Table Of ContentScattering of polarized laser light by an atomic gas in free space: a QSDE approach
Luc Bouten,1 John Stockton,1 Gopal Sarma,1 and Hideo Mabuchi1
1Physical Measurement and Control 266-33, California Institute of Technology, Pasadena, CA 91125
We propose a model, based on a quantum stochastic differential equation (QSDE), to describe
thescattering of polarized laser light by an atomic gas. The gauge terms in theQSDE account for
thedirectscatteringofthelaserlightintodifferentfieldchannels. Oncethemodelhasbeenset,we
canrigorouslyderivequantumfilteringequationsforbalancedpolarimetryandhomodynedetection
experiments,studythestatisticsofoutputprocessesandinvestigateastrongdriving,weakcoupling
limit.
PACSnumbers: 03.65.Ta,42.50.Lc,03.65.Ca
7
0 I. INTRODUCTION interaction strength prefactor, Sz is a Stokes operator
0 measuring the circularity of optical polarization and F
z
2 is the z-component of the collective atomic spin. Under
Many recent experimental [1, 2, 3, 4] and theoretical
n [5, 6, 7, 8, 9, 10] works have been based on a simple ex- this Hamiltonian, photons with a rightcircular polariza-
a tionrotatethecollectiveatomicspinoverapositiveangle
perimental scenario, in which a polarized atomic gas is
J κ along the z-axis,while photons with a left circular po-
continuously probed with a polarized off-resonant opti-
0 larizationrotate the collective spin overa negative angle
cal beam (Fig. 1). By measuring the Faraday rotation
3 κ. With linearlypolarizedlight, the angleof linear po-
ofthe opticalpolarizationresultingfromthe interaction, −
larizationwillFaradayrotatebyadegreeproportionalto
1 one can in principle prepare conditionally spin-squeezed
the z-component of the spin. Note that we entirely ne-
v statesorperformquantummetrologytasks,e.g.estimat-
glect ‘tensor’terms of the interactionHamiltonian (non-
4 ing a magnetic field that rotates the spins.
2 linear in individual spin operators) which are important
Central to the description of these experiments is the
2 near resonance with realistic atoms of spin greater than
quantum filtering equation, which propagates the expec-
1 1/2 [4]. We have also omitted the evolution due to any
0 tationvalueoftheatomicgasobservablesconditionedon driving magnetic field, e.g. H = γBF , purely for rea-
7 prior measurement results. The conditional expectation y
sons of simplicity, it can easily be added at the end.
0 is the mean least squares estimate of an atomic gas ob-
In our QSDE-description, the Faraday interaction is
/ servablegiventhe observationsthus far. Theconditional
h described as a ‘direct’ scattering process, without coher-
p expectations of ‘all’ atomic observables can be summa- ent absorptionand re-emission. This is a consequence of
t- rized in an information state πt. The filtering equation the factthatthe interactionHamiltonianisderivedfrom
n propagates this information state in real time.
anapproximationinwhichtheexcitedstatesareadiabat-
a In quantum optics the filtering equation is often re-
icallyeliminated[4]. Atpresent,however,nomathemat-
u
ferred to as the stochastic master equation [11]. For the
q ically rigorous treatment of this elimination is available
polarimetryexampleconsideredhere,previousmodelling
: in the literature (see [18] for rigorous results on the adi-
v effortshaveeitherproducedanunconditionaldescription
abatic elimination of a leaky cavity mode). Therefore,
Xi [8]orarrivedataconditionaldescriptionbyheuristically we have chosento directly base our QSDE model on the
‘addingtheusualmeasurementterms’[4]inanalogywith
r Faraday Hamiltonian without proceeding through a rig-
a a physicallly different homodyne measurement scheme orous Markov limit [19, 20] followed by adiabatic elimi-
with only a single polarization mode [6, 7]. In this arti-
nation of the excited states. Mathematically, the direct
cle we treat the conditional evolution of the state (due
scattering is represented by gauge-terms in the QSDE
to detection of Faraday rotation with a polarimeter) in
[21].
a rigorous manner, allowing the atomic system to medi-
Having set the underlying model, i.e. the QSDE, we
ateexchangebetweentwoorthogonalopticalpolarization
rigorously derive the quantum filtering equation for the
modes. In particular, we derive the quantum filtering
balanced polarimetry setup and for homodyne detection
equation from an underlying quantum stochastic model,
of the y-polarized channel. We investigate the statistics
i.e.the quantumstochasticdifferentialequation(QSDE)
of the output processes for these two experiments and
governingtheinteractionoftheatomicgaswiththelaser take a limit where the driving laser power α2 goes to in-
light. finity but where the product M =κ2α2 is kept constant
FormalquantumfilteringtheorywaspioneeredbyV.P. (κ is the parameter that couples the field to the atomic
Belavkinin[12,13]usingmartingaletechniques(seealso gas). We show that in this strong driving, weak cou-
[14]). We here employ the reference probability method, pling limit the statistics of the output processes for the
based on the quantum Bayes formula [15, 16], to obtain balanced polarimetry experiment and the homodyne de-
the quantum filter from the QSDE (see also [17]). tectionexperimentareequivalent. Furthermore,weshow
The QSDE model we use here is based on a simple thatinthestrongdriving,weakcouplinglimitweobtain
Faraday Hamiltonian, H = κF S , where κ is a small the quantum filter that has already been intuitively as-
z z
2
sumed in the literature [4]. Ai and Ai∗ are called the annihilation and creation pro-
t t
The remainder of this article is organized as follows. cesses, respectively. The processes Λij are called gauge
t
Section II introduces the fundamental noises and the processes. Formally, we can write the noises as Ai =
t
quantum stochastic calculus, and section III sets our taids, Ai∗ = tai∗ds and Λij = tai∗ajds where ai
QSDEmodel. SectionIVderivesthefilterwhencounting a0ndsaj aretthe u0suasl Bose fieldts. M0athsemsatically, thet
in the 45 degrees rotated xy-basis (balanced polarime- Robjectss ai and aRj are ill-defined andRtherefore we resort
try), and Section V derives the quantum filter for the t s
to the definition of Eq. (1). The formal expressions do
homodyne detection experiment. In sections VI and VII showveryexplicitlythough,thattheoperatorΛii counts
we study the statistics of the observation processes, and t
the number of photons with a polarization in the e di-
i
investigate the strong driving, weak coupling limit. We rection up to time t and that the operator Λij scatters
close the paper with a discussionof the results obtained. t
the polarizationof a photon from the e direction to the
j
e direction.
