Lecture 18 Time-dependent perturbation theory

Transcription

1 Lecture 18 Time-dependent perturbation theory

2 Time-dependent perturbation theory So far, we have focused on quantum mechanics of systems described by Hamiltonians that are time-independent. In such cases, time dependence of wavefunction developed through time-evolution operator, Û = e iĥt/, i.e. for Ĥ n = E n n, ψ(t) = e iĥt/ ψ(0) }{{} P n c n(0) n = n e ie nt/ c n (0) n Although suitable for closed quantum systems, formalism fails to describe interaction with an external environment, e.g. EM field. In such cases, more convenient to describe induced interactions of small isolated system, Ĥ0, through time-dependent interaction V (t). In this lecture, we will develop a formalism to treat such time-dependent perturbations.

3 Time-dependent perturbation theory So far, we have focused on quantum mechanics of systems described by Hamiltonians that are time-independent. In such cases, time dependence of wavefunction developed through time-evolution operator, Û = e iĥt/, i.e. for Ĥ n = E n n, ψ(t) = e iĥt/ ψ(0) }{{} P n c n(0) n = n e ie nt/ c n (0) n Although suitable for closed quantum systems, formalism fails to describe interaction with an external environment, e.g. EM field. In such cases, more convenient to describe induced interactions of small isolated system, Ĥ0, through time-dependent interaction V (t). In this lecture, we will develop a formalism to treat such time-dependent perturbations.

4 Time-dependent perturbation theory: outline Time-dependent potentials: general formalism Time-dependent perturbation theory Sudden perturbation Harmonic perturbations: Fermi s Golden Rule

5 Time-dependent potentials: general formalism Consider Hamiltonian Ĥ(t) =Ĥ0 + V (t), where all time dependence enters through the potential V (t). So far, we have focused on Schrödinger representation, where dynamics specified by time-dependent wavefunction, i t ψ(t) S = Ĥ ψ(t) S However, to develop time-dependent perturbation theory for Ĥ(t) =Ĥ 0 + V (t), it is convenient to turn to a new representation known as the Interaction representation: ψ(t) I = e iĥ 0 t/ ψ(t) S, ψ(0) I = ψ(0) S

6 Time-dependent potentials: general formalism Consider Hamiltonian Ĥ(t) =Ĥ0 + V (t), where all time dependence enters through the potential V (t). So far, we have focused on Schrödinger representation, where dynamics specified by time-dependent wavefunction, i t ψ(t) S = Ĥ ψ(t) S However, to develop time-dependent perturbation theory for Ĥ(t) =Ĥ 0 + V (t), it is convenient to turn to a new representation known as the Interaction representation: ψ(t) I = e iĥ 0 t/ ψ(t) S, ψ(0) I = ψ(0) S

7 Time-dependent potentials: general formalism ψ(t) I = e iĥ0t/ ψ(t) S, ψ(0) I = ψ(0) S In the interaction representation, wavefunction obeys the following equation of motion: We therefore have that i t ψ(t) I = e iĥ 0 t/ (i t Ĥ0) ψ(t) S = e iĥ0t/ (Ĥ Ĥ 0 ) ψ(t) S = e iĥ 0 t/ V (t)e iĥ 0 t/ ψ(t) I }{{} V I (t) i t ψ(t) I = V I (t) ψ(t) I, V I (t) =e iĥ 0 t/ V (t)e iĥ 0 t/

8 Time-dependent potentials: general formalism i t ψ(t) I = V I (t) ψ(t) I, V I (t) =e iĥ0t/ V (t)e iĥ0t/ Then, if we form eigenfunction expansion, ψ(t) I = n c n(t) n, where Ĥ0 n = E n n, i t c n (t) n = e iĥ0t/ V (t)e iĥ0t/ c n (t) n n n i n ċ n (t) n = n c n (t)e iĥ0t/ V (t) e iĥ0t/ n }{{} e ie nt/ n If we now contract with a general state m ċ n (t) m n = c n (t) m e iĥ 0 t/ V (t)e ient/ n }{{}}{{} n n δ mn m e ie mt/ i ċ m (t) = n m V (t) n e i(e m E n )t/ c n (t)

9 Time-dependent potentials: general formalism i t ψ(t) I = V I (t) ψ(t) I, V I (t) =e iĥ0t/ V (t)e iĥ0t/ Then, if we form eigenfunction expansion, ψ(t) I = n c n(t) n, where Ĥ0 n = E n n, i t c n (t) n = e iĥ0t/ V (t)e iĥ0t/ c n (t) n n n i n ċ n (t) n = n c n (t)e iĥ0t/ V (t) e iĥ0t/ n }{{} e ie nt/ n If we now contract with a general state m ċ n (t) m n = c n (t) m e iĥ 0 t/ V (t)e ient/ n }{{}}{{} n n δ mn m e ie mt/ i ċ m (t) = n m V (t) n e i(e m E n )t/ c n (t)

