Anisotropic Quantum Rabi-Stark Model

• The anisotropic Quantum Rabi-Stark Model is a single-qubit, single-mode light-matter Hamiltonian that integrates anisotropic dipole couplings with a photon-number-dependent Stark term, exhibiting both first-order and continuous transitions.
• It employs Bogoliubov-operator techniques and parity symmetry to derive exact spectral solutions and identify critical level-crossing phenomena.
• Its rich phase diagram facilitates studies in quantum thermodynamics, enabling analyses of quantum engines and dissipative dynamics via dressed-state master equations and photon correlations.

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 (Xie et al., 2019, Xu et al., 2024).

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 $U$0. A third parametrization writes

$U$1

so that the single parameter $U$2 controls the relative strength of the counter-rotating contribution (Zhang et al., 31 Aug 2025).

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

Limit	Conditions	Resulting model
Standard quantum Rabi model	$U$3 or $U$4	isotropic quantum Rabi model
Anisotropic quantum Rabi model	$U$5	anisotropic quantum Rabi model
Quantum Rabi-Stark model	$U$6 or $U$7	isotropic Rabi-Stark model
RWA / Jaynes-Cummings-like limit	$U$8 or $U$9	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” (Eckle et al., 2017).

2. Parity symmetry and exact spectral construction

The anisotropic quantum Rabi-Stark model preserves a $r=g_2/g_1$0 parity symmetry. One convenient form of the conserved parity operator is

$r=g_2/g_1$1

while an equivalent thermodynamic notation uses

$r=g_2/g_1$2

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 (Xie et al., 2019).

For $r=g_2/g_1$3, the exact solution is developed through a Bogoliubov-operator approach. After a spin-dependent transformation, one introduces displaced bosonic operators

$r=g_2/g_1$4

and expands the wavefunction in the $r=g_2/g_1$5-basis. The regular spectrum is obtained from the zeros of the transcendental $r=g_2/g_1$6-functions

$r=g_2/g_1$7

with $r=g_2/g_1$8 corresponding to odd and even parity sectors. The same analysis identifies poles of the $r=g_2/g_1$9-function at

$$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$0

together with a special first pole at $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$1. When numerator and denominator conditions are simultaneously satisfied at a pole, the pole is lifted and a doubly degenerate Juddian solution appears (Xie et al., 2019).

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 $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$2, regular singularities at $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$3, 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 (Eckle et al., 2017).

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

$$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$4

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 (Xie et al., 2019).

The dependence of this critical line on $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$5 and $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$6 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 $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$7, which is not allowed in the anisotropic quantum Rabi model; they can also occur for $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$8 if $$\hat{H} = \omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+U\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} + \lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-}) + \lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}),$$9; 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 (Xie et al., 2019).

Later thermodynamic work adopts the same first-order critical point, written in the $r=\lambda_2/\lambda_1$0 notation at $r=\lambda_2/\lambda_1$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 (Xu et al., 2024).

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

A second, qualitatively different transition appears at the special Stark couplings

$r=\lambda_2/\lambda_1$2

At these values, the Bogoliubov-operator construction for $r=\lambda_2/\lambda_1$3 is no longer applicable, and the model can instead be mapped to an effective quantum oscillator. For $r=\lambda_2/\lambda_1$4, introducing

$r=\lambda_2/\lambda_1$5

the lower spinor component obeys an effective harmonic-oscillator Hamiltonian

$r=\lambda_2/\lambda_1$6

with

$r=\lambda_2/\lambda_1$7

The continuous critical couplings are

$r=\lambda_2/\lambda_1$8

and the corresponding upper edges of the lower branch are

$r=\lambda_2/\lambda_1$9

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 (Xie et al., 2019).

The critical gap scales as

$\mathbb Z_2$0

with

$\mathbb Z_2$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 (Xie et al., 2019).

The thermodynamic analysis uses closely related language, describing both the familiar quantum-Rabi-model-like continuous transition at infinite frequency ratio $\mathbb Z_2$2 and a finite-frequency continuous transition at $\mathbb Z_2$3, where low-energy discrete levels converge to a common energy, the gap closes, and gapless excitations emerge (Xu et al., 2024). A separate isotropic Rabi-Stark analysis refines the interpretation of “spectral collapse” itself: at $\mathbb Z_2$4, the isotropic spectrum does not merely collapse into an infinitely degenerate ground state, but instead develops a continuum extending from

