## Abstract

Atom-cavity coupling constant is a key parameter in cavity quantum electrodynamics for describing the interaction between an atom and a quantized electromagnetic field in a cavity. This paper reports a novel way to tune the coupling constant continuously by inducing an averaging of the atomic dipole moment over degenerate magnetic sublevels with elliptic polarization of the cavity field. We present an analytic solution of the stationary-state density matrix for this system with consideration of *F*→ *F* + 1 hyperfine transition under a weak excitation condition. We rigorously show that the stationary-state emission spectra of this system can be approximated by that of a non-degenerate two-level atom with an effective coupling constant as a function of the elliptic angle of the cavity field only. A precise condition for this approximation is derived and its physical meaning is interpreted in terms of a population-averaged transition strength and its variance. Our results can be used to control the coupling constant in cavity quantum electrodynamics experiments with a degenerate two-level atom with magnetic sublevels. Possible applications of our results are discussed.

©2010 Optical Society of America

## 1. Introduction

In cavity quantum electrodynamics (c-QED) the fundamental interaction between an atom and a quantized cavity field is studied. For recent decades, c-QED has attracted much attention owing to its applications to quantum information and quantum computing. Especially, recent progresses on cooling and trapping of single atoms inside a high-Q cavity [1, 2] have made many interesting applications possible, such as deterministic generation of single photons [3, 4], generation of correlated photons via quantum anharmonicity in Jaynes-Cummings ladder [5], observation of quantum jumps and spin dynamics [6], etc.

One of the key parameters for characterizing a c-QED system is a coupling constant between the atom and the cavity. Typically, the coupling constant is considered as a fixed parameter, determined by the geometry of the cavity and the atomic transition involved. This puts some limitations on otherwise possible c-QED studies such as the generation of arbitrary superpositions of Fock states at will [7] and a systematic investigation of singular properties in an atom-cavity composite related to an exceptional point [8]. Several state-of-the-art experiments which tune the coupling constant have recently been performed by utilizing an atomic conveyor belt and the spatial profile of a cavity mode [9, 10]. However, these works still have limitations such as unwanted Stark shifts and difficulties in maintaining the atomic position accurately.

In this paper, we propose a novel method to tune the coupling constant via controlling the induced atomic dipole moment. In order to get a quick understanding of our method, let us consider a two-level atom with Zeeman-sublevel degeneracy, or the so-called *degenerate* two-level atom, coupled to a quantized cavity field driven by an elliptic-polarized weak probe field. We assume that the cavity and the driving field are both near resonant to a hyperfine transition, *F* ↔ *F* +1. In this system the induced dipole moment depends on initial and final sublevels of the transition. Since the stationary-state population of the ground- and excited-state sublevels depends on the elliptic angle of the driving field and since all magnetic sublevels participate in the interaction, the dipole moment or the coupling constant should then be averaged over all transitions. As a result, the coupling constant might then be tuned by varying the elliptic angle.

For this idea to be any useful, however, at least the following questions should be addressed: First, whether the system behaves as if a non-degenerate two-level atom with a single coupling constant, and second, whether there exists a simple analytic expression for the coupling constant. The present work basically provides affirmative answers to these questions.

What we have done in this work is that we have solved the master equation of the Jaynes-Cummings system [11] for a stationary-state density matrix under a theoretical framework similar to that of Ref. [12]. We have found an analytic expression for the stationary-state density matrix by solving the master equation to the dominant order of the driving field intensity. We have also derived analytic expressions for the excitation spectra of atomic spontaneous emission and cavity transmission, respectively. We show that the system can be well approximated as a *non-degenerate* two-level system, unless the atom-cavity coupling is not too strong, with a single effective coupling constant whose analytic expression is obtained as a function of the elliptic angle of the driving field. We present a precise condition for this approximation and discuss its physical meanings in terms of a population-averaged transition strength and its variance. Our results provide a theoretical basis for continuous control of the coupling constant in c-QED experiments. We present possible applications of our work.

This paper is organized as follows. In Sec. 2, we present the stationary-state solution of the density matrix without a proof and by using the solution we calculate excitation spectra of atomic spontaneous emission and cavity transmission and show that both are given in terms of an effective coupling constant whose analytic expression is derived there. Physical meaning of the effective coupling constant is investigated in Sec. 3 with introduction of an eigen-basis called natural basis. Possible applications of our results are provided in Sec. 4. We then present a rigorous derivation of the stationary-state solution of the density matrix, which we used without a proof in Sec. 2, by solving the master equation of the system in Sec. 5. A conclusion is given in Sec. 6.

