Large and well-defined Rabi splitting in a semiconductor nanogap cavity

Mitsuharu Uemoto; Hiroshi Ajiki

doi:10.1364/OE.22.022470

1. Introduction

In the past decade, characteristic features of the cavity quantum electrodynamics (QED), such as the vacuum Rabi splitting [1–4] and the photon blockade effect [4–6], have been observed for a single quantum dot (QD) embedded inside a microcavity. Various applications of such a solid-state cavity QED have been proposed and/or demonstrated: single quantum dot lasers [7], highly efficient entangled-photon generation [8, 9], and quantum information processing mediated by photons [10].

In cavity QED, the interaction between a cavity photon and the material excitation can be classified into two coupling regimes: strong coupling and weak coupling. The coupling frequency g is given by

g = μ \sqrt{\frac{\bar{h} ω}{2 ε V_{m}}},

where μ is the transition dipole moment of material excitation, ω is the excitation frequency of the material, ε is the dielectric constant, and V_m is the volume of the photon-confinement region (cavity-mode volume) [11]. In a simplified representation of coupled harmonic oscillators [12], the criteria to classify these coupling regimes are described using the relationship between the Rabi frequency, Ω_R = 2g, and the effective linewidth γ = (γ_cav + γ₀)/4 of the coupled states, where γ_cav and γ₀ are the spectral widths of the cavity and of the material excitation, respectively. In the weak coupling regime (h̄Ω_R < γ), the radiative decay time of the material excitation is considerably shortened (the Purcell effect) [13]. In the strong coupling regime (h̄Ω_R > γ), the light-matter interaction results in two coupled states exhibiting an anticrossing behavior of the energy levels as a function of the detuning, also called Rabi splitting [11, 14, 15], and their energy difference (h̄Ω_R) at zero detuning is called Rabi splitting energy. The Rabi splitting energy can be increased by reducing V_m, as can be seen from Eq. (1). In order to clearly observe the Rabi splitting, the spectral width of the coupled states should be narrowed by increasing the cavity quality factor (Q factor).

In previous experiments, a micropillar cavity [1], a photonic crystal cavity [2], and a microdisk cavity [3] have been utilized to observe the vacuum Rabi splitting for a single QD (see Table 1). In conventional optical cavities, V_m is restricted to the volume specified by the wavelength of the impinging light (the diffraction limit), and the resulting Rabi splitting energies are on the sub-meV range. However, metal nanogap structures such as plasmonic nanoantennas or plasmonic bow-tie resonators, enable reduction of V_m beyond the diffraction limit and generation of a significantly enhanced and localized field at the gap region called a “hot spot” [17]. Motivated by this fact, Savasta et al. theoretically studied the possibility of observing the Rabi splitting in a zero-dimensional system, constituted by a single quantum emitter placed inside the gap between two metallic nanospheres [14]. Schlather et al. recently observed a giant Rabi splitting, ranging between 230 and 400 meV, for spin-casted molecular excitons of J-aggregates in a metal nanogap structure [16]. However, the spectral widths from those structures are larger than the Rabi splitting energy because of the large plasmon decay. The previous experimental results of the Rabi splitting are summarized in Table 1.

Table 1. Parameters of previously reported zero-dimensional cavity-emitter systems, and those of the proposed semiconductor-nanogap system (“Emitter, SC nanogap”). “MTL” and “SC” stand for metal and semiconductor, respectively. The dipole moment of the J-aggregates is estimated from the metal nanogap structure, by assuming a volume of 15 nm³.

View Table

In this paper, we propose and analyze a new type of optical cavity: a nanogap structure composed of semiconductor nanoparticles. This semiconductor nanogap cavity can generate a strongly enhanced field at the nanogap, and it presents a small cavity-mode volume beyond the diffraction limit, similar to a metal nanogap cavity. The field enhancement is attributed to the exciton. Since the spectral width of the confined exciton is much smaller than that of the surface plasmon, the cavity Q factor is significantly enhanced. We theoretically demonstrate the presence of a large and well-defined vacuum Rabi splitting of a two-level emitter placed inside the semiconductor nanogap cavity.

