Sufficient conditions for adiabaticity in open quantum systems

The adiabatic approximation exhibits wide applicability in quantum mechanics, providing a simple approach for nontransitional dynamics in quantum systems governed by slowly varying time-dependent Hamiltonians. However, the standard adiabatic theorem is specifically derived for closed quantum systems. In a realistic open system scenario, the inevitable system-reservoir interaction must be taken into account, which strongly impacts the generalization of the adiabatic behavior. In this paper, we introduce sufficient conditions for the adiabatic approximation in open quantum systems. These conditions are simple yet general, providing a suitable instrument to investigate adiabaticity for arbitrary initial mixed states evolving under time local master equations. We first illustrate our results by showing that the adiabatic approximation for open systems is compatible with the description of quantum thermodynamics at thermal equilibrium, where irreversible entropy production is vanishing. We also apply our sufficient conditions as a tool in quantum control, evaluating the adiabatic behavior for the Hamiltonians of both the Deutsch algorithm and the Landau-Zener model under decoherence.

Figure 1

Infidelity $\mathcal{I}\left(\omega \tau \right)$ for achieving the open-system adiabatic solution of the Deutsch algorithm for two different values of ${\gamma }_0$. Inset: The behavior of the adiabaticity coefficient ${\mathrm{\Xi }}_1^{\text{DA}}\left(\omega \tau \right)$. Here, we consider the case where the function is balanced, i.e., $F=2$.

Figure 2

Infidelity $\mathcal{I}\left(\omega \tau \right)$ for achieving the open-system adiabatic solution of the Landau-Zener dynamics under bit-phase flip for different values of ${\gamma }_0$. Inset: Adiabaticity coefficient ${\mathrm{\Xi }}_1^{\text{LZ}}\left(\omega \tau \right)$. We have set $\theta =2\pi /5$.

Figure 3

Quantities ${\mathrm{\Xi }}_1^{\left(1\right)}, {\mathrm{\Xi }}_1^{\left(2\right)}$, and ${\mathrm{\Xi }}_1$ as a function of $\omega \tau$ for the Landau-Zener example in log (left) and linear (right) scales. We have set ${\gamma }_0=0.1\omega$ and $\theta =2\pi /5$.