Colloquium: Strong optical forces on atoms in multifrequency light

Optical forces on atoms irradiated with a single frequency of light have been extensively studied for many years, both theoretically and experimentally. The two-level atom model has been used to describe a wide range of optical force phenomena and to successfully exploit a large range of applications. New areas of study were opened up when the multiple levels of real atoms were considered. In contrast, using multifrequency light on a single atomic transition has not been studied as much, but using such light also results in very significant differences in the optical forces. This Colloquium outlines the basic concepts of forces resulting from the use of two-frequency light (bichromatic force) and swept frequency light (adiabatic rapid passage force). Both of these forces derive from stimulated processes only and as a result can produce coherent exchange of momentum between atoms and light. The consequences are impressively larger forces with comparably larger velocity capture ranges and even atom cooling without spontaneous emission.

Figure 1

Energy level diagram for the atom plus field Hamiltonian. Each vertical column has the familiar atomic level scheme, but the columns are vertically displaced by $\hslash {\omega }_{\ell }$ because of the inclusion of additional light energy of $\hslash {\omega }_{\ell }$ in each column. The nearly degenerate pairs are indicated. From [58].

Figure 2

Scheme for the rectification of the dipole force. The solid curve in (a) shows the light shift of the ground state of a two-level atom in the standing wave of field 1, ${\mathrm{\Omega }}_1=40\gamma$ and $\delta =3\gamma$. Note that it is not sinusoidal because $\mathrm{\Omega }\gg \delta$. The dotted curve shows the spatial variation of the detuning caused by the smaller light shift of field 2 that has ${\delta }_2=50\gamma$ and ${\mathrm{\Omega }}_2=200\gamma$. (b) The gradient of the curve in (a), corresponding to the force on a ground state atom from field 1. (c) The total rectified force on the atoms because the sign of the detuning reverses appropriately. Simply taking minus the gradient of the curve in (a) is not appropriate because the atom spends considerable time in the excited state whose light shift is opposite, so it is necessary to calculate $⟨F⟩=-\mathrm{tr}(\rho \nabla \mathcal{H})$. Thus choosing the relative spatial phase of these standing waves carefully results in a rectification of the force. From [58].

Figure 3

Detuning scheme for the bichromatic force. The thickness of the line labeled $|e⟩$ represents the natural width of the excited state $\gamma$, and $|\delta |\gg \gamma$ as shown.

Figure 4

(a) The familiar dressed state pairs of an atom in a single-frequency light field at ${\omega }_{\ell }$, detuned below atomic resonance ${\omega }_a$ ($\delta <0$). Interaction with the light field of quantum number $n$ mixes these bare states $|e,n-1⟩$ and $|g,n⟩$ to form new states split by $\hslash {\mathrm{\Omega }}^{\prime }$ (not shown) found from Eq. (2). (b) With two frequencies, and hence two field quantum numbers $r$ and $b$, there arises a ladder of energy states both up and down from each original dressed state pair. Adapted from [95].

Figure 5

(a) The eigenvalues of the two-frequency dressed state system separated by $\hslash \delta$, where the two fields have equal strength but spatial phase offset of $\lambda /8$, along with a typical path followed by an atom moving to the right. The light field comprises bichromatic standing waves so that the intensities vary with position. The right choice of parameters produces exact crossings at points (A) because one of the standing wave fields has a node as shown in (b). Moving atoms can make Landau-Zener transitions and cross from one energy manifold to another, as shown by the heavy arrow, thereby exchanging optical potential energy for kinetic energy. From [21].

Figure 6

A simulation of the time evolution of the velocity space density of an ensemble of atoms that started with a uniform array of initial velocities. Here $\delta =15\gamma$ and there is a strong accumulation of atoms at speed $-v_c=-7.5\gamma /k$. Behind this high ridge is a deep valley of width $\sim 15\gamma /k\approx 35v_r$ where the atoms began. This simulation was done for the $2{}^{3}S\to 3{}^{3}P$ transition in He at $\lambda =389\mathrm{nm}$, where ${\omega }_r\approx 330\mathrm{kHz}$ so that $t_c\approx 380\mathrm{ns}$. Even at $t=t_c/2\approx 190\mathrm{ns}$ there is evidence of substantial cooling. From [41].

Figure 7

The upper portion shows two pulse pairs with a gap between them. The gap is for experimental reasons and the phase notations will be discussed later. The lower part shows the frequency sweeps, and in this case they are both upward. However, they can be reversed or even in opposite directions. From [83].

