Notions of Adiabatic Drift in the Quantized Harper model

TL;DR

This work investigates how adiabatic drift manifests in a drifting quantum Harper model, a finite-dimensional, torus-confined system closely related to the pendulum. By combining spectral analysis, heuristic two-level considerations, and explicit propagator calculations for time-dependent drift, it reveals that genuine adiabaticity for all states is unattainable at experimentally relevant drift rates due to the vast hierarchy of level spacings, especially near the circulating region separatrix. Two dimensionless quantities—a classical-like adiabatic parameter and a quantum adiabatic parameter—govern the competition between adiabatic and diabatic transitions and the applicability of KNH-based capture probabilities in the quantum setting. The results point to a potential universality for resonant quantum systems with non-local perturbations and have implications for adiabatic quantum computation and control in finite-dimensional platforms.

Abstract

We study a quantized, discrete and drifting version of the Harper Hamiltonian, also called the finite almost Mathieu operator, which resembles the pendulum Hamiltonian but in phase space is confined to a torus. Spacing between pairs of eigenvalues of the operator spans many orders of magnitude, with nearly degenerate pairs of states at energies that are associated with circulating orbits in the associated classical system. When parameters of the system slowly vary, both adiabatic and diabatic transitions can take place at drift rates that span many orders of magnitude. Only under an extremely negligible drift rate would all transitions into superposition states be suppressed. The wide range of energy level spacings could be a common property of quantum systems with non-local potentials that are related to resonant classical dynamical systems. Notions for adiabatic drift are discussed for quantum systems that are associated with classical ones with divided phase space.

Figures (16)

• Figure 1: Level curves in phase space of the classical Harper Hamiltonian of equation \ref{['eqn:Harp']}. Thick red level curves show the two separatrix orbits. Negative energies are shown with dotted contours. A comparison between left and right panels illustrates how the parameter $b$ shifts the center of the librating regions. Parameters for the model are shown on the top of each panel. Librating and circulating regions are annotated on the left panel. The parameter $\epsilon$ sets the width of the librating region.
• Figure 2: Eigenvalues of the Hamiltonian operator in equation \ref{['eqn:h0']} for different values of resonance strength $\epsilon$. Parameters $b=0$, $a=1$ remain fixed and the dimension $N=14$. For $\epsilon>0$, eigenvalues are in nearly degenerate pairs within the circulating region but well separated in the region associated with classically librating orbits. The red dotted line shows the separatrix energy values for the associated classical model (equation \ref{['eqn:Esep']}). As the resonance strength $\epsilon$ increases, the area within the separatrix orbit increases and more eigenstates are in the region associated with librating orbits. The colorbar gives an estimate for the log of the distance to the nearest eigenvalue for each eigenvalue and at each value of $\epsilon$, however the colorbar is truncated at -8 so the spacings can be smaller than shown with navy blue points.
• Figure 3: Eigenvalues of the Hamiltonian operator in equation \ref{['eqn:h0']} for different values of parameter $b$. This figure is similar to Figure \ref{['fig:cc1']} except $b$ varies instead of $\epsilon$, and dimension $N=20$. The parameter $\epsilon =0.3$ is fixed. Energy levels in the librating regions do not vary quickly as $b$ varies. However, energy levels in the circulating regions exhibit a lattice of avoided crossings. The axis on the top shows $b$ in units of $\pi/N$. Avoided crossings in the circulating region occur where $b$ is a multiple of $\pi/N$. A heuristic explanation for the dichotomy is given in section \ref{['sec:heu']}.
• Figure 4: Eigenvalues of the Hamiltonian operator in equation \ref{['eqn:h0']} for different values of parameters $\epsilon, b$ where both $\epsilon, b$ together increase linearly with values shown on the top and bottom axes. This Figure is similar to Figure \ref{['fig:cc1']} except the parameters $b, \epsilon$ both vary and dimension $N= 41$. As the resonance strength (set by $\epsilon$) increases, more eigenstates are associated with librating orbits.
• Figure 5: We show the smallest distance, in color and with colorbar shown on the right, between neighboring eigenvalues as a function of eigenvalue (on the $y$-axis) and as a function of the number of states $N$ in the Hilbert space on the $x$-axis for the Hamiltonian operator of equation \ref{['eqn:h0']} with $a=1,b=0$. The parameter $\epsilon$ is fixed and printed on the plot. The cyan thick lines show the energies of the separatrix classical orbits. For energies between the separatrix orbits there are near degeneracies between pairs of eigenstates. The pairs of eigenvalues are not the same except for $N$ a multiple of 4 and in that case only for a pair of zero eigenvalues.

Theorems & Definitions (37)

• Theorem A.1
• proof
• Corollary A.2
• proof
• Theorem A.3
• proof
• Corollary A.4
• Theorem A.5
• proof
• Corollary A.6