Atomic coherence via an nonsingular-dom reservoir

Transcription

1 Physics Letters A Atomic coherence via an nonsingular-dom reservoir Chunguang Du, Chunfeng Hou, Zhengfeng Hu, Shiqun Li Key Laboratory for Quantum Information and Measurements, Ministry of Education, Department of Physics, Tsinghua University, Beijing , PR China Received 29 July 2002; received in revised form 29 July 2002; accepted 25 November 2002 Communicated by P.R. Holland Abstract An open Λ-type three-level atom in a photonic crystal can become nearly transparent to a resonant probe field even if the density of modes DOM has no mathematical singularity. A high-q defect mode can induce perfect coherent population trapping. The role of singularity/smoothness of DOM is analyzed Elsevier Science B.V. All rights reserved. PACS: Gy; Qs; a; Qk Keywords: Atomic coherence; Nonsingular density of modes; Photonic crystals There has been a growing interest in the investigation of the propagation of electromagnetic EM waves in periodic dielectric structures or photonic band-gap PBG materials in the last decade [1]. It is expected that light emission and propagation can be controlled arbitrarily, and realization of various new optical devices and/or circuits, by utilizing the PBG and/or the dispersion relation between the photon energy and the wave vector. On the other hand, the spontaneous emission in PBG structures can be strongly modified and gives rise to some interesting effects, which has been extensively investigated [2 6]. The propagation of a probe laser field in a photonic crystal doped with threelevel atoms exhibits some interesting features due to the atomic coherence induced by the modified reser- * Corresponding author. address: duchunguang@tsinghua.org.cn C. Du. voir, which has attracted considerable interest recently. A Λ-type atomic system can be transparent to a weak probe laser field whose frequency is near one of the atomic transition frequency when the other atomic transition frequency is near a band edge [7]. Photon photon correlations and entanglement due to this coherent effect can also occur in a photonic band-gap material [8]. The transient properties of this kind of coherent phenomenon including transient gain without population inversion are also investigated [9,10]. This transparency scheme do not require any external driving field to induce atomic coherence, which is an advantage over common EIT [11] schemes. In all of these works, the DOM of the radiation reservoir was assumed to be singular at the band-gap edge and/or defect mode frequencies. The mathematical singularity of the DOM seems to be necessary to these phenomena caused by atomic coherence. However, it can be argued that in reality the DOM will not actually di /02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. doi: /s 转载

2 C. Du et al. / Physics Letters A given by H = Ω g k,λ e iω k ω 12 t 1 2 a k,λ k,λ + H.C. i γ Fig. 1. Energy schematic for analysis. verge e.g., in a real crystal. It is not clear whether these counter-intuitive effects can occur in a real crystal. In this Letter we will analytically investigate the response of an open Λ-type atomic system to a weak laser field in an nonideal photonic crystal with nonsingular DOM, and show that some counter-intuitive effects including the strong reduction of probe absorption are possible. In contrast with Ref. [7], we will consider the effect of the defect modes formed by atomic doping on the dynamics of this system, and will analyze the role of the singularity of both the defect modes and the band-edge. It is interesting that a high-q defect mode formed by atomic doping can induce true population trapping, in which case the ground-state population will not decrease after a transition time even in the presence of the background decay of the excited atomic state. However, in a photonic crystal without any defect, the true population trapping never occurs, although the infinite small absorption on a very weak field is possible. In fact, the total population of this open atomic system is depleted except for the case when a singular distributed high-q defect mode is present. The atomic system under consideration is shown in Fig. 1. Similar to Ref. [7], the transition 1 2 is taken to be nearly resonant with a photonic band edge, while the other transition 0 1 is assumed to be far away from the gap and can therefore be treated as occurring in free space. A probe laser field couples to the transition 0 1. The Hamiltonian of the system h = 1 in the interaction picture and rotating wave approximation is Here g k,λ = i 2πω k /V ɛ k,λ µ 12,whereV is the quantization volume, ɛ k,λ is the unit polarization vector, a k,λ is the photon annihilation operator, and ω k is the angular frequency of the k,λ mode of the modified radiation reservoir vacuum field. Ω = µ 01 ɛet is the Rabbi frequency, with µ 01 being the dipole matrix element of 0 1 transition and ɛ being the polarization unit vector. γ denotes the background decay to all other states of the atom. It is assumed that the transition frequencies to these states are far from the gap-edge frequency so that such back ground decay can be treated as Markovian process. The spontaneous transition 1 2, however,isa non-markovian process due to the strong coupling of it with the modified reservoir. We assume that initially the field is in vacuum state and the atom is in ground state 0. Thewave function can still be written in terms of the bare eigenvectors [7] such that ψt = c 0 t 0, 0 +c 1 te iδt 1, 0 + c k,λ te iω k ω 12 δt 2, k,λ. 2 k,λ We assume that the defect mode formed by atomic doping is localized at the atomic site in a volume V d rl 3 of several r 3 lattice cells L 3 [12 14]. In the dilute regime N< ω12 πcr 3 d >rl, where N is the atomic number density. One can neglected dipole dipole interactions and tunneling hoping of photons between atoms [15,16] and defect modes can serve as high-q cavities. Then, in the vicinity of the upper edge of the PBG, ω g,the DOM function can be written as ρω= ρ g Θω ω g ω ωg + ρ d δω ω d. 3

