Degiovanni I. P.Bondani MPuddu EANDREONI, ALESSANDRAParis M. G. A.mostra contributor esterni nascondi contributor esterni 

Intensity correlations, entanglement properties, and ghost imaging in multimode thermal-seeded parametric down-conversion: Theory

Ivo P. Degiovanni,

¹

Maria Bondani,

²

Emiliano Puddu,

^3,4

Alessandra Andreoni,

^3,4

and Matteo G. A. Paris

^5,6

1Istituto Nazionale di Ricerca Metrologica, Torino, Italy

2National Laboratory for Ultrafast and Ultraintense Optical Science—C.N.R.-I.N.F.M., Como, Italy

3Dipartimento di Fisica e Matematica, Università degli Studi dell’Insubria, Como, Italy

4Consiglio Nazionale delle Ricerche, Istituto Nazionale per la Fisica della Materia (C.N.R.-I.N.F.M.), Como, Italy

5Dipartimento di Fisica e Matematica, Università degli Studi dell’Insubria, Como, Italy

6I.S.I. Foundation, I-10133 Torino, Italy sReceived 12 July 2007; published 14 December 2007

d

We address parametric down-conversion seeded by multimode pseudothermal fields. We show that this process may be used to generate multimode pairwise correlated states with entanglement properties that can be tuned by controlling the seed intensities. Parametric down-conversion seeded by multimode pseudothermal fields represents a source of correlated states, which allows one to explore the classical-quantum transition in pairwise correlations and to realize ghost imaging and ghost diffraction in regimes not yet explored by experiments.

DOI:10.1103/PhysRevA.76.062309 PACS numberssd: 03.67.Mn, 42.50.Dv, 42.65.Lm

I. INTRODUCTION

Ghost imaging f

1

g and ghost diffraction f

2

g consist of the retrieval of an object transmittance pattern or its Fourier transform, respectively, by evaluating a fourth-order correla- tion function at the detection planes between the field that never interacted with the object and a correlated one trans- mitted by the object. A general ghost-imaging–ghost- diffraction scheme involves a source of correlated bipartite fields and two propagation arms usually called test sTd and reference sRd. In the T arm, where the object is placed, a bucket sor a pointliked detector measures the total light trans- mitted by it. The R arm contains an optical setup suitable for reconstructing the image of the object or its Fourier trans- form and a position-sensitive detector f

3

g.

The correlations needed for ghost imaging and ghost dif- fraction may be either quantum, as those shown by entangled states produced by spontaneous parametric down-conversion sPDCd f

1

g or classical, as those present in the fields at the output of a beam-splitter fed with a multimode pseudother- mal beam f4–6g. In recent years several authors discussed analogies and differences between the two cases in terms of the achievable visibility and of the optical configurations needed for image reconstruction. A history of this debate from different points of view may be found in Ref. f3g and references therein. Recently, it has been suggested that the entangled nature of the light source f

7–9

g may be necessary to satisfy the “back-propagating” thin-lens equation, which, indeed, is fulfilled by PDC-based ghost-imaging systems.

Among other things, we prove that this claim is incorrect.

In this paper, we discuss the use of a PDC-based light source for ghost imaging and diffraction. In our scheme ssee Fig.

1d, the nonlinear crystal realizing PDC is seeded by two

multimode thermal sMMTd beams. We show that the en- tanglement properties and the amount of correlation at the output may be tuned by changing the intensities of the seeds, thus leading to a source that can be used to investigate the transition from the classical to the quantum regime. Besides,

our source allows ghost-image reconstruction with the same optical scheme used for ghost imaging based on spontaneous PDC, with the “back-propagating” thin-lens equation that is satisfied irrespective of the entanglement of the state. We notice that the effectiveness of the setup discussed here has already been demonstrated in the case of a crystal seeded with a single MMT beam f

10

g.

The paper is structured as follows. In Sec. II we calculate the state obtained from our PDC source with the injection of MMT seeds on both the T and R arms, thus revealing that the output field on each arm maintains the statistics of the seed.

In Sec. III we analyze both the intensity correlations between the output beams and the entanglement properties of the overall state. We explicitly evaluate separability thresholds in terms of the seed intensities, and show that the condition for the existence of nonclassical correlations in intensity mea- surements subsumes the condition for inseparability, i.e., sub-shot-noise correlations are a sufficient condition for en- tanglement in our system. We also show that entanglement properties of the output field are not affected by losses taking place after the PDC interaction. In Sec. IV we show that the state generated in our scheme satisfies the “back- propagating” thin-lens equation independently on the seed intensities, i.e., independently of being entangled or not, and it is suitable for realizing ghost-imaging and ghost- diffraction experiments. Finally, Sec. V closes the paper with some concluding remarks.