2. Calculation method

EM-field simulations are usually performed using established methods, such as the finite-difference time-domain (FDTD) method, the finite-element method (FEM), and the boundary element method. These methods, however, cannot be directly applied to exciton-active semiconductor nanostructures. The exciton has a strong dispersion relation between the exciton energy and wave vector in contrast to the plasmon in metal, where the plasmon energy is nearly independent of its wave vector. The strong dispersion relation leads to multimode propagation as an exciton polariton inside the semiconductor nanostructure. The dispersion relation of the exciton polariton in a bulk semiconductor is given by: k² = (ω/c)²ε(k, ω). From this equation we can obtain four complex solutions for k, two of which (k_t₁ and k_t₂) have a positive imaginary component, describing physically meaningful wave vectors. These solutions correspond to two exciton polariton modes with transverse character whose electric fields are denoted by E^(t₁) and E^(t₂). A longitudinal exciton also exists, whose dispersion is obtained from ε(k_l, ω) = 0, where k_l is the wave vector of the longitudinal exciton, and its electric field is denoted by E^(l) = −∇⃗Φ, where Φ is a scalar potential. Although the longitudinal exciton cannot be excited in an isotropic bulk semiconductor, longitudinal-transverse mixed exciton modes are formed in the nanostructure, and they can strongly couple with light [18]. Thus, the longitudinal component cannot be ignored. EM-field simulations for exciton-polariton resonances were performed using the FDTD method in [19], where the longitudinal component of the exciton was ignored. A longitudinal field provides an instantaneous Coulomb interaction, and thus, the sequential analysis with respect to time domain in the FDTD method is not suitable for taking account of the longitudinal component of the exciton.

As mentioned above, three excitation modes [E^(t₁), E^(t₂), and E^(l)] are necessary in light scattering problems of semiconductor nanostructures. Because of the multimode excitations, we need a boundary condition [additional boundary condition (ABC)] with respect to the exciton polarization in addition to the Maxwell’s boundary conditions. Here, we use the Pekar-type ABC [20] for which the exciton polarization P(r) vanishes at the surface of the semiconductor. Since the polarization P(r) is given by P(r) = χ(k, ω)E(r) with χ(k, ω) = (1/4π)[ε(k, ω) − ε_bg] being the susceptibility, the Pekar-type ABC at the surface is expressed as

\sum_{μ = 1}^{2} χ (k_{t_{μ}}, ω) E^{(t_{μ})} + χ (k_{l}, ω) (- \vec{\nabla} Φ) = 0 .

We have formulated an FEM to solve the Maxwell’s wave equations under the Maxwell’s boundary conditions and ABC [21]. We apply this method to calculate EM fields for the semiconductor nanogap cavity and its coupling system with the two-level emitter. A detailed procedure of this simulation method can be found in [21].

3. Quality factor

We first evaluate the Q factor of a semiconductor nanogap cavity illustrated in Fig. 1(a). The cavity consists of two semiconductor nanospheres of radius 7 nm separated by 8 nm. We consider CuCl nanospheres and Z₃ excitons, having an exciton band-edge energy of E_G = 3.202 eV, a longitudinal-transverse splitting energy of Δ_LT = 5.7 meV, an exciton translation mass of M = 2.3m_e (where m_e is the free electron mass), and a background dielectric constant of ε_bg = 5.56 [22]. The dielectric function of the CuCl, including the exciton effect, can be expressed as:

ε (k, ω) = ε_{bg} + \frac{ε_{bg} Δ_{LT}}{E_{G} + ({\bar{h}}^{2} / 2 M) k^{2} - \bar{h} ω - i \bar{h} γ_{ex}},

where k is the wave vector, ω is the frequency, and h̄γ_ex is the exciton damping energy. The CuCl exciton has a large oscillator strength, which relates to Δ_LT, and thus, strong enhancement is expected. The exciton damping energy is set to be h̄γ_ex = 100 μeV at the temperature T = 40 K [23]. The surrounding is assumed to be vacuum, i.e., the dielectric constant is 1. The incident electromagnetic (EM) field, E_inc, is taken as a plane wave with linear polarization, parallel to the line connecting the origins of the two spheres (see Fig. 1).