10 Time-dependent potentials: general formalism i ċ m (t) = n m V (t) n e i(e m E n )t/ c n (t) So, in summary, if we expand wavefunction ψ(t) I = n c n(t) n, where Ĥ 0 n = E n n, the Schrödinger equation, i t ψ(t) I = V I (t) ψ(t) I with V I (t) =e iĥ 0 t/ V (t)e iĥ 0 t/ translates to the relation, i ċ m (t) = n V mn (t)e iω mnt c n (t) where V mn (t) = m V (t) m and ω mn = 1 (E m E n )= ω nm.

11 Example: Dynamics of a driven two-level system i ċ m (t) = n V mn (t)e iω mnt c n (t) Consider an atom with just two available atomic levels, 1 and 2, with energies E 1 and E 2. In the eigenbasis, the time-independent Hamiltonian can be written as ( ) E1 0 Ĥ 0 = E E E 2 Note that the two-level atom mirrors a spin 1/2 system. If the system is driven by an electric field, E(r, t) =E 0 (r) cos(ωt), and the states have different parity, close to resonance, ω ω 21 ω 21, the effective interaction potential is given by ( ) V (t) δe iωt δe iω e iωt 2 1 δ e iω where the matrix element, δ = 1 E 2 is presumed real.

12 Example: Dynamics of a driven two-level system i ċ m (t) = n V mn (t)e iω mnt c n (t) Consider an atom with just two available atomic levels, 1 and 2, with energies E 1 and E 2. In the eigenbasis, the time-independent Hamiltonian can be written as ( ) E1 0 Ĥ 0 = E E E 2 Note that the two-level atom mirrors a spin 1/2 system. If the system is driven by an electric field, E(r, t) =E 0 (r) cos(ωt), and the states have different parity, close to resonance, ω ω 21 ω 21, the effective interaction potential is given by ( ) V (t) δe iωt δe iω e iωt 2 1 δ e iω where the matrix element, δ = 1 E 2 is presumed real.

13 Example: Dynamics of a driven two-level system Ĥ 0 + V (t) = ( ) E1 0 0 E 2 + δ ( 0 e iωt e iω ) The electric field therefore induces transitions between the states. If we expand the spinor-like wavefunction in eigenstates of Ĥ0, i.e. ψ(t) I = c 1 (t) 1 + c 2 (t) 2, the equation i ċ m (t) = n V mn (t)e iω mnt c n (t) translates to the quantum dynamics ( 0 e i t c = δ i(ω ω 21)t e i(ω ω 21)t 0 ) c(t), ω 21 = 1 (E 2 E 1 ) where c(t) =(c 1 (t) c 2 (t)).

14 Example: Dynamics of a driven two-level system i t c = δ ( 0 e i(ω ω 21)t e i(ω ω 21)t 0 ) c(t), ω 21 = 1 (E 2 E 1 ) Expanding this equation, we find i ċ 1 = δe i(ω ω 21)t c 2, i ċ 2 = δe i(ω ω 21)t c 1 from which we obtain an equation for c 2, c 2 (t)+ i(ω ω 21 )ċ 2 (t)+ ( δ ) 2 c 2 (t) =0 With the initial conditions, c 1 (0) = 1 and c 2 (0) = 0, i.e. particle starts in state 1, we obtain the solution, c 2 (t) =e i(ω ω 21)t/2 sin(ωt) where Ω = ((δ/ ) 2 +(ω ω 21 ) 2 /4) 1/2 is known as Rabi frequency.

15 Example: Dynamics of a driven two-level system i t c = δ ( 0 e i(ω ω 21)t e i(ω ω 21)t 0 ) c(t), ω 21 = 1 (E 2 E 1 ) Expanding this equation, we find i ċ 1 = δe i(ω ω 21)t c 2, i ċ 2 = δe i(ω ω 21)t c 1 from which we obtain an equation for c 2, c 2 (t)+ i(ω ω 21 )ċ 2 (t)+ ( δ ) 2 c 2 (t) =0 With the initial conditions, c 1 (0) = 1 and c 2 (0) = 0, i.e. particle starts in state 1, we obtain the solution, c 2 (t) =e i(ω ω 21)t/2 sin(ωt) where Ω = ((δ/ ) 2 +(ω ω 21 ) 2 /4) 1/2 is known as Rabi frequency.

