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Controllable quantities for bilinear quantum systems
Gabriel Turinici
To cite this version:
Gabriel Turinici. Controllable quantities for bilinear quantum systems. Control and Decision Confer- ence, 2000, Sydney, Australia. pp.1364-1369. �hal-00723667�
Gabriel TURINICI
ASCI-CNRS Laboratory,Bat. 506, Universite ParisSud, 91405 Orsay Cedex
turiniiasi.fr
Abstrat
This paperis dediated to the searh of tailored on-
trollability onepts for quantum systems interating
with lasers. A negative resultfor innite dimensional
spaes servesasmotivation for anite dimensional a-
nalysis. Weshowthat under physiallyreasonablehy-
pothesisweanloallyontrolsetsofobservables. Asa
remarkablepartiular ase global exatontrollability
isprovenforthepopulationof theeigenstates.
1 Introdution
Controllinghemialreationsatthequantumlevelwas
along-lastinggoal[1,3,4,5,9,11,13,14,16,17℄from
theverybeginningofthelasertehnology. Indeed,due
to the subtle nature of the interations involved, this
kind of manipulation is expeted to allow on the one
handfor muheÆientandnerontrol thanlassial
toolsandontheotherhand fornewphenomenato be
revealed.
The rst experiments have shown that designing the
laserpulse abletoensurethedesiredpropertiesofthe
systemisanon-trivialtaskthatphysialintuitionalone
annot aomplish. It is onlyreentlythat toolsom-
ing from theontrol theorybegan to givesatisfatory
resultsinsomepartiularases.
A legitimate questionarises in this ontext: what are
thenewontrollabilityoneptsthatbesttthisframe-
workandwhiharethequantumquantitiesthatanbe
exatlyontrolled usingsuhanexternal eld? Some
answersaregivenbelow.
2 Innite dimensionalontrollability
Theproblemunderstudyisontrollingthetimeevolu-
tionofquantumsystems. Letus onsidersuhainde-
pendent systemwith internal Hamiltonian H
0
and let
0
(x)beitsinitialstate. Denotingby (x;t)thestate
ShrodingerEquations)forthefreesystemread:
(
ih
t
(x;t)=H
0 (x;t)
(x;t=0)=
0
(x); k
0 k
L 2
(R
)
=1
(2.1)
Theexternal ation expeted to allowfor ontrol is a
lasereldmodeledbyalaserintensity(t)2Randby
aertaintime independent dipole momentoperator B
(seealso[18℄). ThenewHamiltonianisH =H
0 (t)B
andthedynamialequationsread:
(
ih
t
(x;t)=H
(x;t)
(x;t=0)=
0 (x)
(2.2)
Inarstapproximationthegoalmaybeformalizedas
tond(if any)analtime T andanite energylaser
pulse(t),(t)2L 2
([0;T℄)abletosteerthesystemfrom
0
(x)tosomepredenedtarget
(x;T)=
target (x).
Note that theL 2
normof
is onservedthroughout
theevolution:
k
(x;t)k
L 2
x (R
)
=k
0 k
L 2
(R
)
; 8t>0: (2.3)
In general, for any autoadjoint operator O suh that
[H
0
;O℄ and[B;O℄ arebothzero[19℄oneobtains
<
(x;t)jOj
(x;t)>=<
0 jOj
0
>; 8t>0; (2.4)
withtheusualnotation< jOj >=< ;O >
L 2=<
O ; >
L
2. OneremarkablelassofoperatorsareL 2
-
projetionstolosedsubspaes. LetP beaprojetion
to a losed subspae X of L 2
(R
). Then [H
0
;P℄ =
[B;P℄=0meaninpartiularthatXanditsorthogonal
omplementX
?
areinvolutiveforH
0
andB,i.e.
(
8 2X : H
0
2X; B 2X
8 2X
?
: H
0 2X
?
; B 2X
?
(2.5)
Thesystemanthenbeviewedasdeomposedintotwo
independentsubsystemswithwavefuntionstheproje-
tionsofthetotalwavefuntiontoXandX
?
