Entanglement in atomic resonance fluorescence P. Gr¨unwald ∗ and W. Vogel † Arbeitsgruppe Quantenoptik, Institut f¨ur Physik, Universit¨at Rostock, 18055 Rostock, Germany (Dated: received November 3, 2018) The resonance fluorescence from regular atomic systems is shown to represent a continuous source of non- Gaussian entangled radiation propagating in two different directions. For a single atom entanglement occurs under the same conditions as squeezing. For more atoms, the entanglement can be more robust against dephasing than squeezing, hence providing a useful continuous source for various applications of entangled radiation. PACS numbers: 42.50.Dv, 03.67.Bg, 32.50.+d, 03.65.Ud The resonance fluorescence of a single atom played an outstanding role when searching for a radiation source that clearly displays the quantum nature of light. A single two- level atom, after being excited by a driving laser, can only emit a single photon at once and then it must be re-excited, before another photon can be emitted. This feature led to the pre- diction of the nonclassical effect of photon antibunching [1], which could first be demonstrated in the resonance fluores- cence of an atomic beam [2] and later with a trapped ion [3]. Sub-Poissonian photon statistics could also be observed for the first time in resonance fluorescence [4] and squeezing in resonance fluorescence was predicted [5]. Even the resonance fluorescence of many atoms can show squeezing, which re- quires stable phase relations between the emitters. This can be achieved by a regular arrangement of the atoms [6], or by detection of the fluorescence in the forward direction with re- spect to the pump beam [7]. Squeezing from strongly driven regular atomic systems has also been studied [8]. In an early experiment with regular atoms, the interference of the fluores- cence of two trapped ions was demonstrated [9]. Squeezing in resonance fluorescence could be observed for samples of many atoms [10]. A direct demonstration of squeezing in the resonance fluo- rescence of a single atom has not been realized yet. For this purpose it was proposed to apply homodyne correlation mea- surements [11]. The method has been further developed to detect general field correlation functions by balanced homo- dyne correlation techniques [12]. This opens possibilities to study the most general nonclassical features of the atomic res- onance fluorescence radiation [13]. As an example, intensity- field correlation functions could be measured in resonance flu- orescence [14], which is a first step in such a direction. In the context of applications for quantum information pro- cessing, among the manyfold of nonclassical effects entan- glement became of particular importance, for recent reviews see [15, 16]. A variety of possibilities to create entanglement in atoms, e.g. via cooperative fluorescence, has been stud- ied, see [17] and references therein. A cold atom in a cavity can serve as a stable source for entangled EPR-type photons, $^{∗}$Electronic address: peter.gruenwald2@uni-rostock.de $^{†}$Electronic address: werner.vogel@uni-rostock.de with ˆ A ( n ) ab = | a ( n $^{)}$〉〈 b ( n $^{)}$| ( { a, b } =1 , 2) being the atomic flip operator of the n -th atom at the retarded time t ( n ) = t −| r − r ( n $^{)}$| /c . The function g ( r − r ( n $^{)}$) relates the atomic source operators to the source field parts of the radiation field. ˆ E (+) s ,n ( r , t ) = g ( r − r ( n $^{)}$) ˆ A ( n ) 12 ( t ( n $^{)}$) , (2) ˆ E ( ± $^{)}$( r , t ) = ˆ E ( ± ) f ( r , t )+ N ∑ n =1 ˆ E ( ± ) s ,n ( r , t ) , (1) where the indices ’f’ and ’s’ denote the free and source field parts, respectively. The superscript ( ± ) indicates posi- tive/negative frequency field components. For our purposes, the geometry is chosen such that the pump laser does not hit the detectors, hence the corresponding free-field contribution is in the vacuum state, 〈· · · ˆ E (+) f 〉 = 〈 ˆ E ( − ) f · · · 〉 =0 . In the nor- mally ordered correlation functions considered in the follow- ing, we can throughout omit the free field contributions in Eq. (1) and only keep track of the source fields. The source field part of the resonance fluorescence field of a two-level atom in rotating wave approximation is given by For simplicity, we deal with the resonance fluorescence of atomic two-level systems. The theoretical background is given in several textbooks, e.g. [19]. Let us consider an en- semble of N non-interacting two-level atoms, located at the positions r ( n ) ( n = 1 . . . N ) . The electric field strength oper- ator at position r and time t is given in source-quantity repre- sentation as In the present Letter we study the resonance fluorescence of a regular system of atoms, separated from each other by sev- eral wavelengths of the driving laser. Entanglement is found in the electric field strengths of the fluorescence radiation prop- agating in different directions from the atomic sample. Re- markably, the robustness of the entanglement against dephas- ing of the atomic transitions increases with increasing number of atoms. This renders it possible to design continuously ra- diating sources of non-Gaussian entangled light for various applications. by utilizing the coupling between the scattered photons and the quantized atomic center-of-mass motion [18]. To our best knowledge, however, the resonance fluorescence radiation it- self has not been studied yet with respect to entanglement. arXiv:1001.2448v2 [quant-ph] 10 Mar 2010