Quantum theory of dynamic multiple light scattering

0.15 MB·English

by  S. E. Skipetrov

Quantum theory of dynamic multiple light scattering
S.E. Skipetrov∗
Laboratoire de Physique et Mod´elisation des Milieux Condens´es
CNRS and Universit´e Joseph Fourier, 38042 Grenoble, France
(Dated: February 4, 2008)
7 We formulate a quantum theory of dynamic multiple light scattering in fluctuating disordered
0 mediaandcalculatethefluctuationandtheautocorrelation functionofphotonnumberoperatorfor
0 light transmitted through a disordered slab. The effect of disorder on the information capacity of
2 aquantumcommunication channeloperating in adisordered environmentis estimated and theuse
of squeezed light in diffusing-wave spectroscopy is discussed.
n
a
J
Many disordered media (suspensions, emulsions,
3 a b’
2 foams, semiconductor powders, biological tissues, etc.)
scatter light diffusely and appear opaque. Statistical

] properties of light scattered from disordered media are
n
offundamentalimportanceforsuchapplicationsaswire- l
n a’
- lesscommunications[1,2,3]anddifferenttypesofdiffuse
s
optical spectroscopies (diffuse transmissionspectroscopy
i b
d [4], diffusing-wave spectroscopy [5, 6], etc.). With a few
. L
t notable exceptions [7, 8], most of theoretical studies in
a
this field of research have been conducted by treating
m
FIG. 1: An ensemble of mobile scattering particles (circles)
light as a classical wave [9, 10], without accounting for
- is illuminated from the left by a wave in the incoming mode
d the quantum nature of light. Meanwhile, recent exper- a. All other incoming modes are in vacuum states (modes a′
n iments demonstrate that quantum properties of light in ontheleftandb′ ontheright). Weareinterestedinthefluc-
o disorderedmedia(suchasquantumnoise[11]andphoton tuationsof transmitted light in theoutgoing modeb. Dotted
c
statistics [12]) can be of interest from both fundamental line illustrates the multiple scattering with a mean free path
[
and applied points of view. Moreover, the progress in l≪L that light undergoes inside themedium.

2 generation of light with nonclassical properties (single
v
photons and light in number states [13], squeezed light
3 tively. All but one of the incoming modes are assumed
[14],entangledphotons[15])suggeststhatsourcesofnon-
1 to be in vacuum states. The state of the mode a′ =a on
4 classical light could soon become accessible to a wide
the left can be arbitrary and it corresponds to the light
9 scientific community. This calls for a better theoretical
0 understandingofquantumaspectsofmultiple lightscat- that one sends into the medium in an experiment. We
6 areinterestedinthepropertiesofmultiplyscatteredlight
tering.
0 in the transmitted, outgoing mode b.
The purpose of the present Letter is to fill up an im-
/
t To quantize the electromagnetic field we use the same
a portantgapinthequantumtheoryofmultiplelightscat-
method as Beenakker et al. [7] and Lodahl et al. [8]
m tering by developing a quantum description of dynamic
who considereddisorderedmedia with immobile scatter-
- multiple lightscatteringinfluctuating disorderedmedia.
d In contrast to the previous work [7, 8], we take into ac- ing centers. An annihilationoperator aˆi(ω) is associated
n with each mode and standard bosonic commutation re-
count the evolution of disordered medium (e.g., Brown-
o lations are assumed [7, 8]:
ian or more complex motion of scattering centers) in the
c
v: ctoouarnseinotferaesmtinugltiapnled srceaatctherpirnogbleexmpeorfimquenatn.tuTmhdisynleaamds- [aˆi(ω),aˆ†j(ω′)]=δijδ(ω−ω′), (1)
i
X ics, where the quantum fluctuations of light are coupled wherei andj runeither overallincoming modes orover
r with the classical fluctuations of the medium. Our re- all outgoing modes. We now note that, as suggested by
a
sults allow us to estimate the impact of disorder on the Fig.1,thefluctuatingdisorderedmediumactsasabeam
information capacity of quantum communication chan- splitter having M input and M output ports and time-
nels and to lay a foundation for the use of nonclassical dependent transmission and reflection coefficients τa′b(t)
light as a probe of disordered media. and ρb′b(t). By analogy with the work of Hussain et al.
