Motion of an atom in a weakly driven fiber-Bragg-grating cavity: Force, friction, and diffusion

We study the translational motion of an atom in the vicinity of a weakly driven nanofiber with two fiber-Bragg-grating mirrors. We calculate numerically and analytically the force, the friction coefficients, and the momentum diffusion. We find that the spatial dependences of the force, the friction coefficients, and the momentum diffusion are very complicated due to the evanescent-wave nature of the atom-field coupling as well as the effect of the van der Waals potential. We show that the time development of the mean number of photons in the cavity closely follows the translational motion of the atom through the nodes and antinodes of the fiber-guided cavity standing-wave field even though the cavity finesse is moderate, the cavity is long, and the probe field is weak.

Figure 1

An atom in the vicinity of a nanofiber with two fiber-Bragg-grating mirrors driven by a weak guided probe light field.

Figure 2

Radial component $F_r^{(0)}$ and axial component $F_z^{(0)}$ of the force as functions of the atom-to-surface distance $r-a$ (left column) and the axial position $z$ (right column). The fiber radius is $a=200$ nm. The FBG cavity length is $L=10$ cm. The FBG mirror reflectivity is $\left|R\right|{}^2=0.9$. The parameters of the atom correspond to the $D_2$-line transition $6S_{1/2}F=4,M=4\leftrightarrow 6P_{3/2}{F}^{\prime }=5,M^{\prime }=5$ of cesium, with the wavelength ${\lambda }_0=852$ nm and the natural linewidth ${\gamma }_0/2\pi =5.25$ MHz. The van der Waals coefficients are $C_{3g}/2\pi \hslash =1.56$ kHz $\mu$m${}^3$ and $C_{3e}/2\pi \hslash =3.09$ kHz $\mu$m${}^3$. The probe field, the cavity, and the atom are at exact resonance, that is, ${\omega }_p={\omega }_c={\omega }_0$. The input probe power is $P_{\mathrm{in}}=1$ pW. In the left column, the axial position of the atom is ${\beta }_{c}z=\pi /4$. In the right column, the distance from the atom to the fiber surface is $r-a=100$ nm.

Figure 3

Same as Fig. 2, except for ${\Delta }_c/2\pi =-20$ MHz and ${\omega }_c={\omega }_0$.

Figure 4

Same as Fig. 2, except for ${\Delta }_c/2\pi =20$ MHz and ${\omega }_c={\omega }_0$.

Figure 5

Friction coefficients ${\beta }_{\mathit{rr}}$ and ${\beta }_{\mathit{zz}}$ as functions of the atom-to-surface distance $r-a$ (left column) and the axial position $z$ (right column) of the atom for the parameters of Fig. 2, where ${\omega }_p={\omega }_c={\omega }_0$.

Figure 6

Friction coefficients ${\beta }_{\mathit{rz}}$ and ${\beta }_{\mathit{zr}}$ as functions of the atom-to-surface distance $r-a$ and the axial position $z$ of the atom for the parameters of Fig. 2, where ${\omega }_p={\omega }_c={\omega }_0$.

Figure 7

Same as Fig. 5, except for ${\Delta }_c/2\pi =-20$ MHz and ${\omega }_c={\omega }_0$.

Figure 8

Same as Fig. 6, except for ${\Delta }_c/2\pi =-20$ MHz and ${\omega }_c={\omega }_0$.

Figure 9

Same as Fig. 5, except for ${\Delta }_c/2\pi =20$ MHz and ${\omega }_c={\omega }_0$.

Figure 10

Same as Fig. 6, except for ${\Delta }_c/2\pi =20$ MHz and ${\omega }_c={\omega }_0$.

Figure 11

Diffusion coefficient $D$ as a function of (a) the atom-to-surface distance $r-a$ and (b) the axial position $z$ for the parameters of Fig. 2, where ${\omega }_p={\omega }_c={\omega }_0$.

Figure 12

Same as Fig. 11, except for ${\Delta }_c/2\pi =-20$ MHz and ${\omega }_c={\omega }_0$.

Figure 13

Same as Fig. 11, except for ${\Delta }_c/2\pi =20$ MHz and ${\omega }_c={\omega }_0$.

Figure 14

Time dependences of (a) the radial distance $r$, (b) the azimuthal angle $\phi$, (c) the axial position $z$, and (d) the mean intracavity photon number $\mathrm{N̄}$ for a moving atom. The initial position and velocity of the atom are ($r=3a=600\text{nm},\phi =0,z=0$) and ($v_r=0,v_{\phi }=0,v_z=20v_{\mathrm{rec}}\cong 7\text{cm/s}$), respectively. The probe field, the cavity, and the atom are at exact resonance, that is, ${\omega }_p={\omega }_c={\omega }_0$. Other parameters are as in Fig. 2.

Figure 15

Same as Fig. 14, except for the initial velocity $v_z=40v_{\mathrm{rec}}\cong 14\text{cm}/\text{s}$.

Figure 16

Same as Fig. 14, except for the initial position $r=4a=800$ nm.

Figure 17

Same as Fig. 14, except for the cavity length $L=1$ mm.