II. PARAMETRIC DOWN-CONVERSION WITH THERMAL SEEDS

The interaction scheme we are going to consider is

sketched in Fig.

1. It consists of a nonlinear x^s2d

crystal

pumped by a monochromatic nondepleted plane wave propa-

gating along the z axis. The Hamiltonian describing the re-

sulting parametric process is given by

H_I

= E

^d2r

E

₀^Ldz

^x

^s2dEp

^sx,tda

^T

^sx,tda

^R

sx,td + H.c., s1d

L

being the crystal length and x

^s2d

the nonlinear susceptibil- ity. The pump field may be written as

E_p

sx,td

= E

_p

exp fisV

pt

− K

_pz

dg f

11

g.

We can write the interacting quantum fields as

a_j

sx,td ~ o

q_j,n_j

a_j,q

j,n_je^{ifKj,zz+qj·r−sVj+njdtg}

sj = R,Td, s2d where q

_j

is the transverse momentum and K

_j,z

is the magni- tude of the longitudinal momentum, K

_j,z

= Î

K²_j

− q

²_j

, being

K_j

= n

_j

sV

j

+ n

_j

d/c, with n

j

the index of refraction, V

_j

the se- lected central frequency in channel j, n

_j

the frequency dis- placement with respect to V

_j

, and c the speed of light in the vacuum. The commutation relation of the quantum fields are

fa

j,q,n

,a

_j 8^,q8^,n8

†

g = d

_j,j8

d

_q,q8

d

_n,n8

_{sj, j} 8 ^{= R,T} d,

fa

j,q,n

,a

_j8,q8,n8

g = 0. s3d The evolution of a quantum system induced by the interac- tion Hamiltonian in Eq. s

1

d is described by the unitary op- erator U = exp s−i"

⁻¹

eH

Idt

d, where

−

i

" E

^HIdt

^{= i o}

_q,n

^k

^{q,naT,q,naR,−q,−n}

^{+ H.c., s4d}

where k

_q,n

~ sincfsK

p

− K

_T,z

− K

_R,z

dL/2g. To obtain Eq. s4d we have exploited the conservation of energy at the central wavelength V

_p

= V

_T

+ V

_R

obtaining n

_T

= −n

_R

= n, and the con- servation of transverse momentum q

_T

= −q

_R

= q. As, accord- ing to Eq. s

4

d, the extension to the nonmonochromatic case is, in most cases, straightforward, in the following analysis we will focus on the monochromatic emission at the frequen- cies V

_R

and V

_T

and hence we will drop the subscript n from the variables.

The operator U can be rewritten in terms of the operators

S_q

= s k

_qa_T,qa_R,−q

+ H.c.d as U=expsio

qS_q

d. According to the commutation relations in Eq. s

3

d, we have fS

q

, S

_q8

g=0, and therefore U =

^{^}_qe^iSq

, i.e., the interaction establishes pairwise correlations among the modes.

In our analysis we focus on the case in which both the T and R arms are seeded with uncorrelated MMT beams

r

_in

=

^{^}

q

r

_T,q^{^}

r

_R,−q

,

r

_j,q

= o

n=0

`

P_j,q

sndunl

j,qj,q

knu, s5d

where j = R , T and unl

j,q

denotes the Fock number basis for the mode q of the j arm. The thermal probability distribution of the input is given by

P_j,q

snd = m

ⁿ_j,q

_{s1 +} m

_j,q

_d

⁻ⁿ⁻¹

,

m

_j,q

being the average photon number per mode. The density matrix at the output is given by

r

out

= Ur

_inU^†

=

^{^}

q

e^iSq

r

T,q^

r

R,−qe^−iSq

. s6d According to f

12

g, it is possible to “disentangle” e

^iSq

by us- ing the two-boson representation of the SU s1,1d algebra as

e^iSq

= exphz

qa_T,q^† a_R,−q^†

jexph− h

_q

_sa

_T,q^† a_T,q

+ a

_R,−q^† a_R,−q

+ 1dj 3exph− z

q

pa_T,qa_R,−q

j, s7d

where z

_q

= −ie

^−iwq

tanh su k

_q

_ud, h

_q

= ln fcoshu k

_q

_{ug, and e}

^iwq

= k

_q

^/ u k

_q

u.

