Categories:
Nonlinear optics ,
Optics ,
Perturbation ,
Physics ,
Quantum mechanics .
Multi-photon absorption
Consider a quantum system where there are many eigenstates ∣ n ⟩ \Ket{n} ∣ n ⟩ electromagnetic wave
passes by, such that its Hamiltonian gets perturbed by H ^ 1 \hat{H}_1 H ^ 1 electric dipole approximation by:
H ^ 1 ( t ) = − p ^ ⋅ E cos ( ω t ) ≈ − p ^ ⋅ E e − i ω t \begin{aligned}
\hat{H}_1(t)
= -\vu{p} \cdot \vb{E} \cos(\omega t)
\approx -\vu{p} \cdot \vb{E} e^{-i \omega t}
\end{aligned} H ^ 1 ( t ) = − p ^ ⋅ E cos ( ω t ) ≈ − p ^ ⋅ E e − iω t Where E \vb{E} E electric field amplitude,
and p ^ ≡ q x ^ \vu{p} \equiv q \vu{x} p ^ ≡ q x ^ rotating wave approximation
to neglect the e i ω t e^{i \omega t} e iω t
We call the ground state ∣ 0 ⟩ \Ket{0} ∣ 0 ⟩ not be sorted by energy.
However, we demand that the following holds
for all even-numbered states ∣ e ⟩ \Ket{e} ∣ e ⟩ ∣ e ′ ⟩ \Ket{e'} ∣ e ′ ⟩ u u u ∣ u ⟩ \Ket{u} ∣ u ⟩ ∣ u ′ ⟩ \Ket{u'} ∣ u ′ ⟩
⟨ e ∣ H ^ 1 ∣ e ′ ⟩ = ⟨ u ∣ H ^ 1 ∣ u ′ ⟩ = 0 ⟨ e ∣ H ^ 1 ∣ u ⟩ ≠ 0 \begin{aligned}
\matrixel{e}{\hat{H}_1}{e'} = \matrixel{u}{\hat{H}_1}{u'} = 0
\qquad \quad
\matrixel{e}{\hat{H}_1}{u} \neq 0
\end{aligned} ⟨ e ∣ H ^ 1 ∣ e ′ ⟩ = ⟨ u ∣ H ^ 1 ∣ u ′ ⟩ = 0 ⟨ e ∣ H ^ 1 ∣ u ⟩ = 0 This is justified for atomic orbitals thanks to
Laporte’s selection rule .
Therefore, time-dependent perturbation theory
says that the N N N
c e ( N ) ( t ) = − i ℏ ∑ u o d d ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( N − 1 ) ( τ ) e i ω e u τ d τ c u ( N ) ( t ) = − i ℏ ∑ e e v e n ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ e ⟩ c e ( N − 1 ) ( τ ) e i ω u e τ d τ \begin{aligned}
c_e^{(N)}(t)
&= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(N-1)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
\\
c_u^{(N)}(t)
&= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(N-1)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
\end{aligned} c e ( N ) ( t ) c u ( N ) ( t ) = − ℏ i u ∑ odd ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( N − 1 ) ( τ ) e i ω e u τ d τ = − ℏ i e ∑ even ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ e ⟩ c e ( N − 1 ) ( τ ) e i ω u e τ d τ Where ω e u = ( E e − E u ) / ℏ \omega_{eu} = (E_e \!-\! E_u) / \hbar ω e u = ( E e − E u ) /ℏ ∣ 0 ⟩ \Ket{0} ∣ 0 ⟩
c 0 ( 0 ) = 1 c u ( 0 ) = c e ≠ 0 ( 0 ) = 0 \begin{aligned}
c_0^{(0)} = 1
\qquad \qquad
c_u^{(0)} = c_{e \neq 0}^{(0)} = 0
\end{aligned} c 0 ( 0 ) = 1 c u ( 0 ) = c e = 0 ( 0 ) = 0 Finally, we prove the following useful relation for large t t t Dirac delta function δ \delta δ
lim t → ∞ ∣ e i x t − 1 x ∣ 2 = 2 π δ ( x ) t \begin{aligned}
\lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
= 2 \pi \: \delta(x) \: t
\end{aligned} t → ∞ lim x e i x t − 1 2 = 2 π δ ( x ) t
Proof
Proof.
