Continuous control of the coupling constant in an atom-cavity system by
using elliptic polarization and magnetic sublevels

doi:10.1364/OE.18.009286

Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels

Sungsam Kang, Youngwoon Choi, Sooin Lim, Wookrae Kim, Jung-Ryul Kim, Jai-Hyung Lee, and Kyungwon An »View Author Affiliations

Optics Express, Vol. 18, Issue 9, pp. 9286-9302 (2010)
http://dx.doi.org/10.1364/OE.18.009286

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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.

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

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]

] have made many interesting applications possible, such as deterministic generation of single photons [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]

], generation of correlated photons via quantum anharmonicity in Jaynes-Cummings ladder [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]

], observation of quantum jumps and spin dynamics [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]

], 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

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

] and a systematic investigation of singular properties in an atom-cavity composite related to an exceptional point [8

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

]. 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

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]

]. 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

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

] for a stationary-state density matrix under a theoretical framework similar to that of Ref. [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]

]. 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 ê 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_C ) are both near resonant to the atomic transition (F_g = F) ↔ (F_e = F +1) of frequency ω_A [see Fig. 1(b)]. The coupling constant 2g 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

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]

]. Under this condition, as to be seen in Sec. 5, the system can be described by the interaction between manifold ∣a〉_col of (2F + 1) collective dressed states (with the atom in the ground sublevels and with one photon in the cavity) and manifold ∣b〉_col of (2F + 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 (2F + 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:

ρ_{bb} = η 퓔^{2} g_{0}^{2} 퓑 + O (퓔^{4})

(1)

ρ_{aa} = η 퓔^{2} {∣ E_{A} ∣}^{2} {({VV}^{†})}^{- 1} 퓐 + O (𝓔^{4})

(2)

ρ_{00} = η {{∣ E_{A} E_{C} ∣}^{2} {({VV}^{†})}^{- 1} - 2 Re [E_{A} E_{C}] g_{0}^{2} + g_{0}^{4} {VV}^{†}} 퓐 + O (𝓔^{2}),

(3)

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_L ) are the detunings of the atom and the cavity with respect to the driving field, respectively, and E_A = Δ _A − iγ and E_C = Δ _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:

퓑 V^{†} = V^{†} 퓐 .

(4)

and thus these two matrices satisfy the following commutator relations.

[퓐, {VV}^{†}] = 0, [퓑, V^{†} V] = 0 .

(5)

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.

Fig. 1. (a) Elliptic angle ε of arbitrary elliptic polarization in a coordinate frame with quantization axis parallel to the propagation direction of the cavity field. (b) A two-level atom (F ↔ F + 1) with Zeeman sublevel degeneracy with a decay rate γ interacts with a quantized cavity field with a decay rate κ. (c) Collective dressed-state representation with the number of quantum limited up to one.

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

η = \frac{1}{{∣ E_{A} E_{C} ∣}^{2} α_{0} - 2 Re [E_{A} E_{C}] α_{1} g_{0}^{2} + α_{2} g_{0}^{4}} + O (퓔^{2}),

(6)

with coefficients

α_{0} = Tr [{({VV}^{†})}^{- 1} 퓐], α_{1} = Tr [퓐] = Tr [퓑], α_{2} = Tr [{VV}^{†} 퓐] .

(7)

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

T_{cav} = Tr [a^{†} aρ] = Tr [ρ_{aa}]

= \frac{퓔^{2} {∣ E_{A} ∣}^{2}}{{∣ E_{A} E_{C} ∣}^{2} - 2 Re [E_{A} E_{C}] (α_{1} / α_{0}) g_{0}^{2} + (α_{2} / α_{0}) g_{0}^{4}} + O (퓔^{4}),

(8)

T_{sp} = Tr [\sum_{q} D_{q}^{†} D_{q} ρ] = Tr [ρ_{bb}]

= \frac{퓔^{2} (α_{1} / α_{0}) g_{0}^{2}}{{∣ E_{A} E_{C} ∣}^{2} - Re [E_{A} E_{C}] (α_{1} / α_{0}) g_{0}^{2} + (α_{2} / α_{0}) g_{0}^{4}} + O (퓔^{4}) .

