Tuning the coupling between quantum dot and microdisk with photonic crystal nanobeam cavity

Yanhui Zhao; Li-Heng Chen; Xue-Hua Wang

doi:10.1364/OE.27.020211

1. Introduction

To realize quantum information processing [1], manipulating quantum states and transferring quantum states between quantum nodes are required [2]. Cavity QED scheme [3, 4] provides a solution for coherent optical manipulation of quantum states, which benefits from the hybridization of photon and matter quantum states in the strong coupling regime [5]. Furthermore, the photon-mediated interactions make it possible to couple distant quantum emitters together [6, 7].

To date, various types of microcavities are employed to realize the strong coupling between quantum emitters and photons, such as micro-pillar [8], microdisk [9–11], photonic crystal cavities [12–15] and 1D photonic crystal nanobeam cavities [16]. Among them, microdisk supports whispering gallery modes (WGMs) which have large quality factors (Q factors) [17]. Especially, the whispering gallery like field distribution of the resonant mode provides advantages for integrating multiple quantum emitters together, which can be utilized to generate entanglement between separate quantum emitters [18–20]. Photonic crystal nanobeam cavity has also been proposed to enhance the light-matter interactions [21]. Such type of cavity possesses ultra-high Q factor [22] and particularly ultra-small mode volume [23, 24]. Based on the ultrahigh Q factor-to-mode volume ratio in such cavities, strong coupling between the nanobeam cavity and a single quantum dot has been demonstrated [16]. Based on the natural scalability, integrating one nanobeam cavity to waveguide [25] and coupling separate nanobeam cavities [26] have been proposed to realize different functionalities.

To realize the strong coupling between cavity modes and quantum emitters, much effort has been devoted to tune them into resonance both spectrally [27–32] and spatially [33–37], as well as to fabricate microcavities with high Q factor and small mode volume [5, 8, 12, 14]. Among various platforms for implementing Cavity QED, semiconductor self-assembled QDs are good candidates because it is convenient to integrate them with microcavities. However, there are still great challenges to realize strong coupling between photons and QDs because of the random nucleation locations of individual QDs [33] and the difficulties of controlling the emission wavelength of QDs.

In this work, we propose a scheme with coupling nanobeam cavity to microdisk on SiO₂ substrate to modulate the local density of photonic states (LDOS) in the microdisk. As the nanobeam cavity is adjusted to be resonant with the microdisk, coupling between them can be realized through evanescent field when they are placed adjacently. Thus, the LDOS in the microdisk can be modulated by adjusting the distance between the two cavities, which is calculated by FDTD method. Then, the quantum dynamics of a single QD embedded in the microdisk is studied based on the evolution spectrum which can be obtained from the LDOS.

2. Coupled system with nanobeam cavity and microdisk

We designed the structure with a microdisk coupled to a nanobeam cavity on the SiO₂ substrate. We referred to Ref. [22] to design the nanobeam cavity structure. A five-hole tapered one-dimensional photonic crystal mirror is used to form high-Q cavity. In this paper, however, some parameters are changed in order to make the cavity resonant wavelength around 1.3 μm which is near the emission wavelength of InGaAs QD [38,39]. The width of the nanobeam is 0.425 μm and the thickness is 0.187 μm. The center of the nanobeam cavity is at the origin of coordinate and the structure is symmetric about the Y axis (see details in Fig. 1(a)). A microdisk with radius 1.4745 μm and thickness 0.187 μm is located near the center of the nanobeam cavity. The center of the microdisk is denoted as (x_c, y_c) with x_c = 0 and adjustive y_c.