## 2. Stationary state of the system

#### 2.1. Our model system and its stationary-state density matrix

Let us consider a degenerate two-level atom which is coupled to a cavity with a representative coupling constant g0 (half width), whose precise meaning is given below. We assume that the cavity is weakly driven by a classical field of arbitrary-elliptic-polarization **e**̂ and an amplitude 𝓔 with an elliptic angle defined as Fig. 1(a). We also assume that the cavity mirror birefringence is negligibly small so that the polarization of the cavity field matches that of the driving field. The decay rates of the cavity and the atom are denoted by *κ* and *γ* (both half widths), respectively. We assume that the frequency of the driving field (*ω _{L}*) and that of the cavity (

*ω*) are both near resonant to the atomic transition (

_{C}*F*=

_{g}*F*) ↔ (

*F*=

_{e}*F*+1) of frequency

*ω*[see Fig. 1(b)]. The coupling constant 2

_{A}*g*

_{0}is the vacuum Rabi frequency for the cycling transition, (

*mF*=

_{g}*F*) ↔ (

*mF*=

_{e}*F*+ 1).

An analytic solution for this system in the stationary state does not exist in general because the cavity field have an infinite number of bases. However, in a weak excitation limit, the field bases can be truncated to a finite number of low-quantum bases and that the system can be understood approximately in terms of a few low orders of driving-field amplitude 𝓔 [13, 14]. Under this condition, as to be seen in Sec. 5, the system can be described by the interaction between manifold ∣*a*〉_{col} of (2*F* + 1) collective dressed states (with the atom in the ground sublevels and with one photon in the cavity) and manifold ∣*b*〉_{col} of (2*F* + 3) collective dressed states (with the atom in the excited sublevels with zero photon in the cavity), for both of which the number of quantum is one. These manifolds are connected to manifold ∣0 〉_{col} of (2*F* + 1) collective ground dressed states (with the atom in the ground sublevel and with zero photon in the cavity) through two decay channels as shown in Fig. 1(c).

With these considerations, the diagonal elements of the stationary-state density matrix can be obtained as follows:

where *η* is a normalization constant, *ρ _{ii}* = ∣

*i*〉

_{col}〈

*i*∣

_{col}for {

*i*} = {0,

*a*,

*b*}, 𝓐 and 𝓑 are Hermitian matrices which describe the diagonal blocks in

*ρ*,

_{ii}*V*(

*V*

^{†}) is the atomic lowering (raising) operator, Δ

*= (*

_{A,C}*ω*−

_{A,C}*ω*) are the detunings of the atom and the cavity with respect to the driving field, respectively, and

_{L}*E*= Δ

_{A}*−*

_{A}*iγ*and

*E*= Δ

_{C}*−*

_{C}*iκ*are the complex energies of the atom and the cavity, respectively. Rigorous derivations of Eqs. (1)–(3) and Hermitian matrices 𝓐 and 𝓑 will be given in Sec. 5. The matrix elements of 𝓐 and 𝓑 are listed in Appendix A.

As to be shown in Sec. 5, the Hermitian matrices 𝓐 and 𝓑 satisfy the following relation:

and thus these two matrices satisfy the following commutator relations.

In consequence, 𝓐 (𝓑) can be diagonalized simultaneously with *VV*
^{†} (*V*
^{†}
*V*). This suggests that the stationary-state density matrix in Eqs. (1)–(3) can also be diagonalized in the eigen-bases of
*VV*
^{†} and *V*
^{†}
*V*.

The normalization constant *η*, determined by the condition Tr[*ρ*] = 1, is given by

with coefficients

The analytic expressions for *α*
_{0}, *α*
_{1} and *α*
_{2} are given by Eqs. (60)–(62), respectively, in Appendix B.

#### 2.2. Excitation spectra and equivalence to a non-degenerate two-level system

Once the stationary-state density matrix is obtained, the emission from the system can be easily calculated. There exist two emission channels in our system as shown in Fig. 1(c): Cavity transmission *T*
_{cav} is characterized by the cavity decay rate *κ* while the atomic spontaneous emission *T*
_{sp} is characterized by the atomic decay rate *γ*. These two types of emission can be described by using the results from Eqs. (1)–(3) and (6) as

$$\phantom{\rule{16.5em}{0ex}}=\frac{{\ud4d4}^{2}{\mid {E}_{A}\mid}^{2}}{{\mid {E}_{A}{E}_{C}\mid}^{2}-2\mathrm{Re}\left[{E}_{A}{E}_{C}\right]({\alpha}_{1}/{\alpha}_{0}){g}_{0}^{2}+({\alpha}_{2}/{\alpha}_{0}){g}_{0}^{4}}+O\left({\ud4d4}^{4}\right),$$

$$\phantom{\rule{14em}{0ex}}=\frac{{\ud4d4}^{2}({\alpha}_{1}/{\alpha}_{0}){g}_{0}^{2}}{{\mid {E}_{A}{E}_{C}\mid}^{2}-\mathrm{Re}\left[{E}_{A}{E}_{C}\right]({\alpha}_{1}/{\alpha}_{0}){g}_{0}^{2}+({\alpha}_{2}/{\alpha}_{0}){g}_{0}^{4}}+O\left({\ud4d4}^{4}\right).$$

where *a*(*a*
^{†}) is a photon annihilation (creation) operator and *D _{q}* (

*D*

_{q}^{†}) is an atomic downward (upward) transition operator defined in Sec. 5. Equations (8) and (9) describe the excitation spectra for the cavity and the atomic emission channels, respectively. In other words, each describes the normalized emitted power from the corresponding output channel as a function of the frequency and the elliptic angle of the driving laser.