(9)

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

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

], which we can show by using Eq. (65) in Appendix B.

T_{cav} (ε = \pm π / 4) = 퓔^{2} {∣ \frac{E_{A}}{E_{A} E_{C} - g_{0}^{2}} ∣}^{2} + O (𝓔^{4}),

(10)

T_{sp} (ε = \pm π / 4) = 퓔^{2} {∣ \frac{g_{0}}{E_{A} E_{C} - g_{0}^{2}} ∣}^{2} + O (퓔^{4}) .

(11)

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

g' = \sqrt{\frac{α_{1}}{α_{0}}} g_{0},

(12)

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

{∣ E_{A} E_{C} - {g'}^{2} ∣}^{2} + δ (ε) {g'}^{4},

(13)

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

{∣ E_{A} E_{C} - {g'}^{2} ∣}^{2} = {(Δ_{C} Δ_{A} - {g'}^{2})}^{2} + {(Δ_{A} κ)}^{2} + {(Δ_{C} γ)}^{2} + [{(κγ + {g'}^{2})}^{2} - {g'}^{4}] .

(14)

The lower bound of the above expression is 4g′² κγ occurring when Δ _A = Δ _C = 0 and κγ = g′². Therefore, the condition for neglecting the last term of Eq.(13) is

\frac{4 κγ}{{g'}^{2}} \geq \frac{4 κγ}{g_{0}^{2}} ≫ δ (ε),

(15)

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 4κγ/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).

Table 1. Maximum values of δ(ε) for various angular momenta F and corresponding atomic species. They have D₂ lines of (F_g = F) ↔ (F_e = F + 1) closed transition.

F	δ _max	relevant atomic species
1	0.043	¹H
2	0.027	⁷Li, ²³Na, ³⁹K, ⁴¹K, ⁸⁷Rb
3	0.016	⁸⁵Rb
4	0.010	¹³³Cs

Fig. 2. (Color Online) (a) Plot of δ(ε) for various F. (b) Variation of g′ /g ₀ as elliptic angle ε. (c) Excitation spectrum of the cavity transmission for linear polarization (ε = 0) when F = 3. For a fixed value of γ/g ₀ = 0.1, the value of κ/g ₀ is varied. Solid line represents the spectrum given by Eq. (8) while the dotted line shows the equivalent non-degenerate two-level system with a coupling constant equal to g′.

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

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]

]. Degenerate two-level atoms driven by a classical field in free space were analyzed in the natural basis in Ref. [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]

, 16

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

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

{VV}^{†} = \sum_{i = 1}^{2 F + 1} λ_{i, (a)}^{2} ∣ (a) i 〉 〈 (a) i ∣, V^{†} V = \sum_{j = 1}^{2 F + 3} λ_{j, (b)}^{2} ∣ (b) j 〉 〈 (b) j ∣ .

(16)

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

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]

, 16

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

]. Furthermore, we can write the dipole interaction operators V and V ^† in this natural basis as

V = \sum_{i} λ_{i} ∣ (a) i 〉 〈 (b) i ∣, V^{†} = \sum_{i} λ_{i} ∣ (b) i 〉 〈 (a) i ∣,

(17)

where λ _{i (a)} = λ _i,(b) = λ_i for i = 1,2,…, 2F +1 and λ _j,(b) = 0 for j = 2F + 2 and 2F + 3, which implies that no transition is allowed for ∣(b)2F + 2〉 and ∣ (b)2F + 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_i [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.

Similarly, in the natural basis the Hermitian matrices 𝓐 and 𝓑 are also diagonalized as

퓐 = \sum_{i = 1}^{2 F + 1} ν_{i} ∣ (a) i 〉 〈 (a) i ∣, 퓑 = \sum_{j = 1}^{2 F + 3} ν_{j} ∣ (b) j 〉 〈 (b) j ∣,

(18)

where ν_i ’s are eigenvalues of 𝓐 for i = 1,2,…,2F +1 and ν _2F+2 = ν _2F+3 = 0 [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]

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 :

Fig. 3. (a) Conventional Zeeman basis and the transition linkage by the elliptical polarization ê in Fig. 1. (b) Natural basis and the corresponding transition strength λ_i . Two rightmost sublevels of the upper state manifold are uncoupled in the natural basis.

