Hong-Ou-Mandel interference of unconventional temporal laser modes

Abstract

The Hong-Ou-Mandel (HOM) effect ranks among the most notable quantum interference phenomena, and is central to many applications in quantum technologies. The fundamental effect appears when two independent and indistinguishable photons are superimposed on a beam splitter, which achieves a complete suppression of coincidences between the two output ports. Much less studied, however, is when the fields share coherence (continuous-wave lasers) or mode envelope properties (pulsed lasers). In this case, we expect the existence of two distinct and concurrent HOM interference regimes: the traditional HOM dip on the coherence length time scale, and a structured HOM interference pattern on the pulse length scale. We develop a theoretical framework that describes HOM interference for laser fields having arbitrary temporal waveforms and only partial overlap in time. We observe structured HOM interference from a continuous-wave laser via fast polarization modulation and time-resolved single photon detection fast enough to resolve these structured HOM dips.

Full text

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20943
Hong-Ou-Mandel interference of unconventional
temporal laser modes
SASCHA AGNE,1,2,7 JEON GWAN JI N,1,3 KATAN YA B. KUNT Z,1
FILIPPO M. MI ATTO,1,4 JEAN -PHILIPPE BOURGOIN,1,5 AND THOMAS
JENNEWEIN1,6,8
1Institute for Quantum Computing, Department of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada
2
Present Address: Max-Planck Institute for the Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany
3Present Address: National Research Council of Canada, 1200 Montreal Road, Ottawa, ON K1A 0R6,
Canada
4Present Address: Institut Polytechnique de Paris and Télécom Paris, LTCI, 19 Place Marguerite Perey,
Palaiseau 91120, France
5Present Address: Aegis Quantum, Waterloo, ON, Canada
6Quantum Information Science Program, Canadian Institute for Advanced Research, Toronto, Ontario
M5G 1Z8, Canada
7sascha.agne@mpl.mpg.de
8thomas.jennewein@uwaterloo.ca
Abstract:
The Hong-Ou-Mandel (HOM) effect ranks among the most notable quantum interfer-
ence phenomena, and is central to many applications in quantum technologies. The fundamental
effect appears when two independent and indistinguishable photons are superimposed on a beam
splitter, which achieves a complete suppression of coincidences between the two output ports.
Much less studied, however, is when the fields share coherence (continuous-wave lasers) or
mode envelope properties (pulsed lasers). In this case, we expect the existence of two distinct
and concurrent HOM interference regimes: the traditional HOM dip on the coherence length
time scale, and a structured HOM interference pattern on the pulse length scale. We develop a
theoretical framework that describes HOM interference for laser fields having arbitrary temporal
waveforms and only partial overlap in time. We observe structured HOM interference from a
continuous-wave laser via fast polarization modulation and time-resolved single photon detection
fast enough to resolve these structured HOM dips.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Interference of light from two sources rests on the superposition principle of electromagnetic
waves, and remains valid even when the sources emit light independently. If we consider
electromagnetic waves from two independent optical emitters without a phase relationship,
any first-order interference washes out when detector electronics are too slow to trace rapidly
fluctuating electric fields [1]. However, correlation measurements can recover interference effects,
in particular when time-resolved single photon detectors are used [2–5].
Single photons in Fock states are naturally phase-independent and mixing them on an optical
beam splitter results in a bunching effect, whereby detectors in the two output ports of a beam
splitter register fewer coincidence detections than a fair Bernoulli process would allow. This
effect of genuine two-photon interference [6] was first observed by Hong, Ou, and Mandel
(HOM) [7]. HOM interference is observed by varying the optical delay between the two input
photons, resulting in an anti-correlation signal whose shape depends on the photons’ spectra, or
equivalently, temporal waveforms.
#396183 https://doi.org/10.1364/OE.396183
Journal © 2020 Received 24 Apr 2020; revised 8 Jun 2020; accepted 15 Jun 2020; published 30 Jun 2020

