Polarization of free electromagnetic field and electromagnetic radiation in quantum domain

/:.mer f>liysic:s, \.'bl. ·9, No.· I; 1999 •. 1'/1· 26--, . .i?9. Orfi:im,I Te.ti Capyrig/Jt © l~.99 by A.irro. l.Jd, ..

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MODERN TRENDS,

========IN LASER

PHYSICS

Polarization of Free Electromagnetic Field and Electromagnetic

Radiation in Quantum

Domain

A.

S.

Sbumovsky

Physics Depart111e11t, Bi1ke11t Uiriver.~if)~ BUke11t, Ankara; 06533 T(trkey

e-mail: shumo@fen.bi1kent.cdu.tr

Received July 6, 1998

. . . . ·~ ',,,l.-;:: . .

Abstract-:Il

is

shown that conventional disreg;ird.of lo.ngitudinal component, which always e~ists iil radiation

ofa localized source, can lead to some. qualitative errors in the description 9f polarization within the quantum

domain. A new definition of gcneral\zed Sto.kes pijrameters an:d corresponding operators is discussed.

The two main qbjectives of this pa.per are, on the one

harid. to .show that conventional definition of polariza-tion based on.the nopolariza-tion of completely transversal free field is not valid in the quantum clomain, and, on the other hand. to demonstrate how the. spin states of pho-tons may be employed to determine the generalized

Stokes operators. In view of our aim, let us begin with a brief discussion of the conventional definition of pol~rizadon; both classical and quantum. ··

PolarizatiOn is usually

defined

as a physical measure

of Jransversal anisotropy of light [l]. This. definition is based on consideration of a completely transversal electromagnetic field provided by a. solution of the homogeneous wave equation

V2.A· . . l

a

2A

-2-2· = 0.

C

c)t

(1)

For such a solution; the electric field E = -c1_{iJA/at is} always orthogonaltothe magnetic field

B

=

V

xA.

and bot_h E

and

B are orth<>gmial t.o the ditecliort

of

prqpa .. gationk This. means thatthere is no axial symmetry of

a transversal wave with respect to k; i.e., the wave is po1arized. Be.cause of orthogonality of E and B, either of these two vectors can _be chosen to.spec{fy the polar .. ization. The electric field

E

is

usually rise'd

[lJ.

The simplest way to d~termin~ t.he quantitative mea-sure of polarization consists in consideration .of

tlie

Hermitian tensor of polarization, which is defined as having components [2]

Ppp· = EpE; = E011Etw, (2)

which are

the slowly varying bilinear forms

with

respect to the field amplitudes

E

E ( )(

+ . ) .

.;..j«jf

=. o t E1 _ 1£1.

e .

H~re Ep is the

unit

polarization vector.

Because

of

com-plete ttansversality of the wave, index ~ takes only two different values;·111e rank 2 Hermitian tensor (2) can be determined by three reai parameters, forming the con .. ventional

set

of Stokes J?arameters. The notation

for

the

Stokes parameters is µnf ortunately not uniform. The• definitions given in [ 1] in temis of the circular pofa.riza~ lion basis read

s~ =

IEt·Er+IE:·Er

;f

=

2Re[ (E! •.E)

*(E! ·

E)],

(3)

s~ =

2Im[(:t: ·

E)*(E: · E)]~

I 12 I

12.

s;

=

e!'

··E -

E~ • E .

Here E:1: == (E 1

±

iE2)/

Ji

and (s~)2

=

_.(s;>)2

+

_(:s{)2

+

C~{J2. The "¢rameter s~ in (3) measures· the relative. intensity of the wave, the pararrieter

s;

gives

the

pre-ponderance of positive helicity

over

negative helidty,,

and the parameters

sf..

s;

give the phase information fo · terms of the cosine and sine

of

the phase difference

between two circularly polarized compo.nents.