Weconsideraslabofdisorderedmediumthat,forcon- [16] who dealt with a ‘regular’ beam splitter (M = 2)
creteness,weassumetobeasuspensionofmobile,elasti- havingtime-dependentproperties,weassumethatτa′b(t)
callyscatteringparticles(seeFig.1). Foreachfrequency and ρb′b(t) are frequency-independent and that they do
ω, light incident on the slab can be decomposed in M not vary significantly during the typical time it takes for
transverse modes that we denote by indices a′ and b′ for lighttocrossthedisorderedsample. Theformerassump-
modes on the left and on the right of the slab, respec- tionrequiresasufficientlynarrowbandwidthB ofthein-
2
cident light, whereas the latter one is justified provided around the central frequency of a perfectly monochro-
thatthedynamicsofthe mediumisslow. Withtheseas- maticmeanfieldA0exp(−iω0t+iφ)[19],lightinagener-
sumptions in hand, the input-output relations for aˆ (ω) alized continuum number state |n,ξi with a box-shaped
i
read envelope function ξ(t) = (tmax − tmin)−1/2exp(−iω0t),
where tmin ≤ t1,t2 ≤ tmax [21], and chaotic light with
aˆb(t)= τa′b(t)aˆa′(t)+ ρb′b(t)aˆb′(t), (2) a long coherence time tcoh ≫ T. Making use of Eqs.
Xa′ Xb′ (1)and(2), we findthatthe normalizedvarianceofpho-
where aˆ (t) denotes a Fourier transform of aˆ (ω). The ton number fluctuations in the outgoing mode b can be
i i
use of aˆ (t) is permissible provided that B is much written as
i
smaller than the central frequency ω0 of the incident
Var[nˆ ]
light [16, 17]. Similarly to the case of immobile scatter- δ2 = b =
b 2
ers [7, 8], Eq. (2) conserves commutation relations (1), hnˆ i
b
which is due to the unitarity of the scattering matrix 1 Q
2 a 2
tcioomn.poHseodwoevfeτra,′bi(nt)caonndtraρsb′tb(tto)tinhethpereavbisoeunscewoorfkab[7so,r8p]-, = hnˆbi +δclass+ hnˆai(cid:0)1+δclass(cid:1). (3)
Eq. (2) implies a possibility of energy exchange between 2
Here Var[nˆ ] = hnˆ2i−hnˆ i , hnˆ i is the average photon
different spectral components upon transmission of light b b b a
number and Q = Var[nˆ ]/hnˆ i−1 is the Mandel pa-
through a fluctuating disorderedmedium. This becomes a a a
rameter [18] corresponding to the incident light beam.
clear when we back Fourier-transform Eq. (2) because
products of time-dependent quantities become convolu- δc2lass = (2/T) 0T dt(1 − t/T)CTab(t) is the normalized
tions in the frequency space. Such an energy exchange variance of timeR-integrated intensity in the mode b that
is by no means a nonlinear phenomenon, but is a man- one obtains from Eq. (2) by ignoring the quantum na-
ifestation of Doppler effect due to scattering of light on ture of light and dropping the operator ‘hats’ of aˆi(t);
2
moving particles. CTab(t)=Tab(t1)Tab(t1+t)/Tab−1istheautocorrelation
Toestablishalinkbetweenaˆ (t)andmeasurablequan- function of T (t). δ2 represents the ‘classical’ part of
i ab class
tities, we define a flux operator Iˆ(t) = aˆ†(t)aˆ (t) that, δ2inthesensethatitisdetermineduniquelybythe(clas-
i i i b
when integratedfrom t to t+T, yields a photon number sical) fluctuations of transmission coefficient Tab(t) due
operator nˆ (t,T) corresponding to the number of photo- to the motion of scattering centers in the medium.
i
counts measured by an ideal, fast photodetector illumi- The main conclusion following from Eq. (3) is that, in
nated by light in the mode i during some sampling time the general case, the variance of photon number fluctu-
T [17]. A relation between the average values of opera- ations in transmission through a fluctuating disordered
tors nˆ (t,T) and nˆ (t,T) corresponding to the outgoing medium is not simply a sum of quantum and classical
b a
mode b and the incoming mode a, respectively, is read- (due to disorder) contributions, but rather a much more
ily found from Eqs. (1) and (2): hnˆ i = T hnˆ i. Here complicated object. In particular, the last term of Eq.
