Multiple scattering of light in cold atomic clouds with a magnetic field
Olivier Sigwarth, Guillaume Labeyrie, Dominique Delande, Christian Miniatura
MMultiple scattering of light in cold atomic clouds with a magnetic field
Olivier Sigwarth , Guillaume Labeyrie , Dominique Delande , Christian Miniatura , , , Laboratoire de Photonique, LPO Jean Mermoz, 53 rue du Dr Hurst, 68300 Saint-Louis, France Institut Non Lin´eaire de Nice, UMR 6618, UNS, CNRS; 1361 route des Lucioles, 06560 Valbonne, France Laboratoire Kastler Brossel, Ecole Normale Sup´erieure, CNRS, UPMC; 4 Place Jussieu, 75005 Paris, France Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore Institute of Advanced Studies, NEC, Nanyang Technological University, 60 Nanyang View Singapore 639673, Singapore (Dated: November 10, 2018)Starting from a microscopic theory for atomic scatterers, we describe the scattering of light by asingle atom and study the coherent propagation of light in a cold atomic cloud in the presence ofa magnetic field B in the mesoscopic regime. Non-pertubative expressions in B are given for themagneto-optical effects and optical anisotropy. We then consider the multiple scattering regime andaddress the fate of the coherent backscattering (CBS) effect. We show that, for atoms with nonzerospin in their ground state, the CBS interference contrast can be increased compared to its valuewhen B = 0, a result at variance with classical samples. We validate our theoretical results by aquantitative comparison with experimental data. PACS numbers: 42.25.Dd, 33.57.+c, 05.60.Gg
I. PHYSICAL CONTEXT
Multiple scattering of waves is an important topic in-volved in many branches of physics, from atomic physicsto astronomy via condensed matter physics [1, 2]. Oneof its most fascinating aspects occurs when interfer-ence effects persist in the presence of disorder. Theseinterference effects manifest themselves through devia-tions from the usual diffusive behaviour obtained at longtimes (weak localization). Under suitable conditions theycan even completely inhibit transport (strong localiza-tion) reaching an insulating regime. During the last twodecades, both the weak and strong localization regimeshave been extensively studied with systems ranging fromelectronic waves to ultracold atoms [3–12].Coherent multiple scattering of light in cold atomicclouds has been experimentally and theoretically studiedsince ten years in the context of the coherent backscat-tering (CBS) effect [13]. Compared to classical scatteringmedia, such as semi-conducting powders, cold atoms con-fined in a magneto-optical trap (MOT) constitute quitea unique sample of identical strongly-resonant point-likescatterers [14]. A key feature of the atomic scatterersis the presence of quantum internal degrees of freedomwhen their ground state is degenerate i.e. possesses anon-zero angular momentum [15]. These internal degreesof freedom are coupled to the polarization of light wavesduring single scattering events. As they are traced outwhen one observes the interference pattern of light, lightexperiences decoherence during its diffusive propagationin the atomic cloud. This leads to a reduction of the in-terference between scattering amplitudes as exemplifiedby the CBS contrast [16]. The saturation of the atomictransition [17] and the finite temperature of the atomicgas [18] are two additional mechanisms reducing the co-herence of light. To restore the contrast of the interfer- ences, it has been suggested to use a polarized atomicgas [19]. Unfortunately, this method is efficient only inoptically thin clouds. Another possibility is to break thedegeneracy of the atomic ground state by applying anexternal magnetic field B , which is one of the topics weaddress here.In classical scattering media, the presence of an exter-nal magnetic field is known to modify the interferencepattern and to reduce the CBS interference contrast [20].The mechanism at work here is the modification of thepolarization of light during its coherent propagation dueto the Faraday effect [21, 22]. As this modification is notthe same along a scattering path and its reversed part-ner, a dephasing process takes place between the scatter-ing amplitudes associated to these two paths, scramblingin turn their interference. Although the magneto-opticaleffects are very strong in cold atomic gases [23], theyactually have a negligible impact on the shape of the in-terference pattern itself [24]. More surprisingly, a fullrestoration of the interference contrast is even possibleunder well chosen conditions [25]. In this article, we givea detailed explanation of this result. We present a gen-eral, non-perturbative study of the interaction betweena cold atomic cloud and quasi-resonant light in the pres-ence of a magnetic field.The paper is organized as follows. In section II, wederive the scattering operator of a single atom exposedto monochromatic quasi-resonant light in the presence ofa magnetic field B and we analyze the differential andtotal cross-section. In section III, we address the coher-ent propagation of light and derive the refraction indextensor of the effective medium. In particular, birefrin-gence and magneto-optical effects are studied in III Dand III F. The CBS interference effect under a magneticfield is discussed in section IV and the restoration of theCBS contrast is studied in section V. a r X i v : . [ phy s i c s . a t o m - ph ] M a r II. SCATTERING OF LIGHT BY A SINGLEATOM IN THE PRESENCE OF A UNIFORMMAGNETIC FIELDA. Physical setting and basic assumptions
We consider the situation where a gas of identicalatoms, with spatial density n , is both exposed to a uni-form external magnetic field B and a monochromaticlight plane wave with wavelength λ , wavevector k ( k =2 π/λ ), angular frequency ω = k (we use units such that (cid:126) = c = 1) and transverse polarization (cid:15) .We first assume that the average distance betweenatoms is much larger than the wavelength λ of the lightwave, nλ (cid:28)
1, meaning that multiple scattering pro-cesses take place in the dilute regime. In this case asemi-classical description is appropriate and propagationof light in the atomic medium is well described by par-tial waves propagating along classical multiple scatteringpaths. Each path consists in a succession of independentscattering events by a single atom, separated by propa-gation in an effective medium with refractive index N r .To further properly describe scattering of light by onesingle atom, we assume that ω is very close to the tran-sition frequency ω between the atom internal ground-state (with total angular momentum F ) and an inter-nal excited state (with angular momentum F e ), herebyconsidering the case of quasi-resonant scattering. Intro-ducing the light detuning δ = ω − ω , this conditionreads | δ | (cid:28) ω . We further assume that this transition is closed (the excited state with life time 1 / Γ can only de-cay by spontaneous emission into the groundstate) andwell isolated from any other allowed transition connect-ing the groundstate to any other hyperfine levels. Thisis experimentally achieved for instance in the case of the F = 3 → F e = 4 transition of the D2 line of Rb [13].The total atom-light Hamiltonian H = H + U is thesum of the free atom-light Hamiltonian H and of theinteraction term U describing the coupling of the lightmodes to the atomic degrees of freedom. When no mag-netic field is applied, the interaction between light andone single atom is well documented [26, 27] and we thussimply need to incorporate the effect of the magneticfield. We will assume here that weak fields are appliedto the gas so that the linear Zeeman effect is the relevantphysical description. As the magnetic field only affectsthe energy levels of the atom, the free atom-light Hamil-tonian H just reads: H = ω ˆ P e + µg B · F + µg e B · F e + (cid:88) k , (cid:15) ω k a † k(cid:15) a k(cid:15) . (1)Here µ ≈ . g and g e are the Land´e factors of the ground and excitedstates respectively. In the following we will choose B tobe along axis 0 z which we choose as the quantificationaxis for the internal Zeeman states of the atom. Thenˆ P e = (cid:80) F e m e = − F e | F e m e (cid:105) (cid:104) F e m e | is the projector onto the excited state where | F e m e (cid:105) denotes an atomic state ofangular momentum F e and magnetic quantum number m e referred to axis 0 z .To be consistent with our approximations, the Zeemanshifts (of order µB ) must be much smaller than the en-ergy difference between any other atomic hyperfine levels.Still these shifts can be sufficiently large to fully split theZeeman structure and strongly modify the light scatter-ing properties of the atom. Indeed, as exemplified bythe case of Rubidium atoms, Zeeman shifts comparableto the excited-state frequency width (Γ / π = 5 . a k(cid:15) and a † k(cid:15) in (1) are the annihilationand creation operators of the electric field mode withwavevector k , polarization (cid:15) ⊥ k and frequency ω k . Wefurther assume that the light wavelength is much largerthan the size of the relevant atom electronic wavefunc-tions. The interaction between light and an atom locatedat position r is then accounted for in the dipolar approx-imation and reads U ( r ) = − d e · E ⊥ ( r ). The atomicelectric dipole operator writes d e = d + d † where d † = ˆ P e d e ˆ P g (2)is the operator describing transitions from the groundstate to the excited state. Here ˆ P g = (cid:80) Fm = − F | F m (cid:105) (cid:104)