i
II. THE QUANTUM CALCULUS We will usually work in the basis ~ex,~ey which phys-
{ }
ically corresponds to an orthonormal basis in the plane
orthogonal to the direction of propagation of the light.
One polarized photon in a beam of light can be de-
Apart from this basis, we also use the circular basis
scribed by the one particle space
given by e+ = (~ex +i~ey)/√2, e− = (~ex i~ey)/√2 ,
{ − − }
H=C2⊗L2(R)∼=L2 R;C2 , (a~end+th~ee)/4√52d,eg~eree=s r(o~etated~e x)/y√-b2as.isGgiivveenn tbhye {d~eeξfin=i-
x y η x y
of C2-valued quadratically integra(cid:0)ble fun(cid:1)ctions on the tions in Eq. (1) it is eas−y to work}out how the noises
real line. The polarized light field is described by the transform under basis transformations. For example, we
bosonic Fock space ( ) over have
F H H
∞
( )=C ⊗sn, Λ++ = 1 Λxx+Λyy iΛyx+iΛxy .
F H ⊕ H t 2 t t − t t
n=1
M
(cid:0) (cid:1)
which enables arbitrary superpositions between states
Denote by h the Hilbert space of the atomic gas. The
with a different number of photons. Note that photons
space of the combined system of atomic gas and field
are bosons and therefore need to be described by sym-
together is then given by h ( ). Define = C2
metric wavefunctions. For an f we can define the t]
⊗F H H ⊗
exponential vector e(f) in ( ) b∈yH L2( ,t]and [t =C2 L2[t, ). Forallt the bosonic
F H Fock−∞space spliHts in a n⊗atural w∞ay as a tensor product
∞
e(f)=1 1 f⊗n. F(H) = F(Ht]) ⊗ F(H[t). A process Ls, (s ≥ 0) on
h ( )iscalledadapted ifL actsnontriviallyonlyon
⊕Mn=1√n! h⊗F(Hs]) and is the identityson ( [s) for all s 0.
⊗F H F H ≥
We call the spanof the exponentialvectors the exponen- Hudson and Parthasarathy [22] defined stochastic in-
tial domain. The exponential domain is a dense set in tegrals of adapted processes L against the fundamen-
s
( ) and we allow ourselves the freedom to only pro- tal noises, i.e. they gave meaning to the expression
FvidHe the definition of the fundamental noises (a little X =X + tL dM whereM isoneofthefundamental
t 0 0 s s s
further below) on this domain. If we normalize the ex- noises Ai, Ai∗ or Λij. The expression can be written in
ponential vectors then we obtain the coherent vectors s R s s
shorthand as dX = L dM . More importantly, Hudson
t t t
ψ(f) = exp( 1 f 2)e(f). An important vector is the
−2|| || andParthasarathy[22] providedthe calculus withwhich
vacuum vector, givenby Φ=ψ(0)=e(0)=1 0 0....
these stochastic integrals can be manipulated in calcula-
⊕ ⊕
The vacuum state φ = Φ, Φ is obtained by taking
tions. Thecalculusconsistsofthefollowing. SupposeX
h · i t
inner products with the vacuum vector. and Y are stochastic integrals, i.e. dX = L1dM1 and
weIfcawnedcehcooomsepaonseoervtehroynofrmaLl2b(aRs;isC{2)e1a,leo2n}ginthCis2btahseisn, danYdt M=t1La2tdnMd Mt2 w2haerreefuLn1daamndenLta2lanroeisaedst,atphteendttphreotcpersosdes-
i.e. f = f e +f e with f∈and f in L2(R). We now
1 1 2 2 1 2 uct XtYt is itself a stochastic integral. Moreover, the
introducethe fundamentalnoisesAit, Ait∗ andΛitj onthe product XtYt satisfies the following quantum Ito rule,
exponential domain by (see also [21, 22, 23]) (partial integration rule)
t
Aite(f)= fi(s)ds e(f), d(XtYt)=XtdYt+(dXt)Yt+dXtdYt,
(cid:18)Z0 (cid:19)
t
e(g),Ai∗e(f) = g (s)ds e(g),e(f) , (1) wheretoevaluatedX dY weusethattheincrementdM
t i t t t
(cid:18)Z0 (cid:19) of a fundamental noise commutes with all adapted pro-
(cid:10) (cid:11) t (cid:10) (cid:11) cesses,andproductsdM1dM2 aregivenbythefollowing
e(g),Λije(f) = g (s)f (s)ds e(g),e(f) . t t
t i j quantum Itoˆ table [22]
(cid:18)Z0 (cid:19)
(cid:10) (cid:11) (cid:10) (cid:11)
3
dM1 dM2 dAi∗ dΛij dAi
dAk∗\ 0 t 0 t 0 t
t
dΛkl δ dAk∗ δ dΛkj 0
t li t li t
dAk δ dt δ dAj 0
t ki ki t
and all products dM dt and dtdM are zero. As an ex-
t t
ample, suppose dX = L1dAi and dY = L2dAi∗, then
t t t t t t
d(X Y )=X L2dAi∗+L1Y dAi +L1L2dt.
