Truncated states obtained by iteration

1.4 MB·English

by  W. B. Cardoso

Truncated states obtained by iteration
W. B. Cardoso1,∗ and N. G. de Almeida2,1
1Instituto de F´ısica, Universidade Federal de Goi´as, 74.001-970, Goiˆania (GO), Brazil
2Nu´cleo de Pesquisas em F´ısica, Universidade Cat´olica de Goi´as, 74.605-220, Goiˆania (GO), Brazil.
Quantum states of the electromagnetic field are of considerable importance, finding potential
application in various areas of physics, as diverse as solid state physics, quantum communication
and cosmology. In this paper we introduce the concept of truncated states obtained via iterative
processes(TSI)andstudyitsstatisticalfeatures,makingananalogywithdynamicalsystemstheory
(DST).Asaspecificexample,wehavestudiedTSIforthedoublingandthelogisticfunctions,which
are standard functions in studying chaos. TSI for both the doubling and logistic functions exhibit
certainsimilarpatternswhentheirstatistical featuresarecomparedfromthepointofviewofDST.
A general method to engineer TSI in the running-wave domain is employed, which includes the
errors dueto thenonidealities of detectors and photocounts.
7
0
PACSnumbers: 42.50.Dv,42.50.-p
0
2

n I. INTRODUCTION nonchaotic) according to DST, and, for some iterating
a
functions, we found properties of TSI very sensitive (re-
J
Quantum state engineering is an area of growing im- sembling chaos) to the first coefficient C0, which is used
6
as a seed to obtain the remaining C .
portanceinquantumoptics,itsrelevancelyingmainlyin n
This paper is organized as follows. In section 2 we in-
1 the potential applications in other areas of physics, such
v as quantum teleportation [1], quantum computation [2], troducetheTSIandinsection3weanalyzethebehavior
6 quantum communication [3], quantum cryptography [4], ofsomeofitspropertiesastheHilbertspacedimensionis
2 increased. In section 4 we show how to engineer the TSI
quantum lithography [5], decoherence of states [6], and
0 inthe running-wavefielddomain,andthe corresponding
so on. To give a few examples of their usefulness and
1
engineering fidelity is studied in section 5. In section 6
relevance, quantum states arise in the study of quan-
0
we present our conclusions.
7 tum decoherence effects in mesoscopic fields [7]; entan-
0 gled states and quantum correlations [8]; interference in
/ phasespace[9];collapsesandrevivalsofatomicinversion
h
II. TRUNCATED STATES OBTAINED VIA
[10]; engineering of (quantum state) reservoirs [11]; etc.
p
ITERATION (TSI)
- Also,itisworthmentioningtheimportanceofthestatis-
nt ticalpropertiesofonestateindeterminingsomerelevant
a properties of another [12], as well as the use of specific We define TSI as
u quantum states as input to engineer a desired state [13].
N
q
DynamicalSystemsTheory(DST),ontheotherhand, TSI = C n (1)
v: isacompletelydifferentareaofstudy,whoseinterestlies | i X n| i
n=0
i mainly in nonlinear phenomena, the source of chaotic
X
phenomena. DSTgroupsseveralapproachestothestudy where Cn is the normalized complex coefficient obtained
ar of chaos, involving Lyapunov exponent, fractal dimen- as the nth iteration of a previously given generating
sion, bifurcation, and symbolic dynamics among other function. For example, given C0,Cn can be the nth it-
elements[14]. Recently,otherapproacheshavebeencon- erate of the quadratic functions: Cn(µ) = Cn2−1 + µ;
sidered,suchasinformationdynamicsandentropicchaos sine functions: Cn(µ) = µsin(Cn−1); logistic functions
degree [15]. Cn(µ)=µCn−1(1 Cn−1); exponential functions: Cn =
−
The purpose of this paper is twofold: to introduce µexp(Cn−1); doubling function defined on the interval
novel states of electromagnetic fields, namely truncated [0,1): Cn = 2Cn−1 mod 1, and so on, µ being a param-
states having coefficients obtained via iterative process eter. It is worth recalling that all the functions in the
(TSI), and to study chaos phenomena using standard above list are familiar to researchers in the field of dy-
techniquesfromquantumoptics,makingananalogywith namical systems theory (DST). For example, for some
DST. We note that, unlike previousstatesstudied inthe values of µ, it is known that some of these functions can
literature [16], each coefficient of the TSI is obtained behave in quite a chaotic manner [14]. Also, note that
from the previous one by iteration of a function. Fea- by computing all the Cn we are in fact determining the
tures of this state are studied by analyzing several of its orbit ofagivenfunction,andbecausetheCn andPn,the
statistical properties in different regimes (chaotic versus photon number distribution, are related by Pn = Cn 2,
| |
fixed or periodic points of a function will correspond to
fixed or periodic P . Rather than studying all the func-
n
tions listed in this section, we will focus on the doubling
∗Correspondingauthor: [email protected] function and the logistic function. These two functions
2
have been widely used to understand chaos in nature.
