April 27, 2009 22:50 Contemporary Physics ”Freezing Light v2”

Contemporary PhysicsVol. 00, No. 00, June 2009, 1–16

Freezing Light with Cold Atoms

Mark D. Havey∗

Department of Physics, Old Dominion University, Norfolk, VA 23529

(Date Received)

The impact of disorder and localization in electronic conduction was introduced more than half a century agoby Philip Anderson. In a much broader context of disorder-mediated wave dynamics it remains an importantresearch area, and surprises abound. Meanwhile, research in ultracold atomic physics has led to phenomenallydetailed elucidation of properties, including changes in phase, of quantum degenerate Bosonic and Fermionicgases. For example, beautiful experiments have recently demonstrated, in quasi one-dimensional systems, An-derson localization of matter waves. In this brief essay, we describe and discuss research on wave localizationin the context of ultracold atomic physics, with a particular emphasis on light localization in ultracold andhigh-density atomic gases. Essential ideas are reviewed, along with the current experimental status of the field,and promising avenues for future research are discussed.

Keywords: ultracold atomic physics; light localization; Anderson Localization; quantum optics

1 Introduction

1.1 Historical prelude

More than a century ago, Lord Rayleigh published [1] a theoretical treatment of elastic scat-tering of light from atmospheric atoms and molecules. This research described what we calltoday Rayleigh Scattering, or the elastic scattering of light from particles much smaller than thewavelength, and predicted the essential λ−4 dependence of the scattered intensity and variationsof this dependence due to dispersion. Even though the research has long been superseded byquantum descriptions of light scattering [2, 3], the main ideas remain useful in the framework ofsemiclassical description of atomic light scattering. In the context of the present essay, RayleighScattering of nearly monochromatic electromagnetic radiation is a fundamental element of thephysics of light localization in atomic gases.

Indeed, the past year has witnessed the fiftieth anniversary of the classic paper by Philip An-derson, Absence of diffusion in certain random lattices [4], which introduced the fundamentallyimportant idea of disorder into condensed matter physics. This paper dealt specifically with thetransport properties of electrons on a lattice, where the binding of electrons to lattice sites wastaken as a randomly distributed quantity, and predicted that, for a certain critical amount ofdisorder, electrons would become spatially localized, and diffusive transport would cease. Theessential idea of disorder-induced localization has germinated into an extraordinary volume ofresearch in many areas of physics. In fact, the scientific field of Anderson Localization is immensein scope, and the general role of disorder in wave dynamics is being played out in fields rangingfrom electrons in solids, to propagation of matter waves, the study of localization of ultrasonicwaves, propagation of light in photonic materials, and waves in the solid earth. For this reasonat least, a brief encounter, even of limited scope and focused on light localization in ultracoldatomic gases, is doomed from the outset to be incomplete. Fortunately, there already exist a

∗Corresponding author. Email: mhavey@odu.edu

ISSN: 0010-7514 print/ISSN 1366-5812 onlinec© 2009 Taylor & FrancisDOI: 10.1080/0010751YYxxxxxxxxhttp://www.informaworld.com

April 27, 2009 22:50 Contemporary Physics ”Freezing Light v2”

2 M.D. Havey

good number of excellent reviews [5] and books [6, 7] of a pedagogical and of a technical natureon many aspects of disorder-induced localization, and we refer readers who become intrigued bythe science to pursue more details there.

The merger of the ideas of quasielastic scattering of light and localization by disorder weredescribed by Sajeev John a quarter century ago [8]. In a later article, Localization of Light[9], John considers light propagating in disordered dielectric structures which are often termedphotonic band gap materials. He pointed out that there can exist localized photonic modes inthese systems which are analogous to the localized, rather than extended, electronic states incondensed systems. For the purpose of our brief essay, this gets to the heart of the matter; threedimensional light propagation in a sufficiently disordered but finite medium can be localized byinterferences in multiple coherent scattering of the field by a disordered arrangement of scatterers.

Study of Anderson localization with light is very attractive, and offers unique features includingthe fact that light can be localized only through interference in multiple scattering. This is incontrast to electrons which may be trapped in local electrostatic potentials. In addition, photonshave different spin statistics, and the light polarization and scattering cross sections may readilybe controlled by exploiting the resonance frequency dependence. In spite of these possibilities,comprehensive experimental exploration of Anderson localization in three dimensions remainselusive.

Before we proceed further it will be useful to address a few questions one might have aboutthe subject of the present essay. These are: What is localization not? What is a useful definingconcept for light localization? And, finally, what is the big deal for atomic physics and what newphysics might experimental and theoretical examination of atomic systems offer to the alreadyextensive studies of wave localization?

