Speed of evolution in qutrit systems

Abstract

The speed of evolution between perfectly distinguishable states is thoroughly analyzed in a closed three-level (qutrit) quantum system. Considering an evolution under an arbitrary time-independent Hamiltonian, we fully characterize the relevant parameters according to whether the corresponding quantum speed limit is given by the Mandelstam-Tamm, the Margolus-Levitin, or the Ness-Alberti-Sagi (dual) bound, thereby elucidating their hierarchy and relative importance. We revisit the necessary and sufficient conditions that guarantee the evolution of the initial state towards an orthogonal one in a finite time, and pay special attention to the full characterization of the speed of evolution, offering a speed map in parameter space that highlights regions associated with faster or slower dynamics. The general analysis is applied to concrete physical settings, particularly a pair of bosons governed by an extended Bose-Hubbard Hamiltonian, and a single particle in a triple-well potential. Our findings provide a framework to explore how the energetic resources and the initial configurations shape the pace of the dynamics in higher-dimensional systems.

Figures (10)

• Figure 1: Region in the space $(\alpha,\beta)$ that is consistent with the Bhatia-Davis inequality decomposes into three (colored) regions, each one delimited according to (\ref{['cotas']}), thus determining the QSL as either one of the bounds $\tau_{\textrm{MT}}$, $\tau_{\textrm{ML}}$, and $\tau_{\textrm{MT*}}$.
• Figure 2: All (normalized) initial states $\left|{\psi_0}\right\rangle$ given by Eq. \ref{['Psi0']} are mapped to the region $\mathcal{P}$, conformed by the points $(r_1,r_2,r_3)$ that define a probability distribution. The projections on the plane $(r_2,r_3)$ are also shown.
• Figure 3: Regions in the space $(\Omega,r_3=\frac{1}{2}-r)$ characterized by the different bounds that determine $\tau_{\textrm{qsl}}$ along points lying on the edge $\overline{CA}$. For $\Omega=1$ the MT bound dominates along the entire segment.
• Figure 4: Panel (a): Regions in the plane $(r_2,r_3)$, at different values of $\Omega$, colored according to the bound that determines $\tau_{\textrm{qsl}}$ (blue: $\tau_{\textrm{qsl}}=\tau_{\textrm{ML*}}$, orange: $\tau_{\textrm{qsl}}=\tau_{\textrm{ML}}$, and green: $\tau_{\textrm{qsl}}=\tau_{\textrm{MT}}$). Panel (b): Area of each of the three regions of panel (a) as a function of $\Omega$ (semi-logarithmic scale).
• Figure 5: Triangle formed by the three vectors $\boldsymbol{r}_i(\tau)$ satisfying the condition (\ref{['vector']}), indicating that an orthogonal state has been reached.