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20944
The HOM interference effect is not restricted to single photon states, and has been observed
with both thermal [8] and laser light [9–11]. It is counter-intuitive that two independent
continuous-wave (CW) lasers can show a HOM interference pattern (at a reduced visibility).
However, this pattern is observed by correlating the detection times of the two outputs, and the
dip width depends on the coherence lengths of the two input fields [9]. Though the quantum
optical photon picture is frequently employed to explain the reduction in detected coincidences,
the effect also has a semi-classical explanation [12], allowing for new optical measurement
techniques [13,14].
We build on the theoretical and experimental work that has been conducted with single photons
in long temporal wavepackets [15]. The result is a theoretical framework that describes HOM
interference of arbitrarily-shaped pulses with long coherence length and partial overlap in time.
We then use well-defined pulsed coherent states that have a long coherence time (2
µ
s) to explore
whether it is possible to recover full HOM visibility between partially overlapped pulses. By
using a polarization-modulated CW laser to generate 2.83 ns square wave shaped pulses, we
create time-dependent wavepackets that have distinguishable polarization states. We can resolve
structured HOM dips by adjusting the time delay between the two modulated inputs until there is
a region of overlap where the polarization states are indistinguishable. Since the time resolution
of our detectors (
<
0.1 ns) is significantly shorter than the pulse duration, we can recover the full
HOM visibility within the region of temporal overlap for both triangle and square wave HOM
patterns.
2. Theoretical framework
We first derive the standard HOM dip theory, apply it to coherent states, and then study the
effect of polarization modulation on a timescale much shorter than the coherence time. Consider
two input fields
ˆ
E−
1(
t
)=ζ1(
t
)ˆ
a†
1(
t
)
and
ˆ
E−
2(
t
)=ζ2(
t
)ˆ
a†
2(
t
)
to a symmetric beam splitter. Here,
ˆ
a†
i(
t
)
and
ζi(
t
)
are, respectively, the time-dependent creation operators and envelope functions for
optical modes i
=
1, 2. In these expressions, we assume a narrow-band approximation and omit
overall constants. Hence, the beam splitter output fields are given by
ˆ
E−
3(t)=1
√2ζ1(t)ˆ
a†
1(t)+ζ2(t)ˆ
a†
2(t)
ˆ
E−
4(t)=1
√2ζ1(t)ˆ
a†
1(t) − ζ2(t)ˆ
a†
2(t).
(1)
The two-photon coincidence rates measured in HOM interference are described by the second-
order cross-correlation function
G(2x)(t3,t4):=ˆ
E−
3(t3)ˆ
E−
4(t4)ˆ
E+
4(t4)ˆ
E+
3(t3), (2)
which in classical optics quantifies intensity correlations of a field at two space-time coordinates.
Inserting Eqs. (1) into (2), we obtain sixteen terms. However, for sources with independent phase
fluctuations, only six terms are non-zero [16], yielding
G(2x)(t3,t4)=1
4|ζ1(t3)ζ1(t4)|2hˆ
a†
1(t3)ˆ
a†
1(t4)ˆ
a1(t4)ˆ
a1(t3)i
+1
4|ζ2(t3)ζ2(t4)|2hˆ
a†
2(t3)ˆ
a†
2(t4)ˆ
a2(t4)ˆ
a2(t3)i
+1
4|ζ1(t3)|2|ζ2(t4)|2hˆ
a†
1(t3)ˆ
a1(t3)ˆ
a†
2(t4)ˆ
a2(t4)i
+1
4|ζ1(t4)|2|ζ2(t3)|2hˆ
a†
1(t4)ˆ
a1(t4)ˆ
a†
2(t3)ˆ
a2(t3)i
−1
2Renζ∗
1(t3)ζ1(t4)ζ2(t3)ζ∗
2(t4)hˆa†
1(t3)ˆa1(t4)ˆa†
2(t4)ˆa2(t3)io.
(3)

