4 They are however not unique: Indeed, the subspace of the fixed points of the evolution operator for a fixed in- teraction time τ$_{0}$ has infinite dimension, being at least composed by all states | g, m 〉 , | e, m − 1 〉 , with m = ℓ $^{2}$m$_{0}$ and ℓ = 1 , 2 , . . . . These states have been denoted in the literature as trapping states [32, 34]. beam of atoms, of which one controls only the initial in- ternal state and the arrival rate. This procedure does not require atomic detection nor control of the atomic veloc- ity, but only the mean value and the variance of the veloc- ity distribution, so to warrant the weak-coupling regime. Following the line of reasoning presented in Ref. [13], this strategy could have also been identified on the basis of the observation that the state | g, 0 〉 is an eigenstate of Hamiltonian ˆ H such that ˆ H$_{JC}$ g, 0 = 0, and it is the unique dark state of this dynamics. On this basis, one can construct a master equation which has as steady state | g, 0 〉 , and which has the form given in Eq. (6) for R = 0. A useful benchmark for the quantum state preparation is given by the fidelity F ( t ) for preparing the system in the vacuum state at time t since the beginning of the experiment. The fidelity F ( t ) = 〈 0 | ˆ ρ ( t ) | 0 〉 corresponds to the population of the vacuum state at time t . The solution ˆ ρ ( t ) can be exactly evaluated using the damping basis [42], and reads The theory of trapping states has been reported in [34]. These properties have been used in milestone experi- ments [32] in order to generate photon number states of the cavity field. The effect of noise on this dynamics have been theoretically analyzed in [43, 44]. Using simple considerations we now discuss how the vacuum state of the electromagnetic field can be the unique asymptotic state of the dynamics, and determine the corresponding preparation fidelity. For this purpose, we assume that all atoms are initially prepared in their ground state ( R = 0) and that their velocity (interaction time) is distributed according to a function p ( φ ) with finite width. The time- evolution of the diagonal elements of the field density ma- trix, 〈 n | ˆ ρ ( t ) | n 〉 ≡ c$_{n}$ ( t ) (with ∑ n c$_{n}$ ( t ) = 1 as Tr { ˆ ρ } = 1) is found from Eq. (5), and is given by the set of coupled differential equations where ˆ ρ (0) n are the right eigenvector of the Liouvillean de- fined in Eq. (6) corresponding to the eigenvalue nγ , and α (0) n is the n -th moment of the expansion in the number operator, taken over the initial state of the cavity field. Using the explicit form one finds [17] where 〈 f (ˆ a $^{†}$ˆ a ) 〉 0 = Tr { f (ˆ a $^{†}$ˆ a )ˆ ρ (0) } . When the initial state is thermal, with average photon number ¯ n = 〈 ˆ a $^{†}$ˆ a ˆ ρ (0) 〉 , then expression (8) takes the compact form Their solution reads with B$_{0}$ = 0 and showing that the vacuum state is always a trivial station- ary solution. Other stationary solutions can be found if there exist n > 0 for which the coefficients B$_{n}$ = 0. and in this case one sees that the time scale for reaching the ground state is determined by the damping rate γ and by the initial occupation number ¯ n . In other words, the achieved fidelity is given by initial occupation number ¯ n , the average Rabi angle ¯ φ , and the average number of atoms N = rt which have crossed the cavity. The strong-coupling limit is characterized by g ¯ τ ≥ 1. In this regime, one can see from Eqs. (3) and (4) that, for a given interaction time τ$_{0}$ = π/g √ m$_{0}$ (and any in- teger multiple of τ$_{0}$ ), with m$_{0}$ > 0 integer number, there is no net exchange of energy between cavity field and dipole when the initial state of the system is a coherent (or incoherent) superposition of the states | g, m$_{0}$ 〉 and | e, m$_{0}$ − 1 〉 . In other words, these states are fixed points of the resonator dynamics, in the absence of dissipation. such that for σ > 0 only B$_{0}$ vanishes. In this limit, hence, the only stationary state is the ground state. Neverthe- less, for sufficiently small values of the width σ there exist coefficients B$_{n}$ , with n > 0, whose value can be very small, so that the occupation of the oscillator ground state may converge very slowly toward unity as a function of time. A limiting case is found when σ → 0, such that p ( φ ) = δ ( φ − φ$_{0}$ ). This is the situation in which trapping states may exist. For instance, if we choose φ$_{0}$ = π/ 2, then the coefficients B$_{0}$ and B$_{4}$$_{n}$ 2 (for n = 1 , 2 , . . . ) van- ish, indicating that the Fock states | 0 〉 and the states of In order to study the behaviour of the coefficients B$_{n}$ , let us assume that the velocities follow a Gaussian distri- bution, such that p ( φ ) = exp $^{(}$− ( φ − φ$_{0}$ ) $^{2}$/ 2 σ 2 ) $_{/}$√$_{2}$$_{πσ}$ $^{2}$, where φ$_{0}$ is the center and σ the width. The coefficients B$_{n}$ then read D. The strong-coupling limit ˆ ρ ( t ) = ∞ ∑ n =0 e − $^{nγt}$ˆ ρ (0) n α (0) n (7) F ( t ) = 1 − 〈 n 〉 $_{0}$e − γt + 〈 n ( n − 1) 〉 0 2 e − 2 γt + . . . (8) F $_{wc}$( t ) = 1 1 + ¯ n e − γt = 1 1 + ¯ n e − ¯ φ $^{2}$rt (9) B$_{n}$ = 1 2 $^{(}$1 − e − 2 nσ 2 cos $^{(}$2 $_{φ$_{0}$}$√$_{n}$ ) $^{)}$, (13) c$_{0}$ ( t ) = c$_{0}$ (0) + rB$_{1}$ ∫ t 0 d τc$_{1}$ ( τ ) (12) c$_{n}$ ( t ) = e − B$_{n}$$^{rt}$c$_{n}$ (0) + rB$_{n}$$_{+1}$ ∫ t 0 d τ e − B$_{n}$r ( t − τ $^{)}$c$_{n}$$_{+1}$ ( τ ) , B$_{n}$ = ∫ ∞ 0 d φ sin 2 ( $_{φ}$√$_{n}$ $^{)}$p ( φ ) . (11) ˙ c$_{n}$ ( t ) = − rB$_{n}$c$_{n}$ ( t ) + rB$_{n}$$_{+1}$c$_{n}$$_{+1}$ ( t ) (10)