Generating macroscopic-quantum-superposition states in momentum and internal-state space from Bose-Einstein condensates with repulsive interactions

Resonant Raman coupling between internal levels can create double-well momentum-space potentials for multilevel “periodically-dressed” atoms. We develop a many-body formalism for a weakly interacting, trapped periodically dressed Bose gas which illustrates how a tunable exchange interaction yields correlated many-body ground states. In contrast to the case of a position-space double well, the ground state of stable periodically-dressed Bose gases with repulsive interactions tends toward a macroscopic superposition state in the regime where interactions dominate the momentum-space tunneling induced by the external trapping potential. We discuss how real-time control of experimental parameters can be used to create macroscopic quantum superpositions of either momentum or internal states, and how these states could be dynamically controlled, opening the way toward highly sensitive interferometry and frequency metrology.

Figure 1

Creating a double-well momentum-space potential with Raman excitation. (a) Laser beams of frequency ${\omega }_1$ and ${\omega }_2$ may induce Raman transitions between internal states $\mid A⟩$ and $\mid B⟩$. $\delta =({\omega }_1-{\omega }_2)-{\omega }_0$ is the detuning from the Raman resonance. (b) Such a Raman transition imparts a momentum transfer of $\hslash \mathbf{k}=\hslash ({\mathbf{k}}_1-{\mathbf{k}}_2)$, where ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ are the wave vectors of the Raman coupling lasers. (c) Atoms exposed to continuous Raman excitation can be described by a two-branch dispersion relation. Energies (scaled by $E_k$) for the lower $\left(\hslash {\omega }_{\mu }\right)$ and upper $\left(\hslash {\omega }_{\pi }\right)$ dressed states are shown for wave vectors (scaled by $k$) in the direction of the Raman momentum transfer. In the case of exact Raman resonance $\left(\delta =0\right)$ and small Raman Rabi frequency (shown for $\hslash \Omega ∕E_k=1∕8$), the lower dispersion relation takes the form of a parity-symmetry double-well potential.

Figure 2

The two lowest-energy eigenfunctions (solid line for the ground state and dashed for the excited state) which define the right- and left-well modes for the two-mode approximation. Component wave functions in the lower $\left[\mu \left(q_z\right)\right]$ and upper [$\pi \left(q_z\right)$, shown in insets] periodically dressed states are shown for $M=8$ and $M=50$. As the spatial confinement is weakened (larger $M$), the wave functions become further localized in the minima of the double-well potential, and the population in the upper dressed state diminishes. Note the different parity of the $\mu$ and $\pi$ components, as discussed in the text. Here $\hslash \Omega ∕E_k=1∕8$, wave vectors are scaled by $k$, and the one-dimensional wave functions ${\vec{\varphi }}_{0,1}\left(q_z\right)$ are shown.

Figure 3

Numerical calculations of tunneling and interaction strengths. (a) The tunnel splitting $J$ (open circles, solid line) becomes much smaller than the spacing between the second and third excited states (open triangles, solid line) for moderate values of $M$, establishing the validity of the two-mode approximation for weak confinement. (b) Tunneling energies $J$ (open circles, solid line), $J^{\left(1\right)}$ (open squares, dashed line), and $J^{\left(2\right)}$ (filled circles, dashed line) are suppressed for weaker confinement (exponentially with large $M$), while the strength of the momentum exchange energy $U$ ($\times$’s, solid line) remains large. This provides a route to creating correlated many-body states adiabatically from uncorrelated states by gradually weakening the spatial confinement. The energies in (a) are scaled by $E_k$. Dimensionless interaction parameters are plotted as $-U∕g⟨n⟩$, $J^{\left(1\right)}∕2Ng⟨n⟩$, $J^{\left(2\right)}∕\left(g⟨n⟩∕2\right)$. A Rabi frequency $\hslash \Omega ∕E_k=1∕8$ is chosen.