Quantum optical master equation for solid-state quantum emitters

We provide an elementary description of the dynamics of defect centers in crystals in terms of a quantum optical master equation which includes spontaneous decay and a simplified vibronic interaction with lattice phonons. We present the general solution of the dynamical equation by means of the eigensystem of the Liouville operator and exemplify the usage of this damping basis to calculate the dynamics of the electronic and vibrational degrees of freedom and to provide an analysis of the spectra of scattered light. The dynamics and spectral features are discussed with respect to the applicability for color centers, especially for negatively charged nitrogen-vacancy centers in diamond.

Figure 1

(a) The Coulomb interaction of the ground-state electronic orbital (shaded gray) of a defect center lets the surrounding atoms take on a certain lattice position. (b) The excited electronic orbital pushes the adjacent crystal atoms into a displaced equilibrium configuration. (c) Model of the Franck-Condon-like lattice-defect center interaction in single mode and harmonic approximation including damping: In the excited state $|e\rangle$, the wave packet of the single vibrational mode experiences a potential displaced by $x_0$ compared to the potential in the ground state $|g\rangle$. The interaction causes a renormalized transition frequency between the states $|e\rangle$ and $|g\rangle$, shifted by the relaxation frequency ${\omega }_R$. The single vibrational mode is coupled to a temperature bath modeling the phononic modes of the crystal and thermally relaxes with rate $\gamma$. The electronic two-level system decays with rate $\Gamma$ due to spontaneous emission.

Figure 2

(a) The time evolution of the excited-state population (dashed line) and the modulus of the coherences (solid lines) for the initial atomic state $\left(\right|g\rangle +|e\rangle )/\sqrt{2}$ in units of the oscillator period $\tau =2\pi /\nu$ when the oscillator is initially in thermal equilibrium. The parameters are $\gamma =0.2\nu$, $\Gamma =0.1\nu$, and $\eta =1\nu$. The black curves show the case $\overline{m}=0.05$, while the gray one corresponds to $\overline{m}=1$. (b) The density plots show the Wigner function of the oscillator state at the times indicated by the circles in (a). The dashed blue ring represents the variance and location of the $|g\rangle \langle g|$ projection of the oscillator state in phase space, whereas the solid red ring analogously represents the $|e\rangle \langle e|$ projection which follows a trajectory along the spiral around $\beta$. The excited-state population dies off and the system ultimately ends up in the stationary state $|g\rangle \langle g|{\mu }_{\mathrm{th}}$.

Figure 3

The absorption spectrum of the defect center under weak excitation, according to Eq. (82), for the parameters $\gamma =0.2\nu$, $\Gamma =0.01\nu$, $\eta =1.5\nu$, and $\overline{m}=0.05$. The gray dotted curves show the main contributions of the eigenvalue decomposition (82) of the spectrum, namely, the terms corresponding to $n=0$ and $l=-5,...,1$. ZPL denotes the position of the zero-phonon line shifted by ${\tilde{\omega }}_R$ from the bare transition frequency of the electronic states.

Figure 4

(a) The temperature dependence of the absorption spectrum for $\overline{m}=0$ (solid line), $\overline{m}=0.25$ (dashed line), and $\overline{m}=1$ (dotted line). (b) Dependence of the absorption on the coupling strength for $\eta =1.5\nu$ (solid line), $\eta =1.2\nu$ (dashed line), and $\eta =0.9\nu$ (dotted line). The vertical lines are drawn at the corresponding zero-phonon lines, i.e., at the renormalized atomic frequency $\tilde{\omega }$. The other parameters are the same as in Fig. 3.