Quantum Computation III Atomic Spectra
fengxiaot Lv4

Atomic spectra are defined as the spectrum of the electromagnetic radiation emitted or absorbed by an electron during transitions between different energy levels within an atom.

Two-level atom

Classical Model

  • N1N_1 : Number of atoms in g\ket{g} per unit volume.

  • N2N_2 : Number of atoms in e\ket{e} per unit volume.

  • Γ\Gamma : Rate of spontaneous emission. The probability of transitioning of a single atom from e\ket{e} to g\ket{g} due to spontaneous emission per unit time.

  • II : Intensity. I=12cn2ε0E2I = \frac{1}{2} c n^2 \varepsilon_0 |E|^2. Intensity has a dimension of Jm2s1\mathrm{J\cdot m^{-2} \cdot s^{-1}}.

  • σ(ω)\sigma(\omega) : Absorption cross-section, defined such that σ(ω)I\sigma(\omega) I is the energy absorbed by a single atom per unit time when irradiated by intensity II with frequency ω\omega .

  • ΓSR\Gamma_\text{SR} : Rate of stimulated radiation. The probability of stimulated absorption equals the probability of stimulated emission, so that we can define such a universal rate.

Einstein coefficient

The change in number of the excited atoms dN2dt\frac{\mathrm{d}N_2}{\mathrm{d}t} (per unit volume) comes from three sources:

  1. Spontaneous emission: In a unit time, there is a probability of Γ\Gamma for a single atom to transition to g\ket{g}. For N2N_2 atoms, the total transition number is N2ΓN_2 \Gamma on average.

  2. Stimulated absorption: According to the definition of absorption cross-section, σ(ω)I\sigma(\omega) I is the energy absorbed by a single atom per unit time. Since the incident beam has a frequency ω\omega, each time the medium absorb a photon ω\hbar \omega , an atom will be excited into e\ket{e} (although the energy gap is ω0\hbar \omega_0) . Therefore, in a unit time, σ(ω)IN1/ω\sigma(\omega) I N_1 / \hbar \omega is the total number of atoms in the g\ket{g} state being excited into e\ket{e}.

    /* Note: ΓSR=σ(ω)I/ω\Gamma_\text{SR} = \sigma(\omega) I / \hbar \omega */

  3. Stimulated emission: Since stimulated absorption and stimulated emission have the equal probability, the mechanism is similar. We have a negative term σ(ω)IωN2-\frac{\sigma(\omega) I}{\hbar \omega} N_2 .

We write down

dN2dt=N2Γσ(ω)Iω(N2N1)\frac{\mathrm{d}N_2}{\mathrm{d}t} = -N_2 \Gamma -\frac{\sigma(\omega) I}{\hbar \omega} (N_2 - N_1)

For steady state, spontaneous emission + stimulated emission = stimulated absorption, dN2/dt=0\mathrm{d} N_2/\mathrm{d}t = 0. We obtain the steady state equation

σ(ω)Iω(N2N1)=N2Γ-\frac{\sigma(\omega) I}{\hbar \omega} (N_2 - N_1) = N_2 \Gamma

Thus, the steady state of a two-level system always has N2<N1N_2 < N_1, no matter how strong the incident beam is. Which means, it is impossible to use a two-level system as the gain medium of laser, since optical pumping cannot induce population inversion.

Intensity gain

When a laser beam with a frequency of ω\omega and an intensity of II is incident on the medium, we have

dIdz=α(ω)I\frac{\mathrm{d}I}{\mathrm{d} z} = \alpha(\omega) I

where dIdz\frac{\mathrm{d}I}{\mathrm{d} z} is the gain (caused by pure emission) of energy per unit volume per unit time. At the same time, σ(ω)I(N2N1)\sigma(\omega) I(N_2 - N_1) is also the gain of energy of incident beam per unit volume per unit time, where N2N1N_2 - N_1 means stimulated emission minuses stimulated absorption, leaving the pure gain of energy of incident beam. So

α(ω)=σ(ω)(N2N1)\alpha(\omega) = \sigma(\omega)(N_2 - N_1)

Combining N2+N1=NN_2 + N_1 = N and the steady state equation, we could finally obtain

α(ω)=σ(ω)N1+2σ(ω)IωΓ\alpha(\omega) = \frac{\sigma(\omega) N}{1 + \dfrac{2\sigma(\omega)I}{\hbar \omega \Gamma}}

Line shape function

We have found that for a fixed number of particles N1,N2N_1,N_2 and a constant incident light intensity II, the energy absorbed by the medium per unit time varies depending on the frequency of the incident light ω\omega. This is reflected in the dependence of absorption cross-section σ(ω)\sigma(\omega) on ω\omega. Furthermore, we can extract a dimensionless line shape function g(ω)g(\omega) from it, describing the absorption efficiency of the medium for different frequencies of light.

