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Notes on Perturbation Theory for a Two-Ensemble Dicke Model

(Donors + Acceptors coupled to a single quantised mode)

These notes build up the machinery we need to do effective-Hamiltonian perturbation theory properly, including the appearance of folded-diagram (derivative) terms at 4th order. We keep the discussion general first, then later we will plug in our specific Dicke Hamiltonian and basis.

1) The model

We consider a Hamiltonian split into “bare” and “interaction” parts:

$H = H_0 + V$.

A convenient Dicke-like choice is:

  • Single bosonic mode: $a, a^\dagger$ with frequency $\omega_c$.
  • Two ensembles of two-level systems (TLS): donors $D$ and acceptors $A$.
  • Collective spin operators (Dicke operators) for each ensemble:
    • $S_D^\pm, S_D^z$ for donors, size $N_D$.
    • $S_A^\pm, S_A^z$ for acceptors, size $N_A$.

A standard “Rabi/Dicke” style Hamiltonian is:

$$ H_0

\omega_c a^\dagger a + \omega_0 \big(S_D^z + S_A^z\big) $$

$$ V

(a+a^\dagger) \Big[ v_D (S_D^+ + S_D^-) + v_A (S_A^+ + S_A^-) \Big]. $$

Here $v_D$ and $v_A$ are the single-particle couplings for donors and acceptors (not assumed equal).

2) Choice of basis and the P/Q split

Dicke basis (conceptual)

A convenient label for basis states is:

$|N_D^, N_A^, n\rangle$,

where:

  • $N_D^*$ is the number of donor excitations in the symmetric Dicke ladder ($0,1,\dots,N_D$),
  • $N_A^*$ is the number of acceptor excitations ($0,1,\dots,N_A$),
  • $n$ is the photon number.

The bare energies take the schematic form:

$$ E^{(0)}(N_D^,N_A^,n)

\omega_0 (N_D^+N_A^) + \omega_c n \quad \text{(up to constant offsets)}. $$

The two-dimensional P space

We choose a “model space” $P$ consisting of the two single-excitation states at a chosen photon sector (often $k=0$, i.e. photon number fixed to some reference $n$):

  • Donor-like state:

$|D\rangle \equiv |1,0,n\rangle$

  • Acceptor-like state:

$|A\rangle \equiv |0,1,n\rangle$

So $P$ is 2D and is spanned by ${|D\rangle,|A\rangle}$. Let $Q$ be the complement:

$Q = 1 - P$.

We will write operator blocks like:

  • $H_{PP} \equiv PHP$
  • $H_{PQ} \equiv PHQ$
  • $H_{QP} \equiv QHP$
  • $H_{QQ} \equiv QHQ$

and similarly for $H_0$ and $V$.

3) Exact elimination of Q space: Bloch–Horowitz effective Hamiltonian

Start from the Schrödinger equation:

$H|\Psi\rangle = E|\Psi\rangle$.

Decompose the state:

$|\Psi\rangle = |\psi_P\rangle + |\psi_Q\rangle$,

where $|\psi_P\rangle = P|\Psi\rangle$ and $|\psi_Q\rangle = Q|\Psi\rangle$.

Projecting onto $P$ and $Q$ gives the coupled block equations:

$$ (H_{PP} - E),|\psi_P\rangle + H_{PQ},|\psi_Q\rangle = 0 $$

$$ H_{QP},|\psi_P\rangle + (H_{QQ}-E),|\psi_Q\rangle = 0. $$

Assuming $(E-H_{QQ})$ is invertible on $Q$, solve the second equation:

$$ |\psi_Q\rangle

(E-H_{QQ})^{-1} H_{QP},|\psi_P\rangle. $$

Insert into the $P$ equation to obtain the exact energy-dependent effective Hamiltonian:

$$ H_{\mathrm{eff}}(E)

H_{PP} + H_{PQ},(E-H_{QQ})^{-1},H_{QP}. $$

This is often called the Bloch–Horowitz (BH) or Feshbach effective Hamiltonian.

Key point: $H_{\mathrm{eff}}(E)$ depends on $E$, so the eigenvalue problem is nonlinear:

$$ H_{\mathrm{eff}}(E),|\psi_P\rangle

E,|\psi_P\rangle. $$

Solving this self-consistently reproduces the exact full-space eigenvalues associated with the chosen $P$ sector.

4) Perturbative expansion: resolvent and “unfolded” series

We now expand in powers of $V$.

Write:

$H = H_0 + V$,

and correspondingly:

$H_{QQ} = (H_0){QQ} + V{QQ}$, etc.