1.2 What is it not?

One might consider localization of light to refer to situations where electromagnetic radiationis confined to, or temporarily stored in, some region of space. Localization might also refer tothe alteration of spontaneous emission rates due to modification of boundary conditions on theradiation field. Although these are useful and important ideas, there are many means by whichmanipulation of light can be accomplished without disorder, and none of them related, other thanthrough context, to the main topics of this paper. For a physics audience, for example, we mightimagine electromagnetic waves confined as quasi one dimensional modes established between twomirrors forming a laser cavity. This certainly results in a buildup of intracavity electromagneticenergy, but it is not localization of light. Alternatively, light might be confined to move within, orpractically to be guided by, any one of a variety of fiber optic arrangements which transverselyconfine light energy. But this is not localization by disorder either. A researcher in quantumoptics, or an aficionado of the popular science press, might argue that light slowed or stopped bymeans of coherent population trapping, or by electromagnetically induced transparency, shouldqualify. The resulting polariton formed among the spins of the ground state of a system surelyserves as a means to localize light. Unfortunately, it does not. In fact, in these illustrations,it is orderly structures that lead to spatial manipulation of the electromagnetic fields. On theother hand, a biologist once insisted to me that nature had long ago solved the problem of lightlocalization, and that was evident through the phosphorescence of bioluminescent plankton. AndI had a fascinating discussion with a chemist about how very long-lived, but eventually radiating,metastable chemical states, even in small molecules, must be the same thing as localized, orat least stored, light. In these cases, it is the operation of various optical selection rules, orthe influence of propensities for changes in chemical structure, rather than randomness, thataccomplish relatively long energy storage times.

The key point here is that there are many mechanisms through which waves may be spatiallyconfined. However, localization refers specifically to those cases where the confinement developsas a result of wave scattering in a medium consisting of a disordered arrangement of scattering

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Contemporary Physics 3

rr

Figure 1. Schematic diagram illustrating recurrent multiple scattering. The solid and dashed lines represent one possiblescattering path and its time-reversed twin. When the phase accumulated along these two paths is the same, there is increasedprobability for the scattered wave to return to its starting position, labelled r.

centers.

1.3 What is it?

Anderson localization occurs as a result of coherent multiple wave scattering from a disorderedarrangement of scatterers. The collective interference of the multiply scattered waves, for asufficiently strong disorder, leads to exponential spatial localization of what otherwise would bean extended wave propagating in the medium. We may consider the wave intensity to have acharacteristic length scale ξ for exponential decrease within a medium. Evidently, if the mediumis finite and has size L in one dimension, then L >> ξ is required.

There are generally two regimes of localization discussed in the literature. In the so-calledweak localization regime the density of scatterers is relatively low, and multiple scattering maybe considered as a sequential process forming a chain of scattering and propagation elements.This limiting case, and some of the associated observables, is discussed in Section 2.3. As thedensity of scatterers increases, a second and qualitatively different behaviour emerges throughrecurrent scattering; this indicates the approach to the so-called strong localization regime. Inrecurrent scattering a wave scattered from an object has some amplitude to return to that samescattering center. This amplitude, and its time reversed twin, can interfere constructively, thenreducing overall wave transport away from the spatial neighborhood of recurrent scattering.A schematic illustration of this process is given in Figure 1, where the solid and dotted linesrepresent pairs of reciprocal scattering paths which together lead to interference in the returnprobability to the point r. It is important to note here that for purely diffusive transport inone and two dimensions there is always finite probability for return to the neighborhood of theinitial scattering center [7]. On the other hand, in three dimensions this does not a priori occur,and localization would have to occur by another mechanism, which is the subject of this essay.It was the cessation of diffusive transport [10] with increasing disorder and sample size, whichwas the title emphasis in Anderson’s original paper [4].

Even though it suffers from some ambiguities in the near-localization regime, the Ioffe-Regelcondition [11] is commonly used to estimate the conditions where multiple scattering mightlead to interesting physics in the context of localization. This condition is kl = 1, where in theelectromagnetic wave case k = 2π

λ is the magnitude of the light wave vector and l is the meanfree path. For a dilute atomic vapor of uniform density ρ and light scattering cross section σ,

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4 M.D. Havey

the mean free path is given by l = 1ρσ . Although these quantities are well defined in the limit of

a dilute scattering sample, the concept of a mean free path is of limited value when the lengthscale for scattering is much smaller than the wave length itself. Nevertheless, to give a densityscale for localization effects, we consider a classical oscillator having λ = 780 nm (as appropriatefor the D2 resonance transition in atomic 87Rb) and a light scattering cross-section σ ∼ 3λ2

2π ,which gives a typical density ∼ 1014 atoms/cm3.

Although the concept of diffusion and the Ioffe-Regel criterion provide useful guides to thepresent discussion, I should point out that there are alternative formulations that are very fruitfulin discussions of wave localization. In the specific context of light localization, detailed elabo-ration of these and other ideas have been presented by van Tiggelen in Localization of Waves[5].