The quantum counterpa.rt of (3) is provided by con"'.. ·veritiona1 Stokes operators that can be obtained by stan~

dard quaritizaiion of free electromagnetic field in the

representation of "plan·e ph<;,tons" (the photons having given energy and linear mo,mentum) as follows [3;, 4]:

•

C . .+ + ·. S0 = (a+a~ + a_a_), •. ·+ + . Si

=

(aC:a++a+aJ,

(4)

·c. ... + + . S2 = -1(a_a+-ll+lL),

Apart from a factor of 2, the operators

Sf ( I=

1, 2,3)

fomi a representation of the .SU(2) subalgebra in the

Wey1-Heisenberg

algebra

of plane · photons. Since

POLAIUZA TION

OF

FREE ELECTROMAGNETIC FIELD 27

two

operators cannot

be

measµred at

once.;

In

view

of

/tp.e stanqard interpretation of the S.tokes parameters (3),

:\vbich

are ex~cted

to

be the

averages

of the· ope1.1:1-to~

!:{4),. this Jl)eans th.at

the..

cosii-.e· and sine ·of the phase

dif-)Jerence between two component~, with opposite

helici-\ties

cannot be measured simultaneously.

?t Let us now note that the quantum ·electrodynamics

Hr.eats the

polarization as

a given spin state of

a beam

of

{photons

(see,

fot example, [61). The

spin

of a

photon

is

]¢own to be equal to l. Th-µs, it may existin the states

?witll

projectionsm

=

0, ±l. In the case of a free electro-ifri~gnetic fie]d, the ;sta,te

m

=

0 is forbidden ~ecause of

/i'#i:#ripJete tr~sversality of thefield [6]. Atfirst glance,

mus·means that there "is no ·contradiction between the

f~ifinition of polarization via spin states

pf

photons~d

IWiough the use of

Stokes

operators (4).

I1Ji1?in

seems relevant to ask: has anyone observed.free

/iji~~tromagnetic field ?Asfaras

we

know. the. fields:that

I:ti#fe

been detected so far have been. emitted by some

!I/~:t~rce.

· " -!'Jn

the CJ:1.se of i!.-real radiation. t_he; &pin: ·s.tate j-=

1..

)0: cannot

b~, e;itcluded a .. .priori In fact, ·a phpt1,:m

,4 ~~ated

l:ly

-a dipole transition gains the spin state

~;,,·V,7.i.).:.w.ithj=je~i

8

=

land any m b~tw.een-lie-j8

1

and

~~

\j~I,

where}e...:. j.₁ i~ the··.difference ·pf tbe.-angulur :·.:··.-¢.ita. of.excite.d and ground levels [6, 7]. Moreover;

···foal dip.ale.radiation, ejther electric. ot

magnetic,

·,s:.,has, a longitudinal, lineady polarjzed

compo-foge.ther with two transversal components [81.

ifhe·intensity ofthe 1ongituclinal co~ponent fa_lls

( ·:clistance

quite rapjdly, it_ is cus~omary to ignore , .. :·:(fa.r.·zonewhere the radiation field .is

approxi-?,lj/tfcinsverse to t~e fadius ve.cto~. cert~ered ·a1 trye :c:e::To be sure that this. approx1mat10n 1s also vahd

._ '~'.qtiatiturndomain; it is necessary to gerierali:ze the

~~¢ffotion

of polarization

and

ex~ine

carefully

how

'''")l~Jigitudinal

contponent can

influence the qmmtum

:1, __

:fhiyi_pr

of two transversal components. This means

/<'i"'-'"'\ve

have to consider polarization of a dipole

radia-'.;~f

!'lrt

arbitrary. s~acec:-time point that

c.ui

possess

ilii#tion of polaitzation or can be locatedin any

•_::;:.-·=-.· :.

'"611r first

step, wehavei:o consiruct cdttespondiii.g

'Hzation

of classical tensor of polarization [9J.

i~ssjcal

radiation of any localized source is

'"ij·by

the rnu1tipole·fie1d

[8]

).=:\:::\-:· ..

