Transient Dynamics of Light Propagation in Λ-Atom EIT Medium

Transcription

1 Commun. Theor. Phys. (Beijing, China) 44 (2005) pp c International Academic Publishers Vol. 44, o. 2, August 15, 2005 Transient Dynamics of Light Propagation in Λ-Atom EIT Medium LI Yong 1,2 and LIU Xu-Feng 3 1 Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing , China 2 Interdisciplinary Center of Theoretical Studies, the Chinese Academy of Sciences, Beijing , China 3 Department of Mathematics, Peing University, Beijing , China (Received January 7, 2005) Abstract We investigate the transient phenomenon or property of the propagation of an optical probe field in a medium consisting of many Λ-type three-level atoms coupled to this probe field and a classical driven field. We observe a hidden symmetry and obtain an exact solution for this light propagation problem by means of the spectral generating method. This solution enlightens us to propose a practical protocol implementing the quantum memory robust for quantum decoherence in a crystal. As a transient dynamic process this solution also manifests an exotic result that a wave-pacet of light will split into three pacets propagating at different group velocities. It is argued that super-luminal group velocity and sub-luminal group velocity can be observed simultaneously in the same system. This interesting phenomenon is expected to be demonstrated experimentally. PACS numbers: Gy, a, Bs Key words: electromagnetically induced transparency, group velocity, collective excitation, polariton The recent experiments [1,2] have demonstrated many exotic natures of light pulse propagation in the solid and gas systems with electromagnetically induced transparency (EIT), [3] such as the phenomena of superluminal group velocity and sub-luminal group velocity. Physically they reflect the quantum coherence effects to correlate the quantum fluctuations. [4] The concepts of atomic coherence and interference have also been applied to lasing without inversion [5] and the enhancement of linear or nonlinear susceptibilities. [6 8] At present the sub-luminal phenomena including the light stopping in an atomic medium have been extensively studied from various points of view. [9] The recent experiments have also shown the ultra-slow group velocity of light in the solid state systems, such as 3-mm thic crystal of Y 2 SiO 5. [10] Most recently, it is found that when a few rubidium atoms are loaded into high-q optical micro-cavity, the superluminal and sub-luminal phenomena can be observed in a refined version of the above mentioned experiments. [11] With possible applications in quantum information field developed in the past decade, [12] some studies on the sub-luminal problem are closely associated with an ideal and reversible transfer technique for the quantum state between light and metastable collective states of matter. [13] Theoretically, the basic idea is based on the control of light propagation in a coherently driven 3-level atomic medium. The exciting fact that the group velocity is adiabatically reduced to zero [13] means the possibility of proposing a more practical protocol to store and transfer the quantum information of photons in the collective excitations in an ensemble of Λ-type 3-level atoms. [14 17] This may open up an interesting prospective for quantum information processing. The studies show that the excitations of matter coupled by light are more stable under some circumstances and thus form the so-called dar-state polaritons. Physically speaing, this is the essence of the problem. Extended to the multi-atom case, the above idea about adiabatic transfer of the state of a single photon into that of an individual atom [18] provides a conceptually simplest approach for the implementation of the quantum memory of photon information. Technically, this approach combines the enhancement of the absorption cross section in multi-atom systems with dissipation-free adiabatic passage techniques. In addition, this significant investigation motivates the protocol for long distance quantum communication [19] based on atomic ensemble. Recently we have deeply investigated the robustness of this ind of quantum memory. [20] To avoid the spatialmotion-induced decoherence in an ensemble of free atoms, we have naturally proposed a protocol that each Λ-type atom is fixed on a lattice site of a crystal. [21] As quantum memories robust for the spatial-motion-induced decoherence, the quasi-spin wave collective excitation of many Λ-type atoms forms a two-mode exciton system with a dynamic symmetry (or a hidden symmetry) depicted by the semi-direct product algebra SU(2) h 2 (h 2 is an algebra of two-mode boson operator). Physically, this hidden symmetry guarantees the stable spectral structure of such dressed two-mode exciton system while its decoherence implies a symmetry breaing. Based on the The project supported by ational atural Science Foundation of China under Grant o and the ational Fundamental Research Program of China under Grant o. 2001CB

