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Atom-radiation Interaction • Einstein’s coefficients represent a phenomenological description of the matter-radiation interaction • Prescription for computing the values of the A and B coefficients • No definition of the validity limits • Need for a more general theory for transition probabilities F. De Matteis Quantum Optics L2/1 Atom-radiation Interaction Ηˆ i d dt Atom with no radiation. No explicit time-dependency Atomic Hamiltonian n r , t exp iEnt / y n r Hˆ E y n r En y n r yn stationary states Average value of any observable is time independent. Let’s look at a two level system 1 r , t exp iE1t / y 1 r 2 r , t exp iE 2t / y 2 r F. De Matteis Quantum Optics 0 E2 E1 2 Atom-radiation Interaction Atom in presence of an em radiation field. ˆ Η ˆ Η ˆ t Η ˆ Η E I R HE Atomic Hamiltonian HI Interaction Hamiltonian HR Radiation Hamiltonian HR describes the em field energy HI describes the interaction of the em field with the two level atomic system yn no more stationary states Linear combination of stationary states with time dependent coefficients Ci(t) r , t C1 t 1 r,t C2 t 2 r,t r, t F. De Matteis 2 dV C1 t C2 t 1 2 Quantum Optics 2 3 Interaction Hamiltonian Atom in a radiation field Ηˆ E d2 dC dC2 d Ηˆ I C11 C2 2 i C1 1 C2 1 1 2 dt dt dt dt r , t C1 t 1 r,t C2 t 2 r,t r, t 2 dV C1 t C2 t 1 2 2 Let’s left-multiply by the atomic problem solutions i and integrate dCi * ˆ Η I C11 C2 2 dV i dt i C1P11 C2 exp i0t P12 i dC1 dt C1 exp i0t P21 C2 P22 i dC2 dt F. De Matteis Quantum Optics Pij y i r H Iy j r dV * 4 Atom-radiation Interaction 40 2 11 a0 5 10 m 2 me 10 4 ka0 1 Dipole Approximation Eo cos t N d e rj j 1 k Hˆ I d Eo cos t Pij y i r H Iy j r dV * Bo cos t C2 exp i0t P12 i dC1 dt C1 exp i0t P21 i dC2 dt r, t F. De Matteis 2 P12 P21 * P11 P22 0 P12 E0 d12 cos t V12 cos t dV C1 t C2 t 1 2 Quantum Optics 2 5 Transition Rate V cos t exp i0t C2 i dC1 dt C1 0 1 * V cos t exp i0t C1 i dC2 dt C2 0 0 C2 t t 2 B12W V V12 E0 d12 12 Transition Rate for absorption V 0 E2 E1 Small perturbation Solve by successive approximations Ci≈Ci(0) C1 t 1 * 1 exp i0 t 1 exp i 0 t V C t 2 2 0 0 Let’s substitute in the first member of equation F. De Matteis Quantum Optics 6 Transition Rate V cos t exp i0t C2 i dC1 dt C1 0 1 * V cos t exp i0t C1 i dC2 dt C2 0 0 Small perturbation Second order approx. V 0 E2 E1 2 C1 t 1 V f t V * 1 exp i 0 t 1 exp i 0 t C2 t 2 0 0 C2 t t 2 B12W 2 2 W d E V 0 Einstein’s theory of absorption and emission apply to not too-strong fields F. De Matteis Quantum Optics 7 Rotating wave Approximation * V C t 2 2 C1 t 1 1 exp i0 t 1 exp i 0 t 0 0 Second term in C2(t) is much larger than first C2 t 2 sin 2 0 t 2 1 2 2 2 0 t 2 V V t sinc 2 4 0 1 2 2 V t 0 4 2 Increasing t Max increases Nodes approach the origin of axes F. De Matteis Quantum Optics 8 Rotating wave Approximation Transition Frequency 0 ± 1 2 W d E 0 0 2 V12 E0 D12 E0eX 12 C2 t 2 e X12 0 2 2 2 1 2 0 2 1 2 0 sin 2 0 t 2 W d 2 0 Einstein’s theory rely on the assumption of broad-band illumination of the atomic system (the transition line is totally covered) W()~W(0) in the interval t C2 t 2 2e 2 X12 W 0 t 2 0 2 4 2 t 4 sin 2 x dx 2 x 2e 2 X12 W 0 t 2 2 C2 t t 1 2 0 4 2 2 2 e X 2 12 W 0 t C2 t t 1 2 0 2 2 F. De Matteis Quantum Optics 9 Einstein’s coefficients System of N identical two-level atoms eX 12 y 1*ε Dy 2 d 3r y 1*ε ery 2 d 3r Electric Polarization Vector X12 ε D12 2 D12 cos 2 2 C2 t 2 t B12W D12 3 0 2 2 B12 2 1 2 D12 3 Spatial orientation of dipole moment is random in an atomic or molecular gas. Let’s take an average over the possible orientations t 1 A semiclassic theory doesn’t include spontaneous emission C2 t 2 A21 2 / D12 W 0 t 2 0 2 2/ g103 D12 2 g B12 cg 30 g 2 c 