Abstract

The mechanical action of light on atoms is currently a tool used ubiquitously in cold atom physics. In the semiclassical regime, where atomic motion is treated classically, the computation of the mean force acting on a two-level atom requires numerical approaches in the most general case. Here we show that this problem can be tackled in a purely analytical way. We provide an analytical yet simple expression of the mean force that holds in the most general case, where the atom is simultaneously 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.

© 2017 Optical Society of America

1. INTRODUCTION

With the advent of lasers, the mechanical action of light has become an extraordinary tool for controlling the motion of atoms. The first evidence of this control with laser light was demonstrated in the early 1970s with the deflection of an atomic beam by resonant laser radiation pressure [1,2]. One of the most remarkable achievements was made a decade later when the first cold atomic cloud in a magneto-optical trap was observed [3]. These initial experiments gave birth to a vast array of cold atom physics experiments, with each more spectacular than the last. Regimes (for example, Bose–Einstein condensation [4]) that were thought to forever remain in the realm of Gedankenexperiments became reality in labs.

As long as atomic motion can be treated classically, i.e., in regimes where the atomic wave packets are sufficiently localized in space, the resonant laser radiation acts mechanically as a force on the atomic center-of-mass. In a standard two-level approximation and modeling the laser electromagnetic field as a plane wave, the mean force F exerted by a single laser, averaged over its optical period, reaches a stationary regime shortly after establishment of the laser action, where it takes the well-known simple expression [5]

F=Γ2s1+sk.

Here, Γ is the rate of spontaneous decay from the upper level of the transition, k is the laser photon momentum, and s=(|Ω|2/2)/(δ2+Γ2/4) is the saturation parameter, where Ω is the Rabi frequency, and δ=ωω0 is the detuning between the laser and the atomic transition angular frequencies, ω and ω0, respectively. The maximal force the laser can exert on the atom is (Γ/2)k.

The question naturally arises of how Eq. (1) generalizes when several lasers of arbitrary intensities, wave vectors, and phases act simultaneously on the atom. Surprisingly, to date no general exact analytical expression of the resulting force can be found in the scientific literature, and one is often reduced to using numerical approaches [613]. In the low-intensity regime, a generalized version of Eq. (1) provides an approximation of the incoherent action of each laser field. Each individual laser j (j=1,,N with N the total number of lasers) is characterized by an individual detuning δj, a photon momentum kj, a Rabi frequency Ωj, and an individual saturation parameter sj=(|Ωj|2/2)/(δj2+Γ2/4). When all sj are much smaller than 1, the mean resulting force exerted incoherently by all lasers on the atom can be approximated by F=ΣjFj, where

Fj=Γ2sj1+seffkj
is the mean force exerted by each individual laser j, and seff=Σjsj [14]. For larger values of sj, Eq. (2) loses validity, as it is derived in a rate-equation approximation [14]. It also neglects any coherent action of the lasers. Coherence effects can lead to huge forces that vastly exceed the maximal value of (Γ/2)kj per laser, as is observed with the stimulated bichromatic force [6]. An exact expression of the force, including coherent effects, that holds at high intensity can be found in the particular case of a pair of counterpropagating lasers of the same intensity [15].

In this paper, we solve analytically the most general case and provide an expression for the force exerted by an arbitrary number of lasers with arbitrary intensities, phases, detunings, and directions acting on the same individual two-level atom. We show how to express this force in strict generality in the form of Eq. (2), even including coherent effects, with a generalized definition of the saturation parameter sj. We thus provide a unified formalism that holds in any configuration of lasers and enables the engineering of the mechanical action of light on individual atoms.

The paper is organized as follows: In Section 2, the exact expression of the radiation pressure force in the most general configuration is calculated. In Section 3, some specific regimes are investigated, where some interesting simplifications hold. Finally, we draw conclusions in Section 4.

