Harmonic analysis and P-representation of generalized Gaussian fields

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Harmonic analysis and P-representation of generalized Gaussian fields

M Rousseau

To cite this version:

M Rousseau. Harmonic analysis and P-representation of generalized Gaussian fields. Journal de

Physique, 1974, 35 (3), pp.193-198. �10.1051/jphys:01974003503019300�. �jpa-00208142�

LE JOURNAL DE PHYSIQUE

HARMONIC ANALYSIS AND P-REPRESENTATION OF GENERALIZED GAUSSIAN FIELDS

M. ROUSSEAU

Laboratoire d’étude des Phénomènes

Aléatoires,

Université de

Paris-Sud,

^Centre

d’Orsay,

^Bâtiment

210,

91405

Orsay,

^France

(Reçu

^{le 8}

février

^1973, révisé le 2 octobre

1973)

Résumé. ²⁰¹⁴ Nous établissons classiquement la distribution de probabilité (P-distribution) ^des amplitudes complexes ^de

champs

optiques multimodes et monomodes formant la classe des champs

Gaussiens généralisés. Ce résultat donne immédiatement la

représentation-P

unique pour ^cette classe de champs. Mais, excepté ^le

champ

chaotique, ^{ils ne sont} pas totalement stationnaires ; ^on peut néanmoins construire un champ stationnaire équivalent ^{à chacun} d’eux, qui ^a la même inten- sité. Nous exprimons ^la représentation-P ^{de ce}

champ

stationnaire équivalent ^{dans le cas monomode.}

Abstract. ²⁰¹⁴ We establish classically ^the probability distribution (P-distribution) ^{for the} complex amplitudes of multimodal and unimodal

optical

^fields

belonging

^to the class of generalized ^Gaussian fields recently introduced. From this result follows immediately ^the unique P-representation ^for

this class of fields. However, all the generalized ^Gaussian fields, except the chaotic field, ^{are not} fully stationary. ^To each of them we can associate a stationary equivalent field which possesses the ^same

intensity.

^We give ^the P-representation ^{of this} stationary equivalent field in the monomode case.

Classification

Physics ^Abstracts

1.140 ^- 1.650 ^- 2.400

1. Introduction. ^- The class of

generalized

^Gaus-

sian fields was

recently presented [1] from

^{a classical}

point

^{of view. Let us recall that these}

electromagne-

tic

(e. m.)

^{fields are}

^Gaussian,

^{so that the chaotic}

field is a

particular

^element of this class.

They present

the

advantage

^of

being experimentally

realizable

[2]

and _possess some

interesting properties

^when

they

are detected and also in information

theory [3].

^In

fact,

^{it was} established

[1] that,

among the

generalized

Gaussian

fields,

the chaotic field _possesses the smallest’ ^°

bunching

^effect

(h

⁼

2)

^for

photoelectrons,

^whereas

the real Gaussian field _possesses the greatest

(h = 3) [2].

In this _paper we

study

^{the harmonic}

analysis

^of

generalized gaussian

^{fields. We are} interested in

finding ,the unique P-representation

of such fields.

So far as we

know,

^this

problem

^{is still}

unsolved,

^{even .} for the well-known chaotic,field for which

Glauber [6]

’

gave ^a

P-representation,

each mode

being

^chaotic

and

independent

of the others. But Mollow

[9] proved

later that when the

length

^{of the}

cavity

^{tends to}

infinity,

^{it is not} necessary ^to

suppose

any

particular property

^{for each}

mode,

the sufficient condition for the field to be chaotic is

independence

between the modes.

So,

^{we can ask about the}

uniqueness

^{of the}

P-representation, furthermore,

^this quantum ^des-

cription

^{does not seem to be well}

adapted

^{when the}

length

^{of the}

cavity

^{tends to}

infinity.

To establish the

unique P-representation

^{of each}

generalized gaussian field,

^we

adopt

^the artifice of

solving

^the

problem classically by using

^{the well-}

known harmonic

analysis

^of

gaussian

random func- tions

[r. f.’s].

^-

First of all we consider the e. m. field in free _space, with a continuous

spectrum.