First, observe that we can rewrite the fraction using an integral:
e i x t − 1 x = e i x t / 2 e i x t / 2 − e − i x t / 2 x = i e i x t / 2 ∫ − t / 2 t / 2 e i x τ d τ \begin{aligned}
\frac{e^{i x t} - 1}{x}
= e^{i x t / 2} \frac{e^{i x t / 2} - e^{-i x t / 2}}{x}
= i e^{i x t / 2} \int_{-t/2}^{t/2} e^{i x \tau} \dd{\tau}
\end{aligned} x e i x t − 1 = e i x t /2 x e i x t /2 − e − i x t /2 = i e i x t /2 ∫ − t /2 t /2 e i xτ d τ By taking the limit t → ∞ t \to \infty t → ∞
lim t → ∞ e i x t − 1 x = lim t → ∞ i e i x t / 2 2 π 2 π ∫ − ∞ ∞ e i x τ d τ = lim t → ∞ i 2 π e i x t / 2 δ ( x ) \begin{aligned}
\lim_{t \to \infty} \frac{e^{i x t} - 1}{x}
= \lim_{t \to \infty} i e^{i x t / 2} \frac{2 \pi}{2 \pi} \int_{-\infty}^{\infty} e^{i x \tau} \dd{\tau}
= \lim_{t \to \infty} i 2 \pi e^{i x t / 2} \: \delta(x)
\end{aligned} t → ∞ lim x e i x t − 1 = t → ∞ lim i e i x t /2 2 π 2 π ∫ − ∞ ∞ e i xτ d τ = t → ∞ lim i 2 π e i x t /2 δ ( x ) Consequently, the absolute value squared is as follows:
lim t → ∞ ∣ e i x t − 1 x ∣ 2 = 4 π 2 δ 2 ( x ) \begin{aligned}
\lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
= 4 \pi^2 \delta^2(x)
\end{aligned} t → ∞ lim x e i x t − 1 2 = 4 π 2 δ 2 ( x ) However, a squared delta function δ 2 \delta^2 δ 2
δ 2 ( x ) = δ ( x ) lim t → ∞ 1 2 π ∫ − t / 2 t / 2 e i x τ d τ = δ ( x ) lim t → ∞ t 2 π \begin{aligned}
\delta^2(x)
= \delta(x) \lim_{t \to \infty} \frac{1}{2 \pi} \int_{-t/2}^{t/2} e^{i x \tau} \dd{\tau}
= \delta(x) \lim_{t \to \infty} \frac{t}{2 \pi}
\end{aligned} δ 2 ( x ) = δ ( x ) t → ∞ lim 2 π 1 ∫ − t /2 t /2 e i xτ d τ = δ ( x ) t → ∞ lim 2 π t Where we have set x = 0 x = 0 x = 0
lim t → ∞ ∣ e i x t − 1 x ∣ 2 = 4 π 2 δ 2 ( x ) = 2 π δ ( x ) t \begin{aligned}
\lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
= 4 \pi^2 \delta^2(x)
= 2 \pi \: \delta(x) \: t
\end{aligned} t → ∞ lim x e i x t − 1 2 = 4 π 2 δ 2 ( x ) = 2 π δ ( x ) t
One-photon absorption
To warm up, we start at first-order perturbation theory.
