Towards photostatistics from photon-number - The Adami Lab

12 Pages·2004·0.13 MB·English

journal of modern optics, 15 june–10 july 2004
vol. 51, no. 9–10, 1517–1528
Towards photostatistics from photon-number discriminating
detectors
HWANG LEEy, ULVI YURTSEVERy, PIETER KOKyz,
GEORGE M. HOCKNEYy, CHRISTOPH ADAMIy,
SAMUEL L. BRAUNSTEIN} and JONATHAN P. DOWLINGy
yQuantum Computing Technologies Group, Section 367,
Jet Propulsion Laboratory, California Institute of Technology,
MS 126-347, 4800 Oak Grove Drive, CA 91109-8099, USA
zHewlett Packard Laboratories, Bristol BS34 8QZ, UK
}Computer Science, University of York, York YO10 5DD, UK
(Received27October 2003)
Abstract. We study the properties of a photodetector that has a number-
resolving capability. In the absence of dark counts, due to its finite quantum
efficiency,photodetectionwithsuchadetectorcanonlyeliminatethepossibility
thattheincidentfieldcorrespondstoanumberofphotonslessthanthedetected
photon number. We show that such a non-photon number-discriminating
detector, however, provides a useful tool in the reconstruction of the photon
number distribution of theincident fieldeven inthepresence ofdark counts.
1. Introduction
With the recent advent of linear optical quantum computation (LOQC),
interest in photon number-resolving detectors has been growing. The ability
to discriminate the number of incoming photons plays an essential role in the
realizationofnonlinearquantumgatesinLOQC[1–4]aswellasinquantumstate
preparation[5–9].Suchdetectorsareusedtopost-selectparticularquantumstates
of a superposition and consequently produce the desired nonlinear interactions of
LOQC.Inaddition,themostprobingattacksaneavesdroppercanlaunchagainsta
typicalquantumcryptographysystemexploitphoton-numberresolvingcapability
[10, 11]. Recent efforts in the development of such photon number-resolving
detectors include the visible light photon counter [12, 13], fibre-loop detectors
[14–18], and superconducting transition edge sensors [19]. Standard photodetec-
tors can measure only the presence or absence of light (single-photon sensitivity),
andgenerallydonothavethecapabilityofdiscriminatingthenumberofincoming
photons (single-photon number resolution). There have been suggestions for
accomplishingsingle-photonresolutionusingmanysingle-photonsensitivedetec-
tors arranged in a detector array or detector cascade [20–23]. For example, the
VLPC (visible light photon counter) is based on a confined avalanche breakdown
in a small portion of the total detection area and, hence, can be modelled as a
detector cascade [24]. Then again, fibre-loop detectors may be regarded as a
detector cascade in the time domain.
JournalofModernOpticsISSN0950–0340print/ISSN1362–3044online#2004Taylor&FrancisLtd
http://www.tandf.co.uk/journals
DOI:10.1080/09500340410001670901
1518 H. Lee et al.
Let us suppose a number-resolving detector detects two photons in a given
time interval. If the quantum efficiency of the detector is one, we can be certain
that two photons came from the incident light during that time interval. If the
quantum efficiency is, say, 0.2, what can we say about the incident light? In the
absenceofdarkcounts,wecansafelysayonlythattherewereatleasttwophotons
intheincidentbeam.Inthiscase,photodetectionratherhasanon-photonnumber
discriminating feature, since the conditional probability of not having zero, or one
photonintheincidentpulseiszero.Canwesaymorethanthat?Forexample,what
is the probability that the incident pulse actually corresponds to two photons, or
three?Inthispaperweattempttoanswerthesequestionsanddiscusstheeffectof
quantum efficiency on the photon-number resolving capability.