How then, might experimentalists observe effects associated with Anderson Localization? Oneapproach, which has an appealing physical nature, is direct imaging of the spatial distributionassociated with the localized mode. This has been very recently realized in beautiful experiments[12, 13] on localization of matter waves in a one-dimensional spatially disordered potential. InBilly, et al. [12] a quasi one dimensional Bose-Einstein Condensate (BEC) is allowed to expand inthe presence of disorder generated by an optically speckled dipole potential. The expanding BECevolves into an exponentially localized distribution which is imaged by fluorescence with a chargecoupled device camera. Among other things, the length scale ξ associated with the exponentialspatial distribution is directly extracted. An alternative approach is to directly measure thebreakdown of classical diffusive transport in a very well characterized physical sample. In thiscase, solutions to the diffusion equation should be an excellent description of wave transportin the absence of localization. As kl decreases towards the critical regime, a slowdown of theemergence of diffusive flux from a sample would be a strong indicator of Anderson localization.This less-direct approach has been taken by a number of researchers, and has recently led to quiteconvincing experimental demonstration by Maret’s group [14, 15] of just such slowing down.This general experimental technique, although less direct than the matter wave localizationmeasurements, seems promising for study of light localization in ultracold atomic systems aswell.

1.4 Why atomic physics?

Atomic physics with ultracold atoms has provided a fascinating laboratory for studying changesof phase previously identified in condensed matter systems, and for elucidating new phases andmechanisms. For instance, realization of a Bose-Einstein condensates in ultracold atomic gaseshas generated an immense volume of research on quantum degenerate Bosonic systems [16].Deeper exploration of these, and emergent studies of quantum degenerate Fermionic systems,have led to discovery of a constellation of new phenomena. For the study of disorder effects,for which quantum degeneracy is not required, one of the attractions of ultracold quantumgases is that they are nearly monodisperse and have well understood heavy particle and lightscattering properties. For ultracold collisions, the atom-atom scattering properties can be also bemanaged through magnetically, optically, or electrically tuned Feshbach resonances. Consideringinteractions of atoms with electromagnetic radiation, the individual scattering properties ofatoms in weak and strong radiation fields are well understood, and collective radiative processesincluding Dicke super- and sub-radiance have been studied for a wide range of conditions. Theatom-field interaction and collective properties of a medium can also be controlled through, forexample, coherent population trapping via two photon processes. Overall, the mutual coupling oflight and atomic degrees of freedom can be controlled with remarkable precision through high Qatomic resonances. Further control and dynamic manipulation of light scattering can be usefullyachieved by more traditional means through modification of the density and temperature of anultracold gas.

Studies of disorder in atomic systems also provide unique features that are not readily acces-

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sible in condensed matter samples. One of these characteristics is that interactions among thescatterers themselves may be significantly modified through application of magnetic and electricfields or, more interestingly, via the scattering process itself. For example, the presence of a singlenear-resonant photon in the vicinity of a scattering atom strongly modifies the scattering of asecond photon. One then may consider that significant photon correlations, and other stronglynonlinear quantum optical effects, may appear in disorder-induced localization when the numberof radiative quanta is comparable to the number of atoms in the sample. A second intriguingconsequence of using atomic systems is that the internal states of the scatterers may be probedon transitions that are not directly affected by the high optical depth (and resulting disorder)on other radiative transitions. One can thus learn about not only the state of the light emergingfrom an open system of atoms, but also about the resulting state of the atomic scatterers. Ingeneral, these together are expected to form a complex set of relationships in which the atomicand radiative degrees of freedom are entangled.

2 Ultracold atomic physics

2.1 Ultracold gases for light localization

In this section we briefly review some aspects of ultracold atomic samples that are importantto studies of localization in atomic physics. Excellent books [17] and reviews [18] exist whichprovide many practical aspects of laser-based cooling of atoms and molecules.

As estimated earlier, the Ioffe-Regel condition suggests that an atomic density of 1014

atoms/cm3 is necessary to be in the vicinity of the light localization regime. Among other things,this means that ordinary magneto optical traps (MOT) will not be suitable. For this type oftrap, radiation pressure due to rescattering of the trapping light limits the density to be ∼ 1011

atoms/cm3. This can be increased somewhat by using a so-called dark spot MOT, but probablynot to the required densities. Dynamic compression of a dark spot MOT may transiently give therequired densities, but static traps seem the best choice. With this, optical dipole traps are anattractive option, as they can confine all ground level Zeeman states, and they can be switchedon and off relatively quickly. They also permit the option of creating an optical lattice, withwhich the interplay between disorder and lattice site transport could be studied. Beyond this, asa key element in localization studies is the density of scatterers, it is also important to be ableto measure well the spatial atom distribution. Optical dipole techniques permit characterizationof the energetically deepest part of the trap through, for example, measurement of parametricresonance frequencies.

So why should ultracold gases be a sample of choice? There are two main reasons. The firstis that for warm vapors the Doppler broadening reduces the peak light scattering cross section,demanding a two order of magnitude increase in atomic density. Second, the motion of theatoms means that multiply scattered light will be dephased and spectrally redistributed as well.These together would be difficult or impossible to overcome in warm atomic vapors, and presentchallenges of their own even in ultracold samples.