Thanks to our choice of initial condition,
nothing at all happens to any of the even-numbered states ∣ e ⟩ \Ket{e} ∣ e ⟩
c e ( 1 ) ( t ) = − i ℏ ∑ u o d d ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( 0 ) e i ω e u τ d τ = 0 \begin{aligned}
c_e^{(1)}(t)
&= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(0)} \: e^{i \omega_{eu} \tau} \dd{\tau}
= 0
\end{aligned} c e ( 1 ) ( t ) = − ℏ i u ∑ odd ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( 0 ) e i ω e u τ d τ = 0 While the odd-numbered states ∣ u ⟩ \Ket{u} ∣ u ⟩ c u ( 1 ) c_u^{(1)} c u ( 1 ) p u 0 = ⟨ u ∣ p ^ ∣ 0 ⟩ \vb{p}_{u0} = \matrixel{u}{\vu{p}}{0} p u 0 = ⟨ u ∣ p ^ ∣ 0 ⟩
c u ( 1 ) ( t ) = − i ℏ ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ 0 ⟩ c 0 ( 0 ) e i ω u 0 τ d τ = i p u 0 ⋅ E ℏ ∫ 0 t e i ( ω u 0 − ω ) τ d τ = i p u 0 ⋅ E ℏ [ e i ( ω u 0 − ω ) τ i ( ω u 0 − ω ) ] 0 t \begin{aligned}
c_u^{(1)}(t)
&= -\frac{i}{\hbar} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{0} \: c_0^{(0)} \: e^{i \omega_{u0} \tau} \dd{\tau}
\\
&= i \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \int_0^t e^{i (\omega_{u0} - \omega) \tau} \dd{\tau}
\\
&= i \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \bigg[ \frac{e^{i (\omega_{u0} - \omega) \tau}}{i (\omega_{u0} - \omega)} \bigg]_0^t
\end{aligned} c u ( 1 ) ( t ) = − ℏ i ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ 0 ⟩ c 0 ( 0 ) e i ω u 0 τ d τ = i ℏ p u 0 ⋅ E ∫ 0 t e i ( ω u 0 − ω ) τ d τ = i ℏ p u 0 ⋅ E [ i ( ω u 0 − ω ) e i ( ω u 0 − ω ) τ ] 0 t Consequently, the first-order correction
(in the rotating wave approximation) is given by:
c u ( 1 ) ( t ) ≈ p u 0 ⋅ E ℏ e i ( ω u 0 − ω ) t − 1 ω u 0 − ω \begin{aligned}
\boxed{
c_u^{(1)}(t)
\approx \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \frac{e^{i (\omega_{u0} - \omega) t} - 1}{\omega_{u0} - \omega}
}
\end{aligned} c u ( 1 ) ( t ) ≈ ℏ p u 0 ⋅ E ω u 0 − ω e i ( ω u 0 − ω ) t − 1 Since ∣ c u ( 1 ) ( t ) ∣ 2 \big| c_u^{(1)}(t) \big|^2 c u ( 1 ) ( t ) 2 ∣ u ⟩ \Ket{u} ∣ u ⟩ R u ( 1 ) ( t ) R_u^{(1)}(t) R u ( 1 ) ( t ) t = 0 t = 0 t = 0
R u ( 1 ) ( t ) = ∣ c u ( 1 ) ( t ) ∣ 2 t = 1 t ∣ p u 0 ⋅ E ℏ ∣ 2 ⋅ ∣ e i ( ω u 0 − ω ) t − 1 ω u 0 − ω ∣ 2 \begin{aligned}
R_u^{(1)}(t)
= \frac{\big| c_u^{(1)}(t) \big|^2}{t}
= \frac{1}{t} \bigg| \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \bigg|^2
\cdot \bigg| \frac{e^{i (\omega_{u0} - \omega) t} - 1}{\omega_{u0} - \omega} \bigg|^2
\end{aligned} R u ( 1 ) ( t ) = t c u ( 1 ) ( t ) 2 = t 1 ℏ p u 0 ⋅ E 2 ⋅ ω u 0 − ω e i ( ω u 0 − ω ) t − 1 2 For large t → ∞ t \to \infty t → ∞ Fermi’s golden rule :
R u ( 1 ) = 2 π ∣ p u 0 ⋅ E ℏ ∣ 2 δ ( ω u 0 − ω ) \begin{aligned}
\boxed{
R_u^{(1)}
= 2 \pi \bigg| \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \bigg|^2 \delta(\omega_{u0} - \omega)
}
\end{aligned} R u ( 1 ) = 2 π ℏ p u 0 ⋅ E 2 δ ( ω u 0 − ω ) This well-known formula represents one-photon absorption :
it peaks at ω u 0 = ω \omega_{u0} = \omega ω u 0 = ω ℏ ω \hbar \omega ℏ ω ℏ ω u 0 \hbar \omega_{u0} ℏ ω u 0 ⟨ u ∣ p ^ ∣ 0 ⟩ ≠ 0 \matrixel{u}{\vu{p}}{0} \neq 0 ⟨ u ∣ p ^ ∣ 0 ⟩ = 0 ∣ u ⟩ \Ket{u} ∣ u ⟩
Two-photon absorption
Next, we go to second-order perturbation theory.
