Quantum Zeno tomography

0.32 MB·English

by  P. Facchi

Quantum Zeno tomography
P. Facchi,1,2 Z. Hradil,3 G. Krenn,1 S. Pascazio,1,2 and J. Rˇeh´aˇcek3
1Dipartimento di Fisica, Universit`a di Bari I-70126 Bari, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy
3Department of Optics, Palack´y University, 17. listopadu 50, 772 00 Olomouc, Czech Republic
(Dated: February 1, 2008)
We show that the resolution “per absorbed particle” of standard absorption tomography can be
outperformed by a simple interferometric setup, provided that the different levels of “gray” in the
samplearenot uniformly distributed. Thetechniquehinges uponthequantumZenoeffect andhas
beentestedinnumericalsimulations. Theschemeweproposecouldbeimplementedinexperiments
with UV-light,neutronsor X-rays.
4 PACSnumbers: 03.65.Xp
0
0
2 I. INTRODUCTION preserving the resolution? We show that this is indeed
n possible. Closelyrelatedquestionshavebeenrecentlyin-
a vestigated by other authors [6, 7]. Our conclusions are
Absorption tomography is an important experimen-
J somewhat optimistic: we show that standard absorption
tal technique revealing the internal structure of mate-
2 tomography can be outperformed by a Zeno setup, pro-
rial bodies. By measuring the attenuation of a beam
1 vided that the frequency of occurrences of the different
of particles passing through a sample one infers the ab-
levels of “gray” in the sample is not uniform. More to
2 sorption coefficient (density) of the sample in the beam
this,theZenosetup,unlikethestandardone,isendowed
v section. Thepossibilityofdistinguishingtwoslightlydif-
withtwodetectionchannels: asweshallsee,thisfeature,
1 ferent densities of the material is often of vital impor-
2 if properly exploited, leads to even better performances
tance. Under otherwise ideal conditions the shot noise
0 in the Zeno case.
associatedwiththediscretecharacteroftheilluminating
4
beam sets an upper limit to the resolution of absorption
0
1 tomography: for instance, the shadow cast by a brain II. QUANTUM ZENO EFFECT IN A
0 tumor might become totally lost in the noisy data. One
MACH-ZEHNDER INTERFEROMETER
/ possibilitytoovercomethe fluctuationsistoincreasethe
h
intensity of the beam. However, in many situations, like
p Weintroducenotationandsketchthefundamentalfea-
- inmedicineforexample,theintensityoftheilluminating
nt beamcannotbemadearbitrarilyhighduetothedamage tZuerhensdoefrtihneteqrufearnotmumetrZicen(MoeZffIe)cstc.hWemeecwonitshidfeeredthbeacMkadcihs--
a provokedby the absorbed radiation.
playedinFig.1a. Asemitransparentobject,whosetrans-
u
q A significant step towards an “absorption-free tomog- mission amplitude is τ (assumed real for simplicity) is
: raphy”camefromquantumtheory. Itwasdemonstrated, placed in the lower arm of the interferometer. The par-
v both theoretically [1, 2] and experimentally [3], that to- ticle is initially injected from the left, crosses the inter-
Xi tally transmitting and absorbing bodies can be distin- ferometer L times and is finally detected by one of two
r guished without absorbing any particles, by using an detectors. The two semitransparent mirrors M are iden-
a interferometric setup. This idea is in fact a clever im- tical and their amplitude transmissionand reflection co-
plementation of the quantum Zeno effect [4] and hinges efficients are
upon the notion of “interaction-free” measurement [5].
