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Atom interferometry

Atom interferometry relies on the quantum superposition of different position and momentum states for the center-of-mass motion of atoms. In fact, in its currently most widespread and successful version, namely *light-pulse atom interferometers*, the roles of light and matter are reversed compared to traditional optical interferometers: they consist in the interference of matter waves with laser pulses acting as beam-splitters and mirrors.

Quantum sensors based on atom interferometry have demonstrated excellent capabilities for high-precision measurements of fundamental constants, such as the fine structure constant, and as highly accurate inertial sensors (*accelerometers* and *gyroscopes*). The basic idea behind the sensitivity of atom interferometers to accelerations is illustrated in the figure below and can be understood as follows. The wave packets evolving along the different branches acquire a phase that depends on their central position (relative to the laser wavefronts) when diffracted by the laser pulses. One can easily find that the phase shift between the two branches in a typical Mach-Zehnder interferometer configuration is not affected by a constant velocity of the atoms, but it is sensitive to accelerations along the direction of the laser beams.

Furthermore, since the displacement of the wave packets due to the acceleration is proportional to \( aT^2 \) (the acceleration \( a \)* *and the square of the time \( T \) between laser pulses), the phase shift between the interferometer branches is proportional to \( k_\mathrm{eff} a T^2 \), where \(k_\mathrm{eff}\) is the effective wave number associated with the laser pulses. Finally, setups where the atoms have an initial velocity transverse to the laser beams can act as gyroscopes by measuring the corresponding Coriolis acceleration along the beam direction.

In our group we have developed a useful description of the state evolution in atom interferometers in terms of *wave packets* and their *central trajectories*. We have also derived exact results for the phase shift under very general conditions including arbitrary pulse-sequences, gravity gradients and branch-dependent forces as well as rotations and other non-inertial effects.

Since the sensitivity of atom interferometers to accelerations scales quadratically with time, it can be dramatically enhanced by employing long interferometer times (a higher \( k_\mathrm{eff} \) can also be used for a sensitivity increase). Such long times (several seconds and even tens of seconds) can be naturally achieved with compact setups in microgravity environments. This can be exploited in high-precision *tests of the equivalence principle* with quantum systems by performing differential acceleration measurements with two different atomic species in order to check that different objects fall with the same acceleration in a gravitational field in the absence of other external forces. Indeed, dedicated space missions have already been proposed for carrying out this kind of tests at the \( 10^{-15} \) level (i.e. improving by two orders of magnitude the best bounds to date, obtained from torsion balance experiments with macroscopic masses).

Achieving these long interferometer times requires substantially reducing the expansion velocity of the freely evolving atomic cloud so that its size does not become too large. This can be accomplished by combining the use of Bose-Einstein condensates and atomic lensing. Moreover, rotations and gravity gradients give rise to unwanted effects and contributions to the phase shift that also grow with time, in some cases even cubically, and pose a major challenge. We have investigated these problems in detail with particular emphasis on *gravity gradients* (for rotations several effective countermeasures already exist) and developed promising schemes and mitigation strategies to overcome them.

Although the analytic studies mentioned above mainly consider idealized laser pulses, we have also been investigating the effects of *realistic laser pulses* by employing a combination of semi-analytical techniques and numerical simulations. Our treatment accounts for velocity selectivity and dispersion effects in general (momentum dependence of the diffraction process) as well as excitations of off-resonant diffraction orders. In addition, we have been analyzing the impact of wave-front distortions of the laser beams.

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References

[1] E. Giese, A. Roura, G. Tackmann, E. M. Rasel, W. P. Schleich, *Double Bragg diffraction: A tool for atom optics*, Phys. Rev. A **88**, 053608 (2013)

[2] A. Roura, W. Zeller, W.P. Schleich, *Overcoming loss of contrast in atom interferometry due to gravity gradients*, New J. Phys. **16**, 123012 (2014)

[3] D. Schlippert, J. Hartwig, H. Albers, L. L. Richardson, C. Schubert, A. Roura, W. P. Schleich, W. Ertmer, E. M. Rasel, *Quantum Test of the Universality of Free Fall*, Phys. Rev. Lett. **112**, 203002 (2014)

[4] S. Kleinert, E. Kajari, A. Roura, W. P. Schleich, *Representation-free description of matter-wave interferometry*, Phys. Reports **605**, 1 (2015)

[5] A. Roura, *Circumventing Heisenberg’s uncertainty principle in atom interferometry tests of the equivalence principle*, Phys. Rev. Lett. **118**, 160401 (2017)

[6] A. Roura, *Gravitational redshift in quantum-clock interferometry*, arXiv:1810.06744 [physics.atom-ph]

[7] S. Loriani et al., *Interference of Clocks: A Quantum Twin Paradox*, Science Adv. **5**, eaax8966 (2019)

[8] K. Bongs, M. Holynski, J. Vovrosh, P. Bouyer, G. Condon, E. Rasel, C. Schubert, W. P. Schleich, A. Roura, *Taking atom interferometric quantum sensors from the laboratory to real-world applications*, Nature Reviews Physics **1**, 731 (2019)