F m | is the projector onto the groundstate sector with | F m (cid:105) an atomic state of angular momentum F and magneticquantum number m referred to axis 0 z . The transverseradiation field operator reads E ⊥ ( r ) = D ( r ) + D † ( r )where: D ( r ) = i (cid:88) k , (cid:15) E ω k (cid:15) k a k(cid:15) e i k · r (3)describes photon annihilation in all possible field modes.The field strength is E ω k = (cid:112) ω k / ε V , where V is thequantization box volume (it disappears at the end of allphysical calculations).The dipolar interaction operator U describes the pos-sibility for the atom to absorb or emit a photon changingat the same time its internal state | F m (cid:105) → | F e m e (cid:105) or | F e m e (cid:105) → | F m (cid:105) . As the incident light is nearly resonantwith the atomic transition, we only need to consider res-onant contributions where the atom in its ground statecan only absorb a photon, and the atom in its excitedstate can only emit a photon: U ( r ) ≈ D ( r ) · d † + D † ( r ) · d . (4)This is known as the rotating wave approximation.It has to be noticed that the photons considered in thisarticle are associated to the transverse electric field, i.e.to the electric displacement vector D and not to the elec-tric field E (hence our notation) [26]. This point is im-portant for two reasons: first, due to the magnetic field,light propagates in an anisotropic medium where D and E are no longer colinear and, second, former theoreticalstudies treated the propagation of E [22]. B. Scattering amplitude
Starting from the incident state | i (cid:105) = | k ω (cid:15) ; F m (cid:105) , de-scribing a photon | k ω (cid:15) (cid:105) impinging an atom located at r with initial ground state | F m (cid:105) , we consider the scatter-ing process leading to the final state | f (cid:105) = | k (cid:48) ω (cid:48) (cid:15) (cid:48) ; F m (cid:48) (cid:105) ,describing a scattered photon | k (cid:48) ω (cid:48) (cid:15) (cid:48) (cid:105) leaving the atomin the final ground state | F m (cid:48) (cid:105) . The probability ampli-tude for such a transition is the matrix element (cid:104) f | S | i (cid:105) of the scattering operator S = − iπT acting on theatom-photon Hilbert space H = H at ⊗ H L . The transi-tion operator for an atomic point-dipole scatterer writes T = 12 V | r (cid:105)(cid:104) r | ⊗ T (5)where T couples the photon polarization to the atomicinternal degrees of freedom. The (on-shell) matrix ele-ments of T are: (cid:104) f | T | i (cid:105) = 12 V e i ( k − k (cid:48) ) · r δ ( E (cid:48) − E ) T fi ( E ) (6)where the delta distribution ensures energy conserva-tion. For the initial and final states under consideration, E (cid:48) = ω (cid:48) + µgBm (cid:48) and E = ω + µgBm . It is important tonote at this point that, contrary to the case where thereis no magnetic field, scattering has now become inelastic ,a feature that will complicate greatly the analysis of themultiple scattering situation. Indeed as soon as m (cid:48) (cid:54) = m ,the atom experiences a net Zeeman energy change. Inturn, the angular frequency of the scattered photon isaccordingly modified to ω (cid:48) = ω + µgB ( m − m (cid:48) ). It shouldbe noted that we simplified here the problem further byneglecting momentum transfer to the atom during scat-tering. This approximation, supposedly valid when therecoil energy is negligible compared to the energy widthof the excited state, becomes nevertheless questionablewhen considering the multiple scattering regime as stud-ied (without magnetic field) in [18].The matrix element T fi ( E ) is calculated from T fi ( E ) = (cid:104) f | U G e ( E ) U | i (cid:105) where G e ( E ) = ˆ P e ( E − H ) − ˆ P e is theGreen’s function of the system when the atom is in itsexcited state. This operator can be computed by usingresolvent techniques [27] and we find: G e ( E ) = F e (cid:88) m e = − F e | F e m e (cid:105) (cid:104) F e m e | E − g e m e µB − ω + i Γ / , (7)where Γ is the angular width of the atomic excited statedue to coupling to vacuum fluctuations. Technicallyspeaking, there is also a modification of the atomic fre-quency (Lamb-shift) that we get rid off by a proper re-definition of ω . It is noteworthy that neither the Lamb-shift nor the linewidth depend on B . This is so becausethe Zeeman operators do not couple to the dipole inter-action U .Introducing the reduced atomic dipole operator ˆ d e = d e /d e with d e ω = 3 π(cid:15) c (cid:126) Γ [27], the internal transition matrix element then reads: T fi ( E ) = 6 πk Γ / δ + i Γ / (cid:15) (cid:48) · t m (cid:48) m · (cid:15) , (8)where the dyadic transition operator in polarizationspace t m (cid:48) m is: t m (cid:48) m = F e (cid:88) m e = − F e (cid:104) F m (cid:48) | ˆ d e | F e m e (cid:105) (cid:104) F e m e | ˆd e | F m (cid:105) − iφ ( gm − g e m e ) . (9)The dimensionless parameter φ = φ B − iδ/ Γ , (10)where φ B = 2 µB Γ , (11)quantifies the impact of B on the atomic scattering prop-erties and is generally complex-valued except at reso-nance ( δ = 0). One can note in passing that, when B = 0, t m (cid:48) m = (cid:104) F m (cid:48) | dd † | F m (cid:105) which describes the usual ab-sorption and emission cycle from the degenerate ground-state [28].
C. Scattering cross section
Probability conservation assures that, under the scat-tering process, any incident photon is either transmittedin the same mode or scattered in another mode. Thisis the essence of the optical theorem, which relates thetotal scattering cross section of the atom to the forwardscattering amplitude: σ = − V Im (cid:104) k ω (cid:15) | (cid:104) T (cid:105) int | k ω (cid:15) (cid:105) (12)where (cid:104)•(cid:105) int = Tr( • ρ at ) indicates an average over theinitial atomic internal degrees of freedom. As such2 V (cid:104) T (cid:105) int = | r (cid:105)(cid:104) r | ⊗ (cid:104)T (cid:105) int only acts on the photon de-grees on freedom. At this point, we make the impor-tant simplifying assumption that the initial atomic den-sity operator ρ at describes a complete incoherent mix-ture of the groundstate Zeeman sub-states. This is gen-erally the situation for cold atoms prepared in an opti-cally thick magneto-optical trap (MOT) where all Zee-man states are uniformly populated and no Zeeman co-herence is achieved. The initial atomic density operatorthen reads ρ at = ˆ P g / (2 F + 1). Using the fact that ˆ d e , (cid:15) and (cid:15) (cid:48) are irreducible tensors of rank 1, one can express (cid:15) (cid:48) · (cid:104) t (cid:105) int · (cid:15) in terms of the irreducible components of thetensor (cid:15) (cid:48) i (cid:15) j in the cartesian basis set ( ˆ x , ˆ y , ˆ z ). Introduc-ing the dyadic projector ∆ a = − aa /a onto the planeperpendicular to a , we find: (cid:104)T (cid:105) int = M F F e πk Γ / δ + i Γ / k (cid:48) T ( F F e ) ∆ k (13) T ( F F e ) = 32 F e + 1 F (cid:88) m = − F t mm , (14)where M F F e = (2 F e + 1) / (3(2 F + 1)) is a factor tak-ing care of the degeneracies of the ground and excitedstates. Both projectors ∆ k and ∆ k (cid:48) ensure that polar-ization vectors always remain transverse to the directionof propagation.Using the totally antisymmetric tensor of rank 3 ε ijk ( ε xyz = 1), the dyadic tensor T ( F F e ) reads: T ( F F e ) ij = ζδ ij + ηε ijk ˆ B k + ξ ˆ B i ˆ B j . (15)With the z-axis chosen along ˆ B , the matrix representing T ( F F e ) is: T ( F F e ) = ζ η − η ζ