The semiclassical treatment of photon counting statistics was first derived by
Mandel [25]. He showed that the photon counting distribution is Poissonian if
the intensity of the incident light beam is constant in time. The full quantum
mechanical description has been formulated by Kelley and Kleiner [26], and
Glauber [27]. Here, we start from a formula based on the photon-number state
representation, which was first derived by Scully and Lamb [28]. For a given
quantum efficiency of the photodetector, the photon counting distribution can be
written as
X1 (cid:1)n(cid:2)
PðkÞ¼ (cid:1)kð1(cid:1)(cid:1)Þn(cid:1)kSðnÞ; ð1Þ
k
n¼k
where P(k) is the probability of detecting k photons, (cid:1) is the quantum efficiency
and S(n) is the probability that the source (incident light) corresponded to n
photons. It is assumed that the quantum efficiency (cid:1) does not depend on the
intensityoflight.ThisdistributionthenleadstoanaturaldefinitionofP(k|n),i.e.
as the conditional probability of detecting k photons given n photons being in the
source, as
(cid:1)n(cid:2)
PðkjnÞ¼ (cid:1)kð1(cid:1)(cid:1)Þn(cid:1)k: ð2Þ
k
Contrariwise, suppose we detect k photons. If the detector has a quantum
efficiency much less than unity, very little can be said about the actual photon
number associated with the source. We may, however, deduce that the source
actually corresponded to n photons with a certain probability. This probability
should be high if the quantum efficiency is high. In the following, we will use
Bayes’s theorem in order to extract information about the source from the
conditional photon-counting statistics. Armed with the conditional probability
P(k|n), we may write
PðkjnÞSðnÞ PðkjnÞSðnÞ
QðnjkÞ¼ ¼ : ð3Þ
P
PðkÞ PðkjiÞSðiÞ
i
Now we will identify the conditional probability, Q(n|k), with the probability
that an ideal detector would have detected n photons, given that k photons are
detected by the imperfect detector. It is given as the product of the probability of
thesourcecorrespondingtonphotonsbeforethemeasurementandtheprobability
of k-photon detection, given that n photons are in the source, normalized to the
probability of having k photons detected.
Towards photostatistics from photon-number discriminating detectors 1519
perfect detector
mode 1
mode 2
imperfect detector
Figure 1. Schematic ofthe heralded-photon set-up.
It is evident that generally we cannot find Q(n|k) if we do not know S(n)—the
distribution of the source. The exceptions are: (i) if n<k, then PðkjnÞ¼0.
Therefore, QðnjkÞ¼0 for n<k no matter what S(n) is. (ii) If the quantum
efficiency is unity, that is (cid:1)¼1, then PðkjnÞ¼0 when n6¼k, and Pðkjn¼kÞ¼1.
Since PðkÞ¼Sðn¼kÞ, it leads to Qðn¼kjkÞ¼1 and Qðn6¼kjkÞ¼0. Hence,
without knowing S(n), we only have information about Q(n|k) when n<k or
(cid:1)¼1. Such a detector, then, readily excludes certain numbers of photons in the
source rather than determining them. Without a prior knowledge of the source, if
(cid:1)6¼1, we call this a non-photon number-discriminating detector.
Obviously,thedifficultyintalkingaboutthisconditionalprobabilityliesinthe
fact that the photodetection is destructive. We will elaborate this subtlety using
aheralded-photonset-updepictedinfigure1.TheunitaryoperatorUU^ transforms
H
the input state in the following way:
1 1
X X
UU^ (cid:2) jni¼ (cid:2) jni jni : ð4Þ
H n n 1 2
n¼0 n¼0
Suppose now the photon number in mode 1 is measured by the perfect detector
andthephotonnumberinmode2ismeasuredbytheimperfectdetector.Itisnow
simple to understand the conditional probability of finding n photons in mode 1,
given that k photons are detected in mode 2. In this way, we can have an
operational definition of the conditional probability Q(n|k). It is easy to show
that the conditional probability of finding n photons in mode 1 given that k
photons are detected in mode 2 is given by precisely the same expression for the
quantity Qðn;kÞ as equation (3). (Note that the transformation given by UU^ is far
H
fromquantumcloning,sincetheresultofUisfarfrombeingequaltothecloning
transformation ðP1 (cid:2) jni Þ(cid:2)ðP1 (cid:2) jni Þ.)
