Emission Spectrum Property of Modulated Atom-Field Coupling System

Transcription

1 Commun. Theor. Phys. 6 (213) Vol. 6, No. 2, August 15, 213 Emission Spectrum Property of Modulated Atom-Field Coupling System GAO Yun-Feng (Ô ô), 1, FENG Jian (ú ), 1 and LI Yue-Ke (Ó ) 2 1 School of Physical Science and Information Engineering, Liaocheng University, Liaocheng 25259, China 2 School of Medium and Communication Technology, Liaocheng University, Liaocheng 25259, China (Received November 28, 212; revised manuscript received March 18, 213) Abstract The emission spectrum of a two-level atom interacting with a single mode radiation field in the case of periodic oscillation coupling coefficient is investigated. A general expression for the emission spectrum is derived. The numerical results for the initial field in pure number stare are calculated. It is found that the effect of the coupling coefficient modulation on the spectral structure is very obvious in the case of a low modulation frequency and larger amplitude when the initial field is vacuum, which is potentially useful for exploring a modulated light source. PACS numbers: 42.5.Md, 42.5.Pq Key words: emission spectrum, coupling coefficient modulation, two-level atom 1 Introduction The Jaynes Cummings model [1] (JCM) is a milestone in the theory of coherent interaction between a two-level atom and quantized single-mode light field. Using the JCM, some purely quantum phenomena such as collapses and revivals of atomic population inversion, [2] vacuum Rabi splitting [3] has been predicted and observed [4 5] successfully. Since the JCM has been achieved, a variety of generalized JCMs are proposed, such as the two-photon JCM, [6] three-level JCM, [7 8] the JCMs dealing with a Kerr-like medium filling the cavity, [9] intensity-dependent coupling JCM, [1] the phase shifted frequency modulation JCM, [11] etc. In 1993, Joshi and Lawande [12] proposed a generalized JCM with a time-dependent atom-field coupling coefficient and investigated the dynamical as well as field statistical properties for the case of linear variation of the coupling parameter. Although a complex variation of the atom-field coupling can be achieved via controlling the axial movement of atom theoretically, this is virtually impossible for manipulating a natural atom in practice. If the JCMs is realized with artificial atom, [13 16] such as the superconducting Josephson junction circuits, semiconductor quantum dots, etc., the coupling coefficient can be modulated experimentally. Recently, Brüggemann [17] et al. have been achieved the modulation of the coupling coefficient of semiconductor quantum dots and field in a micro-cavity using strain pulses by ultrafast acoustics techniques, and got the emission output enhanced more than two orders of magnitude. The emission spectrum is one of the most important methods in study of atomic structure. But the characteristics of the spectrum depend strongly on the environment of the atom. [18 19] For example, if an atom is placed in a high-finesse optical cavity, the spectral structure of the spontaneous emission can be modified. Since the time-dependent physical spectrum based on the counting rate of a photodetector was introduced by Eberly and Wodkiewicz, [2] a lot of interest has been focused on emission spectrum of generalized JCMs. [21 26] However, the effect of the coupling coefficient modulation on the spectrum was not taken into account. In this paper, we study the emission spectrum of the JCM with a periodic oscillation coupling coefficient. Our attention was focused on the influence of the coupling coefficient modulation on the spectra. We believe this is significant for exploring a new modulated light source by using the artificial atoms. 