Adiabatic-Quadratic Approximation

• Adiabatic-quadratic approximation is an asymptotic scheme for the generalized quantum Rabi model with a quadratic (A^2) interaction, producing explicit eigenvalues and Schrödinger-cat-like states.
• The method employs an adiabatic reduction combined with a Hopfield–Bogoliubov transformation to eliminate the quadratic photon self-interaction and renormalize model parameters.
• It reveals bias-dependent behavior: without the A^2-term, cat-like states appear only at zero bias, while its inclusion restores universal spin-boson entanglement across all bias conditions.

Searching arXiv for the relevant paper and closely related uses of the term. Adiabatic-quadratic approximation denotes the approximation scheme introduced for the generalized quantum Rabi model with a quadratic interaction, specifically the usual $A^2$-term, in the deep-strong coupling regime. In Hirokawa’s formulation, the procedure combines an adiabatic treatment of the spin-boson dynamics with a preliminary elimination of the quadratic photon self-interaction by a Hopfield–Bogoliubov transformation, yielding explicit approximate eigenvalues and Schrödinger-cat-like eigenstates both without and with the $A^2$-term (Hirokawa, 2018).

1. Definition and problem setting

In the relevant formulation, the generalized quantum Rabi Hamiltonian describes the interaction of a two-level artificial atom and a one-mode microwave photon in circuit QED. The model without the quadratic interaction is written as

$$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$

with

$$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$

Here $\omega_a$ is the tunnel splitting of the atom, $\varepsilon$ is the energy bias, $\omega_c$ is the bosonic-mode frequency, and $g$ is the atom-photon coupling strength. The quadratic interaction is introduced through

$$H_{A^2}(\varepsilon) = H_{\rm GQR}(\omega_c,g)+\hbar g\,C_g(a+a^\dagger)^2,$$

where $C_g>0$ satisfies

$A^2$0

Within this framework, the adiabatic approximation is taken in the deep-strong coupling regime $A^2$1, where the oscillator is treated as the fast sector and the two-level atom as the slow sector. The designation “quadratic” refers to the quadratic photon self-interaction $A^2$2, not to a complexity-theoretic speedup or to a quadratic trial action in another model class.

2. Adiabatic reduction without the $A^2$3-term

For the Hamiltonian

$A^2$4

the approximation begins with a conditional displacement of the photon field,

$A^2$5

with $A^2$6.

After this transformation one obtains

$A^2$7

The key adiabatic step is the statement that, in the limit $A^2$8, the displaced off-diagonal factors $A^2$9 in norm-resolvent sense. Dropping these terms gives the leading-order adiabatic Hamiltonian

$$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$0

This reduction is structurally important. It shows that, at leading order and without the $$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$1-term, the tunnel-splitting term $$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$2 is suppressed by the displacement-induced off-diagonal operators, so the remaining effective dynamics is diagonal in the $$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$3 sector. This is the basic mechanism behind the bias-dependent appearance or disappearance of cat-like superpositions in the approximate eigenstates (Hirokawa, 2018).

3. Approximate eigenstates in the bias-free and biased cases

The approximate spectrum and eigenstates depend qualitatively on whether the atomic bias vanishes.

When $$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$4, the two spin levels are degenerate. Undoing the conditional displacement yields the approximate eigenstates

$$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$5

with almost-degenerate energies

$$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$6

These are Schrödinger-cat-like entangled states: each state is an even or odd superposition of opposite spin branches correlated with opposite coherent displacements of the oscillator.

When $$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$7, the effective Hamiltonian remains diagonal in $$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$8, and one finds instead

$$H_{\rm GQR}(\omega_c,g) = H_{\rm atm}(\varepsilon)+H_{\rm ptn}(\omega_c) +\hbar g\,\sigma_z(a+a^\dagger),$$9

with

$$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$0

The resulting distinction is central. Without the $$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$1-term, cat-like superposition appears for $$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$2 but not for $$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$3. The abstract statement that “whether each bare eigenstate forms a Schrodinger-cat-like entangled state or not depends on whether the energy bias of the atom is zero or non-zero” is implemented precisely by these two families of approximate eigenstates.

