# Robust Optimized Pulse Schemes for Atomic Fountain Interferometry

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## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Analytical Pulse Schemes

- 1.
- Split the initial state $|0\rangle $ into a superposition of $|0\rangle $ and $|N\rangle $. In the example here, we use $N=10$, corresponding to a momentum state separation of $20\hslash k$. This is the analog of a beamsplitter in a classic interferometer. The splitting may further be subdivided as:
- (a)
- Perform an initial splitting, from $|0\rangle $ to a superposition of $|0\rangle $ and $|1\rangle $.
- (b)
- Amplify the momentum state separation by transferring population from $|1\rangle $ to $|N\rangle $, resulting in a superposition of $|0\rangle $ and $|N\rangle $.

- 2.
- Let the atoms evolve freely as they travel up the tower. During this time, an external gravitational field or acceleration may introduce a differential phase between the states $|0\rangle $ and $|N\rangle $.
- 3.
- Swap the complex amplitudes of the $|0\rangle $ and $|N\rangle $ states. This is the analog of a mirror in a classic interferometer. It may be subdivided as in step 1:
- (a)
- De-amplify the population from $|N\rangle $ to $|1\rangle $, so that the interferometer is in a superposition of $|0\rangle $ and $|1\rangle $.
- (b)
- Swap the amplitude of $|0\rangle $ and $|1\rangle $.
- (c)
- Amplify the population from $|1\rangle $ to $|N\rangle $, which again brings the interferometer into a superposition of $|0\rangle $ and $|N\rangle $, with swapped amplitudes relative to the end of step 2.

- 4.
- Let the atoms continue to evolve freely as they descend the tower, potentially accumulating a further differential phase.
- 5.
- Recombine the state into a superposition of $|0\rangle $ and $|1\rangle $. This would be subdivided as:
- (a)
- De-amplify the population from $|N\rangle $ to $|1\rangle $, resulting in a superposition of $|0\rangle $ and $|1\rangle $.
- (b)
- Coherently recombine the amplitudes of $|0\rangle $ and $|1\rangle $ by applying the inverse process of step 1 (a). For a phase $\varphi $ accumulated in steps 2 and 4 that is a multiple of $\pi $, this results in the population returning to the ground state $|0\rangle $. More generally, the final state is a superposition of $|0\rangle $ and $|1\rangle $, depending on the accumulated phase $\varphi $, with the population in $|0\rangle $ as$${P}_{0}\left(\varphi \right)\equiv {\left|\u23290\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\Psi \left(T\right)\u232a\right|}^{2}={cos}^{2}\left(\frac{\varphi}{2}\right)$$

#### 3.1. Train of $\pi /2$ and $\pi $ Pulses

#### 3.2. Rapid Adiabatic Passage

## 4. Robustness

## 5. Optimal Control for Robust Pulse Schemes

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Interaction of an atom with interferometer laser fields in the Bragg regime. (

**a**) Two-level atom described as $|\Psi (z,t)\rangle =a(z,t)|g\rangle +b(z,t)|e\rangle $ in the coordinate representation with the two-level transition $|g\rangle \leftrightarrow |e\rangle $ off-resonantly driven by two fields counter-propagating along the z axis, with a detuning $\Delta $ and time-dependent amplitudes ${\Omega}_{1}\left(t\right)$ and ${\Omega}_{2}\left(t\right)$ and time-dependent phases ${\Phi}_{1}\left(t\right)$ and ${\Phi}_{2}\left(t\right)$. (

**b**) Momentum space ladder $|n\rangle $ corresponding to momentum $n\xb72\hslash k$, with transitions between neighboring levels being driven by the effective pulse amplitude $\Omega \left(t\right)$.

**Figure 2.**(

**a**) Error $1-{\left|\u23290\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\Psi \left(T\right)\u232a\right|}^{2}$ of the population in the ground state after a sequence of $\pi /2$ and $\pi $ pulses for a scaled pulse amplitude. The factor $\Omega \left(t\right)/{\Omega}_{TLS}\left(t\right)$ is the ratio of the pulse amplitude $\Omega \left(t\right)$ used in the simulation and the analytic amplitude ${\Omega}_{TLS}\left(t\right)$ for Rabi cycling in a two-level system with a pulse area of $\pi /2$ or $\pi $, Equation (9). The minimum error is reached for a ratio of ≈1.01, indicated by the dashed vertical line. (

**b**) Momentum space dynamics for an interferometric scheme using a sequence of $\pi /2$ and $\pi $ pulses. Each pulse has a Blackman shape, drawn at the top of the panel, with an amplitude adjusted by the correction determined in panel (

**a**). The time unit $1/{\omega}_{k}$ corresponds to roughly ${10}^{-5}$ s for Rb-87 atoms and a laser wavelength of 780 nm, making the duration of the scheme (excluding the free time evolution) roughly 5.9 ms. (

**c**) The final time population in $|0\rangle $ if an instantaneous phase kick is applied to the $|10\rangle $ component of the wave function at maximum separation to account for the free time evolution starting at $t=150\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ and $t=450\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ (not shown).

