Light scattering and localization in an ultracold and dense atomic system
aa r X i v : . [ qu a n t - ph ] M a y Light scattering and localization in an ultracold and dense atomic system
I.M. Sokolov, M.D. Kupriyanova , D.V. Kupriyanov Department of Theoretical Physics, State Polytechnic University, 195251, St.-Petersburg, Russia ∗ M.D. Havey
Department of Physics, Old Dominion University, Norfolk, VA 23529 (Dated: October 30, 2018)The quantum optical response of high density ultracold atomic systems is critical to a wide range offundamentally and technically important physical processes. These include quantum image storage,optically based quantum repeaters and ultracold molecule formation. We present here a microscopicanalysis of the light scattering on such a system, and we compare it with a corresponding descriptionbased on macroscopic Maxwell theory. Results are discussed in the context of the spectral resonancestructure, time-dependent response, and the light localization problem.
PACS numbers: 34.50.Rk, 34.80.Qb, 42.50.Ct, 03.67.Mn
Early studies of formation and dynamics of ultracoldatomic samples were largely focused on obtaining thehigh density and low temperatures necessary to attainBose-Einstein Condensation in atomic gas samples [1].At the same time, a large number of other research ar-eas emerged from studies of ultracold atomic gas sam-ples. Among these, there are several where physicalprocesses are importantly modified at high atomic den-sity. These include efforts in ultracold molecule forma-tion [2], image storage in high-optical depth samples [3],and light storage and manipulation for possible atomic-physics based quantum repeaters [4] and other quantuminformation applications. Another little explored areawhich may have significant impact in a range of scien-tific or technical areas is study of quantum optical pro-cesses at high density n , where n λ ∼ n λ > ∼ ∼ λ and coherently in-teracting with the system of atomic dipoles located ina relevant mesoscopic volume. The dipoles interactingwith the field are locally indistinguishable and the in-teraction process is assumed to be well approximatedby the cooperative coherent response [11] for the sam-ple susceptibility χ ( ω ) at frequency ω . Then the per-mittivity ǫ ( ω ) = 1 + 4 πχ ( ω ) can be extracted via a self-consistent approximation of the collective dipole dynam-ics driven by the transverse electric field and modifiedby the Lorentz-Lorenz effect coming from interference ofproximal atomic dipoles. If incoherent losses have only aradiative nature and atoms equally populate the groundstate Zeeman sublevels we obtain a self-consistent expres-sion for ǫ ( ω ) for an infinite sample ǫ ( ω ) = 1 − πn h | d F F | F +1) 1 ω − ω + i √ ǫ ( ω ) γ/ πn h | d F F | F +1) 1 ω − ω + i √ ǫ ( ω ) γ/ (1)This equation can be analytically solved and ǫ ( ω ) ex-pressed in terms of well defined external parameters, suchas the natural decay rate γ and n . The reduced matrixelement d F F for a given dipole transition can be also ex-pressed by γ . We consider here a closed transition, suchthat for hyperfine levels only the selected lower and up-per states with angular momenta F and F respectivelycan be radiatively. The above result can be applied tocalculation of the scattering cross section of light by aspherical sample of large radius a ( a ≫ λ ) in the stan-dard formalism of the Debye-Mie problem. Q S = π k ∞ X J =1 (2 J + 1) (cid:20)(cid:12)(cid:12)(cid:12) − S ( e ) J (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) − S ( m ) J (cid:12)(cid:12)(cid:12) (cid:21) Q A = π k ∞ X J =1 (2 J + 1) (cid:20) − (cid:12)(cid:12)(cid:12) S ( e ) J (cid:12)(cid:12)(cid:12) + 1 − (cid:12)(cid:12)(cid:12) S ( m ) J (cid:12)(cid:12)(cid:12) (cid:21) (2)where the scattering matrix components for the T M and
T E modes are respectively given by S ( e ) J = − ǫ ( ω ) j J ( ka )[ rh (2) J ( ωc r )] ′ r = a − h (2) J ( ωc a )[ rj J ( kr )] ′ r = a ǫ ( ω ) j J ( ka )[ rh (1) J ( ωc r ))] ′ r = a − h (1) J ( ωc a )[ rj J ( kr )] ′ r = a S ( m ) J = − j J ( ka )[ rh (2) J ( ωc r )] ′ r = a − h (2) J ( ωc a )[ rj J ( kr )] ′ r = a j J ( ka )[ rh (1) J ( ωc r ))] ′ r = a − h (1) J ( ωc a )[ rj J ( kr )] ′ r = a (3)Here j J ( . . . ), h (1) J ( . . . ) and h (2) J ( . . . ) are spherical Besselfunctions of J -th order, and k = p ǫ ( ω ) ω/c . Q S isthe elastic part of the cross section responsible for thecoherent scattering of light from the sample boundary.The absorption part Q A is responsible for diffusely scat-tered light via the incoherent channel and the sum Q = Q S + Q A is the total cross section for the entire scat-tering process. Light emerging the sample via incoher-ent channels can be recovered in the Maxwell theory byconsidering secondary and multiply scattered waves gen-erated by fluctuations of χ ( ω ). This can be simulatedby a Monte-Carlo scheme and crucially requires that thedipole sources generating these waves be indistinguish-able on a mesoscopic distance ∼ λ . Then secondary andmultiply scattered waves can be simulated