$\mathbb Z_2$5

to $\mathbb Z_2$6, with normalizable bound states embedded in that continuum (Braak et al., 2024). 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 $\mathbb Z_2$7, adiabatic expansion from $\mathbb Z_2$8 to $\mathbb Z_2$9, isochoric cooling at fixed $U a^\dagger a\,\sigma_z$0, and adiabatic compression back to $U a^\dagger a\,\sigma_z$1. In the steady state,

$U a^\dagger a\,\sigma_z$2

and the cycle energetics are

$U a^\dagger a\,\sigma_z$3

$U a^\dagger a\,\sigma_z$4

$U a^\dagger a\,\sigma_z$5

The same framework also distinguishes heat-engine, refrigerator, heater, and accelerator operational modes by the signs of $U a^\dagger a\,\sigma_z$6, $U a^\dagger a\,\sigma_z$7, and $U a^\dagger a\,\sigma_z$8 (Xu et al., 2024).

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

$U a^\dagger a\,\sigma_z$9

and for $g_1(a^\dagger\sigma_-+a\sigma_+)$0 the harmonic Otto efficiency is $g_1(a^\dagger\sigma_-+a\sigma_+)$1. By contrast, the anisotropic quantum Rabi-Stark engine reaches a maximum efficiency around $g_1(a^\dagger\sigma_-+a\sigma_+)$2 in the low-temperature regime and still about $g_1(a^\dagger\sigma_-+a\sigma_+)$3 in a higher-temperature regime, close to Carnot limits of $g_1(a^\dagger\sigma_-+a\sigma_+)$4 and $g_1(a^\dagger\sigma_-+a\sigma_+)$5, 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 (Xu et al., 2024).

Finite-time operation introduces quantum friction and convergence to a limit cycle. The adiabatic strokes are assigned durations $g_1(a^\dagger\sigma_-+a\sigma_+)$6, the isochoric strokes $g_1(a^\dagger\sigma_-+a\sigma_+)$7, and convergence is quantified by the fidelity

$g_1(a^\dagger\sigma_-+a\sigma_+)$8

Quantum friction is measured by a Kullback-Leibler divergence,

$g_1(a^\dagger\sigma_-+a\sigma_+)$9

and the total entropy production obeys

$g_2(a^\dagger\sigma_+ + a\sigma_-)$0

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 (Xu et al., 2024).

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 $g_2(a^\dagger\sigma_+ + a\sigma_-)$1 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

$g_2(a^\dagger\sigma_+ + a\sigma_-)$2

where the transition rates are dressed-state matrix elements of $g_2(a^\dagger\sigma_+ + a\sigma_-)$3 and $g_2(a^\dagger\sigma_+ + a\sigma_-)$4. This formulation is used to compute steady-state nonclassical correlations and squeezing in regimes where bare-operator input-output theory fails (Zhang et al., 31 Aug 2025).

Photon correlations are expressed through the dressed output operator $g_2(a^\dagger\sigma_+ + a\sigma_-)$5, and the zero-delay normalized correlators are

$g_2(a^\dagger\sigma_+ + a\sigma_-)$6

The interpretation is standard: $g_2(a^\dagger\sigma_+ + a\sigma_-)$7 indicates bunching, $g_2(a^\dagger\sigma_+ + a\sigma_-)$8 antibunching, and $g_2(a^\dagger\sigma_+ + a\sigma_-)$9 coherent or Poissonian statistics. The reported phenomenology is substantially richer than in the anisotropic quantum Rabi model without Stark coupling. Positive $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$0 enlarges and strengthens antibunching regions, can produce a secondary antibunching transition near $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$1, and can generate double-switching behavior of the form antibunching $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$2 bunching $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$3 antibunching $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$4 bunching. Negative $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$5 suppresses antibunching, moves the primary antibunching feature toward the deep-strong-coupling regime, and strongly expands bunching regions (Zhang et al., 31 Aug 2025).

The same dressed-state framework is used to analyze photon quadrature squeezing. With

$$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$6

the squeezing measure is

$$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$7

with squeezing when $$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$8. Using parity symmetry, the criterion simplifies to

$$\lambda_1(\hat a \hat\sigma^+ + \hat a^\dagger \hat\sigma^-)$$9

The calculations show that positive $U$00 shrinks the squeezing region, whereas negative $U$01 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 $U$02 intersects $U$03 (Zhang et al., 31 Aug 2025).

These dissipative observables also serve as criticality diagnostics. The first-order quantum phase transition at

$U$04

appears in $U$05 as a successive antibunching $U$06 bunching $U$07 antibunching structure, with a near-divergent peak when the gap $U$08 closes. The third-order correlator $U$09 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 (Zhang et al., 31 Aug 2025).