4. Incorporation of the quadratic interaction

With the quadratic interaction present, the first step is not a direct adiabatic reduction but an exact elimination of the $$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$4-term by a Hopfield–Bogoliubov unitary $$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$5. This renormalizes the oscillator frequency and the coupling according to

$$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$6

and yields

$$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$7

The transformed problem is again of generalized quantum Rabi type, but now with renormalized parameters. Repeating the displacement step gives

$$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$8

where $$H_{\rm atm}(\varepsilon) = \frac{\hbar}{2}\bigl(\omega_a\sigma_x+\varepsilon\sigma_z\bigr), \qquad H_{\rm ptn}(\omega_c) = \hbar\omega_c\Bigl(a^\dagger a+\tfrac12\Bigr).$$9 is a small counter-term chosen so that $\omega_a$0.

After dropping the vanishing off-diagonal terms, one arrives at the effective adiabatic Hamiltonian

$\omega_a$1

This is the core of the adiabatic-quadratic approximation in the strict sense. The approximation is “adiabatic” because nonadiabatic transitions are neglected in the deep-strong coupling limit, and “quadratic” because the quadratic photon self-interaction is incorporated through the renormalization step rather than discarded.

5. Approximate spectrum and universal cat-like structure with the $\omega_a$2-term

Under the approximate decoupling, $\omega_a$3 and $\omega_a$4 commute, and one obtains product-eigenstates in the renormalized frame. In the original variables, the approximate eigenstates become

$\omega_a$5

with coefficients

$\omega_a$6

The corresponding approximate energies are

$\omega_a$7

The notable feature is that these states are superpositions of the two spin branches for every bias $\omega_a$8. The paper describes them as universal Schrödinger-cat-like states. In contrast with the bias-sensitive structure of the approximation without the quadratic term, the $\omega_a$9-renormalized theory preserves spin-boson entanglement across the full bias range. The stated conclusion is that this behavior “comes from the effect of the tunnel splitting” (Hirokawa, 2018).

For compact comparison, the two regimes can be organized as follows.

Setting	Approximate state structure	Bias dependence
Without $\varepsilon$0-term	Displaced-spin products for $\varepsilon$1; even/odd superpositions for $\varepsilon$2	Cat-like only at $\varepsilon$3
With $\varepsilon$4-term	Superpositions weighted by $\varepsilon$5	Cat-like for every $\varepsilon$6

A plausible implication is that the quadratic interaction qualitatively alters which terms survive the strong-coupling reduction: rather than suppressing the tunnel splitting from the effective description, the renormalized formulation reintroduces it into the approximate Hamiltonian.

6. Physical interpretation, validity, and conceptual boundaries

The physical interpretation given in the source distinguishes between bare and dressed descriptions. Without the $\varepsilon$7-term, the adiabatic states contain displaced Fock clouds of virtual photons. After the Hopfield–Bogoliubov renormalization, the model with quadratic interaction yields dressed physical eigenstates. This distinction matters because the $\varepsilon$8-term is not handled as a perturbation; it is absorbed into renormalized parameters before the adiabatic limit is taken.

The validity regime is explicitly the deep-strong coupling regime $\varepsilon$9, together with the technical conditions on $\omega_c$0 stated above. The approximation neglects nonadiabatic transitions, and its limiting statements are formulated through norm-resolvent convergence. These features place the method within a mathematically controlled asymptotic framework rather than a purely heuristic strong-coupling argument.

A common misunderstanding would be to treat the approximation as merely the standard adiabatic approximation applied to the quantum Rabi model. In the construction with the quadratic interaction, that is incomplete. The $\omega_c$1-term is first eliminated by the Hopfield–Bogoliubov unitary, producing the renormalized Hamiltonian $\omega_c$2; only then is the adiabatic displacement analysis carried out. Another possible misunderstanding is to assume that the tunnel splitting is always negligible at deep strong coupling. The results show a more nuanced picture: without the quadratic interaction it drops out at leading order, whereas with the $\omega_c$3-term it re-enters the effective Hamiltonian and changes the structure of the eigenstates.

In this sense, the adiabatic-quadratic approximation is best understood as a specific asymptotic scheme for the generalized quantum Rabi model with quadratic photon self-interaction, producing explicit formulas

$\omega_c$4

$\omega_c$5

together with the approximate energies and cat-like eigenstates quoted above. These formulas constitute the adiabatic-quadratic approximation in the strict terminology of the paper (Hirokawa, 2018).