16 Example: Dynamics of a driven two-level system c 2 (t) =Ae i(ω ω 21)t/2 sin(ωt), Ω= ( ( ) 2 δ + ( ) ) 2 1/2 ω ω21 2 Together with c 1 (t) = i δ ei(ω ω 21)tċ 2, we obtain the normalization, δ A = and /4 δ+ 2 (ω ω 2 21 c 2 (t) 2 = δ 2 δ (ω ω 21 ) 2 /4 sin2 Ωt, c 1 (t) 2 =1 c 2 (t) 2 Periodic solution describes transfer of probability between states 1 and 2. Maximum probability of occupying state 2 is Lorentzian, c 2 (t) 2 max = δ 2 δ (ω ω 21 ) 2 /4, taking the value of unity at resonance, ω = ω 21.

17 Example: Dynamics of a driven two-level system c 2 (t) =Ae i(ω ω 21)t/2 sin(ωt), Ω= ( ( ) 2 δ + ( ) ) 2 1/2 ω ω21 2 Together with c 1 (t) = i δ ei(ω ω 21)tċ 2, we obtain the normalization, δ A = and /4 δ+ 2 (ω ω 2 21 c 2 (t) 2 = δ 2 δ (ω ω 21 ) 2 /4 sin2 Ωt, c 1 (t) 2 =1 c 2 (t) 2 Periodic solution describes transfer of probability between states 1 and 2. Maximum probability of occupying state 2 is Lorentzian, c 2 (t) 2 max = δ 2 δ (ω ω 21 ) 2 /4, taking the value of unity at resonance, ω = ω 21.

18 Rabi oscillations: persistent current qubit It is different to prepare and analyse ideal atomic two-level system. However, circuits made of superconducting loops provide access to two-level systems. These have been of great interest since they (may yet) provide a platform to develop qubit operation and quantum logic circuits. By exciting transitions between levels using a microwave pulse, coherence of the system has been recorded through Rabi oscillations.

19 Rabi oscillations: persistent current qubit It is different to prepare and analyse ideal atomic two-level system. However, circuits made of superconducting loops provide access to two-level systems. These have been of great interest since they (may yet) provide a platform to develop qubit operation and quantum logic circuits. By exciting transitions between levels using a microwave pulse, coherence of the system has been recorded through Rabi oscillations.

20 Time-dependent perturbation theory For a general time-dependent Hamiltonian, Ĥ = Ĥ0 + V (t), an analytical solution is usually infeasible. However, as for the time-independent Schrödinger equation, we can develop to a perturbative expansion (in powers of interaction): ψ(t) I = n c n (t) n, c n (t) =c (0) n + c (1) n (t)+c (2) n (t)+ where Ĥ 0 n = E n n, c n (m) O(V m ), and c n (0) (time-independent) initial state of the system. represents some As with the Schrödinger representation, in the interaction representation, ψ(t) I related to inital state ψ( ) I, at time, through a time-evolution operator, ψ(t) I = Û I (t, ) ψ( ) I

21 Time-dependent perturbation theory For a general time-dependent Hamiltonian, Ĥ = Ĥ0 + V (t), an analytical solution is usually infeasible. However, as for the time-independent Schrödinger equation, we can develop to a perturbative expansion (in powers of interaction): ψ(t) I = n c n (t) n, c n (t) =c (0) n + c (1) n (t)+c (2) n (t)+ where Ĥ 0 n = E n n, c n (m) O(V m ), and c n (0) (time-independent) initial state of the system. represents some As with the Schrödinger representation, in the interaction representation, ψ(t) I related to inital state ψ( ) I, at time, through a time-evolution operator, ψ(t) I = Û I (t, ) ψ( ) I

22 Time-dependent perturbation theory ψ(t) I = Û I (t, ) ψ( ) I Substituted into Schrödinger equation i t ψ(t) I = V I (t) ψ(t) I, i t Û I (t, ) ψ( ) I = V I (t)ûi(t, ) ψ( ) I Since this is true for any initial state ψ( ) I, we must have i t Û I (t, )=V I (t)ûi(t, ) with the boundary condition U I (, )=I. Integrating to t, i dt t Û I (t, )=i (ÛI(t, ) I), i.e. Û I (t, )=I i dt V I (t )ÛI(t, ) provides self-consistent equation for U I (t, ),