00 0 ζ + ξ . (16)The ζ , η and ξ coefficients depend on φ and on theClebsch-Gordan coefficients of the atomic transition [29,30]: ζ = 12 (cid:18) F (cid:88) m = − F (cid:104) F F e − mm + 1 | (cid:105) − iφ (( g − g e ) m − g e )+ (cid:104) F F e − mm − | − (cid:105) − iφ (( g − g e ) m + g e ) (cid:19) η = − i (cid:18) F (cid:88) m = − F (cid:104) F F e − mm + 1 | (cid:105) − iφ (( g − g e ) m − g e ) − (cid:104) F F e − mm − | − (cid:105) − iφ (( g − g e ) m + g e ) (cid:19) ξ = − ζ + F (cid:88) m = − F (cid:104) F F e − mm | (cid:105) − iφ ( g − g e ) m (17)As is expected from Onsager’s reciprocity relations [31],one can check that ζ and ξ are even function of φ , while η is an odd function of φ . It can be checked that allthese three coefficients are real at resonance ( δ = 0) since φ = φ B is then real. In the case of a F = 0 → F e = 1transition ( g = 1), these coefficients read ζ = 11 + ( g e φ ) η = − g e φ g e φ ) (18) ξ = ( g e φ ) g e φ ) The dyadic tensor T ( F F e ) embodies the effect of themagnetic field on the photon polarization degrees offreedom and gives rise to the usual magneto-opticaleffects. The ζ term is responsible for normal extinc-tion (Lambert-Beer law). The η term describes themagnetically-induced rotation of the atomic dipolemoment (Hanle effect) [24, 32] and induces Faradayrotation and dichroism effects observed when k (cid:107) B [33].The ξ term is responsible for the Cotton-Mouton effect observed when k ⊥ B [33].From expression (13) and the optical theorem (12) wededuce the total scattering cross section of a photon byan atom initially prepared in an incoherent mixture ofZeeman internal ground states: σ ( φ ) = σ Re (cid:18) (1 + 2 i δ Γ ) ( (cid:15) · T ( F F e ) · (cid:15) ) (cid:19) (19)= σ Re (cid:18) (1 + 2 i δ Γ )( ζ + η ( (cid:15) × (cid:15) ) · ˆ B + ξ | (cid:15) · ˆ B | (cid:19) where σ = σ δ/ Γ) σ = M F F e πk . (20)In the case of a F = 0 → F e = 1 transition ( g = 1), andfor resonant light ( δ = 0), the scattering cross sectionboils down to the simpler form: σ ( φ ) = σ g e φ B ) | (cid:15) · ˆ B | g e φ B ) . (21)As one can see, in the presence of a magnetic field, thetotal scattering cross section depends explicitly on theincident polarization (cid:15) . More precisely, it depends onthe relative direction of (cid:15) with respect to B . We recoverhere the well-known fact that an external magnetic fieldinduces optical anisotropy in otherwise isotropic media.In the absence of a magnetic field, T ( F F e ) reduces to theidentity matrix and we get σ ( φ = 0) = σ , giving backthe result found in [28]. D. Impact of optical pumping
Our previous calculation in fact just considered thescattering of one quasi-resonant photon by one atom as-sumed to be initially prepared in an incoherent mixtureof Zeeman internal ground states. In a real experimenthowever, many quasi-resonant photons are shone. Fora given incident polarization, the repeated scattering ofphotons by the atom induces changes in the populationsof the groundstate Zeeman sublevels and creates coher-ences between them. This effect is known as opticalpumping and is enhanced in the presence of a magneticfield. Our previous calculations thus applies to the casewhere optical pumping can be neglected. This is the sit-uation considered in section V. One can note howeverthat our results can be easily extended to include opticalpumping. Indeed, the most general tensor of rank 2 thatcan be written with the components of B is still given byexpression (15) but with arbitrary coefficients now. Thusequations (13), (16) and (19) remain valid. Only the de-tailed expressions (17) of ζ , η and ξ will be modified. Inthe following, all our analytical results are expressed interms of these three coefficients. They thus remain validin the presence of optical pumping if the coefficients aregiven their appropriate expressions and values. E. Differential scattering cross section
We now turn to the impact of B on the radiation pat-tern of the atom. The intensity of light scattered in thedirection k (cid:48) with polarization (cid:15) (cid:48) while the atom changesits internal state from | F m (cid:105) to | F m (cid:48) (cid:105) is proportional tothe differential scattering cross section dσ m (cid:48) m d Ω ( k(cid:15) → k (cid:48) (cid:15) (cid:48) ) = V ω (2 π ) | (cid:104) k (cid:48) (cid:15) (cid:48) , F m (cid:48) | T | k(cid:15) , F m (cid:105) | = 9 σ π F + 12 F e + 1 | (cid:15) (cid:48) · t m (cid:48) m · (cid:15) | (22)Starting with an atom prepared in state | F m (cid:105) the differ-ential cross-section for a photon to be scattered in mode | k (cid:48) (cid:15) (cid:48) (cid:105) is thus: dσ m d Ω ( k(cid:15) → k (cid:48) (cid:15) (cid:48) ) = F (cid:88) m (cid:48) = − F dσ m (cid:48) m d Ω ( k(cid:15) → k (cid:48) (cid:15) (cid:48) ) (23)The total photon scattering cross-section is then obtainedby averaging over the initial internal atomic density ma-trix, which we assumed to describe a fully incoherentmixture of groundstate Zeeman sublevels. We thus ar-rive at: dσd Ω ( k(cid:15) → k (cid:48) (cid:15) (cid:48) ) = 12 F + 1 F (cid:88) m,m (cid:48) = − F dσ m (cid:48) m d Ω ( k(cid:15) → k (cid:48) (cid:15) (cid:48) )= 9 σ π F (cid:88) m,m (cid:48) = − F | (cid:15) (cid:48) · t m (cid:48) m · (cid:15) | F e + 1 . (24)When the ground state is non-degenerate ( F = 0, F e =1, g = 1), the tensors t and T (01) coincide and thedifferential cross section takes the simple form: dσd Ω ( k(cid:15) → k (cid:48) (cid:15) (cid:48) ) = 3 σ π | (cid:15) (cid:48) · T (01) · (cid:15) | , (25)with (cid:15) (cid:48) ·T (01) · (cid:15) = ζ (cid:15) (cid:48) · (cid:15) + η ( (cid:15) (cid:48) × (cid:15) ) · ˆ B + ξ ( (cid:15) (cid:48) · ˆ B )( (cid:15) · ˆ B ) , (26)the coefficients ζ , η and ξ being given by (18).Since the resonant denominator in (9) explicitlydepends on the magnetic quantum numbers m and m e ,standard irreducible tensor methods [28] are of littlepractical use to boil down the total differential cross-section (24) to a much simpler form. However it can beeasily and efficiently computed for any transition lineby using the symbolic calculus software Maple TM . As ahumorous note, the reader is invited to appreciate thepower of the optical theorem by deriving the total scat-tering cross section (19) by direct computation from (24). III. COHERENT PROPAGATION OF LIGHT
The propagation of an incident light field mode | k(cid:15) (cid:105) ina scattering medium, also known as the coherent propa-gation, is efficiently described by replacing the scattering medium by a homogeneous effective medium having acomplex refractive index tensor N r . When B = 0, N r isa scalar and bears no action on the incident polarization.Its imaginary part gives rise to an exponential attenua-tion (Lambert-Beer law), its characteristic length scalebeing known as the extinction length. Since true ab-sorption (i. e. conversion of electromagnetic energy intoanother form of energy) is absent in our case, depletionof the incident mode can only occur through scattering.The extinction length thus identifies with the scatteringmean free path (cid:96) and one has Im( N r ) = 1 / (2 k(cid:96) ).The presence of a magnetic field B modifies the inter-action between light and atoms and the coherent prop-agation of light in the atomic cloud will be accord-ingly altered. As B induces a preferential orientationof space, it will set an optical anisotropy in the atomicgas. In this case, the refractive index tensor is no longera scalar and will act on the polarization space: the co-herent propagation of light will exhibit birefringence andmagneto-optical effects. These magneto-optical effectsare well documented in the literature [21–23]: for a givenwavevector k , there are two eigen-polarization modespropagating in (possibly) different directions, with dif-ferent velocities and attenuations.In this paragraph, we extend the techniques developedin [22] and [34] to the case of atoms with a degenerateground state. We will always assume that the atomic gasis dilute. In this case, nλ (cid:28) n being the number den-sity), and (cid:96) = 1 / ( nσ ). Furthermore, since for resonantscatterers σ is at most of the order of λ , (cid:96) will be al-ways much larger than the average interatomic distance n − / . As a whole, for point-dipole resonant scatterers,the dilute medium condition nλ (cid:28) k(cid:96) (cid:29)
1. The properties of the effective mediumwill then be directly related to the individual scatteringproperties of the atoms under the magnetic field.
