Photon-mediated interaction between two distant atoms

Stefan Rist,1 Jürgen Eschner,2 Markus Hennrich,2 and Giovanna Morigi11Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

2ICFO—Institut de Ciències Fotòniques, 08860 Castelldefels, Barcelona, Spain�Received 6 May 2008; published 8 July 2008�

We study the photonic interactions between two distant atoms which are coupled by an optical element �alens or an optical fiber� focusing part of their emitted radiation onto each other. Two regimes are distinguisheddepending on the ratio between the radiative lifetime of the atomic excited state and the propagation time of aphoton between the two atoms. In the two regimes, well below saturation the dynamics exhibit either typicalfeatures of a bad resonator, where the atoms act as the mirrors, or typical characteristics of dipole-dipoleinteraction. We study the coherence properties of the emitted light and show that it carries signatures of themultiple scattering processes between the atoms. The model predictions are compared with the experimentalresults of Eschner et al. �Nature �London� 413, 495 �2001��.

DOI: 10.1103/PhysRevA.78.013808 PACS number�s�: 42.50.Ct, 42.50.Ar, 32.70.Jz

I. INTRODUCTION

Control of photon-atom interactions lies at the heart ofquantum technologies based on atomic and photonic systems�1�. Recent experiments have demonstrated the quantum cor-relations between atoms and emitted photons �2–4�. Atom-photon entanglement was then applied for entangling distantatoms by photon measurements �5�. Further experimentsdemonstrated the possibility of spatially confining atomswith nanometric precision inside resonators �6–8�, and hencecontrolling their coupling with the electromagnetic fieldmodes of cavities. Such precision has permitted the realiza-tion of quantum light sources with a high degree of control�9–13�, and hence posed the basis for the realization of quan-tum networks based on atom-photon interfaces �1�.

Parallel to these experimental efforts, studies are also fo-cusing on achieving strong coupling between atoms and pho-tons by means of optical elements, such as lenses of largenumerical aperture �14–19� or optical fibers �20–22�. In par-ticular, in �14� two distant atoms in front of a mirror werecoupled by means of a lens, focusing the radiation emitted byone atom onto the other. In this setup, the first-order coher-ence was experimentally studied, showing an interferencepattern when the optical path length between the atoms wasvaried. In earlier experiments with two trapped ions, far-fieldinterference of their scattered light �23� and their near-fieldinteraction �24� were studied.

In this paper, we present an extensive theoretical study ofthe radiative properties of two distant atoms when they arecoupled via an optical element, which could be an opticalfiber or a lens, as sketched in Fig. 1. In this situation radia-tion is multiply scattered between the atoms, until it is finallydissipated into the external modes of the electromagneticfield. Our model is based on the theory developed in �25,26�for the case of a single atom interacting with itself via amirror, and extends it to the situation of two coupled atoms.The theoretical predictions of our model reproduce the ex-perimental results of �14� and allow us to identify possiblemeasurements that highlight the multiple-scattering features.Moreover, the scattered photons are correlated with the scat-tering atoms, thereby establishing correlations and, in certain

cases, entanglement between their internal excitations.This paper is organized as follows. In Sec. II we make

some preliminary considerations of the system. In Sec. III weintroduce the model in detail and solve the basic equationsdescribing the coupled dynamics of the internal atomic statesand a few photons of the electromagnetic field. In Sec. IV weinvestigate in detail the radiative properties of the system,and in Sec. V we provide the details of the first- and second-order coherence of the light scattered by the atoms when theyare weakly driven by a laser. In Sec. VI we provide someoutlook from the present work, and in the appendixes wereport details of the calculations.

II. PRELIMINARY CONSIDERATIONS

The scattering cross section of an atomic dipole transitionin free space is on the order of the square of its wavelength ��27�. Consequently, the free-space photonic interaction be-tween two atomic dipoles at distance d is determined by theratio � /d �28�: when d��, strong modifications of theatomic emission spectrum of one atom due to the presence ofanother one are observable �24,28,29�; when d��, these ef-fects are negligible, and the atoms scatter photons indepen-

Atom 2Atom 1

Laser Laser

FIG. 1. Dipolar transitions of two atoms, which are several op-tical wavelength apart, coupled by a lens focusing part of theiremitted radiation onto each other. In �14� a similar situation wasrealized, coupling two atoms via a mirror and a lens. Analogousdynamics can be observed when the atoms are trapped close to anoptical fiber; see, for instance, �22�.

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dently. This behavior is dramatically modified if an opticalsystem, like a lens with large numerical aperture or an opti-cal fiber, focuses a significant fraction of the radiation emit-ted by one atom onto the other. This latter situation issketched in Fig. 1 for the case of a lens that images the atomsonto each other.

When the photonic interaction between the atoms is me-diated by an optical element, its strength is characterized bythe fraction � of modes of the electromagnetic field whichthe optical system transforms into each other. Thus � re-places the scaling with � /d of the free-space case, and cou-pling over much larger distances than � may be achieved.

The atom-atom distance d, or more precisely the propaga-tion time for a photon from one atom to the other via theoptical element,

� =d

c�1�

remains an important physical parameter of the photonic in-teraction, since it has to be compared with the radiative life-time of the atomic dipolar transition 1 /�, which determinesthe time scale on which the photonic excitation is dissipatedinto free space, as well as the length of the emitted photonicwave packet. When ���1, the process of photon scatteringby each atom is well localized in time and space: a photonicexcitation is exchanged between the atoms at integer mul-tiples of the delay time �, until its amplitude is damped tozero by emission into the external modes of the electromag-netic field. When ���1, in contrast, multiple scatteringevents add up coherently during the excitation time of eachatom, causing the spontaneous emission rate to be enhancedor suppressed, depending on the interatomic distance�modulo the wavelength�. This regime is equivalent todipole-dipole interaction with a delay time �.

In all cases, the system of two atoms confining radiationby multiple scattering shows some analogies with an opticalresonator with low-reflectivity mirrors. This analogy is ap-propriate when the atomic transition is not saturated. Indeed,in this regime the radiative properties are very similar tothose of a single atom interacting with itself via a mirror,studied in �14,15,26�. The peculiarity of the two-atom systembecomes more evident when saturation effects are relevant.Some important properties, such as the creation of correla-tions and entanglement between the atoms via the multiplyscattered photons, are identified when studying intensity-intensity correlation of the light scattered by the two-atomsystem, as discussed in Sec. V.

III. THE MODEL

In this section we develop the theoretical model for de-scribing the dynamics of two atoms in presence of an opticalelement which focuses the radiation emitted by each atominto the other, as sketched in Fig. 1. In particular, we use thetheoretical formalism in �26� for one atom in front of a mir-ror, and generalize it to the case of two coupled atoms.

The system consists of two identical atoms of mass M,which are trapped at the positions r1 and r2, and whose rel-evant electronic degrees of freedom are the ground state �g�

and the excited state �e� forming a dipole transition withdipole moment D, frequency �0, and wavelength �=2�c /�0. The interatomic distance d= �r2−r1� is such thatd��; thus free-space dipole-dipole interaction between theatoms is negligible. We assume, however, that a lens �or anequivalent optical system� is placed between the atoms,which collects a fraction of the radiation from each atom andfocuses it onto the other one. We use the plane wave decom-position for these modes and label them with , in order todistinguish them from the external modes which do notcouple the atoms; the latter are labeled with �see also Fig.2�. The Hamiltonian of the system describes the interactionbetween the dipoles and the modes of the electromagneticfield, and can be decomposed into the sum

H = H0 + Vemf, �2�

where H0 gives the self-energy and Vemf the interaction be-tween the dipoles and the modes of the electromagnetic field.In detail,

H0 = �j=1,2

��0� j+� j

− + ��=,

���a�†a�, �3�

where the first term describes the energy of the atoms, with� j = �g� j�e� and � j

† its adjoint, and subscript j=1,2 labelingthe atom. The second term is the free Hamiltonian of thetransverse photon field where the summation runs over allfield modes. We label by � the mode with wave vector k�

and polarization ��k�, while a�† and a� are the creation and

annihilation operators for a photon in that mode, obeying thecommutation relation �a� ,a��

† �=��,��. In particular, the modeswith label �= are the ones which couple the atoms via thelens.