Fig. 1 Schematic illustrations of (a) the empty semiconductor nanogap cavity and (b) the nanogap cavity with a spherical two-level emitter (a QD or a cluster of molecules) placed at r₀. The polarization direction of an incident electromagnetic field E_inc is also depicted.

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Figure 2(a) shows the near-field enhancement |E/E_inc|² at the center of the semiconductor nanogap structure (r₀) as a function of the incident field frequency. At the peak energy of E_cav = 3.2086 eV, the near field is enhanced by a factor of 10² due to the exciton resonance of the CuCl nanospheres, and hence, the semiconductor nanogap structure is considered as an optical cavity. The spectral full width at half maximum (FWHM) is evaluated as 227 μeV. The corresponding Q factor of Q ∼ 1.3 × 10⁴ is much larger than the upper bound of the plasmonic cavity (Q ∼ 10²) [24–26]. Figure 3(a) shows the calculated near-field enhancement distribution of a semiconductor nanogap structure for an incident field of energy h̄ω = 3.2086 eV. This large near-field enhancement appears both at the nanogap region and at the center of semiconductor nanospheres.

Fig. 2 Calculated near-field enhancement |E/E_inc|² at r₀ (see Fig. 1) for (a) the empty semiconductor nanogap cavity and (b) the semiconductor nanogap cavity with the two-level emitter under zero detuning condition. The transition dipole moment of the emitter is μ = 10 Debye.

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Fig. 3 The near-field enhancement distributions for the empty semiconductor nanogap cavity in resonant condition (h̄ω = 3.2086 eV) (a), the semiconductor nanogap cavity with the two-level emitter at h̄ω = 3.2083 eV (b), h̄ω = 3.2086 eV (c), and h̄ω = 3.2088 eV (d). The energies in (b) and (d) correspond, respectively, to the lower and the higher peak energies in Fig. 2(b).

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4. Rabi splitting

Next, we consider the situation where a spherical two-level emitter (e.g. a QD or a cluster of aggregated dye molecules) with a radius of 2 nm is placed at r₀, as shown in Fig. 1(b). The dielectric function of the emitter can be described by a Lorentz oscillator model:

ε (ω) = ε_{bg}^{em} - \frac{2 E_{0} μ^{2} / V_{em}}{{(\bar{h} ω)}^{2} - E_{0}^{2} + i (\bar{h} ω) (\bar{h} γ_{0})},

where

ε_{bg}^{em}

is the background dielectric constant, E₀ is the resonant excitation energy, V_em is the volume, and h̄γ₀ is the damping energy. The spectral peak energy of the two-level emitter, denoted by E_ex, is slightly blue-shifted from E₀ due to the exciton-light interaction.

Figure. 2(b) shows the near-field enhancement |E/E_inc|² of the cavity-emitter system at r₀ under zero detuning condition (E_ex = E_cav). The transition dipole moment of the emitter is set to be μ = 10 Debye. The spectrum exhibits two distinct peaks, indicating the resonant excitations of the bonding and anti-bonding coupled states of the cavity photon and the two-level emitter. It is worth noting that the spectral profile is highly asymmetric, because the empty nanogap cavity has a Fano line profile, as shown in Fig. 2(a). The right-hand skirt of the peak rapidly drops to zero, and consequently, the higher peak is strongly reduced. This Fano line profile originates from the interference between the resonant scattering field, due to size-quantized exciton in the CuCl spheres, and a spectrally broad scattering field, due to background dielectric or to the incident field [27]. Figure 3 shows the distributions of |E/E_inc|² for an incident field of: (b) h̄ω = 3.2083 eV (lower peak energy), (c) h̄ω = 3.2086 eV (energy between two peaks), and (d) h̄ω = 3.2088 eV (higher peak energy).