Fig. 1 (a) The coupled GaAs nanobeam cavity and microdisk structure on the substrate of SiO₂. The thickness of the coupled structure is 0.187 μm. The length and width of the nanobeam is 15 μm and 0.425 μm, respectively. The cavity are formed by one-dimensional photonic crystal mirror. The photonic mirror pitch a = 0.3655μm is linearly tapered over a five hole section to a = 0.2805μm at the cavity center. The hole radius is r = 0.28a. The radius of the microdisk is 1.4745 μm. The center of the microdisk is denoted as (x_c, y_c). The electric field densities of the cavity modes of nanobeam (b) and microcavity (c), respectively. (d) The LDOS of the two cavity modes around the resonant wavelength 1.29225 μm. The quality factors of the nanobeam cavity and the microdisk are 5.9 × 10⁴ and 6.1 × 10⁵, respectively.

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We firstly calculated the cavity modes and the LDOS of the individual nanobeam cavity and microdisk using the FDTD method (Lumerical Solutions, Inc.). Both the resonant wavelengths of the two cavities are designed to be at 1.29225 μm. The Q factors of nanobeam cavity and microdisk are 5.9 × 10⁴ and 6.1 × 10⁵, respectively. The electric field densities of the two cavities are shown in Figs. 1(b) and 1(c), respectively. For microdisk, the maximum of the electric field is at the middle of the slab and 1.25 μm away from the center of the disk cylinder. In this paper, all the dipole sources with E_y polarization are fixed at this field maximum position. Generated by a point source oriented along Y direction at point r₀ with dipole moment μ, the dyadic Green’s function G_yy is [40],

G_{y y} (r, r_{0}) = \frac{E_{y} (r) c^{2} ε_{0}}{ω^{2} μ},

where ε₀ is the permittivity of vacuum. Then the LDOS can be obtained with the Green’s function,

ρ (r_{0}, ω) = \frac{2 ω}{π c^{2}} Im [G_{y y} (r_{0}, r_{0}; ω)] .

Using the Green’s function method, the LDOS of the two individual cavities are calculated and shown in Fig. 1(d).

When the nanobeam cavity is located adjacently to the microdisk, their cavity modes may be coupled through evanescent field. Thus, the LDOS of the microdisk may be alternated. And the resonant wavelength and the resonant linewidth can be changed correspondingly. We changed the microdisk center y_c from −2.3 μm to −1.8 μm with 0.1 μm step and calculated the LDOS with the y-polarized dipole source located 1.25 μm away from the center of the microdisk and in the middle of the microdisk slab. The LDOS of the coupled system are shown in Fig. 2. At y_c = −2.3 μm, there are two close peaks in the LDOS spectrum, which indicate the strong coupling between the nanobeam cavity and the microdisk. As the two cavity modes are coupled strongly, symmetric and anti-symmetric supermodes are formed [41,42]. When the microdisk is placed closer to the nanobeam cavity, the field overlap between the individual cavity modes increases, resulting in stronger coupling strength. Consequently, the mode splitting becomes larger as y_c increases. It indicates that the LDOS can be controlled through adjusting the distance between nanobeam cavity and microdisk, which plays a vital role on tailoring the light-matter interactions.

Fig. 2 The LDOS of the coupled nanobeam cavity and microdisk structure with y_c changed from −1.8 to −2.3 μm. The dipole is located 1.25 μm away from the center of the microdisk and at the middle of the microdisk slab.

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3. A single QD coupled to the photonic nano-structure

We now consider the interactions between a single QD and the photonic reservoir of nano-structures [43]. Taking the QD as a two-level atom and using the rotating-wave approximation, the Hamiltonian of the coupled system reads (ħ = 1),

H = H_{0} + V,

H_{0} = ω_{0} σ^{+} σ^{-} + \sum_{n k} ω_{n k} a_{n k}^{†} a_{n k},

V = \sum_{n k} [g_{n k} (r) a_{n k}^{†} σ^{-} + c . c .],

where σ⁺ and σ⁻ are the raising and lowering operators of the QD, respectively.

a_{n k}^{†}

and a_nk are the photonic creation and annihilation operators, respectively. ω₀ is the transition frequency of a bare QD. ω_nk is the frequency of the electromagnetic eigenmode. g_nk(r) denotes the coupling coefficient between QD and photons, which is expressed as [44],

g_{n k} (r) = i ω_{0} {(2 ∊_{0} ℏ ω_{n k})}^{- 1 / 2} E_{n k} (r) \cdot u_{d},

where ∊₀ is the vacuum permittivity and u_d is the transition dipole moment of the QD.

| ψ (t) 〉 = C_{e} (t) | a 〉 + \sum_{n k} C_{1, n k} | b_{n k} 〉 \equiv U (t) | a 〉 .