. Ofourse
thisdeompositionanbefurtherrenedforanyaddi-
tionalprojetionoperatorthatommuteswithH
0 and
B. Inordernottointrodueunneessaryompliations,
wewillsuppose inallthat followsthat thesystemhas
only a nite number of independent subsystems (al-
thoughthetheoryanbeaommodatedtotaount-
thegeneralase),eahbeingassoiateditsL -projetor
P
1 ,...,P
K
suhthat:
[H
0
;P
i
℄=[B;P
i
℄=0; 8i=1;:::;K (2.6)
Moreoveroneanprovethattheprojetorsanbeho-
sentofulll thefollowingonditions:
K
X
i=1 P
i
=I; P
i P
j
=0; 8i6=j; i;j=1;:::K (2.7)
Denote by S
0
the produt of hyper-spheres: S
0
=
ff 2 L 2
(R
);kP
i fk
L 2
(R
)
= kP
i 0 k
L 2
(R
)
;i =
1;:::;Kg By using 2.4 for the projetors P
1 ,...,P
k one
anprovethatthesystemevolvesonS
0 .
Letuspointoutthatduetothequantumnatureofthe
systemitfollowsbytheunertaintypriniplethat one
willneverbeabletoexperimentallyverify,neitherfully
exploit, the exat ontrollability. In fat even if one
obtainsexatlythedesiredtargetstate
target
thefree
evolution(i.e. whenlaserisswithedo(t)=0;tT)
of the quantum system instantaneously modies
thisstate(byatimedependentphaseshiftif
target is
aneigenfuntionofH
0
andbythe(2.1)formulaingen-
eral). Inthisontextanegativeontrollabilityresultis
thereforenotreallyrestritive. Infatusingarguments
asin [2℄ wemayprove(seealso[20℄):
Theorem2.1 Let B be a bounded operator from
H 2
x (R
) to itself and let H
0
generate a C 0
semigroup
of bounded linear operators on H 2
x (R
). Denote by
(x;t)thesolutionof(2.2). Thenthesetofattainable
statesfrom
0
denedby
AS=[
T>0 f
(x;T);(t)2L 2
([0;T℄)g (2.8)
is ontained in a ountable union of ompat subsets
of H 2
x (R
)\S
0
. In partiular its omplement with
respet to S
0
: N = S
0
n AS is everywhere dense
on S
0
. The sameholds truefor the omplementwith
respettoS
0
\H 2
x (R
).
Proof: Toprove the rst partof the theorem one
applies Thm. 3.6 from [2℄ on the spae H 2
x (R
) for
the operators iH
0
and iB (and restrits(t) to L 2
funtions). Denoteforanyset A:
A
r
1
;:::;r
K
=f K
X
i=1 s
i P
i
f; 0s
i r
i
; f2Ag
Then for any ompatsubset C of X C
r
1
;:::;r
K is also
ompat. Applyingthis totheompatomponentsC
ofAS onenotes that
[
r10;:::;rK0 AS
r1;:::;rK
=[
n2N AS
n;:::;n
is also a ountable union of ompats subsets of
H 2
(R
). It follows by the Baire ategory theorem
that [
r10;:::;rK0 AS
r1;:::;rK
hasdenseomplementin
H 2
x (R
); in partiular theomplementofAS withre-
spet toS
0
\H 2
x (R
)hasto beeverywheredenseon
S
0
\H 2
x (R
).
Given this result the searh for exatly ontrollable
quantities has to bedireted to the nite dimensional
setting.
3 Finitedimensional ontrollability
LetD =f
i
(x);i =1;::;Ngbean orthonormalbasis
foranitedimensionalsub-spaeF ofL 2
(R
)[21℄and
A and B be the matries of the operators H
0 and B
withrespettothisbase.
Denote C =(
i )
N
i=1
asthe oeÆientsof
i
(x) in the
formulaoftheevolvingstate (x;t)= P
N
i=1
i (t)
i (x).
