Systematic pathway toPT-symmetry breaking in scattering systems

Abstract

Recently [Phys. Rev. Lett. 106, 093902 (2011)] it has been shown that PT -symmetric scattering systems with balanced gain and loss, undergo a transition from PT -symmetric scattering eigenstates, which are norm preserving, to symmetry broken pairs of eigenstates exhibiting net amplification and loss. In the present work we derive the existence of an invariant non-local current which can be directly associated with the observed transition playing the role of an "order parameter". The use of this current for the description of the PT -symmetry breaking allows the extension of the known phase diagram to higher dimensions incorporating scattering states which are not eigenstates of the scattering matrix.

Nonlocal discrete continuity and invariant currents in locally symmetric effective Schrödinger arrays

“…( 53) also for non-S-eigenstates in dependence of the incoming amplitudes c+ < ,c − > for a given S(E). The crossing from zero to nonzero NLC thus provides an extended identification of a PT 'phase transition' in scattering beyond S-matrix eigenstates [28], identical to that of bound systems (as discussed above in Sec.V B). More importantly, the vanishing of NLCs can be employed to identify configurations with locally PT -invariant states, that is, where Eq.…”

“…They provide a mapping of field amplitudes between LS related points, thus generalizing the mapping through parity or Bloch factors for global symmetry [25]. The NLCs were further linked to perfectly transmitting states [27] and proposed as a natural order parameter for spontaneous symmetry breaking in non-Hermitian PT -symmetric systems [28,29], as well as observed experimentally in lossy acoustic setups [30].…”

Exposing local symmetries in distorted driven lattices via time-averaged invariants

“…Note that, whereas PCC and/or TRI are sufficient for locally invariant Q and Q c in a LS domain, they are not necessary conditions. Indeed, Q and Q c have been shown to be spatially constant also for non hermitian static potentials with antisymmetric [20] and symmetric [25] imaginary part, respectively, for a given symmetry transformation. For such potentials PCC and TRI generally do not apply, but the transfer matrix of a localized scatterer still retains its unimodularity (det(M) = 1).…”

“…In complete analogy to the above, the invariance of the complementary, time-dependent two-point quantity Q c (with Ψ n replacing Ψ † n in Eqs. (18), (20) for the plane wave representation) can also be linked to PCC and/or TRI. Opposite to the case of Q though, we now obtain Q c n = Q c n+1 in an inversion LS domain from PCC alone and in a translation LS domain from combined PCC and TRI.…”

“…These symmetry-induced spatial invariants were shown to enable a mapping of wave amplitudes of scattering states [17] or basis functions [18] between symmetryrelated points within any local symmetry (LS) domain. They have also been used to classify perfect transmission resonances [19] and as an order parameter for spontaneous symmetry breaking in parity-time symmetric scattering [20] and apply to a large variety of systems described by a spatial Helmholtz equation as e. g. in classical optics [21], quantum scattering [19], or even acoustic wave propagation [22][23][24] where it was readily corroborated by experiments [25]. As the invariant two-point currents are calculated from the stationary states, the LS formalism was so far -by construction-restricted to time-independent setups, thus making its generalization to time-dependent systems an intriguing yet challenging task.…”