Atom-light interactions are one of the most intriguing research fields in modern physics. The interactions between atoms with a light field significantly modify the atomic dynamics and lead to many interesting phenomena, such as spontaneous decay-induced coherences [1–4]. The situation becomes even more exciting in the case of interacting atomic ensembles. The pioneering work by Dicke [5] shows that dipole–dipole interactions can induce energy exchange among different atoms and generate superradiance, leading to essential deviations from the single-atom dynamics. Besides the subradiance and superradiance phenomena [6–11], the dipole–dipole interaction can cause the modification of the frequency of the emitted light in spontaneous emission and resonance fluorescence [12–15]. In recent years, dipole–dipole interacting systems are demonstrated to be suitable candidates for the implementation of gate operations between qubits in the context of quantum computation and quantum information [16–20].

Besides the atom–light interaction in a vacuum environment, the interaction in solids has also been investigated [21–23]. The seminal paper given by K Huang and A Rhys [24] studied the emission and absorption of photons in F-Centers. They pointed out that the processes of photon emission and absorption are modified by lattice vibrations, where phonons can be emitted or absorbed.

In this paper, we investigate the dipole–dipole interaction between two defects in a solid. These two solid defects are modeled as two identical two-level atoms. Due to the solid environment experienced by these defects, the defects are assumed to interact with a local phonon mode with discrete energy spectrum. The Hamiltonian for the total system of electrons and lattice is given bywhere H_e is the Hamiltonian for electrons, H_L for the lattice and H_eL for the interaction between phonons and electrons. As is widely used in solid state physics, the adiabatic approximation is adopted. The adiabatic approximation wave function ${\mathrm{\Psi }}_{en}(x,Q)={\varphi }_e(x;Q){\chi }_{en}(Q)$ satisfies the following equations [25]: where φ_e(x; Q) stands for electronic wave function. Here, Q denotes collective lattice coordinates and specifies the lattice configuration, and χ_en(Q) denotes the lattice wave function with the electrons in the state designated by subscript e. With the assumptions of harmonic approximation of the lattice and linear interaction between electrons and phonons, Huang shows the lattice wave function can be written as a product of harmonic oscillator wave functions with displaced origin and the transitions between excited and ground states of the atom is characterized by a key parameter S relating to the lattice relaxation energy [24].When ignoring the relaxation processes of the phonons and focusing on a local phonon mode, the combined defects–phonon systems are regarded as two multilevel atoms, which is shown schematically in figure 1(a). When the two multilevel atoms are so close that their separation is shorter than or comparable to the relevant transition wavelength, the dipole–dipole interaction will influence the dynamics of the multilevel atoms significantly. With the master equation method, the dipole–dipole interacting electron–phonon systems are analyzed. The results show the subradiance and superradiance phenomena also exist in this model, while the population transfer process demonstrates quite different behaviors from those of the simple two-level systems.

In this model, the two defects are assumed to be fixed in the solid. The defect A is located at the origin, with defect B displaced from A by $\mathit{R}=R(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$, where θ and φ are the polar and azimuthal angles of the displacement vector, respectively. The separation distance R between the two defects is assumed to be much smaller than the relevant atomic transition wavelength λ, in which case the dynamics of the two coupled systems are more interesting.

The two defects interact with the vacuum radiation field, which mediates the electric dipole–dipole interaction between them via an exchange of virtual photons. The Hamiltonian of the two coupled atom–phonon systems plus radiation field iswhere H_R = ${\sum }_{\mathit{k},s}{\omega }_k{a}_{\mathit{k}s}^{†}{a}_{\mathit{k}s}$ is the free Hamiltonian of radiation field in 3-dimensional space with $a_{\mathit{k}s}$ and $a_{\mathit{k}s}^{†}$ being the annihilation and creation operator of photons with wave vector $\mathit{k}$ and polarization s, and $V={\sum }_{i=1}^2{V}_i$ describes the electric dipole interaction between the radiation field and each defect. The Hamiltonian $H_{A+L}$ for the defect–phonon system readswhere $|g,m{⟩}_{\mu }$ and $|e,n{⟩}_{\mu }$ is a tensor product of the wave function of the electrons and phonons with eigenenergy $E_{gm}^{\mu }$ and $E_{en}^{\mu }$, respectively. Here, M_g and M_e specify the numbers of phonons in ground and excited states. In the electric dipole approximation, the interaction between radiation field and the multilevel atoms is written aswhere μ = 1, 2 denotes different multilevel atoms. Due to the lattice relaxation, the matrix elements of the electric dipole between the energy levels $|e,n⟩$ and $|g,m⟩$ in the Condon approximation [25, 26] is given bywhere $\mathit{d}$ is the matrix element of the electric dipole moment for the two-level atom with no electron–phonon interactions, S is the dimensionless lattice relaxation parameter indicating the coupling strength between the defects and the lattice, and ${\mathrm{L}}_n^{m-n}$ is the associated Laguerre polynomials.