When the field is circularly polarized [*ε*= ±*π*/4 in Fig. 1(a)], the two spectra coincide with those of the non-degenerate two-level case [13, 14], which we can show by using Eq. (65) in Appendix B.

Even for arbitrary *ε*, the structures of Eqs. (8) and (9) resemble those of the non-degenerate two-level case. In fact, by letting

the denominators of Eqs. (8) and (9) can be written as

where *δ* (*ε*) = *α*
_{0}
*α*
_{2} / *α*
_{1}
^{2} − 1, which is much smaller than unity as to be shown below. Only when the last term in Eq. (13) is negligible, the excitation spectra for two output channels become of the same form as those of the non-degenerate two-level system with *g*′ interpreted as an effective coupling constant of the system.

We can find the precise condition under which we can neglect the last term of Eq. (13). The first term of Eq. (13) can be expanded by using the definition of *E _{A,C}* as

The lower bound of the above expression is 4*g*′^{2}
*κγ* occurring when Δ* _{A}* = Δ

*= 0 and*

_{C}*κγ*=

*g*′

^{2}. Therefore, the condition for neglecting the last term of Eq.(13) is

where for the inequality *g*
_{0} ≥ *g*′ we used the inequality *α*
_{0} ≥ *α*
_{1} from Eq. (69) in Appendix B.

#### 2.3. Accuracy of the non-degenerate two-level approximation

The function *δ*(*ε*) in the condition in Eq. (15) can be calculated by using the analytic expressions of *α*
_{0}, *α*
_{1} and *α*
_{2}. In Table 1, we summarize the maximum value of *δ*(*ε*) for various *F* numbers with relevant atomic species. The variation of *δ*(*ε*) as a function of *ε* is shown in Fig. 2(a) for some *F* values. For circular polarization (*ε* = ±*π*/4), *δ*(*ε*) becomes zero, consistent with the fact that the system becomes a non-degenerate two-level system in the steady state due to optical pumping. We also present the effective coupling constant *g*′ as a function of elliptic angle e in Fig. 2(b). As mentioned before, when *ε* = ±*π*/4, *g*′ coincide with *g*
_{0}, and *g*′ has its minimum value when *ε* = 0 (linear polarization).

Since *δ* is of order of 0.01, the condition in Eq. (15) can be safely satisfied if 4*κγ*/*g*
^{2}
_{0} is order of 0.1 or larger. This requirement can be met in a weak coupling regime and in intermediate and moderately strong coupling regimes as well. Some examples are given below. If the coupling is too strong, however, the condition is not satisfied.

In Fig. 2(c), the excitation spectrum of the cavity transmission, given by Eq. (8), of the degenerate two-level system when driven by linear polarization (*ε* = 0) is compared with that of the non-degenerate two-level system whose coupling constant is the same as *g*′ of the degenerate two-level system. Both exhibit a double-peak structure, also known as the vacuum Rabi splitting. For the same 𝓔 = 1 and *γ*/*g*
_{0} = 0.1, the two spectra are compared for three different values of *κ* when *F* = 3, for which *δ*(0) ~ 0.01. When *κ*/*g*
_{0} = 0.01 (*i.e.*, too strong coupling), the condition in Eq. (15) is not satisfied, and consequently, we see a noticeable discrepancy between the two spectra (red solid line vs. red dotted line). For *κ*/*g*
_{0} =0.1 (modestly strong coupling) and 1 (intermediate coupling), however, the condition is well satisfied, and thus the two spectra agree well with each other in Fig. 2(c).

## 3. Effective coupling constant in natural basis

#### 3.1. Morris-Shore transformation into natural basis

As shown in Appendix A, the Hermitian matrices 𝓐 and 𝓑 have very complicated forms with non-zero off-diagonal elements. In Sec. 2.1 we discussed that 𝓐 (𝓑) can be diagonalized simultaneously with *VV*
^{†} (*V*
^{†}
*V*) and that the stationary-state density matrix in Eqs. (1)–(3) can also be diagonalized in the eigen-bases of *VV*
^{†} and *V*
^{†}
*V*. The transformation from the atomic Zeeman basis into the eigen-basis of *VV*
^{†} and *V*
^{†}
*V*, also known as a natural basis, is called Morris-Shore transformation [15]. Degenerate two-level atoms driven by a *classical field in free space* were analyzed in the natural basis in Ref. [12, 16].

