Decomposed description of Ramsey spectra under atomic interactions

We introduce a description of Ramsey spectra under atomic interactions as a sum of decomposed components with differing dependence on interaction parameters. This description enables intuitive understanding of the loss of contrast and asymmetry of Ramsey spectra. We derive a quantitative relationship between the asymmetry and atomic interaction parameters, which enables their characterization without changing atom density. The model is confirmed through experiments with a Yb optical lattice clock.

Figure 1

Ramsey spectrum with (blue dashed line) and without (red solid line) atomic interactions. Without interactions, the spectrum is symmetric about $\delta =0$. Atomic interactions shift the $n\mathrm{th}$ peak to ${\delta }_n^{\text{I}}$. The separations from the center fringe $d_n^{\text{I}}$ and $d_{-n}^{\text{I}}$ then become unequal, resulting in asymmetry. As an example, $d_3^{\text{I}}$ and $d_{-3}^{\text{I}}$ are shown. Plotted spectrum is obtained by a sequence of two $\tau =16$ ms $\frac{\pi }{2}$ pulses separated by a 60 ms dark interval.

Figure 2

Energy shifts of two-atom states due to the atomic interactions whose strength is denoted by interaction parameters $V^{\alpha \beta }$ and $U^{\alpha \beta }$. Straight arrows show energy differences in terms of detuning $\delta$, and curved arrows indicate coupling strengths expressed as Rabi frequencies.

Figure 3

(a) $\varphi$ (red solid line) plotted together with the approximation $\delta \tau$ (blue dashed line) for $\tau =16$ ms $\frac{\pi }{2}$ pulse. (b) $A_1$ and $A_2$, plotted as red solid and blue dashed lines, respectively, for the same parameters as (a). As a reference, $A_1+A_2$, which corresponds to the envelope of the Ramsey spectra in the absence of atomic interactions, is also plotted as dotted gray lines.

Figure 4

Overview of the experimental setup of our ${}^{171}\mathrm{Yb}$ optical lattice clock. The upper left inset shows the relevant electronic states of the ${}^{171}\mathrm{Yb}$ atom, along with the wavelengths and natural linewidths of the transitions between them.

Figure 5

Experimental Ramsey spectrum (blue dashed line) for $\tau =30$ ms $\frac{\pi }{2}$ pulses and $T=150$ ms with an atom number of $\sim 1500$, which corresponds to an average number of 1.5–3 atoms per lattice site. To increase atomic interactions, the trap depth is set to a large value of $650E_{\text{r}}$. Gray line indicates the theoretical spectrum without atomic interactions. Red line includes interactions with $\tilde{W}=2\pi \times 1.6$ Hz obtained from experimental data by the asymmetry-based measurement.

Figure 6

(a) Asymmetry $a_4$ plotted as a function of the atom number with a linear fit. For our Ramsey sequence with $\tau =16$ ms, $T=100$ ms, and $\overline{\mathrm{\Omega }}\tau =\frac{\pi }{2}$, the fourth peak is located at ${\delta }_4/2\pi =34$ Hz. (b) Asymmetry $a_n$ measured for different $\tau$ with pulse areas $\overline{\mathrm{\Omega }}\tau =0.8\pi ,1.0\pi ,1.2\pi ,1.4\pi$ as indicated. All the points are measured with the atom number of $\sim 3000$. The black dashed lines indicate the position of the $n\mathrm{th}$ peak for each pulse length. (c) Collisional frequency shift of the central peak ${\delta }_0^{\text{I}}$ is measured for different excitation probability $p_1$ after the first Ramsey pulse. All error bars represent 1-$\sigma$ statistical uncertainties without accounting for instability of experimental parameters.