Cumulative Fidelity of LMT Clock Atom Interferometers in the Presence of Laser Noise

TL;DR

This work resolves concerns about laser-noise limitations for large-momentum-transfer clock atom interferometers by showing that, with pulses from alternating directions, cumulative population loss scales linearly as $\sim n$, while parasitic-path contributions remain bounded and negligible. The authors derive a frequency-noise transfer function $H_1(f)$ and show $H_n(f)=nH_1(f)$, enabling quantitative fidelity predictions for long sequences; with realistic parameters ($\Omega$ on the order of $1$ kHz and RMS laser noise $\Delta u$ near 10 Hz), they find per-pulse loss $\alpha^2\approx4.7\times10^{-5}$ and an optimal $n$-dependent LMT enhancement $nC\approx3.3\times10^3$ at $n\approx8.8\times10^3$, suggesting laser-frequency noise is not a practical barrier to $n\hbar k$ clock interferometers. Parasitic paths do not accumulate to appreciable contrast loss, and the analysis extends to more general pulse schemes with proposed mitigations, reinforcing the viability of ultra-high-LMT clocks for precision tests and gravitational sensing. The results support pursuing high-fidelity LMT clock interferometers toward multi-$10^3$ to $10^4\hbar k$ regimes in upcoming experiments like MAGIS-100.

Abstract

Clock atom interferometry is an emerging technique in precision measurements that is particularly well suited for sensitivity enhancement through large momentum transfer (LMT). While current systems have demonstrated momentum separations of several hundreds of photon momenta, next-generation quantum sensors are targeting an LMT enhancement factor beyond $10^4$. However, the viability of LMT clock interferometers has recently come into question due to the potential impact of laser frequency noise. Here, we resolve this concern by analyzing the cumulative fidelity of sequential state inversions in an LMT atom interferometer. We show that the population error from $n$ pulses applied from alternating directions scales linearly with $n$. This is a significant advantage over the $n^2$ scaling that occurs when probing a two-level system $n$ times from the same direction. We further show that contributions to the interferometer signal from parasitic paths generated by imperfect pulses are negligible, for any loss mechanism. These results establish that laser frequency noise is not a practical limitation for the development of high-fidelity LMT clock atom interferometers.

Figures (5)

• Figure 1: (a) Space-time diagram of a narrowband clock atom interferometer with exaggerated pulse infidelities of $0.05$. The vertical lines represent light pulses traveling in opposite directions indicated by the arrows, with the light travel time exaggerated. The first and last pulses are $\pi/2$ pulses while all others are $\pi$ pulses. The internal levels are color-coded in blue (ground) and red (excited), with opacity representing the wavepacket amplitudes of the paths. The position distribution of all path amplitudes after a finite drift time is also shown, with a wavepacket size such that the two output ports are resolved. (b) Bloch sphere representation of an LMT clock atom interferometer with velocity-selective pulses. Each $\pi$ pulse couples two states with opposite internal levels and an external momentum difference of $\hbar k$. The alternation of pulse direction (slanted arrows) builds up momentum in the wavefunction, moving it to a new Bloch sphere after each pulse. Laser frequency noise leads to untransferred population that is not carried over to the next Bloch sphere but merely decreases its size, resulting in a reduction in the radius as shown.
• Figure 2: Momentum-time and space-time diagrams of the upper arm of a narrowband $4\hbar k$ clock atom interferometer for different numbers of pulse errors $m$. The internal atomic levels are indicated as blue (ground) and red (excited). The vertical lines represent $\pi$ pulses (black) and $\pi/2$ pulses (gray), with arrows indicating their directions and detunings. We assume equal pulse spacing $\delta t$ and a non-zero interrogation time $T$. The vertical axes are dimensionless, with momentum in units of $\hbar k$ and position in units of $v_r\,\delta t$, where $v_r$ is the recoil velocity. (a) The main upper path without pulse error ($m=0$). Note that the second half of the sequence addresses the lower arm and is off-resonant with the main upper path, indicated by the open arrows. (b-c) Example parasitic paths, where the cross marks indicate pulses with transition errors. Due to these errors, the parasitic paths fail to follow the main path (shown for reference in lower opacity). The yellow shaded area indicates the relative position error of the parasitic path with respect to the main path, which is also shown explicitly in the space-time diagram. (c) Example paths with even numbers of errors, increasing from top to bottom with $m=2,4,6$. We have selected an $m=4$ example such that the first two errors are the same as the $m=2$ example, and the subsequent errors lead to additional integrated relative position error (indicated by additional yellow shading). A similar choice is made for the $m=6$ example. (b) Example paths with odd numbers of errors $m=1,3,5$. To satisfy the momentum constraint with an odd $m$, an error must occur at the center mirror pulse. Note that the examples in (b) and (c) are only illustrative, and many other parasitic paths are possible. For a single-loop narrowband Mach-Zehnder interferometer sequence, $m=6$ is the largest possible number of errors, independent of $n$.
• Figure 3: (a) Number of parasitic paths that satisfy the momentum constraint $N_p(m,n)$ as a function of LMT order $n$, obtained by numerically enumerating all paths with a fixed number of pulse errors $m$, and only including those with final momentum $p=0$ or $p=1$. The lines are monomial fits to the power of $\lfloor m/2\rfloor$, showing good agreement with the predicted $\mathcal{O}(n^{\lfloor m/2\rfloor})$ growth for sufficiently large $n$ (e.g., $n\geqslant7$). (b) Parasitic-path-induced error $\varepsilon$ as a function of LMT order $n$, obtained by constructively summing over amplitudes of all parasitic paths that satisfy both the momentum constraint and the position constraint. We assume densely-packed pulses at $\Omega=2\pi\times1~\text{kHz}$, $w=1~\text{mm}$, and $v_r=6.6~\text{mm/s}$. We show two pulse inefficiency values $\alpha^2=10^{-4}$ (filled orange markers) and $\alpha^2=10^{-5}$ (open blue markers), with interrogation time $T=0~\text{ms}$ (squares) or $T=50~\text{ms}$ (circles). The induced error does not grow with $n$ for large enough $n$.
• Figure 4: (a) Laser frequency noise transfer function $H_n(f)$ for an augmentation zone with $n$ pulses applied from alternating directions. The analytic expression agrees well with the data (black crosses), which are numerically calculated by solving the Schrödinger equations in the Hilbert space with all possible external momentum states. (b) Augmentation zone fidelity $F_{\textup{aug}}$ as a function of the RMS laser frequency noise $\Delta\nu$ for various sequence lengths $n$, at a Rabi frequency of $\Omega=2\pi\times1~\text{kHz}$. Inset: Frequency noise spectral density at $\Delta\nu=10~\text{Hz}$ (indicated by dashed line). We vary the overall amplitude of this spectrum to set the RMS noise amplitude.
• Figure 5: LMT enhancement of a clock atom interferometer in the presence of laser frequency noise as a function of LMT order $n$. Other loss mechanisms are assumed to introduce an additional pulse inefficiency of $10^{-5}$. (a) Comparison of different RMS laser frequency noise amplitudes $\Delta\nu$ for the same Rabi frequency $\Omega=2\pi\times1~\text{kHz}$. (b) Comparison of different Rabi frequencies using the same RMS noise amplitude $\Delta\nu = 10~\text{Hz}$. In both panels, densely packed pulses and zero interrogation time are assumed.