Figure 1

Ordinary dressed states in a single-frequency field tuned to red of atomic resonance. The symbol ($g$ or $e$) refers to the state (ground or excited) that connects in the limit of $\Omega \to 0$ and the number is the optical field quantum number, which differs by $\pm 1$ for ground- or excited-state atoms.Reuse & Permissions

Figure 2

A schematic diagram of the infinite set of dressed levels in bichromatic light that arise from a single pair of the usual dressed levels in monochromatic light. Each of the usual pairs in Fig. 1 produces such a set. Starting from $\mid g,b,r⟩$, atoms can be excited by red or blue light, raising or lowering the energy of the atom-light system. Stimulated emission by the beam that did not produce the excitation extends the ladder in each direction, with corresponding changes in the field quantum numbers $b$ and $r$. This figure is for equal detuning of the blue and red light, but that is not required.Reuse & Permissions

Figure 3

Example of the spatial dependence of the eigenenergies for different Rabi frequencies and $\chi =\pi ∕4$. We see that for small Rabi frequencies there are no intersections and no avoided crossings. For larger Rabi frequencies there are real crossings and avoiding crossings. The calculations are done for a matrix much larger than that of Eq. (11), and only the central few levels are plotted here. This is to avoid “edge effects” in the diagonalization. The heavy line in the middle panel shows a possible history of a moving atom through the energy levels of the system via Landau-Zener (LZ) transitions indicated by the short, vertical lines. The picture is appropriate as long as the velocity is slow enough for the atom to pass adiabatically through the kinds of anticrossings indicated by the vertical ellipse.Reuse & Permissions

Figure 4

Avoided and real crossings for Rabi frequency equal to the first critical value and $\chi =\pi ∕4$. (N.B. For the light intensity this phase is twice as large, $\pi ∕2$, because the intensity has twice the spatial frequency of the amplitude.) The upper part shows the variation of the Rabi frequencies of the two waves plotted over two wavelengths, and the lower part shows the eigenvalues of Eq. (11). The arrow shows the first point where ${\Omega }_2$ vanishes, and the dashed vertical line shows that the “A” states are maximally split at this point, by $2\delta$ and they cross exactly with the “B” states.Reuse & Permissions

Figure 5

Optimal phases as function of Rabi frequency in units of $\delta$.Reuse & Permissions

Figure 6

The eigenenergies of the two-level atom in the field of the standing bichromatic wave as function of the Rabi frequency ${\Omega }_1$ of the first field for various Rabi frequencies ${\Omega }_2$ of the second field. (a)–(d) Numerical diagonalization (exact). (e)–(h) First-order perturbation theory. (i)–(l) Second-order perturbation theory.Reuse & Permissions

Figure 7

The eigenenergies of the two-level atom in the field of the standing bichromatic wave as function of the position (in units of wavelength) for different values of the Rabi frequency ${\Omega }_0$ with the optimal spatial phase shift for each case. (a)–(d) Numerical diagonalization (exact). (e)–(h) First-order perturbation theory. (i)–(l) Second-order perturbation theory.Reuse & Permissions

Figure 8

The adiabatic functions (in units $kv∕2\pi \delta$) for transitions from the state $⟨{\psi }_N^{\left(2\right)}\mid$ ($N$ is an even integer) to the other states as function of the position (in units of wavelength) for the case of optimal force $(\chi =\pi ∕4,{\Omega }_0=\sqrt{6}\delta )$. Note that the two curves for $N\pm 2$ overlap. Ideally the curves for “$+$” and “$-$” should be mirror images, and the deviation here is indicative of the approximation.Reuse & Permissions

Figure 9

The force as function of the velocity for $\delta =500\gamma$ in optimal case of ${\Omega }_0=500\sqrt{6}\gamma$ and ${\chi }_0=\pi ∕4$.Reuse & Permissions