Population Dynamics and Emission Spectrum of a Cascade Three-Level Jaynes Cummings Model with Intensity-Dependent Coupling in a Kerr-like Medium

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1 Commun. Theor. Phys. (Beijing China) 45 (006) pp c International Academic Publishers Vol. 45 No. 4 April Population Dynamics and Emission Spectrum of a Cascade Three-Level Jaynes Cummings Model with Intensity-Dependent Coupling in a Kerr-like Medium ZHOU Qing-Chun Department of Mathematics and Physics Jiangsu University of Science and Technology Zhenjiang 1003 China (Received August ) Abstract By using the method of eigenvectors the atomic populations and emission spectrum are investigated in a system that consists of a cascade three-level atom resonantly interacting with a single-mode field in a Kerr-like medium. The atom and the field are assumed to be initially in the upper atomic state and the Fock state respectively. Results for models with intensity-dependent coupling and with intensity-independent coupling are compared. It is found that both population dynamics and emission spectrum show no indications of atom-field decoupling in the strong field limit if the intensity-dependent coupling is taken into account. PACS numbers: p 4.50.Ct Key words: population emission spectrum cascade three-level atom Kerr-like medium 1 Introduction The Jaynes Cummings model (JCM) describes the interaction of a two-level atom with a single mode of the electromagnetic field and it has been realized in experiment by using a superconducting cavity and Rydberg atoms. [1 3] Theoretical and experimental investigations of the JCM in the past several decades have helped us in understanding the quantum nature of the interaction between light and matter. [4] Much physical information of the light-matter interaction such as the energy-level structure of the system instabilities and relative populations of the energy levels are provided by spectrum. So the calculation of spectrum has been an important aspect of quantum optics. The emission spectra of the JCM were studied in Refs. [5] [7]. It is well known that the atomic emission spectrum is sensitive to environment. For instance when a two-level atom is driven by a strong coherent field in free space the spectrum exhibits the Mollow triplet. In the weak excitation limit a singlet is observed instead. When an excited two-level atom is placed in a cavity a doublet spectrum is obtained due to the vacuum-field Rabi splitting. In this context the JCM was extended to the situation where the cavity contains a Kerr-like medium as well as a two-level atom. [8] The dynamics and emission spectrum of this model have been studied. [89] Recently a three-level atom interacting with radiation fields in a nonlinear medium has attracted attention. For example Li et al. investigated dynamic behaviors of the atomic occupation probabilities and transfer phenomena in a three-level atom interacting with one- or two-mode fields in a Kerr-like medium. [1011] Abdel-Aty et al. explored the degree of entanglement of a three-level atom and cavity fields in the presence of nonlinearities of fields. [113] In this paper we discuss the atomic emission spectrum and dynamics of a cascade three-level atom bound in a cavity filled with a Kerr-like medium. The dependence of the coupling coefficients on field intensity is taken into account in our system. We will see that this intensity-dependent coupling makes the atom behave quite differently from the intensity-independent coupling case and that the emission spectrum displays new features. The paper is organized as follows. In Sec. we present the theoretical model and its eigenenergies and eigenstates. In Sec. 3 we study the time evolutions of atomic populations for an initial Fock state field. The emission spectrum is calculated in Sec. 4. Section 5 is devoted to a short summary of the main results of the paper. Model and Its Solution Fig. 1 Energy level structure for the three-level atom. Consider a cascade three-level atom interacting resonantly with a single-mode electromagnetic field in a high- Q cavity filled with a Kerr-like medium. The energy-level structure of the system is sketched in Fig. 1 where a b and c denote the ground the intermediate and the The project supported by the Qing Lan project of Jiangsu Province of China under Grant No. 005SL00J