n¼0 n 1 n¼0 n 2
2. POVM description
Suppose an initial quantum state j i of incident light is represented in the
in
photon number states as
1
X
j i¼ (cid:2) jni: ð5Þ
in n
n¼0
1520 H. Lee et al.
The probability of finding n photons is then given by SðnÞ¼jhnj ij2 ¼j(cid:2) j2.
n
We may describe the detection process of the perfect detector by entangling the
source with the detector ancilla j0i
X
j i¼UU^j ij0i¼ (cid:2) jni ja i ; ð6Þ
f in n Q n M
n
where
X 1
UU^ ¼ jjihjj(cid:2) ðaa^yÞj ð7Þ
ðj!Þ1=2
j
is the unitary operator implementing the von Neumann measurement, and
we identified the quantum (Q) and measurement device (M) Hilbert spaces with
appropriate subscripts.
The probability to detect k photons, P(k), can now be calculated as
PðkÞ¼Tr ðja i ha jj ih jÞ¼j(cid:2) j2; ð8Þ
Q k M k f fi k
as expected. Of course, the same result would have been obtained as
PðkÞ¼TrðM (cid:3) Þ using a positive operator valued operator measure (POVM)
k in
M ¼jkihkj: ð9Þ
k
The present unitary treatment, however, is more transparent and allows a
straightforward extension to imperfect detectors.
For an imperfect photodetector with a finite quantum efficiency, the POVM
can be written as
1
X
M ¼ PðkjnÞja iha j; ð10Þ
k n n
n¼k
with P(k|n) given by equation (2). We can verify this by calculating as before
j i¼Uj ij0i j0i ; ð11Þ
f in M E
except that the entanglement operator U now acts on the joint Hilbert space of
the quantum system, a measurement device and an environment j0i , which is
E
where our lost photons go. Thus, the unitary operator can be given by
UU^ ¼Xjjihjj(cid:2) 1 (cid:1)(cid:1)1=2aa^y(cid:2)1þexpði’Þð1(cid:1)(cid:1)Þ1=21(cid:2)bb^y(cid:2)j;
ðj!Þ1=2
j
where’isanarbitraryphaseandby isthecreationoperatorfortherelevantmode
of the environment. Similar to equation (8), we now have
(cid:3) (cid:4)
PðkÞ¼Tr ja i ha jj ih j
QE k M k fi fi
X1 (cid:1)n(cid:2)
¼ j(cid:2) j2 (cid:1)kð1(cid:1)(cid:1)Þn(cid:1)k: ð12Þ
n k
n¼k
supporting our equation (10) as the correct POVM. Note that any relative phase
information between the photon number states disappeared.
ThePOVMeffectivelytransferstheinitialquantumstateoflightintoacertain
quantum state (as a result of k-photon detection) by the transformation
(cid:3) (cid:3)M (cid:3) ; ð13Þ
f(cid:1)k k in
Towards photostatistics from photon-number discriminating detectors 1521
where (cid:3) ¼j ih j. The probability that k photons are detected can then also
in in in
be written as
(cid:3) (cid:4)
PðkÞ¼Tr (cid:3)
f(cid:1)k
¼TrðM j ih jÞ
k in in
X
¼ PðkjnÞSðnÞ: ð14Þ
n
On the other hand, the joint probability that the source state is jni and k photons
are detected can be expressed by the probability
Pðn;kÞ¼Tr ðhn;a j ih jn;a iÞ
E k f f k
¼hnj(cid:3) jni¼PðkjnÞSðnÞ; ð15Þ
f(cid:1)k
which defines the quantity Q(n|k) as
ha j(cid:3) ja i
QðnjkÞ(cid:3) n f(cid:1)k n : ð16Þ
(cid:3) (cid:4)
Tr (cid:3)
f(cid:1)k
In particular, the quantity Qðn¼kjkÞ corresponds to the confidence of the
state preparation in a situation that we have more than two modes for the input
and the photon number in one of the modes is measured [22].