As we shall see in the following, although very different
fromeachother,thesefunctionsgiverisetodifferentTSI
having similar patterns.
III. STATISTICAL PROPERTIES OF TSI USING
THE DOUBLING AND THE LOGISTIC
FUNCTIONS
A. Photon Number Distribution
Since the expansion of TSI is known in the number
state n , we have
| i
P = C 2. (2)
n n
| |
Figs. 1 and 2 show the plots of the photon-number dis-
tribution P versus n for TSI using the doubling func-
n
tion. The Hilbert space dimension is N = 50. In order FIG.1: Photonnumberdistributionforthedoublingfunction
to illuminate the behavior of TSI for different values of with C0 =0.3. Note a four-period type for the probabilities;
this regular behavior coincides with nonchaotic behavior of
C ,we takeC as0.3and0.29711,respectivelyshownin
0 0 thedoubling function in theDST sense.
Figs. 1 and 2. Note the regular behavior for C = 0.3
0
and rather an irregular, or chaotic, behavior for C =
0
0.29711. Figs. 3 and 4 show P for the logistic func-
n
tion. For C = 0.2 and µ = 3.49 the logistic function
0
behavesregularly(Fig.3), showingclearly(as in the case
ofthe doubling-function)four valuesforP ;by contrast,
n
for µ = 4 and C = 0.2, P oscillates quite irregularly
0 n
(Fig.4). This is so because the photon number distri-
bution is equivalent to the orbit of the TSI dynamics
[14]. Thus, once a fixed - attracting or periodic - point
is attained, the subsequent coefficients, and hence the
subsequent P , will behave in a regular manner. Con-
n
versely, when no fixed point exists, P will oscillate in
n
a chaotic manner. Therefore, by choosing suitable C
0
and/or µ, we can compare the properties of TSI when
differentregimes(chaoticversus nonchaotic)in the DST
sense are encountered. Note the similarity between the
propertiesofthelogisticandthedoublingfunctionswhen
the DST regimes are the same. Interestingly, these simi-
larities are observedwhen other properties areanalyzed,
as we shall see in the following.
FIG.2: Photonnumberdistributionforthedoublingfunction
with C0 = 0.29711. This irregular behavior for Pn coincides
B. Even and Odd Photon number distribution withthechaoticbehaviorofthedoublingfunctionintheDST
sense.
The functions P and P represent the photon
odd even
number distribution for n odd and even, respectively,
givenbyEq. (2). Itiswellestablishedinquantumoptics in DST), note that TSI has a classical analog as N in-
[17]thatifP >0.5theGlauber-SudarshanP-function creases. From Figs. 6 and 8 (corresponding to a chaotic
odd
assumes negative values, prohibited in the usual proba- regime in DST), TSI can behave as a nonclassical state,
bility distribution function, and the quantum state has depending on N. More interestingly, note the following
no classical analog. Since P +P = 1, the same is pattern: whenever the coefficients of TSI correspond to
odd even
true whenP <0.5. Figs. 5 and6showthe behavior the nonchaotic regime in DST, P (and so P ) will
even odd even
of P for C = 0.3 and C = 0.29711 for the doubling remain above or below 0.5 on a nearly monotonic curve,
odd 0 0
function the Hilbert space N is increased. Figs. 7 and 8 asseeninFigs. 5and7; wheneverthe coefficientsofTSI
refer to the the logistic function for µ=3.49 and µ=4. correspond to the chaotic regime in DST, P (and so
odd
In Figs. 5 and 7 (corresponding to a nonchaotic regime P ) will tend to oscillate around 0.5 (Figs. 6 and 8).