2 78 ZHOU Qing-Chun Vol. 45 excited states of the atom respectively and ω c represents the resonant frequency of the cavity. Transitions between a and b and between b and c are dipole allowed while the transition between a and c are dipole forbidden. In the dipole and rotating-wave approximations the Hamiltonian for the system is H hω c a a hω c σ aa + hω c σ cc + hχa a + h{[g bc af(a a)σ cb + g ab af(a a)σ ba ] + h.c.}. (1) Here a (a) is the creation (annihilation) operator of the field mode χ is related to the third-order nonlinearity of the Kerr-like medium σ ij i j (i j a b c) are atomic operators f(a a) a a g ab f(a a) and g bc f(a a) play the roles of intensity-dependent coupling coefficients between the atom and the field. We keep the form of f(a a) in Eq. (1) so as to make it easy to compare between intensity-dependent atom-field coupling and the intensity-independent atom-field coupling cases by merely replacing f(a a) with a a and 1 in the following results. For convenience we will take the coupling constants to be real and let g ab g bc g. The eigenenergies and eigenvectors of H can be easily obtained by solving the eigenequation H φ n j Ej n φ n j (j 1 3). () The eigenvectors can be written as φ n j Cjm ψ n m n (3) where m1 ψ n 1 c; n ψ n b; n + 1 ψ3 n a; n + (4a) with a b c denoting the atomic states and n n + 1 n + representing the photon numbers in the cavity mode. The expansion coefficients in Eq. (3) are as follows: (n + 1)(n + ) Cj1 n g f(n + 1)f(n + ) n + f(n + ) Cj n g (nχ u n j ) C n j3 [un j (un j nχ) g (n + 1) f(n + 1)] {g 4 (n + 1)(n + )f (n + 1)f (n + ) + g (n + )f (n + )(nχ u n j ) + [g (n + 1)f (n + 1) u n j (u n j nχ)] } 1/.(4b) The egenvalues of H are Ej n h[(n + 1)ω c + n(n + 1)χ u n j ] (5) where u n 1 R cos ϕ 3 A 3 un R cos π + ϕ A 3 3 u n j R cos π ϕ A 3 3 (6) with ϕ arccos q R q A3 7 AB 6 + C R Sign(q) B 3 A 1/ A χ 9 B 4n(n + 1)χ g [(n + 1)f (n + 1) + (n + )f (n + )] C g χ[n(n + )f (n + ) (n + 1) f (n + 1)]. (7) 3 Dynamics of Atomic Populations With the help of Eqs. () (7) we are able to calculate atomic populations in atomic states. In this section and the next we assume that initially the atom is in the upper state c and the cavity field in a Fock state with photon number n. Thus the initial state of the system can be expressed by state vector Ψ AF (0) c; n ψ n 1. (8) The atomic populations of states c b and a are respectively given by W c (t) Ψ AF (0) σ cc (t) Ψ AF (0) Cj1C n j1 n e iun j t W b (t) Ψ AF (0) σ bb (t) Ψ AF (0) Cj1C n j n e iun j t W a (t) Ψ AF (0) σ aa (t) Ψ AF (0) Cj1C n j3 n e iun j t. (9) In Fig. we present relationships between the atomic populations and the scaled time for different values of χ and n. A comparison of Figs. (b) (c) (e) and (f) shows that the presence of a Kerr-like medium in the cavity reduces the oscillation amplitudes of the populations on the upper atomic level and the ground level for both intensitydependent and intensity-independent coupling cases. This phenomenon indicates that the Kerr-like medium tends to weaken the atom-field coupling. The effect is due to the frequency shift of the field in the presence of the medium and it can be easily seen from the motion equation of the field operator a. If we drop terms connected with the atom in Eq. (1) we have i d dt a (ω c + nχ)a which means the field frequency is shifted from ω c to ω c + nχ. In the presence of the Kerr-like medium the atom actually interacts with this field and the field is not resonant with the atomic transitions anymore because of the non-vanishing frequency shift nχ of the field thus reduces the coupling between the atom and the field. When we change the initial field intensity in a cavity filled with Kerr-like medium different features appear in the two

3 No. 4 Population Dynamics and Emission Spectrum of a Cascade Three-Level Jaynes Cummings Model with 79 coupling cases i.e. for large n the atom is almost decoupled from the field and tends to staying in the excited state for the intensity-independent case while the coupling of the atom to the field does not reduce much as n increases [see Figs. (b) and (c)]. As n becomes larger the atom experiences a greater detuning of the field so the atom is reluctant to emit a photon to transit to a lower level. But for the intensity-dependent coupling this is not the case. The effect of increasing the initial field is twofold: on the one hand the equivalent detuning nχ weakens the coupling; on the other hand intensity-dependent factor n strengthens the coupling. The competition of the two processes cancel each other for large n thus the amplitudes of Rabi oscillations change little for χ g n 5 and n 50 in Figs. (b) and (c). We will see that these different dependencies of coupling strength on n make the atomic emission spectra for the two kinds of coupling situations quite unalike. Fig. Time evolutions of atomic populations as functions of scaled time for intensity-dependent coupling ((a) (b) and (c)) and for intensity-independent coupling ((d) (e) and (f)). Parameters: Γ 0.1g T 40/g. 4 Atomic Emission Spectrum The emission spectrum of the cascade three-level atom is defined as [14] S(ν) Γ T T dt dt exp[ (Γ iν)(t t ) 0 0 (Γ + iν)(t t)] Ψ AF (0) [σ ab (t ) + σ bc (t )] [σ ab (t) + σ bc (t)] Ψ AF (0) (10) where T is the interaction time at which the spectrum is measured Γ is the bandwidth of the filter. The initial state of the system is still assumed to be given by Eq. (8). By making use of Eqs. () (3) and (8) we arrive at the final expression for the atomic emission spectrum: S n (ν) Γ l1 G n jlz n jl (11)