The interaction of the atoms with the electromagneticfield, Vemf, is given in the electric dipole and rotating waveapproximation, and takes the form

Modes ρ

Det. µ'

Det. 1,ρ

ϑ

Laser Laser

Atom 2at r2

Atom 1at r1 ϑL

Modes µ

Det. 1,µ

Det. 2,ρ

FIG. 2. Detailed schematic of the physical system showing thedetectors that correspond to the various measurements described inthe text. Another detector 2 , would be placed in a location equiva-lent to that of detector 1 ,, to measure the emission of atom 2individually.

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Vemf = − i��j=1,2

� j+ �

�=,g�a�eik�·rj + H.c., �4�

where

g� = �D · ����/�2�0�V�

with the vacuum electric permittivity �0 and the quantizationvolume V. In the presence of a laser driving the atoms theHamiltonian will be given by

H� = H + VL�t� , �5�

where the term VL describes the atom-laser coupling andreads

VL = ���j=1,2

� j+ei�kL·rj−�Lt� + H.c. �6�

Here, the laser is a classical field at frequency �L �27�, � isthe coupling strength, and kL is the wave vector of the inci-dent laser beam.

The dynamics of the system is studied by solving theSchrödinger equation treating the interaction of the atomswith the electromagnetic field as a perturbation. For this pur-pose, the wave function ���t�� of the atoms and the field attime t, in the interaction picture with respect to H0, is de-scribed by

���t�� = be�1��t��e1,g2,0� + be

�2��t��g1,e2,0� + �

bg���t�

��g1,g2,1,0� + �

bg���t��g1,g2,0,1� , �7�

where the state �0� corresponds to the vacuum state of theelectromagnetic field, and the state �n� ��n�� to n photons inmode ��. In Eq. �7� we have assumed that at most oneexcitation is present in the system. In particular, the coeffi-cients be

�j��t� are the probability amplitudes at time t for atomj being in the excited state, while the coefficient bg

����t� givesthe probability amplitude to find a photon in the field mode �at time t, with both atoms in the ground state. For later con-venience, we also introduce the probability amplitudesbg

�j,���t�, with

bg����t� = bg

�1,���t� + bg�2,���t� ,

and which distinguish which atom has emitted the photoninto mode �.

We will solve the Schrödinger equation using this ansatzfirst in the absence and then in the presence of a laser drivingthe atoms. In particular, we will study the dynamics as afunction of two important physical quantities which charac-terize the system. The first is the time delay � for light topropagate from one atom to the other, defined in Eq. �1�. Asnoted before, we consider the case c���. The second im-portant quantity is the strength of the photonic coupling be-tween the atoms mediated by the lens, which is definedthrough the fraction of 4� solid angle within which the ra-diation from one atom is focused onto the other. This corre-sponds to the fraction of modes labeled with , which propa-gate from one atom to the other via the lens. We denote thecoupling by the dimensionless parameter �,

� = �n

�1 − �D · n�2/�D�2� =3

8���

d�0�1 − �D · n�2/�D�2� ,

�8�

where n=k /k and �� is the solid angle collected by thelens. The value of � lies in the interval 0���1, whereby�→0 corresponds to the limit without the lens and �→1would describe an ideal optical system that maps all radia-tion from one atom onto the other.

A. Perturbative solution of the Schrödinger equationin the absence of the laser

When the atom-laser coupling is set to zero, then in thereference frame of the atoms the coefficients be

�j� ,bg�j,� ,bg

�j,�

obey the differential equations

be�j��t� = − �

geik·rjei��0−��tbg

���t�

− �

geik·rjei��0−��tbg�j,��t� , �9a�

bg�j,��t� = ge

−ik·rje−i��0−��tbe�j��t� , �9b�

bg�j,��t� = ge−ik·rje−i��0−��tbe

�j��t� , �9c�

where in the regime �r2−r1��� we have neglected processesin which a photon emitted into a mode by one atom isreabsorbed by the other one.

A closed form for the coefficients of the dipole excitationsis found by summing over the modes of the electromagneticfield and by applying the Wigner-Weisskopf approximationas in �26�. The details of the calculation are reported in Ap-pendix A. The resulting equations take the form

be�1��t� = −

�

2be

�1��t� − ��

2ei�0�be

�2��t − ����t − �� , �10a�

be�2��t� = −

�

2be

�2��t� − ��

2ei�0�be

�1��t − ����t − �� . �10b�

Equations �10a� and �10b� show different behavior depend-ing on whether t�� or t��. For t�� these equations aredecoupled and describe exponential damping at the rate � ofthe single-atom excited-state occupation, as in free space.After the time �, coupling by light scattering from each atomonto the other appears, its strength being set by the parameter�.

We proceed by solving Eqs. �10a� and �10b� for an arbi-trary initial state with a single atomic excitation,

���0�� = �1�e,g,0� + �2�g,e,0� . �11�

A simple solution is then found by using the decompositioninto symmetric and antisymmetric coefficients C��t�,

be�1��t� = �C+�t� + C−�t��/2, �12�

be�2��t� = �C+�t� − C−�t��/2, �13�

obeying the differential equations

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C��t� = −�

2C��t�� �

�

2ei�0�C��t − ����t − �� , �14�

whose solution is �30�

C��t� = C��0��k=0

�

��1�kIk�t� ,

with

Ik�t� =�− ��2 ei�0��k

k!�t − k��ke−��/2��t−k����t − k�� . �15�

Correspondingly, the probability amplitudes for the excitedstates are

be�1��t� = �1�

k

I2k�t� + �2�k

I2k+1�t� , �16a�

be�2��t� = �1�

k

I2k+1�t� + �2�k

I2k�t� , �16b�

while the probability amplitudes bg�j,��t� for the emission of

a photon into mode by atom j are given by

bg�1,��t� =

ge−ik·r1

�2 + i�

�k=0

�

��1H2k�t,�� + �2H2k+1�t,��� ,

�17�

bg�2,��t� =

ge−ik·r2

�2 + i�

�k=0

�

��1H2k+1�t,�� + �2H2k�t,��� ,

�18�

with

� = �0 − �. �19�

In Eqs. �17� and �18� we assumed that the electromagneticfield is initially in the vacuum state, bg

�0�=0, and we intro-duced the function

Hk�t,�� =�− ��2 ei���k

k!�tk�kGk„�i� + �/2�tk…��tk� , �20�

with tk= t−k� and

Gk�s� = 1F1�k,k + 1,− s� − e−s, �21�

where 1F1�k ,k+1,−s� is the confluent hypergeometric func-tion �31�. In the limit �→0, i.e., when there is no couplingbetween the atoms, Eqs. �17� and �18� reduce to the usualfree-space decay spectrum of two independent dipoles withlinewidth � �32�.

