BRAMBILLA, ENRICOGATTI A.BACHE M.LUGIATO, LUIGImostra contributor esterni nascondi contributor esterni 

Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato

INFM, Dipartimento di Scienze CC FF MM, Universita` dell’Insubria, Via Valleggio 11, 22100 Como, Italy 共Received 15 May 2003; published 9 February 2004兲

We study the spatial correlations of quantum fluctuations that can be observed in multimode parametric down-conversion in the regime of high gain. We investigate both a type-I and a type-II phase-matching configuration: in the latter case spatial correlations at the quantum level are shown to exist both in the near-field and in the far-field zones of the down-converted light. In the stationary and plane-wave approxima- tion we treat the problem analytically. A stochastic model is solved numerically to obtain quantitative results beyond this approximation. The finite transverse size and pulse duration of the pump beam and other features of the system, such as spatial walk-off and diffraction are taken into account, and we show that correlations beyond the standard quantum limit exist for values of parameters consistent with realistic experiments.

DOI: 10.1103/PhysRevA.69.023802 PACS number共s兲: 42.50.Dv, 42.65.⫺k

I. INTRODUCTION

The spatial aspects of correlations of quantum optical fluctuations have been the object of several studies in the past 关1–11兴. In general they show up in nonlinear optical processes, typically wave-mixing phenomena which involve a large number of spatial modes of the electromagnetic field.

From the middle of the 1990s there has been a renewal of attention because of new potential applications which exploit the quantum properties of the field for image processing or multichannel operations. Examples are quantum holography 关12兴, the quantum teleportation of optical images 关13兴, and the measurement of small displacements beyond the Ray- leigh limit关14兴. An overview of this relatively new branch of quantum optics, for which the name quantum imaging was coined, can be found in Ref. 关15兴.

The process of frequency down-conversion is particularly suitable for this kind of application because of its large emis- sion bandwidth in the spatial frequency domain. We consider parametric down-conversion共PDC兲 taking place in a crystal with a second-order nonlinearity set in a traveling-wave con- figuration. In this process the photons of a high-intensity pump field are split into pairs of photons of lower energy and momentum through the nonlinear interaction with the me- dium. Since no signal field is injected, down-conversion is initiated only by vacuum fluctuations that equally cover all spatial and temporal frequencies. The fluorescence pattern that arises has therefore the angular spectrum determined by phase-matching conditions, and depends only on the linear dispersion properties of the nonlinear material.

The spatial aspects of quantum correlations in PDC have been initially studied mainly in the low-gain regime. In this case, the mean number of photons per mode is small com- pared to unity and single-photon pairs can be resolved in time by the detectors. Information on spatial quantum corre- lations can therefore be obtained directly from coincidence measurements. A detailed theory has been developed in order to evaluate the two-photon coincidence rate using a pertur- bative method to determine the two-photon entangled state generated in the nonlinear process 关3兴. This theory has been

applied to explain the outcomes of ‘‘two-photon entangled imaging’’ experiments. These exploit the entanglement of the two-photon state in order to retrieve information from an object inserted in the path of one of the photons by detecting the position of its twin关4兴. Both geometrical and coherence aspects of the physics underlying these two-photon imaging experiments have been carefully examined关5–10兴.

In this paper we focus on spatial correlations in the oppo- site regime, where the number of photons per mode in the emitted field can be quite large, and their detection gives rise to continuous photocurrents. Under these conditions the per- turbative approach used in Refs.关3,4兴 fails and an analytical expression of the output state can be formulated only within the stationary and plane-wave pump approximation共PWPA兲.

Recently, in the PWPA framework we used a multimode theory in order to demonstrate that PDC is able to display correlation effects in the spatial domain, both when photon- number关16,17兴 and polarization 关18兴 measurements are con- sidered. In these papers it is shown that the correlations of quantum origin, usually investigated in the photon-counting regime, survives in a regime of high gain where a large num- ber of photons are emitted in each mode. In particular, the theory predicts noise reduction well below the shot-noise level for the difference in the number of photons measured in the far-field from two detection areas R₁ and R₂correspond- ing to couples of phase-conjugate 共signal and idler兲 modes.

In other terms, the photon numbers measured over the two detection areas are identical even at the quantum level. This phenomenon finds its explanation at the microscopic level in the conservation of the photon transverse momentum which is fulfilled in each elementary down-conversion process: for each photon detected in, say, area R₁, the detection of its twin in R₂ is ensured by this law. The same kind of spatial correlations have also been extensively investigated in the continuous wave共cw兲 pumped optical parametric oscillators, where the down-conversion efficiency is enhanced by en- closing the nonlinear crystal inside an optical resonator关11兴.