even
3
FIG. 5: Even (solid) and odd (dots) photon number distri-
butions for the doubling function. We haveused C0 =0.3 to
FIG. 3: Photon number distribution for the logistic function coincide with nonchaotic behavior in theDSTsense.
with C0 = 0.2 and µ = 3.49. This regular or “four-period”
type behavior for the probabilities coincides with nonchaotic
behavior in the DSTsense.
FIG.6: Even(solid)andodd(dots)photonnumberdistribu-
tions for the doubling function. C0 =0.29711 was chosen to
coincide with chaotic behavior in theDST sense.
FIG. 4: Photon number distribution for the logistic function
with C0 = 0.2 and µ = 4. This irregular behavior of Pn Fig. 9 shows the plot of hnˆi and Fig. 10 the plot of
coincideswith thechaoticbehaviorofthelogistic functionin ∆nˆ as functions of the dimension N of Hilbert space,
h i
theDST sense. for the doubling function. Note the near linear behav-
ior of the average photon number and its variance as
N increases for C = 0.3 (nonchaotic regime in DST);
0
C. Average number and variance
this is not seen when C = 0.29711 (chaotic regime in
0
DST). Figs. 11 and 12 for the logistic function show es-
The averagenumber nˆ andthe variance ∆nˆ inTSI sentiallythesamebehaviorwhenthesetwoDSTregimes
h i h i
are obtained straightforwardlyfrom are shown together.
N
nˆ = P(n)n, (3)
h i X
n=0 D. Mandel parameter and second order correlation
function
and
∆nˆ =q nˆ2 nˆ 2. The Mandel Q parameter is defined as
h i h i−h i
4
FIG.7: Even(solid)andodd(dots)photonnumberdistribu- FIG. 9: Average photon number for the doubling function.
tions for the logistic function. Here we chose C0 = 0.2 and Here we chose C0 = 0.3 (dots) and C0 = 0.29711 (solid),
µ = 3.49 to coincide with nonchaotic behavior in the DST allowing theHilbert space N to increase to 50.
sense.
FIG. 10: Variance of the photon number for the doubling
FIG.8: Even(solid)andodd(dots)photonnumberdistribu- function. Here we chose C0 = 0.3 (dots) as well as C0 =
tions for the logistic function. Here we chose C0 = 0.2 and 0.29711(solid), allowing the Hilbert space N to increase to
µ=4 to coincide with chaotic behavior in theDSTsense.
50.
Ifg(2)(0)<0,thenthe Glauber-SudarshanP-function
assumes negative values, outside the range of the usual
∆nˆ2 nˆ
Q= (cid:0) −h i(cid:1), (4) probability distribution function. Moreover, by Eq. (5)
nˆ it is readily seen that g(2)(0) < 1 implies Q < 0. As
h i
for a coherent state Q = 0, a given state is said to be a
while the second order correlation function g(2)(0) is
“classical” one if Q>0.
Figs. 13 to 16 show the plots of the Q parameter and
nˆ2 nˆ
g(2)(0)= (cid:0)(cid:10) (cid:11)−h i(cid:1), the correlation function g(2)(0) versus N , for both the
nˆ 2 doubling and the logistic functions. Note that TSI is
h i
predominantly super-Poissonian for these two functions
and for Q < 0 (Q > 0) the state is said to be sub- (Q > 0 and g(2)(0)˙ > 1), thus being a “classical” state
Poissonian (super-Poissonian). Also, the Q parameter
in this sense for N & 12, while for small values of N
andthesecondordercorrelationfunctiong(2) arerelated
(N < 12), the Q parameter is less than 0, showing
by [18]
sub-Poissonian statistics and is thus associated with a
“quantum state”. From Figs.13 and 15, note that us-
Q=hg(2)(0)−1ihnˆi. (5) ing C0 = 0.3 for the doubling function, C0 = 0.2 and
5
FIG. 11: Average photon number for the logistic function. FIG. 13: Q parameter for the doubling function. Here we
HerewechoseC0 =0.2withµ=3.49(dots)orµ=4(solid), chose C0 = 0.3 (dots) and C0 = 0.29711 (solid), allowing
allowing theHilbert space N to increase to 50. the Hilbert space N to increase to 50. Note the irregular
behavior of the Q parameter when C0 = 0.29711, coinciding
with chaotic behavior in the DSTsense.
FIG.12: Varianceofthephotonnumberforthelogisticfunc- FIG. 14: Second order correlation function for the doubling
tion. Here we chose C0 = 0.2 with µ= 3.49 (dots) or µ= 4 function. Here we chose C0 = 0.3 (dots) and C0 = 0.29711
(solid), allowing the Hilbert space N to increase to 50. (solid), allowing the Hilbert space N to increase to 50. Note
the irregular rise of the g(0) function when C0 = 0.29711,
coinciding with chaotic behavior in theDST sense.
µ = 3.49 for the logistic function (nonchaotic regime in
E. Quadrature and variance
DST), Q shows a linear dependence on N. However, us-
ingC =0.29711forthedoublingfunction,C =0.2and
0 0
Quadrature operators are defined as
µ = 4 for the logistic function (chaotic regime in DST),
Q oscillates irregularly. Similarly, from Figs. 14 and 16,
X = 1 a+a† ;
using C0 = 0.3 for the doubling function, C0 = 0.2 and 1 2(cid:0) (cid:1)
µ = 3.49 for the logistic function, we see that g(2)(0) X = 1 a a† . (6)
2 2i(cid:0) − (cid:1)
increases smoothly, while using C = 0.29711 for the
0
doubling function, C = 0.2 and µ = 4 for the logistic where a (a†) is the annihilation (creation) operator in
0
function, the rise of g(2)(0) is rather irregular. This pat- Fock space. Quantum effects arise when the variance of
ternisobservedforotherC asinputaswellasforother one of the two quadratures attains a value ∆X < 0.5,
0 i
valuesofthe parameterµ,andwheneverthedynamicsis i = 1,2. Figs. 17 and 18 show the plots of quadrature
chaotic(regular),theQ parameterandthe g(2)(0)corre- variance ∆X versus N. Note in this figures that vari-
i
lation function oscillate irregularly (regularly). ances increase when N is increased.
6
FIG.15: Qparameterforthelogisticfunction. Herewechose FIG. 17: Averaged quadratures for the doubling function
C0 =0.2,withµ=3.49(dots)andµ=4(solid),allowingthe whentheHilbert spaceN is increased to50. Doted (dashed-
Hilbert space N to increase up to 50. Note the irregular be- doted) line refer to quadrature variance 1 (2) for C0 = 0.3;
havior of theQ parameter when µ=4, exactly when chaotic solid (dashed) line refer to quadrature variance 1 (2) for
behavior occurs according to DST. C0 =0.29711.
FIG. 16: Second order correlation function for the logistic FIG. 18: Averaged quadratures for thelogistic function with
function. Here we chose C0 = 0.2, with µ = 3.49 (dots) and C0 =0.2 when theHilbert space N is increased to 50. Doted
µ=4 (solid), allowing theHilbert space N to increase up to (dashed-doted)linerefertoquadraturevariance1(2)forµ=
50. Notetheirregularriseofg2(0)whenµ=4,exactlywhen 3.49; solid (dashed) refer to quadrature variance 1 (2) for
chaotic behavior occurs according to DST. µ=4.
