---
title: Anisotropic Quantum Rabi-Stark Model
url: https://www.emergentmind.com/topics/anisotropic-quantum-rabi-stark-model
type: topic
---

# Anisotropic Quantum Rabi-Stark Model

The anisotropic quantum Rabi-Stark model, also termed the anisotropic Rabi-Stark model in part of the literature, is a single-qubit, single-mode light-matter Hamiltonian that combines anisotropic rotating-wave and counter-rotating-wave dipole couplings with a nonlinear Stark-type term proportional to photon number. In the notation used for the exact quantum-phase-transition analysis, it is
\[
H=\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+\omega a^\dagger a +g_1\left(a^\dagger \sigma_-+a \sigma_+\right) +g_2\left(a^\dagger \sigma_+ + a \sigma_-\right),
\]
where \(U\) is the nonlinear Stark coupling and \(r=g_2/g_1\) quantifies anisotropy. In an equivalent notation used in thermodynamic studies,
\[
\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),
\]
with \(r=\lambda_2/\lambda_1\). The model supports both first-order and continuous quantum phase transitions, retains a \(\mathbb Z_2\) parity symmetry, admits exact or effectively exact spectral constructions in several parameter regimes, and has been used both as a testbed for critical spectral phenomena and as a working medium in quantum thermodynamics and dissipative quantum-optical analyses [1912.10042] [2407.09027].

## 1. Model definition within the Rabi-model family

The defining feature of the model is the simultaneous presence of anisotropy in the linear dipole coupling and a nonlinear Stark coupling \(U a^\dagger a\,\sigma_z\). The rotating-wave and counter-rotating-wave sectors appear separately as \(g_1(a^\dagger\sigma_-+a\sigma_+)\) and \(g_2(a^\dagger\sigma_+ + a\sigma_-)\), or equivalently as \(\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)\) and \(\lambda_2(\hat a^\dagger \hat\sigma^+ + \hat a \hat\sigma^-)\). A third parametrization writes
\[
H_{\textnormal{AQRSM}}=\left(\frac{1}{2}\Delta +Ua^{\dag }a\right)\sigma _{z}+\omega _{0}a^{\dag }a
+g\big[(a\sigma _{+}+a^{\dag }\sigma _{-})+r(a\sigma _{-}+a^{\dag }\sigma _{+})\big],
\]
so that the single parameter \(r\) controls the relative strength of the counter-rotating contribution [2509.00821].

Several established models are recovered as limits of the same Hamiltonian family.

| Limit | Conditions | Resulting model |
|---|---|---|
| Standard quantum Rabi model | \(U=0,\ g_1=g_2\) or \(\lambda_1=\lambda_2,\ U=0\) | isotropic quantum Rabi model |
| Anisotropic quantum Rabi model | \(U=0,\ g_1\neq g_2\) | anisotropic quantum Rabi model |
| Quantum Rabi-Stark model | \(g_1=g_2\) or \(\lambda_1=\lambda_2,\ U\neq 0\) | isotropic Rabi-Stark model |
| RWA / Jaynes-Cummings-like limit | \(g_2=0\) or \(r=0\) | rotating-wave limit |

This placement within the Rabi hierarchy is important because the anisotropic model inherits exact-spectral techniques from the isotropic quantum Rabi-Stark model while extending its phase structure. The earlier isotropic Stark-modified model was solved exactly by a Braak-type Bargmann-space construction, and that work explicitly identified the anisotropic quantum Rabi model as a natural nearby generalization and stated that examining the anisotropic generalization of the Rabi-Stark model “will be very interesting” [1706.02687].

## 2. Parity symmetry and exact spectral construction

The anisotropic quantum Rabi-Stark model preserves a \(\mathbb Z_2\) parity symmetry. One convenient form of the conserved parity operator is
\[
\Pi=\exp(i\pi \widehat N),\qquad \widehat N=\frac{1+\sigma_z}{2}+a^\dagger a,
\]
while an equivalent thermodynamic notation uses
\[
\hat{\pi}=\exp\!\left(i\pi\hat N\right),\qquad \hat N=\hat a^\dagger \hat a+\hat\sigma^+\hat\sigma^-.
\]
The spectrum therefore decomposes into even and odd parity sectors, and this parity resolution is central both to exact spectral equations and to the identification of phase transitions [1912.10042].

