Traceable metrology for characterizing quantum optical communication devices

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Traceable metrology for characterizing quantum optical communication devices
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2014 Metrologia 51 S258
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Bureau International des Poids et Mesures
Metrologia
Metrologia 51 (2014) S258–S266
doi:10.1088/0026-1394/51/6/S258
Traceable metrology for characterizing
quantum optical communication devices
C J Chunnilall, G Lepert, J J Allerton, C J Hart and A G Sinclair
National Physical Laboratory, Teddington, TW11 0LW, UK
E-mail: [email protected]
Received 9 June 2014, revised 1 August 2014
Accepted for publication 13 August 2014
Published 20 November 2014
Abstract
Industrial technologies based on the production, manipulation, and detection of single and
entangled photons are emerging. Quantum key distribution is one of the most commercially
advanced, and among the first to directly harness the peculiar laws of quantum physics.
To assist the development of this quantum industry, the National Physical Laboratory is using
traditional, and quantum, methods to develop traceable metrology for the devices used in these
technologies. We report on the instrumentation we have developed to characterize such
systems.
Keywords: metrology, quantum key distribution, single-photon
(Some figures may appear in colour only in the online journal)
1. Introduction
A focus of current work at the National Physical Laboratory
(NPL) is to develop a measurement infrastructure for
characterizing the optical components of emerging quantum
technologies, such as quantum key distribution (QKD)
[1–3]. This ranges from ‘traditional’ measures such as
photon number, and detection efficiency, to others such as
indistinguishability that determine device utility.
QKD is underpinned by a physical, as opposed to
algorithmic, process. Its security relies upon encoding each bit
of information sent between the parties seeking to establish a
common secret key onto a single photon. If a hacker intercepts
these photons, (s)he will disturb their encoding in a way that
can be detected. QKD does not prevent hacking, but reveals
whether a hacker has been able to compromise the key.
Current algorithmic key distribution is vulnerable to
advances in classical computing power and quantum
computing, as well as new mathematical insights. QKD,
if faithfully implemented, is guaranteed to be secure by the
laws of physics. This security depends on the physical
performance of the system at the time of key creation. Random
number generators (RNGs) are also essential components of
QKD systems. Optical true quantum RNGs operate at the
single photon level, and depend on the performance of their
constituent single-photon sources and detectors [4].
Figure 1 shows a schematic of a phase-encoding QKD oneway system. Attenuated laser pulses are used to approximate
0026-1394/14/060258+09$33.00
single photons, hence a small fraction will contain more than
one photon. Information is encoded on the phase of the
photons [5]. In the transmitter, the laser pulses are split in
an asymmetric Mach–Zehnder interferometer (AMZI), with
the path difference between the two arms being approximately
equal to half the time separation between pulses from the
laser, resulting in two pulse trains emerging at the output of
the interferometer. An additional phase delay is imparted to
pulses in one arm [5]. An attenuator reduces these pulses to
the single-photon level. Schemes have been developed [6, 7]
to reduce the power of photon-number-splitting (PNS) [8]
attacks on multi-photon pulses. In the decoy-state protocol [7],
the mean photon number, µ, of each pulse is randomly switched
between two or three values which, if set incorrectly, opens a
route to undetectable eavesdropping. These decoy states are
usually implemented with an intensity modulator, as shown
in figure 1. The phase and intensity modulators are driven in
independent random sequences. The receiver has a matching
AMZI, together with (usually) two single-photon detectors.
The performance of all of these components will affect the
performance and security of the QKD system. A classification
of these effects is suggested in table 1.
Characterizing the optical components of a QKD system
can establish: (i) whether they satisfy the assumptions and
requirements of the security proofs [9, 10]; (ii) the expected
secure bit-rate and range of the system; (iii) immunity
from known side-channel attacks; (iv) whether component
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Metrologia 51 (2014) S258
C J Chunnilall et al
Figure 1. Main components of a one-way QKD system using phase encoding.
Table 1. Classification of how physical parameters of a QKD
system may affect its bit-rate, range and security.