t t t t t t t t t t
It can be shown [19, 20] that in the weak coupling
limit(aMarkovlimit)QEDmodelsconvergetoquantum
stochastic models, i.e. in the limit the unitary time evo-
lutionU satisfiesaquantumstochasticdifferentialequa-
t
tion in the sense of Hudson and Parthasarathy. Usually,
the QSDE obtained via the weak coupling limit can be
simplified further by adiabatic elimination of degrees of
freedomoftheinitialsystem. Seeforinstance[18]forrig-
orousresults onthe adiabaticeliminationofa leakycav-
ity. We,however,areinterestedinthe adiabaticelimina-
tion of the excited states of the atoms in the atomic gas.
Unfortunately, at present, no rigourous results on this
kindofadiabatic eliminationareavailable. Thereforewe
choosenottogothroughaweakcouplinglimit/adiabatic
eliminationprocedurehere,butratherwritedownaphe-
FIG. 1: Schematic depicting balanced polarimetric detection
nomenological QSDE based on the Faraday interaction,
oflaserlightafterinteractingwithapolarizedcloudofatomic
seeEqs.(3) and(6)below. See [4]for aderivationofthe spins via the Faraday Hamiltonian. The light is initially lin-
Faraday Hamiltonian of Eq. (2) via usual non-rigorous early polarized along the x direction. After the interaction,
adiabatic elimination methods. thelightcarriesoffinformationabouttheatomicgasencoded
inasmallopticalpolarizationrotation. Thelightismeasured
in the ξ-η basis rotated 45 degrees from the x-y basis, such
III. THE MODEL thatwithouttheatomicgasthemeanoutputofthepolarime-
ter is balanced to zero. The change of measurement basis is
achievedwiththewaveplatelocatedjustbeforethepolarizing
The interaction between the laser light and the spin
beamsplitter.
polarizedatomic gasis governedby the Faradayinterac-
tion given by
Hdt=2κF S dt=κF dΛ++ dΛ−− . (2) Ifweexpressthegaugeprocessesinthelinearlypolarized
z z z t − t
xy-basis, then Eq. (3) reads (U0 =I)
Here κ is a coupling parameter(cid:0), F is the z-co(cid:1)mponent 0
z
oaf+∗tah+e coal−le∗cat−iveisstphienzv-ceocmtoproonfenttheofatthoemsst,okaensdve2cStozr=S dUt0 = cos(κFz)−1 dΛxtx+dΛyty −
oftthet p−olatrizetdlight. The time evolution of the coupled n(cid:0) (cid:1)(cid:0)sin(κF ) dΛx(cid:1)y dΛyx U0.
z t − t t
system of light and atomic gas together is given by the
exponential (the superscript 0 distinguishes U0 from U (cid:0) (cid:1)o (4)
t t
to be introduced later) Thesecondtermshowsthattheinteractioncanscatterx-
polarizedphotons to y-polarizedphotons andvice versa.
t
U0 =exp i κF dΛ++ dΛ−− . Initially,theatomicgasisinanx-spinpolarizedstate,
t z s − s denotedρ,andthefieldisinanx-polarizedcoherentstate
(cid:18) Z0 (cid:19)
(cid:0) (cid:1) ψx(f) which represents the driving laser. The function
SinceΛ+t +andΛ−t −arejumpprocesses,theItoˆruleleads f L2(R+)givesthephaseandamplitudeofthedriving
to a quantum stochastic differential equation (QSDE) las∈er field at every time t R+. In computations it
[22]thatcontainsthefollowingdifferenceterms(U00 =I) is often convenient to work ∈with respect to the vacuum
stateφ= Φ, Φ forthefield. Wecanobtainacoherent
dU0 = eiκFz 1 dΛ+++ e−iκFz 1 dΛ−− U0. state by ahctin·g iwith a displacement or Weyl operator
t − t − t t
n(cid:16) (cid:17) (cid:16) (cid:17) o (3) Wx(f) on the vacuum vector
That is, right circular polarized photons rotate the col- ψx(f)=Wx(f)Φ.
lectivespinoftheatomsoveranangleκalongthez-axis,
whereas left circular polarized photons rotate the collec- If we work with respect to the vacuum state φ, then we
tive spinof the atoms overanangle κ alongthe z-axis. have to sandwich all operators with the Weyl operator
−
4
Wx(f). An observable S of the combined system of Furthermore, it can be shown that [j (X),Yα] = 0 for
t s
atomic gas and field up to time t is therefore at time allt s 0andα ξ,η . Thisis calledthe nondemo-
≥ ≥ ∈{ }
t given by lition property. Together the self-nondemolition and the
nondemolition property ensure the existence of the con-
jt(S)==WWxx((ff))∗∗UUt00∗∗SSUUt00WWxx((ff)). (5) dtiimtieontaolnextpheectoabtsioenrvPat(ijotn(Xs u)|pYtt)ootfimaesyts.teSminocepetrhaetocroant-
t t t t
ditionalexpectationis linearin the atomic gasoperators
Here, ft denotes the function f truncated at time t, i.e. X, we can define an information state πt on the atomic
ft(s) = f(s) for all s t and ft(s) = 0 for all s > t. gas system by
≤
The relation Eq. (5) follows since all the operators split
as a tensor product at time t and U0 and S act as the π (X)=P(j (X) ).
t t t |Yt
identity operatorafter time t. Since Wx(f) is unitary, it
then cancels against its adjoint for the part that is after Note that πt is a stochastic state since it depends on the
time t. observations Yξ and Yη up to time t.