Aclassicalmeasuringapparatus(heretheblacksample), c≡cosθ , s≡sinθ (θ =π/4L), (1)
L L L
placed in one arm of the interferometer, projects the il-
luminating particle into the other arm, destroying inter- respectively. Notice that both coefficients depend on L,
ference,freezingthe evolutionandforcingthe particle to the number of “loops” in the MZI.
exit through a different channel from that it would have The incoming state of the particle (coming from the
chosen had both arms been transparent (white sample). source at initial time) is
In practical applications, however, samples are nor-
1
mally neither black nor white: they are gray. In this |ini= (2)
0
paper we endeavorto understandwhether applicationof (cid:18) (cid:19)
the quantum Zeno effect, which turns out to be ideal and we call “Zeno” and “orthogonal” channels the ex-
for discriminating black and white, might be advanta- traordinary 1 and ordinary 0 channels of the MZI,
0 1
geous also for the more practical task of discriminating respectively. The total effect of the interferometer is
two gray bodies with different transmission coefficients. (cid:0) (cid:1) (cid:0) (cid:1)
Morespecifically,weask: isitpossiblebyquantumZeno c −s 1 0
V =BA B, B = , A = . (3)
effect to reduce the number of absorbed particles while τ τ s c τ 0 τ
(cid:18) (cid:19) (cid:18) (cid:19)
2
IntheinfiniteLlimittheinitialstateis“frozen”andthe
particle ends up in the Zeno channel.
C. Gray sample
What happens if 0<τ <1? We easily get
(1+τ)c2−τ −sc(1+τ)
V = . (10)
τ sc(1+τ) τ −(1+τ)s2
(cid:18) (cid:19)
FIG. 1: a) Scheme of the Zeno interferometric setup. b)
Standard transmission experiment. S – source; M – semi- The computation of VL is straightforward but lengthy
τ
transparent mirror; o– orthogonal channel; z–Zenochannel; and yields a final expression which is elementary but
D – detector. complicated. However, we are mainly interested in the
large-L limit, that for
In general, τ <τZ ≡(1−sinπ/2L)/(1+sinπ/2L) (11)
L
B =exp(−iθ σ ), BB† =B†B =1, (4)
L 2 reads [13]
where σ is the second Pauli matrix, while A is not
unitary (2if τ < 1 there is a probability loss). Tτhe final VL = 1− 8πL2 11−+ττ O(L−1) +O(L−2). (12)
state,aftertheparticlehasgonethroughLloops,reads τ O(L−1) τL[1+O(L−1)]
(cid:18) (cid:19)
|outi=VτL|ini=(BAτB)L|ini. (5) This is an interesting result: indeed
1 0
V ≡ lim VL = , 0≤τ <1 (13)
A. White sample τ L→∞ τ 0 0
(cid:18) (cid:19)
analogouslyto(8). Thisshowsthatevenforasemitrans-
The choice of the angle θ in (1) is motivated by our
L
parentobject,withtransmissioncoefficientτ 6=1,abona
requirement that if τ = 1 (“white” sample, i.e. no semi-
fide QZEtakesplaceandtheparticleendsupintheZeno
transparent object in the MZI) the particle ends up in
channel with probability one:
the “orthogonal”channel:
VτL=1 =B2L =e−i2LθLσ2 =e−iπσ2/2 =−iσ2, (6) |outi=Vτ|ini=|ini, τ 6=1. (14)
so that
III. DISTINGUISHING DIFFERENT SHADES
0
|outi=VL |ini= . (7) OF GRAY
τ=1 1
(cid:18) (cid:19)
This is easy to understand: each loop “rotates” the par- Aquestionarises[7]: isitpossibletodistinguishdiffer-
ticle’s state by 2θ = π/2L and after L loops the final entvaluesofτ (different “shades”or“levels”ofgray)by
L
state is “orthogonal”to the initial one (2). the technique outlined above? This is not a simple task,
for after a large number of loops L the particle ends up
in the orthogonal channel only if τ = 1 [see Eq. (7)]; by
B. Black sample contrast, for any value of τ 6= 1, the particle ends up in
the Zeno channel [see (14)] irrespectively of the partic-
Let us now look at the case τ = 0, corresponding to ular value of τ. However, the asymptotic correction in
a completely opaque (“black”) object in the MZI. We the (1,1) element of VL in (12) is τ dependent: the de-
τ
obtain tails of the convergence to the limit (13)-(14) depend on
the grayness of the sample. By exploiting this feature,
VL = B(A B2)LB−1
τ=0 0 we shall now show that it is indeed possible to resolve
1 −tan2θ different gray levels by QZE, within a given statistical
= BcosL2θ L B−1
L 0 0 accuracy.
(cid:18) (cid:19)
We start by observingthat if one performs a standard
L→∞ 1 0
−→ ≡V . (8) transmission experiment, by shining a particle beam on
0 0 τ=0
(cid:18) (cid:19) a semitransparent object in order to measure the trans-
This yields QZE: mission coefficient τ, see Fig. 1b, the detection and ab-
sorption probabilities read
1
|outi=V |ini=|ini= . (9)
τ=0 0 p′(τ)=τ2, p′(τ)=1−τ2. (15)
(cid:18) (cid:19) d a
3
The statistics is binomial. numberofparticles,absorbingasfewparticlesaspossible
On the other hand, if one uses the Zeno configuration for the requested precision. To perform this task in an
sketched in Fig. 1a, the final state of the particle after L optimal way one should find optimal estimators for each
loops in the MZI is, from (12), scheme. A lower bound on the variance of an unbiased
estimator Tˆ of the parameter T (here T = τ2) is the
VL 1 = uz = 1− 8πL2 11−+ττ +O(L−2) , (16) Cram´er-Raolower bound (CRLB) [8],
τ 0 uo O(L−1)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 2 −1
1 ∂
where u and u are the amplitudes in the Zeno and (∆Tˆ)2 ≥ ≡ lnp(n|T) , (18)
z o
F ∂T
orthogonal channels, respectively. Both these quantities *(cid:20) (cid:21) +
are real. Let (0≤τ <1)
where F is the Fisher information, p(n|T) the probabil-
π2 1+τ ityofobservingnparticlesconditionedbythe valueT of
pz(τ) = u2z =1− 4L1−τ +O(L−2), the unknown parameter and h...i denotes ensemble av-
erage with respect to n. The probability p is binomially
p (τ) = u2 =O(L−2), (17)
o o distributed inboth the standardcaseandthe Zeno case,
π2 1+τ which yields
p (τ) = 1−p (τ)−p (τ)= +O(L−2)
a z o
4L1−τ
τ2(1−τ2)
(∆Tˆ )2 ≥ ,
be the probabilities that the particle is detected in the st N
Zeno, orthogonal channel or is absorbed by the semi- 4τ2(1−τ)3(1+τ)L
transparentobject, respectively. We assume that a fixed (∆TˆZe)2 ≥ π2N (19)
numberofparticlesN issentintheMZIduringanexper-
imentalrun. Inthissituationthedistributionofparticles forstandardandZenotomography,respectively,N being
in the Zeno, orthogonalor absorption channels follows a the (fixed) number of input particles in both cases. Ex-
trinomial statistics with probabilities (17). pressingthe aboveinequalities (19) in terms ofthe num-
It seems natural to think that, since by increasing the ber of absorbed particles N(st) =Np′ and N(Ze) =Np ,
a a a a
number of loops L, p vanishes much faster than p , for they both reduce to the same bound
o a
largeL the distributionis practicallybinomialwithp +
a τ(1−τ2)
pz ≈ 1 [14]. However, as we shall see, the presence of ∆Tˆsot,pZte ≥ , (20)
a small trinomial component, po, will play an important N(st),(Ze)
a
role, enabling the Zeno method to perform much better
q
than the standard one. showing that the CRLB’s for standard and Zeno tomog-
raphy are the same, given the number of absorbed par-
ticles. Further, it is trivial to show that the unbiased
IV. AN INTRINSIC LIMIT FOR BINOMIAL estimator given by the relative frequency of transmit-
STATISTICS: THE CRAME´R-RAO BOUND ted particles, Tˆ = n /N, saturates the CRLB (20).
