Colloquium: Physics of optical lattice clocks

Recently invented and demonstrated optical lattice clocks hold great promise for improving the precision of modern time keeping. These clocks aim at the ${10}^{-18}$ fractional accuracy, which translates into a clock that would neither lose nor gain a fraction of a second over an estimated age of the Universe. In these clocks, millions of atoms are trapped and interrogated simultaneously, dramatically improving clock stability. Here the principles of operation of these clocks are discussed and, in particular, a novel concept of magic trapping of atoms in optical lattices. Recently proposed microwave lattice clocks are also highlights and several applications that employ the optical lattice clocks as a platform for precision measurements and quantum information processing.

Figure 1

Illustration of the essential elements of optical lattice clocks. (a) A spatial interference of laser beams creates an egg-carton-like optical potential that traps clock atoms. The atoms are confined to regions much smaller than the laser wavelength ${\lambda }_L$. (b) Atoms are probed on the ${}^{1}S_0\mathrm{\text{−}}{}^{3}P_0$ clock transition. The wavelength ${\lambda }_L$ is tuned to its magic value so that the clock ${}^{1}S_0$ and ${}^{3}P_0$ states are equally energy shifted by the lattice potential, leaving the transition frequency unperturbed.

Figure 2

A diagram of the low-lying energy levels for Mg ($n=3$), Ca ($n=4$), Sr ($n=5$), and Yb ($n=6$). The relative position of the levels above the ${}^{3}P_J$ fine-structure manifold depends on the atom. This diagram reflects energy levels of Yb (core-excited states are not shown). The clock transition is between the ground and the lowest-energy ${}^{3}P_0$ state.

Figure 3

Dynamic polarizabilities $\alpha$ of the two clock levels in Yb as a function of laser frequency $\omega$. The solid curve is the polarizability of the $6s^2{}^{1}S_0$ lower clock state, and the dashed line is $\alpha (\omega )$ of the $6s6p{}^{3}P_0$ upper clock state. The unit a.u. stands for atomic units. Conversion factors are $\alpha /h[\mathrm{Hz}/(\mathrm{V}/\mathrm{m}{)}^2]=2.48832\times {10}^{-8}\alpha [\mathrm{a}.\mathrm{u}]$ for polarizability and $\omega /(2\pi )[\mathrm{Hz}]=4.1341\times {10}^{16}\omega [\mathrm{a}.\mathrm{u}.]$ for frequency. Magic frequencies of the laser field are marked by small circles on the plot.

Figure 4

(a) A one-dimensional optical clock is realized by a standing wave of light tuned to the magic wavelength. Multiply trapped spin-polarized fermions in a single pancake potential may be protected from collisions by the Pauli blocking. (b) Electric and magnetic field amplitudes in a standing wave.

Figure 5

Schematic of the experimental setup for Sr spectroscopy used in an early experiment. Ultracold ${}^{87}\mathrm{Sr}$ atoms are loaded into a 1D optical lattice produced by the standing wave of a Ti-sapphire laser tuned to the magic wavelength. The atoms interact with the clock laser propagating along this axis, and the Lamb-Dicke condition is satisfied. AOM stands for the acousto-optic modulator. From 88.

Figure 6

(a) The ${}^{87}\mathrm{Sr}$ magic wavelength determined by investigating the spectral line broadening of the clock transition, as shown in the inset. This broadening revealed the vibrational frequency differences in two states of the clock transition, which is plotted as a function of lattice laser wavelength to determine the degenerate wavelength to be ${\lambda }_L=813.5\pm 0.9\mathrm{nm}$. (b) The first spectrum of the clock transition in the magic lattice. The spectrum consists of the heating and cooling sidebands at $\approx \pm 64\mathrm{kHz}$ and the recoilless spectrum (the carrier component) with a linewidth of 700 Hz (full width half maximum) as shown in the inset.

Figure 7

(a) Energy levels for alkaline-earth atoms relevant to blue-detuned magic wavelengths for the ${}^{1}S_0-{}^{3}P_0$ clock transition. By applying the lattice laser detuned slightly above the nearby resonant state, $nsnp{}^{1}P_1$ and $ns(n+1)d{}^{3}D_1$, atoms can be trapped near the nodes of the standing wave. (b) The light shifts for the ${}^{1}S_0$ (dashed) and ${}^{3}P_0$ (solid) states of Sr as a function of the lattice laser wavelength for a laser intensity of $I=10\mathrm{kW}/{\mathrm{cm}}^2$. Intersections of the curves indicate blue magic wavelengths, which include ${\lambda }_b\approx 360$ and 390 nm.