Itcanbeshown[23]thatWx(f )satisfiesthefollowing It is the goal of quantum filtering theory to obtain a
t
QSDE (Wx(f )=I) recursivestochasticdifferentialequationthatpropagates
0
the information state π in time. Our approach here is
t
1 based on the reference probability method [15, 16]. In
dWx(f )= f(t)dAx∗ f(t)dAx f(t)dt Wx(f ).
t t − t − 2| | t these references, the interested reader can find further
(cid:26) (cid:27)
details on the exposition below.
DefiningUt =Ut0Wx(ft)andusingthe quantumItoˆrule Ourfirststepisoneofmereconvenience. Itisachange
[22], we obtain (U0 =I) ofpicture thatwill simplify subsequentcalculations. Let
W be given by W =I and
t 0
dU = cos(κF ) 1 dΛxx+dΛyy
t z − t t − 1
sin(κFnz)(cid:0)dΛxty−dΛytx(cid:1)(cid:0)+f(t)cos(κF(cid:1)z)dAxt∗ − (6) dWt = f(t)dAxt −f(t)dAxt∗− 2|f(t)|dt Wt.
(cid:26) (cid:27)
1
f(t)dAxt +(cid:0) f(t)sin(κFz(cid:1))dAyt∗− 2|f(t)|2dt Ut. Note that Wt is the adjoint of Wx(ft). Now define Ut′ =
o WtUt,whereUt isgivenbyEq.(6). Iteasilyfollowsfrom
Summarizing, we work in the state P :=ρ φ, the time
⊗ the quantum Itoˆ rule [22] that
evolution of (adapted) observables S is given by j (S)=
t
Ut∗SUt, with Ut given by Eq. (6). dU′ = cos(κF ) 1 dΛxx+dΛyy
t z − t t −
sin(κFn)(cid:0)dΛxy dΛyx(cid:1)(cid:0)+f(t) cos(κF(cid:1) ) 1 dAx∗
IV. THE QUANTUM FILTER z t − t z − t −
f(t) cos((cid:0)κFz)−1 dAxt(cid:1)+f(t)s(cid:0)in(κFz)dAyt∗ (cid:1)−
After the interaction, the light carries off information f(t)(cid:0)sin(κF )dAy +(cid:1) f(t)2 cos(κF ) 1 dt U′.
z t | | z − t
about the atomic gas. Therefore, measuring the field
(cid:0) (cid:1) o (9)
will enable us to make inference about the atomic gas
observables. Let us suppose that we are counting the Defineanewstateonthecombinedsystemofatomicgas
photonswithapolarizationalongthe~eξ =1/√2(~ex+~ey) andfieldbyQt(S)=P(U′∗SU′). Tocompleteourchange
t t
axis, and that we are separately counting the photons of picture we need to sandwich the observables with the
with a polarization along the ~eη = 1/√2(~ex ~ey) axis, opposite rotation. That means that the system Eq. (8)
−
see Fig. 1. That is, our observations are given by is now simply given by U′U∗XU U′∗ = W XW∗ = X.
t t t t t t
In the last step we used that X acts on the atoms and
1
Ytξ =Ut∗ΛξtξUt = 2Ut∗ Λxtx+Λyty+Λxty+Λytx Ut, is the identity on the field and vice versa for Wt. In the
(7) new picture the observations read
Yη =U∗ΛηηU = 1U∗(cid:0)Λxx+Λyy Λxy Λyx(cid:1)U .
t t t t 2 t t t − t − t t Zξ =U′YξU′∗ =W ΛξξW∗,
t t t t t t t
(cid:0) (cid:1)
Let X be an atomic gas operator, its time evolution is Zη =U′YηU′∗ =W ΛηηW∗.
t t t t t t t
given by
Using the quantum Itoˆ rule, it easily follows that
j (X)=U∗XU . (8)
t t t
1
Eq. (8) is called the system. Together Eqs. (8) and (7) dZtξ =dΛξtξ+ 2 f(t)(dAxt +dAyt) +
form a system-observations pair. f(t)(cid:0)(dAx∗+dAy∗)+ f(t)2dt ,
ξ,Itηisaenadsilyforchaelclketd,sthat0[Y.tα,TYhsβis] =is 0caflolerdaltlhαe,sβel∈f- dZη =dΛηη+ 1 f(tt)(dAx tdAy)|+ | (cid:1) (10)
{nond}emolition property≥and ensures that our observa- t t 2 t − t
tions are simultaneously observable classical processes. f(t)((cid:0)dAx∗ dAy∗)+ f(t)2dt .
