z-oriented dipoles. The configuration is shown in Fig. 2(b). All the waveguide parameters are the same as in Fig. 3(b).

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4. SOI microring resonator for three-dipole interaction

In free space or PPW, the electromagnetic field radiated by a quantum emitter will disperse to infinity eventually. The field intensity has to decrease with distance to preserve the total energy, which results in the weakening of DDI at a long distance. An effective way to mediate DDI while maintaining its strength is to use low-loss waveguides [11–13] or high-Q cavities [27–29]. Here, we propose a system of three quantum emitters embedded in a SOI microring resonator, which is schematically depicted in Fig. 4(a). The equilateral triangle connecting three dipoles is inscribed in the ring. The radius of the ring R is much bigger than its width w so that we can neglect the difference between inner and outer radii and treat the ring as a bent waveguide with the same width w and height h. Assuming that the transition frequency of quantum emitter is in resonance with the microring resonator, the tangential wavenumber of ring mode k satisfies$k\cdot 2\pi R=n\cdot 2\pi$, where n can be any integer. The noncoherent mutual interaction between dipoles γ_i now depends on the interference between clockwise- and counterclockwise-propagating waves and reads [11]

 

Fig. 4 (a) Schematic of three dipoles embedded in a SOI microring resonator. The ring radius, width and height are R, w and h, respectively. Mode profile in (b) the microring designed for the configuration in Fig. 2(a) and (c) the counterpart microring designed for the configuration in Fig. 2(b). The dipole orientations are indicated in the figures. (d) Spectrum of the noncoherent mutual interaction strength γ_i/γ₀ (red dashed line) and Purcell factor (blue solid line) in the SOI microring resonator shown in Fig. 4(b). (e) Counterpart of Fig. 4(d) but in the SOI microring resonator shown in Fig. 4(c).

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For the configuration in Fig. 2(a), the parameters of microring are designed as follows: R = 5 μm, w = 500 nm and h = 250 nm. The electric field profile in the corresponding bent waveguide is shown in Fig. 4(b). The horizontal component is dominant in this mode, which leads to good couplings with the transition dipoles of quantum emitters. The DDI and Purcell factor are displayed in Fig. 4(d) around λ₀ = 1.5 μm. The peaks in the Purcell factor appear at 1.48, 1.50 and 1.52 μm, which correspond to the microring resonances with n = 52, 51 and 50, respectively. As predicted, γ_i/γ₀ reaches −0.486, 0.981 and −0.475 at 1.48, 1.50 and 1.52 μm. The discrepancy from the ideal value of −0.5 and 1 happens because the β factor is less than unity and the material and bend losses are finite in the system. For the configuration in Fig. 2(b), a similar device can be used to facilitate long-range DDIs. The parameters of microring are modified as: R = 5.625 μm, w = 250 nm and h = 500 nm. As shown in Fig. 4(c), the mode profile derived in the corresponding bent waveguide resembles that in Fig. 4(b) except for a rotation of 90 degrees. So now the vertical component becomes principal and couples well with the z-oriented dipoles. Moreover, the microring width w is reduced by half compared to our previous design, making it more vulnerable to bends. This feature is reflected in the DDI and Purcell factor drawn in Fig. 4(e). The Purcell factor peaks at 1.50, 1.52 and 1.54 μm, corresponding to the microring resonances with n = 58, 57 and 56. Their values are much smaller than those in Fig. 4(d), which is attributed to the higher bending loss and hence lower Q factor of the cavity. It also weakens DDI as γ_i/γ₀ now equals to −0.432, 0.957 and −0.436 at the resonance wavelengths and the peaks and dips are much wider than those in Fig. 4(d). However, strong DDI can still be observed between emitters separated by several wavelengths. In essence, SOI microring resonator is an excellent platform for long-range and nearly undamped DDIs between three dipoles in both configurations.

5. Silicon microshell/silica core structure for four-dipole interaction

The idea of using SOI microring to mediate long-range DDI can be extended to higher dimensions, giving rise to the silicon microshell/silica core architecture plotted in Fig. 5(a). The configuration of four quantum emitters is arranged as in Fig. 2(c), with the regular tetrahedron inscribed in the shell. Since the emitters are embedded in the shell layer which has a higher refractive index than the core, the energy radiated by emitters would be confined in silicon, rendering strong DDIs between distant emitters. The parameters of the shell/core structure are designed as follows: the radius of the core sphere R = 1 μm and the thickness of the shell t = 250 nm. The DDI and Purcell factor in the structure are displayed in Fig. 5(b) around λ₀ = 1.56 μm. Three resonant modes are found at 1.564, 1.566 and 1.569 μm where the Purcell factor peaks and γ_i/γ₀ reaches its extrema of −0.076, 0.937 and −0.285. It is also evident that a higher Purcell factor is always related to a stronger DDI, i. e., γ_i/γ₀ is closer to −0.5 or 1and the peak or dip is narrower. It can be explained as follows: disregarding the minor variation of mode volume, the high Purcell factor results from the low loss and thus high Q factor of the mode, which will also lead to the nearly non-fading DDI in a long distance. As a result, the silicon microshell/silica core structure is verified to be a promising candidate for mediating and manipulating DDIs between four quantum emitters.