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20945
Note that here we treat mode envelopes as independent of intrinsic field coherence properties.
Consequently,
ζk(
t
)
is deterministic and can be taken out of the expectation values. This means
that field modulations and statistical properties of the field are independent, which simplifies the
problem and may be justified on the ground that many well-established experimental techniques
for control over ζk(t)exist.
The complexity of Eq. (3) is greatly reduced for independent coherent states, which represent
lasers. Two independent lasers are best described by a mixture of coherent states
αk=
|αk|exp(iΘk), represented by [17,18]
ˆρ=Ì
k=1,2 ∫Ck
dαkP(αk)|αkihαk|, (4)
where the integration region extends over an area in the complex plane and P
(αk)
are probability
distributions. Making use of the optical equivalence theorem [18], we can convert quantum-
mechanical expectation values for
ˆ
a
and
ˆ
a†
into classical expectation values for coherent state
(laser) amplitudes
αk
. Furthermore, the expectation values involving modes 1 and 2 factorize
(due to their independence), and we obtain
G(2x)(t3,t4)=1
4|ζ1(t3)ζ1(t4)|2|α1(t3)|2|α1(t4)|2α1
+1
4|ζ2(t3)ζ2(t4)|2|α2(t3)|2|α2(t4)|2α2
+1
4|ζ1(t3)|2|ζ2(t4)|2|α1(t3)|2α1|α2(t4)|2α2
+1
4|ζ1(t4)|2|ζ2(t3)|2|α1(t4)|2α1|α2(t3)|2α2
−1
2Reζ∗
1(t3)ζ1(t4)ζ2(t3)ζ∗
2(t4)G(1)
1(τ)G∗(1)
2(τ),
(5)
where the expectation values are defined as ensemble averages
hfαk,α∗
kiαk=∫Ck
dαkP(αk)f(αk,α∗
k). (6)
For CW lasers, the following three statistical assumptions are valid. First, statistical stationarity
implies that the first-order autocorrelation functions
G(1)
k(τ)=α∗
k(t)αk(t+τ)αk
(7)
only depend on the detection time difference
τ
:
=
t
4−
t
3
. Second, intensities are statistically
constant and identical,
|αk(t)|2αk
=G(1)
k(0) ≡ I0. (8)
Third, intensity fluctuations are constant (for sufficiently small τ),
|αk(t)|2|αk(t0)|2αk
=I2
0. (9)
In the following, we set I
0=
1 for convenience. In our experiment, we generate two fields from a
single laser, and consequently G
(1)
1(τ)=
G
(1)
2(τ) ≡
G
(1)(τ)
. Our laser’s spectrum is best described

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20946
by a Voigt line [19], for which
G(1)(τ)=exp −|τ|
τcoh −τ2
τ2
coh !exp(−iω0τ), (10)
where
τcoh
and
ω0
are the coherence time and central frequency of the laser, respectively. The
mode envelope for an unmodulated CW laser is constant, ζk(t)=1, which gives
G(2x)(τ)=1−1
2exp "−2|τ|
τcoh −2τ2
τ2
coh #. (11)
This equation describes the Hong-Ou-Mandel dip with a visibility
VHOM =G(2x)(τ)max −G(2x)(τ)min
G(2x)(τ)max
(12)
of 50 % for phase-randomized coherent states. It is common to measure a coincidence rate in
two-photon interference experiments, which can be described by
R(2x)(t0,τ,∆T)=η1η2∫t0+∆T
t0
dt1∫t0+τ+∆T
t0+τ
dt2G(2x)(t1,t2), (13)
where
η1
and
η2
denote the detection efficiencies of the single photon detectors, and t
0
is the
initial detection time. We are interested in the time-resolved regime where the detector’s time
resolution
∆
T
τG
, where
τG
is the width of G
(2x)(
t
1
,t
2)
, i.e. the correlation length. In this case,
the measured coincidences are proportional to the fundamental quantity (correlation function),
R(2x)(t0,τ,∆T) ≈ η1η2(∆T)2G(2x)(t0,t0+τ), (14)
which means we can analyze G
(2x)(
t
1
,t
2)
in terms of the initial detection times, and thereby
resolve correlation features within a time window ∼τG.
When the mode envelopes
ζk(
t
)
are modulated, a structured HOM dip is obtained. Here, we
are interested in polarization modulation, which leads to coincidence detection probabilities
that are dependent on the optical delay
τOpt
between the inputs. For this we need to consider
two polarization modes (designated as Hfor horizontal polarization and Vfor vertical) for
each input mode to the HOM beam splitter. Hence, four modes with their corresponding mode
envelopes
ζσ,k(
t
)
,
σ∈ {
H,V
}
, need to be considered. The generalization of the second-order
cross-correlation function, which takes into account polarization modes, is given by
G(2x)(t3,t4)=G(2x)
H,H(t3,t4)+G(2x)
H,V(t3,t4)+G(2x)
V,H(t3,t4)+G(2x)
V,V(t3,t4), (15)
where the first and second polarization subscripts refer to first and second spatial modes,
respectively. The correlation functions with equal polarization in both modes takes the same form
as Eq. (5). For the correlation functions with unequal polarization in both modes, the interference
term, which is the last one in Eq. (5), vanishes. Thus, from knowledge of Eq. (5) we can find an
expression for Eq. (15). To do this, we first simplify Eq. (5) using assumptions Eqs. (7)-(10). In
addition, we consider the zero time-delay case (
τ=
0, i.e. t
3=
t
4≡
t
0
), for which G
(1)(
0
)=
1.
Focusing on only true coincidences allows us to isolate the dependence of the HOM interference