Probability distribution
Mean photon number
Temporal purse jitter, duration
Phase randomization
(pulse-to-pulse)
Spectral indistinguishability
(base and bit encoding)
Detection efficiency
Dark count probability
After-pulse probability
Dead-time and recovery time
Detector output signal jitter
Back-flash
Detector indistinguishability
(mufti-detector)
Bit-rate
Range
Security
Y
Y
Y
Y
Y
2. Experiment
The characterization of detectors is fundamental to characterizing single-photon devices, since current traceability of optical
scales, i.e. the SI system, is based on cryogenic radiometers, detectors of optical power. Free-space monochromatic
radiation at the 100 µW power level can be measured with this
approach, with an uncertainty around 0.005% (k = 2) [14, 15].
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2.1. Calibration of QKD receiver
Y
Y
Y
performance has changed, either from natural ageing, or from
hacking.
The most important properties of the photons are their
mean photon number(s) and timing jitter. Furthermore, phase
encoding should impart no discriminating information onto
the photons (e.g. on their spectral properties) which an
eavesdropper can probe.
Single-photon avalanche photodiodes (SPADs) operating
in Geiger mode are used in the most commercially advanced
QKD systems. These are non-photon-number-resolving
detectors; devices for the 1550 nm spectral region exhibit
after-pulsing, and are normally gated to reduce their dark
count rate (see [11–13] and references therein). The photon
detection, dark count and after-pulse probabilities, dead-times,
and recovery times of the photon receivers impact directly on
the achievable range and quantum bit error rate (QBER) of the
QKD system. Their spectral and temporal distinguishability
are also important, as is monitoring the loss of the optical
channel.
NPL has developed a test-bed to characterize the photon
transmitter (‘Alice’) and single photon receiver (‘Bob’)
modules of faint-pulse QKD systems operating at clock rates
up to 1 GHz over optical fibre in the 1550 nm telecom band.
This work will enable QKD systems to be evaluated and
inform the definition of standard measures, thus helping to
shape a validation and certification process for the technology.
This paper will focus on the traceability for characterizing
the mean photon number emitted by the transmitter and the
detection efficiency of the photon counters within the receiver,
measurements which require traceability to the SI unit for
optical radiation.
The calibration of a QKD receiver is essentially the calibration
of (normally two) single-photon SPADs. The ‘effective’ values
of parameters such as detection efficiency, timing jitter, and
spectral responsivity will be modified by the optical path the
signals take in travelling from the input port of the receiver to
the SPADs (see figure 1). Photons will, in addition, be directed
at random into either the short or long paths of the asymmetric
Mach–Zehnder interferometer unless, as in some systems [16],
polarization encoding is used to select these paths. Below we
describe our work to characterize the SPAD module of a QKD
receiver, i.e. without the associated optics discussed above.
A three-step process is used to calibrate a fibrecoupled telecom wavelength detector at the single-photon
level (figure 2). Step 1: a fibre-optic power meter is
calibrated traceably to the cryogenic radiometer and a fibreoptic attenuator is also calibrated over 6 orders of attenuation.
Step 2: the output of a pulsed laser is then transmitted through
the attenuator and measured with the power meter. Step 3: the
output of the attenuator is then further reduced to the singlephoton level (step 3) and used to calibrate the photon counters
in the QKD receiver.
2.2. Calibration of power-meter
The calibration of the fibre-optic power meter is crucial to
these measurements. The traceability chain used at NPL for the
calibration of the spectral response of a fibre-optic power meter
at power levels of 0 dBm (1 mW) is shown in figure 3 [17].
The NPL mechanically cooled cryogenic radiometer
[18] is used to measure the optical power in a collimated,
monochromatic, laser beam, and the responses of silicon trap
detectors [19, 20] are calibrated with reference to this beam.
The longest wavelength normally used is 799 nm, although
measurements have on occasion been carried out at 833 nm
and 852 nm. A thermopile detector [21] with a spectrally
flat response (Laser Instrumentation Ltd; model 14BT) is
then used to transfer the trap detector calibrations to the
850 nm–1650 nm spectral region required for fibre optic power
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Metrologia 51 (2014) S258
C J Chunnilall et al
Primary Standard
Calibrated Fibre Optic
1
Cryogenic Radiometer
Power Meter
2
Pulsed Laser
Calibrated Fibre Optic
QKD Receiver
3
at 1550 nm
Attenuator
(Photon Counter)
Figure 2. Process for calibrating a fibre-coupled QKD receiver (photon counter).