 

Fig. 5 (a) Schematic of four dipoles embedded in the silicon microshell layer on the silica core. The core sphere radius is R, and the thickness of shell layer is t. The dipole orientations are indicated in the figure. (b) Spectrum of the noncoherent mutual interaction strength γ_i/γ₀ (red dashed line) and Purcell factor (blue solid line) in the silicon microshell/silica core structure. The configuration of four dipoles is shown in Fig. 2(c).

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6. Demonstration of W state generation

To prove the feasibility of controlling DDIs with nanophotonic structures for W state generation, we choose the three-qubit system in Fig. 4(d) and suppose that the transition frequency of emitters matches the resonant frequency of n = 52 eigen mode of the SOI microring resonator. Also, we assume that initially only one of the emitters is excited, e. g., ${\rho }_{11}=1,{\rho }_{00}={\rho }_{22}={\rho }_{33}=0$(${\rho }_{00}=0,{\rho }_{AA}=\frac{1}{2},{\rho }_{BB}=\frac{1}{6},{\rho }_{WW}=\frac{1}{3}$). Given the strong DDIs in this case, i. e., γ_i/γ₀ = −0.486, the time evolution of the state populations${\rho }_{WW}$and${\rho }_{AA}+{\rho }_{BB}$as well as the purity of W state$P={\rho }_{WW}/({\rho }_{WW}+{\rho }_{AA}+{\rho }_{BB})$is plotted in Fig. 6(a). Despite that${\rho }_{AA}+{\rho }_{BB}$is the double of${\rho }_{WW}$at the initial time, it decays very fast to less than 0.01 within a time of$3{\gamma }_0^{-1}$while${\rho }_{WW}$drops only by 8.05% in that time, resulting in a high purity of W state of 97.54%. The purity can rise up to 99.86% in a time of$5{\gamma }_0^{-1}$when${\rho }_{AA}+{\rho }_{BB}$drops to 0.0004 and${\rho }_{WW}$is still 0.29. In Fig. 6(b), we display the purity of W state P as a function of DDI γ_i/γ₀. The purity drops from 97.83% (99.89%) for γ_i/γ₀ = −0.5 to 94.82% (99.51%) for γ_i/γ₀ = −0.4 at t = 3γ₀⁻¹ (5γ₀⁻¹), indicating that relatively high purity can still be achieved with moderate DDIs at the cost of longer evolution time. More efficient W state generation is envisioned as novel nanophotonic devices are being developed to achieve extremely strong DDIs between multiple qubits. Another factor that influences the W state generation is the presence of nonradiative transition in emitters. It can be accounted for by adding the corresponding interaction term into the Hamiltonian. As a result, the term γ₀ in Eqs. (4)-(6) are replaced by γ₀ + γ_nr, where γ_nr is the rate of non-radiative transitions [30]. Since$-{\gamma }_0/2\le {\gamma }_i\le {\gamma }_0$ ($-{\gamma }_0/3\le {\gamma }_i\le {\gamma }_0$) for three (four) qubits, the zero decay rate of W state is no longer available in theory. Stronger DDIs are required to compensate the effect of nonradiative transitions in the W state generation. It is also worth mentioning that our scheme of W state generation can be extended to more than four emitters as long as the conditions γ_jj = γ_kk = γ₀, γ_jk = γ_kj = γ_i and g_jk = g_kj = g are satisfied, which might be implemented in judiciously designed asymmetric nanophotonic structures.

 

Fig. 6 (a) Time evolution of the state populations ρ_WW (red solid line) and ρ_AA + ρ_BB (blue dashed line) as well as the purity of W state P (green dashed line) . The initial state is$|001〉$and γ_i/γ₀ = −0.486. (b) Purity of W state P as a function of noncoherent mutual interaction strength γ_i/γ₀ at t = 3γ₀⁻¹ (red solid line) and 5γ₀⁻¹ (blue dashed line).

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7. Conclusion

In conclusion, we have studied the dynamics of three and four quantum emitters coupled via DDIs. Nanophotonic architectures such as PPW, SOI microring resonator and silicon microshell/silica core structure have also been developed to mediate long-range and strong DDIs between emitters. Numerical results show that desired γ_i/γ₀ can be achieved on a distance scale of several free space wavelengths. Thanks to that, high-purity W state will be generated in a short time on the order of the lifetime of single emitter after we excite only one of the emitters initially. The proposed systems can serve as an ideal platform to engineer long-range DDIs between multiple quantum emitters, showing promising potential for basic research of light-matter interactions and/or quantum information processing.