Quantum Optics III Light-Matter Interaction
fengxiaot Lv4
  2023-08-10 11:49:11 2023-08-10 11:49  

Semi-classical description of matter-light interaction means that treating atoms quantum mechanically while treating electromagnetic field classically.

Electrodynamics

Lagrangian and Hamiltonian

It is well known that the Maxwell equations are

{∇⋅E=ρε0∇⋅B=0∇×E=−∂B∂t∇×B=μ0∂D∂t+μ0J\begin{cases} \nabla \cdot \boldsymbol{E} = \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \boldsymbol{B} = 0 \\ \nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t} \\ \nabla \times \boldsymbol{B} = \mu_0 \frac{\partial \boldsymbol{D}}{\partial t} + \mu_0 \boldsymbol{J} \end{cases}

which implies the existence of vector potential A(r,t)\boldsymbol{A}(\boldsymbol{r},t) and scalar potential ϕ(r,t)\phi(\boldsymbol{r},t)

E=−∂A∂t−∇ϕ,B=∇×A\boldsymbol{E} = -\frac{\partial \boldsymbol{A}}{\partial t} - \nabla \phi, \quad \boldsymbol{B} = \nabla \times \boldsymbol{A}

The Lagrangian and Hamiltonian of a particle with mass mm and charge qq in the electromagnetic fields are

L=−mc21−β2+qAμdxμdt≈12mv2+qv⋅A−qϕL = -mc^2 \sqrt{1-\beta^2} +qA_\mu \frac{\mathrm{d}x^\mu}{\mathrm{d}t} \approx \frac{1}{2} m \boldsymbol{v}^2 + q \boldsymbol{v} \cdot \boldsymbol{A} - q\phi

H=12mv2+qϕ=12m(p−qA)2+qϕH = \frac{1}{2}m\boldsymbol{v}^2 +q\phi = \frac{1}{2m} (\boldsymbol{p}-q\boldsymbol{A})^2 +q\phi

where p=mv+qA\boldsymbol{p} = m\boldsymbol{v} + q\boldsymbol{A} is the canonical momentum.

In canonical quantization, it is always the canonical momentum (along with the generalized coordinate) that is promoted to an operator P\boldsymbol{P}, not the mechanical momentum. Mechanical momentum does not even exist in quantum mechanics; it only appears as an expectation value md⟨R⟩/dtm \mathrm{d} \langle \boldsymbol{R} \rangle / \mathrm{d}t, as described by the Ehrenfest theorem.

Gauge transformation

The choice of scalar potential ϕ(r,t)\phi(\boldsymbol{r},t) and vector potential A(r,t)\boldsymbol{A}(\boldsymbol{r},t) is not unique. Perform a gauge transformation as follows[1]

A(r,t)→A(r,t)−∇χ(r,t)\boldsymbol{A}(\boldsymbol{r},t) \to \boldsymbol{A}(\boldsymbol{r},t) - \nabla \chi(\boldsymbol{r},t)

ϕ(r,t)→ϕ(r,t)+∂χ∂t\phi (\boldsymbol{r},t) \to \phi (\boldsymbol{r},t) + \frac{\partial \chi}{\partial t}

where χ(r,t)\chi(\boldsymbol{r},t) is an arbitrary scalar function, the electric field E\boldsymbol{E} and magnetic induction B\boldsymbol{B} and other observables will not change.


Atom-Light Interaction

Consider an electron in an atom interacting with an external electromagnetic field. Its Hamiltonian can be expressed as

H=12m[P−qA(R,t)]2+qϕ(R,t)−qmS⋅B+V(R)H = \frac{1}{2m} [\boldsymbol{P}-q\boldsymbol{A}(\boldsymbol{R},t)]^2+q \phi(\boldsymbol{R},t) - \frac{q}{m} \boldsymbol{S} \cdot \boldsymbol{B} + V(\boldsymbol{R})

where q=−eq=-e is the charge of electron. R\boldsymbol{R} is the position vector of the electron, pointing from the nuclei to the electron, which means, the origin of the coordinate system has been set as the center of the nuclei. P\boldsymbol{P} is the canonical momentum of the electron. V(R)V(\boldsymbol{R}) is the interaction potential between the nuclei and the electron, mainly the Coulomb potential −14πε0q2∣R∣-\frac{1}{4\pi\varepsilon_0} \frac{q^2}{|\boldsymbol{R}|}.