3 3 0 1 2 3 2 03 D12 me10 8 6 108 s 1 3 5 6 3 90c 3 40 c 2 Hydrogen atom F. De Matteis Aps A 1 1.6 ns Quantum Optics 10 Optical Bloch Equations Monochromatic em field at frequency V cos t exp i0t C2 i dC1 dt C1 0 1 * V cos t exp i0t C1 i dC2 dt C2 0 0 Quadratic terms of coefficients Ci 11 C1 2 N1 N 2 22 C2 N 2 N 21 C1C2* 12 C1*C2 d 22 dt d11 dt i cos t V * exp i0t 12 V exp i0t 21 d dt d * dt iV cos t exp i t 21 0 11 22 12 d d d ij Ci C *j C *j Ci dt dt dt 11 22 1 According to the rotating-wave approximation 21 12* d 22 dt d11 dt i 1 V * exp i0 t 12 V exp i0 t 21 2 1 * d dt d dt i V exp i 0 t 11 22 12 21 2 F. De Matteis Quantum Optics Optical Bloch Equations 11 Rabi Oscillations Let’s take the particular case 2200 1200 The solution of Bloch eq is: 2 0 V 2 2 2 2 2 t V / sin 22 2 exp i t V / 2 sin t sin t i cos t 12 0 0 2 2 2 Rabi Oscillation where |V| is Rabi frequency The solutions are derived assuming strictly monochromatic light (frequency ) i.e. A width of the incident frequency distribution smaller than the atomic transition linewidth. The broadening of the transition line modifies the Bloch equations. |V|<<(0) limit yields the Einstein coefficient treatment F. De Matteis Quantum Optics 12 Radiative broadening The theory of absorption and emission contains an intrinsic line broadening mechanism linked to spontaneous emission. Effect of spontaneous emission in the derivation of the susceptibility of a gas of 2-level atoms E (t ) E0 cos t 1 E0 exp it exp it 2 P(t ) 0 E t 1 0 E0 exp it exp it 2 Z d t t e ex j t dV j 1 * X N P(t ) d t V Take a two level system d t e C1*C2 d12 exp i0t C2*C1d 21 exp i0t X 22 X 11 0 * X 12 X 21 F. De Matteis Dipole moment d is an observable real quantity Quantum Optics 13 Radiative Broadening The rate equations lead to steady-state wavefunction time-independent in the absence of any applied em field. If initial state is the excited state, a way must exist to relax to fundamental state. Let’s introduce a term to represent the spontaneous emission V cos t exp i0t C2 i dC1 dt V E0 d12 V * cos t exp i0t C1 iC2 i dC2 dt With no-incident radiation we easily get the solution C2 t C2 0exp t N 2 t N C2 t N 2 0 exp 2 t 2 2 A21 1 R The susceptibility is found by substitution of the rate eq. solutions Correct to first perturbative order in V C1=1 e C2(t) 1 expi0 t expi0 t C2 t V * 2 0 i 0 i F. De Matteis Quantum Optics C1 t 1 o ( V ) 2 14 Radiative Broadening Substitute the C2(t) expression in the atomic dipole moment exp it 1 d 21E0 exp it exp it exp it d t d12 2 i i i i 0 0 0 0 Pt N 1 d t 0 E0 exp it exp it V 2 2 N d12 1 1 * 3 0 V 0 i 0 i K ' ' ( ) c Making again the rotating wave approximation, we end up with N d12 0 K 3 0 cV 0 2 2 2 F. De Matteis Lorentzian Lineshape. Natural linewidth of the spectral line Quantum Optics 15 Lorentzian lineshape FL 0 2 2 Lorentzian lineshape around the angular frequecy 0. Unit Area FWHM 2 A21 1 FL 0 Hydrogen Atom 03 D12 me10 8 6 108 s 1 3 5 6 3 90 c 3 40 c 2 Ap s A 1 r F. De Matteis r 1.6 ns 108 3 1015 Minimal transition linewidth. The linear time dependency of the transition probability ( expression for B) is valid for times sustantially longer than r Quantum Optics 16 Power Broadening 2 N d12 1 1 3 0 V 0 i 0 i * Correct result to the second order in the dipole matrix element D12 for the linear response of the atomic gas to the incident beam. Generalized maintaining the rotating wave approx and including the effects of spontaneous emission d 22 dt i 1 V * exp i0 t12 V exp i0 t 21 222 2 1 d dt i V exp i0 t11 22 12 12 2 Only rate eq. for C2(t) has been modified. The spontaneous emission introduces a damping term in the solutions which are no longer purely obscillatory. After a sufficiently long time the system reaches a stedy