F. Husimi -Q function are compared, Husimi Q-functions show essentially no
difference from each other.
The Husimi Q-function for TSI is given by
1
Q (β)= β TSI 2, (7) IV. GENERATION OF TSI
|TSIi π |h | i|
where β is a coherent state. Figs. 19 to 22 show the TSIcanbe generatedinvariouscontexts,asforexam-
HusimiQ-functionforboththe doublingandthe logistic ple trappedions[19], cavityQED[13,20],andtravelling
functions for N = 15. For the doubling function, we use wave-fields[21]. Butduetoseverelimitationimposedby
C = 0.29711 and C = 0.3, respectively, and for the coherence loss and damping, we will employ the scheme
0 0
logisticfunctionweuseµ=3.49andµ=4,respectively, introduced by Dakna et al. [21] in the realm of run-
for the chaotic and nonchaotic regimes. Interestingly, ningwavefield. Forbrevity,the presentapplicationonly
evenwhenthechaoticandnonchaoticregimesoftheDST shows the relevant steps of Ref.[21], where the reader
7
FIG.19: HusimiQfunctionfordoublingfunction. HereC0 = FIG. 21: Husimi Q function for logistic function. Here C0 =
0.3. 0.2 and µ=3.49.
FIG. 22: Husimi Q function for logistic function. Here C0 =
FIG.20: HusimiQfunctionfordoublingfunction. HereC0 = 0.2 and µ=4.
0.29711.
where Dˆ(β ) stands for the displacement operator and
n
the β are the roots of the polynomial equation
n
willfindmoredetails. Inthis scheme,adesiredstate Ψ
| i N
composed of a finite number of Fock states n can be
written as | i XCnβn =0. (9)
n=0
AccordingtotheexperimentalsetupshownintheFig.1
N N
Ψ = C n = CN aˆ+ β∗ 0 of Ref.[21], we have (assuming 0-photon registered in all
| i X n| i √N! Y(cid:0) − n(cid:1)| i detectors) that the outcome state is
n=0 n=1
N N
C
= N Dˆ(β )aˆ+Dˆ(β ) 0 , (8) Ψ D(α )aˆ+TnˆD(α ) 0 , (10)
√N! Y k k | i | i∼ Y k+1 k | i
k=1 k=1
8
TABLE I: The roots βk∗ = |βk|e−iϕβk of the characteristic TABLE III: The roots βk∗ = |βk|e−iϕβk of the characteristic
polynomialandthedisplacementparametersα∗k =|αk|e−iϕαk polynomialandthedisplacementparametersα∗k =|αk|e−iϕαk
are given for TSI using the doubling function for C0 = 0.3 are given for TSI using the logistic function for C0 = 0.2,
(coincidingwithnonchaoticbehaviorintheDSTsense),N = µ = 3.49 (coinciding with nonchaotic behavior in the DST
5 and T = 0.862. The probability of producing the state is sense), N = 5 and T = 0.893. The probability of producing
0.22%. thestate is 0.11%
N |β | ϕ |α | ϕ N |β | ϕ |α | ϕ
k βk k αk k βk k αk
1 2.169 2.638 1.187 -0.220 1 3.948 3.141 2.794 0.051
2 2.169 -2.638 1.155 1.570 2 0.609 2.566 2.195 -3.045
3 0.545 3.141 1.096 -2.483 3 0.609 -2.566 0.472 1.570
4 1.460 1.084 1.323 -2.331 4 1.828 1.373 1.830 -1.959
5 1.460 -1.084 2.225 1.570 5 1.828 -1.373 3.202 1.570
6 1.460 -1.084 6 1.828 -1.373
pToAlyBnLoEmiIaIl:aTnhdethroeodtisspβlak∗ce=m|eβnkt|pe−ariϕaβmketoefrsthαe∗kc=ha|αrakc|tee−riiϕstαikc TpoAlyBnLoEmiIaVl:anTdhethreodoitsspβlak∗ce=me|βnkt|pea−riaϕmβketoefrsthαe∗kc=ha|αrakc|tee−riiϕstαikc
