# Absolute efficiency estimation of photon-number-resolving detectors using twin beams.

**ABSTRACT** A nonclassical light source is used to demonstrate experimentally the absolute efficiency calibration of a photon-number-resolving detector. The photon-pair detector calibration method developed by Klyshko for single-photon detectors is generalized to take advantage of the higher dynamic range and additional information provided by photon-number-resolving detectors. This enables the use of brighter twin-beam sources including amplified pulse pumped sources, which increases the relevant signal and provides measurement redundancy, making the calibration more robust.

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- Zachary H. Levine, Thomas Gerrits, Alan L. Migdall, Daniel V. Samarov, Brice Calkins, Adriana E. Lita, Sae Woo Nam[Show abstract] [Hide abstract]

**ABSTRACT:**Improving photon-number resolution of single-photon sensitive detectors is important for many applications, as is extending the range of such detectors. Here we seek improved resolution for a particular superconducting transition-edge sensor (TES) through better processing of the TES output waveforms. With that aim, two algorithms to extract number resolution from TES output waveforms are compared. The comparison is done by processing waveform data sets from a TES illuminated at nine illumination levels by a pulsed laser at 1550 nm. The algorithms are used to sort the individual output waveforms and then create clusters associated with individual photon numbers. The first uses a dot product with the waveform mean (for each illumination level), while the second uses K-means clustering modified to include knowledge of the Poisson distribution. The first algorithm is shown to distinguish adjacent peaks associated with photon numbers up to 19, whereas the second algorithm distinguishes photon numbers up to 23, using the same data.Journal of the Optical Society of America B 07/2012; 29(8):2066-2073. · 2.21 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present an ab-initio derivation of a hierarchy of efficient nonclassicality criteria for sources of photons. The operational criteria are explicitly proposed for the linear optical multiport and feasible multiplexed single-photon detectors. The lowest criterion is equivalent to the frequently used g^{(2)}(0) autocorrelation measurement testing the anticorrelation effect of photons. We also derive a hierarchy of criteria for the detectors capable to at least partially estimate photon-number distribution. We prove the usefulness of both hierarchies to detect the nonclassical states from the noisy multimode single-photon sources and the nonclassical states from the multiphoton sources.Physical Review A 10/2013; 88(4). · 3.04 Impact Factor - H. B. Coldenstrodt-Ronge, A. P. Worsley, J. S. Lundeen, A. Feito, K. L. Pregnell, C. Silberhorn, P. J. Mosley, B. J. Smith, G. Puentes, N. Thomas-Peter, T. C. Ralph, J. Eisert[Show abstract] [Hide abstract]

**ABSTRACT:**Full characterization of detectors in the quantum regime is presented. We determine the POVM elements of a photon-number-resolving detector. A method for precise calibration of detector efficiency is demonstrated.01/2009;

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arXiv:0906.2182v1 [quant-ph] 11 Jun 2009

Absolute efficiency estimation of

photon-number-resolving detectors

using twin beams

A. P. Worsley, H. B. Coldenstrodt-Ronge*, J. S. Lundeen, P. J. Mosley,

B. J. Smith, G. Puentes, N. Thomas-Peter, and I. A. Walmsley

University of Oxford, Clarendon Laboratory, Parks Road,

Oxford, OX1 3PU, United Kingdom

*Corresponding author: h.coldenstrodt-ronge1@physics.ox.ac.uk

Abstract:

mentally the absolute efficiency calibration of a photon-number-resolving

detector.Thephoton-pairdetectorcalibrationmethoddevelopedbyKlyshko

for single-photon detectors is generalized to take advantage of the higher

dynamic range and additional information provided by photon-number-

resolving detectors. This enables the use of brighter twin-beam sources

including amplified pulse pumped sources, which increases the relevant

signal and provides measurement redundancy, making the calibration more

robust.

