Numerical Optical Centroid Measurements

0.59 MB·English

by  Qurrat-ul-Ain Gulfam

Numerical Optical Centroid Measurements
Qurrat-ul-Ain Gulfam1, and J¨org Evers1,
∗ †
1Max-Planck-Institut fu¨r Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany
(Dated: December 11, 2013)
Optical imaging methods are typically restricted to a resolution of order of the probing light
wavelength λp by the Rayleigh diffraction limit. This limit can be circumvented by making use of
multiphotondetectionofcorrelatedN-photonstates,havinganeffectivewavelengthλp/N. Butthe
requiredN-photondetectionusuallyrenderstheseschemesimpractical. Toovercomethislimitation,
recently, so-called optical centroid measurements (OCM) have been proposed which replace the
multi-photon detectors by an array of single-photon detectors. Complementary to the existing
approximate analytical results, we explore the approach using numerical experiments by sampling
andanalyzingdetectioneventsfromtheinitialstatewavefunction. Thisallowsustoquantitatively
3
study the approach also beyond the constraints set by the approximate analytical treatment, to
1
compare different detection strategies, and to analyze other classes of input states.
0
2
PACSnumbers: 42.65.Sf,42.50.Nm,42.60.Da,04.80.Nn
n
a
J
I. INTRODUCTION all events rather than only N-photon events contribute

4 to the signal. Recently, a proof-of-principle experiment

Conventionalopticallithographyschemesareimpeded ofOCMfor two photon entangledstate has been carried
]
h by the Rayleighdiffractionlimit [1, 2]owing to the wave out by [17]. The experiment used two kinds of detec-
p nature of light. Not least due to the continuous and tors: One is a pair of single photon detectors for OCM
t- increasing demand of faster devices in microelectronics, whiletheotheristwo-photonabsorptiondetectorforQL.
n OCMachievedthe samequantumsuperresolutionasQL
smaller structures in semiconductors need to be written.
a But highly energetic photons can lead to enhanced ma- but much more efficiently.
u
q terial damage and demand the development of new op- Motivated by this, here, we study the OCM using nu-
[ tical imaging systems. Motivated by this, several alter- merical experiments. In contrast to existing approxi-
native routes have been explored. A prominent example mate analytical approaches, the numerical study allows
1
is quantum lithography [3–6] based on number-path en- us to quantitatively study OCM over a broader parame-
v
tangled states of light. But generation of the required ter range, to compare different detection strategies, and
4
7 entangledstates,decoherencedue tophotonlossandde- to analyze other classes of input states. In particular,
7 tection remain a challenge. Alternatively, classical light in addition to the NOON and jointly Gaussian input
0 fields can be used. One tool is post-selection of certain states proposed in [16], we analyze cat states made out
1. detectioneventswhichcontainimagedatawithhighres- of superpositions of coherent states. For our analysis,
0 olution [7, 8]. Other methods aim at facilitating a N- we first generate random detection events from position
3 photon absorption process leading to a reduced effective spacewavefunction. Afterwards,wemodelthedetection
1 wavelength[9–13]. These,however,requiremultiphoton by a suitable discretization using different methods, and
v: transitions,whichcomplicateapracticalimplementation. apply the centroid method. We characterize the error
i This requirementcanbe relaxedbymeans ofdarkstates in the recovery of the theoretical centroid distribution
X
in multi-level systems [14] or of Rabi oscillations in two- by root mean square (rms) deviation from the original
r level systems [15]. wavefunction. Weconfirmourmethodbyrecoveringkey
a
Recently, a different method has been proposed to re- predictions from the analytical calculations [16] quanti-
move the requirement of multi-photon detection [16]. In tatively, such as the dependence of multiphoton count
the so-called optical centroid method (OCM), the N- rates for jointly Gaussian states on the properties of the
photon detector is replaced by an array of single-photon input state.
detectors. For each input pulse, the results of the indi- We then study the dependence of the rms deviation
vidual single-photon detectors are then combined into a on the size of the individual detectors in the array. We
singlevariable,thecentroid. Theprobabilitydistribution find severalfeatures which may appear counter-intuitive
of the centroid obtained from many repetitions of the at first sight, but can be explained in terms of the mea-
measurement then contains the desired sub-wavelength surementprocedurechosenandofstatisticaleffectsmost
resolution. Thus, no N-photonabsorberis required,and pronouncedinthe smalldetectorlimit. Forintermediate
atthesametime,thedetectionbecomesmoreefficientas detector sizes, the rms deviation is nearly a linear func-
tionofthe detectorsize. We analyzethis dependence for
NOON-states with different N and show how the slope
ofthelineardependence canbetracedbacktothestruc-
∗ [email protected] ture of the underlying wave function. We then extract
† [email protected] single-and multi-photon detection events, and verify the
2
improvedefficiencyexpectedfortheOCMoverN-photon forany k >2π/λ. Thespatialannihilationoperatoris
n
| |
absorption schemes. For the jointly Gaussian state, we defined as
augment this analysis by studying the rms deviation for 1
different N as a function of the detector size while keep- Aˆ(x) dkaˆ(k)eikx. (5)
≡ √2π
ingthefeaturesizeconstant. We finallyturntocoherent Z
A corresponding N-photon state can be constructed as
cat states, and show that the OCM can also be applied
to them. These states are of interest, as the feature size 1
x ,x ,...,x Aˆ (x )...Aˆ (x )0 . (6)
of the cat states can continuously be tuned via the co- | 1 2 Ni≡ √N! † 1 † N | i
herent state parameter α. Furthermore, the structure of
The position space wave function is obtained by taking
thecatstatewavefunctionalongthecentroidaxiscanbe
theN-dimensionalFouriertransformofφ(k ,k ,...,k )
modified via the phase of α such that it differs from the 1 2 N
and is given by
regular patterns observedfor NOON or Gaussian states.
