Origin of maximal symmetry breaking in evenPT-symmetric lattices

Abstract

By investigating a parity and time-reversal (PT ) symmetric, N -site lattice with impurities ±iγ and hopping amplitudes t0(t b ) for regions outside (between) the impurity locations, we probe the origin of maximal PT -symmetry breaking that occurs when the impurities are nearest neighbors. Through a simple and exact derivation, we prove that the critical impurity strength is equal to the hopping amplitude between the impurities, γc = t b , and the simultaneous emergence of N complex eigenvalues is a robust feature of any PT -symmetric hopping profile. Our results show that the threshold strength γc can be widely tuned by a small change in the global profile of the lattice, and thus have experimental implications.Introduction: The discovery of "complex extension of quantum mechanics" by Bender and coworkers [1,2] set in motion extensive mathematical [3][4][5] and theoretical investigations [6] of non-Hermitian Hamiltonians H PT =K +V that are symmetric with respect to combined parity (P) and time-reversal (T ) operations. Such continuum or lattice Hamiltonians [7-10] usually consist of a Hermitian kinetic energy part,K =K † , and a non-Hermitian, PT -symmetric potential part,V = PTV PT =V † . Although it is not Hermitian H PT has purely real eigenvalues E = E * over a range of parameters, and its eigenfunctions are simultaneous eigenfunctions of the combined PT -operation; this range is defined as the PT -symmetric region. The breaking of PT -symmetry, along with the attendant non-reciprocal behavior, was recently observed in two coupled optical waveguides [11,12] and has ignited further interest in PT -symmetric lattice models. These evanescently coupled waveguides provide an excellent realization [13] of an ideal, one-dimensional lattice with tunable hopping [14], disorder [15], and non-Hermitian, on-site, impurity potentials [16,17].Recently nonuniform lattices with site-dependent hop-α/2 and a pair of imaginary impurities ±iγ at positions (m,m) have been extensively explored [17][18][19][20], wherem = N + 1 − m and N 1 is the number of lattice sites. The PT -symmetric phase in such a lattice is robust when α ≥ 0, the loss and gain impurities ±iγ are closest to each other, and γ ≤ γ c where the critical impurity strength is proportional to the bandwidth of the clean lattice, γ c ∝ 4t 0 (N/2) α . For a generic impurity position m, when the impurity strength γ > γ c (m) increases the number of complex eigenvalues increases sequentially from four to N − 1 when N is odd and to N when it is even. In an exceptional contrast, when m = N/2 -nearest neighbor impurities on an even lattice -all eigenvalues simultaneously become complex at the onset of PT -symmetry breaking. This maximal symmetry breaking is accompanied by unique signatures in the time-evolution of a wavepacket [20].

Optical waveguide arrays: quantum effects and PT symmetry breaking

“…The remarkable exception to this rule is the case of nearest-neighbor loss and gain waveguides in an even N array. In this case, since the array can be effectively divided into two systems, one with the loss and the other with the gain, all N eigenvalues of H P T α become complex simultaneously [74,77]. Thus, the implications of PT -symmetry breaking are determined by both the threshold loss-and-gain strength γ P T and the location and number of eigenvalues that become complex at the threshold.…”

“…The European Physical Journal Applied Physics the center of the cosine-band become complex for m = 1, whereas all eigenvalues simultaneously become complex when m = N/2 [73,74,77]. The top-left and bottom-left panels show that in the PT -symmetric phase, γ/γ P T < 1, the net intensity I(t) oscillates but remains bounded, and its time-average increases monotonically with its proximity to the PT -symmetric phase boundary.…”

Systems of coupledPT-symmetric oscillators

“…The simplest case to study is that for which γ(x) = γδ(x); that is, the case of a localized point-like PT -symmetric loss-gain impurity at the origin. Studies of this type have been performed for tight-binding models by Joglekar et al [17,18] and Longhi [19].…”

Optical mesh lattices withPTsymmetry

“…In other words the index distribution must be an even function of position whereas the gain/loss must be anti-symmetric. Thus far, several works have pointed out that PTsymmetry can lead to altogether new optical dynamics which are otherwise impossible in standard passive optical arrangements [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. These may include for example the occurrence of abrupt phase transitions along with the appearance of the so-called exceptional points [24][25][26], power oscillations [18], breaking left-right symmetry and the occurrence of secondary emissions [18].…”