Absorption imaging of a quasi 2D gas: a multiple scattering analysis
AAbsorption imaging of a quasi 2D gas:a multiple scattering analysis
L Chomaz , L Corman , , , T Yefsah , R Desbuquois andJ Dalibard Laboratoire Kastler Brossel, CNRS, UPMC, Ecole normale sup´erieure,24 rue Lhomond, 75005 Paris, France Ecole polytechnique (member of ParisTech), 91128 Palaiseau cedex, France Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Abstract.
Absorption imaging with quasi-resonant laser light is a commonly usedtechnique to probe ultra-cold atomic gases in various geometries. Here we investigatesome non-trivial aspects of this method when it is applied to in situ diagnosis of aquasi two-dimensional gas. Using Monte Carlo simulations we study the modificationof the absorption cross-section of a photon when it undergoes multiple scattering inthe gas. We determine the variations of the optical density with various parameters,such as the detuning of the light from the atomic resonance and the thickness of thegas. We compare our results to the known three-dimensional result (Beer–Lambertlaw) and outline the specific features of the two-dimensional case.PACS numbers: 42.25.Dd,37.10.-x,03.75.-b a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec bsorption imaging of a quasi 2D gas
1. Introduction
The study of cold atomic gases has recently shed new light on several aspects of quantummany-body physics [1, 2, 3, 4]. Most of the measurements in this field of research arebased on the determination of the spatial density of the gas [5]. For instance one can usethe in situ steady-state atomic distribution in a trapping potential to infer the equationof state of the homogenous gas [6]. Another example is the time-of-flight method, inwhich one measures the spatial density after switching off the trapping potential andallowing for a certain time of ballistic expansion. This gives access to the momentumdistribution of the gas, and to the conversion of interaction energy into kinetic energyat the moment of the potential switch-off.To access the atomic density n ( r ), one usually relies on the interaction of the atomswith quasi-resonant laser light. The most common method is absorption imaging, inwhich the shadow imprinted by the cloud on a low intensity probe beam is imagedon a camera. The simplest modelling of absorption imaging is based on a mean-fieldapproach, in which one assumes that the local value of the electric field driving anatomic dipole at a given location depends only on the average density of scatterers.One can then relate the attenuation of the laser beam to the column atomic density n (col) ( x, y ) = (cid:82) n ( r ) dz along the line-of-sight z . The optical density of the cloud D ( x, y ) ≡ ln[ I in ( x, y ) /I out ( x, y )] is given by the Beer–Lambert law D BL ( x, y ) = σ n (col) ( x, y ) , (1)where σ is the photon scattering cross-section, and I in (resp. I out ) are the incoming (resp.outgoing) intensity of the probe laser in the plane xy perpendicular to the propagationaxis. For a closed two-level atomic transition of frequency ω = ck , σ depends on thewavelength λ = 2 π/k associated to this transition and on the detuning ∆ = ω − ω between the probe light frequency ω and the atomic frequency: σ = σ δ , σ = 3 λ π , δ = 2∆Γ . (2)Here Γ represents the natural line width of the transition ( i.e. , Γ − is the natural lifetime of the excited state of the transition). Eq. (2) assumes that the intensity of theprobe beam is much lower than the saturation intensity of the atomic transition. Quasi-resonant absorption imaging is widely used to measure the spatial distribution of atomicgases after a long time-of-flight, when the density has dropped sufficiently so that themean-field approximation leading to Eq. (1) is valid.One can also use absorption imaging to probe in situ samples, at least in the casewhere σ n (col) is not very large so that the output intensity is not vanishingly small.This is in particular the case for low dimensional gases. Consider for example a 2Dgas, such that the translational degree of freedom along z has been frozen. For a probebeam propagating along this axis, one can transpose the Beer–Lambert law of Eq. (1)by simply replacing the column density by the surface density n (2D) of the gas. This 2DBeer–Lambert law can be heuristically justified by treating each atom as a disk of area σ bsorption imaging of a quasi 2D gas A (cid:29) σ containing N = An (2D) (cid:29) − σ/A ) N ≈ exp( − σn (2D) ).In a quasi-2D gas there is however an important limitation on the optical densities towhich one may apply the Beer-Lambert prediction of Eq. (1). Already for σ n (2D) = 1the mean interparticle distance is only 0 . λ and one may expect that the opticalresponse of an atom strongly depends on the precise location of its neighbours. Moreprecisely the exchange of photons between closely spaced atoms induces a resonantvan der Waals interaction that significantly shifts the atomic resonance frequency withrespect to its bare value ω . The optical density of the gas at resonance may then bereduced with respect to Eq. (1), and this was indeed observed in a series of experimentsperformed with a degenerate Rb gas [7, 8].The general subject of the propagation of a light wave in a dense atomic sample,where multiple scattering plays an essential role, has been the subject of numerousexperimental and theoretical works (see e.g. [9, 10] in the context of cold atoms, and[11] for a review). Here we present a quantitative treatment of the collective effects thatappear when a weak probe beam interacts with a quasi-2D atomic gas. We consider anensemble of N atoms at rest with random positions and we investigate the transmissionof quasi-resonant light by the atom sheet. We model the resonance transition betweenthe atomic ground ( g ) and excited ( e ) states by a J g = 0 ↔ J e = 1 transition. We presenttwo equivalent approaches; the first one is based on the calculation of the field radiatedby an assembly of N dipoles, where each dipole is driven by an external field plus the fieldradiated by the N − T matrix formalismof scattering theory. We show