Let ∣(*a*)*i*〉 and ∣(*b*)*j*〉 be the eigenvectors of *VV*
^{†} and *V*
*†*
*V* with *λ*
^{2}
^{i,(a)} and *λ*
^{2}
_{j,(b)} corresponding eigenvalues, respectively, and then *VV*
^{†} and *V*
^{†}
*V* are diagonalized as

It can be shown that the eigenvalues are real non-negative numbers and that the sets {*λ*
^{2}
_{i,(a)}} and {*λ*
^{2}
_{j,(b)}} of non-zero eigenvalues coincide with each other [12, 16]. Furthermore, we can write the dipole interaction operators *V* and *V*
^{†} in this natural basis as

where *λ*
_{i (a)} = *λ*
_{i,(b)} = *λ _{i}* for

*i*= 1,2,…, 2

*F*+1 and

*λ*

_{j,(b)}= 0 for

*j*= 2

*F*+ 2 and 2

*F*+ 3, which implies that no transition is allowed for ∣(

*b*)2

*F*+ 2〉 and ∣ (

*b*)2

*F*+ 3〉 sublevels [see Fig. 3(b)]. In the conventional Zeeman basis, all degenerate sublevels are linked together by dipole transitions [see Fig.3(a)]. In the natural basis, however, one ground-state sublevel ∣(

*a*)

*i*〉 is coupled to only one excited-state sublevel ∣(

*b*)

*i*〉 with a transition strength

*λ*[see Fig. 3(b)], analogous a transition by linear polarization in the Zeeman basis. In fact, for linear or circular polarization, the Zeeman basis coincides with the natural basis.

_{i}Similarly, in the natural basis the Hermitian matrices 𝓐 and 𝓑 are also diagonalized as

where *ν _{i}*’s are eigenvalues of 𝓐 for

*i*= 1,2,…,2

*F*+1 and

*ν*

_{2F+2}=

*ν*

_{2F+3}= 0 [12].

#### 3.2. Physical interpretation of effective coupling constant g′ and small parameter δ

The analytic expressions for *g*′ and *δ*(*ε*) can be obtained explicitly by using Eqs. (60)–(62) in Appendix B. However, we gain more physical insights by expressing *g*′ and *δ*(*ε*) in the natural basis. With substitution of Eqs. (16) and (18) in Eq. (2), we find that *ρ _{aa}* is proportional to 𝓐/(

*VV*

^{†}) and thus its eigenvalue or the population in ∣(

*a*)

*i*〉 sublevel is proportional to

*ν*/

_{i}*λ*

_{i}^{2}. Let us define

*π*

_{i}^{(a)}be the ratio of the population in ∣(

*a*)

*i*〉 sublevel to the total ground-state population

*ρ*:

_{aa}From Eqs. (7), (16) and (18), we have

Then, the *g*′, and *δ*(*ε*) can be rewritten as,

$$\phantom{\rule{7em}{0ex}}=\frac{\left({\sum}_{i}{\lambda}_{i}^{4}{\pi}_{i}^{\left(a\right)}\right)-{\left({\sum}_{i}{\lambda}_{i}^{2}{\pi}_{i}^{\left(a\right)}\right)}^{2}}{{\left({\sum}_{i}{\lambda}_{i}^{2}{\pi}_{i}^{\left(a\right)}\right)}^{2}}$$

$$\phantom{\rule{1em}{0ex}}={\left[\frac{\left(\Delta {\lambda}^{2}\right)}{{\lambda}^{2}}\right]}^{2},$$

where *x̄* = ∑_{i}π_{i}^{(a)}
*x _{i}* the population-weighted average of

*x*in the natural basis and (Δ

*x*)

^{2}=

*x*

^{2}̄ −

*x*̄

^{2}the variance of

*x*. Therefore, the effective coupling constant

*g*′ is scaled with respect to

*g*

_{0}by the root-mean-square of the transition strength

*λ*while the mean is given by the population-weighted average in the natural basis. Likewise,

_{i}*δ*(

*ε*) is the ratio of the variance to the square of the population-weighted average of transition

*probability*

*λ*

_{i}^{2}. In the statistics terminology,

*δ*(

*ε*) is the same as the square of the

*coefficient of variation*for the transition probability. With this interpretation of

*δ*(

*ε*), we can first give an alternative explanation why

*δ*(±

*π*/4) = 0 for circular polarization. It is due to the vanishing dispersion Δ

*λ*

^{2}since only one

*π*

_{i}^{(a)}is nonzero in this case.