b ab a
T (t)=|τ (t)|2, the angular brackets denote the quan- (3) mixes classical fluctuations and quantum noise in a
ab ab
tum mechanical expectation value, and the bar denotes multiplicative way. Only for incident light in the coher-
averaging over disorder. By the ergodicity hypothesis, ent state Qa = 0 and Eq. (3) reduces to a sum of shot
the simultaneous averaging h...i over both disorder and noise 1/hnˆbi and classical noise δc2lass. This is the mini-
quantum fluctuations can be replaced by time averaging mumvalue of δ2 that canbe obtainedfor lightin astate
b
2
in a realistic experiment with a fluctuating disordered admitting classical description. An ‘excess noise’ δexcess
medium, provided that the incident light is stationary. — defined as a difference between δb2 and 1/hnˆbi+δc2lass
On the contrary, there is no simple way of obtaining av- — arises for other states. For chaotic light, for example,
erages over either disorder or quantum fluctuations sep- Qa = hnˆai and δe2xcess = 1+δc2lass. Note that for both
arately. coherent and chaotic states we do not really need the
Although the relation between hnˆ i and hnˆ i is inde- quantum model developed here, but can instead use the
b a
pendent of the quantum state of the incident light, the semiclassical Mandel’s formula [18] as it has been done
latter is crucial for the second order statistics of the in Ref. [12].
photon number operator. To illustrate this, we con- We now turn to nonclassical light — light in states
sider a particularly simple case of a stationary, quasi- that cannot be described classically and to which the
monochromaticincidentbeamforwhichbothhIˆ (t)iand Mandel’s formula does not apply. For incident light in
a
h: Iˆa(t1)Iˆa(t2) :i, with : ... : denoting normal ordering the number state |na,ξi, Qa = −1 and the excess noise
[18], can be considered time-independent on the scale of is negative: the fluctuations of transmitted photon num-
T. This includes a number of practically important ex- ber are suppressed below the simple sum of shot noise
amples on which we will dwell in more detail: coherent 1/hnˆbi and classical fluctuation δc2lass. δb2 becomes even
light, continuous-wave quadrature squeezed light with lessthanthebareshotnoise1/hnˆ iifn <1+δ−2 . An
b a class
a rectangular squeezing spectrum of width B ≪ 1/T interestingsituationoccursforthesingle-photonnumber
3
FIG.2: Excessnoiseofphotonnumbernˆ inthetransmitted FIG. 3: Autocorrelation function of photon number fluctu-
b
modebfortheincidentlight(modea)inthecontinuous-wave ations in transmission of amplitude-squeezed light through a
amplitude-squeezed state with a narrow rectangular squeez- suspensionofsmalldielectricparticlesinBrownianmotionfor
ing spectrum of width B ≪1/T. The excess noise is defined the same values of parameters as in Fig. 2 and t0 = 0.1/B.
asadifferencebetweenthevarianceofphotonnumberfluctu- Thedashedredlineillustrates thatCb(t)isindependentofr
n
ations δb2, the shot noise 1/hnˆbi and the classical noise δc2lass. for t=1/B.
It is shown as a function of sampling time T and squeez-
ing parameter r for a suspension of small dielectric particles
in Brownian motion (diffusion coefficient D ) as depicted in ration T and containing up to nmax photons, in the in-
B a
Fig. 1. The unit of time is t0 = l2/6k2DBL2, where l is the comingmodea. BobreconstructsthemessageofAliceby
meanfreepath,L≫listhesamplethickness,andk≫1/lis countingphotonsintheoutgoingmodebduringthesame
thewave numberof light. The intensity carried by themean
time intervals T. By using the theory of linear bosonic
field A20 is set equal to B. Thick black lines show the excess
communication channels [20] and Eq. (3), we find that
noise as a function of sampling time for r = 0, 0.2, 0.4, 0.6,
the information capacity of such a channel is
0.8 and 1.0.