that in both cases the optical density of the medium canbe determined by solving the same 3 N × N linear system. A similar formalism hasbeen previously used for the study of light propagation in small 3D atomic samples, inthe presence of multiple scattering (see e.g. [12, 13, 14, 15, 16, 17, 18, 19]). Howeverits application to quasi-2D samples has (to our knowledge) not yet been investigated,except in the context of Anderson localisation of light [13]. Our numerical calculationsare performed for N = 2048 atoms, which is sufficient to reach the ‘thermodynamic limit’for the range of parameters that is relevant for experiments. We show in particular thateven for moderate values of σ n (col) , the optical density is notably reduced comparedto what is expected from the Beer-Lambert law ( e.g. , more than 20 % reduction for σ n (col) = 1). We investigate how the absorption line shape is modified by the resonantvan der Waals interactions and we also show how the result (1) is recovered when oneincreases the thickness of the gas, for a given column density n (col) .The paper is organised as follows. In section 2, we detail the modelling of the atom-light interaction with the two-level and rotating wave approximations. Then we explainthe principle of the calculation for the absorption of a weak probe beam crossing theatom slab (section 3). The ensemble of our numerical results are presented in section 4.Finally in section 5 we discuss some limitations to our model and draw some concludingremarks. bsorption imaging of a quasi 2D gas
2. Modelling the atom-light interaction
We use the standard description of the quantised electromagnetic field in the Coulombgauge [20], and choose periodic boundary conditions in the cubic-shaped quantisationvolume V = L x L y L z . We denote a q , s the destruction operator of a photon with wavevector q and polarisation s ( s ⊥ q ). The Hamiltonian of the quantised field is H F = (cid:88) q , s (cid:126) cq a † q , s a q , s , (3)and the transverse electric field operator reads E ( r ) = E (+) ( r ) + E ( − ) ( r ) with E (+) ( r ) = i (cid:88) q , s (cid:114) (cid:126) cq ε V a q , s e i q · r s , (4)and E ( − ) ( r ) = (cid:16) E (+) ( r ) (cid:17) † . The wave vectors q are quantised in the volume V as q i = 2 πn i /L i , i = x, y, z , where n i is a positive or negative integer. We consider a collection of N identical atoms at rest in positions r j , j = 1 , . . . N . Wemodel the atomic resonance transition by a two-level system with a ground state | g (cid:105) with angular momentum J g = 0 and an excited level of angular momentum J e = 1. Wechoose as a basis set for the excited manifold the three Zeeman sublevels | e α (cid:105) , α = x, y, z ,where | e α (cid:105) is the eigenstate with eigenvalue 0 of the component J α of the atomic angularmomentum operator. We denote (cid:126) ω the energy difference between e and g . The atomicHamiltonian is thus (up to a constant) H A = N (cid:88) j =1 (cid:88) α = x,y,z (cid:126) ω | j : e α (cid:105)(cid:104) j : e α | . (5)The restriction to a two-level approximation is legitimate if the detuning ∆ betweenthe probe and the atomic frequencies is much smaller than ω . The modelling of thistransition by a J g = 0 ↔ J e = 1 transition leads to a relatively simple algebra. Thetransitions that are used for absorption imaging in real experiments often involve moreZeeman states ( J g = 2 ↔ J e = 3 for Rb atoms in [7, 8]), but are more complex to handle[21, 22] and they are thus out of the scope of this paper. However we believe that themost salient features of multiple scattering and resonant Van der Waals interactions arecaptured by our simple level scheme. We treat the atom-light interaction using the electric dipole approximation (lengthgauge), which is legitimate since the resonance wavelength of the atoms λ is much bsorption imaging of a quasi 2D gas V = − (cid:88) j D j · E ( r j ) , (6)where D j is the dipole operator for the atom j . We will use the rotating waveapproximation (RWA), which consists in keeping only the resonant terms in the coupling: V ≈ − (cid:88) j D (+) j · E (+) ( r j ) + h.c. , (7)where h.c. stands for Hermitian conjugate. Here D (+) j represents the raising part of thedipole operator for atom j : D (+) j = d (cid:88) α = x,y,z | j : e α (cid:105)(cid:104) j : g | ˆ u α , (8)where d is the electric dipole associated to the g − e transition and ˆ u α is a unit vectorin the direction α .When a single atom is coupled to the electromagnetic field, this coupling resultsin the modification of the resonance frequency (Lamb shift) and in the fact that theexcited state e acquires a non-zero width ΓΓ = d ω πε (cid:126) c . (9)For simplicity we will incorporate the Lamb shift in the definition of ω . Note that theproper calculation for this shift requires that one goes beyond the two-level and therotating wave approximations. The linewidth Γ on the other hand can be calculatedfrom the above expressions for V using the Fermi golden rule.The RWA provides a very significant simplification of the treatment of the atom-light coupling, in the sense that the total number of excitations is a conserved quantity.The annihilation (resp. creation) of a photon is always associated with the transitionof one of the N atoms from g to e (resp. from e to g ). This would not be the case ifthe non-resonant terms of the electric dipole coupling D (+) i · E ( − ) and D ( − ) i · E (+) werealso taken into account. The small parameter associated to the RWA is ∆ /ω , which isin practice in the range 10 − − − ; the RWA is thus an excellent approximation.Formally the use of the electric dipole interaction implies to add to the Hamiltonianan additional contact term between the dipoles (see e.g. [23, 12]). This term will playno role in our numerical simulations because we will surround the position of each atomby a small excluded volume, which mimics the short range repulsive interaction betweenatoms. We checked that the results of our numerical calculations (see Sec. 4) do notdepend on the size of the excluded volume, and we can safely omit the additional contactterm in the present work.