Second, as shown in Figs. 2(a) and 2(b), *δ*(*ε*) decreases as the angular momentum *F* increases while the change of *g*′ is negligible. We can now explain this by the reduction of the variance of the transition probability *λ _{i}*

^{2}as the

*F*increases whereas the average of

*λ*

_{i}^{2}does not change much. For simplicity, let us consider the case of linear polarization, for which the dependence on

*F*is quite pronounced. When

*ε*= 0, for which the natural basis coincides with the Zeeman basis, the transition probability

*λ*

_{i}^{2}becomes,

with *i* = 1,2,…,2*F*+ 1. It has a maximum (*F* + 1)/(2*F* + 1) for *i* =*F* + 1, which corresponds to *m _{F}* = 0 in the Zeeman basis. The ratio of the second largest value to the maximum is 1 − 1/(

*F*+ 1)

^{2}, which converges to unity as

*F*increases. Likewise, the ratio of the second largest value of

*π*

_{i}^{(a)}to its maximum also converges to unity as

*F*increases. Therefore, the variance of the transition probability decreases as

*F*increases. Fig. 4 shows the plot of

*λ*

_{i}^{2}, and

*π*

_{i}^{(a)}as

*F*is varied. As

*F*increases, the differences in

*λ*

_{i}^{2}among three most-contributing sublevels (

*i*=

*F*,

*F*+ 1,

*F*+ 2) decrease rapidly.

## 4. Possible applications to cavity-QED experiments

We have shown that a degenerate two-level Jaynes-Cummings system under a weak excitation can be approximated in the steady-state by a non-degenerate two-level Jaynes-Cummings system with a single effective coupling constant. Since the effective coupling constant is a monotonous-varying function of only the elliptic angle of the cavity field, it can be continuously tuned by the elliptic angle, and thus it can serve as a new control parameter in c-QED experiments.

In particular, with a suitable choice of *g*
_{0} with respect to *κ* and *γ*, it should be possible to control the effective coupling constant from a weak coupling to a strong coupling regime continuously. Here the weak and strong coupling regimes are specified by *g* < *γ*
_{−} and *g* > *γ*
_{−}, respectively, where *γ*
_{−} ≡ ∣*κ*− *γ*∣/2. This specification is based on the fact that the real and imaginary parts of quasi-eigenfrequencies of the atom-cavity system exhibit a dramatic change across a critical coupling of *g* = *γ*
_{−} under the condition *ω _{A}* =

*ω*. When

_{C}*g*>

*γ*

_{−}, the real parts of two quasi-eigenenergies exhibit an avoided crossing while the imaginary parts show a simple crossing as the atom-cavity detuning is scanned. When

*g*<

*γ*

_{−}, we have the opposite, the real parts showing a simple crossing whereas the imaginary parts undergoing an avoided crossing. Therefore, one can define the case of

*g*>

*γ*

_{−}as a strong coupling regime and the case of

*g*<

*γ*

_{−}as a weak coupling regime. Figure 5 illustrates the transition between avoided crossing and simple crossing in the atom-cavity system, which can be studied by using our

*g*control method.

The capability that *g* can be varied across the critical coupling would make it possible to study the quasi-eigenstates of an atom-cavity system near a topological exceptional point [8], where a transition from the weak to the strong coupling regime occurs in the system parameter space. Exceptional points are of considerable interest related to open quantum systems described by non-Hermitian Hamiltonian [17].

Another possible application is related to the stationary-state solution itself. In order to describe a c-QED system, it is necessary to know the stationary-state populations in the system. The density matrix solution presented in this paper can provide this knowledge for many cavity QED experiments done with a degenerate two-level atom. In particular, if one employs special types of cavity such as microspheres [18], microtoroids [19] and photonic crystal nanocavities [20], the optical pumping to a cycling transition cannot be fulfilled because circular polarization is not supported there due to the special geometry of supported modes. Only linear polarization is supported there. Therefore, it is necessary to consider the multi-sublevels of atom and coherence among them, making the analysis very complicated. However, by using our results, these complicated systems can be described easily in the natural basis as demonstrated in Sec. 3. In addition, recent studies on the quantum state engineering often deal with the magnetic sub-levels in c-QED systems [21, 22], and for these experiments, our results can provide a useful theoretical background.

It should be pointed out that our results do not take into account mirror birefringence, which could be a serious problem in an experiment in a very strong coupling regime. However, the birefringence might not be an issue in a moderate-strong coupling regime which our scheme is applicable to. In fact, it is possible to design a cavity with negligible birefringence in the moderate-strong coupling regime. For example, for the strong coupling regime in Fig. 5, we have *g* = 4*γ* and *κ* = 8*γ*, so *g* > *γ*
_{−} = 3.5*γ*. For D2 transition of atomic rubidium, we have *γ*/2*π*=3 MHz and thus *κ*/2*π*=24 MHz, which we can obtain with mirrors with a finesse of about 10^{4}. Since the phase retardation per reflection for supermirrors is known to be in the order of 10^{-6} [23, 24], the effect of birefringence would be negligible in such a cavity.