nmaxT 1
C = a ab 1+ nmaxT
Teln2 (cid:26) 2 a ab
state (n = 1): the relative fluctuation of transmitted
photon naumber becomes independent of δc2lass. Another × (cid:20)nm1ax − 1e −δc2lass(cid:18)1− nm1ax(cid:19)(cid:21)(cid:27) bists, (4)
importantexampleofnonclassicallightisthecontinuous- a a
wave squeezed light for which Qa = hnˆaif(r,φ,A20/B,1) wherewedrophigher-ordertermsinnmaxT ≪1,which
a ab
[19], where r is the squeezing parameter and we define
isjustifiedbytheconditionT ≪1typicalfortransmis-
an auxiliary function f(r,φ,x,y) = [xy(e−2rcos2φ + ab
sion of light through a diffusely scattering medium. As
e2rsin2φ − 1) + y2sinh2rcosh2r]/(x + sinh2r)2. For could be expected, the (classical) fluctuations δ2 re-
class
the amplitude-squeezed light (φ = 0) the Mandel pa-
ducethecapacityofthechannel. However,thisreduction
rameter is negative if r < r0, with r0 depending on isonlysecond-orderinnmaxT ≪1. Thedropofcapac-
A2/B, leading to δ2 < 0, similarly to the num- a ab
0 excess ity in the presence of disorder is therefore mainly due to
ber state. This is illustrated in Fig. 2 for the case of
the reduced averagevalue of transmission coefficient T
CTab(t)=(t/t0)/sinh2 t/t0 which corresponds to a sus- rather than to the fluctuations of the latter. Interesatb-
pensionofsmallBrownpianparticles[5,6](the character- ingly, C appears to be independent of δ2 if nmax = 1
class a
istic decay time t0 is defined in the caption of Fig. 2). (Alice sends either 1 or no photons).
With A2/B =1 chosen for this figure the excess noise is
0 Correlationofphotonnumbersinthesamemodebbut
negative for r<r0 ≃0.6. in two different, nonoverlapping time intervals,
Equation (3) can be used to estimate the effect of dis-
order on the information capacity of quantum commu- Cb(t)= hnˆb(t1,T)nˆb(t1+t,T)i −1, (5)
n
nication channels operating in disordered environments. hnˆb(t1,T)ihnˆb(t1+t,T)i
As anexample,let us considera communicationchannel
isalsoaffectedbythequantumstateofincidentlightand
connecting Alice on the left of the disorderedmedium in
can be reduced to
Fig. 1 and Bob on the right. Alice communicates with
Bobby sending a sequence ofnumber states, eachofdu- Cb(t)=[1+Ca(t)]Cab(t)+Ca(t), (6)
n n T n
4
where Ca(t) is the autocorrelation function of photon troscopy with squeezed light. Our results suggest that
n
number fluctuations in the incident mode a. In order using nonclassical light (single photons, light in number
to put Eq. (6) in a simple form, the typical decay time states, squeezed light) opens interesting perspectives in
of Cab(t) is assumed to be much longer than T. Equa- optics of disordered media.
T
tion(6)ismoregeneralthanEq.(3)andrequiresneither The author acknowledges fruitful discussions with R.
stationarity nor monochromaticity of incident light. For Maynard and F. Scheffold. This work is supported by
lightinthecoherentstateCa(t)=0andCb(t)=Cab(t). the French ANR and the French Ministry of Education
n n T
This result actually lies at the heart of diffusing-wave and Research.
spectroscopy (DWS) and other photon correlation tech-
niques that identify the measuredCb(t) with Cab(t) and
n T
interpret it in terms of classical wave scattering.
For nonclassical states of incident light Cb(t) is not
equal to Cab(t). For the number state |n ,ξni, Ca(t) = ∗ [email protected]
T a n [1] A.L. Moustakas et al.,Science 287, 187 (2000).
−1/n and Cb(t) can take negative values and becomes
a n [2] M.R. Andrews, P.P. Mitra, and R. deCarvalho, Nature
independent of time for n = 1. These strange proper-
a 409, 316 (2001).
ties are direct consequences of the conservation of total [3] S.E. Skipetrov,Phys. Rev. E 67, 036621 (2003).
number ofphotons —a conservationlawwhich doesnot [4] P.D. Kaplan et al., Phys. Rev. E 50, 4827 (1994).
apply to the coherent state where the number of pho- [5] D.A. Weitz and D.J. Pine, in Dynamic Light Scattering:
tons is not a good quantum number. For squeezed light The Method and Some Applications, ed. by W. Brown
we find Ca(t) = f[r,φ,A2/B,sinc(πBt)]. As illustrated (OxfordUniversityPress, Oxford,1993), Vol.49,p.652.