2 o. 2 Transient Dynamics of Light Propagation in Λ-Atom EIT Medium 357 above method of atomic quasi-spin wave collective excitation, we have studied the quantum information memory of light based on 3-level or 4-level atomic ensemble in the case of the 2-photon or 3-photon resonant (or nearly resonant) EIT. [22] We have also studied the light propagation in such atomic ensemble systems under the steady state solution. [23] If one only considers the quantum memory, one can focus on the case of single mode light field. However, if one considers the propagation of light pulse in this EIT medium, physically it may be necessary to concern the multi-mode light field [24] since a light pulse can be understood as a superposition of the components of different frequencies. The aim of the present paper is to analyze the transient phenomenon or property of the multi-mode light field propagation in this EIT medium, especially to emphasize the role of the generalized hidden symmetry. In this paper, our proposed system still consists of the quasi-spin wave collective excitation of many Λ-type 3-level subsystems (similar as in Ref. [21]). The metastable state still interacts with an exactly resonant classical field, but the approximately resonant quantized light field, which couples to the transition between the excited state and the relative ground state, is no longer of single mode. We will prove that, in the large limit, one needs to introduce a pair of exciton operators for each mode of quantized light field. These realize the infinite boson algebra local in the mode space (or the frequency domain), but the collective quasi-spin operators intertwining between the meta-stable states and the excited ones generate a single global SU(2) algebra. With the help of this hidden symmetry and the corresponding spectral generating algebra method, we exactly obtain the dressed spectra of the total system formed by the infinite twomode excitons coupling to a multi-mode quantized electromagnetic field in the large limit. The exact solutions for the eigen-states obtained in this way also include the multi-mode dar states extrapolating from photon to one mode quasi-spin wave exciton. Actually, these dressed states describe the polaritons only coupling a quasi-spin wave exciton to a photon for some special case. With the EIT mechanism the external classical field can be artificially manipulated to change the dispersion properties of the quasi-spin wave excitonic medium dramatically so that the quantized probing light can propagate in exotic ways. Our studies in this paper are substantially related to the hidden symmetry and its breaing. The hidden symmetry leads to an exact class of solutions for the light propagation problem, showing that a wave-pacet of light will split into three wave-pacets which propagate at different group velocities, namely, the super-luminal group velocity, the sub-luminal group velocity, and the usual light group velocity. We consider the transient process for the wea light propagation in a medium consisting of many three-level subsystems. It can be an ensemble of free Λ-type atoms, or a crystal with lattice sites attached by Λ-type subspaces. Recently the similar exciton system in a crystal slab with spatially fixed two-level atoms has been extensively discussed with the emphasis on fluorescence process and relevant quantum decoherence problem. [25,26] Fig. 1 Configuration of the quantum memory with Λ- type atoms. (a) Attached on lattice sites of crystal; (b) Resonantly coupled to a control classical field and a quantized probe field. As shown in Fig. 1, the Λ-type subsystem possesses an excited state a, a ground state b, and a meta-stable state c. The transition frequency ω ac from a to c of each atom is resonantly driven by a classical field of Rabi-frequency Ω. The transition frequency ω ab from a to b is coupled to a multi-mode quantized field with the annihilation operator a, and the coupling constant g for the optical mode of wave vector. Strictly speaing, a multi-mode field cannot be resonantly coupled to an atomic transition since it is a superposition of many components of different frequencies. We should emphasize here that we only consider the case that the quantized field is a Gaussian wave pacet in the frequency domain and the center frequency ω 0 just equals ω ab. When the frequency width of the wave pacet is very tiny compared with Ω ( ω Ω), from the viewpoint of approximation it is reasonable to assume that the multi-mode quantized field couples resonantly to the atomic transition from a to b. Therefore, the interaction Hamiltonian of total system reads H = + Ω g a exp(i r j )σ (j) ab exp(iq r j )σ (j) ac + h.c., (1) where r j (j = 1, 2,..., ) denotes the position of the j-

