Atomic Coherent Trapping and Properties of Trapped Atom

Commun. Theor. Phys. (Beijing, China 46 (006 pp. 556 560 c International Academic Publishers Vol. 46, No. 3, September 15, 006 Atomic Coherent Trapping and Properties of Trapped Atom YANG Guo-Jian, XIA Li-Xin, and XIE Min Department of Physics, Beijing Normal University, Beijing 100875, China (Received September 30, 005 Abstract Based on the theory of velocity-selective coherent population trapping, we investigate an atom-laser system where a pair of counterpropagating laser fields interact with a three-level atom. The influence of the parametric condition on the properties of the system such as velocity at which the atom is selected to be trapped, time needed for finishing the coherent trapping process, and possible electromagnetically induced transparency of an altrocold atomic medium, etc., is studied. PACS numbers: 4.50.Vk, 3.80.-t Key words: velocity selective coherent population trapping, ultracold atom It has long been known that atomic coherent trapping [1] is closely related to many unexpected effects and techniques in modern optics, such as electromagnetically induced transparency (EIT, [] lasering without inversion, [3] and light speed controlling, [4] etc. It usually occurs in multi-level atomic systems in which absorption cancellation, the essence for all new things above, will be induced by the destructive interference between different coherent routes of atomic transitions. On the other hand, the atomic coherent trapping technique has been successfully used for the purpose of obtaining cold or even ultracold atoms, with which one can test again the wave-particle duality that an atom at very low temperatures, besides behaving as a particle, can be viewed as a de Broglie matter wave with its wavelength even larger than that of an electromagnetic field. The recent development in this field has been outlined in the books on atomic optics. [5,6] The applications of matter wave can be understood by reversing the roles of light and atom from their familiar roles in conventional optics. The relative analysis, of course, becomes complicated due to the fact that in the ultracold circumstance the quantized motion of atomic center-of-mass comes into effect, and due to a strong correlation between the atomic internal and the external degree of freedom. The three-level atomic-electromagnetic interaction system is one of the simplest models that can be used for understanding the facts mentioned above. The model under consideration in this letter shares a configuration similar to the velocity selective coherent population trapping (VSCPT, a well-known laser cooling technique. [7] We analyze the problems at hand based on the VSCPT theory and observe slowing down of the atomic trapping process due to the nondegenerance of the two atomic ground states, atomic population transform and EIT effect appearing in an ultracold atom, etc. The detailed model arrangement can be described schematically by Fig. 1. A three-level atom is assumed to have one excited level e o and two ground levels g 1,. A laser field of frequency ω 1 (or wave number k 1 propagates toward +Oz, coupling with the atomic transition between e o and g 1, while a counterpropagating laser field of ω (or wave number k couples with another transition between e o and g. As atomic temperature is low enough that the atomic coherent length becomes longer than the laser wavelength, the description of the quantized center-of-mass motion must be adopted. Thus, for the present consideration, the description of all the atomic states should be replaced by e o, p, g 1, p ħk 1, g, p + ħk. Here each state is labeled by both atomic internal and translational quantum numbers. For example, e o, p denotes the atomic state with a momentum p along