n 0 [6] G. Maret, Curr. Opin. Colloid Int. Sci. 2, 251 (1997).
in Fig. 3, squeezing induces oscillations of Cb(t) and de-
n [7] C.W.J. Beenakker, Phys. Rev. Lett. 81, 1829 (1998); M.
creasesitsvalueatshortt. Thelatterreachesaminimum
Patra and C.W.J. Beenakker, Phys. Rev. A 60, 4059
atsomer =r1 (r1 ≃0.3inFig.3)andthengrowsagain, (1999); ibid. 61, 063805 (2000); C.W.J. Beenakker, M.
exceeding 1 at large r. The results presented in Fig. 3 Patra, and P.W. Brouwer, ibid. 61, 051801 (2000).
open a way to performing DWS with squeezed light; the [8] P.Lodahl,A.P.Mosk,andA.Lagendijk,Phys.Rev.Lett.
necessary experimental setup is completely analogous to 95, 173901 (2005); P.Lodahl, Opt. Lett. 31,110 (2006);
Optics Express 14, 6919 (2006).
the ‘traditional’ DWS one [5, 6], except that a source of
[9] M.C.W. van Rossum and Th.M. Nieuwenhuizen, Rev.
squeezed light should be used instead of a laser.
Mod. Phys. 71, 313 (1999).
Provided that T (t) is replaced by T (t) = T (t)
ab a b ab [10] E. Akkermans and G. Montambaux, Physique
in the definition of CTab(t), the main results ofPthis Let- M´esoscopique des E´lectrons et des Photons (EDP
ter — Eqs. (3) and (6) — also hold for the variance and Sciences/CNRS Editions, Paris, 2004).
the autocorrelation function of nˆ = nˆ — a photon [11] P.LodahlandA.Lagendijk,Phys. Rev. Lett.94,153905
b b
number operator corresponding to tPhe ‘total transmis- (2005).
sion’ measurement. Equations (3) and (6) thus implic- [12] S. Balog et al.,Phys. Rev. Lett. 97, 103901 (2006).
[13] B. Lounis and M. Orrit, Rep. Prog. Phys. 68, 1129
itly account for all possible long-range (in space or in
(2005); J.K. Thompson et al., Science 313, 74 (2006).
time) correlations of T (t) that were extensively stud-
ab [14] G.Breitenbach,S.Schiller,andJ.Mlynek, Nature, 387,
ied previously [9, 10]. The effect of long-range correla-
471 (1997).
tionsonphotonnumberfluctuationsintotaltransmission [15] O.Benson et al.,Phys. Rev. Lett. 84,2513 (2000); R.M.
througha suspension of scattering particles in Brownian Stevenson et al.,Nature 439, 179 (2006).
motion has been recently studied by Balog et al. [12] [16] N.Hussain, N.Imoto, and R.Loudon,Phys. Rev. A45,
for incident light in the coherent state, and the result 1987 (1992).
[17] K.J. Blow et al.,Phys. Rev. A42, 4102 (1990).
following for this case from Eq. (3) has been confirmed
[18] L.MandelandE.Wolf,OpticalCoherence andQuantum
experimentally.
Optics (Cambridge University Press, Cambridge, Eng-
In conclusion, we developed a quantum description of
land, 1995).
dynamic multiple light scattering in fluctuating disor- [19] C. Zhu and C.M. Caves, Phys. Rev. A42, 6794 (1990).
deredmedia. Wecalculatedthefluctuationandtheauto- [20] C.M. Caves and P.D. Drummond, Rev. Mod. Phys. 66,
correlationfunctionofphotonnumberoperatorintrans- 481 (1994).
missionthroughaslabofmobilescatteringparticles. We [21] The generalized continuum number state with an enve-
lopeξ(t)isconstructedfromthevacuumstate|0iaccord-
applied our results to estimate the effect of disorder on
ingto|n,ξi=(n!)−1/2[Aˆ†]n|0i,whereAˆ†= dtξ(t)aˆ†(t)
the information capacity of a quantum communication
[16, 17]. R
channeloperatinginadisorderedenvironmentandtodis-
cuss the possibilities of performing diffusing-wave spec-