Equation s

7

d implies that

e^iSq

unl

T,q^

uml

R,−q

= o

k=0 minhm,nj

o

l=0

`

C_q

sm,n,k,ldun − k + ll

T,q

^

um − k + ll

R,−q

, s8d with

C_q

sm,n,k,ld = e

^{−hqsn+m−2k+1d}

3 Î

n

! m ! sn − k + ld ! sm − k + ld!

k

! l ! sn − kd ! sm − kd! z

_q^l

s− z

q p

d

^k

.

s9d By substituting Eq. s

8

d in Eq. s

6

d we obtain the output state in the Schrödinger picture as

r

out

=

^{^}

q

o

nm

P_T,q

sndP

R,−q

smd o

k₁,k₂=0 minhm,nj

l₁

o

,l₂=0

`

C_q

sm,n,k

1

,l

₁

d 3C

_q

sm,n,k

2

,l

₂

d

^p

un − k

1

+ l

₁

l

T,qT,q

kn − k

2

+ l

₂

u

^

um − k

1

+ l

₁

l

R,−qR,−q

km − k

2

+ l

₂

u. s10d Meanwhile, in the Heisenberg description, the modes after the interaction with the crystal are given by b

_j,q

= U

^†a_j,qU,

i.e.,

b_j,q

= U

_qa_j,q

+ e

^iwq

V

_qa_j 8^,−q

†

sj, j 8 = R,T, j Þ j 8 d, s11d where U

_q

= coshu k

_q

u and V

q

= sinhu k

_q

u sand obviously U

q

= U

_−q

, V

_q

= V

_−q

, and w

_q

= w

_−q

d. Equation s11d represents the quantum dynamical evolution of the system i.e., the input- output relations of the parametric process. Of course, they are independent on the initial states and are valid in any working regime, i.e., for any input state and for any value of the pump intensity within the parametric approximation. This

NLC

Pump

FIG. 1. sColor onlined Schematic diagram of the nonlinear interaction. T and R are the test and reference arms of the setup.

includes the two-photon regime as well as the continuous variable regime. On the other hand, the entanglement prop- erties of the output state strongly depend on the initial state.

In the following section we discuss in detail the entangle- ment properties for the case of thermal light at the input.

As expected, the first moments of the photon distribution for each mode are those of a thermal statistics

kn

T,q

l = m

_T,q

+ n

_PDC,q

s1 + m

_T,q

+ m

_R,−q

d, kn

R,−q

l = m

R,−q

+ n

_PDC,q

s1 + m

T,q

+ m

R,−q

d,

ksDn

T,q

d

²

l = kn

T,q

lskn

T,q

l + 1d,

ksDn

R,−q

d

²

l = kn

R,−q

lskn

R,−q

l + 1d, s12d where kOl=TrfOr

out

g=TrfU

^†OUr_in

g sSchrödinger and Heisenberg picture, respectively d, DO=O−kOl, and n

PDC,q

= sinh

²

u k

_q

u is the average number of photons due to sponta- neous PDC.

Notice that the case of vacuum inputs, r

_in

= u0lk0u

T

^

u0lk0u

R

, corresponds to spontaneous down-conversion, i.e., to the generation of twin-beam, whereas the case of a single MMT on one arm and the vacuum on the other, r

_in

=

^{^}_q

su0lk0u

T,q^

r

R,−q

d, corresponds to the state considered in Ref. f10g.

III. ENTANGLEMENT AND INTENSITY CORRELATIONS

In this section we address intensity correlations and en- tanglement properties of the beams generated in our scheme.

As we will see, the amount of nonclassical correlations and entanglement can be tuned upon changing the intensity of the thermal seeds and there exist thresholds for the appearance of those nonclassical features. On the other hand, the index of total correlations seither classical or quantumd is a mono- tonically increasing function of both the seed and the PDC energy.

A. Entanglement and separability

The down-conversion process is known to provide pair- wise entanglement between signal and idler beams. In our notations the spossiblyd entangled modes are a

T,q

and a

_R,−q

. In the spontaneous process the output state is entangled for any value of the parametric gain si.e., for any value of the crystal susceptibility, length, etc.d, whereas in the case of a thermally seeded crystal the degree of entanglement crucially depends on the intensity of the seeds.

Since thermal states are Gaussian and the PDC Hamil- tonian is bilinear in the field modes, the overall output state is also Gaussian. Therefore, the entanglement properties may be evaluated by checking the positivity of the partial trans- pose sPPT conditiond, which represents a sufficient and nec- essary condition for separability for Gaussian pairwise mode entanglement f

13

g. Gaussian states are completely character- ized by their covariance matrix. In this context let us intro- duce the “position”s-liked operators X and “momentum”

s-liked operators Y

X_j,q

=

aj,q

+ a

_j,q^†

Î 2 ,

Y_j,q

=

a_j,q

− a

^†_j,q

i

Î 2 sj = R,Td. s13d

Introducing the vector operator j = sX

T,q₁

,Y

_T,q

1

,X

_R,−q

1

,Y

_R,−q

1

, . . .d

^T

, s14d with m = 1 , 2 , . . . , `, from the commutation relations in Eq.