23 Time-dependent perturbation theory ψ(t) I = Û I (t, ) ψ( ) I Substituted into Schrödinger equation i t ψ(t) I = V I (t) ψ(t) I, i t Û I (t, ) ψ( ) I = V I (t)ûi(t, ) ψ( ) I Since this is true for any initial state ψ( ) I, we must have i t Û I (t, )=V I (t)ûi(t, ) with the boundary condition U I (, )=I. Integrating to t, i dt t Û I (t, )=i (ÛI(t, ) I), i.e. Û I (t, )=I i dt V I (t )ÛI(t, ) provides self-consistent equation for U I (t, ),

24 Time-dependent perturbation theory ψ(t) I = Û I (t, ) ψ( ) I Substituted into Schrödinger equation i t ψ(t) I = V I (t) ψ(t) I, i t Û I (t, ) ψ( ) I = V I (t)ûi(t, ) ψ( ) I Since this is true for any initial state ψ( ) I, we must have i t Û I (t, )=V I (t)ûi(t, ) with the boundary condition U I (, )=I. Integrating to t, i dt t Û I (t, )=i (ÛI(t, ) I), i.e. Û I (t, )=I i dt V I (t )ÛI(t, ) provides self-consistent equation for U I (t, ),

25 Time-dependent perturbation theory Û I (t, )=I i dt V I (t )ÛI(t, ) If we substitute ÛI(t, ) on right hand side, Û I (t, )=I i dt V I (t ) ( + i ) 2 dt V I (t ) dt V I (t )Û I (t, ) Iterating this procedure, Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) where term n = 0 translates to I.

26 Time-dependent perturbation theory Û I (t, )=I i ( dt V I (t ) I i dt V I (t )Û I (t, ) ) If we substitute ÛI(t, ) on right hand side, Û I (t, )=I i dt V I (t ) ( + i ) 2 dt V I (t ) dt V I (t )Û I (t, ) Iterating this procedure, Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) where term n = 0 translates to I.

27 Time-dependent perturbation theory Û I (t, )=I i dt V I (t )ÛI(t, ) If we substitute ÛI(t, ) on right hand side, Û I (t, )=I i dt V I (t ) ( + i ) 2 dt V I (t ) dt V I (t )Û I (t, ) Iterating this procedure, Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) where term n = 0 translates to I.

28 Time-dependent perturbation theory Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) Remark: Since operators V I (t) appear as a time-ordered sequence, with t n t n 1 t 1 t this expression is sometimes written as [ Û I (t, )=T e i R t dt V I (t ) ] where T denotes the time-ordering operator and is understood as the identity above. Note that, for V independent of t, ÛI(t, )=e i Vt reminiscent of the usual time-evolution operator for time-independent Ĥ.

29 Time-dependent perturbation theory Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) If a system is prepared in an initial state, i at time t =, at a subsequent time, t, the system will be in a final state, ÛI(t, ) i. Using the resolution of identity, n n n = I, we therefore have Û I (t, ) i = n n c n (t) {}}{ n Û I (t, ) i From relation above, the coefficients in the expansion given by c n (t) =δ ni i 1 2 dt dt n V I (t ) i dt n V I (t )V I (t ) i +

30 Time-dependent perturbation theory Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) If a system is prepared in an initial state, i at time t =, at a subsequent time, t, the system will be in a final state, ÛI(t, ) i. Using the resolution of identity, n n n = I, we therefore have Û I (t, ) i = n n c n (t) {}}{ n Û I (t, ) i From relation above, the coefficients in the expansion given by c n (t) =δ ni i 1 2 dt dt n V I (t ) i dt n V I (t )V I (t ) i +

31 Time-dependent perturbation theory Û I (t, )= n=0 ( i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) If a system is prepared in an initial state, i at time t =, at a subsequent time, t, the system will be in a final state, ÛI(t, ) i. Using the resolution of identity, n n n = I, we therefore have Û I (t, ) i = n n c n (t) {}}{ n ÛI(t, ) i From relation above, the coefficients in the expansion given by c n (t) =δ ni i 1 2 dt dt n V I (t ) i dt m n V I (t ) m m V I (t ) i +

32 Time-dependent perturbation theory c n (t) =δ ni i 1 2 dt dt n V I (t ) i dt n V I (t ) m m V I (t ) i + m Recalling the definition, V I (t) =e iĥ 0 t/ V (t)e iĥ 0 t/, the matrix elements entering the coefficients are then given by n V I (t) m = n e iĥ0t/ V (t)e iĥ0t/ m [ ] i = n V (t) m exp }{{} (E n E m ) V }{{} nm e iω nmt where V nm (t) = n V (t) m denote matrix elements between the basis states of Ĥ0 on the perturbation, and ω nm =(E n E m )/.

