Counterdiabatic driving for random-gap Landau-Zener transitions

Abstract

The Landau--Zener (LZ) model describes a two-level quantum system that undergoes an avoided crossing. In the adiabatic limit, the transition probability vanishes. An auxiliary control field $H_\text{CD}$ can be reverse-engineered so that the full Hamiltonian $H_0 + H_\text{CD}$ reproduces adiabaticity for all parameter values. Our aim is to construct a single control field $H_1$ that drives an ensemble of LZ-type Hamiltonians with a distribution of energy gaps. $H_1$ works best statistically, minimizing the average transition probability. We restrict our attention to a special class of $H_1$ controls, motivated by $H_\text{CD}$. We found a systematic trade-off between instantaneous adiabaticity and the final transition probability. Certain limiting cases with a linear sweep can be treated analytically; one of them being the LZ system with Dirac $δ(t)$ function. Comprehensive and systematic numerical simulations support and extend the analytic results.

Figures (11)

• Figure 1: (Left panel) 3D graph of the transition probability generated by the Hamiltonian in Eq. (\ref{['eq:HCD']}). The black curve, $b = 0$, corresponds to the LZ formula, $\mathcal{P}(a,0) = \mathcal{P}_\text{LZ}(a) = \text{e}^{-\pi a^2}$. The blue curve corresponds to zero gap, $a = 0$ and the red curve corresponds to infinite control parameter, $b\rightarrow \infty$. Along the green curve, $b=1/a$, we have CD driving with $\mathcal{P}(a,1/a) = 0$. For large $a$, we can ignore the correction term and $\mathcal{P}(a,b) \stackrel{a\rightarrow \infty}{\longrightarrow}0$. Around $(0,0)$, $\mathcal{P}(a,b) \stackrel{}{\approx}\text{e}^{-\pi(a^2+b^2/4)}$. $\forall a,b$, $\mathcal{P}(a,b)\in[0,1]$ and $\mathcal{P}(-a,-b) = \mathcal{P}(a,b)$, Eq. (\ref{['eq:sym2']}). Slices of $\mathcal{P}(a,b)$ are shown for constant $a$ (upper right panel) and $b$ (lower right panel), respectively.
• Figure 2: (Left panel) Time dependence of $\mathcal{P}(t;a,b)$. Typically, for unrelated parameters $a,b$, $\mathcal{P}(t,\cdot)$ behaves as in the original LZ system. $\mathcal{P}(a,b)$ in Fig. \ref{['fig:1']} denotes the final (asymptotic) value of the $\mathcal{P}(t;a,b)$. (Right panel) $\mathcal{P}$ as a function of the gap coupling $a$, for $b = 2$ (blue line). The minimum at $a=1/b =0.5$ with $\mathcal{P}(0.5,2) = 0$, corresponds to CD driving. Keeping only the the first term in Eq. (\ref{['eq:bstar']}) is equivalent to aligning the center of distribution with the minimum. The shaded area weighted by the Normal distribution (shown by the green curve that is not in scale), gives the optimal average transition probability $\mathcal{P}^\ast$.
• Figure 3: Comparison between $\sigma_1\ (\varphi = 0)$ (blue line) and $\sigma_2\ (\varphi = \pi/2)$ (orange line). (Left panel) Time dependence of the transition probability $\mathcal{P}(t;a,b_0(a))$ for $a= 0.5$. For the control $\sigma_2$, the counterdiabatic evolution guarantees vanishing transition probability for all times while for the control $\sigma_1$ the probability only vanishes asymptotically. (Right panel) Characteristic Curves, Eq. (\ref{['eq:CC']}). For the control $\sigma_1$, we observe irregular behavior for large $a$.
• Figure 4: 3D graph of $\mathcal{P}(a,b)$ for $\varphi = 0$, in analogy to the left panel in Fig. \ref{['fig:1']}. The limits $a\rightarrow0$ (blue curve) and $b\rightarrow 0$ (black curve) are unaffected by the choice of $\varphi$. The green curve is the Characteristic Curve, see the right panel in Fig. \ref{['fig:3']}. Comparing with Fig. \ref{['fig:1']}, the limit $b\rightarrow \infty$ (red curve) is better behaved, reducing the transition probability.
• Figure 5: (Left panel) Plot of Eq. (\ref{['eq:redprob']}) as a function of the gap coupling $a$, for different values of $\varphi$, superimposed with the Normal distribution, centered at $\mu = 0$ (not in scale). (Right panel) Average probability over $a\sim\mathcal{N}(0,\sigma^2)$ as a function of $\sigma$. Colors are the same in both figures according to the legend in the right panel. The dashed lines correspond to the LZ formula $\mathcal{P}_\text{LZ}(a) = \text{e}^{-\pi a^2}$, Eq. \ref{['eq:PLZ']} (left panel), and $\langle\mathcal{P}_\text{LZ} \rangle(0,\sigma) = 1/\sqrt{1+2\pi\sigma^2}$, Eq. \ref{['eq:PLZav']} (right panel). The inset shows a zoom around the origin with the black lines calculated via Eq. \ref{['eq:redAs0']}.