Based on the previous result, this time
all odd-numbered states ∣ u ⟩ \Ket{u} ∣ u ⟩
c u ( 2 ) ( t ) = − i ℏ ∑ e e v e n ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ e ⟩ c e ( 1 ) ( τ ) e i ω u e τ d τ = 0 \begin{aligned}
c_u^{(2)}(t)
&= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(1)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
= 0
\end{aligned} c u ( 2 ) ( t ) = − ℏ i e ∑ even ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ e ⟩ c e ( 1 ) ( τ ) e i ω u e τ d τ = 0 While the even-numbered states ∣ e ⟩ \Ket{e} ∣ e ⟩ ω e u + ω u 0 = ω e 0 \omega_{eu} \!+\! \omega_{u0} = \omega_{e0} ω e u + ω u 0 = ω e 0
c e ( 2 ) ( t ) = − i ℏ ∑ u o d d ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( 1 ) ( τ ) e i ω e u τ d τ = i ∑ u o d d ( p e u ⋅ E ) ( p u 0 ⋅ E ) ℏ 2 ( ω u 0 − ω ) ∫ 0 t e i ( ω e u + ω u 0 − 2 ω ) τ − e i ( ω e u − ω ) τ d τ = i ∑ u o d d ( p e u ⋅ E ) ( p u 0 ⋅ E ) ℏ 2 ( ω u 0 − ω ) [ e i ( ω e 0 − 2 ω ) τ i ( ω e 0 − 2 ω ) − e i ( ω e u − ω ) τ i ( ω e u − ω ) ] 0 t \begin{aligned}
c_e^{(2)}(t)
&= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(1)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
\\
&= i \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)}
\int_0^t e^{i (\omega_{eu} + \omega_{u0} - 2 \omega) \tau} - e^{i (\omega_{eu} - \omega) \tau} \dd{\tau}
\\
&= i \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)}
\bigg[ \frac{e^{i (\omega_{e0} - 2 \omega) \tau}}{i (\omega_{e0} - 2 \omega)}
- \frac{e^{i (\omega_{eu} - \omega) \tau}}{i (\omega_{eu} - \omega)} \bigg]_0^t
\end{aligned} c e ( 2 ) ( t ) = − ℏ i u ∑ odd ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( 1 ) ( τ ) e i ω e u τ d τ = i u ∑ odd ℏ 2 ( ω u 0 − ω ) ( p e u ⋅ E ) ( p u 0 ⋅ E ) ∫ 0 t e i ( ω e u + ω u 0 − 2 ω ) τ − e i ( ω e u − ω ) τ d τ = i u ∑ odd ℏ 2 ( ω u 0 − ω ) ( p e u ⋅ E ) ( p u 0 ⋅ E ) [ i ( ω e 0 − 2 ω ) e i ( ω e 0 − 2 ω ) τ − i ( ω e u − ω ) e i ( ω e u − ω ) τ ] 0 t The second term represents one-photon absorption between ∣ u ⟩ \Ket{u} ∣ u ⟩ ∣ e ⟩ \Ket{e} ∣ e ⟩
c e ( 2 ) ( t ) ≈ ∑ u o d d ( p e u ⋅ E ) ( p u 0 ⋅ E ) ℏ 2 ( ω u 0 − ω ) e i ( ω e 0 − 2 ω ) t − 1 ω e 0 − 2 ω \begin{aligned}
\boxed{
c_e^{(2)}(t)
\approx \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)}
\frac{e^{i (\omega_{e0} - 2 \omega) t} - 1}{\omega_{e0} - 2 \omega}
}
\end{aligned} c e ( 2 ) ( t ) ≈ u ∑ odd ℏ 2 ( ω u 0 − ω ) ( p e u ⋅ E ) ( p u 0 ⋅ E ) ω e 0 − 2 ω e i ( ω e 0 − 2 ω ) t − 1 As before, we can define a rate R e ( 2 ) ( t ) R_e^{(2)}(t) R e ( 2 ) ( t )
R e ( 2 ) ( t ) = ∣ c e ( 2 ) ( t ) ∣ 2 t = 1 t ∣ ∑ u o d d ( p e u ⋅ E ) ( p u 0 ⋅ E ) ℏ 2 ( ω u 0 − ω ) ∣ 2 ⋅ ∣ e i ( ω e 0 − 2 ω ) t − 1 ω e 0 − 2 ω ∣ 2 \begin{aligned}
R_e^{(2)}(t)
= \frac{\big| c_e^{(2)}(t) \big|^2}{t}
= \frac{1}{t} \bigg| \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)} \bigg|^2