Define an “unperturbed” $Q$-space resolvent:

$$ G_0(E) \equiv (E-(H_0)_{QQ})^{-1}. $$

Then we expand the full resolvent:

$$ (E-H_{QQ})^{-1}

\big(E-(H_0){QQ}-V{QQ}\big)^{-1}

G_0(E) + G_0(E)V_{QQ}G_0(E) + G_0(E)V_{QQ}G_0(E)V_{QQ}G_0(E)+\cdots $$

Insert into BH:

$$ H_{\mathrm{eff}}(E)

H_{PP} + V_{PQ}G_0(E)V_{QP} + V_{PQ}G_0(E)V_{QQ}G_0(E)V_{QP} + V_{PQ}G_0(E)V_{QQ}G_0(E)V_{QQ}G_0(E)V_{QP} +\cdots $$

This produces an expansion in powers of $V$:

  • 2nd order: $V_{PQ}G_0(E)V_{QP}$
  • 3rd order: $V_{PQ}G_0(E)V_{QQ}G_0(E)V_{QP}$
  • 4th order: $V_{PQ}G_0(E)V_{QQ}G_0(E)V_{QQ}G_0(E)V_{QP}$
  • etc.

If we evaluate these terms at a fixed reference energy $E=E_0$ (typically the degenerate bare energy of the $P$ states), we get what we will call the unfolded contributions (no intermediate return to $P$ inside the resolvent expansion).

5) Why energy dependence matters: self-consistency vs energy-independent expansions

Self-consistent (BH) viewpoint

In BH, you compute $H_{\mathrm{eff}}(E)$ and solve the nonlinear problem

$H_{\mathrm{eff}}(E)\psi_P = E\psi_P$.

This automatically includes “folding” effects because you never expand away the $E$ dependence: it is treated exactly (within the chosen truncation of $Q$).

Energy-independent perturbation theory viewpoint

Often we want an energy-independent effective Hamiltonian expanded order-by-order (Rayleigh–Schrödinger style). Then we must handle the fact that $H_{\mathrm{eff}}$ depends on $E$.

This is where folded-diagram (derivative) terms appear.

6) The origin of folded terms: Taylor expanding the self-energy

Define the BH self-energy operator:

$\Sigma(E) \equiv H_{PQ}(E-H_{QQ})^{-1}H_{QP}$.

Then:

$H_{\mathrm{eff}}(E) = H_{PP} + \Sigma(E)$.

Choose a reference energy $E_0$ (for our degenerate $P$ space this is the common bare energy of $|D\rangle$ and $|A\rangle$), and write:

$E = E_0 + \delta E$.

Now Taylor expand:

$$ \Sigma(E)

\Sigma(E_0) + \delta E,\Sigma'(E_0) + \frac{(\delta E)^2}{2}\Sigma''(E_0) + \cdots $$

Crucial power counting:

  • $\Sigma(E_0)$ starts at order $V^2$.
  • Typically $\delta E$ is also order $V^2$ (it is an energy shift generated by the interaction).

Therefore the term

$\delta E,\Sigma'(E_0)$

is of order $V^4$.

That is exactly the leading folded contribution at 4th order.

Why derivatives appear

The derivative comes from differentiating the resolvent:

$$ \frac{d}{dE}(E-H_{QQ})^{-1}

(E-H_{QQ})^{-2}. $$

So folded terms correspond to squared denominators in the intermediate-state sums. Diagrammatically, they are associated with intermediate returns to the model space $P$ that would otherwise produce zero denominators in naive time-ordered perturbation theory.

7) What “folded” means when P is two-dimensional

Here $P$ contains two states, $|D\rangle$ and $|A\rangle$.

A process can “return to P” in the middle by landing on either state (with the correct photon sector).

  • Returning early to $|D\rangle$ corresponds to “dressing the donor leg”.
  • Returning early to $|A\rangle$ corresponds to “dressing the acceptor leg”.

This is why, in a 2D $P$ space, the folded correction is naturally a matrix product structure (schematically):

$$ H_{\mathrm{eff}}^{(\le 4)} \approx H_{PP} + \Sigma^{(2)}(E_0) + \Sigma^{(4)}_{\mathrm{unfolded}}(E_0) + \Sigma^{(2)\prime}(E_0),\Sigma^{(2)}(E_0) \quad \text{(up to convention-dependent symmetrisation)}. $$

When we look specifically at the off-diagonal element $AD$, the folded term contains contributions corresponding to “hit $D$ early” and “hit $A$ early”:

  • “hit $D$ early” dressing contributes via the $P$-index $p=D$
  • “hit $A$ early” dressing contributes via the $P$-index $p=A$

This is exactly how “getting to the end point early” enters analytically: it is one of the allowed intermediate $P$ states in the folded correction.

8) Stop point for the general formalism

At this stage we have:

  1. A clear $P/Q$ reduction to an exact $H_{\mathrm{eff}}(E)$ (BH/Feshbach).
  2. A perturbative expansion of $H_{\mathrm{eff}}(E)$ in powers of $V$ that yields “unfolded” contributions at fixed $E_0$.
  3. A Taylor expansion in $E$ showing that, at 4th order, we must include derivative terms (folded diagrams) because $\delta E$ is itself order $V^2$.

Next step (in our Dicke model):

  • Choose the explicit Dicke basis states and compute $\Sigma^{(2)}{DD}(E_0)$, $\Sigma^{(2)}{AA}(E_0)$, $\Sigma^{(2)}_{AD}(E_0)$ keeping full $(n,n+1)$ dependence.
  • Then compute $\Sigma^{(2)\prime}(E_0)$ and assemble the 4th-order folded correction.
  • Compare “static at $E_0$” vs “self-consistent BH” and show the cancellations we observed numerically.