Oz. Similar notations apply to g 1, p ħk 1 and g, p + ħk. Furthermore, we assume that the atom may jump from the excited state e o, p to the unexcited state g 1, p ħk 1 or g, p+ħk at the rate γ 1 or γ via spontaneous emission. The state g, p + ħk is also assumed to be a metastable one with small decay rate γ. The Hamiltonian for the coherent part of this system is ( Ĥ = ˆρ ˆρ m e 0, p e 0, p + m + ħ 1 + p ( ˆρ g 1, p ħk 1 g 1, p ħk 1 + [ ħω1 e 0, p g 1, p ħk 1 + ħω e 0, p g, p + ħk g, p + ħk g, p + ħk m + ħ ] + H.c. (1 The project supported by National Natural Science Foundation of China under Grant No. 10174007 and the State Key Basic Research Programs under Grant No. 004CB719903

No. 3 Atomic Coherent Trapping and Properties of Trapped Atom 557 where we have expressed Ĥ in the rotating wave approximation and assumed that the two laser fields are classical. The quantity m stands for the atomic mass, i denotes the detuning between the i-th atomic transition and laser field. The quantity Ω i = d 0i E i /ħ is the corresponding Rabi frequency with the definition of the transition matrix element d 0i and the amplitude of the laser field E i. It is easily seen that there exists a family F (p = { g 1, p ħk 1, g, p + ħk, e o, p } that is closed under the coherent action of Ĥ. Spontaneous emission has been known to play a key role in VSCPT. [7] When it is taken into consideration, the master equation method is effective in dealing with the problem at hand. By defining the matrix elements in the family F (p as ρ 00 = e 0, p ρ e 0, p, ρ 01 = e 0, p ρ g 1, p ħk 1, ρ 0 = e 0, p ρ g, p + ħk, ρ 11 = g 1, p ħk 1 ρ g 1, p ħk 1, ρ = g, p + ħk ρ g, p + ħk, etc., we arrive at optical Bloch equations, ρ 00 ρ 01 ρ 0 ρ 1 ρ 11 ρ = γ 1 + γ ρ 00 i [Ω 1 (ρ 10 ρ 01 + Ω (ρ 0 ρ 0 ], ( γ1 = + i 01 ( γ = + i 0 ρ 01 + Ω ρ 1 + Ω 1 (ρ 11 ρ 00, ρ 0 + Ω 1 ρ 1 + Ω (ρ ρ 00 γρ 0, = i 1 ρ 1 + Ω 1 ρ 0 Ω ρ γρ 1, = γ 1 k1 k 1 duw 1 (u ρ 00 (p k 1 + u iω 1 (ρ 01 ρ 10, = γ k k duw (u ρ 00 (p + k + u iω (ρ 0 ρ 0 γρ, ( and their conjugations ρ 10 / = ( ρ 01 /, ρ 0 / = ( ρ 0 /, ρ 1 / = ( ρ 1 /. For simplicity we have defined here the new atom-field detunings with Doppler shift effect introduced, for the one-photon process 0i (i = 1, and for the two-photon Raman process 1, as 01 = pk 1 m ħk 1 /m 1, pk 0 = m ħk /m, 1 = ħk 1 m ħk m + p (k 1 + k + 1. (3 m In Eq. ( W i (u is a momentum distribution function describing the probability of a photon emitting along Oz direction, given by W i (u = (3/8 ( 1 + u /ħ k i /ħki, (i = 1,. [8] The following two superposition states Ψ + (p = 1 Ω [Ω g 1, p k 1 + Ω 1 g, p + k ], Ψ (p = 1 Ω [Ω 1 g 1, p k 1 Ω g, p + k ] (4 are of central importance in understanding the atomic trapping process, where Ω = [ Ω 1 + Ω ] 1/. Among them the state Ψ + (p is optically non-absorptive, called the dark state. [7] The atom will oscillate between Ψ + (p and Ψ (p at the Doppler shift related frequency 1 under the coherent action of Ĥ. If the atom falls into Ψ + (p with a momentum p = p 0 = ħ ( k1 k + m ( 1, (5 (k 1 + k however, it will stop oscillation and remain in Ψ + (p 0 infinitively. Furthermore, spontaneous emission allows the atom to jump unilaterally from e o, p 0 +ħu into Ψ + (p 0 with the probability W i (u for u k i. In this way it will accumulate there. The state Ψ ± (p has limited lifetime that is proportional inversely to its loss rates Γ ± (p. As the momentum p takes the value in the area very closed to p 0, the function Γ ± (p can be worked out through Ĥ as analyzed in Ref. [7]. Specifically, as Ω i i, γ i, Ω 1 = Ω, and γ 1 = γ, as well as γ being too small to be concerned, Γ + (p has a simple form as Γ + (p = 4 1 (p, k 1, k, 1, γ 1 Ω, (6 where 1 has been defined in Eq. (3. The function Γ (p can be given approximately by its expression at the point p 0 and reads where Γ p p0 = γ (Ω/ (p 0, k 1, k, 1, + (γ /, (7 (p 0, k 1, k, 1, = ħk 1k m k 1 + k 1 k 1 + k and it can be testified (p 0, k 1, k, 1, = 01 (p 0 = 0 (p 0. Obviously, as p = p 0 we have Γ + p=p0 = 0, which means that the state Ψ + (p 0 is indeed dark. On the other hand, the atom trapped in Ψ + (p 0, because its temperature is even lower than its recoil temperature in the theory of VSCPT, can be viewed as two atomic matter waves corresponding respectively to the components of g 1, p 0 ħk 1 and g, p 0 + ħk. These two waves counterpropagate and form a matter-wave beat with a beat frequency proportional to p 0 + ħk p 0 ħk 1. If we make zero atomic velocity selection (p 0 = 0 and apply the laser fields with the same wave numbers (k 1 = k,

558 YANG Guo-Jian, XIA Li-Xin, and XIE Min Vol. 46 the difference between the frequencies of the two components disappears and the beat pattern will change into a standing matter wave. Fig. 1 Schematic drawing of a three-level atom interacting with a pair of counterpropagating laser fields. Each state is labeled by its quantized internal and external (its momentum along Oz numbers. The efficiency of the atomic coherent trapping process can be illustrated by the atomic momentum distribution along Oz. As an example, we display in Fig. the situation at time τ = 160, obtained by directly simulating the equations Eq. ( under the assumption that the atom is initially in g 1 and obeys Gaussian momentum distribution with a standard half-width at p = 3ħk i (Note: In fact, in simulating Eq. ( for Fig. 1 and all other figures in the rest part of the paper, we have scaled the variables and the parameters by tω γj = τ, p /mħω γj = p, ħki /mω γ j = ki, Ω i/ω γj = Ω i, i /ω γj = i, γ i /ω γj = γ i, γ/ω γj = γ, ħ/mω γj u = u, where ω γj = ħkj /m (j = 1, is the recoil frequency related to the wave number k j. Obviously as i = j, it has ħki s/mω γ j = 1. In each figure there are two peaks located respectively at p z at = ±ħ(k 1 + k /. The height of the peak indicates the probability of finding the atom trapped in Ψ + (p 0, and its width determined by Γ + is proportional to the atomic temperature. Beside these two peaks, a wider but lower packet at the right side for k /k 1 < 1 can be seen, and it moves right further as k /k 1 decreases or, as our numerical simulation has demonstrated, as time develops. This phenomenon can be attributed mainly to the fact that the atom, after interacting with a pair of photons coming from the two counterpropagating laser fields of different wave numbers, will get net recoil momentum proportional to ħ (k 1 k in the same direction as that of photon with larger wave number. Obviously this phenomenon is harmful to the trapping process interested. The simplest way to avoid it is to choose the two laser fields fulfilling the condition k 1 = k. Compared to the configuration with the degenerate atomic ground states, however, more time is needed now for nondegenerate case to finish trapping process, and the larger 1 is, the longer the time will be (see Fig. 3. Fig. Atomic momentum distribution P (p = ρ 11 (p + ħk 1 + ρ (p ħk + ρ 00 (p at time τ = 160 for the ratio of the wave-number k /k 1 taking values (a 1.0, (b 0.75, and (c 0.50. Other parameters are Ω 1 = Ω = 5, γ 1 = γ = 5, γ = 0.0, 1 = = 0. The dashed curve corresponds to the initial Gaussian momentum distribution. Equation (4 indicates that the atomic population trapped in the component g 1, p ħk 1 (or g, p + ħk of Ψ + (p 0 is determined by its weight Ω /Ω (or Ω 1 /Ω, which implies that the atomic population transfer between these two components can be readily realized by adiabatically changing the laser field. [9] As an example, figure 4 shows what happens in this process in which the corresponding Rabi frequency Ω i is swept in the way introduced in Ref. [10].