Fromnowwewillworkin atomi units only(h=1) ;
theequations(2.2)read
i
t C
=AC
(t)BC
; C
(t=0)=C
0
(3.1)
C
0
=(
0i )
N
i=1
;
0i
=<
0
;
i
> (3.2)
Theontrollability of(3.1) hasbeendealt with in the
literature(f. [12℄)byderivingresultsfromtheontrol-
lability of asystem posed onthe spae of theunitary
matriesofdimensionN. Thisapproahhasthebenet
of grantingaess to the generaltoolson the ontrol-
lability of bilinear systems on Lie groups. However,
these results give only suÆient onditions for exat
ontrollability (due to the setting whih is moregen-
eral). Finally there exists a lass of simple quantum
systemsontrollableinasensetobedenedfurtheron
that do not verify the riteria emerging from the Lie
group analysis. We have therefore judged instrutive
to study this issue in a new framework; we were thus
leadintoidentifyingsimpleneessaryand suÆient
onditionsforthenitedimensionalontrollability(see
also[5℄foranintrodutiontothistopi).
IntheaseofourmodelingtheAmatrixisdiagonaland
B is symmetrialwith nulldiagonalelements(see [15℄
forthegeneralase). Letusdenote by
i
; i=1;::;N
thediagonal elementsof A (the energies of thestates
i
). Before presenting the theoretial results we will
introduetheontrollabilityoneptused.
LetO
1 ,...,O
p
bepositivequantumobservables(positive
autoadjointoperators). Wesaythatthedistributionof
observablesÆ=(Æ
i )
p
i=1 ,Æ
i
0,i=1;:::;pisreahable
fromtheinitialstateC
0
ifforany>0thereexistsa
naltimeT
d
>0andaneletrield(t)2L 2
([0;T
d
℄)
suhthat thesolutionof(3.1)satises:
j< (x;T
d )jO
i j (x;T
d )> Æ
2
j<; i=1;:::;p
Ifthisisalsotruefor=0wesaythatthedistribution
of observablesÆ anbe exatly reahed fromthe initial
stateC
0 .
Aspeialaseofpositiveobservablesaretheprojetions
ontheeigenstatesP
i
denedbyP
i
=< ;
i
>
L 2
i
,i=1;:::N. Theobservablequantities< jP
i j >
orresponding to this operators are alled populations
of theeigenstates. Inourase these arej
k (T
d )j
2
. A
remarkablepropertyof these observablesisthat when
the systemis evolving freely((3.1)with (t) =0) the
populationsoftheeigenstatesdonothange.
As it was previously seen the system evolves on the
unit sphereofL 2
x (R
)whihin nitedimensionalrep-
resentation reads P
N
i=1 j
i (t)j
2
=1; 8t 0. We all
populationdistributionforthesystem(3.1)anyN-tuple
d2R N
suhthat
N
X
i=1 d
2
i
=1; d
i
0; i=1;:::;N (3.3)
Apopulationdistributionbeingapartiularaseofdis-
tributionofobservablesweextendthereahabilityon-
eptsdened abovetothis asealso.
4 Transfer graph and neessaryonditions
Wedene asin [15℄thenon-orientedtransfergraphof
thesystemG=(V;E)whihorrespondsto theintu-
itive image of population ow among dierent eigen-
statesofthesystem. ThesetV ofvertiesisthesetof
eigenstates
i
and the set of edges E is theset of all
pairsofeigenstatesoupledbythematrixB:
G=(V;E); V =f
1
;:::;
N
gE=f(
i
;
j );B
ij 6=0g
(4.1)
Thisgraphanbedeomposed intoonnetedompo-
nentsG
=(V
;E
),a=1;::;K that orrespondto a
blo-diagonal struture of the matrixB (modulo per-
mutationsontheindies). Itisworthwhilementioning
that this operation is the disrete version of the de-
ompositionusingprojetionoperatorsthatwasunder-
takenfortheinnitedimensionalase;indeed,foreah
onneted omponent G
, =1;:::;K, oneanasso-
iatethelinearspaespannedbytheeigenfuntionsin
V
andprovethatthe(disrete)projetionoperatoron
thisspaeP
ommuteswithA andB.