The electric field operator is [27]where ω_k and ${ϵ}_{\mathit{k}s}$ (s = 1, 2) are the frequency and polarization vector of the quantized electric field with wave vector $\mathit{k}$, v is the quantization volume, ε₀ is the vacuum dielectric constant, and ε_r is the relative dielectric constant of the solid. Here, we take ε_r ≈ 1 in the following subsections.

The dynamics of the two coupled atoms are mediated by the vacuum electromagnetic field and can be well described by the master equation, where the degrees of freedom of the radiation are traced out leaving an effective dipole–dipole interaction between the two multilevel atoms. With the Born–Makov and rotating wave approximations, the master equation for the reduced density operator ρ(t) of the multilevel-atom systems is given by [28]where

The master equation is written in the rotating frame defined by the optical frequency ω₀, which corresponds to the energy level difference between $|e,0⟩$ and $|g,0⟩$, and the relevant unitary transformation operator ishere, we have used a single index i or j to denote the associated transitions between the energy levels $|g,m⟩$ and $|e,n⟩$, as shown in figure 1(b) with the corresponding transition matrix elements ${\mathit{d}}_i$ or ${\mathit{d}}_j$ determined by equation (7). H_I is the free Hamiltonian of the electron–phonon system, where ${\omega }_p$ is the frequency of the phonon and m, n represents phonon numbers in the ground state g and excited state e, respectively. The radiation-mediated electric dipole coupling between the two atoms is described by H_Ω with coherent coupling strength [29]with $\overleftrightarrow{\chi }$ a complex tensor whose cartesian components have the formwhere $x=k_{0}R$ (with $k_0={\omega }_0/c$). Equation (12) describes the spontaneous emission of the two atoms due to their coupling to the radiation field. The spontaneous emission rates γ_ij are given byFinally, equation (13) accounts for the modification of spontaneous emission of one atom due to the presence of the other atom, where the parameter ${\mathrm{\Gamma }}_{ij}^{\mu \upsilon }$ is written aswhere $\mathrm{I}\mathrm{m}\overleftrightarrow{\chi }(\mathit{R})$ is the imaginary part of the coupling tensor. We shall omit the superscript μ, &ugr; in the subsections, since two identical systems are considered.

In this section, we analyze the dynamics of the two coupled electron–phonon systems. In the case where the coupling strength Ω_ij is on the order of the phonon frequency, the evolution of the system is more complicated and interesting as well. By solving the master equation using numerical computation, we find a new feature in the population transfer process: a slow decaying oscillation due to the participation of phonons in the coupling process, which is significantly different from that of the two coupled two-level atoms in vacuum (see figure 3). The system configuration and calculation parameters are shown in figure 2.

The coherent coupling strength and cross-damping are essential for the dynamics of the two coupled systems. They are determined by the real part and imaginary part of the complex tensor $\overleftrightarrow{\chi }(\mathit{R})$ and the parameter S in our model as well, due to the modification of the elements of electric dipole moment as is shown is equation (7). Furthermore, the dipole–dipole interaction is anisotropic [30]. So  Ω_ij and Γ_ij depend on the alignment of the dipole moments and their orientations to the separation vector $\mathit{R}$.

In in paper, we consider a simple case where the dipoles of the two atoms are all in the $\hat{x}$ direction and the separation vector in the $\hat{z}$ direction. In this configuration, Ω_ij and Γ_ij are expressed as where γ₀ is the spontaneous emission rate associated with the bare two-level systems defined byand p(x) and q(x) are functions of the dimensionless quantities xThe functions F_ij(S) can be evaluated from equations (7) and (15) for different transitions i, j (see figure 1(b)). Take the $|e,0⟩\leftrightarrow |g,0⟩$ transition for example,which can be evaluated from equation (7) with m = n = 0. Figure 3 shows the dependence of Ω₂₂ and Γ₂₂ on the dimensionless parameter x in this configuration. In the region of small separation distance the coherent coupling strength Ω₂₂ and cross-damping rate Γ₂₂ are large enough such that they are comparable to the spontaneous radiation rate of the single atom. This indicates strong coupling between the two systems and may change the quantum dynamics significantly. While in the long distance limit, Ω₂₂ and Γ₂₂ approach zero, implying the two individual atoms interacting with the vacuum filed, respectively. In this case, collective effects are negligible.

The coupled electron–phonon systems in our model are essentially two multilevel atoms with infinite energy levels. To make proper truncation of the energy levels in order to reveal important physical results, relevant parameters are chosen such that the multilevel atom is reduced to four-level atoms and the coherent coupling between the two multilevel atoms are strong enough to generate large population transfer between the two systems. The dynamical evolution is then given by calculating the master equation numerically.