A nonclassical light source is used to demonstrate experi-

© 2009 Optical Society of America

OCIScodes: (030.5630) Coherence and statistical optics, radiometry; (040.5570) Quantum de-

tectors; (120.3940) Metrology; (120.4800) Optical standards and testing; (270.5290) Quantum

optics, photon statistics; (270.6570) Quantum optics, squeezed states

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1.Introduction

Quantum optics enables one to make measurements that are more precise than the fundamental

limits of classical optics [1]. Central to this capability are quantum optical detectors, those that

are sufficiently sensitive to discern the inherent discreteness of light. These detectors are key to

emerging quantum technologies such as quantum imaging and lithography, in which the stan-

dard wavelength limit to resolution is surpassed by using quantum states of light and photon-

numbersensitivity.However,themajorityofquantumopticaldetectorshavea responsethat sat-

Page 3

uratesatonlyonephoton,imposingasignificantlimitationonthebrightnessoftheopticalfields

that can be used for such quantum technologies. The result of this binary detector response is

a measurement that can discriminate only between zero photons and one or more photons ar-

riving simultaneously; the detector produces an identical response for any number of photons

greater than zero. To allow increased brightness of the light sources used in the technologies

outlined above (and the concomitant improvement in accuracy that this brings), it is necessary

to use detectors that can discern the number of photons incident simultaneously on the detector

— a photon-number-resolving detector (PNRD). Indeed, the development of PNRDs is an ac-

tive area of research including photomultiplier tubes, the extension of avalanche photodiodes

(APDs) to higher photon number through time multiplexing [2, 3] or spatially-multiplexed ar-

rays [4], transition-edgesensors (TES)[5, 6, 7], charge-integrationphotondetectors(CIPS) [8],

superconducting single-photon detectors [9, 10], visible-light photon counters (VLPCs) [11],

and quantum dot field effect transistors [12, 13].

Calibration of optical detectors is a difficult problem. The standard approach uses a previ-

ously calibrated light source. The drawback of this approach is that errors in the source bright-

ness translate directly into errors in the detector efficiency calibration. The converse is also

true, leading to a detector and light-source calibration dilemma. To get around this, brightness

calibration is typically based on a fundamentalphysical process, for example,the luminosity of

blackbody radiation of gold at its melting point [14], or heating of a cryogenic bolometer [15].

Such methods are suitable for calibrating bright light sources. In contrast, using this method

to calibrate detectors operating at the quantum level (i.e. fields containing only a few photons)

requires sources with powers on the order of a femtowatt to avoid saturation. Such sources are

impractical.

Nonclassical states of light allow us to circumvent the horns of the dilemma due to their

behavior in the presence of loss. Realistic detectors can be modeled by optical loss (i.e. at-

tenuation) followed by unit efficiency detectors. The transformation of classical states (e.g. a

coherent state, thermal state, etc.) under loss is parameterized only by source brightness, which

scales linearly with the loss. In contrast, nonclassical states change their statistical character

upon experiencing loss: correlations in optical phase, intensity, photon number, and electric

field transform in a non-trivial way under loss. Based on this fact, Klyshko proposed a way to

calibrate detectors based on the statistical character of light rather than its brightness [16]. This

approachreliesonspontaneousparametricdownconversion(SPDC) as a lightsourcein whicha

photonis simultaneouslycreatedin eachof two opticalmodes, usuallydenotedsignal andidler.

Since the photons are created in pairs the two output modes are perfectly correlated in photon

number. Thus detection of the idler without the simultaneous detection of the signal photoncan

only be caused by loss in the signal arm. Measuring the detected rate of idler photons Riand

photon pairs Rcthen allows a calibration of the detector efficiency ηs

ηs=Rc

Ri,

(1)

and vice versa with i ↔ s. This efficiency estimation was shown experimentally in [17, 18, 19,

20].

The Klyshko scheme is limited by three factors. First, it relies on the implicit assumption

that at most one photon pair is emitted at a time, a feature which can be violated if the SPDC

is pumped strongly. Indeed more than one pair is often desirable, as in continuous-variable

experiments. In this case, the detector efficiency can be obtained by first lowering the pump

power and using the Klyshko method [21]. However, the detector is now calibrated outside the

regime of its intended use, which could necessitate subsequent assumptions such as the inde-

pendence of the estimated efficiency from the pump power. Thus, the direct in situ calibration

of detectors would be desirable. Second, the Klyshko scheme is primarily designed for single-

Page 4

photon detectors and is not directly translated to PNRDs. And third, this calibration method is

highly sensitive to the input state quality and measurement uncertainties. Because the number

of measurements taken is exactly the number needed to determine the efficiency, any errors in

the measurements will propagate directly into the efficiency estimation.