Here, U(t) is the evolution operator and can be deduced from the advanced and retarded Green’s function,

U (t) = \frac{1}{2 π i} \int_{- \infty}^{+ \infty} d ω [G^{-} (ω) - G^{+} (ω)] exp (- i ω t),

where G^±(ω) = lim_η→0⁺ G(z = ω ± iη) and the resolvent operator is G(z) = 1/(z − H).

We expand the resolvent operator G(z) in the Hilbert space of H₀ eigenstates |a〉, |b_nk〉,

(z - ω_{0}) G_{a a} (z) = 1 + \sum_{n k} V_{a b_{n k}} G_{b_{n k} a} (z),

(z - ω_{n k}) G_{b_{n k} a} (z) = V_{b_{n k} a} G_{a a} (z) .

Then G_aa(z) can be calculated as,

G_{a a} (z) = \frac{1}{z - ω_{0} - \sum_{n k} \frac{V_{a b_{n k}} V_{b_{n k} a}}{z - ω_{n k}}} = \frac{1}{z - ω_{0} - \sum_{n k} \frac{{| g_{n k} (r) |}^{2}}{z - ω_{n k}}} .

Using the identity ${lim}_{η \to 0^{+}} \frac{1}{x \pm i η} = P (\frac{1}{x}) \mp i π δ (x)$ , where P represents the integral principal value and δ(x) the delta function, we obtain,

G_{a a}^{\pm} (ω) = lim_{η \to 0^{+}} G_{a a} (z = ω \pm i η) = \frac{1}{ω - ω_{0} - Δ (r, ω) \pm i Γ (r, ω) / 2},

where

Γ (r, ω) = 2 π \sum_{n k} {| g_{n k} (r) |}^{2} δ (ω - ω_{n k}) = \frac{π ω_{0}^{2} d^{2}}{ℏ ∊_{0} ω} ρ (r, ω, \hat{d}),

Δ (r, ω) = \frac{P}{2 π} \int_{0}^{\infty} \frac{Γ (r, ω^{'})}{ω - ω^{'}} d ω^{'} .

Inserting Eq. (12) into Eq. (8), we have

U_{a a} (t) = \int_{- \infty}^{+ \infty} d ω C_{e} (r, ω) exp (- i ω t),

with the evolution spectrum expressed as,

C_{e} (r, ω) = \frac{1}{2 π i} [G_{a a}^{-} (ω) - G_{a a}^{+} (ω)] = \frac{1}{π} \frac{Γ (r, ω) / 2}{{[ω - ω_{0} - Δ (r, ω)]}^{2} + Γ {(r, ω)}^{2} / 4},

where Γ(r, ω) represents the local coupling strength, and Δ(r, ω) the level shift. ρ(r, ω, d̂) in Eq. (13) is the LDOS [45], which is defined as,

ρ (r, ω, \hat{d}) = \sum_{n k} {| E_{n k} (r) \cdot \hat{d} |}^{2} δ (ω - ω_{n k}) .