state. F. De Matteis Quantum Optics 17 Power Broadening d 22 dt i 1 V * ~12 V~21 222 2 ~ 1 ~ ~ d12 dt i V 11 22 12 i0 12 2 Bloch equations become in virtue of the substitution: ~12 expi0 t12 ~ * 21 12 and complex conjugate for 21 Setting all the rate to zero, the equilibrium steady-state solutions d t 21d12 exp i0t 12d 21 exp i0t Pt N d t V 2 V 4 22 0 2 2 V 2 2 exp i t 12 V 0 i 0 12 0 2 2 V 2 2 1 0 E0 exp it exp it 2 2 No more linear because of |V|2 N d12 0 i * in the denominator 2 2 2 3 0V 0 V 2 2 0 V 2 F. De Matteis 2 2 2 FWHM 2 2 V 2 Quantum Optics 2 18 Collision Broadening The collisions between atoms in a gas during the interaction with the em radiation produce further line-broadening p( )d 1 0 exp / 0 d 4a 2 N 0 V 1 k BT Probability of free flight time between collisions 0 free flight mean time 4a2 collision cross-section M Instantaneous Collision (<< 0) Collisions influence optical processes only indirectly by changing the state of the system before and after the collision. Inelastic Collisions Additional decay terms for the atomic level populations besides to the radiative one Elastic Collisions the atom remains in the initial state with a phase change of the wavefunction An additional decay rate in the off-diagonal Bloch equations d12 dt iV exp i 0 t11 22 ' 12 2 ' V 2 4 22 0 2 '2 V 2 ' 2 0 i 'V 2 exp i t 0 2 2 2 12 ' 2 ' V 0 F. De Matteis Quantum Optics ' coll coll 1 0 FWHM 2 '2 V ' 2 2 0 3 10 11 s R 100 19 Doppler Broadening During the absorption (and emission) of a photon by means of an atomic gas, the total momentum must be conserved too E 1 Mv 2 E 1 Mv 2 E2 E1 v1z c 2 / 2 Mc 2 5 v1z v1z c 10 0 1 c / 2 Mc 2 10 9 0 1 1 Mv1 k Mv 2 2 2 2 2 The distribution of the absorption frequencies mirrors the Maxwell distribution exp M v z / 2k BT dvz 2 exp / 2 c d exp Mc 2 0 / 202 k BT c 0 d 2 2 2 0 0 Gaussiana lineshape FWHM 2 2 2 ln 2 FG F. De Matteis 1 2 2 20 c k BT / M 2 ln 2 exp 0 / 2 2 2 Quantum Optics 20 Composite Broadening Mechanism When different lineshape broadening mechanismes are simultaneously present, the lineshapes combine by convolution of the functions F ; 0 F1 ; 0 F2 ; 0 d The combination of two lines of the same type gives a lineshape with a width equal to the sum of the widths 1 2 2 21 22 The combination of two lines of different type gives a composite (Voigt) lineshape Lorentzian lineshape Homogeneous broadening mechanism Gaussian lineshape Inhomogeneous broadening mechanism For homogeneous broadening the uncertainty principle applies (consequence of the finite time during which the atom can emit/absorb undisturbed) property of the Fourier transform t 1 F. De Matteis Quantum Optics 21 Optical BlochEq. / Rate eq. d i *~ d V 12 V~21 222 11 dt 22 dt 2 d i ~12 V 11 22 i 0 '~12 2 dt ' coll 2 1 22 0 0 12 0 0 2 ' ' exp 2 t 2 '2 2 2 '2 0 0 2 2 V 2 0 ' 2 ' cos0 t 40 ' sin 0 t 22 4 0 2 '2 0 2 2 '2 exp ' t 12 iV 2 exp ' t exp i0 t 12 0exp ' t i 0 ' d d N1 N 2 N 2 A21 N1 B12W N 2 B21W dt dt N2 0 0 F. De Matteis N 2 t NBW A1 exp At Quantum Optics Weak incident beam 22 Optical Bloch Eq./ Rate eq. Two distinct regimes in which the solutions of Bloch eq. Resembles the time dependence of the excited-state population obtained by the rate equations (Einstein coefficients ) 2 ' ' 2 '2 2 2 '2 exp 2t 0 0 2 2 V 2 0 ' 2 ' cos0 t 40 ' sin 0 t 22 4 0 2 '2 0 2 2 '2 exp ' t 2. Collision broadening much greater than radiative one ’ t 1 1 ' 22 V 2 ' 1 exp 2t 2 2 4 0 ' N 2 t NBW A1 exp 2t 1. Broad band incident light 2’ 22d V / 4 1 exp 2t 2 ' The rate equations are valid in general when: • The bandwidth of the incident light exceeds the atomic transition linewidth • The dephasing broadening (collision + Doppler) greatly exceeds the radiative linewidth of the transition F. De Matteis Quantum Optics 23