Coupled Oscillator Systems Having Partial PT Symmetry

This paper examines chains of $N$ coupled harmonic oscillators. In isolation, the $j$th oscillator ($1\leq j\leq N$) has the natural frequency $\omega_j$ and is described by the Hamiltonian $\frac{1}{2}p_j^2+\frac{1}{2}\omega_j^2x_j^2$. The oscillators are coupled adjacently with coupling constants that are purely imaginary; the coupling of the $j$th oscillator to the $(j+1)$st oscillator has the bilinear form $i\gamma x_jx_{j+1}$ ($\gamma$ real). The complex Hamiltonians for these systems exhibit {\it partial} $\mathcal{PT}$ symmetry; that is, they are invariant under $i\to-i$ (time reversal), $x_j\to-x_j$ ($j$ odd), and $x_j\to x_j$ ($j$ even). [They are also invariant under $i\to-i$, $x_j\to x_j$ ($j$ odd), and $x_j\to- x_j$ ($j$ even).] For all $N$ the quantum energy levels of these systems are calculated exactly and it is shown that the ground-state energy is real. When $\omega_j=1$ for all $j$, the full spectrum consists of a real energy spectrum embedded in a complex one; the eigenfunctions corresponding to real energy levels exhibit partial $\mathcal{PT}$ symmetry. However, if the $\omega_j$ are allowed to vary away from unity, one can induce a phase transition at which {\it all} energies become real. For the special case $N=2$, when the spectrum is real, the associated classical system has localized, almost-periodic orbits in phase space and the classical particle is confined in the complex-coordinate plane. However, when the spectrum of the quantum system is partially real, the corresponding classical system displays only open trajectories for which the classical particle spirals off to infinity. Similar behavior is observed when $N>2$.