are given for a TSI using the doubling function for C0 = are given for TSI using the logistic function for C0 = 0.2,
0.29711 (coincidingwith chaoticbehaviorin theDSTsense), µ = 4 (coinciding with chaotic behavior in the DST sense),
N =5andT =0.867. Theprobabilityofproducingthestate N =5andT =0.879. Theprobabilityofproducingthestate
is 0.21% is 0.15%
N |βk| ϕβk |αk| ϕαk N |βk| ϕβk |αk| ϕαk
1 2.306 2.692 1.372 -0.198 1 3.290 3.141 2.027 0.094
2 2.306 -2.692 1.130 1.570 2 0.563 2.708 1.665 -3.056
3 0.543 3.141 1.193 -2.563 3 0.563 -2.708 0.321 1.570
4 1.489 1.089 1.357 -2.321 4 1.893 1.255 1.787 -2.064
5 1.489 -1.089 2.289 1.570 5 1.893 -1.255 3.165 1.570
6 1.489 -1.089 6 1.893 -1.255
now take into account the quantum efficiency η at the
where T is the transmittance of the beam splitter and
photodetectors. For this purpose, we use the Langevin
α are experimental parameters. After some algebra,
k operator technique as introduced in [22] to obtain the
the Eq.(8) and Eq.(10) can be connected. In this
fidelity to get the TSI.
way, one shows that they become identical when α =
1 Output operators accounting for the detection of a
N T−lα and α = T∗N−k+1(β β ) for
−Pl=1 l+1 k k−1 − k given field αˆ reaching the detectors are given by [22]
k = 2,3,4...N. In the present case the coefficients C
n
aregivenbythoseofthe TSI.Therootsβk∗ =|βk|e−iϕβk
of the characteristic polynomial in Eq.(9) and the dis- α =√ηα +L , (11)
out in α
TplaabcleemseIn;ItI;pIIaIraamndetIeVrs,αfokr=N|α=k5|e.−iϕαk are shownin the where η stands forbthe efficiebncy ofbthe detector and Lα,
For N = 5, the best probability of producing TSI is actingontheenvironmentstates,isthenoiseorLangebvin
0.22% when the doubling function is used, and 0.15% operator associated with losses into the detectors placed
when the logistic funcion is used. The beam-splitter inthepathofmodesα=a,b. Weassumethatthedetec-
transmittancewhichoptimizesthis probabilityisaround tors couple neither dibfferent modes a,b nor the Langevin
T =0.878. operatorsLα,sothefollowingcommutationrelationsare
readily obtbained from Eq.(11):
V. FIDELITY OF GENERATION OF TSI L ,L† = 1 η, (12)
h α αi −
b b
Until now we have assumed all detectors and beam- L ,L† = 0. (13)
h α βi
splitters asideal. Althoughverygoodbeam-splittersare b b
available by advanced technology, the same is not true Theground-stateexpectationvaluesforpairsofLangevin
for photo-detectors in the optical domain. Thus, let us operators are
9
for the doubling function, we find F 0.9983, 0.9943
≃
and 0.9909, respectively, and for the logistic function,
DLαL†αE = 1−η, (14) starting with C0 = 0.2, µ = 3.49 and µ = 4, we find
b b F 0.9986, 0.9944 and 0.9911, respectively. These high
DLαL†βE = 0, (15) fid≃elities show that efficiencies around 0.9 lead to states
b b whose degradation due to losses is not so dramatic for
which are useful relations specially for optical frequen- N =5.
