Quantum state engineering of spin-orbit coupled ultracold atoms in a Morse potential

TL;DR

To address fast, high-fidelity control of a spin-orbit-coupled Bose-Einstein condensate in a Morse potential, the paper develops two invariant-based inverse-engineering protocols that coherently couple internal spin and motional degrees of freedom. The first scheme designs time-dependent Raman coupling $\Omega(t)$ and detuning $\Delta(t)$ via a Lewis-Riesenfeld invariant to drive a two-level transition between adjacent vibrational states; the second scheme tunes the SO coupling direction and an effective magnetic field $\beta(t)$, with nonlinear compensation for interacting condensates. The authors verify that a two-level description remains accurate for narrow gaps and demonstrate robustness to laser-noise and systematic errors, offering a practical framework for fast, robust state engineering in SO-coupled trapped atoms. The approach suggests extensions to bound-to-continuum transitions and has potential implications for metrology and quantum information tasks that exploit spin-orbit physics.

Abstract

Achieving full control of a Bose-Einstein condensate can have valuable applications in metrology, quantum information processing, and quantum condensed matter physics. We propose protocols to simultaneously control the internal (related to its pseudospin-1/2) and motional (position-related) states of a spin-orbit-coupled Bose-Einstein condensate confined in a Morse potential. In the presence of synthetic spin-orbit coupling, the state transition of a noninteracting condensate can be implemented by Raman coupling and detuning terms designed by invariant-based inverse engineering. The state transfer may also be driven by tuning the direction of the spin-orbit-coupling field and modulating the magnitude of the effective synthetic magnetic field. The results can be generalized for interacting condensates by changing the time-dependent detuning to compensate for the interaction. We find that a two-level algorithm for the inverse engineering remains numerically accurate even if the entire set of possible states is considered. The proposed approach is robust against the laser-field noise and systematic device-dependent errors.

Figures (9)

• Figure 1: (Color Online) Schematic configuration of bosonic atoms trapped in a Morse-potential in the presence of SO coupling and the external effective magnetic field, by which degeneracy was eliminated and the energy gap between $|n,\uparrow \rangle$ to $|l,\downarrow \rangle$ is much less than the neighbouring orbital states.
• Figure 2: (Color online) (a) Time dependence of Raman coupling strength $\Omega$ with different SO coupling strength $\alpha = 0.8$ (solid, blue), $\alpha = 1.2$ (dashed, red), $\alpha = 1.6$ (dot-dashed, black), $\alpha=2$ (dotted, orange). (b) Time dependence of the detuning $\Delta$, which is independent of $\alpha$. Other parameters are $t_f=10$, $c=0.1$ in both plots.
• Figure 3: Time dependence of expectation value of coordinate $\langle x \rangle$ in the unit of $l_\textrm{c}$ for non-interacting atoms, with $t_f = 10$, $\alpha=1.6$, and $c=0.1$. The initial state is $|0,\uparrow \rangle$ and the final one is $|1, \downarrow \rangle$.
• Figure 4: Time evolution of $z$-component of spin polarization $P_z$. The parameters are the same as those in Fig. \ref{['x']}.
• Figure 5: (Color online) Trajectory of state evolution (blue bold line) inside the Bloch sphere in the spin space (a) and on the Bloch sphere in the total space (b).