through thetime dependence of the sample fluorescence when the ex-citation light is turned off. The analysis of the transientprocess and time dependence of the fluorescence is sim-plified by a diffuse approximation if the extinction lengthfor the field penetration inside the sample is ≫ λ . We now turn to the quantum-posed description of thephoton scattering problem. There photon scattering onan atomic system is expressed by the following relationbetween the total cross section and the T -matrix Q = V ¯ h c ω ′ (2 π ) Z X e ′ | T g e ′ k ′ ,g ek ( E i + i | d Ω (4)where V is a quantization volume. Here we keet only theRayleigh channel and assumed that the atomic system isdescribed by the same ground state g before and afterthe scattering process, which includes the averaging overinitial and sum over final Zeeman states. The initial en-ergy of the entire system E i is given by E i = E g + ¯ hω and the incoming and outgoing photons have the samefrequency ω ′ = ω . We consider below the simplest rele-vant example of a ”two-level” atom, which has only onesublevel in its ground state and three Zeeman sublevelsin its excited state, such that F = 0 and F = 1 andin an isotropic situation the total cross section does notdepend on the momentum direction and polarization ofthe incoming photon.The T -matrix is expressed by the total Hamiltonian H = H + V and by its interaction part V as T ( E ) = V + V E − H V. (5)In the rotating wave approximation the internal resolventoperator contributes to (4) only by being projected on thestates consisting of single atom excitation, distributedover the ensemble, and the vacuum state for all the fieldmodes. Defining such a projector as P the projectedresolvent ˜ R ( E ) = P E − H P (6)performs a 3 N × N matrix, where N is the number ofatoms. For the dipole-type interaction between atomsand field this projected resolvent can be found as thereversed matrix of the following operator˜ R − ( E ) = P (cid:18) E − H − V Q E − H QV (cid:19) P (7)where the complementary projector Q = 1 − P , op-erating in the self-energy term, can generate only twotypes of intermediate states: a single photon + all theatoms in the ground state; and a single photon + twodifferent atoms in the excited state and others are inthe ground. For such particular projections there is thefollowing important constraint on the dipole-type inter-action V : P V P = QV Q = 0. Due to this constraintthe series for the reversed resolvent (7) is expressed bya finite number of terms, which would be not the casein a general situation [12]. The resolvent ˜ R ( E ) can benumerically calculated and, for an atomic system con-sisting of a macroscopic number of atoms, when N ≫ any type of collective atomic excitations cancontribute, which are not responsibly cooperative . Onecan then expect significant difference in their predictionsfor the cross section and for the fluorescence behavior.The time dependence of the fluorescence signal can bebuilt up via Fourier expansion of the outgoing photonwavepacket with the S -matrix formalism.Below we consider the spectral dependence of the crosssections, calculated in both the self-consistent mesoscopicmodel and in an exact microscopic approach. Our cen-tral idea is to follow how this dependence is modifiedwhen the dimensionless density of atoms n λ is variedfrom smaller to greater values. In Fig. 1 we reproducesuch a spectral dependence when the density of atomsis small, n λ = 0 .
02. There is excellent agreementbetween the Debye-Mie and microscopically calculateddata and the microscopic result is insensitive to configu-ration averaging over random atomic distributions. Thisjustifies that only cooperative modes, which allow themacroscopic Maxwell description, contribute to the scat-tering process. As a consequence the long term timedynamics of the fluorescence signal can be very well ap-proximated by a Holstein mode. Such a spectral andtemporal behavior is typical for dilute atomic systems aswas verified by our numerical simulations done for theatomic ensembles of different sizes and consisting of dif-ferent numbers of atoms. While varying the density togreater values n λ ∼ n c ∼ .
09 the solution of the self-consistent equation (1) turns the permittivity to nega-tive values in a part of the spectrum where ǫ ′ < ǫ ′′ = 0. As is well known in, e.g. plasma physics [11] thenegative permittivity can be associated with a forbid-den spectral zone, where the radiation cannot penetrateinside the system and can exist only in the form of asurface light-matter wave. In the limit of even higherdensity samples, the light undergoes mainly surface scat-tering and the absorption (incoherent/diffusion scatter-ing) channels suppressed. This is illustrated in Fig. 2 bythose spectral dependencies of the cross section, whichwere calculated in the Debye-Mie model for n λ = 0 . Detuning, -10 -5 0 5 10 Q , Q S , Q A , D/g ( l / p ) FIG. 1: The spectral dependence of the total Q , elastic Q S ,and absorption Q A cross sections for an atomic sample ofsize a = 25 λ and density n λ = 0 .
