Theory of polarization-controlled polariton logic gates
T. Ostatnický,
1,2
I. A. Shelykh,
3
and A. V. Kavokin
1,4
1
School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom
2
Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Praha 2, Czech Republic
3
Science Institute, University of Iceland, Dunhagi 3, IS-107 Reykjavik, Iceland
4
Faculta di Fisica, Universita di Roma II “Tor Vergata,” 1 via della Ricerca Scientifica, 00173 Roma, Italy
Received 3 September 2009; revised manuscript received 23 February 2010; published 15 March 2010
Elastic scattering of exciton polaritons in planar semiconductor microcavities has been used to create
X-NOR logic gates. Polaritons with identical linear polarization scatter preferentially at the right angle and
rotate their polarization by 90°. On the other hand, scattering of polaritons having orthogonal linear polariza-
tions is suppressed. We show that these effects are a consequence of the multiple scattering in microcavities
which involve three and more polaritons. The theory quantitatively reproduces the experimental data of
C. Leyder et al. Phys. Rev. Lett. 99, 196402 2007.
DOI: 10.1103/PhysRevB.81.125319 PACS numbers: 71.36.c, 42.65.Yj, 72.25.Rb
I. INTRODUCTION
Exciton polaritons, also referred to as cavity polaritons,
are elementary excitations in semiconductor microcavities.
1
Being a combination of bosonic crystal excitations quantum
well QW excitons and photons, cavity polaritons posses a
number of peculiar properties, which make them promising
candidates for observation of interesting collective phenom-
ena, including high-T
c
Bose-Einstein condensation BEC
Ref. 2 and superfluidity.
3
An important peculiarity of a polariton system is the spin
structure of a polariton state: being formed usually by heavy-
hole excitons, polaritons have two allowed spin projections
on the structure growth axis 1, corresponding to the right
and left circular polarizations of counterpart photons. The
exciton states having spin projections 2 dark states are
split-off in energy and affect the polariton dynamics in the
second order with respect to perturbation caused by Cou-
lomb interaction between the exciton polaritons.
4
The polar-
ization of light emitted by a microcavity is the same as the
polarization of exciton polaritons. From the formal point of
view the spin structure of cavity polaritons is analogous to
spin structure of the electrons both are two-level systems,
which is why a concept of a pseudospin vector S is suitable
for the description of their polarization.
5
The pseudospin is a
quantum analogy of the Stockes vector of a classical light. It
is linked with a 2 2 spin-density matrix of a polariton
quantum state by a relation,
=
N
2
I + S · , 1
where N is the occupation number of the polariton state, I is
the identity matrix, and is the Pauli-matrix vector. The
orientation of the pseudospin determines the polarization of
emission from a microcavity. The parameter =2S / N is the
total polarization degree of light emitted by the cavity, which
may vary between 0 and 1. The z component of the pseu-
dospin is linked with the circular polarization degree of emit-
ted light
c
=2S
z
/ N, while S
x
and S
y
characterize the linear
polarization degree of emitted light measured in x , y axes
l
and in the diagonal axes
d
, respectively:
l
2S
x
/ N,
d
2S
y
/ N.
In the absence of an external magnetic field the “spin-up”
and “spin-down” states of the exciton polaritons correspond-
ing to their spin projections +1 and -1 to z axis and the
pseudospin parallel and antiparallel to z axis, respectively,
are degenerate. On the other hand, the interaction of polari-
tons in triplet configuration identical spin projections on the
structure growth axis is usually much stronger than that of
polaritons in singlet configuration
6–8
spin projections of dif-
ferent signs. This may lead to lifting of degeneracy of the
spin-up and spin-down polariton states if their populations
are not equal. The spin state which is stronger populated has
a higher energy than the spin state which is less populated.
This interaction-induced spin splitting may be described as
an effective magnetic field applied in z direction which
causes the Larmor precession of the polariton pseudospin in
the x , y plane.
9
This precession, also referred to as the self-
induced Larmor precession, is responsible for the mixing of
linearly polarized polariton states which manifests itself in
remarkable nonlinear effects in polariton spin relaxation.
8
The difference in the polariton-polariton interaction con-
stants in singlet and triplet configurations is also responsible
for the predominantly linear polarization of BECs of exciton
polaritons, recently observed in microcavities at the condi-
tions close to the thermal equilibirum.
10
From the point of view of potential applications, it has
been recently pointed out that the controllable manipulation
of the pseudospin of cavity polaritons can provide a basis for
the construction of optoelectronic devices of the new genera-
tion, referred to as spin-optronic devices,
11
which can be of
importance in various technological implementations includ-
ing the classical or quantum information transfer. With re-
spect to the conventional spintronics operating with electri-
cally charged spin carriers, the spin optronics has an
advantage of strongly reducing the dramatic impact of carrier
spin relaxation or decoherence,
12
which severely limits the
functionality of spintronic devices. Macroscopically large
coherence lengths of exciton polaritons and their bosonic
properties led to formulation of several concepts of spin-
optronic devices based on microcavities. These include in
particular mesoscopic optical interferometers,
13
optical
PHYSICAL REVIEW B 81, 125319 2010
1098-0121/2010/8112/1253199 ©2010 The American Physical Society 125319-1