Atoms and photons. Chapter 1. J.M. Raimond. September 6, J.M. Raimond Atoms and photons September 6, / 36

Size: px

Start display at page:

Download "Atoms and photons. Chapter 1. J.M. Raimond. September 6, J.M. Raimond Atoms and photons September 6, / 36"

Brandon Black
5 years ago
Views:

1 Atoms and photons Chapter 1 J.M. Raimond September 6, 2016 J.M. Raimond Atoms and photons September 6, / 36

2 Introduction Introduction The fundamental importance of the atom-field interaction problem Provides all information we have on the universe Provides the most precise theory so far: QED Provides the best tests of fundamental quantum physics J.M. Raimond Atoms and photons September 6, / 36

3 Introduction Introduction The practical importance of the atom-field interaction problem Lasers Atomic clocks Cold atoms and BEC Quantum simulation J.M. Raimond Atoms and photons September 6, / 36

4 Introduction Outline of this course Chapter 1: Interaction of atoms with a classical field 1 The harmonically bound electron: a surprisingly successful model 2 The Einstein coefficients J.M. Raimond Atoms and photons September 6, / 36

5 Introduction Outline of this course Chapter 2: Quantized atom and classical field 1 Interaction Hamiltonian 2 Free atom and resonant field 3 Relaxing atom and resonant field 4 Optical Bloch equations 5 Applications of the optical Bloch equations J.M. Raimond Atoms and photons September 6, / 36

6 Introduction Outline of this course Chapter 3: Field quantization 1 Field eigenmodes 2 Quantization 3 Field quantum states 4 Field relaxation J.M. Raimond Atoms and photons September 6, / 36

7 Introduction Outline of this course Chapter 4: quantized matter and quantized field 1 Interaction Hamiltonian 2 Spontaneous emission 3 Photodetection 4 The dressed atom 5 Cavity Quantum electrodynamics J.M. Raimond Atoms and photons September 6, / 36

8 Introduction Bibliography Cohen-Tannoudji, Dupont-Roc and Grynberg, An introduction to quantum electrodynamics and Photons and atoms, Wiley, 1992 Cohen-Tannoudji and Guery Odelin Advances in atomic physics: an overview, World Scientific 2012 Schleich, Quantum optics in phase space, Wiley 2000 Vogel, Welsch and Wallentowitz, Quantum optics an introduction, Wiley 2001 Meystre and Sargent Elements of quantum optics, Springer 1999 Barnett and Radmore Methods in theoretical quantum optics, OUP, 1997 Scully and Zubairy Quantum optics, 1997 Loudon Quantum theory of light, OUP 1983 Haroche and Raimond Exploring the quantum, OUP 2006 and, of course, the online lecture notes and slides handouts. J.M. Raimond Atoms and photons September 6, / 36

9 Introduction Online lecture notes following the menu items teaching, Jean-Michel Raimond J.M. Raimond Atoms and photons September 6, / 36

10 Introduction Outline 1 Introduction J.M. Raimond Atoms and photons September 6, / 36

11 Introduction Outline 1 Introduction 2 The harmonically bound electron J.M. Raimond Atoms and photons September 6, / 36

12 Introduction Outline 1 Introduction 2 The harmonically bound electron 3 Einstein s coefficients J.M. Raimond Atoms and photons September 6, / 36

13 The harmonically bound electron A classical model: the harmonically bound electron The simplest classical model for an atom: a single charge (electron) bound to a force center by an harmonic potential. An early atomic theory model (Thomson s plum pudding ) A good guide to identify relevant parameters by dimensional analysis Surprisingly accurate predictions J.M. Raimond Atoms and photons September 6, / 36

14 The harmonically bound electron A classical model: the harmonically bound electron Equations of motion Dynamics d 2 r dt 2 + ω2 0r = 0 (1) Solution r = r 0 exp( iω 0 t) (2) J.M. Raimond Atoms and photons September 6, / 36

15 The harmonically bound electron A classical model: the harmonically bound electron Equations of motion Damping: radiation reaction. Model emitted power by a viscous damping term in the equation of motion. A reasonable approximation for weak damping. Larmor formula for radiated power where P = q2 a 2 6πɛ 0 c 3 = mτa2 (3) τ = 1 6πɛ 0 q 2 mc 3 = s (4) linked to the classical radius of electron r e = r e = 3 2 cτ q2 4πɛ 0 mc 2 = m by J.M. Raimond Atoms and photons September 6, / 36

16 The harmonically bound electron A classical model: the harmonically bound electron Equations of motion Modified equation of motion with d 2 r dt 2 + γ dr dt + ω2 0r = 0 (5) γ = ω 2 0τ (6) being the amplitude damping coefficient obtained by equalling the average dissipated energy to the average radiated power (the energy damping coefficient is obviously 2γ). J.M. Raimond Atoms and photons September 6, / 36

17 The harmonically bound electron A classical model: the harmonically bound electron Equations of motion Order of magnitude estimate for γ: ω 0 R y /, where R = mc 2 α 2 /2 is the Rydberg constant and α = q2 4πɛ 0 c (7) the fine structure constant. Then γ = ω 0 τ = Rτ ω 0 = R 1 q 2 6πɛ 0 mc 3 = α (8) J.M. Raimond Atoms and photons September 6, / 36

