Inverse engineering of fast state transfer among coupled oscillators

Abstract

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.

Dynamical invariant formalism of shortcuts to adiabaticity

“…This solution restricts possible protocols in the inverse engineering because the boundary conditions b˙false(0false)=b˙false(tnormalffalse)=0 and b¨false(0false)=b¨false(tnormalffalse)=0 give ωfalse(0false)=ωfalse(tnormalffalse)=0. This linear invariant was used for momentum or position scaling [25] and for coupled/multi-dimensional harmonic oscillators [26–28].…”

Fast Driving of a Particle in Two Dimensions without Final Excitation

“…Inverse engineering the dynamics of two coupled oscillators is thus a basic and important operation for controlling quantum systems. In a previous publication [ 7 ] we demonstrated that when the coupling is proportional to the product of oscillator coordinates , it is possible to inverse engineer the time dependence of the quadratic potential using a combination of invariants to swap the quantum numbers of any eigenstate of the initial uncoupled oscillators. The process may be faster than the adiabatic one and it is not state-specific, in other words, the initial quantum numbers need not be known.…”

“…The specific application we worked out in ref. [ 7 ] was the swapping of quantum numbers describing a single particle state in a two-dimensional harmonic trap whose final configuration is rotated by with respect to the initial configuration (The intermediate driving though is not a pure trap rotation since the eigenfrequencies are also deformed along the process). Up to a phase factor, which may be manipulated, the final state was a replica, rotated by , of the initial eigenstate.…”

“…Up to a phase factor, which may be manipulated, the final state was a replica, rotated by , of the initial eigenstate. On a rotating basis, the protocol in [ 7 ] facilitates a one-to-one mapping between initial and final eigenstates keeping at the final time the same quantum numbers set initially. In other words, the “swapping” results from defining the quantum numbers instead in non-rotating bases for the initial and final oscillators in x and y directions, which were the principal axes directions of the initial and final trap configurations.…”

“…The first generalization considered here is to allow for rotations by an arbitrary final angle (in the examples we use ). The particular, the angle treated in [ 7 ] is somewhat simpler, because the directions of the initial and final principal axes coincide, and ref. [ 7 ] only hinted at how to deal with other angles.…”