In regions where no charge exists, we can choose Coulomb gauge and temporal gauge at the same time to eliminate the electrostatic potential[2]

∇⋅A=0\nabla \cdot \boldsymbol{A} = 0

ϕ=0\phi = 0

The requirement of ρ=0\rho = 0 comes from Gauss’s law. Imposing the temporal gauge leads to E=−∂A∂t\boldsymbol{E}=-\frac{\partial \boldsymbol{A}}{\partial t}. If we further requires the Coulomb gauge, the Gauss’s law implies −∂∂t(∇⋅A)=ρε0- \frac{\partial}{\partial t} (\nabla \cdot \boldsymbol{A}) = \frac{\rho}{\varepsilon_0}.

The Hamiltonian simplifies to

H=12m[P2−q(P⋅A(R,t)+A(R,t)⋅P)+q2A2]−qmS⋅B+V(R)H = \frac{1}{2m} \left[\boldsymbol{P}^2 - q \left(\boldsymbol{P}\cdot \boldsymbol{A}(\boldsymbol{R},t) + \boldsymbol{A}(\boldsymbol{R},t) \cdot \boldsymbol{P}\right) + q^2 \boldsymbol{A}^2 \right] - \frac{q}{m} \boldsymbol{S} \cdot \boldsymbol{B} + V(\boldsymbol{R})

In the weak coupling regime, The q2A2q^2 \boldsymbol{A}^2 term can be omitted. q2A2q^2 \boldsymbol{A}^2 represents the self-energy of the charged particle due to its interaction with the electromagnetic field. In many cases, this self-energy is renormalized into the mass of the particle or treated as a small correction. Then

H=12m[P2−q(P⋅A(R,t)+A(R,t)⋅P)]−qmS⋅B+V(R)H = \frac{1}{2m} \left[\boldsymbol{P}^2 - q \left(\boldsymbol{P}\cdot \boldsymbol{A}(\boldsymbol{R},t) + \boldsymbol{A}(\boldsymbol{R},t) \cdot \boldsymbol{P}\right) \right] - \frac{q}{m} \boldsymbol{S} \cdot \boldsymbol{B} + V(\boldsymbol{R})

Long-wavelength approximation

In AMO experiments, the laser field is usually described by a plane wave. For instance, consider the following electromagnetic wave

A=A0cos⁡(k⋅r−ωt)\boldsymbol{A} = \boldsymbol{A}_0 \cos (\boldsymbol{k}\cdot\boldsymbol{r} - \omega t)

E=−∂A∂t=−ωA0sin⁡(k⋅r−ωt):=E0sin⁡(k⋅r−ωt)\boldsymbol{E} = -\frac{\partial \boldsymbol{A}}{\partial t} = -\omega \boldsymbol{A}_0 \sin (\boldsymbol{k}\cdot\boldsymbol{r} - \omega t) := \boldsymbol{E}_0 \sin (\boldsymbol{k}\cdot\boldsymbol{r} - \omega t)

B=∇×A=−k×A0sin⁡(k⋅r−ωt):=B0sin⁡(k⋅r−ωt)\boldsymbol{B} = \nabla \times \boldsymbol{A} = -\boldsymbol{k}\times \boldsymbol{A}_0 \sin (\boldsymbol{k}\cdot\boldsymbol{r} - \omega t) := \boldsymbol{B}_0 \sin (\boldsymbol{k}\cdot\boldsymbol{r} - \omega t)

In the atom-light interactions studied in quantum optics, the wavelength λ\lambda of the light is usually very large compared to atomic dimensions. For typical atoms, the size is on the order of angstroms 10−1010^{-10} m, while the wavelength of visible light is on the order of hundreds of nanometers 10−710^{-7} m. Thus, the condition λ≫d\lambda \gg d is usually satisfied for atomic systems interacting with visible or longer-wavelength light.

