6 This is simply saying that when the photon is detected by the photodetector (the term with the η prefactor) the feedback is applied, and when the photon is not detec- tected (the term with the (1 − η ) prefactor) no feedback is applied. A cavity emission event is essentially a lost opportunity to apply feedback, so the photodetector ef- ficiency is a scaling factor on the rate at which feedback is applied. the coupling strengths of the two atoms differ, asymmet- ric elements are introduced into the cavity emission dy- namics. This stops the antisymmetric state being a dark state, and the state can now lose excitations through cou- pling with the environment (decay of the cavity mode), and in this way the concurrence drops. This cavity emis- sion projects the ions into the light, unentangled state. This can be seen in Fig. 7, where there is no spontaneous emission, but there are cavity emission events even when the system is in the maximally entangled anti-symmetric state. A summary of the effects of detection inefficiencies and cavity coupling fluctuations can be seen in Fig. 8. The steady-state concurrence for a range of trap sizes and spontaneous emission rates running the full master equa- tion simulations is shown. For reasonable parameters, γ = 0 . 1 g and with ions trapped within .08 of a wave- length, the concurrence is above 0.88. As the trap size increases past 0.1 wavelengths the concurrence rapidly drops. The mean time before a cavity emission from the dark state | a$_{01}$ 〉 is the inverse of the rate of the decay: In this paper we have shown that a feedback control scheme to generate entanglement using Raman three- level atoms is mostly equivalent to a system using two- level optical transitions. In the regimes where cavity mode and upper level can be adiabatically eliminated, the final entanglement depends on the cooperativity pa- rameter that is the same for Raman or optical schemes. While this equivalence is true while feedback is on, the Raman system has two main advantages. First, the nec- essary feedback bandwidth can be experimentally con- trolled, rather than having to operate at the lifetimes of optical transitions. Second, the Raman scheme has a higher entanglement lifetime if feedback is turned off. This is the mean time if the cavity coupling rates are kept constant. As the coupling constants vary, the rate of decay also varies. For the ions to remain in a highly entangled state, this timescale must be much longer than the timescale over which the feedback drives the ions to the dark state. Feedback prepares unconditioned entangled states with concurrence greater than 0.88 even when taking into account a realistic delocalisation of the atoms within the cavity mode. Entanglement can be even higher, and her- alded, if the system is conditioned on the detection of photons leaving the cavity. Reliably produced pairs of entangled states could be used to make entangled states of more particles [33], with an ultimate aim of produc- ing large cluster states for measurement based quantum computing. This opens the perspective to produce mul- tipartite entangled states directly through a similar feed- back setup, by choosing appropriate measurement and controls. The effect of having photodetectors without perfect ef- ficiency is the same in the Raman system as it is in a two level system. When the photodetector has an efficiency η , the master equation (9) is transformed into [20] ˙ ˆ ρ = − i ℏ [ ˆ H, ˆ ρ ] + η V 2 L κ Δ 2 D [ ˆ U$_{fb}$ ( g$_{1}$ ( t )ˆ σ − 1 + g$_{2}$ ( t )ˆ σ − 2 ]ˆ ρ + (1 − η ) V 2 L κ Δ 2 D [( g$_{1}$ ( t )ˆ σ − 1 + g$_{2}$ ( t )ˆ σ − 2 ]ˆ ρ + ∑ i = s,a ∑ j =0 , 1 ( D $^{[}$R$_{i,j}$$^{]}$ˆ ρ$^{)}$. (16) T − 1 dark = V 2 L κ Δ 〈 a$_{01}$ | $^{(}$g$_{1}$ σ + 1 + g$_{2}$σ + 2 $^{)}$× $^{(}$g$_{1}$ σ − 1 + g$_{2}$σ − 2 $^{)}$| a$_{01}$ 〉 T$_{dark}$ = 2 κ g 2 1 − g 2 . (15) V. CONCLUSIONS B. Non-unity cavity emission FIG. 7: (Color online) The concurrence and cavity emissions for a system with no spontaneous emission and a trap size of 0.2 wavelengths. The solid line shows the concurrence, while the circles show the times when photons leaked out from the cavity. 0 1 2 3 4 5 6 7 8 9 10 x 10 4 0 0.2 0.4 0.6 0.8 1 Time (in units of 1/2 πg ) Concurrence