π_{i}^{(a)} \equiv \frac{ν_{i} / λ_{i}^{2}}{\sum_{i} ν_{i} / λ_{i}^{2}}, \sum_{i} π_{i}^{(a)} = 1,

(19)

From Eqs. (7), (16) and (18), we have

α_{0} = \sum_{i} ν_{i} / λ_{i}^{2}, α_{1} = \sum_{i} ν_{i}, α_{2} = \sum_{i} ν_{i} λ_{i}^{2} .

(20)

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

g' = \sqrt{\frac{α_{1}}{α_{0}}} g_{0} = \sqrt{\sum_{i} λ_{i}^{2} π_{i}^{(a)}} g_{0} = \sqrt{\bar{λ^{2}}} g_{0},

(21)

δ (ε) = α_{0} α_{2} / α_{1}^{2} - 1

= \frac{(\sum_{i} λ_{i}^{4} π_{i}^{(a)}) - {(\sum_{i} λ_{i}^{2} π_{i}^{(a)})}^{2}}{{(\sum_{i} λ_{i}^{2} π_{i}^{(a)})}^{2}}

= {[\frac{(Δ λ^{2})}{λ^{2}}]}^{2},

(22)

where x̄ = ∑ _iπ_i ^(a) x_i the population-weighted average of x in the natural basis and (Δx)² = x ²̄ − x̄² 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_i while the mean is given by the population-weighted average in the natural basis. Likewise, 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 Δλ 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,

λ_{i}^{2} = \frac{(2 F + 2 - i) i}{(F + 1) (2 F + 1)},

(23)

with i = 1,2,…,2F+ 1. It has a maximum (F + 1)/(2F + 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.

Fig. 4. Plot of λ_i ² and π_i ^(a) for various F values when ε = 0. As F increases, the differences in λ_i ² among three most-contributing sublevels (i = F,F + 1,F + 2) decrease, and accordingly, the dispersion Δλ ² decreases.

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_C . When 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

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

], 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

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]

Fig. 5. Real and imaginary parts of quasi-eigenenergies of the atom-cavity system vs. the atom-cavity detuning for three representative coupling constants, with κ= 8γ and g ₀ = 4γ: (a) g = 0.73g ₀ in a weak coupling regime, (b) g = γ ₋ = 0.88g ₀ at the critical coupling, and (c) g = 1.0g ₀ in a strong coupling regime. Also shown are the cavity transmission (T _cav) spectra when ω_A = ω_C . Although the spectra do not change much, the associated real and imaginary parts of the quasi-eigenenergies exhibit a dramatic transition across the critical coupling g = γ ₋. The transition can be observed experimentally by measuring the spectra with high signal-to-noise ratio and then by fitting them to the theory to extract quasi-eigenenergies.

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

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]

], microtoroids [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]

] and photonic crystal nanocavities [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]

], 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

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]

], 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 = 4γ and κ = 8γ, so g > γ ₋ = 3.5γ. For D2 transition of atomic rubidium, we have γ/2π=3 MHz and thus κ/2π=24 MHz, which we can obtain with mirrors with a finesse of about 10⁴. Since the phase retardation per reflection for supermirrors is known to be in the order of 10^-6 [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]

], 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 ê of the driving field as shown in Fig. 1(a). In general it can be expressed as

\hat{e} = \sum_{q = - 1}^{+ 1} e_{q} {\hat{e}}_{q}

(24)

where ê_{{-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, ê can be expressed as

\hat{e} = - \sin (ε + π / 4) {\hat{e}}_{+ 1} + \cos (ε + π / 4) {\hat{e}}_{- 1}

(25)

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

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]

]. In the rotating frame of frequency ω_L , (i.e., in the interaction picture), the Hamiltonian is written as

