Perturbative operator approach to high-precision light-pulse atom interferometry

Light-pulse atom interferometers are powerful quantum sensors, however, their accuracy for example in tests of the weak equivalence principle is limited by various spurious influences like stray magnetic fields or blackbody radiation. Improving the accuracy therefore requires a detailed assessment of the size of such deleterious effects. Here we present a systematic operator expansion to obtain phase shift and contrast analytically in powers of a perturbation potential. The result can either be employed for robust straightforward order-of-magnitude estimates or for rigorous calculations. Together with general conditions for the validity of the approach, we provide a particularly useful formula for the phase including wave-packet effects.

Figure 1

Mach-Zehnder gravimeter. Initially released from an atomic trap, the free fall of the atoms is interrupted by a $\pi /2$ pulse at $t=t_{\mathrm{i}}=0$ to split the wave packet into two components transferring a momentum of $\hslash k$ to one of them, which gains the recoil velocity $v_{\mathrm{r}}=\hslash k/m$. After a time $T$ both components are redirected by a $\pi$ pulse at $t_1=T$ and finally recombined by a second $\pi /2$ pulse at $t_2=2T$. At the detection time $t_{\mathrm{d}}$ the number of particles is measured at one of the exit ports, which forms an interference pattern dependent on the relative phase accumulated between the two branches. The unperturbed branches (thick solid lines) are determined by the analytic expressions shown next to them, where $g$ is the local gravitation acceleration. The perturbation $V^{\left(\alpha \right)}$ slightly disturbs the atoms, leading to the deviating branches of the actual interferometer (thin dashed lines).

Figure 2

Time contour. To merge the two time-evolution operators corresponding to the two branches of the interferometer, we introduce the time contour depicted in the figure over which all integrals extend. Starting at $t_{\mathrm{i}}$ the integrals run to the detection time $t_{\mathrm{d}}$ over the perturbation potential corresponding to the upper branch and subsequently return along the lower branch back to the initial time $t_{\mathrm{i}}$.

Figure 3

Comparison between different scales. (a) Perturbation potential evaluated at the two unperturbed trajectories spanning the interferometer as a function of time (blue and orange curve). Whereas $\mathrm{\Delta }V$ corresponds to the maximal potential difference probed by the atoms, $\delta V$ is the maximal potential difference between the branches of the interferometer, which is generally smaller. (b) The space-time diagram of a given interferometer sequence determines the domain of the potential probed by the atoms (colored in bright red). The suppression factor of consecutive terms in the Magnus expansion as well as wave-packet effects are crucially dependent on the value of $\xi$ which is the length scale on which the potential changes. If we assign a wave number to the oscillatory behavior of the potential shown in the figure, $\xi$ is given by its inverse (colored in darker red). In contrast, in the case of a simple polynomial form of the potential, the value of $\xi$ would scale with the extent of the interferometer itself. A third length scale is determined by the characteristic size of the wave-packet $d$.

Figure 4

Plot of phase shifts caused by the gravitational field of Earth. As shown in the figure, the dominant contribution $kg{T}^2$ to the phase (blue line) is due to the unperturbed interferometer. The phase shift caused by the first gravity gradients [41] scales with $R^{-1}$ (purple line). Phase contributions from the second gravity gradients scale with $R^{-2}$ and can be divided into the first term inside the bracket in Eq. (33) (orange line) and the second term which originates from wave-packet effects (yellow line). For this figure the parameters are: The radius of Earth is $R\simeq 6\times {10}^6\mathrm{m}$, the effective wave vector is given by $k=4\pi /\left(780\mathrm{nm}\right)$, an interferometer time of $T=1$ s is assumed, the initial trapping frequency is $\omega =2\pi \times 60\mathrm{Hz}$, and a mass corresponding to rubidium 87 is chosen.