Phase shift due to atom-atom interactions in a light-pulse atom interferometer

We present a theoretical model allowing precise calculation of the phase shift induced by atom-atom interactions in a light-pulse atom interferometer based on two-photon Raman atom optics. This model is in good agreement with numerical simulations based on solving the Gross-Pitaevskii equation. The atom interferometer exhibits an atom-atom-interaction-induced phase shift when there is asymmetry between the two arms of the interferometer. In the case of a Ramsey-Bordé atom interferometer $(\{\pi /2–\pi /2\}–\{\pi /2–\pi /2\}$ pulse configuration), the asymmetry comes from the fact that the number of atoms in each arm of the interferometer is not constant. In the case of a Mach-Zehnder $(\{\pi /2–\pi –\pi /2\}$ pulse sequence), if the pulses are perfect, the number of atoms is constant in the interferometer. We study the effect due to imperfections of the light pulses as well as the effect due to the expansion of the cloud. Our model leads to precise and simple formulas of the mean-field phase shift as a function of the experimental parameters.

Figure 1

The phase ${\varphi }_i$ generated by interactions in state $|1\rangle$ [(blue) circles] and state $|2\rangle$ [(red) diamonds] plotted versus $\mathrm{\Omega }$: contributions of the ${\omega }_{ii}$ terms (a) ${\omega }_{11}$, (b) ${\omega }_{22}$, and (c) ${\omega }_{12}$, with the same initial state $|1\rangle$ for all atoms. Solid lines are obtained using the formulas derived in the text, whereas symbols are the results of numerical resolution of the two-component coupled GPE.

Figure 2

The phase ${\varphi }_i$ generated by interactions in state $|1\rangle$ [(blue) circles] and state $|2\rangle$ [(red) diamonds] plotted versus $\mathrm{\Omega }$: contributions of the ${\omega }_{ii}$ terms (a) ${\omega }_{11}$, (b) ${\omega }_{22}$, and (c) ${\omega }_{12}$, with the same initial state $|1\rangle \left(|1\rangle +\mathbf{i}|2\rangle \right)/\sqrt{2}$ for all atoms. Solid lines are obtained using the formulas derived in the text, whereas symbols are the results of numerical resolution of the two-component coupled GPE.

Figure 3

Variation of the number of atoms in state $|1\rangle$ [(blue) circles] and state $|2\rangle$ [(red) diamonds] induced by atom-atom interactions. The initial state is $\left(|1\rangle +\mathbf{i}|2\rangle \right)/\sqrt{2}$. Contributions of the ${\omega }_{ii}$ terms (a) ${\omega }_{11}$, (b) ${\omega }_{22}$, and (c) ${\omega }_{12}$. Solid lines are obtained using the formulas derived in the text, whereas symbols are the results of numerical resolution of the two-component coupled GPE.

Figure 4

Mean positions of the two clouds plotted versus $\mathrm{\Omega }$ [(blue) circles for $|1\rangle$ and (red) diamonds for $|2\rangle )$, during a Raman pulse, for ${10}^5$ atoms initially in state $|1\rangle$ forming a wave packet with a spatial extension $w=30$ $\mu \mathrm{m}$. No interaction is applied to the atoms. Numerical simulation results are represented as symbols, whereas results from the model described in the text correspond to solid curves.

Figure 5

Evolution of the phases of the two clouds from paths $A$ [(blue) circles] and $B$ [(red) diamonds] during the first Ramsey interrogation, with $N={10}^5$ atoms and ${\omega }_{12}\simeq 2\pi \times 3$ Hz. Dashed vertical lines show the boundaries of the free propagation phase. Symbols are the results of a numerical resolution of the coupled GPE, and solid curves correspond to the results obtained with the model we develop in this paper.

Figure 6

Trajectories of atom wave packets during the sequence of the interferometer. After each Raman pulse, $\alpha N$ atoms are transferred to the other state. The two arms of the interferometer are defined by paths $A$ [thin (blue) line] and $B$ [thick (red) line]. The proportion of atoms in each cloud at each stage of the sequence is indicated. At the end of the second Raman pulse, the wave function of state $|1\rangle$ is considered null, as all the atoms are assumed to be pushed away.

Figure 7

Position of the central fringe in the interference pattern of the complete sequence as a function of the fraction of atoms transferred from one state to another at the end of each Raman pulse. The effects of each type of interactions have been isolated for ${\omega }_{11}$ [(blue) circles], for ${\omega }_{22}$ [(green) diamonds], and for ${\omega }_{12}$ [(red) squares]. For each case, the full model corresponding to expressions (31) and (32) is represented as a solid line; the approximated expression, (36), as a dashed line; and simulations, as symbols. All results shown are obtained using the parameters $N={10}^5$ atoms, $w=30\mu \mathrm{m},\tau =0.5$ ms, $T=9.5$ ms, $T^{\prime }=10.5$ ms, and ${\omega }_{ij}\simeq 3$ Hz.

Figure 8

Position of the central fringe as a function of the spatial extension of the clouds. Again, each type of interaction has been isolated for ${\omega }_{11}$ [(blue) circles], for ${\omega }_{22}$ [(green) diamonds], and for ${\omega }_{12}$ [(red) squares]. For each case, the full model corresponding to expressions (31) and (32) is represented as a solid line; the approximated expression, (33), using the value of $\alpha$ given by (35), as a dashed line; and simulations, as symbols.

Figure 9

Trajectories of atom wave packets during the $\pi /2-\pi -\pi /2$ light-pulse interferometer. After each Raman pulse, $\alpha N$ atoms are transferred to the other state. The two arms of the interferometer are defined by paths $A$ [thin (blue) line] and $B$ [thick (red) line]. The proportion of atoms in each cloud at each stage of the sequence is indicated.