3 358 LI Yong and LIU Xu-Feng Vol. 44 th subsystem, the total atomic number, the wave vector of quantized light of the -th mode and q the wave vector of the classical light field. The flip operators σ (j) αβ = α jj β (α, β = a, b, c) for α β define the quasi-spin between the given levels α and β. The coupling constant g = c/2 hɛv depends on the matrix element of the electric dipole moment between a and b. For simplicity, g and Ω are considered as real, without loss of generality. Generalizing the definition of the excitation operators of single optical mode in Ref. [21], we introduce a class of collective excitation operators of optical multi-mode A = 1 e i rj σ (j) ba (2) with respect to the transition from b to a for each wave vector. Correspondingly, the collective virtual transition from b to c can be described by another class of collective operators C = 1 e iq() rj σ (j) bc, (3) where Q() = q means the momentum conservation in the virtual process of collective transition from b to c. These collective operators create the general collective excitations m, n = ( 1 ) m!n! A m C n b defined for the set of multi-indices m = (m 1, m 2,...), n = (n 1, n 2,...), and the collective ground state b = b, b,..., b with all atoms staying in the same single particle ground state b. For example, the single particle excitations 1 a A b and 1 c C b. These excitations are easy to understand. Indeed, it is obvious that from the ground state b, the first order perturbation of the interaction creates the so-called one exciton quasi-spin wave state, 1 a = 1 1 c = 1 e i rj b, b,..., j-th e iq() rj b, b,..., {}}{ a,..., b, j-th {}}{ c,..., b. (4) Physically, in the large limit under the low excitation condition, that is, there are only a few atoms occupying the states a or c [25] and the population of state b is approximately, the above two classes of quasispin wave excitations behave as two classes of bosons since they have the following bosonic commutation relations [A, A ] = δ,, [C, C ] = δ,, [A, C ] = [A, C ] = 0. (5) To prove the above basic commutation relations, we first calculate [A, A ] = 1 e i( ) r j (σ (j) bb σ(j) aa ) 1 e i( ) r j. Here we have considered that e i( ) r j (σ (j) bb σ(j) aa ) e i( ) r j σ (j) bb (6) in the case of low excitation. For an optical thic crystal with regular lattice structure (or an optical thic atomic ensemble with homogeneous density under low temperature), according to the theory of solid state, we should have 1 e i( ) r j = δ,. (7) The other commutation relations can be proved in a similar way. As argued above, physically the operators A and C depict the collective-excitation processes of bosonic type. The importance of these operators lies in that they define an invariant subspace V C : span{ m, n m = (m 1, m 2,...), n = (n 1, n 2,...)} (8) for the interaction Hamiltonian (1). This means that driven by this Hamiltonian, any collective state from V C evolves to a new collective state which are still in V C. Thus we can use the collective excitations as basic blocs to describe the quantum dynamic process with a hidden symmetry. Let us introduce the following additional collective operators, T = e iq rj σ ca (j), T + = (T ), (9) concerning the transition from c to a, which is resonantly driven by a classic light. Together with the third collective operator T 3 = 1 2 (σ(j) aa σ cc (j) ), they generate SU(2) algebra globally. The commutation relations between SU(2) and the collective operators A and C are easy to calculate. The non-vanishing commutators are as follows: [T, C ] = A, [T +, A ] = C, [T +, C ] = A, [T, A ] = C. (10)

4 o. 2 Transient Dynamics of Light Propagation in Λ-Atom EIT Medium 359 The above close commutation relations and the expression of the interaction Hamiltonian in terms of these excitation operators: H = g a A + ΩT + + h.c. (11) show that there is a dynamic group (algebra) G d for the light excited system, which is generated by A, C, A, C, T ± and T 3. Using Ξ to denote the Heisenberg Weyl algebra generated by A, C, A, and C, we observe that the dynamic group, G d = SU(2) Ξ is a semi-direct product of SU(2) and Ξ because [SU(2), Ξ] Ξ. (12) Actually the above dynamic symmetry of G d is a straightforward generalization of the symmetry of the single quantized optical mode. With this symmetry the Hamiltonian H can be diagonalized in an elegant