33 Time-dependent perturbation theory c n (t) =δ ni i 1 2 dt dt n V I (t ) i dt n V I (t ) m m V I (t ) i + m Therefore, using the relation, n V I (t) m = n V (t) m e iω nmt, c n (1) (t) = i c n (2) (t) = 1 2 m dt e iω ni t dt V ni (t ) dt e iω nmt +iω mi t V nm (t )V mi (t ) As a result, we obtain transition probability i n i, P i n (t) = c n (t) 2 = c (1) n + c (2) n + 2

34 Example: Kicked oscillator Suppose quantum harmonic oscillator, Ĥ = ω(a a +1/2), prepared in ground state 0 at time t =. If it is perturbed by weak (transient) electric field, V (t) = eex e t2 /τ 2 what is probability of finding it in first excited state, 1, at t =+? Working to first order in V, P 0 1 c (1) 1 2 where c (1) 1 (t) = i dt e iω 10t V 10 (t ) with V 10 (t )= ee 1 x 0 e t 2 /τ 2 and ω 10 = ω

35 Example: Kicked oscillator Suppose quantum harmonic oscillator, Ĥ = ω(a a +1/2), prepared in ground state 0 at time t =. If it is perturbed by weak (transient) electric field, V (t) = eex e t2 /τ 2 what is probability of finding it in first excited state, 1, at t =+? Working to first order in V, P 0 1 c (1) 1 2 where c (1) 1 (t) = i dt e iω 10t V 10 (t ) with V 10 (t )= ee 1 x 0 e t 2 /τ 2 and ω 10 = ω

36 Example: Kicked oscillator c (1) 1 (t) = i dt e iωt V 10 (t ), V 10 (t )= ee 1 x 0 e t 2 /τ 2 Using the ladder operator formalism, with 1 = a 0 and x = 2mω (a + a ), 1 x 0 = 2mω 0 a(a + a ) 0 = With = dt e iωt e t 2 /τ 2 = πτ exp [ 14 ω2 τ 2 ], c (1) π 1 (t )=ieeτ 2 2m ω e ω τ 2 /4 2mω Transition probability, ( P 0 1 c (1) 1 (t) 2 =(eeτ) 2 π ) e ω2 τ 2 /2 2m ω Note that P 0 1 is maximal for τ 1/ω.

37 Example: Kicked oscillator c (1) 1 (t) = i dt e iωt V 10 (t ), V 10 (t )= ee 1 x 0 e t 2 /τ 2 Using the ladder operator formalism, with 1 = a 0 and x = 2mω (a + a ), 1 x 0 = 2mω 0 a(a + a ) 0 = With = dt e iωt e t 2 /τ 2 = πτ exp [ 14 ω2 τ 2 ], c (1) π 1 (t )=ieeτ 2 2m ω e ω τ 2 /4 2mω Transition probability, ( P 0 1 c (1) 1 (t) 2 =(eeτ) 2 π ) e ω2 τ 2 /2 2m ω Note that P 0 1 is maximal for τ 1/ω.

38 Example: Kicked oscillator c (1) 1 (t) = i dt e iωt V 10 (t ), V 10 (t )= ee 1 x 0 e t 2 /τ 2 Using the ladder operator formalism, with 1 = a 0 and x = 2mω (a + a ), 1 x 0 = 2mω 0 a(a + a ) 0 = With = dt e iωt e t 2 /τ 2 = πτ exp [ 14 ω2 τ 2 ], c (1) π 1 (t )=ieeτ 2 2m ω e ω τ 2 /4 2mω Transition probability, ( P 0 1 c (1) 1 (t) 2 =(eeτ) 2 π ) e ω2 τ 2 /2 2m ω Note that P 0 1 is maximal for τ 1/ω.

39 Sudden perturbation quantum quench Suppose there is a switch from Ĥ 0 to Ĥ 0 in a time shorter than any other characteristic scale perturbation theory is irrelevant: If system is initially in eigenstate n of Ĥ 0, time evolution after switch will just follow that of Ĥ 0, i.e. simply expand initial state as a sum over eigenstates of Ĥ 0, n = n n n n, n(t) = n e ie n t/ n n n Non-trivial part of the problem lies in establishing that the change is sudden enough. This is achieved by estimating the actual time taken for the Hamiltonian to change, and the periods of motion associated with the state n and with its transitions to neighbouring states.

40 Sudden perturbation quantum quench Suppose there is a switch from Ĥ 0 to Ĥ 0 in a time shorter than any other characteristic scale perturbation theory is irrelevant: If system is initially in eigenstate n of Ĥ 0, time evolution after switch will just follow that of Ĥ 0, i.e. simply expand initial state as a sum over eigenstates of Ĥ 0, n = n n n n, n(t) = n e ie n t/ n n n Non-trivial part of the problem lies in establishing that the change is sudden enough. This is achieved by estimating the actual time taken for the Hamiltonian to change, and the periods of motion associated with the state n and with its transitions to neighbouring states.