Here we present a technique for measuring the absolute quantum efficiency of a detector

based on the Klyshko method, but explicitly taking into account multiple photon events. This

improves the calibration accuracy and allows for calibration of a PNRD. This approach utilizes

the PNRD capability to measure the photon-numberdistribution of an optical mode. Using two

PNRDs the joint photon-number statistics between the two electromagnetic field modes, in-

cluding photon-number correlations and individual photon-number distributions, of the SPDC

source can be determined. For each element of the resulting joint photon statistics, one can

find a formula giving the detector efficiencies of the two PNRDs. For the zero- and one-click

elements, this reproducesthe Klyshko result. However,the increased dynamic range of PNRDs

allows the use of brighter sources that produce measurable rates in the higher photon-number

elements of the joint statistics. We then use optimization techniques to estimate the detector

efficiencies from the increased number of measurements. This added redundancy improves the

tolerance of our scheme to background light and statistical noise. We experimentally demon-

strate this efficiency estimation method with two time-multiplexed PNRDs [3].

We begin by introducing a general treatment of PNRDs and then use this as a basis for de-

scribing our generalized Klyshko method. This is followed by the description of an experiment

to test the efficiency estimation with PNRDs.

2.Photon-number-resolving detectors

Photon-number-resolving detectors (PNRDs) are a class of photodetectors that have a unique

response for every input photon-numberstate within their range. Ideally these responses can be

perfectly discriminated. However, the less than perfect efficiency of realistic detectors causes

these responses to overlap, and thus does not allow for direct photon-number discrimination.

Overlap of detectorresponses can also arise fromthe detectorelectronics (e.g. amplification)or

the underlyingdetectordesign.Despite this overlap,thelinearrelationshipbetweenthedetector

response and the input state allows for the reconstruction of the input photon statistics from the

measured outcome statistics. This linear relationship is encapsulated by

Pn= Tr?ˆ ρˆΠn

?,

(2)

where ˆ ρ is the input-state densitymatrix, Pnis the probabilityfor the nth measurementoutcome

andˆΠnis the associated positive operator-value measurement (POVM) operator. Consider the

matrices expressed in the photon-number basis. Since PNRDs do not contain an optical phase

reference, the off-diagonal elements ofˆΠnare zero, meaning the photon-number-resolvingde-

tection is insensitive to off-diagonal elements in ˆ ρ. It is thus useful to write the diagonal ele-

ments of ˆ ρ, the photon-number statistics, as a vector ? σ. Similarly, we write the outcome prob-

abilities {Pi} as a vector ? p. In the following, we truncate ? σ at photon number N −1, where N

is the number of detector outcomes, although this is not strictly necessary.

The POVM operators of a general PNRD can be modeled by dividing the detector perfor-

mance into two components: efficiency and detector design, described by matrices F and L

respectively. In the photon number basis, a POVM element can be written as [24]

ˆΠn=∑

m

[F·L(η)]n,m|m??m|,

(3)

where [F·L(η)]n,mcorrespondsto the probabiltyof detectingn out of m incidencephotons.As

pointedoutin the introduction,detectorefficiencyη can be modeledbya precedingopticalloss

Page 5

l of (1−η). In the context of a PNRD, loss causes the photon-number statistics to transform

according to ? σ → L(η)? σ, where

Li,j(η) =

??j

i

?ηi(1−η)j−i

if j ? i

otherwise.

0

(4)

These matrixelementstransformthe state by loweringphotonnumbersfrom j to i, representing

the loss of photons through a binomial process with probability l = (1−η).

Although the detector-design component of the model depends on the detailed functioning

of the device, a large class of PNRDs – mode-multiplexers – can be treated in the same way.

These detectors divide an input optical field mode into many spatial and/or temporal modes

and then use single-photon detection on each mode to achieve number resolution. Examples

include nanowire superconducting detectors [9], VLPCs [11], intensified charge-coupled de-

vices (CCDs) [22], integrated APD arrays [4, 23], and time-multiplexed detectors (TMD). All

of these detectors suffer from detector saturation; the one-photon detector response occurs if

two or more photons occupy the same mode. This saturation effect is modelled by a detector

design matrix F = C, the “convolution” matrix, which has a general but complicated analytic

form given in [24] (in the context of the TMD). The form of C depends on relatively few pa-

rameters comprising the splitting ratios of the input mode into each of the multiplexed modes

and the total number of these modes.

As an example, in a CCD array detector the pixel shape and size defines the detected optical

mode. The spatial overlap of the detector mode of each pixel with the incoming optical mode

then gives the corresponding splitting ratio. The total number of pixels is the total number of

multiplexed modes. As the number of multiplexed modes goes to infinity the C matrix goes to

the identity. All PNR detectors can be described by a POVM, which can be reconstructed by

detector tomography [24, 25]. For PNR detectors that do not rely on mode multiplexing, such

as transition-edge superconducting detectors, and single APD detectors, one must factor the

POVM elements into an F matrix and an L matrix to apply the calibration procedure described

in this paper.

