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

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

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

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

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

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

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

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 >

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> 2 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

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> 2 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 ′

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

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

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

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

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>

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>

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

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

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

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

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

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

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

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