Passive parity-time-symmetry-breaking
transitions without exceptional points in
dissipative photonic systems [Invited]
YOGESH N. JOGLEKAR* AND ANDREW K. HARTER
Department of Physics, Indiana University-Purdue University Indianapolis (IUPUI), Indianapolis, Indiana 46202, USA
*Corresponding author: yojoglek@iupui.edu
Received 11 January 2018; revised 14 June 2018; accepted 28 June 2018; posted 2 July 2018 (Doc. ID 314689); published 24 July 2018
Over the past decade, parity-time (PT)-symmetric Hamiltonians have been experimentally realized in classical,
optical settings with balanced gain and loss, or in quantum systems with localized loss. In both realizatio ns, the
PT-symmetry-breaking transition occurs at the exceptional point of the non-Hermitian Hamiltonian, where its
eigenvalues and the corresponding eigenvectors both coincide. Here, we show that in lossy systems, the PT tran-
sition is a phenomenon that broadly occurs without an attendant exceptional point, and is driven by the potential
asymmetry between the neutral and the lossy regions. With experimentally realizable quantum models in mind,
we investigate dimer and trimer waveguide configurations with one lossy waveguide. We validate the tight-
binding model results by using the beam-propagation-method analysis. Our results pave a robust way
toward studying the interplay between passive PT transitions and quantum effects in dissipative photonic
configurations.
© 2018 Chinese Laser Press
OCIS codes: (080.1238) Array waveguide devices; (270.5585) Quantum information and processing.
https://doi.org/10.1364/PRJ.6.000A51
1. INTRODUCTION
A fundamental principle of traditional quantum theory is that
the observables of a system are Hermitian operators [1]. This
self-adjoint character of observables is defined with respect to
a global (Hamiltonian-independent) Dirac inner product. In
particular, the Hamiltonian of a closed quantum system is
Hermitian. It determines the energy levels of the system and
therefore the experimentally observable transition frequencies.
Thus, it came as a great surprise when Carl Bender and co-work-
ers discovered a broad class of non-Hermitian, continuum
Hamiltonians with purely real spectra [2,3]. The salient feature
of such Hamiltonians is the presence of complex potentials V x
that are invariant under the combined parity (P: x → −x)and
time-reversal (T )operations,i.e.,V
−xV x.
Initial efforts on this subject focused on developing a self-
consistent complex extension of quantum mechanics via a redefi-
nition of the inner product that is used to define the adjoint of
an observable [4,5]. This line of inquiry led to significant
mathematical developments in understanding the properties
of pseudo-Hermitian operators [6–11].Butitdidnotelucidate
a simple physical picture for complex potentials that are a
hallmark of parity-time (PT)-symmetric Hamiltonians.
Experimental progress on the PT-symmetric systems started
with two realizations [12,13]. First, the Schrödinger equation is
isomorphic, with paraxial approximation to the Maxwell’s
equation, where the local index of refraction nxn
R
x
in
I
x plays the role of the potential V x. Second, it is easy to
engineer a lossy index of refraction n
I
x < 0 and not too dif-
ficult to engineer a gain either, i.e., n
I
x > 0. These dual real-
izations provided a transparent, physical insight into the
meaning of complex, PT-symmetric potentials: they represent
balanced, spatially separated loss and gain [14]. Since then, over
the past decade, coupled photonic systems described by non-
Hermitian, PT-symmetric effective Hamiltonians have been
extensively investigated [15–22]. Light propagation in such
systems shows nontrivial functionalities, such as unidirectional
invisibility [23,24], that are absent in their no-gain, no-loss
counterparts. We emphasize that these realizations are essen-
tially classical. Gain at the few-photons level is random due
to spontaneous emission [25,26]; in contrast, loss is linear
down to the single-photon level. Thus, engineering a truly
quantum PT-symmetric system is fundamentally difficult.
Therefore, in spite of a few theoretical proposals [27], there
are no experimental realizations of such systems that show
quantum correlations present.
By recognizing that a two-state loss-gain Hamiltonian is
the same as a two-state loss-neutral Hamiltonian apart from
an “identity-shift” along the imaginary axis, the language of
Research Article
Vol. 6, No. 8 / August 2018 / Photonics Research A51
2327-9125/18/080A51-07 Journal © 2018 Chinese Laser Press