The interaction of light and matter

Transcription

1 Outline The interaction of light and matter Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Elementary processes Elementary processes 1 Elementary processes Einstein relations Optical gain Population dynamics 3 Time-dependent perturbation theory Electric dipole approximation Atomic states Rabi oscillations 4 MCWF-method Relation to Einstein B 5 Spontaneous decay 6 Driving atoms Lorentz oscillator Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, 014 / 3 Einstein relations Einstein relations e Time-derivatives of populations: Absorption B ge ρ(ω) Spontaneous Emission Stimulated Emission A eg B eg ρ(ω) Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 B ge Steady state: B eg A eg g ρ(ω) = Boltzmann s and Planck s law: N g = g ( ) g ω exp N e g e k B T Thus: dn g = dn e = N e A eg N g B ge ρ(ω) N e B eg ρ(ω) A eg (N g /N e )B ge B eg ρ(ω) = ω3 π c 3 1 exp( ω/k B T ) 1 B eg = g ( ) g B ge and ω 3 A g eg = e π c 3 B eg Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3

2 Einstein relations Conditions for optical gain Optical gain Steady-state inversion Einstein relations Optical gain Gain: Thus: Thermal: N e B eg ρ(ω) > N g B ge ρ(ω) N e g e > N g [ N e /g e = exp E ] e E g < 1 N g / k B T Lasing is impossible in thermal equilibrium R e R g N e A eg e g τ e Steady state dn e / = dn g / = 0 or Gain: dn e dn g = R e N e τ e = R g N e A eg N g N e = R e τ e and N g = R g R e τ e A eg R e R g τ e g e [ 1 g ] e A eg > 1 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Einstein relations Optical gain Conditions for inversion Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Einstein relations Population dynamics Atomic population dynamics Gain: Conditions: R e R g τ e g e [ 1 g ] e A eg > 1 Differential equations: dn g = dn e = N e A eg N g B ge ρ(ω) N e B eg ρ(ω) Assuming N g (0) = N 0 and N e (0) = 0 leads to Selective pumping: R e > R g Favourable lifetimes: τ e > Favourable degeneracy: > g e Necessary, but not sufficient A eg < g e 1 and N e (t) = N 0 B ge ρ(ω)τ (1 ) e t/τ 1 τ = B geρ(ω) B eg ρ(ω) A eg. Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3

3 Time-Dependent Perturbation Theory The time-dependent Schrödinger equation Time-dependent perturbation theory Ψ( r, t) HΨ( r, t) = i t Field-free Hamiltonian H 0, eigenvalues E n ω n, eigenfunctions φ n ( r) Total Hamiltonian: H(t) = H 0 H (t) Solution Ψ( r, t): Ψ( r, t) = k c k (t)φ k ( r)e iω kt Time-Dependent Perturbation Theory Substitute and H 0 φ k ( r) = ω k φ k ( r) Time-dependent perturbation theory ( ) t c k(t)e iωkt dck = iω k e iωkt, Multiply on both sides with φ k ( r) and integrate over r using the normalization d r φ k ( r)φ k ( r) = δ k,k Substitution: H(t)Ψ( r, t) = [H 0 H (t)] c k (t)φ k ( r)e iω kt k = i c k (t)φ k ( r)e iω kt t k Result: i dc j(t) = k c k(t)h jk (t)eiω jkt H jk (t) φ j H (t) φ k and ω jk (ω j ω k ) Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Electric dipole approximation Electric dipole approximation Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Atomic states Excitation of Na with laser light 5 Na Neutral system r α q α = 0. Dipole moment µ = q α r α, α α 4 3 5s 4s 5p 4p 4d 3d 4f Using F = q α E and F = U, yields for the Hamiltonian H = q α r α E = µ E α Energy (ev) 3p Dipole moment one-electron atom (nucleus in origin): µ = e 0 e r = e r. 1 Laser cooling λ= nm The electric dipole interaction is the only coupling of an atom with an electro-magnetic field in the long wavelength approximation ( r λ) 0 3s S P D F Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3

