Lecture 4 Diffusing photons and superradiance in cold gases Model of disorder-elastic mean free path and group velocity. Dicke states- Super- and sub-radiance. Scattering properties of Dicke states. Multiple scattering and superradiance. Comparison with experiments: slow diffusion of light. Strong disorder-effective Hamiltonian. Distribution of escape times
Scalar waves in random media Scalar and monochromatic ( k 0 ) electromagnetic wave: ψ(r) is the electric field, solution of the Helmholtz wave equation: 2 ψ k0µ(r)ψ 2 = k0ψ 2 (compare to Schrödingerequation with disorder) Disorder potential is continuous : fluctuations of dielectric constant V (r) = k0µ(r) 2 = δɛ/ɛ Gaussian white noise model: easy to do calculations V (r) = 0 V (r)v (r ) = Bδ(r r ) B is related to scattering properties of individual scatterers
Edwards model : N i identical, localized, randomly distributed scatterers V (r) = N i j=1 v(r r j ) The potential v(r) is short range compared to so that v(r r j ) = v 0 δ(r r j ) k0 1 Weak potential limit (Born approximation) the scattering cross section of a single scatterer is σ = v2 0 4π and B = n i v 2 where n 0 i is the density of scatterers.
Average amplitude of the field Solution of the wave eq. with a source j(r) is given in terms of the Green s function G(r, r ) : ψ(r) = dr j(r )G(r, r ) Solution of ( 2 + k 2 0 V (r) ) G(r, r ) = δ(r r ) G(r,r ') may be expressed in terms of the free Green s function without scattering potential: G 0 (r,r ') G(r,r ') = G 0 (r,r ') dr 1 G(r,r 1 )V(r 1 )G 0 (r 1,r ') Disorder average restores translational invariance and the Fourier transform of the Green s function is G(k)
G(k) is expressed in terms of the self-energy Σ(k) G(k) = G 0 (k) 1+ Σ(k)G(k) the self-energy is given by the sum of irreducible scattering events =... 1 2 3 4 The main contribution to scatterers, In real space: Σ(k) neglects interference effects between 1 interferences : 2
The self-energy Σ is a complex valued function. Its imaginary part defines the elastic mean free path l, k 0 l the first neglected term provides a correction Average Green function: = Im Σ 1 (k) = n i σ Im Σ 2 (k) = π 2k 0 l Im Σ 1(k) 1 identify the small parameter : weak disorder k 0 l 1 G(r,r ') = G 0 (r,r ')e R 2l = 1 e ik 0 R R 2l 4π R e k 0 l 1 without disorder
Σ 1 is proportional to the average polarizability of the scattering medium, so that its real part gives the average index of refraction η = ck/ω η = (1 ( c ω )2 ReΣ 1 ) 1/2 The group velocity v g = dω dk of the wave inside the medium is, c = η + ω dη v g dω = 1 η (1 ( c2 2ω ) d ) dω ReΣ 1
Summary: multiple scattering Characteristic lengths: Wavelength: λ 0 Elastic mean free path: density of scatterers l = 1 n i σ n 1/d i scattering cross section Weak disorder λ 0 l independent scattering events
Multiple and resonant scattering of photons by a cold atomic gas. Γ δ{ J e m e ω 0 Jm
Scattering cross section and elastic mean free path The scattering cross section free path by l = 1/n i σ l σ is related to the elastic mean λ l = n 3λ 3 ia JJe 2π 1 1 + (2δ/Γ) 2 with a JJe = 1 3 2J e + 1 2J + 1 Resonant scattering is much more efficient than Rayleigh scattering 10
We have used a model of disorder where scatterers are independent : Edwards model or white noise In atomic gases, there are cooperative effects (superradiance, subradiance) that lead to an interacting potential between pairs of atoms. Dicke states: g = J g = 0, m g = 0 e = J e = 1,m e natural width Γ Pair of two-level atoms in their ground state + absorption of a photon. Unperturbed 0-photon states : Singlet Dicke state : 00 = 1 2 e 1g 2 g 1 e 2 Triplet Dicke states : 11 = e 1 e 2, 10 = 1 2 e 1g 2 + g 1 e 2, 1 1 = g 1 g 2
Second order in perturbation theory in the coupling to photons Superradiant state ε = +1 εv e (r) = ε Γ 2 cosk 0 r k 0 r Γ (ε ) = Γ 1+ sink r ε 0 k 0 r Superradiance V e (r) 1 r (attractive at short distance) Subradiant state ε = 1 1 2 e g 1 2 + g 1 e 2 1 2 e g 1 2 g 1 e 2 Γ (+1) = 2Γ (for r = 0) Γ ( 1) = 0 Characteristics of superradiance Photon is trapped by the two atoms
