Feasibility of an optical fiber clock

We explore the feasibility of a fiber clock, i.e., a compact, high-precision, optical lattice atomic clock based on atoms trapped inside a hollow-core optical fiber. Such a setup offers an intriguing potential both for a substantially increased number of interrogated atoms (and thereby an improved clock stability) and for miniaturization. We evaluate the sensitivity of the ${{}^{1}S}_0\text{−}{{}^{3}P}_0$ clock transition in Hg and other divalent atoms to the fiber inner core surface at nonzero temperatures. The Casimir-Polder interaction induced ${{}^{1}S}_0{\text{−}}^3{P}_0$ transition frequency shift is calculated for the atom inside the hollow capillary as a function of atomic position, capillary material, and geometric parameters. For $\mathrm{Hg}$ atoms on the axis of a silica capillary with inner radius $\ge 15\mu \mathrm{m}$ and optimally chosen thickness $d\sim 1\mu \mathrm{m}$, the atom-surface interaction induced ${{}^{1}S}_0{\text{−}}^3{P}_0$ clock transition frequency shift can be kept on the level $\delta \nu /{\nu }_{\mathrm{Hg}}\sim {10}^{-19}$. We also estimate the atom loss and heating due to collisions with the buffer gas, lattice intensity noise induced heating, spontaneous photon scattering heating, and residual birefringence induced frequency shifts.

Figure 1

(a) Cold atoms optically trapped in the ground vibrational state inside the hollow-core dielectric cylinder. (b) Energy level diagram for Hg atoms: the clock transition is shown with the dashed straight arrow, and dashed lines show the virtual transitions contributing to the resonant part of the Casimir-Polder interaction potential. (c) Schematic representation of the optical lattice field spatial distribution.

Figure 2

(a) ${}^{199}\mathrm{Hg}{{}^{1}S}_0\text{−}{{}^{3}P}_0$ clock transition frequency shift due to the resonant part of the atomic interaction with the inner surface of a silica capillary as a function of its thickness $d$ and inner radius $R_{\mathrm{in}}$ on the capillary axis $\rho =0$. The rate of the ${{}^{1}S}_0\text{−}{{}^{3}P}_0$ transition $\gamma {{}^{1}S}_0\text{−}{{}^{3}P}_0=0.6{\mathrm{s}}^{-1}$ from [34] was used in our calculations. (b) The normalized value of $A={\rho \mathrm{\Delta }\nu \left(\rho \right)\left|\psi \left(\rho \right)\right|}^2$ (purple solid line) as a function of atom position $\rho$ inside the capillary at the inner core radius $R_{\mathrm{in}}=15\mu \mathrm{m}$ and the thickness $d=1\mu \mathrm{m}$. The short dashed red and blue dashed curves show the normalized values of the frequency shift $\mathrm{\Delta }\nu /\mathrm{\Delta }{\nu }_{\max }$ and the amplitude of the ground vibrational state wave function for the atom inside the red detuned optical lattice with trapping potential depth $U=180\mu \mathrm{K}$.

Figure 3

${}^{199}\mathrm{Hg}{{}^{1}S}_0\text{−}{{}^{3}P}_0$ clock transition frequency shift due to the resonant part of the atomic interaction with the inner surface of a silica capillary as a function of its thickness $d$ at the fixed value of the inner radius $R_{\mathrm{in}}=15.5\mu \mathrm{m}$. The atoms are on the capillary axis $\rho =0$, at two different values of relative permittivity of capillary material: ${\epsilon }_r=2.45$ (blue dashed) and ${\epsilon }_r=89$ (red solid).

Figure 4

(a) Frequency shift of the ${{}^{1}S}_0\text{−}{{}^{3}P}_0$ clock transitions in Hg (blue long-dashed), Cd (green medium-dashed), Mg (orange short-dashed), and in Sr (red solid) atoms due to the (dominant) nonresonant part of the Casimir-Polder interaction potential. The clock atoms are assumed to be located on the capillary axis. The capillary of thickness $d=1\mu \mathrm{m}$ has temperature $T_S=77\mathrm{K}$. (b) Frequency shift of the ${{}^{1}S}_0\text{−}{{}^{3}P}_0$ clock transition in Hg atoms as a function of the distance $R_{\mathrm{in}}-\rho$ between the atom and the capillary inner surface. The capillary has the same parameters as in (a) with the fixed inner radius $R_{\mathrm{in}}=15\mu \mathrm{m}$. The green dashed line shows the value $\delta \nu (\rho =11\mu \mathrm{m}){\left(\frac{11\mu \mathrm{m}}{R_{\mathrm{in}}-\rho }\right)}^4$ expected for CP interaction in the planar geometry.

Figure 5

(a) Frequency shift of the ${{}^{1}S}_0\text{−}{{}^{3}P}_0$ transition in $\mathrm{Hg}$ atoms originating from the nonresonant part of the CP interaction potential, as a function of the capillary inner radius $R_{\mathrm{in}}$ at different values of its thickness $d$. The gray dashed line shows the CP interaction limit for the dielectric half-space interface. (b) Dependence of the ${{}^{1}S}_0\text{−}{{}^{3}P}_0$ transition frequency shift (nonresonant part) in the $\mathrm{Hg}$ atom on the waveguide inner radius $R_{\mathrm{in}}$ and thickness $d$.