st t
Hence, if one neglects the output of the ordinary chan-
We arenow readyto discussthe possibility of a “Zeno nel po and considers the statistics (17) practically bino-
tomography.” The goal is to get information about the mial, the Zeno estimation can be at most as good as the
distribution of the absorption coefficient in the sample, standard one: it cannot be better.
absorbing as few particles as possible. We will accom-
plishthisintwosteps. First,usingestimationtheory,we
showthatif one limits one’s attention tobinomial statis- V. TWO STATISTICAL PROTOCOLS
tics,the Zenoestimationofany levelofgrayofonepixel
(i.e. τ continuously distributed between 0 and 1) cannot A. Binomial (single-channel) protocol
performbetter thanthestandardmethod. Atbest,both
methods are equivalent. This is bad news. However, the Inspite ofthe conclusionsofSec.IV, wewill nowcon-
very proof of the above-mentioned statement will show structa protocolandshow that by QZEone can achieve
that there are two ways out: first, we will see that Zeno aresolutionthatis superior to the “ordinary”resolution
performsbetter whenonewantstodistinguishtwolevels obtained in a standard transmission experiment. Notice
ofgraythatarenotequallypopulatedinthesample(this that p (τ) in (17), unlike p′(τ) in (15), is an increasing
a a
requires some prior knowledge about the distribution of function of τ. Therefore, with respect to absorbed par-
grays in the sample). This is good news, for it enables ticles, the Zeno tomographicimage (for sufficiently large
one to find a method that in some cases works better L) yields a kind of negative of the standard absorption
than the standard one. Second, one is led to think that tomographic image. This can be given a rather intuitive
theintroductionandexploitationofatrinomialstatistics explanation: indeed, the absorption probability in (17)
can enable the Zeno method to perform better. reduces to the same form as the standardone (15), i.e.
Let us start from the estimation of any level of gray.
In this case one tries to estimate τ2 from the counted p (τ)=1− τZe 2, (21)
a eff
(cid:0) (cid:1)
4
by introducing an “effective” transmission coefficient nd]. Anoptimalprotocolisgivenbydeterminingnd that
a a
minimizes the error (24).
τeZffe = 1−pa. (22) Alternatively, one defines the likelihood ratio [9]
For example, if we take pτ = 0.98, τ = 0.99 and
1 2 L(τ |N,n ,α)
1 a
choose L = 12000, then, according to Eq. (17), we get R= , (27)
L(τ |N,n ,1−α)
τZe ≈ 0.99 and τZe ≈ 0.98. The two gray levels are 2 a
eff1 eff2
interchanged by the Zenoapparatus. Ifmostofthe sam-
where
plehastransmissioncoefficientτ theabsorbedenergyis
2
reduced by using the Zeno setup.
L(τ |N,n ,α) = P (H )p(n |H )
i a 0 i a i
A more precise comparison of the performances of the
N
Zenoandstandardtechniquescanbe givenintheframe- = α p (τ )na[1−p (τ )]N−na(28)
a i a i
n
work of decision theory. For simplicity let us focus on (cid:18) a(cid:19)
distinguishing only two gray levels τ and τ (τ < τ )
1 2 1 2
representsthelikelihoodofhypothesisH (i=1,2). The
corresponding to hypotheses H and H that occur in i
1 2 optimum decision level nd is determined by solving for
the sample with frequencies a
equal likelihoods [15]
P (H )=α, P (H )=1−α. (23)
0 1 0 2
R=1. (29)
With this simplification we lose no generality since the
tomographywithM graylevelscanalwaysbesplitintoa In both cases one gets
sequenceofpairwisedecisionsbetweentwoadjacentgray
levels. log 1−α −Nlog 1−pa(τ1)
Wewillproceedintwosteps. Firstwewillassumethat nd = α 1−pa(τ2) (30)
in the Zeno configuration of Fig. 1a all output particles a log(cid:2) pa(τ(cid:3)1) −log h1−pa(τ1)i
pa(τ2) 1−pa(τ2)
are collected at a single detector. In other words, the
h i h i
Zeno and orthogonal channels are considered as a single and, substituting in (24),
output and the statistics (17) is binomial (p +p ,p ).