Thepaperisorganizedasfollows. InSec.II,weprovide ψ(x ,x ,...,x )= x ,x ,...,x N . (7)
1 2 N 1 2 N
h | i
the theoretical background. In particular, we review the
Offundamentalimportanceinthecentroidschemeisthe
OCMmethodinIIA,introducethevariousnon-classical
transformation of the photon position coordinates x to
input states in Sec. IIB, discuss the implementation of i
the centroid coordinate X and the relative position co-
the numerical experiments (Sec. IIC) and the detection
ordinates ξ which are defined as
system (Sec. IID), and describe the procedure to esti- n
mate the error in the OCM recovery in Sec. IIE. In 1 N
Sec. III, our results for different types of non-classical X xn, ξn xn X. (8)
≡ N ≡ −
input states arediscussed. The final Sec.IV summarizes n=1
X
the results. Note that when x = x = = x = x, X = x, that
1 2 N
···
is, if all photons are constrained to arrive at the same
point x, the relative positions ξ become zero and the
n
II. THEORETICAL CONSIDERATIONS centroid variable also lies at the same point x according
to Eq. (8).
Conventional quantum lithography proposals rely on
A. Centroid method
N-photon detection at a single detector at position x.
The input state is a N-photon state ρˆ = N N . The
Webrieflyreviewtheformalismofresolutionenhance- N | ih |
probability density for multi photon absorption is there-
ment developed in [16, 18] using quantum lithography
fore given by
and optical centroid measurements. We consider pho-
tons of wavelength λp incident on a detection plane un- pMP(x)=h:IˆN(x):i=h[Aˆ†(x)]N[Aˆ(x)]Ni
der an angle such that the wavelength associated to the =N ψ(x,x,...,x)2. (9)
wave vector component in the detection plane is given | |
by λ=λ /sinθ. In the following, we restrict the analy- The centroid method, in contrast, relies on an array
p
sis to one spatial dimension in the detection plane. Let of detectors, such that each of the incident N photons
aˆ(k)(aˆ (k))denotethephotonannihilation(creation)op- is detected by an individual detector. Out of the indi-
†
erator in this transverse momentum space. vidual photon detection positions, the centroid position
These operators follow the commutation relation X is calculated. The probability density of the centroid
distribution is obtained from
[aˆ(k),aˆ†(k′)]=δ(k k′). (1) :ΠN Iˆ(x ): (10)
− h n=1 n i
An N-photon momentum eigenvector can be defined us- as a marginal probability density by tracing out all rela-
ing aˆ†(ki) as tive positions ξn using xn =X +ξn,
k ,k ,...,k = 1 aˆ (k )aˆ (k )...aˆ (k )0 , (2) pOCM(X)= dξ1,...,dξN−1h:ΠNn=1Iˆ(X +ξn):i
| 1 2 Ni √N! † 1 † 2 † N | i Z
= dξ ,...,dξ N ψ(X +ξ ,...,X +ξ )2. (11)
1 N 1 1 N
where 0 indicates the state without photons. The mo- Z − | |
| i
mentumspacewavefunctionrepresentationofapureN- The key result of Tsang’s analysis is that the two dis-
photon Fock state N is given by tributions Eq. (11) and Eq. (9) have the same spatial
| i
dependence and thus resolution. Therefore, the techni-
φ(k ,k ,...,k )= k ,k ,...,k N . (3) callychallengingN-photondetectioninasingledetector
1 2 N 1 2 N
h | i
canbereplacedbyN single-photondetections. Afurther
Thetransversemomentaofthephotonsarerestrictedby advantageofthecentroidmethodisthatitispotentially
the Rayleigh diffraction limit [1] such that muchmoreefficientthanmulti-photondetection,i.e.,for
experimentally relevant parameters, p (X) can ex-
OCM
φ(k ,k ,...,k )=0 (4) ceed p (X) significantly [16–18].
1 2 N MP
3
B. Trial states
Throughout our analysis, we apply the optical cen-
troid method using numerical experiments to various
non-classical photon states, which we introduce in the
following.
1. NOON states
Following [18], we consider momentum-correlated N-
photon NOON states
1
NOON = (N 0 + 0 N ) (12)
A B A B
| i √2 | i | i | i | i
inwhichthetwomodesAandB correspondtotwowave
vector directions for the photons incident on the detec-
tion plane, with mean transversecomponentof the wave
vector k and k , respectively. The momentum spread
0 0
−
of the transverse wave vector is denoted by ∆k with
k ∆k. Denoting the (normalized and even) trans-
0
≫
verse wave vector profile as f(q), the momentum space
wave function is
N
1 k k
n 0
φ (k ,k ,...,k )= f −
NOON 1 2 N
√2∆kN " ∆k
n=1 (cid:18) (cid:19)
Y
N
k k
n 0
+ f − . (13)
n=1 (cid:18)− ∆k (cid:19)#
Y
By Fourier transform, assuming f(q) = FIG. 1. (Color online) Momentum (a-c) and the correspond-
π 1/4e q2/(2∆k2), we then obtain the position space ing position (d-f) space probability distributions of a two-
− −
wave function photon NOON state (top row), joint Gaussian state with
N = 2 and B = β = λ−1 (middle row), and correlated cat
ψNOON(x1,x2,...,xN)=√π2N∆/k4N e−21∆k2 PNi=1x2i state with α=i(bottom row).