A. Average Green’s function for light propagation
The first step to find the refractive index tensor N r anddescribe the coherent propagation of light is to determinethe Green’s function for light once all atomic degrees offreedom have been averaged out.Let N be the total number of atoms in the gas andtheir respective positions be labeled by r i ( i = 1 , · · · N ).The Hamiltonian of the system { atoms + light } is H = N (cid:88) i =1 H ( r i ) + N (cid:88) i =1 U ( r i )= H + U , (27)where H and U are given by expressions (1) and (4). TheGreen’s functions of the whole system G ( z ) = ( z − H ) − and of the uncoupled system G ( z ) = ( z − H ) − satisfythe recursive equation G ( z ) = G ( z ) + G ( z ) U G ( z )= G ( z ) + G ( z ) U G ( z ) + . . . (28)Since we are interested in the situation where all atomsstart and end in their groundstate, we merely look forthe Green’s function projected onto the atomic ground-state manifold ˆ P g G ( z ) ˆ P g . In this case, only expansionterms containing U an even number of times can con-tribute. The average over the atomic degrees of free-dom will generate a Dyson equation for the averageGreen’s function G ( z ) = (cid:104) ˆ P g G ( z ) ˆ P g (cid:105) . Under the di-lute medium assumption, and since all atoms are iden-tical and uniformly distributed in space (at least on thescale of the scattering mean free path), the correspond-ing self-energy is given by Σ( z ) = N (cid:104) T i ( z ) (cid:105) int where T i ( z ) = U ( r i ) + U ( r i ) G ( z ) U ( r i ) + . . . is the transitionoperator of the i -th atom. The average over the internaldegrees of freedom is given by (13).The matrix elements of the average photon Green’sfunction are then given by: (cid:104) k (cid:48) (cid:15) (cid:48) |G ( ω ) | k(cid:15) (cid:105) = δ kk (cid:48) (cid:15) (cid:48) · G ( k , ω ) · (cid:15) (29) G ( k , ω ) = ∆ k ω − k − Σ( k, ˆ k ) ∆ k (30)Σ( k, ˆ k ) = 12 (cid:96) Γ / δ + i Γ / k T ( F F e ) ∆ k (31)where (cid:96) = 1 / ( nσ ). The Kronecker symbol δ kk (cid:48) fea-tures the restoration of translation invariance under thespatial average. The self-energy tensor Σ( k, ˆ k ) containsall the information on the effective medium. It has anexplicit dependence on the incident direction because, inthe presence of the magnetic field, the scattering mediumdevelops an optical anisotropy (see section II C). B. Optical anisotropy
The equation (30) receives a simple interpretation ina basis where Σ( k, ˆ k ) is diagonal. It corresponds to po-larization modes which propagate in direction ˆ k withoutdeformation. The poles of G ( k , ω ) then give the corre-sponding dispersion relation for the eigenmode.Trivially, k is an eigenvector of Σ( k, ˆ k ) with eigenvalue0. This is a consequence of the transversality of light,and this eigenvector is not physically relevant. To findthe other complex eigenmodes and complex eigenvalues,Σ( k, ˆ k ) ˆ V ± = Λ ± ˆ V ± , we parametrize ˆ k by its sphericalangles θ and ϕ in a coordinate frame with z-axis parallelto ˆ B . We find:Λ ± (ˆ k ) = ζ + ξ sin θ ± (cid:115) − η cos θ + ξ sin θ θ between ˆ B andˆ k expresses the optical anisotropy of the atomic cloudinduced by the magnetic field. The corresponding eigen- modes are:ˆ V ± (ˆ k ) ∝ (cid:16) η cos θ cos ϕ + ξ sin θ ϕ ∓ sin ϕ (cid:115) − η cos θ + ξ sin θ (cid:17) ˆ x + (cid:16) η cos θ sin ϕ − ξ sin θ ϕ ± cos ϕ (cid:115) − η cos θ + ξ sin θ (cid:17) ˆ y − η cos θ sin θ ˆ z (33)To allow a relative ease of reading, these vectors have notbeen normalized.One difficulty arises when ( − η cos θ + ξ (sin θ ) /
4) =0 for non-vanishing η and ξ . This situation can only hap-pen if η and ξ are real, thus for φ real, i. e. at resonance( δ = 0). The solutions are of the form ( ± θ , π ± θ ),which means, because of the invariance around the z-axis, that the photon has to propagate along a cone ofapex θ . When this is the case, Λ + = Λ − but ˆ V + = ˆ V − ,and the self-energy is non-diagonalizable. However thissituation is largely unphysical in the sense that it arisesfrom the first-order approximation in the atomic densityused to compute Σ( k, ˆ k ). At next order in density, thisdifficulty disappears. However, this could lead to inter-sting effects for the propagation of light near the apexangle. We chose to neglect such effects in the following.Noticeably, ˆ V + and ˆ V − are not orthogonal vectors ingeneral. Indeed, scattering depletes the coherent modeand its energy decays. This is reflected by the fact thatthe self-energy is not a hermitian operator. Thus itseigenvectors have no reason to be orthogonal. When B = 0, T ( F F e ) reduces to the identity and Σ( k, ˆ k ) is thenproportional to the projector ∆ k . In this case alone, allpolarization states orthogonal to k are eigenmodes andit is then possible to choose an orthogonal basis. Onecan however note that ˆ V + and ˆ V − are nearly orthogo-nal when | η | (cid:28) | ξ | or when | η | (cid:29) | ξ | . This happens inthe limit of small or strong magnetic fields ( µB (cid:28) δ, Γ, µB (cid:29) δ, Γ), or at a very large detuning ( δ (cid:29) µB, Γ).
C. Refraction index
In the polarization eigenbasis, the poles of the Green’sfunction (30) give the dispersion relation for ˆ V ± . Wefind: ω ± ( k ) = k + 12 (cid:96) Γ / δ + i Γ / ± (ˆ k ) (34)The refraction index tensor N r is diagonal in the polar-ization eigenbasis and the polarization vectors ˆ V ± prop-agate each with different complex refractive indexes: N ± r ( k ) = kω ± ( k ) ≈ − k(cid:96) Γ / δ + i Γ / ± (ˆ k ) (35)since k(cid:96) (cid:29)
1. As a consequence, the two eigen-polarizations propagate with different phase velocitiesand experience different attenuations (dichroism). Inturn, the effective medium acts as an absorbing polar-ization filter for the incoming light.The index mismatch between the two eigen-polarizations is∆ N r = N + r − N − r = i2 k(cid:96) − δ/ Γ (Λ + − Λ − ) (36)= i2 k(cid:96) − δ/ Γ (cid:115) − η cos θ + ξ sin θ θ = θ , i. e. when the two eigen-polarizations collapse onto each other, rendering the re-fractive index tensor no longer diagonalizable. D. Group velocity and birefringence
Let us consider a polarized monochromatic wavepacket propagating in the atomic cloud with a centralwavevector k , and let us assume the polarization is one ofthe vectors ˆ V ± . Then the maximum of the wave packetpropagates with the group velocity v g = Re( ∇ k ω ( k )).As ω ( k ) depends on the angle θ between B and k , v g pos-sesses a component orthogonal to k : in general, the wavepacket does not propagate parallel to the wavevector.Since each polarization eigen-mode has its own directionof propagation, a birefringence effect takes place, as iswell known in anisotropic media [35]. This magnetically-induced birefringence has already been observed [36],though under conditions differing from the ones describedin the present article.We have checked that, provided | δ | is not much largerthan µB , the deviation of the wave packet from the direc-tion of k is negligible: the walk-off angle remains smallerthan 0 . / ( k(cid:96) ) [37]. Birefringence effects will thus beneglected in the following. E. Propagation in real space
To study the propagation properties of the coherentmode in the atomic cloud, we need the Green’s functionof light in real space, G ω ( r ). It is the Fourier transformof G ( k , ω ) (30): G ω ( r ) = (cid:90) d k (2 π ) ∆ k e i k · r ω − k − Σ( k, ˆ k ) ∆ k (38)Neglecting birefringence effects, the angular part of theintegral can be calculated with a stationary phase ap-proximation around the direction ˆ r : G ω ( r ) = 12 π r ∆ r I ω ( r ) ∆ r (39) I ω ( r ) = (cid:90) ∞ k sin( kr ) dkω − k − Γ / (cid:96) res ( δ + i Γ / ∆ r T ( F F e ) ∆ r (40) The I ω ( r ) integral has an ultra-violet divergence ( k →∞ ) which needs to be regularized. It expresses that theinteraction between two atoms separated by less than oneoptical wavelength cannot be reduced to the exchange ofone resonant photon. The divergent part of the integralcomes from the contact term of the radiated field [26]. Asthe average interatomic distance is much larger than theoptical wavelength, we are interested only in the far-fieldcomponent. The latter can be obtained from the regularpart of (39) and calculated with the help of the residuetheorem. The final result reads: G ω ( r ) = − ω πr ∆ r e ikrN r ∆ r (41)featuring the refractive index tensor N r ( k, ˆ r ) = − ∆ r Σ( k, ˆ r ) k ∆ r . (42)In the presence of a magnetic field, N r is represented by amatrix with an antisymmetric part proportional to η , ascan be seen from (31) and (16). This implies that G ω ( r )is not a symmetric operator: the transposition operationis equivalent to flip the sign of η , or equivalently to flipthe sign of B : G ω ( r , B ) = t G ω ( r , − B ) (43)This property is closely related to the reciprocity theorem[38]. As a simple illustration, assume a light beam isgoing successively through a linear polarizer (cid:15) , the atomiccloud and a linear analyzer (cid:15) (cid:48) . The amplitude of thetransmitted light is then proportional to A dir = (cid:15) (cid:48) · G ω · (cid:15) .If we now consider the reverse situation where the lightbeam is traveling through the system along the oppositedirection, its transmitted amplitude will now be A rev = (cid:15) · G ω · (cid:15) (cid:48) = (cid:15) (cid:48) · t G ω · (cid:15) . We see that A dir = A rev ifand only if G ω ( r ) is symmetric, which is not the case inthe presence of the magnetic field. This is in essence theunderlying principle behind optical isolators (or optical”diodes”), which are devices realizing A rev = 0 while A dir (cid:54) = 0. F. Magneto-optical effects
The impact of B on the coherent propagation of lightis embodied in the refractive index tensor N r (42). Asalready mentioned in III C, it concerns a differential de-phasing and attenuation of the eigenmodes of propaga-tion. These magneto-optical effects have been extensivelystudied when light propagates parallel (Faraday effect) orperpendicular to (Cotton-Mouton effect) B . We presentbelow both effects in terms of our formalism, giving sub-stance to the physical interpretation of the parameters ζ , η and ξ . B (a)B V + V - (b) V + V - ! ! ’ ! ! ’ FIG. 1. (a) Resonant Faraday effect ( δ = 0). A linearly-polarized light beam propagates parallel to B . The polariza-tion eigenmodes are the circular polarizations. In the courseof propagation the polarization of the light beam keeps lin-ear but rotates around B (b) Resonant Cotton-Mouton effect( δ = 0). A linearly-polarized light beam propagates perpen-dicular to B . The polarization eigenmodes are linear, onebeing parallel to B and the other one being perpendicular to B . In the course of propagation the polarization of the lightbeam keeps linear but rotates around the propagation axis.