H / \bar{h} = Δ_{C} a^{†} a + Δ_{A} \sum_{q = - 1}^{+ 1} D_{q}^{†} D_{q} + g_{0} (V_{a}^{†} - a V^{†}) + 퓔 (a + a^{†}),

(26)

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 ê _q polarization, defined as

D_{q} = \sum_{m_{F}, {m'}_{F}} C_{F_{g} m_{F} 1 q}^{F_{e} m_{F}^{'}} ∣ g, m_{F} 〉 〈 e, {m'}_{F} ∣,

(27)

with the Clebsch-Gordan coefficients

C_{jm j' m'}^{JM} = {(- 1)}^{j - j' + M} \sqrt{2 J + 1} (\begin{matrix} j & j' & J \\ m & m' & - M \end{matrix}) .

(28)

Note that D_q satisfies the relation ∑ _q D_q ^† D_q = ∑ _{m′_F} ∣e,m′_F〉〈e,m′ _F which implies the conservation of the total population for a closed transition. Atomic lowering operator V can then be expressed as

V = (\sum_{q = - 1}^{+ 1} D_{q} {\hat{e}}_{q}) \cdot \hat{e} .

(29)

Since m_F = −F, −F + 1,… ,F, and m′ _F = −F − 1, − F,… ,F + 1, both operators D_q and V can be represented by (2F+ 1) × (2F + 3) matrices.

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

{∣ 0 〉}_{col} = \sum_{m_{F}} ξ_{m_{F}}^{0} {∣ g, m_{F} 〉}_{atom} \otimes {∣ 0 〉}_{field}

(30)

{∣ a 〉}_{col} = \sum_{m_{F}} ξ_{m_{F}}^{a} {∣ g, m_{F} 〉}_{atom} \otimes {∣ 1 〉}_{field}

(31)

{∣ b 〉}_{col} = \sum_{{m'}_{F}} ξ_{{m'}_{F}}^{b} {∣ e, {m'}_{F} 〉}_{atom} \otimes {∣ 0 〉}_{field},

(32)

where {ξ_i ^(0,a,b)} are complex amplitudes. There are as many as (2F_g,e + 1) independent sets of {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_i ^a,b } are at least of the first order of 𝓔 while {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 (6F + 5) × (6F + 5) matrix, which can be divided into 9 sub-matrices, as

H = \bar{h} [\begin{matrix} Δ_{A} I_{2 F + 3} & g_{0} V^{†} & 0 \\ g_{0} V & Δ_{C} I_{2 F + 1} & 퓔 I_{2 F + 1} \\ 0 & 퓔 I_{2 F + 1} & 0 \end{matrix}],

(33)

ρ = [\begin{matrix} ρ_{bb} & ρ_{ba} & ρ_{b 0} \\ ρ_{ab} & ρ_{aa} & ρ_{a 0} \\ ρ_{0 b} & ρ_{0 a} & ρ_{00} \end{matrix}],

(34)

where I_n denotes an n×n unit matrix.

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

\dot{ρ} = 퓛 [ρ] = \frac{1}{ih} [H, ρ] + κ (2 aρ a^{†} - a^{†} aρ - ρ a^{†} a)

+ γ \sum_{q = - 1}^{+ 1} (2 D_{q} ρ D_{q}^{†} - D_{q}^{†} D_{q} ρ - ρ D_{q}^{†} D_{q}),

(35)

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

{\dot{ρ}}_{bb} = - 2 γ ρ_{bb} - i g_{0} (V^{†} ρ_{ab} - ρ_{ba} V)

(36)

{\dot{ρ}}_{aa} = - 2 κ ρ_{aa} - i g_{0} (V ρ_{ba} - ρ_{ab} V^{†}) - i퓔 (ρ_{0 a} - ρ_{a 0})

(37)

{\dot{ρ}}_{00} = 2 κ ρ_{aa} + 2 γ \sum_{q} D_{q} ρ_{bb} D_{q}^{†} + i퓔 (ρ_{0 a} - ρ_{a 0})

(38)

{\dot{ρ}}_{ab} = {({\dot{ρ}}_{ba})}^{†} = i (E_{A}^{*} - E_{C}) ρ_{ab} - i g_{0} (V ρ_{bb} - ρ_{aa} V) - i퓔 ρ_{0 b}

