Radiation Pressure on a Two-Level Atom: an Exact Analytical Approach
RRadiation Pressure on a Two-Level Atom: an Exact Analytical Approach
L. Podlecki, R. Glover,
1, 2
J. Martin, and T. Bastin Institut de Physique Nucl´eaire, Atomique et de Spectroscopie,CESAM, University of Liege, Bˆat. B15, Sart Tilman, Li`ege 4000, Belgium Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4111, Australia (Dated: February 17, 2017)The mechanical action of light on atoms is nowadays a tool used ubiquitously in cold atom physics.In the semiclassical regime where the atomic motion is treated classically, the computation of themean force acting on a two-level atom requires in the most general case numerical approaches.Here we show that this problem can be tackled in a pure analytical way. We provide an analyticalyet simple expression of the mean force that holds in the most general case where the atom issimultaneously exposed to an arbitrary number of lasers with arbitrary intensities, wave vectors,and phases. This yields a novel tool for engineering the mechanical action of light on single atoms.
With the advent of lasers, the mechanical action oflight has become an extraordinary tool for controllingthe motion of atoms. The first evidence of this controlwith laser light was demonstrated in the early seven-ties with the deflection of an atomic beam by resonantlaser radiation pressure [1]. One of the most remarkableachievements was made a decade later when the first coldatomic cloud in a magneto-optical trap was observed [2].These initial experiments gave birth to the vast array ofcold atom physics experiments with each more spectac-ular than the last. Regimes that were thought to foreverremain in the realm of Gedankenexperiments became re-ality in labs, for example Bose-Einstein condensation [3].As long as the atomic motion can be treated classi-cally, i.e., in regimes where the atomic wavepackets aresufficiently localized in space, the resonant laser radia-tion acts mechanically as a force on the atomic center-of-mass. In a standard two-level approximation and mod-eling the laser electromagnetic field as a plane wave, themean force F exerted by a single laser, averaged overits optical period, reaches shortly after establishment ofthe laser action a stationary regime where it takes thewell-known simple expression [4] F = Γ2 s s (cid:126) k . (1)Here, Γ is the rate of spontaneous decay from the upperlevel of the transition, (cid:126) k is the laser photon momentum,and s = ( | Ω | / / ( δ + Γ /
4) is the saturation parame-ter, where Ω is the Rabi frequency and δ = ω − ω is thedetuning between the laser and the atomic transition an-gular frequencies, ω and ω , respectively. The maximalforce the laser can exert on the atom is (Γ / (cid:126) k .The question naturally arises of how Eq. (1) general-izes when several lasers of arbitrary intensities, wave vec-tors and phases are acting simultaneously on the atom.Surprisingly, to date no general exact analytical expres-sion of the resulting force can be found in the scientificliterature and one is often reduced to using numericalapproaches [5–7]. In the low intensity regime a gener-alized version of Eq. (1) provides an approximation of the incoherent action of each laser field. Each individuallaser j ( j = 1 , . . . , N with N the total number of lasers)is characterized by an individual detuning δ j , a photonmomentum (cid:126) k j , a Rabi frequency Ω j , and an individualsaturation parameter s j = ( | Ω j | / / ( δ j + Γ / s j are much smaller than 1, the mean resulting forceexerted incoherently by all lasers on the atom can beapproximated by F = (cid:80) j F j , where F j = Γ2 s j s eff (cid:126) k j (2)is the mean force exerted by each individual laser j and s eff = (cid:80) j s j [8]. For larger values of s j , Eq. (2) loses va-lidity as it is derived in a