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 $\mathbf{F}$ 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]

Here, $\mathrm{\Gamma }$ is the rate of spontaneous decay from the upper level of the transition, $\hslash \mathbf{k}$ is the laser photon momentum, and $s=({|\mathrm{\Omega }|}^2/2)/({\delta }^2+{\mathrm{\Gamma }}^2/4)$ is the saturation parameter, where $\mathrm{\Omega }$ is the Rabi frequency, and $\delta =\omega -{\omega }_0$ is the detuning between the laser and the atomic transition angular frequencies, $\omega$ and ${\omega }_0$, respectively. The maximal force the laser can exert on the atom is $(\mathrm{\Gamma }/2)\hslash k$.

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 [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 $j$ ($j=1,\dots ,N$ with $N$ the total number of lasers) is characterized by an individual detuning ${\delta }_j$, a photon momentum $\hslash {\mathbf{k}}_j$, a Rabi frequency ${\mathrm{\Omega }}_j$, and an individual saturation parameter $s_j=({|{\mathrm{\Omega }}_j|}^2/2)/({\delta }_j^2+{\mathrm{\Gamma }}^2/4)$. When all $s_j$ are much smaller than 1, the mean resulting force exerted incoherently by all lasers on the atom can be approximated by $\mathbf{F}={\mathrm{\Sigma }}_j{\mathbf{F}}_j$, where

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 $s_j$. 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 $E_e$ and $E_g$, respectively ($E_e>E_g$). We denote the atomic transition angular frequency $(E_e-E_g)/\hslash$ by ${\omega }_{eg}$. The atom interacts with a classical electromagnetic field $\mathbf{E}(\mathbf{r},t)$ resulting from the superposition of $N$ arbitrary plane waves: $\mathbf{E}(\mathbf{r},t)={\mathrm{\Sigma }}_j{\mathbf{E}}_j(\mathbf{r},t)$, with ${\mathbf{E}}_j(\mathbf{r},t)=({\mathbf{E}}_j/2)e^{i({\omega }_{j}t-{\mathbf{k}}_j·\mathbf{r}+{\varphi }_j)}+\text{c.c.}$ Here, ${\omega }_j$, ${\mathbf{k}}_j$ and ${\varphi }_j$ are the angular frequency, the wave vector, and the phase of the $j$th plane wave, respectively, and ${\mathbf{E}}_j\equiv E_j{\mathit{\epsilon }}_j$, with $E_j>0$ and ${\mathit{\epsilon }}_j$ the normalized polarization vector of the corresponding wave. The quasi-resonance condition is fulfilled for each plane wave: $|{\delta }_j|\ll {\omega }_{eg}$, $\forall j$, where ${\delta }_j={\omega }_j-{\omega }_{eg}$ is the detuning. We define a weighted mean frequency of the plane waves by $\overline{\omega }={\mathrm{\Sigma }}_j{\kappa }_j{\omega }_j$, with $\{{\kappa }_j\}$ an a priori arbitrary set of weighting factors (${\kappa }_j\ge 0$ and ${\mathrm{\Sigma }}_j{\kappa }_j=1$).

In the electric-dipole approximation and considering spontaneous emission in the master equation approach [16], the atomic density operator $\widehat{\rho }$ obeys

The hermiticity and the unit trace of the density operator make all four matrix elements ${\rho }_{ee}$, ${\rho }_{eg}$, ${\rho }_{ge}$, and ${\rho }_{gg}$, with ${\rho }_{kl}=⟨k|\widehat{\rho }|l⟩$ for $k,l=e,g$ dependent variables. We consider here the vector of real independent variables $\mathbf{x}={(u,v,w)}^T$, with $u=\mathrm{Re}({\tilde{\rho }}_{ge})$, $v=\mathrm{Im}({\tilde{\rho }}_{ge})$, and $w=({\rho }_{ee}-{\rho }_{gg})/2={\rho }_{ee}-1/2$, where ${\tilde{\rho }}_{ge}={\rho }_{ge}{e}^{-i\overline{\omega }t}$. In the rotating wave approximation (RWA), the time evolution of $\mathbf{x}$ resulting from Eq. (3) obeys