Quantum Model

  • ρgg\rho_{gg} : The probability of a single atom on the ground state.
  • ρee\rho_{ee} : The probability of a single atom on the excited state.

By solving the Optical Bloch Equation, we can obtain

ρee(t)=Ω2/Γ21+4Δ2Γ2+2Ω2Γ2\rho_{ee} (t\to \infty) = \frac{\Omega^2 / \Gamma^2}{1+\dfrac{4\Delta^2}{\Gamma^2}+\dfrac{2\Omega^2}{\Gamma^2}}

Define a saturation parameter S0=2Ω2/Γ2=I/IsatS_0 = 2\Omega^2/\Gamma^2 = I/I_\text{sat}, the density matrix element becomes

ρee(t)=S0/21+S0+4Δ2Γ2\rho_{ee} (t\to \infty) = \frac{S_0/2}{1+S_0+\dfrac{4\Delta^2}{\Gamma^2}}

When on-resonance and large intensity S01S_0 \gg 1, 1/21/2 of the atoms occupy the excited state, reaches its maximum. It proves again that the steady state of a two-level system always has N2<N1N_2 < N_1, no matter how strong the incident beam is.

Resonance fluorescence

We will now consider the radiation due to a single, isolated atom driven by a monochromatic field. The radiation, viewed as scattered light, is called resonance fluorescence.

The optical Wiener-Хи́нчин theorem writes

Isc(r,ω)=12πcn2ε0+E()(r,t)E(+)(r,t+τ)eiωτdτI_\text{sc}(\boldsymbol{r},\omega) = \frac{1}{2\pi} c n^2 \varepsilon_0 \int_{-\infty}^{+\infty} \langle E^{(-)}(\boldsymbol{r},t) E^{(+)}(\boldsymbol{r},t+\tau) \rangle \, \mathrm{e}^{\mathrm{i}\omega\tau} \, \mathrm{d} \tau

where E^\hat{E} contains σ,σ\sigma, \sigma^\dagger. This will become

Isc(r,ωsc)=ω0Γr2f(θ,ϕ)S(ωsc)I_\text{sc}(\boldsymbol{r},\omega_\text{sc}) = \frac{\hbar \omega_0 \Gamma}{r^2} f(\theta,\phi) S(\omega_\text{sc})

where S(ωsc)S(\omega_\text{sc}) is the spectrum of the scattered light with a dimension of Hz1\mathrm{Hz^{-1}}, which describes the probability of the fluorescence having a frequency between ωsc\omega_\text{sc} and ωsc+dωsc\omega_\text{sc} + \mathrm{d}\omega_\text{sc}. However, S(ωsc)S(\omega_\text{sc}) is not normalized. The total probability is

0S(ωsc)dωsc=ρee(t)\int_0^\infty S(\omega_\text{sc}) \, \mathrm{d}\omega_\text{sc} = \rho_{ee}(t\to \infty)

This results in

Psc=ω0Γr2f(θ,ϕ)dΩS(ωsc)dωsc=Γω0ρeeP_\text{sc} = \int \frac{\hbar \omega_0 \Gamma}{r^2} f(\theta,\phi) \,\mathrm{d} \Omega \int S(\omega_\text{sc}) \, \mathrm{d}\omega_\text{sc} = \Gamma \hbar \omega_0 \rho_{ee}

It’s worth mentioning that the spectrum of the scattered light is named by Mollow Triplet Spectrum

S(ωsc)=S0(1+S0)2δ(ωscω)+S08π(1+S0)Γ[(ωscω)2+(Γ/2)2]+S(\omega_\text{sc})=\frac{S_0}{(1+S_0)^2} \delta\left(\omega_\text{sc}-\omega\right)+\frac{S_0}{8 \pi(1+S_0)} \frac{\Gamma}{\left[(\omega_\text{sc}-\omega)^2+(\Gamma / 2)^2\right]} + \cdots

where S0=2Ω2/Γ2S_0=2\Omega^2/\Gamma^2 is the saturation parameter.

Conclusion: In spite of the fact that scattering spectrum are the superposition of waves at different frequencies Isc(r,ωsc)I_\text{sc}(\boldsymbol{r},\omega_\text{sc}), the total power PscP_\text{sc} of all the scattered beams is exactly equivalent to the power of spontaneous emission, regardless of whether the incident beam is on-resonant ω=ω0\omega = \omega_0 or off-resonant ωω0\omega \neq \omega_0. However, The spectrum of scattered light is centered at the frequency of incident beam ω\omega !