rate-equation approximation [8].It also neglects any coherent action of the lasers. Coher-ence effects can lead to huge forces that vastly exceed themaximal value of (Γ / (cid:126) k j per laser, as is observed withthe stimulated bichromatic force [5]. An exact expressionof the force including coherent effects and that holds athigh intensity can be found in the particular case of apair of counterpropagating lasers of same intensity [9].In this Letter, we solve analytically the most generalcase and provide an expression for the force exerted byan arbitrary number of lasers with arbitrary intensities,phases, detunings and directions acting on the same in-dividual two-level atom. We show that the force can stillbe written in strict generality in the form of Eq. (2), evenincluding coherent effects, provided a generalized defini-tion of the saturation parameter s j is given. We thus pro-vide a unified formalism that holds in any configurationof lasers and enables the engineering of the mechanicalaction of light on individual atoms.To this end, we consider a two-level atom with lev-els | e (cid:105) and | g (cid:105) of energy E e and E g , respectively ( E e >E g ). We denote the atomic transition angular frequency( E e − E g ) / (cid:126) by ω eg . The atom interacts with a classicalelectromagnetic field E ( r , t ) resulting from the superpo-sition of N arbitrary plane waves: E ( r , t ) = (cid:80) j E j ( r , t ),with E j ( r , t ) = ( E j / e i ( ω j t − k j · r + ϕ j ) + c.c. Here, ω j , k j and ϕ j are the angular frequency, the wave vec-tor and the phase of the j -th plane wave, respectively, a r X i v : . [ qu a n t - ph ] F e b and E j ≡ E j (cid:15) j , with E j > (cid:15) j the normal-ized polarisation vector of the corresponding wave. Thequasi-resonance condition is fulfilled for each plane wave: | δ j | (cid:28) ω eg , ∀ j , where δ j = ω j − ω eg is the detuning. Wedefine the weighted mean frequency and detuning of theplane waves by ω = (cid:80) j κ j ω j and δ = (cid:80) j κ j δ j , respec-tively, with κ j a set of weighting factors ( κ j (cid:62) (cid:80) j κ j = 1) that can be taken a priori arbitrarily.In the electric-dipole approximation and consider-ing spontaneous emission in the master equation ap-proach [10], the atomic density operator ˆ ρ obeys ddt ˆ ρ ( t ) = 1 i (cid:126) [ ˆ H ( t ) , ˆ ρ ( t )] + D (ˆ ρ ( t )) , (3)with ˆ H ( t ) = (cid:126) ω e | e (cid:105)(cid:104) e | + (cid:126) ω g | g (cid:105)(cid:104) g | − ˆ D · E ( r , t ) and D (ˆ ρ ) = (Γ /
2) ([ˆ σ − , ˆ ρ ˆ σ + ] + [ˆ σ − ˆ ρ, ˆ σ + ]), where ˆ D is theatomic electric dipole operator, r is the atom position inthe electric field, Γ is the spontaneous deexcitation rate ofthe excited state | e (cid:105) , and ˆ σ − ≡ | g (cid:105)(cid:104) e | and ˆ σ + ≡ | e (cid:105)(cid:104) g | arethe atomic lowering and raising operators, respectively.The hermiticity and the unit trace of the density op-erator make all four matrix elements ρ ee , ρ eg , ρ ge , and ρ gg , with ρ kl = (cid:104) k | ˆ ρ | l (cid:105) for k, l = e, g dependent vari-ables. We consider here the vector of real independentvariables x = ( u, v, w ) T , with u = Re(˜ ρ ge ), v = Im(˜ ρ ge ),and w = ( ρ ee − ρ gg ) / ρ ee − /
2, where ˜ ρ ge = ρ ge e − iωt .In the rotating wave approximation (RWA) the time evo-lution of x resulting from Eq. (3) obeys˙ x ( t ) = A ( t ) x ( t ) + b , (4)with b = (0 , , − Γ / T and A ( t ) = − Γ / δ Im (Ω( t )) − δ − Γ / − Re (Ω( t )) − Im (Ω( t )) Re (Ω( t )) − Γ , (5)where Ω( t ) = (cid:80) j Ω j e i ( ω j − ω ) t , with Ω j ( j = 1 , . . . , N ) thecomplex Rabi frequenciesΩ j = − D ge · E j e i ( − k j · r + ϕ j ) / (cid:126) ≡ Ω R ,j e iφ j , (6)where D ge = (cid:104) g | ˆ D | e (cid:105) , Ω R ,j > j , and φ j its phase. Without loss of generality, theglobal phases of the atomic states | e (cid:105) and | g (cid:105) can alwaysbe chosen so as to have one Rabi frequency real and pos-itive. This is usually considered in all studies where asingle plane wave interacts with the atom. However, with N arbitrary plane waves, we cannot assume without lossof generality that all Rabi frequencies are real and theirphases cannot be ignored.Equation (4) constitutes the so-called Optical BlochEquations (OBE’s) adapted to the present studied case.Here, because of the time dependence of A ( t ), the solu-tion cannot be expressed analytically and the equationmust be integrated numerically. However, within an ar-bitrary accuracy, we can always assume that the N fre-quency differences ω j − ω are commensurable, i.e., that all ratios ( ω j − ω ) / ( ω l − ω ), ∀ j, l : ω l (cid:54) = ω , are ratio-nal numbers. Within this assumption, Ω( t ) and A ( t )are periodic in time with a period T c = 2 π/ω c , where ω c = (cid:0) LCM (cid:2) ( ω j − ω ) − , ∀ j : ω j (cid:54) = ω (cid:3)(cid:1) − , with LCM de-noting the least common multiple conventionally takenas positive. It also follows that the numbers ( ω j − ω ) /ω c ,hereafter denoted by m j , are integer, ∀ j [11]. In the par-ticular case where all frequencies ω j are identical, A ( t ) isconstant in time, or equivalently periodic with an arbi-trary value of ω c (cid:54) = 0, and all integer numbers m j vanish.Within the commensurability assumption where A ( t )is T c -periodic and given initial conditions x ( t ) = x , theOBE’s admit the unique solution (Floquet’s theorem, see,e.g., Ref. [12]) x ( t ) = P I ( t ) e R ( t − t ) ( x − x p ( t )) + x p ( t ) , (7)where R is a logarithm of the OBE monodromy ma-trix divided by T c [13], P I ( t ) is an invertible T c -periodicmatrix equal to X I ( t ) e − R ( t − t ) for t ∈ [ t , t + T c ],with X I ( t ) the matriciant of the OBE’s [13], and x p ( t ) is an arbitrary particular solution of the OBE’s.The real parts of the eigenvalues of the matrix R ,the so-called Floquet exponents, belong to the inter-val [ (cid:82) T c q min ( t ) dt/T c , (cid:82) T c q max ( t ) dt/T c ] with q min ( t ) and q max ( t ) the minimal and maximal eigenvalues of the ma-trix [ A ( t ) + A † ( t )] /
2, respectively [14]. Here, this matrixreads diag( − Γ / , − Γ / , − Γ) and the real parts of the Flo-quet exponents are thus necessarily comprised between − Γ and − Γ /
2, hence strictly negative. This implies firstthat the matrix e R ( t − t ) tends to zero with a characteris-tic time not shorter than Γ − and not longer than 2Γ − .At long times ( t − t (cid:29) − ) x ( t ) (cid:39) x p ( t ). Second,the OBE’s are ensured to admit a unique T c -periodic so-lution [12], that the particular solution x p ( t ) can be setto. This certifies that at long times the solution of theOBE’s is necessarily periodic (periodic regime).The unique T c -periodic solution of the OBE’s can beexpressed using the Fourier expansion x ( t ) = + ∞ (cid:88) n = −∞ x n e inω c t , (8)with x n ≡ ( u n , v n , w n ) T the Fourier components of x ( t ).Since x ( t ) is real, x − n = x ∗ n , and since it is continuousand differentiable, (cid:80) n | x n | < ∞ . Inserting Eq. (8) intothe OBE’s yields an infinite system of equations connect-ing all u n , v n and w n components. The system can berearranged so as to express all u n and v n as a function ofthe w n components. Proceeding in this way yields, ∀ n , u n = − i τ + n N (cid:88) j =1 Ω j w n − m j − τ − n N (cid:88) j =1 Ω ∗ j w n + m j ,v n = − τ + n N (cid:88) j =1 Ω j w n − m j + τ − n N (cid:88) j =1 Ω ∗ j w n + m j , (9)with τ ± n = 1 / [Γ + 2 i ( nω c ± δ )] and w n + (cid:88) m ∈ M W n,m w n + m = −
