496
A. Karabchevsky et al.
biorthogonality relations. Orthogonality can be re-established by modifying the inner
product. Discussion of quantum mechanics based on biorthogonal states is given in
[3–6].
18.2.1 Phase Transition in PT-Symmetric Systems
One of the most intriguing features of PT-symmetric systems is a phase transition
from the PT-symmetric to PT-symmetry-broken phase. If the Hamiltonian of the
system ˆ
H ( ˆ
p, ˆ
r, t, p) depends on some parameter p, the Hamiltonian can have real
as well as complex eigenvalues being, either in PT-symmetric or non-PT-symmetric
states, respectively. When due to variation of the parameter p the system’s spectrum
changes from real to complex and vice-versa, one can study the phase transition
related to the spontaneous breaking of PT symmetry. The point in parameter space
p = p c at which phase transition occurs is named as an exceptional point (EP). At
the EP, both eigenvalues and eigenfunctions coalesce.
18.2.2 PT-Symmetry in Optics
Quantum-mechanical concept of the PT symmetry can be realised in optics due to the
fact that Maxwell’s equations in case of two- and one-dimensional photonic structures can be reformulated into an equation formally coinciding with the Schrödinger
equation.
With light propagation along these structures it is convenient to introduce the
so called slowly varying envelope (SVE) field, where most of the electromagnetic
field variation is extracted by defining a suitably selected reference propagation
constant [7]. Specifically, if the structure is invariant in z, SVE e for electric field is
defined as
E(x, y, z) = e(x, y, z)e
−ik 0 n 0 z
(18.8)
where k 0 = ω/c is the vacuum wavenumber and n 0 is the reference (background)
refractive index. SVE h for magnetic field is defined analogously as
H(x, y, z) = h(x, y, z)e
−ik 0 n 0 z
.
(18.9)
Within the slowly varying envelope approximation (SVEA) it is assumed that
∂
2
∂z 2
e
h
2k 0 n 0
∂
∂z
e
h
(18.10)
and the second-order z-derivative terms are neglected. Due to z-invariance of the
structure transverse and longitudinal components of e and h decouple. The transverse
A. Karabchevsky et al.
biorthogonality relations. Orthogonality can be re-established by modifying the inner
product. Discussion of quantum mechanics based on biorthogonal states is given in
[3–6].
18.2.1 Phase Transition in PT-Symmetric Systems
One of the most intriguing features of PT-symmetric systems is a phase transition
from the PT-symmetric to PT-symmetry-broken phase. If the Hamiltonian of the
system ˆ
H ( ˆ
p, ˆ
r, t, p) depends on some parameter p, the Hamiltonian can have real
as well as complex eigenvalues being, either in PT-symmetric or non-PT-symmetric
states, respectively. When due to variation of the parameter p the system’s spectrum
changes from real to complex and vice-versa, one can study the phase transition
related to the spontaneous breaking of PT symmetry. The point in parameter space
p = p c at which phase transition occurs is named as an exceptional point (EP). At
the EP, both eigenvalues and eigenfunctions coalesce.
18.2.2 PT-Symmetry in Optics
Quantum-mechanical concept of the PT symmetry can be realised in optics due to the
fact that Maxwell’s equations in case of two- and one-dimensional photonic structures can be reformulated into an equation formally coinciding with the Schrödinger
equation.
With light propagation along these structures it is convenient to introduce the
so called slowly varying envelope (SVE) field, where most of the electromagnetic
field variation is extracted by defining a suitably selected reference propagation
constant [7]. Specifically, if the structure is invariant in z, SVE e for electric field is
defined as
E(x, y, z) = e(x, y, z)e
−ik 0 n 0 z
(18.8)
where k 0 = ω/c is the vacuum wavenumber and n 0 is the reference (background)
refractive index. SVE h for magnetic field is defined analogously as
H(x, y, z) = h(x, y, z)e
−ik 0 n 0 z
.
(18.9)
Within the slowly varying envelope approximation (SVEA) it is assumed that
∂
2
∂z 2
e
h
2k 0 n 0
∂
∂z
e
h
(18.10)
and the second-order z-derivative terms are neglected. Due to z-invariance of the
structure transverse and longitudinal components of e and h decouple. The transverse