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 ${\omega }_j-\overline{\omega }$ are commensurable, i.e., that all ratios $({\omega }_j-\overline{\omega })/({\omega }_l-\overline{\omega })$, $\forall j,l\text{:}{\omega }_l\ne \overline{\omega }$, are rational numbers. Within this assumption, $\mathrm{\Omega }(t)$ and $A(t)$ are periodic in time with a period $T_c=2\pi /{\omega }_c$, where ${\omega }_c=(\mathrm{LCM}[{({\omega }_j-\overline{\omega })}^{-1},\forall j\text{:}{\omega }_j\ne \overline{\omega }]{)}^{-1}$, with LCM denoting the least common multiple conventionally taken as positive. It also follows that the numbers $({\omega }_j-\overline{\omega })/{\omega }_c$, hereafter denoted by $m_j$, are integer, $\forall j$ [18]. In the particular case where all frequencies ${\omega }_j$ are identical, $A(t)$ is constant in time, or, equivalently, periodic with an arbitrary value of ${\omega }_c\ne 0$, and all integer numbers $m_j$ vanish.

Within the commensurability assumption where $A(t)$ is $T_c$-periodic and given initial conditions $\mathbf{x}(t_0)={\mathbf{x}}_0$, the OBEs admit the unique solution (Floquet’s theorem; see, e.g., Ref. [19])

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

Here, ${\delta }_{n,0}$ denotes the Kronecker symbol, $M_0$ is the set of all distinct nonzero integers $m_{lj}\equiv m_l-m_j$ ($j,l=1,\dots ,N$), ${\mathcal{W}}_{n,m}={\beta }_{n,m}/({\alpha }_n+{\beta }_{n,0})$ ($m\in \mathbb{Z}$), with ${\alpha }_n=\mathrm{\Gamma }+\text{in}{\omega }_c$ and ${\beta }_{n,m}={\mathrm{\Sigma }}_{j,l\text{:}{m}_{lj}=m}{\mathrm{\Omega }}_j{\mathrm{\Omega }}_l^{*}({\tau }_{n+m_l}^{+}+{\tau }_{n-m_j}^{-})$, and $\tilde{s}={\mathrm{\Sigma }}_j{\tilde{s}}_j$, with

We have ${\tau }_{-n}^{\pm }={\tau }_n^{\mp *}$, ${\alpha }_{-n}={\alpha }_n^{*}$, ${\beta }_{-n,-m}={\beta }_{n,m}^{*}$, and ${\mathcal{W}}_{-n,-m}={\mathcal{W}}_{n,m}^{*}$.

If we define the vector of all $w_n$ components for $n$ ranging from $-\infty$ to $+\infty$, $\mathbf{w}={(\dots ,w_{-1},w_0,w_1,\dots )}^T$, and the infinite matrix $W$ of elements $W_{n,n^{\prime }}={\mathrm{\Sigma }}_{m\in M_0}{\mathcal{W}}_{n,m}{\delta }_{n^{\prime },n+m}$ ($n,n^{\prime }$ ranging from $-\infty$ to $+\infty$), Eq. (10) yields the complex inhomogeneous infinite system of equations

According to the Ehrenfest theorem, the mean power absorbed by the atom from the $j$th plane wave, $P_j(t)\equiv \hslash {\omega }_j{⟨dN/dt⟩}_j(t)$, with ${⟨dN/dt⟩}_j(t)$ the mean number of photons absorbed per unit of time by the atom from that wave, is given by $P_j(t)={\mathbf{E}}_j(\mathbf{r},t)·d⟨\widehat{\mathbf{D}}⟩(t)/dt$. Similarly, the mean force ${\mathbf{F}}_j(t)$ exerted by the $j$th plane wave on the atom reads ${\mathbf{F}}_j(t)={\mathrm{\Sigma }}_{i=x,y,z}⟨{\widehat{D}}_i⟩(t){\nabla }_{\mathbf{r}}{E}_{j,i}(\mathbf{r},t)$, with $⟨{\widehat{D}}_i⟩$ and $E_{j,i}(\mathbf{r},t)$ the $i$th components ($i=x,y,z$) of $⟨\widehat{\mathbf{D}}⟩$ and ${\mathbf{E}}_j(\mathbf{r},t)$, 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 $\widehat{\rho }(t)$, 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 $P(t)={\mathrm{\Sigma }}_j{P}_j(t)$ and $\mathbf{F}(t)={\mathrm{\Sigma }}_j{\mathbf{F}}_j(t)$, respectively. In the RWA approximation, where one neglects the fast oscillating terms, we immediately get $P_j(t)=R_j(t)\hslash \overline{\omega }$ and ${\mathbf{F}}_j(t)=R_j(t)\hslash {\mathbf{k}}_j$, with

It also follows that ${⟨dN/dt⟩}_j(t)=(\overline{\omega }/{\omega }_j)R_j(t)$. In the quasi-resonance condition, where ${\omega }_j\simeq \overline{\omega }$, $\forall j$, we can clearly consider ${⟨dN/dt⟩}_j(t)\simeq R_j(t)$.