B. Perturbative solution of the Schrödinger equationin the presence of the laser

We consider now the situation where the atoms areweakly driven by a laser at intensity �. Hence, we set��0 in the Schrödinger equation and solve the dynamics of

the new Hamiltonian assuming that VL is a weak perturbationto the atomic dynamics. We use the ansatz for the wavefunction in Eq. �7�, where we denote now the probabilityamplitudes by be

�j��t� ,bg�j,���t� ,bg

����t�→ce�j��t� ,cg

�j,���t� ,cg����t�

�with j=1,2 and �= ,�. Let ���0��= �g1 ,g2 ,0� be the initialstate. By solving the coupled differential equations for theprobability amplitudes in first order in � and in the referenceframe rotating at the laser frequency �L, we find

ce�1��t� = − i/�

0

t

dt�ei�Lt�e1,g2,0�e−iH�t−t��/�VL�t��

�e−iHt�/��g1,g2,0�

= − i2�0

t

dt�ei�L�t−t���e1,g2,0�

��e−iH�t−t��/��eikL·r1�e1,g2,0� + eikL·r2�g1,e2,0��/2� .

�22�

Corresponding expressions are derived for ce�2��t� and cg

�j��t�.The term inside the square brackets corresponds to thetime evolution of the state ��0�= �eikL·r1�e1 ,g2 ,0�+eikL·r2�g1 ,e2 ,0�� /2 when there is no laser. Hence we canwrite

ce�j��t� = − i2�

0

t

dt�ei�t�be�j��t�� , �23a�

cg����t� = − i2�

0

t

dt�e−i��t�bg����t�� , �23b�

where �= , and we have introduced the detunings

� = �0 − �L, �24a�

�� = �� − �L. �24b�

The coefficients be�j� and bg

�� are found using the solutionsderived in Sec. III A when the initial state is ��0�. One gets

ce�1��t� =

− i��2 + i�

eikL·r1�k=0

�

�H2k�t,�L� + ei�LH2k+1�t,�L�� ,

�25�

with

�L = kL · �r2 − r1� . �26�

The equation for ce�2��t� results from Eq. �25� by interchang-

ing the indices 1↔2. The probability amplitudes cg�j,��t� are

found using Eq. �17� in Eqs. �23a� and �23b�, assuming thatinitially both atoms are in the ground state and the electro-magnetic field in the vacuum state. One gets

cg�1,��t� = − i

�g�2 + i�

0

t

dt�ei��L−��t�ei�kL−k�·r1

��k=0

�

�H2k�t�,�� + ei�LH2k+1�t�,��� , �27�

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with � given in Eq. �19�. The probability amplitude cg�2,��t�

for atom 2 is obtained by swapping the superscripts 1↔2 inEq. �27�.

C. Discussion

The probability amplitudes of the atomic excited states inthe absence and in the presence of the laser, given in Eqs.�15� and �25�, respectively, are coherent sums over contribu-tions starting at different instants of time �k= t−k�. Thesecontributions correspond to the effect of k exchanges of aphotonic excitation between the two atoms. In particular, forthe case of atom 1, the contributions at �2k correspond to anexcitation which propagated to atom 2 and back. Hence, inEq. �15� this term vanishes when initially only atom 2 isexcited. Similarly, the contributions at t=�2k+1 vanish whenatom 2 is initially in the ground state. Similar considerationsapply for the case in which the laser drives the atom, Eq.�25�.

An important property of these equations is that each termof the sum has a well-defined phase, which is an integermultiple of �0� ��L� with the laser excitation�. At the sametime the contributions are damped by an exponential func-tion at rate �. Consequently, the individual terms show inter-ference if over the time � they do not decay appreciably. Thisshows in more detail how the radiative properties of the sys-tem are determined by the parameter ��, the ratio betweenthe delay time and the excited state lifetime. In particular, for���1 interference plays no role, and the photonic excitationis a wave packet bouncing between the two atoms, until itsintensity is damped to zero by scattering into free space. For���1 the terms in �15� add up coherently and interfere. Theeffect of the interaction hence modifies the radiative proper-ties of the atoms, and the dynamics are analogous to an ef-fective dipole-dipole interaction �28�.

In this perspective, the optical setup composed of twoatoms and the lens can be considered as a resonator, wherethe atoms are mirrors of low reflectivity and reflection band-width � /2, while 2� is the round-trip time. The parameter ��hence gives the number of modes that this peculiar “two-atom cavity” sustains: for ���1 it sustains several modesand can be considered a multimode resonator. Conversely,for ���1 only a single mode of radiation is supported, andwe will denote this case as a single-mode resonator.

Using this insight, we now analyze the probability ampli-tude and the spectrum of the emitted photons in the externalmodes labeled with . Let us first assume that the laser isabsent, and that initially atom 1 is in the excited state, i.e.,�1=1 , �2=0 in Eq. �11�. From Eqs. �17� and �18�, the am-plitude probability for the state of the field reads

bg���t� =

g�2 + i�

�k=0

�

�e−ik·r1H2k�t,�� + e−ik·r2H2k+1�t,��� ,

�28�

where the two terms in the sum account for the respectivecontributions of the two atoms to the emission into the fieldmode. The label k gives the number of photon exchangesbetween the two atoms before the photon is finally emittedinto the external mode .

In the long time limit Eq. �28� reduces to the form

bg���t → �� = ge−ik·r1

� �2 + i�� − ���/2�ei���1+cos ��

� �2 + i��2 − ����/2�ei���2,

where � denotes the angle between the vector r1−r2 and thewave vector k of the mode; see Fig. 2. At �=� /2, in par-ticular, the probability to measure a photon in mode isgiven by

�bg���t → ���2

=g

2

��2/4��1 + � cos ���2 + ��0 − � + ���/2�sin ���2 ,

�29�

showing that the spectrum exhibits a modulation at multiplesof the frequency 1 /2�. In the resonator picture, the modula-tion peaks are at the mode frequencies of the resonator, and1 /2� corresponds to the free spectral range. The spectralmodulation will be visible when ���1, i.e., when the sys-tem is in the multimode-resonator regime. On the other hand,in the single-mode regime ���1, one will observe a changeof the radiative linewidth, which depends on the phase �0�.

When the atoms are laser driven, the probability ampli-tude for the excited state occupation in the long-time limit is

ce�1��t → �� =

− i��2 + i�

eikL·r1�1 − Kei�L

1 − K2 � , �30�

while the probability amplitude that mode is occupied byone photon scattered by atom 1 takes the form

cg�1,��t� = 2���t����

g�e−i�t/2

�i��/2� − ���1 − K2�

�ei�kL−k�·r1�1 − Kei�L� + O�1� . �31�

Here

K = ��

2

ei�L�

�2 + i�

�32�

and ��t����= 1�

sin��t/2�� is the diffraction function �27�. The

second term on the right-hand side gives no contribution tothe rate of emission, so that it is not explicitly reported. Itsspecific form can be found from Eq. �27� and Appendix B.The probability amplitudes for the second atom are obtainedby swapping the indices 1↔2. The detailed derivation ofthese expressions is reported in Appendix B.

These results are discussed for various specific limits inthe following sections.

IV. RADIATIVE PROPERTIES

In this section we discuss the radiative properties of thetwo atoms, when they are observed together or individually,in the absence of laser excitation, and assuming that atom 1is initially excited. The various quantities which will be dis-cussed correspond to measurements with different detectors,as illustrated in the detailed setup in Fig. 2.