3. An example
Letustakeasimpleexampleofaninputcoherentstateinordertoseehowthe
valueofQ(n|k)behaves.Indoingthis,weassumethatthedarkcountrateissmall
enough that we can ignore it, and the quantum efficiency is independent of the
number of incident photons.
A coherent state j(cid:2)i is expressed in terms of number state as
j(cid:2)i¼expð(cid:1)j(cid:2)j2=2ÞP ð(cid:2)n=ðn!Þ1=2Þjni [29]. The probability of finding n photons in
n
j(cid:2)i is then given by a Poisson distribution:
expð(cid:1)nn(cid:1)Þnn(cid:1)n
SðnÞ¼jhnj(cid:2)ij2 ¼ ; ð17Þ
n!
where nn(cid:1) ¼j(cid:2)j2. The probability of detection of k photons with an imperfect
detector is well known to be [28]
X1 (cid:1)n(cid:2) expð(cid:1)nn(cid:1)Þnn(cid:1)n
PðkÞ¼ (cid:1)kð1(cid:1)(cid:1)Þn(cid:1)k
k n!
n¼k
ðnn(cid:1)(cid:1)Þk
¼ expð(cid:1)nn(cid:1)(cid:1)Þ: ð18Þ
k!
That is, the photon counting statistics are simply a Poisson distribution where
the average number of photons detected is the average number of incident
photons multiplied by the quantum efficiency of the detector—given by kk(cid:1) ¼nn(cid:1)(cid:1)
as expected.
1522 H. Lee et al.
1
Q(n|1)
0.8
0.6
n=1
0.4
(a)
(b)
0.2
2 3 4
n
Figure 2. The conditional probability Q(n|k¼1) for a coherent state with nn(cid:1) ¼1 as a
functionof n.(a) Fordetector efficiency (cid:1)¼0.2,Qð1j1Þ¼0:45,Qð2j1Þ¼0:36.(b) For
detector efficiency(cid:1)¼0.9, Qð1j1Þ¼0:90,Qð2j1Þ¼0:09.
NowletusconsidertheconditionalprobabilityQ(n|k).Usingequation(3),itis
simply found as
(cid:1)n(cid:2)
(cid:1)kð1(cid:1)(cid:1)Þn(cid:1)k expð(cid:1)nn(cid:1)Þnn(cid:1)n=n!
k
QðnjkÞ¼
expð(cid:1)nn(cid:1)(cid:1)Þðnn(cid:1)(cid:1)Þk=k!
1
¼ ð1(cid:1)(cid:1)Þn(cid:1)k exp½(cid:1)nn(cid:1)ð1(cid:1)(cid:1)Þ(cid:4)nn(cid:1)n(cid:1)k
ðn(cid:1)kÞ!
¼expð(cid:1)(cid:1)llÞ(cid:1)lll; ð19Þ
l!
where l¼n(cid:1)k and (cid:1)ll¼nn(cid:1)ð1(cid:1)(cid:1)Þ. Therefore, the form of Q(n|k) is again a Poisson
distribution as a function of n(cid:1)k with the average number of photons that fail to
register given as nn(cid:1)ð1(cid:1)(cid:1)Þ.
Figure 2 shows the behaviour of Q(n|k) as a function of n, for the case of
coherent state input with nn(cid:1) ¼1, when one photon is detected. We can see that if
thequantumefficiencyofthedetectoris0.2,withthedetectionofonephoton,then
the chances are more than 50% that the input contains more than two photons
(Qð1j1Þ¼0:45). Obviously, Qð1j1Þ increases for higher values of the quantum
efficiency (Qð1j1Þ¼0:9 for (cid:1)¼0:9).