Within the commensurability assumption, ${\omega }_j-\overline{\omega }=m_j{\omega }_c$, $\forall j$. In the periodic regime, $u(t)$ and $v(t)$ are in addition $T_c$-periodic, and thus so are $R_j(t)$, $P_j(t)$, and ${\mathbf{F}}_j(t)$. In this regime, the Fourier components of $R_j(t)\equiv \sum _{n=-\infty }^{+\infty }{R}_{j,n}{e}^{\text{in}{\omega }_{c}t}$ are easily obtained by inserting the Fourier expansion [Eq. (8)] into Eq. (16). By using further Eq. (9) and $w_n=q_n{w}_0$ with $w_0$ as in Eq. (14), we get

In particular, the temporal mean value ${\overline{R}}_j$ of $R_j(t)$ in the periodic regime is given by the Fourier component $R_{j,0}$, and observing that $s_{j,0}$ is nothing but the parameter $s_j$ of Eq. (15), we get ${\overline{R}}_j=(\mathrm{\Gamma }/2)s_j/(1+s_{\mathrm{eff}})$. The Fourier components of the force ${\mathbf{F}}_j(t)\equiv \sum _{n=-\infty }^{+\infty }{\mathbf{F}}_{j,n}{e}^{\text{in}{\omega }_{c}t}$ in the periodic regime are then given by ${\mathbf{F}}_{j,n}=R_{j,n}\hslash {\mathbf{k}}_j$ [26], and the mean force in this regime reads, consequently,

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 [7–12]. We consider a detuning $\delta =10\mathrm{\Gamma }$, a Rabi frequency amplitude of $\sqrt{3/2}\delta$, and a phase shift of $\pi /2$ for one of the waves. We show in Fig. 1 the amplitude of the resulting bichromatic force (averaged over the $2\pi$ 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 $(2/\pi )(\delta /\mathrm{\Gamma })$ (in units of $\hslash k\mathrm{\Gamma }/2$) and spans a velocity range of the order of $\delta /\mathrm{\Gamma }$ (in units of $\mathrm{\Gamma }/k$) [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.

 

Fig. 1. Stimulated bichromatic force $F$ as a function of the atomic velocity $v$ computed via our formalism for a detuning $\delta =10\mathrm{\Gamma }$, a Rabi frequency of $\sqrt{3/2}\delta$, and a phase shift of $\pi /2$ for one laser.