Let
~
D =f
~
1
;:::;
~
N
gbeanorthonormalbasisforthe
nite dimensional spae F and
~
P
1
;:::;
~
P
N
projetions
operatorson
~
1 ,...,
~
N
respetively. Supposemoreover
that these observablesare ommutingwith P
1
;:::;P
K ,
whih is equivalent to the fat that
~
D is the unionof
orthonormal basis for eah subsystem. Denote by U
the unitary matrix that allow to hange between the
orthonormalbasisD and
~
D:
~
i
=
j U
ij j
. We will
supposeinallthatfollowsthatallentriesinU arereal.
Onean hekby thedenition ofG and using equa-
tions(3.1)thatforall=1;::;K: i d
dt kP
(x;t)k 2
L 2
=
0 ; eah subsystem (onneted omponent) omply
thereforewiththeonservationlaws
X
fi; i2Vg
< (x;t)j
~
P
i
j (x;t)>=onstant;
t>0;=1;::;K (4.2)
Thisallowsusto giveneessaryonditionsforontrol-
lability
Lemma4.1 If the distribution of observables Æ is
reahable fromthe initial onguration C
0 then
X
fi; i2Vg
<
0 j
~
P
i j
0
>=
X
fi; i2Vg Æ
2
i
; =1;::;K :
(4.3)
Asapartiularaseoneobtainsthefollowing
Corollary4.1 If the population distribution d is
reahable fromthe initial onguration C
0 then
X
fi; i2Vg j
0i j
2
= X
fi; i2Vg d
2
i
; =1;::;K : (4.4)
5 Controllability results
Denote!
k l
=
k
l
; k;l=1;:::;N. Letusintrodue
thefollowinghypothesis:
HA The omponents G
; = 1;::;K of G re-
main onneted after elimination of all edge pairs
(
i
;
j );(
a
;
b
)suhthat!
ij
=!
ab
(degeneratetran-
sitions).
Theorem 5.1 (Loal exat ontrollability) Let T >0
be a given nal time,
0
(t) 2 L 2
([0;T℄) a given laser
eldsuhthat:
HB lim
t!T
0 (t)=0,
soin partiular the limit lim
t!T
0
(t) is supposed toexist
(seealsoRemark5.1); let
T
bethestateattimeT of
thesystempropagatedwiththelasereld
0 andÆ
T (d
T )
the distributionof observables (populations) assoiated
tothestate
T :
Æ
T
=( q
<
T j
~
P
i j
T
>) N
i=1
;
d
T
=(j<
T
;
i
>
L 2j)
N
=(j
0i j)
N
:
T i T i
hypothesisHA isveried. Suppose alsothat:
HC For eah onnetedomponent G
; =1;:::;K
ofGitdoesnotexistsapartitionV
=V 1
[V
2
,V
1
\
V 2
=;suhthat
j X
a2V 1
U
jq
<
T
;
a
>j=j X
b2V 2
U
jb
<
T
;
b
>j; 8j2V
(5.1)
orif suhapartitionexiststhen
P
a2V 1
U
ja
<
T
;
a
>
P
b2V 2
U
jb
<
T
;
b
>
=onstant; 8j2V
:
Then thereexistsanopenneighborhoodD of Æ
T onthe
surfae of R N
given by the neessary onditions (4.3)
endowed with the anonial topology suh thatone an
exatly reah any distribution of observables Æ in D
fromC
0 .
