Figure 1

(a) Graphical construction for the determination of bound states of the $\mathcal{PT}$-symmetric Lee Hamiltonian with spectral coupling and dispersion relation defined by Eq. (15). The eigenvalues $m_V$ of bound states are obtained as the intersection of the straight line $m_V-m_{V_0}$ with the curve $-g_0^2\Delta \left(m_V\right)$, where $\Delta \left(m_V\right)$ is given by Eq. (17). In case I there is only one intersection, corresponding to the physical $V$-particle state with a positive norm. In case II there are two intersections, one with a negative norm corresponding to a ghost state. The dashed straight line corresponds to the boundary of $\mathcal{PT}$ symmetry breaking, where the two intersections coalesce. (b) Domain of existence of bound states for the Lee Hamiltonian in the $(m_{V_0},g_0)$ plane. In domain I, there exists solely one bound state, corresponding to the physical $V$-particle state. In domain II there are two bound states, the additional one being a ghost state. In domain III a pair of complex-conjugate energies appears (broken $\mathcal{PT}$ phase). The analytical expression of curves 1 and 2, that define the boundaries of the three domains, are given in the text.Reuse & Permissions

Figure 2

(a) Schematic of the optical realization of the $\mathcal{PT}$-symmetric Lee model discussed in Sec. II B, based on light transport in a semi-infinite array of evanescently coupled optical waveguides. In the optical analog, the longitudinal propagation distance $t$ plays the role of time in the QFT model. The complex coupling $g=i{g}_0$ of the Lee Hamiltonian in the ghost regime requires a complex coupling between the boundary waveguides in the semi-array. Panels (b) and (c) show two possible optical implementations of such a complex coupling. In panel (b), a semi-infinite array of waveguides with a second-order ${\chi }^{\left(2\right)}$ nonlinearity is considered, in which parametric amplification between signal and idler fields at frequencies ${\omega }_1$ and ${\omega }_2$ is allowed in the boundary waveguide via a QPM grating. The signal field is then evanescently coupled to the other waveguides in the array, where propagation is linear. In panel (c) the complex coupling $g=i{g}_0$ is realized by introduction of a fast modulation of both real and imaginary parts of the effective refractive index in the boundary waveguide, with spatial period $2\pi /\Omega$. As discussed in the text, in the fast modulation limit and for appropriate modulation parameters, the waveguide lattice in (c) effectively behaves as the lattice in (a).Reuse & Permissions

Figure 3

Behavior of $|c_0{\left(t\right)|}^2$; that is, evolution of the probability of the bare particle state ${\widehat{V}}^{†}|0\rangle$ versus normalized time ${\kappa }_{0}t$ for the initial condition $c_0\left(0\right)=0$, $\Phi (k,0)=0$, detuning $m_{V_0}-{\omega }_0=-4{\kappa }_0$, and for (a) $g_0/{\kappa }_0=0.91$, (b) $g_0/{\kappa }_0=1.547$, and (c) $g_0/{\kappa }_0=1.638$. The dotted curves are obtained by numerical analysis of Eqs. (24, 25, 26), which exactly reproduce the $\mathcal{PT}$-symmetric Lee model and are implemented by the optical system of Fig. 2b. The solid curves in the figures refer to the numerical analysis of Eqs. (30, 31, 32), which describe the optical implementation of Fig. 2c based on fast modulation of the real and imaginary parts of the refractive index in the boundary waveguide.Reuse & Permissions