No. 3 Atomic Coherent Trapping and Properties of Trapped Atom 559 The total atomic momentum conservation is clearly shown in this figure, where the probability of the atomic population with momentum p = ħk decreases roughly in the time segment 40 τ 100, while that with momentum p = ħk 1 increases during the same period. The interesting feature of this figure lies in the fact that this type of population transfer takes place between the states that are of both the atomic internal and external degrees of freedom. As the atom-field system develops fully enough, it can be taken approximately as having reached the steady state. Therefore, the explicit expression of the matrix element ρ 0i can be derived from the steady state equations of Eq. (. The matrix element ρ 01 of the atomic transition e 0 g 1, for example, reads iω 1 ρ 11 (p 0 [γ (γ + i (p 0 + Ω 1 (Ω 1 Ω ] ρ 01 (p γ (γ + i (p 0 (γ 1 i (p 0 + Ω (γ + i (p 0 + Ω 1 (γ 1 i (p 0, (8 where the relations ρ 11 (p 0 Ω 1 ρ (p 0 Ω and (p 0 = 01 (p 0 = 0 (p 0 have been used. Fig. 3 Atomic momentum distribution at time τ = 160 for (a 1 = 0, = 10, (b 1 = 0, = 5, and (c 1 = 0, = 0. All other parameters are the same as those in Fig.. The complex polarization corresponding to the atomic transition e 0 g 1 is defined through P 1 = d 01 ρ 01. It is known from Eq. (8 that if we neglect the contribution from the decay rate of the metastable state (γ 0, the zero light refraction appears always under the condition of Doppler-shift-related single-photon resonance (p 0 = 0, while the small light absorption can be observed only in the cases Ω Ω 1 and Ω 1 = Ω. Obviously, if Ω Ω 1, equation (8 reduces to P 1 iρ 11 (p 0 Ω 1/Ω (γ + i (p 0, and the atom becomes transparent approximately for small decay rate γ, accompanied by light refraction with refraction index (p 0 /γ times large as absorption one. If Ω 1 = Ω, the total transparency occurs due to the fact that both the real and the imaginary parts of the polarization vanish. The situation for other sets of the parameters can be illustrated representatively by Fig. 5, where small absorption accompanied by a detectable refraction can be seen in the figure. The above observation, which is done with the atom in ultracold situation, is different from those made with the similar laser-atom configuration at the normal temperature, at which, e.g., the laser field must be adjusted resonant with the atom to observe the zero absorption. [] Nowadays, EIT medium in most of the literatures about its applications is declared to be composed of cold or even ultracold atoms in order to allow the relative measurements more precise, or to create better conditions for observing other optical phenomena interested. The present work may throw a little light on how to get this kind of medium. In conclusion, we have studied the properties of an extended VSCPT system, where two counterpropagating laser fields couple respectively to the transitions of a three-level atom. We obtain from Doppler-shift Raman resonance the momentum of the trapped atom as the function of the wave-numbers of the laser fields and the atom-field detunings, and find the slowing down of the atomic coherent trapping process due to the nondegenerance of the atomic ground states. By sweeping the driven field adiabatically, we obtain the atomic population transfer in which not only the internal but also the external states of the atom change. Contrast to the common knowledge of EIT effect that can be

560 YANG Guo-Jian, XIA Li-Xin, and XIE Min Vol. 46 detected in an atomic system at normal temperature, we observed it in an ultracold atomic system. Fig. 4 Time evolution of the atomic momentum distribution obtained by sweeping the Rabi frequency Ω 1 in the way Ω 1 = 5{1 0.5 tanh[0.1(τ 40] + 0.5 tanh[0.1[τ 100]} for k /k 1 = 1.0. All other parameters are the same as those in Fig.. Fig. 5 Dispersive (solid curve and absorptive (dashed curve parts of polarization scaled by d 01Ω 1ρ 11 (p 0 vs. detuning 1 for Ω 1 = 0.1, Ω = 0.8, k /k 1 = 0.5, γ 1 = γ = 1, γ = 0.01, and = 1. References [1] E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17 (1976 333; G.S. Agarwal and N. Nayak, J. Phys. B 19 (1986 3375. [] S.E. Harris, Phys. Today 50 (1997 36; E. Arimondo, in Progress in Optics, ed., E. Wolf, Elsevier, Amsterdam (1996. [3] S.E. Harris, Phys. Rev. Lett. 6 (1989 1033; O. Kocharovskaya and P. Mandel, Phys. Rev. A 4 (1990 53; M. Scully, S. Zhu, and A. Gavrieliedes, Phys. Rev. Lett. 6 (1989 813. [4] S.E. Harris and L.V. Hau, Phys. Rev. Lett. 8 (1999 46; M.D. Lukin, et al., ibid. 8 (1999 1847; L.V. Hau, et al., Nature (London 397 (1999 594; Olga Kocharovskaya, Yuri Rostovtsev, and Marlan Scully, Phys. Rev. Lett. 86 (001 68. [5] Harold J. Metcalf and Peter van der Straten, Laser Cooling and Trapping, Springer, Berlin (1999. [6] P. Meystre, Atom Optics, Springer, Berlin (001. [7] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61 (1988 86; A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 (1989 11; E. Goldstein, P. Pax, K.J. Schernthanner, B. Taylor, and P. Meystre, Appl. Phys. B 60 (1995 161. [8] Javanainen and S. Stenholm, Appl. Phys. 1 (1980 35. [9] Marlan O. Scully and M. Suhail Zubairay, Quantum Opics, Cambridge University Press, Cambridge (1997. [10] M. Fleischhauer and M.D. Lukin, Phys. Rev. Lett. 84 (000 5094.