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A. Multimode-resonator regime

When ���1, then the transient dynamics of the system ischaracterized by the two atoms exchanging a photonic exci-tation well localized in time. The excitation probabilities Pj

= �be�j��t��2, with be

�j��t� given by Eqs. �16a� and �16b�, aredisplayed in Fig. 3 as a function of time. One clearly seesthat a photonic excitation propagates back and forth betweenthe atoms, while its amplitude is damped due to the scatter-ing into the external modes of the electromagnetic field. Theshape of the photonic wave packet exchanged between thetwo atoms changes with time: with each bounce it acquires amore symmetric and broader shape, due to the frequency-dependent reflection by the atoms. The broadened wavepackets increasingly overlap with time, such that interferencebetween subsequent excitations may become visible for longtimes, as shown in the example of Fig. 4.

The effects of the coherent addition of the multiple scat-tering events become more visible by inspecting the time-dependent probability of emitting the photon into the exter-nal modes of the electromagnetic field. In the continuumlimit of Eq. �28�, it takes the form �33�

S��,t� � �bg��,t��2, �33�

which for t→� coincides with the emission spectrum. Anexample of S�� , t� is shown in Fig. 5. For times t��, beforescattering events can interfere, it exhibits a Lorentzian formlike an atom in free space, while after a time t�� it developsspectral modulation with peaks spaced by 1 /2�.

The effect of the distance between the atoms on theirindividual emission spectra is displayed in Figs. 6�a� and6�b�. In particular, the maxima of the spectra, spaced by the“free spectral range” 1 /2�, shift according to the optical dis-tance between the atoms. The visibility of modulation islarger the closer � is to unity.

B. Single-mode-resonator regime

We now analyze the regime ���1, in which several pho-ton excitations are exchanged between the atoms during thenatural lifetime of the excited state. In Fig. 7 the excited statepopulations of both atoms are displayed. As atom 1 is ini-tially excited, atom 2 stays in the ground state until the in-stant t=�, after which its excited state occupation increasesdue to the interaction with the radiation from atom 1; seeFig. 7�b�. The excitation of atom 1 is damped as in free spaceuntil time t=2�, after which the damping rate is attenuated orenhanced depending on the relative interatomic distance, i.e.,

1 2 3 4 5 60t / τ

0.2

0.4

0.6

0.8

1

|b(1)e (t)|

2

0.02

0.01 2.5 x 10-5

0

1 2 3 4 5 6

0.005

0.01

0.015

0.02

0.025

2.2 x 10-4

1.1 x 10-4

3.5 x10-6

0t / τ

|b(1)e (t)|

2

0

(b)

(a)

FIG. 3. Excited state occupation of the atoms as a function oftime in units of �, as given in Eq. �15�, when initially atom 1 isexcited. The other parameters are ��=10, �=0.4, and �0�=n�.Note the change of vertical scale from each maximum to the nextone.

0 2 4 6 810 -13

10 -10

10 -7

t / τ

|b(1)e (t)|2

10 -4

10 -1

FIG. 4. �Color online� Logarithmic plot of the excited state oc-cupation of atom 1 as a function of time in units of �, as given inEq. �15�. The curves correspond to the values �0�=n� �red solidline� and �2n+1 /2�� �green dotted line� for �=0.4 and ��=5. Thedashed line corresponds to �=0 and is plotted for reference.

1

2

3

2

10

-1-2

(ω0−ω)/γ

S(ω,t)

t / τ

0

FIG. 5. �Color online� Probability of emission of a photon, Eq.�33�, as a function of frequency �in units of �� and of time �in unitsof ��, for the emission angle �=� /2 and the phase �0�=2n�.

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on �0�. The effect of the relative phase between the mul-tiple absorption-emission events is more evident when plot-ting the excitation probabilities on a logarithmic scale, asshown in Fig. 8.

Figure 9�a� displays the emission probability S�� , t� as afunction of frequency and time, showing that it is always asingle-peaked curve, whose width varies with time. Figure9�b� displays the emission spectrum of the first atom in com-parison with the one in free space for different values of theparameter �0�, showing that depending on the relative dis-tance one can observe subradiant or superradiant emission.The atomic interaction is hence a retarded dipole-dipole in-teraction, mediated by the photonic excitation over the inter-atomic distance.

0 2 4 6-2-4-6

(ω0-ω)τ/π

S1(ω)

S2(ω)

0 2 4 6-2-4-6

(ω0-ω)τ/π

S1(ω)

S2(ω)

(b)

(a)

FIG. 6. �Color online� Spectrum of the light emitted by atom j,Sj���� limt→��bg

�j��� , t��2, as a function of the frequency �in units of� /�� for �0�= �a� n� and �b� �2n+1��2 . The parameters are �=0.4 and ��=10. The dashed blue line gives the spectrum of theatom when �=0 and is plotted for comparison.

2 4 6 8 10 12

0.2

0.4

0.6

0.8

1

t / τ

|b(1)e (t)|

2

10

0.01

0

-0.01

t / τ0

0

2 4 6 8 10 12

0.005

0.01

0.015

0.02

00

t / τ

|b(2)e (t)|

2

(b)

(a)

FIG. 7. �Color online� Excited state occupation of �a� atom 1and �b� atom 2 as a function of time �in units of ��, as given in Eq.�15�, when initially atom 1 is excited, and for �0�=n� �red solidline� and �2n+1��2 �green dotted line�. The dashed blue line is thesolution for �=0� and is displayed for reference. The other param-eters are ��=0.4 and �=0.4. The red solid and green dotted lines inthe inset of �a� display the difference between the values of �be

�1��t��2

at �0�=n� and at �2n+1��2 , respectively, from the correspondingexcited state occupation at �=0 as a function of time.

0 10 20 30

10 -10

10 -7

t / τ

|b(j) e (t)|2

10 -4

10 -1

FIG. 8. �Color online� Logarithmic plot of the excited state oc-cupation of atom 1 �solid line� and atom 2 �dotted line� as a func-tion of time �in units of �� for the initial state � �0��= �e ,g ,0�. Theother parameters are ��=0.4, �=0.4, and �0�= �2n+1 /2��. Theblue dashed line shows the corresponding atomic excitation for �=0 and is plotted for reference.

3

6

9

12

2

1

0-1

-2

t / τ

(ω0-ω)/γ

0

S1(ω,t)

0 0.5 1 1.5-0.5-1-1.5

(ω0-ω)/γ

S1(ω)

(b)

(a)

FIG. 9. �Color online� �a� Probability of emission of a photon byatom 1, S1�� , t�� �bg

�1��� , t��2, as a function of the frequency of theemitted photon �in units of �� and of time �in units of �� for �0�=n�. �b� Spectrum of emission for atom 1, S1���, for �0�=n� �redsolid line� and �2n+1��2 �green solid line�. The other parameters are��=0.4 and �=0.4. The dashed blue line gives the emission spec-trum in free space and is plotted for comparison.

PHOTON-MEDIATED INTERACTION BETWEEN TWO… PHYSICAL REVIEW A 78, 013808 �2008�

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C. Two atoms vs single atom

The cases studied so far share several analogies with theradiative properties of a single atom in front of a mirror,analyzed, for instance, in �25,26�. In particular, in �25� Alberstudied the dynamics of one photon coupled to one atom atthe center of a spherically symmetric cavity with perfect re-flectivity. Depending on the radius of the cavity mirror, andthus on the time the photon needs to travel to the mirror andback in relation to the atomic decay time, Alber defines thesmall- and large-cavity limits, whereby in the first case theatom-cavity system is characterized by a delocalized excita-tion, while in the second case a photonic wave packet propa-gates back and forth, periodically exciting the atom. Al-though our system is a low-quality resonator, the multimodecavity that the two atoms form for ���1 is analogous to thelarge-cavity limit in �25�.