18 The harmonically bound electron A classical model: the harmonically bound electron Polarizability Response to a classical oscillating field E 0 u z exp( iωt) Equation of motion Steady-state solution d 2 r dt 2 + γ dr dt + ω2 0r = qe 0 m u ze iωt (9) Position: r = r 0 exp( iωt); Dipole: d = d 0 exp( iωt) with d 0 = qr 0 = ɛ 0 α c E 0 u z (10) where α c = q2 mɛ 0 1 ω 2 0 ω2 iγω (11) J.M. Raimond Atoms and photons September 6, / 36

19 The harmonically bound electron A classical model: the harmonically bound electron Diffusion Total power diffused by the atom given by Larmor formula: P = 1 2 mτω4 r 0 2 (12) or P = ɛ 0 12πc 3 α c 2 ω 4 E 2 0 (13) Cross Section Ratio of this power to the incident power per unit surface P i = ɛ 0 ce 2 0 /2: σ c = 1 ( ω ) 4 αc 2 = 8π 6π c 3 r e 2 ω 4 (ω0 2 ω2 ) 2 + γ 2 ω 2 (14) J.M. Raimond Atoms and photons September 6, / 36

20 The harmonically bound electron A classical model: the harmonically bound electron The three diffusion regimes Rayleigh diffusion for ω < ω 0 and ω 0 ω γ σ c = 8π 3 r 2 e Blue sky: σ c m 2, N = m 3 : the attenuation length is L = 1/Nσ c 100 km ω 4 ω0 4 (15) Thomson diffusion for ω > ω 0 σ c = 8π 3 r 2 e. (16) The resonant regime for ω ω 0 σ c = 8π 3 r 2 e ω 2 0 4(ω 0 ω) 2 + γ 2 (17) J.M. Raimond Atoms and photons September 6, / 36

21 The harmonically bound electron A classical model: the harmonically bound electron Resonant diffusion At exact resonance ω 0 = ω: With where λ 0 = 2πc/ω 0 is the wavelength. Hence σ c = 8π 3 r 2 e ω 2 0 γ 2 (18) r e ω 0 γ = 3 2 cτ 1 ω 0 τ = 3 4π λ 0 (19) σ c = 3 2π λ2 0 (20) This model doe not apply for high powers: saturation (about 1 mw/cm 2 ). A quantum effect. More on that in next Chapter. J.M. Raimond Atoms and photons September 6, / 36

22 The harmonically bound electron A classical model: the harmonically bound electron Propagation in matter Apply the model to propagation in matter. Simplifiying hypothesis: Consider harmonic plane wave Linear response theory Dilute matter: no difference between local and global field Equation of propagation with ɛ r = 1 + Nα c Dispersion relation where k 0 = ω/c E + ω2 c 2 ɛ r E = 0 (21) k 2 = k 2 0 ɛ r (22) J.M. Raimond Atoms and photons September 6, / 36

23 The harmonically bound electron A classical model: the harmonically bound electron Propagation in matter Refraction index n = ɛ r = n + in n = 1 ɛ r + ɛ 2 r + ɛ 2 r and n = ɛ r ɛ r + ɛ 2 r + ɛ 2 r (23) Real part: refraction (ordinary index), Imaginary part: absorption. Power released in matter 1 2 Re j 0 E 0 where j 0 = iωp 0. E = 1 2 Re ( iωp 0 E 0) (24) E = 1 2 ɛ 0ωχ E 0 2 = 1 2 ɛ 0ωNα E 0 2 (25) J.M. Raimond Atoms and photons September 6, / 36

24 The harmonically bound electron A classical model: the harmonically bound electron Propagation in matter Imaginary part of polarizability: E = 1 2 ɛ 0ωχ E 0 2 = 1 2 ɛ 0ωNα E 0 2 (26) α = q2 γω mɛ 0 (ω0 2 ω2 ) 2 + γ 2 ω 2 (27) Power released always positive, matter always absorbing. Laser needs a quantum ingredient J.M. Raimond Atoms and photons September 6, / 36

25 Einstein s coefficients Einstein s coefficents Introduction A phenomenological description of energy exchanges between light and matter. A very simple description: Field only described by its spectral energy density u ν. Numerical density of photons between ν and ν + dν : u ν /hν. Total energy per unit volume: u = u ν dν Matter made of two-level atoms e above g energies E e and E g. No degeneracy. (E e E g )/h = ν 0. Transition wavelength λ 0 = c/ν 0. Number (or density) of atoms in the two levels N e and N g, normalized to the total atom number (or density N ) so that N e + N g = 1. Goal: obtain rate equations for the variations of N e and u ν. We shall consider in particular the radiation/matter thermal equilibrium at a temperature T. For that, three process come into play: J.M. Raimond Atoms and photons September 6, / 36