41 Harmonic perturbations: Fermi s Golden Rule Consider system prepared in initial state i and perturbed by a periodic harmonic potential V (t) =Ve iωt which is abruptly switched on at time t = 0. e.g. atom perturbed by an external oscillating electric field. What is the probability that, at some later time t, the system is in state f? To first order in perturbation theory, c (1) f (t) = i dt e iω fit V fi (t ) i.e. probability of effecting transition after a time t, P i f (t) c (1) f (t) 2 = i f V i ei(ω fi ω)t/2 sin((ω fi ω)t/2) (ω fi ω)/2 2

42 Harmonic perturbations: Fermi s Golden Rule Consider system prepared in initial state i and perturbed by a periodic harmonic potential V (t) =Ve iωt which is abruptly switched on at time t = 0. e.g. atom perturbed by an external oscillating electric field. What is the probability that, at some later time t, the system is in state f? To first order in perturbation theory, c (1) f (t) = i dt e iω fit V fi (t ) i.e. probability of effecting transition after a time t, P i f (t) c (1) f (t) 2 = i f V i ei(ω fi ω)t/2 sin((ω fi ω)t/2) (ω fi ω)/2 2

43 Harmonic perturbations: Fermi s Golden Rule Consider system prepared in initial state i and perturbed by a periodic harmonic potential V (t) =Ve iωt which is abruptly switched on at time t = 0. e.g. atom perturbed by an external oscillating electric field. What is the probability that, at some later time t, the system is in state f? To first order in perturbation theory, c (1) f (t) = i 0 dt f V i e i(ω fi ω)t = i f V i ei(ω fi ω)t 1 i(ω fi ω) i.e. probability of effecting transition after a time t, P i f (t) c (1) f (t) 2 = i f V i ei(ω fi ω)t/2 sin((ω fi ω)t/2) (ω fi ω)/2 2

44 Harmonic perturbations: Fermi s Golden Rule Consider system prepared in initial state i and perturbed by a periodic harmonic potential V (t) =Ve iωt which is abruptly switched on at time t = 0. e.g. atom perturbed by an external oscillating electric field. What is the probability that, at some later time t, the system is in state f? To first order in perturbation theory, c (1) f (t) = i 0 dt f V i e i(ω fi ω)t = i f V i ei(ω fi ω)t 1 i(ω fi ω) i.e. probability of effecting transition after a time t, P i f (t) c (1) f (t) 2 = i f V i ei(ω fi ω)t/2 sin((ω fi ω)t/2) (ω fi ω)/2 2

45 Harmonic perturbations: Fermi s Golden Rule Consider system prepared in initial state i and perturbed by a periodic harmonic potential V (t) =Ve iωt which is abruptly switched on at time t = 0. e.g. atom perturbed by an external oscillating electric field. What is the probability that, at some later time t, the system is in state f? To first order in perturbation theory, c (1) f (t) = i 0 dt f V i e i(ω fi ω)t = i f V i ei(ω fi ω)t 1 i(ω fi ω) i.e. probability of effecting transition after a time t, P i f (t) c (1) f (t) 2 = 1 ( ) 2 2 f V sin((ωfi ω)t/2) i 2 (ω fi ω)/2

46 Harmonic perturbations: Fermi s Golden Rule P i f (t) 1 2 f V i 2 ( sin((ωfi ω)t/2) (ω fi ω)/2 ) 2 Setting α =(ω fl ω)/2, probability sin 2 (αt)/α 2 with a peak at α = 0 maximum value t 2, width O(1/t) total weight O(t). ( ) 2 1 sin(αt) For large t, lim = πδ(α) =2πδ(2α) t t α Fermi s Golden rule: transition rate, R i f (t) = lim t P i f (t) t = 2π 2 f V i 2 δ(ω fi ω)

47 Harmonic perturbations: Fermi s Golden Rule P i f (t) 1 2 f V i 2 ( sin((ωfi ω)t/2) (ω fi ω)/2 ) 2 Setting α =(ω fl ω)/2, probability sin 2 (αt)/α 2 with a peak at α = 0 maximum value t 2, width O(1/t) total weight O(t). ( ) 2 1 sin(αt) For large t, lim = πδ(α) =2πδ(2α) t t α Fermi s Golden rule: transition rate, R i f (t) = lim t P i f (t) t = 2π 2 f V i 2 δ(ω fi ω)

48 Harmonic perturbations: Fermi s Golden Rule R i f (t) = 2π 2 f V i 2 δ(ω fi ω) This result shows that, for a transition to occur, to satisfy energy conservation we must have: (a) final states exist over a continuous energy range to match E = ω for fixed perturbation frequency ω, or (b) perturbation must cover sufficiently wide spectrum of frequency so that a discrete transition with E = ω is possible. For any two discrete pair of states i and f, since V fi 2 = V if 2, we have P i f = P f i statement of detailed balance.