(39)

{\dot{ρ}}_{0 b} = {({\dot{ρ}}_{b 0})}^{†} = i E_{A}^{*} ρ_{0 b} + i g_{0} ρ_{0 a} V - i𝓔 ρ_{ab}

(41)

{\dot{ρ}}_{0 a} = {({\dot{ρ}}_{a 0})}^{†} = i E_{C}^{*} ρ_{0 a} + i g_{0} ρ_{0 b} V^{†} + i퓔 (ρ_{00} - ρ_{aa})

(41)

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

ρ_{ab} = {(ρ_{ba})}^{†} = i s^{*} g_{0} (V ρ_{bb} - ρ_{aa} V) + i s^{*} g_{0} 𝓔^{2} ρ_{00} V X^{- 1} + O (𝓔^{4}),

(42)

ρ_{0 b} = {(ρ_{b 0})}^{†} = g_{0} 퓔 ρ_{00} V X^{- 1} + O (𝓔^{3}),

(43)

ρ_{0 a} = {(ρ_{a 0})}^{†} = {- E}_{A}^{*} 퓔 ρ_{00} Y^{- 1} + O (퓔^{3}),

(44)

where s = i/(E_A −E_C ^*) and X and Y are the square matrices defined as

X = E_{A}^{*} E_{C}^{*} I_{2 F + 3} - g_{0}^{2} V^{†} V,

(45)

Y = E_{A}^{*} E_{C}^{*} I_{2 F + 1} - g_{0}^{2} V V^{†} .

(46)

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

X V^{†} = V^{†} Y, VX = YV .

(47)

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

- 2 γ ρ_{bb} - [s^{*} g_{0}^{2} V^{†} (ρ_{aa} - 퓔^{2} ρ_{00} Y^{- 1}) V - s^{*} g_{0}^{2} V^{†} V ρ_{bb} + h . c .] = 0,

(48)

- 2 κ ρ_{aa} + [s^{*} g_{0}^{2} (ρ_{aa} - 퓔^{2} ρ_{00} Y^{- 1}) V V^{†} - s^{*} g_{0}^{2} V ρ_{bb} V^{†} + i E_{A}^{*} 퓔^{2} ρ_{00} Y^{- 1} + h . c .] = 0,

(49)

2 κ ρ_{aa} + 2 γ \sum_{q} D_{q} ρ_{bb} D_{q}^{†} - [i E_{A}^{*} 퓔^{2} ρ_{00} Y^{- 1} + h . c .] = 0,

(50)

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

V ρ_{bb} V^{†} - V V^{†} \sum_{q} D_{q} ρ_{bb} D_{q}^{†} = 0 .

(51)

Let us define Hermitian matrices 𝓐 and 𝓑 as

퓑 = \frac{ρ_{bb}}{η 퓔^{2} g_{0}^{2}}, 퓐 = \sum_{q} D_{q} 퓑 D_{q}^{†},

(52)

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

V퓑 V^{†} = V V^{†} 𝓐,

(53)

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

It is already shown in Ref. [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]

, 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).

] 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

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]

] 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

ρ_{bb} = η 𝓔^{2} g_{0}^{2} 𝓑 + O (𝓔^{4})

(54)

ρ_{aa} = η 𝓔^{2} {∣ E_{A} ∣}^{2} {({VV}^{†})}^{- 1} 𝓐 + O (𝓔^{4})

(55)

ρ_{00} = η {({YY}^{†}) ({VV}^{†})}^{- 1} 𝓐 - O (퓔^{2}) .

(56)

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 𝓔⁰ for ρ ₀₀ and 𝓔² 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.

Appendices

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

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).

], rather than the conventional Zeeman basis. In this frame, the polarization vector can be written as

\hat{e} = \sqrt{\cos 2 ε} {\hat{e}'}_{0} - \sqrt{2} \sin ε {\hat{e}'}_{+ 1} .