Under these conditions, the amplitude of the external field is practically constant over the spatial extent of the atom and the vector potential A(r,t)\boldsymbol{A}(\boldsymbol{r},t) can be replaced by its value at the nucleus. This is called the long-wavelength approximation.

Electric dipole

Under long-wavelength approximation, k⋅r≪1\boldsymbol{k}\cdot\boldsymbol{r} \ll 1. To the zeroth order, the taylor expansion of electromagnetic field is

A≃A0cos⁡ωt\boldsymbol{A} \simeq \boldsymbol{A}_0 \cos \omega t

E≃−E0sin⁡ωt=ωA0sin⁡(ωt)\boldsymbol{E} \simeq -\boldsymbol{E}_0 \sin \omega t = \omega \boldsymbol{A}_0 \sin (\omega t)

B≃−B0sin⁡ωt=k×A0sin⁡ωt\boldsymbol{B} \simeq -\boldsymbol{B}_0 \sin \omega t = \boldsymbol{k}\times \boldsymbol{A}_0 \sin \omega t

and the zeroth-order Hamiltonian is

H=P22m−qmP⋅A0cos⁡ωt+qmS⋅B0sin⁡ωt+V(R)H = \frac{\boldsymbol{P}^2}{2m} - \frac{q}{m} \boldsymbol{P}\cdot \boldsymbol{A}_0 \cos \omega t+ \frac{q}{m} \boldsymbol{S} \cdot \boldsymbol{B}_0 \sin\omega t + V(\boldsymbol{R})

Perform a gauge transformation χ=r⋅A=r⋅A0cos⁡ωt\chi = \boldsymbol{r} \cdot \boldsymbol{A} = \boldsymbol{r} \cdot \boldsymbol{A}_0 \cos \omega t. This gauge is called Göppert-Mayer gauge or length gauge. The scalar potential and vector potential now become

{A′(r,t)=A(r,t)−∇χ(r,t)=0ϕ′(r,t)=ϕ(r,t)+∂χ∂t=r⋅E0sin⁡ωt\begin{cases} \boldsymbol{A}^\prime (\boldsymbol{r},t) = \boldsymbol{A}(\boldsymbol{r},t) - \nabla \chi(\boldsymbol{r},t) = 0 \\ \phi^\prime (\boldsymbol{r},t) = \phi (\boldsymbol{r},t) + \frac{\partial \chi}{\partial t} = \boldsymbol{r} \cdot \boldsymbol{E}_0 \sin \omega t \end{cases}

which implies that E′=−∇ϕ′=−E0sin⁡ωt=E\boldsymbol{E}^\prime = -\nabla\phi^\prime = -\boldsymbol{E}_0 \sin \omega t = \boldsymbol{E} and B′=∇×A′=0\boldsymbol{B}^\prime = \nabla \times \boldsymbol{A}^\prime = 0. The Hamiltonian becomes

H=12m[P2−q(P⋅A′(R,t)+A′(R,t)⋅P)]+qϕ′(R,t)−qmS⋅B′+V(R)H = \frac{1}{2m} \left[\boldsymbol{P}^2 - q \left(\boldsymbol{P}\cdot \boldsymbol{A}^\prime(\boldsymbol{R},t) + \boldsymbol{A}^\prime(\boldsymbol{R},t) \cdot \boldsymbol{P}\right) \right] +q \phi^\prime(\boldsymbol{R},t)- \frac{q}{m} \boldsymbol{S} \cdot \boldsymbol{B}^\prime + V(\boldsymbol{R})

H=P22m−qR⋅E+V(R)H = \frac{\boldsymbol{P}^2}{2m} - q\boldsymbol{R} \cdot \boldsymbol{E} + V(\boldsymbol{R})

the second term is the electric dipole interaction

WDE=−d⋅E=−qR⋅EW_{DE} = -\boldsymbol{d} \cdot \boldsymbol{E} = - q\boldsymbol{R} \cdot \boldsymbol{E}

for an electron in an atom, q=−eq=-e, the dipole potential energy is WDE=eR⋅EW_{DE} = e \boldsymbol{R} \cdot \boldsymbol{E}.