4 Excitation of Na Atomic states Polarisation dependence Atomic states 6 MHz 36 MHz 16 MHz F=3 F= F=1 F=0 3 P 3/ 19 MHz F= F=1 3 P 1/ D nm D nm Linear polarized light Circular polarized light 177 MHz F= F=1 3 S 1/ Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 The Rabi Two-Level Problem Two states (1 g and e): i dc g (t) = c e (t)h ge(t)e iω eg t and Rabi oscillations i dc e(t) = c g (t)h eg (t)e iω eg t Electric-dipole approximation: H (t) = µ E( r, t) Plane wave: E( r, t) = E 0 cos(kz ωt) = 1 E 0 ( e ikz iωt e ikziωt) Substitution (z = 0): H eg (t)e iω eg t = e iω eg t e µ E g = e µ g ( E0 e i(ω eg ω)t E ) 0 e i(ω eg ω)t Rotating wave approximation: H eg (t) = e µ g E 0 e i(ω eg ω)t H ge(t) = g µ e E 0 e i(ω eg ω)t Rabi frequency: Ω = e µ E 0 g / Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Rabi oscillations The Rabi Two-Level Problem (cont.) Final result (δ = ω ω eg ): Uncoupling: d c g (t) dc g (t) iδ dc g (t) Ω = i c e(t)ω e iδt and dc e(t) 4 c g (t) = 0 and d c e (t) Initial conditions (c g (0) = 1 and c e (0) = 0): c g (t) = ( cos Ω t i δ ) Ω sin Ω t e iδt/ where Ω Ω δ. = i c g (t)ωe iδt iδ dc e(t) Ω 4 c e(t) = 0 and c e (t) = i Ω Ω sin Ω t e iδt/ Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3

5 Rabi oscillations Rabi oscillations Probability Rotating frame transformation: c e (t) c e (t)e iδt leads to dc g (t) = iω c e(t) and d c e(t) Introducing the Bloch vector R = (u, v, w) with = iω c g (t) iδ c e (t). u R(c g c e ) v I(c g c e ), and w c e c g Time Probability c e (t) for the atom to be in the excited state for Ω = γ and δ = 0 (solid line), δ = γ (dotted line), and δ =.5γ (dashed line). Time is in units of 1/γ. Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 leads to du = δv dv or with Ω = ( Ω, 0, δ) this becomes = δu Ωw and dw = Ωv. d R = Ω R Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Two level atom t =3π/Ω Optical Bloch Equations (γ A eg ): B u v A t = π/ω B u (a) (b) (c) (a) Application of a π/ω and 3π/Ω pulse with δ = 0 brings the Bloch vector from the south pole to the equator. (b) A subsequent π/ brings the vector either to the north pole (excited state), or the south pole (groud state). (c) If the detuning is non-zero, a more complicated trajectory is induced. v A C C v A du = γ u δv dv = γ v δu Ωw dw = γ(1 w) Ωv. Here γ takes into account the increased damping rate of the optical coherences compared to the populations due to phase-changing collisions (for an isolated atom, γ γ ). Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3

6 Numerical integration of OBE Two level atom (continued) Excitation probability Time Probability c e (t) for the atom to be in the excited state for Ω = γ and δ = γ by numerical integration of the OBEs. The solutions are identical to the Monte Carlo wavefunction method with an infinite number of atom trajectories. Time is in units of 1/γ. Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Scattering rate Scattering rate γp [γ ] s 0 =100.0 s 0 = 10.0 s 0 = 1.0 s 0 = Detuning δ [γ ] Excitation rate γ p as a function of the detuning δ for several values of the saturation parameter s 0. Note that for s 0 > 1 the line profiles start to broaden substantially from power broadening. Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Steady state: c e s = (1 s) = s 0 / 1 s 0 (δ/γ) with the off-resonance saturation parameter s: s Ω / δ γ /4 s 0 1 (δ/γ) and the on-resonance saturation parameter: Saturation intensity: Scattering rate γ p : s 0 Ω γ γ p = γρ ee = I s πhc 3λ 3 τ = I I s s 0 γ/ 1 s 0 (δ/γ) Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, 014 / 3 MCWF-method Monte Carlo Wavefunction Method Start with the atom in the ground state (c g = 1, c e = 0). After a short period t, determine the new amplitudes by a coherent evolution of the state vector c = (c g, c e ): ( c 0 Ω = H c and H = / Ω/ (δ iγ/) with γ the decay rate due to spontaneous emission. The probability p for the atom to decay is given by p = c e γ t. From a random string of numbers take one number x. If x > p, nothing happens. If x < p, the atom decays to the ground state. In the first case, we have to renormalize the wavefunction to c g c e = 1. In the second case, we have c = (1, 0). If we repeat this procedure for one atom, we have a single trajectory of the time-evolution of the atom. This method can be repeated for several atoms. Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 ),