\cdot \bigg| \frac{e^{i (\omega_{e0} - 2 \omega) t} - 1}{\omega_{e0} - 2 \omega} \bigg|^2
\end{aligned} R e ( 2 ) ( t ) = t c e ( 2 ) ( t ) 2 = t 1 u ∑ odd ℏ 2 ( ω u 0 − ω ) ( p e u ⋅ E ) ( p u 0 ⋅ E ) 2 ⋅ ω e 0 − 2 ω e i ( ω e 0 − 2 ω ) t − 1 2 Which for t → ∞ t \to \infty t → ∞
R e ( 2 ) = 2 π ∣ ∑ u o d d ( p e u ⋅ E ) ( p u 0 ⋅ E ) ℏ 2 ( ω u 0 − ω ) ∣ 2 δ ( ω e 0 − 2 ω ) \begin{aligned}
\boxed{
R_e^{(2)}
= 2 \pi \bigg| \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)} \bigg|^2
\delta(\omega_{e0} - 2 \omega)
}
\end{aligned} R e ( 2 ) = 2 π u ∑ odd ℏ 2 ( ω u 0 − ω ) ( p e u ⋅ E ) ( p u 0 ⋅ E ) 2 δ ( ω e 0 − 2 ω ) This represents two-photon absorption , since it peaks at ω e 0 = 2 ω \omega_{e0} = 2 \omega ω e 0 = 2 ω ℏ ω \hbar \omega ℏ ω ℏ ω e 0 \hbar \omega_{e0} ℏ ω e 0 ⟨ e ∣ p ^ ∣ 0 ⟩ = 0 \matrixel{e}{\vu{p}}{0} = 0 ⟨ e ∣ p ^ ∣ 0 ⟩ = 0 ∣ e ⟩ \Ket{e} ∣ e ⟩ ∣ E ∣ 4 |\vb{E}|^4 ∣ E ∣ 4
Three-photon absorption
For third-order perturbation theory,
all even-numbered states ∣ e ⟩ \Ket{e} ∣ e ⟩
c e ( 3 ) ( t ) = − i ℏ ∑ u o d d ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( 2 ) ( τ ) e i ω e u τ d τ = 0 \begin{aligned}
c_e^{(3)}(t)
&= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(2)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
= 0
\end{aligned} c e ( 3 ) ( t ) = − ℏ i u ∑ odd ∫ 0 t ⟨ e ∣ H ^ 1 ( τ ) ∣ u ⟩ c u ( 2 ) ( τ ) e i ω e u τ d τ = 0 And the odd-numbered states ∣ u ⟩ \Ket{u} ∣ u ⟩
c u ( 3 ) ( t ) = − i ℏ ∑ e e v e n ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ e ⟩ c e ( 2 ) ( τ ) e i ω u e τ d τ = i ∑ e e v e n ∑ u ′ o d d ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ∫ 0 t e i ( ω u e + ω e 0 − 3 ω ) τ − e i ( ω u e − ω ) τ d τ = i ∑ e e v e n ∑ u ′ o d d ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) [ e i ( ω u 0 − 3 ω ) τ i ( ω u 0 − 3 ω ) − e i ( ω u e − ω ) τ i ( ω u e − ω ) ] 0 t \begin{aligned}
c_u^{(3)}(t)
&= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(2)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
\\
&= i \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
\frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)}
\int_0^t e^{i (\omega_{ue} + \omega_{e0} - 3 \omega) \tau} - e^{i (\omega_{ue} - \omega) \tau} \dd{\tau}
\\
&= i \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
\frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)}
\bigg[ \frac{e^{i (\omega_{u0} - 3 \omega) \tau}}{i (\omega_{u0} - 3 \omega)}
- \frac{e^{i (\omega_{ue} - \omega) \tau}}{i (\omega_{ue} - \omega)} \bigg]_0^t
\end{aligned} c u ( 3 ) ( t ) = − ℏ i e ∑ even ∫ 0 t ⟨ u ∣ H ^ 1 ( τ ) ∣ e ⟩ c e ( 2 ) ( τ ) e i ω u e τ d τ = i e ∑ even u ′ ∑ odd ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ∫ 0 t e i ( ω u e + ω e 0 − 3 ω ) τ − e i ( ω u e − ω ) τ d τ = i e ∑ even u ′ ∑ odd ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) [ i ( ω u 0 − 3 ω ) e i ( ω u 0 − 3 ω ) τ − i ( ω u e − ω ) e i ( ω u e − ω ) τ ] 0 t Once again, the second term is uninteresting,