(57)

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

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]

] as

퓐_{ij} = \sum_{k = - F - 1}^{Min [i, j]} \frac{1}{(i - k)! (j - k)!}

\times \sqrt{\frac{(2 F + 1 + i - k)! (2 F + 1 + j - k)!}{(2 F + 1 - i + k)! (2 F + 1 - j + k)!}}

\times C_{Fi (F + 1) - k}^{(2 F + 1) (i - k)} C_{Fj (F + 1) - k}^{(2 F + 1) (j - k)} {(\frac{\sin ε}{\sqrt{\cos 2 ε}})}^{i + j - 2 k},

(58)

퓑_{ij} = \sum_{k = - F}^{Min [i, j]} \frac{1}{(i - k)! (j - k)!}

\times \sqrt{\frac{(2 F + 1 + i - k)! (2 F + 1 + j - k)!}{(2 F + 1 - i + k)! (2 F + 1 - j + k)!}}

\times C_{(F + 1) i F - k}^{(2 F + 1) (i - k)} C_{(F + 1) j F - k}^{(2 F + 1) (j - k)} {(\frac{\sin ε}{\sqrt{\cos 2 ε}})}^{i + j - 2 k} .

(59)

The dimension of 𝓐 is (2F +1)×(2F +1) while 𝓑 is (2F +3)×(2F +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 α₀, α₁, and α₂

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

] as

α_{0} = \frac{1}{\cos 2 ε} \sum_{l = 0}^{F} C_{2 l}^{2} P_{2 l} (1 / \cos 2 ε)

(60)

α_{1} = P_{2 F + 1} (1 / \cos 2 ε)

(61)

α_{2} = \cos 2 ε \sum_{l = 2 F}^{2 F + 2} D_{l}^{2} P_{l} (1 / \cos 2 ε),

(62)

where the P_l (x) denotes the Legendre polynomials and the coefficients C_l and D_l are given by Eq. (77) of Ref. [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]

] as

C_{1} = \sqrt{\frac{(2 l + 1) (2 F - 1)! (2 F + l + 1)!}{(2 F + 1) (4 F + 1)!}}

(63)

D_{1} = \sqrt{(2 F + 3) (4 F + 3)} C_{10 2 F + 10}^{l 0} \times (\begin{matrix} l & 1 & 2 F + 1 \\ F & F + 1 & F + 1 \end{matrix}),

(64)

where the last factor denotes the 6j symbol. Since 1/cos2ε ≥ 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:

lim_{ε \to \pm \frac{π}{4}} \frac{α_{1}}{α_{0}} = lim_{ε \to \pm \frac{π}{4}} \frac{α_{2}}{α_{0}} = 1 .

(65)

From the recursion relation of Legendre polynomials

(2 l + 1) x P_{l} (x) = (l + 1) P_{l + 1} (x) + l P_{l - 1} (x),

(66)

the coefficients α ₀ and α ₂ in Eqs. (60)–(62) become

α_{0} = α_{1} + \frac{2 F}{2 F + 1} P_{2 F - 1} (1 / \cos 2 ε) + \frac{1}{\cos 2 ε} \sum_{l = 0}^{F - 1} C_{2 l}^{2} P_{2 l} (1 / \cos 2 ε),

(67)

α_{2} = α_{1} - \frac{2 F \cos 2 ε}{4 F + 1} P_{2 F} (1 / \cos 2 ε),

(68)

from which the following inequality follows,

α_{0} \geq α_{1} \geq α_{2} .

(69)

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).

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(270.5580) Quantum optics : Quantum electrodynamics
(300.6210) Spectroscopy : Spectroscopy, atomic

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: January 19, 2010
Revised Manuscript: February 15, 2010
Manuscript Accepted: April 16, 2010
Published: April 19, 2010

Citation
Sungsam Kang, Youngwoon Choi, Sooin Lim, Wookrae Kim, Jung-Ryul Kim, Jai-Hyung Lee, and Kyungwon An, "Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels," Opt. Express 18, 9286-9302 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9286