1. Faraday effect
The Faraday effect occurs when a light beam with a lin-ear polarization propagates along B in the atomic cloud.For sake of simplicity, we consider here that the frequencyof light is exactly at resonance with the atomic transition( δ = 0) so that φ = φ B , ζ , η and ξ are all real in the fol-lowing discussion. From (32) and (33), the eigenmodesand their associated eigenvalues are:ˆ V ± = ( ˆ x ± i | η | η ˆ y ) Λ ± = ζ ± i | η | (44)The eigenmodes thus identify with the left and rightcircular polarization vectors (see Fig.1a). Their indexmismatch ∆ N r = −| η | / ( k(cid:96) ) is real, meaning that thetwo eigenmodes develop a phaseshift in the course ofpropagation. An elementary calculation shows that thepolarization of the traveling beam remains linear but ro-tates around ˆ B by an angle Θ = ηL/ (2 (cid:96) ) proportional to the traveled distance L . Hence, the parameter η describesthe Faraday effect. As one can also see, the parameter ζ plays the same role for the two eigenpolarizations : itacts as an isotropic refractive index.At low magnetic fields ( µB (cid:28) Γ), η is proportional to B , and so is the rotation angle Θ. The proportionalityconstant between this angle and the product BL is knownas the Verdet constant V B . For the F = 3 → F e = 4transition of Rb, one finds V B = − µ Γ (cid:96) ∼ − . rad/(Tm) (45)with (cid:96) ∼ µ m. This value is three orders of magnitudelarger than those of classical materials, and comparableto the value measured in [23].
2. Cotton-Mouton or Voigt effect
This effect describes the modification of the polariza-tion of a light beam propagating perpendicularly to B .As for the Faraday effect, we assume δ = 0 to simplifythe discussion. The eigenmodes and their eigenvaluesnow read:ˆ V + = ˆ z Λ + = ζ + ξ ˆ V − = ˆ y Λ − = ζ (46)The eigenmodes are now the linear polarization vectors(see Fig.1b). Their index mismatch ∆ N r = i ξ/ (2 k(cid:96) ) isnow purely imaginary, showing that ˆ V + gets more atten-uated than ˆ V − by a factor e − ξL/ (2 (cid:96) ) after traveling thedistance L . Here again, the parameter ζ plays the samerole for the two propagation eigenmodes and acts as anisotropic refractive index.If the incident polarization is linear, it can be writtenas a real linear combination of ˆ V + and ˆ V − . During thepropagation, the component along ˆ V + , i.e. along ˆ B , willdecrease more than the one along ˆ V − , i. e. perpendicularto ˆ B . As a whole, the polarization of the light beamremains linear, but rotates around ˆ k by an angle whichdepends on ξ , which is thus the parameter describing theCotton-Mouton effect.In ref. [35], the Cotton-Mouton effect is rather de-scribed as the transformation of an incident linear po-larization into an elliptical one in the course of propa-gation, this transformation being a consequence of theaccumulated phase shift between the two propagatingeigenmodes. This apparent contradiction can be lifted ifone realizes that, contrary to our discussion, the Cotton-Mouton effect described in [35] is in fact the one usuallyobserved for light frequencies which are far-detuned fromany resonance frequencies. Work out our theory at a verylarge detuning δ , we indeed recover the description givenin [35].
3. General case
When the direction of propagation ˆ k is neither alongnor perpendicular to ˆ B , both effects mix. The Faradayeffect will dominate when ˆ k is roughly parallel or anti-parallel to ˆ B , i. e. essentially when ˆ k is well inside thecone with apex angle θ for which (37) vanishes. On thecontrary, the Cotton-Mouton effect dominates when ˆ k isessentially well outside this cone, i. e. roughly perpen-dicular to ˆ B . To give orders of magnitude, η and ξ arecomparable when φ (cid:39)
1, which corresponds to B (cid:39) Rb atoms. The apex angle is then θ ≈ ◦ . TheCotton-Mouton effect then dominates for directions ofpropagation making an angle between 65 ◦ and 90 ◦ withˆ B . In classical media, this would happen only in a nar-row angular width of order 10 − rad around the direc-tion orthogonal to ˆ B . This big difference in orders ofmagnitude is due to the strong resonant character of theatoms. All in all, the giant Faraday effect and the largezone of preponderance of the Cotton-Mouton effect makeit necessary to take both effects into full account whenstudying the coherent propagation of light.When δ (cid:54) = 0, ζ , η and ξ are complex valued. Thenthe index mismatch is neither purely imaginary nor real.The two polarization eigenmodes still experience differentphase shifts and different extinctions but the calculationsand physical pictures lack the previous simplicity. IV. THE CBS EFFECTA. Independent scattering approximation
At low optical density nλ (cid:28)
1, a semi-classical de-scription of propagation along scattering paths consist-ing of rays between consecutive scatterers is justified.For resonant scatterers, it implies k(cid:96) (cid:29)
1. As a con-sequence, scattering paths involving different scatterersare uncorrelated and the associated interference averagesto zero. Recurrent scattering sequences (visiting a givenscatterer more than once) can also be neglected definingthe independent scattering approximation (ISA) [34]. Inthis regime, the wave amplitude A is constructed as thecoherent superposition A = (cid:80) P A P of the partial wavesscattered along all quasi-classical scattering paths P join-ing the positions of the scatterers. Between two succes-sive scatterers the partial waves experiences the effectivemedium. In the ISA regime, the scattering amplitude A P is thus computed using two building blocks, the scatter-ing by an individual atom and the coherent propagation. B. Multiple scattering and interference
The average intensity of the wave I = (cid:104)| (cid:80) P A P | (cid:105) breaks into an incoherent contribution I i = (cid:80) P (cid:104)| A P | (cid:105) and a coherent contribution I c = 2Re( (cid:80) P , P (cid:48) (cid:104) A P (cid:48) A P (cid:105) ). The incoherent contribution itself breaks into the sum ofa single scattering contribution I s and a diffuse one, I d ,involving scattering paths containing more than 2 scat-terers. All these contributions depend on the polarizationof the incoming light and on the detected polarization ofthe outgoing light.As is well known, the disorder average does not scram-ble two-wave interference effects between scattering loopstraveled in opposite directions [3, 39, 40]. This is at thecore of the CBS effect where interference between ampli-tudes associated to reverse scattering paths P and (cid:101) P (i.e. paths with the same sequence of scatterers but trav-eled in opposite order) contribute a constructive interfer-ence in a narrow angular cone around the backscatteringdirection [8, 34, 40]. C. Backscattered intensity and the CBS contrast
In this section, we compute the amplitude of multiplescattering paths and the backscattered intensity. We usethe contrast of the interferences to determine the degreeof coherence of light, which we express in term of a phasecoherence length.Denoting by ϑ the angle between k and the outgoingwave vector k (cid:48) , the total average backscattered signal is I ( ϑ ) = I s ( ϑ ) + I d ( ϑ ) + I c ( ϑ ) where: I c ( ϑ ) = 2 (cid:88) P≥ Re (cid:68) a P a (cid:101) P e i ( k + k (cid:48) ) · R P (cid:69) . (47)Here P ≥ R p being thevector joining the endpoints of path P . The CBS signal I c varies on a very small angular scale ∼ / ( k(cid:96) ) (cid:28) I s and I d follows theLambert’s law and takes place over an angular range oforder 1 radian. The incoherent contributions appear tobe constant at the angular scale of the CBS cone and canbe safely evaluated at ϑ = 0.As a two-wave interference, the CBS signal gives accessto the degree of coherence of the outgoing wave and, inturn, at the coherence length of the scattering medium.The interference contrast is quantified by the CBS en-hancement factor α = 1+ I c / ( I s + I d ) computed at ϑ = 0.which is the ratio of the total intensity at exact backscat-tering to the total intensity out of the backscatteringcone. As single scattering events do not participate to theinterference process, they decrease the contrast even if nodephasing mechanism is at work. When I s can be madeto vanish or negligible, the coherence loss is directly asso-ciated to the ratio I c /I d . For classical scatterers, I s = 0in the helicity-preserving polarization channel and reci-procity arguments show that I c = I d and α = 2 in theabsence of a magnetic field [13, 28, 34]. When a magneticfield is present, I c < I d and the coherence length of themedium becomes finite [20–22]. The situation for atomswith nonzero spin in the ground state proves more subtleand will be addressed in the next Section.0 D. The intricacies of scattering under a magneticfield
