Reversing adiabatic state preparation in few-level quantum systems

TL;DR

This work addresses reversible, fast, high-fidelity adiabatic state preparation in a three-level quantum system by employing counterdiabatic (CD) driving to accelerate the protocol. It demonstrates that forward preparation from $|1\rangle$ to $|T\rangle$ and a designed inverted sweep to return to $|1\rangle$ trace a closed contour with controllable Berry phase, and that CD enables order-of-magnitude reductions in total time while preserving fidelity. The authors show that CD yields a $t_f$ as small as $50$ ns for a complete cycle and allows repeated cycles with minimal loss, making the approach practical for quantum technologies. They also relate the geometric Berry phase $\gamma_B$ to the population transfer and demonstrate tunability via a complex phase in the coupling and sweep profile, proposing Berry-phase-based diagnostics of transfer quality. The results have implications for fast molecular state control, quantum optics, and programmable quantum gates, and the methodology generalizes to other well-separated avoided crossings in few-level systems.

Abstract

We present a detailed study of an adiabatic state preparation in an effective three-level quantum system. States can be prepared with high speed and fidelity by adding a counterdiabatic (CD) quantum control protocol. As a second step, we invert the preparation protocol to get back to the initial state. This describes an overall cyclic evolution in state space. Using counterdiabatic terms, the resulting composed fast evolution can be repeated many times. We then analyze the control of Berry's phase along the adiabatic cyclic path and show that Berry's phase can act as a sensitive detector of non-perfect state transfer.

Figures (7)

• Figure 1: Sketch of the considered three state system in the bare, undriven basis. The evolution under driving occurs from the situation (a) to (b). The level $\ket{1}$ moves towards level $\ket{T}$, see the red arrow in (a), and both states form a tiny avoided crossing in (b), visible also in the inset of Fig. \ref{['fig:2']}. In a second step of the protocol the inversion of this drive brings the system back to its initial state $\ket{1}$, see the red arrow in (b).
• Figure 2: Instantaneous eigenvalues of the Hamiltonian from Eq. \ref{['eq:Ham-1']}, for both the considered sweep functions in Eqs. \ref{['eq:atan']} and \ref{['eq:fit']}: $f(\tau)$ (a) and $p(\tau)$ (b). The insets show a zoom around the first avoided level crossing we encounter during the evolution. Here no difference is visible between the two sweeps. The adopted protocol, discussed in Sec. II, aims to follow adiabatically the blue line along the time evolution. Please keep in mind that the labels of the bare basis $\{1,T,S\}$ only make sense far way from the avoided crossings, i.e., at $\tau=t/t_f\approx 0$ and $\tau\approx 1$.
• Figure 3: Overlap $F(t=t_f/2)$ with the target state $\ket{T}$, for different $t_f$ and sweeps $f(\tau)$ (blue line) and $p(\tau)$ (orange line).
• Figure 4: Time dependence of the populations of all three states for a total protocol time $t_f=5 \, \mu$s. While the overall fidelity is good, corresponding to the plateau in Fig. \ref{['fig:2']}, tiny oscillations are visible that will only disappear for even longer protocol times.
• Figure 5: Time evolution with the polynomial sweep $p(\tau)$ and the CD control added. Three repeated cycles are shown. The evolution can now be much faster than in Fig. \ref{['fig:3']}, here with a single rotation time of $t_{f}=50$ ns. No appreciable loss in the fidelity occurs.