Engineering Quantum States of Matter for Atomic Clocks in Shallow Optical Lattices

We investigate the effects of stimulated scattering of optical lattice photons on atomic coherence times in a state-of-the art ${}^{87}\mathrm{Sr}$ optical lattice clock. Such scattering processes are found to limit the achievable coherence times to less than 12 s (corresponding to a quality factor of $1\times {10}^{16}$), significantly shorter than the predicted 145(40) s lifetime of ${}^{87}\mathrm{Sr}$’s excited clock state. We suggest that shallow, state-independent optical lattices with increased lattice constants can give rise to sufficiently small lattice photon scattering and motional dephasing rates as to enable coherence times on the order of the clock transition’s natural lifetime. Not only should this scheme be compatible with the relatively high atomic density associated with Fermi-degenerate gases in three-dimensional optical lattices, but we anticipate that certain properties of various quantum states of matter—such as the localization of atoms in a Mott insulator—can be used to suppress dephasing due to tunneling.

Figure 1

Measured (a) lifetimes and (b) coherence times. Experimental data are shown in black points along with $1\sigma$ error bars. Results of the master equation simulation are shown as shaded red regions. The solid red line in (a) represents the fitted decay rate from Ref. [17].

Figure 2

(a) Motional dephasing in a conventional, $a/{\lambda }_{\mathrm{clk}}\approx 0.6$ lattice. An atom in an equal superposition of $|g⟩$ and $|e⟩$ is depicted as the face of a clock where the position of the hand reflects the relative phase between the atomic superposition and the local oscillator. Upon tunneling to an adjacent lattice site, the relative phase changes by an amount $\varphi \approx 1.2\pi$. (b) Motional dephasing can be eliminated by matching the lattice constant to the probe wavelength. An atom then sees the same local oscillator phase at each lattice site. (c) Calculated motional dephasing rates (${\gamma }_t$), kinetic energies ($t^{*}=12t$), and interaction energies ($U$) as a function of lattice spacing, $a$, in a $V=4E_r$ lattice. The horizontal grey line represents the inverse lifetime (${\tau }_0^{-1}$) of the strontium clock transition. Single particle motional effects are suppressed below the natural decay rate for $a\gtrsim 2\mu \mathrm{m}$.

Figure 3

(a) Energy spectrum of Eq. (5) at half-filling. States with zero, one, and two atoms in $|e⟩$ are shown as blue, red, and green lines, respectively. The $|gg⟩$ and $|ee⟩$ states are noninteracting due to the Pauli exclusion principle. The $|eg⟩\pm |ge⟩$ states are spread by twice the Bloch band width at $U=0$. Whereas in the Mott-insulating regime ($U/t\gg 1$), an energy gap $U$ opens up and a pair of weakly interacting states become spectroscopically resolvable. (b) Dephasing rates, as given by Eq. (6), for one (red dashed line) and two (solid blue line) atoms in a double well potential. The tunneling rates and interaction strengths as a function of lattice spacing are taken from Fig. 2 and inserted into the double well Hamiltonian with $\mathrm{\Omega }/2\pi =0.5\mathrm{Hz}$. The curves are not plotted for $4t>\mathrm{\Omega }$ where the analogy between the double well system and an infinite lattice breaks down as the discrete levels of the finite sized system become resolved.