1. Approximation Employed in Spontaneous Emission Theory / D. F
Approximation Employed in Spontaneous Emission Theory / D.F. Walls and C.W. Gardiner (Physics Letters 41A (1972))
Roy Elkabetz

2. Outline Motivation Atom – Field Hamiltonian
Derivation of the Wigner - Weisskopf approximations
Comparison between the R.W.A and the Ladder approximation
Conclusions

3. Motivation
Solving Schrodinger’s equation using non – perturbative methods
Introduce the equivalence of the WWI approximation to the RWA
Show the resemblance between the RWA, the Ladder approximation and the WWI approximation

4. Two level Atom in radiation field Hamiltonian
Background
Two level Atom in radiation field Hamiltonian
radiation
atom
𝐻= 𝐻 0 + 𝐻 1
𝐻 0 = 𝑘 ℏ 𝜔 𝑘 𝑎 𝑘 † 𝑎 𝑘 +ℏ𝜔 𝜎 𝑧
𝑔> <𝑒 atomic operator
𝐻 1 = 𝑗 ℏ 𝜅 𝑗 𝑎 𝑗 † + 𝑎 𝑗 𝜎 𝜎 −
coupling
constant
creation
of photon
annihilation of photon

5. Derivation of the Wigner - Weisskopf Approximations
Wigner – Weisskopf Ι approximation (WWΙ)
|𝑒,0>
excited atom, no photons
|𝑔, 1 𝑘 >
atom in ground state, one photon in mode 𝑘
𝜓 𝑡 > = 𝑐 0 𝑡 𝑒 −𝑖 𝜔 0 𝑡 𝑒,0> + 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘 𝑡 |𝑔, 1 𝑘 >
𝜔 0
is the Bohr frequency
𝜔 𝑘 =𝑐𝑘

6. Derivation of the Wigner - Weisskopf Approximations
Wigner – Weisskopf Ι approximation (WWΙ)
𝜓 𝑡 > = 𝑐 0 𝑡 𝑒 −𝑖 𝜔 0 𝑡 𝑒,0> + 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘 𝑡 |𝑔, 1 𝑘 >
𝑖ℏ 𝑐 𝑛 𝑡 = 𝑐 0 𝑡 𝑒 𝑖 𝜔 𝑛0 𝑡 <𝑔, 1 𝑛 𝐻 1 𝑒,0>
𝑖ℏ 𝑐 0 𝑡 = 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘0 𝑡 <𝑒,0 𝐻 1 𝑔, 1 𝑘 >
𝜔 𝑘0 =: 𝜔 𝑘 − 𝜔 0

7. Derivation of the Wigner - Weisskopf Approximations
Wigner – Weisskopf Ι approximation (WWΙ)
𝜓 𝑡 > = 𝑐 0 𝑡 𝑒 −𝑖 𝜔 0 𝑡 𝑒,0> + 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘 𝑡 |𝑔, 1 𝑘 >
𝑖ℏ 𝑐 𝑛 𝑡 = 𝑐 0 𝑡 𝑒 𝑖 𝜔 𝑛0 𝑡 <𝑔, 1 𝑛 𝐻 1 𝑒,0>
𝑖ℏ 𝑐 0 𝑡 = 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘0 𝑡 <𝑒,0 𝐻 1 𝑔, 1 𝑘 >
The WWΙ Approximation

8. Derivation of the Wigner - Weisskopf Approximations
Solving for 𝑐 𝑘 𝑡
𝑐 0 𝑡=0 =1 𝑐 𝑛 𝑡=0 =0
Initial conditions:
⇒ 𝑐 𝑛 𝑡 = 1 𝑖ℏ 0 𝑡 𝑑 𝑡 ′ 𝑐 0 𝑡 ′ 𝑒 𝑖 𝜔 𝑛0 𝑡 ′ <𝑔, 1 𝑛 𝐻 1 𝑒,0>
𝑖ℏ 𝑐 0 𝑡 = 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘0 𝑡 <𝑒,0 𝐻 1 𝑔, 1 𝑘 >

9. Derivation of the Wigner - Weisskopf Approximations
Solving for 𝑐 𝑘 𝑡
𝑖ℏ 𝑐 0 𝑡 =−𝑖 𝜔 0 𝑐 0 0
− 1 ℏ 2 𝑛 <𝑔, 1 𝑛 𝐻 1 𝑒,0> 𝑡 𝑑 𝑡 ′ 𝑐 0 𝑡 ′ 𝑒 𝑖 𝜔 𝑛0 ( 𝑡 ′ −𝑡)
Defining the next transformation:
𝛼 𝑡 = 𝑐 0 𝑡 𝑒 𝑖 𝜔 0 𝑡 𝒩 𝑡 = 1 ℏ 2 𝑛 <𝑔, 1 𝑛 𝐻 1 𝑒,0> 2 𝑒 𝑖 𝜔 0 − 𝜔 𝑛 𝑡

10. Derivation of the Wigner - Weisskopf Approximations
Solving for 𝑐 𝑘 𝑡
𝑖ℏ 𝑐 0 𝑡 =−𝑖 𝜔 0 𝑐 0 0
− 1 ℏ 2 𝑛 <𝑔, 1 𝑛 𝐻 1 𝑒,0> 𝑡 𝑑 𝑡 ′ 𝑐 0 𝑡 ′ 𝑒 𝑖 𝜔 𝑛0 ( 𝑡 ′ −𝑡) ⇓
𝑑 𝑑𝑡 𝛼 𝑡 =− 0 𝑡 𝑑𝜏𝒩 𝜏 𝛼(𝑡−𝜏)
𝜏=𝑡− 𝑡 ′