Figure 8

(a) The arc of a smoothly swept $\overrightarrow{\mathrm{\Omega }}(t)$ as the detuning varies from one side of the resonance to the other (smooth red curve) using ${\delta }_0=30{\omega }_m$ and ${\mathrm{\Omega }}_0=50{\omega }_m$. It begins at one pole where $\mathrm{\Omega }(0)\ll \delta (0)={\delta }_0$ so that $\overrightarrow{\mathrm{\Omega }}(0)$ is nearly polar, arcs toward the equator where $\delta (t)=0$ and $\mathrm{\Omega }(t)={\mathrm{\Omega }}_0$, and finishes at the other pole at the end of the pulse where $\mathrm{\Omega }(t)$ is again very small. Here $\overrightarrow{R}$ (looped black curve) makes many precession cycles and stays close to $\overrightarrow{\mathrm{\Omega }}(t)$ during the sweep. (b)  ${\delta }_0=1.10{\omega }_m$ and ${\mathrm{\Omega }}_0=1.61{\omega }_m$ show an unusual case where the trajectory of $\overrightarrow{R}$ is a simple arc along a meridian that is $\sim 90°$ away from the path of $\overrightarrow{\mathrm{\Omega }}(t)$ so that $\overrightarrow{R}$ also goes from pole to pole. Adapted from [59] and [83].

Figure 9

A plot of Eq. (2). The $|g,n⟩$ and the $|e,n-1⟩$ dressed states comprise two separated energy sheets except at their conical intersection at the origin. Their upper (lower) state is ground at $\mathrm{\Omega }=0$ for $\delta >0(\delta <0)$. The indicated path is a possible trajectory for ARP. From [54].

Figure 10

Plots showing calculated values of $\overrightarrow{R}$ on the Bloch sphere as viewed from the south pole. Each set of points lies on a circle whose radius and center depend on the sweep parameters. These parameters are $({\delta }_0/{\omega }_m,{\mathrm{\Omega }}_0/{\omega }_m)=$ (a) (2.4, 1.8), (b) (3, 4.4), (c) (7, 7), and (d) (14, 18). The points appear to not be evenly spaced [e.g., (a)] because the rotation angle is large and more than one full rotation is shown. From [54].

Figure 11

Counterpropagating beams, each carrying two frequencies. The frequencies are equally displaced from the atomic frequency ${\omega }_a$ so that the carrier is on resonance. The beat between these frequencies produces the “pulses” that are used in the $\pi$-pulse model of the BF.

Figure 12

Each panel shows the velocity dependence of the bichromatic force calculated for a relative spatial phase offset of 5% larger than $\lambda /8$ (95°) by direct numerical integration of the OBEs (dotted lines), the calculated values convolved with the experimental resolution (solid lines), and the measured values (data points). These measurements and calculations were done with $\delta =2\pi \times 55\mathrm{MHz}=9.1\gamma$. The different values of $\mathrm{\Omega }/\delta$ were set using a half-wave plate and polarizer combination, starting with (a) 1.19, (b) 1.13, (c) 1.07, (d) 0.98, (e) 0.88, and (f) 0.76. The calculated and experimental plots have vertical scales different by the factor 0.83 in all panels. From [90].

Figure 13

Profile of the collimated He* beam. Atoms could be captured from 180 mrad ($2\times \mathrm{FWHM}$) into this 7.5 mrad FWHM peak. The intensity was nearly ${10}^{10}\mathrm{atoms}/\mathrm{s}{\mathrm{mm}}^2$ and the brightness was ${10}^{16}\mathrm{atoms}/\mathrm{s}\mathrm{sr}{\mathrm{mm}}^2$. The BF detuning was $2\pi \times 60\mathrm{MHz}$. From [65].

Figure 14

The atomic distribution measured 63 cm downstream from the interaction region after an average cooling time of 220 ns. The Rabi frequency was $\mathrm{\Omega }=2\pi \times 36\mathrm{MHz}$, a compromise between two optimum values ([20]). There is a large background from the He* source so the raw data here show a change of only a few percent. The velocity smearing caused by the width of the longitudinal velocity distribution is a few m/s, but it affects only the deflected atoms. It does not affect the width of the “hole” because there are no atoms there. Adapted from [21].

Figure 15

Observed slowing of a He* beam with chirped light using the BF with $\delta =74\gamma$ and $\mathrm{\Omega }=\sqrt{3/2}\delta$. The left panel shows the result with no chirp (fixed $\delta$), and the right panel shows the result of chirping the center frequency by 300 MHz. The velocity range is very much larger in the latter case. The dotted curves show the original velocity distributions, the red curves near them show the changes, and the blue curves (lowest ones) are their difference. Adapted from [16].

Figure 16

(a) The frequency spectrum of the amplitude-modulated light with no phase modulation and (b) the frequency spectrum of the light with both phase and amplitude modulation (note the difference in scales). Adapted from [59].

Figure 17

(a) The dependence of the ARP force on the peak Rabi frequency ${\mathrm{\Omega }}_0$ and the range of the frequency sweep $\pm {\delta }_0$, calculated from the solutions of Eq. (6) and using the Ehrenfest theorem ([54]). (b) The results of measuring the ARP force over a similar region of parameter space as (a). The agreement in the strength of the force is qualitatively good and quantitatively acceptable except for a numerical factor slightly larger than 2. From [59].

Figure 18

The bright vertical line is an image from the source UV emission through the collimating slit. The profile of the atoms deflected to the right in the indicated rectangle is analyzed to find the average deflection for plotting in Fig. 17.