"f~J~

=

L

[a;m;1Jk)Aj;,,1(k) + c.c.],

jmkA.

jn,u

Jfii.(~)

= ikAim;.(k), the indices;/, m show the

{lfj¢

multipole field, the index

X

specifies the

Vol. 9 No. l 1999

type ofthe rnultipole field (either electric or. magnetic),

and

aj,r(}._(k) =

I

r2

_{drdnE;/1,. •}

_E.

Here the coeffidents Ai11,'J..(k) are definecl"by the standard combinations of vector spherical hannoriics and Bessel functions [8]. Thus, the .general definition ofthe classi-cal

tensor

of

polarization

reads [9, 10]

P,;,1;,{>.., k)

=

[Ej,.,i(k) · EJ* [E!,,n.(k) ·

EJ.

(5) Consider the case of dipole radiation when j

=

1 and

m

=·

0, ±1. There is no loss in generality in choo!:!iog

fixed A

and k,

but

it enable1:_1

us

to fairly simplify the

notatio_.s. Then, nine components of the rank ,3

Hel'llli-tian tensor (!,) a~ determined by five real parameters

which, must specify three intensitijes.-arid the ··phase

dif-ferences., Amm' such

that

A+o

+

Ao-+ A-+ =

Q.

To.establish

contact

with conventional definition (4), let us choose the generalized Stokes parameters as

fol-lows:

I

S =

~

IE*

•E]

2 ·· 0 ,£,... m

m ;=-l

s1 = 2Rel(E; · E)*(E5' · E) +

(Et ·

E)*(E~ · E)

+(E~ ·

E}*(E; · E)],

s2.=2Im[(E! ··E)*(E; ·

E)

+(Et·

E)*(E! ·

E)

(6)

+

(E~ · E}'"(E; · E)

J,

S4 =

IE:. El

2

+IE!.

El

2

-21E:.

El

2

Just as. in the case of conventional Stokes parameters

(3), the parameters0measuresthe relative intensity, the

parameter s3 measures the preponderance of positive

helicity over negative helicity, and the parameters s 1 ~ .'1'2

give

ihe phase infont1ation. Additional parameter s4

measures the preponderance of circular polarizations

over the linear polarization. Under the assumption that

IE0J

=

0, the set of generalized Stokes.

parameters

(6) is

reduced into (3) with

s

4 - - -

s;r

The generalized Stokes operators. can be obtained

from (6) through the use -of quantization iii tenn.s of

28 .SHUMOVSKY

given energy and angular momentµm. The definitions read [9, IO] . +

S,

= <~.+~·), S2 = -'-i(t - ~+}, S3 = fi~-lL S4 == fi++i(-2n~, (7)

where ii 11, ;=:

a;~,,,

is the number operator of the dipole

photons and

(8)

The. operators (7) and ( 4) are very different in their algebraic structure. In fact, the operat.ors (7) arethe

lin-eaj"'combinations of generators of the S0(3) subalgebra inthe Weyl-Heisenberg algebra of dipole photons [12], while (4) represents some SU(iJ algebra. The mos.t important fact is that [S1,

S

2] = O so that <;orresponding physical quantities can be measured. at once. Sinc!:l

[Su; S0] = 0, their measurement is compatible with simultaneous measurement.of the total photon number.

At .the same time, [S1 2., S3 4]

*

0, which implies that

there ate uncertainty.

re]atib~S

between SI, 2 and

$.J.