Cryogenic Radiometer (799 nm)
± 0.01%
Si trap detectors (799 nm)
± 0.02%
14BT Thermopile (799 nm)
± 0.2%
Working standards (1550 nm)
(integrating spheres)
± 0.5%
Fibre-optic power meter (1550 nm)
± 0.7%
Figure 3. Traceability chain for calibration of fibre-optic power
meters at 0 dBm. The value quoted for the power meter is a
best-measurement value. Adapted from [17].
meters. Thermopile detectors are not suitable for direct use in
calibrating fibre-optic power meters because of the sensitivity
of their response to objects in their field of view, and therefore
working standards comprising integrating spheres and Ge or
InGaAs detectors [22] are calibrated relative to the thermopile.
The calibration of the integrating sphere working
standards includes the transition from free-space collimated
light to fibre-optic light delivery (figure 4) [23]. A pair of
lenses is used to collimate and then refocus the light emitted
by an optical fibre, ensuring that the beam divergence matches
that of SMF-28 single-mode fibre. The waist of the re-focused
beam is positioned at the point corresponding to where the end
of a fibre coupled to the integrating sphere would be, and the
response of the integrating sphere assembly measured. A 14BT
thermopile is then inserted into the converging beam, and the
incident power recorded. Measurements using a collimated
beam incident on the 14BT over a range of angles spanning
the divergence of the fibre beam established the uniformity of
response of the thermopile over this angular range [23]. The
fibre-holder for the input patch cord is blackened on the inside,
and no changes in the signal are observed when it is mounted
over the integrating sphere entrance aperture shown in figure 4.
The response of the power meter below 0.1 mW is carried
out by measuring its linearity of response. This can be
performed by comparing the change in the signal from the
device under test (DUT) with that of a sensor which is known to
be linear. High accuracy measurements use the superposition
technique (figure 5). This is an in-fibre implementation
[24, 25] of the double-aperture technique which was already
in use for many decades preceding the 1970s [26], and is
inherently more accurate than other methods as it is a ratio
measurement.
The output from a source is connected to an in-fibre system
to generate two switchable channels. At each power level
P0 , the DUT can receive the following photon fluxes: PA (A
closed, B open), PB (A open, B closed), (PA + PB ) (A closed,
B closed). Both switches open enable a dark setting. These
settings allow the linearity to be evaluated.
In this measurement, the reference power Z0 (0.1 mW)
is the high power (PA + PB ) value, and the non-linearity NL
can be described as the relative deviation of the responsivity
R(Z) from the responsivity R(Z) at a stated value of the input
quantity Z0 :
R(Z0 ) − R(Z)
(1)
; Z < Z0
R(Z)
where L is the linearity. If NL is zero, the detector is linear
between Z0 and Z. NL > 0 indicates super-linearity, and
NL < 0 indicates sub-linearity. The attenuation in paths A
and B are set to be (ideally) equal, and the power with both
switches closed is set to 0.1 mW with the variable attenuator
before the first splitter. Thus, the linearity L0.1 of response
for an input power of 0.1 mW relative to 0.05 mW can be
established. The power with both switches closed is then set to
0.05 mW, and the linearity L0.05 of response for an input power
of 0.05 mW relative to 0.025 mW can be established. The
linearity at 0.025 mW with respect to 0.1 mW is the product
L0.1 × L0.05 . In this way, linearity measurements relative to
NL (Z0 ) = L(Z0 ) − 1 =
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C J Chunnilall et al
Detector
+
Integrating
sphere
Lenses
Fibre
Thermopile
Figure 4. Coupling to integrating sphere. The arrowed lines indicate the extremal light rays from the fibre. The thermopile is inserted into
the beam to measure the incident power, and then retracted to obtain the response from the detector attached to the integrating sphere.
Path A
PA
50
Source
Device Under Test
Variable fibre
attenuator
50
Variable fibre
attenuator
Path B
PB
Figure 5. Superposition set-up.
the 0.1 mW responsivity can be established down to the noise
floor of the device.
The length of fibre in path A (a few 100 m) is to eliminate
interference fringes at the DUT. A multiline laser of ∼10 nm
bandwidth is used. An isolator placed immediately before the
DUT may also be required to avoid back-reflections.
