The problem of localizing photons

MODERN TIIBNDS

=========:::===

IN LASER. PHYSICS

The Problem of Localizing Photons

A. S. Shum.ovsky and A. A. Klyac:hko

Faculty ofScience, Bilkei1t Universit;)\ Bi1keiit,A11karil, 06S33 Tt11key

e. mail: [email protected].

tr

Received July 17. 2000

\/ttAllstract:,...lt is

shown thauhe boundary conditions, specifying the plane

wa~Js

of photons,

arc

i»capub1c or

f

description of the photon localiiatfon at detectio11 and emission. It is also shown lhal Mllcciion from the plaile

};;Vvavcs causes rin important change (deterioration) of the quantum limJt of precision of different photonic mea,

''{surcmcnts.

:m2e

the eatly days ofquantum theory of racliation,

·<f:ohlem

of localizing photons has attracted

a

great /Jjnterest. The point is that the photon operators :)tion and annihilation ate defined in all space; \}he intensity measurement through the use of . 'd'etector assumes the localization of photons;

at

\rffvicinity

of the sensitive area. of the photodetec-@(for recent discussion, see [2, 3 J).

1_{'"this note we show that the problem of localization}

<Jg'ris

is, directly coimetted with th&·spatial

stmc-Jhe

electtomagnetic

vacuum

State; In particular;

\fthat

the photon localization

is accompanied

by

Hrictease of the zero-point oscillations. Since the 'bi11:t oscillations defines, via the uncertriintyr:ela'."'

m~-quantum limit

e>f

precision of measui;enieiltof ··tphysical observables, [2, 3],

an

adeq1,1ate

'JioiHhe

of photon localization seems to be quite

"hf.

·;:~~\¥

note that the bask concepts of the quantum ?ofiadiation include the notion of the

e.lectro-W{v.icuu m state ( e.g., see [2, 3] ). It is possible to .. Jtfie,dectromagnetic vacuum is consid¢red

asa

·Jtsµbstance" which is able to support any elec ...

'hetic

radiation. The properties of the vacuum

;,;:'-defined geometrically via the boundary

condi-:ofe

precis~. via the:;t:hoice of the class of

sym-'l$of utions of the homogeneous Helmholtz

\fafion

used

in

the canonical quantization

[4)-riiple,

the

plane waves of photons obey the

Bnal

invariance which leads to the description 'h:()perators in terrilS of the linear momentum '.~spatial homogeneity of the

zero-pointosdl-Oturn,

the spherical waves

or

photons

I

SJ

obey

J:;iinvariance, defining the states with given

'foq'111entum and parity. We show here the

spa-mcfaeneity

of the zero.;point oscillations i,n' lat-·:=t~:_:.:.:: .

6Yl'note.·that the simplest example of the

pho-Ti~tion

is provided by the emission or

absorp-'H6ton

by an atom. In fact;

in some

stage of ·'ijfptotesses; tile photon seems to be localized

';i§inity of the atom. Let us remind that the

_;ffsitions emit the mnltipole photons [6, 7]

57

described by the·quantized spherical rather than plane waves [5]. The strong difference between the two rep'-resentations of quantum electromagnetic field can be illustrated by the following argument. Tq describe the monochromatic plane wave of photons

it

is enough to use only four different operators of creation ancl annihi-lation, representing the states with the two different polarizations [2, 3]. In tum, the monochromatic sphel'-ical wave of given parity and angular momentumJ 2:': l isdescdbed by 2(2j

+

I) 2:': 6 different operators of

cre-ation and annihilation [5]. Unlike the plane waves of

photons, the polarization index

µ

is not the, quantum

number here, S4J that the polarization is a local property of the spherical waves of photons [8}.

Consider now the zero-point oscillations bf the elec-troinagnetic field in the presence of atom. Let us stress the growing interest in the field of atom,photon

inter-actions caused by the expedments with the single"atom lasers

and in

the atomic

beams

as well :as by the engi-neered

entanglement

in the trapped Ridberg atoms (e.g., see [9-11]). To describe the zero-point oscilla-tions, it is not necessmy to ~pecify the type of ihe atomic transitio11,.so that the atom is effectively

repre-sented by a singular point at the origin of the reference frame, setting off the S0(3) symmetry of the vacuum

state;

·

The zero-point oscillations of the electric field

amplitude

l

·

i(r)

=

E{r)+ E\r),

where E(r) denotes the qperator positive~frequency part of the electric field str~ngth, are described as fol~ lows [2; 3):

~· ' +

(01~-(r)IO}.

=

(OIE(r) · E (r)IO) {l)

because the averaging of the norma:1~orde1·ed operator constructions over the vacuum state 10) gives zem. To calculate (1), we consider the multipole fleld in the so-calledhelicity basis [l2J

ex± ie;,

58 SHUMOVSKY, KL Y ACHKO

where

E(r) =

I,<-l

lX.-µE,

1(r).