N
cos k x . (14)
0 i
× !
i=1
X
The probability distribution thus follows as Using λ = 2π/k , we have defined the dimensionless
0
width σ = k /∆k. Throughout the numerical analy-
ψ (x ,x ,...,x )2 0
| NOON 1 2 N | sis, we have chosen σ = 4√2π such that ∆k k0 as
≪
=2 ∆k N e−∆k2 PNi=1x2i cos2 k0 N xi . (15) required.
√π !
(cid:18) (cid:19) i=1
X
An example for the position space probability distri-
Note that this expression is only normalized in the limit bution ofa two-photonNOONstate using dimensionless
∆k k0, in which the field operatorsfor the two modes variables is shown in Fig. (1).
≪
A and B satisfy the usual bosonic commutation rules.
Fornumericalsimulationsweemploydimensionlesspo-
sition variables x¯ = x/λ such that the position coordi- 2-photon NOON states can be created using sponta-
nates are measured in the units of λ. In terms of the neous parametric down conversion combined with the
dimensionless variables, Eq. (15) becomes Hong-Ou-Mandel effect [3, 19]. Such NOON states have
been employed in proof-of-principle experiments on res-
N N olution enhancement [5, 17, 20]. Higher order NOON
2 ∆k e−4σπ22 PNi=1x¯2i cos2 2π x¯i . (16) states with larger N have also been created experimen-
√π
(cid:18) (cid:19) i=1 ! tally [21–25].
X
4
2. Jointly Gaussian states states of the form [26]
ψ = (α α + α α ), (22)
ThemomentumspacerepresentationofajointlyGaus- | ccci N | i1| i2 |− i1|− i2
sian state can be written as [18]
in which α is a coherent state in mode i with com-
i
| i
plex parameter α, defined by a α = αα where a
φJG(k1,k2,...,kN)= C e−14Pn,mknBnmkm, (17) is the photon annihilation operia|toiri of m|odiei i. Notei
N
r that other two mode cat states have also been consid-
with components of the matrix B given by eredinthe literatureindifferentcontexts[27–29]. These
states can be seen as an extension of the famous sin-
1 1 1
B = + 1 , gle particle Schr¨odinger coherent cat states [30] given
nn N2B2 (cid:18) − N(cid:19)β2 by α+ α + α [31]. A proposal for the gen-
1 1 erat|ionio≡f S|chir¨odi|n−ger’is cat state has been discussed
B = (n=m). (18)
nm N2B2 − Nβ2 6 in [32]. Experiments performed on the optical cat states
include [33–35]. is a normalization factor which is
C is a normalization constant. B can be interpreted N
given by 1/ 2(1+exp( 4α2).
as the width of the average momentum of the photons, − | |
While Eq. (22) resembles a two-photon NOON state,
whereas β characterizes the width in momenta relative p
there is an important difference between the states. The
to the average[18]. Thesetwo parametersdetermine the
bi photon NOON state consists of exactly two photons.
variance of k given by
n Individualrealizationsofthesamecatstate,however,can
1 contain different numbers of photons. The probability
k2 =B2+(1 )β2. (19)
h ni − N p(n1,n2) to detect ni photons in mode i evaluates to
ThepositionwavefunctionforthejointlyGaussianstate
p(n ,n )= n ,n ψ ψ n ,n
1 2 1 2 ccc ccc 1 2
then follows as h | ih | i
e 2α2
|ψJG(x1,x2,...,xN)i∝exp(− xnBn−m1xm). (20) =2N2n−1!|n2|!|α|2(n1+n2)[1+(−1)n1+n2]. (23)
n,m
X
It follows that the state can only contain even numbers
The multi photon absorption pattern : IˆN(x) : is a
of photons. For a certainrange of α, states with less or
h i
Gaussian with a root mean square width 1/(2NB). If | |
equaltotwophotonsarepopulatedwithhighprobability.
B = β/√N, the distribution becomes classical and the This motivates the analysisof detection events,in which
photonsareindependent. This distributionisa symmet- one photon is detected in each of the two modes. In the
ric Gaussian with equal widths along the diagonal axes following, we will focus on this case.
in the configuration space. The classical multi photon The position representation of a single mode coherent
absorptionwidthisgivenbythestandardquantumlimit state is [36]
W = 1/(2 N k2 ). Increasing B from the classical
C h ni
vmaaluxeimmuamkevsaptluheeodfisBtr,ibBut=ion qhukan2nituremndmereschtahneicuallt.imTahtee hx|αi= x10√πe−12(α2−|α|2)e−12(x/x0−√2α)2 (24)
quantum limit for the width of the multi photonabsorp-
p
tion pattern Wmin =1/(2N hkn2i). withx0 = p~/(mω0). Thepositionspacerepresentation
The quantum mechanical elongated jointly Gaussian
of Eq. (22) then follows as
p
has a shorter width along the centroid direction and is p
stretched along all the other diagonals in the position ψ (x ,x )= x α x α + x α x α
ccc 1 2 1 1 2 2 1 1 2 2
space. h | i h | i h |− i h |− i
The position space probability distribution is = 2N e−2x120(x21+x22)e−2|α|αcos(φ)
√πx
0
|ψJG(x1,x2,...,xN)i|2 ∝exp(−2 xnBn−m1xm). (21) cosh √2α(x +x ) , (25)
Xn,m × x0 1 2 !