way by means of the spectrum generating algebra method. [27] To this end we calculate the commutators of H with the generators of G d and light field operators respectively: [C, H] = ΩA, [A, H] = ΩC, [a, H] = g A. (13) It follows that the dar-state polariton operators D = a cos θ C sin θ (14) commute with the Hamiltonian H for the θ satisfying tan θ = g Ω. (15) It is obvious that the dar-state polariton operators satisfy the bosonic commutation relations and define new dressed excitations mixing the electromagnetic field and collective excitations of quasi spin wave. Especially these new excitations are stable since [D, H] = 0. (16) For the construction of the complete collective space dressed by the quantized light fields, another ingredient is the bright-state polariton operator B = a sin θ + C cos θ, (17) satisfying [D, B ] = 0. It also extrapolates from the light field of a to the exciton of C when one adiabatically changes θ from π/2 to zero. Evidently, the product state 0 = b 0 l is an eigen-state of H with zero eigen-value where 0 l is the vacuum of the electromagnetic field. So we can construct a degenerate class of zero eigen-valuestates or dar states, d(n) d(n, t) = 1 n! D n 0, (18) where the adiabatic time dependence originates from the change of Rabi-frequency Ω for the artificially-controlled classical field. The fact that dar states are cancelled by H means they can trap the electromagnetic radiation from the excited states. Physically this is due to quantum interference cancelling. Maing full use of the hidden symmetry, we have constructed a class of dar states above. That means the coherent optical information of multi-mode can be stored in a medium with collective effect in EIT. The storing and reading-out processes can be controlled by stimulated photon transfer between the classical and quantized light field. In what follows, we will consider the transient dynamic process for the interaction between light fields and the EIT medium. Some exotic natures of light propagation in such a medium will be investigated with the help of the dynamic symmetry analysis. The evolution of the Heisenberg operators corresponding to the optical field and the collective excitations can be described as ȧ = i g A, Ȧ = i g a iωc, Ċ = iωa, (19) for each light mode. The above motion equations of a, A, and C with respect to a given optical mode are coupled together, but there is no coupling of mode operators to those of different optical mode. Due to the semi-direct product property of the dynamic group G = SU(2) Ξ Γ, the generators T +, T, and T 3 of SU(2) algebra do not occur in the above system of Heisenberg equations. Here Γ is the Heisenberg Weyl group generated by the creation and annihilation operators a and a. It leads to the evolution equations of the bright- and dar-state polariton operators, Ȧ = i g 2 + Ω2 B, Ḃ = i g 2 + Ω2 A, Ḋ = 0, (20) in a straightforward way. The above equations manifest the basic features of the dar states: decoupled from other states and stable in the time evolution. With the initial conditions determined by those for C and a, the exact solutions of B and D are obtained as follows: D (t) = D (0) = a (0) cos θ C (0) sin θ, (21) B (t) = O 1 e itθ + O 2 e itθ, (22) where O 1 = 1 2 (B (0) A (0)), O 2 = 1 2 (B (0) + A (0)), a (0), A (0), B (0), and C (0) are the initial Heisenberg operators, and Θ = g 2 + Ω2 is the light field dressed Rabi frequency that measures the effective coupling of excitonic states to the external field.

5 360 LI Yong and LIU Xu-Feng Vol. 44 The dar-state polariton operator D (t) is indeed a time-independent constant (when Ω eeps invariable). That is, in the Heisenberg picture, the operator D (t) always retains the initial -mode dar-state polariton operator D (0) for each and the atomic system always eeps being an EIT medium. Here for simplicity we assume that the quantized light field propagates along the x-axis. Then, the positive frequency part of the quantized light field, (x, t) = hc 2ɛV a (t)e ix ict, (23) can be expressed explicitly in terms of the mode operators of quantized light field a (t) = D (t) cos θ + B (t) sin θ = [a (0) cos θ C (0) sin θ ] cos θ (O 1e itθ + O 2e itθ ) sin θ, (24) which results from the expressions of bright- and darstate polariton operators given above. With the above simple solution we can straightforwardly investigate how the quantized light propagates in the EIT medium. We assume that the collective state of the medium is initially in the ground state b and the initial state of the light field is a wave pacet, which can be depicted by the direct product of many coherent states, or the multi-mode coherent state α = α, (25) where α = exp[ f 2 ( 0 ) 2 ] (up to a