s3d gives

fj

a

,j

_b

g = iV

a,b

, s15d where V =

^%m

v

^%

v and v is the symplectic matrix

v = S − 1 0 ^{0 1} D ^{. s16d}

The covariance matrix

V

is calculated as

V_a,b

= 2

⁻¹

khDj

a

, Dj

_b

jl, where hO

1

, O

₂

j denotes the anticommuta- tor. Uncertainty relation among the position and momentum operators impose a constraint on the covariance matrix, V +

₂ⁱ

V $ 0, corresponding to the positivity of the state. The input-output relations for position and momentum operators are calculated according to Eq. s

11

d, obtaining

U^†X_j,qU

= U

_qX_j,q

+ V

_qX_j8,−q

,

U^†Y_j,qU

= U

_qY_j,q

− V

_qY_j8,−q

sj, j 8 = R,T, j Þ j 8 ^d.

s17d Without any loss of generality, in the derivation of Eqs. s

17

d we set w

_q

= 0, which, in turn, corresponds to a proper choice of the phase or, equivalently, to a proper redefinition of the operators a

_j,q

that amounts to a rotation of the phase space.

From Eqs. s

17

d we calculate the covariance matrix

V = ^%

m=1

`

V_q

m

= 1

^V

^{0 0 ]}

^{q1 V}

^{] 0 0}

^{q2 V}

^{] 0 0}

^q3

^{¯ ¯ ¯} 2 ^s18d

with

V_q

= 1 ^{A C 0 0}

^qq

^{− C A 0 0}

^qq

^{B C 0 0}

^qq

^{− C B 0 0}

^qq

2 ^{, s19d}

where

A

_q

= fU

q

2

s2 m

_T,q

+ 1 d + V

q

2

s2 m

_R,−q

+ 1 dg/2, B

_q

= fU

q

2

s2 m

_R,−q

+ 1d + V

q

2

s2 m

_T,q

+ 1dg/2,

C

_q

= U

_q

V

_q

s m

_T,q

+ m

_R,−q

+ 1 d. s20d

V satisfies the uncertainty relations ensuring the positivity of

r

_out

.

In order to check whether and when the state r

_out

is en- tangled we apply the PPT criteria for Gaussian entanglement f

13

g. For instance, we apply the positive map L

R,−q8

to the state r

_out

. L

_R,−q8

sr

out

d is the transposition scomplex conjuga- tion d only of the subspace H

R,−q8

corresponding to the mode

R

, −q 8 . Simon showed that this corresponds to calculation of the covariance matrix V

˜ , where all the matrix blocks V_q

remain the same except for the matrix V

_q8

→ V

˜

q8

. V

˜

q8

is calculated with a sign change in the R , −q 8 momentum vari- able sY

R,−q8

→ −Y

_R,−q8

d, while the other momentum and po- sition variables remain unchanged sX

T,q8

→ X

_T,q8

, Y

_T,q8

→ Y

_T,q8

, and X

_R,−q8

→ X

_R,−q8

d. Thus we obtain

V˜

q8

= 1 ^{A C 0 0}

^qq88

^{A C 0 0}

^qq88

^{B C 0 0}

^qq88

^{B C 0 0}

^qq88

2 ^{, s21d}

where A

_q8

, B

_q8

, and C

_q8

are defined in Eqs. s20d. According to PPT criteria, the separability of r

_out

is guaranteed by the positivity of L

_R,−q8

sr

out

d, i.e.,

V˜ + i

2 V $ 0. s22d

Inequality in Eq. s

22

d corresponds to

m

_T,q8

m

_R,−q8

− n

_PDC,q8

s1 + m

_T,q8

+ m

_R,−q8

_{d $ 0. s23d} We observe that the spontaneous PDC corresponds to the situation with m

_T,q8

= m

_R,−q8

= 0; thus r

_out

is entangled. Also the case considered in Ref. f

10

g, a MMT seeded PDC only on one arm si.e., m

_R,−q8

= 0 d, is always entangled. On the contrary, in the case of MMT seeded PDC on both arms, inequality in Eq. s

23

d introduces a threshold. For instance, if we consider a MMT seed with the same mean number of photon per mode, m , only when the inequality m

²

$ n

PDC,q

s1+2 m d is satisfied, r

out

is separable. It is notewor- thy to observe that if the PPT is applied to any other sub- spaces the inequalities obtained are analogous to Eq. s

23

d, and thus the result is the same.

B. Separability and losses

Here we address the problem of the effect of the losses on the separability of the state in Eq. s

10

d. In fact the presence of losses, e.g., internal reflection or absorption in the nonlin- ear crystal, may modify the quantum properties of the state, in particular the transition from entanglement to separability, which, in the absence of losses, is marked by the condition in Eq. s

23

d.