49 Harmonic perturbations: second order transitions Although first order perturbation theory often sufficient, sometimes f V i = 0 by symmetry (e.g. parity, selection rules, etc.). In such cases, transition may be accomplished by indirect route through other non-zero matrix elements. At second order of perturbation theory, c (2) f (t) = 1 2 dt m dt e iω fmt +iω mi t V fm (t )V mi (t ) If harmonic potential perturbation is gradually switched on, V (t) =e εt Ve iωt, ε 0, with the initial time, c (2) f (t) = 1 2 f V m m V i m dt dt e i(ω fm ω iε)t e i(ω mi ω iε)t

50 Harmonic perturbations: second order transitions Although first order perturbation theory often sufficient, sometimes f V i = 0 by symmetry (e.g. parity, selection rules, etc.). In such cases, transition may be accomplished by indirect route through other non-zero matrix elements. At second order of perturbation theory, c (2) f (t) = 1 2 dt m dt e iω fmt +iω mi t V fm (t )V mi (t ) If harmonic potential perturbation is gradually switched on, V (t) =e εt Ve iωt, ε 0, with the initial time, c (2) f (t) = 1 2 f V m m V i m dt dt e i(ω fm ω iε)t e i(ω mi ω iε)t

51 Harmonic perturbations: second-order transitions From time integral, c (2) n = 1 2 ei(ω fi 2ω)t e 2εt ω fi 2ω 2iε m f V m m V i ω mi ω iε Leads to transition rate (ε 0): d dt c(2) n (t) 2 = 2π 4 m f V m m V i ω mi ω iε 2 δ(ω fi 2ω) This translates to a transition in which system gains energy 2 ω from harmonic perturbation, i.e. two photons are absorbed Physically, first photon takes effects virtual transition to short-lived intermediate state with energy ω m. If an atom in an arbitrary state is exposed to monochromatic light, other second order processes in which two photons are emitted, or one is absorbed and one emitted are also possible.

52 Harmonic perturbations: second-order transitions From time integral, c (2) n = 1 2 ei(ω fi 2ω)t e 2εt ω fi 2ω 2iε m f V m m V i ω mi ω iε Leads to transition rate (ε 0): d dt c(2) n (t) 2 = 2π 4 m f V m m V i ω mi ω iε 2 δ(ω fi 2ω) This translates to a transition in which system gains energy 2 ω from harmonic perturbation, i.e. two photons are absorbed Physically, first photon takes effects virtual transition to short-lived intermediate state with energy ω m. If an atom in an arbitrary state is exposed to monochromatic light, other second order processes in which two photons are emitted, or one is absorbed and one emitted are also possible.

53 Harmonic perturbations: second-order transitions From time integral, c (2) n = 1 2 ei(ω fi 2ω)t e 2εt ω fi 2ω 2iε m f V m m V i ω mi ω iε Leads to transition rate (ε 0): d dt c(2) n (t) 2 = 2π 4 m f V m m V i ω mi ω iε 2 δ(ω fi 2ω) This translates to a transition in which system gains energy 2 ω from harmonic perturbation, i.e. two photons are absorbed Physically, first photon takes effects virtual transition to short-lived intermediate state with energy ω m. If an atom in an arbitrary state is exposed to monochromatic light, other second order processes in which two photons are emitted, or one is absorbed and one emitted are also possible.

54 Time-dependent perturbation theory: summary For a general time-dependent Hamiltonian, Ĥ = Ĥ 0 + V (t), in which all time-dependence containing in potential V (t), the wavefunction can be expressed in the interaction representation, ψ(t) I = e iĥ 0 t/ ψ(t) S, ψ(0) I = ψ(0) S In this representation, the time-dependent Schrödinger equation takes the form, i t ψ(t) I = V I (t) ψ(t) I, V I (t) =e iĥ 0 t/ V (t)e iĥ 0 t/ If we expand ψ(t) I = n c n(t) n in basis of time-independent Hamiltonian, Ĥ0 n = E n n, the Schrödinger equation translates to i ċ m (t) = n V mn (t)e iω mnt c n (t) where V mn (t) = m V (t) m and ω mn = 1 (E m E n )= ω nm.