normalized factor) since the probe light is a Gaussian wave pacet in frequency domain. This is because the expectation value of the free quantized electromagnetic field operator (x, t) = hc 2ɛV a (t)e ix ict resembles a classical wave pacet as the superposition of infinite components of different frequencies. In interaction with EIT medium, the mean of a (t) over the initial state ψ(0) = b α is a (t) = a (0) cos 2 θ + sin θ ( O 2 e itθ + O 1 e itθ ) = a (0) (cos 2 θ + sin 2 θ cos Θ t), (26) where we have used A (0) = C (0) = 0, B (0) = a (0) sin θ. So the mean of (x, t) is decomposed into three parts: (x, t) = hc 2ɛV a (0) (cos 2 θ + sin 2 θ cos Θ t)e i(x ct) = 0 (x, t) + + (x, t) + (x, t), (27) where ± (x, t) = 0 (x, t) = hc 2ɛV a (0) sin 2 θ e i[(x ct) Θt], hc 2 2ɛV a (0) cos 2 θ e i(x ct). (28) It should be noticed that the light field dressed Rabi frequency Θ modifies the dispersion relations of light propagation in the exciton medium. The three parts in the above decomposition of (x, t) can be understood as three wave pacets spreading in the coordinate space. We can see this point by considering the initial quantized light field in the frequency domain as a Gaussian wave pacet with the center frequency ω 0 (= 0 c = ω ab ). For the first wave pacet + (x, t), its position center is determined by maximizing the wave-vector-dependent phase: (ct + Θ t x) 0 = 0. (29) This determines the group velocity of the wave pacet + (x, t) as V g+ = x = c + Θ. (30) t 0 0 The term on the right-hand side simply leads to a modification of the group velocity ( V g+ = c 1 + Ω n ), (31) 2 0 c 1 + n where n = g 2 0 /Ω 2. This is one of the central results of this paper. It is evident that the first wave pacet + (x, t) represents the super-luminal light propagation with the group velocity larger than the light speed c. It is also emphasized that the super-luminal light propagation can be understood according to the classical theory of wave propagation in an anomalous dispersion medium. It is the interference between different frequency components that results in this rather counterintuitive effect. In this sense, we thin that the super-luminal light pulse propagation observed in experiments is not too odd and there is no direct connection between this phenomenon and the causality in relativity. It is not surprising to see the usual group velocity V g0 = c for the second wave pacet 0 (x, t). But for the third wave pacet (x, t) we find the sub-luminal group velocity similarly: V g = c Θ ( = c 1 Ω n ). (32) c 1 + n This sub-luminal group velocity phenomenon is expected to find applications in quantum information processing. In fact, it is a common sense to believe that the system stopping and slowing light can be used to store quantum information of photon qubits and in the storage time quantum information processing may be possible if we can suf-

6 o. 2 Transient Dynamics of Light Propagation in Λ-Atom EIT Medium 361 ficiently reduce the dissipative loss during the reading and writing operations. We can also analytically integrate out 0 (x, t), + (x, t), and (x, t) and obtain an explicit depiction of the propagating wave pacet. Since g = c/2 hɛv, it is convenient to write g 2 as G2 with G = c/2 hɛv. Replacing the summation over by integral and using a (0) = α, we obtain the normal part 0 (x, t) = 1 2π 2L me c 2ɛV Ω 2 Ω 2 + G 2 e f 2 ( 0) 2 e i(x ct) d, (33) and the super-luminal and sub-luminal (or negative) parts ± (x, t) = 1 c G 2 L me 2π 2ɛV Ω 2 + G 2 e f 2 ( 0) 2 e ix it(c±θ ) d, (34) where L me is the scale of integration along the propagation direction (x-axis) of the probe light. We then expand the terms, Ω 2 /(Ω 2 + G 2 ), G 2 /(Ω 2 + G 2 ), and Θ around the central value 0 of the wave vector of the input probe light. We can neglect the higher-order terms of = 0 since is very close to 0. Then, with the defined parameters A = G 2 /Ω 2, Ω 0 = Ω 1 + A 0, the approximate expressions of field components are obtained analytically as ± (x, t) A 0 J 0 {2f 2 + id 0 [x (c ± E 0 )t]} [ exp 1 4f 2 [x (c ± E 0)t] 2] e i0(x c±t), (35) 0 (x, t) 2J 0 [2f 2 + if 0 (x ct)] [ exp 1 4f 2 (x ct)2] e i0(x ct), (36) where c ± = c ± (Ω 0 / 0 ) and D 0 = 3 + A (1 + A 0 ), E 0 = F 0 = 1 A (1 + A 0 ), Ω 0 A 2(1 + A 0 ), J 0 = L me c0 /(2πɛV ) 4f 2. (1 + A 0 ) The above result proves again the coexistence phenomena of both the super-luminal and sub-luminal (or negative) group velocities. From Eqs. (35) and (36), it is obvious that the group velocity of the part 0 (x, t) is V g0 = c, and the group velocity of wave pacet E ±(x, + t) is ( V g± = c ± E 0 = c 1 ± Ω n ), (37) 2 0 c 1 + n which is the same as Eqs. (31) and (32). Through the above theoretical analysis based on dynamic algebraic