Figure 4(a) shows the near-field spectra of the cavity-emitter system at r₀ for various detunings (E_ex − E_cav). Because the temperature dependences of E_ex and E_cav are different [1–3], the detuning can be controlled with the temperature. In the calculations, the detunings are set only by changing E_ex. Figure 4(b) summarizes the spectral peak energies of Fig. 4(a), as a function of the detuning. We find a clear anti-crossing behavior, characteristic of the strong-coupling regime. The coupled states of the cavity photon and the exciton can be described by a Jaynes-Cummings model, whose Hamiltonian is given by:

\hat{ℋ} = E_{cav} {\hat{a}}^{†} \hat{a} + \frac{E_{ex}}{2} {\hat{σ}}_{z} + \frac{\bar{h} Ω_{R}}{2} ({\hat{a}}^{†} {\hat{σ}}_{-} + \hat{a} {\hat{σ}}_{+}),

where â^† and â are the creation and destruction operators for the cavity photon, respectively, and σ̂_z and σ̂_± are the pseudospin operators for the two-level emitter. The energies of the coupled states are calculated as

E = \frac{1}{2} (E_{cav} + E_{ex}) \pm \frac{1}{2} \sqrt{{(E_{cav} - E_{ex})}^{2} + {(\bar{h} Ω_{R})}^{2}} .

The Rabi splitting energy h̄Ω_R can be estimated by fitting the calculated energy curves using Eq. (6). The energy curves, indicated by solid lines in Fig. 4(b), are well fitted using h̄Ω_R = 580 μeV. It is noteworthy that this Rabi splitting energy is very large considering the small transition dipole moment of 10 Debye.

Fig. 4 (a) The near-field spectra of the semiconductor nanogap cavity with two-level emitter at r₀. From the bottom to the top, the detuning changed from negative (E_ex < E_cav) to positive (E_ex > E_cav) values. (b) The peak energies in (a) are plotted with triangles as a function of the detuning. The solid curves are obtained from Eq. (6) with h̄Ω_R = 580 μeV.

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Figure 5 shows h̄Ω_R as a function of the transition dipole moment μ. The cavity-mode volume is evaluated as V_m = 1.5 × 10⁻² μm³ from Eq. (1), a value smaller than that of conventional cavities (see Table 1). This small cavity-mode volume beyond the diffraction limit originates from the resonant excitation of an exciton. For a metal nanogap cavity, giant Rabi splitting ranging between 230 and 400 meV has been reported [16], in which the cavity-mode volume is estimated to be ∼ 10⁻⁵ μm³, much smaller than that of the semiconductor nanogap cavity. This can be related to the appreciable EM-field intensity appearing inside the semiconductor nanostructures, as shown in Fig. 3(a). However, the EM field cannot penetrate inside metallic nanostructures, so the photon confinement region is limited to only the gap region. As a consequence, the Rabi splitting energy for a metal nanogap cavity is much larger than for a semiconductor nanogap cavity. Unfortunately, the large losses due to the plasma oscillation prevents the observation of a clear Rabi splitting. The decay constant of the metal nanogap cavity is h̄γ_cav > 100 meV >> h̄γ₀, and h̄Ω_R is at most comparable to h̄γ_cav. In contrast, the semiconductor nanogap cavity with two-level emitter has h̄γ_cav ≈ 200 μeV, h̄γ₀ ≈ 100 μeV, and a small cavity-mode volume beyond the diffraction limit. Therefore, large and well-defined Rabi splitting can be seen even for a small dipole moment.