Using this approach, the response of the Hewlett-Packard
fibre-optic power meter (Optical head model: HP81524A;
plug-in module model: HP81533B, power unit model
HP8153A) was calibrated with an uncertainty of 1.0%
(k = 2) at 0.1 mW, and when combined with linearity
measurements, resulted in a combined uncertainty of 1.02%
at 1 nW, increasing to 1.04% at 50 pW. Below 50 pW, noise
and digitizing coarseness lead to the uncertainty increasing
rapidly, e.g. 1.6% at 10 pW, 8% at 1 pW.
2.3. Calibration of attenuator
The calibrated power meter is used to calibrate the attenuator
(model HP 8158B, figure 2) which is used to span the range
from nW to single photon flux. Only the linearity calibration
of the power meter is required. The attenuator has a single
continuously variable filter (0–10 dB), plus 5 metallic neutral
density filters which provide attenuation from 10 dB to 50 dB in
steps of 10 dB, enabling total attenuation (including insertion
loss) up to 64 dB [27]. A range of input power levels (1 mW–
100 nW) are used to check that there are no significant heating
effects caused by absorption in the filters. Any such effects
would be reduced at the power levels (∼nW) used in calibrating
the photon counter. The linearity of the attenuator is measured
between the 3 dB and 64 dB settings, since 3 dB exceeds the
insertion loss of the attenuator. With this method, the linearity
of the attenuator can be calibrated with an uncertainty of 0.4%
(k = 2). The attenuator is equipped with Diamond® HMS10/HP connectors, and the corresponding fibre patch-cords are
not disconnected from the attenuator between its calibration
and use.
The set-up for calibrating the SPADs in a QKD receiver
is shown in figure 6. The measurement approach for a gated
detector which exhibits after-pulsing is to measure dark count
probability, followed by after-pulse probability [28, 29] and
then detection efficiency, as described by Yuan et al [30]. The
arrival of the laser pulses at the detector is synchronized to
occur within the detector gates with an overall rms jitter of less
than 10 ps, using a low-jitter programmable electronic delay
line. This is essential for characterizing systems with narrow
detector gates operating at GHz rates [31].
For pulses from an attenuated laser operating above
threshold with a mean number of photons per pulse µ, the
probability of there being n photons in a pulse is assumed
to follow the Poisson distribution. The probability of a true
detection, i.e. one due to the detection of a photon, and not a
dark count or after-pulse, is given by [32, 33]:
P true = 1 − exp (−µηD )
(2)
1 ln 1 − P true ,
µ
(3)
therefore
ηD = −
where ηD is the desired detection efficiency.
The dark count probability, P true , of the detector is
measured by recording the number of detection events per
gate in the absence of incident photons. The after-pulse and
detection probabilities are then measured by illuminating the
detector with a pulsed laser source attenuated to the single
photon level. A timer/counter (FAST Comtec MCS6A), with
a time-resolution of 100 ps, and operating in multi-stop mode
with zero dead time after stops, is used to record temporal
histograms of laser triggers and detections w.r.t. an initial
laser trigger. The laser pulse frequency flaser is stepped down
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Metrologia 51 (2014) S258
C J Chunnilall et al
Master clock
Frequency
divider
Variable
delay line
Trigger in
QKD receiver /
photon counter
Signal out
Counter / timer
Optical
input
Trigger In
Pulsed
laser source
Calibrated
optical power
meter
Calibrated
optical
attenuator
Uncalibrated
optical
attenuator
Figure 6. Set-up for calibration of QKD photon counter(s). Electronic connections are shown in dashed lines, fibre-optic connections in
solid lines. The output of the calibrated optical attenuator can be connected to either the calibrated power meter or the QKD receiver/photon
counter. The master clock may be part of the QKD receiver.
by an integer factor R compared to the detector gate rate using
a frequency divider. At zero-time delay with respect to the
laser pulse, the histogram peak is composed of detection events
observed under laser light illumination (plus after-pulses and
dark counts). Peaks at a time delay in this histogram not
corresponding to an illuminated gate are due to after-pulses
and dark counts. By normalizing the detected count rate to the
total number of applied gates, the after-pulse probability can
be calculated using equation (4), where Ci , Cni are the average
number of counts per illuminated and non-illuminated gate,
and Cdark is the number of dark counts, calculated from the
previously established dark count probability.