Fig. (1) shows the position space probability distri-
bution for a quantum mechanical jointly Gaussian state where α= α exp(iφ).
| |
in terms of dimensionless position variables defined in For the further discussion, we now specialize to the
Sec. (IIB1). case of α = iα (i.e., φ = π/2) to obtain a probability
| |
density
3. Cat states ψ (x ,x ,α=iα)2
ccc 1 2
| | | |
The third class of states we investigate are correlated = 4πNx22e−x120(x21+x22)cos2 √x2|α|(x1+x2) . (26)
coherent cat states. We in particular analyze two mode 0 0 !
5
Thestructureofthisexpressionresemblestheprobability 1. Random number generation
density for the two photon NOON states. Other choices
for φ allow to rotate the fringe pattern in the position The firststep is to generaterandomphotonevents ac-
space with respect to the centroid axis. This way, the cording to the respective position space wave function.
condition for the centroid method that the probability As can be seen from Eqs. (15), (21) and (27), the po-
density should separate into a function of the centroid sition variables are correlated. To facilitate the random
coordinate times a function of all other coordinates can event generation, we apply unitary transformations to a
be continuously violated by modifying φ from π/2. set of uncorrelatedvariables. For example, the probabil-
As before, for the numerical calculations we define ity distribution
dimensionless position variables x¯. For x = 2πx x¯,
Eq. (26) becomes 0 2 ∆k 2 e−∆k2(x21+x22) cos2[k0(x1+x2)] . (28)
√π
(cid:18) (cid:19)
ψ (x ,x ,α=iα)2
ccc 1 2 ofEq.(15)specializedtothecaseoftwophotons(N =2)
| | | |
= 4πNx22e−4π2(x¯21+x¯22)cos2 2π√2|α|(x¯1+x¯2) . (27) ddoenestlyn.otAaplplolywintgotdhreawtrarannsfdoormmaptoiosintions x1,x2 indepen-
0
(cid:16) (cid:17)
Y~ =M X~ , (29)
2 2 2
Comparing Eq. (27) with Eq. (15), we find that the ex-
ponential parts of the two expressions agree for σ = 1, with X~ =(x ,x )T, Y~ =(y ,y )T, and
2 1 2 2 1 2
whereas the cosine-parts differ in the sense that the cat
state allows to continuously tune the structure size of 1 1 1
the fringe pattern by varying α. This way, the sub- M2 = √2 1 −1 ,
| | (cid:20) (cid:21)
wavelengthresolutioncapabilitiesofthecentroidmethod
can be probed independent of the number of detected the probability distribution Eq. (28) becomes
photons.
Notethatalternatively,onecouldchoosedimensionless 2 ∆k 2 e−∆k2y12 e−∆k2y22 cos2 √2k0y2 . (30)
variables x=√2πx0x¯/|α|, such that the cosine-parts of (cid:18)√π(cid:19) (cid:16) (cid:17)(cid:16) h i(cid:17)
the NOON-state and the cat-state probability densities
In this form, random values can be drawn for y and y
1 2
become the same. However, in this scaling, neither the
independently. In a similar way, also 3- and 4-photon
overall scaling nor the width σ of the cat state are inde-
events can be handled, with coordinate transforms
pendent of the choice of α.
| |
The probabilty distribution for φ = π/2 in configura- 1 1 1
1
tion space is shown in Fig. (1). Increasing φ from π/2 M3 = 0 3/2 3/2
√3 − 
(Fig.(1f)resultsinanincreaseofthewidthofeachfringe √2 1/√2 1/√2
−p −p
along the centroid direction such that the fringes merge  
intoeachother. Furtherincreasecausesthemiddlefringe and
to become smallerand smalleruntileventually atφ π,
≈ 1 1 1 1
the two modes of the coherent states no more interfere
1 0 0 √2 √2
and can be seen as separate circles. M4 = √4 0 2 2/3 2/3 − 2/3 ,
−
 √6 1/√3 1/√3 1/√3 
 −p −p −p 
 
respectively.
C. Numerical experiments After application of the variable transform, for the
NOON and correlated coherent cat states, N 1 of the
−
newvariablesfollowunivariatenormaldistributions. The
In order to simulate the centroid method numerically,
only exception is one variable which coincides with the
wefirstgeneratealargenumberofrandomphotonevents
centroid coordinate except for an overall scaling factor
distributedandcorrelatedaccordingtotherespectivepo- of √N. For the random number generation in this vari-
sition space wave function. Next, we apply a discretiza-
able,weapplythecumulativedistributionfunction. This
tionschemetomodeltheexperimentallyaccessiblesignal
function for a continuous random variable V is obtained
fordifferentdetectorcharacteristics. Finally,weperform
from the probability density function via
thecentroidanalysisonthisdatatorecoverawavefunc- I
tion. We then compare this wave function to the origi- v
(v)= (t)dt. (31)
nal position space wave function and calculate the root- FV I
mean-square deviation to access the predictive power of Z−∞
the centroid method for the given conditions. In the fol- If x are random numbers drawn from the cumulative
i
lowing, we describe all steps in more detail. distribution, 1(x ) is a random sample from .
− i
F I
6
d then follow from Eq. (8) as
0
(a)
N
d
0
X = i . (32)
n
2d N
n=1
(b) 0 X
If the detector size is changed, also the set of possible
2d detectionoutcomesischanged. Forexample,inFig.2(b),
(c) 0 detectorsofdoublethesizecomparedtothosein(a)lead
toasetofpossiblemeasurementoutcomeswhichcontain
x=0 only every second possibility compared to that of (a).
In addition, the detection array can be shifted, with an
example for shift d shown in Fig. 2(c). The measure-
FIG. 2. (Color online) Model for the detection setup. (a) 0
ments of (b) and (c) together allow for a set of potential
showsanarrayofdetectorsofsized0. Theredarrowsindicate
measurementoutcomes which coincides with that of (a).
thecentralpositionsoftheindividualdetectors. (b)showsthe
Note, however, that the detectors in (b) and (c) overlap
corresponding array with double detector size d0. Compared
to(a),onlyhalfofthepossible detectionpositions occur. (c) spatially, such that the combined measurements of (b)
shows the array of (b) suitably shifted that now the possible and (c) are not equivalent to measurement (a).
detection eventscoverthose of (a) which are missing in (b). Throughout the later analysis, we will compare the
performance with different detector sizes. In order to
obtain comparable predictions for the wave function, we
proceed as follows. For the smallest detector size d , we
0
The situation is simpler in the case of a jointly Gaus-
apply the centroid method for a single detection array
sian distribution. In this case, all transformed variables
position, which we denote as shift 0. Next, we double
followunivariateGaussiandistributions,eventhoughone
the detector size to 2d , and obtain centroid data at the
0
of the new variables is proportionalto the centroid vari-
two shift positions 0 and d . Analogously, for detectors
0
able.
of size md , we calculate centroid data for m suitable
0
A back-transformation from the variables y to the shifts. This way, the wave function is estimated at the
i
original coordinates xi via the inverse Mi−1 then yields same set of positions for all detector sizes.
thedesiredcorrelatedphotonevents. Inallcases,wever- For this analysis,we employ two different methods for
ified that the position distribution of the obtained ran- the calculations for different shifts. First, we generate
dom numbers agree to the original respective position N realizations of the correlated N-photon state as de-
0
space distribution functions. scribed in Sec. IIC1. In method I, we then use the total
number of N events for each of the m required shifts.
0
In method II, we divide the total number of events N
0
by the number of shifts m, and evaluate each shift with
N /m events only. In terms of an experimental realiza-
0
D. Modeling of the detection system
tion, in method I, the number of measurements required
increases with detector size, whereas in method II, the
number of measurements is independent of the detector
The positions of the correlated N-photon states ob-
size.
tained in the previous section are continuously dis-
tributed. Any measurement, however,employs detectors Due to the statistical nature of the measurement,
of finite size, which yield a discretized position informa- in principle, also a limited number of individual non-
tion. To model this discretization, we assume an array adjacent detectors can be used, if their positions are
of identical detectors, as shown in Fig. 2(a). Each de- shifted such that the combinations of all measurements
tector has size d . We assume that the detectors do not coverthe entire beam correlationarea. This method has
0
overlap,andthatthereis nospacebetweentwoadjacent beenadoptedin[17,37]. Whilethemeasurementtimein-
detectors. Thus, each photon position of the continuous creases, the complicated fabrication process for an array
position distribution obtained from the random number ofdetectorscanbeavoidedinproof-of-principlemeasure-
generationcan uniquely be assignedto a single detector. ments.
The numerical discretization procedure thus amounts to
replacingthepositionofeachindividualphotonwiththe
central position of the detector which registers the re- E. Error estimation
spective photon.
In a suitable coordinate system, the possible measure- To evaluate the performance of the centroid method
ment outcomes are id , with i Z. The possible out- for given detection parameters, we first fit the centroid
0
∈
comes for the centroid coordinate in a N-photon detec- data to the original distribution in order to obtain the
tion with photons hitting the i th detector (1 n N) optimumoverallscalingfactorforthe centroiddata,and
n
≤ ≤
7
1.0
0.200
0.8 n 0.100
o
2LX 0.6 eviati 00..002500
HΨ 0.4 sd 0.010
È m 0.005
0.2 r
0.002
0.0
0.0 0.2 0.4 0.6 0.8 1.0
-1.0 -0.5 0.0 0.5 1.0
detector size Hunits of ΛL
XHunits of ΛL
FIG. 4. (Color online) Root-mean-square deviation of the
FIG. 3. (Color online) Result of the centroid method for
recovered two-photon centroid distribution from the original
NOON states with different N, shown against the centroid
wavefunctionposition spacedistributionasafunctionofthe
position in units of the single-photon transverse wavelength
detector size. Note thelogarithmic scale.