26 Einstein s coefficients Einstein s coefficents Three processs Spontaneous emission Deexcitation of e with a constant probability per unit time, A eg = Γ. ) dn e dt spon = A eg N e (28) Absorption Transfer from g to e by absorption of photons. Rate proportional to the photon density (a cross-section approach). ) dn e dt abs = B geu ν0 N g (29) J.M. Raimond Atoms and photons September 6, / 36

27 Einstein s coefficients Einstein s coefficents Something is lacking Absorption and spontaneous emission are not enough. At infinite temperature, all atoms in the upper state. Not the prediction of thermodynamics (50% in each state). Einstein adds a third process: Stimulated emission Transition from e to g and emission of a photon at a rate proportional to the photon density. ) dn e dt stim = B eg u ν0 N e (30) Einstein s rate equations dn e dt = A eg N e B eg u ν0 N e + B ge u ν0 N g (31) J.M. Raimond Atoms and photons September 6, / 36

28 Einstein s coefficients Einstein s coefficents Relations between the three coefficients At thermal equilibrium (temperature T ) N e N g = e (Eg Ee)/k bt = e hν 0/k b T (32) k b : Boltzmann constant. And (Planck s law) u ν0 = 8πhν 0 3 c 3 1 exp(hν 0 /k b T ) 1 (33) J.M. Raimond Atoms and photons September 6, / 36

29 Einstein s coefficients Einstein s coefficents Relations between the three coefficients Steady state for T i.e. u ν0 and N e /N g 1. Neglect spontaneous emission. B ge = B eg = B (34) Noting A eg = A, steady state at a finite temperature T : Hence A + Bu ν0 = Bu ν0 e hν 0/k b T u ν0 = A B Comparing with Planck s law 1 exp(hν 0 /k b T ) 1 (35) (36) A B = 8πhν 0 c 3 3 = 8πh λ 3 0 (37) J.M. Raimond Atoms and photons September 6, / 36

30 Einstein s coefficients Einstein s coefficents Case of degenerate atomic levels g e and g g degeneracies of energies E e and E g. At thermal equilibrium N e /N g = (g e /g g ) exp( hν 0 /k b T ) (38) From the infinite temperature limit: B ge /B eg = g e /g g (39) A purely algebraic complication, not to be considered any further here. J.M. Raimond Atoms and photons September 6, / 36

31 Einstein s coefficients The Laser Light Amplification Stimulated emission: addition of energy to the incoming wave. A simple situation: plane wave at frequency ν 0 on a thin slice of atoms. Incoming power per unit surface P, outgoing P + dp. Obviously: dp P(N e N g ) = P D (40) where we define the population inversion density: D = N e N g (41) The power increases when D > 0: gain when population inversion J.M. Raimond Atoms and photons September 6, / 36

32 Einstein s coefficients The Laser Population inversion Conditions to achieve D > 0 No thermal equilibrium No two-level system (in the steady state) Three or four level system Case of a four level system (f : ground state, i intermediate, plus e and g: Fast incoherent pumping from f to i Fast relaxation from i to e Stimulated emission from e to g Extremely fast relaxation from g to f J.M. Raimond Atoms and photons September 6, / 36

33 Einstein s coefficients The Laser Principle Gain + feedback = oscillation A laser is composed of an amplifying medium (gain) and of an optical resonant cavity (feedback). When the gain exceeds the losses in the feedback, a self-sustained steady-state oscillation occurs. J.M. Raimond Atoms and photons September 6, / 36

34 Einstein s coefficients The Laser A simple model Catpures the main physical ideas without any complication. Forget about all details and proportionality constants. Variables Population inversion density D. If g strongly damped, D = N e. Intra-cavity intensity I (photon density) Evolution of intensity di = κi + gid (42) dt κ: rate of internal or coupling cavity losses. J.M. Raimond Atoms and photons September 6, / 36

35 Einstein s coefficients The Laser A simple model Evolution of population inversion with dd dt Λ : pumping rate in the upper level e = Λ ΓD gid (43) Γ : relaxation rate of e (spontaneous emission in modes other than the cavity one, other sources of atomic losses) J.M. Raimond Atoms and photons September 6, / 36

36 Einstein s coefficients The Laser Steady state Laser off solution I = 0 always a solution D = Λ/Γ Laser on solution D = κ/g Relevant if I 0 Threshold condition I = 1 κ ( Λ Γκ ) g Λ Λ t = Γκ g (44) (45) J.M. Raimond Atoms and photons September 6, / 36

37 Einstein s coefficients The Laser Steady state Stability of the solutions: Λ < Λ t : only solution I = 0 Λ Λ t : two possible solutions, but I = 0 unstable J.M. Raimond Atoms and photons September 6, / 36

38 Einstein s coefficients The Laser Main properties of laser radiation Directive Intense Narrow band and extremely long coherence length or... Extremely short pulses (as) J.M. Raimond Atoms and photons September 6, / 36

MODERN OPTICS. P47 Optics: Unit 9

MODERN OPTICS. P47 Optics: Unit 9 MODERN OPTICS P47 Optics: Unit 9 Course Outline Unit 1: Electromagnetic Waves Unit 2: Interaction with Matter Unit 3: Geometric Optics Unit 4: Superposition of Waves Unit 5: Polarization Unit 6: Interference