^{Such a field cannot be} studied in

quantum mechanics,

^unless it is considered

as the limit of a field in a

cavity ;

but what is then the

meaning

^{of the}

P-representation ?

On the other

hand, free-space

^{e. m. fields can} be studied classi-

cally owing

^{to the}

theory

^{of r.}

f., which

^{has been}

developed

^{since 1944}

[ 11 ].

Let us recall the most

important

^{result. about the}

decomposition

^{of a}

temporal

random function

E(x, t),

its harmonic

analysis.

^{It can} be written for x ⁼

0,

^as

where

de(v)

^{is a} differential element of a r. f.

e(v)

^that

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01974003503019300

194

can be obtained from

E(t) by

^linear

filtering [4]

through

^{a filter that cuts out all the}

frequencies greater

^{than v,}

This new r. f.

e(v)

is very

special,

^{for it has no deri-}

vative

(even

^if

E(x, t)

^is

differentiable),

unlike deter- ministic

temporal functions,

^{so that we can}

only

consider its increments

e(v

⁺

Av) - e(v).

Through

^linear

filtering theory,

the statistical _pro-

perties

^of

E(O, t)

^are

equivalent

^{to those of}

e(v).

When

E(O, t)

^is

^Gaussian, e(v)

is Gaussian too since any

linearly

filtered Gaussian function is still Gaus- sian. For

generalized

Gaussian fields we

specify

^some’

of the

properties

^of

e(v) [§§ 2 .1

^and

3 . 2 .1 ].

After that we treat the discrete

spectrum

^{as a} par- ticular case of the continuous

spectrum

^{for which} the

frequencies

^are distributed on Dirac masses.

When an e. m. field is inserted in a

cavity,

^{it can}

be written as

where L is the

length

^{of the}

cavity

^{and c the}

speed

^of

light.

The statistical

properties

^{of the} classical field described

by

eq.

(3)

^are all defined when we know the multidimensional distribution P

{ Cn }

^{of the}

random variables

(r. v.) Cn ;

^we call it the P-distribu- tion.

Let us show that P

{ Cn }

^{can be} obtained from the harmonic

analysis

^{of the}

corresponding

^Gaussian

field in the free space. In

fact,

eq.

(1)

^and

(3)

^are

equivalent

^{when we set}

Integrating

^from

obtain

Hence the

complex amplitudes

of the different modes of an e. m. field enclosed in a

cavity

^can be obtained

by

^{a linear}

filtering

^of

E(t)

^since

they

^are

only

^linear

combinations of

e(vi).

For each

generalized

Gaussian field we set up the

unique

P-distribution

(§§ 2.2, 3.2.2, 4).

Now let us retum to thé quantum

description

^of

such

fields which can be _very useful for

solving

^some

physical problems [3]

^{- for}

instance,

the interac- tion of radiation with matter, the ^Raman

effect, multiphoton absorption

^{and so on... It is now well}

known that every classical field can be described quantum

mechanically by taking

^the

corresponding

P-distribution as its

P-representation.

^Thus

by using

the harmonic

analysis,

^{we have} established the

unique P-representation

^{for the set of}

generalized

Gaussian fields.

In section 5 we discuss some aspects ^{of the} quantum

description

^of

generalized

^Gaussian

fields, especially

the chaotic fields

(§ 5.1)

^{and their}

stationarity (§ 5.2).

Any generalized

Gaussian field other than a chaotic

one is

only quasi-stationary

^{and not}

fully stationary,

but

by introducing

^a

uniformly

distributed

phase

^we

can construct a

fully stationary.

^{e. m.} field that has the same

intensity

^as the Gaussian one but which is

no

longer

^Gaussian. We call it the

stationary equi-

valent

field

^{and we}

give

^its

P-representation

^{in the.}

unimodal case.

2. The P-distribution for the chaotic field. - 2.1 CHAOTIC FIELD IN THE FREE SPACE. ^- We are

going

^to show that the random function

e(v) appearing

in the harmonic

analysis

^{of an}

electromagnetic

^field

in free _space

(eq. (2)) ^is,

^{for a chaotic}

^field,

^{a random}

Gaussian function with

independent increments,

^which

is

commonly

^{known in}

probability theory

^{as a}

Wiener-Lévy

process. In

fact,

^{the function}

e(v)

^is

obviously

^{Gaussian since it} is filtered from

E(t)

^which

is Gaussian

[4]. Therefore,

every increment

is a Gaussian ^{r. v.}

Moreover,

^{as the function}

E(t)

^is

fully stationary [7],

the second moments of

de(v) satisfy

^the

equations

Let us

decompose

^the

complex

^increment

Ae(v)

^into

its real and

imaginary parts :

From eq.

