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0Huygens-Fresnel principle for N-photon states of light

E. Brainis!

Service OPERA, Universite libre de Bruxelles, Avenue F. D. Roosevelt 50, B-1050 Bruxelles, Belgium(Dated: May 17, 2013)

We show that the propagation of a N-photon field in space and time can be described by ageneralized Huygens-Fresnel integral. Using two examples, we then demonstrate how familiar Fourieroptics techniques applied to a N-photon wave function can be used to engineer the propagation ofentanglement and to design the way the detection of one photon shapes the state of the others.

PACS numbers: 42.50.Ar, 42.50.Dv, 42.30.-dKeywords: Quantum Optics, Di!raction

In quantum optics and quantum information science,one often deals with systems in which the total pho-ton number N is known, the challenge being to preparethe required N -photon state and engineer its evolution.When N is small, a wave-function formalism provides amore compact and intuitive physical description than theusual quantum field formalism. This has been advocatedin the recent years [1, 2] and justifies the renewed interestin photon wave-mechanic.

Single photons propagate in space and di!ract on ob-stacles exactly as classical waves do. How a N -photonwave function propagates is less obvious. In Ref. 1, theauthors used a two-photon Maxwell-Dirac equation tostudy the disentanglement of a photon pair. In thisLetter, we would like to lay the grounds of a propaga-tion theory of N -photon wave-packets. Instead of usingdi!erential equations, we generalize the Huygens-Fresnelprinciple to obtain an integral formulation relating thevalues of the N -photon wave function at a given time toits values on a fixed reference surface (usually a plane)at previous times. Familiar Fourier optics techniques canthen be applied in order to engineer the propagation ofentanglement and shape the photon state through thedetection process. We illustrate this point using two ex-amples. Our approach generalizes previous works on the“ghost imaging” properties of photon pairs produced byparametric down conversion [3, 4] and is relevant to manyapplications of modern quantum optics, including quan-tum super-resolution imaging, quantum lithography, aswell as spatial quantum communication.

The very idea of using position wave functions to de-scribe the state of N -photon systems relies on the ex-istence (but not uniqueness) of a photon position op-erator r, the Cartesian components of which are com-muting Hermitian operators satisfying [rk, pl] = ih!kl,p being the photon momentum [5, 6]. The eigenfunc-tions of r are transverse waves that can be interpreted aslocalized-photon states [7]. Any admissible single-photonwave function is obtained as a linear combination of theselocalized states. N -photon wave functions are symmet-ric elements of the tensor product of N single-particleHilbert spaces. Because the definition of the position op-erator r is not unique, there is more than one way to

assign a position wave function to a single photon. Themost popular one is probably the so-called Bialynicki-Birula-Sipe wave function "(r, t) = [!+(r, t) !"(r, t)]which has two vector components corresponding to pho-tons with positive and negative helicity [8, 9]. Each vec-tor component has a Fourier expansion that reads

!±(r, t) =

!

d3k!hkc e±(k) f±(k)

ei(k·r"kc t)

(2#)3/2, (1)

where k = |k| and e±(k) are the unit circular polariza-tion vectors for photons propagating in the k-direction.Normalization is such that the complex coe"cients f±satisfy

"

h=±

#

d3k |fh(k)|2 = 1. This wave functiontransforms as an elementary object under Lorentz trans-formation and can be easily connected to Maxwell fields.In this Letter, we write the wave function is a slightlydi!erent (but equivalent) way that consists in summingboth helicity components together:

!(r, t) = !+(r, t) +!"(r, t). (2)

This provides a vector representation instead of the bi-vector one [10]. Since !+ and !" are orthogonally po-larized, they never mix: if ! is given, !+ and !" can bededuced. Therefore the information content in the vectorfunction ! is the same as in the bi-vector field ".Working with wave functions (‘first quantization” for-

malism) is fully equivalent to working with quantumfields (“second quantization” formalism). Replacingthe complex coe"cients f±(k) by annihilation operatorsa±(k) in Eq. (1), the fundamental quantum field is foundto be proportional to the positive frequency part of theelectric field:

!(r, t) =$

h=±

!

d3k!hkc eh(k) ah(k)

ei(k·r"kc t)

(2#)3/2

= "i!2$0 E(+)(r, t).