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20947
on τOpt. Using only real modulation functions, the correlation function Eq. (5) gives us
G(2x)
σ,σ(t0)=1
4ζσ,1(t0)4+ζσ,2(t0)4
G(2x)
σ,σ0(t0)=1
4ζσ,1(t0)2ζσ0,1(t0)2+ζσ,2 (t0)2ζσ0,2(t0)2
+ζσ,1(t0)2ζσ0,2(t0)2+ζσ,2 (t0)2ζσ0,1(t0)2,
(16)
for the equal and unequal polarization cases, respectively. We choose mode functions in the
shape of square waves, which for Hpolarization are described by
ζH,1(t)=SW1
0t−τOpt
TMod 
ζH,2(t)=SW1
0t
TMod ,
(17)
where
SWb
a(
t
)
denotes a square wave alternating between lower level aand upper level b(duty
cycle 50 %, period is T
Mod
, and
SWb
a(
0
)=
b). Accordingly, the mode functions for Vpolarization
are shifted by TMod/2,
ζV,1(t)=SW1
0t−TMod/2−τOpt
TMod 
ζV,2(t)=SW1
0t−TMod/2
TMod .
(18)
Using our modulation functions Eq. (17) and Eq. (18), we arrive at a simple expression for the
second-order cross-correlation function Eq. (15),
G(2x)(t0,τOpt)=1
43−SW1
−1t0
TMod SW1
−1t0−τOpt
TMod . (19)
Hence, the coincidence rate can follow a square shape when analyzed with respect to the
initial detection time t
0
. The duty cycle of this HOM square wave (i.e. the duration over
which coincidences are suppressed during a period T
Mod
) depends on
τOpt
. In particular, when
τOpt =
T
Mod/
4, the period of the coincidence rate is half of the modulator’s period T
Mod
. This
frequency doubling effect of the oscillations, which is shown in Fig. 1(b), is the result of the
HOM interference between indistinguishable polarization states of the input beams.
3. Experimental realization
Figure 1(a) shows a sketch of the experimental setup. We use a CW grating-stabilized laser diode
(785 nm single mode light with
>
2 mW intensity and microsecond-scale coherence time), and
employ a variable fiber attenuator (VFA) to decrease the laser intensity down to the single-photon
counting regime. The laser signal is then sent into a polarization modulator (PMOD) that acts as
a fast switch between Hand Vpolarization. The PMOD consists of a polarization Mach-Zehnder
interferometer with a phase modulator in each path, and has been discussed elsewhere [20,21].
It is driven by an arbitrary waveform generator (AWG) at 353MHz, which corresponds to a
modulation period of T
Mod =
2.83 ns. One output of the PMOD is passed through a 2 km fixed
fiber delay line (FFD) that acts as a relative dephasing channel for the two output modes because
this delay is longer than the laser coherence length. The other output mode from the PMOD is
sent through a variable free-space delay line (VFSD), which consists of a hollow retro-reflector
moving along an optical rail. The VFSD can provide optical delays up to 3 ns with negligible