2. GENERAL AND EXACT EXPRESSION OF THE RADIATION PRESSURE FORCE

We consider a two-level atom with levels |e and |g of energy Ee and Eg, respectively (Ee>Eg). We denote the atomic transition angular frequency (EeEg)/ by ωeg. The atom interacts with a classical electromagnetic field E(r,t) resulting from the superposition of N arbitrary plane waves: E(r,t)=ΣjEj(r,t), with Ej(r,t)=(Ej/2)ei(ωjtkj·r+ϕj)+c.c. Here, ωj, kj and ϕj are the angular frequency, the wave vector, and the phase of the jth plane wave, respectively, and EjEjεj, with Ej>0 and εj the normalized polarization vector of the corresponding wave. The quasi-resonance condition is fulfilled for each plane wave: |δj|ωeg, j, where δj=ωjωeg is the detuning. We define a weighted mean frequency of the plane waves by ω¯=Σjκjωj, with {κj} an a priori arbitrary set of weighting factors (κj0 and Σjκj=1).

In the electric-dipole approximation and considering spontaneous emission in the master equation approach [16], the atomic density operator ρ^ obeys

ddtρ^(t)=1i[H^(t),ρ^(t)]+D(ρ^(t))
with H^(t)=ωe|ee|+ωg|gg|D^·E(r,t) and D(ρ^)=(Γ/2)([σ^,ρ^σ^+]+[σ^ρ^,σ^+]), where D^ is the atomic electric dipole operator, r is the atom position in the electric field, Γ is the spontaneous de-excitation rate of the excited state |e, and σ^|ge| and σ^+|eg| are the atomic lowering and raising operators, respectively.

The hermiticity and the unit trace of the density operator make all four matrix elements ρee, ρeg, ρge, and ρgg, with ρkl=k|ρ^|l for k,l=e,g dependent variables. We consider here the vector of real independent variables x=(u,v,w)T, with u=Re(ρ˜ge), v=Im(ρ˜ge), and w=(ρeeρgg)/2=ρee1/2, where ρ˜ge=ρgeeiω¯t. In the rotating wave approximation (RWA), the time evolution of x resulting from Eq. (3) obeys

x˙(t)=A(t)x(t)+b
with b=(0,0,Γ/2)T and
A(t)=(Γ/2δ¯Im(Ω(t))δ¯Γ/2Re(Ω(t))Im(Ω(t))Re(Ω(t))Γ),
where δ¯=ω¯ωeg and Ω(t)=ΣjΩjei(ωjω¯)t, with Ωj (j=1,,N) the complex Rabi frequencies
Ωj=Dge·Ejei(kj·r+ϕj)/ΩR,jeiϕj,
where Dge=g|D^|e, ΩR,j>0 denotes the modulus of Ωj, and ϕj its phase. Without loss of generality, the global phases of the atomic states |e and |g can always be chosen so as to have one Rabi frequency real and positive. This is usually considered in all studies where a single plane wave interacts with the atom. However, with N arbitrary plane waves, we cannot assume without loss of generality that all Rabi frequencies are real, and their phases cannot be ignored. In the quasi-resonance condition, the RWA is fully justified as long as ΩR,j/ωj1, j [17].

Equation (4) constitutes the so-called optical Bloch equations (OBEs) adapted to the present studied case. Here, because of the time dependence of A(t), the solution cannot be expressed analytically, and the equation must be integrated numerically. However, within an arbitrary accuracy, we can always assume that the N frequency differences ωjω¯ are commensurable, i.e., that all ratios (ωjω¯)/(ωlω¯), j,l:ωlω¯, are rational numbers. Within this assumption, Ω(t) and A(t) are periodic in time with a period Tc=2π/ωc, where ωc=(LCM[(ωjω¯)1,j:ωjω¯])1, with LCM denoting the least common multiple conventionally taken as positive. It also follows that the numbers (ωjω¯)/ωc, hereafter denoted by mj, are integer, j [18]. In the particular case where all frequencies ωj are identical, A(t) is constant in time, or, equivalently, periodic with an arbitrary value of ωc0, and all integer numbers mj vanish.