## 5. Derivation of the stationary-state density matrix

In the preceding sections, in order to make our discussion as comprehensible as possible, we presented the stationary-state solution of the density matrix in Eqs. (1)–(3) without any proof and go ahead to use the solution for calculating the excitation spectra of the system and showing the equivalence to a nondegenerate two-level atom with an effective coupling constant in an analytic form. In this section, we will derive Eqs. (1)–(3) by solving the master equation of the system to the lowest order of the driving field intensity.

Let us first consider the elliptical polarization **e**̂ of the driving field as shown in Fig. 1(a). In general it can be expressed as

where **e**̂_{{-1 ,0,+1}} stand for the unit polarization vector of {*σ*
_{−},*π*,*σ*
_{+}} polarization, respectively, and *e _{q}* are their coefficients. By choosing the quantization axis as parallel to the propagating direction of the field,

**e**̂ can be expressed as

with a single parameter *ε*, an elliptic angle as defined in Fig. 1(a), satisfying −*π*/4 ≤ *ε* ≤ +*π*/4.

Under the rotating wave approximation, the Hamiltonian for the model system of Sec. 2 can be described by a multi-level expansion of the Jaynes-Cummings Hamiltonian [25, 26]. In the rotating frame of frequency *ω _{L}*, (

*i.e*., in the interaction picture), the Hamiltonian is written as

where the amplitude 𝓔 of the classical driving field is scaled in such a way that 𝓔 /*κ* represents a dimensionless injected photon flux. Operator *D _{q}* is an atomic downward-transition operator for

**e**̂

*polarization, defined as*

_{q}with the Clebsch-Gordan coefficients

Note that *D _{q}* satisfies the relation ∑

_{q}*D*

_{q}^{†}

*D*= ∑

_{q}*∣*

_{m′F}*e*,

*m*′

_{F}〉〈

*e*,

*m*′

*which implies the conservation of the total population for a closed transition. Atomic lowering operator*

_{F}*V*can then be expressed as

Since *m _{F}* = −

*F*, −

*F*+ 1,… ,

*F*, and

*m*′

*= −*

_{F}*F*− 1, −

*F*,… ,

*F*+ 1, both operators

*D*and

_{q}*V*can be represented by (2

*F*+ 1) × (2

*F*+ 3) matrices.

The collective dressed-state bases in Fig. 1(c) can be written as

where {*ξ _{i}*

^{(0,a,b)}} are complex amplitudes. There are as many as (2

*F*+ 1) independent sets of {

_{g,e}*ξ*

_{i}^{(0,a,b)}}. In the absence of the driving field (𝓔 = 0), only {

*ξ*,

_{i}^{0}} can have non-zero values in the steady state. Therefore, we expect that the dominant terms of {

*ξ*} are at least of the first order of 𝓔 while {

_{i}^{a,b}*ξ*

_{i}^{0}} are of the zeroth order of 𝓔. With these bases the Hamiltonian and the density matrix can respectively be expressed by a (6

*F*+ 5) × (6

*F*+ 5) matrix, which can be divided into 9 sub-matrices, as

where *I _{n}* denotes an

*n*×

*n*unit matrix.

The time evolution of the system is then described by the master equation

$$+\gamma \sum _{q=-1}^{+1}\left(2{D}_{q}\rho {D}_{q}^{\u2020}-{D}_{q}^{\u2020}{D}_{q}\rho -\rho {D}_{q}^{\u2020}{D}_{q}\right),$$

and the stationary-state density matrix is obtained by solving 𝓛[*ρ _{ss}*] = 0. By substituting Eqs. (33) and (34) into Eq. (35), we obtain a set of first-order differential equations

To solve for a stationary-state solution, we should find 9 sub-matrices which make the time derivatives in the left-hand side of Eqs. (36) – (41) vanish, under the normalization condition Tr[*ρ*] = 1. By rearranging terms and removing the higher order terms of 𝓔 than the second order, we obtain

where *s* = *i*/(*E _{A}* −

*E*

_{C}^{*}) and

*X*and

*Y*are the square matrices defined as

One can easily prove *X* and *Y* satisfy the following identities:

By substituting Eqs. (42) – (47) into Eqs. (36) – (38), we then obtain

where ‘h.c.’ denotes Hermitian conjugate. By substituting Eqs. (48), (49) into Eq. (50), Eq. (50) can be expressed in a simpler form as

Let us define Hermitian matrices 𝓐 and 𝓑 as

and substitute it into Eq. (51), which then becomes

which is exactly the same as Eq. (4).

It is already shown in Ref. [12, 27] that the matrices 𝓐 and 𝓑 are determined uniquely by the polarization (elliptic angle *ε*) and the atomic level structure (angular momentum *F*). Moreover, although we are dealing with *F* ↔ *F* + 1 transition, 𝓐 and 𝓑 can be found for arbitrary transitions. The details of finding 𝓐 and 𝓑 are well documented in Ref. [12] and references therein, and therefore, in this paper we only give the explicit expressions for matrix elements of 𝓐 and 𝓑 in Appendix A.