Here we shall give a description of the photon-number spatial correlations that can be observed in a traveling-wave configuration, both in type-I and in type-II phase-matching conditions. In the second case, we shall focus our attention

on the spatial correlation property of PDC displayed in the near field, a problem that we never analyzed before. The signal and idler beams are found to exhibit quantum corre- lated photon-number fluctuations when measured from de- tection areas that image the same portion of the beam cross section. Twin photons are indeed generated simultaneously and they remain localized in a limited region of space as long as they are observed close to the crystal. This ‘‘position en- tanglement’’ of the generated photon pairs can be seen as the near-field counterpart of the momentum entanglement which can be observed in the far field. However, we shall see that for a realistic crystal length the measurement of near-field correlation is strongly affected by propagation effects, in par- ticular diffraction and spatial walk-off. We shall propose a procedure to overcome at least partially this problem.

It should be noticed that the existence of correlations both in the near and in the far field of PDC is known in the literature in the coincidence counting regime 关10兴. It has been used in two-photon imaging experiments 关4兴 and has been recently emphasized in a more general context关19兴. In this work we focus on aspects that are specific of the high- gain regime, because we study the spatial correlation func- tions of the photon number, determining the condition under which the signal-idler correlation beats the standard quantum limit.

The observation of photon-number correlation phenom- ena in the high-gain regime of PDC is also the aim of an experiment presently performed at the University of Insubria in Como. In this experiment photodetection takes place by means of a high quantum efficiency charged coupled device 共CCD兲 camera, which is able to resolve photon number fluc- tuations that are below the standard quantum limit关20兴. The pump field is a high-power picosecond laser pulse that pro- vides energy for a large number of down-converted photons, in a configuration such that the plane-wave and cw pump approximations are very raw. Here we shall present a realis- tic description of the system, based on a numerical model that includes the finite duration and the transverse size of the pump pulse. Other features of the system that are relevant from an experimental point of view, such as spatial and tem- poral walk-off, different kinds of linear dispersion and phase- matching 共type-I and type-II crystals兲, are included in the model. It is important to investigate how they affect the spa- tial quantum correlation phenomena predicted by the plane- wave pump theory, also in order to identify the best condi- tions under which they can be observed in the experiment. A numerical evaluation of the far-field photon-number correla- tion function is presented in Ref.关21兴 in the case of a type-I crystal at degeneracy. In Ref. 关21兴, which treats down- conversion within a classical framework, the shape of the pump pulse included into the numerical model is taken from experimental data and the obtained signal-idler correlation peak displayed between symmetrical point reproduces well the correlation measured experimentally.

The paper is organized as follows. In Sec. II we briefly introduce the theoretical model used to describe PDC within a classical framework. The quantum description of the sys- tem is illustrated in Sec. III, where a fully analytical treat- ment is developed in the framework of the plane-wave and

cw pump approximation. It is based on a multimode input- output formalism, first introduced in Ref. 关22兴 for a type-I crystal at degeneracy, which is here extended to a type-II phase-matching configuration.

In Sec. IV we give a qualitative description of the phase- matching mechanism that determines both the photon- number distribution and the characteristic bandwidths of the down-converted field, illustrating thereby the differences be- tween type-I and type-II phase matching.

In Sec. V we define the quantities that can be measured experimentally and that put in evidence the quantum nature of the spatial correlations in which we are interested. Their analytical expressions are derived within the PWPA, which will be used to interpret the results of the numerical model.

The last part of the paper共Sec. VI兲 is devoted to present the numerical results obtained for two particular crystals with different phase matching 共type I and type II兲. The amount of correlation that may be achieved is evaluated as a function of different parameters that can be varied experi- mentally, such as the size of the pump beam waist and the size of the detectors.

II. CLASSICAL DESCRIPTION OF THE PROCESS We decompose the electric field in the superposition of three quasimonochromatic wave packets 共denoted with E0, E₁, and E₂) of central frequencies

␻

0,

␻

1, and

␻

2, corre- sponding to the pump, signal, and idler fields, respectively.

These frequencies are taken to satisfy the energy- conservation condition

␻

1⫹

␻

2⫽

␻

0. Assuming the mean di- rection of propagation is the z direction, and denoting with x

ជ

⫽(x,y) the coordinate vector in the transverse plane, we can write

E_j共z,x

ជ

,t兲⬀Aj共z,x

ជ

,t兲e^{ikjz⫺i␻jt}⫹c.c. 共 j⫽0,1,2兲, 共1兲 where k_j⫽nj

␻

j/c is the wave number of wave j at the car- rier frequency along the z axis共for an extraordinary wave the refraction index n_j depends on the propagation direction, a property leading to spatial walk-off兲. To simplify the notation we have ignored the vectorial character of the three fields, their polarization being determined by the kind of phase- matching conditions that are met inside the crystal.

