Ramsey spectroscopy, matter-wave interferometry, and the microwave-lensing frequency shift

We derive the general frequency shift of a microwave atomic clock due to resonant dipole forces acting on the atomic wave functions during the Ramsey interactions. We explicitly demonstrate that only dressed-state populations in position space contribute significantly to the frequency shift and that the de Broglie phases of the dressed-state wave functions do not contribute. In addition, we show that momentum changes in the second Ramsey interaction normally produce negligible frequency shifts in comparison to those from the first Ramsey interaction.

Figure 1

(a) The spatial gradients of the dipole energy in Ramsey spectroscopy deliver impulses to the atomic dressed-state wave functions, acting as positive lens for state $|2\rangle$ and negative lens for $|1\rangle$. The phase of the Ramsey field is shifted in the second interaction, giving dressed states $|1^{\prime }\rangle$ and $|2^{\prime }\rangle$, and four interferometer paths. The exaggerated deflections are of the order of nanometers, compared to centimeter widths of the atomic cloud. Apertures lead to a greater fraction of dressed state $|2\rangle$, ψ${}_2$($\vec{r}$,$t_d$) or equivalently ψ${}_{21}$($\vec{r}$,$t_d$) and ψ${}_{22}$($\vec{r}$,$t_d$), being detected than $|1\rangle$. (b) A clock's transition probability versus temporal phase χ of the second Ramsey interaction. The transition probability is most sensitive to χ at χ = ±π/2. For χ = +π/2 the second Ramsey interaction transfers the incident dressed state $|2\rangle$ to the excited state and, for χ = −π/2, $|1\rangle$ to the excited state. Because more of ψ${}_2$($\vec{r}$,$t_d$) passes through an aperture than of ψ${}_1$($\vec{r}$,$t_d$), the microwave lensing of wave functions produces a frequency shift of clocks.

Figure 2

The spatial difference of dressed-state populations $|2\rangle$ and $|1\rangle$ of a centered cloud in the detection region. Apertures after the first Ramsey interaction transmit the center of the wave functions, leading to a frequency shift. Normally, the momentum changes in the second Ramsey interaction lead to small separations and do not produce a frequency shift.