Focusing on the specific case of mode-multiplexeddetectors, one can rewrite the Eq. (3) as,

? p = C·L(η)·? σ,

(5)

where the elements of the ith row of C·L(η) are the diagonals of the ith operatorin the POVM

set?ˆΠi

number statistics ? σ1and ? σ2, but also the joint photon-numberdistribution of these two beams.

This distribution is written as the joint photon statistics matrix σ, where σm,nis the probability

of simultaneously having m photons in mode 1 and n photons in mode 2. We extend Eq. (5) to

relate the probability Pm,n, of getting outcome m at detector 1 and outcome n at detector 2, to

the joint photon statistics σ,

P = C1·L(η1)·σ ·LT(η2)·CT

where subscripts indicate the relevant detector and ATis the transpose of A. Joint photonstatis-

tics are a measureofphoton-numbercorrelationsinbeams 1 and2,and arethus sensitiveto loss

as discussed in the introduction. We use this description of PNRDs to generalize the Klyshko

calibration.

?.

With PNRDs intwo beams,denoted1 and2,onecanmeasurenotonlythe individualphoton-

2,

(6)

3.Generalizing the Klyshko method

The assumption that only a single photon pair is generated at a time in the Klyshko efficiency-

estimation scheme is only valid for very low SPDC pump powers. We can determine in what

Page 6

manner the Klyshko scheme breaks downwhen more than one pair is produced.Simply assum-

ing an additional contribution of pairs σ2,2to the pair rate σ1,1changes the efficiency estimate

of Eq. (1) to

˜ Rc

˜Ri

σ1,1+(2−ηi)σ2,2

This overestimatestheefficiencyforσ2,2?=0.Similarly,sensitivitytothe detailedphotonstatis-

tics of the input states acts to degrade the accuracy of the efficiency estimate. Let us consider

the effect of background light (e.g. fluorescence from the optical elements and detector dark

counts) on the efficiency estimation of Eq. (1). Background photons are uncorrelated between

the signal and idler beams, increasing the singles rate, and thus increasing Ri. This will make

the estimated efficiency of the detector ˜ ηslower than the actual efficiency ηs.

To generalize the Klyshko scheme to PNRDs and avoid the above limitations we need to

determine the true state generatedby SPDC photon-pairsources. Ideally,these sources produce

a ‘two-mode vacuum squeezed state’

˜ ηs=

=σ1,1+(ηsηi−2ηs−2ηi+4)σ2,2

ηs.

(7)

|Ψ? =

?

1−|λ|2∑

n

λn|n?s|n?i,

(8)

where λ is proportional to the pump beam energy, and |n?s(i)is an n-photon state of the signal

(idler) mode. Having only one free parameter, it is tempting to use this state in the generalized

efficiencyestimation.However,this state canonlybe generatedwith carefulsourcedesign[26].

Instead, photons are typically generated in many spectral and spatial modes in the signal and

idler beams [27]. Dependingon the numberof modes in the beams, the thermal photon-number

distribution in Eq. (8) changes continuously to a Poisson distribution [28]. Consequently, it

would be incorrect to assume the source produces an ideal two-mode vacuum squeezed state.

Still, the number of photons remains perfectly correlated between the two beams. Without

access to the number of generated modes we can make only the following assumption about

the joint photon statistics of the source

σm,n= cm·δm,n,

(9)

where {ci} are arbitrary up to a normalization constant and δm,nis the Kronecker delta.

Insertingthejointphotonstatistics definedbyEq.(9)intoEq.(6)wearriveatthebasisforour

generalized Klyshko method. Since C1and C2of the detectors are known the predicted joint

outcome probabilities P are highly constrained having N2elements uniquely defined by the N

parameters in {ci} and the two efficiencies η1and η2. Consequently, a measurement of the

joint outcome statistics specifies η1and η2with a large amount of redundancy; the efficiencies

are overdetermined. In order to correctly incorporate all measured outcome statistics into the

efficiency estimates a numerical optimization approach is used. We minimize the difference G

between the measured outcome statistics R and the predicted outcome statistics P, which are

determined by {ci},η1, and η2

G = R−C1·L(η1)·σ ·LT(η2)·CT

2.