λ. Continuous blue, dashed red, and dotted green curves
correspondrespectivelytoN =2,3and4. Theincreaseinthe
number of fringes clearly shows the resolution enhancement
with a detector size small enough to resolve all features
with 1/N.
of the wave function. It can be seen that there are 2N
fringesperwavelengthforN photons,confirmingthe ex-
multiplythecentroiddatawiththisfactor. Thisisneces- pected scaling of the obtained resolution with 1/N. We
sary,asweevaluatethe centroiddataonlyoveralimited thusfindthatoursimulationtechnique isableto recover
positionrange,suchthattheoverallnormalizationofthe the predicted resolution enhancement.
measured centroid data a priori is unknown. We then
employ the weighted root mean square deviation given
by
2. Dependence on the detector size
b
1
ψ(X )2 z 2, (33) Wethusturntoadetailedanalysisofthecentroidtech-
√bvui=1|| i | − i| nique. As our first step, Fig. 4 shows the dependence of
uX the rms deviation of the recovered two-photon centroid
t
where b is the number of centroid data points, and distributionfromthe originalpositionspace distribution
ψ(Xi)2 and zi are the reference value from the orig- as a function of the detector size. As expected, the de-
| |
inal probability distribution and the estimate obtained viation is low for small detectors, and initially increases
from the centroid method, respectively, at positions Xi. with growing detector size. Starting from detector sizes
Note that the rms deviation in part depends on the of about λ/2, the rms deviation saturates. Next to this
spatial extent over which the centroid data is compared generalstructure,anincreaseofthedeviationatdetector
to the originalwavefunction. If a larger rangeis consid- sizeλ/4and3λ/4,aswellassuddendropsinthermsfor
ered, positions are included into the analysis for which λ/2 and λ can be observed. Finally, towards very low
both the original probability distribution and the cen- detectorsizes,the rmsdeviationstartstoincreaseagain.
troid prediction are very low, such that they in essence Inthefollowing,wewillexplainandinterpretallofthese
donotincreasethe suminthe rmsdeviation,butonlyb. features.
This is particularly important if rms values for different First, we analyze the structures at λ/4, λ/2 and λ.
wave functions are compared. For this, we choose a detector size, and then calculate
the rms deviation as a function of a shift of the whole
detectorarrayalongthecentroidaxiswithoutmodifying
III. RESULTS
the detector size. The results areshownin Fig.5. It can
be seen that for detector sizes other than λ/4, λ/2 and
In the following, we present our results for three dif- λ, the rms deviation is approximately constant over the
ferent trial states. We start with NOON states. whole range of shifts. For these detector sizes, we con-
clude that the naive expectation that a smaller detector
leadsto abetterrecoveryofthe wavefunctionis correct.
A. NOON States
In contrast, at detector size λ/2, the rms deviation
strongly depends on the shift. Even more surprisingly,
1. Resolution enhancement whilethereisaslightdependenceonthedetectorshiftfor
detectorsizes λandλ/2, the rms deviationremainsvery
Fig.3depictsthethecentroidprobabilitydistributions low for all shifts despite the large detector size. We will
for two, three and four photon NOON states, obtained shownowthattheseresultareartifactsofthecalculation
8
1.0 æ æ æ
HaL
on 51×.1100--21 æàæàæàæàæàæàæææææàæàæàæàæàæàæàæàæàHæàvæàæàLæàæàæàæàæàæàæàæàHæàiiæiæLæææàæàHæàiæàvæàLæà 00..68
i à à à à
eviat 51.×1100--23 ìììììììàìàìììHìiìLìììììììììììììììììììàìàìììììììì 00..24æ æ æ æ
rmsd 51.×1100--34 òôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôHòôiòôiòôLòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòôòô 2HLΨX 01..00 H-bL0.3-0.2-0.1 0.0 0.1 0.2 0.3
-0.10 -0.05 0.00 0.05 0.10 È 0.8
detector shift Hunits of ΛL æ æ æ æ æ æ
0.6 * * * * * *
0.4
FIG. 5. (Color online) Root-mean-square deviation of the
recovered two-photon centroid distribution from the original 0.2
wavefunctionposition spacedistributionasafunctionofthe
0.0
detectorshift. Thedifferentcurvesshowresultsforthedetec- -0.3-0.2-0.1 0.0 0.1 0.2 0.3
torsizes(i)trianglesfacedrightλ,(ii)trianglesfacedleftλ/2,
(iii) blueblobs0.3λ,(iv)greensquaresλ/4,and(v)magenta XHunits of ΛL
rhombuses λ/10. Note thelogarithmic scale.
FIG.6. (Coloronline)Centroidanalysisfordetectorsizeλ/4.
The continuous red curve shows the probability distribution
fora2-photonNOONstate. Thebluedotsindicatetheresults
procedure, which arise due to the particular structure of
of a centroid measurement with particular detector settings.
the wave function to be recovered.
In (a), the detection array is not shifted with respect to the
Fig. 6 explains the situation for detector size λ/4. In origin of the probability distribution, such that one of the
(a), the original position distribution along the centroid centroiddatapointscoincideswiththecenterpositionX =0.
directionforatwo-photonNOONstateisshowntogether (b) The detection array is shifted to s = 13λ/200, which
−
with the centroid data obtained for detector size λ/4 corresponds to a minimum in the rms deviation in Fig. 5.