t − t | |
(cid:1)
5
Denote =U′ U′∗, i.e. consists of the processesZξ We can write Eq. (13) in integral form and approximate
andZη uCtptottimYtett. ItcanCteasilybeshownthatthecon- thestochasticintegralsintheusualwaywithsimplepro-
ditionalexpectations in the two differentpictures arere- cesses. If we proceed by taking the conditional expec-
lated by P(j (X) ) = U′∗Qt(X )U′. That completes tation P( ), then we can pull the integrators which
t |Yt t |Ct t ·|Ct
ourdiscussionofthechangeofpicture. Wewillnowfocus are elements of out of the expectation. Furthermore,
t
on deriving an equation that propagates Qt(X ). the conditional Cexpectation P(L ),(0 s t) of an
t s t
|C |C ≤ ≤
Atthe heartofthe referenceprobabilitymethodisthe adaptedprocessLequalsP(Ls s). Inthiswayweobtain
|C
following quantum Bayes formula [15, 16]. Let V be an
afoonprderaatlhtloa0rttP≤h(aVst∗cV≤o)mt=.mu1Mt.eoTsrhewoeivnthewr,Zessαcuapfnoprodsaeelfilntαehaa∈ts{Vtξa∗,tVηe}Q>anbdy0 dP(cid:0)Vt′∗XVt′(cid:12)(cid:12)CPt(cid:1)V=t′∗PL(cid:0)VηtX′∗(cid:0)LLηξ−XXLξ−Vt′XC(cid:1)tVdt′Z(cid:12)(cid:12)Ctηt.(cid:1)dZtξ +
Q(S)=P(V∗SV),andforalloperatorsX thatcommute Now define σ (X(cid:0)) =(cid:0)U′∗P(V′∗XV(cid:1)′ (cid:12))U′(cid:1)for all atomic
with Zα (α ξ,η ,0 s t), we have (see [15, 16] for operators X.tUsing thetquanttum Ittoˆ|Crt(cid:12)ulet, we obtain the
s ∈{ } ≤ ≤
a proof) linear version of the quantum filtering equation
Q(X )= P(V∗XV|Ct). dσt(X)=σt L(X) dt +
|Ct P(V∗V|Ct) σt LξXLξ−(cid:0)X d(cid:1)Ytξ− 21|f(t)|2dt + (15)
(cid:18) (cid:19)
We would like to apply the quantum Bayes formula to (cid:0) (cid:1) 1
Qt, i.e. with V = U′. However, Eq. (9) shows that U′ σ LηXLη X dYη f(t)2dt ,
{tishξe,dηrZ}iv,αe0’ns.≤bysn≤oist)e,sit.hte.atUdt′oitnseoltf dcoomesmnuottecwoimthmZutsαe(wαit∈ht where thet(cid:0)Lindblad−gen(cid:1)er(cid:18)atortL−is2g|iven|by(cid:19)
s
The following trick [17] solves this problem. Suppose (X)= f(t)2 sin(κF )Xsin(κF ) +
z z
V′ satisfies the QSDE L | | (16)
t (cid:16)
cos(κF )Xcos(κF ) X ,
z z
−
dVt′ = cos(κFz)+sin(κFz)−1 dZtξ + forallatomicoperatorsX. NowrecallfromEq(cid:17).(12)that
n(cid:0) cos(κFz)−sin(cid:1)(κFz)−1 dZtη Vt′. πclta(sXsi)ca=l Kσta(lXlia)n/σput(rI-)S,trwiehbiechl fiosramquulaa.ntUusmingvetrhsieonItoˆofrtuhlee
o(11)
(cid:0) (cid:1) once more, we obtain the following quantum filter
Then, the coefficients ofdAxt∗, dAyt∗ anddt are the same dπt(X)=πt (X) dt +
as in Eq. (9). Since dAα and dΛαβ (α,β x,y ) are L
zero when acting on thetvacuum vtector Φ,∈w{e the}refore πt LξXLξ π (X(cid:0)) d(cid:1)Yξ 1 f(t)2π LξLξ dt +
have that for all operators S [17] π(cid:0)t LξLξ (cid:1) − t !(cid:18) t − 2| | t (cid:19)
(cid:0) (cid:1)
Qt(S)=P Ut′∗SUt′ =P(Vt′∗SVt′). ππt(cid:0)t(cid:0)LLηηXLLη(cid:1)η(cid:1) −πt(X)!(cid:18)dYtη− 21|f(t)|2πt LηLη dt(cid:19).
Moreover, since Vt′ i(cid:0)s driven(cid:1)by Ztξ and Ztη, it com- The p(cid:0)rocess(cid:1)es dYξ 1 f(t)2π LξLξ dt (cid:0)and d(cid:1)Yη
mutes with t, and we can therefore apply the Bayes t − 2| | t t −
formula withCV = Vt′. That is, summarizing what we 21|f(t)|2πt LηLη dt are called the(cid:0)innov(cid:1)ations or inno-
have achieved thus far vating martingales. It can indeed be shown [13, 14] that
(cid:0) (cid:1)
the innovations are martingales with respect to the fil-
U′∗P V′∗XV′ U′ tration and the measure induced by P.
P j (X) =U′∗Qt(X )U′ = t t t|Ct t. Yt
t |Yt t |Ct t U′∗P V′∗V′ U′
t (cid:0) t t|Ct (cid:1)t
(cid:0) (cid:1) (12)
(cid:0) (cid:1) V. A DIFFERENT SETUP: HOMODYNE
The next step is to find the equation that propagates
P(V′∗XV′ ) in time. Using the quantum Itoˆ rule we DETECTION
t t|Ct
find
For the remainder ofthe paper we assume that f(t)=
dV′∗XV′ =V′∗ LξXLξ X V′dZξ + αeiφt with α real and φt in [0,2π). Now suppose that
t t t − t t (13) instead of the balanced polarimetry setup described in
(cid:0) Vt′∗ LηXL(cid:1)η−X Vt′dZtη, the previoussection,weuse ahomodyne detectionsetup
to measurethe y-componentofthe output light, seeFig.
