Cooperative emission from two atoms localized in a standing wave field

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(1)Home. Search. Collections. Journals. About. Contact us. My IOPscience. Cooperative emission from two atoms localized in a standing wave field. This content has been downloaded from IOPscience. Please scroll down to see the full text. 2012 Phys. Scr. 2012 014007 (http://iopscience.iop.org/1402-4896/2012/T147/014007) View the table of contents for this issue, or go to the journal homepage for more. Download details: IP Address: 146.155.28.40 This content was downloaded on 23/05/2016 at 16:36. Please note that terms and conditions apply..

(2) IOP PUBLISHING. PHYSICA SCRIPTA. Phys. Scr. T147 (2012) 014007 (5pp). doi:10.1088/0031-8949/2012/T147/014007. Cooperative emission from two atoms localized in a standing wave field N Ciobanu1,2 , N A Enaki2 and M Orszag1 1. Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile Institute of Applied Physics, Academy of Sciences of Moldova, Academiei 5, Chisinau MD-2028, Republic of Moldova 2. E-mail: [email protected] (N Ciobanu) and [email protected] (N Enaki). Received 9 August 2011 Accepted for publication 27 October 2011 Published 17 February 2012 Online at stacks.iop.org/PhysScr/T147/014007 Abstract The effect of standing wave amplitude on the resonance fluorescence of a three-level system of radiators is discussed. The cooperative emission and the fluorescence spectrum of emitted photons from two atoms situated in the nodes and anti-nodes of the laser field are investigated. A realizable model for producing an entangled state, that can be controlled via the distance of localization between two atoms in the anti-nodes, is discussed. PACS numbers: 42.50.−p, 32.30.−r, 32.50.+d, 42.50.Hz, 03.65.Ud. coupling. Here, we discuss the resonance fluorescence from a three-level system of radiators dressed in the standing wave of a laser field. Using the dressed Hamiltonian and equation of motion we investigate the Stark dynamics and the exchange integrals between two atoms situated in the anti-nodes of the standing wave. The correlation functions and the fluorescence spectrum of emitted photons at two dressed frequencies are obtained. A realizable model for producing an entangled state that can be controlled via the distance of localization between two atoms in the anti-nodes is discussed.. 1. Introduction The study of cooperative spontaneous emission and manipulation with photons from atoms dressed in the standing wave of coherent field plays an important role in quantum optics [1–5]. Since the photons emitted in the cooperative effect can produce entangled states, they are applied in many areas of quantum information or the investigations of strongly correlated systems [6]. Quantum interference significantly affects the spectral features of the emitted radiation [7] and such kinds of interactions are mediated by the electromagnetic field or by localization of the atoms [8]. The implementation of entanglement between the atoms is based on achieving and controlling the effective interaction between the atoms that are entangled [9, 10]. For a system of two cold atoms placed in a vacuum field, the back-action of emitted photons on the wave-packet evolution about the relative position of the two cold atoms was discussed in [11]. It was shown that the photon recoil resulting from atomic spontaneous emission can induce localization of the relative position of the two atoms, through the entanglement between the spatial motion of individual atoms and their emitted photons. Cooperative emission from two or more atoms localized in the standing wave of a laser field can generate new effects connected with the non-homogeneity of the field, where the interference of two photons represents a crucial component of the scheme to entangle atomic qubits, based on photonic 0031-8949/12/014007+05$33.00. 2. Theoretical model In this section we describe the cooperative emission from a system of three-level atoms dressed by a standing wave that is in resonance with one atomic transition (see figure 1(a)). The excitation diagram corresponds to the situation for which the atoms are prepared in the second state |2i, and excited by a strong coherent field relative to |2i ↔ |3i transition. The system is periodically in an excited state, so we can detect the spontaneous emission of dressed atomic states relative to |3i ↔ |1i transition. It is supposed that the intensity of the electromagnetic field is larger than the spontaneous decay rates γ32 and γ31 . In the dipolar and rotating wave approximation the interaction of such an atomic ensemble with the dressed standing wave and electromagnetic field vacuum is described 1. © 2012 The Royal Swedish Academy of Sciences. Printed in the UK.

