Robust Optimized Pulse Schemes for Atomic Fountain Interferometry

Atoms 11, 36 (2023) · DOI: 10.3390/atoms11020036 · arXiv: 2212.12602

Paper license: CC BY 4.0. Data: Hosted with the authors' permission; no licence granted. Ask the authors before reuse beyond citation.

Abstract

The robustness of an atomic fountain interferometer with respect to variations in the initial velocity of the atoms and deviations from the optimal pulse amplitude is examined. We numerically simulate the dynamics of an interferometer in momentum space with a maximum separation of 20ℏk and map out the expected signal contrast depending on the variance of the initial velocity distribution and the value of the laser field amplitude. We show that an excitation scheme based on rapid adiabatic passage significantly enhances the expected signal contrast, compared to the commonly used scheme consisting of a series of π/2 and π pulses. We demonstrate further substantial increase of the robustness by using optimal control theory to identify splitting and swapping pulses that perform well on an ensemble average of pulse amplitudes and velocities. Our results demonstrate the ability of optimal control to significantly enhance future implementations of atomic fountain interferometry.

Figures

20 panels with data across 4 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF. Page 3 of the PDF.

Fig. 2

• panel (a): Error $1 - |\langle 0|\Psi(T)\rangle|^2$ of the ground-state population after a sequence of $\pi/2$ and $\pi$ pulses against the amplitude ratio $\Omega(t) / \Omega_{TLS}(t)$, on a logarithmic error axis. The dashed vertical line marks the minimum error; the solid line marks a ratio of 1. data.csv
• panel (b-1): Pulse envelope $\Omega(t)$ ($\omega_k$) against time ($1/\omega_k$) for the sequence of Blackman-shaped $\pi/2$ and $\pi$ pulses, the strip at the top of panel (b). The print draws it without an axis; here it has a labelled amplitude axis, and the pulse labels are not shown. data.csv
• panel (b-2): Momentum-space dynamics of the interferometric scheme with $\pi/2$ and $\pi$ pulses: population of each momentum level ($2 \hbar k$) against time ($1/\omega_k$), with dashed lines at the boundaries between pulses. The view spans the printed levels $-3$ to 11; the data run from $-5$ to 15. The phase-kick arrows and label of the print are not shown. data.csv
• panel (c): Final-time population $P_0$ in $|0\rangle$ against the phase kick $\phi$ ($\pi$) applied to the $|10\rangle$ component of the wave function at maximum separation, in place of the free time evolution (not shown), for the scheme of panel (b). data.csv

Fig. 3

• panel (a-1): Top plot of panel (a): population (pop) of the momentum states $|2\rangle$ to $|10\rangle$ against time ($1/\omega_k$) during rapid adiabatic passage (RAP) from $|2\rangle$ to $|10\rangle$, nine curves. Dashed lines mark the crossing times of neighbouring levels; the $t_n$ and $\tau_B$ annotations of the print are not shown. data.csv
• panel (a-2): Centre plot of panel (a): dynamic energy levels $E_n$ ($\hbar\omega_k$) against time ($1/\omega_k$), one for each state $|2\rangle$ to $|10\rangle$, with the crossings of neighbouring levels marked and dashed lines at the crossing times. The legend naming each level by its state is added here; the print pairs them by colour only. data.csv
• panel (a-3): Bottom plot of panel (a): envelope $\Omega$ ($\hbar\omega_k$) of the RAP pulse against time ($1/\hbar\omega_k$), with a half-Blackman switch-on and switch-off. The green dotted line marks the offset time $t_c$, the blue dotted lines the switch-on and switch-off time $t_r$, and the dashed lines the level-crossing times. data.csv
• panel (b-1): Pulse envelope $\Omega(t)$ ($\omega_k$) against time ($1/\omega_k$) for the full scheme with rapid adiabatic passage ($\pi/2$, $\pi$, RAP and central $\pi$ pulses), the strip at the top of panel (b). The print draws it without an axis; here it has a labelled amplitude axis, and the pulse labels are not shown. data.csv
• panel (b-2): Momentum-space dynamics of the full interferometric scheme with rapid adiabatic passage: population of each momentum level ($2 \hbar k$) against time ($1/\omega_k$), with dashed lines at the boundaries between pulses. The view spans the printed levels $-3$ to 11; the data run from $-5$ to 15. The phase-kick arrows and label of the print are not shown. data.csv
• panel (c): Final-time population $P_0$ in $|0\rangle$ against the phase kick $\phi$ ($\pi$) applied to the $|10\rangle$ component of the wave function at maximum separation, in place of the free time evolution (not shown), for the scheme of panel (b). data.csv

Fig. 4

The underlying data is not available. The figure is shown in the paper PDF. Page 8 of the PDF.

Fig. 5

The underlying data is not available. The figure is shown in the paper PDF. Page 9 of the PDF.

Fig. 6

• panel (a): Ensemble points used for the optimization, $\beta$ against $\mu$: 64 batches of 16 points each, told apart by the combination of colour and marker shape. Solid lines mark $\mu = 1$ and $\beta = 0$. data.csv
• panel (b-1): Top plot of panel (b): amplitude, ampl. ($\omega_k$), of the optimized initial splitting pulse $|0\rangle \to (|0\rangle + i|1\rangle)/\sqrt{2}$ against time ($1/\omega_k$). data.csv
• panel (b-2): Middle plot of panel (b): phase ($\pi$) of the optimized initial splitting pulse $|0\rangle \to (|0\rangle + i|1\rangle)/\sqrt{2}$ against time ($1/\omega_k$). data.csv
• panel (b-3): Bottom plot of panel (b): spectrum (arb. units) of the optimized initial splitting pulse $|0\rangle \to (|0\rangle + i|1\rangle)/\sqrt{2}$ against frequency ($\omega_k$). The view is the printed frequency window; the data extend beyond it. data.csv
• panel (c-1): Top plot of panel (c): amplitude, ampl. ($\omega_k$), of the optimized central swap pulse between $|0\rangle$ and $|1\rangle$ against time ($1/\omega_k$). data.csv
• panel (c-2): Middle plot of panel (c): phase ($\pi$) of the optimized central swap pulse between $|0\rangle$ and $|1\rangle$ against time ($1/\omega_k$). data.csv
• panel (c-3): Bottom plot of panel (c): spectrum (arb. units) of the optimized central swap pulse between $|0\rangle$ and $|1\rangle$ against frequency ($\omega_k$). The view is the printed frequency window; the data extend beyond it. data.csv

Fig. 7

• panel (a-1): Pulse envelope $\Omega(t)$ ($\omega_k$) against time ($1/\omega_k$) for the scheme of optimized pulses (OCT) and RAP pulses, the strip at the top of panel (a). The print draws it without an axis; here it has a labelled amplitude axis, and the pulse labels are not shown. data.csv
• panel (a-2): Momentum-space dynamics of the interferometric scheme using optimized pulses (OCT) with rapid adiabatic passage (RAP): population of each momentum level ($2 \hbar k$) against time ($1/\omega_k$), with dashed lines at the boundaries between pulses. The view spans the printed levels $-3$ to 11; the data run from $-5$ to 15. The phase-kick arrows and label of the print are not shown. data.csv
• panel (b): Final-time population $P_0$ in $|0\rangle$ against the phase kick $\phi$ ($\pi$) applied to the $|10\rangle$ component of the wave function at maximum separation, in place of the free time evolution (not shown), for the scheme of panel (a). data.csv

Cite

Michael H. Goerz, Mark A. Kasevich, Vladimir S. Malinovsky. Robust Optimized Pulse Schemes for Atomic Fountain Interferometry. Atoms 11, 36 (2023). https://doi.org/10.3390/atoms11020036

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