cies, when the state of the environment can be very well
approximated by the vacuum state, even for room tem-
perature. VI. COMMENTS AND CONCLUSION
Let us now apply the scheme of the Ref.[21] to the
present case. For simplicity we will assume all detectors In this paper we have introduced new states of the
having high efficiency (η & 0.9). This assumption al- quantized electromagnetic field, named truncated states
lowsustosimplify the resultingexpressionbyneglecting withprobabilityamplitudesobtainedthroughiterationof
terms of order higher than (1 η)2. When we do that, afunction(TSI).AlthoughTSIcanbebuildingusingvar-
−
instead of TSI , we find the (mixed) state ΨFE de- iousfunctionssuchaslogistic,sine,exponentialfunctions
| i | i
scribing the field plus environment, the latter being due andsoon,wehavefocusedourattentiononthedoubling
to losses coming from the nonunit efficiency detectors. and the logistic functions, which, as is well known from
We have, dynamicalsystemstheory,canexhibitachaoticbehavior
intheinterval(0,1]. TocharacterizetheTSIforthedou-
ΨFE RND(αN+1)aˆ†TnˆD(αN)aˆ†Tnˆ blingandlogisticfunctionswehavestudiedvariousofits
| i ∼
D(α )...aˆ†TnˆD(α ) 0 L† features, including some statistical properties, as well as
+× RN−N1D−1(α )aˆ†TnˆD1(α| i)aˆb†0Tnˆ the behavior of these features when the dimension N of
N+1 N Hilbert space is increased. Interesting, we found a tran-
× D(αN−1)...L†1TnˆD(α1)|0i sition from sub-poissonian statistics to super-poissonian
+ RN−1D(α b)aˆ†TnˆD(α )L† Tnˆ statistics when N is relatively small (N 12). Besides,
N+1 N N−1 photon number distribution, which is an∼alogous to con-
D(αN−1)...aˆ†TnˆD(α1) 0 b ceptoforbitsinthestudyofthedynamicofmaps,shows
× | i
+ RN−1D(α )L† TnˆD(α )aˆ†Tnˆ aregularorrathera“chaotic”behaviordependingonex-
N+1 N N
istingornotfixedorperiodicpointsinthefunctiontobe
D(α )...aˆ†TbnˆD(α ) 0 , (16)
× N−1 1 | i iterated. Interestingly enough, we have found a pattern
whenthepropertiesof TSIforlogisticfunctionarecom-
where, for brevity, we have omitted the kets correspond-
pared with that of TSI for the doubling function from
ing to the environment. Here R is the reflectance of the
the point of view of dynamical systems theory (DST).
beam splitter, L† = 1 is the identity operator and L ,
0 k Forexample, asP has ananalogwith orbits fromDST,
k =1,2..N standbsforlossesinthefirst,second...N bth n
− itisstraightforwardtoidentifyrepetitions(orperiods)in
detector. Although the L s commute with any system
k P , if there are any,when the Hilbert space is increased.
′ n
operator, we have maintbained the order above to keep Surprisingly,althoughthedoublingandthelogisticfunc-
clear the set of possibilities for photo absorption: the
tion aredifferent from eachother, when other properties
first term, which includes L†0 = 1, indicates the proba- such as even and odd photon number distribution, the
bility for nonabsorption; thbe second term, which include average number and its variance, the Mandel parameter
L+, indicates the probability for absorption in the first and the second order correlation function were studied,
1
bdetector; and so on. Note that in case of absorption at they presented the same following pattern: if, from the
the k-thdetector, the annihilationoperatora is replaced pointofview ofDST, the coefficients ofTSI forthe dou-
by the L† creation Langevin operator. Other possibili- blingandthelogisticfunctioncorrespondtoanonchaotic
k
tiessuchbasabsorptioninmorethanonedetectorleadto (chaotic) regime, all those properties increases smoothly
a probability of order lesser than (1 η)2, which will be (irregularly) when the Hilbert space is increased.
−
neglected.
Next, we have to compute the fidelity [23], F =
Ψ Ψ 2, where Ψ is the ideal state given by VII. ACKNOWLEDGMENTS
FE
kh | ik | i
Eq.(10), here corresponding to the TSI characterized by
the parameters shown in Tables I-IV, and Ψ is the NGA thanks CNPq, Brazilian agency, and VPG-
FE
| i
state given in the Eq.(16). Assuming η = 0.99, 0.95 Universidade Cat´olica de Goi´as, and WBC thanks
and 0.90 and starting with C = 0.3 and C = 0.29711 CAPES, for partially supporting this work.