Magnetic dipole and electric quadrupole

To the higher order, the Hamiltonian can be written as[3][4]

H=P22m+V(R)+WDE+WQE+WMDH= \frac{\boldsymbol{P}^2}{2m} + V(\boldsymbol{R}) + W_{DE} + W_{QE} + W_{MD}

H=P22m+V(R)−d⋅E+Qij∂iEj−μ⋅BH= \frac{\boldsymbol{P}^2}{2m} + V(\boldsymbol{R}) - \boldsymbol{d}\cdot \boldsymbol{E} + Q_{ij} \partial_i E_j -\boldsymbol{\mu}\cdot\boldsymbol{B}

H=P22m+V(R)−qR⋅E(0)+q2RR:∇E(0)−q2m(L+2S)⋅B(0)H =\frac{\boldsymbol{P}^2}{2m} + V(\boldsymbol{R}) - q \boldsymbol{R}\cdot \boldsymbol{E}(0) + \frac{q}{2} \boldsymbol{R}\boldsymbol{R}:\nabla \boldsymbol{E}(0) - \frac{q}{2m} (\boldsymbol{L}+2\boldsymbol{S}) \cdot \boldsymbol{B}(0)

where the electric dipole d=qR\boldsymbol{d}=q\boldsymbol{R}, electric quadrupole Qij=q2(RiRj−13R2δij)Q_{ij}= \frac{q}{2} (R_i R_j - \frac{1}{3} R^2 \delta_{ij}), magnetic dipole μ=q2m(L+2S)\boldsymbol{\mu} = \frac{q}{2m}(\boldsymbol{L}+2\boldsymbol{S}).

For an electron, q=−eq=-e, thus

  • The electric dipole term becomes: WDE=eR⋅EW_{DE} = e\boldsymbol{R} \cdot\boldsymbol{E}
  • The electric quadrupole term becomes: WQE=−e2(RiRj−13R2δij)∂iEjW_{QE}= -\frac{e}{2} (R_i R_j - \frac{1}{3} R^2 \delta_{ij}) \partial_i E_j
  • The magnetic dipole becomes: WMD=e2m(L+2S)⋅BW_{MD} =\frac{e}{2m} (\boldsymbol{L}+2\boldsymbol{S}) \cdot \boldsymbol{B}

To solve for the exact form of magnetic dipole moment and electric quadrupole moment, one need to perform Power–Zienau transformation. The detailed derivation is very tricky, and involves the fundamental discussion of measurable quantities[5][6][7][8].


  1. Cohen-Tannoudji, Claude, Bernard Diu, and Franck Laloë. Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications. 2nd ed., Wiley, 2019, pp. 321-334. Complement HIII: Gauge Invariance. ↩︎

  2. 颜波. 原子分子与光物理[M]. 北京: 高等教育出版社, 2023 :241-243. ↩︎

  3. Tokmakoff, A. (2024). Magnetic Dipole and Electric Quadrupole Transitions. LibreTexts. https://doi.org/10.6082/uchicago.2772 ↩︎

  4. Steck, Daniel A. Quantum and Atom Optics. Revision 0.16.2, 15 Nov. 2024, pp. 473-474, http://steck.us/teaching. ↩︎

  5. Cohen-Tannoudji, Claude, Bernard Diu, and Franck Laloë. Quantum Mechanics, Volume 2: Angular Momentum, Spin, and Approximation Methods. 2nd ed., Wiley, 2019, pp. 1339-1349. Complement AXIII: Interaction of an Atom with an Electromagnetic Wave. ↩︎

  6. Steck, Daniel A. Quantum and Atom Optics. Revision 0.16.2, 15 Nov. 2024, pp. 473-474, http://steck.us/teaching. ↩︎

  7. Scully, Marlan O., and M. Suhail Zubairy. Quantum Optics. Cambridge University Press, 1997, pp. 178-180. ↩︎

  8. Sakurai, J. J., and Jim Napolitano. Modern Quantum Mechanics. 3rd ed., Cambridge University Press, 2021, pp. 126-131. Gauge Transformations in Electromagnetism. ↩︎