# Production of entanglement in Raman three-level systems using feedback 

Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, ACT 0200, Australia 

R. N. Stevenson and A. R. R. Carvalho 

J. J. Hope 

Australian Centre for Quantum-Atom Optics, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, ACT 0200, Australia 

(Dated: November 21, 2018) 

We examine the theoretical limits of the generation of entanglement in a damped coupled ion- cavity system using jump-based feedback. Using Raman transitions to produce entanglement be- tween ground states reduces the necessary feedback bandwidth, but does not improve the overall effect of the spontaneous emission on the final entanglement. We find that the fidelity of the result- ing entanglement will be limited by the asymmetries produced by vibrations in the trap, but that the concurrence remains above 0.88 for realistic ion trap sizes. 

The production and manipulation of entangled states has been a major feature of quantum information re- search in the last several years. The efforts in this di- rection were rewarded with extraordinary experimental advances that led to the realisation of entangled states in a variety of physical systems [1–5], including entan- glement involving multiple particles [6, 7], and long-lived entangled states [8, 9]. The latter are achieved either by reducing the experimental imperfections and undesirable interactions with the environment that are responsible for entanglement deterioration, or by encoding entangle- ment in a decoherence-free subspace [10, 11]. A promis- ing approach to deal with the decoherence problem is the use of active quantum feedback control [12–15]. In fact, quantum feedback has been recently proposed and used to improve entanglement production and stability in both continuous [16, 17] and discrete [18–22] variable systems. 

Recently, two of us have shown that direct feedback under the appropriate detection strategy leads to the ro- bust production of highly entangled states of two atoms or ions in a cavity [19, 20]. Motivated by the perspec- tive of experimental implementations, and the possibility of improving even further the proposed scheme, in this paper we analyse the use of Raman transitions in place of the optical dipole ones considered in [19, 20]. A Ra- man scheme has also been considered by Metz et al. in a different approach to generate entanglement through the detection of macroscopic quantum jumps [23] without feedback. The entanglement is conditioned on the detec- tion of certain spontaneous emission events that transi- tion the system from a light to a dark state. Moreover, as in [19, 20], entanglement is heralded as prolonged periods of no cavity photon detection indicate that the entangled state has been prepared. 

In this paper we will show that the use of feedback in the Metz et al. setup will allow the desired state to 

PACS numbers: 03.67.Mn, 42.50.Lc, 03.65.Yz 

## I. INTRODUCTION 

be achieved much quicker, with less time spent in the light state as the transition to the dark state is forced by the feedback, rather than switched through sponta- neous emission. We will also show that, although the Ra- man scheme produces entanglement between metastable levels, which have orders of magnitude smaller decay rates than optical transitions, this does not translate into higher or more stable entanglement in our scheme while feedback is on. In fact, the effect of spontaneous emis- sion from the upper level used for the Raman transition can be detrimental enough to put optical and Raman schemes at the same level in terms of entanglement pro- duction. However, the Raman scheme will be shown to significantly reduce the necessary feedback bandwidth. 

The paper is organised as follows: In Section II we de- scribe the Raman model and derive the equivalent two- level description after the elimination of the upper level. We establish the equivalence between this model and the one in [19] in the absence of spontaneous emission, and present the important effective parameters for the entan- glement dynamics. In Section III we analyse the effect of spontaneous emission, while in Section IV we include important experimental limitations as the delocalisation of the ions within the standing wave of the cavity and de- tection inefficiency. The latter seems to be an important limiting factor for current experimental parameters. 

The systems consists of a pair of atoms (or ions) with the Raman level configuration shown in Fig. 1. The two target levels are labelled | 0 〉 and | 1 〉 , and have en- ergies ℏ ω$_{0}$ and ℏ ω$_{1}$ respectively, with the intermediate upper level labelled | 2 〉 , with energy ℏ ω$_{2}$ . The parti- cles are coupled to a single mode of an optical cavity which is detuned from the | 0 〉 → | 2 〉 transition by Δ, i.e. ω$_{C}$ = ( ω$_{2}$ − ω$_{0}$ ) − Δ. The transition from level | 1 〉 to level | 2 〉 is pumped using light detuned by Δ − δ from reso- 

## II. MODEL 