Within the commensurability assumption where A(t) is Tc-periodic and given initial conditions x(t0)=x0, the OBEs admit the unique solution (Floquet’s theorem; see, e.g., Ref. [19])

x(t)=PI(t)eR(tt0)(x0xp(t0))+xp(t),
where R is a logarithm of the OBE monodromy matrix divided by Tc [20], PI(t) is an invertible Tc-periodic matrix equal to XI(t)eR(tt0) for t[t0,t0+Tc], with XI(t) the matriciant of the OBEs [20], and xp(t) is an arbitrary particular solution of the OBEs. The real parts of the eigenvalues of the matrix R, the so-called Floquet exponents, belong to the interval [0Tcλmin(t)dt/Tc,0Tcλmax(t)dt/Tc] with λmin(t) and λmax(t) the minimal and maximal eigenvalues of the matrix [A(t)+A(t)]/2, respectively [21]. Here, this matrix reads diag(Γ/2,Γ/2,Γ), and the real parts of the Floquet exponents are thus necessarily comprised between Γ and Γ/2, hence, strictly negative. This implies, first, that the matrix eR(tt0) tends to zero with a characteristic time not shorter than Γ1 and not longer than 2Γ1. At long times (tt02Γ1) x(t)xp(t). Second, the OBEs are ensured to admit a unique Tc-periodic solution [19] that the particular solution xp(t) can be set to. This certifies that at long times the solution of the OBEs is necessarily periodic (periodic regime).

The unique Tc-periodic solution of the OBEs can be expressed using the Fourier expansion

x(t)=n=+xneinωct,
with xn(un,vn,wn)T the Fourier components of x(t). Since x(t) is real, xn=xn*, and since it is continuous and differentiable, Σn|xn|2<. Inserting Eq. (8) into the OBEs yields an infinite system of equations connecting all un, vn and wn components. The system can be rearranged so as to express all un and vn as a function of the wn components. Proceeding in this way yields, n,
un=i(τn+j=1NΩjwnmjτnj=1NΩj*wn+mj),vn=(τn+j=1NΩjwnmj+τnj=1NΩj*wn+mj),
with τn±=1/[Γ+2i(nωc±δ¯)] and
wn+mM0Wn,mwn+m=12(1+s˜)δn,0.

Here, δn,0 denotes the Kronecker symbol, M0 is the set of all distinct nonzero integers mljmlmj (j,l=1,,N), Wn,m=βn,m/(αn+βn,0) (mZ), with αn=Γ+inωc and βn,m=Σj,l:mlj=mΩjΩl*(τn+ml++τnmj), and s˜=Σjs˜j, with

s˜j=Re[ΩjΓ/2iδjl=1l:δl=δjNΩl*Γ].

We have τn±=τn*, αn=αn*, βn,m=βn,m*, and Wn,m=Wn,m*.

If we define the vector of all wn components for n ranging from to +, w=(,w1,w0,w1,)T, and the infinite matrix W of elements Wn,n=ΣmM0Wn,mδn,n+m (n,n ranging from to +), Eq. (10) yields the complex inhomogeneous infinite system of equations

(I+W)w=c,
with I the infinite identity matrix and c the infinite vector of elements cn=δn,0/[2(1+s˜)]. W is an infinite centrohermitian band-diagonal matrix with as many bands as the cardinality of M0. Its main diagonal is zero. Since the OBEs admit a unique Tc-periodic solution, the infinite system admits a unique solution w with the property Σn|wn|2<. In these conditions and observing that Σn|cn|2< and that the series Σn,nWn,n=ΣnΣmM0Wn,m is absolutely convergent whatever the values of the Rabi frequencies Ωj, the detunings δj, and the de-excitation rate Γ, the infinite system [Eq. (12)] can be solved via finite larger and larger truncations, whose solutions are ensured to converge in all cases to the unique sought solution w [22]. This solution is necessarily such that w00; otherwise, all other coefficients wn would solve a homogeneous system of equations and vanish, in which case the equation for n=0 could not be satisfied. This allows us to define the ratios qn=wn/w0, n. In particular q0=1, and the reality condition yields qn=qn*. We have (Cramer’s rule)
qn=limkΔk(n)Δk(0),
with Δk(n) the determinant of the I+W matrix truncated to the lines and columns k,,k and where the n-indexed column is replaced by the vector c, correspondingly truncated. As argued above, the limit is ensured to exist in all cases [23]. Inserting wm=qmw0 in Eq. (10) for n=0 allows for expressing w0 in the form
w0=1211+seff,
where seff=Σjsj, with newly defined parameters sj as
sj=Re[ΩjΓ/2iδjl=1NΩl*Γqmlj].

