Time Dependent Perturbation Theory. Andreas Wacker Mathematical Physics Lund University

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1 Time Dependent Perturbation Theory Andreas Wacker Mathematical Physics Lund University

2 General starting point (t )Ψ (t ) Schrödinger equation i Ψ (t ) = t ^ (t ) has typically no analytic solution for Ψ (t ) amiltonian (t )= + V (t) Decompose known eigenstates a a =E a a hopefully small Eigenstates form basis Ψ (t) = ψa (t )a a Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

3 Time-dependent perturbation theory + V (t) ]Ψ (t ) with known a = E a i Ψ (t ) = [ a t V (t )= : eigenstates evolve as e i E a t / a Finite perturbation creates contributions from other basis states Determine P b (t)= b Ψ(t ) for Ψ () =a Transition probability Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

4 Interaction picture (Dirac 197) + V (t) ]Ψ (t ) with known a = E a i Ψ (t ) = [ a t t Ψ (t ) =exp i Ψ (t=) V=: ( ) ( Ansatz for finite V: Ψ (t ) =exp i t ) Ψ D (t) i Ψ D (t ) =V^ D (t )Ψ D (t ) t ^ t ^ t D with V^ (t )=exp i V^ (t ) exp i ( ) ( ) Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

5 Born approximation t / i Ψ (t ) =e Ψ D (t ) with i Ψ D (t ) =V D (t ) Ψ D (t ) t Determine Pb (t)= b Ψ (t) = b Ψ D (t) for a = Ψ () = Ψ D () Formal solution t 1 Ψ D (t ) = a + dt ' V D (t ') Ψ D (t ') i [ t t' 1 1 D dt ' V (t ') a + ds V D (s) Ψ D (s) i i Lowest order in V = a + t 1 Pb (t )= b Ψ D (t) = dt ' b V D (t ' ) a i Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se / ]

6 Application: Laser pulse on hydrogen atom in ground state For <t<τ V (t )=e F sin(ω t ) z t 1 P b (t)= dt ' b V D (t ') a i t t D with V (t )=exp i V (t) exp i ( ) ( ) i t / i t / n, l, mv D (t )1,, =e F sin(ωt ) n, l, me z e 1,, =ef sin(ω t )e i (E n E 1 ) t / n, l, mz 1,, Matrix elements n, l, mz 1,, = for m n,,z1,, = Selection rules For n=, there is only e =n=, l=1, m= 7 Integral povides,1,z1,, = 5 ab 3 Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

7 Result: Probability to find atom in excited state e> at time t ( Integral povides P e (t )= 7 efab 35 ) i (ωeg +ω) τ i (ωeg ω) τ e 1 e 1 i (ωeg +ω) i (ωeg ω) E e E g with ωeg = = / s Note P~ F Sharp in ω for long τ Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

8 Fermi's golden rule (Dirac 197) V (t)= F e i ω t t 1 P b (t)= dt ' b V D (t ') a i provides t t D with V (t )=exp i V (t) exp i ( ) ( ) with Math: Define Transition rate Γ a b= delta-function in energy integral if prefactor is const on energy scale ħ/t P b (t ) t π = b Fa δ( E b E a ω) Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

9 Fermi's golden rule, continuation V (t )= F e i ω t provides Γ a b= P b (t ) t = π b Fa δ ( E b E a ω) Constant potential V^ (t )=V^ : π Γ a b= b V a δ( E b E a ) Periodic time-dependence V (t )= F e i ω t + F ei ω t : Γ a b= π b Fa δ( E b E a ω) absorption of quanta from the field π + b F a δ( E b E a + ω) emission Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

10 Comments π π Γ a b= b Fa δ(e b E a ω)+ b F a δ(e b E a + ω) Result is lowest order in perturbation theory: Perturbation F so small that Γ τ<1 within observation time Delta function asks for integral: Requires continuum of final states of frequencies Finite lifetime τ for states: π δ( E b E a ω) / τ (E b E a ω) +(/ τ) Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

Example: Optical transition in semiconductor heterostructures Conduction band valence band Typical energy difference: 1 mev Corresponds to

11 Example: Optical transition in semiconductor heterostructures Conduction band valence band Typical energy difference: 1 mev Corresponds to light with ω=e /= π 4 Tz or λ=1 μ m Infrared Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

12 Apply Fermi s golden rule π π Γ a b= b Fa δ(e b E a ω)+ b F a δ(e b E a + ω) Perturbation V (t )=ef z cos(ωt ) 1 π ef ϕzϕ Γ1 = δ(e E 1 ω) emission π ef ϕ1z ϕ Γ 1= δ(e 1 E + ω) absorption planar layer with n1/ electrons/cm in level 1/ Change in photon number R photon= n1 Γ1 +n Γ 1=(n n1 )Γ1 Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

13 Balance for the photons R photon= n1 Γ1 +n Γ 1=(n n1) Γ1 Common: Electrons are dominantly in lower level: n<n1 Photons are absorbed by the material Photons are generated (gain) if upper level has higher electron density Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

14 Drive current by electric bias via tunneling and scattering transitions Tunneling injection ħωlo Phonon extraction First ideas: Kazarinov and Suris 1971 Quick if level difference matches the optical phonon energy (36 mev in GaAs) Realized: Faist, Capasso et al, Science 1994 Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

times Cascading 46 periods, see Kumar APL9 operates at 4.

15 Waveguide with cascaded structure Minimal QCL design z Mode confinement requires amplification over µm range while module thickness ~4 nm Repeat the structure several times Cascading 46 periods, see Kumar APL9 operates at 4.6 Tz for T<11 K similar Scalari et al. OE 1 Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

Quantum Cascade Laser is now a commercial device

realized by the EU project MIRIFISENS presented last

697μm (3Tz, 14 mev) From Daylight Solutions Lund

16 Quantum Cascade Laser is now a commercial device used for IR spectroscopy ighly tunable IR-QCL realized by the EU project MIRIFISENS presented last Tuesday in Paris Ozon at 9.697μm (3Tz, 14 mev) From Daylight Solutions Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

Perturbation Theory. Andreas Wacker Mathematical Physics Lund University

Perturbation Theory. Andreas Wacker Mathematical Physics Lund University Perturbation Theory Andreas Wacker Mathematical Physics Lund University General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates