Theory of nonlinear sub-Doppler spectroscopy taking into account atomic-motion-induced density-dependent effects in a gas

We develop a field-nonlinear theory of sub-Doppler spectroscopy in a gas of two-level atoms, based on a self-consistent solution of the Maxwell-Bloch equations in the mean-field and single-atom density-matrix approximations. This makes it possible to correctly take into account the effects caused by the free motion of atoms in a gas, which lead to a nonlinear dependence of the spectroscopic signal on the atomic density even in the absence of a direct interatomic interaction (e.g., dipole-dipole interaction). Within the framework of this approach, analytical expressions for the light field were obtained for an arbitrary number of resonant waves and arbitrary optical thickness of a gas medium. Sub-Doppler spectroscopy in the transmission signal for two counterpropagating and copropagating waves has been studied in detail. A previously unknown redshift of a narrow sub-Doppler resonance is predicted in a counterpropagating waves scheme, when the frequency of one wave is fixed and the frequency of the other wave is varied. The magnitude of this shift depends on the atomic density and can be more than an order of magnitude greater than the known shift from the interatomic dipole-dipole interaction (Lorentz-Lorenz shift). The found effects, caused by the free motion of atoms, require a significant revision of the existing picture of spectroscopic effects depending on the density of atoms in a gas. Apart from the fundamental aspect, obtained results are important for precision laser spectroscopy and optical atomic clocks.

Figure 1

(a) Scheme of a two-level atom. (b) Illustration of a spatially attenuated plane wave in a gas, when the field amplitude becomes time dependent from the viewpoint of a moving atom.

Figure 2

Scheme illustrating the case of two counterpropagating waves. Here, the wave with frequency ${\omega }_1^{}$, propagating from left to right, at the entrance to the medium ($z=0$) has an oscillating amplitude $E_{10}^{}{e}^{-i{\omega }_1^{}t}$, while the wave with frequency ${\omega }_2^{}$, propagating from right to left, at the entrance to the medium ($z=L$) has an oscillating amplitude $E_{20}^{}{e}^{-i{\omega }_2^{}t}$.

Figure 3

The line shape in the case of counterpropagating waves for a monochromatic field: (a) total transmission signal (42); (b) nonlinear correction ($\propto \mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\left\{C_{12.2}^{}\right\}$), describing sub-Doppler resonance for an optically thin medium [see Eq. (44)], and classical expression $W(\delta ,\delta )$ [see Eq. (48)]. Calculation parameters: $\mathcal{N}{k}^{-3}=0.01$, $k{v}_0^{}=50\gamma$, $kL=2\pi \times 5$, $S_1^{}=S_2^{}=0.2$.

Figure 4

The line shape of the transmission signal (42) in the case of counterpropagating waves, when the frequency ${\omega }_1^{}$ is scanned (i.e., ${\delta }_1^{}$ is varied) at the fixed frequency ${\omega }_2^{}$ for different thicknesses of the medium: (a) $kL=2\pi \times 100$; (b) $kL=2\pi \times 400$. Calculation parameters: $\mathcal{N}{k}^{-3}=0.001$, $k{v}_0^{}=50\gamma$, $S_1^{}=S_2^{}=0.2$, ${\delta }_2^{}=-10\gamma$.

Figure 5

Comparison of the sub-Doppler resonance line shape for an optically thin medium in our theory [see $\mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\left\{C_{12.2}^{}\right\}$ in Eq. (44)] (red solid line) with the known expression [see $W({\delta }_1^{},{\delta }_2^{})$ in Eq. (48)] (blue dashed line) in the case of counterpropagating waves, when the frequency ${\omega }_1^{}$ is scanned (i.e., ${\delta }_1^{}$ is varied) at a fixed frequency ${\omega }_2^{}$ for different values of ${\delta }_2^{}$: (a) ${\delta }_2^{}=-50\gamma$; (b) ${\delta }_2^{}=50\gamma$. Calculation parameters: $\mathcal{N}{k}^{-3}=0.02$, $k{v}_0^{}=50\gamma$.

Figure 6

The frequency dependence $\mathcal{R}(\delta ,\gamma ,k{v}_0^{})$ [see Eqs. (51) and (53)] for different values of $\gamma /k{v}_0^{}$.

Figure 7

(a) The dependence ${\mathrm{\Phi }}_{12}^{}({\delta }_2^{}/k{v}_0^{},\gamma /k{v}_0^{})$ for different values of $\gamma /k{v}_0^{}$. (b) The dependence ${\mathrm{\Phi }}_{21}^{}({\delta }_1^{}/k{v}_0^{},\gamma /k{v}_0^{})$ for different values of $\gamma /k{v}_0^{}$. (c) The general functional dependence $\tilde{\mathrm{\Phi }}({\delta }_{1,2}^{}/k{v}_0^{})$ in the limit of $(\gamma /k{v}_0^{})\to 0$.