According to the Ehrenfest theorem, the mean power absorbed by the atom from the jth plane wave, Pj(t)ωjdN/dtj(t), with dN/dtj(t) the mean number of photons absorbed per unit of time by the atom from that wave, is given by Pj(t)=Ej(r,t)·dD^(t)/dt. Similarly, the mean force Fj(t) exerted by the jth plane wave on the atom reads Fj(t)=Σi=x,y,zD^i(t)rEj,i(r,t), with D^i and Ej,i(r,t) the ith components (i=x,y,z) of D^ and Ej(r,t), respectively (see, e.g., Refs. [24,25]). Of course, each plane wave does not act independently of each other on the atom, since the mean value of the atomic electric dipole moment is at any time determined by the atomic state ρ^(t), which in turn is fully determined by the simultaneous action of all plane waves through the OBEs [Eq. (4)]. The total mean power absorbed from all plane waves and the net force exerted on the atom are then given by P(t)=ΣjPj(t) and F(t)=ΣjFj(t), respectively. In the RWA approximation, where one neglects the fast oscillating terms, we immediately get Pj(t)=Rj(t)ω¯ and Fj(t)=Rj(t)kj, with

Rj(t)=Re[Ωj(v(t)+iu(t))ei(ωjω¯)t].

It also follows that dN/dtj(t)=(ω¯/ωj)Rj(t). In the quasi-resonance condition, where ωjω¯, j, we can clearly consider dN/dtj(t)Rj(t).

Within the commensurability assumption, ωjω¯=mjωc, j. In the periodic regime, u(t) and v(t) are in addition Tc-periodic, and thus so are Rj(t), Pj(t), and Fj(t). In this regime, the Fourier components of Rj(t)n=+Rj,neinωct are easily obtained by inserting the Fourier expansion [Eq. (8)] into Eq. (16). By using further Eq. (9) and wn=qnw0 with w0 as in Eq. (14), we get

Rj,n=Γ2sj,n1+seff
with sj,n=(σj,n+σj,n*)/2, where
σj,n=ΩjΓ/2+i(nωcδj)l=1NΩl*Γqn+mlj.

In particular, the temporal mean value R¯j of Rj(t) in the periodic regime is given by the Fourier component Rj,0, and observing that sj,0 is nothing but the parameter sj of Eq. (15), we get R¯j=(Γ/2)sj/(1+seff). The Fourier components of the force Fj(t)n=+Fj,neinωct in the periodic regime are then given by Fj,n=Rj,nkj [26], and the mean force in this regime reads, consequently,

F¯j=Γ2sj1+seffkj,
in support of our introductory claim. Here, nevertheless, in contrast to the saturation parameter sj of Eq. (2), the newly defined parameter sj in Eq. (15) can be negative depending on the different phases of Ωl*qmlj with respect to Ωj. This accounts for two important physical effects. First, it can make R¯j negative, in which case the atom acts as a net mean photon emitter in the jth plane wave (stimulated emission) and the force exerted by that wave is directed opposite to kj (the atom is pushed in the direction opposite to the direction of the incident photons). Second, the ratio |sj/(1+seff)| can exceed 1, and the force exerted by the individual laser j can exceed the maximal spontaneous force (Γ/2)kj, as is expected for coherent forces such as the stimulated bichromatic force [6,27].

We illustrate our formalism with the calculation of the stimulated bichromatic force in a standard four traveling-wave configuration [6]. This force is subtle, since it relies on both coherent and high-intensity effects. Except for some rough approximations, it has so far only been modeled numerically [712]. We consider a detuning δ=10Γ, a Rabi frequency amplitude of 3/2δ, and a phase shift of π/2 for one of the waves. We show in Fig. 1 the amplitude of the resulting bichromatic force (averaged over the 2π range of the spatially varying relative phase between the opposite waves) acting on a moving atom as a function of its velocity v. As expected, the peak value of the bichromatic force is of the order of (2/π)(δ/Γ) (in units of kΓ/2) and spans a velocity range of the order of δ/Γ (in units of Γ/k) [28,29]. The narrow peaks are due to velocity-tuned “Doppleron” resonances [6]. A direct numerical integration of the OBEs completed with a numerical average in the periodic regime yields identical results.

 figure: Fig. 1.