3. Interaction of a probe laser beam with a dense quasi-2D atomic sample
We present in this section the general formalism that allows one to calculate theabsorption of a quasi-resonant laser beam by a slab of N atoms. We address this bsorption imaging of a quasi 2D gas N oscillating dipoles [12]. The equation of motionfor each dipole is obtained using the Heisenberg picture for the Hamiltonian presentedin section 2. It contains two driving terms, one from the incident probe field and onefrom the field radiated by all the other dipoles at the location of the dipole under study.The steady-state of this assembly of dipoles is obtained by solving a set of 3 N linearequations. The second approach uses the standard quantum scattering theory [24],which is well suited for perturbative calculations and partial resummations of diagrams.We suppose that one photon is incident on the atomic medium and we use resummationtechniques to take into account the multiple scattering events that can occur beforethe photon emerges from the medium. The relevant quantity in this approach is theprobability amplitude T ii that the outgoing photon is detected in the same mode as theincident one [14, 17], and we show that T ii is obtained from the same set of equationsas the values of the dipoles in the first approach. In this section we assume that the incident field is prepared in a coherent statecorresponding to a monochromatic plane wave E L (cid:15) e i ( kz − ωt ) . We choose the polarization (cid:15) to be linear and parallel to the x axis ( (cid:15) = ˆ u x ). Since we consider a J g = 0 ↔ J e = 1transition, this choice does not play a significant role and we checked that we recoveressentially the same results with a circular polarisation. Note that the situation would bedifferent for an atomic transition with larger J g and J e since optical pumping processeswould then depend crucially on the polarisation of the probe laser.The amplitude E L is supposed to be small enough that the steady-state populationsof the excited states e j,α are small compared to unity. This ensures that the response ofeach atomic dipole is linear in E L ; this approximation is valid when the Rabi frequency dE L / (cid:126) is small compared to the natural width Γ or the detuning ∆.Using the atom-light coupling (6), the equations of motion for the annihilationoperators a q , s in the Heisenberg picture read:˙ a q , s ( t ) = − i cq a q , s ( t ) + (cid:114) cq (cid:126) ε V (cid:88) j (cid:48) s ∗ · D j (cid:48) ( t ) e − i q · r j (cid:48) . (10)This equation can be integrated between the initial time t and the time t , and theresult can be injected in the expression for the transverse field to provide its value atany point r : E α ( r , t ) = E free ,α ( r , t ) + (cid:88) j (cid:48) ,α (cid:48) (cid:88) q , s (cid:90) t − t dτ cq ε V (cid:2) iD j (cid:48) ,α (cid:48) ( t − τ ) e i q · ( r − r j (cid:48) ) − icqτ s α s ∗ α (cid:48) + h.c. (cid:3) , (11)where E free stands for the value obtained in the absence of atoms. We now take thequantum average of this set of equations. In the steady-state regime the expectationvalue of the dipole operator D j ( t ) can be written d j e − iωt + c.c., and the average of bsorption imaging of a quasi 2D gas E free ( r , t ) is the incident field E L (cid:15) e i ( kz − ωt ) + c.c. . We denote the average value of thetransverse field operator in r as (cid:104) E ( r , t ) (cid:105) = ¯ E ( r ) e − iωt + c.c., and we obtain after somealgebra (see e.g. [12, 25])¯ E α ( r ) = E L (cid:15) α e ikz + k πε (cid:88) j (cid:48) ,α (cid:48) g α,α (cid:48) ( u j (cid:48) ) d j (cid:48) ,α (cid:48) , (12)where we set u j = k ( r − r j ) (with k ≈ k ), g α,α (cid:48) ( u ) = δ α,α (cid:48) h ( u ) + u α u α (cid:48) u h ( u ) , (13)and h ( u ) = 32 e iu u ( u + iu − , h ( u ) = 32 e iu u ( − u − iu + 3) . (14)The function g α,α (cid:48) ( k r ) is identical to the one appearing in classical electrodynamics [26],when calculating the field radiated in r by a dipole located at the origin.We proceed similarly for the equations of motion for the dipole operators D ( − ) j andtake their average value in steady-state. The result can be put in the form [12]( δ + i ) d j,α + (cid:88) j (cid:48) (cid:54) = j, α (cid:48) g α,α (cid:48) ( u jj (cid:48) ) d j (cid:48) ,α (cid:48) = − πε k E L (cid:15) α e ikz j , (15)where the reduced detunig δ = 2∆ / Γ has been defined in Eq. (2) and u j,j (cid:48) = k ( r j (cid:48) − r j ).This can be written with matrix notation[ M ] | X (cid:105) = | Y (cid:105) (16)where the 3 N vectors | X (cid:105) and | Y (cid:105) are defined by X j,α = − k π(cid:15) E L d j,α , Y j,α = (cid:15) α e ikz j , (17)and where the complex symmetric matrix [ M ] has its diagonal coefficients equal to δ + i and its off-diagonal coefficients (for j (cid:54) = j (cid:48) ) given by g α,α (cid:48) ( u jj (cid:48) ). This matrix belongs tothe general class of Euclidean matrices [27], for which the ( i, j ) element can be writtenas a function F ( r i , r j ) of points r i in the Euclidean space. The spectral properties ofthese matrices for a random distribution of the r i ’s (as it will be the case in this work,see Sec. 4) have been studied in [27, 28, 29, 30].Eq. (15) has a simple physical interpretation: in steady-state each dipole d j isdriven by the sum of the incident field E L and the field radiated by all the other dipoles.This set of 3 N equations was first introduced by L. L. Foldy in [31] who named it,together with Eq. (12), “the fundamental equations of multiple scattering”. Indeed fora given incident field, the solution of (16) provides the value of each dipole d j , whichcan then be injected in (12) to obtain the value of the total field at any point in space. bsorption imaging of a quasi 2D gas Lens% Screen%Atoms% (a)%
Screen%2% f" f" Lens%Atoms% (b)% ` ` + f z F% Figure 1.