(10)

This is done by minimizing the Frobenius norm F =

and η2– our estimates of the PNRD efficiencies. Using the Frobenius norm makes this a least-

squares optimizationproblem over {ci},η1and η2, where 0≤ηi≤1. However,this methodof

efficiency estimation is amenable to other optimization techniques such as maximum-entropy

or maximum-likelihood estimation. We use the Matlab® function ‘lsqnonneg’ (with the con-

straints σm,m≥ 0).

?

Tr

?

(G)2??1/2

to find the optimal η1

Page 7

1.0 0.80.6 0.40.2 0.0

1.0

0.8

0.6

0.4

0.2

0.0

0

5

10

15

20

25

η2, Detector efficiency

η1, Detector efficiency

Optimization residual (x 10-3)

Fig. 1. A typical optimization residual (F) for simulated joint outcome statistics as a func-

tion of the two PNRD efficiencies η1and η2. There is only one minimum, suggesting that

the problem is convex.

Thisefficiencyestimationis similartotheoptimizationsperformedinstate orprocesstomog-

raphy, which are known to be convex problems (i.e. there are no local minima) [29]. However,

we are now estimating parameters in both our state ˆ ρ and the POVM set?ˆΠi

single minimum. Using C1and C2for the PNRDs in our experiment(TMDs), with several sets

of photon statistics {ci}, and a range of efficiencies η1and η2, we simulated various outcome

statistics. In all cases, the optimization reproduced the correct efficiencies and only a single

minimum was observed. Figure 1 shows the result of a typical simulation displaying a global

minimum in the optimization residual (the minimum value of F for a given pair of efficiencies

η1and η2). This is a good indication that efficiency estimation is a convex problem.

?. Because this

is not guaranteed to be convex [29], we simulated a variety of measured statistics to test for a

4. Experimental setup

To experimentallydemonstrate and test our efficiency estimation method, time-multiplexedde-

tectors (one possible realization of a PNRD) were employed. In a TMD, the input optical state

is containedin a pulsed wavepacket mode.The pulse is split into two spatial and several tempo-

ral modes by a network of fiber beam splitters and then registered using two APDs [3]. APDs

produce largely the same response for one or more incident photons. The TMD overcomes this

binary response by making it likely that photons in the input pulse separate into distinct modes

and are thus individually registered by the APDs [3]. The TMD is a well-developed technol-

ogy, which makes this an ideal detector to test our approach to detector efficiency estimation.

The convolution matrix C for this detection scheme is calculated from a classical model of the

detector using the fiber splitting ratios [3], and is also reconstructed using detector tomography

Page 8

BBO

DM

RFDM

HWP

PBS

FPGA

electronics

TMD

2

SMF

KDP

SMF

BF

Coax

TMD

1

SHG

SPDC

PD

Coax

Pulsed?Laser

Fig. 2. The experimental setup. A KDP crystal two-beam source is pumped by an amplified

Ti:Sapph laser. The two generated beams propagate collinearly and are orthogonally po-

larized. Each beam is measured by a time-multiplexed photon number resolving detector.

Details are given in the text.

[24, 25]. Loss effects in TMDs have also been thoroughly investigated [30]. In our experi-

ments, the TMDs have four time bins in each of two spatial modes, giving resolution of up to

eight photons, with a possible input pulse repetition rate of up to 1 MHz. Field-Programmable-

Gate-Array (FPGA) electronics are used to time gate the APD signals with a window of 4 ns,

which significantly cuts background rates. The joint count statistics R are accumulated by the

electronics and transferred to a computer for data analysis.

The experimental setup is centered on a nearly-two-mode SPDC source [26] as depicted

in Fig. 2. The twin beam state produced by SPDC in a potassium dihydrogen phosphate

(KDP) nonlinear crystal consists of two collinear beams with orthogonal polarizations. The

pulsed pump (415 nm central wavelength) driving the SPDC is a frequency-doubledamplified

Ti:Sapphire laser operating with a 250 kHz repetition rate. A pick-off beam is sent to a fast

photodiode (PD) that is used to trigger the detection electronics. Dichroic mirrors (DM) and a

red-pass color glass filter (RF) are used to separate the blue pump from the near-infrared (830

nm central wavelength) SPDC light. A polarizing beam splitter (PBS) is used to separate and

direct the two co-propagating downconversion beams into separate single-mode fibers (SMFs)

connected to the time-multiplexed detectors (TMD1 and TMD2). The joint statistics R of the

two TMDs are recorded for a range of pump powers between 1 and 55 mW in order to estimate

the two TMD efficiencies at each power. To examine the spectral response of the detector effi-

ciency, one could tune the wavelength of the SPDC source by adjusting the pump wavelength

and the crystal orientation.