method, several interesting properties of the light pulse propagation in an EIT medium are observed. The central result is the possibility of coexistence of sub-luminal and super-luminal group velocity phenomena, occurring as a triple split of wave pacet when spreading in such an EIT medium. We also observe that if we inject coherent light into the EIT medium for a certain time, there may appear three light pulses. One of these three light pulses still propagates at the velocity c, but the other two have super-luminal and sub-luminal group velocities respectively. The sub-luminal group velocity might even have negative value in some cases. This result is different from that of stable process in which only one single group velocity appears. This is because we here only consider the transient phenomenon or property. It is also remared that both super-luminal and sub-luminal (even or negative) group velocities are natural properties of light [2] and could be realized experimentally. Fig. 2 The dynamical evolution of three parts of the light pulse in normalized units. (a) The normal part E 0 with velocity v g0 = c; (b) the super-luminal part E + with v g+ 1.75c; (c) The sub-luminal part E with v g 0.25c. The system parameters are given as: Ω K 0c Hz, n = g 2 0 /Ω 2 = 3. ow we numerically simulate the dynamic process of the light pulse propagating in the above described EIT medium. According to Eqs. (35) and (36), we calculate the shape evolutions of the wave pacets of the three parts of the light field, respectively. Let λ 0 = 2π/ 0 = 2πc/ω 0 be the central wavelength of the initial wave pacet of the light pulse. Define E i (x, t) = i (x, t) + E ( ) i (x, t), for i = 0, ±. It is shown as in Fig. 2 that the three parts E 0 (x, t) and E ± (x, t) of light field indeed have different group velocities. The group velocities of E 0,± (x, t) are v g0 = c, v g+ 1.75c, and v g 0.25c, respectively. It is noted that in order to satisfy the near-resonance condition, the frequency width of the experimental probe light pulse should be very narrow and the spatial width contains large numbers of wavelengths, much more than those given

7 362 LI Yong and LIU Xu-Feng Vol. 44 above, where it is convenient to see the pulse splitting. Fig. 3 The dynamical evolution of three parts of the light pulse in normalized units. (a) The normal part E 0 with velocity v g0 = c; (b) the super-luminal part E + with v g+ 2.36c; (c) The part E with a negative group velocity v g 0.36c. The system parameters are given as Ω K 0c Hz, n = g 2 0 /Ω When the system parameters are changed so that Ω n > 1, (38) 2 0 c 1 + n the group velocity of E (x, t) will be less than 0, v g < 0. This seem-to-be exotic phenomenon of light pulse propagation is also illustrated in Fig. 3 under satisfying Eq. (38). Seen from Fig. 3, these three parts will possess the normal group velocity v g0 = c, super-luminal group velocity (v g+ 2.36c), and negative group velocity (v g 0.36c), respectively. In fact the amplitude of these field components depends on the system parameters. It is obviously seen from Eq. (28) that the intensity of E + (x, t) is equal to that of E (x, t), but is not equal to that of E 0 (x, t). The ratio of the amplitude of E ± (x, t) to that of E 0 (x, t) is approximately sin 2 θ 0 2 cos 2 = g2 0 θ 0 2Ω 2 = n 2. (39) When the ratio n/2 is very small ( 1), most part of the light pulse will propagate in the EIT medium at the group velocity c. If the ratio n/2 is very large ( 1), then most part of the light pulse propagates at super-luminal or sub-luminal (even negative) group velocity. To see the dynamic details more clearly we also plot the caves of the spatial wave pacets for different fixed instances in Figs. 4 and 5. Fig. 4 Evolution of three splitting wave pacets of light pulse (in normalized units). (a) The initial un-split wave pacet with the spatial width of light pulse at time t = 0; (b) [(c)] The shapes of the three splitting wave pacets at times t = 20 [t = 40]. The cases with normal, super-luminal, and sub-luminal are depicted respectively by solid line, dotted line, and dashed line. The parameters are the same as those in Fig. 2. Fig. 5 Evolution of three splitting wave pacets of light pulse (in normalized units). (a) The initial un-split wave pacet with the spatial width of light pulse at time t = 0; (b) [(c)] The shapes of the three splitting wave pacets at times t = 20 [t = 40]. The cases with normal, superluminal and negative-luminal are depicted respectively by solid line, dotted line, and dashed line. The parameters are the same as those in Fig. 3. Figure 4 shows the time evolution of these three wavepacets at t = t 0 (= 0), t 1, t 2 (0 < t 1 < t 2 ) according to Fig. 2. The field components E + (x, t) and E (x, t) with