Two possible setups for measuring the absorption of an incident probebeam by a slab of atoms using a lens of focal f . a) Global probe. b) Local probe. From the expression of the average value of the dipoles we now extract the absorptioncoefficient of the probe beam and the optical density of the gas. We suppose that the N atoms are uniformly spread in a cylinder of radius R along the z axis and locatedbetween z = − (cid:96)/ z = (cid:96)/
2. We can consider two experimental setups to address thisproblem. The first one, represented in Fig. 1a, consists in measuring after the atomicsample the total light intensity with the same momentum k = k ˆ u z as the incident probebeam. This can be achieved by placing a lens with the same size as the atomic sample,in the plane z = (cid:96) (cid:48) > (cid:96)/ F gives the desired attenuation coefficient. We refer to this method as ‘global’,since the field E ( F ) provides information over the whole atomic cloud. One can also usethe setup sketched in Fig. 1b, which forms an image of the atom slab on a camera andprovides a ‘local’ measurement of the absorption coefficient. In real experiments localmeasurements are often favored because trapped atomic sample are non homogeneousand it is desirable to access the spatial distribution of the particles. However for ourgeometry with a uniform density of scatterers, spatial information on the absorptionof the probe beam is not relevant. Therefore we only present the formalism for globalmeasurements, which is simpler to derive and leads to slightly more general expressions.We checked numerically that we obtained very similar results when we modelled thelocal procedure.We assume that the lens in Fig. 1a operates in the paraxial regime, i.e. , its focallength f is much larger than its radius R . We relate the field at the image focal pointof the lens to the field in the plane z = (cid:96) (cid:48) just before the lens: E ( F ) = − ie ikf λ f (cid:90) L E ( x, y, (cid:96) (cid:48) ) dx dy, (18)where the integral runs over the lens area. Since the incident probe beam is supposed tobe linearly polarised along x , we calculate the x component of the field in F . Plugging bsorption imaging of a quasi 2D gas T ≡ E x ( F ) | with atoms E x ( F ) | no atom = 1 − e − ik(cid:96) (cid:48) πR (cid:88) j,α X j,α (cid:90) L g x,α [ k ( r − r j )] dx dy . (19)This result can be simplified in the limit of a large lens by using an approximatedvalue for the integral appearing in (19). We suppose that k(cid:96) (cid:48) (cid:29) g x,α is the e iu /u contribution to h . More precisely the domain inthe lens plane contributing to the integral for the dipole j is essentially a disk of radius (cid:112) λ ( (cid:96) (cid:48) − z j ) ∼ √ λ(cid:96) (cid:48) centered on ( x j , y j ). When this small disk is entirely included inthe lens aperture, i.e. , the larger disk of radius R centered on x = y = 0, we obtain (cid:90) L g x,α [ k ( r − r j )] dx dy ≈ iπk δ x,α e ik ( (cid:96) (cid:48) − z j ) . (20)We use the result (20) for all atoms, which amounts to neglect edge effects for the dipoleslocated at the border of the lens, and we obtain: T = 1 − i σ n (col) Π , (21)with n (col) = N/πR and where the coefficient Π is defined byΠ = 1 N (cid:88) j X j,x e − ikz j . (22)This coefficient captures the whole physics of multiple scattering and resonant van derWaals interactions among the N atoms. Indeed one takes into account all possiblecouplings between the dipoles when solving the 3 N × N system [ M ] | X (cid:105) = | Y (cid:105) . Once T is known the optical density is obtained from D ≡ ln |T | − . (23)As an example, consider the limit of a very sparse sample where multiple scatteringdoes not play a significant role ( σ n (col) (cid:28) M ]are then negligible and [ M ] is simply the identity matrix, times i + δ . Each X j,x solutionof the system (16) is equal to e ikz j / ( i + δ ), and we obtain as expected: σ n (col) (cid:28) T ≈ − − iδ ) σ n (col) , D ≈ σ n (col) δ . (24) In order to study the attenuation of a weak probe beam propagating along the z axiswhen it crosses the atomic medium, we can also use quantum scattering theory. TheHamiltonian of the problem is H = H + V , H = H A + H F , (25)and we consider the initial state where all atoms are in their ground state and where asingle photon of wave vector k = k ˆ u z and polarisation (cid:15) = ˆ u x is incident on the atomicmedium | Ψ i (cid:105) = |G(cid:105) ⊗ | k , (cid:15) (cid:105) , (26) bsorption imaging of a quasi 2D gas |G(cid:105) ≡ | g, g, . . . , N : g (cid:105) . The state | Ψ i (cid:105) is an eigenstate of H withenergy (cid:126) ω . The interaction of the photon with the atomic medium, described by thecoupling V , can be viewed as a collision process during which an arbitrary number ofelementary scattering events can take place. Each event starts from a state |G(cid:105) ⊗ | q , s (cid:105) and corresponds to:(i) The absorption of the photon in mode q , s by atom j , which jumps from its groundstate | j : g (cid:105) to one of its excited states | j : e α (cid:105) . The state of the system is then |E j,α (cid:105) = | g, . . . , j : e α , . . . , N : g (cid:105) ⊗ | vac (cid:105) , (27)where | vac (cid:105) stands for the vacuum state of the electromagnetic field. The subspacespanned by the states |E j,α (cid:105) has dimension 3 N .