Remark5.1 ThehypothesisHB isnotreally restri-
tive. In allpratial ases
0
(t) isontinuous(at least
atnal/initial time)whih assuresthe existeneof the
limit. The requirement that the limit of
0
(t) in T
be exatly 0 an be readily satised by replaing the
triplet (
0
;A;B) by (
0
(T);A+
0
(T)B;B), where
0
(T)= lim
t!T
0
(t). Note that in this situation the hy-
pothesis HA has to be veried for the eigenvalues of
A+
0
(T)B whihareingeneral dierentfromthoseof
A. Finally, notethat the setof nal states
T
that do
notomply withthehypothesisHC isofnullanonial
measurefor any(real) unitary matrixU.
Remark5.2 The result above may be somehow sur-
prising due to the spei onept of loality used. In
fat, supposethatthe evolutionofthesystemhasended
in somenal statep
T
with the orresponding distribu-
tion of observablesÆ
T
. Then, in order toobtain some
other admissible distributionÆ
lose to Æ
T
onehas to
gobakintimeandmodifythe eletrieldratherthan
tostartfromp
T
andgoforÆ
! Tounderstandthisone
hastorememberthattheobservablesdonotneessar-
ilyommutewiththehamiltoniansothefreeevolution
(fromp
T
) dragsthedistributionof observablestowards
thediretion givenbytheevolutionequations2.2;there
is therefore no reason tohope that smallperturbations
(after the timeT) analways ounter-balanethis bias
andatthe sametimelloutaneighborhood ofÆ
T .
Remark5.3 The tehnial onditions (Æ
T )
i
6=0; i =
1;:::;N analso beintuitively justied. Indeed ifsome
(Æ
T )
i
=0onehavetotake arewhenhoosing thegood
tions" having some stritly negative observables, as
anyprojetion-likeobservableisapositiveoperator.
Proof: Forthesakeof simpliity wetreat onlythe
ase!
ij 6=!
ab
; 8(i;j)6=(a;b),thegeneralasebearing
no new onepts. Let us denote A = iA and B =
iB. Then(3.1)beome:
t C
=(A+(t)B)C
; C
(t=0)=C
0
(5.2)
Denoteby (;C
0
;t) =(
a (;C
0
;t)) N
a=1
thesolutionat
thetime t of(3.1) for theinitial (t =0) data C
0 and
eletri eld (t). Denote also w(t) = (
0
;C
0
;t) and
onsidertheanonialbase fe
1
;:::;e
N gofR
N
.
Wedene theappliation M:L 2
(R)!R N
givenby
M()=(<(;C
0
;T)j
~
P
a j(;C
0
;T)>) N
a=1
(5.3)
Note that by theneessaryonditions (4.3)the range
ofM isasubsetof
(x
i )
N
i=1 2R
N
; X
fi;
i 2V
g
x
i
= X
fi;
i 2V
g
<
0 j
~
P
i j
0
>
=1;::;K
Theloalontrollabilityisinfatapartiularsurjetiv-
ity property ofM. We will provethat thedierential
DMofMhasthesurjetivitypropertywedesireandby
theimpliitfuntiontheoremtheonlusionwillfollow
thenforM itself. MorepreiselyweprovethatDM is
onto the linearmanifold (P)(produt of hyper-planes
ofR
ardinality(S)
; =1;::;K):
(x
i )
N
i=1 2R
N
; X
fi;
i 2V
g
x
i
=0; =1;::;K
whoseM(
0
)-translationistangentto therangeofM.
Denotebyf
a
; a=1;:::;N theomponentsof DM:
DM()j
=
0
~= <f
a
;~>
L 2
N
a=1
(5.4)
Dueto the nitedimensionalityof oursetting wejust
haveto showthat the rangeof DM()j
=0
hasanull
orthogonalwithrespetto (P),that isany vetork=
(k
a )
N
a=1 2R
N
suhthat
X
fi; i2Vg k
i
=0; =1;::;K (5.5)
N
X
i=1 k
i <f
i
;~>
L
2=0; 8 ~2L 2
([0;T℄) (5.6)
isneessarythenullvetor. Equation(5.6)analsobe
written
X
i=1 k
i f
i
(s)=0; 80sT (5.7)