(7),

^{we can} prove that the real increments

are

independent

and that the

complex

increments

Ae(v)

and

Ae(v’)

^{for two}

disjoint

^intervals

(Av, Av’)

^are

also

independent..

.

Therefore, if

^the

r. f. e(v)

^{is a Gaussian}

r. f.

^with

independent

increments, ^then it is a

complex

^Wiener-

Lévy

process whose real and

imaginary

parts ^are

independent Wiener-Lévy

processes.

2.2 NECESSARY AND SUFFICIENT CONDITION FOR A FIELD ENCLOSED IN A CAVITY TO BE CHAOTIC. ^- Neces- sary condition. ^- An

electromagnetic

^{field enclosed in}

a

cavity

^can be described

by

çq.

(3),

^or

by

^the group

of eq.

(1), (2)

^and

(5)

^as well. If the function

E(t)

^is

stationary ’and Gaussian,

^{the results}

of paragraph

^{2 .1}

are

valid,

^{and the}

complex

^{r. v.}

Cn

^are circular inde-

pendent

^{Gaussian r. v.}

(with uniformly

distributed

phase).

^Hence their multidimensional

probability

distribution

(P-distribution)

^is

where

Ck = C kx

⁺

iCk, and 2013

^{is the common}

variance of the r. v.

Ckx

^and

Cky.

The

sufficient

condition is evident. When we start from _eq.

(9),

^{we can} show after a brief calculation that the field has the same moments as the chaotic field.

z

3.

P-representation

of the real Gaussian field. - 3. 1 Let us recall that the real Gaussian field was

experimentally

^obtained

[2] by

modulation of a laser beam

passing through

^{a KD* P}

crystal

^inserted

between two crossed

polarizers.

^The

intensity

^{of the}

emerging light,

^{when a}

voltage x(t)

^is

applied along

the

optical

^{axis of the}

crystal,

^{is then}

proportional

to the _square of the

voltage only

^{for small}

voltages,

Here a is a constant related to the

crystal

^and

Io

^is

the

intensity

^{of the} incident laser

light.

^{When the}

electrical source

x(t)

^{is a Gaussian}

noise,

^the

intensity

of the

light

^{is the} square of a Gaussian r.

f.,

^{so that}

the

electromagnetic

^field

produced by

^{such a device}

can be described

by

^the

analytic signal

whose

amplitude

^{is a} real Gaussian function. We therefore we call it a real

Gaussian field.

^{Let us recall}

that the

analytic signal

^{of a} chaotic field is

where

z(t)

^{is a}

complex

^{Gaussian r. f. whose real and}

imaginary parts

^are Hilbert transforms of each other.

Let us recall too, ^that the conditional

probability density

^of

detecting

^one

photon

^{at time t + T, if}

another one has been detected ^at time t is propor- tional to

which is

equal

^to

3 I >

^{for a real Gaussian}

field,

whereas it

equals only 2 I >

^{for a chaotic}

field ;

the real Gaussian field therefore has a

bunching

effect greater ^{than the} chaotic field.

The r. f. described

by

eq.

(11)

^{is not}

fully stationary sine x(tl) x(t 2) > =1=

^{0. Let us} consider the

fully stationary

^{r. f. that has the same}

intensity

^as

E(t)

given by

eq.

( 11 ),

where ₉ is a

uniformly

distributed

phase.

^{The e. m.}

field

E(t)

^{is no}

longer Gaussian ;

^{we shall} present its P-distribution in section 5.

3.2 HARMONIC ANALYSIS OF THE REAL GAUSSIAN

FIELD. ^- 3 .2.1 The real Gaussian

field

^{in the}

free

space. ^- Let us

decompose x(t)

^{into its}

negative

^and

positive frequency parts :

where

de(v)

^{is the} differential increment

appearing

in the harmonic

analysis

^{of the} real Gaussian field described

by

eq.