4 730 ZHOU Qing-Chun Vol. 45 with G n jl C n j1c n jc n 1 l3 + C n j1c n j1c n 1 l Zjl n exp[i(ν + En 1 l / h Ej n / h)t ] exp[ ΓT ] Γ + i(ν + E n 1 l / h Ej n/ h). (1) Equation (9) holds for n 1. The calculation of emission spectrum S 0 (ν) for the initial vacuum field needs C 1 jm and E 1 j which are not included in Eqs. (4) and (5) so we have to find them separately. The results are E 1 E 1 3 hgf(1) (13) C 1 C3 1 1 C 1 3 C (14) S 0 (ν) Γ l G 0 jlz 0 jl. (15) In Fig. 3 we display atomic spectra for various values of χ and n. In the absence of the Kerrr-like medium (χ 0) the emission spectrum is symmetric about the resonant frequency. It can be observed from Figs. 3(a) and 3(b) that the presence of the Kerr-like medium shifts the whole spectrum to a higher frequency and suppresses spectral lines at higher frequencies while enhances spectral lines at lower frequencies. In the large-n limit there are only three spectral peaks in the emission spectrum. The third peak on the rightmost side of the three has so small a height that it is difficult to be observed. This peak is due to the atomic transition a b and the other two visible peaks come from contributions of both dipole-allowed transitions with the transition b c having a heavier weight. From Fig. (c) we know that the probability of the transition a b is small when n is large but it never vanishes and this explains why the corresponding third peak is quite low. The three peaks reach their asymptotic peak values rapidly and recede away from the resonant frequency as n increases. For comparison we also depict the atomic spectrum in the case of intensity-independent coupling for χ g and various values of n in Fig. 4. In contrast to the intensity-dependent coupling case only one spectral peak remains in the spectrum in the large-n limit and the peak moves to the resonant frequency ω c as the initial field gets stronger. All the other spectral peaks including the visible peak for n 5 and n 10 on the right-hand side of the spectrum in Fig. 4 disappear if n is large enough. The physics behind this phenomenon is that the atom is nearly decoupled from the field for strong initial field in the intensity-independent coupling case. So the atom tends to remaining in the excited atomic state and this tendency results in a longer decay time which is seen in the spectrum as a single narrow central peak. [315] The non-single peak structure of the emission spectrum in the strong field limit for the intensity-dependent coupling case is a reflection of a persistent coupling of the atom and the field. Fig. 3 Atomic emission spectra for fields of various initial photon numbers in the intensity-dependent coupling case in the absence of a Kerr-like medium (a) and in the presence of a Kerr-like medium (b). The atom is initially in the upper state. Parameters: Γ 0.1g T 40/g. Fig. 4 The same as Fig. 3 but for the intensityindependent coupling case. 5 Conclusion We have investigated the dynamics of an atom with equally spaced three levels resonantly interacting with a single-mode radiation field in a cavity filled with a Kerrlike medium. Atomic populations and atomic emission

5 No. 4 Population Dynamics and Emission Spectrum of a Cascade Three-Level Jaynes Cummings Model with 731 spectra for intensity-dependent coupling and those for intensity-independent coupling are calculated and compared. We find that in the limit of the strong initial Fock-state field the atom does not decouple from the field in the intensity-dependent coupling case while it does in the intensity-independent coupling case and this leads to a non-single peak atomic emission spectrum for the intensity-dependent coupling case. References [1] D. Meschede H. Walther and G. Muller Phys. Rev. Lett. 54 (1985) 551. [] G. Rempe H. Walther and N. Klein Phys. Rev. Lett. 58 (1987) 353. [3] M. Brune F. Schmidt-Kaler A. Maali J. Dreyer E. Hagley J.M. Raimond and S. Haroche Phys. Rev. Lett. 76 (1996) [4] B.W. Shore and P.L. Knight J. Mod. Opt. 40 (1993) [5] J.J. Sanchez-Mondragon N.B. Narozhny and J.H. Eberly Phys. Rev. Lett. 51 (1983) 550. [6] J. Gea-Banacloche R.R. Schlincher and M.S. Zubairy Phys. Rev. A 38 (1988) [7] J.H. Eberly C.V. Kunasz and K. Wodkiewicz J. Phys. B 13 (1980) 17. [8] V. Buzek and I. Jex Opt. Commun. 78 (1990) 55. [9] V. Buzek and I. Jex J. Mod. Opt. 38 (1991) 987. [10] W.D. Li Y.Z. Lai and J.O. Liang Opt. Commun. 186 (000) 303. [11] W.D. Li Y.Z. Lai and J.O. Liang J. Mod. Opt. 48 (001) [1] M. Abedel-Aty J. Mod. Opt. 50 (003) 161. [13] M. Abedel-Aty and A.-S.F. Obada Eur. Phys. J. D 3 (003)155. [14] J.H. Eberly and K. Wodkiewicz J. Opt. Soc. Am. 67 (1991) 15. [15] D. Kleppner Phys. Rev. Lett. 47 (1981) 33.