Substituting Eqs. (52) into Eqs. (48) – (50) and after some straightforward algebra, we finally obtain the diagonal elements of the density matrix in closed forms as

These are exactly the same as Eqs. (1)–(3) that we used in Sec. 2 without proof. Furthermore, it is unique solution because the matrices 𝓐 and 𝓑 are unique. Note that as mentioned before the dominant order of 𝓔 is 𝓔^{0} for *ρ*
_{00} and 𝓔^{2} for *ρ _{bb}* and

*ρ*.

_{aa}## 6. Conclusion

We have shown that the effective coupling constant of the coupled system of a degenerate two-level atom and an elliptic-polarized cavity mode can be controlled continuously by varying the elliptic angle of the driving field. For this, the stationary-state density matrix is obtained up to the first-order of the driving-field intensity. We have also calculated the excitation spectra for the cavity transmission and the atomic spontaneous emission, respectively. Unless the atom-cavity coupling is too strong, the precise meaning of which we specify, the excitation spectra are shown to be the same as those of a non-degenerate two-level system with an effective coupling constant *g*′. We show that the effective coupling constant can be interpreted as a population weighted average of the transition strength in the natural basis. The fact that the effective coupling constant can be continuously controlled suggests a new venue in c-QED experiments such as the study of an exceptional point and its associated singular properties in the atom-cavity composite described by a non-Hermitian Hamiltonian. Our results also provide a simplified description of a c-QED systems of a degenerate two-level atom in a non-Fabry-Perot-type cavity such as microsphere, microtoroid and photonic crystal nanocavity, where optical pumping to cyclic transition cannot be fulfilled due to the cavity geometry.

## Appendix A: Expression for matrix elements of 𝓐 and 𝓑

For calculating the matrix elements of 𝓐 and 𝓑, it is more convenient to execute the calculation in a basis called natural coordinate frame [28], rather than the conventional Zeeman basis. In this frame, the polarization vector can be written as

and with the assumption of *F* ↔ *F* +1 transition the elements of the matrices 𝓐 and 𝓑 can be calculated by using Eq. (B8) of Ref. [12] as

$$\phantom{\rule{.5em}{0ex}}\times \sqrt{\frac{\left(2F+1+i-k\right)!\left(2F+1+j-k\right)!}{\left(2F+1-i+k\right)!\left(2F+1-j+k\right)!}}$$

$$\phantom{\rule{.5em}{0ex}}\times {C}_{\mathrm{Fi}\left(F+1\right)-k}^{\left(2F+1\right)\left(i-k\right)}{C}_{\mathrm{Fj}\left(F+1\right)-k}^{\left(2F+1\right)\left(j-k\right)}{\left(\frac{\mathrm{sin}\epsilon}{\sqrt{\mathrm{cos}2\epsilon}}\right)}^{i+j-2k},$$

$$\phantom{\rule{.5em}{0ex}}\times \sqrt{\frac{\left(2F+1+i-k\right)!\left(2F+1+j-k\right)!}{\left(2F+1-i+k\right)!\left(2F+1-j+k\right)!}}$$

$$\phantom{\rule{.5em}{0ex}}\times {C}_{\left(F+1\right)i\phantom{\rule{.2em}{0ex}}F-k}^{\left(2F+1\right)\left(i-k\right)}{C}_{\left(F+1\right)j\phantom{\rule{.2em}{0ex}}F-k}^{\left(2F+1\right)\left(j-k\right)}{\left(\frac{\mathrm{sin}\epsilon}{\sqrt{\mathrm{cos}2\epsilon}}\right)}^{i+j-2k}.$$

The dimension of 𝓐 is (2*F* +1)×(2*F* +1) while 𝓑 is (2*F* +3)×(2*F* +3). The values of *ν _{i}*’s, which are diagonal elements of 𝓐 and 𝓑 in the natural basis, can then be obtained by finding the eigenvalues of the above matrices.

## Appendix B: Analytic expressions for coefficients α_{0}, α_{1}, and α_{2}

The coefficients *α*
_{0}, *α*
_{1}, and *α*
_{2} in Eq. (7) depend on the elliptic angle **e**̂ and the angular momentum *F* only. For arbitrary polarization **e**̂ and for the transition *F* ↔ *F* +1, these coefficients can be expressed in explicit forms by using Eq. (79) of Ref. [12] as

where the *P _{l}* (

*x*) denotes the Legendre polynomials and the coefficients

*C*and

_{l}*D*are given by Eq. (77) of Ref. [12] as

_{l}where the last factor denotes the 6*j* symbol. Since 1/cos2*ε* ≥ 1, *α*
_{1} and *α*
_{2} should have positive values. When ε = ±*π*/4, these coefficients diverge to infinity, but the ratios between them converge to unity:

From the recursion relation of Legendre polynomials

the coefficients *α*
_{0} and *α*
_{2} in Eqs. (60)–(62) become

from which the following inequality follows,

The equality holds for *ε* = ±*π*/4 as shown in Eq. (65).

