Inverse engineering of fast state transfer among coupled oscillators

Xiao-Jing Lu1,2, Ion Lizuain3,4, and J. G. Muga2,4

1School of Science, Xuchang University, Xuchang 461000, China
2Departamento de Química Física, University of the Basque Country UPV/EHU, Apdo. 644, 48080 Bilbao, Spain
3Department of Applied Mathematics, University of the Basque Country UPV/EHU, Donostia-San Sebastián, Spain
4EHU Quantum Center, University of the Basque Country UPV/EHU

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We design faster-than-adiabatic state transfers (switching of quantum numbers) in time-dependent coupled-oscillator Hamiltonians. The manipulation to drive the process is found using a two-dimensional invariant recently proposed in S. Simsek and F. Mintert, Quantum 5 (2021) 409, and involves both rotation and transient scaling of the principal axes of the potential in a Cartesian representation. Importantly, this invariant is degenerate except for the subspace spanned by its ground state. Such degeneracy, in general, allows for infidelities of the final states with respect to ideal target eigenstates. However, the value of a single control parameter can be chosen so that the state switching is perfect for arbitrary (not necessarily known) initial eigenstates. Additional 2D linear invariants are used to find easily the parameter values needed and to provide generic expressions for the final states and final energies. In particular we find time-dependent transformations of a two-dimensional harmonic trap for a particle (such as an ion or neutral atom) so that the final trap is rotated with respect to the initial one, and eigenstates of the initial trap are converted into rotated replicas at final time, in some chosen time and rotation angle.

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Cited by

[1] Kazutaka Takahashi, "Dynamical invariant formalism of shortcuts to adiabaticity", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 2239, 20220301 (2022).

[2] Xiao-Jing Lu, Mikel Palmero, Ion Lizuain, and Juan Gonzalo Muga, "Fast Driving of a Particle in Two Dimensions without Final Excitation", Entropy 24 11, 1694 (2022).

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