3 198 C. Du et al. / Physics Letters A The probability amplitude equations can be obtained from Eqs. 1 and 2 as follows: i d dt c 0t = Ωtc 1 t, i d dt c 1t = Ωte iδt c 0 t t i 0 δ + i γ c 1 t 2 dt Gt t c 1 t, Gt t = G g t t + G d t t, G g t t = β 3/2 e i[π/4+δ g δt t ] πt t t <t, G d t t = βd 2 e iδ d δt t t <t, 4 where δ d = ω d ω 12, δ g = ω g ω 12, G g and G d are Green functions which correspond to the band-gap edge and the defect mode, respectively. Here β d and β are coupling constants of the atom with the structured reservoir [3], whose main contributions are near the defect mode frequency ω d and the gap-edge frequency ω g, which are defined as β 3/2 = 2ω7/2 12 µ c 3, βd 2 = 2ω4 12 µ 12 2 π 2 cr 3. In general, this open lambda atomic system has no true steady-state due to the background spontaneous decay of the excited state to other states besides what are illustrated in Fig. 1. In fact, the population in the ground state cannot approach a nonzero steady-state value, which can be easily to see if we do not assume c 0 t = 1. Hence, the above mentioned steady-states are quasi-steady states. However, there is a exceptional case when, in the presence of a singular distributed defect mode with frequency ω d. In this case, when δ = δ d, the population trapping will occur, which can be shown by use of Laplace transformation on Eq. 4, which yields, C 0 s = 1 s If δ = δ d, 1 c 0 t = lim s 0 [ sc0 s ] = iω 2 s δ + i γ 2 + i Gs + is + iω 2 β 2 d Ω 2 + βd 2,. 6 7 where Gs is the Laplace transformation of Gt. However, in this case c 1 t = 0, i.e., the population is trapped in atomic level 0 and 2. The population trapping arises from the strong singularity of the DOM at the defect mode frequency. In practical, the population trapping can occur only in the case where the Q-factor of the defect mode is very high. However, the population trapping cannot occur in the absence of defect modes, even in the case the DOM is singular at band-gap edge. We will further to investigate the more general situation where the density of modes DOM is a nonsingular function. We assume the DOM has the form of ρω= c ω ωg 8 π ω ω g + ε Θω ω g, with Θ being the Heaviside step function and ω g being the gap frequency. Here ε is the cut-off-smoothing parameter and c is a suitable constant [17]. According to the tables of integrals in Ref. [18] the Green function can be given by { G g t t = β 3/2 e iδ g δt t e i π 4 πt t [ εe iεt t 1 erf ] } εt t e i π 4, 9 where erf is the error function. The Laplace transformation of G g t can be written as G g s = iβ 3/2 e iπ/4 s + iδg δ + iε. 10 In the case of weak probe field and the atom is in ground state initially, c 0 t = 1 for all times can be assumed, and Eqs. 4 can be solved analytically by means of Laplace transformation and using final-value theorem. We can also treat with the case where the DOM distribution is nonsingular at both the PBG-edge frequency and the defect-mode frequency. In this case the DOM function can be written as ρω= c ω ωg π ω ω g + ε Θω ω g γ d + K π ω ω d 2 + γd 2, 11

4 C. Du et al. / Physics Letters A a Fig. 3. The steady-state absorption Im[χδ] spectra for a nonsingular DOM in the presence of a defect mode, where β d = 1, d = 1, ɛ = 0.01, the thick solid curve is for γ d = 0.01, the solid curve for γ d = 0.05, the dashed curve for γ d = 0.1, and the dotted curve for γ d = 10. Other parameters are the same as that in Fig. 2. b Fig. 2. The steady-state absorption Im[χδ] a and dispersion spectra Re[χδ] b in arbitrary units for a nonsingular DOM in the absence of defect mode. It is calculated under the weak-field approximation for parameters γ = 1andδ g = 0. The solid curve is for ɛ = 0, the dot-dashed curve for ɛ = 0.01, the dashed curve for ɛ = 0.1, and the dotted curve for ɛ = 10. All parameters are in unit of β. where γ d is the cut-off-smoothing parameter, γ d = ω d /2Q, Q represent the Q-factor of the defect mode, c, K are suitable constants. Then G d t t = β 2 d e iδ d δ+γ d t t Laplace transformation of G d t is βd 2 G d s =. s + iδ d δ + γ d Thus the linear susceptibility [7] can be given by χδ= 4πN µ δ i γ 2 + β 3/2 δg δ+ ε + β 2 d δ d δ+iγ d The imaginary part of χ determines the absorptive properties and the real part of it determines the dispersive properties of this system. It is easy to see that for a high-q defect mode, χ is very small at δ = δ d as well as δ = δ g. Fig. 2 shows the steady-state absorption and dispersion spectra where the DOM is nonsingular at defect mode frequency as well as at PBG-edge frequency. From this figure one can see that the absorption closes to zero when γ d 0. Fig. 3 shows the steady-state absorption and dispersion spectra. It is obvious that the absorption and dispersion is nearly zero at δ = δ g,wherethe absorption increases with ε. The absorption spectra is faraway from Lorentz profile except for very large ε case. When ε, the absorption spectra becomes Lorentz profile. It is interesting that in the presence of the defect mode with also nonsingular DOM with nonzero γ d, the absorption at δ = δ d will decrease with ɛ, i.e., the DOM s nonsingularity at δ = δ g is profitable for the probe transparency at δ = δ d see Fig. 4. In conclusion, we have investigated the steady-state properties of a Λ-type atom with one transition coupled with a modified radiation reservoir. It was shown that the coherent phenomena, e.g., the strong reduction of the steady-state absorption can occur in nonsingular DOM case as well as in singular DOM case. Although the perfect transparency Imχ = 0 requires ε 0orγ d 0, the steady-state absorption can be decreased to be nearly zero in a wide region