02. The configurationaveraged microscopic result, shown as a gray solid line, is ev-idently the same as the data taken for particular atomic con-figuration, shown as a black line. The results of a mesoscopicself-consistent approach (red curves) reproduce the exact mi-croscopic spectra. Color online.
Detuning, -40 -20 0 20 40 Q , Q S , Q A , D/g ( l / p ) FIG. 2: Same as Fig.1, but for a sample size a = 10 λ and den-sity n λ = 0 .
5. The results of the self-consistent mesoscopicapproach (red curves) perceptibly differ from the exact mi-croscopic spectra. In turn, the microscopic calculation givenfor a particular configuration (black curve) indicates specklemicro-cavity structure generated by the resolvent poles. Coloronline. distributed atomic scatterers. Any distribution createsa specific quantization problem for the incoming field,whose mode structure can be properly defined in termsof standard scattering theory. The main difficulty for thequantization procedure is description of the complicatedstructure of the resonance states. The resonances are de-
Time, g t I n t e n s it y ( A r b . U n it s ) -5 -4 -3 -2 -1 (l/2p) = 0.015Diffuse Holstein modeMicro n (l/2p) = 0.5 FIG. 3: Time dependent fluorescence decay for samples ofdifferent densities and with relatively equal optical depths.The dilute system has a sample radius a = 27 λ and density n λ = 0 . a = 5 λ and n λ = 0 .
5. Color online. scribed by the resolvent poles and can be specified byvarious superpositions of atomic states, which transformthe reversed projected resolvent (7) to diagonal form.The micro-cavity structure manifests itself by finely re-solved speckle dependence in the scattering spectrum andthe scattering process becomes extremely sensitive to thespatial configuration of atomic scatterers, as illustratedin Fig. 2. Some of these resonance states have a sub-radiant nature and manifest themselves via significantlyslowed long time decay of the fluorescence in compari-son with a classical Holstein mode. Fig. 3 illustrates thedifference in the fluorescence decay for dilute and denseatomic systems initially excited by a short probe pulse.The external parameters for the systems are such thatthe optical depth for both the systems is nearly the same.For a dilute system the long term asymptote, extractedthrough exact microscopic calculations, is well describedby a diffuse Holstein-type mode evaluated via the samplemacroscopic characteristics. For the dense system thereis evident deviation of the asymptotic behavior from suchbehavior. Such a deviation should be associated with thepresence of sub-radiant resonance states in the resolventpoles.The analogy of sub-radiant states with a localizationprocess, usually discussed as a multiple wave scatteringproblem, suggests evolution with density of the qualita-tive scattering properties of the atomic gas. The dis-cussed characteristics, such as the spectral dependenceof the sample cross-section or the time-dependent fluo-rescence, have no direct relations with static thermody-namic properties of the system, which could be evaluatedbased on its partition function. However, they describehow the system in equilibrium responds to optical exci-tation near the atomic resonance transition. The atomictransition is dressed by cooperative dipole-field interac-tions and at higher densities, the disordered atomic sys- tem reveals a micro-cavity structure. Such a micro-cavityhas unique properties for any particular configurationand its mode splitting is competitive with the mode de-cay rate. The modification of the dynamic response to anexternal field with density is an intrinsic property of thesystem, which may be visualized in terms of a cross overor a change of phase. In this sense our calculations showthat the transformation of the system behavior from oneof individual, independent atomic scatterers to the co-operatively organized micro-cavity structure mainly de-velops within a narrow and critical density zone. Theconfiguration sensitive speckle structure of the spectralcross section manifests itself in those conditions whenthe self-consistent permittivity becomes negative in partof the spectrum. The description of the atomic systemwith the macroscopic Maxwell theory preferably yieldssurface scattering of light in this case. More precise mi-croscopic description shows that part of the excitationcan penetrate the system and be converted into a long-time decay of the fluorescence signal.We appreciate the financial support by NSF (GrantNo. NSF-PHY-0654226) and INTAS (Project No 7904). ∗ Electronic address: [email protected]; currently atV.A. Fock Physics Institute, St.-Petersburg University,198504 Stary Petershof, St.-Petersburg, Russia[1] C.J. Pethick and H. Smith, Bose-Einstein Condensa-tion in Dilute Gases, Cambridge University Press (Cam-bridge, UK, 2002).[2] R. Wester, S.D. Kraft, M. Mudrich, M.U. Staudt, J.Lange, N. Vanhaecke, O. Dulieu and M. Weidemller,Appl. Phys. B: Lasers and Optics , 993 (2004).[3] Ryan M. Camacho, Curtis J. Broadbent, Irfan Ali-Khan,and John C. Howell, Phys. Rev. Lett. , 043902 (2007).[4] D. Bouwmeester, A. Ekert, and A. Zeilinger, The physicsof quantum information (Springer-Verlag Berlin, 2000)[5] R. H. Dicke, Phys. Rev. , 99 (1954).[6] P.W. Anderson, Phys. Rev. , 1492 (1958).[7] Hui Cao, Lasing in Random Media , Waves Random Me-dia, v.12, p. R1 (2003).[8] E. Akkermans and G. Montambaux,
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