55 Time-dependent perturbation theory: summary For a general time-dependent Hamiltonian, Ĥ = Ĥ 0 + V (t), in which all time-dependence containing in potential V (t), the wavefunction can be expressed in the interaction representation, ψ(t) I = e iĥ 0 t/ ψ(t) S, ψ(0) I = ψ(0) S In this representation, the time-dependent Schrödinger equation takes the form, i t ψ(t) I = V I (t) ψ(t) I, V I (t) =e iĥ 0 t/ V (t)e iĥ 0 t/ If we expand ψ(t) I = n c n(t) n in basis of time-independent Hamiltonian, Ĥ0 n = E n n, the Schrödinger equation translates to i ċ m (t) = n V mn (t)e iω mnt c n (t) where V mn (t) = m V (t) m and ω mn = 1 (E m E n )= ω nm.

56 Time-dependent perturbation theory: summary For a general time-dependent Hamiltonian, Ĥ = Ĥ 0 + V (t), in which all time-dependence containing in potential V (t), the wavefunction can be expressed in the interaction representation, ψ(t) I = e iĥ 0 t/ ψ(t) S, ψ(0) I = ψ(0) S In this representation, the time-dependent Schrödinger equation takes the form, i t ψ(t) I = V I (t) ψ(t) I, V I (t) =e iĥ 0 t/ V (t)e iĥ 0 t/ If we expand ψ(t) I = n c n(t) n in basis of time-independent Hamiltonian, Ĥ0 n = E n n, the Schrödinger equation translates to i ċ m (t) = n V mn (t)e iω mnt c n (t) where V mn (t) = m V (t) m and ω mn = 1 (E m E n )= ω nm.

57 Time-dependent perturbation theory: summary For a general time-dependent potential, V (t), the wavefunction can be expanded as a power series in the interaction, ψ(t) I = n c n (t) n, c n (t) =c (0) n + c (1) n (t)+c (2) n (t)+ The coefficents can be expressed as matrix elements of the time-evolution operator, c n (t) = n Û I (t, ) i, where ( Û I (t, )= i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) n=0 From first two terms in the series, we have c n (1) (t) = i c n (2) (t) = 1 2 m dt e iω ni t dt V ni (t ) dt e iω nmt +iω mi t V nm (t )V mi (t )

58 Time-dependent perturbation theory: summary For a general time-dependent potential, V (t), the wavefunction can be expanded as a power series in the interaction, ψ(t) I = n c n (t) n, c n (t) =c (0) n + c (1) n (t)+c (2) n (t)+ The coefficents can be expressed as matrix elements of the time-evolution operator, c n (t) = n Û I (t, ) i, where ( Û I (t, )= i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) n=0 From first two terms in the series, we have c n (1) (t) = i c n (2) (t) = 1 2 m dt e iω ni t dt V ni (t ) dt e iω nmt +iω mi t V nm (t )V mi (t )

59 Time-dependent perturbation theory: summary For a general time-dependent potential, V (t), the wavefunction can be expanded as a power series in the interaction, ψ(t) I = n c n (t) n, c n (t) =c (0) n + c (1) n (t)+c (2) n (t)+ The coefficents can be expressed as matrix elements of the time-evolution operator, c n (t) = n Û I (t, ) i, where ( Û I (t, )= i ) n n 1 dt 1 dt n V I (t 1 )V I (t 2 ) V I (t n ) n=0 From first two terms in the series, we have c n (1) (t) = i c n (2) (t) = 1 2 m dt e iω ni t dt V ni (t ) dt e iω nmt +iω mi t V nm (t )V mi (t )

60 Time-dependent perturbation theory: summary c n (1) (t) = i dt e iω ni t V ni (t ) For a harmonic perturbation, V (t) =Ve iωt, turned on at t = 0, the leading term in series translates to transition rate, R i f (t) = lim t P i f (t) t = 2π 2 f V i 2 δ(ω fi ω) Fermi s Golden rule. If this term vanishes by symmetry, transitions can be effected by second and higher order processes through intermediate states. In the next lecture, we will apply these ideas to the consideration of radiative transitions in atoms.

61 Time-dependent perturbation theory: summary c n (1) (t) = i dt e iω ni t V ni (t ) For a harmonic perturbation, V (t) =Ve iωt, turned on at t = 0, the leading term in series translates to transition rate, R i f (t) = lim t P i f (t) t = 2π 2 f V i 2 δ(ω fi ω) Fermi s Golden rule. If this term vanishes by symmetry, transitions can be effected by second and higher order processes through intermediate states. In the next lecture, we will apply these ideas to the consideration of radiative transitions in atoms.