Note that ! # E(+) holds information about bothelectric and magnetic field. Therefore, it provides acomplete information about electromagnetic configura-tion. To show this explicitly, one decomposes ! againinto its helicity components !+ and !" and subtract

2

them: this yields B(+) =%

µ0/2(!+ " !"), the pos-itive frequency part of the magnetic field. In the sec-ond quantization formalism, the state of a single-photonwave packet writes: |#$ =

"

h=±

#

d3k fh(k) |1k,h$,where |1k,h$ = a†h(k)|0$ and f±(k) are the same spec-tral amplitudes that appear in Eq. (1). The connec-tion between the first and second quantization formal-ism is given by the relation !(r, t) = %0|!(r, t)|#$ ="i

!2$0 %0|E(+)(r, t)|#$. Since %$|!(r, t)|#$ = 0 for all

|$$ &= |0$, we have #!i!(q

#)#i(q) = %#|#†i!(q

#)#i(q)|#$ =2$0%E(")

i! (q#)E(+)i (q)$ for any pair of points q = (r, t) and

q# = (r#, t#), where the indexes (i, i#) ' {x, y, z}2 repre-sent Cartesian components. This relates the Bialynicki-Birula-Sipe wave-function to the usual first-order cor-relation functions of coherence theory. In particular,|!(r, t)|2 is proportional to the probability to detect thephoton energy at point r at time t. This gives to theBialynicki-Birula-Sipe wave function the usual interpre-tation of a probability amplitude to find the photon en-ergy at some position.The generalization to N -photon states

|#$ =$

h1,...,hN

!

d3k1 . . .

!

d3kN

fh1,...,hN(k1, . . . ,kN )|1k1,h1

, . . . , 1kN ,hN$

is straightforward. The connection between wave func-tions and fields is given by

#i1...iN (q1, . . . , qN ) = ("i)N (2$0)N/2

( %0|E(+)iN

(qN ) . . . E(+)i1

(q1)|#$ (3)

and

#!i!1...i!

N

(q#1, . . . , q#N )#i1...iN (q1, . . . , qN ) = (2$0)

N

( %E(")i!1

(q#1) . . . E(")i!N

(q#N )E(+)iN

(qN ) . . . E(+)i1

(q1)$. (4)

Eq. (4) shows that any field correlation function of aN -photon system can be computed as a product of twotensor elements of the N -photon wave function. The bestway to compute the propagation of the N -photon wavefunction may depend on the situation. Often, the pho-tonic state is prepared in such a way that the wave func-tion is known at all times, but only on a specific surface%. Therefore, a di!raction theory of N -photon states isneeded. An important example is the generation of en-tangled photon pairs in a nonlinear crystal. In that case,% is the output face of the crystal and di!raction fromthat plane leads to phenomena the phenomena of “ghostimaging” [11].To understand how N -photon detection correlations

spread in space and time, we first consider free-spacepropagation. We also make the simplifying assump-tion that we deal with paraxial states of light, in which

case polarization does not change much during propaga-tion. Therefore, we drop the polarization-related indexes.Considering photons propagating along the z-axis, weuse the Huygens-Fresnel principle [12] to express E(+)

at some point as a function of its values on the referenceplane % at z-coordinate %,

E(+)(r, t) =1

2#c

!!

d2&$ddt E

(+)(", t" |r"!|c )

|r" "|, (5)

and inject this in Eq. (3). This yields

#(r1, t1, . . . , rN , tN ) =1

(2#c)N

!!

d2&$1 . . .

!!

d2&$N

ddt1

· · · ddtN

#("1, t1 " |r1"!1|c , . . . ,"N , tN " |rN"!N |

c )

|r1 " "1| · · · |rN " "N |.

(6)

We call this integral the generalized Huygens-Fresnel (GHF) principle for N -photon wave func-tions. In Eqs. (5) and (6), "j = ('j , (j , %)(j ' {1, . . . , N}) are points in the %-plane and"$j = ('j , (j) are their transverse components. Inthe optical domain, photons can usually be con-sidered as quasi-monochromatic. Therefore, thewave function can be written #(r1, t1, . . . , rN , tN ) =a(r1, t1, . . . , rN , tN ) exp ("i2#c( t1

!1+ · · ·+ tN

!N)), where

a(r1, t1, . . . , rN , tN ) is a slowly varying function of timeand )j (j ' {1, . . . , N}) are the central wavelengthsof the photons. Note that nothing prevents photonsfrom having the same central wavelength or even beingindistinguishable. Inserting that anzats in Eq. (6) andtaking into account that a(r1, t1, . . . , rN , tN ) is slowlyvarying in time, one obtains

a(r1, t1, . . . , rN , tN ) =("i)N

)1 . . .)N

!!

d2&$1 . . .