(ii) The emission of a photon in the mode ( q (cid:48) , s (cid:48) ) by atom j , which falls back into itsground state.Finally a photon emerges from the atomic sample, and we want to determine theprobability amplitude to find this photon in the same mode | k , (cid:15) (cid:105) as the initial one.The T matrix defined as T ( E ) = V + V E − H + i + V , (28)where 0 + is a small positive number that tends to zero at the end of the calculation,provides a convenient tool to calculate this probability amplitude. Generally T if = (cid:104) Ψ f | T ( E i ) | Ψ i (cid:105) (29)gives the probability amplitude to find the system in the final state | Ψ f (cid:105) after thescattering process. The states | Ψ i (cid:105) and | Ψ f (cid:105) are eigenstates of the unperturbedHamiltonian H , with energy E i . Here we are interested in the element T ii of the T matrix, corresponding to the choice | Ψ f (cid:105) = | Ψ i (cid:105) . Using the definition (28) we find T ii = (cid:126) ωd ε V (cid:88) j,j (cid:48) e ik ( z j − z (cid:48) j ) (cid:104)E j (cid:48) ,x | (cid:126) ω − H + i + |E j,x (cid:105) . (30)We now have to calculate the (3 N ) × (3 N ) matrix elements of the operator 1 / ( z − H ),with z = (cid:126) ω + i + , entering into (30). We introduce the two orthogonal projectors P and Q , where P projects on the subspace with zero photon, and Q projects on theorthogonal subspace. We thus have P |E j,α (cid:105) = |E j,α (cid:105) P |G(cid:105) ⊗ | k , (cid:15) (cid:105) = 0 , (31) Q |E j,α (cid:105) = 0 Q |G(cid:105) ⊗ | k , (cid:15) (cid:105) = |G(cid:105) ⊗ | k , (cid:15) (cid:105) . (32)We define the displacement operator R ( z ) = V + V Qz − QH Q − QV Q V (33)and use the general result [20] P z − H P = Pz − H eff , (34) bsorption imaging of a quasi 2D gas H eff is H eff = P ( H + R ( z )) P. (35)For the following calculations, it is convenient to introduce the dimensionless matrix[ M ] proportional to the denominator of the right hand side of (34):[ M ] ( j (cid:48) ,α (cid:48) ) , ( j,α ) = 2 (cid:126) Γ (cid:104)E j (cid:48) ,α (cid:48) | z − H eff |E j,α (cid:105) . (36)It is straightforward to check ‡ that for z → (cid:126) ω this matrix coincides with the symmetricmatrix appearing in (16). Indeed the matrix elements of R ( z ) are (cid:104)E j (cid:48) ,α (cid:48) | R ( z ) |E j,α (cid:105) = (cid:126) d ε V (cid:88) q , s cq s ∗ α s α (cid:48) e i q · ( r j (cid:48) − r j ) z − (cid:126) ω , (37)which can be calculated explicitly. For j = j (cid:48) , the real part of this expression is theLamb shift that we reincorporate in the definition of ω , and its imaginary part reads: (cid:104)E j,α (cid:48) | R ( z ) |E j,α (cid:105) = − i (cid:126) Γ2 δ α,α (cid:48) . (38)For j (cid:54) = j (cid:48) , the sum over ( q , s ) appearing in (37) is the propagator of a photon from anatom in r j in internal state | e α (cid:105) , to another atom in r j (cid:48) in internal state | e α (cid:48) (cid:105) . This isnothing but (up to a multiplicative coefficient) the expression that we already introducedfor the field radiated in r j (cid:48) by a dipole located in r j : (cid:104)E j (cid:48) ,α (cid:48) | R ( z ) |E j,α (cid:105) = − (cid:126) Γ2 g α,α (cid:48) ( u j,j (cid:48) ) , (39)where the tensor g α,α (cid:48) is defined in Eqs. (13-14).Suppose now that the atoms are uniformly distributed over the transverse area L x L y of the quantisation volume. We set n (col) = N/ ( L x L y ) and we rewrite the expression(30) of the desired matrix element T ii as T ii L z (cid:126) c = 12 N σ n (col) (cid:88) j,j (cid:48) e ik ( z j − z j (cid:48) ) [ M − ] ( j,x ) , ( j (cid:48) ,x ) = 12 σ n (col) Π , (40)where the coefficient Π has been defined in (22). The result (40) combined with (21)leads to T = 1 − i T ii L z (cid:126) c , (41)which constitutes the ‘optical theorem’ for our slab geometry, since it relates theattenuation of the probe beam T to the forward scattering amplitude T ii .The emergence of resonant van der Waals interactions is straightforward in thisapproach. Let us consider for simplicity the case where only N = 2 atoms are present.The effective Hamiltonian H eff is a 6 × | e (cid:105) and one in | g (cid:105) , form in this particular case anorthogonal basis, although H eff is non-Hermitian [32, 33]. For a short distance r between ‡ As for the derivation leading from Eq. (10) to Eq. (12), one must take into account the non-resonantterms that are usually dropped in the RWA, in order to ensure the proper convergence of the sum (37)and obtain the tensor g αα (cid:48) . bsorption imaging of a quasi 2D gas kr (cid:28) h ( u ) and h ( u ) is u − and the energies (realparts of the eigenvalues) of the six eigenstates vary as ∼ ± (cid:126) Γ / ( kr ) (resonant dipole-dipole interaction). The imaginary parts of the eigenvalues, which give the inverse ofthe radiative lifetime of the states, tend either to Γ or 0 when r →
0, which correspondto the superradiant and subradiant states for a pair of atoms, respectively [34].For
N > H eff are ingeneral non orthogonal, which complicates the use of standard techniques of spectraltheory in this context [29, 30]. More precisely, one could think of solving the linearsystem (16), or equivalently calculating T ii in Eq. (30), by using the expansion of thecolumn vector | Y (cid:105) defined in Eq. (17) on the left ( | α j (cid:105) ) and right ( (cid:104) β j | ) eigenvectorsof H eff . Then one could inject this expansion in the general expression of the matrixelement T ii , to express it as a sum of the contributions of the various eigenvalues of H eff . However the physical discussion based on this approach is made difficult by thefact that since H eff is non-Hermitian, the {| α j (cid:105)} and the {| β j (cid:105)} bases do not coincide.Hence the weight (cid:104) β j | Y (cid:105)(cid:104) Y | α j (cid:105) of a given eigenvalue in the sum providing the value of T ii is not a positive number, and this complicates the interpretation of the result. For a sparse sample, we already calculated the optical density at first order in density(Eq. (24)) and the result is identical for a strictly 2D gas and a thick one. Theapproach based on quantum scattering theory is well suited to go beyond this firstorder approximation and look for differences between the 2D and 3D cases. The basisof the calculation is the series expansion of Eq. (34), which gives P z − H P = Pz − H + ∞ (cid:88) n =1 Pz − H (cid:18) P R ( z ) P z − H (cid:19) n . (42)Consider the case of a resonant probe δ = 0 for simplicity. The result T ≈ − σ n (col) / P/ ( z − H )] ofthis expansion. Here we investigate the next order term and explain why one can stillrecover the Beer-Lambert law for a thick (3D) gas, but not for a 2D sample. Double scattering diagrams for a thick sample ( k(cid:96) (cid:29) ). We start our study by addingthe first term ( n = 1) in the expansion (42) to the zero-th order term already takeninto account in Eq. (24). This amounts to take into account the diagrams where theincident photon is scattered on a single atom, and those