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/To the interaction frame

To the interaction frame

Contents9 sections
  1. The Interaction Hamiltonian between light and matter
  2. Moving into a spinning basis (The Interaction Picture)
  3. Atomic Excitation Operators
  4. Photon Field Operators
  5. Combined Operators in the Interaction Picture
  6. Invariants
  7. Co-rotating Terms (Difference Frequency)
  8. Counter-rotating Terms (Sum Frequency)
  9. Rotating Wave Approximation

The Interaction Hamiltonian between light and matter§

We begin with the standard electric dipole interaction Hamiltonian:

H^I=−d^⋅E^(r)\hat{H}_I = -\hat{\mathbf{d}}\cdot \hat{\mathbf{E}}(\mathbf{r})

Expanding the dipole operator d^=der(σ^++σ^−)\hat{\mathbf{d}} = \mathbf{d}_{er}(\hat{\sigma}^+ + \hat{\sigma}^-) and the quantized multi-mode electric field operator yields the full interaction expression:

H^I=−∑k,λEk(σ^++σ^−)der⋅(b^k,λek,λeik⋅r+b^k,λ†ek,λ∗e−ik⋅r)=−∑k,λEk(σ^+b^k,λder⋅ek,λeik⋅r+σ^+b^k,λ†der⋅ek,λ∗e−ik⋅r+σ^−b^k,λder⋅ek,λeik⋅r+σ^−b^k,λ†der⋅ek,λ∗e−ik⋅r)\begin{align*} \hat{H}_I &= -\sum_{\mathbf{k},\lambda}\mathcal{E}_k \left(\hat{\sigma}^+ + \hat{\sigma}^-\right) \mathbf{d}_{er} \cdot \left(\hat{b}_{\mathbf{k},\lambda}\mathbf{e}_{\mathbf{k},\lambda}e^{i\mathbf{k}\cdot\mathbf{r}}+\hat{b}^\dagger_{\mathbf{k},\lambda}\mathbf{e}^*_{\mathbf{k},\lambda}e^{-i\mathbf{k}\cdot\mathbf{r}}\right) \\ &= -\sum_{\mathbf{k},\lambda}\mathcal{E}_k \Big( \hat{\sigma}^+\hat{b}_{\mathbf{k},\lambda}\mathbf{d}_{er}\cdot\mathbf{e}_{\mathbf{k},\lambda}e^{i\mathbf{k}\cdot\mathbf{r}} + \hat{\sigma}^+\hat{b}^\dagger_{\mathbf{k},\lambda}\mathbf{d}_{er}\cdot\mathbf{e}^*_{\mathbf{k},\lambda}e^{-i\mathbf{k}\cdot\mathbf{r}} \\ &\qquad\qquad\quad + \hat{\sigma}^-\hat{b}_{\mathbf{k},\lambda}\mathbf{d}_{er}\cdot\mathbf{e}_{\mathbf{k},\lambda}e^{i\mathbf{k}\cdot\mathbf{r}} + \hat{\sigma}^-\hat{b}^\dagger_{\mathbf{k},\lambda}\mathbf{d}_{er}\cdot\mathbf{e}^*_{\mathbf{k},\lambda}e^{-i\mathbf{k}\cdot\mathbf{r}} \Big) \end{align*}

Moving into a spinning basis (The Interaction Picture)§

Take our unperturbed baseline Hamiltonian as:

H^0=ℏωegσ^e+∑k,λℏωkb^k,λ†b^k,λ\hat{H}_0 = \hbar\omega_{eg}\hat{\sigma}_e+\sum_{\mathbf{k},\lambda}\hbar\omega_{k}\hat{b}^\dagger_{\mathbf{k},\lambda}\hat{b}_{\mathbf{k},\lambda}

To isolate the dynamics driven purely by the interaction, we transform our operators into the rotating frame via:

A^(t)=eiH^0ℏtA^e−iH^0ℏt\hat{A}(t)=e^{i\frac{\hat{H}_0}{\hbar}t}\hat{A}e^{-i\frac{\hat{H}_0}{\hbar}t}

Atomic Excitation Operators§

By the Hadamard Lemma, the time-evolution of the operators depends entirely on their commutation relations with H^0\hat{H}_0. For the atomic system, we have:

[σ^e,σ^+]=σ^+[\hat{\sigma}_e, \hat{\sigma}^+]=\hat{\sigma}^+

Using the identity eab^B^e−ab^=eaλB^e^{a\hat{b}}\hat{B}e^{-a\hat{b}}=e^{a\lambda}\hat{B} which holds when [b^,B^]=λB^[\hat{b},\hat{B}]=\lambda \hat{B}, we evaluate the time-dependent raising operator:

σ^+(t)=σ^+eiωegt\hat{\sigma}^+(t)=\hat{\sigma}^+e^{i\omega_{eg}t}

Taking the Hermitian conjugate yields the lowering operator:

σ^−(t)=σ^−e−iωegt\hat{\sigma}^-(t)=\hat{\sigma}^-e^{-i\omega_{eg}t}

Photon Field Operators§

Using the identical algebraic structure for the bosonic field modes where [b^†b^,b^]=−b^[\hat{b}^\dagger\hat{b}, \hat{b}] = -\hat{b}, we obtain:

b^k,λ(t)=b^k,λe−iωkt\hat{b}_{\mathbf{k},\lambda}(t)=\hat{b}_{\mathbf{k},\lambda}e^{-i\omega_{k}t} b^k,λ†(t)=b^k,λ†eiωkt\hat{b}^\dagger_{\mathbf{k},\lambda}(t) =\hat{b}^\dagger_{\mathbf{k},\lambda} e^{i\omega_{k}t}

Combined Operators in the Interaction Picture§

Substituting these time-dependent forms back into H^I\hat{H}_I reveals that the terms separate into two distinctly behaving categories based on their phase frequencies.

Invariants§

b^†b^=n^b^b^†=n^+1σ^+σ^−=σ^eσ^−σ^+=σ^g\hat{b}^\dagger\hat{b}=\hat{n} \qquad \hat{b}\hat{b}^\dagger=\hat{n}+1 \qquad \hat{\sigma}^+\hat{\sigma}^-=\hat{\sigma}_e \qquad \hat{\sigma}^-\hat{\sigma}^+=\hat{\sigma}_g

Co-rotating Terms (Difference Frequency)§

These terms represent energy-conserving processes where an atomic transition is balanced by the creation or destruction of a photon:

b^k,λ†σ^−ei(ωk−ωeg)t\hat{b}^\dagger_{\mathbf{k},\lambda}\hat{\sigma}^-e^{i(\omega_k-\omega_{eg})t} b^k,λσ^+e−i(ωk−ωeg)t\hat{b}_{\mathbf{k},\lambda}\hat{\sigma}^+e^{-i(\omega_k-\omega_{eg})t}

Counter-rotating Terms (Sum Frequency)§

These terms represent virtual fluctuations where a photon is created alongside an atomic excitation, or destroyed alongside a de-excitation:

b^k,λ†σ^+ei(ωk+ωeg)t\hat{b}^\dagger_{\mathbf{k},\lambda}\hat{\sigma}^+e^{i(\omega_k+\omega_{eg})t} b^k,λσ^−e−i(ωk+ωeg)t\hat{b}_{\mathbf{k},\lambda}\hat{\sigma}^-e^{-i(\omega_k+\omega_{eg})t}

Rotating Wave Approximation§

When the coupling parameter gg (which collects the dipole moments and field amplitudes) is small relative to the optical frequencies, the counter-rotating terms oscillate at extreme speeds (ωk+ωeg\omega_k + \omega_{eg}) and average out to zero over macroscopic timescales.

Assuming a near-resonant scenario where ωk≈ωeg\omega_{k} \approx \omega_{eg}, the phase clocks of the co-rotating terms slow down completely, dominating the physical evolution. Dropping the counter-rotating terms leaves us with the final RWA Interaction Hamiltonian:

H^int(t)=∑k,λ(ℏgk,λb^k,λσ^+e−i(ωk−ωeg)t+ℏgk,λ∗b^k,λ†σ^−ei(ωk−ωeg)t)\hat{H}_{\text{int}}(t) = \sum_{\mathbf{k},\lambda} \left( \hbar g_{\mathbf{k},\lambda}\hat{b}_{\mathbf{k},\lambda}\hat{\sigma}^+ e^{-i(\omega_k-\omega_{eg})t} + \hbar g^*_{\mathbf{k},\lambda}\hat{b}^\dagger_{\mathbf{k},\lambda}\hat{\sigma}^- e^{i(\omega_k-\omega_{eg})t} \right)

This will be established with rigour in the next article.

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