Consider a scattering path P containing s atoms, lo-cated at r i , whose initial and final magnetic numbersare m i and m (cid:48) i ( i = 1 , ..., s ). The incoming light angu-lar frequency is ω and its polarization vector (cid:15) . Alongpath P , light propagates to the first atom, is scattered,propagates to the second atom and so on. After the s -thscattering event, light exits the medium and is detectedin the polarization channel (cid:15) (cid:48) . One crucial point is thatthe internal Zeeman state of an atom with a degenerateground state can change under scattering, which means,when a magnetic field is present, that the scattered pho-ton can have a different frequency that the incoming one.In other words, single scattering under a magnetic fieldis inelastic , the frequency change being of the order of µB . When µB is larger than or comparable to δ and/orΓ, this effect is not negligible: the effective medium andthe magneto-optical effects felt by the photon depend onits frequency.If ω (cid:48) denotes the outgoing frequency, the amplitudeassociated to path P reads: A P = (cid:15) (cid:48) · G ω (cid:48) ( r (cid:48) − r s ) t m (cid:48) s m s G ω s − ,s ( r s − r s − ) . . . G ω , ( r − r ) t m (cid:48) m G ω ( r − r ) · (cid:15) (48)where ω i,i +1 = ω + gµB i (cid:88) a =1 ( m a − m (cid:48) a ) (49)is the frequency of light between the i -th and the ( i + 1)-th scatterers. The tensor t m (cid:48) m has been defined in (9).For the reverse path (cid:101) P , the incoming angular frequencyis still ω and, by energy conservation, the outgoing an-gular frequency is still ω (cid:48) . The amplitude associated to (cid:101) P reads: A (cid:101) P = (cid:15) (cid:48) · G ω (cid:48) ( r − r ) t m (cid:48) m G ω , ( r − r ) . . .G ω s,s − ( r s − − r s ) t m (cid:48) s m s G ω ( r s − r (cid:48) ) · (cid:15) (50)with ω i +1 ,i = ω + gµ B B p (cid:88) a = i +1 ( m a − m (cid:48) a ) (51)The angular frequency of light traveling between atoms i and ( i +1) is ω i,i +1 for path P and ω i +1 ,i for path (cid:101) P . Ingeneral, they differ and satisfy ω i,i +1 + ω i +1 ,i = ω + ω (cid:48) .As the magnetic field introduces an explicit differencebetween A P and A (cid:101) P , the interference contrast will bereduced unless the change of frequency does not occur oris unlikely. This happens for example when the atom doesnot change its internal state under scattering ( m i = m (cid:48) i ).The conditions for this situation will be examined in thenext Section.As a consequence of this frequency change, the averageover the internal degrees of freedom involves the whole scattering path. Indeed, because of the magneto-opticaleffects and the frequency change under scattering, theatomic internal and external degrees of freedom are intri-cated in a complicated way: after a scattering event takesplace, the location of the next one depends on the valueof the scattering mean free path, hence on the frequencyof the emitted photon, hence on the change or not of in-ternal state. In such a situation, I s , I d and I c can only becomputed numerically. In practice one calculates the di-mensionless bistatic coefficients γ x = 4 π V ω I x / (4 π A )( x = s, d, c ), where A is the illuminated area. E. Monte-Carlo simulation
Analytical results about the properties of the CBS conecan be obtained only in specific cases. In the absence ofa magnetic field, the problem has been exactly solved forvector waves in a random medium of Rayleigh scattererswith a uniform density and a slab geometry [41]. Fol-lowing the same lines, the solution has been extended toquasi-resonant atomic scatterers with degenerate groundstates [42]. For other geometries, numerical calculationsare necessary [43].In the presence of a magnetic field, reference [22] con-tains a generalization of the analytical methods devel-oped in [41] for Rayleigh scatterers but unfortunatelyfails to describe some aspects of the experimental resultsreported in [20]. This has to be related to the approxi-mations done to compute the CBS cone, and in fine tothe complexity of the exact calculation. For atomic scat-terers, the intrication between the external and internaldegrees of freedom makes it almost impossible.The average intensity (cid:104)| A ( s ) P | (cid:105) contributed by ascattering path P with s scatterers contains an averageover the positions of the scatterers, i.e. a 3 s -upleintegral, and an average over the internal degrees offreedom, i.e. a sum over the final and initial Zeemansub-levels of each scatterer in the ground state. Tocompute this multiple integral and these sums, we usea Monte-Carlo simulation able to extract at the sametime A ( s ) P and A ( s ) (cid:101) P for a large number of paths. Thisnumerical simulation allows us to take into account someexperimental constraints, such as the shape and thedensity profile of the atomic cloud or the finite spectralwidth of the laser probe. Its principle is as follows: Step 1.
A photon with frequency ω , wave vector k = k ˆ u and polarization (cid:15) enters the atomic cloud. It propagateson a distance r along ˆ u chosen according to the proba-bility distribution P ( r ) = 1 (cid:96) ( r ) exp( − (cid:90) r dr (cid:48) (cid:96) ( r (cid:48) ) ) . (52)Here (cid:96) − ( r ) = n ( r ) σ ( φ ) is the inverse local scatteringmean free path. It depends on the position of the photonif the atomic number density n is non-uniform. The1propagator is the Green’s function in the real space (41). Step 2.
The photon at position r ˆ u from the entrancepoint in the cloud is scattered by an atom. The atomicinitial and final states | F m (cid:105) and | F m (cid:48) (cid:105) are chosenrandomly with a uniform probability distribution. Thescattering operator is t m (cid:48) m , eq.(9). It transforms theincident polarization into the scattered polarization. Step 3.
The photon angular frequency is changed by gµB ( m − m (cid:48) ). Step 4.
A contribution to the single scattering bistaticcoefficient γ s is computed: the scattered photon ispropagated along the backscattering direction ϑ until itexits the atomic gas yielding the amplitude from whicha contribution to γ s is obtained. Step 5.
The scattered photon is propagated towards asecond scatterer. The direction of propagation is chosenaccording to an isotropic probability distribution tosave computation time. The propagation distance iscomputed with the help of the distribution law eq.(52),the change of frequency and polarization being takeninto account.
Step 6.
The doubly scattered photon is propagatedalong the backscattering direction ϑ until it exits theatomic gas yielding the amplitude from which a contri-bution to γ (2) d is obtained. Step 7.
A photon, identical to the incident one, entersthe atomic cloud, propagates along the previous doublescattering path in reverse order, and exits the medium inthe backscattering direction ϑ . The scatterers experienceexactly the same atomic transitions. This yields theamplitude associated to the reverse previous doublescattering path from which, together with Step 6, acontribution to γ (2) c is obtained. Step 8.
The process is continued (triple scattering, etc)until the photon finally exits the atomic cloud.
Step 9.