2 Theoretical Model and Equations We consider a composite atom-field model in which the coupling coefficient is modulated by vibrating of the atom along the cavity axis. Assume that one or both of the amplitude and frequency of the atom vibration is small enough so that the atomic kinetic energy can be ignored compared to its internal energy. The Hamiltonian for such a system in the rotating-wave approximation is written as [12] (we use = 1) H(t) = ω a a ω aσ z + g(t)(aσ + + a σ), (1) where a (a ) is the annihilation (creation) operator for the cavity field mode. σ z, σ, and σ + are the pseudospin operators for the atom. ω is the frequency of the cavity field and ω a is the frequency of the atomic transition. We study the resonant process, i.e. ω a = ω. g(t) is the atom-field coupling coefficient. We consider the coupling coefficient is periodic oscillation with time, i.e. g(t) = g o + sin(ω m t), (2) where g is the average coupling coefficient of atom and field, and ω m is the amplitude and frequency of the modulation respectively. Although the expression of the Supported by the Shandong Province Natural Science Foundation under Grant No. ZR29AM19 gaoyf@lcu.edu.cn c 213 Chinese Physical Society and IOP Publishing Ltd

2 218 Communications in Theoretical Physics Vol. 6 coupling coefficient in equation (2) is very simple compared to the actual case in Ref. [17], we believe that it can characterize the key features of a periodic variation. Select an orthonormal basis of this system as Ψ +,k = +, k = + k, Ψ,k =, k + 1 = k + 1, where the quantum number +( ) represents the atom in upper level (lower level), k represents the photon-number state of the field. To export the time evolution of the system state, we first derive the time evolution of the base vectors. Let Ψ +,k (t) = C +,k (t) +, k + C,k (t), k + 1 (3) with the initial conditions C +,k () = 1, C,k () =. Substitute (3) into the Schrödinger equation ( Ċ +,k (t) +, k + Ċ,k(t), k + 1 = i kω + 1 ) [ 2 ω a C +,k (t) +, k i (k + 1)ω 1 ] 2 ω a C,k (t), k + 1 ig(t) k + 1C +,k (t), k + 1 ig(t) k + 1C,k (t) +, k, ( Ċ +,k (t) = i kω + 1 ) 2 ω a C +,k (t) ig(t) k + 1C,k (t), Ċ,k (t) = ig(t) [ k + 1C +,k (t) i (k + 1)ω 1 ] 2 ω a C,k (t). We get the solution of this equations is C +,k (t) = 1 2 e ie kt (e i(g kt+ω k ) + e i(g kt+ω k ) ) = e ie kt cos(g k t + Ω k ), C,k (t) = 1 2 e ie kt (e i(g kt+ω k ) e i(g kt+ω k ) ) = i e ie kt sin(g k t + Ω k ), (4) where E k = (k + 1/2)ω, g k = g k + 1, Ωk = k + 1( /ω m )(1 cosω m t). Similarly, for Ψ,k (t) = D +,k (t) +, k + D,k (t), k + 1 (5) with D +,k () =, D,k () = 1, we get D +,k (t) = 1 2 e ie kt (e i(g kt+ω k ) e i(g kt+ω k ) ) = i e ie kt sin(g k t + Ω k ), D,k (t) = 1 2 e ie kt (e i(g kt+ω k ) + e i(g kt+ω k ) ) = e ie kt cos(g k t + Ω k ). (6) If the initial state of the system is the atom in the upper level and light field in an arbitrary state, namely, ϕ() = q k +, k = q k Ψ +,k, (7) k k where q k is the field number-state expansion coefficient. The time evolution of the system state is ϕ(t) = k q k Ψ +,k (t) = k q k [C +,k (t) +, k + C,k (t), k + 1 ]. (8) 3 Expression of the Atomic Emission Spectrum Following the definition of the physical spectrum, [2] the atomic emission spectrum may be written S(ω) = 2Γ dt exp[ (Γ iω)(t t )] dt exp[ (Γ + iω)(t t)] ϕ() σ + (t )σ(t) ϕ(), (9) where Γ is the bandwidth of the spectrometer, T is the time at which the measurement takes place, ϕ() is the initial state of the system, σ + (t ) and σ(t) is the atomic raising and lowering operator in Heisenberg picture respectively. In order to calculate the emission spectrum of the time-dependent Hamiltonian system, we transform the two-time correlation function ϕ() σ + (t )σ(t) ϕ() to Schrödinger picture representation: ϕ() σ + (t )σ(t) ϕ() = ϕ() U (t )σ + U(t )U (t)σu(t) ϕ() = ϕ(t ) σ + U(t )U + (t)σ ϕ(t) = l,k ϕ(t ) σ + U(t ) Ψ l,k Ψ l,k U + (t)σ ϕ(t) = l,k ϕ(t ) σ + Ψ l,k (t ) Ψ l,k (t) σ ϕ(t), (1) where U(t) is the unitary time-evolution operator, Ψ l,k is an orthonormal basis, i.e. Ψ l,k Ψ l,k = δ ll δ kk and Ψ l,k Ψ l,k = 1. l,k Substituting (1) into (9) yields S(ω) = 2Γ dt exp[ (Γ iω)(t t )] dt exp[ (Γ + iω)(t t)] l,k ϕ(t ) σ + Ψ l,k (t ) Ψ l,k (t) σ ϕ(t),

