Systems of coupled $\mathcal{PT}$-symmetric oscillators

The Hamiltonian for a $\mathcal{PT}$-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $\gamma$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken $\mathcal{PT}$ symmetry disappears entirely. However, if $\gamma$ is localized in the sense that it decreases for more distant oscillators, then the unbroken $\mathcal{PT}$-symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a $\mathcal{PT}$-symmetric pair of wave equations, and a localized loss-gain impurity leads to a pseudobound state. It is also shown that a planar configuration of coupled oscillators can have multiple disconnected regions of unbroken $\mathcal{PT}$ symmetry.

Figure 1

Infinite $\mathcal{PT}$-symmetric array of identical particles coupled by springs. The masses in the top row, whose position coordinates are $x_n\left(t\right)$, experience loss, and the masses in the bottom row, which are located at $y_n\left(t\right)$, experience gain.

Figure 2

Imaginary parts of the frequencies $\lambda$ for $N=1,$ 2, 3, and 4 as functions of the coupling constant $\epsilon$ for $\omega =1$ and $\gamma =0.1$. Frequencies are the zeros of the polynomials $P_N$ in (31). Observe that the extent of the unbroken $\mathcal{PT}$-symmetric region (where the frequencies are all real) decreases as $N$ increases and disappears entirely when $N=4$.

Figure 3

Plot of ${\gamma }_{\mathrm{crit}}$ as a function of $1/N$. This sequence evidently converges to 0 with increasing $N$. Thus, a system of coupled oscillators with a uniform loss-gain parameter $\gamma >0$ has no unbroken $\mathcal{PT}$-symmetric region if $N$ is sufficiently large.

Figure 4

Analog of Fig. 3: Oscillatory convergence of ${\gamma }_{\mathrm{crit}}$ when the loss-gain parameter ${\gamma }_n$ decreases like $\gamma /n$ (left) and like $\gamma /n^2$ (right). Evidently, if the loss-gain parameter decays to 0 for more distant oscillators, a region of unbroken $\mathcal{PT}$ symmetry can persist as $N\to \infty$.

Figure 5

Regions in the space of parameters (${\epsilon }_1$ horizontal axis; ${\epsilon }_2$, vertical axis) for which the $\mathcal{PT}$ symmetry is unbroken; that is, the roots of $P\left(\lambda \right)$ in (46) are all real and positive. The frequency $\omega =0.8$ and the damping parameter has the values $\gamma =0.02$, 0.06, 0.10, 0.20, 0.28, 0.34, 0.40, and 0.50. As $\gamma$ increases, the unbroken $\mathcal{PT}$-symmetric regions in $({\epsilon }_1,{\epsilon }_2)$ space decrease in size and eventually disappear. Unlike the case of linear chains of $\mathcal{PT}$-symmetric coupled oscillators, as ${\epsilon }_2$ increases from 0 for fixed ${\epsilon }_1$, there is a range of $\gamma$ and $\omega$ such that one can observe five regions of broken, unbroken, broken, unbroken, and broken $\mathcal{PT}$ symmetry. For example, there are five regions when $\gamma =0.10,{\epsilon }_1=0.10$, and $0\le {\epsilon }_2\le 0.70$.

Figure 6

Same as Fig. 5, but with $\omega =0.9$.

Figure 7

Same as Fig. 5, but with $\omega =1.0$.

Figure 8

Same as Fig. 5, but with $\omega =1.1$.

Figure 9

Imaginary parts of the frequencies $\lambda$ plotted as a function of ${\epsilon }_2$ for various values of the parameters $\omega ,\gamma$, and ${\epsilon }_1$. The regions of unbroken $\mathcal{PT}$ symmetry occur when the imaginary parts vanish and all frequencies are real.