Fig. 1. Stimulated bichromatic force F as a function of the atomic velocity v computed via our formalism for a detuning δ=10Γ, a Rabi frequency of 3/2δ, and a phase shift of π/2 for one laser.

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3. SPECIFIC REGIMES

At low intensity, i.e., for ΩTΣjΩR,j/Γ1, we have |s˜|1 and Σn|Wn,n|2ΩT21, n. It implies that the resolvent RW,1(I+W)1 identifies to I+k=1(W)k, and the solution w of the infinite system [Eq. (12)] is such that w01/[2(1+s˜)]1/2 and |wn|2ΩT2, n0. Hence, for mlj0, |qmlj|4ΩT21, and it follows that sjs˜j. If all plane waves have different frequencies, the sum over l in Eq. (11) only contains the single term l=j and thus

sj|Ωj|2/2δj2+Γ2/4.

If, in contrast, some plane waves have identical frequencies, coherent effects can be observed. However, if we are only interested in the incoherent action of the plane waves, an average ·ϕ over all phase differences must be performed. In the low-intensity regime, the statistical delta method [30] yields R¯jincR¯jϕ(Γ/2)s˜jϕ/(1+s˜ϕ). Since the average s˜jϕ is simply the sj of Eq. (20), we get again R¯jinc(Γ/2)sj/(1+seff) with sj as in Eq. (20). In all cases, the incoherent and low-intensity limit of our newly defined parameter sj in Eq. (15) reduces to the standard expression (20), known to hold in this regime [14].

If all plane waves have the same frequency, we get j sj=Re[(Ωj/[Γ/2iδ])(Ω*/Γ)], with Ω=ΣlΩl and δδj. For N=1, this reduces to the standard expression (20) of the saturation parameter. For two counterpropagating plane waves of identical intensity and polarization, the mean net resulting force F¯=(Γ/2)(s1s2)/(1+s1+s2)k1 reduces to the well-known phase-dependent dipole force in a stationary monochromatic wave F¯=[4ΩR2δsin(Δϕ)]/[Γ2+4δ2+8ΩR2cos2(Δϕ/2)]k1, with Δϕ=ϕ2ϕ1, and ΩRΩR,1=ΩR,2 [5].

Very generally, in a configuration with two plane waves of different frequencies, the required solution of the infinite system [Eq. (12)] can be obtained in a continued fraction approach (see also Ref. [15] for the particular case of two waves of identical intensity and polarization in a counterpropagating configuration). For N=2 and ω1ω2, the commensurability assumption implies that n2κ1=n1κ2, with n1 and n2 two positive coprime integers. It follows that ωc=|ω1ω2|/ns with ns=n1+n2, m1=sgn(ω1ω2)n2, m2=sgn(ω2ω1)n1, m12=sgn(ω1ω2)ns, and M0={±ns}. The infinite system [Eq. (10)] reads in this case, n,

wn+Wn,nswn+ns+Wn,nswnns=12(1+s˜)δn,0.

The system only couples together the Fourier components wn with n=kns (kZ). All other components are totally decoupled from these former components and thus vanish, since they satisfy a homogeneous system. Hence, the only a priori nonvanishing ratios qn and Fourier components Rj,n and Fj,n are for these specific values of n, and the periodic regime is rather characterized by the beat period Tc/ns=2π/|ω1ω2|. For n0, Eq. (21) implies wn/wnns=Wn,ns*/[1+Wn,ns(wn+ns/wn)]. Applying recursively this relation for n=ns,2ns,3ns, yields qns=Wns,ns*/[1+Kk=1(pk/1)], with pk=Wkns,nsW(k+1)ns,ns* and Kk=1(pk/1) the continued fraction p1/(1+p2/(1+p3/)). Dividing further Eq. (21) for n=kns (k>0) by w0 yields the recurrence relation q(k+1)ns=[qkns+Wkns,ns*q(k1)ns]/Wkns,ns, which allows for computing all remaining qkns with k>1, and hence along with qns all nonzero Fourier components Rj,kns and Fj,kns (k0). The set {Wkns,ns,kZ} proves to be independent of the actual value of ns, and thus, so is the set of numbers {qkns,kZ}. The continued fraction and, as expected, all Fourier components with respect to the beat periodicity are independent of the arbitrary choice of the weighting factors κ1 and κ2 that set ns.