P after =
Cni − Cdark
· R.
Ci − Cni
(4)
The photon detection probability P true can be obtained from
equation (5). P i is the probability of detecting a photon at each
illuminated gate, and is given by Ci /Ni , where Ni is the total
number of illuminated gates.
P true = P i − P dark ·
1
1 + P after
(5)
Equation (4) yields the total after-pulse probability for all
(R) gates subsequent to an illuminated gate. However,
the probability can be evaluated as a function of time after
illumination—see figure 10.
For a required µ, the mean power in the stream of
pulses emitted by the laser is given by P = nµhνm , where
n = 1/flaser , h is Planck’s constant, and νm is the mean spectral
frequency of the emitted photons. The output of the laser
(PicoQuant LDH-P-F-N-1550) is attenuated by two fibrecoupled attenuators in series. The second attenuator is the
calibrated attenuator, and its attenuation is set to 3 dB. The
first attenuator, an OZ Optics (OZ 560151-18) attenuator is
used at minimum attenuation in order to measure the spectral
profile of the pulses in the high power regime with a calibrated
spectrum analyser. From this the mean spectral frequency
can be calculated. The attenuators exhibit no significant
spectral variation with attenuation over the narrow width of the
optical pulses (few nanometres). The output of the calibrated
attenuator is then connected to the calibrated power meter, and
the first attenuator used to set the measured output power in
the nW regime, between 30 dB and 50 dB above the required
single photon level. The calibrated attenuator can then be used
to reduce the power to the single-photon level, and the output
connected to the DUT.
The detector gate needs to be wider than the temporal
extent of the optical pulse in order to ensure that all of
the incident photons can be detected. Strictly speaking, the
method described in the previous paragraphs only measures
the detection efficiency for the specific combination of incident
pulse profile and jitter in the system. These may be different
when the receiver detects pulses transmitted over fibre from the
QKD transmitter. Measurement of the detector gate profile is
accomplished by sending CW radiation (at the single-photon
level) to the detector, and recording the counts which are timecorrelated with the detector gate trigger. Figure 7(a) shows the
detector gate profile of a Princeton Lightwave photon counter
(PGA-602) for a specific setting, which was measured using
a PicoQuant timer/counter (HydraHarp) to obtain data at 4 ps
time-intervals. For measurements involving pulsed sources,
the optical pulses are centred within the detector gate by
recording detection events as the time-delay between the laser
pulse and the detector gate is varied. The optimum time-delay
for synchronization is the value at which the resulting number
of detected events is a maximum (see figure 7(b)).
When the peak of the laser pulse is exactly aligned with
the gate detection efficiency maximum, the number of detected
events will fall as the laser temporal profile broadens (while
maintaining constant power). Figure 7(c) shows two extreme
pulse temporal profiles (obtained using different settings on
the laser driver, and measured at high power with a 65 GHz
bandwidth optical head and oscilloscope) with the same
overall output power. The corresponding synchronization data
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Metrologia 51 (2014) S258
Probability (after-pulses + darks)
1.0
0.8
Probability (after-pulses + darks)
Relative detection efficiency
(a)
C J Chunnilall et al
0.6
0.4
0.2
0.0
5
5.2
5.4
5.6
5.8
6
Time after gate trigger (ns)
(b)
3.0E+05
Detector counts
A
B
2.0E+05
Figure 10. Probability of after-pulses plus dark counts.
The power meter is calibrated with CW radiation, but is
used to measure pulsed radiation. Figure 8 shows the response
of the power meter and its electronics to the optical pulses,
which are ∼90 ps wide (full-width half-maximum). At a pulse
repetition rate of 250 kHz, there is a ripple of ±0.05%, while
at 500 kHz the ripple is ±0.01%. These make insignificant
contributions to the overall uncertainty budget.
Another potential source of uncertainty is due to the timer
missing counts, or jitter leading to signals being recorded in
the wrong time-bins. Tests with reference clock signals show
that neither of these factors is significant, and that the timer
exhibits zero dead-time for stop events.