Let us recall that the Ioffe-Regal criterion, kl ∼ 1, implies that a quite small light scatteringmean free path l is essential. As the mean free path is inversely proportional to the light scat-tering cross section, the atomic transition should be driven near resonance, so that σ is near itsmaximum value. In the case of an ultracold atomic gas, where the width of the atomic resonanceis very nearly the natural width of the resonance transition, an offset from resonance of no morethan a few MHz sets a reasonable scale. In addition to this, an atomic transition with a verysmall transition rate (a so-called leak rate) to other levels is required. This is because, even ina diffusive regime, a single photon injected into the center of a dense sample can scatter manythousands of times before emerging. In the case of the experiments in our group, this along withother factors led to a choice of the 87Rb F = 2 → F ′ = 3 hyperfine transition.

We point out that the role of frequency redistribution of the scattered light may be crucial in

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6 M.D. Havey

Absorption Shadow I(x,y)Probe Profile Io(x,y)

_____________________ ____

____

____

____

____

_

x

y

Figure 2. Images of the intensity profile of a probe laser beam in the presence and absence of an ultracold atomic gassample. The shadow of the atomic sample is the darker region of the image on the right.

localization studies. All of these either shift near resonance light away from resonance, or broadenthe overall spectrum, reducing the scattering cross section. There are a number of main sourcesof redistribution, the first being residual Doppler shifts due to the finite temperature of theatomic gas. This can be reduced, for example, by evaporatively cooling the atomic gas. A secondsource is the effect of recoil in the light scattering process. This both heats the gas, and leadsto a progressively increasing frequency shift to lower energy of the injected photons. Althoughthe heating effect should be small, the accumulating recoil shifts will limit how many times aphoton can scatter before shifting significantly off resonance. Third, as pointed out by Berman[2], optical pumping of Zeeman transitions in the ground state leads to frequency shifts in thescattered light spectrum. In addition to these, if the optical field within the sample becomestoo high, then the light scattering spectrum develops inelastic components. For a single atomin a near resonance radiation field, these would form the well-known Mollow triplet [19]. In adense gas where the radiation undergoes multiple scattering, the field in the sample is not so wellcharacterized, but consists of spatially distributed random interfering components. Treatmentof this case seems to be challenging, even in the more dilute gas case.

To close this section, we provide some useful formulas which approximately describe the spatialdistribution of atoms in an ultracold gas and the light scattering in such a gas. For an atomconfined in a MOT, in a crossed dipole trap, or in a lattice site for a 3-D optical lattice, theatom spatial distribution is often close to a Gaussian in form, and is given by

n(r) = noe− r2

2r2o (1)

where the Gaussian radius is ro and the peak number density at the center of the trap (r = 0)is no. The total number of atoms in the trap is given by N = (2π)3/2nor

3o

One way to characterize such a sample is to measure the shadow it forms in the transmittedlight from a spatially broad and near resonance probe propagating in the z-direction. Such ashadow is illustrated in Figure 2. The shadow is characterized by the Beer-Lambert Law, whichgives the transmitted intensity I(x,y) relative to the incident intensity Io as

I(x, y)Io

= exp(−b(x, y)). (2)

Here b(x,y) is the spatially distributed optical depth, which has a peak value given by b0 =

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√2πnoσoro, where σo is the weak field resonance light scattering cross section. The dependence

of σ on the shift of the light frequency f from resonance, given by δ = f − fo, is

σ =σo

1 + (2δ/γ)2. (3)

The quantity γ is the natural width of the atomic transition, which for the atomic Rb resonancetransition is 5.9 MHz.

2.2 Light scattering in dilute ultracold gases

In this section I briefly and qualitatively review the main physical principles of light scatteringin ultracold atomic gases. A very nice discussion of this subject entitled Light Scattering by PaulBerman has recently appeared in this journal [2], and I refer the reader to that article and othersources [3, 20] for a quantitative discussion.

As mentioned in the introduction, a necessary ingredient to understand multiple light scatter-ing is to first learn about quasi elastic Rayleigh and Raman scattering of light from single atoms.In the present instance our main concern is the near-resonance interaction of quasi monochro-matic radiation with a two level atom; the two levels are considered to have Zeeman degeneracy.

First consider a stationary two-level atom which is resonantly excited by a weak light pulse thatis temporally short on the scale of the lifetime of the excited state of the two level system. Theatomic response consists of an initial transient buildup of amplitude in the excited level, followedby exponential decay of the excited state. The spectrum of the scattered light is determined bythe squared magnitude of the Fourier transform of the radiated field, and is well described bya Lorentzian spectral shape [21]. If the atom is instead excited by a weak and monochromaticlight source whose bandwidth is much narrower than the natural spectral width of the transition,then the scattered light is also monochromatic, and at the same frequency as the driving field,so long as recoil of the atom can be neglected.

The distinction between these two cases is important for studies of multiple scattering in anultracold atomic gas. The reason for this is that transport of multiply scattered quasimonochro-matic radiation is well described, over a wide range of atomic densities, as a diffusion process.In diffusion we can define a mean free path, l = 1

ρσ and a (3-dimensional) diffusion coefficient,D = 1

3vl, and the spatial distribution of intensity may be obtained through solution of thediffusion equation with appropriate initial and boundary conditions. On the other hand, whenthe spectral distribution of scattered light is broad, then the light transport is more correctlydescribed by Levy statistics, and the mean free path diverges for an infinite three dimensionalsample [22, 23]. In fact, the path length distribution falls off asymptotically as a typical Levydistribution for the Cauchy problem as 1

r3/2 , where r is the distance from the atom. For studiesof light localization, one wants the possibility of very many orders of multiple scattering, anda Levy distribution is not desirable. Quasimonochromatic radiation of a spectral breadth muchsmaller than the natural radiative width is then clearly required.