A very close analogy exists between our system and thesystem discussed in �26�, where Dorner and Zoller investi-gated the case of an atom interacting with its own light back-reflected by a distant mirror �26�. In particular, the dynamicsof two atoms exchanging photons via the lens share stronganalogies with the one of an atom interacting with its mirrorimage, if one restricts the Hilbert space to only one excita-tion, and if the atoms are initially prepared in a symmetricstate with �1=�2=1 /2 in Eq. �11�. For this initial state, thetime evolution of the excitation of one of the atoms is thesame as the one of the atom in front of the mirror. Theprobability amplitude for photon emission, however, showssome differences between the two cases. Figure 10 displays

the emission spectrum as a function of the emission angleand of the frequency. Here, the oscillation of the intensity asa function of the angle of emission � is indeed an exclusiveproperty of the two-atom case, arising from the fact that thelight emitted from the two scatterers interferes in the farfield.

V. LIGHT SCATTERING

In this section we analyze the scattering properties of thesystem when the atoms are driven by a laser below satura-tion. In this case the time evolution of the excited state am-plitudes, Eqs. �25�, describes the photon exchange betweenthe two atoms, which now additionally interferes with theincident laser light. For ���1, stepwise dynamics with thecharacteristic time step � are visible in the excited state oc-cupation of each atom, as displayed in Figs. 11�a� and 11�b�.For different distances between the atoms, and hence differ-ent phases of the various contributions, the discontinuities inthe curves at multiples of � show constructive or destructiveinterference, while for long times t�� the excited statepopulation tends to a steady state value. In the limit ���1,displayed in Figs. 11�c� and 11�d�, the curves are smooth andtend to the same steady state values. This stationary valuedepends on the two phases �L �the laser direction� and �L��the optical path length between the atoms�, according to

�ce�1��t → ���2 =

�2

� �2

4 + �2��1 − K2�2�1 − Kei�L�2, �34�

where K is given in Eq. �32� and is proportional to the cou-pling strength � between the two ions. Equation �34� doesnot depend on the parameter ��, which affects only the tran-sient dynamics. When the atoms are not coupled, �=0, onerecovers the free-space steady state value, as found for anatom hat is driven by a weak laser �27�.

We note that for the specific value �L= �2n+1�� we ob-tain

�ce�1�����2 =

�2

�L2/4 + �2

, �35�

which is the free-space formula with modified decay rate anddetuning,

� = ��1 − � cos �L�� ,

� = � − ��

2sin �L� .

This result coincides with the excited state population of asingle atom subject to interference between the laser excita-tion and the light backscattered from a mirror �26�. Thisequivalence holds only for the particular value �L= �2n+1�� but not in the general case.

In order to get some more insight, we analyze Eq. �34� for��1. At first order in � the excited state population of thefirst atom takes the value

-3 -2 -1 0 1 2 3

ϑ

-3 -2 -1 0 1 2 3

(ω − ω)τ/π0

0

π/4

π/2

3π/4

π

ϑ

0

π/4

π/2

3π/4

π

FIG. 10. Contour plots of the emission spectrum from both at-oms S���, Eq. �33�, as a function of the frequency � �in units of� /�� and of the angle of emission �, for the initial state � �0��= �1 /2���e ,g ,0�+ �g ,e ,0��. The other parameters are ��=10, �=0.4, and �0�=2n� �top panel�, �0�= �2n+1�� �bottom panel�.

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�ce�1��t → ���2

�2

�2/4 + �2 �1 + 2�A cos��L� + �L − !�� ,

�36�

with

A = �2/4�2/4 + �2 ,

tan ! =2�

�. �37�

The result for atom 2 is found from Eq. �36� by swapping thesubscripts 1↔2, i.e., by changing the sign of �L. Equation�36� shows how the excited state population is enhanced orsuppressed as the parameter �L� is changed. This change inthe atomic spontaneous emission rate, as well as a shift ofthe atomic resonance frequency, both controlled by the pa-rameter �L�, are manifestations of the modification of theradiative properties of the atoms due to their mutual interac-tion. Analogous frequency shifts in a single atom interactingwith itself via a mirror have been experimentally observedby Wilson et al. �15�.

A. Intensity of the scattered light

Let us now consider the intensity of the light scattered bythe laser-driven atoms, for several of the measurement setupsillustrated in Fig. 2. First we consider the situation where thedetection apparatus resolves the atomic positions and there-fore sums up incoherently the photons emitted by the atoms�detector 1 , plus detector 2 ,�. Then the detection rate is"="1,+"2,, whereby " j,=limt→��cg

�j,��t��2 / t. Using Eq.�27� we find

"1, = "2, = 2���� − �L�g

2�2

� �2

4 + �2 �

�1 − Kei�L�2

�1 − K2�2, �38�

where K is given in Eq. �32�. For K=0, i.e., in absence of theoptical element coupling the two atoms, the signal repro-duces the free-space resonance curve of the atomic dipole.For K�0 it shows two modulations, with the phase 2�L��through K2 in the denominator� and with the phase �L�+�L �in the numerator�. The first one corresponds to previ-ously emitted light returning to the same atom after scatter-ing from the other one; the other modulation is produced byscattered laser light arriving from the other atom. In general,the modulations show how the scattering of a single atom ismodified by the presence of another identical scatterer at afixed distance. The maximum enhancement, when all scatter-ing terms add up coherently, is found for 2�L�=2�, �L=0,and �=0, and is equal to �1+�� / �1−��. For �=0.2 it givesan enhancement of the signal of the order of 150%, as dis-played in Fig. 12. For the case of a single atom interactingwith itself via a distant mirror, analogous signals have beenexperimentally observed in Refs. �14,15�.

When the light emitted by the two atoms is super-posed coherently on a detector �labeled � in Fig. 2�, the

system is analogous to a double-slit setup �23� with the im-portant difference that the atoms additionally interact by pho-ton exchange via the lens. In this case, the correspondingdetection rate is "� =limt→��cg

���t��2 / t and takes the explicitform

1 2 3 4 5 6

0.01

0.02

0.03

00

t / τ

|c(1)e (t)|

2

ω τ=2nπL

ω τ=(2n+1)πL

ω τ=(2n+1)π/2L

1 2 3 4 5 6

0.01

0.02

00

t / τ

|c(1)e (t)|

2

ω τ=nπL

ω τ=(2n-1/2)πL

ω τ=(2n+1/2)πL

4 8 12 16

0.01

0.02

0.03

|c(1)e (t)|

2

00

t / τ

ω τ=2nπL

ω τ=(2n+1)πL

ω τ=(2n+1)π/2L

4 8 12 16

0.01

0.02

00

t / τ

|c(1)e (t)|

2

ω τ=nπL

ω τ=(2n-1/2)πL

ω τ=(2n+1/2)πL

(b)

(a)

(c)

(d)

FIG. 11. �Color online� Excited state population of atom 1 whenboth atoms are driven by the laser, as evaluated from Eq. �25�, as afunction of time �in units of ��. The figures are evaluated for �=0.4, �=0.05�, �=0, and for ��=20 �upper row� and 1 �lowerrow�. �a� and �c� refer to the case �L= �2n+1��, �b� and �d� to thecase �L= �2n−1 /2��. The value of the phase �L� for each curve isexplicitly given in the plots.