(11)

^and 2 nvo ⁼ coo. The ^r. f.

e(v)

^.

filtered from

E(t)

^{is then}

Gaussian ;

moreover,

x(t)

is a real

function,

^{so that}

Hence the differential increments are not

indepen- dent,

^but

symmetrically

correlated in

pairs.

^The

optical

spectrum is indeed

symmetric

^since

x(t)

^and

its correlation function

Fx(-r)

^are real functions.

Let us suppose that

x(t)

^{is a}

stationary

^r.

f. ;

^thus

From eq.

(17)

^{it follows that}

Ae(vo

⁺

v) given by

eq.

(6)

^satisfies

Therefore the r. v.

Ae(vo

⁺

v)

^are circular Gaussian

r. v.

3.2.2 Field in a

cavity.

^- From the last results of

paragraph 3.2,

^{and from} eq.

(5)

^it follows that the

complex amplitudes Cn

^{are also} circular Gaussian

r. v.,

correlated, by pairs, symmetrically,

^with respect

Hence their P-distribution is

given by

196

where

( n >no+m

^is the variance

of Cno+m I2.

^When

the real Gaussian field is

unimodal,

eq.

(20)

^{reduces to}

where C ⁼

Cx

⁺

iCy.

4. Generalized Gaussian fields. ^- These are defined in classical

theory [1] ] by

^eq.

(12)

^where

z(t)

is any

stationary

^{Gaussian r. f. whose} correlation functions

are

rzz( 7:)

^and

rzz*( 7:).

We are

going

^to

give

^their P-distribution in terms of these second moments.

When the

generalized

Gaussian field is enclosed in

cavity,

^the

complex amplitude

^{of the}

analytic signal E(t)

^{can be}

expanded

^as

Since

zL(t)

^{is a}

stationary

^r.

f.,

Hence,

^{as for the} real Gaussian

field,

^for any genera- lized Gaussian field

except

^{the chaotic}

field,

^{the r. v.}

{ Cn }

^are circular and correlated

by pairs,

symme-

trically

^with respect ^{to the mean}

frequency

VO.

We shall express the characteristic function of the ^{r. v.}

in terms of the Fourier transform of

TZZ(i)

^and

Ti (z)

where (k, j) = (z, z*).

Identifying

^{the moment of}

zi(t)

^{we obtain}

where

Eq. (23)

^to

(26)

^{lead to the}

following

relations :

where we set

fn

= ocn +

if3 n.

The characteristic function of the set of Gaussian

r. v. can thus be written as

where _wm is the colum vector

(um vm

u-m

v-m)

^and

F.

the correlation matrix of

(xm ym X-m y -m)

The P-distribution is thus the Fourier transform of this characteristic function :

where

Vm

⁼

(xm ym x _ m Y - m)

^{and from} eq.

(12)

^we

conclude that

where the function on the

right

^{hand side is}

given by

eq.

(30).

We can

verify for y

^{= 0}

(or y

⁼

kx),

^{which cor-}

responds

^{to the real Gaussian}

^field,

that eq.

(30)

and

(31)

^{are identical to the} P-distribution of the real Gaussian field

(eq. (20)).

Moreover,

^{if we introduce the relations}

which characterize the chaotic field

(i.

^e. a,, ⁼

/3n

⁼

0),

we obtain

Thus the characteristic function can be

factorized,

the modes are

independent,

^and eq.

(30)

^{leads to the}

P-distribution of the chaotic field _{in eq.}

(9).

5. Extension to quantum mechanics. ^- We

explain-

ed in the introduction

why

^we have established these last results from the classical

theory,

^{and we}

have now obtained the P-distribution of the

complex

amplitudes

^for

chaotic,

^real

Gaussian,

^and genera- lized Gaussian fields

(eq. (9), (20)

^and

(30)).

^{If we}

consider as

equivalent

^two

optical

fields that have the same set of moments,

determining

the P-distri- bution in classical mechanics is the same

problem

as

finding

^the

P-representation

^{in quantum mecha-}

nics

[5], [6].