Figure 8

Laser-induced fluorescence of atoms (a) confined in a 1D optical lattice and (b) in free fall. The dashed line shows a Gaussian fit to the data points (b). The confinement suppressed the Doppler width of 83 kHz and gave a narrow Lorentzian linewidth of 11 kHz, which was limited by the saturation broadening. A slight blue shift of the center frequency in (b) is caused by the photon recoil shift.

Figure 9

Energy levels for ${}^{88}\mathrm{Sr}$ and ${}^{87}\mathrm{Sr}$ atoms. Spin-polarized ultracold ${}^{87}\mathrm{Sr}$ atoms were prepared by optical pumping on the ${}^{1}S_0(F=9/2)-{}^{3}P_1(F=9/2)$ transition at $\lambda =689\mathrm{nm}$ with circularly polarized light. The first-order Zeeman shift and the vector light shift on the clock transition at $\lambda =698\mathrm{nm}$ were eliminated by averaging the transition frequencies $f_{\pm }$.

Figure 10

Two optical lattice clocks with different isotopes and lattice configurations were operated to investigate their beat note.

Figure 11

Differential polarizability for Al $\mu \mathrm{\text{Magic}}$ clock in the $\mathbf{B}\parallel \widehat{k}$ geometry as a function of the lattice laser frequency. Dotted line: contribution from the scalar term; dashed line: contribution from the tensor term; solid line: total polarizability. Total clock shift vanishes at two magic values of the laser frequency.

Figure 12

Left panel: Zeeman effect (Breit-Rabi diagram) for the hyperfine manifold in the ground state of $I=3/2$ isotopes of alkali atoms. Two clock levels $|F^{\prime }=2,M_F^{\prime }=+1⟩$ and $|F=1,M_F=-1⟩$ are shown. Clock transition at the magic $B$ field is indicated by a vertical doubly headed arrow. Right panel: Geometry of laser-atom interaction; degree of circular polarization, angle $\theta$, and laser wavelength may be varied.

Figure 13

Magic conditions for ${}^{133}\mathrm{Cs}$. A dependence of the product $M_{F^{\prime }}A\cos \theta$ on trapping laser frequency (in atomic units) is plotted. The shaded regions are bound by $-|M_{F^{\prime }}|$ and $+|M_{F^{\prime }}|$ lines. Magic trapping for a $|F^{\prime }=4,M_F^{\prime }⟩\to |F=3,-M_F^{\prime }⟩$ clock transition is only possible when the computed curve lies inside the corresponding shaded region.

Figure 14

Fractional clock shifts for Sr as a function of separation from a gold surface at $T=300\mathrm{K}$. Individual points represent shifts in individual trapping sites of the optical lattice. First well is placed at ${\lambda }_m/4\approx 200\mathrm{nm}$ and subsequent points are separated by ${\lambda }_m/2\approx 400\mathrm{nm}$. Inset: Idealized setup for measuring atom-wall interactions with optical lattice clocks.

Figure 15

Schematic of the entanglement process. The transport lattice is created by a superposition of two displaced circularly polarized standing wave lattices: ${\sigma }_{+}$ and ${\sigma }_{-}$ lattices. (a) A single head atom (light circle) and several clock atoms (dark circles) are trapped in the minima of a 1D optical lattice, with one or fewer atoms per site. Because of an intensity differential of the underlying lattices, the clock atoms couple strongly to the ${\sigma }_{+}$ lattice (solid line). The head atom is placed in a superposition of atomic states: one which couples strongly to the ${\sigma }_{+}$ lattice and one which couples strongly to the ${\sigma }_{-}$ lattice (dashed line). (b) As the displacement between the two circularly polarized lattices increases, the ${\sigma }_{-}$ state is spatially separated and is transported along the lattice. (c) This portion of the head atom is then brought into contact with a clock atom to entangle the two atoms. (d) The head atom is transported further to obtain entanglement with the remaining clock atoms in a similar manner.