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
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 . 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 ) 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 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 
Here, is the rate of spontaneous decay from the upper level of the transition, is the laser photon momentum, and is the saturation parameter, where is the Rabi frequency, and is the detuning between the laser and the atomic transition angular frequencies, and , respectively. The maximal force the laser can exert on the atom is .
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 [6–13]. 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 ( with the total number of lasers) is characterized by an individual detuning , a photon momentum , a Rabi frequency , and an individual saturation parameter . When all are much smaller than 1, the mean resulting force exerted incoherently by all lasers on the atom can be approximated by , where14]. For larger values of , Eq. (2) loses validity, as it is derived in a rate-equation approximation . It also neglects any coherent action of the lasers. Coherence effects can lead to huge forces that vastly exceed the maximal value of per laser, as is observed with the stimulated bichromatic force . 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 .
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 . 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 and of energy and , respectively (). We denote the atomic transition angular frequency by . The atom interacts with a classical electromagnetic field resulting from the superposition of arbitrary plane waves: , with Here, , and are the angular frequency, the wave vector, and the phase of the th plane wave, respectively, and , with and the normalized polarization vector of the corresponding wave. The quasi-resonance condition is fulfilled for each plane wave: , , where is the detuning. We define a weighted mean frequency of the plane waves by , with an a priori arbitrary set of weighting factors ( and ).
In the electric-dipole approximation and considering spontaneous emission in the master equation approach , the atomic density operator obeys
The hermiticity and the unit trace of the density operator make all four matrix elements , , , and , with for dependent variables. We consider here the vector of real independent variables , with , , and , where . In the rotating wave approximation (RWA), the time evolution of resulting from Eq. (3) obeys17].
Equation (4) constitutes the so-called optical Bloch equations (OBEs) adapted to the present studied case. Here, because of the time dependence of , the solution cannot be expressed analytically, and the equation must be integrated numerically. However, within an arbitrary accuracy, we can always assume that the frequency differences are commensurable, i.e., that all ratios , , are rational numbers. Within this assumption, and are periodic in time with a period , where , with LCM denoting the least common multiple conventionally taken as positive. It also follows that the numbers , hereafter denoted by , are integer, . In the particular case where all frequencies are identical, is constant in time, or, equivalently, periodic with an arbitrary value of , and all integer numbers vanish.
Within the commensurability assumption where is -periodic and given initial conditions , the OBEs admit the unique solution (Floquet’s theorem; see, e.g., Ref. )20], is an invertible -periodic matrix equal to for , with the matriciant of the OBEs , and is an arbitrary particular solution of the OBEs. The real parts of the eigenvalues of the matrix , the so-called Floquet exponents, belong to the interval with and the minimal and maximal eigenvalues of the matrix , respectively . Here, this matrix reads , and the real parts of the Floquet exponents are thus necessarily comprised between and , hence, strictly negative. This implies, first, that the matrix tends to zero with a characteristic time not shorter than and not longer than . At long times () . Second, the OBEs are ensured to admit a unique -periodic solution  that the particular solution can be set to. This certifies that at long times the solution of the OBEs is necessarily periodic (periodic regime).
The unique -periodic solution of the OBEs can be expressed using the Fourier expansion8) into the OBEs yields an infinite system of equations connecting all , and components. The system can be rearranged so as to express all and as a function of the components. Proceeding in this way yields, ,
Here, denotes the Kronecker symbol, is the set of all distinct nonzero integers (), (), with and , and , with
We have , , , and .
If we define the vector of all components for ranging from to , , and the infinite matrix of elements ( ranging from to ), Eq. (10) yields the complex inhomogeneous infinite system of equations12)] can be solved via finite larger and larger truncations, whose solutions are ensured to converge in all cases to the unique sought solution . This solution is necessarily such that ; otherwise, all other coefficients would solve a homogeneous system of equations and vanish, in which case the equation for could not be satisfied. This allows us to define the ratios , . In particular , and the reality condition yields . We have (Cramer’s rule) 23]. Inserting in Eq. (10) for allows for expressing in the form
According to the Ehrenfest theorem, the mean power absorbed by the atom from the th plane wave, , with the mean number of photons absorbed per unit of time by the atom from that wave, is given by . Similarly, the mean force exerted by the th plane wave on the atom reads , with and the th components () of and , 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 , 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 and , respectively. In the RWA approximation, where one neglects the fast oscillating terms, we immediately get and , with
It also follows that . In the quasi-resonance condition, where , , we can clearly consider .