On the other hand, for a given quantum efficiency of the detector, Qð1j1Þ gets
large as the average number of photons in the input decreases. For nn(cid:1) ¼0:2,
Qð1j1Þ¼0:85 even if the detector efficiency is 0.2, as depicted in figure 3. In
this case, with the detection of one photon the chances are 85% that the detection
result corresponds to the correct value.
4. Reconstruction of photon-number distribution
So far we have seen that we cannot say much about Q(n|k) if we do not know
atleasttheform ofS(n).Inotherwords,using animperfectdetector,even aftera
detection of k photons, we do not gain much information about the number of
incident photons without a prior knowledge of the incident field. Hence, the
Towards photostatistics from photon-number discriminating detectors 1523
1
Q(n|1)
0.8
0.6
n=0.2
0.4
(a)
0.2
(b)
2 3 4
n
Figure 3. The conditional probability Q(n|k¼1) for a coherent state with nn(cid:1) ¼0:2
as a function of n. (a) For detector efficiency (cid:1)¼0:2, Qð1j1Þ¼0:85, Qð2j1Þ¼0:14. (b)
Fordetector efficiency (cid:1)¼0:9,Qð1j1Þ¼0:98,Qð2j1Þ¼0:02.
problem comes back to reconstruction of photon number distribution of the
incident field, S(n), from the photon counting probability, P(k) by a sequence of
identical inputs and measurements. For this purpose, we may write equation (1)
in a matrix form as
P¼PS; ð20Þ
where
0 1 0 1
Pð0Þ Sð0Þ
BPð1ÞC BSð1ÞC
B C B C
B C B C
P¼BPð2ÞC; S¼BSð2ÞC ð21Þ
B C B C
B (cid:5) C B (cid:5) C
@ A @ A
(cid:5) (cid:5)
and each component of the matrix P represents the conditional probability
P(k|n) as
(cid:1)n(cid:2)
P (cid:3)PðkjnÞ¼ (cid:1)kð1(cid:1)(cid:1)Þn(cid:1)k: ð22Þ
kn k
Then the inverse of P can be found explicitly; it is given by
(cid:1)m(cid:2)
ðP(cid:1)1Þ ¼ (cid:1)(cid:1)mð(cid:1)(cid:1)1Þm(cid:1)n; ð23Þ
nm n
which,in the absence of dark counts, determines the photon-number distribution
of the source from the photon-counting distribution. Note, however, that for a
fixed n there are alternating signs in ðP(cid:1)1Þ as the index m changes, so that this
mn
inversion formula can yield unphysical results depending on the probability
distribution P(k). Put another way, for small efficiencies the matrix P is ill-
conditioned and P(cid:1)1 will contain matrix elements that can become very large in
absolute value and can have either sign. This means that even if the calculated
S(n)’s are all non-negative, a good inversion is still not guaranteed, since the
inversion formula (23) is highly sensitive to small changes of P(k), amplifying the
statistical and other noise which is inevitably present in any actual measurement.
1524 H. Lee et al.
0.2
(a)
y
t0.15
i
l
i
b
a
b 0.1
o
r
P (b)
0.05
10 20 30 40
Number of photons
Figure 4. Reconstruction of the input-state photon-number distribution: (a) photon
counting distribution from the NIST experiment [19]; (b) reconstruction of the
inputstatephotondistribution(nn(cid:1) (cid:7)20:15),assumingaquantumefficiencyof0.2and
nodark counts.
ThefactthattheinversematrixP(cid:1)1 isalsoanuppertriangularmatriximpliesthat
S(n)isdeterminedbyP(k)withk(cid:6)n.Inotherwords,thetailofthemeasurement
result, P(k), where the statistical noise can be expected to be greatest, affects
the inversion significantly. The validity conditions for the inversion formula shall
be discussed elsewhere.