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References

J. Ye, D.W. Vernooy, and H. J. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987-4990 (1999). [CrossRef]
P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, “Trapping an atom with single photons,” Nature 404, 365-368 (2000). [CrossRef] [PubMed]
A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon sources for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002). [CrossRef] [PubMed]
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]
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]
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]
. C. K. Law and J. H. Eberly, “Arbitrary control of a quantum electromagnetic field,” Phys. Rev. Lett. 76, 1055-1058 (1996). [CrossRef] [PubMed]
T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966).
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]
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]
. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).
A. V. Ta˘ıchenachev, A.M. Tuma˘ıkin, V. I. Yudin, and G. Nienhuis, “Steady state of atoms in a resonant field with elliptical polarization,” Phys. Rev. A 69, 033410 (2004). [CrossRef]
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]
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]
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]
G. Nienhuis, “Natural basis of magnetic substates for a radiative transition with arbitrary polarization,” Opt. Commun. 59353-356 (1986). [CrossRef]
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. 86787-790 (2001). [CrossRef] [PubMed]
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 57R2293-R2296 (1998). [CrossRef]
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 71013817 (2005). [CrossRef]
J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2001). [CrossRef]
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]
T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488-490 (2007). [CrossRef] [PubMed]
D. Jacob, M. Vallet, F. Bretenaker, A. L. Floch, and M. Oger, “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671-673 (1995). [CrossRef]
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]
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]
K. M. Birnbaum, A. S. Parkins, and H. J. Kimble, “Cavity QED with multiple hyperfine levels,” Phys. Rev. A 74, 063802 (2006). [CrossRef]
A. V. Taichenachev, A. M. Tumaikin, 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).
A. M. Tumaikin 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).

Ahmadi, P.
Alt, W.
Birnbaum, K. M.
Boca, A.
Boozer, A. D.
Brecha, R. J.
Bretenaker, F.
Buck, J. R.
Carmichael, H. J.
Chapman, M. S.
Cummings, F.W.
Dembowski, C.
Dotsenko, I.
Eberly, J. H.
Fischer, T.
Floch, A. L.
Fortier, K. M.
Fuhrmanek, A.
Furusawa, A.
Georgiades, N. Ph.
Gibbons, M. J.
Goh, K. W.
Gräf, H. -D.
Hahn, J. W.
Harney, H. L.
Heine, A.
Heiss, W. D.
Hennrich, M.
Ilchenko, V. S.
Jacob, D.
Jaynes, E. T.
Kampschulte, T.
Khudaverdyan, M.
Kim, J. W.
Kim, S. Y.
Kimble, H. J.
Kippenberg, T. J.
Kubanek, A.
Kuhn, A.
Kuzmich, A.
Law, C. K.
Lee, H-W.
Lee, J. Y.
Lenhard, K.
Loncar, M.
Mabuchi, H.
Maunz, P.
McKeever, J.
Meschede, D.
Miller, R.
Morris, J. R.
Murr, K.
Nienhuis, G.
Northup, T. E.
Oger, M.
Parkins, A. S.
Pinkse, P. W. H.
Puppe, T.
Rauschenbeutel, A.
Rehfeld, H.
Reick, S.
Rempe, G.
Rice, P. R.
Richter, A.
Scherer, A.
Schorner, K.
Schuster, I.
Shore, B. W.
Spillane, S. M.
Ta?ichenachev, A. V.
Taichenachev, A. V.
Thobe, A.
Tuma?ikin, A.M.
Tumaikin, A. M.
Vahala, K. J.
Vallet, M.
Vernooy, D. W.
Vernooy, D.W.
Vuckovic, J.
Webster, S. C.
Widera, A.
Wilcut, E.
Wilk, T.
Xiao, M.
Ye, J.
Yoo, Y. S.
Yudin, V. I.

Applied Optics

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]

JETP

A. V. Taichenachev, A. M. Tumaikin, 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).

Nature

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

Nature Phys.

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]

New J. Phys.

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]

Opt. Commun.

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]
G. Nienhuis, “Natural basis of magnetic substates for a radiative transition with arbitrary polarization,” Opt. Commun. 59353-356 (1986). [CrossRef]

Opt. Lett.

