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Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004, pp. 303∼307 A Full-Quantum Three-Dimensional Analysis of the Dynamics of a Single Atom in an All-Optical Trap Young-Tak Chough∗ School of Computers, Electronics & Communications Engineering, Gwangju University, Gwangju 503-703 Kyungwon An School of Physics, Seoul National University, Seoul 151-742 (Received 12 November 2003) We analyze two recent “state-of-the-art” experiments on the dynamics of a single atom in an all-optical trap by using computer simulations based on the theory of quantum trajectories. We compare the so-called “quasi-classical” simulations of the experiments with our full-quantum calculations and discuss how far the two models agree. We point out that the atomic wave packet spreading in quantum calculations is mimicked by an interesting behavior of atom skipping around from a potential minimum to another in the quasi-classical model. PACS numbers: 42.50, 32.80.-t, 42.62.Fi Keywords: All-optical-trap, Quantum trajectory, Quasi-classical, Tracking atomic trajectories I. INTRODUCTION momentum uncertainty principle, and this is indeed the issue that the present article addresses. That is, in quantum mechanics, an atom is after all to be described as a wave packet, and it is well known that the wave packet undergoes rapid changes in morphology in the course of the atom-field exchanges of energy-momentum quanta. Then, without a specific mechanism which continuously localizes the atomic wave packet in the cavity, it may be difficult to think about such a classical trajectory of an atom inside the cavity. Nevertheless, when an atom gets out of the interaction region and hits a scintillation screen, everything about the wave packet changes all of a sudden, and we do see a single atom as a shining spot on the screen. Thus, it is indeed tempting to think of such a corpuscular atom flying along a unique trajectory, on the other hand. Figure 1 contrasts these two different concepts schematically. In (a), the transmission signal at a given instant of time is the outcome of the collective effect of every “fraction” of the atom within the wave packet whereas in (b), the signal depends on the particular location of the “whole” atom. No doubt, the quantum mechanical picture is (a), so if one readily admits that (a) is ultimately the correct view, the immediate question would be to what extent the (much simpler) classical description, (b), can be valid. It would be also very meaningful to ask whether these two different views might give different outcomes that would be experimentally discernible. We wish to present some interesting clues in answering these questions in this article. We will perform a fully quantum-mechanical, threedimensional treatment of the system, and for this pur- Recently two major workgroups in cavity-quantum electrodynamics(cavity-QED) announced successful capture of a single atom by the radiation pressure force of a single photon in the so-called “all-optical trap” [1, 2] within the setting of the cavity-QED, demonstrating an advance in the ability to observe and manipulate the dynamical processes of individual quantum systems. They even claimed the capability of tracking the trajectories of the atoms inside the trap cavity from the experimentally observable variations of the intracavity field. The idea is that the strong atom-field coupling will cause the presence of an atom to significantly modify the intracavity field, thereby providing a means to track the atomic motion by way of the light emerging from the cavity. By introducing what they called “quasi-classical” model, Doherty et al. reported atomic trajectories reconstructed from the cavity transmission signals [3]. They solved the Ito-type stochastic equations derived from the FokkerPlanck equation for the atomic center-of-mass motion, approximating the dynamics of the Wigner operator of the system by adiabatically separating the atomic motion from the photonic dynamics. Thus, the solution was essentially a classical stochastic trajectory of an atom moving in the trap. One can, however, raise a legitimate question about this idea of “determining the trajectory of an individual atom” when one rigorously adheres to the position∗ E-mail: [email protected] -303- -304- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 Fig. 1. Two different viewpoints: (a) wave packet point of view and (b) particle point of view. γ) enters a standing-wave cavity (resonance frequency ωC , photon loss rate 2κ) driven by an external field (amplitude E, frequency ωL ), along a direction transverse to the cavity axis and encounters a spatially varying interaction energy ~g(~r) = ~g0 exp −ρ2 /w2 cos(kx) ≡ ~g0 ψ(ρ) eikx + e−ikx , where k = 2π/λ and ρ2 = y 2 + z 2 . As the atom approaches the interaction region, the cavity transmission will change since the eigenfrequencies of the atom-cavity system depend on the atomic position ~r via the coupling g(~r), thereby signaling the arrival of the atom. Then, one raises the potential as illustrated in Fig. 3 by increasing the drive field amplitude E. The entire system is, therefore, described by the following master equation: 1 [H, %] + κ(2a%a† − a† a% − %a† a) i~ γ + (2σ− %σ+ − σ+ σ− % − %σ+ σ− ), 2 %̇ = Fig. 2. Scheme of the actual experiments. (1) where % is the density operator of the atom-cavity system, a† (a) the photon creation (annihilation) operator, σ+ (σ− ) the atomic excitation(de-excitation) operator such that [σ+ , σ− ] = σz . H is the Hamiltonian of the system in the interaction picture and is given by p2y p2x p2 + + z 2M 2M 2M ikx + i~g0 ψ(ρ) e + e−ikx eiδt σ− a† − e−iδt σ+ a + i~E ae−i∆t − a† ei∆t , (2) H = Fig. 3. How the trap works. pose, we will make direct comparisons with the aforementioned “quasi-classical” description. where δ = ωC − ωA and ∆ = ωC − ωL [4]. We will work in the framework of the quantum trajectory theory [5] by expanding the state vector of the system as ∞ X Z X |Ψi = d3 p Ca (n, px , py , pz )|n, px , py , pz , ai. n=0a=e,g II. 3-D FULL-QUANTUM MODEL Figure 2 shows the schemes of the experiments in Refs. 1 and 2. The essential mechanism of the all-optical trap is as simple as depicted in Fig. 3. When an atom comes into the initial optical potential, the potential depth is raised so that the atom has no other choice but to be trapped inside. In the actual setup, a two-state atom (mass M , transition frequency ωA , radiative decay rate (3) However, the radial and the axial motions can be easily be shown to be separable in the fashion Ca (n, px , py , pz ) = Ca (n, px )R(py , pz ), (a = e, g), (4) which is one of the useful findings in this work. Thus, the axial motion in the coherent evolution turns out to be simply given by √ √ √ d De (n, qx ) = E n + 1ei∆τ De (n + 1, qx ) − E ne−i∆τ De (n − 1, qx )ψρ n + 1e−iδτ e−i(2qx +1)µτ dτ √ γ × Dg (n + 1, qx + 1)ψρ n + 1e−iδτ ei(2qx −1)µτ Dg (n + 1, qx − 1) − nκ + De (n, qx ) 2 √ √ √ d Dg (n, qx ) = E n + 1ei∆τ Dg (n + 1, qx ) − E ne−i∆τ Dg (n − 1, qx )ψρ neiδτ e−i(2qx +1)µτ dτ (5) A Full-Quantum Three-Dimensional Analysis of the Dynamics· · · – Young-Tak Chough and Kyungwon An √ × De (n − 1, qx + 1)ψρ neiδτ ei(2qx −1)µτ De (n − 1, qx − 1) − nκDg (n, qx ), where τ = g0 t, px = qx ~k, and µ = ~k 2 /2M g0 , if transformations such that Ca (n, qx ) = exp −iµqx2 τ Da (n, qx ), (7) d i hqξ i = − dτ 2π ∂ψ iδt † −iδt e σ− a − e σ+ a , ∂ξ -305- (6) (9) are used, whereas the radial motion is given by d µ hξi = hqξ i dτ π (8) where ξ = y, z. Note here that Z Z √ ∗ 2hξi X 0 0 0 ψ dq ρ x dqx nDg (n, qx )De (n−1, qx )hqx | cos kx|qx i w2 n Z √ hξi X dqx nDg∗ (n, qx ) [De (n−1, qx −1) + De (n−1, qx +1)] = − 2 ψρ w n ∂ψ σ− a† ∂ξ =− (10) since 1 [δ(qx , qx0 + 1) + δ(qx , qx0 − 1)] . (11) 2 The three-dimensional motion has thus been essentially reduced to a one-dimensional problem. As to the implementation of the discontinuous evolution of the quantum trajectory, let us just briefly summarize it. The atomic decay process is implemented by axial jumps such that hqx | cos kx|qx0 i = 2 Dg (n, qx ) −→ e−iµτ (2qx ηx +ηx ) Dg (n, qx + ηx ), De (n, qx ) −→ 0, (12) (13) and by radial jumps such that R(qy , qz ) −→ R(qy + ηy , qz + ηz ) , (14) where ηi (i = x, y, z) are the direction cosines of the random spontaneous emissions. The cavity decay is simply described by the jumps √ Da (n, qx ) −→ n + 1Da (n + 1, qx ), (a = e, g). (15) III. RESULT We did separate computations for the experimental situations of Refs. 1 and 2. The important experimental parameters in Ref. 1 are (δ, ∆) = 2π(−47, 78)MHz and (g0 , γ, κ) = 2π(110, 2.6, 14.2)MHz with a mean empty-cavity photon number n̄ ∼ 0.3 for 55 Cs atoms whereas in Ref. 2, (δ, ∆) = 2π(−35, 5)MHz, (g0 , γ, κ) = 2π(16, 3, 1.4)MHz and n̄ ∼ 0.9 for 85 Rb atoms. As seen in Figs. 4 (a)–(d), the two experiments gave characteristically different