Observation of Parity-Time Symmetry
in Optical Systems
C.E. Rüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides,
M. Segev and D. Kip
Quantum mechanics demands that each physical observable must be
hermitian. In the case of the Hamiltonian operator, this axiom not only
implies real eigen-energies but also
guarantees probability conservation.
Hermiticity is believed to be an absolute must in order to have real eigenvalues. Interestingly, however, a wide class
of non-hermitian Hamiltonians can
still exhibit entirely real spectra. Among
these, are Hamiltonians respecting
parity-time (PT) symmetry. 1
In general, such PT reflection requires
that the associated complex potentials
obey the condition V (x)=V *(-x). Even
though the hermiticity of quantum
observables was never in doubt, such concepts have motivated discussion in theoretical physics. They have led to a critical
re-examination of hermiticity in many
disciplines, including quantum field theo-ries, non-hermitian Anderson models,
and open quantum systems. While the
impact of PT symmetry is still debated,
optics provides a fertile ground for PT-related notions to be investigated. 2-4
Early this year, we reported observation of PT symmetry in an optical coupled system. 5 Such PT “optical potentials” can be realized through a judicious
inclusion of index guiding and gain/loss
regions. Given that the complex refractive index distribution n(x)=nR(x)+inI (x)
plays the role of an optical potential, one
can then design a PT-symmetric system
where the refractive index profile must
be an even function of position x while
the gain/loss distribution should be odd.
In our experiments, we used two Ti
in-diffused parallel waveguide channels in Fe-doped LiNbO3. One of the
channels was optically pumped from the
top via photorefractive two-wave mixing
to provide the necessary gain for the
guided light, while the neighboring arm
experienced loss. In this single-cell PT
1 t/τ 2 C
(a) Front (top) and top (bottom) view of the PT-symmetric coupled system fabricated in LiNbO3.
(b) Measured (normalized) intensities I1, 2 at the output facet during optical pumping as a func-
tion of time t (normalized by the time constant τ for build-up of gain). The upper/lower panel
shows the situation when light is coupled into channel 1 and 2, respectively. Clearly, with
increasing gain, the system behaves in a nonreciprocal manner. Blue dashed lines mark the
symmetry-breaking threshold. Above that, light is predominantly guided in channel 1—thus
experiencing gain—and the intensity in both channels depends solely on the magnitude of the
gain. The power evolution is also depicted (last column) at various times.
system, we observed both spontaneous
PT symmetry breaking and power oscillations violating left-right symmetry. The
experimental response (intensities I1, 2 of
channels 1 and 2, as well as their phase
relation) of our optical system, when
exciting either the gain or loss channel,
is in excellent agreement with solutions
of the corresponding wave equations. 5
At t = 0, the system evolves from zero
gain and shows a reciprocal response.
However, as the photorefractive gain
builds up for recording times t > 0, optical wave propagation becomes strongly
nonreciprocal. At threshold, the system’s
supermodes become degenerate. From
there on, power in the gain channel
monotonically increases, while power in
the loss channels decays.
Our results, when extended to trans-
versely periodic media (photonic lattices,
waveguide arrays), pave the way toward a
new class of PT-synthetic optical materi-
als with intriguing properties that rely
on nonreciprocal light propagation and
tailored transverse energy flow. Nonlin-
earities can be used to fabricate novel
functional systems like PT lattices. This
may provide a platform to investigate,
for example, the fascinating behavior of
phase transition. t
C.E. Rüter and D.Kip ( firstname.lastname@example.org) are with
the department of electrical engineering, Helmut
Schmidt University, Hamburg, Germany. K.G.
Makris, R. El-Ganainy and D.N. Christodoulides
are with the University of Central Florida, School of
Optics - CREOL, Orlando, Fla., U.S.A. M. Segev is
with the department of physics at Technion - Israel
Institute of Technology, Haifa, Israel.
1. C.M. Bender and S. Böttcher. Phys. Rev. Lett. 80, 5243-
2. K.G. Makris et al. Phys. Rev. Lett. 100, 103904 (2008).
3. S. Klaiman et al. Phys. Rev. Lett. 101, 080402 (2008).
4 . A. Guo et al. Phys. Rev. Lett. 103, 093902 (2009).