Quantum Light-Matter Interactions

Quantum Light-Matter Interactions QIC 895: Theory of Quantum Optics David Layden June 8, 2015

Outline Background Review Jaynes-Cummings Model Vacuum Rabi Oscillations, Collapse & Revival Spontaneous Emission Wigner-Weisskopf Theory Photoelectric Effect Semi-Classical vs. Quantized EM Field Predictions Summary 2 / 30

Semi-classical L-M Interactions Semi-classically, an electron in an EM field evolves as i ψ [ ( t = 2 ie A( r, 2m ) 2 ] t) + ˆV (r) ψ A is the magnetic potential in the Coulomb (radiation) gauge V is the atomic binding potential In the dipole approximation, we define ϕ( r, t) = exp(ie A r/ )ψ( r, t) and find where r 0 is the location of the nucleus. i ϕ [ ˆp 2 t = 2m + ˆV (r) eˆ r E( r 0, t) }{{} H A } {{ } H int ] ϕ 3 / 30

Quantum L-M Interactions We d like to treat the EM field quantum mechanically. This will require two changes: 1. ψ will need to represent the joint atom-field state 2. We ll need a Hamiltonian that acts on H A H F Generically, we want H = H A + H F + H int : We know H F = k ν k( a k a ) k + 1 2 Let s use semi-classical H int = e r E and promote E ˆ E For an atom at the origin: ˆ E(t) = k ɛ k E k â k e iν kt + h.c. In the Schrödinger picture we ll use ˆ E := ˆ E(0) in place of ˆ E(t). 5 / 30

Quantum L-M Interactions (2) In analogy to our semi-classical analysis, we ll focus on an electron transition that is (nearly) resonant with some mode: Consider two electron energy eigenstates g and e with energy gap ω In the { g, e } subspace up to a multiple of I. H A = ω 2 e e ω 2 g g = ω 2 σ z As in the semi-classical picture, we take the dipole operator eˆ r e g + g e = σ x }{{}}{{} σ + σ 6 / 30

Quantum L-M Interactions (3) Putting it all together, we get the multi-mode Rabi Hamiltonian: H = ν k a k a k + ω 2 σ z + g k (σ + + σ )(a k + a k ) k k }{{}}{{}}{{} H F H A H int = eˆ r ˆ E From here we re going to make two more approximations: 1. Single-mode approximation 2. Rotating wave approximation These will get us to the well-known Jaynes-Cummings model. 7 / 30

Interaction Picture Refresher If the Schrödinger Hamiltonian has the form H 0 + H 1 then: States transform as Operators transform as ψ I (t) = e ih0t/ ψ S (t) A I (t) = e ih0t/ A S e ih0t/ N.b. Both states and operators have time dependence in this picture! States evolve according to i d dt ψ I(t) = H 1,I (t) ψ I (t) 8 / 30

Rotating Wave Approximation If we move into the interaction picture (rotating frame) with H 0 = H A + H F we find: H int,i = g ( aσ e i(ω+ν)t + a σ + e i(ω+ν)t + aσ + e i(ω ν)t + a σ e i(ω ν)t ) }{{}}{{} High Frequency Low Frequency Low freq. terms: g n e n 1 e n g n + 1 conserve energy High freq. terms: g n e n + 1 e n g n 1 no classical analogue 9 / 30

Rotating Wave Approximation (2) The solution to the interaction picture SE will involve t 0 H int,i(τ)dτ. This will give terms of the form t 0 ei(ω±ν)τ dτ. In general: t 0 e iξτ dτ = i ξ ( e iξt 1 ) Low freq. terms (ξ = ω ν small): t 0 e i(ω ν)τ dτ = i ξ ( ) 1 + iξt + O(ξ 2 ) 1 = t + O(ξ) High freq. terms (ξ = ω + ν big): t 0 e i(ω+ν)τ = i ω + ν }{{} small ( e i(ω+ν)t 1 ) } {{ } bounded 10 / 30

Jaynes-Cummings Model Discarding the high frequency (counter-rotating) terms and returning to the Schrödinger picture we find H = νa a + ω 2 σ z + g ( aσ + + a σ ) This Hamiltonian can be diagonalized exactly: E ±,n = ν ( n + 1 2 ) ± 2 + 4g 2 2 (n + 1) n, ± = a n,± e n + b n,± g n + 1 where := ν ω is the detuning. 11 / 30

Jaynes-Cummings Model (2) Scully writes a general state in the uncoupled energy eigenbasis: ψ(t) = ( ) α n (t) g n + β n (t) e n n Plugging into the Schrödinger equation, we get β n (t) = ig n + 1e i t α n+1 (t) α n+1 (t) = ig n + 1e i t β n (t) The probability of finding n photons in the cavity is P (n) = α n (t) 2 + β n (t) 2, which depends non-trivially on t (see animation). 12 / 30

Vacuum Rabi Oscillations Consider W (t) = σ z, the atomic inversion. Semi-classically, when an EM wave is incident on an atom, W (t) oscillates (Rabi oscillations) This does not happen in the classical vacuum (when E = B = 0) With a quantized field W (t) = ] n [ β n (t) 2 α n (t) 2 For an initial state e 0 : W (t) = 1 [ 2 + 4g 2 2 + 4g 2 cos ( t 2 + 4g 2)] Key feature of quantum L-M interactions: Rabi oscillations occur even in the vacuum. 13 / 30

Collapse and Revival Semi-classically, an EM wave induces Rabi oscillations at a single frequency (a) Semi-classical Rabi oscillations (b) Fully quantum Rabi oscillations A fully quantum description predicts that the oscillations will vanish and then re-appear. 14 / 30

Collapse and Revival (2) If the atom and field mode are initially uncorrelated with = 0: W (t) = α n 2 cos ( gt n + 1 ) n=0 where n α n n is the initial field state. Notice that a single mode can produce oscillations at many frequencies These different contributions interfere to create complex collapse/revival behaviour (observed experimentally) 15 / 30

Collapse and Revival (3) Scully argues that if n were continuous there would be collapse, but no revival. Let s estimate the timescale of revival (with = 0): Each n contributes with frequency Ω n := g n + 1 For a mode with photon number sharply peaked around n, the frequencies near Ω n = g n + 1 will dominate How quickly do Ω n and Ω n 1 move in and out of phase? ( Ω n Ω n 1 ) tr = 2πm m N = t r 2πm n g Revivals keep occurring periodically. 16 / 30

Spontaneous Emission Collapse/revival observed experimentally, but only in certain settings (e.g., reflective cavities) A more common phenomenon is for an atom to relax (irreversibly) by emitting a photon To analyse this scenario, we ll need to consider multiple modes again: H = ν k a k a k + ω 2 σ z + g k (σ + + σ )(a k + a k ) k k }{{}}{{} H 0 H int In the interaction picture, under the RWA: H int,i = k g k ( a k σ + e i(ω ν k)t + a k σ e i(ω ν k)t ) 18 / 30

Spontaneous Emission (2) If, initially, the atom is excited and the field is in the vacuum state, the general solution has the form ψ(t) = α(t) e 0 + k β k (t) g 1 k under the RWA. Here 0 is the total field vacuum, and 1 k = a k 0. The Schrödinger equation yields α(t) = i k g k e i(ω ν k)t β k (t) (1) β k (t) = ig k e i(ω ν k)t α(t) (2) Substituting (2) into (1), we find α(t) = k t g k 2 dt e i(ω ν k)(t t) α(t ) 0 19 / 30

Spontaneous Emission (3) Approximation 1 The modes form a continuum: ( ) 3 L d 3 k ρ(k) where ρ(k)d 3 k = 2 k 2 dk dφ sin θdθ 2π k Substituting in the value of g k and integrating over (φ, θ) we get α(t) 0 dν k ν 3 k t 0 dt e i(ω ν k)(t t) α(t ) Approximation 2 The integral is only important when ω ν k (resonance), so we can replace ν 3 k with ω3, and integrate over ν k R. 20 / 30