Within the paraxial and slowly varying envelope approxi- mation, the propagation equations for the signal and idler 共S-I兲 field envelopes and the pump field envelope can be written in the form 关23兴

⳵

^Aj

⳵

^{z ⫹k}⬘^j

^⳵ _⳵

^A_t^j⫹i 2k⬙_j

^⳵

2A_j

⳵

^{t2 ⫺}

␳

j

⳵

^Aj

⳵

^{y ⫺} i 2k_jⵜ_⬜²A_j

⫽

␴

^A0A_l*e^⫺i⌬0z 共 j,l⫽1,2; j⫽l兲, 共2a兲

⳵

^A0

⳵

^{z ⫹k0}⬘

^⳵ _⳵

^A_t⁰⫹ i 2k₀⬙

^⳵

2A₀

⳵

^{t2 ⫺}

␳

0

⳵

^A0

⳵

^{y ⫺} i 2k₀ⵜ_⬜²A₀

⫽⫺

␴

^A1A₂e^i⌬0z 共2b兲

The driving terms on the right-hand side describe the wave- mixing process due to the second-order nonlinearity of the medium, the coupling constant

␴

being proportional to the effective second-order susceptibility

␹

e f f

(2) characterizing the down-conversion process. ⌬0⫽k1⫹k2⫺k0 is the collinear phase mismatch of the central frequency components.

Linear propagation is described by the lhs of these equa- tions: the terms proportional to k_j⬘⫽(

⳵

^kj/

⳵␻

⁾␻⫽␻_j and k⬙_j

⫽(

⳵

^2kj/

⳵␻

²⁾␻⫽␻_j lead to temporal walk-off between the different waves and group-velocity dispersion, respectively, while the terms containing the first- and second-order deriva- tives in the transverse coordinates (x,y ) are responsible for spatial walk-off and diffraction, respectively.

␳

jindicates the walk-off angle of wave j, determined by the anisotropy of the crystal 共the walk-off direction is taken along the y axis兲.

Linear losses are neglected, so that the three waves exchange energy but their total energy is conserved.

In Sec. VI A we shall also consider the special case of a type-I phase-matched crystal where the signal and the idler fields have the same polarization and are observed close to the degenerate frequency

␻

1⫽

␻

2⫽

␻

0/2. Under these con- ditions the signal and idler fields are no more distinguishable and the down-converted field must be described by a single slowly-varying envelope A(z,x

ជ

,t) satisfying the following propagation equation:

⳵

^A

⳵

^z⫹k⬘

^⳵ _⳵

^A_t ^⫹₂^ik⬙

^⳵

2A

⳵

^{t2 ⫺}

␳

1

⳵

^A

⳵

^y⫺ i

2kⵜ_⬜²A⫽

␴

^A0A*e^⫺i⌬0z, 共3兲 which is readily obtained from Eqs. 共2a兲 by dropping the S-I indices j ,l, which denote different polarizations and/or carrier frequencies in the nondegenerate case.

In a single-pass configuration with crystal length on the order of a few millimeters, the amplitudes of the down- converted field remain small with respect to the pump am- plitude and the nonlinear driving term in the rhs of Eq. 共2b兲 can be neglected. The pump depletion due to down- conversion and absorption is indeed of small entity, unless extremely high intensity laser sources are used. We shall therefore work within the parametric approximation that treats the pump as a known classical field which propagates linearly inside the crystal, while the down-converted fields are quantized according to rules that are briefly illustrated in the following section.

III. QUANTUM DESCRIPTION IN THE PARAMETRIC APPROXIMATION

We need now to substitute the classical signal and idler fields with operators. Making the formal substitution for the field envelopes A_j(z,x

ជ

,t)→aj(z,x

ជ

,t) ( j⫽1,2), we impose the following commutation rules at equal z关1兴:

关ai共z,x

ជ

,t兲,aj

†共z,x

ជ

⬘^,t⬘兲兴⫽

␦

i j

␦

共x

ជ

_⫺x

ជ

⬘兲

␦

共t⫺t⬘兲,

关ai共z,x

ជ

,t兲,aj共z,x

ជ

⬘^,t⬘兲兴⫽0 共i, j⫽1,2兲, 共4兲

valid within the framework of the paraxial and quasimono- chromatic approximations. With this definition

I_j共z,x

ជ

,t兲⫽aj

†共z,x

ជ

,t兲aj共z,x

ជ

,t兲 共 j⫽1,2兲 共5兲 is the photon flux density operator associated with wave j: its expectation value gives the mean number of photons cross- ing a region of unit area in the transverse plane. In the linear regime the field operators obey the same equations as the corresponding classical quantities. For our purpose, it is use- ful to introduce the Fourier transforms of the field envelopes with respect to time and to the transverse plane coordinates:

a_j共z,q

ជ

,⍀兲⫽冕^2dx

␲ ^ជ

冕 _冑^dt2

␲

^aj共z,x

ជ

,t兲e⫺iqជ•xជ⫹i⍀t 共 j⫽1,2兲.