4. CONCLUSION

In conclusion, we have provided a unified and exact formalism to calculate within the usual RWA the mechanical action of a set of arbitrary plane waves acting simultaneously on a single two-level atom. We have shown how to write the steady mean light forces acting in strict generality in the form of Eq. (2), including coherent and high-intensity effects, with a generalized definition of the parameter sj [Eq. (15)]. We have provided within a commensurable assumption similar expressions for all Fourier components of the light forces in the steady periodic regime that is established after a transient. These results provide a novel tool for engineering the mechanical action of lasers on individual atoms. They can offer an alternative to purely numerical approaches, where the extraction of the mean force and their Fourier components can be subjected to instabilities, especially when the forces vary very slowly. Our results always converge to the exact values. In the case of two lasers, we have shown in strict generality how to convert the limit of Eq. (13) into a continued fraction computable straightforwardly. This work admits a natural extension where multilevel atoms are instead considered to account for the Zeeman degeneracy of the atomic levels.

Funding

Fonds de la Recherche Scientifique - FNRS (FNRS) (4.4512.08); Federaal Wetenschapsbeleid (BELSPO) (BriX network P7/12).

Acknowledgment

T. Bastin acknowledges financial support of the Belgian FRS-FNRS through IISN and, with R. D. Glover, of the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (BriX network P7/12). L. Podlecki acknowledges an FNRS grant and the Belgian FRS-FNRS for financial support.

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18. All nonzero mj integers are setwise coprime (the greatest common divisor of these integers is 1) but not necessarily pairwise coprime (each pair of nonzero mj is not necessarily setwise coprime).

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20. The monodromy matrix C of a set of equations x˙=A(t)x+b(t), where A(t) and b(t) are T-periodic (comprising the case b(t) constant) is the matriciant XI(t) of the set of equations evaluated at time t=t0+T: C=XI(t0+T). The matriciant is the unique matrix solution X(t) to the matricial equation X˙=A(t)X subscribed to the initial conditions X(t0)=I, with I the identity matrix and where X has the same dimension as A(t). The matriciant and monodromy matrices are both invertible. A logarithm of an invertible matrix B is a matrix L such that eL=B. Every invertible matrix admits a logarithm, which is not unique but with invariant real parts of the eigenvalues.

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26. The specific links between the unknowns wn in the linear system [Eq. (12)] [23] can make the ratios qn+mlj and hence the Fourier components Fj,n not necessarily significant for the first n>0 values. In this case, the forces Fj(t) could be better described in the periodic regime by an almost periodic behavior with an almost period smaller than Tc depending on each specific case.

27. V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

28. M. Cashen and H. Metcalf, “Optical forces on atoms in nonmonochromatic light,” J. Opt. Soc. Am. B 20, 915–924 (2003). [CrossRef]  

29. L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004). [CrossRef]  

30. G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

References

  • View by:

  1. R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
    [Crossref]
  2. P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
    [Crossref]
  3. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
    [Crossref]
  4. M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
    [Crossref]
  5. R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
    [Crossref]
  6. J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
    [Crossref]
  7. M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
    [Crossref]
  8. M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
    [Crossref]
  9. S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
    [Crossref]
  10. C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
    [Crossref]
  11. C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
    [Crossref]
  12. X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
    [Crossref]
  13. D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
    [Crossref]
  14. V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).
  15. V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
    [Crossref]
  16. G. S. Agarwal, “Quantum statistical theories of spontaneous emission and their relation to other approaches,” in Springer Tracts in Modern Physics (Springer, 1974), Vol. 70, pp. 1–129.
  17. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).
  18. All nonzero mj integers are setwise coprime (the greatest common divisor of these integers is 1) but not necessarily pairwise coprime (each pair of nonzero mj is not necessarily setwise coprime).
  19. L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs (American Mathematical Society, 1995).
  20. The monodromy matrix C of a set of equations x˙=A(t)x+b(t), where A(t) and b(t) are T-periodic (comprising the case b(t) constant) is the matriciant XI(t) of the set of equations evaluated at time t=t0+T: C=XI(t0+T). The matriciant is the unique matrix solution X(t) to the matricial equation X˙=A(t)X subscribed to the initial conditions X(t0)=I, with I the identity matrix and where X has the same dimension as A(t). The matriciant and monodromy matrices are both invertible. A logarithm of an invertible matrix B is a matrix L such that eL=B. Every invertible matrix admits a logarithm, which is not unique but with invariant real parts of the eigenvalues.
  21. V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.
  22. F. Riesz, Les Systèmes d’Équations Linéaires à une Infinité d’Inconnues (Gauthier-Villars, 1913).
  23. As a consequence of the linear system [Eq. (12)] that connects w0 to wmlj, ∀l,j, then to wmlj+ml′j′, ∀l′,j′, and so on, the convergence in Eq. (13) is highly optimized by reordering the vector of unknowns w (and accordingly the lines and columns of I+W) so as to have w0 symmetrically surrounded by w±|mlj| for all distinct nonzero mlj, then by w±|mlj+ml′j′| for all additional distinct mlj+ml′j′, and so on.
  24. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).
  25. H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).
  26. The specific links between the unknowns wn in the linear system [Eq. (12)] [23] can make the ratios qn+mlj and hence the Fourier components Fj,n not necessarily significant for the first n>0 values. In this case, the forces Fj(t) could be better described in the periodic regime by an almost periodic behavior with an almost period smaller than Tc depending on each specific case.
  27. V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).
  28. M. Cashen and H. Metcalf, “Optical forces on atoms in nonmonochromatic light,” J. Opt. Soc. Am. B 20, 915–924 (2003).
    [Crossref]
  29. L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
    [Crossref]
  30. G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

2016 (1)

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

2015 (2)

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

2013 (1)

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

2011 (1)

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

2004 (1)

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

2003 (1)

2000 (1)

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

1999 (1)

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

1997 (1)

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

1995 (1)

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

1988 (1)

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

1986 (1)

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

1979 (2)

R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
[Crossref]

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

1973 (1)

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

1972 (1)

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Adrianova, L. Y.

L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs (American Mathematical Society, 1995).

Agarwal, G. S.

G. S. Agarwal, “Quantum statistical theories of spontaneous emission and their relation to other approaches,” in Springer Tracts in Modern Physics (Springer, 1974), Vol. 70, pp. 1–129.

Aldridge, L.

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

Anderson, M. H.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Anisimov, P. M.

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Arnold, B.

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

Ashkin, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Berger, R. L.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

Bjorkholm, J. E.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Bouyer, Ph.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Cable, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Casella, G.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

Cashen, M.

Cashen, M. T.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

Chi, F.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

Chu, S.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

Cook, R. J.

R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
[Crossref]

Corder, C.

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

Cornell, E. A.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Danileiko, M. V.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

Elgin, J.

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Enshner, J. R.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Eyler, E. E.

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

Galica, S. E.

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

Grimm, R.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

Hua, X.

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

Iakubovich, V. A.

V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.

Jacquinot, P.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Letokhov, V. S.

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).

Liberman, S.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Matthews, M. R.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Metcalf, H.

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

M. Cashen and H. Metcalf, “Optical forces on atoms in nonmonochromatic light,” J. Opt. Soc. Am. B 20, 915–924 (2003).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).

Minogin, V. G.

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).

Negriiko, A. M.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Ovchinnikov, Yu. B.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Picqué, J. L.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Pinard, J.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Riesz, F.

F. Riesz, Les Systèmes d’Équations Linéaires à une Infinité d’Inconnues (Gauthier-Villars, 1913).

Romanenko, V. I.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Salomon, Ch.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Schieder, R.