(3) Phys. Scr. T147 (2012) 014007. N Ciobanu et al. q ˜ j )/2, β 2j = (1 − ˜ j = 2j + 12 and α 2j = (1 + 4/ where ˜ j )/2, describe the probability of population on the atomic 4/ levels. Here, the operators U31j = C3 j a1† j , U21j = C2 j a1† j and U13j = C3† j a j1 , U12j = C2† j a j1 are the transition operators between the quasi-energy levels of the excited and ground states, which satisfy the following commutation relation: γ γ [Uβα , Uδ ] = Uδα δγβ − Uβ δαδ . The operators a3 j and a2 j are expressed: a3 j = α j C3 j + β j C2 j , a2 j = γ j C3 j + δ j C2 j , where |γ j | = |β j |, |α j | = −|δ j |. The above transformation is done after the dressed states of the atoms, and can be explained as a new three-level system with two excited quasi-levels from which the spontaneous emission occurs on the ground state |1i. In accord with the method of elimination of electromagnetic field operators relative to the atomic transitions |3i → |1i and |2i → |1i, one can write the following equation for the generalized dressed atomic operator O(t):. Figure 1. (a) Energetic representation of a three-level atomic ensemble and (b) schematic localization of its atoms in the standing wave.. by the Hamiltonian H = H0 + HD + HI where H0 =. N X 3 X j=1 α=1. HD = ih̄. N X. h̄ω3α aα† j aα j +. X. h̄ωk bk† bk ,. k. j (a3† j a2 j e−iω0 t − h.c.),. j=1 N iω31 d31 X X E gk (bk a3† j a1 j ei(k·Er j ) − h.c.). HI = c k j=1. Ȯ(t) = i. ˜ j [U33j (t) − U22j (t), O(t)] . j=1. (1). N nn X + α 2j [U13j (t), O(t)]U3l1 (t). Here, h̄ω3α is the energy of α level; aα† j , aα j are the generation and annihilation operator for level α and atom j; d31 and d32 represent the dipole momentum transitions between the states |3i ←→ |1i and |3i ←→ |2i, respectively. bk† (bk ) describes the generation (annihilation) of the photons with energy h̄ωk ; p E 31 /d31 ) represents the interaction gk = 2πc2 h̄/V ωk (Eeλ · d constant with the fluctuations of electromagnetic vacuum; E 31 the dipole eEλ depicts the photon polarization vector and d momentum; V is the volume and 1 = ω32 − ω0 the detuning between the transition frequency ω32 and the frequency of the external laser field. In equation (1) H0 represents the free electromagnetic field and free atomic Hamiltonians. The second Hamiltonian describes the atoms dressed in the external standing wave of the coherent field, and HI represents the cooperative spontaneous emission from third to first level (d31 is larger than d32 ). The Rabi frequency is dependent on the position of the atoms in the nodes and anti-nodes j = 0 sin(kE0 · rEj ), E h̄ while E E is the amplitude of the where 0 = (dE32 · E)/ electromagnetic field intensity. We assume that the atoms are continuously driven by a resonant laser field with the wave vector kE0 and the frequency ω0 . Excluding the time dependence and diagonalizing the non-perturbed part of the Hamiltonian (1) using a Bogoliubov transformation as in [12], the following dressed Hamiltonian is obtained: H = h̄. N X. l, j=1. ˜ l ) + β 2j [U12j (t), O(t)]U2l1 (t) × ζ (ω − o o ˜ l ) I j,l + h.c. , ×ζ (ω + . (3). ˜ l ) = i P + π δ(ω ± ˜ l ), ω = ωk − ω31 − where ζ (ω ± ˜l ω± 1/2 and I j,l represents the exchange integral between the atoms j and l. Here, we consider only the case when atoms situated at points l and j occupy a similar position in the standing wave so that α j = αl and β j = βl . In general, the dressed state of the atom situated at position j differs from point l of the standing wave and the correlations between the Stokes and anti-Stokes scattering processes are realized if the atoms are situated in non-equivalent locations of the standing wave. The light scattered at the interaction of the standing wave field and the atomic ensemble, depends on the position of the ensemble relative to the standing wave nodes and anti-nodes. In order to study the behaviour of atomic system in the standing wave, we will consider the case of two indistinguishable atoms situated at a distance comparable with the wave length rab > λ for which the exchange integral has the form: sin ξ 3 + (1 − 3 cos2 θ ) Iab = τ0−1 (1 − cos2 θ ) 4 ξ cos ξ sin ξ × − 3 . (4) ξ2 ξ ξ =ωk rab /c. N X X 1 † ˜j bk bk + h̄ (U33j − U22j ) ωk − ω31 − 2 k j=1. Here, ωk describes the photon emitted at the Stokes ω31 − j or anti-Stokes ω31 + j frequencies, and θ is the angle formed between the atomic dipolar momentum of the atoms and the direction of propagation of emission. If the atoms are located at a distance less than the wave length of the. N o iω31 d31 X X n E + gk bk eik·Er j (α j U13j + β j U12j ) − h.c. , c k j=1. (2) 2.

(4) Phys. Scr. T147 (2012) 014007. N Ciobanu et al. As follows from figure 2, this is, of course, a manifestation of the familiar photon-bunching effect. The higher-order correlation properties of the emitted photons may be affected significantly by atomic correlation and distance of localization. For atoms situated at the distance rab = 3λ/2, the correlations decrease in comparison with λ/2. Photon-bunching generated by two atoms, localized at a distance comparable with the wave length of the cavity for significant atom–atom interactions, may instead be entangled via their emitted photons and can be used in quantum processing and transmission in a crucial way [15]. The correlation function between the atoms a and b describes an entangled state of the photons emitted at Stokes ω31 − j and anti-Stokes ω31 + j frequencies. Supposing that the atoms are situated around the anti-nodes of the standing wave and in equivalent positions, the spectrum of cooperative emission is given by. 1.0. G2. 0.8 0.6 0.4 0.2 0.0 0. 1. 2. 3. 4. 5. delay time. Figure 2. Second-order correlation function G (2) as function of the detection time delay τ/τ0 for 4 = 0.5, 0 = 10, t/τ0 = 0.1 and rab = λ/2 (solid curve), 3λ/2 (dashed curve).. S(ω) = 4K ( ) 3 1 2 1 α 2 I α U1a (t)U3b (t) β 2 I β U1a (t)U2b (t) × + , (ω − ω31 −)2 +(2I α )2 (ω−ω31 +)2 + (2I β )2. field, the exchange integral (4) coincides with one half of 3 2 the spontaneous emission rate Iab = τ0−1 /2 = 2ω31 d31 /(3c3 h̄) and does not depend on the atom positions in the standing wave.. (6). where the correlated terms have the form:. 3. The fluorescence spectrum emitted from two atoms dressed in the anti-nodes of the laser field. 