1.0E+05
0.0E+00
1
1.5
2
2.5
3
Delay (ns)
Intensity (arbitrary units)
(c)
3.0E-03
A
B
2.0E-03
1.0E-03
2.4. Calibration of QKD transmitter
0.0E+00
0.0
0.2
0.4
0.6
0.8
1.0
Time (ns)
Figure 7. (a) Measured detector gate profile of a Princeton
Lightwave photon counter (PGA-602). (b) Detector counts as a
function of time-delay for two different incident pulse profiles.
(c) Pulse profiles corresponding to laser driver settings A and B.
obtained with the Princeton Lightwave photon counter are
shown in figure 7(b). The areas under the curves are equal
to within 0.5% (ideally, they should be equal), and therefore
such comparisons can be used to rescale measured detection
efficiency values if the calibration laser has a different temporal
profile to that in the QKD system. Ideally, the output of the
QKD transmitter should be the pulsed laser source used in
the calibration scheme shown in figure 6. This would require
that its attenuation can be reduced to ensure an output power
level measureable with the power meter, and accounting for
any temporal broadening caused by the fibre link between the
transmitter and receiver. We note that the laser pulse profile can
be measured at the single-photon level, by deconvolving the
detector gate profile and system jitter from the synchronization
curve.
Calibration of the output of the QKD transmitter uses the setup shown in figure 9. The output of the transmitter is sent to
a calibrated gated photon counter, and the optical pulses are
synchronized to arrive at the detector during detector gates by
using the variable electronic delay line. A Princeton Lightwave
photon counter (PGA-602), was calibrated using the same
process described above for the QKD photon counter, and used
for these measurements.
Figure 10 shows the after-pulse plus dark count probability
of the photon counter, measured as a function of time after
a pulse. At long times, the probability approaches the dark
count level. The detector was gated at 4 MHz, and the laser
at 4 MHz/256 = 15.625 kHz. The maximum time after a
pulse of 63.75 ms corresponds to 255 gates. It was previously
confirmed that the detection efficiency was independent of gate
frequency in the 4 MHz to 250 kHz regime. In view of this, the
detector was gated at 250 kHz for measurements of the QKD
transmitter, in order to minimize the effect of after-pulses while
maintaining a reasonable gate frequency. A frequency divider
was used to reduce the QKD master clock frequency to the
required rate.
The number of gates, Ngates and the corresponding number
of detections, Ndet is recorded by the counter/timer. The mean
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Metrologia 51 (2014) S258
C J Chunnilall et al
Fractional deviation from mean value
Power meter response to input pulses (~ 90 ps pulsewidth)
0.0010
Freq. (kHz)
250
0.0005
500
0.0000
-0.0005
-0.0010
0
5
10
15
20
Time (µs)
Figure 8. Response of power meter to pulsed radiation (measured with an 80 GHz bandwidth sampling oscilloscope).
Frequency
divider
Master clock
Variable
delay line
QKD transmitter
Optical
input
Calibrated
photon
counter
Trigger in
Signal out
Counter / timer
Figure 9. Set-up for calibration of QKD transmitter.
photon number is calculated from (see appendix)
µ=
1
p det
dark
,
ln 1 −
+
p
ηD
1 + p after
where
pdet =
Ndet
.
Ngates
(6)
(7)
As for the calibration of the QKD receiver, correction for the
effective detection efficiency may need to be made if the QKD
transmitter pulses and jitter are different from those used in
calibrating the single-photon detector.
3. Uncertainties
Table 2 presents simplified uncertainty budgets for the
calibration of the QKD receiver SPAD detection efficiencies
and the mean photon number of the pulses emitted by the
QKD transmitter. The after-pulse and dark count probabilities
are three to four orders of magnitude lower than the detection
efficiency. The fibre-coupling component is a best estimate of
the inequivalence of the fibre coupling between the calibrated
attenuator and the power meter or photon counter, respectively,
in the case of receiver calibration. A similar consideration
applies in the case of the transmitter; here the inequivalence
is between the measuring system and the link to the rest of
the QKD system. For the receiver, additional uncertainties
can result from: (i) a mismatch between the laser profile used
for calibrating the device, and that of the QKD transmitter it
will be coupled to (the calibration correction); (ii) measuring
losses through the associated optics in the receiver module
(not included in the uncertainty budget). In the case of the
QKD transmitter, additional uncertainties can result, as with
the receiver, from a mismatch between the laser profile used
for calibrating the measuring detector and the QKD transmitter,
as well as any additional optical components that may be used
to separate the two pulse trains generated by the asymmetric
Mach–Zehnder interferometer. If a decoy state protocol is
implemented, some of the pulses may have a mean photon
number ∼0.001, and therefore the fractional uncertainty in the
detected counts will be much larger.