共6兲 A similar definition holds also for the Fourier component A₀(z,q

ជ

,⍀) of the classical pump field envelope. The propa- gation equations共2a兲 take then the form

⳵

^aj共z,q

ជ

,⍀兲

⳵

^z

⫽i

␦

j共q

ជ

,⍀兲aj共z,q

ជ

,⍀兲⫹

␴

^e⫺i⌬0z冕^dq2

^ជ ␲

^⬘冕 _冑^d⍀2

␲

^⬘

⫻A0共z,q

ជ

_⫺q

ជ

⬘^,⍀⫺⍀⬘兲al

†共z,⫺q

ជ

⬘^,⫺⍀⬘兲

共 j,l⫽1,2; j⫽l兲, 共7兲

where

␦

j共q

ជ

,⍀兲⫽k⬘j⍀⫹1

2k⬙_j⍀²⫹

␳

jq_y⫺ 1 2k_j共qx

2⫹qy

2兲 共 j⫽1,2兲, 共8兲

is the quadratic expansion of k_jz(

␻

j⫹⍀,q

ជ

)⫺kj around q

ជ

⫽0

ជ

,⍀⫽0, and kjz(

␻

j⫹⍀,q

ជ

)⫽冑^kj

2(

␻

j⫹⍀,q

ជ

)⫺q²denotes the z component of the k vector associated with the (q

ជ

,⍀)j

plane-wave mode. In particular the walk-off angle

␳

j can be identified as

⳵

^kj/

⳵

^qycalculated for q

ជ

⫽0,⍀⫽0. A more de- tailed derivation can be found in 关1,24兴.

Equations 共7兲 contain the convolution integral in Fourier space of the S-I field envelope with the pump field envelope.

Within the undepleted pump approximation, the latter can be expressed as

A₀共z,q

ជ

,⍀兲⫽e^{i␦0(qជ,⍀)z}A₀共z⫽0,q

ជ

,⍀兲, 共9a兲

␦

0共q

ជ

,⍀兲⫽k0⬘⍀⫹1

2k₀⬙⍀²⫹

␳

0q_y⫺ 1 2k₀共qx

2⫹qy 2兲,

共 j⫽1,2兲 共9b兲

the z⫽0 plane being taken at the input face of the crystal. In the following we shall assume that the pump pulse has a Gaussian profile both in space and time, of beam waist w₀ and time duration

␶

0 at z⫽0:

A₀共z⫽0,x

ជ

,t兲⫽共2

␲

兲^3/2A_pe^{⫺(x2⫹y2)/w02}e^⫺t2/␶02. 共10兲 In Fourier space we have then the expression

A₀共z⫽0,q

ជ

,⍀兲⫽2冑^{2 Ap}

␦

^q0 2

␦␻

0

e^{⫺(qx2⫹qy2)/␦q02}e^{⫺⍀2/␦␻02}, 共11兲

where

␦

^q0⫽2/w0,

␦␻

0⫽2/

␶

0 共12兲

denote the bandwidths of the pump in the spatial frequency domain and in the temporal frequency domain respectively.

Let us now consider the limit of the PWPA approxima- tion, in which w₀ and

␶

0 tend to infinity and

A₀共z,q

ជ

,⍀兲→共2

␲

兲^3/2A_p

␦

共q

ជ

_兲

␦

共⍀兲. 共13兲 Under this condition, Eqs. 共7兲 couple only pairs of phase- conjugated modes (q

ជ

,⍀)1 and (⫺q

ជ

,⫺⍀)2 and can be solved analytically. The unitary input-output transformations relating the field operators at the output face of the crystal a^out_j (q

ជ

,⍀)⬅aj(z⫽lc,q

ជ

,⍀) to those at the input face aⁱⁿ_j (q

ជ

,⍀)⬅aj(z⫽0,q

ជ

,⍀) take the following form:

a₁^out共q

ជ

,⍀兲⫽U1共q

ជ

,⍀兲a1

in共q

ជ

,⍀兲⫹V1共q

ជ

,⍀兲a2

in†共⫺q

ជ

,⫺⍀兲,

a₂^out共q

ជ

,⍀兲⫽U2共q

ជ

,⍀兲a2

in共q

ជ

,⍀兲⫹V2共q

ជ

,⍀兲a1

in†共⫺q

ជ

,⫺⍀兲, 共14兲 with

U₁共q

ជ

,⍀兲⫽exp

冋

ⁱ

^␦

^1共q

^ជ

^,⍀兲⫺

^␦

^2共⫺q2

^ជ

^{,⫺⍀兲⫺⌬0lc}

册

⫻

冋

^{cosh关⌫共q}

^ជ

^,⍀兲lc兴

⫹i ⌬共q

ជ

,⍀兲

2⌫共q

ជ

,⍀兲sinh共⌫共q

ជ

,⍀兲lc兲

册

^,

V₁共q

ជ

,⍀兲⫽exp

冋

ⁱ

^␦

^1共q

^ជ

^,⍀兲⫺

^␦

^2共⫺q2

^ជ

^{,⫺⍀兲⫺⌬0lc}

册

⫻

␴

p

⌫共q

ជ

,⍀兲sinh共⌫共q

ជ

,⍀兲lc兲,

U₂共q

ជ

,⍀兲⫽exp

冋

ⁱ

^␦

^2共q

^ជ

^,⍀兲⫺

^␦

^1共⫺q2

^ជ

^{,⫺⍀兲⫺⌬0lc}

册

⫻

冋

^{cosh关⌫共⫺q}

^ជ

^{,⫺⍀兲lc兴}

⫹i ⌬共⫺q

ជ

,⫺⍀兲

2⌫共⫺q

ជ

,⫺⍀兲sinh共⌫关⫺q

ជ

,⫺⍀兴lc兲

册

^,

V₂共q

ជ

,⍀兲⫽exp

冋

ⁱ

^␦

^2共q

^ជ

^,⍀兲⫺

^␦

^1共⫺q2

^ជ

^{,⫺⍀兲⫺⌬0lc}

册

⫻

␴

p

⌫共⫺q

ជ

,⫺⍀兲sinh共⌫关⫺q

ជ

,⫺⍀兴lc兲, 共15兲 and

⌫共q

ជ

,⍀兲⫽

冑 ^␴

^p2⫺⌬共q

^ជ

4^,⍀兲2, 共16a兲

⌬共q

ជ

,⍀兲⫽⌬0⫹

␦

1共q

ជ

,⍀兲⫹

␦

2共⫺q

ជ

,⫺⍀兲⬇k1z共q

ជ

,⍀兲

⫹k2z共⫺q

ជ

,⫺⍀兲⫺k0, 共16b兲

␴

p⫽

␴

^Ap. 共16c兲

It is important to note that the gain functions U_jand V_jgiven by Eq. 共15兲 satisfy the following unitarity conditions:

兩Uj共q

ជ

,⍀兲兩²⫺兩Vj共q

ជ

,⍀兲兩²⫽1 共 j⫽1,2兲 共17a兲 U₁共q

ជ

,⍀兲V2共⫺q

ជ

,⫺⍀兲⫽U2共⫺q

ជ

,⫺⍀兲V1共q

ជ

,⍀兲,

共17b兲 which guarantee the conservation of the free-field commuta- tion relations共4兲 after propagation.

IV. MEAN INTENSITY DISTRIBUTION

In the following we shall consider measurements either in the near-field or in the far-field zones of the nonlinear crystal.

In order to simplify the notation we shall omit the explicit dependence of the fields on the z coordinate: when specifi- cation is explicitly needed, the measured quantities will be labeled with

␲

^or

␲

⬘, which will denote the near-field and the far-field detection planes, respectively 关see scheme of Fig. 1共a兲兴. The analytical results given here and in the next sections are all obtained within the PWPA: on the one hand, they generalize those illustrated in Ref.关17兴 for a type-I crys- tal at degeneracy to a type-II phase-matching configuration;

on the other hand, they provide a good starting point to in- terpret the results of the numerical model that includes the pulse shape and the finite cross section of of the pump beam.

A more general input-output formalism that goes beyond the PWPA is developed in Appendix A.

With a stationary and plane-wave pump the near-field in- tensity distribution in the output plane of the crystal clearly does not depend on x

ជ

and t, because of the system invariance

with respect to translation in time and in the transverse plane.

Using input-output relations共14兲 and recalling that the input fields are the vacuum states, we obtain easily

具^Ij共x

ជ

,t兲典␲⫽冕^d2⍀

␲

冕 _共2^dq

_␲ ^ជ

_兲^2兩Vj共q

^ជ

^{,⍀兲兩2 共 j⫽1,2兲.}

共18兲 The function 兩Vj(q

ជ

,⍀)兩² gives the contribution of mode (q

ជ

,⍀)j to the total photon flux of beam j, and is usually referred to as its spectral gain. On the other hand, in the far-field plane

␲

⬘the spatial Fourier modes are resolved spa- tially and the photon distribution reflects the q

ជ

dependence of these spectral functions. From the expression of ⌫(q

ជ

,⍀) given by Eqs.共16兲, we see that down-conversion occurs most efficiently for the modes satisfying the condition ⌬(q

ជ

,⍀)

⬍2

␴

p. Using Eqs.共16b兲 and 共8兲, the phase mismatch accu- mulated during propagation can be written in the form

⌬共q

ជ

,⍀兲lc⫽⌬0l_c⫹sgn关k1⬘⫺k2⬘兴⍀

⍀0⬘^⫹

⍀²

⍀0⬙^2⫺

^␳

^2qy⫺ q_x²⫹qy

2

q₀² , 共19兲 where we assumed that the signal wave is ordinarily polar- ized, so that

