Beijing Computational Science Research Center,Beijing 100193,China

Abstract We investigate the quantum dynamics of two defect centers in solids,which are coupled by vacuum-induced dipole–dipole interactions.When the interaction between defects and phonons is taken into account,the two coupled electron–phonon systems make up two equivalent multilevel atoms.By making Born–Markov and rotating wave approximations,we derive a master equation describing the dynamics of the coupled multilevel atoms.The results indicate the concepts of subradiant and superradiant states can be applied to these systems and the population transfer process presents different behaviors from those of the two dipolar-coupled two-level atoms due to the participation of phonons.

Keywords:dipole–dipole interaction,electron–phonon interaction,superradiance

1.Introduction

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 by

Figure 1.(a)The energy levels of the coupled electron–phonon systems.∣e,j〉represents the electron is in the excited state with j phonons and∣g,i〉represents the electron is in the ground state with i phonons.(b)Transition labels.The indices are designated according to energy level difference(from low to high).

where φe(x; Q)stands for electronic wave function.Here,Q denotes collective lattice coordinates and specifies the lattice configuration,and χen(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.

2.Method

2.1.Hamiltonian of the system plus reservoir

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 byR=R(sinθcosφ,sinθsinφ,cosθ),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 is

where∣g,m〉μand∣e,n〉μis a tensor product of the wave function of the electrons and phonons with eigenenergyandrespectively.Here,Mgand Mespecify 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 as

where 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 Condonapproximation[25,26]is given by

wheredis 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,andis the associated Laguerre polynomials.

The electric field operator is[27]

where ωkand?ks(s=1,2)are the frequency and polarization vector of the quantized electric field with wave vectork,v is the quantization volume,?0is the vacuum dielectric constant,and ?ris the relative dielectric constant of the solid.Here,we take ?r≈1 in the following subsections.

2.2.Master equation

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 is

here,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 elementsdiordjdetermined by equation(7).HIis the free Hamiltonian of the electron–phonon system,whereωpis 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 HΩwith coherent coupling strength[29]

wherex=k0R(withk0=ω0c).Equation(12)describes the spontaneous emission of the two atoms due to their coupling to the radiation field.The spontaneous emission rates γijare given by

Finally,equation(13)accounts for the modification of spontaneous emission of one atom due to the presence of the other atom,where the parameteris written as

3.Results and analysis

In this section,we analyze the dynamics of the two coupled electron–phonon systems.In the case where the coupling strength Ωijis 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.

3.1.Coupling strength and cross-damping rate

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 tensorand 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 Ωijand Γijdepend on the alignment of the dipole moments and their orientations to the separation vectorR.

In in paper,we consider a simple case where the dipoles of the two atoms are all in thedirection and the separation vector in thedirection.In this configuration,Ωijand Γijare

Figure 2.System configuration and calculation parameters.Defect A is located at the origin and defect B is placed in the z-axis with a displacement vector R=(0,0,50)nm.The electric dipoles are all along the direction which are denoted by dA and dB.The magnitude of the dipole is taken to be∣dA∣=∣ dB∣=∣d∣=ea0,where e is the electron charge and a0 is the Bohr radius.The parameter S is chosen to be S=0.1.

expressed as

where γ0is the spontaneous emission rate associated with the bare two-level systems defined by

and p(x)and q(x)are functions of the dimensionless quantities x

The functions Fij(S)can be evaluated from equations(7)and(15)for different transitions i,j(see figure 1(b)).Take the∣e,0〉 ?∣g,0〉transition for example,

which can be evaluated from equation(7)with m=n=0.Figure 3 shows the dependence of Ω22and Γ22on the dimensionless parameter x in this configuration.In the region of small separation distance the coherent coupling strength Ω22and cross-damping rate Γ22are 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,Ω22and Γ22approach zero,implying the two individual atoms interacting with the vacuum filed,respectively.In this case,collective effects are negligible.

3.2.Time evolution of the coupled systems in single excitation regime

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 stateρ(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 Pegis[31]:

where Γsand Γacorrespond 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.

Figure 3.(a)Coherent coupling strength Ω22/γ0.(b)Cross-damping rate Γ22/γ0.This graph is drawn as a function of the dimensionless parameter x=k0 r with two electric dipoles parallel to the x-axis.The relative displacement vector r is along the direction.The parameter S is chosen to be S=0.1.

with matrix representation

The form of HTcan be greatly simplified by introducing a basis consisting of symmetry and antisymmetry states defined within the degenerate energy levels:

In the new basis,HThas the form

where

and Equations(28)–(30)show the energy degeneracy is partially eliminated and the energy splittings are determined by Ωii.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 ∣a1〉and ∣a4〉is much larger the corresponding coupling strength Ω14,so the population transfer can be ignored between them.Furthermore,the coupling strengths{Ωij}have the following relations:

This indicates that we can make further symmetrization and antisymmetrization in the degenerate subspace{∣a2〉,∣a3〉}.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:

Figure 4.(a)Time evolution of the population Pe0g 0 of the coupled electron–phonon systems.(b)Time evolution of the population Peg of the coupled simple two-level atoms.(c)Population evolutions of symmetric state ∣s〉1(the red curve)and antisymmetric state ∣a〉1(the black curve).(d)Power spectral density of Pe0g 0.

We find the energy level difference betweenandiswhich is approximately equal to that of the slow decaying oscillation in figure 4(a).As is shown in equation(32),the statesandinvolve one phonon and two phonons,respectively.These states come into the coupling process through the coherent coupling strength Ω12,Ω13and 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.

4.Conclusions

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.

Figure 5.(a)Couplings among antisymmetry states The blue lines denote the energy-level difference between two antisymmetry states:andThe red lines denote the coherent coupling strength between corresponding energy levels.In this figure,the coupling betweenandis not drawn because theenergy difference is much larger than the coupling strength between them(b)Couplings in the new basisBlue lines denote the energy difference between corresponding states. The red lines specify the coherent coupling strength in the new basis. is totally decoupled from other energy levels.

Figure 6.(a)Couplings among the symmetry states in basis Blue lines denote the energy difference between corresponding states.andThe statesandare degenerate.The red lines represent the associated coupling strength between two involved energy levels.The coupling betweenandis not drawn as in figure 5(b).Couplings among the symmetry states in basisThe energy level differences are and

Acknowledgments

This work has been supported by the NSFC(Grant No.11 534 002),and the NSAF(Grant No.U1930402 and Grant No.U1730449).