Fig. 5 The Rabi splitting energy h̄Ω_R as a function of the transition dipole moment μ of the emitter. The red points are obtained by fitting the calculated energy curves using Eq. (6), and the dashed line indicates h̄Ω_R = 64.6 (μeV/Debye)×μ. R denotes the radius of CuCl spherical QDs, with the transition dipole moment indicated by the vertical dotted lines.

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5. Conclusions

We have proposed a new optical cavity composed of semiconductor nanoparticles, called semiconductor nanogap cavity. A strong field enhancement occurs at the nanogap region because of the resonant excitation of excitons in the semiconductor nanoparticles. To evaluate the cavity-mode volume, which determines the magnitude of interaction between a cavity photon and the material excitation, the vacuum-Rabi splitting of a two-level emitter placed inside the nanogap has been calculated. The resulting cavity-mode volume of 1.5 × 10⁻² μm³ is very small and goes beyond the diffraction limit. The cavity Q factor is determined by the exciton damping energy, and reaches Q ∼ 1.3 × 10⁴ for a CuCl nanogap structure at a temperature 40 K. Although very high-Q factors have been achieved in conventional microcavities (such as in micropillars, photonic crystals, or in microdisk cavities), the cavity-mode volume is restricted by the diffraction limit. Metal nanogap structures, instead, can act as an optical cavity due to the presence of surface plasmons, and they can exhibit a much smaller cavity-mode volumes, but their Q factor is very low because of the large plasmon damping. In contrast, the semiconductor nanogap cavity has both (i) a smaller cavity-mode volume than the conventional microcavities and (ii) a larger Q factor than a metal nanogap cavity. As a result, this cavity provides a large and well-defined Rabi splitting. Recently, fast and highly efficient entangled-photon generation has been predicted for such a QD-cavity system having large Rabi splitting energy and high-Q factor [28]. In this way, the semiconductor nanogap cavity is suitable for development of new applications as well as for fundamental studies of cavity QED.

Acknowledgments

The authors thank Dr. H. Ishihara and Dr. H. Yoshida for fruitful discussions and comments. This work was supported by ”Core to Core” program of the Japan Society for Promotion of Science (JSPS), Advanced Low Carbon Technology Research and Development Program (ALCA) of the Japan Science and Technology Agency (JST), and the Grant-in-Aid for Scientific Research (C), No. 25400325 of the Japan Society for Promotion of Science (JSPS).

References and links

1. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004). [CrossRef] [PubMed]

2. T. Yoshie, A. Scherer, and J. Hendrickson, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 9–12 (2004). [CrossRef]

3. E. Peter, P. Senellart, D. Martrou, a. Lemaître, J. Hours, J. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. 95, 067401 (2005). [CrossRef] [PubMed]

4. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot–cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

5. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]

6. A. Faraon, A. Majumdar, and J. Vučković, “Generation of nonclassical states of light via photon blockade in optical nanocavities,” Phys. Rev. A 81, 033838 (2010). [CrossRef]

7. O. Benson and Y. Yamamoto, “Master-equation model of a single-quantum-dot microsphere laser,” Phys. Rev. A 59, 4756 (1999). [CrossRef]

8. H. Ajiki and H. Ishihara, “Entangled-photon generation from a quantum dot in cavity QED,” Phys. Stat. Solidi C 6, 276–279 (2009). [CrossRef]

9. H. Ajiki, H. Ishihara, and K. Edamatsu, “Cavity-assisted generation of entangled photons from a V-type three-level system,” New J. Phys. 11, 033033 (2009). [CrossRef]

10. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2010). [CrossRef]

11. G. Khitrova, H. Gibbs, M. Kira, S. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nat. Phys. 2, 81–90 (2006). [CrossRef]

12. S. Rudin and T. Reinecke, “Oscillator model for vacuum Rabi splitting in microcavities,” Phys. Rev. B 59, 10227 (1999). [CrossRef]

13. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

14. S. Savasta, R. Saija, A. Ridolfo, and O. D. Stefano, “Nanopolaritons: vacuum Rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano 4, 6369–6376 (2010). [CrossRef] [PubMed]

15. The complex eigenfrequency, whose imaginary part represents the spectral width, of the coupled states are given by $Ω_{\pm} = ω_{0} - i (γ_{cav} + γ_{0}) / 4 \pm \sqrt{g^{2} - {(γ_{cav} - γ_{0})}^{2} / 16}$ at the zero detuning where the cavity-mode frequency agrees with that of material excitation. Although two real parts of Ω_± appear for g > (γ_cav − γ₀)/4, the splitting energy should be larger than the spectral width, i.e., g > (γ_cav + γ₀)/4 for observing an evident energy splitting [14].

16. A. E. Schlather, N. Large, A. S. Urban, P. Nordlander, and N. J. Halas, “Near-field mediated plexcitonic coupling and giant Rabi splitting in individual metallic dimers,” Nano Lett. 13, 3281–3286 (2013). [CrossRef] [PubMed]

17. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011). [CrossRef]

18. H. Ajiki and K. Cho, “Longitudinal and transverse components of excitons in a spherical quantum dot,” Phys. Rev. B 62, 7402–7412 (2000). [CrossRef]

19. Y. Zeng, Y. Fu, M. Bengtsson, X. Chen, W. Lu, and H. Ågren, “Finite-difference time-domain simulations of exciton-polariton resonances in quantum-dot arrays,” Opt. Express 16, 4507–4519 (2008). [CrossRef] [PubMed]

20. S. I. Pekar, “The theory of electromagnetic waves in a crystal in which excitons are produced,” Sov. Phys. JETP 6, 785–796 (1958).

21. M. Uemoto and H. Ajiki, “Simulation method for resonant light scattering of exciton confined to arbitrary geometry,” Opt. Express 22, 9450–9464 (2014). [CrossRef] [PubMed]

22. K. Cho and M. Kawata, “Theoretical analysis of polariton interference in a thin platelet of CuCl. I-Additional boundary condition,” J. Phys. Soc. Jpn. 54, 4431–4443 (1985). [CrossRef]

23. M. Kalm and C. Uihlein, “Investigations on the temperature-dependent TPA linewidth of the Z3-exciton in CuCl,” Phys. Stat. Solidi B 87, 575 (1978). [CrossRef]

24. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]

25. D. K. Gramotnev, A. Pors, M. Willatzen, and S. I. Bozhevolnyi, “Gap-plasmon nanoantennas and bowtie resonators,” Phys. Rev. B 85, 045434 (2012). [CrossRef]

26. A. Raman, W. Shin, and S. Fan, “Upper bound on the modal material loss rate in plasmonic and metamaterial systems,” Phys. Rev. Lett. 110, 183901 (2013). [CrossRef] [PubMed]

27. H. Ajiki, T. Tsuji, K. Kawano, and K. Cho, “Optical spectra and exciton-light coupled modes of a spherical semiconductor nanocrystal,” Phys. Rev. B 66, 245322 (2002). [CrossRef]

28. K. Shibata and H. Ajiki, “Entangled-photon generation from a quantum dot in a microcavity through pulsed laser irradiation,” Phys. Rev. A 89, 042319 (2014). [CrossRef]

	Rabi Splitting (meV)	FWHM (meV)	Mode Volume (μm³)	Dipole Moment (Debye)
InGaAs QD Micropillar [1]	0.16	0.09	0.30	58
InAs QD Photonic crystal [2]	0.17	0.08	0.04	29
GaAs QD Microdisk [3]	0.40	0.25	0.07	87
J-Aggregates MTL nanogap [16]	230	300	3.3×10⁻⁵	380
Emitter SC nanogap	1.7	0.14	0.015	25

Large and well-defined Rabi splitting in a semiconductor nanogap cavity

Abstract

1. Introduction

2. Calculation method

3. Quality factor

4. Rabi splitting

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (6)

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