Three-level superadiabatic quantum driving

The superadiabatic quantum driving, producing a perfect adiabatic transfer on a given Hamiltonian by introducing an additional Hamiltonian, is theoretically analyzed for transfers within a three-level system. Our starting point is the stimulated Raman adiabatic passage, realized through different schemes of laser pulses. We determine the superadiabatic correction for each scheme. The fidelity, robustness, and transfer time of all the superadiabatic transfer schemes are discussed. We derive that all superadiabatic corrections are based on a $\pi$- (or near-$\pi$)-area pulse coupling between the initial and final states. The benefits in the protocol robustness overcome the difficulties associated to the actual implementation of the three-level superadiabatic transfer.

Figure 1

Time dependencies of the ${\Omega }_p$ and ${\Omega }_s$ STIRAP exponential pulses (line 2 of Table 1) and of the ${\Omega }_d$ sa-STIRAP correction. ${\Omega }_T=1$ in the dimensionless units of Eq. (26).

Figure 2

In (a) and (b) sa-STIRAP realization in a $\Lambda$ system between the $J_g=1$ Zeeman sublevels and an excited $J_e=0$ state. (a) Laser configuration defined within a Cartesian $|i\rangle ,(i=X,Y,Z)$ basis set with linearly polarized lasers and static magnetic field $B_x$. (b) Laser configuration defined in the basis of the $J_z$ eigenstates with $\pi$/${\sigma }^{-}$ lasers and a ${\sigma }^{+}$ circularly polarized magnetic radio frequency field. (c) sa-STIRAP in a ladder system with the the detuning pulse produced by an additional two-photon transition.

Figure 3

Time dependence of the $|1,2,3\rangle$ populations for the STIRAP and sa-STIRAP exponential protocols in (a) and (b), respectively, for ${\Omega }_T=1$. ${\Gamma }_2=0$ in all cases.

Figure 4

Fidelity vs ${\Omega }_T$ for STIRAP Gaussian and sin-cos protocols of Table 1, the Gaussian ones with different values of ${\tau }_T$. No loss (${\Gamma }_2=0$) and with loss at ${\Gamma }_T=4$.

Figure 5

$L$ STIRAP losses vs ${\tau }_T$ and ${\Omega }_T$ for Gaussian pulses with ${\Gamma }_T=10$.

Figure 6

$F$ and $L$ vs ${\tau }_T$ and ${\Omega }_T$ for STIRAP and sa-STIRAP Gaussian pulses at ${\Gamma }_T=10$. Fidelity $F$ in (a) and (b) two-dimensional (2D) plots, and in 3D surfaces of (c). Losses $L$ in 2D plot (d). In (a) and bottom surfaces of (c) for STIRAP protocols; in (b), top surface of (c) and (d) for sa-STIRAP protocols. For the sa-STIRAP's $F$ deviates from one and $L$ from zero because the detuning pulse is blocked at its value for ${\tau }_T=1/2$.

Figure 7

2D plot of the sa-STIRAP fidelity vs ${\Omega }_T$ and the detuning pulse area around the $\pi$ optimal value, for Gaussian pulses with ${\tau }_T=0.5$ and ${\Gamma }_T=1$.

Figure 8

Three-level $T^{0.9}$ transfer time vs a characteristic $\tilde{\Omega }$ Rabi frequency. In the top vs $\tilde{\Omega }={\Omega }_{\mathrm{peak}}$ for the Gaussian STIRAP protocol with optimized $T$ and $\tau$ parameters. In the center line vs $\tilde{\Omega }={\Omega }_d$ peak value for the Gaussian or exponential sa-STIRAP protocols. In bottom line vs the $\tilde{\Omega }=\Omega$ Rabi frequency of a $|1\rangle$ and $|3\rangle$ direct coupling for the quantum speed limit, this limit reached also by the sa-sin-cos protocol with $\tilde{\Omega }={\Omega }_d$. ${\Gamma }_2=0$ everywhere.