Losses in a quantum channel can be modeled by a beam splitter, in one port of which the quantum channel is injected while the vacuum enters the other port. The model implies that Gaussian states after interaction are still Gaussian states due to the bilinearity of the beam-splitter Hamiltonian. Thus, also in the presence of losses, the covariance matrix com- pletely describes the quantum state. If we consider an overall transmission factor t on both channels we obtain the covari-

ance matrix V

_t

= t

V +

s1− t d1/2. The form of the covariance matrix V

_t

is completely analogous to Eq. s

18

d, where the block matrices V

_q

are substituted with the block matrices

V_t,q

. V

_t,q

has the same structure of V

_q

in Eq. s

19

d, where A

_q

, B

_q

, and C

_q

are substituted by

A

_t,q

= h1 + 2 t fU

q

2

m

_T,q

+ V

_q²

s m

_R,−q

+ 1dgj/2,

B

_t,q

= h1 + 2 t _fU

_q²

m

_R,−q

+ V

_q²

s m

_T,q

+ 1 dgj/2, and

C

_t,q

= t C

_q

,

respectively. Thus, following the same line of thought of Sec.

III A we obtain the covariance matrix V

˜

t

, corresponding to the partial transposition of the state. According to PPT sepa- rability criteria, the state is separable if and only if the inequality V

˜

t

+

₂ⁱ

V $ 0 is fulfilled. This condition can be rewritten as

t

²

f m

T,q8

m

R,−q8

− n

_PDC,q8

s1 + m

T,q8

+ m

R,−q8

dg $ 0. s24d Since Eq. s

24

d is fully equivalent to Eq. s

23

d, we conclude that losses do not affect the entanglement properties of the state in Eq. s

10

d.

C. Intensity correlations

We now evaluate the pairwise intensity correlations pos- sessed by the generated beams. In addition, we analyze the connections between threshold for separability and the threshold required to have nonclassical correlations. As we will see a state obtained by thermally seeded PDC that ex- hibits sub-shot-noise correlations is entangled, whereas the converse is not necessarily true. In other words, the existence of nonclassical intensity correlations is a sufficient condition for entanglement.

The normalized index of intensity correlation between a pair of modes a

_j,q

and a

_j8,q8

is defined as

g

_j,j8

_sq,q 8 d = G

j,j8

sq,q 8 d

Î _ksDn

_T,q

_d

²

_lksDn

_R,−q

_d

²

_l ^{, s25d}

where the correlation term is given by

G

_j,j8

sq,q 8 ^{d = kDn}

j,q

Dn

_j8,q8

l. s26d Upon evaluating the first moments as we did in Eq. s12d we have, for the pair of modes a

_T,q

and a

_R,−q

,

G

_T,R

sq,− qd = n

PDC,q

s1 + n

PDC,q

ds1 + m

_T,q

+ m

_R,−q

d

²

= C

_q²

. s27d A nonzero value of G

_T,R

, and hence of g

T,R

, indicates the presence of correlations between the considered modes. Per- fect correlations correspond to g

_T,R

= 1. Note that g

_T,R

is an increasing function of n

_PDC

and does not undergo any thresh- old. In Fig.

2

we plot g

_T,R

ssolid linesd as a function of n

PDC

in two different conditions, namely, m

_T

= 0 and m

_R

Þ 0 fpanel

sadg and m

_T

= m

_R

Þ 0 fpanel sbdg. As expected, g

_T,R

ap-

proaches unity irrespective of the mean values of the seeding

thermal fields as soon as n

_PDC

becomes relevant.

For large n

_PDC,q

the index of correlation approaches unity as follows:

g

_T,R

sq,− qd . 1 − 1 2

m

_T,q

+ m

_R,−q

+ 2 m

_T,q

m

_R,−q

s1 + m

_T,q

+ m

_R,−q

d

²

1

n_PDC,q²

.

s28d

In the two cases m

_T,q

= m ≫ 1 and m

_R,−q

= 0 sor vice versad and m

_T,q

= m

_R,−q

= m ≫ 1 we have, respectively,

g

T,R

sq,− qd . 1 − 1 s1 + n

PDC,q

dn

PDC,q

1 2 m ^,

g

_T,R

sq,− qd . 1 − 1

s1 + 2n

PDC,q

d

²

+ O S m ¹

²

D ^{. s29d}

The nonclassical nature of this pairwise correlation may be assessed by the quantity f

14

g

NRF

_T,R

sqd = ksDn

T,q

d

²

l + ksDn

R,−q

d

²

l − 2G

T,R

sq,− qd kn

T,q

l + kn

R,−q

l ,

s30d which is usually referred to as “the noise reduction factor.” A noise reduction, NRF