2 1 U1a (t)U2b (t) =. Here, we discuss the second-order correlation function and the resonance fluorescence spectrum from two indistinguishable atoms localized at different position in the standing wave of the field. The photon statistics of light describes whether it originated from a source of classical or quantum character [13]. The interference of photons emitted by two sources projects the state of atoms a and b into an entangled state. In such models the photons involved do not need to be initially entangled, and can even be emitted by different sources, but they need to be indistinguishable. In general, it is not easy to realize several independent sources emitting indistinguishable photons. A realizable scheme set up for our model could allow us to observe collective excitation of two individual atoms in the Rydberg blockade regime [14]. Considering that the cooperative spontaneous emission is spherically distributed, and with the same probabilities in all directions, the frequencies ω31 ± do not coincide with the frequencies of the cavity modes, and the second-order correlation function of two atoms localized at the distance rab is given by 2 2 (2) 2 2 2 G (Er , t, t + τ ) = K α β 4α 2 β 2 e−[2t+(Iab +1)τ ]/τ0 (eα + eβ ). 1 j ˜ 2j 4 2. . α β I α + I0α I α − I0α − + α I +I E + E − I E 0 2α 2(I0 α − I β ) β 2(I0α + I β ) β I0 − I 2β β. β. +α. I β + I0. 4. β. 2(I0 −. E β+ I β). −. 2I0 I β 2β. I0 − I 2β. e. β. . β. −4I0 t. +. !. I β − I0 β. 2(I0 +. E β− I β). ,. 3 1 U1a (t)U3b (t) =. 1 j ˜ 2j 4 2. +β. β. ". I β − I0 β. 2(I0 + 4. . E α− I α). −. β I0. Iα + Iβ 2β. I0 − I 2α. β. E+. I β + I0 β. 2(I0 −. #. E α+ I α). I α + I0α 2I0α I α −4I α t I α − I0α − + 0 e + E − 2α E . 2(I0α + I α ) α 2(I0α − I α ) α I0 − I 2α α(β). ± I α(β) )t], E = exp[−2(I0α + β I0 )t] and the exchange integrals between the same atoms β I0α = αa2 Iaa = α 2 τ0−1 /2 and I0 = β 2 τ0−1 /2 do not depend α on the distance rab , while I = αa αb Iab and I β = βa βb Iab Here,. ± E α(β) = exp[−2(I0. depend on the distance of localization. As the atoms are indistinguished and occupy the similar position in the standing (5) wave, we assume that |αa | = |αb | = α and |βa | = |βb | = β. In equation (6), only the incoherent component and the 4 2 where K = ω31 d31 /c4r 2 . Here, we consider that the detector is cooperative part of the spectrum are considered, and the situated perpendicular to the distance rab between the atoms, Rabi frequency associated with the driving field becomes and the photon is detected in the time t = t̄ − r/c for which the comparable with the spectral width of the atoms. Thus, the Born–Markov approximation is applicable. It is well known excited level is split into a doublet of dressed states and that the second-order correlation function of the radiation spontaneous emission is realized at the frequency ω31 ± . emitted from a source of atoms has a maximum when τ → 0 The resonance fluorescence spectrum for two atoms localized (see figure 2). at the distances λ/2 and 3λ/2 is plotted in figure 3. 2 −α2 (Iab +1)(t+τ )/τ0 −(Iab +1)(α2 t+β 2 τ )/τ0 + [e ] , +e ˜2 . 3.