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20948
Fig. 1. Square HOM Waves: Experimental Setup and Concept. (a)
A continuous-wave
laser is attenuated and polarization modulated before being divided into two optical modes
for the HOM interferometer. As explained in the main text, the fixed and variable time delays
in the two paths are responsible for isolating the HOM interference and scanning the optical
delay, respectively.
(b)
Adjusting the optical delay
τOpt
changes the polarization overlap
pattern between the two input modes at the HOM beam splitter (blue dotted and black dashed
traces represent the two inputs), which results in changes to the two-photon coincidences
(red solid traces). Y-axes for plots in (b) refer to the normalized coincidences (red solid
traces) only. VFA: variable fiber attenuator, FFD: fixed 2 km fiber delay line, VFSD: variable
free-space delay line, BS: (HOM) 50:50 non-polarizing beam splitter, PMOD: polarization
modulator, FBS: fiber beam splitter, TRSPD: time-resolving single-photon detectors. Note
that the coherence time of the laser (microseconds) is much larger than the modulation
period (∼2.8 ns).
error (
∼
10
−3
ns). The two modes are then optically combined on a 50:50 non-polarizing beam
splitter (BS) in free-space (HOM beam splitter). The time-resolving single-photon detectors
(TRSPD) consist of silicon avalanche photodiodes in series with time tagging units that assign
time stamps to detection events with a 78.125 ps time resolution. We also record trigger signals
from the AWG that drives our PMOD.

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20949
Fig. 1(b) illustrates the conceptual idea of how both traditional HOM dip and structured
HOM interference are controlled through the relative optical delay between the two HOM input
modes. Three special cases are shown. In the first case, no optical delay is imposed (
τOpt =
0),
which results in varying yet identical polarization states of the two modes (blue dotted and black
dashed traces represent the two inputs) at any given time that are in-phase polarization patterns.
Consequently, these modes interfere perfectly at the HOM beam splitter, which reduces the
normalized coincidences (red solid trace) to 0.5. In the next case, the optical delay is equal to one
quarter of the modulation period (
τOpt =
T
Mod/
4). Thus, half the time the polarization states are
identical between the input modes, and half the time they are orthogonal. Now the normalized
coincidences follow a square wave pattern at twice the modulation frequency.
In the third case, the optical delay equals one half the modulation period, causing orthog-
onal polarization states at any given time (out-of-phase polarization patterns). Therefore, no
interference ensues and the normalized coincidences trace remains at the baseline of 1. In
general, for optical delays between 0 and T
Mod/
2, the regions of overlap and non-overlap alternate.
Correspondingly, the normalized coincidences oscillates between 0.5 and 1.
4. Experimental results
Figure 2shows our experimental results. The mean photon number per modulation period was
3.4
·
10
−3
with a slight count rate difference between the two detectors, which contributed less
than 1 % to the visibility loss. The measured average rate of single photon detection during a
measurement time of 6 s for each retroreflector position was 1.2 million per second. The total
number of two-photon coincidences (coincidence window T
coin =
312.5 ps) for each step was on
average 4000.
In Fig. 2(a) we illustrate how to distill the square wave predicted by Eq. (19). Per modulation
period, we either detect none or
∼
1 coincidence. To form the correlation function Eq. (19),
which is a statistical quantity, we need to add up coincidences coherently over time. We utilize
Fig. 2. Experimental Square Wave HOM Interference with Polarization Modulated
CW Lasers. (a)
Using time stamps of the modulation trigger as a stable time reference
(indicated by the arrows in the lower plot), together with the photon detection times, we can
selectively add up coincidences over time to form the resolved HOM square wave (see main
text).
(b)
As expected from Eq. (19), a square wave pattern with double the modulation
frequency emerges for certain optical delays
τOpt
. Coincidences are detected within a
T
coin =
312.5 ps window, which approximates
τ=
0 ns. The histogram bin size is 156.25 ps.

