Theory of ghost scattering with biphoton states
Jing Cheng
Department of Physics, South China University of Technology, Guangzhou 510640, China (phjcheng@scut.edu.cn)
Received October 21, 2016; revised December 1, 2016; accepted December 13, 2016;
posted December 15, 2016 (Doc. ID 279196); published January 6, 2017
We show how quantum entangled biphoton states can be used to realize ghost scattering, a nonlocal scheme to
obtain scattering information of an unknown object through the correlation measurement of the scattering photons
in two different optical paths. We present a framework to describe the biphoton ghost scattering process from the
T-matrix formula of the scattering theory. We find the scattering information of a test object can be retrieved from
either the test arm or the reference arm. By adjusting the biphoton states, the ghost scattering patterns may be
varied from the scattering pattern of the object in the test arm to the object in the reference arm. © 2017
Chinese Laser Press
OCIS codes: (270.0270) Quantum optics; (290.5825) Scattering theory; (110.2990) Image formation theory.
https://doi.org/10.1364/PRJ.5.000041
1. INTRODUCTION
Ghost imaging (GI) has been an active research topic in recent
years [1–37]. Depending on the spatial correlation properties of
fluctuating light fields (either thermal or entangled), research-
ers can divide the source into two correlated beams and send
one of them through a test path, which contains an unknown
test object, and the other through a reference path having no
interaction with the test object. An image of the object can be
obtained from the correlation measurement between the inten-
sity recorded in the test and reference detectors. Historically,
entangled two-photon (or biphoton) states were first used to
generate a ghost image via coincident measurement, and the
entanglement properties are claimed to be the key factor to
realize GI. However, further studies revealed that classically in-
coherent light sources also can be used to realize GI. Since
then, there have been a lot of works on classical GI.
Recently, we have followed the idea of GI to investigate
light scattering with classically correlated light sources [38].
In our work, we use the scatterers to replace the objects in
the GI scheme and place the detectors far from the scatterers
to collect the scattering data. We also perform nonlocal cor-
relation measurement of the differential scattering cross-
section fluctuations in the test and reference paths, and then
we find that the scattering information of the test scatterer can
be obtained in the reference arm with the help of a fixed point-
like detector in the test arm, just like the case in GI. It is natu-
ral to ask whether and how entangled light sources can be
used in the research of ghost scattering. The purpose of this
paper is to present a theoretical analysis of the quantum ghost
scattering with biphoton quantum states.
In fact, there already exists some literature on light scatter-
ing with quantum light sources [39–43]. Notably, Schotland
et al. [43] used the formal scattering theory to study the scat-
tering of biphoton states from collections of small particles,
paying special attention to the variation of quantum correla-
tion properties when the light is scattered by only one scat-
terer. Instead, here we will use the mathematical tools in
[43] to study how biphoton ghost scattering patterns can be
formed when two optical paths are involved.
The paper is organized into four sections. In Section 2,we
give the theoretical description on biphoton ghost scattering
from the formal scattering theory. We present numerically
simulated results in Section 3. Conclusions are given in
Section 4.
2. BIPHOTON GHOST SCATTERING
THEORY
We show a typical biphoton ghost scattering configuration in
Fig. 1. Just like in the case of an entangled photon GI, a non-
linear crystal is pumped by an incident laser beam to generate
entangled photon pairs. These biphoton states then are sepa-
rated in space by a beam splitter. The output test and refer-
ence photons are entangled in propagating directions and
travel in two paths: the test path and reference path. In the
test path, the test photons are scattered by an unknown test
object and are detected by a point-like detector D
t
. In the
other reference path, we place a known object as a reference
scatterer and measure the scattered reference photons from
this scatterer via a point-like detector D
r
. As done in the GI
experiments, D
t
and D
r
can be scanned over a large enough
space, and we can use the coincidence measurement to
obtain the second-order coherence function Γ
2
~
x
t
;
~
x
r
hψj
ˆ
E
−
~
x
t
ˆ
E
−
~
x
r
ˆ
E
~
x
r
ˆ
E
~
x
t
jψi (with
ˆ
E
the positive/nega-
tive frequency component of the electric-field operator).
We will show that the scattering information of the unknown
test object can be retrieved from this kind of coherence
function.
The initially incident biphoton quantum state can be de-
scribed by
jψ
i
i
ZZ
d
~
kd
~
k
0
A
i
~
k;
~
k
0
ˆ
a
†
~
k
ˆ
a
†
~
k
0
j0i
ZZ
d
~
xd
~
x
0
A
i
~
x; ω;
~
x
0
; ω
0
ˆ
a
†
~
x
ˆ
a
†
~
x
0
j0i; (1)
where
ˆ
a
†
~
x
(
ˆ
a
†
~
k
) is the creation operator in the real space
(momentum space), A
i
~
x; ω;
~
x
0
; ω
0
and A
i
~
k;
~
k
0
are the two-
photon amplitude of the incident field in the real space and
J. Cheng Vol. 5, No. 1 / February 2017 / Photon. Res. 41
2327-9125/17/010041-05 © 2017 Chinese Laser Press