For \(|U|<1\), the exact solution is developed through a Bogoliubov-operator approach. After a spin-dependent transformation, one introduces displaced bosonic operators
\[
A^\dagger=a^\dagger+w,\qquad B^\dagger=a^\dagger-w,\qquad
w=\frac{\beta}{\sqrt{1-U^2}},\qquad \beta=\sqrt{g_1g_2},
\]
and expands the wavefunction in the \(A\)-basis. The regular spectrum is obtained from the zeros of the transcendental \(G\)-functions
\[
G_\mp(E)=\sum_{n=0}^{\infty}(e_n\pm f_n)w^n=0,
\]
with \(\mp\) corresponding to odd and even parity sectors. The same analysis identifies poles of the \(G\)-function at
\[
E_m^{\rm pole}=(1-U^2)m-\lambda_+ - \frac{U\Delta}{2}, \qquad m>0,\qquad
\lambda_\pm=\frac{g_1^2\pm g_2^2}{2},
\]
together with a special first pole at \(m=0\). When numerator and denominator conditions are simultaneously satisfied at a pole, the pole is lifted and a doubly degenerate Juddian solution appears [1912.10042].

This parity-resolved spectral logic is closely related to the exact treatment of the isotropic Rabi-Stark model, where Bargmann-space methods yield parity-sector transcendental functions \(G_\pm(E;0)=0\), regular singularities at \(z=\pm w\), and a Frobenius classification into regular and exceptional spectra. The anisotropic problem uses a different construction, but the underlying strategy remains parity decomposition plus analytic continuation of sector-resolved spectral equations [1706.02687].

## 3. First-order transition line and pole-structure diagnostics

A central result for the anisotropic model is the existence of a first-order quantum phase transition detected by a level crossing between the ground state and the first excited state. In the exact solution, the transition occurs when the first pole is regularized, which yields the critical coupling
\[
g_{1c}= \sqrt{ \frac{\Delta(1-U^2)} {U(1+r^2)+1-r^2} }.
\]
At this coupling, the lowest even- and odd-parity levels cross, the ground-state energy becomes nonanalytic, and the first derivative of the ground-state energy with respect to coupling is discontinuous; these are the defining features used to classify the transition as first-order [1912.10042].

The dependence of this critical line on \(U\) and \(r=g_2/g_1\) is a distinctive feature of the anisotropic Stark problem. The first-order transition arises from the combined action of the Stark coupling, the anisotropy, and the interplay between rotating and counter-rotating terms. The exact analysis emphasizes three parameter-space consequences: first-order quantum phase transitions can occur even for \(r>1\), which is not allowed in the anisotropic quantum Rabi model; they can also occur for \(U<0\) if \(r<1\); and the parameter region supporting first-order criticality is therefore larger than in either the anisotropic quantum Rabi model or the isotropic Rabi-Stark model [1912.10042].

Later thermodynamic work adopts the same first-order critical point, written in the \(\lambda_1,\lambda_2\) notation at \(\omega=1\), and treats it as the organizing line for engine performance. In that setting, the abrupt rearrangement of low-lying levels around the crossing is directly tied to changes in work extraction and operational mode [2407.09027].

## 4. Continuous criticality, effective-oscillator reduction, and collapse phenomena

A second, qualitatively different transition appears at the special Stark couplings
\[
U=\pm 1.
\]
At these values, the Bogoliubov-operator construction for \(|U|<1\) is no longer applicable, and the model can instead be mapped to an effective quantum oscillator. For \(U=1\), introducing
\[
\alpha=\frac{g_1+g_2}{2},\qquad \kappa \alpha=\frac{g_1-g_2}{2},\qquad
\kappa=\frac{1-r}{1+r},
\]
the lower spinor component obeys an effective harmonic-oscillator Hamiltonian
\[
H_{\rm eff}=2\left(1+\frac{2\kappa^2\alpha^2}{E+\Delta/2}\right)
\left[\frac{p^2}{2}+\frac{1}{2}\omega_{\rm eff}^2 x^2\right],
\]
with
\[
\omega_{\rm eff} = \sqrt{ \frac{1+\frac{2\alpha^2}{E+\Delta/2}}
{1+\frac{2\kappa^2\alpha^2}{E+\Delta/2}} }.
\]
The continuous critical couplings are
\[
\alpha_c^+=\sqrt{\frac{1-\Delta+\kappa}{2}},\qquad
\alpha_c^-=\sqrt{\frac{1+\Delta-\kappa}{2}},
\]
and the corresponding upper edges of the lower branch are
\[
E_c^+=-\frac{\Delta}{2}-2\alpha^2,\qquad
E_c^-=\frac{\Delta}{2}-2\alpha^2.
\]
At the critical point, all low-lying levels close toward the same limiting energy and the excitation gap vanishes, which is the basis for classifying the transition as continuous [1912.10042].

