Temporal Pulse Origins in Atom Interferometric Quantum Sensors

TL;DR

The paper introduces the temporal pulse origin as a unifying, time-based descriptor for the inertial phase response of finite-duration pulses in atom interferometers, enabling straightforward calculation of the measurement scale factor $\mathcal{S}$ and its stability under perturbations. By treating the pulse origin as a design parameter and applying GRAPE optimization, it demonstrates shorter, more robust shaped beamsplitters that maintain symmetry and reduce sensitivity to laser-intensity fluctuations and Doppler compensation errors. The approach yields up to orders-of-magnitude reductions in scale-factor error for practical $T$-values and helps suppress biases arising from phase-continuous and phase-discontinuous frequency jumps. The work has practical implications for current and next-generation inertial and gravitational sensors, including mobile platforms, by enabling shorter pulses with higher fidelity and more stable performance across varying experimental conditions.

Abstract

Quantum sensors based upon atom interferometry typically rely on radio-frequency or optical pulses to coherently manipulate atomic states and make precise measurements of inertial and gravitational effects. An advantage of these sensors over their classical counterparts is often said to be that their measurement scale factor is precisely known and highly stable. However, in practice the finite pulse duration makes the sensor scale factor dependent upon the pulse shape and sensitive to variations in control field intensity, frequency, and atomic velocity. Here, we explore the concept of a temporal pulse origin in atom interferometry, where the inertial phase response of any pulse can be parameterized using a single point in time. We show that the temporal origin permits a simple determination of the measurement scale factor and its stability against environmental perturbations. Moreover, the temporal origin can be treated as a tunable parameter in the design of tailored sequences of shaped pulses to enhance scale factor stability and minimize systematic errors. We demonstrate through simulations that this approach to pulse design can reduce overall sequence durations while increasing robustness to realistic fluctuations in control field amplitude. Our results show that the temporal pulse origin explains a broad class of systematic errors in existing devices and enables the design of short, robust pulses which we expect will improve the performance of current and next-generation interferometric quantum sensors.

Figures (10)

• Figure 1: (a) Bloch sphere trajectories for a range of detunings during a rectangular $\pi/2$ pulse. (b) The superposition phase during (blue shaded region) and immediately after a rectangular $\pi/2$ pulse is depicted for a range of detunings. The non-zero phase dispersion causes differently detuned atoms to accumulate different phases during the pulse. Since these phases depend linearly on detuning, they can all be traced back to a common origin in time.
• Figure 2: The amplitude profile (a) and interferometer phase (b) as functions of time during a three-pulse Mach-Zehnder sequence comprised of rectangular $\pi/2$ (beamsplitter) and $\pi$ (mirror) pulses. The interferometer phase is shown for a range of detunings near resonance. The diagonal dotted lines show how the change in phase due to each pulse can be traced back to a common point in time - the pulse origin. (c) shows the response function for a Mach-Zehnder sequence composed of rectangular pulses. The scale factor is given by the area under the response function, which may be approximated as triangular in the limit where $T\gg\tau$. (d) shows the fractional scale factor error $\mathcal{S}/\mathcal{S}_{\mathrm{analytic}}$ for the same sequence as a function of the interrogation time $T$. $\mathcal{S}_{\mathrm{analytic}}=k\int_{-\infty}^{+\infty} h(t)\, \mathrm{d}t$ is computed using the analytic expression for $h(t)$ from bonnin2015characterization. The dashed red and solid blue lines correspond to the scale factor obtained by approximating the response function as a triangle whose vertices occur at the temporal centers and temporal origins of each pulse, respectively. Using the pulse origins results in a more accurate value for the scale factor. The beamsplitter pulse duration $\tau$ was 10 $\mu$s. The Rabi frequency was held constant for all pulses.
• Figure 3: (a) and (b) show the phase sensitivity and inertial response functions for a three-pulse Mach-Zehnder sequence composed of arbitrary shaped pulses. For small detunings, the area under the sensitivity function during each pulse is given by the gradient $m_1$, $m_2$ or $m_3$ of the phase dispersion for that pulse near resonance. The interferometer scale factor $\mathcal{S}$ - given by the area under the response function - can be approximated by considering all three pulses to occur instantaneously at their respective temporal origins ($\tau_1^o, \tau_2^o, \tau_3^o$) - shown by the solid dashed curves in (a) and (b). The precise shape of the sensitivity function shown within each pulse is arbitrary.
• Figure 4: The superposition phase (a) and excited state probability (d) as functions of detuning after three different beamsplitter pulses: rectangular $\pi/2$, point-to-point ($\tau^o=\tau$) and optimized origin ($\tau^o<\tau$). (b) and (e) show the amplitude profiles for each optimized pulse overlayed with the superposition phases computed as a function of time immediately after each pulse for a range of detunings (in units of $\Omega_0$). The pulse origins are found for each pulse by tracing back the superposition phases accumulated during each pulse (grey dotted lines). (c) and (f) show Bloch sphere trajectories during each optimized pulse for the same range of detuning. The optimized pulse in (b) refocuses all atomic spins to a single point on the equator of the Bloch sphere, while the shorter pulse (e) places them on the equator with a linear spread in phase.
• Figure 5: (a) Terminal infidelity of optimized beamsplitters as the pulse duration is varied (shown as multiples of the duration of a $\pi$ pulse). 28 optimizations of each beamsplitter class (point-to-point and optimized $\tau^o$) were performed for each choice of $\tau$. (b) Interferometer contrast for sequences composed of different beamsplitter pulses and the total beamsplitter pulse area. For each sequence, we assumed the mirror was ideal with zero Doppler sensitivity and that the atomic momentum distribution was Gaussian with $1\sigma=0.4\ \hbar k$. The blue empty circles correspond to point-to-point pulses where the origin is at the end of the pulse ($\tau^o=\tau$). The red filled circles correspond to optimized origin pulses which are designed such that the origin lies within the pulse ($\tau^o<\tau$). The black square corresponds to a sequence of rectangular beamsplitters and the purple diamond a sequence formed using the point-to-point beamsplitter from saywell2021can. The spread in points in (a) and (b) is due to the fact that a different random guess was used to optimize each pulse.