The set-ups described above were used to characterize the
receiver and transmitter of a QKD system operating at 1 GHz,
and also provide real-time monitoring of the output of the QKD
transmitter during a field trial of a QKD-secured 10 Gb s−1
DWDM transmission system over a single installed fibre [34].
4. Conclusion
A test-bed has been constructed which can be synchronized
with the pulse emission, or detector gates, of a faint-pulse
QKD system, with low jitter (<10 ps rms). Pulses of mean
photon number µ, traceable to the primary scale realized by
cryogenic radiometry, can be used to characterize the response
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C J Chunnilall et al
Table 2. Simplified uncertainty budgets for calibration of QKD
receiver (a) and QKD transmitter (b).
Source of
uncertainty
(a) Calibration of QKD receiver
Power meter
Power meter stability
Attenuator
Fibre coupling
Synchronisation stability
Laser power stability
Dark count probability
After-pulse probability
True detection counts
Calibration correction
Combined standard uncertainty
Expanded uncertainty (k = 2)
(b) Calibration of QKD transmitter
DE of photon counter
Fibre coupling
Synchronisation stability
Dark count probability
After-pulse probability
Detection counts
Calibration correction
Extinction of adjacent pulses
Losses through optics
Combined standard uncertainty
Expanded uncertainty (k = 2)
Appendix: derivation of equation (6)
Ndet includes Ndark dark counts and Nafter after-pulses:
100 ×
Relative standard
uncertainty
0.5
0.2
0.2
0.2
0.5
0.3
0.1
0.1
0.1
0.5
1.0
2.0
Ndark = Ngates p dark
(A.1)
Nafter = (Ntrue + Ndark ) pafter ,
(A.2)
where Ntrue is the number of true detection events, and afterpulses from after-pulses are ignored. Hence
(A.3)
Ndet = (Ntrue + Ndark ) 1 + p after
Ntrue =
Ndet
− Ndark .
1 + p after
(A.4)
The probability of a true detection is therefore given by:
p det
Ntrue
=
− p dark .
Ngates
1 + p after
1.0
0.2
0.5
0.1
0.1
0.1
0.5
0.5
0.5
1.4
2.9
(A.5)
For a Poisson distribution with a mean number µ of
photons in a pulse:
Ntrue
= 1 − exp (−µηD ) .
Ngates
of the photon counters in a QKD receiver with an uncertainty of
∼2% (k = 2). A photon counter (Princeton Lightwave PGA602), operating at 250 kHz, was traceably calibrated using the
same set-up, and used to measure the µ values of the pulses
from a QKD transmitter operating at up to 1 GHz, with an
uncertainty of ∼3% (k = 2). This latter ability was used
to provide real-time measurements during a field trial of a
QKD-secured 10 Gb s−1 DWDM transmission system. This
demonstrates the ability to provide traceable measurements
for new quantum optical technologies, of which QKD is the
most commercially advanced, and supports work to develop a
metrology and standards infrastructure [35–38] to accelerate
the commercial uptake of these technologies.
Therefore, the mean photon number is given by:
Ntrue
1
µ = − ln 1 −
ηD
Ngates
1
p det
dark
= − ln 1 −
.
+
p
ηD
1 + p after
(A.6)
(A.7)
© 2014 Queen’s Printer and Controller of HMSO
References
Acknowledgments
This work was funded by: the National Measurement Office
of the UK Department of Business, Innovation and Skills;
UK Technology Strategy Board Trusted Services Project TP
1913–19252; project MIQC (contract IND06) of the European
Metrology Research Programme (EMRP). The EMRP is
jointly funded by the EMRP participating countries within
EURAMET and the European Union. David J Szwer (NPL)
provided the data in figure 7. The authors acknowledge
discussions with Andrew J Deadman and David A Humphreys
(NPL), as well as colleagues participating in the above projects
and in the European Telecommunications Standard Institute’s
Industry Specification Group on QKD.
S265
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