The blue dots show the centroid data with the same scaling
and zero shift. For this shift value, one of the centroid
as in (a). The green asterisks show this data fitted to the
datapointscoincideswiththepositiondistributionmax-
original distribution.
imum at X =0. It can be seen that half of the centroid
data points coincide perfectly with the original distribu-
tion - but the other half strongly deviates. The reason
is that the position distribution is zero only at single detectors of size λ or λ/2 cannot recover the wave func-
points, whereasthe large detectors cover a rangearound tion. To show this, we have taken a set of N0 =106 de-
these zeros with non-zero photo detection probability. tectionevents,calculatedthe centroiddistributionsfor a
As a result, the centroid data is non-zero, in contrast largenumber of shifts, and joined the correspondingdis-
to the probability distribution. This deviation for half tributions for the respective shifts without further pro-
of the points explains the large rms value at zero shift cessing them beforehand. Only the total set obtained
in Fig. 5. We next consider the same setting, but with from all shifts was then fitted in amplitude to the orig-
shift s= 13λ/200,corresponding to a minimum in the inal position distribution. The result was a structure-
−
rmsvalue fordetector sizeλ/4inFig.5. Fig.6(b) shows less Gaussian which did not contain the sub-wavelength
thecorrespondingcentroidresultobtainedwiththesame fringe features of the original distribution. Therefore, in
scaling of the centroiddata as in (a) as blue dots. It can conclusion,itisnotpossibletorecoverthewavefunction
be seen that the deviation to the probability distribu- with detectors of specific larger sizes, despite the small
tion is strong. But if we fit the overall scaling factor of rms deviation for specific detector shifts, which has to
the centroid data to the original distribution, then good be interpreted as an artifact of the analysis. Note, how-
agreement is obtained. This is the reason for the mini- ever,thatsuchmeasurementswithlargerdetectorscould
muminFig.5,inwhichthe centroiddata isfitted tothe potentially be sufficient to analyze the periodicity or the
original distribution for each shift position individually. symmetry of a given wave function.
The analogous analysis for detector size λ is shown in Next,weanalyzewhythermsdeviationincreasesagain
Fig. 7. In this case, due to the λ-periodicity of the wave at very small detector sizes. We found that this is a sta-
function, a perfect fit of the centroid data is possible for tistical effect. If the detector size is decreased while the
any detector shift. detection area is kept constant, the number of detectors
However, in an actual measurement, results from all increases. Then, the mean number of events per detec-
shiftswouldhavetobeaccountedforwiththesameover- tor decreases with the detector size. At some point, the
allprefactor,asanindividualfittingtothe unknowndis- statistical fluctuations due to the decreasing number of
tributiontobedeterminedisimpossible. Forthisreason, events become large enough to dominate the rms devia-
9
1.0 æ
æ æ 0.050 æìàò
0.8 HaL æìàò æìàò æìàò
æìàò
00..46 viation 0.020 HiìòiiLìò HòivòL ò ò ò ìò æìàò æìàò æìàò æìàò æìàò
0.2 msde 0.010 æà à ìàHiiìàL ìà æìà æìà æà
2LX0.0 -0.6-0.4-0.2 0.0 0.2 0.4 0.6 r 0.005 æ æ æ æ HiL
1.0 æ
HΨ æ æ
È0.8 HbL 0.00 0.05 0.10 0.15
detector size Hunits of ΛL
0.6
* * *
0.4 FIG.8. (Coloronline)Effectofstatisticalfluctuationsonthe
centroid methodsfor small detector sizes. Therms deviation
0.2 was evaluated (i) black blobs, once for 106 events, (ii) red
squares, twice for two subsets with 5 105 events, (iii) blue
0.0 -0.6-0.4-0.2 0.0 0.2 0.4 0.6 rhombuses,5timesforsubsetsof2 10×5events,and(iv)green
XHunits of ΛL triangles, 10 times for subsets of 1×05 events. In all cases, the
same 106 events were analyzed, and the respective subsets
were chosen disjunct. The figure shows the results averaged
FIG. 7. (Color online) Centroid analysis for detector size λ.
overthe respectivesubsets. Note thelogarithmic scale.
ThefigureisanalogoustoFig.6,exceptforthedetectorsize.
tor size is already limited by the low number of events
tion. In order to verify this interpretation, we evaluated
the rms deviation (i) once for 106 events, (ii) twice for per detection bin. For the next higher detector size 2d0,
two subsets with 5 105 events, (iii) 5 times for subsets inmethodI,thenumberofeventsperdetectionbinisin-
of 2 105 events, ×and (iv) 10 times for subsets of 105 creased,asthesamenumberofN0 eventsaredistributed
event×s. In all cases, the same 106 events were analyzed, over a lower number of bins. In contrast, for method II,
the number of events per bin is not increased, as with
and the respective subsets were chosen disjunct. After
increasingdetectorsize,the number ofeventsconsidered
this, we averagedthe results over the respective subsets,
for each detector shift is reduced. As a consequence,
such that in every case, the same events were analyzed.
the rms deviation for method II monotonously increases
The result is shown in Fig. 8. It can be seen that the
with detector size, whereas that for method I can de-
rms deviation of all cases approximately agrees towards
crease with increasing detector size. In the latter case,
largerdetector sizes. But towards smaller detector sizes,
the reduction of the rms due more events per bin on av-
the smaller the corresponding subsets are, the more the
erage outweighs the increase in rms due to the increase
deviation increases. Note that also the difference in rms
of the detector size.
deviation between different subsets of equal size is much
smaller than the difference in deviation between subsets This interpretation is further supported by the two
of different size. other data sets in Fig. 9. These show method II, but
with half the number of events (green diamonds) or one
third of the events (black triangles). It can be seen that
the rms results for the smallest detector size d roughly
3. Comparison of the two methods 0
agree with those for method II with half the data points
atdetectorsize2d . Further,thermsformethodIIwith
0
In this section, we compare the two analysis methods
onethirdofthedatapointsagreestothis valueatdetec-
IandIIintroducedinSec.IID.Fig.9showsresultsfora
tor size 3d . This shows that for such small detectors,
0
2-photon NOON state using the two methods. The two
the rms deviation is dominated by the low number of
curvesformethodI(bluesquares)andII(reddots)agree
counts per bin.
for larger detectors, but deviate for small detectors. In
We obtained qualitatively similar results also for N
particular, the unexpected increase in the rms deviation ∈
3,4 .
towards low detector sizes observed in Fig. 8 and inter- { }
preted as statistical fluctuations due to a low number of
events per detection bin is absent for method II.