results, primarily due to the different Fig. 4. Radial orbits in the yz-plane. (a) and (b) are for the case of Hood et al. [1] (λ = 852 nm), and (c) and (d) are for case of Pinkse et al. [2] (λ = 780 nm). Axes are in units of λ. potential depths in comparison to the heating strength. From Eqs. (8) – (10), one can easily guess that the radial motion is essentially subject to a central force except for the diffusion introduced by quantum jumps. Just like in the quasi-classical calculation, mostly the radial motion has rose-shaped orbits with quite long periods in the case of Ref. 1 as shown in Figs. 4(a) and (b). However, in the case of Ref. 2, the orbits are much more diffusive, as shown in Figs. 4(c) and (d). The axial dynamics in quantum and quasi-classical models, however, are apparently quite different. First, Figs. 5(a)–(c) show the typical time evolutions of the atomic wave packet in the coordinate space for the experimental situation of Ref. 1, starting from the initial width ∆x = λ/4π. In (a), the initial mean atomic position is hxi = 0, i.e., at an antinode, in (b) hxi = λ/8, halfway between the antinode and the node, and in (c) -306- Journal of the Korean Physical Society, Vol. 44, No. 2, February 2004 Fig. 5. Axial motions for the case of Hood et al.: Quantum description (a)–(c) and quasi-classical description (i)– (iii). The horizontal axes are in units of λ. The time span in the top figures is 0–80 µsec, and in the bottom, it is 0– 200 µsec. hxi = λ/4, at a node. These figures are compared with the quasi-classical counterparts, Figs. 5(i)–(iii). Note that in Figs. 5(iii), the initial mean position of the atom is not exactly at the node, but is hxi = 0.2499λ since if hxi = λ/4, there will be no dynamics whatsoever. It is seen that the quantum wave packet tends to migrate into the nearby potential minima to settle down there, ending up being split mostly into two pieces, quite reminiscent of a viscous liquid drop fallen into a double-well potential. In the quasi-classical case, the atom described by a mass point behaves similarly; i.e., instead of the probability peaks, it shows a fast oscillation around a potential minimum. However, one of the differences in principle is that in the quantum model, the cavity transmission at a given time is determined by the atom-cavity coupling strength averaged over the extent of the wave packet whereas in the quasi-classical model, it is determined by the definite atomic position. Also, one can easily expect the fluctuations in the cavity transmission signal to be greater, at least in principle, in the quasiclassical model than in the quantum counterpart due to the fast oscillations in the axial direction. One may, however, argue that a meaningful quantumclassical comparison should be made between the quantum wave packet and the distribution of the classical trajectories obtained by evolving an ensemble of initial positions corresponding to the initial wave packet. In this viewpoint, it is obvious that the quantum and the quasiclassical predictions are by no means in contradiction. That is, in an ensemble in which atoms in both dressed states are present, some atoms will go to the right and some to the left, resulting in an overall distribution essentially the same as the quantum wave packet. It would be very comforting if this were the case because, then, Fig. 6. Axial motions for the case of Pinkse et al.: Quantum descriptions (a) and (b), and quasi-classical descriptions (i) and (ii). The horizontal axes are in units of λ, and the vertical axes are in units of µsec. the formidable quantum mechanical calculation could be replaced by a number of much simpler classical calculations without such an ambiguous wavy picture of atoms. However, one can easily see that this type of argument loses its logical ground when one considers the case of an atom-cavity on resonance. As we have shown in our previous publication [6], the wave packet splits even in the case of an exact resonance, in general. The point is that a single atom on resonance with the cavity is always in a 50-50 combination of the two dressed states; and therefore, the net force that a single atom gets from the optical potential is simply zero, so no center-of-mass motion occurs, no matter where the “corpuscular” atom is located in the potential. Indeed, this is precisely why the optical dipole force vanishes when the atom-cavity detuning is zero [7]. Thus, it appears that