where the photon ‘bounces’on two atoms before leaving the atomic sample. Injecting the first two terms of theexpansion (42) into (40), we obtain T ii L z (cid:126) c = 12 σ n (col) (cid:34) − i + 1 N (cid:88) j (cid:88) j (cid:48) (cid:54) = j e ik ( z j − z j (cid:48) ) g xx ( u jj (cid:48) ) (cid:35) . (43)We now have to average this result on the positions of the atoms j and j (cid:48) . Thereare N ( N − ≈ N couples ( j, j (cid:48) ). Assuming that the gas is dilute so that the average bsorption imaging of a quasi 2D gas | z j − z j (cid:48) | ) is much larger than k − , the leadingterm in g xx is the e iu /u contribution of h ( u ) in Eqs. (13)-(14). We thus arrive at T ii L z (cid:126) c = 12 σ n (col) (cid:20) − i + 3 N k (cid:104) e ik ( z − z (cid:48) ) e ik | r − r (cid:48) | | r − r (cid:48) | (cid:105) (cid:21) , (44)where the average is taken over the positions r and r (cid:48) of two atoms. We first calculatethe average over the xy coordinates and we get (cf. Eq. (20)) T ii L z (cid:126) c = 12 σ n (col) (cid:20) − i + i σ n (col) (cid:104) e ik ( z − z (cid:48) ) e ik | z − z (cid:48) | (cid:105) (cid:21) . (45)For a thick gas ( k(cid:96) (cid:29)
1) the bracket in this expression has an average value of ≈ / z < z (cid:48) , which occurs in half of thecases, and it oscillates and averages to zero in the other half of the cases, where z > z (cid:48) .We thus obtain the approximate value of the transmission coefficient: k(cid:96) (cid:29) T = 1 − i T ii L z (cid:126) c ≈ − σ n (col) + 18 (cid:0) σ n (col) (cid:1) , (46)where we recognize the first three terms of the power series expansion of T =exp( − σ n (col) / D = σ n (col) . Double scattering diagrams for a 2D gas ( (cid:96) = 0 ). When all atoms are sitting in thesame plane, the evaluation of the second order term (and the subsequent ones) in theexpansion of T ii in powers of the density is modified with respect to the 3D case. Thecalculation starts as above and the second term in the bracket of Eq. (43) can now bewritten 1 N (cid:88) j (cid:88) j (cid:48) (cid:54) = j g xx ( u jj (cid:48) ) = n (2D) (cid:90) g xx ( u ) d u . (47)If we keep only the terms varying as e iu /u in h and h (Eq. (14)), we can calculateanalytically the integral in (47) and find the same result as in 3D, i.e. , iσ n (2D) /
4. If thiswas the only contribution to (47), it would lead to the Beer–Lambert law also in 2D, atleast at second order in density. However one can check that a significant contribution tothe integral in (47) comes from the region u = kr <
1. In this region, it is not legitimateto keep only the term in e ikr /kr in h , h , since the terms in e ikr / ( kr ) , correspondingto the short range resonant van der Waals interaction, are actually dominant. Thereforethe expansion of the transmission coefficient T in powers of the density differs from (46),and one cannot recover the Beer–Lambert law at second order in density. Calculatinganalytically corrections to this law could be done following the procedure of [12]. Herewe will use a numerical method to determine the deviation with respect to the Beer–Lambert law (see section 4.2). Remark.
For a 3D gas there are also corrections to the second term in Eq. (45) duethe 1 /r contributions to h and h . However these corrections have a different scalingwith the density and can be made negligible. More precisely their order of magnitudeis ∼ n (3D) k − , to be compared with the value ∼ n (col) k − of the second term in Eq. (45). bsorption imaging of a quasi 2D gas n (3D) k − (cid:28) n (col) k − (cid:38)
1, if the thickness (cid:96) of the gas along z is (cid:29) /k .
4. Absorption of light by a slab of atoms
In order to study quantitatively the optical response of a quasi-2D gas, we haveperformed a Monte Carlo calculation of the transmission factor T given in Eq. (21),and of the related optical density D = ln |T | − . We start our calculation by randomlydrawing the positions of the N atoms, we then solve numerically the 3 N × N linearsystem (16), and finally inject the result for the N dipoles in the expression of T .The atoms are uniformly distributed in a cylinder of axis z , with a radius R and athickness (cid:96) . The largest spatial densities considered in this work correspond to a meaninter-particle distance ≈ k − . Around each atom we choose a small excluded volumewith a linear size a = 0 . k − . We varied a by a factor 10 around this value and checkedthat our results were essentially unchanged. Apart from this excluded volume we donot include any correlation between the positions of the atoms. This choice is justifiedphysically by the fact that, in the case of large phase space densities which motivates ourstudy, the density fluctuations in a 2D Bose gas are strongly reduced and the two-bodycorrelation function g ( r , r (cid:48) ) is such that g ( r , r ) ≈ N that is needed to reach the‘thermodynamic limit’ for our problem: for a given thickness (cid:96) , D should not be anindependent function of the number of atoms N and the disk radius R , but shoulddepend only of the ratio N/πR = n (col) . We will see that this imposes to use relativelylarge number of atoms, typically N > N = 2048. We then study thedependence of D with the various parameters of the problem: the column density n (col) ,the thickness of the gas (cid:96) , and the detuning ∆. In particular we show that for a given n (col) we recover the 3D result (1) when the thickness (cid:96) is chosen sufficiently large. We start our study by testing the minimal atom number that is necessary to obtain afaithful estimate of the optical density. We choose a given value of n (col) = N/πR andwe investigate how D depends on N either for a strictly 2D gas ( (cid:96) = 0) or for a gasextending significantly along the third direction ( (cid:96) = 20 k − ). We consider a resonantprobe for this study (∆ = 0). We vary N by multiplicative steps of 2, from N = 8 upto N = 2048 and we determine how large N must be so that D is a function of n (col) only.The results are shown in Fig. 2a and Fig. 2b, where we plot D as a function of N . We perform this study for four values of the density n (col) , corresponding to σ n (col) = 0 . , , σ n (col) = 0 .