_T,R

sqd,1, indicates the presence of nonclassical correlations. The value NRF

_T,R

sqd=1 is usually called “shot-noise limit” and corresponds to the case of a pair of uncorrelated coherent signals. By substituting the result for our system, we get

NRF

_T,R

sqd = m

_T,q

s1 + m

_T,q

d + m

_R,−q

s1 + m

_R,−q

d m

_T,q

+ m

_R,−q

+ 2n

_PDC,q

s1 + m

_T,q

+ m

_R,−q

_d ^.

s31d We have NRF

_T,R

sqd,1 if

nPDC,q

. 1 2

m

_T,q²

+ m

_R,−q²

1 + m

_T,q

+ m

_R,−q

^{, s32d} which subsumes the separability threshold of Eq. s

23

d and individuates the same region for m

_T,q

= m

_R,−q

. Therefore, for thermally seeded PDC, sub-shot-noise correlations imply en- tanglement f

15

g. In Fig.

2

we also plot NRF

_T,R

as a function of n

_PDC

for the same parameters used for g

_T,R

. As expected, the figure shows that NRF

_T,R

crosses the shot-noise level at different values of n

_PDC

that depend on the mean values of the thermal seeds, thus confirming the intuition that, in order to achieve sub-shot-noise correlations in the presence of two thermal seeds, we need to have a PDC process strong enough.

IV. MMT-PDC BASED GHOST IMAGING AND GHOST DIFFRACTION

The bipartite state obtained by the nonlinear process de- scribed above is suitable for applications to ghost-imaging–

ghost-diffraction protocols. Ghost-imaging and ghost- diffraction protocols rely on the capability of retrieving an object transmittance pattern and its Fourier transform, re- spectively, by the evaluation of a fourth-order correlation function at the detection planes of a light field that has never interacted with the object and a correlated one transmitted by the object. We consider the schemes depicted in Fig.

3. An

object, described by the transmission function t sx

_T

9 d, is in- serted in the T arm on the plane x

_T

9 and a bucket detector measures the total light, transmitted by the transparency. The

R

arm contains an optical setup suitable for reconstructing either the image of the object or its Fourier transform and a position-sensitive detector that measures the local intensity map. The procedure for calculating the correlation function between the light detected in the two arms of the setup is equivalent to evaluating first the correlation function be- tween the intensity operators, which in the Heisenberg pic- ture corresponds to

G^s2d

sx

R

,x

_T

d = TrfDI

R

sx

R

dDI

T

sx

T

dr

in

g

= Tr fI

R

sx

R

dI

T

sx

T

dr

in

g

− TrfI

R

sx

R

dr

in

gTrfI

T

sx

T

dr

in

g, s33d

(b) (a)

FIG. 2. sColor onlined Index of total correlations gT,R ssolid lines; legend correlates to lines from top to bottom in figured and noise reduction factor NRF_T,R sline plus symbold as a function of n_PDC in the cases sad mT= 0 and mRÞ 0 andsbd mT= mRÞ 0. The values chosen for the parameters are indicated in the figures.

and then integrating over all the values of x

_T

G

^s2d

sx

R

d = E

^dxTGs2d

^sx

^R

^,x

^T

^{d. s34d}

In Eq. s

33

d we have defined the intensity operators as I

j

sx

j

d

= c

^†_j

sx

j

dc

j

sx

j

d sj=R,Td, with c

j

sx

j

d the field operators at the detection planes. Thus the average operation in Eq. s33d is taken on the initial state r

_in

fsee Eq. s5dg. Note that Tr fI

j

sx

j

dr

in

g are proportional to the detected intensities and that Tr fI

R

sx

R

dI

T

sx

T

dr

in

g is the second-order correlation func- tion f

11

g.

In order to calculate the mean values in Eq. s

33

d, we exploit the connection between the field operators at the de- tection planes and those at the output of the crystal

c_j

sx

i

d = E

^dx

⁸

^jhj

^sx

^j

^{,x 8}

^j

^db

^j

^{sx 8}

^j

^{d, s35d}

where b

_j

sx

_j

8 d are the reference and test field operators at the output face of the crystal fsee Eq. s

11

dg and h

R

sx

R

, x

_R

8 d and

h_T

sx

T

, x

_T

8 d are the two response functions describing the propagation of the field in the two arms of the setup f16g.