Instead, we can carry out the following recipe. (1) Seek an underlying distri-
bution S for which PS matches the observed frequencies according to a (cid:4)2
goodness-of-fit test. (2) Then, of all such distributions, seek the one with maxi-
mum entropy; i.e. pick the most likely underlying distribution, which is statisti-
callyconsistentwiththedata.Wenotethatentropyisthelogarithmofthenumber
of underlying configurations corresponding to a given distribution. Thus, such a
strategy seeks the ‘most-likely’ distribution satisfying the constraint that, when
filtered by imperfect detection, the resulting distribution is consistent within the
statisticalnoiseoftheactualdata[30].Thisapproachhasbeenusedformanyyears
in the imaging community for deconvolution [31]. Here we ensure a global
maximumentropydistributionisfoundbyimplementingthisstrategyasagenetic
algorithm, which we will discuss in a forthcoming paper [32]. Using this method,
the reconstruction of S with a data set from an actual experiment is depicted
in figure 4. The plot of figure 4, curve (a) shows the probability distribution
of the detected photons. The raw count data was normalized by the total number
of counts (data provided courtesy of NIST [19]). The result of reconstruc-
tion is shown by the plot of figure 4, curve (b) and yields a nn(cid:1) ¼20:15 for the
incidentcoherentstate—agoodagreementwiththedetectedexperimentalvalueof
nn(cid:1) ¼4:02 [19].
exp
5. Dark counts
In addition to the finite quantum efficiency, the dark counts further make
the results extracted from the experiment obscure. For a constant dark-count
rate, however, a generalization of the inversion may be handled analytically.
Towards photostatistics from photon-number discriminating detectors 1525
Suppose in general that the probability of d dark counts is D(d), a discrete
probability distribution, which satisfies the normalization condition
1
X
DðdÞ¼1: ð24Þ
d¼0
We assume that the distribution D(d) represents probabilities of dark counts in a
fixedintervaloftime(cid:5)(durationofasinglephoton-countingexperiment),andthat
these probabilities are independent of all other relevant observables such
as incoming photon number, number of detected photons, etc. Therefore in the
presence of dark counts, the conditional probability of detecting k photons, given
n photons in the incident field equation (2), is modified accordingly:
Xk (cid:1) n (cid:2)
P ðkjnÞ¼ DðdÞ (cid:1)k(cid:1)dð1(cid:1)(cid:1)Þn(cid:1)kþd; ð25Þ
D k(cid:1)d
d¼0
which expresses the probability P ðkjnÞ as a sum (over all possible d) of the
D
probability of registering k(cid:1)d photons from the incident field via true detection
while the remaining d are registered via dark-count events.
As long as the physical time scale for the emergence of a dark-count event is
much smaller than (cid:5) and the events arrive at a constant rate (cid:6), the probability
distributionD(d)(ofthearrivalofddark-counteventsinthefixedtimeinterval(cid:5))
has a Poissonian-distribution limit:
ð(cid:6)(cid:5)Þd
DðdÞ¼expð(cid:1)(cid:6)(cid:5)Þ ; ð26Þ
d!
where the average number of dark count is dd(cid:1) ¼(cid:6)(cid:5). Since after interchanging the
summation index d with k(cid:1)d we can rewrite equation (25) as
Xk (cid:1)n(cid:2)
P ðkjnÞ¼ Dðk(cid:1)dÞ (cid:1)dð1(cid:1)(cid:1)Þn(cid:1)d; ð27Þ
D d
d¼0
where the summation over d is in the form of a matrix multiplication (equation
(22)),
X
ðP Þ ¼ D P ; ð28Þ
D kn kd dn
d
or in matrix notation,
P ¼DP; ð29Þ
D
where ðP Þ (cid:3)P ðkjnÞ, and
D kn D
D (cid:3)Dðk(cid:1)dÞ
kd
ð(cid:6)(cid:5)Þk(cid:1)d
¼expð(cid:1)(cid:6)(cid:5)Þ : ð30Þ
ðk(cid:1)dÞ!