z o a
Eachparticleistheneitherabsorbedortransmitted(and P = α{1−B [N −n˜d,1+n˜d;1−p (τ )]}
e I a a a 1
detected) by the Zeno apparatus. Obviously, by merg- +(1−α)B [N −n˜d,1+n˜d;1−p (τ )], (31)
ing the two output channels together, some information I a a a 2
about the sample is wasted. We know, however, that
whereB (a,b;z)istheregularizedincompleteBetafunc-
I
this strategy will be optimal if the number of loops L tion [10] andn˜d is the greatestinteger less than or equal
a
is very large. Then there are almost no particles exit- to nd. The mean number of absorbed particles is
a
ing via the ordinary channel n ≈ 0 which can then be
o
safely ignored. A better and more general strategy will N(Ze) =N[αp (τ )+(1−α)p (τ )]. (32)
be studied later. a a 1 a 2
Since both experiments obey the same (binomial)
By plugging Eqs. (30) and (32) in Eq. (31), the average
statistics we use the notation of the Zeno experiment.
probability of error (31) can be expressed as a function
The analysisof the standardexperimentis similar. If no of α, τ ≡τ , dτ =τ −τ and N(Ze). The probability of
distinctionismadebetweentheZenoandordinarychan- 1 2 1 a
error for the standard setup is obtained in a completely
nels, the decision is based on the number of absorbed
analogous way.
particles n . If n is smaller than or equal to a decision
a a The performances of the Zeno and standard methods
level nd, then H is chosen; otherwise H is chosen. The
a 1 2 are compared in Fig. 2. First, the (exact) Lth power of
probability of making an error in identifying the gray
the matrix V in (10) is evaluated, then for a givenerror
level of a given pixel is τ
rate P , the number of absorbed particles is calculated
e
P =αP(H |H )+(1−α)P(H |H ), (24) by solving numerically Eq. (31). Their ratio is shown as
e 2 1 1 2
a function of α for a few values of the transmission co-
where efficient τ. Notice that the exposition of the sample can
be significantly reduced if the distribution of gray lev-
P(H |H )= p(n |H ) (25)
1 2 a 2 els in the sample is not uniform. For instance, a reduc-
naX≤nda tion factor of 2.5 is obtained when the sample consists
of 97% of dense material and 3% of the less absorbing
is the probability of choosing H when H is true and
1 2 one, α = 0.97. Such parameters are typical for struc-
N tural analyses: a small structural defect (crack) inside a
p(n |H )= p (τ )na[1−p (τ )]N−na (26)
a 2 a 2 a 2 thinsamplewouldtypicallyshowsmallcontrast(δτ ≪1)
n
(cid:18) a(cid:19) with the surrounding almost transparent (τ ≈ 1) mate-
isthebinomialprobabilityofabsorbingn particleswhen rial, while its area would be small compared to the area
a
H is true. [For P(H |H ) the summation is over n > of the sample (α≈1).
2 2 1 a
5
FIG. 2: Ratio of the number of absorbed particles in the
Zeno (N(Ze)) and standard (N(st)) setup. The smaller the
a a
ratio in the graph, the less irradiation in the Zeno appa-
ratus (for the same resolution). Pe = 0.5%; dτ = 0.02; FIG.3: Typicaldecisionlevelsofthebinomial(dashedlines)
τ ={0.8,0.9,0.95,0.97}; L=2000.
and trinomial (solid lines) decision strategies when a ≈ 1.