^{Hence eq.}

(9), (20)

^and

(30) give

^the

P-representation

^of

^chaotic,

^real

^Gaussian,

^{and gene-}

ralized Gaussian

fields, ( nk >

ⁱⁿ eq.

(9) being

^the

mean number of

photons

ⁱⁿ the kth mode.

Let us now consider two further

points.

^{For the}

chaotic field we shall

give

^three

equivalent definitions,

and then we shall present ^a

stationary

^{field with the}

same

intensity

^{as each}

generalized

Gaussian field.

5. 1 THE CHAOTIC FIELD : 1 THREE EQUIVALENT ^DEFI- NITIONS. ^- We started

by defining

^{the chaotic field}

as the

only

^{one that} is Gaussian and

fully stationary [1] ]

and in section

2.2,

^we

proved

that the modes of a

chaotic field are

necessarily independent

^{and Gaussian.}

We shall now show that an

equivalent

^definition

of this field related to the entropy concept, ^{is the}

following.

^{For a} chaotic Gaussian field the modes are

independent

^{and for}

given

^mean numbers of

pho- tons

nk

>,

^the

entropy

^{in each} mode is minimum.

To prove that this definition is

equivalent

^{to the}

previous

^{one, we can write the}

density

^{matrix in two}

différent _ways, on the basis of the

photon

^number

states

[6] 1 { nk } >,

^{or on} the basis of the coherent states

[6] 1 { llk } >.

or

when the

P-representation

^exists.

The coherent states are linear combinations of the

photon

^{number states as :}

The identification of _eq.

(33)

^and

(35)

^{enables us to}

obtain the matrix elements p{nk,mk} ^{when the}

P-repre-

sentation is known.

For the chaotic

field,

^{since the modes are}

indepen- dent,

^the identification is

easy and

^{we obtain}

This

expression corresponds

^to the maximum of the function

Sk

^{= -}

Tr {

Pk

log Pk }

which characterizes the entropy ^{in the k mode}

[8].

Therefore

^a

stationary

^{Gaussian e. m.}

field

^{in a}

cavity

^{is a}

field

whose modes are

independent

^and

for

which the entropy ^{is maximum in} each

mode ;

^{we can}

define

^{in a}

unique

way such a chaotic

field

ⁱⁿ quantum

mechanics,

^either

by

^its

P-representation (eq. (9))

^or

by

^the distribution

given

ⁱⁿ eq.

(36)

^{on the basis}

of

^the

n

photons

^states.

5.2 STATIONARY EQUIVALENT ^{FIELD. -} Let us now treat the consequences of

nonstationarity

^{for the}

generalized

Gaussian fields

(§ ^3,

^eq.

(11».

Any generalized

Gaussian field except ^{the chaotic}

one is

only quasi-stationary

^{and not}

fully stationary,

i. e. second ^moments such

as ( E(t1) E(t2) +

^{are not}

functions

of , (tl - t2).

^We

supposed only

^{that the}

moments E(tl) £*(t2) >

^are functions of

(tl

^-

t2).

We said

previously

^{that it is}

possible

^to obtain from such a

quasi-stationary

^field

E(t),

^a

fully stationary

field

É(t) by introducing

^{a uniform}

phase (see

^eq.

(14))

Hence every ^moment such

as E(t)p >

vanishes. This

new field

£(1)

^{has the same}

intensity

^as

E(t)

^{but is not}

Gaussian

(cf. § 5.2,

eq.

(45)).

^{All the fields}

^differing

from

E(t) only by

^a

phase

^fàctor

belong

^{to the same}

equivalence

^class

(J)

^with respect ^{to the}

intensity.

In quantum

mechanics,

^the

stationarity

condition is

which

corresponds

^{to the classical one of invariance}

of all the moments under time

translation,

^{i. e. the}

stationarity

in the strict sense. We shall examine the

particular

^case of unimodal fields.

The relation

(38)

^{for an unimodal e. m.} field reduces

[10]

^{to the}

following

^{condition on the}

density

^matrix

We can prove that this relation is

equivalent

^{to the}

condition that the

P-representation

^{is a function}

only

of the modulus r of a ⁼ r ejqJ.

The matrix elements in the basis of the

photon

states are

This

expression

vanishes for all

(n, m) only

^if

P(r, ç)

is

independent

^of ç because _eq.

