nance, i.e. of frequency ω$_{L}$ = ( ω$_{2}$ − ω$_{1}$ ) − (Δ − δ ), where δ << Δ, and with a coupling strength V$_{L}$ . This laser pumps the internal levels directly, and does not pump the cavity mode. The transition from level | 0 〉 to level | 1 〉 is pumped using microwaves detuned from resonance by δ , with coupling strength V$_{M}$ , or equivalently using a different Raman transition, as in [24]. We neglect sponta- neous emission for now, but will include it further down. Here the particles are assumed to have constant and equal cavity coupling strengths g . In section IV A, the position of the particles will be allo wed to vary in time and therefore the coupling strengths with the cavity mode, g$_{a}$ ( t ) and g$_{b}$ ( t ), will be time-dep endent and dif- ferent for each particle. tors of the cavity mode, respectively. D is the decoher- ence superoperator and ˆ U is a unitary matrix describing the effect of the control Hamiltonian ˆ F applied for a finite time δt : Following the results in [19], we use an asymmetric form of control, acting on either of the particles. We c hoose ˆ F = α ( | 1 〉 1 〈 2 | + | 2 〉 1 〈 1 | ), i.e. ˆ F = α ( σ − 1 + σ + 1 ) on the first particle, where α represents the strength of the con trol. In the regime of large detuning, i.e. we can adiabatically eliminate the upper level and obtain an effective two-level dynamics given by the Hamiltonian Light that escapes through one of the cavity mirrors is monitored using a photodetector. When a photon is detected, a finite amount of evolution is imposed on the system using control lasers or microwaves. A schematic of this feedbac k setup is shown in Fig. 2. where ˆ H is the Hamiltonian The master equation for this system, in the interaction picture, using the rotating wave approximation, is with the op erators ˆ A defined as ˆ A$_{i,j}$ = | i 〉 1 〈 j | + | i 〉 2 〈 j | , and ˆ a and ˆ a † being the annihilation and creation opera- Note that by choosing the second detuning appropriately , δ = V 2 L $_{4Δ}$, the last two terms cancel. Now all the terms in the Hamiltonian are on the order of 1 Δ or smaller, so the cavity decay rate κ is the fastest rate in the dynamics of the system, and the cavity mode spends most of the time in the zero occupation state. This means the ca vity mode can also be adiabatically eliminated in a similar way to [18]. The resulting master equation after elimi- nating the cavity is D Ufb ω$_{M}$ ω$_{L}$ 1 2 ω$_{L}$ Δ ω$_{C}$ ω$_{M}$ δ 0 | 2 〉 | 1 〉 FIG. 2: A diagram representing the model. The particles are pumped from the side of the cavity. Light leaks from the cavity at a rate κ and is de- tected using a photodetector. When a photon is detected a finite evolution represented by the uni- tary matrix ˆ U$_{fb}$ is applied. FIG. 1: A Raman level scheme. Electro ns are excited from level | 0 〉 to level | 1 〉 by a two photon transition via level | 2 〉 . The transition with frequency ω$_{L}$ is excited by a laser, the transition with frequency ω$_{C}$ is excited by the ca vity mode and a direct transition between level | 0 〉 and | 1 〉 transition with frequency ω$_{M}$ could be directly excited by microwaves. 2 D $^{[}$ˆ c $^{]}$ˆ ρ = ˆ c ˆ ρ c † − 1 2 $^{[}$ˆ c $^{†}$ˆ c ˆ ρ + ˆ ρ ˆ c $^{†}$ˆ c $^{]}$, (3) ˆ U = Exp$^{[}$ i ˆ F δt ℏ ] . (4) Δ >> V$_{L}$, g, κ > V$_{M}$, (5) ˆ H$_{2}$ = − ℏ V$_{L}$g 2Δ ( ˆ A$_{0}$$_{,}$$_{1}$ ˆ a † + ˆ A$_{1}$$_{,}$$_{0}$ ˆ a ) − ℏ g 2 Δ ˆ a $^{†}$ˆ a ˆ A$_{0}$$_{,}$$_{0}$ + ℏ V$_{M}$ 2 ( ˆ A$_{0}$$_{,}$$_{1}$ + ˆ A$_{1}$$_{,}$$_{0}$ ) + ℏ V 2 L 4Δ ( ˆ A$_{1}$$_{,}$$_{1}$ − ˆ A$_{0}$$_{,}$$_{0}$ $^{)}$− ℏ δ ( ˆ A$_{1}$$_{,}$$_{1}$ − ˆ A$_{0}$$_{,}$$_{0}$ $^{)}$. (6) ˙ ˆ ρ = − i ℏ [ ˆ H, ˆ ρ $^{]}$+ κ D [ ˆ U ˆ a $^{]}$ˆ ρ, (1) ˆ H = − ℏ δ ˆ A$_{1}$$_{,}$$_{1}$ + ℏ Δ ˆ A$_{2}$$_{,}$$_{2}$ + ℏ V$_{L}$ 2 ( ˆ A$_{1}$$_{,}$$_{2}$ + ˆ A$_{2}$$_{,}$$_{1}$ ) + ℏ V$_{M}$ 2 ( ˆ A$_{1}$$_{,}$$_{0}$ + ˆ A$_{0}$$_{,}$$_{1}$ ) + ℏ g ( ˆ A$_{0}$$_{,}$$_{2}$ ˆ a † + ˆ A$_{2}$$_{,}$$_{0}$ ˆ a $^{)}$, (2) ˙ ρ = − iV$_{M}$ 2 [( ˆ A$_{0}$$_{,}$$_{1}$ + ˆ A$_{1}$$_{,}$$_{0}$ $^{)}$, ρ$^{]}$+ V 2 $_{L}$g 2 Δ $^{2}$κ D [ ˆ U ˆ A$_{0}$$_{,}$$_{1}$ ] ρ. (7)