5. Foreach value of N we perform a number of draws that is sufficient to bring the standard bsorption imaging of a quasi 2D gas D N a) k ‘ = 0 D N b) k ‘ = 20 Figure 2.
Variation of the optical density D = ln |T | − calculated from (21) asfunction of the number of atoms N , for (cid:96) = 0 (a) and (cid:96) = 20 k − (b), and for 4 valuesof the density: σ n (2D) = 0 . N ( N = 2048). The results have been obtained at resonance (∆ = 0). error below 2 × − and we find that the calculated optical density is independent of N (within standard error) already for N (cid:38) (cid:96) . Consider now ourlargest value σ n (col) = 4; for a strictly 2D gas ( (cid:96) = 0), D reaches an approximatelyconstant value independent of N for N (cid:38) σ n (col) = 4 and a relativelythick gas ( (cid:96) = 20 k − , blue squares in Fig. 2b), reaching the thermodynamic limit ismore problematic since there is still a clear difference between the results obtained with1024 and 2048 atoms. This situation thus corresponds to the limit of validity of ournumerical results. In the remaining part of the paper we will show only results obtainedwith N = 2048 atoms for column densities not exceeding σ n (col) = 4. The number ofindependent draws of the atomic positions (at least 8) is chosen such that the standarderror for each data point is below 2%. We now investigate the variation of the optical density D = ln |T | − as function ofthe column density of the sample n (col) , or equivalently of the Beer–Lambert prediction D BL = n (col) σ . We suppose in this section that the probe beam is resonant (∆ = 0),and we address the cases of a strictly 2D gas ( (cid:96) = 0) and a thick slab ( (cid:96) = 20 k − ).Consider first the case of a strictly 2D case, (cid:96) = 0, leading to the results shown inFig. 3a. We see that D differs significantly ( ∼ D BL already for D BL around1. A quadratic fit to the calculated variation of D for σ n (2D) < bsorption imaging of a quasi 2D gas D D BL a) k ‘ = 0 D D BL b) k ‘ = 20 Figure 3.
Variations of the optical density D as function of the Beer–Lambertprediction D BL for (cid:96) = 0 (a) and (cid:96) = 20 k − (b). The black dotted line is the straightline of slope 1. In (a) the continuous red line is a quadratic fit D = D BL (1 − µ D BL )with µ = 0 .
22 to the data points with D BL ≤
1. The calculations are done for N = 2048, ∆ = 0 and the bars indicate standard deviations. gives D ≈ D BL (1 − . D BL ) . (48)The discrepancy between D and D BL increases when the density increases: for D BL = 4,the calculated D is only ≈ .
4. For such a large density the average distance betweennearest neighbours is ≈ k − and the energy shifts due to the dipole-dipole interactionsare comparable to or larger than the linewidth Γ. The atomic medium is then muchless opaque to a resonant probe beam than in the absence of dipole-dipole coupling.Consider now the case of a thick sample, (cid:96) = 20 k − (Fig. 3b). The calculatedoptical density is then very close to the Beer–Lambert prediction over the whole rangethat we studied. This means that in our chosen range of optical densities, the mean-fieldapproximation leading to D BL is satisfactory as soon as the sample thickness exceeds afew optical wavelengths λ = 2 π/k .It is interesting to characterize how the optical density evolves from the value for astrictly 2D gas to the expected value from the Beer–Lambert law D BL when the thicknessof the gas increases. We show in Fig. 4 the variation of D as function of (cid:96) for threevalues of the column density corresponding to D BL = 1 , D = α + β exp( − (cid:96)/(cid:96) c ) to these data for 2 k − ≤ (cid:96) ≤ k − gives a good account of theobserved variation over this range, and it provides the characteristic thickness (cid:96) c neededto recover the Beer–Lambert law. We find that (cid:96) c ≈ . k − for D BL = 1, (cid:96) c ≈ . k − for D BL = 2, and (cid:96) c ≈ . k − for D BL = 4. bsorption imaging of a quasi 2D gas D k ‘ Figure 4.
Variation of D with the thickness (cid:96) of the gas for various column densitiescorresponding to D BL = 1 (red), 2 (green), 4 (blue). The continuous lines areexponential fits to the data. The dotted lines give the Beer–Lambert result. Thecalculations are done for N = 2048, ∆ = 0 and the bars indicate standard deviations. Remark.
For the largest value of the column density considered here ( n (col) σ = 4)we find that D increases slightly above the value D BL when (cid:96) is chosen larger than20 k − (upper value considered in Fig. 4). We believe that this is a consequence of theedge terms that we neglected when approximating Eq. (19) by Eq. (21). These termsbecome significant for D BL = 4 because for our atom number N = 2048, the sampleradius R ≈ k − is then not very large compared to its thickness for (cid:96) (cid:38) k − . Inorder to check this assumption, we also calculated numerically the result of Eq. (19)(instead of Eq. (21)) for practical values of the parameters (position and radius) of thelens represented in Fig. 1a. The results give again D ≈ D BL , but now with D remainingbelow D BL . Since our emphasis in this paper is rather put on the 2D case, we will notexplore this aspect further here. Resonant van der Waals interactions manifest themselves not only in the reduction of theoptical density at resonance but also in the overall line shape of the absorption profile.To investigate this problem we have studied the variations of D with the detuning of theprobe laser. We show in Fig. 5a the results for a strictly 2D gas ( (cid:96) = 0) for n (col) σ = 1 , n (2D) and reaches ∆ ≈ Γ / σ n (2D) = 4. We alsonote a slight broadening of the central part of the absorption line, since the full-widthat half maximum, which is equal to Γ for an isolated atom, is (cid:39) .