Spontaneous Emission (4) Changing the order of integration, we have t α(t) ω 3 dt α(t ) dν k e i(ω ν k)(t t) 0 t = ω 3 dt α(t ) 2πδ(t t ) 0 = 2πω 3 α(t) In other words, dα dt = Γ 2 α(t) for some constant Γ, or e a 2 = exp( Γt) This is a good description of energy relaxation, where Γ = 1/T 1. It is called the Weisskopf-Wigner approximation. 21 / 30

Spontaneous Emission (5) What about the emitted photon? We can plug in α(t) to our ODE for β k (t) to get β k (t) = i t 0 dt g k e i(ω ν k)t Γt /2 The frequency spectrum of the resulting radiation is given by P (ν k ) = ρ(ν k ) dω β k (t) 2 λ=1,2 Once the transients have vanished, we find lim P (ν 1 k) t Γ 2 /4 + (ω ν k ) 2 i.e., a Lorentzian centred around ω of width Γ (see plot). 22 / 30

Fermi s Golden Rule Refresher System evolves according to free Hamiltonian H 0, subject to perturbation λh (t). We ll be interested in the case where H (t) = V e iωt + V e iωt What is the probability of λh inducing a transition between unperturbed energy eigenstates i and f? P i f = f T e iλ t 0 dτ H I (τ) i 2 = f i t 2 + λ 2 dτ f e ih0τ H (τ)e ih0τ i 2 + O(λ 3 ) 0 24 / 30

Fermi s Golden Rule Refresher (2) For long times t, the transitions probability i f for a perturbation of the form H (t) = V e iωt + V e iωt is approximately P i f = { λ 2 2πt λ 2 2πt where ρ(e) is the density of final states. f V i 2 ρ(e i + Ω) f V i 2 ρ(e i Ω) (absorption) (emission) Notice that both cases grow unbounded with t! Instead of P we re going to work with P, which is bounded. 25 / 30

Photoelectric Effect (Semi-Classically) Consider a hydrogenic atom with an electron initially in the n energy eigenstate. Suppose it is subject to a classical EM wave A( r, t) = A 0 ɛ (e ) i( k r ωt) + e i( k r ωt) which ejects it from the atom to a free state p. The electron s evolution is generated by H = ˆ p 2 2m e m ˆ p A + e2 2m A 2 + e ˆφ( r) The A 2 term is suppressed by the e2 2m factor and will not contribute to our first order transition rate. Therefore, we ll take the perturbation to be λh (t) = e m ˆ p A = ea 0 m }{{} λ (ˆ p ɛ) e i k r e iωt + h.c. } {{ } V 26 / 30

Photoelectric Effect (Semi-Classically, 2) Let s look at the rate of transitions for photoabsorption n p : P n p = λ 2 2π p (ˆ p ɛ)e i k r n 2 ρ(e n + ω) For a free electron in a box of side length L, the density of states is where E = p 2 2m. ρ(e) = ( ) 3 L m p dω 2π From here we could compute various cross-sections, average over final electron states etc. Instead though, let s set up an equivalent scenario with quantized radiation. 27 / 30

Photoelectric Effect (Quantized EM field) The electron-field system evolves according to ( H = 1 ˆ p e ˆ A 2m c )2 + k, ɛ ω(a k, ɛ a k, ɛ + 1 2 ) + e ˆφ( r) where ˆ A = 2π [ ] L 3 a k, ɛ e i k r + h.c. ɛ k ω k k, ɛ Consider the initial state n 1 k, ɛ := n a k, ɛ 0 and the final state p 0. As before, we ll omit the A 2 term as it won t contribute to our first-order transition rate. 28 / 30

Photoelectric Effect (Quantized EM field, 2) After some work, Fermi s Golden Rule gives with the same λ as before. P i f = λ 2 2π f (ˆ p ɛ)e i k r i 2 ρ(e f ) The density of final states ρ(e) is the same as in the semi-classical case. P i f here is the same as in the semi-classical analysis. To first order in λ, semi-classical and fully quantum descriptions give the same transition rates, cross sections etc. for the photoelectric effect! 29 / 30

Summary Present in semi-classical description: Rabi oscillations Photoelectric effect (!) Require quantized EM field: Vacuum Rabi oscillations Periodic collapse and revival of Rabi oscillations Spontaneous emission 30 / 30