5 200 C. Du et al. / Physics Letters A a is very weak so that the time evolution of the ground state population can be neglected. However, there is a exceptional case where the true population trapping can occur when the probe field is tuned to be Ramanresonant to a high-q defect mode. Our results suggest that such structured-reservoirinduced coherent phenomena can occur in real photonic crystals with nonsingular DOM and exhibit many interesting properties. This kind of system have many advantages over ordinary EIT and promises to be applicable to a wide variety of quantum optical or nonlinear optical phenomena [8]. The dispersive and transient properties of this system, such as the group velocity reduction and the probe pulse reshaping, is another interesting topic which will be investigated in the future. Acknowledgements This work is supported by the National Nature Science Foundation of China Grant No and funded by the National Fundamental Research Program 2001 CB References b Fig. 4. The dependence of the steady-state absorption at δ = δ d on the smoothing parameter ɛ for different γ d, the other parameters are the same as that in Fig. 2. of the cut-off-smoothing parameters. It is interesting that, in the case of δ = δ d, the probe absorption will decrease when the smoothing parameter ɛ increases, i.e., the nonsingularity of the DOM at band-edge is profitable to the transparency for the probe field being Raman resonant with the defect mode. In general, this open atomic system has no true steady state due to the background spontaneous emission of the excited atomic level unless the probe field [1] See, for example, J. Mod. Opt , special issue on photonic band structures; E.N. Ecomonou Ed., Photonic Band Gap Materials, NATO Advanced Study Institute, Series E: Applied Science, Vol. 315, Plenum, New York, 1996; C.M. Soucoulis Ed., Photonic Band Gaps and Localization, NATO Advanced Study Institute, Series B: Physics, Vol. 38, Plenum, New York, [2] E. Yablonovitch, Phys Rev. Lett [3] S. John, T. Quang, Phys. Rev. A [4] H. Huang, X.H. Lu, S.Y. Zhu, Phys. Rev. A [5] H. Huang, X.H. Lu, S.Q. Li, Phys. Rev. A [6] S.Y. Zhu, H. Chen, H. Huang, Phys. Rev. Lett [7] E. Paspalakis, N.J. Kylstra, P.L. Knight, Phys. Rev. A R33. [8] D. Petrosyan, G. Kurizki, Phys. Rev. A [9] D.G. Angelakis, E. Paspalakis, P.L. Knight, Phys. Rev. A [10] C.G. Du, Z.F. Hu, C.F. Hou, S.Q. Li, Chin. Phys. Lett [11] For reviews see: E. Arimondo, Coherent Population Trapping in Laser Spectroscopy, in: E. Wolf Ed., Progress in Optics, Vol. XXXV, Elsevier Science, Amsterdam, 1996, p. 257;

6 C. Du et al. / Physics Letters A M.O. Scully, M.S. Zubeairy, Quantum Optics, Cambridge Univ. Press, Cambridge, England, 1997, Chapters 7 and 14; See also: H. Lee, M. Fleischhauter, M.O. Scally, Physics A , and references therein. [12] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton Univ. Press, Princeton, NJ, [13] P.R. Villeneuve, S. Fan, J.D. Joannopoulos, Phys. Rev. B ; E. Yablonovitch, T.J. Gmitter, R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, Phys. Rev. Lett [14] B. Sherman, G. Kurizki, A. Kadyshevitch, Phys. Rev. Lett ; G. Kurizki, B. Sherman, A. Kasdyshevitch, J. Opt. Soc. Am. B [15] S. John, J. Wang, Phys. Rev. Lett ; S. John, J. Wang, Phys. Rev. B [16] S. Bay, P. Lambropoulos, K. Mølmer, Phys. Rev. A [17] A.G. Kofman, G. Kurizki, B. Sherman, J. Mod. Opt [18] I.S. Gradstein, I.M. Ryshik, Tables of Series, Products, and Integrals, Verlag Harri Deutsch, Thun, 1981, Vol. 1, p. 511.