# Creation of a quantum "memory"
in the spatially varying phase structures of photons via
polarization transformation

### Cody Leary

(a) Spatially
varying polarization distribution of a "hedgehog" mode.
(b)–(d) Predicted intensity distributions resulting
from the interference of two hedgehog modes, after one
of them has undergone a transformation of its
polarization state from linear to circular and back
again to its original state. Surprisingly, a "memory"
of the polarization transformation persists in the
overall phase structure of the light, so that the
initial interference pattern (b) is quite different than
the final one (d), even though the interfering modes are
identical with respect to one another in both cases.

Information may be encoded into the electromagnetic
field of a photon in three fundamentally distinct ways:
(i) in the spatial electric field distributions E_{x}(r)
and E_{y}(r) associated with the two polarization
components of the light; (ii) in the relative phase Φ
(r) between these polarization components; and (iii) in
the overall phase Θ(r) of the electromagnetic field. It
is important to note that each of these objects is
generally a function of location across a light beam,
through the the spatial coordinate (r). Mathematically,
the entire electromagnetic field of the photon may be
expressed in terms of these parts as: e^{iΘ(r)} [ E_{x}(r) u_{x}
+ e^{iΦ(r)} E_{y}(r) u_{y} ]. Means for manipulating the amplitudes
E_{x}(r) and E_{y}(r) and relative phase distribution Φ(r)
for single photons are currently under development in
our lab and many others, for applications in quantum
information processing. However, the third way of
encoding information into photons -- by manipulating the
overall phase distribution Φ -- has been studied less
extensively. One possibility for achieving this is shown
in the figure above. Using devices available in the lab,
a so-called "hedgehog" mode of light may be produced,
which has a spatially varying linear polarization
structure with a rotational symmetry, as shown. If the
polarization properties of light modes of the above type
are continuously manipulated, they may record a "memory"
of the manipulation process in the spatial distribution
properties of the phase Φ. This means that there is a
measurable effect on the light, even if its polarization
is first changed and then brought back to its original
state -- the overall phase structure Θ(r) can vary
dramatically depending on the details of the
polarization transformation (see figure for details).
This project will involve developing theoretical and
experimental capabilities for modeling and achieving the
controllable impartation of overall phase structures
onto single photons. It will involve the construction
and improvement of a new experimental apparatus for
imparting and measuring these phase structures, in
addition to theoretical modeling of the predicted
results in *Mathematica*.