!!

d2&$N

a("1, t1 "|r1 " "1|

c, . . . ,"N , tN "

|rN " "N |c

)

exp&

i 2"!1|r1 " "1|

'

|r1 " "1|. . .

exp&

i 2"!N|rN " "N |

'

|rN " "N |. (7)

Eq. (7) is only valid in free space. If propagation from "ito ri is through an optical system, the free space propa-gator

hfs(ri,"i) ="i

)i

exp&

i 2"!i|ri " "i|

'

|ri " "i|(8)

must be replaced by the appropriate one hi(ri,"i). Withthis generalization, Eq. (7) becomes

a(r1, t1, . . . , rN , tN ) =

!!

d2&$1 . . .

!!

d2&$N

a("1, t1 "l(r1,"1)

c, . . . ,"N , tN "

l(rN ,"N)

c)

h1(r1,"1) . . . hN(rN ,"N ), (9)

3

where l(ri,"i) is the optical path length from "i to ri.Formula (9) assumes that there is only one optical pathfrom "i to ri. However, interferometers with arms hav-ing di!erent path lengths can be placed between "i andri. To take this into account, we generalize (9) in thefollowing way:

a(r1, t1, . . . , rN , tN ) =

!!

d2&$1 . . .

!!

d2&$N

$

k1,...,kN

a("1, t1 "lk1

(r1,"1)

c, . . . ,"N , tN "

lkN(rN ,"N )

c)

h(k1)1 (r1,"1) . . . h

(kN )N (rN ,"N). (10)

The indexes ki label the di!erent paths from "i to ri.We illustrate formula (10) using two examples. Con-sider the system in Fig. 1. The scheme is similar to theoriginal “ghost imaging” scheme [11], with the di!erencethat two di!raction masks (called M1 and M2) are placedin the arms of a balanced Mach-Zenhder interferometer.Co-propagating energy-degenerated ()1 = )2 ) )) en-tangled photon pairs are generated by parametric down-conversion in a nonlinear crystal and split using polariz-ing beam splitter (PBS). The generation process entan-gles the photons in position and time. The detector Db

clicks whenever the photon going upward passes throughM1 orM2 and is detected on axis. The di!raction patternof the second photon is detected by moving the point-likedetector Da. Assuming a Gaussian pump and broadbandphase-matching, the wave function amplitude at the out-put of the nonlinear crystal is well approximated by

a("1, t1,"2, t2) =exp

&

" t22

4T 2

'

(2#T 2)1/4

exp&

" #22+$2

2

4S2

'

(2#S2)1/2

( !("$1 " "$2 ) !(t1 " t2),

where T and S are the time and space rms-half-width ofthe pump pulse. The propagation in the optical system

FIG. 1: Interference e!ect in quantum imaging with two sep-arated di!raction masks.

can be computed using formula (10), with

h1(ra,"1) ="i

)

exp(

i 2"! |ra " "1|)

|ra " "1|,

h(k)2 (rb,"2) =

("1)k!2)2

!!

Mk(r$)

exp(

i 2"! |r" "2|)

|r" "2|

(exp

(

i 2"! |rb " r|)

|rb " r|d2r$

where Mk(r$) (k ' {1, 2}) are the mask transfer func-tions. The GHF integral (10) simplifies if 4S2 * )(s0 +min(s1, s2)) (i.e. the wavefront curvature of the gener-ated photons can be neglected) and if the far-field condi-tions d2/) + s3 and d2/) + (s0 + s2) apply, where d isthe relevant length of the mask profile functions. Drop-ping constant and quadratic phase factors, one finds

a(ra, ta, rb, tb) = exp

*

tb " (s0 + s2)/c)

4T 2

+

!(ta " tb " *)

exp [iQ]2

$

k=1

Mk

*

2#

)

,

xa

L+

xb

s3

-

,2#

)

,

yaL

+ybs3

-+

,

where L = (2s0+s1+s2), * = (s1"s2"s3)/c, ri = (xi, yi)and Mk(k$) =

#

Mk(r$) exp["ik$ · r$] dr$ is theFourier transform of the mask function Mk(r$). Theexponential factor indicates that the detection of a pho-ton in Db can only happen a propagation time (s0+s2)/cafter the photon has been created and within the initialtime uncertainty T . The !-factor shows that the di!er-ence * in the detection times of Da and Db is just dueto di!erent propagation distances from the crystal to thedetectors, as expected for time-entangled photons. Thewave-front curvature of both photons are related throughthe quadratic phase Q = (#/))[(x2

a+y2a)/(2s0+s1+s2)+(x2

b + y2b )/s3]. If detector Db, placed on axis (r$b = 0),detects a photon a time tb = t!, the wave function ampli-tude of the photon travelling towardsDa is automaticallyprojected on

a(ra, ta) = ! (ta " [t! + * ]) exp(iQa)2

$

k=1

Mk

&

2#xa

)L, 2#

ya)L

'