(cid:0) (cid:1)
with
2. It is well known [11, 24] that for such a homodyne
detection setup the observations are given by
Lξ =cos(κF )+sin(κF ),
z z
(14)
Lη =cos(κFz) sin(κFz). Yt =Ut∗(e−iφtAyt +eiφtAyt∗)Ut. (17)
−
6
That is, for homodyne detection of the y-channel the
system-observationspairisgivenbyEqs.(8)and(17). It
iseasilycheckedthatthe homodynesystem-observations
pair satisfies the self-nondemolition and nondemolition
properties, meaning that the conditional expectation
P(j (X) ) is well-defined. Here denotes the homo-
t t t
|Y Y
dyne observations of Eq. (17) from time 0 up to time
t. We will now derive the filter for the corresponding
information state π (X)=P(j (X) ).
t t t
|Y
Our first step is again one of convenience. We change
totheSchr¨odingerpicturebydefiningthefollowingstate
on the combined system of atomic gas and field together
Qt(S) = P(U∗SU ). In the Schr¨odinger picture our sys-
t t
tem is simply givenby U j (X)U∗ =X and the observa-
t t t FIG. 2: In the homodyne detection setup, the detection ap-
tions are given by paratus in the dashed box of Fig. 1 should be replaced with
Z =U Y U∗ =e−iφtAy +eiφtAy∗. the apparatus depicted schematically here. As in Fig. 1, the
t t t t t t lightisinitiallypolarizedalongxandthepolarizationrotates
Denote =U U∗, i.e. consists of the process Z up slightlyduetotheinteractionwiththeatoms. Here,however,
Ct tYt t Ct thestrongxcomponentissplitoffandignoredwhiletheweak
to time t. It can easily be shown that the conditional
y component is sent to a standard homodyne setup. The y
expectations in the Heisenberg and Schr¨odinger pictures
component is mixed at a 50/50 (non-polarizing) beamsplit-
are related by P(j (X) )=U∗Qt(X )U .
To compute Qt(tX |Y)twe wotuld like|Ctto utse the Bayes ter with a strong local oscillator beam also polarized along
t they direction and derived from thesame laser as theprobe
|C
formula. Suppose Vt satisfies the following QSDE (V0 = beam. The photocurrent representing the interference signal
I) isthenderivedfromthedifferencebetweentheoutputsofthe
photodetectors.
α2
dV = eiφtαcos(κF )dAx∗+αsin(κF )dZ dt V .
t z t z t− 2 t
n o(18)
Then,the coefficientsofdAx∗, dAy∗ anddt arethe same Using π (X)=σ (X)/σ (I) andthe Itoˆ rule, we find the
t t t t t
as in Eq. (6). Therefore we have Qt(S) = P(Ut∗SUt) = following homodyne quantum filter
P(V∗SV ). The equation for V is driven by Z and Ax∗,
t t t
both commute with t, i.e. Vt commutes with t. That dπt(X)=πt (X) dt +
C C L
means we can now apply Bayes formula with V = V to
t
α π sin(κF(cid:0))X +(cid:1)Xsin(κF ) 2π sin(κF ) π (X)
obtain t z z − t z t
P jt(X)|Yt =Ut∗Qt(X|Ct)Ut = UUt∗t∗PP(cid:0)VVt∗t∗XVVt|tC|CttU(cid:1)Utt. ×(cid:16)(cid:16)d(cid:0)Yt−2απt(cid:0)sin(κFz)(cid:1)dt(cid:17).(cid:1) (cid:0) (cid:1) (20(cid:17))
(cid:0) (cid:1)
Using the quantum Itoˆ rule we find (cid:0) (cid:1)
TheprocessdY 2απ sin(κF ) dtisagaincalledthein-
t t z
dV∗XV =V∗ (X)V dt + novations or the−innovating martingale. It can be shown
t t t L t (cid:0) (cid:1)
αe−iφtV∗cos(κF )XV dAx + [13,14] thatthe innovationsarea continuousmartingale
t z t t withrespecttothefiltration andthemeasureinduced
t
αeiφtVt∗Xcos(κFz)VtdAxt∗ + byP. ItfollowsfromLevy’sthYeoremthattheinnovations
αV∗ sin(κF )X +Xsin(κF ) V dZ . forthehomodynedetectionsetupformaWienerprocess.
t z z t t
where is given b(cid:0)y Eq. (16). Since dAx(cid:1)and dAx∗
L t t
are independent of t and since vacuum expectations of VI. STRONG DRIVING, WEAK COUPLING
stochastic integralsCwith respect to dAx and dAx∗ are
t t
zero, we find in an analogous way as before Define the measurement strength as the product M =
dP V∗XV =P V∗ (X)V dt + α2κ2. In a typical experimental setting α will be very
t t Ct t L t Ct large (strong driving) and κ will be very small (weak
αP(cid:0)V∗ sin(cid:12)(κ(cid:1)F )X(cid:0)+Xsin(κF(cid:12))(cid:1)V dZ . coupling). The idea in this section will be to exaggerate
t (cid:12) z (cid:12)z t Ct t
thisbytakingthelimitα whilekeepingtheproduct
Nowintro(cid:16)duc(cid:0)eσt(X)=Ut∗P(Vt∗XVt|Ct(cid:1))Ut(cid:12)(cid:12)(cid:12)fo(cid:17)rallatomic M =α2κ2 constant. →∞
gas operators X. Using the quantum Itoˆ rule, we obtain Let us introduce the following scaled sum and differ-
the linear homodyne filtering equation ence processes
dσ (X)=σ (X) dt +
t t L (19) Y+ = Ytξ+Ytη, Y− = Ytξ−Ytη. (21)
ασ(cid:0)t sin((cid:1)κFz)X +Xsin(κFz) dYt. t α2 t α
(cid:0) (cid:1)
7
Note that we scaled the sum by α2 and the difference that the above system of differential equation is closed
by α. We will see that with these scalings we get finite and consists only of a finite number of equations. In the
output processes in the limit. In practice the scalings limitαtoinfinity(whilekeepingM =α2κ2constant),we
aredeterminedbytheexperiment,i.e.theyarechosenin obtain the following finite system of coupled differential
suchawaythatthephotocurrentsnicelyfillthescaleson equations (n 0)
≥
the read out devices. We are interested in the statistics
of the processes Yt+ and Yt−. Therefore, following [24], dΦ− (√MF )n,k,t = k(t)2Φ− (√MF )n,k,t
we introduce their characteristic functionals dt z − 2 z −
(cid:16) (cid:17) (cid:16) (cid:17)
t 2ik(t)Φ− (√MF )n+1,k,t .