The critical gap scales as
\[
E_g=E_1-E_0\propto |\alpha-\alpha_c|^{z\nu},
\]
with
\[
z\nu=2 \quad \text{if counter-rotating terms are present } (\kappa<1),\qquad
z\nu=1 \quad \text{if the counter-rotating term is absent } (\kappa=1).
\]
The counter-rotating interaction is therefore not merely quantitative: it changes the critical exponent. In the rotating-wave approximation, the literature further suggests that gapless Goldstone-mode-like excitations can appear above the critical coupling [1912.10042].

The thermodynamic analysis uses closely related language, describing both the familiar quantum-Rabi-model-like continuous transition at infinite frequency ratio \(\Delta/\omega\) and a finite-frequency continuous transition at \(U=\pm1\), where low-energy discrete levels converge to a common energy, the gap closes, and gapless excitations emerge [2407.09027]. A separate isotropic Rabi-Stark analysis refines the interpretation of “spectral collapse” itself: at \(\gamma=\omega\), the isotropic spectrum does not merely collapse into an infinitely degenerate ground state, but instead develops a continuum extending from
\[
E_{\mathrm{thr}}=-\Delta-\frac{2g^2}{\omega}
\]
to \(+\infty\), with normalizable bound states embedded in that continuum [2403.16758]. This suggests caution in reading collapse phenomena in the anisotropic problem: the gap-closing and level-condensation statements are established, whereas the full continuum structure requires a separate analysis.

## 5. Criticality as a thermodynamic resource

The anisotropic quantum Rabi-Stark model has been used as the working substance of a quantum Otto engine in both ideal and finite-time settings. The cycle consists of isochoric heating at fixed \(H^h\), adiabatic expansion from \(\omega_h\) to \(\omega_c\), isochoric cooling at fixed \(H^c\), and adiabatic compression back to \(\omega_h\). In the steady state,
\[
\hat{\rho}_\mathrm{ss}=\sum_n \frac{e^{-E_n/T}}{\mathcal{Z}} |\phi_n\rangle\langle \phi_n|,\qquad
\mathcal{Z}=\sum_n e^{-E_n/T},
\]
and the cycle energetics are
\[
Q_h = \sum_n E_n^h\,[P_n^{ss}(T_h)-P_n^{ss}(T_c)],
\]
\[
Q_c = \sum_n E_n^c\,[P_n^{ss}(T_c)-P_n^{ss}(T_h)],
\]
\[
W = \sum_n (E_n^h-E_n^c)\,[P_n^{ss}(T_h)-P_n^{ss}(T_c)],\qquad
\eta=\frac{W}{Q_h}.
\]
The same framework also distinguishes heat-engine, refrigerator, heater, and accelerator operational modes by the signs of \(Q_h\), \(Q_c\), and \(W\) [2407.09027].

The main conclusion is that the model’s critical spectral structure strongly modulates work, efficiency, and power. Near the first-order transition, the level crossing and associated rearrangement of low-energy states can produce enhanced work output, high efficiency, and abrupt changes between operational modes. Near the continuous transition, the compressed low-energy spectrum associated with spectral collapse can make the efficiency approach Carnot-like values under ideal conditions, but work output becomes harder to optimize and power is significantly suppressed in finite-time operation. The comparison with harmonic working media is explicit: for decoupled harmonic limits, the Otto efficiency is
\[
\eta_\alpha = 1-\frac{1}{\alpha},
\qquad
\eta_{\lambda=U=0}=1-\frac{\omega_c}{\omega_h},
\qquad
\xi_{\lambda=U=0}=\frac{\omega_c}{\omega_h-\omega_c},
\]
and for \(\omega_h=2\omega_c\) the harmonic Otto efficiency is \(0.5\). By contrast, the anisotropic quantum Rabi-Stark engine reaches a maximum efficiency around \(0.75\) in the low-temperature regime and still about \(0.64\) in a higher-temperature regime, close to Carnot limits of \(0.8\) and \(0.75\), respectively; under the stated low-temperature parameters, its maximum work is reported to be roughly twice that of the decoupled qubit-plus-oscillator reference engine [2407.09027].

Finite-time operation introduces quantum friction and convergence to a limit cycle. The adiabatic strokes are assigned durations \(\tau_1=\tau_3\), the isochoric strokes \(\tau_2=\tau_4\), and convergence is quantified by the fidelity
\[
F_h = F\big(\rho^{(N-1)}(\tau_0),\rho^{(N)}(\tau_0)\big).
\]
Quantum friction is measured by a Kullback-Leibler divergence,
\[
W_{\text{fric}}=\frac{1}{\beta_{h(c)}}D(\rho_{1(3)}\Vert\rho_{1(3)}^{\text{qe}}),
\]
and the total entropy production obeys
\[
\eta = \eta_{\text{Carnot}} - \frac{\langle\Sigma_{\text{total}}\rangle}{\beta_c \langle Q_h\rangle}.
\]
In the reported simulations, the first-order critical region remains relatively favorable even in finite time, whereas near the continuous transition the power is often suppressed by about an order of magnitude compared with the first-order case [2407.09027].