0 0
[1] C. H. Bennett et al 1993 Phys. Rev. Lett. 70 1895. [2] B.E.Kane1998Nature 393143,andreferencestherein.
10
[3] T. Pellizzari 1997 Phys. Rev. Lett. 79 5242, and refer- [14] R. L. Devaney, An Introduction to Chaotic Dynami-
ences therein. cal Systems, Second Edition, Addison-Wesley, Redwood
[4] N.Gisin,G.Ribordy,W.TitelandH.Zbinden2002Rev. City, Calif. (1989).
Mod. Phys. 74 145. [15] M. Ohya 1998 International Journal of Theoretical
[5] see,e.g.,G.Bj¨orkandL.L.Sanchez-Soto2001Phys.Rev. Physics 37 No.1 495.
Lett. 86 4516; M. Mu¨tzel et al 2002 Phys. Rev. Lett. 88 [16] For an excelent review, see V. V. Dodonov 2002 J. Opt.
083601, and refs. therein. B: Quantum Semiclass. Opt. 4 R1-R33, and references
[6] W.H. Zurek 1991 Phys. Today 44 36; C.C. Gerry and therein.
P.L. Knight (1997) Am. J. Phys. 65 964; B.T.H. Varcoe [17] Leonard Mandel, Emil Wolf, Optical Coherence and
et al 2000 Nature 403 743. Quantum Optics, Cambridge UniversityPress (1995).
[7] J. M. Raimond, M. Brune, and S. Haroche 1996 Phys. [18] D. F. Walls, G. J. Milburn, Quantum Optics, Springer-
Rev. Lett. 79 1964; S. Osnaghi, P. Bertet, A. Auffeves, Verlag, (Berlin, 1994).
P. Maioli, M. Brune, J. M. Raimond, and S. Haroche [19] R.M.Serra,P.B.Ramos,N.G.deAlmeida,W.D.Jos´e,
2001 Phys. Rev. Lett. 87 37902. and M. H. Y.Moussa 2001 Phys. Rev. A63 053803.
[8] M. Bruneet al 1996 Phys. Rev. Lett. 77 4887. [20] K. Vogel, V. M. Akulin, and W. P. Schleich 1993 Phys.
[9] C. H. Bennett, D. P. Vicenzo 2000 Nature 404 247; A. Rev. Lett.711816; M.H.Y.MoussaandB.Baseia1998
K.Ekert 1991 Phys. Rev. Lett. 67 661. Phys. Lett. A238 223.
[10] N. B. Narozhny, J. J. Sanchez-Mondragon, J. H. Eberly [21] M. Dakna, J. Clausen, L. Kn¨oll and D.-G. Welsch 1999
1981 Phys. Rev. A 23 236; G. Rempe, H. Walther, N. Phys. Rev. A59 1658.
Klein 1987 Phys. Rev. Lett. 58 353. [22] C.J.Villas-Boas,N.G.deAlmeidaandM.H.Y.Moussa
[11] J. F. Poyatos, J. I. Cirac, and P. Zoller 1996 Phys. Rev. 1999 Phys. Rev. A 60 2759.
Lett. 77 4728. [23] The expression F =khTSI|Ψ ik2 stands for usual ab-
FE
[12] S.M.Barnett,D.T.Pegg1996Phys.Rev.Lett.764148; breviation in the literature. Actually, this is equivalent
G. Bjork, L. L. Sanchez-Soto, J. Soderholm 2001 Phys. to hΨ |Tr ̺ˆ |Ψ i where̺ˆ =|Ψ ihΨ |and
TSI E FE TSI FE FE FE
Rev. Lett. 86 4516. ̺ˆ =Tr ̺ˆ .
F E FE
[13] R. M. Serra, N. G. de Almeida, C. J. Villas-Bˆoas, and
M. H.Y. Moussa 2000 Phys. Rev A. 62 43810.