This figure corresponds to the simulation shown in the last
row of Fig. 5c, where M = 3, the triangle is divided into
B. Trinomial (two-channel) protocol
M =3regionsandthethreegrayshadesarelabelledaswhite,
gray and black, respectively.
Thebinomialdecisionstrategyoutlinedintheprevious
subsection is not the optimum one. Unavoidable losses
and other imperfections of real experimental devices set Ina two-dimensionalrepresentationeachpossible exper-
a strict limit on the maximum number of loops that can imental outcome (nz,no) is represented by a point lying
be achieved in a laboratory. In such case the ordinary inside the triangle {0 ≤ no +nz ≤ N} shown in Fig. 3.
channel can no longer be ignored. The data consist then Equation (35) divides this triangle into two regions. All
of the two component vector (n ,n ) of the numbers of experimental outcomes that fall within the same region
z o
particles counted in the Zeno and ordinaryoutput chan- issuethesamedecision. InthegeneralcaseofM different
nels. The decision will be based on both these numbers. graylevelsthereareM−1equations(35)definingM−1
Thedecisionlevelsarereadilyobtainedfromtheequal in general non-parallel lines dividing the square into M
likelihood criterion strip-like regions. This is shown in Fig. 3 for M =3.
An interesting situation arises when the coefficient a
L(τ |N,n ,n ,α)
1 z o
R= =1, (33) in (35) becomes close to unity. In that case,the decision
L(τ |N,n ,n ,1−α)
2 z o level is the line nd −nd = const. Let us recall that the
z o
where decisionlevelsofthebinomialdecisionstrategydiscussed
αN! intheprevioussubsectionwerenda =const.(see(30)),or,
L(τ|N,nz,no,α)= n !n !n !pz(τ)nzpo(τ)nopa(τ)na. equivalently, ndz +ndo = const. Hence if L, τ1 and τ2 are
z o a such that a ≈ 1, the decision levels of the binomial and
(34)
trinomialdecisionstrategiesareorthogonal toeachother.
This equation is to be solved for the decision vector
(nd,nd). Equation (33) is just one condition for the two This is shown in Fig. 3. Under such conditions one can
z o
expectfurthergainintheprecisionoftheZenoapparatus
unknowns n and n , so there exists a one-parametric
z o
as compared to standard absorption tomography. This
family ofsolutions. Byplugging(34)into (33)oneeasily
regime was chosen for our computer simulations of the
finds
following section.
ndz −a(τ1,τ2)ndo =b(τ1,τ2,α), (35) Notice that the steepness of the decisionlines (35) de-
pends only on the absorptionof the corresponding adja-
where the coefficients a and b read
cent graylevels. It depends neither on their frequencies,
log po(τ1)pa(τ2) nor on the total number of incident particles.
po(τ2)pa(τ1)
a = , (36) Finally, let us discuss the limit L → ∞. When the
loghpz(τ2)pa(τ1)i number of loops L increases, p → 0 much faster than
pz(τ1)pa(τ2) o
Nlohg ppaa((ττ21)) +ilog 1−αα Npa−[seneda(i1n7)E]qa.n(d35o)n,ewihsicahllorwedeudcteosptout ndo = 0 and ndz =
b = . (37)
lohg pz(τi2)pa(τ1)h i nd =N −b(τ ,τ ,α). (38)
pz(τ1)pa(τ2) a 1 2
h i
6
of the illuminating beam is increased further, in frame
(d), all the reconstructed images become visually hard
to tell from the sample, but the error rates of the Zeno
apparatuses are still much better (by a factor three or
more), as shown by the number of misinterpretedpixels.