## 6. Dressed-state correlations, squeezing, and experimental diagnostics

In dissipative settings, the anisotropic quantum Rabi-Stark model has been analyzed with a quantum dressed master equation designed for the ultrastrong-coupling and deep-strong-coupling regimes. The key point is that dissipation is formulated in the eigenbasis \(\{|\varphi_n\rangle\}\) of the interacting Hamiltonian rather than in the bare cavity or qubit basis. With separate qubit and cavity baths, the dressed-state master equation is
\[
\frac{\partial \rho }{\partial t}=-i[H_{\textnormal{AQRSM}},\rho]
+\sum_{j,k>j}^{u=\textnormal{q,c}}
\left\{\Gamma _{u}^{j,k}n_{u}(\Delta _{k,j})\mathcal{D}[\left\vert \varphi _{k}\right\rangle \left\langle \varphi _{j}\right\vert ,\rho ]
+\Gamma _{u}^{j,k}[1+n_{u}(\Delta _{k,j})]
\mathcal{D}[\left\vert \varphi _{j}\right\rangle \left\langle \varphi _{k}\right\vert ,\rho]\right\},
\]
where the transition rates are dressed-state matrix elements of \(\sigma_-+\sigma_+\) and \(a+a^\dagger\). This formulation is used to compute steady-state nonclassical correlations and squeezing in regimes where bare-operator input-output theory fails [2509.00821].

Photon correlations are expressed through the dressed output operator \(X^+\), and the zero-delay normalized correlators are
\[
G_{2}(0)=\frac{\left\langle X^{-}X^{-}X^{+}X^{+}\right\rangle _{ss}}
{\left\langle X^{-}X^{+}\right\rangle _{ss}^{2}},
\qquad
G_{3}(0)=\frac{\left\langle X^{-}X^{-}X^{-}X^{+}X^{+}X^{+}\right\rangle _{ss}}
{\left\langle X^{-}X^{+}\right\rangle _{ss}^{3}}.
\]
The interpretation is standard: \(G_2(0)>1\) indicates bunching, \(G_2(0)<1\) antibunching, and \(G_2(0)=1\) coherent or Poissonian statistics. The reported phenomenology is substantially richer than in the anisotropic quantum Rabi model without Stark coupling. Positive \(U\) enlarges and strengthens antibunching regions, can produce a secondary antibunching transition near \(r\to1\), and can generate double-switching behavior of the form antibunching \(\to\) bunching \(\to\) antibunching \(\to\) bunching. Negative \(U\) suppresses antibunching, moves the primary antibunching feature toward the deep-strong-coupling regime, and strongly expands bunching regions [2509.00821].

The same dressed-state framework is used to analyze photon quadrature squeezing. With
\[
X_\theta=ae^{-i\theta }+a^{\dagger }e^{i\theta },
\qquad
P_\theta=\frac{1}{i}(ae^{-i\theta }-a^{\dagger }e^{i\theta }),
\]
the squeezing measure is
\[
\xi _{\mathrm{B}}^{2}=\min_{\theta \in [0,2\pi )}(\Delta X_{\theta })^{2},
\]
with squeezing when \(\xi_{\mathrm{B}}^{2}<1\). Using parity symmetry, the criterion simplifies to
\[
\xi _{\mathrm{B}}^{2}=2[\langle a^{\dagger }a\rangle -\langle a^{2}\rangle ]+1.
\]
The calculations show that positive \(U\) shrinks the squeezing region, whereas negative \(U\) can generate new squeezing regions in ultrastrong- and deep-strong-coupling parameter space where the anisotropic quantum Rabi model has none. Repeated transitions between squeezed and unsqueezed regimes occur when \(\langle a^\dagger a\rangle\) intersects \(\langle a^2\rangle\) [2509.00821].

These dissipative observables also serve as criticality diagnostics. The first-order quantum phase transition at
\[
g_c=\sqrt{\frac{\Delta (1-U^{2})}{U(1+r^{2})+1-r^{2}}}
\]
appears in \(G_2(0)\) as a successive antibunching \(\to\) bunching \(\to\) antibunching structure, with a near-divergent peak when the gap \(\Delta_{1,0}\) closes. The third-order correlator \(G_3(0)\) additionally detects an excited-state crossing between the second and third excited states. In this sense, parity-constrained dressed-state transition pathways turn photon statistics into a probe of both ground-state and excited-state spectral reorganizations [2509.00821].

Source: https://www.emergentmind.com/topics/anisotropic-quantum-rabi-stark-model