For single scattering of near-resonant and weak monochromatic light on the atomic 87RbF → F ′, total light cross-section is given by

σ(δ) =2F ′ + 12F + 1

λ2

2π

11 + (2δ/γ)2

. (4)

Here σ(δ) is the total cross section and the shift of the light frequency f from resonance at fo

is given by δ = f − fo. The quantity γ is the natural width of the atomic transition, which forthe atomic Rb resonance transition is 5.9 MHz, while the wavelength λ = 780 nm. F and F ′are the total angular momenta of the excited and ground levels of the transition of interest. In

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8 M.D. Havey

our case F = 2 and F ′ = 3. The total cross section includes both Rayleigh and elastic Ramanscattering, but ignores any effects due to recoil or to atomic motion of the scatterers. These areexcellent approximations for single scattering of light from atoms in an ultracold atomic gas. Ishould point out that the cross-section differential in incident and scattered light polarization,and angles is considerably more complex, but will not be discussed further here.

For weak field multiple scattering in a dilute gas, one may envision the scattering processas a chain of scattering and propagation elements. This is a good approximation because thescatterers are on the average well separated, and so secondary and further scattering takesplace in the far field of the initial scattering process. Propagation may be described by theappropriate Green’s function propagator for the effective medium associated with the atomicgas. For a comprehensive review of the theoretical aspects of this process, with a specific focuson coherent backscattering, we refer to Kupriyanov, et al. [24]. Many aspects of experimentalcomponents of this research are found in [25, 26].

2.3 Weak localization and coherent backscattering

To introduce the idea of multiple coherent scattering, first consider a coherent light wave incidenton a static slab of disordered scatterers. We consider this to be a single configuration of scatterers,and by static we mean that the scatterers are not moving or otherwise changing with time. Ingeneral, the electric field of the light transmitted through the slab will have a spatially randomdistribution of phases and amplitudes. When these amplitudes are combined to form the intensityat spatial points external to the sample, interferences among the multiply scattered waves leadto strong fluctuations in the spatially varying intensity. If we alternatively examine multipleconfigurations at a single spatial point, then the resulting intensity distribution F (I) will begiven by

F (I) =1

Icaexp

(−I

Ica

)(5)

where Ica is the configuration averaged intensity. This represents a speckle distribution, andillustrates that such speckle patterns are characteristically noisy, with a mean intensity equal tothe standard deviation [7]. Such speckle patterns emerge through the interference in multiplyscattered waves, and appear in wave scattering in any direction from a random distribution ofscattering centers. Laser speckle is a commonly observed example of this phenomenon. If oneaverages over the speckle, then one obtains the mean intensity Ica, and the interference structureis lost.

A natural question to ask is whether there are situations where interferences in multiplescattering survive the configuration averaging and thus produce observable consequences at thelevel of the measured mean intensity. The answer is yes, which may be seen with reference toFigure 3, which schematically illustrates a wave multiply scattered from a gas of stationaryatoms considered as classical scatterers. In the picture, there is a plane wave incident from theleft, and one scattering path that might occur starts from the spatial point r1 along the linesshown to the point r2. Alternatively, scattering may just as likely be initiated at r2 and end atr1. The effect of multiple scattering depends on the accumulated geometrical phase differencefor this path, which for stationary scatterers is given by

∆φ = (k + k′) · (r2 − r1) (6)

.When the accumulated phase ∆φ is near zero, then one may expect interference effects to sur-

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Contemporary Physics 9

k

k

r1

r2

k

k'

r1

r2

Figure 3. Schematic diagram illustrating a multiple scattering geometry for coherent backscattering and weak localization.

vive configuration averaging over all possible scattering paths. As seen from the above equation,this may occur in two distinct ways. In one, which gives rise to the coherent backscattering ef-fect, observation is made of the scattered wave intensity in the nearly backwards direction, wherek ∼ −k′. For measurements in this direction, each multiple scattering path will have a smallaccumulated phase, and so an interferometric enhancement of the intensity will be observed.This is not a small effect, producing a factor of two enhancement over the incoherent albedo forexact backscattering. In the second case, the multiply scattered wave returns to the scatteringcenter from which it originated (r1 = r2), giving rise to what is described as weak localization[6]. Recurrent scattering in three dimensions, and its increasing frequency with the density ofscattering centers, is an important element in the approach to strong wave localization.

Coherent backscattering of light, which can be understood in terms of classical wave scatter-ing, was initially reported in 1984 [27–29]. These measurements, and the resulting theoreticaltreatments, focused on scattering from generally slab-type geometries of polydisperse condensedsamples. The most striking feature is a spatially narrow and cusp-shaped enhancement of thebackscattered intensity by a factor of two, in the most favorable cases. The angular width ofthe cone is estimated by δθ

′ ∼ 1kl , this turning out to be on the order of a few milliradians.