Figure 4(a) demonstrates the time evolution of the population $|e0,g0⟩$ of the multilevel systems with initial state $\rho (0)=|e0,g0⟩⟨e0,g0|$. A slow decaying oscillation appears in the time evolution of $|e0,g0⟩$, besides a rapid oscillation with much larger decay rate. This is quite different from the two coupled two-level atoms in the vacuum. The time evolution of the population P_eg is [31]:where Γ_s and Γ_a correspond to the spontaneous emission rate of the superradiance and subradiance state, respectively, and Ω is the coherent coupling strength between these two-level atoms. Equation (24) and figure 4(b) show there is only one oscillation frequency 2Ω, which is nothing but the energy difference between the superradiance and subradiance state.

The oscillating characteristic of $P_{e0g0}$ can be traced back to the coherent evolution part $-\mathrm{i}/\hslash [H_I+H_{\mathrm{\Omega }},\rho ]$, since ${\mathcal{L}}_{\gamma }\rho$ and ${\mathcal{L}}_{\mathrm{\Gamma }}\rho$ only give rise to decay of the total population of the two coupled systems. This indicates the diagonalization of the effective Hamiltonian $H_T=H_I+H_{\mathrm{\Omega }}$ may account for this slow oscillation behavior.

The state space of the reduced total system is $\mathcal{E}={\mathcal{E}}_A⨂{\mathcal{E}}_B$, where ${\mathcal{E}}_i$ is the state space of the i subsystem. A natural basis of $\mathcal{E}$ is $\{|\alpha ,i{⟩}_1⨂|\beta ,j{⟩}_2\}$ with α, β = e, g and i,j = 0, 1. The states $\{|gi,gj⟩,|ei,ej⟩\}$ are already the eigenstates of H_T, so the diagonalization of H_T can be realized in the subspacewith matrix representationThe form of H_T can be greatly simplified by introducing a basis consisting of symmetry and antisymmetry states defined within the degenerate energy levels:In the new basis, H_T has the formwhereandEquations (28)–(30) show the energy degeneracy is partially eliminated and the energy splittings are determined by Ω_ii. More importantly, the symmetry-states subspace are totally decoupled from the antisymmetry-states subspace. Therefore, the dynamical evolution in each subspace can be analyzed separately. Similar to the case of simple two-level systems where the symmetry state has a much larger decay rate than that of the antisymmetry state, figure 4(c) shows the same results for the multilevel atoms. This indicates the coupling between antisymmetry states may account for the slow oscillation characteristic in figure 4(a).

The coupling among the antisymmetry states is depicted in figure 5 according to equation (30). The energy level difference between $|a_1⟩$ and $|a_4⟩$ is much larger the corresponding coupling strength Ω₁₄, so the population transfer can be ignored between them. Furthermore, the coupling strengths {Ω_ij} have the following relations:This indicates that we can make further symmetrization and antisymmetrization in the degenerate subspace $\{|a_2⟩,|a_3⟩\}$. This case is particularly similar to that of electromagnetically induced transparency, where one superposition state is totally decoupled to other states. In the new basis:The sub-block matrix in the lower right corner of equation (28) now has the form:The state $|\overline{S}⟩$ is decoupled from other states, as is demonstrated in figure 5(b). Since the energy level differences are much larger than relevant coupling strengths, the energy of $E_{\overline{S}},E_{A_1},E_{A_4},E_{\overline{A}}$ can be evaluated by nondegenerate perturbation theory [32]:We find the energy level difference between $|A_4⟩$ and $\overline{S}$ is $\delta =E_{A_4}-E_{\overline{S}}\approx 36\times {10}^6\mathrm{r}\mathrm{a}\mathrm{d}$/s which is approximately equal to that of the slow decaying oscillation in figure 4(a). As is shown in equation (32), the states $|\overline{A}⟩$ and $|A_4⟩$ involve one phonon and two phonons, respectively. These states come into the coupling process through the coherent coupling strength Ω₁₂, Ω₁₃ and Ω₁₄. This indicates that phonons can take part in the population transfer process and make the dynamics of the coupled systems much more complicated.

The couplings among symmetry states are shown in figure 6 and give the rapid decaying oscillation in figure 4(a). The analysis is similar to the discussion above.

In this paper, we investigate the dynamics of two coupled defect centers in a solid. With adiabatic and harmonic approximation, the electron–phonon systems are regarded as two multilevel atoms. The master equation derived with the Born–Markov and the rotating wave approximations describes the time evolution of the two coupled systems under a single excitation condition. The numerical results demonstrate that superradiance and subradiance phenomena exist in the coupled electron–phonon systems, where the superradiant states decay faster than those of the subradiant states. When the coherent coupling strengths are strong enough, the population transfer between adjacent defects are remarkable, and the slow decaying oscillation component appears due to the participation of phonons in the solid, which distinguishes it from the case of two simple two-level atoms.