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20950
the modulator trigger as a stable time reference (indicated by the arrows in the lower plot): when
a coincidence is found, the time-difference to the previous trigger is calculated. We refer to
this time difference as the “modulator phase”, which is the variable t
0
in Eq. (19). Moreover,
since the time resolution of our time tagging units is two orders of magnitude higher than the
modulator period, we are able to resolve the square wave pattern by taking a histogram over t0.
In Fig. 2(b) we show the outcome of the analysis for optical delays corresponding to the
three cases in Fig. 1(b). We obtain the expected flat coincidence line for the case
τOpt =
0 ns
. Experimentally, we identify the zero optical delay case by finding the retroreflector position
in the VFSD (see Fig. 1(a)) that maximizes the HOM dip visibility. We achieve a maximum
visibility of 0.35
±
0.03, which is lower than the theoretical maximum of 0.5. To assess the upper
bound for the visibility in our setup, we bypass the AWG-driven polarization controller, and use
manual polarization control via a half-wave plate, and achieved a maximum HOM dip visibility
of 0.42
±
0.02. Thus, due to imperfections in the AWG-driven polarization modulator and its
finite rise and fall times (∼500 ps) for generating square waves, we incur a 7% visibility drop.
Next, we move the retroreflector to a position corresponding to a 0.68 ns optical delay,
which approximates T
Mod/
4
≈
0.7 ns. We now observe a square wave of coincidences while
simultaneously the HOM dip visibility reduces to roughly half the maximum (
∼
18 %). This
reduction is because half the time the coincident photons have orthogonal polarization within the
dip. Finally, we set the delay to 1.36 ns, corresponding to T
Mod/
2
≈
1.4 ns. We can clearly see
that the coincidence counts now oscillate between 70 and 120 coincidences. An explanation for
this modulation pattern is given in Fig. 4, and is discussed in detail in the next section. While
one may have expected a flat line corresponding to maximum coincidences, two experimental
effects cause the resolved HOM interference to modulate.
Our setup can also be used to generate triangle HOM waves. If we ignore the modulator phase,
and thus integrate Eq. (19) over t0for a measurement time TM, we obtain
G(2x)(τOpt)=TM1−1
2TW1
0τOpt
TMod −3
4, (20)
where
TWb
a(
t
)
is a triangle wave with minimum a, maximum b, and
TWb
a(
0
)=
0. Thus, without
a phase reference from the modulation trigger, the correlation function follows a triangular shape
with the same period as the modulator, T
Mod
. Note that this is true for
τ≈
0 as before (i.e. true
Fig. 3. Experimental Triangle Wave HOM Interference with CW Lasers. (a)
Tradi-
tional HOM dips extracted for optical delays (retroreflector positions) ranging over one
modulator period T
Mod
. Black dots: data points, red solid line: fit.
(b)
Plot of HOM dip
visibilities for optical delays ranging from 0 to 7.14 ns. The red curve is a fit described by
Eq. (20) for
τ≈
0. The period of the fitted waveform is T
=(
2.80
±
0.04) ns, which matches
the modulator period as expected. Compared with the measurements presented in Fig. 2,
here our setup had an offset of the optical delay of (0.85
±
0.02) ns, which the fit takes into
account.

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20951
coincidences). We carried out another measurement, using the following simple way to extract
the triangle wave. In Fig. 3(a) we show the HOM dip measurement for various optical delays, as
described by Eq. (11). Note that the normalized coincidences at
τ=
0 are identical to the dip
depth. Furthermore, a plot of the visibility Eq. (12) as a function of optical delay
τOpt
represents
a triangle function of optical delay, as expected from Eq. (20). Our results plotted in Fig. 3
confirm this is indeed the case. The visibility does not reach zero because the optical delay is not
matched for perfect anti-overlap of the square waves. Also, the finite rise and fall times of the
polarization modulator in conjunction with a non-symmetric duty cycle reduces the visibility.
5. Discussion
The traditional HOM dip is a feature of the full HOM interference pattern that appears on the
coherence length time scale of the fields. Traces of high-speed field modulations are present in
HOM interference, but are not discernible at the coherence length time scale. We have shown
this in our experiment for square wave modulation, where the ratio of modulation period to
coherence length is
∼
10
−9/
10
−6=
10
−3
. Crucially, though, any other modulation function
would have resulted in identical HOM dips, which is determined entirely by the unmodulated
field’s autocorrelation function, or spectrum. In other words, infinitely many “HOM waves”
give rise to the same HOM dip, and alleviation of this uncertainty (and revelation of hidden
information) required, in our case, single-photon detectors with sub-nanosecond time-resolution.
There are two effects that currently limit the performance. The first effect arises from the
non-symmetric duty cycle of our polarization modulator. It deviates substantially from the
symmetric value of 50 %, in which case we should get, for instance, H-polarized photons during
the first half of the modulation period, and V-polarized photons the other half. However, a
polarization modulator with a duty cycle of 70 % produces, for instance, H-polarized photons
during nearly 3
/
4 of the modulation period. The theory plot for the case where the duty cycle
equals 0.7 in Fig. 4shows the polarization patterns (black dashed lines) and resulting coincidences
(solid red line). We see that, even though the optical delay is set correctly to half the modulation
period, the coincidences do not follow a flat line because the two polarization patterns are not
complementary. This duty cycle effect does not affect the case where the optical delay is zero
because non-shifted polarization patterns overlap perfectly no matter what their shape. This is
Fig. 4. Effect of non-ideal optical delay and asymmetrical duty cycle settings.
(a) and
(b) are theory plots showing the predicted normalized coincidences (red solid line) and
the polarization patterns (black dashed lines). (a) Non-ideal (offset) optical delay with a
symmetrical 0.5 duty cycle, (b) non-ideal optical delay and an asymmetrical duty cycle of
0.7 are taken into account in Eq. (19) for the
τOpt =
T
Mod/
2 case. As a consequence, the
extracted experimental HOM interference pattern (c) does not show a flat line at maximum
coincidences. Note that for clarity, the x-axis for (a) and (b) graphs extend only over one
modulator period, whereas the data in (c) extends over two modulator periods.