PHOTON-MEDIATED INTERACTION BETWEEN TWO… PHYSICAL REVIEW A 78, 013808 �2008�

013808-9

"� = 8���� − �L�g

2�2

� �2

4 + �2 ��1 − K2�2

��cos��kL − k� · �r1 − r2�/2�

− K cos��kL + k� · �r1 − r2�/2��2. �39�

For K=0, the observed spatial interference is that of adouble-slit setup with two coherently driven sources �23,34�.When K�0, these properties are modified by the multiplescattering. An important special case is when the direction ofemission is k=−kL, corresponding to the coherent back-scattering direction �35�, where one always finds a spatialmaximum of the scattered intensity.

We now consider a setup in which one observes themodes through which the atoms interact �detectors 1 , and2, in Fig. 2�. This corresponds to the measurement arrange-ment in �14�. In this case, the rate at detector 1 , is given by

"1,=limt→��cg�1,��t�+cg

�2,���t��2 / t, superposing the lightemitted by atom 1 directly into the detector with the lightemitted by atom 2 toward atom 1, and then into the detector,whereby k and k� are transformed into each other by thelens. It reads

"1, = 8���� − �L�g

2�2

� �2

4 + �2��1 − K2�2

��cos���L − �L��/2� − K cos���L + �L��/2��2,

�40�

where we used that k� ·r2−k ·r1→�L� via the optical ele-ment. The corresponding rate "2, is found by changing thesign of �L in Eq. �40�. The total rate "

�0�="1,�0� +"2,

�0� wasmeasured in Ref. �14�. in an optical setup, which was char-acterized by small values of �. Taking ��1, the total rate"

�0� reads

"�0� � 1 + cos �L cos �L�

− �A�cos ! + 2 cos �L cos��L� − !�

+ cos�2�L� − !�� + O��2� , �41�

where we omitted global constant factors, and ! and A aredefined in Eq. �37�. We observe that at zero order in � aninterference pattern appears as a function of �L�, i.e., bychanging the optical path between the ions. This is the clas-sical interference of the light elastically scattered from bothatoms into the same detector. The interference has visibility

V1 = �cos �L� , �42�

which is maximum when �L=n�, with n integer, and whichvanishes when �L= �2n+1�� /2. This is a consequence ofsumming the signals from the two detectors, whose indi-vidual interference patterns may be shifted depending on �L.It provides an explanation for the low-contrast interferenceobserved in the experiment of Ref. �14�.

The vanishing contrast when the two signals are perfectlyanticorrelated provides a condition where the higher-ordereffects in �, and thereby the interaction of the atoms, areparticularly evident. Choosing this specific condition, byvarying the optical path length between the ions one ob-serves an interference pattern at twice the frequency of theclassical interference, i.e., oscillating with 2�L�, with a shift! determined by the detuning �, and whose visibility isgiven by

V2 = �A =�

1 + 4�2/�2, �43�

where we used Eq. �37�. The visibility is maximum at atomicresonance. The doubled frequency of the interference �com-pared to the classical one� with the interatomic distanceshows that it is caused by two partial waves originating fromthe same atom, one reaching directly the detector and theother being backscattered once by the other atom. Analo-gously, processes where the same wave is scattered n timesby the atoms give rise to interference terms with frequencyn�L� and at higher order in �.

B. Intensity-intensity correlations

We now study the intensity-intensity correlations in thissetup, assuming that two detectors are placed in the far fieldof the scattered light at positions x1 and x2 �corresponding totwo detectors � in Fig. 2 at angles �1 and �2�. We denoteby G�2��x1 , t ;x2 , t+ t�� the �unnormalized� intensity-intensitycorrelation function for measuring a photon at time t andposition x1, and another at x2 after an interval t�. It reads �34�

G�2��x1,t;x2,t + t�� = �#,$,�,%=1,2

eik��r�−r$�·x1+�r%−r#�·x2�

����+�t��%

+�t + t���$�t + t���#�t�� ,

�44�

where k= �kL� and x=x / �x�. Assuming that the atoms aredriven by the laser and have reached the steady state, Eq.�44� depends solely on the time t� elapsed between the twodetection events. In this limit, we evaluate its explicit form inperturbation theory for the atom-photon interaction, and findthe expression

2 3 4 5

0.6

1

1.4

Γ (κ) / Γ (0)1,µ 1,µ

ω τ/πL

10

FIG. 12. �Color online� Intensity of the light emitted by oneatom into free space, as a function of 2�L� and for �=0.2, when thesystem is driven by a laser at frequency �L=�0 and at �L=0. Theintensity is normalized to the value obtained without coupling,�=0 �blue dashed line�.

RIST et al. PHYSICAL REVIEW A 78, 013808 �2008�

013808-10

G�2���1,�2;t�� =16�4

��2 + 4�2�2

1

�1 − K2�2��1 + ei��1+�L� − K�ei�1 + ei�2����1 + ei��2+�L���k

H2k�t�,�L� + �ei�L + ei�2��k

H2k+1�t�,�L��+ ei��L−�t���1 − K cos �L���ei�1 + ei�2��

k

I2k + �1 + ei��1+�2���k

I2k+1��2, �45�

where we have set

� j = k�r2 − r1� · x j = kd cos � j �46�

and defined G�2���1 ,�2 ; t��=G�2��x1 , t ;x2 , t+ t��. The detailedderivation of Eq. �45� is reported in Appendix C. For �=0,i.e., in the absence of coupling, G�2� exhibits an interferencepattern as a function of the distance �x2−x1� between thedetectors. Such interference emerges from two indistinguish-able paths of two-photon emission. It was first predicted in�36� for the case of two independent quantum sources, andgeneralized in �34� for the light scattered by two trappedatoms illuminated by a laser. In particular, the result of �34�for weak laser intensity is obtained from Eq. �45� by takingthe limit �→0 and setting �=0 �37�.

We now consider this spatial interference pattern at t�=0,for �L=0 and �=0, but keeping ��0. For these parametersit takes the form

G�2���1,�2;0� =64�4

�4

cos2��1 − �2�/21 + �2 + 2� cos �0�

. �47�

In particular, Eq. �47� vanishes for ��1−�2�= �2n+1��,showing strong antibunching at these points. Similarly,

bunching is encountered whenever the condition ��1−�2�=2n� is satisfied. We note, moreover, that since this signaldepends only on the difference �1−�2, there exists a finiteprobability of measuring two photons simultaneously at thepositions of the screen which are the dark fringes of thefirst-order interference pattern, � j = �2n+1��. This behaviorhas been discussed in �38�; it is connected to the fact thatsaturation effects diminish the contrast of the first-order cor-relation function, leading to a nonvanishing probability ofmeasuring a photon at these detector positions. The probabil-ity to measure the first photon in the dark fringe is essentiallyproportional to the occupation of the collective state �e1 ,e2�,and the first detection projects the atoms into the antisym-metric Dicke state ���0��= 1

2��e ,g�− �g ,e��, which is an en-

tangled state of the two distant atoms. The denominator ofEq. �47� shows how the spatial interference pattern is modi-fied due to the atom-atom interaction by multiple photonscattering, and how this modification depends on the phase�L�.

Figures 13 and 14 display the intensity-intensity correla-tion function versus t� and �2, for the situation ���1. Thetwo figures correspond to the bright ��1=2n�� and dark��1= �2n+1��� fringes of the first-order correlation function,and both show the cases �=0 and 0.4, for comparison. One

0

2

3

4

1

0.0007

0.0015

1

0

ϕ / π2

t / τ

2

G (ϕ ,ϕ ;t)21

(2)

0

2

3

4

1

2

3

0.005

0.01

0

1

ϕ / π2

t / τ

G (ϕ ,ϕ ;t)21

(2)

(b)

(a)

FIG. 13. �Color online� G�2���1 ,�2 ; t� as a function of time t �inunits of �� and �2 for �1=2n�. �= �a� 0 �no interaction� and �b�0.4. The other parameters are �=0.05�, �=0, �L=0, ��=20, and�0�= �2n+1��.