so we drop it and look at the first term only:
c u ( 3 ) ( t ) ≈ ∑ e e v e n ∑ u ′ o d d ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) e i ( ω u 0 − 3 ω ) t − 1 ω u 0 − 3 ω \begin{aligned}
\boxed{
c_u^{(3)}(t)
\approx \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
\frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}
{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)}
\frac{e^{i (\omega_{u0} - 3 \omega) t} - 1}{\omega_{u0} - 3 \omega}
}
\end{aligned} c u ( 3 ) ( t ) ≈ e ∑ even u ′ ∑ odd ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ω u 0 − 3 ω e i ( ω u 0 − 3 ω ) t − 1 The resulting transition rate R u ( 3 ) ( t ) R_u^{(3)}(t) R u ( 3 ) ( t )
R u ( 3 ) ( t ) = ∣ c u ( 3 ) ( t ) ∣ 2 t = 1 t ∣ ∑ e e v e n ∑ u ′ o d d ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ∣ 2 ⋅ ∣ e i ( ω u 0 − 3 ω ) t − 1 ω u 0 − 3 ω ∣ 2 \begin{aligned}
R_u^{(3)}(t)
= \frac{\big| c_u^{(3)}(t) \big|^2}{t}
= \frac{1}{t} \bigg| \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
\frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}
{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)} \bigg|^2
\cdot \bigg| \frac{e^{i (\omega_{u0} - 3 \omega) t} - 1}{\omega_{u0} - 3 \omega} \bigg|^2
\end{aligned} R u ( 3 ) ( t ) = t c u ( 3 ) ( t ) 2 = t 1 e ∑ even u ′ ∑ odd ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) 2 ⋅ ω u 0 − 3 ω e i ( ω u 0 − 3 ω ) t − 1 2 Applying our formula to this yields the following analogue of Fermi’s golden rule:
R u ( 3 ) = 2 π ∣ ∑ e e v e n ∑ u ′ o d d ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ∣ 2 δ ( ω u 0 − 3 ω ) \begin{aligned}
\boxed{
R_u^{(3)}
= 2 \pi \bigg| \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
\frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}
{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)} \bigg|^2 \delta(\omega_{u0} - 3 \omega)
}
\end{aligned} R u ( 3 ) = 2 π e ∑ even u ′ ∑ odd ℏ 3 ( ω u ′ 0 − ω ) ( ω e 0 − 2 ω ) ( p u e ⋅ E ) ( p e u ′ ⋅ E ) ( p u ′ 0 ⋅ E ) 2 δ ( ω u 0 − 3 ω ) This represents three-photon absorption , since it peaks at ω u 0 = 3 ω \omega_{u0} = 3 \omega ω u 0 = 3 ω ℏ ω \hbar \omega ℏ ω ℏ ω u 0 \hbar \omega_{u0} ℏ ω u 0 ⟨ u ∣ p ^ ∣ 0 ⟩ ≠ 0 \matrixel{u}{\vu{p}}{0} \neq 0 ⟨ u ∣ p ^ ∣ 0 ⟩ = 0 ∣ E ∣ 6 |\vb{E}|^6 ∣ E ∣ 6
N-photon absorption
A pattern has appeared in these calculations:
in N N N N N N ∣ E ∣ 2 N |\vb{E}|^{2N} ∣ E ∣ 2 N E \vb{E} E
If N N N ∣ u ⟩ \Ket{u} ∣ u ⟩ ∣ 0 ⟩ \Ket{0} ∣ 0 ⟩ N N N ∣ e ⟩ \Ket{e} ∣ e ⟩ ∣ 0 ⟩ \Ket{0} ∣ 0 ⟩ ⟨ f ∣ p ^ ∣ i ⟩ \matrixel{f}{\vu{p}}{i} ⟨ f ∣ p ^ ∣ i ⟩
References
R.W. Boyd,
Nonlinear optics , 4th edition,
Academic Press.
R. Shankar,
Principles of quantum mechanics , 2nd edition,
Springer.