(5) Phys. Scr. T147 (2012) 014007. N Ciobanu et al. the measurement of the atomic protocol in the appropriate bases, and application of single-qubit rotation to the logic atoms based on the measurement results. We consider that good entanglement fidelity can be achieved by manipulating the distance between the logic atom and the atom responsible for coupling with the next anti-node.. 0.06 0.05. S. 0.04 0.03 0.02. 4. Conclusions. 0.01. Cooperative spontaneous emission from an extended ensemble of 3-type three-level atoms dressed by a standing wave field was investigated. Using the equation of motion, we studied the time evolution of fluorescence emission from two indistinguishable atoms dressed with the standing wave of the field. We observed that, for large values of laser field intensity, the excited level was splitting and the control of spontaneous emission was possible at two frequencies ω31 ± j . The Stark splitting was directly proportional to the atomic localization, and achieved maximal value in the anti-nodes and minimal value in the nodes of the standing wave. Thus, the photons resulting from atomic spontaneous emission can induce an entangled state, through the correlations between the individual trapped atoms. The entanglement function of emitted photons at Stokes and anti-Stokes dressed frequencies for various atomic separation was obtained. A realizable model for producing an entangled state that can be controlled via the distance of localization between two atoms in the anti-nodes was discussed. Controlling the positions of the atoms, we can neglect the cooperative interaction with other atoms of the cavity, and with photon–photon entanglement, good entanglement fidelities can be obtained. To which extent this can be performed in practice depends on the future development of cavity QED.. 0.00 20. 10. 0 31. 10. 20. τ0. Figure 3. Resonance fluorescence spectrum for the same values as in figure 2.. The dashed peaks correspond to a case where the atom a is placed in the centre of one anti-node kra = π/2 and the atom b in the next anti-node centre krb = 3π/2, and the distance between them is rab = λ/2 (see the localization scheme from figure 1(b)). In this situation the Rabi frequency is maximal = 0 |sin(kra )| = 0 |sin(krb )| = 0 . The continued peak corresponds to the distance rab = 3λ/2, one atom is located at the left side and the other at the right side of the node, the position of the node being considered in π. As the atoms are dressed near the node, the frequency of the field tends to zero, and the amplitude of the spectrum increases with decreasing distance between them. In this case, we have only one peak that corresponds to the superposition of two lines given by the cooperative terms of S(ω). In general, for the atoms located symmetrically with respect to a node or anti-node, the individual atomic positions, determine the distance between the spectral peaks, but the distance rab determines the cooperative effects (amplitude of the spectrum). By applying laser beams, one manipulates the atomic positions relative to the anti-nodes, so that the resonance fluorescence spectrum from two atoms can be achieved and described by (6). In this model, cooperative spontaneous emission can occur with in a ‘well-defined’ time. The second correlation function G (2) is not zero and represents an entangled state between the photons emitted at Stokes and anti-Stokes frequencies. One approach to scalable quantum computation based on probabilistic entangling gates is to have an array of trapping zones, each containing two atoms: a logic atom a that encodes the quantum information and a transferring atom b that is responsible for coupling to another atomic pair a1 , b1 via a probabilistic entanglement protocol [15]. The atoms a and b should be separated at a given distance in the dressed field through the laser action. In order to achieve scalability we must realize the quantum gates between two arbitrary atoms from the pairs (for example a and a1 ). This can be done, realizing an entangled state from the spontaneous emitted photons of those atoms. After the preparation of the entanglement, we can achieve local deterministic motional controlled-NOT gates on each logic-transferring atoms from the zones (C-NOT for a − b and a1 − b1 pair). The next step after realizing the gates’ operation at each pair consists of. Acknowledgments This work was supported by Chilean grands Fondecyt (numbers 3110148 and 1100039), as well as by the Academy of Sciences of Moldova (no.11.819.05.11F).. References [1] Nussmann S, Hijlkema M, Weber B, Rohde F, Rempe G and Kuhn A 2005 Phys. Rev. Lett. 95 173602 [2] Kreuter A, Becher C, Lancaster G P, Mundt A B, Russo C, Hauffner H, Roos C, Eschner J, Schmidt-Kaler F and Blatt R 2004 Phys. Rev. Lett. 92 203002 [3] Bienert M, Torres J M, Zippilli S and Morigi G 2007 Phys. Rev. A 76 013410 [4] Beugnon J, Jones M P A, Dingjan J, Darquiere B, Messin G, Browaeys A and Grangier P 2006 Nature 440 779 [5] Miroshnychenko Y, Alt W, Dotsenko I, Förster L, Khudaverdyan M, Meschede D, Reick S and Rauschenbeutel A 2006 Phys. Rev. Lett. 97 243003 [6] Hanbury-Brown R and Twiss R Q 1956 Nature 178 1046 [7] Das S, Agarwal G and Scully M 2008 Phys. Rev. Lett. 101 153601 [8] Kimble H J, Dagenais M and Mandel L 1977 Phys. Rev. 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