Light-Matter Interactions

Transcription

1 Light-Matter Interactions Paul Eastham February 15, 2012

2 The model = Single atom in an electromagnetic cavity Mirrors Single atom Realised experimentally Theory: Jaynes Cummings Model Rabi oscillations energy levels sensitive to single atom and photon get inside the mechanics of emission and absorption

3 Where we re going = One field mode, two atomic states Energy of photon in field mode Ĥ = ( /2) ( e e g g ) + ω â â + Ω 2 (â e g + â g e ). Dipole coupling energy Energy difference between atomic levels

4 How to get there Show that Ĥ = Ĥatom + Ĥfield Ê.(eˆr) Write electron position operator ˆr in basis eigenstates of Ĥatom == atomic orbitals Approximate to one mode of field and two atomic levels Neglect non-resonant wrong-way terms (like electron drops down orbital and emits photon)

5 How to get there Show that Ĥ = Ĥatom + Ĥfield Ê.(eˆr) Write electron position operator ˆr in basis eigenstates of Ĥatom == atomic orbitals Approximate to one mode of field and two atomic levels Neglect non-resonant wrong-way terms (like electron drops down orbital and emits photon)

6 Hamiltonian for Atom+Field Field modes ω i â i âi Atom Ĥ = ĤEM + Ĥatom + Ĥint [ 2 2 e 2 ] e 2m e 4πɛ 0 r e R 0 (nucleus R 0 ) Interaction energy?

7 Interaction energy: classical field Ĥ atom = 1 2m ˆp 2 + V (ˆr). With vector potential A, ˆp (ˆp ea) and scalar potential φ, Ĥatom Ĥatom + eφ (minimal coupling) Ĥ atom field = 1 2m (ˆp ea) 2 + eφ + V (ˆr).

8 Interaction energy: A.p form Choose the Coulomb gauge where.a = 0, φ = 0 Ĥ atom field = ˆp 2 2m e m A.ˆp + e2 2m A2 + V (r) = Ĥatom + Ĥint (Can use this form directly not in this course)

9 Interaction energy: dipole approximation Interested in interaction with light waves A = A 0 e i(k.r ωt) + c.c. For an atom the wavefunction extends over about 1Å For light k = 2π/λ (500nm) 1 A approximately constant in space over the atom, A(r, t) A(r = 0, t)

10 Interaction energy: E.r form Coulomb gauge form: Ĥ atom field = 1 2m (ˆp ea) 2 + eφ(= 0) + V (r). Now change gauge : so A A + χ(r, t) φ φ χ(r, t), t Ĥ atom field = 1 2m (ˆp e(a + χ)) 2 e χ t + V (r).

11 Interaction energy: E.r form Ĥ atom field = 1 2m (ˆp e(a + χ)) 2 e χ t + V (r). Choose χ(r, t) = A.r so that χ = A and χ t = A.r = E.r t [ E = φ A = A ] t t (A, φ-coulomb gauge) Ĥ atom field = 1 2m (ˆp 2 + V (r)) eˆr.e(t).

12 Hamiltonian: E.r form Quantum form: E(r) Ê(r 0) at the position of the atom So for our cavity quantization + one electron atom: Ĥ = Ĥatom + ω n â nân n + ω n ɛ n,s 0 V sin(k nz at )(â n + â n)e s.( eˆr).

13 How to get there Show that Ĥ = Ĥatom + Ĥfield Ê.(eˆr) Write electron position operator ˆr in basis eigenstates of Ĥatom == atomic orbitals Approximate to one mode of field and two atomic levels Neglect non-resonant wrong-way terms (like electron drops down orbital and emits photon)

14 Second quantization: general Generally have Z indistinguishable electrons Atomic eigenstates labelled by occupation of orbitals (1s 2 2s 1 etc) These states i form a complete set (for Z electrons) This allows us to formally write atomic operators in terms of transition operators i j

15 Second quantization: Hamiltonian ˆ1 = i i i. Formal representation of Ĥ : Ĥ = ˆ1Ĥˆ1 = i i i Ĥ j j j = i,j = i i i E j j j (Ĥ j = E j j, i j = δ ij ) E i i i.

16 Second quantization: One-body operators Eigenstates of Ĥ i form a complete set for Z electrons, so ˆ1 = i i. i Formal representation of ˆD = i eˆr i : ˆD = ˆ1ˆDˆ1 = i i i ˆD j j j = i,j i ˆD j i j

17 Second quantization: One-body operators ˆD = ˆ1ˆDˆ1 = i i i ˆD j j j = i,j i ˆD j i j If we know the orbitals in real-space, can calculate D ij = i ˆD j = dxdydzψi (r)(er)ψ j(r).

18 Dipole matrix elements D ij = i ˆD j = dxdydzψ i (r)(er)ψ j(r). D ij only non-zero between some states selection rules i.j different parity. l = ±1 (if l good quantum number) Magnitude D e 1 Å

19 Atom-field Hamiltonian So for our cavity problem Ĥ = n ω n â nân + i + n,s E i i i E n sin(k n z at )(â n + â n)e s.d ij i j. ij

20 How to get there Show that Ĥ = Ĥatom + Ĥfield Ê.(eˆr) Write electron position operator ˆr in basis eigenstates of Ĥatom == atomic orbitals Approximate to one mode of field and two atomic levels Neglect non-resonant wrong-way terms (like electron drops down orbital and emits photon)

21 Why two levels? Light-matter interactions weak (cf. energy differences in uncoupled problem) small effects which can be treated in perturbation theory except: if ω, degeneracy between n, g, n 1, e can focus on the physics of one mode + nearly resonant atomic transition

22 Rotating-Wave approximation Interaction is Ω 2 [ â e g + â g e + â g e + â g e ]. Energy changes 0 ±2 drop these terms Jaynes-Cummings Model in Rotating-Wave Approx Ĥ = ( /2)( e e g g ) + ωâ â + Ω 2 (â e g + â g e ).

23 Summary = One field mode, two atomic states Energy of photon in field mode Ĥ = ( /2) ( e e g g ) + ω â â + Ω 2 (â e g + â g e ). Dipole coupling energy Energy difference between atomic levels