Scattering properties of Dicke states Scattering amplitudes of a photon by pairs of atoms in superradiant T + or subradiant states are: T T + = Ae i(k k ) R cos ( ) k r 2 cos ( ) k r 2 G + k = k = k 0 ( ) ( ) k r T = Ae i(k k ) R k r sin sin 2 2 R = r 1 + r 2 2 r = r 1 r 2 G ± = 1 δ + i Γ 2 ± Γ 2 e i k 0 r At short distance k 0 r 1 between the two atoms, the subradiant term T δ = ω ω 0 (detuning) is negligible compared to Photon frequency two-level spacing the superradiant term T + G k 0 r
Superradiance and Cooperon The scattering diagram of a photon on a superradiant state is analogous to a Cooperon k ' k 1 k infinite number of exchanges of a virtual photon 2 k ' I 2 = 2 t 2 G 0 1 t 2 G 2 0 2 [1 + cos(k + k ) (r 1 r 2 )] 14
Multiple scattering and superradiance Multiple scattering of a photon by atoms in superradiant states, i.e. coupled by the attractive potential V e (r) 1/r Use Edwards model to calculate the self-energy disorder limit k 0 l 1 Σ (1) e = 6πn i k 0 r m rm 0 dr δ Γ + 1 2k 0 r + i Σ (1) e in the weak n i atomic density r m maximum separation between the two atoms. k 0 l = ImΣ (1) e c v g = 1 η (1 c2 2ω ) d dω ReΣ(1) e Elastic mean free path Index of refraction: η = ( 1 ( c ω ) ) 2 1/2 ReΣ (1) e Group velocity
Absence of divergence of the group velocity The group velocity at resonance is c v g (0) = 1 + 4πn i k 3 0 v g (0) c 3.84 10 5 4 x 10!3 ω 0 Γ (k 0r m ) 2 10 5 for 85 Rb 0.26 Divergence of the group velocity for scattering by independent atoms 3 v g / c v 0 / c 2 1 0!1!2 Group velocity for superradiant states!3!4!5!4!3!2!1 0 1 2 3 4! / " 16
Weak disorder: k 0 l 2 10 3 1 v g (0) 3.1 10 5 c D 0.66m 2 /s k 0 r m 0.51 17
Transport time is defined by: 4 τ tr (δ) = l v g = 3 D v 2 g 1 τ tr Γ 3.5 3 k 0 r m = 0.1 # tr!1 / " 2.5 k 0 r m = 0.07 Transport time τ tr depends weakly on the frequency ω 2 k 0 r m = 0.05 1.5!4!3!2!1 0 1 2 3 4! / " τ tr Γ 18
Strong disorder Onset of a photon localization transition: for a critical amount of disorder (density of atoms), there is a phase transition from delocalized to localized photon states. Ioffe -Regel criterion: λ l where l = 1/n i σ For resonant atom-photon scattering: At resonance, Cold vapor of δ = 0 so that, λ 85 Rb : λ l 10 4 λ l n iλ 3 l = n 3λ 3 ia JJe 2π (far from localization) 1 1 + (2δ/Γ) 2 while n i 10 14 cm 3 and λ l 3.4 19
Cooperative effects and dipole-dipole resonant interation Dipolar atoms are not independent scatterers. There are long range dipole-dipole interactions. Moreover, when two resonant scatterers are close enough, collective states appear (Dicke states) that change substantially the nature of the localization transition. Characterize the onset of localization transition by mean of the distribution P(t) of escape times.
Effective Hamiltonian Atoms = collection of resonant two-level systems: Ground state: g i Excited state: e i H int = i h Γ 2 N i i=1 e i e i h Γ 2 i j e ik r i r j k r i r j d+ i d j with d + i = d e i g i and d i = (d + i ) Diagonal elements: spontaneous emission of isolated atoms Off-diagonal terms: modification of the spontaneous emission due to collective effects and dipole-dipole resonant interaction.
Distribution of escape times Probability π(t) that a photodetector placed outside the atomic cloud will detect a photon at time t. π(t) = Γ i, j sin(k r ij ) k r ij d + i (t)d j (t) Probability to detect a photon between times 0 and t : t 0 P(t)= π (t ')dt ' Laplace transform P (Γ)
0.05 Distribution of escape rates 0.012 Distribution of escape rates Commoness 0.045 0.04 0.035 0.03 0.025 0.02 P (Γ) sinc / cube a = 60 V = 216000! 3 N = 216 " = 0.001 n!1/3 = 10! kl e = 9869.6044 M = 50 0.01 0.008 density n Commoness 0.006 0.004 sinc / cube a = 10 V = 1000! 3 N = 216 " = 0.216 n!1/3 = 1.6667! kl e = 45.6926 M = 50 0.015 0.01 0.002 Diffusive limit Commoness 0.07 0.06 0.05 0.04 0.03 0.005 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 # / # 0 Distribution of escape rates sinc / cube a = 4 V = 64! 3 N = 216 " = 3.375 n!1/3 = 0.66667! kl e = 2.9243 M = 50 Γ/Γ 0 Commoness 0 0 0.5 1 1.5 2 2.5 # / # 0 12000 10000 8000 6000 Distribution of escape rates Subradiant mode sinc / cube a = 0.01 V = 1e!006! 3 N = 216 " = 216000000 n!1/3 = 0.0016667! kl e = 4.5693e!008 M = 50 0.02 Anderson localization 4000 Superradiant mode 0.01 2000 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 # / # 0 0!50 0 50 100 150 200 250 23 # / # 0