3 No. 2 Communications in Theoretical Physics 219 S(ω) = 2Γ l,k dt exp[ (Γ iω)(t t )] ϕ(t ) σ + Ψ l,k (t ) dt exp[ (Γ + iω)(t t)] Ψ l,k (t) σ ϕ(t). then Let A lk = dt exp[ (Γ + iω)(t t)] Ψ l,k (t) σ ϕ(t), (11) S(ω) = 2Γ l,k A lka lk. (12) For the system components of a two-level atom and a single-mode field, the above equation can be written as [ ] S(ω) = 2Γ A, ( A +,k 2 + A,k 2 ) with A ±,k = k= (13) dt exp[ (Γ + iω)(t t)] Ψ ±,k (t) σ ϕ(t). (14) Substituting the equations (3) (8) into (14), we obtain A +,k 2 = q k+1 2{ dt e Γ(T t) cosνt sin(g k t + Ω k )cos(g k+1 t + Ω k+1 ) dt e Γ(T t) sin νt sin(g k t + Ω k )cos(g k+1 t + Ω k+1 ) }, (15a) A,k 2 = q k+1 2{ dt e Γ(T t) cosνt cos(g k t + Ω k )cos(g k+1 t + Ω k+1 ) dt e Γ(T t) sin νt cos(g k t + Ω k )cos(g k+1 t + Ω k+1 ) }, (15b) where ν = ω ω. Using (13) and (15), we can calculate the atomic emission spectrum finally. 4 Numerical Results and Discussion Equations (13) and (15) show that the spectrum of a superposition state field is the sum of the pure number state field spectrum, and the weight factors are just the photon number distribution in the superposition state. The spectral characteristics of arbitrary state can be obtained by analyzing the number state spectrum. Therefore, we only discuss the case of the number state initial field. Consider that the atom is initially in its upper level and the field is in a pure number state with the photonnumber n, i.e. { 1, i = n, q i =, i n, equation (15) shows A ±,k only in conditions of k + 1 = n. Therefore, equation (13) can be simplified as S(ω) = dt e Γ(T t) cosνt sin(g n 1 t + Ω n 1 ) dt e Γ(T t) sin νt sin(g n 1 t + Ω n 1 ) dt e Γ(T t) cosνt cos(g n 1 t + Ω n 1 ) dt e Γ(T t) sin νt cos(g n 1 t + Ω n 1 ). (16) To analyze the influence of the modulation amplitude and frequency on the spectral structure, we give the numerical results by equation (16). In this paper, we always select T = 4/g, Γ =.2g. Firstly, we analyze the effects of modulation amplitude under the conditions of a lower modulation frequency (ω m =.1g ). Clearly, the range of the modulation amplitude is g. Where, = corresponds to the case that the atom does not move (the standard JCM), and = g corresponds to the case that the atom moves furthest to the node of the wave field so that the instantaneous coupling strength become zero.

4 22 Communications in Theoretical Physics Vol. 6 Fig. 1 Influence of modulation amplitude of coupling coefficient on the atomic emission spectrum for a pure number state field when (a) = ; (b) =.2g ; (c) =.5g ; (d) = g. Other parameters ω m =.1g, T = 4/g, Γ =.2g. Fig. 2 Influence of modulation frequency of coupling coefficient on the atomic emission spectrum for a pure number state field when (a) ω m =.2g ; (b) ω m =.5g ; (c) ω m = g. Other parameters =.2g, T = 4/g, Γ =.2g. Figure 1 shows the influence of modulation amplitude on the spectrum. For purposes of comparison, Figure 1(a) gives the emission spectrum for the zero modulation amplitude (i.e. the standard JCM). In the condition of the vacuum initial field (n = ), with the modulation amplitude increases, the number of Rabi peaks from one pair of the standard JCM increase to six pairs gradually. The pair of the highest peaks is always located in the outside and their splitting increases as increases. When the initial field is not vacuum, the peak number of the two sidebands increases and their height reduce rapidly with the increasing of. But the pair of center peaks is almost not affected. When the modulation amplitude is larger and the initial field is strong, the two sidebands disappear and the pair of central peaks remains only. Their height is unchanged and their splitting is broadened slightly with increasing. Modulating the coupling can make an increase in the emission spectral lines, and this effect is most obvious when initial field is a vacuum. Secondly, we analyze the influence of modulation fre-