## Acknowledgments

This work was supported by NRL and WCU Grants.

## References and links

**1. **J. Ye, D. W. Vernooy, and H. J. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. **83**, 4987–4990 (1999). [CrossRef]

**2. **P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, “Trapping an atom with single photons,” Nature **404**, 365–368 (2000). [CrossRef] [PubMed]

**3. **A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon sources for distributed quantum networking,” Phys. Rev. Lett. **89**, 067901 (2002). [CrossRef] [PubMed]

**4. **J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, “Deterministic generation of single photons from one atom trapped in a cavity,” Science **303**, 1992–1994 (2004). [CrossRef] [PubMed]

**5. **I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nature Phys. **4**, 382–385 (2008). [CrossRef]

**6. **M. Khudaverdyan, W. Alt, T. Kampschulte, S. Reick, A. Thobe, A. Widera, and D. Meschede, “Quantum jumps and spin dynamics of interacting atoms in a strongly coupled atom-cavity system,” Phys. Rev. Lett. **103**, 123006 (2009). [CrossRef] [PubMed]

**7. **C. K. Law and J. H. Eberly, “Arbitrary control of a quantum electromagnetic field,” Phys. Rev. Lett. **76**, 1055–1058 (1996). [CrossRef] [PubMed]

**8. **T. Kato, *Perturbation Theory for Linear Operators* (Springer, New York, 1966).

**9. **K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, “Deterministic loading of individual atoms to a high-finesse optical cavity,” Phys. Rev. Lett. **98**, 233601 (2007). [CrossRef] [PubMed]

**10. **M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schorner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. **10**, 073023 (2008). [CrossRef]

**11. **H. J. Carmichael, *An Open Systems Approach to Quantum Optics* (Springer, Berlin, 1993).

**12. **A. V. Taĭchenachev, A. M. Tumaĭkin, V. I. Yudin, and G. Nienhuis, “Steady state of atoms in a resonant field with elliptical polarization,” Phys. Rev. A **69**, 033410 (2004). [CrossRef]

**13. **H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. **82**, 73–79 (1991). [CrossRef]

**14. **R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A **59**, 2392–2417 (1999). [CrossRef]

**15. **J. R. Morris and B. W. Shore, “Reduction of degenerate two-level excitation to independent two-state systems,” Phys. Rev. A **27**, 906–912 (1983). [CrossRef]

**16. **G. Nienhuis, “Natural basis of magnetic substates for a radiative transition with arbitrary polarization,” Opt. Commun. **59**353–356 (1986). [CrossRef]

**17. **C. Dembowski, H. -D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. **86**787–190 (2001). [CrossRef] [PubMed]

**18. **D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A **57**R2293–R2296 (1998). [CrossRef]

**19. **S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A **71**013817 (2005). [CrossRef]

**20. **J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E **65**, 016608 (2001). [CrossRef]

**21. **A. D. Boozer, R. Miller, T. E. Northup, A. Boca, and H. J. Kimble, “Optical pumping via incoherent Raman transitions,” Phys. Rev. A **76**, 063401 (2007). [CrossRef]

**22. **T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science **317**, 488–490 (2007). [CrossRef] [PubMed]

**23. **D. Jacob, M. Vallet, F. Bretenaker, A. L. Floch, and M. Oger, “Supermirror phase anisotropy measurement,” Opt. Lett. bf **20**, 671–673 (1995). [CrossRef]

**24. **J. Y. Lee, H-W. Lee, J. W. Kim, Y. S. Yoo, and J. W. Hahn, “Measurement of ultralow supermirror birefringence by use of the polarimetric differential cavity ringdown technique,” Applied Optics **39**, 1941–1945 (2000). [CrossRef]

**25. **E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE **51**, 89–109 (1963). [CrossRef]

**26. **K. M. Birnbaum,, A. S. Parkins, and H. J. Kimble, “Cavity QED with multiple hyperfine levels,” Phys. Rev. A **74**, 063802 (2006). [CrossRef]

**27. **A. V. Taĭchenachev, A. M. Tumaĭkin, and V. I. Yudin, “An atom in an elliptically polarized resonant field: exact stationary solution for closed J ↑ J + 1 transitions,” JETP **83**, 949–961 (1996).

**28. **A. M. Tumaĭkin and V. I. Yudin, “Coherent stationary states under the interaction of atoms with polarized resonant light in a magnetic field,” Sov. Phys. JETP **71**, 43–47 (1990).