The narrowness of this feature may be what was responsible for the relatively recent discoveryof this classical optical effect. In 1999, first reports of measurements of coherent backscatteringfrom atomic samples were made by the Kaiser group [30, 31]. One attraction of atomic sampleswas that they are essentially point scatterers, and that the samples are monodisperse. The ex-periments were done on ultracold samples of atomic 85Rb thus avoiding the dephasing effects ofatomic motion on the fidelity of the coherent backscattering cone. Surprisingly, the enhancementof the backscattered intensity was far less than a factor two, being instead on the order of 15%. This was soon recognized to be due to the deleterious effects of the Zeeman structure of theatomic transition studied [32]. Studies on the 1S0 → 1P1 resonance transition in ultracold atomicstrontium showed recovery of the factor of two enhancement [33]. These studies were elaboratedlater by experiments and theoretical studies of atomic coherent backscattering which illustratedpolarization, magnetic field, and thermal dephasing effects on the process [24, 26, 34–38]. Acharacteristic image of a coherent backscattering cone generated by backscattering of near res-onance radiation from a gas of ultracold 85Rb atoms, along with comparison to a Monte Carlosimulation of the cone profile [39, 40], is shown in Figure 4.

A number of aspects of studies of atomic coherent backscattering anticipate investigation oflocalization in atomic gases. One of these is the role of decrease in the fidelity of enhancement,either due to atomic degeneracy or due to dephasing effects, as previously mentioned. An in-triguing effect is the role of the degree of electromagnetic excitation of the sample; when the

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10 M.D. Havey

-3 -2 -1 0 1 2 3

Enha

ncem

ent F

acto

r

0.90

0.95

1.00

1.05

1.10

1.15

1.20

θ (mrad)Figure 4. False color image of the coherent backscattering cone on the F = 3 → F ′ = 4 hyperfine transition in ultracold85Rb. White indicates the highest intensity. The graph is a horizontal cross section through the image, and the solid lineindicates a theoretical calculation.

number of excitation quanta is comparable to the number of atoms in the sample, we mightexpect strong correlations in the scattered light intensity, and a reduction of interference effects.As shown in Figure 5, we (and others [36]) have observed [41] this in the lower atomic densityregime of coherent backscattering. As shown in the figure, when the amount of optical excitationis increased, the fidelity of the coherent backscattering cone is decreased. The degree of opticalsaturation is measured by the parameter s, where s = 1 corresponds to an incident light intensityequal to the atomic saturation parameter.

Finally, an important aspect of the coherent backscattering effect is that the observables, thecone angular width and shape of the cone near the peak, can show a sample size dependency.For example, in spatially finite samples long multiple scattering paths are cut off, and do notcontribute to the cone. As long path lengths have a correspondence with smaller angular dis-placements from exact backscattering, this results in a rounding of the cone peak. For extendedsamples another characteristic observable, the cone angular width, is inversely proportional tothe mean-free-path. In contrast, for spatially localized samples such as atomic clouds, the coher-ent excitation fills the entire sample volume, and the relevant length scale for the backscatteringis the atomic sample size ro, viz. Eq. (1), which gives a cone angular width δθ ∼ 1

kro.

3 Anderson localization of light

In the previous sections, multiple coherent light scattering has been introduced and discussedfor the general case where the light scattering and propagation parts of the process may beconsidered practically distinct. The most widely studied observable in this case is the coherentbackscattering cone, and its dependence on many physical parameters. For the relatively lowscatterer densities in this regime, the role of recurrent scattering (see Section 1.3) is small. In

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(a) s = 0.08 (b) s = 0.3

(c) s = 3.0 (d) s = 9.0

Figure 5. False color images of the probe-intensity dependence of the coherent backscattering cone on the F = 3 → F ′ = 4hyperfine transition in ultracold 85Rb. White indicates the highest intensity. The spatially integrated intensity is the samein each of the four images. Loss of cone contrast is due to dephasing by the increased optical saturation of the atomictransition. S indicates the saturation parameter for the transition.

the present section we consider the higher density regime, where recurrent scattering becomesincreasingly important. In particular, this section is devoted to experiments aimed at observationof Anderson localization in disordered samples. The focus is on three dimensional localization oflight in the optical or near infrared regime, as these cases are most closely related to experimentson ultracold gases. At present there are, to my knowledge, two groups with specific researchefforts directed towards light localization in ultracold atomic gases. One is my own, and theother is Robin Kaiser’s Group in Nice, France. Although it is not a focus of this paper, a numberof groups have complementary efforts in Anderson localization of matter waves in ultracold gases(see, for examples, [12, 13]).