Power broadening

Therefore, by measuring the power of resonance fluorescence, which is actually measuring ρee\rho_{ee} since PscρeeP_\text{sc} \propto \rho_{ee} , we can draw a graph

To explain the line shape, we recall the expression of ρee\rho_{ee} :

ρee(t)=S0/21+S0+4Δ2Γ2\rho_{ee} (t\to \infty) = \frac{S_0/2}{1+S_0+\dfrac{4\Delta^2}{\Gamma^2}}

Define a parameter Γ=Γ1+S0\Gamma^\prime = \Gamma \sqrt{1+S_0} where S0=2Ω2/Γ2=I/IsatS_0 = 2\Omega^2/\Gamma^2 = I/I_\text{sat} evaluates the intensity of the incident beam. The ρee\rho_{ee} is written

ρee=S02(1+S0)11+4Δ2Γ2\rho_{ee} = \frac{S_0}{2(1+S_0)} \frac{1}{1+\dfrac{4\Delta^2}{\Gamma^{\prime 2}}}

The full width at half maximum (FWHM) is exactly Γ\Gamma^\prime (When Δ=±Γ/2\Delta = \pm \Gamma^\prime/2 we have 11+4Δ2Γ2=12\frac{1}{1+\frac{4\Delta^2}{\Gamma^{\prime 2}}} = \frac{1}{2}). Two mechanism of broadening could be found

  • In the weak-field limit, S00S_0 \simeq 0, ΓΓ\Gamma^\prime \simeq \Gamma. This is called natural broadening.
  • In the strong-field limit, Γ=Γ1+S0\Gamma^\prime = \Gamma \sqrt{1+S_0} . The FWHM of the transition is effectively larger due to the strong coupling to the field. This phenomenon is called power broadening of the transition.

Connection with classical model

Warning: Only self-consistent when on-resonance ω=ω0\omega=\omega_0.

Optical Bloch Equation writes

ρ˙ee=ΓρeeΩ2/Γ1+4Δ2Γ2(ρeeρgg)\dot{\rho}_{ee} = -\Gamma \rho_{ee} - \frac{\Omega^2/\Gamma}{1 + \dfrac{4\Delta^2}{\Gamma^2}} (\rho_{ee}-\rho_{gg})

Naturally ρee=N2/N,ρgg=N1/N\rho_{ee} = N_2/N,\,\rho_{gg} = N_1 / N. Recall that in classical model

dN2dt=N2Γσ(ω)Iω(N2N1)\frac{\mathrm{d}N_2}{\mathrm{d}t} = -N_2 \Gamma -\frac{\sigma(\omega) I}{\hbar \omega} (N_2 - N_1)


σ(ω)=ωIΩ2/Γ1+4Δ2Γ2\sigma(\omega) = \frac{\hbar \omega}{I} \frac{\Omega^2/\Gamma}{1 + \dfrac{4\Delta^2}{\Gamma^2}}


Ω2=gd^e22E2Γ=ω03πε0c3gd^e2I=12cε0E2\Omega^2 = \frac{|\langle g |\hat{d}|e\rangle|^2}{\hbar^2} |E|^2 \qquad \Gamma=\frac{\omega^3_0}{\pi \varepsilon_0 \hbar c^3} |\langle g |\hat{d}|e\rangle|^2 \qquad I = \frac{1}{2}c \varepsilon_0 |E|^2

The absorption cross-section now becomes

σ(ω)=2πc2ωω0311+4Δ2Γ2λ024Γ22π(Γ2/4+Δ2)\sigma(\omega) = \frac{2 \pi c^2 \omega}{\omega_0^3} \frac{1}{1 + \dfrac{4\Delta^2}{\Gamma^2}} \approx \frac{\lambda^2_0}{4} \frac{\Gamma^2}{2\pi (\Gamma^2/4 +\Delta^2)}

where λ0=2πc/ω0\lambda_0 = 2\pi c/\omega_0. If we define a line shape function

g(ω)=12πΓΓ2/4+Δ2g(\omega) = \dfrac{1}{2\pi} \dfrac{\Gamma}{\Gamma^2/4 +\Delta^2}

we find σ(ω)=Γλ024g(ω)\sigma(\omega) = \Gamma \frac{\lambda^2_0}{4} g(\omega). We can obtain the same result taking advantage of energy conservation:

σ(ω)I(N2N1)=Psc=NΓω0ρee    σ(ω)=Γλ024g(ω)-\sigma(\omega) I (N_2 - N_1) = P_\text{sc} = N\Gamma \hbar \omega_0 \rho_{ee} \implies \sigma(\omega) = \Gamma \frac{\lambda^2_0}{4} g(\omega)

which means, given the same incident intensity and the same population, the efficiency of energy absorption by atoms varies with the changing incident light frequency ω\omega, and reaches its maximum at resonance Δ=0\Delta = 0.