11. Derivation of the Wigner - Weisskopf Approximations
Wigner – Weisskopf ΙΙ approximation (WWΙΙ)
Markov Process:
A Markov process is a process where a future outcome of a system depends only on the present state of itself and not on former states from the past.
Memoryless Process

12. Derivation of the Wigner - Weisskopf Approximations
Wigner – Weisskopf ΙΙ approximation (WWΙΙ)
𝑑 𝑑𝑡 𝛼 𝑡 =− 0 𝑡 𝑑𝜏𝒩 𝜏 𝛼(𝑡−𝜏)
Recall:
WWΙΙ approximation ~ Markov process
⟹ 𝑑 𝑑𝑡 𝛼 𝑡 ≃−𝛼(𝑡) 0 ∞ 𝑑𝜏𝒩 𝜏

13. Derivation of the Wigner - Weisskopf Approximations
Wigner – Weisskopf ΙΙ approximation (WWΙΙ)
⟹ 𝑐 0 𝑡 = 𝑒 − Γ 𝑏 2 𝑡 𝑒 −𝑖 𝜔 0 + Δ 𝑏 𝑡
The probability to stay in |𝑒,0> :
P 𝑡 = 𝑐 0 𝑡 2 = 𝑒 − Γ 𝑏 𝑡
Exponential Decay

14. R.W.A and Ladder Approximation
Rotating Wave approximation (RWA)
𝐻 1 𝑅𝑊𝐴 =ℏ 𝑗 𝜅 𝑗 ( 𝑎 𝑗 𝜎 𝑎 𝑗 † 𝜎 − )
Annihilate a photon
and excite the atom
create a photon and atom in ground state
The RWA holds for weak intensity and for 𝜔≈ 𝜔 0

15. R.W.A and Ladder Approximation
Equivalence of WWΙ and RWA
𝐻 1 𝑅𝑊𝐴 =ℏ 𝑗 𝜅 𝑗 ( 𝑎 𝑗 𝜎 𝑎 𝑗 † 𝜎 − ) ⇓
𝑖ℏ 𝑐 𝑛 𝑡 = 𝑐 0 𝑡 𝑒 𝑖 𝜔 𝑛0 𝑡 <𝑔,𝑛 𝐻 1 𝑅𝑊𝐴 𝑒,0>
𝑖ℏ 𝑐 0 𝑡 = 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘0 𝑡 <𝑒,0 𝐻 1 𝑅𝑊𝐴 𝑔,𝑘>

16. R.W.A and Ladder Approximation
Equivalence of WWΙ and RWA
We get the same set of equations as in the WWΙ approximation
The WWΙ approximation and the RWA are equivalent
𝑖ℏ 𝑐 𝑛 𝑡 = 𝑐 0 𝑡 𝑒 𝑖 𝜔 𝑛0 𝑡 <𝑔,𝑛 𝐻 1 𝑅𝑊𝐴 𝑒,0>
𝑖ℏ 𝑐 0 𝑡 = 𝑘 𝑐 𝑘 𝑡 𝑒 −𝑖 𝜔 𝑘0 𝑡 <𝑒,0 𝐻 1 𝑅𝑊𝐴 𝑔,𝑘>

17. R.W.A and Ladder Approximation
𝐻= 𝐻 0 + 𝐻 𝑖𝑛𝑡
𝐻 0 = 𝑘 ℏ 𝜔 𝑘 𝑎 𝑘 † 𝑎 𝑘 +ℏ𝜔 𝜎 𝑧
𝐻 𝑖𝑛𝑡 =− 𝒑 ∙ 𝑨 ⊥
Atom basis: |1>, |0>
Field basis: |0>, | 1 a >,| 2 a >,…,| n a −1>, | n a >
Single photon approximation

18. R.W.A and Ladder Approximation
Ladder Approximation: Diagrams
Single photon approximation: | n a −1>, | n a >
zero photon scattering, virtual transitions
single photon scattering + virtual transitions
Figure 1: Diagrams included in Ladder approximation

19. R.W.A and Ladder Approximation
Ladder Approximation: Diagrams
Figure 2: Diagrams excluded from Ladder approximation

20. R.W.A and Ladder Approximation
Ladder Approximation: Another Diagrams
Figure 3: a) Diagrams included in Ladder approximation.
b) Diagrams excluded from Ladder approximation.

21. R.W.A and Ladder Approximation
The Ladder and the WWI Approximations
We sum over infinite number of terms
Notice that |1, n a −1>, |0, n a > ~
|𝑒,0>, |𝑔, 1 𝑘 >
Ladder basis
WWI basis
⇒ The Ladder approximation and the WWI approximation has close resemblance

22. R.W.A and Ladder Approximation
Ladder Approximation and RWA
Under the assumptions:
Single photon process
Close to resonance
⇒ There is a close resemblance between the RWA and the Ladder approximation

23. For spontaneous emission of a two level atom in radiation field
Conclusions
For spontaneous emission of a two level atom in radiation field
The RWA and the WWI approximation are equivalent
Using the approximations above has a clear superiority over finite perturbation techniques
Under proper assumptions the RWA and the Ladder approximation are with close resemblance

24. Thank You For Listening
The End
Thank You For Listening