3.1 Experiments on light localization in three dimensions

Localization of light in a disordered medium [42] is the title of a recent paper reporting experi-mental observation of Anderson localization of light in three dimensions. In the experiment byWiersma, et al., a combination of measurements of transmitted intensity and coherent backscat-tering were analyzed to provide strong evidence of light localization in polydisperse GaAs pow-ders. Although the interpretation of these results has been questioned, and the questions rebutted[43], the experimental results remain intriguing and a subject of discussion in the community.The main points of discussion are concerned with the role of absorption, and whether eventhe quite large absorption lengths in the experiments can mimic localization behaviour in theobservables.

Unambiguous experimental observation of light localization is challenging, and is plagued byconfounding effects due to absorption. One reason for this is the scaling properties of the trans-mission of strongly scattering samples with sample length L. For an idealized medium with noabsorption, the intensity of light transmitted through a diffusive medium decays inversely withthe sample length; the analog of this for electron transport is Ohm’s Law for the electrical con-ductivity. As disorder is increased towards the localization threshold, the transmission scalingevolves through a L−2 dependence, and deep into the localized regime, transmission should nearlycease as extended waves become localized and display an exponentially decaying spatial charac-

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12 M.D. Havey

ter, viz. exp(−L/ξ), where ξ is a localization length. Unfortunately, realistic condensed samplesalways have some degree of absorption, the effect of which also scales exponentially with samplelength. Absorption can thus mimic or mask localization, and so more distinctive characteristicsare desirable. For instance, Chabanov, et al. [44] in Statistical signatures of photon localizationpresented a novel and attractive approach to dealing with absorption. In this, the variance in thetotal normalized transmission of a sample is found to be a clear measure of localization indepen-dent of the amount of absorption. This conclusion was demonstrated through measurements ofmicrowave transmission in quasi one dimensional localizing and nonlocalizing systems. Recently,beautiful experiments and analysis on localization of ultrasonic waves have used this method,and others, to provide comprehensive evidence for wave localization in a three dimensional elasticnetwork [45].

Recent experiments by Storzer, et al. [14, 15] have taken another approach and examined thetime evolution of light transmitted through a slab of strongly scattering TiO2 particles. Themeasurements are done over a range of Ioffe-Regel parameters 2.5 ≤ kl ≤ 30 and reveal a strongdeparture from diffusive intensity transmission as kl is lowered through a critical regime aroundkl ∼ 4. By using a model in which the diffusion coefficient evolves in time as D ∼ t−a, they extractthe exponent a and its variation with kl from a value near zero for more dilute samples to a ∼ 1in the localized regime. In the diffusive transport regime, the mean square displacement withtime from some starting location is given by < r2 >= Dδt. Thus if D develops an inverse lineardependence with δt, the mean square displacement becomes constant, and diffusive transportceases. The role of absorption in these experiments, while not negligible, is considered to beunder sufficient control that the evolution towards strongly subdiffusive transport seems clear.

3.2 Experiments on ultracold gases

As mentioned earlier, there are currently two experimental groups with cold-atom based lightlocalization efforts. One of these is the group of Robin Kaiser in Nice, France. At present, Iam kindly informed [46] that they have two experimental approaches based on compression ofatomic samples into optical dipole traps. One of these achieves a Ioffe-Regel parameter kl ∼ 50for the case of ultracold Rb. This sample contains an attractively large number of atoms N ∼108. In a second project they obtain kl ∼ 20 for Sr, but at a smaller total number of atomsN ∼ 105 − 106. These cases are characterized by a sufficiently small kl that recurrent scatteringshould be important in the light transport, and we look forward to yet smaller values of theIoffe-Regel parameter and very interesting results from these experiments.

In the remainder of this section I focus on research in my laboratories at Old Dominion, wherewe have an ongoing project to demonstrate unambiguously, and then to further study, lightlocalization by disorder in an ultracold gas of atomic 87Rb atoms. Such a clear experiment hasto our knowledge not yet been successful performed in any laboratory. So what we describe hereis the current experimental situation in our group in this fascinating area of ultracold atomicphysics research. In the current experiments, the atomic 87Rb sample is formed in an opticaldipole trap and has a peak density of 5 · 1013atoms/cm3. The temperature of the atomic cloudis near 65µK, while the total number of confined atoms is N ∼ 107. The sample approximatelyforms an ellipsoid of revolution, which has a radius of about 10µm and a long axis of about250µm. Light scattering experiments are done on the nearly closed F = 2 −→ F ′ = 3 atomictransition at a wavelength near 780 nm. For this transition, and for our experimental condi-tions, the Ioffe-Regel parameter kl ∼ 1, and so effects due to disorder are expected to be quitepronounced.