01

2

3

4

1

2

0.0002

0.0004

0ϕ / π2

t / τ

G (ϕ ,ϕ ;t)21

(2)

0

1

2

3

4

1

20.0005

0.001

0

ϕ / π2

t / τ

21

(2)G (ϕ ,ϕ ;t)

(b)

(a)

FIG. 14. �Color online� Same as Fig. 13 but with �1= �2n+1��. The curves in �b� at t�� are magnified by a factor of 30.

PHOTON-MEDIATED INTERACTION BETWEEN TWO… PHYSICAL REVIEW A 78, 013808 �2008�

013808-11

clearly observes the effect due to multiply scattered photons,giving rise to abrupt changes in the slope of the correlationfunction at multiples of �. Hence, the interference due tomultiple scattering enhances or suppresses the probability ofmeasuring the second photon at a certain time interval. Re-lated effects have been observed in a single-atom interfer-ence experiment in Ref. �39�. We now analyze this latterproperty, setting �1= �2n+1��, i.e., setting the first detectorat a dark fringe of the first-order correlation function. In Fig.14�a� one sees that for �=0 the second-order correlationfunction is different from zero at t�=0 and for �2�2n�, andit vanishes after a transient time of the order 1 /2�, corre-sponding to the lifetime of the collective state �e ,e� �38�. For�=0.4, Fig. 14�b�, one observes “revivals” when the timebetween the two detections is a multiple of �, and whoseamplitude is strongly damped as a function of time. Inspect-ing Eq. �45� for these specific parameters, we find that atshort times the correlation function behaves as

�G�2���1,�2;t����1=�2n+1��,t���

32�4�1 − cos �2�

�4�1 + �2 + 2� cos �0����k

�− 1�kIk�2, �48�

and it is essentially proportional to the probability of mea-suring the atoms in the state ���t���, obtained by freelyevolving the initial state ���0��= 1

2��e ,g�− �g ,e��, according

to Eq. �15�. For longer times t���, the second-order corre-lation function scales with �2 and takes the form

�G�2���1,�2;t����1=�2n+1��,t���

64�4�1 − cos2 �2�

�4�1 + �2 + 2� cos �0���K�k

Hk��L,t���2, �49�

which is essentially proportional to the stationary excitedstate occupation of the atoms given in Eq. �25�.

While the limit ���1 is characterized by revivals of thecorrelation function versus the time t� between two photondetections, in the limit ���1 one observes a smooth decayof the correlation function with t�, whereby its decay rate ismodified depending on whether the multiply scattered wavesinterfere constructively or destructively at the atom. A com-parison between the two regimes is shown in Fig. 15.

To conclude this section, one of the main features associ-ated with photon-mediated atom-atom interaction is that theintensity-intensity correlation exhibits an enhanced or sup-pressed probability to measure a second photon as a functionof the time after the first detection. This behavior is due tointerference between the various paths of multiple scattering,and can be interpreted as a combined photonic-atomic exci-tation which is stored inside the system. In view of the inter-pretation in �40,41�, one can say that for a transient time thesystem develops and stores entanglement and correlations,determined by the strength of the interaction �, until theatoms finally dissipate the excitation into free space. In thefuture, it would be interesting to consider these dynamics ina quantum jump picture �40�. This could open the possibilityof implementing schemes for entangling atoms in this kindof setup, as proposed in �41�.

VI. CONCLUSIONS

We have studied the photonic properties of two atomswhich are coupled by radiation via an optical element suchas, e.g., a lens or an optical fiber focusing a relevant fraction� of electromagnetic field modes from one atom to the other.Signatures of multiple scattering of photons between the at-oms are observed in the first- and second-order coherence ofthe scattered light. These features show that the presence ofthe second atom substantially modifies the radiative proper-ties of the first one, even when the atoms are separated by adistance d much larger than the light wavelength �.

The efficiency of the interaction, which for two atoms infree space scales with � /d, is determined by � when anoptical system mediates the coupling. The atom-atom dis-tance d plays a new role, separating two regimes where thedelay of the interaction �=d /c is smaller or larger than theatomic decay time 1 /�. In these two regimes ���1 and���1, the coupled two-atom system shows characteristicsof a single- or multimode resonator, respectively, with mir-rors of low reflectivity � and bandwidth �.

In this paper we considered the limit in which the atomsare weakly driven by the laser, and we neglected the effect of

1 2 3

0.0002

0.0006

0.001

9 x 10-7

2 x 10-8

0t / τ

21

(2)G (ϕ ,ϕ ;t)

1 2 3 4 5

0.001

0.0005

0.0002

0.0001

00

t / τ

21

(2)G (ϕ ,ϕ ;t)

(b)

(a)

FIG. 15. �Color online� G�2���1 ,�2 ; t� as a function of time for�1=�2= �2n+1�� and �=0.4 when ��= �a� 20 �note the changeof vertical scale from each maximum to the next one� and �b� 0.4.The red solid �bottom� and green solid �top� curves are evaluated at�0�=2n� and �2n+1��, respectively. The blue dashed line repre-sents the behavior at �=0 and is plotted for reference. The otherparameters are �=0.05�, �=0, and �L=0.

RIST et al. PHYSICAL REVIEW A 78, 013808 �2008�

013808-12

atomic motion. It should be remarked that localization of theatoms within a wavelength of the scattered light is a relevantrequirement for observing the interference effects arisingfrom multiple scattering. In fact, atomic motion gives rise toa dephasing in the signal, which can be interpreted as which-way information imprinted by the photon recoil on the scat-tering atom �35,42,43�. Nevertheless, when the recoil of theatom in each photon scattering event is taken into account,correlations between the atomic motion and the light are es-tablished �44�. In particular, mechanical effects between thedistant atoms arise, which are mediated and retarded by theoptical coupling �16,45�. It is interesting to consider whethersuch effects may lead to novel collective behavior of atomiccenter-of-mass and photonic variables, in analogy to collec-tive dynamics predicted for cold atoms inside resonators�46�.

ACKNOWLEDGMENTS

The authors acknowledge discussions with and helpfulcomments from Endre Kajari, Georgina Olivares-Renteria,and Wolfgang Schleich. This work was supported by theEuropean Commission �EMALI, Grant No. MRTN-CT-2006-035369; Integrated Project SCALA, Contract No.015714� and by the Spanish Ministerio de Educación yCiencia �Consolider-Ingenio QOIT Grant No. CSD2006-00019; QLIQS Grant No. FIS2005-08257; QNLP GrantNo. FIS2007-66944; Ramon-y-Cajal; Acción IntegradaGrant No. HA2005-0001�.

APPENDIX A

We formally integrate Eqs. �9b� and �9c� using the initialcondition bg

����0�=0. Inserting the result into Eq. �9a� yields

be�1��t� = − �

�=,g�

20

t

dt�ei��0−����t−t��be�1��t��

− �

g2

0

t

dt�ei��0−���t−t��eik·�r1−r2�be�2��t�� , �A1�

where the equation for be�2��t� is found by swapping the indi-

ces 1 and 2 in Eq. �A1�. We see that the differential equationfor the probability amplitude be

�1��t� depends linearly on theprobability amplitude be

�2��t�� for the excited state of atom 2.Such dependence is due to the common modes which me-diate the interaction between the two atoms. In absence ofthe second atom, the equation reduces to the well-knownequations describing the radiative decay of a two-level atomin free space �27�.