µ

Here £± describe the circular polarized components, while £11 gives the linearly polarized "longitudinal"

component (in the z-direction). In the case of plane waves, the third component is forbidden due to the transversal invariance. Nevertheless, this component is allowed for the electric multipole radiation both classi-cal and quantum [5, 12]. The components are repre-sented as follows

.i

E11(r)

=

iky

L

L,

V1111,1(r)a1111 • (2)

j

=

1 m

=

-.i

Here y is the normalization factor, j describes the total angular momentum, 111 gives the projection of the total

angular momentum on the quantization axis and ai,,, is the multipole photon annihilation operator. The mode functions in (2) have the form [5, 12]

I V ~;mµ(r)

=

J2j +

r-X [J}fi+ 1(1,

j +I,µ, 111-µljm)Y_;+

1.m-µ

-JT+tfj-1(1,j- l,µ,m-µljm)Y;_1.,,,-µ], where the radial functionfi(kr) is represented either by the spherical Bessel functionji(kr), in the case of stand-ing waves in a spherical cavity, or by the spherical Han-kel functions of the first and the second kind, describing the outgoing and converging spherical waves respec-tively. Here( ... 1/m) denotes the Clebsch-Gordon coef-ficient of vector addition of the spin and orbital parts of the total angular momentum and

Y

1_,,, -µ is the spherical

harmonics. It is now seen that (I) takes the form

2 · 2"" "" ~

(01~ (r)IO) = (ky) k-1,°')V;,,,µ(rW (3) µ "· Ill

Tims, the zero-point oscillations of the electric field strength of the electric multipole field are the functions of position with respect to the singular point (atom) in very contrast to the case of plane waves. Similar contri-bution can also be calculated for the magnetic induc-tion. To make a comparison, we note that the standard level of the zero-point oscillations of the monochro-matic plane waves of photons has the form

(4)

at any point r [2, 3]. As an example of some consider-able interest, we examine the zero-point oscillations (3) in the case of the atom located at the center of an ideal spherical cavity when the radial dependence of the mode function is represented by the spherical Bessel

functionj1(kr). It is a straightforward matterto arrive at

conclusion that the first (electric dipole) term in (3)

lim (Ol(~(r))2_IO)i=

1

=

12(ky)2 kr-'> 0

exceeds (4) in six times. More detailed examination shows that (3) in the case of the dipole field exceeds (4) at kr ~ 2.4 which means that r ~ 0.38A, where A is the wavelength of the cavity mode. Let us stress that such a distance correcponds to the standard order of the interatomic separation of the trapped Rydberg atoms

[ 11 ]. Similar result can be calculated for all other j as well. Then, summation over j in (3) shows that the zero-point oscillations in vicinity of the origin strongly exceed the standard level (4). In fact, (3) converges to

(4) at kr ~ I which corresponds to the so-called far or wave zone.

Consider now the model of the complete Hertz-type optical experiment where the source and detector are the two identical atoms separated by some distance d.

The "source" atom is prepared initially in excited state of some multipole transition, while the "detector" atom is in the ground state. The measurements are then pro-vided by the emission and successive absorption of the photon. Taking into account, that the photon seems be localized initially at the source location and finally at the detector location, we have to describe the radiation field in terms of linear superposition of outgoing and converging spherical waves, focused on the source and detector re~pectively. This superposition should obey the boundary conditions for real radiation. In view of the recent results [ 13], one can anticipate that this model obeys the causality principle. Then, it follows from the above discussion that the measurement is accompanied by the strong zero-point oscillations. In fact, it follows from (3) that the photon in vicinity of the detecting atom is subjected to the strong vacuum noise due to the presence of this atom.

If

d

<

A, this noise is increased due to the influence of the source atom.

We stress that similar increase of the vacuum noise accompanies the localization of photons in vicinity of the sensitive area of photodetector. Since the sensitive area O' is finite, the measurement of the plane wave of photons validates some kind of the photon localization [I, 2]. To describe such a localization, we have to assume that the incident wave focuses on

o

which means that near

o

the radiation should be described by a superposition of converging waves deflected with respect to the incident k. In other words, instead of the translational invariant plane wave, we have to choose the superposed state, corresponding to the boundary conditions on o. In view of the above discussion, such a superposed state should manifest much more strong zero-point oscillations than (4).

Let us briefly discuss the results.

It

has been shown that the photon localization needs the presence of some material object, interacting with the photons (atom, photodetector etc.). The localization is then defined via

THE PROBLEM OF LOCALIZING PHOTONS 59

the boundary conditions for the radiation field.

In

tum, 1he boundary conditions cause the spatial structure of the electromagnetic vacuum state and of the zero-point oscillations. Hence, the presence of the material objects (atoms, molecules, detectors etc.) leads to the increase of the vacuum noise in the optical measurements which can strongly exceed the standard level (4). It is now clear that the model of plane waves of photons consid-ered in empty space is incapable of calculation of a realistic estimations of the quantum limit of precision of optical measurements.

The obtained results seem to be important for any quantum optical measurement when very few photons are detected, especially in the engineered entanglement in the atom-photon systems.

REFERENCES

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LASER PHYSICS Vol. 11 No. I 2001

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(New York: Dover).

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