␳

1⫽0, and we introduced the parameters

q₀⫽

冑

^¯l^k_c, ⍀0⬘⫽ 1 兩k1⬘⫺k2⬘兩lc

, ⍀0⬙⫽

冑

共k1⬙⫹k² 2⬙兲lc

, 共20兲 where k¯⫽2k1k₂/(k₁⫹k2). These determine the characteris- tic bandwidths of PDC both in the temporal frequency do- main and in spatial frequency domain. In the type-I phase- matching configuration we will consider in Sec. VI A, both the signal and the idler waves are ordinarily polarized and are observed close to degeneracy, i.e., for

␻

1⫽

␻

2⫽

␻

0/2. In this special case the temporal bandwidth is determined by

⍀0⬅⍀0⬙⫽冑^1/兩k1⬙^lc兩, since k1(

␻

⁾⫽k2(

␻

) implies that ⍀0⬘

⫽⬁. Far from frequency degeneracy the emission spectrum has a much narrower bandwidth, on the order of ⍀0⬅⍀0⬘ which is about two to three orders of magnitude smaller than

⍀0⬙ 共for a typical crystal length of few millimeters兲. On the

other hand, in type-II crystals the signal and idler waves are characterized by different polarizations and frequency dis- persion relations, so that ⍀0⬅⍀0⬘ remains finite even for

␻

1⫽

␻

2. The spatial bandwidth q₀ gives the range of trans- verse wave vectors for which the gain spectrum 兩Vj(q

ជ

,⍀)兩² is close to its maximum value, sinh²(

␴

pl_c), when the j th field is observed at a given frequency

␻

j⫹⍀ 关the gray region shown schematically in Figs. 1共b兲 and 1共c兲 for ⍀⫽0]. We remark that 1/

␳

2l_cand q₀ are about the same order of mag- nitude as long as l_cremains in the millimeter range.

For definiteness we assume that the far field is observed in the focal plane of a thin lens of focal length f which performs the Fourier transformation of the field from the output face of the crystal共the so-called f -f system兲. The field operators in the focal plane

␲

⬘ ^{at z}⫽lc⫹2 f 关see Fig. 1共a兲兴, which we denote with b_1,2(x

ជ

,t), are related to those in the output plane of the crystal by the following Fresnel transformation:

b_j共x

ជ

,t兲⫽冕^dx

^ជ

^⬘hj共x

^ជ

^,x

^ជ

^{⬘兲aoutj 共x}

^ជ

^{⬘,t兲, 共21a兲}

h_j共x

ជ

,x

ជ

⬘兲⫽⫺i

␭jfexp^{⫺共2␲i/␭jf兲xជ•xជ⬘} 共 j⫽1,2兲, 共21b兲 where␭j⫽2

␲

^c/

␻

j ( j⫽1,2) are the free-space wavelengths corresponding to the carrier frequencies. Using the input- output relations 共14兲 and unitarity relations 共17兲 we can evaluate the mean intensity distribution of the two fields with the following approximate expression:

具^Ij共x

ជ

,t兲典␲⬘⬇ 1

S_{di f f}^{( j)} 冕^d2⍀

␲

^兩V¯j共x

ជ

,⍀兲兩² 共 j⫽1,2兲, 共22兲 where we introduced the barred gain functions defined in real space

U¯

j共x

ជ

,⍀兲⫽Uj

冉

^␭2

^␲

^jfx

^ជ

^,⍀

冊

^{, V¯j共x}

^ជ

^,⍀兲⫽Vj

冉

^2␭

^␲

^jfx

^ជ

^,⍀

冊

^,

共 j⫽1,2兲, 共23兲

and S_{di f f}^{( j)} ⫽(␭jf )²/S_A ( j⫽1,2), denotes the resolution areas in the far-field plane at the S-I wavelengths, with S_A being the area characterizing the dimension of the system in the FIG. 1. Scheme for the observation of down-conversion in the far-field zone共a兲. The lens 共not shown in the figure兲 is located at z⫽lc

⫹ f . 共b兲 and 共c兲 display the phase-matching curves 共25兲 in the spatial frequency plane for a type-II 共b兲 and a type-I 共c兲 crystal, respectively.

The symmetrical black squares R1 and R2 indicate the locations of the detectors from which maximal signal-idler correlation can be measured.

transverse plane. As is shown in Ref. 关17兴, Eq. 共22兲 can be obtained by assuming that a pupil of area S_AⰇ1/q0

2 is put on the crystal exit face. Assuming that the transverse dimen- sions of the crystal are large compared to the pump waist, S_A can be identified with the effective cross section area of the pump beam. Equation共22兲 represents a good approximation provided the pump beam shape changes negligibly during propagation in the crystal and behaves therefore as a plane wave. This happens when the Rayleigh length characterizing the Gaussian pump beam divergence, z_R⁰⫽

␲

^w0

2/␭0, and its analog characterizing dispersion, z_{dis p}⁰ ⫽

␶

0

2/2k₀⬙^{, are much} longer than the crystal length l_c. The same conditions can also be written in terms of the pump spatial and temporal bandwidths共12兲 as