Page 9

12

0 10 20 3040 5060

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Pump power, (mW)

η, Detector efficiency

3

Fig. 3. The estimated detector efficiencies for TMD1(?) and TMD2(?) as a function of

the average SHG power pumping the SPDC. Three distinct regimes (regions 1, 2, and 3)

are indicated by shading.

5. Estimated efficiencies

For each pump power we determine the optimum detector efficiencies that are consistent with

the measured statistics at both PNRDs, shown in Fig. 3. Three different regimes are observed.

At low powers (up to 6mW) the estimated efficiency increases with power. Between powers of

6 and 40mW the estimates appear constant. At 40 mW there is a sudden jump in the estimated

efficiency (approximately twice the previous value); above this power the estimates remain

constant. Continued investigation revealed that the second-harmonic generation (SHG) pro-

cess qualitatively changed its behavior at 40mW: the increased pump power induced unwanted

higher-order nonlinear effects, resulting in the generation of additional frequency components

other than the second harmonic and a change in the spatial mode structure. This changed both

the transmission of the short wave pass filter (DM and BF) and the efficiency of the fiber cou-

pling into our detectors. Since this behavior is outside the scope of our investigation we omit

this data from further discussion.

The TMD efficiency should be independent of the average photon number of the state and

thus the pump power. This is true in the second regionof Fig. 3 but not the first. By reconstruct-

ing the joint photon-number distribution of the input state (σm,n) using the estimated efficien-

cies, one can gain insight into the estimation accuracy of the detector efficiency. This serves

as a partial check for our assumption that the number of photons in the two beams is equal.

In Fig. 4, we show the reconstructed joint photon statistics for two pump powers in regimes 1

and 2. In the second regime, the photon-number distribution is largely diagonal: only 10% of

the incident photons arrive without a partner in the other beam. In contrast, the state in the low

power regime has significant off-diagonalcomponents,with 43% of the photons arrivingalone.

This suggests that at low powers the reference state is corruptedby backgroundphotons, possi-

Page 10

0

1

2

3

4

0

1

2

n2

3

4

0

0.2

0.4

0.6

0.8

1.0

n1

Probability

0

1

2

3

4

0

1

2

3

4

0

0.05

0.10

0.15

0.20

0.25

n1

n2

Probability

a)b)

Fig. 4. The reconstructed photon statistics σ for n1and n2photons in beam 1 and 2, respec-

tively, measured at a pump power of (a) 1.5 mW and (b) 30 mW. The presence of significant

off-diagonal elements indicate that the input state is corrupted by background.

bly fluorescence from optics in the pump beam path, pump photons leaking through our filters,

or scattered pump photons penetrating our fiber coatings. Contributions from dark counts are

expected to be negligible, since the specified dark count rates of our detectors are significantly

lower than these other effects. In the next section we attempt to remove this background in

order to better estimate the efficiency.

6.Compensation for background light

We investigate two methods of dealing with background light. In the first, a parameterized

background contribution is incorporated into the efficiency estimation procedure. The second

attempts to measure the outcome distributions due to the background alone and then subtract

these from the efficiency data.

As pointed out in Section 3, the efficiency estimation is greatly overdetermined. One ex-

pects that a modeled background could be added to our input state model, Eq. (9), without

jeopardizing the convergence of the optimization. This background would be entirely uncor-

related, possibly making its contribution to the joint outcome statistics easily distinguishable

from the SPDC contribution.We model the photon-numberstatistics of the backgroundin each

beam by a Poisson photon-number distribution d(n) = αnexp(−α)/n! [31], which is fixed by

a single parameter – the average background photon number, α = ?nB?. Consequently, only

two additional parameters (one for each beam) enter into the efficiency estimation, keeping

it overdetermined. Unfortunately, we theoretically found that the loss in one beam transforms

the outcome statistics in a manner similar to background in the other beam. Thus, the prob-

lem is no longer convex; there is a set of equally optimal points {(?nB?,η)}. To show this we

compare two different two-mode number-correlatedstates σA(B), similar to the state in Eq. (9),

that undergo the addition of uncorrelated background light and loss respectively. The addition

of background to beam 1 of state σAis given by the convolution of the Poisson distribution

with input state, σA(defined by arbitrary?cA?). The elements of the background-added joint

the Poisson background, d, to the first beam of the initial state, σA, that add up to a particular

photon-number statistics ˜ σAare given by the sum of all possible ways to add photons from

Page 11

number of photons in the final state, ˜ σA

˜ σA

k,l=

k

∑

m=0

d(m)·σA

k−m,l.