It is worth commenting on the distribution of misin-
terpreted pixels. Clearly, in all the cases analyzed, it is
not uniform. In general, when the distribution of gray
levels in the sample is not uniform, any reconstruction
technique tends to perform better in the “background,”
while making more mistakes in the region where the
“structure” is present. The improvement due to the
FIG. 4: The object to be reconstructed: a cell of Giardia
Zeno method becomes apparent if one looks in partic-
lamblia, one of the most primitive eukaryotes. The original
ularatFigures5(b)and(c): inthesecases,interestingly,
picturehasbeenreducedforsimplicitytothreelevelsofgray:
the standard method yields more mistakes in the back-
white, gray and black, occurring with frequencies αw =0.02,
ground; this is an unpleasant feature, if one is interested
αg =0.07 and αb =0.93 respectively.
in detecting small irregular structures in a more or less
uniform background. The features of the distribution of
By substituting p (τ)=1−p (τ) in (37)-(38), one reob- misinterpretedpixelsrequiremorecarefulstudyandtheir
z a
tains the binomial condition (30): the binomial strategy comprehension might lead to additional ideas.
becomes optimal in this limit. Any increase in the number of loops L in the interfer-
ometer makes the difference between standard and Zeno
tomographyevenbigger. Clearly,thisismoredemanding
in terms of experimental realization.
VI. SIMULATIONS
We have seen that the Zeno technique can reduce the
VII. CONCLUSIONS
level of absorption without losing resolution (compared
to the standard technique). (Alternatively, the Zeno
setupcanyieldanimprovedresolution,whilekeepingthe We have shown that a quantum Zeno tomography is
absorptionatthe samelevelofthe standardsetup.) The possible and performs better than standard tomography
object in Fig. 4 is a cell of giardia lamblia, a protist, one ifagivenpriorknowledgeaboutthedistributionofgrays
ofthemostprimitiveeukaryotes. Giardiahasbeencalled inthe sample is available. This is a commonsituationin
a “missing link” in the evolution of eukaryotic cells from radiography,where one is often interested in detecting a
prokaryoticcells. The numberofgraylevelsinthe figure smallstructureinauniformbackground,likeforinstance
has been reduced to three to make the analysis simpler: in the analysis of small structural defects.
white, gray and black, τ = 0.99, τ = 0.96, τ = 0.8, In our numerical simulations we have illustrated some
w g b
occurring with frequencies α = 0.02, α = 0.07, and situationsinwhichtheresolutionisimprovedbytheZeno
w g
α = 0.93 respectively. Figure 5 shows the results of a method, for a given number of absorbed particles. Alter-
b
numerical simulation, performed with the standard and natively, for a given resolution, the Zeno method per-
Zenomethods,thelatterforL=10andL=165,fordif- forms better, absorbing less particles. This can be inter-
ferent numbers of absorbed particles N . In each frame esting in applications, for instance if one wants to limit
a
the standard and the two Zeno reconstructions are com- thedamageprovokedbytheabsorptionofradiationwith-
pared, together with the pixels that have been misinter- out losing in resolution.
preted. Figure 5 confirms the expectation based on the It is obvious from Fig. 2 that an even larger improve-
asymptotic formulas (17): in general, provided that the mentispossibleforalmosttransparentsamples,provided
objectcontains a smallfractionofmore transparentpix- thatα isclose to unity. This meansthat thereis no fun-
els and a larger fraction of more absorbing material, the damental limit ontheimprovementthatcanbeachieved
Zeno setup yields a better resolution for a given irradia- overthe standardsetup: inother words,there is no “op-
tion. Clearly, a significant improvement with respect to timal” configuration.
standardabsorptiontomographyisachievedforasfewas There are additional issues that deserve careful study.