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

Phys. Rev. A

A. V. Ta˘ıchenachev, A.M. Tuma˘ıkin, V. I. Yudin, and G. Nienhuis, “Steady state of atoms in a resonant field with elliptical polarization,” Phys. Rev. A 69, 033410 (2004). [CrossRef]
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]
K. M. Birnbaum, A. S. Parkins, and H. J. Kimble, “Cavity QED with multiple hyperfine levels,” Phys. Rev. A 74, 063802 (2006). [CrossRef]
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 57R2293-R2296 (1998). [CrossRef]
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 71013817 (2005). [CrossRef]
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]
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]

Phys. Rev. E

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

Phys. Rev. Lett.

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. 86787-790 (2001). [CrossRef] [PubMed]
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]
. C. K. Law and J. H. Eberly, “Arbitrary control of a quantum electromagnetic field,” Phys. Rev. Lett. 76, 1055-1058 (1996). [CrossRef] [PubMed]
A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon sources for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002). [CrossRef] [PubMed]
J. Ye, D.W. Vernooy, and H. J. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987-4990 (1999). [CrossRef]
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]

Proc. IEEE

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]

Science

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

Sov. Phys. JETP

A. M. Tumaikin 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).

Other

T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966).
. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).

2009, Khudaverdyan, Phys. Rev. Lett.
2008, Schuster, Nature Phys.
2008, Khudaverdyan, New J. Phys.
2007, Fortier, Phys. Rev. Lett.
2007, Boozer, Phys. Rev. A
2007, Wilk, Science
2006, Birnbaum, Phys. Rev. A
2005, Spillane, Phys. Rev. A
2004, Ta?ichenachev, Phys. Rev. A
2004, McKeever, Science
2002, Kuhn, Phys. Rev. Lett.
2001, Vuckovic, Phys. Rev. E
2001, Dembowski, Phys. Rev. Lett.
2000, Pinkse, Nature
2000, Lee, Applied Optics
1999, Ye, Phys. Rev. Lett.
1999, Brecha, Phys. Rev. A
1998, Vernooy, Phys. Rev. A
1996, Law, Phys. Rev. Lett.
1996, Taichenachev, JETP
1995, Jacob, Opt. Lett.
1991, Carmichael, Opt. Commun.
1990, Tumaikin, Sov. Phys. JETP
1986, Nienhuis, Opt. Commun.
1983, Morris, Phys. Rev. A
1963, Jaynes, Proc. IEEE

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Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels

Article Tools

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Abstract

1. Introduction

2. Stationary state of the system

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

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

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

3. Effective coupling constant in natural basis

3.1. Morris-Shore transformation into natural basis

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

4. Possible applications to cavity-QED experiments

5. Derivation of the stationary-state density matrix

6. Conclusion

Appendices

Appendix A: Expression for matrix elements of 𝓐 and 𝓑

Appendix B: Analytic expressions for coefficients α₀, α₁, and α₂

Acknowledgments

References and links

References

Applied Optics

JETP

Nature

Nature Phys.

New J. Phys.

Opt. Commun.

Opt. Lett.

Phys. Rev. A

Phys. Rev. E

Phys. Rev. Lett.

Proc. IEEE

Science

Sov. Phys. JETP

Other

Cited By

Figures

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OpticsInfoBase

Optics Express

Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels

Article Tools

Article Navigation

Abstract

1. Introduction

2. Stationary state of the system

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

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

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

3. Effective coupling constant in natural basis

3.1. Morris-Shore transformation into natural basis

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

4. Possible applications to cavity-QED experiments

5. Derivation of the stationary-state density matrix

6. Conclusion

Appendices

Appendix A: Expression for matrix elements of 𝓐 and 𝓑

Appendix B: Analytic expressions for coefficients α0, α1, and α2

Acknowledgments

References and links

References

Applied Optics

JETP

Nature

Nature Phys.

New J. Phys.

Opt. Commun.

Opt. Lett.

Phys. Rev. A

Phys. Rev. E

Phys. Rev. Lett.

Proc. IEEE

Science

Sov. Phys. JETP

Other

Cited By

Figures

Appendix B: Analytic expressions for coefficients α₀, α₁, and α₂