␦

^q0

q₀ Ⰶ1,

␦␻

0

⍀0 Ⰶ1. 共24兲

At the considered carrier frequencies

␻

1 and

␻

2 the gain functions V_j(q

ជ

,⍀⫽0) are maximal and perfect phase match- ing is achieved when the equations ⌬(⫾q

ជ

,⍀⫽0)⫽0 are satisfied, with the plus sign for field 1, and the minus sign for field 2 关see Eqs. 共15兲 and 共16b兲兴. More explicitly, as can be seen using expression共19兲, they can be written as

q_x²

q₀²⫹

冉

^qqy0⫾12

^␳

^2lcq0

冊

^{2⫽⌬0lc⫹}

冉

¹²

^␳

^2lcq0

冊

^{2. 共25兲}

Provided that ⌬0l_c⬎⫺¹4

␳

2

2l_c²q₀², we have therefore two circles of radius q_R and centered at (q_x⫽0,qy⫽⫾qC), with q_C⫽¹2

␳

2l_cq₀²⫽¹2¯k

␳

2, 共26a兲

q_R⫽q0

冑

^{⌬0lc⫹qqC2}0

2⫽冑^{¯k⌬0⫹14共k¯}

^␳

^{2兲2. 共26b兲}

They are plotted in Figs. 1共b,c兲, respectively, for a type-II and a type-I phase-matching configuration. Modes close to these circles within q₀ 共gray annuli in the figure兲 give a non-negligible contribution to the down-converted field. In the detection plane they give rise to characteristic couples of rings which have been observed in many experiments on parametric down-conversion 共see, e.g., Refs. 关25–27兴兲. It should be stressed that, without any spectral filtering, emis- sion occurs on a very wide range of wavelengths and emis- sion angles, as allowed by the phase-matching conditions 共see, e.g., Refs. 关26,9兴兲. However, from an experimental point of view, a particular couple of rings can always be selected with the use of frequency filters centered at the cho- sen frequencies

␻

1and

␻

2⫽

␻

0⫺

␻

1. Noting that there is the following mapping between the spatial frequency plane and the far field plane:

共qx,q_y兲→␭1f

2

␲

^共qx,qy兲 for field 1, 共27兲

共qx,q_y兲→␭2f

2

␲

^共qx,qy兲 for field 2,

it is easily seen that the ring radii, x_R^(1,2)⫽␭1,2f /2

␲

^qR, and their distance from the z axis, y_C^(1,2)⫽␭1,2f /2

␲

^qC, are gener- ally different except when observation is performed at fre- quency degeneracy 共i.e., for ␭1⫽␭2). In a type-I phase- matching configuration these rings are concentric, since there is no spatial walk-off between the two fields (

␳

2⫽0) and the radial symmetry of the system is preserved.

Figures 2 and 3 illustrate the kind of far-field patterns that can be obtained from a single pump pulse at frequency de- generacy in a type-I and a type-II crystal, respectively. They are obtained by numerical integration of the classical field equations 共2兲, with a white input noise which simulates the vacuum fluctuations that trigger the process, as will be de- scribed in Sec. VI. The pump pulse duration is 1.5 ps and the large waist condition

␦

^q0Ⰶq0 is fulfilled. In the type-II case the width of the rings is determined by the interval of fre- quencies of the numerical grid. In the examples shown in Fig. 2 the grid acts as a 15 nm box-shaped interference filter.

In the type-I phase-matching case, being at frequency degen- FIG. 2. Typical far-field pattern from down-conversion in a type-II crystal, assuming observation is performed at the degenerate frequency共i.e., at ␭1⫽␭2). They are obtained for decreasing values of the collinear phase-mismatch parameter ⌬0, which makes the radius of the rings shrinks to zero. The pump pulse duration is␶0

⫽1.5 ps, the pump beam waist is w0⫽664␮m (␦^q0/q₀⫽0.05), and the parametric gain is␴pl_c⫽4.

FIG. 3. Far-field pattern in a type-I crystal at the degenerate frequency for collinear 共a兲 and noncollinear 共b兲 phase-matching.

␶0⫽1.5 ps,␦^q0/q0⫽0.05 (w0⫽920␮m) and ␴plc⫽4.

eracy, the two rings merge into one that contains both signal and idler modes. Moreover, in this case the width of the rings in the spatial frequency plane is determined by the natural bandwidth q₀, since the temporal frequency interval used in the numerical simulations is small compared to the charac- teristic temporal bandwidth⍀0关as pointed out in the discus- sion following Eqs. 共20兲, in the type-I configuration at de- generacy the temporal bandwidth is much larger than in the type-II configuration兴.

V. NEAR- AND FAR-FIELD CORRELATIONS We now define explicitly the quantities that can be mea- sured in an experiment in order to put in evidence the S-I correlations in the spatial domain we are investigating. We assume that the signal and idler beams are spatially separated in the detection plane and are measured over two detection areas, which we denote with R₁ and R₂.