L=10loops. TheimprovementisverylargeforL=165. For instance, the effects due to a Poissonianbeam (total
The number of absorbed particles increases from (a) numberofincomingparticlesN notfixed)andacomplex
to (d) in Fig. 5. Observe that in (a) the standard re- transmission coefficient τ.
constructionfails completely,while the outline andbasic Let us also comment on experimental feasibility. Fig-
shapeoftheobjectcanberecognizedalreadyintheZeno ure 5 shows that an experimentaltest of the Zeno tomo-
reconstruction with L = 10. In (c) the Zeno reconstruc- graphictechnique shouldnotbe as difficult asone might
tions are quite good, while standard tomography does think: simulations have been performed for as few as
notdetectwhitepixelsintheobject. Whentheintensity L = 10 loops in the interferometer, giving better results
7
FIG. 5: Comparison of standard and Zeno tomographic techniques. In each frame: top left=reconstruction by standard
technique; top right=misinterpreted pixels by the standard technique; center left=reconstruction by Zeno technique with
L=10; centerright=misinterpreted pixels bytheZeno techniquewith L=10; bottom left=reconstruction by Zenotechnique
withL=165;bottomright=misinterpretedpixelsbytheZenotechniquewithL=165. Themeannumberofabsorbedparticles
per pixel (irradiation) is Na =1.7, 2.3, 4 and 13 for frames (a), (b), (c) and (d) respectively. The total numberof particles N
(total energy) scales approximately like 3Na, 1.8Na, and 6.5Na for top, center and bottom reconstructions, respectively. We
usedτw =0.99, τg =0.96, andτb =0.8. Thesampleconsistsof10,000 (=100×100) pixels,wherewhite,grayandblackoccur
with frequencies αw =0.02, αg =0.07 and αb =0.93, respectively. The number of misinterpreted pixels are (top to bottom):
(a) 968, 786, 315; (b) 942, 596, 212; (c) 717, 382, 68; (d) 205, 69, 0.
thanthe standardmethod. Itisreasonabletothink that mirrors several thousands times. This would lead us to
a Zeno setup with a much largernumber of loops can be the full asymptotic (L ≫ 1) regime considered in Fig. 2
built for UV light (highly absorbed by some biological and last row of Fig. 5, where the Zeno method can per-
samples). Also, by changing the light wavelength, one form much better. There is hopefully more to come.
couldefficiently“observe”differentregionsofthesample
(orslightlydifferentsamples). Moreover,the experimen-
tal configuration we have proposed (photons in a MZI,
Acknowledgments
like in Fig. 1) is certainly not the only conceivable one.
Phaseimagingandtomographyhavebeendemonstrated
for both X-rays and neutrons [11]. More to this, Rauch ThispaperisdedicatedtoProfessorJanPeˇrinaonthe
and collaborators,with the VESTA apparatus [12], have occasion of his 65th birthday. This work is supported
been able to keep neutrons in a one-meter long perfect by the TMR-Network of the European Union “Perfect
crystal storage system (“resonator”) for a few seconds, Crystal Neutron Optics” ERB-FMRX-CT96-0057. J. Rˇ.
so that the neutrons bounce back andforth betweentwo andZ.H.acknowledgesupportbytheprojectLN00A015.
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8
[10] M. Abramovitz and I. A. Stegun, “Handbook of mathe- the apparatus starts operating in the Zeno regime. We
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L
26.5.7. cording to the simple expression τZ > 1 − π/L. Al-
L
[11] F. Dubus, U. Bonse, T. Biermann, M. Baron, F. Beck- ternatively, one can rephrase these conditions in terms
mann, and M. Zawisky, Proceedings SPIE 4503, 359 of L: the asymptotic expansion (12) is then valid for
(2002). L>LZ =π/2arcsin[(1−τ)/(1+τ)].
τ
[12] E.Jericha, D.E.Schwab,M.R.J¨akel,C.J.Carlile, and [14] Moreso,sincepa isalsosmallinthislimit,thedetection
H.Rauch,PhysicaB283,414(2000);H.Rauch,Physica statistics is almost Poissonian.
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[13] It is worth stressing that when τ > τZ the asymptotic binomial) statistics and can be easily generalized to the
L
expansion has a completely different form and tends to case of more than two gray levels.
−iσ whenL→∞.Uponcrossingthethresholdτ =τZ,
2 L