z
Φ+(k,t)=P exp i k(s)dY+
− s (cid:16) (cid:17) (22)
(cid:18) (cid:18) Z0 (cid:19)(cid:19)
t k(s)
=P U∗exp i (dΛxx+dΛyy) U , Inprinciplewecouldnowtrytosolvethissystemofequa-
t − α2 s s t
(cid:18) (cid:18) Z0 (cid:19) (cid:19) tions. However, instead of finding an explicit solution,
t let us compare this with the statistics of the homodyne
Φ−(k,t)=P exp i k(s)dY−
− s observations Yt defined in Eq. (17). In analogy to the
(cid:18) (cid:18) Z0 (cid:19)(cid:19) discussion above, we define for all atomic gas operators
t k(s)
=P U∗exp i (dΛxy+dΛyx) U , X
t − α s s t
(cid:18) (cid:18) Z0 (cid:19) (cid:19)
Φ(X,k,t)=
where k is an arbitrary function in L2(R+). The char-
t
acteristic functionals Φ+ and Φ− faithfully encode the P U∗Xexp i k(s)d(e−iφtAy+eiφtAy∗) U .
complete statistics of the processes Yt+ and Yt−. (cid:18) t (cid:18)− Z0 s s (cid:19) t(cid:19)
Using the quantum Itoˆ rule and the fact that vacuum
Using the quantum Itoˆ rule and the fact that vacuum
expectations of stochastic integrals are zero, we find the
expectations of stochastic integrals are zero, we find the
following differential equation for Φ+(k,t)
following system of differential equations (n 0)
≥
dΦ+ k(t)
dt (k,t)=α2 exp −i α2 −1 Φ+(k,t), dΦ αnsinn(κFz),k,t =
(cid:18) (cid:18) (cid:19) (cid:19) dt
In the limit α to infinity (while keeping M = α2κ2 con- (cid:16) k(t)2Φ αn(cid:17)sinn(κF ),k,t
z
stant), we therefore obtain − 2 −
(cid:16) (cid:17)
2ik(t)Φ αn+1sinn+1(κF ),k,t .
t z
Φ+(k,t)=exp i k(s)ds .
(cid:16) (cid:17)
−
(cid:18) Z0 (cid:19) Taking the limit α (while M = α2κ2 is held con-
→ ∞
This is the characteristic functional of the deterministic stant)thenagainleadstothesystemofdifferentialequa-
timeprocesst. Inshort,asαtendstoinfinity,dY+ tends tions Eq. (22). Therefore we conclude that in the limit
to dt. t theprocessesYt− andYt haveexactlythesamestatistics!
To calculate Φ−(k,t), define for all atomic gas opera- This means that from the point of view of statistical in-
tors X ference of the atomic gas system from the observations,
the balanced polarimetry experiment and the y-channel
Φ−(X,k,t)= homodyne detection experiment are equivalent.
t k(s) Rearranging terms, we can write the linear quantum
P Ut∗Xexp −i α (dΛxsy+dΛysx) Ut . filtering equation Eq. (15) as
(cid:18) (cid:18) Z0 (cid:19) (cid:19)
NotethatΦ−(k,s)=Φ−(I,k,s). UsingthequantumItoˆ
dσ (X)=α2 σ sin(κF )Xsin(κF ) +
rule and the fact that vacuum expectations of stochastic t t z z
integrals are zero, we find the following system of differ- (cid:0) (cid:1)
ential equations (n 0)
dΦ− ≥ σt cos(κFz)Xcos(κFz) −σt(X)!dYt+ +
αnsinn(2κF ),k,t = (cid:0) (cid:1)
z
dt
(cid:16) k(t) (cid:17) α σt cos(κFz)Xsin(κFz) +
α2 cos 1 Φ− αnsinn(2κF ),k,t
z
α − − (cid:0) (cid:1)
(cid:16) (cid:18) (cid:19) (cid:17) (cid:16) (cid:17)
k(t) σ cos(κF )Xsin(κF ) dY−.
iαsin Φ− αn+1sinn+1(2κF ),k,t . t z z t
α z !
