Exponential energy growth due to slow parameter oscillations in quantum mechanical systems

It is shown that a periodic emergence and destruction of an additional quantum number leads to an exponential growth of energy of a quantum mechanical system subjected to a slow periodic variation of parameters. The main example is given by systems (e.g., quantum billiards and quantum graphs) with periodically divided configuration space. In special cases, the process can also lead to a long period of cooling that precedes the acceleration, and to the desertion of the states with a particular value of the quantum number.

Figure 1

Schematic diagram for the adiabatic evolution on lowest energy levels. (a) Particle in a periodically divided segment, Eq. (1), for $a\left({\tau }_1\right)=1$ and $a\left({\tau }_2\right)=3$. During the separation phase (from ${\tau }_1$ to ${\tau }_2$) the $n\mathrm{th}$ left state (solid lines) has constant energy $E=-1+{\pi }^2{n}^2$, while the $m\mathrm{th}$ right state (dashed lines) cools from $E={\pi }^2{m}^2$ down to ${\pi }^2{m}^2/9$. During the reconnection phase (from ${\tau }_2$ to ${\tau }_1+T$) there is no division to left and right states, and the instantaneous energy levels (dotted lines) do not cross. This completely defines the new energy level number $k$ at the next separation moment ${\tau }_1+T$ (the ground state remains the ground state, etc.); the process repeats with period $T$. One can observe the fast energy increase for the left states and desertion of the right states with time. (b) Nonrelativistic spin-$\frac{1}{2}$ particle in a spatially inhomogeneous, strongly oscillating magnetic field, Eq. (4), for $B\left({\tau }_1\right)=-1/4$ and $B\left({\tau }_2\right)=3/4$. The field is spatially homogeneous for the time intervals $[{\tau }_1,{\tau }_2]\mathrm{mod}T$, so the instantaneous energy eigenstates separate into spin-up (dashed lines) and spin-down (solid lines) states. During the separation phase the energy of the spin-up states decreases from $E=m-1/4$ to $E=m-5/4$, while the energy of the spin-down states increases from $E=n-3/4$ to $E=n+1/4$. At the reconnection phase the field is inhomogeneous, which destroys the division of the energy eigenstates (dotted lines) into two groups, so the level crossing does not happen. As a result, the level number of the spin-down states at the separation moments ${\tau }_1\mathrm{mod}T$ increases by 2 with each period. The level number of the spin-up states decreases by 2 with each period until the lowest energy spin-up state $4E=3$ is reached. Then, after one more period $T$, it evolves to the spin-down ground state $4E=1$, from which the energy growth starts.