3Γ for n (col) σ = 4.Finally we note the emergence of large, non-symmetric wings in the absorption profile. bsorption imaging of a quasi 2D gas
012 -10 0 10 D Reduced detuning δ a) σ n (col) = 1 σ n (col) = 2 σ n (col) = 4 D / D B L Reduced detuning δ b) k ‘ = 0 , k a = 0 . k ‘ = 20 , k a = 0 . k ‘ = 0 , k a = 1 Figure 5. (a) Variation of D with the reduced detuning δ = 2∆ / Γ of the probe laserin the case of a 2D gas ( (cid:96) = 0), for three values of the column density σ n (col) = 1(red), 2 (green), and 4 (blue). (b) Blue full squares: same data as in (a), now plottedfor D / D BL as function of δ . Blue open squares: D / D BL for a thick gas ( (cid:96) = 20 k − ).Black stars: D / D BL for a 2D gas ( (cid:96) = 0) and a large exclusion region around eachatom ( a = k − ). All data in (b) correspond to σ n (2D) = 4. The calculations are donewith N = 2048 atoms and the bars indicate standard deviations. This asymmetry is made more visible in Fig. 5b, where we show with full blue squaresthe same data as in Fig. 5a for n (col) σ = 4, but now plotting D / D BL as function of δ . For a detuning δ = ±
15, the calculated optical density exceeds the Beer–Lambertprediction by a factor 4.1 (resp. 2.8) on the red (resp. blue) side.In order to get a better understanding of these various features, we give in Fig. 5btwo additional results. On the one hand we plot with empty blue squares the variationsof D / D BL for a thick gas ( (cid:96) = 20 k − ) with the same column density n (col) σ = 4. Thereare still some differences between D and D BL in this case, as already pointed out in[17], but they are much smaller than in the (cid:96) = 0 case. This indicates that the strongdeviations with respect to the Beer–Lambert law that we observe in Fig. 5a are specific2D features. On the other hand we plot with black stars the variations of D / D BL fora 2D gas ( (cid:96) = 0) in which we artificially increased the exclusion radius around eachatom up to a = k − instead of a = 0 . k − (blue full squares) for the other results inthis paper. This procedure, which was suggested to us by Robin Kaiser, allows one todiscriminate between effects due to isolated pairs of closely spaced atoms, and many-body features resulting from multiple scattering of photons among larger clusters ofatoms. The comparison of the results obtained for a = 0 . k − and a = k − suggeststhat the blue shift of the resonance line, which is present in both cases, is a many-bodyphenomenon, whereas the large amplitude wings with a blue-red asymmetry, whichoccurs only for a = 0 . k − , is rather an effect of close pairs. bsorption imaging of a quasi 2D gas r (cid:28) k − , the levels involving one ground and one excited atomhave an energy (real part of the eigenvalues of H eff ) that is displaced by ∼ ± (cid:126) Γ / ( kr ) .A given detuning δ can thus be associated to a distance r between the two membersof a pair that will resonantly absorb the light. To be more specific let us consider apair of atoms with kr (cid:28)
1, and suppose for simplicity that it is aligned either alongthe polarization axis of the light ( x ) or perpendicularly to the axis ( y ). In both casesthe excited state of the pair that is coupled to the laser is the symmetric combination( | g ; 2 : e x (cid:105) + | e x ; 2 : g (cid:105) ) / √
2. If the pair is aligned along the x axis, thisstate has an energy (cid:126) ω − (cid:126) Γ / kr ) , hence it is resonant with red detuned light suchthat δ = − / ( kr ) . If the pair axis is perpendicular to x , the state written abovehas an energy (cid:126) ω + 3 (cid:126) Γ / kr ) , hence it is resonant with blue detuned light such that δ = 3 / kr ) . This clearly leads to an asymmetry between red and blue detuning; indeedthe pair distance r needed for ensuring resonance for a given δ > r blue = (3 / | δ | ) / , issmaller than the value r red = (3 / | δ | ) / for the opposite value − δ . Since the probabilitydensity for the pair distance is P ( r ) ∝ r in 2D for randomly drawn positions, we expectthe absorption signal to be stronger for − δ than for + δ . In a 3D geometry the variationof the probability density with r is even stronger ( P ( r ) ∝ r ), but it is compensated bythe fact that the probability of occurrence of pairs that are resonant with blue detunedlight is dimensionally increased. For example in our simplified modelling where the pairaxis is aligned with the references axes, a given pair will be resonant with blue detunedlight in 2/3 of the cases (axis along y or z ) and resonant with red detuned light onlyin 1/3 of the cases (axis along x ). This explains why the asymmetry of the absorptionprofile is much reduced for a 3D gas in comparison to the 2D case.
5. Summary
We have presented in this paper a detailed analysis of the scattering of light by adisordered distribution of atoms in a quasi-two dimensional geometry. The particleswere treated as fixed scatterers and their internal structure was modeled as a two-levelsystem, with a J = 0 ground state and a J = 1 excited state. In spite of these simplifyingassumptions the general trend of our results is in good agreement with the experimentalfinding of [7], where a variation of the measured optical density similar to that of Fig. 3was measured.Several improvements in our modeling can be considered in order to reach aquantitative agreement with theory and experiment. The first one is to include therelatively complex atomic structure of the alkali-metal species used in practice, witha multiply degenerate ground state; this could be done following the lines of [21, 22].A second improvement consists in taking into account the atomic motion. This is inprinciple a formidable task, because it leads to a spectacular increase in the dimensionof the relevant Hilbert space. This addition can however be performed in practice in bsorption imaging of a quasi 2D gas ∼
10 microseconds only). Eachatom scatters only a few photons in this time interval and its displacement is then smallerthan the mean interatomic spacing for the spatial densities encountered in practice. Theacceleration of the atoms under the effect of resonant van der Waals interaction shouldalso have a minor effect under relevant experimental conditions. Finally another aspectthat could be valuably studied is the interaction of the gas with an intense laser beam[38]. One could thus validate the intuitive idea that saturation phenomena reduce theeffects of resonant van der Waals interactions [39, 8], and are thus helpful to provide afaithful estimate of the atomic density from the light absorption signal.
Acknowledgments
We thank I. Carusotto, Y. Castin, K. G¨unter, M. Holzmann, R. Kaiser, W. Krauth andS.P. Rath for helpful discussions and comments. This work is supported by IFRAF andANR (project BOFL).
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