(11)

This can be written in terms of a matrix product

˜ σA= D(?nB?)·σA,

(12)

where the elements of D(?nB?), Dm,n= d(n−m) are the probabilities of having an additional

n−m photons from the Poissonian background. To compare this background-addedcase with

the situation in which there is no background, but there is loss, we assume a loss l = (1−η)

in beam 2 of the second state σB(defined by arbitrary?cB?). We then attempt to show that for

equal, that is

D(?nB?)·σA= σB·LT(η).

Elimination of?cB?and?cA?from the resulting equations is facilitated using a computer al-

that it is not possible to simultaneously fit for background and efficiency. This emphasizes the

factthat onecannotdistinguishbetweenloss andPoissonbackgroundforthenumber-correlated

states. Figure 5 shows l as a function of ?nB? for PNRDs with a maximum photon number of

M = 1 to 20. The standard Klyshko case corresponds to M = 1. Note that as M becomes large

some σAand σB(each with perfectly correlated photon statistics) the two resulting states are

(13)

gebra program,and we find that there is a range of l and ?nB? ≤1 that solve Eq. (13), indicating

0.10.20.30.40.50.6 0.70.80.91.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

<n>,Average background photon number

l, Loss

Maximum PNR

photon number er

M = 1

M = 2

3

20

4

0

Fig. 5. The loss l in beam 1 of a twin-beam state σBthat results in the same joint outcome

statistics as the addition of a Poissonian background, with average photon number ?nB?,

to beam 2 of another twin-beam state, σA, for some σAand σB. M is the photon number

range of PNRDs in beams 1 and 2.

Page 12

the loss curve converges to line of slope 0.068 suggesting that even PNRDs with an infinite

photon number range would not allow one to distinguish loss and background. This suggests

that another approach to backgroundlight should be used.

Another approach to address the background contribution to the efficiency estimation at-

tempts to subtract an independently measured background contribution from the click statis-

tics. For each pump power the pump polarization is rotated by 90 degrees, extinguishing the

SPDC and allowing the measurement of the joint outcome statistics due to background alone.

Generally, the statistics of two concurrent but independent processes is the convolution of the

statistics of the two processes. However, the measured outcome probabilities PMof both light

sources combined is not a simple convolution of the background outcome probabilities PBand

the twin-beam outcome probabilities PS

PM?= PS∗PB.

(14)

0

1

2

3

4

0

1

2

3

4

0

0.2

0.4

0.6

0.8

1.0

n1

n2

Probability

0

1

2

3

4

0

1

2

3

4

0

0.1

0.2

0.3

n1

n2

Probability

a)b)

Fig. 6. The reconstructed photon statistics σ for n1and n2photons in beam 1 and 2, re-

spectively, after subtracting an independently measured background. For pump powers (a)

1.5 mW and (b) 30 mW, as in Fig. 4. The reduction of significant off-diagonal elements

indicates that the background subtraction method works.

This is due to the fact that there is an effective interaction between the background and

twin-beam signal detection events due to the strong detection nonlinearity of the APDs (i.e.

the detectors saturate at one photon). However this saturation effect can be eliminated by first

applying the inverse of the C matrices to the measured statistics

P′

P′

M= C−1

B= C−1

1PM(CT

1PB(CT

2)−1,

2)−1,

(15)

(16)

which gives the estimated photon-numberstatistics prior to the mode multiplexing. These non-

mode-multiplexedstatistics can be assumed independent so that,

P′

M=?C−1

1PS(CT

2)−1?∗P′

B.

We then use the convolution theorem to find

PS = C1F−1

?F{P′

M}

B}

F{P′

?

CT

2,

(17)

Page 13

where F indicates the Fourier transform, and the matrix division is element by element. Using

PSwe estimate the efficiency as previously by the method in Section 3.

