Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Floquet Engineering Clock Transitions in Magnetic Molecules

Andrew Cupo, Shuanglong Liu, Silas Hoffman, X. -G. Zhang, Hai-Ping Cheng

Abstract

We theoretically study Floquet engineering of magnetic molecules via a time-periodic magnetic field that couples to the emergent total electronic spin of the metal center. By focusing on the low-lying energy levels using an $S = 1$ spin Hamiltonian containing the zero-field and Zeeman terms, we demonstrate their continuous tunability under the Floquet field. Remarkably, under the action of linearly polarized Floquet controls, the energy levels of a clock transition qubit retain their stability against variations in an external static magnetic field. This property is closely linked to having a net-zero total Zeeman shift, which results from both static and effective dynamical contributions. Further, using second-order Van Vleck degenerate perturbation theory, we derived analytically an effective Hamiltonian, which explicitly shows the dependence of the renormalized zero-field tensor on the driving field. Based on our theoretical predictions, experimentalists will be able to dynamically tune qubit energy gaps to values that are useful in their specific laboratory settings, while retaining the spin decoherence suppressing effect of maintaining a clock transition.

Floquet Engineering Clock Transitions in Magnetic Molecules

Abstract

We theoretically study Floquet engineering of magnetic molecules via a time-periodic magnetic field that couples to the emergent total electronic spin of the metal center. By focusing on the low-lying energy levels using an spin Hamiltonian containing the zero-field and Zeeman terms, we demonstrate their continuous tunability under the Floquet field. Remarkably, under the action of linearly polarized Floquet controls, the energy levels of a clock transition qubit retain their stability against variations in an external static magnetic field. This property is closely linked to having a net-zero total Zeeman shift, which results from both static and effective dynamical contributions. Further, using second-order Van Vleck degenerate perturbation theory, we derived analytically an effective Hamiltonian, which explicitly shows the dependence of the renormalized zero-field tensor on the driving field. Based on our theoretical predictions, experimentalists will be able to dynamically tune qubit energy gaps to values that are useful in their specific laboratory settings, while retaining the spin decoherence suppressing effect of maintaining a clock transition.

Paper Structure

This paper contains 20 sections, 77 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory of Floquet-Driven Magnetic Molecules
  3. Static System
  4. Floquet Engineering
  5. Assessment of Static Magnetic Field Stability
  6. Exact Numerical Effective Hamiltonian
  7. The Dynamical Cancellation Problem for $S = 1$
  8. Approximate Analytical Effective Hamiltonian
  9. Results and Analysis
  10. Floquet Renormalized Clock Transitions
  11. Comparison to the Dynamical Cancellation Problem
  12. Clock Transition Gap Modification via Floquet Renormalization of the Zero-Field Tensor
  13. Realization of the Floquet Control
  14. Operation of Floquet Qubits
  15. Conclusions and Outlook
  16. ...and 5 more sections

Figures (9)

  • Figure 1: Renormalization of the three energy levels of an $S = 1$ magnetic molecule by Floquet driving. The zero-field splitting parameters are $D =$ 5 $\mu$eV and $E/D =$ 0, and the external static magnetic field is zero. The top left inset of each panel indicates the polarization of the Floquet drive, with linear and circular polarizations in the top six and bottom three plots, respectively. Specific details about the polarizations are provided between Eqs. \ref{['polarization_vector_conversion']} and \ref{['unfolded']} in the main text. The labeling of each of the energy curves is explained in Appendix \ref{['sec:app-state']}. The essential finding of our work is that for all linear polarizations and amplitudes of the Floquet control, the three energy levels feature static magnetic field stability (SMFS) as defined by Eq. \ref{['smfs_condition']}, which is crucial for suppressing spin decoherence.
  • Figure 2: Renormalization of the three energy levels of an $S = 1$ magnetic molecule by Floquet driving. Same as Fig. \ref{['fig:energy_levels_ED0']} except $E/D =$ 0.1.
  • Figure 3: Renormalization of the three energy levels of an $S = 1$ magnetic molecule by Floquet driving. Same as Fig. \ref{['fig:energy_levels_ED0']} except $E/D =$ 1/3.
  • Figure 4: Renormalization of energy levels for circularly polarized Floquet control with possible SMFS. Taking the parameters from Fig. \ref{['fig:energy_levels_ED0.1']} ($E/D =$ 0.1), the external static magnetic field vector components (see next figure) are varied until all three energy levels are as close as possible to displaying SMFS. If the gradient magnitude from Eq. \ref{['smfs_condition']} is less than $10^{-2} \mu \textrm{eV} / \textrm{mT}$ for a given energy level, then we assign SMFS in this figure. If this condition is violated, the points on the energy curves are marked in yellow to indicate that there is no SMFS. Note that for the linear polarizations in Figs. \ref{['fig:energy_levels_ED0']}-\ref{['fig:energy_levels_EDone-third']}, the computed gradient magnitudes in Eq. \ref{['smfs_condition']} are all less than $10^{-9} \mu \textrm{eV} / \textrm{mT}$.
  • Figure 5: Corresponding external static magnetic field vector components required to realize the energy curves in the previous figure. Only one component will be non-zero and is along the axis perpendicular to the plane of the circulation. The direction is reversed between the different helicities.
  • ...and 4 more figures