7 MCWF-method Excited State Decay and its Effects Excitation probability (a) (b) (c) Time Trajectories for atoms is a radiation field with Ω = γ and δ = γ, where γ is the natural wih. The number of atoms averaged over is 1 (a), 10 (b), and 100 (c). Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Relation to Einstein B Calculating the Einstein B-coefficient Relation to Einstein B Calculating the Einstein B-coefficient Probability in excited state (Rabi flopping) c e (t) = Ω sin Ω Small excitations (Ω Ω = Ω δ ): c e (t) = Ω t 4 sinc Broad band excitation using U = 1 ε 0E 0 = leads to a transition rate R: where δ = ω eg ω. R = c e(t) t = µ t ε 0 0 ( Ω t ). ( ) δt. ρ(ω)dω ( ) δt ρ(ω)sinc dω, Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Relation to Einstein B Validity of the rate-equation approach Assuming bandwih much larger than wih of the transition and using ( ) δt sinc dω = π t leads to or 0 R B ge ρ(ω eg ) = πµ ε 0 ρ(ω eg ) Ω δ, γ B ge = πµ ε 0 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3

8 Spontaneous decay Spontaneous decay (emission) Spontaneous decay Spontaneous decay (emission) We need Quantum Electro Dynamics (QED)! INTERAKTIE r ATOOM - VACUUM VELD p 1s A ge = ( ωeg 3 π c 3 A eg = λ 3 ψ e r ψ g [s 1 ] ge ) B eg = 4αω eg 3 3c ψ e r ψ g ω eg = (E e E g )/ A eg (E e E g ) 3 and A eg ψ e r ψ g W s ba E b E a ψ e r ψ g = (0.74) a.u. λ ge = 11.67nm W s eg (p 1s) = (11.67) 3 (0.74) = s 1 Lifetime: τ p = 1/W s eg = 1.63ns state 1s s p 3s 3p 3d 4s 4p 4d 4f τ(ns) Metastable (τ 0.15 s) λ ge in nm (=10 9 m) Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Driving atoms Classical atom-field interaction Solution: ω 0 Substitution: m ω a iγωa ω 0 a = ee 0 m Lorentz oscillator Lorentz equation: x(t) = ae iωt a e iωt Approximation (ω 0 ω, δ = ω ω 0 ω): a ee 0/mω 0 δ iγ Final result: ẍ γẋ ω 0 x = ee 0 m or a = 1 / x 0 e iφ with x 0 = ee 0/mω 0 γ 4δ x = x 0 cos(ωt φ) cos ωt ee 0 /m (ω 0 ω ) iγω and φ = arctan ( γ δ Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 ) Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3 Driving atoms Classical atom-field interaction Power absorbed: Lorentz oscillator P = F ẋ = eeẋ = eωx 0 E 0 [ cos ωt sin ωt cos φ cos ωt sin φ ] Time average: Define: I = 1 ε 0cE 0 Scattering rate R: and P = e γe 0 m(γ 4δ ) I s = ε 0mcγ ω e or s 0 I = e E 0 I s mγ ω R = P ω = s 0 γ/ 1 (δ/γ) OBE Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, / 3