4. Multiphoton NOON states
The reason for this qualitative difference is as follows.
For the lowest detector size d , no shift is required, and
0
the two methods are equivalent. Thus, the same rms de- WenowturntoNOONstateswithhigherphotonnum-
viation is obtained. As can be seen from the increase in ber. Fig.10showsthe rmsdeviationasafunction ofthe
the rms value for method I, the prediction at this detec- detector size for N = 2, 3 and 4 photons. Note that
10
0.06
0.012 ò æìàò
0.05 0.008 æìà æìò æìò æìàò æìàò 0.20
tion 0.04 0.004 à à æìàò æìàò ion 0.15
evia 0.03 0.01 0.02 0.03 æ æìàò æìàò viat
d æ ìàò e 0.10
s 0.02 æ ìàò d
rm 0.01 æìàò æìàò æìàò æìàò æìàò æìàò æìàò æìàò æìàò ìàò rms 0.05
0.00
0.00 0.05 0.10 0.15 0.00
0.0 0.1 0.2 0.3 0.4
detector size Hunits of ΛL
detector sizeHunits of ΛL
FIG. 9. (Color online) (a) Root-mean-square deviation as a
functionofdetectorsize. Resultsareshownforatwo-photon FIG. 10. (Color online) Root-mean-square deviation versus
NOON state. The blue squares are obtained using method I detector size for NOON states using method I. Results are
discussedinthetext,whereasthereddotsareobtainedusing compared for number of photons N = 2 (lower blue curve),
method II. The green diamonds and the black triangles are N = 3 (middle red curve), and N = 4 (upper green curve).
obtained using method II with one half and one third of the The lines indicate linear fits to the curves over the indicated
total number of events used in (a), respectively. Inset shows detector size range.
a magnification at small detector sizes.
δxisagainanormaldistribution,withwidth√Nδx. Ap-
Fig. 10 has been generated including all detector shifts proximately, it also holds for other distributions. To-
required for the respective detector sizes, such that the gether with the prefactor 1/N in the centroidvariable, a
spurious maxima or minima in the rms deviation at spe- scaling of 1/√N in the centroid coordinate uncertainty
cificdetectorssizesfoundinFig.4duetothedependence isobtained. Next,weusethatanincreaseofthedetector
on the shift position shown in Fig. 5 do not appear. All size fromthe low-sizelimit leads to mostsignificantcon-
datapointsaregeneratedusingthesameN =106events tributions to the rms deviation in the regions of highest
0
with method I introduced in Sec. IIIA3. As the spatial slopeofthe centroidprobabilitydistribution. We denote
extent over which the probability distribution is signifi- such a centroid coordinate with highest slope as x ,
max
cantly larger than zero decreases with increasing photon and note that the value of the cos2() part in the proba-
number N, we adjust the position range over which the bilitydistributionevaluatesto1/2atthispoint. Wethus
wave function is matched to the centroid measurement can estimate the scaling of the rms deviation via
accordingly. Thus, the 3- and 4-photon cases are eval-
uated over 2/3 and 2/4 = 1/2 the range of that of the 1 xmax+δX/2 1 2
N =2 case, respectively. cos2(2πNx) dx. (34)
It can be seen that at small detector sizes, the rms √δXsZxmax−δX/2 (cid:12)(cid:12) − 2(cid:12)(cid:12)
deviation is low and approximately independent of N. To leading order in the(cid:12)(cid:12)detector size, th(cid:12)(cid:12)is expression
In this limit, the detectors are chosen small enough to scales as √Nd , i.e., linearly with the detector size, and
0
recover all features even of the N = 4 wave function. with slope proportional to √N, as observed in the nu-
With increasing detector size, the rms deviation starts merical data.
differing from the small-detector limit value first for the
N =4,thenfortheN =3,andfinallyfortheN =2case.
Increasingthedetectorsizefurther,thermsdeviationhas 5. Single- and multi-photon detection
an approximately linear dependence of the detector size,
until it eventually saturates for the large-detector limit.
One motivation for the centroid measurement method
In the near-linear region, we estimate slopes α = is the technical difficulty to achieve multi-photon or
∆(rms deviation)/∆(detectorsize)forN =2,3and4as photon-number resolving detection, as required, e.g.,
0.57, 0.72 and 0.87, as indicated in Fig. 10, respectively. for sub-wavelengthcorrelatedmultiphoton measurement
Thesevaluesareconsistentwithascalingwith√N. One schemes. If the size of the individual detectors in the
possibleinterpretationofthisscalingcouldbeasfollows. centroid method is small enough, the number of events
If the detection positions of each of the N photons ac- in which two or more photons hit the same detector is
quiresanuncertaintyoforderd0duetothefinitedetector negligibly small, such that no photon number resolution
size, then an estimate of the uncertainty of the centroid is required.
coordinate X = 1 N x is given by δX = d /√N. Motivated by this, we have analyzed the percentage
N i=1 i 0
ThisestimateismotivatedbythefactthatthesumofN of two-photon events in our numerical calculations, as a
P
normally distributed independent variables with widths function of detector size. We denote the position range