8 o. 2 Transient Dynamics of Light Propagation in Λ-Atom EIT Medium 363 super-luminal and sub-luminal group velocities change their phase-shapes of wave-pacet during the evolution, but the part E 0 (x, t) always propagates at c with an unchanged shape of wave-pacet. Due to the intrinsic quantum coherence the EIT medium does not change the height of each wave-pacet. Strictly speaing, it is not c but c 0 (= c/n 0 ) that is the vacuum velocity of light, where n 0 is the index of refraction. [28] Here for simplicity, we set n 0 = 1 in our numerical simulation in this wor. To see the exotic phenomenon with negative group velocity more clearly, we tae the two-dimensional curves at certain time in Fig. 5 according to Fig. 3. Then, an inverse-direction light propagation can be seen clearly. In conclusion, we have investigated how the quantized light propagates in a three-level Λ type EIT medium by directly solving the Heisenberg evolution equation of this light field based on the dynamic group method. For simplicity, the quantized light field is considered as resonantly coupling to the Λ type subsystem even though this light field contains a series of light modes and cannot be resonant simultaneously. The physical reason for this consideration is that the light wave pacet has a small width in frequency domain. In fact, the quantized light is considered as a quasi-plane wave and propagating in the medium without the boundary effect. It should be noticed that the present treatment is valid only when the EIT medium is prepared in the low density excitation situation. We wish to emphasize again that it is also owing to the wave nature of light that the coexistence phenomena of both the super-luminal and sub-luminal (or negative) group velocities appears as predicted in this paper. It is very interesting to observe this coexistence phenomena experimentally and explore its potential application in quantum memory and quantum information process. To this end we need more details of physical considerations on the experimental techniques. For instance, we need to compare the size of the sample of the EIT medium and the split distances of three wave pacets resulting from the light pulse. In principle due to the different group velocities the evolution of sufficiently-long time will distinguish the wave pacets, which might not preserve their shape because of wave pacet spreading or the dissipation and decoherence due to the coupling to environment. On the other hand, what we predict is the case of transient phenomenon, which is only effective in short time. So it is somehow difficult to observe this coexistence phenomenon of both the super-luminal and sub-luminal (or negative) group velocities in a practical experiment. Acnowledgments We sincerely than Y. Wu, P. Zhang, and L. You for useful discussions with them. Appendix: Analytic Result from Sub-dynamics Mehtod In general we consider a physical system with a dynamic symmetry characterized by a Lie group G. This means that the Hamiltonian of the considered system H = H[G ] H(g 1, g 2,...) (A1) is a function of the generators g 1, g 2,... of G. These generators can be understood as the basic dynamic variables of the system. Suppose there exists a subgroup S G such that [H, G ] S. (A2) Let s 1, s 2,... be the generators of S. Then the system of the Heisenberg equations i d dt s = [s, H ] S (A3) about s 1, s 2,... are closed to form the so-called subdynamics. Through the subset {s 1, s 2,...} of the complete set of dynamic variables g 1, g 2,..., the sub-dynamics depict the main features of the considered system. The excitonic system in this paper serves as a practical example for G = SU(2) Ξ Γ, (A4) and S = Ξ Γ, where Γ is the Heisenberg Weyl group generated by the creation and annihilation operators a and a of light field. References [1] L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, ature 397 (1999) 594. [2] L.J. Wang, A. Kuzmich, and A. Dogariu, ature 406 (2000) 277; Y. Shimizu,. Shioawa,. Yamamoto, M. Kozuma, T. Kuga, L. Deng, and E.W. Hagley, Phys. Rev. Lett. 89 (2002) [3] S.E. Harris, Physics Today 50 (1997) 36. [4] M.O. Scully, Phys. Rev. Lett. 55 (1985) [5] S.E. Harris, Phys. Rev. Lett. 62 (1989) [6] M.O. Scully, Phys. Rev. Lett. 67 (1991) [7] S.E. Harris, et al., Phys. Rev. Lett. 64 (1990) 1107; M. Fleischhauer, M.D. Luin, A.B. Matso, and M.O. Scully, Phys. Rev. Lett. 82 (1999) [8] Y. Gu, Q. Sun, and Q. Gong, Phys. Rev. A 67 (2003) ; Y. Gu, Q. Sun, and Q. Gong, Phys. Rev. A 69 (2004) [9] Ö.E. Müstecaplioglu and L. You, Phys. Rev. A 64 (2001) [10] A.V. Turuhin, et al., Phys. Rev. Lett. 88 (2002)

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