Therefore, using the LDOS of the nano-structure calculated by FDTD method, we can obtain the evolution spectrum of a single QD and the quantum dynamics of its excited state. A single InGaAs QD with transition dipole moment d = 1.12955 × 10⁻²⁸ C · m is located at the middle of the microdisk slab and 1.25 μm away from the center of the microdisk. The spontaneous emission rate of QD in vacuum is given by $Γ_{0} = ω_{0}^{3} d^{2} / (3 π ∊_{0} ℏ c^{3})$ , which corresponds to QD with life time of about 6ns in vacuum given the transition frequency of QD f₀ = 231.993 THz. We assume the experiment is carried out at low temperature and ignore the dephasing of QD for simplicity. The direction of the QD transition dipole moment is along the Y axis. Considering a single QD coupled to an individual microdisk, the LDOS is given in Fig. 1(d). The evolution spectrum of the QD can be obtained from Eq. (16) and is shown in Fig. 3(a). It can be seen that vacuum Rabi splitting appears in the evolution spectrum, indicating strong coupling between QD and photon. From the splitting, the coupling strength between QD and photons can be evaluated as g = 10.580 GHz. The time evolution of the excited state population |U_aa(t)|² can be calculated with Eq. (15), which is shown in Fig. 3(b). The population evolution exhibits Rabi oscillations which indicates coherent excitation exchange between photon and QD.

Fig. 3 (a) The evolution spectrum of the QD in the coupled nanobeam cavity and microdisk structure. The splitting indicates that the QD and photon enter into the strong coupling regime. (b) The population of the excited states of the QD. Rabi oscillation appears due to strong coupling.

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The evolution spectra of QD are given in Fig. 4 with transition frequency f₀ denoted by dot lines, which are calculated from the corresponding LDOS in Fig. 2. At y_c = −1.8 μm, both the strong coupling between QD and individual supermodes are observed when the QD transition frequency f₀ is tuned into resonance with one of the two supermodes. The detunings between the emission wavelengths of QDs and the bare microdisk mode (1.29225 μm) are +1.27 nm and −1.44 nm, respectively. Another peak far away from the Rabi splitting peaks exists in the evolution spectra. However, it does not affect the dynamics of the QD states evidently because it is several orders smaller than the Rabi splitting peaks.

Fig. 4 The evolution spectra with different y_c and transition frequencies of QD f₀ which are signified by dot lines.

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As y_c decreases, the mode splitting between the supermodes shrinks. The strong coupling between QD and individual supermodes are observed until y_c = −2.0 μm. However, at y_c = −2.1 μm the QD begins to couple the two supermodes simultaneously, because the three peaks in the evolution spectra become comparable in magnitude. Particularly, at y_c = −2.2 μm and with QD transition frequency f₀ = 231.999 THz, the QD couples to the two supermodes equally. The simultaneously coupling between two cavity modes and a single QD will give remarkable different quantum dynamics of the QD quantum states comparing to the normal Jaynes-Cummings model (JC model) [46]. It can be inferred that, we can modulate the splitting between the supermodes continuously by adjusting the distance between the two cavities. Therefore, the supermodes can be tuned into strong coupling with a QD with emission wavelength varied in several nm range.

Figure 5 gives the evolution spectra with sweeping the transition frequency of QD f₀ across the two supermodes of the coupled nanobeam cavity and microdisk at y_c = −1.8 μm and y_c = −2.2 μm, respectively. At y_c = −1.8 μm, the two photonic supermodes are far away from each other spectrally. As the transition frequency of QD f₀ sweep across the supermodes, the QD couples to one of the two supermodes without coupling to the other one. Two anti-crossing points appears in Fig. 5(a). However, at y_c = −2.2 μm, the two photonic supermodes are close to each other spectrally. When a single QD couples to one of the supermodes, the coupling to the other one can not be neglected. Therefore, one QD can simultaneously couple to two supermodes when the QD transition frequency are around the two close supermodes. Anti-crossing between distinct triplet modes around the resonance of the superomodes is observed in Fig. 5(b).

Fig. 5 The contour plot of the evolution spectra with sweeping the transition frequencies of QD across the splitting supermodes of the coupled nanobeam cavity and microdisk system at (a) y_c = −1.8 μm and (b) y_c = −2.2 μm, respectively.