12 (1 + ˜ s ) δ n, . (10)Here, δ n, denotes the Kronecker symbol, M is the setof all distinct nonzero integers m lj ≡ m l − m j ( j, l =1 , . . . , N ), W n,m = β n,m / ( α n + β n, ) ( m ∈ Z ), with α n =Γ + inω c and β n,m = (cid:80) j,l : m lj = m Ω j Ω ∗ l ( τ + n + m l + τ − n − m j ),and ˜ s = (cid:80) j ˜ s j , with˜ s j = Re Ω j Γ / − iδ j N (cid:88) l =1 l : δl = δj Ω ∗ l Γ . (11)We have τ ±− n = τ ∓ n ∗ , α − n = α ∗ n , β − n, − m = β ∗ n,m , and W − n, − m = W ∗ n,m .If we define the vector of all w n components for n ranging from −∞ to + ∞ , w = ( . . . , w − , w , w , . . . ) T ,and the infinite matrix W of elements W n,n (cid:48) = (cid:80) m ∈ M W n,m δ n (cid:48) ,n + m ( n, n (cid:48) ranging from −∞ to + ∞ ),Eq. (10) yields the complex inhomogeneous infinite sys-tem of equations ( I + W ) w = c , (12)with I the infinite identity matrix and c the infinite vec-tor of elements c n = − δ n, / [2(1 + ˜ s )]. W is an infi-nite centrohermitian band-diagonal matrix with as manybands as the cardinality of M . Its main diagonal iszero. Since the OBE’s admit a unique T c -periodic so-lution, the infinite system admits a unique solution w with the property (cid:80) n | w n | < ∞ . In these conditionsand observing that (cid:80) n | c n | < ∞ and that the series (cid:80) n,n (cid:48) W n,n (cid:48) = (cid:80) n (cid:80) m ∈ M W n,m is absolutely conver-gent whatever the values of the Rabi frequencies Ω j , thedetunings δ j , and the deexcitation rate Γ, the infinite sys-tem (12) can be solved via finite larger and larger trunca-tions, whose solutions are ensured to converge in all casesto the unique sought solution w [15]. This solution is nec-essarily such that w (cid:54) = 0, otherwise all other coefficients w n would solve an homogeneous system of equations andvanish, in which case the equation for n = 0 could not besatisfied. This allows us to define the ratios r n = w n /w , ∀ n . In particular r = 1 and the reality condition yields r − n = r ∗ n . We have (Cramer’s rule) r n = lim k →∞ ∆ ( n ) k ∆ (0) k , (13)with ∆ ( n ) k the determinant of the I + W matrix trun-cated to the lines and columns − k, . . . , k and where the n -indexed column is replaced by the vector c correspond-ingly truncated. As here-above argued, the limit is en-sured to exist in all cases [16]. Inserting w m = r m w in Eq. (10) for n = 0 allows for expressing w in the form w = −
12 11 + s eff , (14)where s eff = (cid:80) j s j , with newly defined parameters s j as s j = Re (cid:34) Ω j Γ / − iδ j N (cid:88) l =1 Ω ∗ l Γ r m lj (cid:35) . (15)According to Ehrenfest theorem, the mean power ab-sorbed by the atom from the j -th plane wave, P j ( t ) ≡ (cid:126) ω j (cid:104) dN/dt (cid:105) j ( t ), with (cid:104) dN/dt (cid:105) j ( t ) the mean number ofphotons absorbed per unit of time by the atom fromthat wave, is given by P j ( t ) = E j ( r , t ) · d (cid:104) ˆ D (cid:105) ( t ) /dt . Simi-larly, the mean force F j ( t ) exerted by the j -th plane waveon the atom reads F j ( t ) = (cid:80) i = x,y,z (cid:104) ˆ D i (cid:105) ( t ) ∇ r E j,i ( r , t ),with (cid:104) ˆ D i (cid:105) and E j,i ( r , t ) the i -th components ( i = x, y, z )of (cid:104) ˆ D (cid:105) and E j ( r , t ), respectively (see, e.g., Ref. [17]). Ofcourse each plane wave does not act independently ofeach other on the atom since the mean value of the atomicelectric dipole moment is at any time determined by theatomic state ˆ ρ ( t ), which in turn is fully determined by thesimultaneous action of all plane waves through the OBE’s(4). The total mean power absorbed from all plane wavesand the net force exerted on the atom are then given by P ( t ) = (cid:80) j P j ( t ) and F ( t ) = (cid:80) j F j ( t ), respectively. Inthe RWA approximation where one neglects the fast os-cillating terms, we immediately get P j ( t ) = R j ( t ) (cid:126) ω and F j ( t ) = R j ( t ) (cid:126) k j , with R j ( t ) = Re (cid:104) Ω j ( v ( t ) + iu ( t )) e i ( ω j − ω ) t (cid:105) . (16)It also follows that (cid:104) dN/dt (cid:105) j ( t ) = ( ω/ω j ) R j ( t ). In thequasi-resonance condition where ω j (cid:39) ω , ∀ j , we canclearly consider (cid:104) dN/dt (cid:105) j ( t ) (cid:39) R j ( t ).Within the commensurability assumption, ω j − ω = m j ω c , ∀ j . In the periodic regime, u ( t ) and v ( t ) are inaddition T c -periodic and thus so are R j ( t ), P j ( t ) and F j ( t ). In this regime the Fourier components of R j ( t ) ≡ (cid:80) + ∞ n = −∞ R j,n e inω c t are easily obtained by inserting theFourier expansion (8) into Eq. (16). By using furtherEq. (9) and w n = r n w with w as of Eq. (14), we get R j,n = Γ2 s j,n s eff , (17)with s j,n = (cid:0) σ j,n + σ ∗ j, − n (cid:1) /
2, where σ j,n = Ω j Γ / i ( nω c − δ j ) N (cid:88) l =1 Ω ∗ l Γ r n + m lj . (18)In particular, the temporal mean value R j of R j ( t ) in theperiodic regime is given by the Fourier component R j, and observing that s j, is nothing but the parameter s j of Eq. (15), we get R j = (Γ / s j / (1 + s eff ). The Fouriercomponents of the force F j ( t ) ≡ (cid:80) + ∞ n = −∞ F j,n e inω c t inthe periodic regime are then given by F j,n = R j,n (cid:126) k j [18]and the mean force in this regime reads consequently F j = Γ2 s j s eff (cid:126) k j , (19)in support of our introductory claim. Here nevertheless,in contrast to the saturation parameter s j of Eq. (2), thenewly defined parameter s j in Eq. (15) can be negativedepending on the different phases of Ω ∗ l r m lj with respectto Ω j . This accounts for two important physical effects.First, it can make R j negative, in which case the atomacts as a net mean photon emitter in the j -th plane wave(stimulated emission) and the force exerted by that waveis directed oppositely to k j (the atom is pushed in the di-rection opposite to the direction of the incident photons).Second, the ratio | s j / (1+ s eff ) | can exceed 1 and the forceexerted by the individual laser j can exceed the maximalspontaneous force (Γ / (cid:126) k j , as is expected for coherentforces such as the stimulated bichromatic force [5].At low intensity, i.e., for Ω (cid:48) T ≡ (cid:80) j Ω R ,j / Γ (cid:28)
1, wehave | ˜ s | (cid:28) (cid:80) n (cid:48) | W n,n (cid:48) | (cid:46) (cid:48) (cid:28) ∀ n . It impliesthat the resolvent R W, − ≡ ( I + W ) − identifies to I + (cid:80) ∞ k =1 ( − W ) k and the solution w of the infinite system(12) is such that w (cid:39) − / [2(1 + ˜ s )] (cid:39) − / | w n | (cid:46) (cid:48) , ∀ n (cid:54) = 0. Hence, for m lj (cid:54) = 0, | r m lj | (cid:46) (cid:48) (cid:28) s j (cid:39) ˜ s j . If all plane waves have differentfrequencies, the sum over l in Eq. (11) only contains thesingle term l = j and thus s j (cid:39) | Ω j | / δ j + Γ / . (20)If in contrast some plane waves have identical frequencies,coherent effects can be observed. However, if we are onlyinterested in the incoherent action of the plane waves,an average (cid:104)·(cid:105) ϕ over all phase differences must be per-formed. In the low intensity regime, the statistical deltamethod [19] yields R inc j ≡ (cid:104) R j (cid:105) ϕ (cid:39) (Γ / (cid:104) ˜ s j (cid:105) ϕ / (1+ (cid:104) ˜ s (cid:105) ϕ ).Since the average (cid:104) ˜ s j (cid:105) ϕ is simply the s j of Eq. (20), weget again R inc j (cid:39) (Γ / s j / (1 + s eff ) with s j as of Eq. (20).In all cases, the incoherent and low intensity limit of ournewly defined parameter s j in Eq. (15) reduces to thestandard expression (20) known to hold in this regime [8].In the particular case where all plane waves have thesame frequency, ω = ω j , δ ≡ δ = δ j and m j = 0, ∀ j . It follows that Ω( t ) = (cid:80) j Ω j ≡ Ω and A ( t ) is nottime dependent, or equivalently is T c -periodic with anarbitrary value of T c >