Another incident photon is sent in the cloudand the whole process is repeated as many times as nec-essary to obtain a good signal-to-noise ratio. Typically,one needs to launch between 10 and 10 photons toobtain well converged values for γ s , γ d = (cid:80) s γ ( s ) d and γ c = (cid:80) s γ ( s ) c .Up to the statistical errors, this method is quasi-exactand limited only by computer resources in the limit k(cid:96) (cid:29)
1. When the magnetic field vanishes, the results of theMonte-Carlo simulations reported in [43] are recovered.At large magnetic fields, the modulus of the amplitudesassociated to reverse paths are very sensitive to the scat-tering parameters of the paths. Any change in a directionof propagation modifies significantly the refractive indexof the effective medium. As a consequence, the Monte-Carlo simulation needs to average over more and morefluctuating quantities when the magnetic field increases.The statistical error on the total diffuse intensity can beestimated by its standard deviation. It remains smallerthan 1% for small magnetic fields ( µB/ Γ < µB/ Γ (cid:39) V. RESTORATION OF THE CBS CONTRAST
In the following, we apply the results of the previousSections to compute the CBS cone for quasi-resonantlight propagating in a cold Rb cloud. The frequencyof light is chosen close to the frequency of the F = 3 → F e = 4 transition of the D2 line (wavelength λ = 780nm,linewidth Γ / (2 π ) = 5 . g = 1 / g e = 1 / µB = Γcorresponds to B = 4 . B = 0, CBS experiments have reported very low en-hancement factors, e. g. α ≈ .
05 in the helicity preserv-ing channel [44, 45]. This is in marked contrast with ex-periments with spherically-symmetric classical scattererswhere reciprocity guarantees α takes its maximal value 2in the same polarization channel [46]. A detailed analy-sis shows that the low α value observed with cold atomscomes from an imbalance between the amplitudes asso-ciated to reverse paths [13]. This imbalance is noticeablyabsent for a F = 0 → F e = 1 transition where α = 2 isrecovered [47]. It is our goal in this section to show thatthe interference contrast can be fully restored with thehelp of an external magnetic field. A. Filtering out a closed transition
The key idea to restore the CBS contrast in thehelicity-preserving channel is simply to lift the degener-acy of the atomic ground state and to filter out a closedtransition. This is done by splitting the Zeeman sub-levels with an external magnetic field (Zeeman effect)and by shining the atomic cloud with a light wave whichis resonant with the | F = 3 , m = 3 (cid:105) → | F e = 4 , m e = 4 (cid:105) transition. To achieve this, one needs to impose δ =(4 g e − g ) µB = µB . This transition is closed since anatom in the excited state | (cid:105) can only make a transitionto the ground state | (cid:105) . At sufficiently large B , the otherZeeman sub-levels of the ground and excited states aresufficiently split away and are out of resonance, meaningthat the | (cid:105) → | (cid:105) transition is isolated. Thus, at large B , the atomic cloud consists of (i) atoms which are inthe sub-state | (cid:105) and can scatter light, and (ii) atomswhich are not in the sub-state | (cid:105) and cannot scatterlight because the frequency is too far-detuned from theother transitions. These | (cid:105) -scatterers behave like effec-tive two-level atoms which can only absorb and emit σ + radiation, i. e. light with positive helicity along ˆ B .Under these circumstances, it makes no difference forlight to travel a scattering path in one direction or theother. The multiple scattering amplitudes associated toany path P and to its reverse partner (cid:101) P are equal andthe CBS contrast is restored. This restoration is expected2to be most spectacular in the helicity-preserving channel,because it is in this channel that the contrast is the lowestwithout any magnetic field. If the incident light beamis parallel to B and B is sufficiently large, it is easyto see that one only gets a non-vanishing CBS signalin the helicity non-preserving polarization channel. Wewill thus choose in the following the ”Cotton-Mouton”configuration where B is perpendicular to the incidentlight beam and analyze the CBS signal in the helicity-preserving polarization channel. B. Small magnetic fields
From the previous discussion, it seems that the con-trast restoration only occurs at sufficiently large B . Infact, it turns out that the contrast restoration even startsat small magnetic fields and gets larger as B is increased.To demonstrate this, we study analytically the singleand double scattering signal in a uniform, semi-infinitemedium in the limit µB (cid:28) Γ (meaning B (cid:28) . Rb). It is then possible to neglect the magneto-opticaleffects, and to propagate photons with the propagator(41) evaluated at B = 0. This will be justified be-low in Section V D. Expanding the scattering matrices t m (cid:48) m at second order in φ B = 2 µB/ Γ, and noticing that φ ≈ φ B + iφ B at same order, we then use expressions(48) and (50) to compute the single and double scatter-ing amplitudes amplitudes. In the chosen geometry, theaverage over the external degrees of freedom when com-puting γ s , γ (2) d and γ (2) c can be done analytically [28].The calculations have been made here with the symboliccalculation software Maple TM and yields: γ s /γ (2) d = 0 .
305 + 0 . φ B (53) γ (2) c /γ (2) d = 0 .
217 + 0 . φ B (54) α (2) = 1 .
166 + 0 . φ B (55)As one can see, both quantities increase with B , meaningthat the coherence length of the system is increased. C. Monte-Carlo simulations
When the magnetic field is neither small nor large,there is no simple approximation that allows to computeanalytically the bistatic coefficients, but they can at leastbe computed numerically with the help of the Monte-Carlo simulation described in Section IV E. It is thenpossible to take into account a more realistic model of theatomic cloud than a semi-infinite uniform medium. Wepresent results here for a spherically-symmetric atomiccloud with gaussian density and optical thickness b = 31(measured at B = 0 and δ = 0). We take a laser probebeam with spectral width equal to 0 . α are FIG. 2. Plot of the CBS enhancement factor α as measuredin the parallel helicity channel h (cid:107) h for different values of B (circles) for light backscattered by a cold Rb atomic cloudin the Cotton-Mouton configuration k ⊥ B . For each valueof B , the light is tuned on resonance with the | (cid:105) → | (cid:105) transition ( δ = µB ). The spherically-symmetric atomic cloudis characterized by a gaussian density and an optical thick-ness b = 31 when δ = 0 and B = 0. One witnesses a dramatic increase of the CBS contrast compared to the situtation at B = 0 despite the fact that the time-reversal symmetry isbroken. The solid line is the result of the Monte-Carlo simu-lation with no adjustable parameters. compared to the experimental ones for various values of B in Fig.2. As one can see, α increases with B , startingfrom α = 1 .
05 at B = 0 up to α (cid:39) .