Figure 8

The line shape of the nonlinear correction $\propto \mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\left\{C_{21.2}^{}\right\}$ in Eq. (44) for an optically thin medium in the case of counterpropagating waves when the frequency ${\omega }_1^{}$ is scanned (i.e., ${\delta }_1^{}$ is varied) at a fixed frequency ${\omega }_2^{}$ for different values of ${\delta }_2^{}$: (a) ${\delta }_2^{}=0$; (b) ${\delta }_2^{}=-50\gamma$; (c) ${\delta }_2^{}=50\gamma$. Calculation parameters: $\mathcal{N}{k}^{-3}=0.001$, $k{v}_0^{}=50\gamma$.

Figure 9

The case of counterpropagating waves, when the frequency ${\omega }_2^{}$ is scanned (i.e., ${\delta }_2^{}$ is varied) at a fixed frequency ${\omega }_1^{}$ for different values of ${\delta }_1^{}$. (a) The line shape of the transmission signal (42) for ${\delta }_1^{}=-10\gamma$, $\mathcal{N}{k}^{-3}=0.001$, $S_1^{}=S_2^{}=0.2$, $kL=2\pi \times 100$. (b), (c) Comparison of the sub-Doppler resonance line shape for an optically thin medium in our theory [see $\mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\left\{C_{12,2}^{}\right\}$ in Eq. (44)] (red solid line) with the well-known expression [see $W({\delta }_1^{},{\delta }_2^{})$ in Eq. (48)] (blue dashed line): (b) ${\delta }_1^{}$=$-50\gamma$, $\mathcal{N}{k}^{-3}=0.02$; (c) ${\delta }_1^{}=50\gamma$, $\mathcal{N}{k}^{-3}=0.02$. Calculation parameters: $k{v}_0^{}=50\gamma$.

Figure 10

The case of copropagating waves, when the frequency ${\omega }_1^{}$ is scanned (i.e., ${\delta }_1^{}$ is varied) at a fixed frequency ${\omega }_2^{}$ for different values ${\delta }_2^{}$. (a) The line shape of the transmission signal (58) for ${\delta }_2^{}=10\gamma$, $\mathcal{N}{k}^{-3}=0.001$, $S_1^{}=S_2^{}=0.2$, $kL=2\pi \times 400$. (b), (c) Comparison of the sub-Doppler resonance line shape for an optically thin medium in our theory [see $\mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\{C_{12,2}^{}+C_{21,2}^{}\}$ in Eq. (44)] (red solid line) with the known expression $[W({\delta }_1^{},{\delta }_2^{})+V({\delta }_1^{},{\delta }_2^{})]$ [see Eq. (59)] (blue dashed line): (b) ${\delta }_2^{}=50\gamma$, $\mathcal{N}{k}^{-3}=0.02$; (c) ${\delta }_2^{}=-50\gamma$, $\mathcal{N}{k}^{-3}=0.02$. Calculation parameters: $k{v}_0^{}=50\gamma$.

Figure 11

The case of copropagating waves, when the frequency ${\omega }_1^{}$ is scanned (i.e., ${\delta }_1^{}$ is varied) at a fixed frequency ${\omega }_2^{}$. The frequency dependences $\mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\left\{C_{12,2}^{}\right\}$ (green solid line) and $\mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\left\{C_{21,2}^{}\right\}$ (blue dashed line) are shown. Calculation parameters: ${\delta }_2^{}=50\gamma$, $\mathcal{N}{k}^{-3}=0.02$, $k{v}_0^{}=50\gamma$.

Figure 12

The case of copropagating waves, when the frequency ${\omega }_2^{}$ is scanned (i.e., ${\delta }_2^{}$ is varied) at a fixed frequency ${\omega }_1^{}$ for different values ${\delta }_1^{}$. (a) The line shape of the transmission signal (58) for ${\delta }_1^{}=-10\gamma$, $\mathcal{N}{k}^{-3}=0.001$, $S_1^{}=S_2^{}=0.2$, $kL=100$. (b, c) Comparison of the sub-Doppler resonance line shape for an optically thin medium in our theory [see $\mathrm{Re}\left\{{\mathcal{K}}_2^{}\right\}\mathrm{Re}\{C_{12,2}^{}+C_{21,2}^{}\}$ in Eq. (44)] (red solid line) with the known expression $[W({\delta }_1^{},{\delta }_2^{})+V({\delta }_1^{},{\delta }_2^{})]$ [see Eq. (59)] (blue dashed line): (b) ${\delta }_1^{}=-50\gamma$, $\mathcal{N}{k}^{-3}=0.02$; (c) ${\delta }_1^{}=50\gamma$, $\mathcal{N}{k}^{-3}=0.02$. Calculation parameters: $k{v}_0^{}=50\gamma$.

Figure 13

The line shape of the amplitude $|A_{11,2}^{}|\propto \mathcal{N}{k}^{-3}\left|C_{11,2}^{}\right|$ of the nonlinear correction oscillating at the combination frequency ${\omega }_{11,2}^{}=2{\omega }_1^{}-{\omega }_2^{}$, in the case when the frequency ${\omega }_2^{}$ is scanned (i.e., ${\delta }_2^{}$ is varied) at the fixed frequency ${\omega }_1^{}$: (a) for counterpropagating waves; (b) for copropagating waves. Calculation parameters: $\mathcal{N}{k}^{-3}=0.01$, $k{v}_0^{}=50\gamma$, ${\delta }_1^{}=0$.