0. The periodic regime corre-sponds in this case to a stationary regime where R j ( t ), P j ( t ) and F j ( t ) are constant and determined by the zerocomponent of their Fourier expansion. Since m lj = 0for any pairs of integers l and j , the s j parameters(15) simplify to s j = Re [(Ω j / [Γ / − iδ ])(Ω ∗ / Γ)]. For N = 1, this reduces to the standard expression (20)of the saturation parameter. For 2 counterpropagating plane waves, the mean net resulting force acting on theatom reads F = (Γ / s − s ) / (1 + s + s ) (cid:126) k . If bothplane waves have identical intensity and polarization, themean force F reduces to the well known phase depen-dent dipole force in a stationary monochromatic wave F = [4Ω R δ sin(∆ φ )] / [Γ + 4 δ + 8Ω R cos (∆ φ/ (cid:126) k ,with ∆ φ = φ − φ , Ω R ≡ Ω R, = Ω R, [4].Very generally, in a configuration with 2 plane waves ofdifferent frequencies, the required solution of the infinitesystem (12) can be obtained in a continued fraction ap-proach (see also Ref. [9] for the particular case of 2 wavesof identical intensity and polarization in a counterprop-agating configuration). For N = 2 and ω (cid:54) = ω , thecommensurability assumption implies that n κ = n κ ,with n and n two positive coprime integers. It followsthat m = sgn( ω − ω ) n , m = sgn( ω − ω ) n , m =sgn( ω − ω ) n s , and M = {± n s } , with n s = n + n .The infinite system (10) reads in this case, ∀ n , w n + W n,n s w n + n s + W n, − n s w n − n s = −
12 (1 + ˜ s ) δ n, . (21)The system only couples together the Fourier compo-nents w n with n = kn s ( k ∈ Z ). All other componentsare totally decoupled from these former and thus van-ish since they satisfy an homogeneous system. Hence,the only a priori non-vanishing ratios r n and Fouriercomponents R j,n and F j,n are for these specific val-ues of n and the periodic regime is rather character-ized by the period T c /n s . For n (cid:54) = 0, Eq. (21) implies w n /w n − n s = −W ∗− n,n s / [1 + W n,n s ( w n + n s /w n )]. Ap-plying recursively this relation for n = n s , n s , n s , . . . yields r n s = −W ∗− n s ,n s / [1 + K ∞ k =1 ( p k / p k = −W kn s ,n s W ∗− ( k +1) n s ,n s and where K stands for the con-tinued fraction ∞ K k =1 (cid:16) p k (cid:17) ≡ p p p . (22)Dividing further Eq. (21) for n = kn s ( k >
0) by w yields the recurrence relation r ( k +1) n s = − [ r kn s + W ∗− kn s ,n s r ( k − n s ] / W kn s ,n s that allows for computing allremaining r kn s with k >
1, and hence along with r n s allnonzero Fourier components R j,kn s and F j,kn s ( k ≥ δ = 10Γ, a Rabi frequency amplitudeof (cid:112) / δ and a phase shift of π/ (cid:80) j F j acting on a moving atom as a func-tion of its velocity v . As expected, the peak value of the − − v (Γ /k ) F ( ~ k Γ / ) FIG. 1. Stimulated bichromatic force F computed via ourformalism for a detuning δ = 10Γ, a Rabi frequency of (cid:112) / δ and a phase shift of π/ bichromatic force is of the order of (2 /π )( δ/ Γ) (in unitsof (cid:126) k Γ /
2) and spans a velocity range of the order of δ/ Γ(in units of Γ /k ) [20]. A direct numerical integration ofthe OBE’s completed with a numerical average of R j ( t )in Eq. (16) in the periodic regime produces identical re-sults.In conclusion, we have provided a unified and exact for-malism to calculate the mechanical action of a set of arbi-trary plane waves acting simultaneously on a single two-level atom. We have shown that the light forces can stillbe written in strict generality in the form of Eq. (2), in-cluding coherent and high intensity effects, provided thegeneralized definition (15) of the parameter s j is given.These results provide a novel tool for engineering the me-chanical action of lasers on individual atoms.T.B. acknowledges financial support of the BelgianFRS-FNRS through IISN grant 4.4512.08 and, withR.G., of the Interuniversity Attraction Poles Programmeinitiated by the Belgian Science Policy Office (BriX net-work P7/12). L.P. acknowledges an FNRS grant and theBelgian FRS-FNRS for financial support. [1] R. Schieder, H. Walther, and L. W¨oste, Opt. Comm. ,337 (1972); P. Jacquinot, S. Liberman, J. L. Picqu´e, andJ. Pinard, Opt. Comm. , 163 (1973).