A major experimental challenge in these experiments is that the sample optical depth is verylarge. In the shortest direction through the sample the peak optical depth is b ∼ 200, meaningthat an incident beam of intensity Io is attenuated by a factor of e−200 in traversing the sample.In order to inject photons into the center of the sample, we use an auxiliary laser source which,for a very short time, reduces the peak optical depth to b ∼ 1− 5. Simultaneous application of

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Contemporary Physics 13

)-40 -20 0 20 40

Q (

)2

0

200

400

600

800

1000

Figure 6. Fully quantum calculation of the spectral response of a spherical sample consisting of a very dense and ultracoldgas of classical scatterers. The black curve shows the result for a single configuration, while the lighter but thicker line showsthe average over many configurations. For these results, the spherical sample radius is 1.2 · 10−4 cm, while the uniformsample density is 2.6 · 1014 atoms/cm3.

a probe field results in some photons being trapped in the sample when the auxiliary laser israpidly turned off. The general procedure is illustrated in Figure 6, where the elliptical sampleof ultracold atoms has been allowed to expand, and then illuminated with a short pulse ofnear resonance light. At the same time, the focused auxiliary beam is shined diagonally onthe atom cloud; the modification of the light scattering properties is clear. What one sees inthe leftmost panel (-8.0 GHz) is that the auxiliary laser has shifted the atomic resonance somuch that no light is scattered in the central zone of the sample). In the other two panels, theauxiliary laser is detuned to larger values of -80 GHz and -130 GHz. At the same time, the probelaser is left on the unperturbed atomic resonance transition. For these larger detunings, thereis significant light scattering, as evidenced by the brighter central zones in these images. Thiseffect, and the evident loading of probe photons has been demonstrated. The most importantmeasurements are those where the time evolution of light emerging from the sample after theprobe and auxiliary beams are removed, and such signals have in fact been recently measured.Comprehensive mapping of these transients in the parameter space consisting of atomic density,sample size, atomic temperature, and probe light intensity and detuning are now underway.

How does one know that the ultracold gas in these experiments is localizing light? To answerthat question, we use a technique similar to that described earlier [14], in which we measure thetime evolution of light emerging from the atomic sample. For kl >> 1, one should see a diffusivedecay of the intensity of light emerging from the sample. As the sample becomes increasinglydense, a critical slowing down should be observed, and the behaviour of the slowing down withsample density, detuning from exact atomic resonance, or increasing temperature will providequantitative measures of evolution of the system towards localization. Finally, trapping of toomany photons in the atomic cloud can also destroy localization. When the number of photonsbecomes significant compared with the number of atoms in the sample, then any one atommay be exposed to significant amplitude from different photon wave packets; this is expected tobroaden and split the atomic resonance, leading to a reduction in the scattering cross sectionand a corresponding increase in kl.

An intriguing physical picture of the process is the following. In any given static realization ofthe atomic cloud, theoretical simulations [47] have shown that there will be a number of excitationfrequencies for which there exist narrow resonances; the spectrally narrow widths correspond tolong lived excitations within the volume of the atomic cloud. A characteristic result for thesenarrow resonances, and the associated configuration-averaged spectrum, is shown in Figure 7.When the probe field excites these narrow resonances, then the emergent intensity from the

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14 M.D. Havey

-8.0 GHz -130 GHz-80 GHz

Figure 7. False color images of a dense and ultracold sample of 87Rb atoms probed on the F = 2 → F ′ = 3 hyperfinetransition. The smaller diagonal area is the overlap with an auxiliary laser beam which assists in loading photons into theotherwise optically opaque sample. White indicates the highest intensity. The detuning of the auxiliary laser from atomicresonance is given below each image.

sample will have long lived components. In any different realization, the narrow resonances occurin different spectral regimes; the narrow resonances are inhomogeneously distributed over thefull spectral profile of the sample optical response. It is this that requires us to make time domainmeasurements, rather than spectroscopic ones. In any case, from this point of view, one mightconsider the entire atomic cloud to be a weakly interacting many-atom molecule with a spectralresponse that includes very narrow resonances, and some very broad features corresponding tomore rapidly decaying configurations of atoms [46].

4 Future prospects and opportunities

It might be rash to forecast where studies of light localization and related physics could leadus over the next few years. Nevertheless, I will close here with a few observations. First, exper-imental and theoretical research on ultracold and dense gases will likely provide us with someinsights into the physics of random lasing [48, 49] in atomic systems. The physics of randomlasers appears to be related to that of light localization, and recent theoretical work on localiza-tion in inverted opals indicated just that [50]. We might hope that such insights will translateinto better understanding of random lasing in condensed systems, and lead to some new thingsalong the way. Second, at this writing, mechanisms and clear observables identifying light lo-calization in atomic gases continue to be of great interest. It is likely that there is an intimateinterplay between disorder and extended Dicke-type sub- and super-radiant states that bearsclose connection to light localization [51]. Finally, the role of disorder on matter wave propaga-tion in quasi one dimensional degenerate quantum gases has been explored in some beautifulrecent experiments; Anderson localization of matter waves has in fact been reported recentlyin Bose-Einstein condensates subject to controllable disorder [12, 13]. Perhaps we should beexploring the boundaries of phase diagrams where quantum degeneracy, light localization, andmatter wave localization grow close, and the productive friction that could result.

4.1 Acknowledgements

I would like to thank my many students and colleagues at Old Dominion who have made this re-search possible, and my colleagues Dmitriy Kupriyanov and Igor Sokolov, who have contributedimmeasurably to the theoretical aspects of the science discussed in this paper. Extensive andvaluable discussions with Robin Kaiser and Eric Akkermans are greatly appreciated. We ac-knowledge the financial support of the National Science Foundation.

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