4,•

To underline. the difference between. the physical

quantities described by (?)and (4). consider theaverag~ ing with respect to a state of the .radiation field in which the longitudinal component m = Q is in the vacuum state. Then (S1}

=

(S;-)

(/.= 0, 1, 2, 3) and (S

4)

=

(S0)

=

( S~ ). Thus. both sets .of the .. Stokes operators (7) and (4)

le:ad to the same set of S~okes parairieters,

represented

by the averages of corresponding operators. Consider ·now.the variances ·of the operators S~ and S1• It is a

str~.ightforward. matter to,arrive

at

the. relations

V(S1) = '2Re V(a~a+) + 2( (ii+ii_}

-l(a~a+>l

2

J

+

2(S0)

+

2Re(a~a+.),

V(S~)

=

2Re

V(!1~a+) + 2( (iz+iL)

-l(a~a:1-}j2)

+

(Sq}.

Here V(X) ::!

(X":} -

(X)2 .. As can be seen from the above

relations; th~ ptiysical quantities describe.d by the oper-ators (7) J.mdergo much stro11geni)1antum fluctuations than those described by the operators (4)because.ofthe influence of the component in= 0 even

if

it is consid-ered in the vacllum state. Similar results can be

obtained for

S

2 and

s;

iis well. As a particular example

of considerable interest, we ,now inve~tigate the case

when two circularly polarized components are in the

coherent states

lex+)

and

lo:_)

such that

lex+

I

=

Jtt-1

=

l<xl.'

The h;,ngitudinal c.omponent is still supposed to be

fo.

the vacuum state .. Then, the above reJ~tions take

the

fonn· . ~ (Si) = (S2) = 2lcxl~cosA, V(S1)

=

2lal2_r2

₊

_cos.i.\), . V(Sf) =

2lal

2, ... _!~~~·::. .

Here .6. ;:· arga+ -arga_. These relatiQns snow the quaH itative difference of fluctuations be.cause of the pres;

ence

of phase dependence .in V(S1),

This

difference is very

important in

the

quantum

domain when

ltxl

2 "" 1.

i

Thus we see that the description of pol~rization:

based on the use

of

completely transvei:sal free electro-magnetic field can lead to a wrong picture of fluctua~ tions in the quantum domain, at any distance, including the far zone, as fa, as the radiation

of

a localized source

is. considered~ The contribution of the longitudinai ,componerit can be taken into account through the use of

generalized Stokes operators (7). The above results cari

pe

very

important for

arty

polarization meaflun;ment with small intensities. An importl;lnt example Js pro~ vided

by

the atoniic physics experiments with colJerent and rionclassicaJ light focµsed

on

th~ cJetection of spin;

polarized atomic States via polarization rotation of

probing light [J 3-'16]. .

Another example of high importance is provided 'by the.quanturil:-phase problem. Perhaps the most impot.c tant result ofseventy years ofefforts to find a quantun1 object describing universally the pha,se properties of light, is that such an object does nqt ~xist a,t all [17}; lf! addition to the ope111tioilal phases determined by inter.; action

of

light with different mactoscopfo detecting

devices [17]; there cC>uld be s9me inherent quantum phases; obtained by photons in the

process

of gei1era~

tion. [1 SJ: The azimuthal phase of spin of a p,hotorii introduced

by

the cosine and sine operators

I18,

·J9]····

m Ket> a,+ K

·~ = -(0+ 0 )

_{2 ·}

_·

:::-S

_{2 ·}

1,

(9)

I( '- + K.

~ =

2/~-~

)=2S2,

where

K

= (('~~+)t'(i, is very interesting · inherent

phas~ with qui~e nontrivial quantum ptopeities (also see [:20]).. The existenc~ of w~ll~defined eigenstates and e1genvah1es of the operators (9) in whole Hilbert

space

[12] makes it possible to build a bridge between thf!

radi~tiori phase (9) and the results of the method baseq

on the

use

ofra:dius-integrated quasiprobability distrh l,utioi1s, describing the homodyne .detection of phase

[21-25].

.·

· Thus, it is shown that the contribution of longitudi:

nal component.into the qoantuin fluctuations of

polar.;;

izatfon js important even if this component is taken iri the vacuum state (for example, in the case of a

dipole

Ji/>:\'':'> · POLARIZATION OF FREE ELECTROMAdNETIC FIELD 29

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