5 No. 2 Communications in Theoretical Physics 221 quency on the emission spectrum with fixed modulation amplitude, shown in Fig. 2. For spontaneous emission (n = ), the significant effect on the spectrum only occurs in ω m = =.2g. When the modulation frequency is not equal to the amplitude modulation, the emission spectrum is roughly the same as the one of the standard JCM the difference is some additional small peaks only [see Figs. 1(a), 1(b), and 2]. When n >, the increase of modulation frequency does not affect the central two peaks, but leads to changes in the two sidebands. The number of sideband peaks increases and the peak height decreases gradually as ω m increasing from zero to.2g. However, the further increasing of ω m causes the spectral structure changes at a reverse trend. When ω m = g, the sidebands are almost three peaks as shown in Fig. 2(c). 5 Conclusion We have studied the atomic emission spectra for a twolevel atom in a high-q cavity with a modulated coupling coefficient. The expression for emission spectrum with arbitrary initial field was obtained. The numerical result of spectral structure for the atom in upper level with the field in number state initially was given. Our main attention was focused on the influence of the coupling coefficient modulation on the spectra. When the modulation frequency ω m of the coupling coefficient is less than or approximately equal to the modulation amplitude, the modulation will affect the spectral structure. When ω m, the modulation has no effect on the emission spectrum. In the case of vacuum field initially, the Rabi peak number of spontaneous emission spectrum and spacing of the highest pair peaks both increase with the modulation amplitude increase. In the case of strong initial field, the increasing of modulation amplitude makes the peak number of sidebands increasing and the peak height reducing gradually until disappearing. However, with the increasing of modulation amplitude, the height of a pair of center peaks remains unchanged and the peak spacing increased slightly. In the case of weak initial field, the modulation of interaction affects the atomic emission spectral structure sensitively. For a system composed by an artificial two-level atom and single-mode cavity with resonance, the number of emission spectral line will increase result in the periodic vibration of the atom. This effect is most obvious when the initial field is a vacuum. In order to obtain a rich emission spectrum, we should select the vibration with low frequency and larger amplitude. Taking into account the characteristic of that the artificial atom coupling coefficient can be controlled in real time, our result is believed to have potential applications for modulated light sources. References [1] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. [2] J.H. Eberly, N.B. Narozhny, and J.J. Sanchez- Mondragon, Phys. Rev. Lett. 44 (198) [3] J.J. Sanchez-Mondragon, N.B. Narozhny, and J.H. Eberly, Phys. Rev. Lett. 51 (1983) 55. [4] G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [5] R.J. Thompson, G. Rempe, and H.J. Kimble, Phys. Rev. Lett. 68 (1992) [6] D.J. Gauthier, Q.L. Wu, S.E. Morrin, and T.W. Mossberg, Phys. Rev. Lett. 68 (1992) 464. [7] H.I. Yoo and J.H. Eberly, Phys. Rep. 118 (1985) 239. [8] N.N. Bogolubov, F.L. Kien, and A.S. Shumovsky, Phys. Lett. A 11 (1984) 21. [9] V. Buzek and I. Jex, Opt. Commun. 78 (199) 425. [1] B. Buck and C.V. Sukumar, Phys. Lett. A 81 (1981) 132. [11] K.V. Priyesh and R.B. Thayyullathil, Commun. Theor. Phys. 57 (212) 468. [12] A. Joshi and S.V. Lawande, Phys. Rev. A 48 (1993) [13] Y. Yu, S. Han, X. Chu, S.I. Chu, and Z. Wang, Science 296 (22) 889. [14] J.Q. You and F. Nori, Phys. Rev. B 68 (23) [15] T. Stauber, R. Zimmermann, and H. Castella, Phys. Rev. B 62 (2) [16] J. Luo, W. Lai, D. Lu, C.L. Du, Y.W. Liu, S.Q. Gong, D.N. Shi, and C.L. Guo, J. Phys. B: At. Mol. Opt. Phys. 45 (212) [17] C. Bruggemann, A.V. Akimov, A.V. Scherbakov, M. Bombeck, C. Schneider, S. Hofling, A. Forchel, D.R. Yakovlev, and M. Bayer, Nat. Photonics 6 (212) 3. [18] E.M. Purcell, Phys. Rev. 69 (1946) 681. [19] H.B. Casimir and D. Polder, Phys. Rev. 73 (1948) 36. [2] J.H. Eberly and K. Wodkiewicz, J. Opt. Soc. Am. 67 (1977) [21] J. Gea-Banacloche, R.R. Schlicher, and M.S. Zubairy, Phys. Rev. A 38 (1988) [22] J.H. Li, A.X. Chen, J.B. Liu, and X.X. Yang, Opt. Commun. 278 (27) 124. [23] J.H. Li, Eur. Phys. J. D 42 (27) 467. [24] A.S.F. Obada, A.A. Eied, and G.M. Abd Al-Kade, J. Phys. B: At. Mol. Opt. Phys. 41 (28) [25] P. Tang and H. Guo, Chin. Phys. B 18 (29) [26] H.J. Wang and Y.F. Gao, Chin. Phys. B 19 (21) 1429.