Experimental Study of the Validity of Quantitative Conditions in the Quantum Adiabatic Theorem

The quantum adiabatic theorem plays an important role in quantum mechanics. However, counterexamples were produced recently, indicating that their transition probabilities do not converge as predicted by the adiabatic theorem [K. P. Marzlin et al., Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong et al., Phys. Rev. Lett. 95, 110407 (2005)]. For a special class of Hamiltonians, we examine the standard criterion for adiabatic evolution experimentally and theoretically, as well as three newly suggested adiabatic conditions. We show that the standard criterion is neither sufficient nor necessary.

Figure 1

Measured fidelity when ${\omega }^{\prime }={\omega }_0$ compared to the fidelity when ${\omega }^{\prime }=10{\omega }_0$. The solid black curve and dashed red curve are the theoretical results of ${\omega }^{\prime }={\omega }_0$ and ${\omega }^{\prime }=10{\omega }_0$, respectively. The solid black circles and open red squares are the experimental results of ${\omega }^{\prime }={\omega }_0$ and ${\omega }^{\prime }=10{\omega }_0$, respectively.

Figure 2

$F_{\min }$, $C_1$, $C_2$, $C_3$, and $C_4$ as functions of ${\omega }^{\prime }/{\omega }_0$ and ${\omega }_1/{\omega }_0$. (a) The minimum fidelity in the process of evolution as a function of $\ln ({\omega }^{\prime }/{\omega }_0)$ and ${\omega }_1/{\omega }_0$. (b)–(e) The expression of a traditional QAC, the QAC in Ref. 13, the QAC in Ref. 14, and the QAC in Ref. 15 as a function of the same parameters.

Figure 3

(a) Fidelity of the evolution measured as a function of ${\omega }^{\prime }/{\omega }_0$. The solid black circles and open red squares are the experimental minimum fidelities for ${\omega }_1=0.05{\omega }_0$ and ${\omega }_1=0.1{\omega }_0$, respectively. The solid black curve and the dashed red curve are the theoretical results. (b) The minimum fidelity as a function of ${\omega }_1/{\omega }_0$. The solid black circles and open red squares represent the experimental data for ${\omega }^{\prime }=10{\omega }_0$ and ${\omega }^{\prime }={\omega }_0$, respectively. The solid black curve and the dashed red curve are the theoretical results.