different choices of jump operators can be interpreted as different ways to monitor the system in the framework of quantum trajectories [25, 26]. In Eq.(9), we followed ref- erence [24] and broke up the spontaneous emission into four jump terms R$_{i,j}$ This is completely equivalent to the feedback equation for a two-level system derived in [19], but with an effec- tive cavity coupling constant of G$_{eff}$ = V$_{L}$g 2Δ . The system will therefore evolve towards the maximally entangled Bell state | a$_{01}$ 〉 = 1 √ $_{2}$( | 01 〉 − | 10 〉 ), as shown previously in [19, 20]. This jump-based feedback has been shown to be fairly robust against spontaneous emission effects as well as to detection inneficiencies. In the next sec- tions we will investigate the impact of these phemomena on the Raman system, as well as looking at the effect of imperfect trapping of the particles within the cavity mode. Spontaneous emission in a Raman system is signifi- cantly different from spontaneous emission in a two-level system. The rate of emission between the two lower levels | 0 〉 and | 1 〉 is negligible, as the energy difference between them is small. The only significant spontaneous emis- sion is from the upper level | 2 〉 to the other two levels. The rates of spontaneous emission are γ$_{0}$ from | 2 〉 to | 0 〉 , and γ$_{1}$ from | 2 〉 to | 1 〉 , with γ$_{0}$ + γ$_{1}$ = γ . We choose the parameter regime given in Eq.(5), so that the rate of spontaneous emission is on the order of the other main rates in the system. The full master equation including the spontaneous emission is with the Hamiltonian ˆ H given by Eq.(2). One can per- form the adiabatic elimination of the upper level and the cavity mode in the same way it was done in Section II. In these calculations it is useful to consider a differ- ent set of basis states: The symmetric states | 00 〉 , | 11 〉 and | s$_{01}$ 〉 = 1 √ $_{2}$( | 01 〉 + | 10 〉 ), and the antisymmetric state | a$_{01}$ 〉 = 1 √ $_{2}$( | 01 〉 − | 10 〉 ). These states are chosen because the Hamiltonian and cavity emission terms are symmet- ric in the two ions, so they don’t couple between the symmetric and antisymmetric subspaces. We can write the final master equation as where ˆ H$_{red}$ is the adiabatically reduced Hamiltonian as in Eq.(7): Note that the decoherence term can be written in many different ways that are equivalent when it comes to solv- ing the full master equation. Note also, however, that With the feedback on, the entanglement is robust enough that the system doesn’t need to be conditioned on the monitoring of spontaneous emission events. Mon- itoring particular kinds of spontaneous emission events would have additional experimental complications and a finite quantum efficiency, which that could degrade the conditional entanglement, so achieving high entan- glement without this process is a useful design feature. Figure 4 shows simulations for the system when the evo- lution is conditioned only on the cavity emissions, and the spontaneous emission is averaged out. We used the experimental parameters for single atoms in a cavity from [30] ( { g, κ, γ } = 2 π × { 1 . 61 , 0 . 054 , 11 . 1 } MHz) (top plot), and from [31] ( { g, κ, γ } = 2 π × { 10 , 0 . 4 , 2 . 6 } MHz) (bottom plot). The detuning Δ was chosen to be large We have investigated the effect of feedback in this sit- uation and the results are shown in Fig. 3. The entan- glement (as measured by concurrence [29]) and jumps are shown for the cases with (bottom) and without (top) feedback. In the simulations without feedback there is a clear distinction between the dark state, where there are no cavity emissions and the concurrence is 1, and the light state, where there are many cavity emissions and the concurrence oscillates rapidly, much faster than the sampling rate. When feedback is turned on, the duration of the dark periods stays the same but the lengths of the light periods are vastly reduced, and the system spends a greater portion of the time in the desired state. This choice reflects the cooperative effects in the fluores- cence [27] if one could monitor the quantum jumps using an electron shelving technique [28]. In [23], Metz et al. considered the evolution of the system conditioned on the observation of cavity photons and the spontaneous emis- sion jumps in Eq.(11) to show that entanglement can be produced in this system, but with no feedback. Their re- sults show that some spontaneous emission events bring the system to an entangled dark state, namely the maxi- mally entangled antisymmetric Bell state | a$_{01}$ 〉 , and that the system changes from dark (entangled) to light (unen- tangled) periods according to the measurement events. It was shown that the system spends approximately a quar- ter of the time in the dark state. R$_{0}$$_{,s}$ = $^{√}$γ$_{0}$ V 2 L 4Δ 2 ( | 00 〉〈 s$_{01}$ | + | s$_{01}$ 〉〈 11 | ) R$_{0}$$_{,a}$ = $^{√}$γ$_{0}$ V 2 L 4Δ 2 ( | 00 〉〈 a$_{01}$ | − | a$_{01}$ 〉〈 11 | ) R$_{1}$$_{,s}$ = $^{√}$γ$_{1}$ V 2 L 8Δ 2 ( | a$_{01}$ 〉〈 a$_{01}$ | + | s$_{01}$ 〉〈 s$_{01}$ | + 2 | 11 〉〈 11 | ) R$_{1}$$_{,a}$ = $^{√}$γ$_{1}$ V 2 L 8Δ 2 ( | s$_{01}$ 〉〈 a$_{01}$ | − | a$_{01}$ 〉〈 s$_{01}$ | ) . (11) ˙ ˆ ρ = − i ℏ [ ˆ H, ˆ ρ $^{]}$+ κ D [ ˆ U ˆ a $^{]}$ˆ ρ + ∑ i =1 , 2 ∑ j =0 , 1 γ$_{j}$$^{(}$ D $^{[}$| j 〉 i 〈 2 | $^{]}$ˆ ρ$^{)}$, (8) ˙ ˆ ρ = − i ℏ [ ˆ H$_{red}$, ˆ ρ $^{]}$+ V 2 $_{L}$g 2 Δ $^{2}$κ D [ ˆ U ˆ A$_{0}$$_{,}$$_{1}$ $^{]}$ˆ ρ + ∑ i = s,a ∑ j =0 , 1 ( D $^{[}$R$_{i,j}$$^{]}$ˆ ρ$^{)}$, (9) ˆ H$_{red}$ = ℏ V$_{M}$ 2 ( ˆ A$_{0}$$_{,}$$_{1}$ + ˆ A$_{1}$$_{,}$$_{0}$ $^{)}$. (10) III. SPONTANEOUS EMISSION EFFECTS 3