Within the commensurability assumption, , . In the periodic regime, and are in addition -periodic, and thus so are , , and . In this regime, the Fourier components of are easily obtained by inserting the Fourier expansion [Eq. (8)] into Eq. (16). By using further Eq. (9) and with as in Eq. (14), we get
In particular, the temporal mean value of in the periodic regime is given by the Fourier component , and observing that is nothing but the parameter of Eq. (15), we get . The Fourier components of the force in the periodic regime are then given by , and the mean force in this regime reads, consequently,2), the newly defined parameter in Eq. (15) can be negative depending on the different phases of with respect to . This accounts for two important physical effects. First, it can make negative, in which case the atom acts as a net mean photon emitter in the th plane wave (stimulated emission) and the force exerted by that wave is directed opposite to (the atom is pushed in the direction opposite to the direction of the incident photons). Second, the ratio can exceed 1, and the force exerted by the individual laser can exceed the maximal spontaneous force , 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 . 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 [7–12]. We consider a detuning , a Rabi frequency amplitude of , and a phase shift of for one of the waves. We show in Fig. 1 the amplitude of the resulting bichromatic force (averaged over the range of the spatially varying relative phase between the opposite waves) acting on a moving atom as a function of its velocity . As expected, the peak value of the bichromatic force is of the order of (in units of ) and spans a velocity range of the order of (in units of ) [28,29]. The narrow peaks are due to velocity-tuned “Doppleron” resonances . A direct numerical integration of the OBEs completed with a numerical average in the periodic regime yields identical results.
3. SPECIFIC REGIMES
At low intensity, i.e., for , we have and , . It implies that the resolvent identifies to , and the solution of the infinite system [Eq. (12)] is such that and , . Hence, for , , and it follows that . If all plane waves have different frequencies, the sum over in Eq. (11) only contains the single term and thus
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 over all phase differences must be performed. In the low-intensity regime, the statistical delta method  yields . Since the average is simply the of Eq. (20), we get again with as in Eq. (20). In all cases, the incoherent and low-intensity limit of our newly defined parameter in Eq. (15) reduces to the standard expression (20), known to hold in this regime .
If all plane waves have the same frequency, we get , with and . For , 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 reduces to the well-known phase-dependent dipole force in a stationary monochromatic wave , with , and .
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.  for the particular case of two waves of identical intensity and polarization in a counterpropagating configuration). For and , the commensurability assumption implies that , with and two positive coprime integers. It follows that with , , , , and . The infinite system [Eq. (10)] reads in this case, ,
The system only couples together the Fourier components with (). 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 and Fourier components and are for these specific values of , and the periodic regime is rather characterized by the beat period . For , Eq. (21) implies . Applying recursively this relation for yields , with and the continued fraction . Dividing further Eq. (21) for () by yields the recurrence relation , which allows for computing all remaining with , and hence along with all nonzero Fourier components and (). The set proves to be independent of the actual value of , and thus, so is the set of numbers . 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 and that set .
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 [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.
Fonds de la Recherche Scientifique - FNRS (FNRS) (4.4512.08); Federaal Wetenschapsbeleid (BELSPO) (BriX network P7/12).
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.
REFERENCES AND NOTES
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 integers are setwise coprime (the greatest common divisor of these integers is 1) but not necessarily pairwise coprime (each pair of nonzero 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 of a set of equations , where and are -periodic (comprising the case constant) is the matriciant of the set of equations evaluated at time : . The matriciant is the unique matrix solution to the matricial equation subscribed to the initial conditions , with the identity matrix and where has the same dimension as . The matriciant and monodromy matrices are both invertible. A logarithm of an invertible matrix is a matrix such that . 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 to , , then to , , and so on, the convergence in Eq. (13) is highly optimized by reordering the vector of unknowns (and accordingly the lines and columns of ) so as to have symmetrically surrounded by for all distinct nonzero , then by for all additional distinct , 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 in the linear system [Eq. (12)]  can make the ratios and hence the Fourier components not necessarily significant for the first values. In this case, the forces could be better described in the periodic regime by an almost periodic behavior with an almost period smaller than 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).