(cid:18) (cid:19) (cid:16) (cid:17) (cid:0) (cid:1)
Note thatalthoughthe atomic gassystemmightbe very Writing Y for the limit process of Y−, and taking the
t t
high dimensional, the dimension is finite. That means limitoftheaboveequation,weobtainthefollowinglinear
8
quantum filtering equation Now, if we replace κ by √M/α and take the limit α to
infinity, then the time evolution satisfies the following
dσ (X)=σ (X) dt + √Mσ F X+XF dY , (23) QSDE
t t t z z t
L
(cid:0) (cid:1) (cid:0) (cid:1) M
where dU = √MF e−iφtdAy eiφtdAy∗ F2dt U .
t z t − t − 2 z t
1 (cid:26) (cid:27)
(X)=M F XF F2X +XF2 . (cid:0) (cid:1)
L z z− 2 z z That is, the x-channel has been decoupled from the in-
(cid:16) (cid:0) (cid:1)(cid:17) teraction, see also [9, 25].
Moreover, we obtain the following normalized quantum Inasimilarwayweeasilyseethatinthestrongdriving,
filter weak coupling limit we have dY+ = dt and for Y , the
t t
limit of Y−, we obtain
t
dπ (X)=π (X) dt+√M π (F X +XF )
t t t z z
L − (24) U∗(Λxy+Λyx)U
2πt(Fz)πt(X(cid:0)) dY(cid:1)t−2√Mπ(cid:16)t(Fz)dt . Yt = t t α t t
(cid:17)(cid:16) (cid:17) U∗(αe−iφtAy+αeiφtAy∗)U (25)
Since dY 2√Mπ (F )dt is a continuous martingale = t t t t
t t z α
−
[13, 14], it follows from Levy’s theorem that it is a =U∗(e−iφtAy +eiφtAy∗)U .
Wiener process. That is, we find that dY = dW + t t t t
t t
2√Mπ (F )dt, with W a Wiener process.
t z t This shows once more the equivalence of the balanced
Furthermore, note that if we start from Eq. (19), tak- polarimetryexperimentandthey-channelhomodynede-
ing α to infinity while M = α2κ2 is held constant, then tection experiment. It is easy to see that the charac-
we also obtain the linear filter Eq. (23). Likewise, the teristic functional of the process Y satisfies the set of
homodyne filter Eq. (20) converges to the filter in Eq. coupled differential equations of Eq. (22). Moreover,
(24) when α is taken to infinity while M = α2κ2 is held after decoupling the x-channel the system is given by
constant. ∗
j (X) = U XU and the observations by Eq. (25). Fol-
t t t
lowing essentially the same steps as in Section V, we
again obtain Eqs. (23) and (24) as the linear and nor-
VII. DECOUPLING THE X-CHANNEL malized filters, respectively.
Letusgiveabriefformaldiscussiontoshowwhathap-
pens in the strong driving, weak coupling limit. As α VIII. DISCUSSION
increases and κ = √M/α decreases, the relative effect
of the atoms on the x-polarized channel also decreases. We haveprovidedaquantumstochasticmodelEq.(6)
Therefore, we can reasonably expect that the x-channel to describe recent polarimetry experiments in which po-
remains in a coherent state. Instead of working with re- larized laser light interacts with an atomic gas via the
specttothestateP=ρ φwewillnowworkwithrespect Faraday interaction. In our description the gauge pro-
⊗
to the state cess plays a prominent role. It represents the scattering
between different channels in the field and it provides us
Q=ρ ψx(f), ψx(f) , withcountingprocessesthatcanbeobserved. Asin[21],
⊗ ·
our quantum stochastic model presents a novel applica-
D E
tion to quantum optics of the gauge terms in a QSDE.
with f(t) = αeiφt. Note that this means that the y-
Once we set the model, we derived quantum filter-
channelisstillinthevacuumstate. Workingwithrespect
ing equations for balanced polarimetry and homodyne
to the coherent state on the x-channel means that the
detection experiments, studied the statistics of output
timeevolutionisgivenbyEq.(4). Formallywecanwrite
processesandobtainedfiltersinthestrongdriving/weak
for β,γ x,y
∈{ } coupling limit. Our results in the limit confirm the ad
hoc filter for the balanced polarimetry experiment that
dΛβγ =aβ∗aγdt, dAβ =aβdt, dAβ∗ =aβ∗dt,
t t t t t t t hasalreadybeeninuseintheliterature[4]. Moreover,we
showedthatfromthepointofviewofstatisticalinference
andsinceforlargeαandsmallκthex-channelisapprox-
the balanced polarimetry experiment and the homodyne
imately in the coherentstate ψ(f), we can replace ax by
t detection experiment are equivalent.
αeiφt and ax∗ by αe−iφt. This means that we obtain for
t Usingformalargumentswehaveseenthatinthestrong
large α and small κ
driving, weak coupling limit the x-channel decouples
fromthedescription. Rigorousresultsonthisdecoupling
dUt0 = cos(κFz)−1 α2dt+dΛyty − are still to be obtained.
n(cid:0)sin(κFz)α e−i(cid:1)φ(cid:0)tdAyt −eiφtdA(cid:1)yt∗ Ut0. derHivaavtiniognsacnanunddeperalryti,ngis mlikoedlyeltofrobme awdvhaicnhtagreigooursouins
(cid:0) (cid:1)o
9
future further investigations. In particular, combining the ARO under Grant W911NF-06-1-0378. H.M. and
the modelpresentedinthis paperwiththe resultsin[26] J.S. are supported by the ONR under Grant N00014-05-
could prove useful for investigating the situation where 1-0420. H.M. and G.S. are supported by the NSF under
the laser beam passes through the gas multiple times Grant CCF-0323542.
[10, 27].
Acknowledgments
L.B.thanksMikeArmen,RamonvanHandelandTony
Miller for stimulating discussion. L.B. is supported by
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