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3. SPECIFIC REGIMES

At low intensity, i.e., for ${\mathrm{\Omega }}_{\mathrm{T}}^{\prime }\equiv {\mathrm{\Sigma }}_j{\mathrm{\Omega }}_{\mathrm{R},j}/\mathrm{\Gamma }\ll 1$, we have $|\tilde{s}|\ll 1$ and ${\mathrm{\Sigma }}_{n^{\prime }}|W_{n,n^{\prime }}|\lesssim 2{\mathrm{\Omega }}_{\mathrm{T}}^{\prime 2}\ll 1$, $\forall n$. It implies that the resolvent $R_{W,-1}\equiv {(I+W)}^{-1}$ identifies to $I+\sum _{k=1}^{\infty }{(-W)}^k$, and the solution $\mathbf{w}$ of the infinite system [Eq. (12)] is such that $w_0\simeq -1/[2(1+\tilde{s})]\simeq -1/2$ and $|w_n|\lesssim 2{\mathrm{\Omega }}_{\mathrm{T}}^{\prime 2}$, $\forall n\ne 0$. Hence, for $m_{lj}\ne 0$, $|q_{m_{lj}}|\lesssim 4{\mathrm{\Omega }}_{\mathrm{T}}^{\prime 2}\ll 1$, and it follows that $s_j\simeq {\tilde{s}}_j$. If all plane waves have different frequencies, the sum over $l$ in Eq. (11) only contains the single term $l=j$ 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 ${⟨·⟩}_{\varphi }$ over all phase differences must be performed. In the low-intensity regime, the statistical delta method [30] yields ${\overline{R}}_j^{\mathrm{inc}}\equiv {⟨{\overline{R}}_j⟩}_{\varphi }\simeq (\mathrm{\Gamma }/2){⟨{\tilde{s}}_j⟩}_{\varphi }/(1+{⟨\tilde{s}⟩}_{\varphi })$. Since the average ${⟨{\tilde{s}}_j⟩}_{\varphi }$ is simply the $s_j$ of Eq. (20), we get again ${\overline{R}}_j^{\mathrm{inc}}\simeq (\mathrm{\Gamma }/2)s_j/(1+s_{\mathrm{eff}})$ with $s_j$ as in Eq. (20). In all cases, the incoherent and low-intensity limit of our newly defined parameter $s_j$ 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 $\forall j$ $s_j=\text{Re}[({\mathrm{\Omega }}_j/[\mathrm{\Gamma }/2-i\delta ])({\mathrm{\Omega }}^{*}/\mathrm{\Gamma })]$, with $\mathrm{\Omega }={\mathrm{\Sigma }}_l{\mathrm{\Omega }}_l$ and $\delta \equiv {\delta }_j$. 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 $\overline{\mathbf{F}}=(\mathrm{\Gamma }/2)(s_1-s_2)/(1+s_1+s_2)\hslash {\mathbf{k}}_1$ reduces to the well-known phase-dependent dipole force in a stationary monochromatic wave $\overline{\mathbf{F}}=[4{\mathrm{\Omega }}_R^2\delta \sin (\mathrm{\Delta }\varphi )]/[{\mathrm{\Gamma }}^2+4{\delta }^2+8{\mathrm{\Omega }}_R^2{\cos }^2(\mathrm{\Delta }\varphi /2)]\hslash {\mathbf{k}}_1$, with $\mathrm{\Delta }\varphi ={\varphi }_2-{\varphi }_1$, and ${\mathrm{\Omega }}_R\equiv {\mathrm{\Omega }}_{R,1}={\mathrm{\Omega }}_{R,2}$ [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 ${\omega }_1\ne {\omega }_2$, the commensurability assumption implies that $n_2{\kappa }_1=n_1{\kappa }_2$, with $n_1$ and $n_2$ two positive coprime integers. It follows that ${\omega }_c=|{\omega }_1-{\omega }_2|/n_s$ with $n_s=n_1+n_2$, $m_1=\mathrm{sgn}({\omega }_1-{\omega }_2)n_2$, $m_2=\mathrm{sgn}({\omega }_2-{\omega }_1)n_1$, $m_{12}=\mathrm{sgn}({\omega }_1-{\omega }_2)n_s$, and $M_0=\{\pm n_s\}$. The infinite system [Eq. (10)] reads in this case, $\forall n$,

The system only couples together the Fourier components $w_n$ with $n=k{n}_s$ ($k\in \mathbb{Z}$). 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 $q_n$ and Fourier components $R_{j,n}$ and ${\mathbf{F}}_{j,n}$ are for these specific values of $n$, and the periodic regime is rather characterized by the beat period $T_c/n_s=2\pi /|{\omega }_1-{\omega }_2|$. For $n\ne 0$, Eq. (21) implies $w_n/w_{n-n_s}=-{\mathcal{W}}_{-n,n_s}^{*}/[1+{\mathcal{W}}_{n,n_s}(w_{n+n_s}/w_n)]$. Applying recursively this relation for $n=n_s,2n_s,3n_s,\dots$ yields $q_{n_s}=-{\mathcal{W}}_{-n_s,n_s}^{*}/[1+{\mathcal{K}}_{k=1}^{\infty }(p_k/1)]$, with $p_k=-{\mathcal{W}}_{k{n}_s,n_s}{\mathcal{W}}_{-(k+1)n_s,n_s}^{*}$ and ${\mathcal{K}}_{k=1}^{\infty }(p_k/1)$ the continued fraction $p_1/(1+p_2/(1+p_3/\dots ))$. Dividing further Eq. (21) for $n=k{n}_s$ ($k>0$) by $w_0$ yields the recurrence relation $q_{(k+1)n_s}=-[q_{k{n}_s}+{\mathcal{W}}_{-k{n}_s,n_s}^{*}{q}_{(k-1)n_s}]/{\mathcal{W}}_{k{n}_s,n_s}$, which allows for computing all remaining $q_{k{n}_s}$ with $k>1$, and hence along with $q_{n_s}$ all nonzero Fourier components $R_{j,k{n}_s}$ and ${\mathbf{F}}_{j,k{n}_s}$ ($k\ge 0$). The set $\{{\mathcal{W}}_{k{n}_s,n_s},k\in \mathbb{Z}\}$ proves to be independent of the actual value of $n_s$, and thus, so is the set of numbers $\{q_{k{n}_s},k\in \mathbb{Z}\}$. 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 ${\kappa }_1$ and ${\kappa }_2$ that set $n_s$.

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 $s_j$ [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.