Research Article Vol. 28, No. 14 / 6 July 2020 / Optics Express 20952
why we still obtain a flat line at minimum coincidences in Fig. 2(b). A second effect arises when
the optical delay does not match precisely, for example, T
Mod/
2. Then the polarization patterns
also do not perfectly overlap, as the theory plot for a duty cycle of 50 % shows, though the effect
is much smaller than that for the asymmetric duty cycle. Hence, if Eq. (19) is used with both
a non-ideal optical delay and asymmetric duty cycle, the detected coincidences, and thus the
HOM square wave, follows the predicted shape. The theory could also be expanded to include a
varying input intensity to explore how it may affect the coincidence signal but this is beyond the
scope of our present work.
6. Conclusion
In summary, we have modelled and observed Hong-Ou-Mandel square and triangle wave
interference patterns using polarization-modulated continuous-wave lasers. The basic equation
is Eq. (5), which shows that any kind of laser field modulation affecting the mode envelopes
ζk(
t
)
will result in visible HOM interference patterns. Our work demonstrates a striking feature
of Hong-Ou-Mandel interference between coherent states: despite different envelope functions
ζk(
t
)
, the full interference visibility can still be obtained within the region of temporal overlap.
As in our case, we experimentally recovered clear HOM interference between wavepackets that
are identical in shape but shifted in time. This seems counterintuitive, as traditional HOM
experiments are based on well-defined wavepackets and careful path alignment to ensure they
overlap at the interfering beam splitter.
Our results can apply to various modulation schemes to encode information. For example, a
sender could encode information in the polarization state of photons, and a receiver could use
HOM dip visibility levels to decode the message. This will be particularly useful in cases where
alignment of the reference frames is difficult or not possible (communication without a shared
Cartesian frame) [22,23]. Our findings could be applicable to HOM-based quantum protocols
which are sensitive to temporal alignment where fluctuating optical path lengths can degrade the
HOM effect. In particular, free-space quantum communication links with moving platforms, such
as between vehicles, aircraft and/or satellites [21,24], may have rapidly changing path lengths that
are difficult to compensate for in real time. For instance, the measurement-device independent
quantum key distribution implementation of Yuan et al. [25] had a 50 ps misalignment that
prevented HOM interference. Our work is a proof-of-concept system which, after improvements
to the polarization modulator and using independent lasers, could be utilized for such protocols.
Funding
Industry Canada; Natural Sciences and Engineering Research Council of Canada (RGPIN-
386329-2010); Canadian Institute for Advanced Research; Ontario Research Foundation (098,
RE08-051); Canada Foundation for Innovation (25403, 30833); Canadian Space Agency; Office
of Naval Research.
Acknowledgments
The authors would like to thank Matteo Mariantoni for lending us the high-speed arbitrary
waveform generator.
Disclosures
The authors declare no conflicts of interest.
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