The first term on the right-hand side �RHS� of Eq. �A1�can be rewritten in a compact way by converting the sumover the modes into an integral. Using the Wigner-Weisskopfapproximation one obtains �47�

be�1��t� = −

�

2be

�1��t� − 0

t

dt�be�2��t���

g2eik·�r1−r2�ei��0−���t−t��,

�A2�

where �=D2�03 / �3��0�c3� is the free-space decay rate �27�.

The second term on the RHS of Eq. �A2� corresponds to the

sum of the modes mapped from one atom into the other bythe optical setup. Let us consider a lens between the atomscollecting a solid angle ��0 of modes with aperture &0. Con-verting the sum over the modes into an integral, the sum overthe modes in Eq. �A2� can be rewritten as

�

g2eik·�r1−r2�ei��0−���t−t��

→1

2�2��3�0�c30

�

d� �3ei��0−���t−t��

���0

d� eik·�r1−r2��D2 −�D · k�2

k2 � . �A3�

Since optics compensate for the phase difference between thevarious modes, we take

eik·�r1−r2� → ei��, �A4�

where � is defined in Eq. �1�, and is the time a photon emit-ted inside the solid angle ��0 needs to cover the distancebetween one atom and the other via the optical setup. UsingEq. �A3� in Eq. �A2�, we can now make the Wigner-Weisskopf approximation and obtain Eqs. �10a� and �10b�.

APPENDIX B

We consider the terms of Eq. �27�. Let us introduce thesimple relation

I = 0

t

dt�ei��L−��t�Hk�t�,��

=�− ��2 �k

k!ei�Lk���t − k��

0

t−k�

dx ei��L−��xxkGk��x� ,

�B1�

with �= �2 +i� and where we used Eq. �20�. Using the defi-

nition of the confluent hypergeometric function �31�,

1F1�a,b,z� = 1 +a

bz +

a�a + 1�b�b + 1�

z2

2!+ ¯ , �B2�

we rewrite Eq. �21� as

Gk�s� = �1 − e−s if k = 0,

− �n=0

�n

n + k

�− s�n

n!if k � 0.� �B3�

or equivalently

Gk�s� =k!

sk − �− 1�ks�k

�sk

e−s

s, �B4�

whereby

�− 1�ks�k

�sk

e−s

s= e−s�

n=0

kk!

�k − n�!s−n. �B5�

We use Eq. �B4� in Eq. �B1�, and obtain

PHOTON-MEDIATED INTERACTION BETWEEN TWO… PHYSICAL REVIEW A 78, 013808 �2008�

013808-13

I =�− ��2 ei�L��k

�k ��t − k���0

t−k�

dx ei��L−��x − Sk� ,

�B6�

with

Sk�t� =�− 1�k

k!

0

t−k�

dx e−i�xxk+1 �k

�xk

e−�x

x. �B7�

As we are interested in evaluating the scattering processesfor long times, t→�, we neglect the term k� in the upperbound of the integral and take the Heaviside function to beone. The first term inside the parentheses on the RHS of Eq.�B6� gives

0

t

dx e−i�x = e−i�t/22���t���� . �B8�

In order to evaluate the term �B7� we use the relations Eq.�B5� and

0

t

dx xje−�x =j!

� j+1 −j ! e−�t

� j+1 �l=0

j��t�l

l!,

in Eq. �B7�, which then reads

Sk�t� = �j=0

k� j

�� + i�� j+1�1 − e−��+i��t�l=0

j�� + i��ltl

l!� .

In particular,

�k=0

�K�2kS2k

K�2k+1S2k+1=

1

2��

k=0

�

K�kSk� �− 1�kK�kSk� , �B9�

where

�k=0

�

��1�kK�kSk 1

�1� K���� + i� � �K��, �B10�

with K�=−��2 ei�L� / ��2 + i��, and where we used the Cauchyproduct for an absolute convergent series, thereby neglectingthe vanishing exponentials as we consider the long-timelimit. In the long-time limit, using that limt→+� �

�t��x�=��x�,we finally arrive at the relation in Eq. �31�.

APPENDIX C

Starting from Eq. �44� we calculate the second-order cor-relation function of two atoms, which are weakly driven bythe laser and both scatter toward the detector. In the refer-ence frame rotating at the laser frequency, assuming that theinitial state is the atomic ground state, we rewrite Eq. �44� as

G�2��x1,t;x2,t + t�� = ���1 + �2ei�2�U�t����1 + �2ei�1�U�t�

��g,g,0��2, �C1�

where the correlation function is evaluated at lowest order inperturbation theory in the atom-photon interactions. The op-erator U�t� is the total evolution operator,

U�t� = exp�− iH�t/�� ,

with H� given in Eq. �5�, which is to be expanded in powerseries of the interactions Vemf and VL. At lowest nonvanish-ing order, Eq. �C1� can be rewritten as

G�2��x1,t;x2,t + t�� = ��A1�t� + A2�t�ei�1��A1�t�� + A2�t��ei�2�

+ B�t��C11�t�� + ei�1C21�t��

+ ei�2�C12�t�� + C22�t��ei�1���2, �C2�

where the coefficients Aj and B are the transition amplitudes

A1�t� = �e,g,0�U�t��g,g,0� = ce�1��t� , �C3�

A2�t� = �g,e,0�U�t��g,g,0� = ce�2��t� , �C4�

B�t� = �e,e,0�U�t��g,g,0� , �C5�

where ce�j��t� is given in Eq. �25�, while the probability am-

plitudes Cji are defined as

C11 = �e,g,0�U�t��e,g,0� = e−i�tbe�1��t� with be

�1��0� = 1,

C12 = �g,e,0�U�t��e,g,0� = e−i�tbe�2��t� with be

�1��0� = 1,

C21 = �e,g,0�U�t��g,e,0� = e−i�tbe�1��t� with be

�2��0� = 1,

C22 = �g,e,0�U�t��g,e,0� = e−i�tbe�2��t� with be

�2��0� = 1,

with be�j��t� given in Eq. �15�. We evaluate the coefficient B�t�

in second-order perturbation theory, hence obtaining

B�t� = − i�0

t

dt�e−�2i�+���t−t���eikL·r2ce�1��t�� + eikL·r1ce

�2��t��� .

�C6�

Inserting the explicit value of the coefficients, Eq. �25�, afterpartial integration B�t� reads

B�t� = −�2

�2 �2eikL·�r1+r2��k

F2k + �e2ikL·r1 + e2ikL·r2��k

F2k+1� ,

�C7�

where

Fk = ��2

− �ei�L�

��k 1

k!��t − k���"„k + 1,��t − k��…

+ �− 1�ke−2��t−k��"„k + 1,− ��t − k��…� ,

with �= �2 + i� and "�k ,�� the generalized Gamma function

�31�. In the long-time limit the second term in the squarebrackets is negligible, and we find

B�t� =−�2

2� �2 + i��2

1

1 − K2 �2eikL·�r1+r2� − K�e2ikL·r1 + e2ikL·r2�� ,

�C8�

while the form Aj�t� valid in the long-time limit is given byEq. �30�. Inserting the explicit values of the coefficients inEq. �C2� we finally obtain Eq. �45�.

RIST et al. PHYSICAL REVIEW A 78, 013808 �2008�

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013808-15