By using Eqs. s

35

d and s

33

d we can rewrite G

^s2d

sx

R

, x

_T

d as

G^s2d

sx

R

,x

_T

d = E

^dxR

⁸

^dxR

⁹

^dxT

⁸

^dxT

⁹

^hR

^sx

^R

^,x

^R

^{8 dh}

^Rp

^sx

^R

^,x

^R

^{9 d}

3h

_T

sx

T

,x

_T

8 dh

_T^p

sx

T

,x

_T

9 d

3hTrfb

_R^†

sx

_R

9 db

R

sx

_R

8 db

_T^†

sx

_T

9 db

T

sx

_T

8 dr

in

g

− Tr fb

R

†

sx

R

9 db

R

sx

R

8 dr

in

gTrfb

T

†

sx

T

9 db

T

sx

T

8 dr

in

gj.

s36d Equation s36d can be simplified by calculating the factoriza- tion rule for Trfb

_R^†

sx

_R

9 db

R

sx

_R

8 db

_T^†

sx

_T

9 db

T

sx

_T

8 dr

in

g, that is by re- writing the four-points correlation function in terms of the two-points correlation functions f

17

g. To do this we rewrite

b_j

sxd in terms of plane waves as b

j

sxd=No

qe^iq·xb_j,q

sN is the normalization coefficient d and then substitute the input- output relations of Eq. s

11

d. After some algebra we obtain

Tr fb

R

†

sx

R

9 db

R

sx

R

8 db

T

†

sx

T

9 db

T

sx

T

8 dr

in

g

= N

⁴

o

q,q8^,q9^,q-

e^{−isq·xR9−q8·xR8+q9·xT9−q-·xT8d}

3Trfb

R,q

† b_R,q8b_T,q 9

† b_T,q-

r

in

g

= N

⁴

o

q,q8

e^{−ifq·sxR9−xR8d+q8·sxT9−xT8dg}

fU

q 2

m

_T,q

+ V

_q²

s1 + m

_R,−q

dgfU

_q²₈

m

_R,q8

+ V

_q 8

2

s1 + m

_T,−q8

dg + N

⁴

o

q,q8

e^{−ifq·sxR9−xT9d−q8·sxR8−xT8dg}

fU

q

V

_−q

m

_T,q

+ V

_q

U

_−q

s1 + m

R,−q

dgfU

q8

V

_−q

8

s1 + m

T,q8

d + V

_q8

U

_−q

8

m

_R,−q8

g. s37d

Performing the same calculation for Tr fb

R

†

sx

R

9 db

R

sx

R

8 dr

in

gTrfb

T

†

sx

T

9 db

T

sx

T

8 dr

in

g + Tr fb

R

†

sx

R

9 db

T

†

sx

T

9 dr

in

gTrfb

R

sx

R

8 db

T

sx

T

8 dr

in

g, s38d we obtain the same result obtained in Eq. s

37

d. Thus we obtained the noteworthy factorization rule

Tr fb

R

†

sx

R

9 db

R

sx

R

8 db

T

†

sx

T

9 db

T

sx

T

8 dr

in

g

− Tr fb

R

†

sx

R

9 db

R

sx

R

8 dr

in

gTrfb

T

†

sx

T

9 db

T

sx

T

8 dr

in

g

= Tr fb

R

†

sx

R

9 db

T

†

sx

T

9 dr

in

gTrfb

R

sx

R

8 db

T

sx

T

8 dr

in

g. s39d Note that the factorization rule in Eq. s39d obtained for MMT-PDC state is exactly the same obtained for spontane- ous PDC f

17

g and multithermal one-arm-seeded PDC f

10

g.

According to Eq. s

39

d, and in complete analogy with the case of spontaneous PDC f17g, also in the case of the MMT- seeded PDC we obtain

G^s2d

sx

R

,x

_T

d = U E

^dxR

⁸ E

^dxT

⁸

^hR

^sx

^R

^,x

^R

^{8 dh}

^T

^sx

^T

^,x

^T

^{8 d}

3Tr fb

R

sx

R

8 db

T

sx

T

8 dr

in

g U

²

^{, s40d}

where

Tr fb

R

sx

R

8 db

T

sx

T

8 dr

in

g

= N

²

o

q,q8

e^{isq·xR8+q8·xT8d}

Tr fb

R,qb_T,q8

r

in

g

~ o

q,q8

e^{ifq·sxT8−xR8d+wqg}

sU

q

V

_q

8

Tr fa

R,qa_R,−q 8

†

r

in

g

+ V

_q

U

_q

8

Tr fa

T,−q

† a_T,q8

r

_in

gd

= o

q

e^{ifq·sxT8−xR8d+wqg}

C

_q

. s41d By using Eq. s41d, Eq. s40d can be rewritten as

G^s2d

sx

R

,x

_T

d ~ U ^o

q

˜h

R

sx

R

,− q dh˜

T

sx

T

,q dC

q

U

²

^{, s42d}

where h

˜

j

sx

j

, q d=edx

j

8

^eiq·xj8hj

sx

j

, x 8

_j

d.

