Interference of quantum beats in Hong–Ou – Mandel
interferometry
Jing Qiu,
1
Jun-Heng Shi,
1
Yong-Sheng Zhang,
2,3
Shen-Sheng Han,
1
and You-Zhen Gui
1,4
1
Key Laboratory for Quantum Optics and Center for Cold Atom Physics, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
2
Lab of Quantum Information, University of Science and Technology of China, Hefei 230026, China
3
e-mail: yshzhang@ustc.edu.cn
4
e-mail: yzgui@siom.ac.cn
Received December 24, 2014; revised February 20, 2015; accepted February 24, 2015;
posted February 27, 2015 (Doc. ID 231297); published April 10, 2015
Quantum beats can be produced in fourth-order interference such as in a Hong–Ou–Mandel (HOM) interferometer
by using photons with different frequencies. Here we present theoretically the appearance of interference of quan-
tum beats when the HOM interferometer is combined with a Franson-type interferometer. This combination can
make the interference effect of photons with different colors take place not only within the coherence time of
downconverted fields but also in the region beyond that. We expect that it can provide a new method in quantum
metrology, as it can realize the measurement of time intervals in three scales. © 2015 Chinese Laser Press
OCIS codes: (270.1670) Coherent optical effects; (270.5290) Photon statistics; (270.5570) Quantum
detectors.
http://dx.doi.org/10.1364/PRJ.3.000082
1. INTRODUCTION
Interference of two photons has been widely studied because
it provides important information about the optical field, such
as the properties of photon statistics. Since Hong–Ou–Mandel
(HOM) interferometry was first presented in 1987 [
1], it has
been used in many areas such as testing the violation of Bell’s
inequality [
2,3], dispersion cancellation [4–7], quantum com-
puting [
8,9], quantum communication [7,10–12], quantum
metrology [
13], and quantum imaging [5,14,15].
Usually, HOM interference experiments are carried out
with two incident photons at the same frequencies. However,
quantum beats will arise when the two photons have different
frequencies [
16–20]. This information can be used to study the
nondegenerate spontaneous parametric downconversion
(SPDC), which is very useful for quantum communications
[
21–23]. In this paper we will investigate the interference
effect of quantum beats when the HOM interferometer is
combined with a Franson-type interferometer [
24–26]. With
this combination, we can show that photons with different
colors can not only interfere within their coherence lengths
but also interfere beyond their coherence lengths. In this case,
we can realize the measurement in three scales, i.e., the
coherence time of the pump photons, the coherence time
of downconverted photons, and a much smaller time interval
shown in the beat, which can improve the measurement
sensitivity in experiments.
2. MODEL AND ANALYTICAL SOLUTION
Our proposed scheme is sketched in Fig. 1. A type II degen-
erate nonlinear crystal is pumped by a continuous-wave (CW)
laser [
27] and generates pairs of frequency anticorrelated pho-
tons, referred to as the signal and the idler. The photon pairs
are sent into an HOM interferometer. In each arm, there is an
unbalanced Mach–Zehnder (MZ) interferometer, so that both
the signal and the idler arms are divided into two paths. Before
the MZ interferometer in the signal arm, we introduce a tun-
able time delay τ
1
through which we can control the fourth-
order interference. The lengths of the shorter (longer) paths in
the signal and the idler arms have the same value when τ
1
0.
The difference between the longer path τ
2
and the shorter
path τ
3
is much greater than the coherence time of the down-
conversion photon pairs τ
c
, i.e., τ
2
–τ
3
≫ τ
c
. Two filters IF1 and
IF2 with different central frequencies are placed in front of
detectors D1 and D2, respectively.
The biphoton state that is generated from the SPDC process
can be given by [
28,29]
jψ i
Z
dω
s
dω
i
Φω
s
; ω
i
ˆ
a
†
s
ω
s
ˆ
a
†
i
ω
i
j0i; (1)
where Φω
s
; ω
i
is the biphoton spectral function, which is de-
termined by the phase-matching conditions. As we introduce a
tunable time delay τ
1
in the signal arm and an MZ interferom-
eter in each arm, it generates a phase shift,
exp−iω
s
τ
1
1 exp−iω
s
τ
2
1 exp−iω
i
τ
2
; (2)
if we assume the lengths of the shorter paths τ
3
in each arm
have a value of 0. Then the biphoton state that interferes on
the beam splitter should be rewritten as
jψ i
Z
dω
s
dω
i
Φω
s
; ω
i
exp−iω
s
τ
1
1
exp−iω
s
τ
2
1 exp−iω
i
τ
2
ˆ
a
†
s
ω
s
ˆ
a
†
i
ω
i
j0i: (3)
The positive electrical field operators at detectors D1 and
D2 are defined by
82 Photon. Res. / Vol. 3, No. 3 / June 2015 Qiu et al.
2327-9125/15/030082-04 © 2015 Chinese Laser Press