Spin Squeezing in a Rydberg Lattice Clock

We theoretically demonstrate a viable approach to spin squeezing in optical lattice clocks via optical dressing of one clock state to a highly excited Rydberg state, generating switchable atomic interactions. For realistic experimental parameters, these interactions are shown to generate over 10 dB of squeezing in large ensembles within a few microseconds and without degrading the subsequent clock interrogation.

Figure 1

(a) Energy level diagram, labeled for the specific example of strontium atoms. Transitions between the clock states $|g⟩$ and $|e⟩$ are laser driven with Rabi frequency $\mathrm{\Omega }$. A second laser off resonantly couples $|e⟩$ to a high-lying Rydberg state $|r⟩$ with Rabi frequency $\tilde{\mathrm{\Omega }}$. For a large laser detuning $\mathrm{\Delta }\gg \tilde{\mathrm{\Omega }}$, the system reduces to effective two-level atoms with binary interactions shown by the solid curve in (b). The potential resembles the van der Waals interaction $\sim 1/r^6$ (dotted curve) at large separation $r$, but saturates below a critical distance $R_{\mathrm{c}}$ (vertical dashed line). For typical parameters, the interaction potential in a lattice [60] (dashed curve) is virtually identical to the continuum case (solid line), given by Eq. (3). (c) Spin-echo type squeezing protocol, consisting of linear spin rotations around the $x$ axis and nonlinear rotations around the $z$ axis, driven by the two laser fields. The resulting evolution of the total spin is illustrated on a generalized Bloch sphere. For clock operation, this spin-echo sequence is followed by a conventional Rabi or Ramsey scheme.

Figure 2

Time dependence of the squeezing parameter ${\xi }^2$ in a $d$-dimensional optical lattice for $R_c/a=3$. Optimal squeezing is indicated by the symbols.

Figure 3

Optimal squeezing parameter in a 2D lattice as a function of $N$ for $R_c/a=5$. The dashed curve shows the result of one-axis twisting by infinite-range interactions [12], which agrees with the exact calculations (circles) for small system sizes $N<N_{\mathrm{c}}=\pi (R_{\mathrm{c}}/2a{)}^2$ for which all atoms are blocked. However, the squeezing parameter decreases well below the corresponding value ${\xi }_{\min }^2(N_{\mathrm{c}})$ (dotted line). For large $N$, ${\xi }_{\min }^2$ approaches a limiting value ${\xi }_{\infty }^2$ (dash-dotted line) as $\sim N^{-1/2}$ (thick solid curve).

Figure 4

(a) Minimal squeezing parameter ${\xi }_{\infty }^2$ attainable in $d$-dimensional lattices as a function of $R_c/a$. For large $R_{\mathrm{c}}$, ${\xi }_{\infty }^2$ display a power-law decay with an exponent $\alpha \propto d$ (inset). Panel (b) shows the corresponding interaction time ${\tau }_{\min }$ to realize optimal squeezing.