(40)

represents ^the Fourier coefficients of the

periodic

^function

defined in

(0,

²

n).

^{The condition} pnm = 0

V(n, m)

^is

equivalent

^to

Ck

^{= 0 for all k =1= 0 or}

f«({J) = 2,

i. ^e.

P(a)

^is

independent

^of ç. Hence for an unimodal

e. m.

field,

^{there is an}

equivalence

^{between the three}

relations

r" -1 - a 1

.

198

Now we show that the

P-representation

^{of one}

of the elements of this

equivalence

^class

(J)

^is

subject

to the condition

where

Ps(1 ri. ¹⁾

^{is the}

P-representation

^of the statio- nary field

É(t)

^{in this}

class,

^and

P( ^{oc 1,} ç)

represents the

P-representation

ⁱⁿ

polar

coordinates.

As a matter of

fact,

^the statistical

properties

^{of the}

intensity

^{of an e. m. field are described}

by

^{the set of}

moments 6’"’"

{ tb ti }

We see from _eq.

(43)

that all fields whose

P-represen-

tation satisfies _eq.

(42)

^{have the same set of moments}

G n,n(j ti ti })

^{i. e.}

^they

^are

equivalent

^{as far as the}

statistical

properties

^{of the}

intensity

^{are concerned.}

The

only

^one among them that is

fully stationary

^has

a

P-representation

^{that is}

independent

^{of 0 in}

polar

coordinates and

depends only

^{of the} modulus of a.

Hence each

generalized

^{Gaussian field that is not}

fully stationary

^{has the same}

intensity properties

^as

a

fully stationary

^{field which we call the}

stationary equivalent

^{field and whose}

P-representation

^is

given by

eq.

(42).

For

example,

^the

P-representation

^{of a unimodal}

generalized

^{Gaussian field is}

given by

^{the two dimen-}

sional Gaussian

probability

distribution for correlated

r. v. X = a cos ç and Y =

LJ, sin cp :

where

The

P-representation

^{of the}

stationary equivalent field,

^for (J i ⁼ Q2 ⁼ 6, ^can be obtained from _eq.

(42) :

where dt is the modified Bessel

function of order zero.

References

[1] PICINBONO, B. and ROUSSEAU, M., Phys. ^{Rev. A 1} (1970) ^635.

[2] BENDJABALLAH, C. and PERROT, F., Opt. ^Commun. 3 (1971) ^21.

PERROT, F., Thèse de 3e Cycle, Université de Paris XI (1971).

[3] ROUSSEAU, M., Thèse de Doctorat d’Etat, Université de Paris XI (1972).

[4] BLANC-LAPIERRE, A. et FORTET, R., Théorie des fonctions aléatoires (Masson) ^1953.

[5] GLAUBER, ^R. J., Proceeding of ^the 10th session of the Scottish Universities Summer School in Physics ^{1969. Ed.} by

S. M. Kay ^{and A. Maitland} (Academic Press).

[6] GLAUBER, ^R. J., Quantum Optics and Electronics (Les Houches) 1964, ^Ed. by ^{C. de} Witt, A. Blandin and C. Cohen-

Tannoudji (Gordon and Breach Science Publ. inc, ^N. Y.) 1965.

[7] ^We show in ref. [1] ^{that the} chaotic field is, among all the Gaussian fields, ^the only ^{one which is} fully stationary.

This property derives : from the fact that the moments

like E*(t1) E*(t2) > ^{are identical to zero for all} (t1, t2).

[8] LANDAU, ^L. D. and LIFSCHITZ, ^E. M., Statistical Physics (Pergamon Press), 1959, p. 25.

See also YvoN, J., Les corrélations et l’entropie ^en mécanique statistique classique (Dunod), 1965, p. 83.

[9] MOLLOW, ^B. R., Phys. ^{Rev. 175} (1968) ^1555.

[10] KANO, Y., ^Ann. Phys. ³⁰ (1964) ^127.

[11] RICE, S., Bell. Syst. Tech. Journal 23 (1944) 282 and 24 (1945) 46.

LOÈVE, M., ^{C. R. Hebd. Séan. Acad. Sci. 220} (1945) ^380.