In the far field, correlations find their origin in the conser- vation of the transverse momentum of the generated photon pairs. Therefore, in order to find maximal correlation, R₁and R₂ must correspond to couples of phase-conjugate modes, such as those indicated with the black squares in Figs. 1共b兲 and 1共c兲. For simplicity, in order to avoid the heavy notations that arise if ␭1⫽␭2, we shall restrict our analysis to the frequency degenerate case, indicating with␭ both ␭1and␭2. Phase-conjugate modes are then mapped by the lens into symmetrical points in plane

␲

⬘ according to relation 共27兲 and R₁ and R₂ must be taken symmetrical.

On the other side, near-field correlations arising from the position entanglement of the twin photons are expected to be observed if R₁ and R₂ occupy the same region of the near- field plane. In practice a type-II phase-matching configura- tion should be considered, so that the use of a polarizing beam splitter and lens systems allows the imaging of the S-I near fields on two physically separated detection planes共see detection scheme illustrated in Fig. 4兲.

If the detectors are the pixels of a CCD camera, as in the

experiment described in Ref. 关26兴, they do not allow any spectral measurement due to the very low resolution power of the device in the time domain. They simply measure the total number of incoming photons down-converted in each single pump shot, and the measurement time T_d can be iden- tified with the pump pulse duration. We introduce therefore the operators corresponding to the number of photons col- lected by the two detectors in the finite-time window 关⫺Td/2,T_d/2兴:

N_j⫽冕R_j

dx

ជ

冕⫺Td/2 T_d/2

dt I_j共x

ជ

,t兲 共 j⫽1,2兲. 共28兲

The measurable quantity capable of displaying the quantum nature of the photon-number statistics in the spatial domain is the variance of the photon-number difference, N_⫺⫽N1

⫺N2, which can be written in the form

具共

␦

^N⫺兲²典⫽具^N⫹典⫹具^:共

␦

^N1兲²:典⫹具^:共

␦

^N2兲²:典⫺2具

␦

^N1

␦

^N2典^. 共29兲

␦

^Nj⫽Nj⫺具^Nj典 ^and

␦

^N_⫺⫽N_⫺⫺具^N⫺典 denote the photon- number fluctuation operators associated with N_jand N_⫺and the colon ‘‘:’’ denotes normal ordering 共n.o.兲 for the expec- tation values. In Eq.共29兲 the shot-noise contribution, i.e., the total number of photons intercepted by the two detector 具^N⫹典⫽具^N1典⫹具^N2典, has been explicitly separated from the term that describes the field correlations. We define

具^:

␦

^Ni

␦

^Nj:典⫽冕R_i

dx

ជ

冕R_j

dx

ជ

⬘冕⫺Td/2 T_d/2

dt冕⫺Td/2 T_d/2

⫻dt⬘^Gi j共x

ជ

,t,x

ជ

⬘^,t⬘兲 共i, j⫽1,2兲, 共30兲 where

G_{i j}共x

ជ

,t,x

ជ

⬘^,t⬘兲⫽具^:Ii共x

ជ

,t兲Ij共x

ជ

⬘^,t⬘兲:典^⫺具^Ii共x

ជ

,t兲典

⫻具^Ij共x

ជ

⬘^,t⬘兲典 共i, j⫽1,2兲 共31兲 are the n.o. self- and cross-photon number correlation func- tions of the S-I beams. Notice that in the nondegenerate case 共type II or type I far from frequency degeneracy兲 we have 具^:I1(x

^ជ

^,t)I2(x

^ជ

⬘^,t⬘^):典^⫽具^I1(x

^ជ

^,t)I2(x

^ជ

⬘^,t⬘⁾典^.

We now focus on the analytical results that can be de- duced from the PWPA. We shall consider explicitly only the case of type-II phase matching. The case of type I, at least in the far field, can be described with a similar treatment and has been already discussed in Ref. 关17兴. Assuming that the detection time T_d is large compared to the coherence time

␶

coh⫽⍀0⫺1, as is usually the case, we have

具^:

␦

^Ni

␦

^Nj:典⬇Td冕^Ri

dx

ជ

冕^Rj

dx

ជ

⬘ ^Gi j共x

ជ

,x

ជ

⬘^,⍀⫽0兲 共i, j⫽1,2兲, 共32兲 where G_{i j}(x

ជ

,x

ជ

⬘^,⍀⫽0) is the Fourier transform of function 共31兲 with respect to t⫺t⬘ 共notice that in a cw regime this FIG. 4. Detection scheme to measure spatial correlations in the

near field. A polarizing beam splitter共PBS兲 separates the S-I beams.

Their near fields, at the plane ␲:z⫽lc⫺⌬z, are imaged by two lenses (L and L⬘) onto the pixel detectors R₁and R₂, which lie in the plane conjugate to plane ␲. ⌬z and ⌬y indicate the spatial shifts applied to the optical devices that are necessary to optimize the measurement.