NLC ^t(xT)

CCDsensor

x_T

x_R

NLC

CCDsensor

x_T xR

(a)

(b)

FIG. 3. sColor onlined Setup for ghost-imaging and ghost- diffraction experiments: t, object transmission function; f_T,R, focal length of lenses. sad Experimental configuration with detection plane coinciding with the object plane, xT= x_T

9

^. sbd Experimental configuration with detection plane coinciding with the Fourier plane of the collecting lens in the test arm.

According to Fig.

3, we consider the following two dif-

ferent schemes for the collection optics in the test arm of the setup.

sad The detection plane coincides with the plane of the transparency, x

_T

= x

_T

9 , and hence

˜h

T

sx

T

,q d ~ e

^{−isld1/4pdq2}e^−iq·xTt

sx

T

d s43d only describes free propagation over a distance d

₁

.

sbd A collection lens f

T

is located beyond the transparency at a distance f

_T

from the detection plane, so that the detection plane coincides with that of the Fourier transform, and hence

˜h

T

sx

T

,q d ~ e

^{−isld1/4pdq2}t˜

S ^{− q −} lf ^2p

_Tx_T

D ^{. s44d}

The optical scheme in the reference arm contains a lens sfo- cal length f

_R

d located at x

l,R

and thus the Fourier transform of the impulse response functions can be written as

˜h

R

sx

R

,− q d ~ E

^dxR

⁸

^e−iq·xR8

E

^{dxl,Reisp/ld2dsxl,R− xR8d2e−isp/lfRdxl,R2 eisp/ld3dsxl,R− xRd2}

~ e

^{−isld2/4pdq2}

E

^{dxl,Re−ifs2p/ld3dxR+qg·xl,Reisp/lds1/d3−1/fRdxl,R2}

^{. s45d}

If f

_R

Þ d

₃

, Eq. s

45

d becomes

h˜

R

sx

R

,− q d ~ e

^{−isl/4pdfd2+1/s1/d3−1/fRdgq2}e^{−si/d3df1/s1/d3−1/fRdgq·xR}

, s46d while if f

_R

= d

₃

, Eq. s45d becomes

˜h

R

sx

R

,− q d ~ e

^{−isld2/4pdq2}

d S ld ^2p

₃x_R

+ q D ^{. s47d}

Depending on the chosen geometrical configuration, these schemes allow realization of either a ghost-imaging or a ghost- diffraction experiment f

18

g, as explained below.

A. Ghost imaging

To perform a ghost-imaging experiment we choose f

_R

Þ d

₃

.

Let us first consider case sad. Substituting Eq. s

43

d and Eq. s

46

d into Eq. s

42

d yields the expression

G^s2d

sx

R

,x

_T

d ~ utsx

T

du

²

U ^o

q

C

_qe^{−iq·hxT+s1/d3df1/s1/d3−1/fRdgxRj}e−isl/2pdfsd₁+d₂d/s1/d₃−1/f_Rdgf1/sd₁+d₂d+1/d₃−1/f_Rgq²

U

²

^{. utsx}

^T

^du

²

^uC

^q

^u

²

^d S

^xT

⁺

^xM^R

D ^,

s48d which, once integrated over the bucket detector,

G

^s2d

sx

R

d = E

^dxTGs2d

^sx

^R

^,x

^T

^{d .} U

^t

S ⁻

^xM^R

D U

²

^uC

^q

^u

²

^{, s49d}

gives the image of the object. Note that in passing from Eq. s48d to Eq. s49d we have assumed that C

q

is virtually independent of q. Moreover, we have chosen distances d

₁

, d

₂

, and d

₃

that satisfy the so-called “back-propagating thin lens equation,”

1 / sd

1

+ d

₂

d+1/d

3

= 1 / f

_R

f

19

g, so that we obtain an imaging system with magnification factor M =d

3

/ sd

1

+ d

₂

d.

In case sbd, that is with a collection lens in the test arm that forms the Fourier transform on the detection plane, we proceed similarly by substituting Eq. s

44

d and Eq. s

46

d into Eq. s

42

d and making the same assumption or choice as before and obtain

G^s2d

sx

R

,x

_T

d ~ U ^o

q

C

_qt˜

S ^{− q − lf 2p}

T

x_T

D

^{e−is1/d3df1/s1/d3−1/fRdgq·xRe}−isl/2pdfsd₁+d₂d/s1/d₃−1/f_Rdgf1/sd₁+d₂d+1/d₃−1/f_Rgq²

U

²

^. U

^t

S ⁻

^xMR

D U

²

^uC

^q

^u

²

^,

s50d