Effect of phonon-bath dimensionality on the spectral tuning of single-photon emitters in the Purcell regime

We develop a theoretical frame to investigate the spectral dependence of the brightness of a single-photon source made of a solid-state nanoemitter embedded in a high-quality factor microcavity. This study encompasses the cases of localized excitons embedded in a one-, two-, or three-dimensional matrix. The population evolution is calculated based on a spin-boson model, using the noninteracting blip approximation. We find that the spectral dependence of the single-photon source brightness (hereafter called spectral efficiency) can be expressed analytically through the free-space emission and absorption spectra of the emitter, the vacuum Rabi splitting, and the loss rates of the system. In other words, the free-space spectrum of the emitter encodes all the relevant information on the interaction between the exciton and the phonon bath to obtain the dynamics of the cavity-coupled system. We compute numerically the spectral efficiency for several types of localized emitters differing by the phonon bath dimensionality. In particular, in low-dimensional systems where this interaction is enhanced, a pronounced asymmetric energy exchange between the emitter and the cavity on the phonon sidebands yields a considerable extension of the tuning range of the source through phonon-assisted cavity feeding, possibly surpassing that of a purely resonant system.

Figure 1

Localized emitter coupled to a 1D (a), 2D (b), or 3D (c) phonon bath embedded in a cavity.

Figure 2

(a) Time evolution of the modulus of the phonon kernel function $\left|K\right(t\left)\right|$ for different bath dimensionalities. The temperature is set to 5 K. The parameters are $\alpha =0.29$, $\hslash {\omega }_c=8.9\mathrm{meV}$ in 1D [9], $\alpha =0.2$, $\hslash {\omega }_c=5\mathrm{meV}$ in 2D, and $\alpha =0.17$, $\hslash {\omega }_c=1.09\mathrm{meV}$ in 3D [22] (see the main text for details). Parts (b), (c), and (d) are the corresponding emission spectra in 1D, 2D, and 3D calculated with Eq. (6), and using $\gamma +{\gamma }^{*}=200\mu \mathrm{eV}$ in 1D, $\gamma +{\gamma }^{*}=100\mu \mathrm{eV}$ in 2D, and $\gamma +{\gamma }^{*}=50\mu \mathrm{eV}$ in 3D. The absorption spectra can be deduced from the emission spectra through a mirror symmetry with respect to the ZPL position (set to 0 eV in these plots). The dashed lines in panels (c) and (d) correspond to the emission spectra in the absence of phonon coupling.

Figure 3

(a) Schematic output spectra for different detuning. The envelope (red dashed line) is proportional to the single-photon spectral efficiency $\beta \left(\omega \right)$. Panels (b), (c), and (d) are the single-photon spectral efficiency as a function of the cavity detuning for different bath dimensionalities at $5\mathrm{K}$. Panels (b), (c), and (d) correspond, respectively, to the 1D, 2D, and 3D cases presented in Fig. 2. In each panel, the five plots correspond to a coupling strength $\hslash g=$ 50 (purple line), 100 (cyan line), 150 (blue line), 200 (green line), and $250\mu \mathrm{eV}$ (red line). The horizontal dashed black line is the resonant limit given by $\kappa /(\kappa +\gamma )$ (see the main text). The gray dashed line is the asymptotic behavior at large coupling $g$. We use a cavity lifetime of $1/\kappa =15\mathrm{ps}$ for the 1D [19] and 2D cases, and $6\mathrm{ps}$ in 3D [22]. The lifetime $1/\gamma$ is $70\mathrm{ps}$ in 1D [19], $1\mathrm{ns}$ in 2D [14, 15, 16, 17], and $1.3\mathrm{ns}$ in 3D [22]. In panel (b) (the 1D case), the gray area corresponds to the regime where $0.1({\delta }^2+{\gamma }_{\text{all}}^2)\le g^2\le {\delta }^2+{\gamma }_{\text{all}}^2$ (see the main text).

Figure 4

Evolution of the single-photon source spectral efficiency as a function of the detuning for several temperatures between 5 and 100 K in the 1D (a) and 2D (b) cases calculated for $g=100\mu \mathrm{eV}$.

Figure 5

Sketch of the system as a set of two coupled boxes representing the emitter and the cavity, respectively. The exchanges between these two boxes are governed by the coupling rate $g$ and the free-space absorption/emission probability ${\tilde{S}}_{{\omega }_{\text{cav}}}^{\text{abs}}$/${\tilde{S}}_{{\omega }_{\text{cav}}}^{\text{emi}}$, taken at the cavity frequency. The emitter can also decay toward the environment at a rate $\gamma$, while a photon in the cavity can leak out at a rate $\kappa$.

Figure 6

Time decay of the exciton population $\langle {\widehat{\sigma }}^{+}{\widehat{\sigma }}^{-}\rangle$ [panels (a), (c), and (e) for the 1D, 2D, and 3D cases, respectively] and related cavity population $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ [panels (b), (d), and (f), respectively] for ${\langle {\widehat{\sigma }}^{+}{\widehat{\sigma }}^{-}\rangle }_{t=0}=1$ and for several detunings (expressed in units of $\kappa$). The black curves correspond to a numerical resolution of Eqs. (8) and (9). The red dashed lines are obtained in the Markovian approximation from Eq. (12). The coupling is set to $g=250\mu \mathrm{eV}$ for the 1D case and to $g=50\mu \mathrm{eV}$ for the 2D and 3D cases. Note that the smallest detuning considered in the 1D case corresponds to the maximum efficiency (see Fig. 3). Beyond this value, one retrieves a behavior similar to that of larger detunings (i.e., as a function of $\delta$, the dynamics is symmetrical with respect to the point yielding the highest $\beta$).