[2] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys.Rev. Lett. , 314 (1986).[3] M. H. Anderson, J. R. Enshner, M. R. Mathews, C. E.Wieman, and E. A. Cornell, Science , 198 (1995).[4] R. J. Cook, Phys. Rev. A , 224 (1979).[5] J. S¨oding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer,and Ch. Salomon, Phys. Rev. Lett. , 1420 (1997).[6] M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf,Phys. Rev. A , R1763 (1999); M. R. Williams, F. Chi,M. T. Cashen, and H. Metcalf, Phys. Rev. A , 023408(2000); S. E. Galica, L. Aldridge, and E. E. Eyler, Phys. Rev. A , 043418 (2013); C. Corder, B. Arnold, X. Hua,and H. Metcalf, J. Opt. Soc. Am. B , B75 (2015); C.Corder, B. Arnold, and H. Metcalf, Phys. Rev. Lett. ,043002 (2015); X. Hua, C. Corder, and H. Metcalf, Phys.Rev. A , 063410 (2016).[7] D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, Phys.Rev. A , 013420 (2011).[8] V. G. Minogin and V. S. Letokhov, Laser Light Pressureon Atoms (Gordon & Breach, New York, 1987).[9] V. G. Minogin and O. T. Serimaa, Opt. Comm. , 373(1979).[10] G. S. Agarwal, Quantum Statistical Theories of Sponta-neous Emission and Their Relation to Other Approaches ,Springer Tracts in Modern Physics Vol. 70 (Springer,Berlin, 1974), p. 1.[11] All nonzero m j integers are setwise coprime (the greatestcommon divisor of these integers is 1) but not necessar-ily pairwise coprime (each pair of nonzero m j are notnecessarily setwise coprime).[12] L. Y. Adrianova, Introduction to linear systems of differ-ential equations , Am. Math. Soc. , 1 (1995).[13] The monodromy matrix C of a set of equations ˙ x = A ( t ) x + b ( t ) where A ( t ) and b ( t ) are T -periodic (com-prising the case b ( t ) constant) is the matriciant X I ( t )of the set of equations evaluated at time t = t + T : C = X I ( t + T ). The matriciant is the unique matrixsolution X ( t ) to the matricial equation ˙ X = A ( t ) X sub-scribed to the initial conditions X ( t ) = I , with I theidentity matrix and where X has the same dimension as A ( t ). The matriciant and monodromy matrices are bothinvertible. A logarithm of an invertible matrix B is a ma-trix L such that e L = B . Every invertible matrix admitsa logarithm, which is not unique but with invariant realparts of the eigenvalues.[14] V. A. Iakubovich et V. M. Starzhinskii, Linear differen-tial equations with periodic coefficients , Halsted Press ,(1975).[15] F. Riesz, Les syst`emes d’´equations lin´eaires `a une infinit´ed’inconnues , Gauthier-Villars, Paris, 1913.[16] As a consequence of the linear system (12) that con-nects w to w m lj , ∀ l, j , then to w m lj + m l (cid:48) j (cid:48) , ∀ l (cid:48) , j (cid:48) , andso on, the convergence in Eq. (13) is highly optimizedby reordering the vector of unknowns w (and accord-ingly the lines and columns of I + W ) so as to have w symmetrically surrounded by w ±| m lj | for all distinctnonzero m lj , then by w ±| m lj + m l (cid:48) j (cid:48) | for all additional dis-tinct m lj + m l (cid:48) j (cid:48) , and so on.[17] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applica-tions (Wiley Science, New York, 1992); H. Metcalf andP. Van der Straten,
Laser cooling and trapping , Springer,New York, (1999).[18] The specific links between the unknowns w n in the linearsystem (12) [16] can make the ratios r n + m lj and hencethe Fourier components F j,n not necessarily significantfor the first n > F j ( t )could be better described in the periodic regime by analmost periodic behavior with an almost period smallerthan T c depending on each specific case.[19] G. Casella and R. L. Berger, Statistical Inference, 2ndEd. (Duxbury, Pacific Grove (CA, USA), 2002).[20] M. Cashen and H. Metcalf, J. Opt. Soc. Am. B20