Throughout the paper, we adopt the convention that k!¼(cid:2)ðkþ1Þ(cid:3)1 for
(cid:1)a(cid:2)
negative integer k, and correspondingly (cid:3)0 whenever b>a. In equation
b
(30), this convention implies that D ¼0 for d>k.
kd
1526 H. Lee et al.
Now by inverting equation (27) in a matrix form, from the fundamental
equation,
X
PðkÞ¼ P ðkjnÞSðnÞ; ð31Þ
D
n
wecanreconstructthephotonnumberdistributionS(n)oftheincidentfieldgiven
thevectorofobservedphoton-countprobabilitiesP(k),whoserelationshiptoS(n)
(givenbyequation(31)),canagainbewritteninmatrixformasP¼P S,justasin
D
equations (20) and (21). The inverse of the coefficient matrix P ðkjnÞ¼ðP Þ is
D D kn
found from equation (29),
P(cid:1)1 ¼P(cid:1)1D(cid:1)1: ð32Þ
D
Let us first consider the inverse of the matrix D defined as in equation (30). It is
not difficult to prove that the inverse of D is given by
ð(cid:1)(cid:6)(cid:5)Þi(cid:1)j
ðD(cid:1)1Þ ¼expð(cid:6)(cid:5)Þ
ij ði(cid:1)jÞ!
ð(cid:6)(cid:5)Þi(cid:1)j
¼expð(cid:6)(cid:5)Þð(cid:1)1Þi(cid:1)j : ð33Þ
ði(cid:1)jÞ!
Combining equation (33) with equation (32) and equation (23), we find
X(cid:5)i(cid:6) ð(cid:1)(cid:6)(cid:5)Þi(cid:1)n
ðP (cid:1)1Þ ¼expð(cid:6)(cid:5)Þ (cid:1)(cid:1)ið(cid:1)(cid:1)1Þi(cid:1)k
D kn k ði(cid:1)nÞ!
i
ð(cid:1)1Þkþn expð(cid:6)(cid:5)ÞX1 (cid:8)(cid:5)i(cid:6)(cid:7) (cid:9)(cid:8)ð1(cid:1)(cid:1)Þ(cid:6)(cid:5)(cid:9)i
¼ ði(cid:1)nÞ! : ð34Þ
ð1(cid:1)(cid:1)Þkð(cid:6)(cid:5)Þn k (cid:1)
i¼0
The infinite power series over i in equation (34) is absolutely convergent for all
values of the argument
ð1(cid:1)(cid:1)Þ(cid:6)(cid:5)
z(cid:3) ; ð35Þ
(cid:1)
andcanbeexpressedintermsoftheconfluenthypergeometricfunction F ða;b;zÞ
1 1
(whichisanentirefunctionoftheargumentzforallparametervaluesa,b),giving
a compact expression for the inverse matrix elements in the form:
expð(cid:6)(cid:5)Þ
ðP (cid:1)1Þ ¼ð(cid:1)1Þkþn
D kn ð1(cid:1)(cid:1)Þkð(cid:6)(cid:5)Þn
8 n!
>><k!ðn(cid:1)kÞ!zn1F1ðnþ1;n(cid:1)kþ1;zÞ; if k(cid:9)n;
(cid:8) ð36Þ
1
>>:ðk(cid:1)nÞ!zk1F1ðkþ1;k(cid:1)nþ1;zÞ; if k>n:
The inversion of equation (32) can now be effected by substituting equations (35)
and (36) into the matrix formula
X
SðkÞ¼ ðP (cid:1)1Þ PðnÞ: ð37Þ
D kn
n
Thissuggeststhatthereconstructionofthephotonstatisticsofthesourcemaybe
possiblewithanygivenquantumefficiencyofthedetector,eveninthepresenceof