,

with Qa = (#/))(x2a + y2a)/(2s0 + s1 + s2). The system

of Fig. 1 can be used to produce heralded single pho-tons with on-demand or adaptive spatial profile. Onecan mimic a complex modulation (amplitude and phase)using phase modulators at M1 and M2. Such a systemwould be useful in the context of earth-satellite quan-tum cryptography [13] if true single photons had to beused. Even with simple masks, such as the double slitsin the inset of Fig. 1, non trivial shaping can be done. If2& = !, changing the interferometric phase + from 0 to #doubles the spatial frequency of the interference fringesseen by Da. Single photons with very di!erent spatialprofiles can be created by tuning only one parameter.

4

As a second example, consider the system in Fig. 2. Anonlinear device produces photon triplets with time andspace entanglement. (Such a device do not exist yet, buttime-entangled triplets have been demonstrated recently[14]. A bulk version of that experiment would exhibitspace entanglement as well.). We assume that, in theoutput plane of the device, the three-photon amplitudeis

a("1, t1,"2, t2,"3, t3) =exp

&

" t23

4T 2

'

(2#T 2)1/4

exp&

" #23+$2

3

4S2

'

(2#S2)1/2

( !("$1 " "$3 ) !("$2 " "$3 ) !(t1 " t3) !(t2 " t3).

We also assume that two photons have a common wave-length )1, while the third one has a di!erent one )2.They are separated using the dichroic beam splitter(DBS). In the )1 output of the DBS, we place a thinlens (focal f1) that images the output of the photonsource in the plane of detector D1 with a magnificationM = s2/(s0 + s1). The detector Db is placed on theoptical axis. Using formula (9), one can calculate thethree-photon amplitude in the detector planes. If a pho-ton is detected by Db at time t!, the wave function of theremaining )1 photons is projected on

a(ra, ta, r#a, t

#a) = !(ta " (t! + *)) !(ta " t#a)

exp&

" x2

a+y2

a

4S2M2

'

(2#S2)1/2

!(xa " x#a) !(ya " y#a) exp

,

i2#x2a + y2a)1s2

(1 + 1/M)

-

h2(0,"ra/M).

where * = (s1 + s2 " s3 " s4)/c and h2(rb,"3) is thepropagator from the nonlinear device to the detector Db.In the plane of detector Da, the photons exhibit spatialbunching. The field is a linear superposition of local-ized two-photon Fock states of light. Due to the mag-nification factor M , the spatial extension of that linearsuperposition can be much larger than S. Such a quan-tum state of light is interesting in the context of coher-ent super-resolution imaging [15]: The object to be im-aged would be placed in the plane of detector Da in or-der to be illuminated with that special quantum state.

FIG. 2: Generation of heralded linear superposition of local-ized two-photon states of light.

Resolution enhancement only matters in optical systemsin which geometrical aberration have been eliminated.Wavefront curvature must also be under control to avoiddistortions due to non-isoplanatism [16]. The scheme ofFig. 2 makes it possible to control the wavefront cur-vature of the )1 photons by tailoring the detection ofthe )2 photon. For instance, one can make the wave-front of )1 photons flat in the plane of Da by placinga lens (focal f2) in the path to D2 (see Fig. 2) andchoosing s3, s4 and f2 such that 2()2/)1)M(M + 1) =s2/((1/f2 " 1/s4)"1 " (s0 + s3)). This solution exists ifthe inequalities s4(s0 + s3)/(s0 + s3 + s4) < f2 < s4 aresatisfied.In summary, we developed a formalism that allows us

to analyse many interesting issues related to the propa-gation of arbitrary number states of light using the wavefunction formalism. We derived a generalization of theHuygens-Fresnel principle that accounts for the propa-gation of field correlations (including entanglement) inspace and time and showed how to applied it in practice.The formalism is very helpful to design sources of her-alded N -photons states with engineered spatial profile.This research was supported by the Interuniversity At-

traction Poles program, Belgium Science Policy, undergrant P6-10 and the Fonds de la Recherche Scientifique- FNRS (F.R.S.-FNRS, Belgium).

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