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The populations of the excited state of QD, corresponding to the evolution spectra in Fig. 4, are calculated with Eq. (15), and are shown in Fig. 6. From y_c = −1.8 μm to y_c = −2.0 μm, normal Rabi oscillations are observed due to the strong coupling between a single QD and an individual supermodes. The Rabi oscillations come from the fact that the polarition states between QD and photon are formed. However, at y_c = −2.1 μm, the oscillation is not a typical decayed sine type in Fig. 6(g), because the QD couples not only to the lower frequency supermode but also to the higher frequency supermode partially. It can be seen from the evolution spectrum in Fig. 4 with y_c = −2.1 μm and f₀ = 231.980 THz that the third peak at higher frequency is comparable to the Rabi splitting peaks, which will affect the excited state of QD evidently.

Fig. 6 The population of the excited states of QD with different y_c and QD transition frequency f₀.

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We next analyze the quantum dynamics in the case with QD simultaneously coupled to two cavity supermodes. In Fig. 6(i), the population of the excited state of QD begins from 1 and goes down along with time. Counterintuitively, the population does not rise up fully but exhibits a very small oscillation subsequently. After that, the excited state of QD climbs up to a large population. This process appears periodically. Actually, the excitation transfers from the QD to the microdisk mode firstly. And then, when the excitation comes back, it mainly goes into the nanobeam cavity mode and partially into the excited state of QD. Following, the excitation returns back to the excited state of QD with both the microdisk and nanobeam cavity modes empty. Due to the strong coupling between QD and the two supermodes, eigen-states of the coupled system are formed by the superposition of the bases |e0_m0_n〉, |g1_m0_n〉 and |g0_m1_m〉, where e(g) signifies the excited (ground) state of QD, 1_m (1_n) indicates one photon in the microdisk (nanobeam cavity). Therefore, the excitation will oscillate between QD, microdisk and nanobeam cavity if the initial state was prepared in |e0_m0_n〉. This type of excitation oscillation provides a new opportunity for the optical coherent manipulation of quantum states, i.e. the quantum states of QD in the microdisk can be controlled by the nanobeam cavity which couples to waveguide naturally. At y_c = −2.3 μm, the population evolution begins to return to the normal Rabi oscillation. It is because that as the two supermodes are very close, the middle peak in the evolution spectrum is very small. We can imagine that the system will degrade to the normal JC model as the distance between the two cavities becomes larger.

4. Conclusion

In conclusion, we have proposed a scheme to tune the coupling between a single QD and a microdisk with 1D photonic crystal nanobeam cavity. When a nanobeam cavity is located adjacently to the microdisk, the coupling between the two cavity modes can be realized through evanescent field. By adjusting the distance between the two cavities, the LDOS in the microdisk can be modified which is vital for the light-matter interactions. Two splitting supermodes appears when the two cavity modes enters into the strong coupling regime. We observed strong coupling between a single QD and one of the supermodes when the splitting between the supermodes is large. By modulating the splitting between the two supermodes, strong coupling between QD and photon can be realized even when the QD is several nm spectrally detuned with the bare microdisk mode. Particularly, simultaneous coupling between QD and the two supermodes arises when the splitting between the two supermodes is small, which exhibits distinct quantum dynamics comparing to the normal Rabi oscillation. Such coupling between QD and two supermodes enriches the optical coherent control methods of quantum states. The coupled system can be used to control the interaction between QD and microdisk. And the nanobeam cavity can be coupled to waveguide naturally, which is beneficial for the realizing of quantum internet.

Funding

National Natural Science Foundation of China (Grants No. 11704423, No. 91750207); The Key R&D Program of Guangdong Province (Grant No. 2018B030329001).

Acknowledgments

This work was supported by the National Supercomputer Center in Guangzhou.

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Tuning the coupling between quantum dot and microdisk with photonic crystal nanobeam cavity

Abstract

1. Introduction

2. Coupled system with nanobeam cavity and microdisk

3. A single QD coupled to the photonic nano-structure

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (6)

Equations (17)

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