To test the accuracy of this backgroundsubtraction method we reconstruct the joint statistics

at the same powers as in Fig. 3. We find that in both cases the off-diagonal components are

significantly reduced. In the low power regime, now only 16% of incident photons are not

part of a pair, and at higher power this becomes just 4%. With the background subtracted, the

estimated efficiencies are plotted in Fig. 7 and are now in better agreement with the expected

constant detector efficiency through the first two regions. However, the estimates still drop off

as the power goes very low. Although we measure the background, we do not do so in situ; we

need to change the apparatus by rotating a half wave plate (HWP). Thus, we are not guaranteed

that this is equalto the backgroundpresentduringtheefficiencyestimation.Furthermore,errors

in background measurements are more significant at low powers as background then forms a

larger component of the outcome statistics.

Also plotted in Fig. 7 is the Klyshko efficiency.In contrast to the increased dynamic range of

our method the standard Klyshko efficiency increases with pump power, evidence that higher

photon numbers in the input beams distort the estimated efficiency.

The average efficiency across the second region was found to be 9.4%±0.4% for detector 1

and 8.0%±0.4% for detector 2, where the errors are the standard deviations. These relatively

0 102030 40

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Pump power (mW)

η, Detector efficiency

Fig. 7. The estimated detector efficiencies for TMD1(?) and TMD2(?) determined from

background subtracted outcome statistics, plotted as a function of the average pump power.

Also plotted is the Klyshko efficiencies that would have been estimated for single photon

detector 1 (♦) and 2 (?). The standard Klyshko method overestimates the efficiencies for

high powers. The dotted lines indicate the average efficiencies of the two PNRDs.

low efficiencies are only partly due to the quantum efficiency of the avalanche photodiodes

themselves, which is specified to be 60%±5% at our wavelength. Bulk crystal SPDC sources

ordinarily emit into many spatial modes, which makes coupling into a single-mode fiber diffi-

cult and inefficient. Typical coupling efficiencies are less than 30% [32].

Page 14

0 5 10 15202530 3540

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Power (mW)

n , Reconstructed

< >

mean photon number

Fig. 8. The reconstructed average photon number (+) as a function of the pump power.

SPDC theory predicts a linear relationship. The line is a linear fit.

As a final check of the reconstructed photon statistics, we plotted the average reconstructed

photon number as a function of pump power in Fig. 8. The relationship is linear, as expected

when taking into account the higher dynamic range of a TMD in comparison to standard APDs

[33].

7.Conclusion

Despite being one of the seven base SI physical quantities, the working standards for luminous

intensity have a relative accuracy of only 0.5% [34], compared to 10−12and better for the sec-

ond [35]. Relying on a light beam of a known intensity, the efficiency calibration of detectors

is similarly limited. Quantum states of light give us the opportunityto bypass the working stan-

dards and calibrate detectors directly. Indeed, the photon has been suggested as an alternative

to the candela as the definition of luminous intensity [36]. We use twin-beam states and their

perfect photon-numbercorrelationsto measurethe efficiencyof a photon-number-resolvingde-

tector. This type of state can be produced at a wide range of wavelengths from many different

types of source including nonlinear crystals, optical fibers, periodically poled waveguides, and

atomic gases. This method has the advantage that it only assumes perfect photon-number cor-

relation and does not assume the state has a specific photon-number distribution, nor even that

it is pure. Despite these seemingly detrimental assumptions,the efficiency estimation presented

has a large amount of redundancyleading to a relatively small absolute error of 0.4%. We show

that this measurement is independent of the average photon number of the state, unlike the

Klyshko method, making it more widely applicable. In particular, it is ideal for characterizing

photon-numberresolving detectors, and for use with bright reference states. PNR detectors are

Page 15

undergoing rapid development via a number of competing technologies. These detectors will

play an important role in precision optical measurements and optical quantuminformationpro-

tocols where photon-number resolution is necessary for large algorithms. Future development

of direct efficiency calibration should focus on the issue of background, which can corrupt the

state and thus the calibration.

Acknowledgments

The disclosure in this paper is the subject of a UK patent application (Ref: 3960/rr). Please

contact Isis Innovation,the Technology Transfer arm of the University of Oxford, for licensing

enquires (innovation@isis.ox.ac.uk).

This work has been supported by the European Commission under the Integrated Project

Qubit Applications (QAP) fundedby the IST directorate Contract Number 015848,the EPSRC

grant EP/C546237/1, the QIP-IRC project and the Royal Society. HCR has been supported by

the European Commission under the Marie Curie Program and by the Heinz-Durr Stipendien-

progamm of the Studienstiftung des deutschen Volkes.

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