in this situation the wave packet never corresponds to the statistical ensemble of anything in so far as there is no classical theory that gives a nonzero force to each atom in a cavity on exact resonance. It is, however, true that in the present setting with nonzero detunings, the quasi-classical treatment appears to be an excellent approximation to the quantum description. In Figs. 6, we contrast the results of the quantum and quasi-classical calculations for the case of Ref. 2. The quantum description says that, in this parameter region, the wave packet rapidly spreads and tends to settle down into every slot of the potential minima (at antinodes) within its span. The quasi-classical description, on the other hand, suggests quite an interesting atomic behav- A Full-Quantum Three-Dimensional Analysis of the Dynamics· · · – Young-Tak Chough and Kyungwon An Fig. 7. Shape of the wave packet at a given time. ior. That is, the atom appears to be skipping around from one potential minimum to another. Such skipping happens when an atom is heated up and kicked out of a potential well, followed by cooling down in another potential minimum. The heated atom will move around rather freely along the cavity axis with its acquired high energy, but when an appropriate sequence of spontaneous emissions happens, it falls into the nearby potential minimum. This type of behavior has also been mentioned by Pinkse et al. [2] and Doherty et al. [3]. Let us, however, note that such skipping, in fact, happens only in rare occasions because, for it to happen, the process of heating and cooling should occur in just the right sequence and at just the right strengths. Thus, the fact is that mostly the atoms are trapped in a single potential minimum until they leave the interaction region. Nevertheless, it is interesting to see how the classical picture mimics the quantum description. IV. DISCUSSION At this stage, a meaningful question would be whether one can experimentally probe the difference between these two pictures. Again, if one compares the ensemble of the classical point atoms with the quantum wave packet description, it does not seem easy at all to tell any difference between the two. However, as we mentioned before, the very first question of “whether the quantum wave packet only and always represents a statistical ensemble, and whether there really is no way to describe a single atom in the quantum-mechanical ideology” is still there. Thus, the questions cast by these experiments are, indeed, massive. If one could find a practical way -307- and actually probe the existence, or absence, of the fast axial oscillation and/or the skipping behavior of atoms predicted in the quasi-classical description—for example, by measuring the multi-order correlation functions as suggested in Ref. 2, it would mean a lot. The goal of our next work is, therefore, none other than a theoretical investigation of such a possibility. Let us finally point out another piece of interesting quantum-classical correspondence by using Fig. 7. This is the shape of the wave packet shown in Fig. 6 (a) at time t ≈ 200 µsec. It shows that every probability density peak has a dip at the center. This fact can be directly connected to the quasi-classical oscillations around the potential centers; i.e., the probability density to find a particle at the bottom of a potential well is lowest because it moves fastest there. However, as the atom moves, the depths of the dips change in time, but we find that, when a peak has a dip, so do all the others, implying the dynamics around one potential minimum is correlated to the rest, quite reminiscent of the spatial coherence of an optical wave front. ACKNOWLEDGMENTS This work was supported by a Korea Research Foundation grant (KRF-2002-070-C00044). REFERENCES [1] C. J. Hood, T. W. Linn, A. C. Doherty, A. S. Parkins and H. J. Kimble, Science 287, 1447 (2000). [2] P. W. H. Pinkse, T. Fischer, P. Maunz and G. Rempe, Nature 404, 365 (2000). [3] A. C. Doherty, T. W. Linn, C. J. Hood and H. J. Kimble, Phys. Rev. A 63, 013401 (2000). [4] Young-Tak Chough, Sun-Hyun Youn, Hyunchul Nha, Sang Wook Kim and Kyungwon An, Phys. Rev. A 65, 023810 (2002). [5] H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993). [6] Young-Tak Chough, Keon-Kee Kim, Hyunchul Nha, JaiHyung Lee, Kyungwon An and Sun-Hyun Youn, J. Korean Phys. Soc. 42, 106 (2003). [7] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).