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Serimaa, O. T.

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

Shore, B. W.

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).

Söding, J.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Stack, D.

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Starzhinskii, V. M.

V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.

Van der Straten, P.

H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).

Voitsekhovich, V. S.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Walther, H.

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Wieman, C. E.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Williams, M. R.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

Wöste, L.

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Yatsenko, L.

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

Yatsenko, L. P.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

Phys. Rev. A (7)

R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Phys. Rev. Lett. (3)

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

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[Crossref]

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[Crossref]

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All nonzero mj integers are setwise coprime (the greatest common divisor of these integers is 1) but not necessarily pairwise coprime (each pair of nonzero mj is not necessarily setwise coprime).

L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs (American Mathematical Society, 1995).

The monodromy matrix C of a set of equations x˙=A(t)x+b(t), where A(t) and b(t) are T-periodic (comprising the case b(t) constant) is the matriciant XI(t) of the set of equations evaluated at time t=t0+T: C=XI(t0+T). The matriciant is the unique matrix solution X(t) to the matricial equation X˙=A(t)X subscribed to the initial conditions X(t0)=I, with I the identity matrix and where X has the same dimension as A(t). The matriciant and monodromy matrices are both invertible. A logarithm of an invertible matrix B is a matrix L such that eL=B. Every invertible matrix admits a logarithm, which is not unique but with invariant real parts of the eigenvalues.

V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.

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As a consequence of the linear system [Eq. (12)] that connects w0 to wmlj, ∀l,j, then to wmlj+ml′j′, ∀l′,j′, and so on, the convergence in Eq. (13) is highly optimized by reordering the vector of unknowns w (and accordingly the lines and columns of I+W) so as to have w0 symmetrically surrounded by w±|mlj| for all distinct nonzero mlj, then by w±|mlj+ml′j′| for all additional distinct mlj+ml′j′, and so on.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).

The specific links between the unknowns wn in the linear system [Eq. (12)] [23] can make the ratios qn+mlj and hence the Fourier components Fj,n not necessarily significant for the first n>0 values. In this case, the forces Fj(t) could be better described in the periodic regime by an almost periodic behavior with an almost period smaller than Tc depending on each specific case.

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Figures (1)

Fig. 1.
Fig. 1. Stimulated bichromatic force F as a function of the atomic velocity v computed via our formalism for a detuning δ=10Γ, a Rabi frequency of 3/2δ, and a phase shift of π/2 for one laser.

Equations (21)

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F=Γ2s1+sk.
Fj=Γ2sj1+seffkj
ddtρ^(t)=1i[H^(t),ρ^(t)]+D(ρ^(t))
x˙(t)=A(t)x(t)+b
A(t)=(Γ/2δ¯Im(Ω(t))δ¯Γ/2Re(Ω(t))Im(Ω(t))Re(Ω(t))Γ),
Ωj=Dge·Ejei(kj·r+ϕj)/ΩR,jeiϕj,
x(t)=PI(t)eR(tt0)(x0xp(t0))+xp(t),
x(t)=n=+xneinωct,
un=i(τn+j=1NΩjwnmjτnj=1NΩj*wn+mj),vn=(τn+j=1NΩjwnmj+τnj=1NΩj*wn+mj),
wn+mM0Wn,mwn+m=12(1+s˜)δn,0.
s˜j=Re[ΩjΓ/2iδjl=1l:δl=δjNΩl*Γ].
(I+W)w=c,
qn=limkΔk(n)Δk(0),
w0=1211+seff,
sj=Re[ΩjΓ/2iδjl=1NΩl*Γqmlj].
Rj(t)=Re[Ωj(v(t)+iu(t))ei(ωjω¯)t].
Rj,n=Γ2sj,n1+seff
σj,n=ΩjΓ/2+i(nωcδj)l=1NΩl*Γqn+mlj.
F¯j=Γ2sj1+seffkj,
sj|Ωj|2/2δj2+Γ2/4.
wn+Wn,nswn+ns+Wn,nswnns=12(1+s˜)δn,0.

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