**Figure 3.**(

**a**) Transfer of population from momentum state $|2\rangle $ to $|10\rangle $ using rapid adiabatic passage (RAP) with a constant linear chirp rate $\alpha =0.1\phantom{\rule{0.166667em}{0ex}}{\omega}_{k}$ with an offset time of ${t}_{c}=5.93\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ in Equation (11) (green dotted line, bottom). The pulse envelope $\Omega \left(t\right)$, shown in the bottom of panel (

**a**), has a switch-on and switch-off time of ${t}_{r}=19.25\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ (blue dotted line), using half of a Blackman shape. The center of panel (

**a**) shows the dynamic energy levels of the Hamiltonian according to Equation (11). Neighboring levels cross at intervals of ${\tau}_{B}=2/\left|\alpha \right|$, resulting in the transfer of population shown at the top of panel (

**a**). (

**b**) Momentum space dynamics for a full interferometric scheme using an initial $\pi /2$ and $\pi $ pulse to achieve momentum state separation, cf. Figure 2. Then, the RAP pulse from panel (

**a**) first amplifies and then de-amplifies the momentum state separation. The momentum components are swapped with three central $\pi $ pulses. In the second half of the scheme, two additional RAP pulses amplify and de-amplify again. Finally, a $\pi $ and $\pi /2$ pulse perform the recombination. The amplitude of the envelope $\Omega \left(t\right)$ in each pulse is drawn to scale at the top of the panel. (

**c**) The final time population in $|0\rangle $, if an instantaneous phase kick is applied to the $|10\rangle $ component of the wave function at maximum separation, to account for the free time evolution starting at $t=201.8\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ and $t=590.6\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ (not shown).

**Figure 4.**Contrast achieved with both analytical and optimized pulse schemes for a full $20\hslash k$ interferometer scheme. In each panel, the expectation value of the signal contrast is shown for a fixed amplitude scaling factor $\mu $ of the ideal pulse amplitude, and assuming a Gaussian distribution with width $\Delta \beta $ for the atom’s initial momentum relative to the rest frame in units of $2\hslash k$. The schemes are (

**a**) a train of $\pi /2$ and $\pi $ Rabi pulses, (

**b**) a combination of $\pi /2$ and $\pi $ pulses with rapid adiabatic passage (RAP), and (

**c**) a scheme using optimized control pulses in combination with rapid adiabatic passage, cf. Figure 2, Figure 3 and Figure 7. The value of the contrast for each point is obtained from the average populations in the ground state, see Equation (13).

**Figure 5.**Contrast improvement between different schemes. Panel (

**a**) shows the difference between Figure 4a,b, that is, between a scheme using a train of $\pi /2$ and $\pi $ Rabi pulses and a scheme using rapid adiabatic passage (RAP). Panels (

**b**,

**c**) show the difference between Figure 4a,c, respectively Figure 4b,c; that is, between a scheme using pulses derived from optimal control theory (OCT) and the two analytic schemes (Rabi, RAP). The light gray points mark a (negligible) loss of contrast, $\left|\Delta C\right|<0.04$ in panel (

**a**) and $\left|\Delta C\right|<0.01$ in panels (

**b**,

**c**).

**Figure 6.**(

**a**) Ensemble points used for the optimization. The sampling points were chosen from a normal distribution around $\mu =1$ and $\beta =0$, with a width of $\Delta \mu =\Delta \beta =0.025$, divided into 64 batches with 16 points per batch. The different batches are distinguished by the combination of color and marker shape. (

**b**) Optimized pulse amplitude, phase, and spectrum for the initial splitting pulse $|0\rangle \to (|0\rangle +i|1\rangle )/\sqrt{2}$. (

**c**) Optimized pulse amplitude, phase, and spectrum for the central swap pulse between $|0\rangle $ and $|1\rangle $. For Rb-87 and a laser wavelength of 780 nm, the two-photon recoil frequency is ${\omega}_{k}=2\pi \xb715.1$ kHz. The unit of time $1/{\omega}_{k}$ corresponds to roughly ${10}^{-5}$ s. Thus, the duration of the shown pulses is on the order of 150 µs.

**Figure 7.**(

**a**) Momentum space dynamics for an interferometric scheme using optimized pulses (OCT) in combination with rapid adiabatic passage (RAP). The optimized pulses are those shown in Figure 6b,c and implement the initial splitting, the center swap, and the final recombination between levels $|0\rangle $ and $|1\rangle $. These are combined with RAP pulses similar to the one shown in Figure 3a, transferring population between $|1\rangle $ and $|10\rangle $. All pulse amplitudes are shown to scale at the top of the panel. (

**b**) The final time population in $|0\rangle $ if an instantaneous phase kick is applied to the $|10\rangle $ component of the wave function at maximum separation to account for the free time evolution starting at $t=206.8\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ and $t=605.6\phantom{\rule{0.166667em}{0ex}}/{\omega}_{k}$ (not shown). The maximum population at $\varphi =0$ or $\varphi =\pi $ is 0.934 and the minimum population at $\varphi =\frac{\pi}{2}$ is 0.001.

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**MDPI and ACS Style**

Goerz, M.H.; Kasevich, M.A.; Malinovsky, V.S.
Robust Optimized Pulse Schemes for Atomic Fountain Interferometry. *Atoms* **2023**, *11*, 36.
https://doi.org/10.3390/atoms11020036

**AMA Style**

Goerz MH, Kasevich MA, Malinovsky VS.
Robust Optimized Pulse Schemes for Atomic Fountain Interferometry. *Atoms*. 2023; 11(2):36.
https://doi.org/10.3390/atoms11020036

**Chicago/Turabian Style**

Goerz, Michael H., Mark A. Kasevich, and Vladimir S. Malinovsky.
2023. "Robust Optimized Pulse Schemes for Atomic Fountain Interferometry" *Atoms* 11, no. 2: 36.
https://doi.org/10.3390/atoms11020036