35 at B = 40 G.The agreement between theory (solid line) and experi-ment (circles) is quite satisfactory. This shows that theMonte-Carlo simulation contains the essential ingredientsthat play a role in the restoration of the contrast. In thefollowing, we will rest on the results of the Monte-Carlosimulation to elucidate the mechanisms at work by com-puting quantities which are not accessible to experiment,e. g. the bistatic coefficients for each scattering order.In Fig.3a, we plot γ c /γ d as a function of B . This ratiois a measure of the degree of coherence of the outgoinglight. This ratio grows when B increases, and tends to 1for large B (not shown in the figure). This confirms thatthe contrast of the interference, and in turn the coherenceof the outgoing light, is actually restored by a magneticfield.In Fig.3b we show how γ s (triangles), γ d (crosses) and γ c (circles) change with B . γ d strongly decreases stronglywith B because the atomic scattering cross section itselfdecreases. Meanwhile, γ c increases for magnetic fields upto 8 G. This shows the efficiency of the mechanism restor-ing interference. At larger fields however, the decrease ofthe scattering cross section takes over and γ c decreasesagain, although slower than γ d . At large B , γ c and γ d tends to the same value (yielding a perfect coherence)but are outgrown by γ s , meaning α <
2. This behavior3
FIG. 3. Results of the Monte-Carlo simulations under thesame experimental conditions as Fig.2. (a) The coherence ra-tio γ c /γ d increases with B . (b) Plots of the bistatc coefficients γ s (triangles), γ d (crosses) and γ c (circles) in the backwarddirection as a function of B . The solid lines are drawn toguide the eyes. As one can see, γ c first increases than de-creases with B without varying too much. At the same time, γ d decreases strongly. At large B , γ c = γ d and the coherenceis restored. However the enhancement factor α < γ s is not negligible. is not generic: at larger optical thickness, γ s would havebeen smaller than γ d .We also mention that the values of the bistatic coeffi-cients are independant of the value of k(cid:96) (cid:29)
1. However,the angular width of the backscattering cone depends on k(cid:96) . D. Impact of Faraday and Cotton-Mouton effects
To study the impact of magneto-optical effects on theenhancement factor, the simplest way is to discard themin the Monte-Carlo simulation, and compare the obtainedresult with the experimental data in Fig.2. The magneto-optical effects distort the polarization of a propagatingwave. Discarding them means here that a polarizationis propagated without distorsion, only with attenuation.To ensure energy conservation, the attenuation length(i.e. the scattering mean free path) must be equal to (cid:96) =1 / ( n ( r ) σ ( φ )) where n ( r ) is the local density of atoms and σ ( φ ) the total scattering cross section of an atom givenby equation (19). All other parameters in the simulationare left unchanged.Fig.4 shows the plot of this expurgated enhancementfactor ˜ α as a function of B (dotted line), together withits quantitative comparison to the real theoretical curveborrowed from Fig.2 (solid line). For B (cid:46)
4G ( µB (cid:46) Γ),the impact of Faraday and Cotton-Mouton effects is neg-ligible. For larger fields, the true value α is slightly lowerthan ˜ α , the two curves being roughly parallel to eachother. This means that magneto-optical effects do de-crease the phase coherence of the sample but this detri- FIG. 4. Impact of magneto-optical effects occurring dur-ing propagation (Faraday and Cotton-Mouton effects) on theCBS enhancement factor α . The solid line is the theoreticalcurve obtained in Fig.2. The dotted line is the enhancementfactor ˜ α calculated by discarding the magneto-optical effects.as one can see, the Faraday and Cotton-Mouton effects dodecrease the contrast but their detrimental effect is counter-balanced and beaten by an efficient mechanism restoring thecontrast. This mechanism is the modification of the scatter-ing properties of the atoms which, because of the Zeemansplitting, behave more and more as effective two-level atomswhen B is increased and the light is tuned on resonance witha closed transition. mental effect is counter-balanced and beaten by a morepowerful mechanism restoring coherence. As a matter offact, at large B and after the first scattering event, a sin-gle eigenmode can propagate in the scattering medium,exemplifying why phase or extinction differences betweenthe polarization eigenmodes cannot scramble the con-trast. Thus, the phenomenon explaining the restorationof the CBS contrast with B is really the modification ofthe scattering properties of the atoms which, because ofthe Zeeman splitting, behave more and more as effectivetwo-level atoms when B is increased and the light is tunedon resonance with a closed transition. This is in markedcontrast with classical scatterers where no such mecha-nism counter-balancing the detrimental magneto-opticaleffects does exist. E. Influence of higher and higher scattering orders
When B = 0, the CBS effect observed with atomshaving a degenerate ground state is dominated by dou-ble scattering paths, while higher-order scattering pathscontribute significantly to the diffuse background [45].This is shown in the first line of Table I. As B increases,the contrast is restored and higher and higher scatteringorders contribute significantly both to γ c and γ d , see sec-ond line of Table I. However, at the same time, the opti-cal thickness of the atomic cloud decreases. Higher-order4 Scattering order 2 3 4 5 γ c /γ d B=0G 0.21 0.08 0.04 0.02B=30G 0.81 0.75 0.69 0.38TABLE I. CBS coherence factor γ c /γ d for the first scatteringorders when B = 0G and B = 30G. The incoming light andatomic cloud parameters are given in Fig.2. The atomic cloudis spherically symmetric with gaussian density.FIG. 5. Influence of higher and higher scattering orderson the CBS enhancement factor. The incoming light andatomic cloud parameters are given in Fig.2. We compare thetheoretical enhancement factor borrowed from Fig.2) with theone obtained by only considering single and double scatteringorders. At small B , high scattering orders contribute mostlyto the diffuse signal. At large B , they contribute equally tothe coherent and diffuse signals, making the CBS cone heightincrease. scattering paths become less and less probable and thetwo effects compete. The dotted line in figure 5 showsthe enhancement factor calculated from the single anddouble scattering contributions alone. This approxima-tion overestimates the height of the backscattering coneat small magnetic fields, but underestimates it at largemagnetic fields. This shows that high scattering ordersdo contribute to the CBS cone and cannot be discardedfor a quantitative comparison. F. Role of optical pumping
In section II D, we mentioned that our theory does nottake optical pumping into account (though it could be ex-tended to do so). In the present section, we give conclu-sive evidence that optical pumping is indeed negligible inour experiment by measuring the coherent transmissionof the atomic cloud. The results are presented in Fig.6,for an incoming wavevector perpendicular to B and a cir-cular incoming polarization. If δ = 0, the coherent trans-mission varies in time and its stationary value is shown in FIG. 6. Coherent transmission of the atomic cloud as a func-tion of B for an initial optical thickness b = 31 measured at B = 0 and δ = 0. Crosses: experiment for δ = 0. Dashedline: theory for δ = 0. Solid circles: experiment for δ = µB .Solid line: theory for δ = µB . When δ = µB the incominglight is kept on resonance with the closed atomic transition | (cid:105) → | (cid:105) . When δ = 0, the coherent transmission increasesin time and converges to the plotted value. In this case, opti-cal pumping is at work, a situation not accounted for by ourtheory. For δ = µB , no time variation of the coherent trans-mission is observed. This is a strong indication that opticalpumping is negligible in this case. Fig.6 as a function of B (crosses). Our Monte Carlo sim-ulation (dotted line) is unable to reproduce these resultsfor B >
10 G, indicating that optical pumping is indeedpresent in our sample when δ = 0. However, when theincident light beam is kept at resonance with the atomictransition | (cid:105) → | (cid:105) (i.e. δ = µB ), no time evolution ofthe coherent transmission is observed. This shows thatthe populations of the various Zeeman substates are al-most constant. The experimental data (circles) are wellreproduced by the Monte-Carlo simulation (solid line).This is a strong indication that optical pumping is in-deed negligible when the incoming light is continuouslykept at resonance with the closed atomic transition. G. Coherence length
The notion of phase coherence length L φ is a very im-portant concept in mesoscopic physics. It is the lengthscale at which, because of some dephasing mechanisms,the interference effects as produced by the medium areeffective. The larger is L φ , the stronger is the impact ofinterference, and, in the case of the CBS effect, the largeris the CBS contrast. In the case of cold atoms, at B = 0,the degeneracy of the atomic ground state causes a loss ofphase coherence between reversed scattering paths givingrise to a finite value of L φ of the order of few mean freepaths (cid:96) [13]. The increase of the CBS contrast when amagnetic field is applied is accordingly accompanied bya growth of L φ . The Monte-Carlo simulation allows us5 FIG. 7. Plot of s φ = L φ /(cid:96) as a function of B , as extractedfrom our Monte-Carlo calculation. The incoming light andatomic cloud parameters are given in Fig.2. The points dis-persion reflects the numerical accuracy. As one can see, s φ i ncreases with B , roughly linearly (the solid line is drawn toguide the eye). This is due to the lifting of the Zeeman degen-eracy which make atoms behave like effective two-level atomswhen the light is tuned on resonance with a closed transition.This behavior is in sharp contrast with classical scattererswhere s φ decreases when B increases. to estimate the phase coherence length in the followingway. In the presence of dephasing, the interference term γ c associated to two reversed scattering paths of length L is related to the diffuse term γ d by: γ c (cid:39) γ d e − L/L φ = γ d e − s/s φ , (56)where L/(cid:96) = s , s being the scattering order, and L φ /(cid:96) = s φ . In Fig.7 we plot s φ as a function of B as obtainednumerically. It increases roughly linearly.One should note that our definition of the coherencelength differs from the usual one where the distance trav-elled diffusively by the light inside the disordered sampleis introduced [3]. With this convention, L ∝ √ s and L φ ∝ √ s φ . H. Analogy with paramagnetic impurities insolid-state physics
The surprising fact that a magnetic field can restoreweak localization effects under well chosen circumstancesalthough it breaks time-reversal invariance is alreadyknown in solid-state physics [48, 49]. In this context,one considers the propagation of electrons inside a metalat low (but finite) temperature, containing paramagneticimpurities. Thermal fluctuations make the spin of theseimpurities fluctuate in time. The scattering of an electronby such a fluctuating impurity randomizes the electronspin and the weak localization correction to electronic transport are reduced. This is similar to the loss of con-trast due to the degeneracy of the atomic ground state.When a large enough magnetic field is applied to themetal, the spins of the impurities are all aligned along B .The fluctuations of the spin component of the electronsare suppressed and the weak localization corrections totransport are restored.Finally, in both cases, the magnetic field freezes the in-ternal degrees of freedom and restores interference effect.The main difference with our case is that in solid-statephysics, the magnetic field populates a unique spin state,whereas the Zeeman substates of the atoms are equallypopulated. It could be possible to realize an atomic cloudalmost containing only atoms in the ground state | (cid:105) , byusing optical pumping. However, this would only restorethe interference between reverse double scattering paths[19]. Nevertheless, it would be possible to first populatethe | (cid:105) state, and then to apply an external magneticfield. This would increase the number of atoms partic-ipating to the scattering of light. Multiple scatteringwould then play a more important role and the enhance-ment factor would be larger than reported in the presentarticle. VI. CONCLUSION
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