Open Quantum Rabi-Stark Model

• The open quantum Rabi–Stark model is a dissipative extension of the quantum Rabi–Stark Hamiltonian featuring nonlinear Stark coupling and preserved parity symmetry, particularly in ultrastrong-coupling regimes.
• It employs a dressed-basis approach for modeling dissipation, ensuring accurate treatment of light–matter interactions where bare decay approximations fail.
• Experimental signatures such as photon correlations and quadrature squeezing provide measurable probes for underlying quantum phase transitions and multiphoton processes.

Searching arXiv for recent work on the open quantum Rabi–Stark model and closely related AQRSM studies. arxiv_search(query="open quantum Rabi-Stark model", max_results=10) The open quantum Rabi–Stark model is the dissipative extension of the quantum Rabi–Stark Hamiltonian, in which a single bosonic mode interacts with a two-level system through the usual Rabi light–matter coupling together with a nonlinear Stark-type term proportional to $a^\dagger a\,\sigma_z$, while the composite light–matter system is coupled to external environments. In current arXiv literature, the term covers both the isotropic quantum Rabi–Stark model (QRSM) and its anisotropic generalization (AQRSM), where rotating-wave and counter-rotating-wave couplings are unequal. The open problem is technically nontrivial because the regimes of primary interest are the ultrastrong-coupling and deep-strong-coupling regimes, where bare-subsystem dissipators become unreliable and dissipation must be expressed in the dressed eigenbasis of the full interacting Hamiltonian (Zhang et al., 31 Aug 2025).

1. Hamiltonian content and model family

A standard anisotropic form used for the open problem is

$$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$

where $a^\dagger,a$ are cavity creation and annihilation operators, $\Delta$ is the qubit splitting, $g$ is the rotating-wave coupling strength, $r$ is the anisotropy parameter controlling the relative weight of the counter-rotating-wave term, and $U$ is the nonlinear Stark coupling (Zhang et al., 31 Aug 2025). In the isotropic limit $r=1$, this reduces to the quantum Rabi–Stark model; at $U=0$ it reduces to the anisotropic quantum Rabi model; and for $r=0$ in the weak-coupling limit it reduces to the Jaynes–Cummings model (Zhang et al., 31 Aug 2025).

The isotropic QRSM is also written in the form

$$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$0

or, in another convention,

$$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$1

with $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$2, $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$3, or $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$4 denoting the Stark coupling in different notational conventions (Ye et al., 3 Jul 2026, Eckle et al., 2017). A related convention writes the Stark term as $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$5, which shifts several critical formulas by a factor of two (Xie et al., 2018).

Despite the counter-rotating-wave terms, these models preserve a discrete $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$6 or parity symmetry. In the anisotropic formulation one may write

$$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$7

with $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$8 (Zhang et al., 31 Aug 2025). In the isotropic and anisotropic closed formulations, parity organizes the spectrum into even and odd sectors, permits crossings between opposite-parity states, and remains central for any open extension because dressed dissipative transitions inherit parity-based matrix-element structure (Xie et al., 2019).

2. Closed-system spectral structure underlying the open model

The open model is built on a closed Hamiltonian backbone with unusually rich exact and near-exact spectral structure. For the isotropic QRSM, exact solutions were obtained by adapting Braak’s method, yielding parity-resolved spectral functions $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$9 and revealing two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases (Eckle et al., 2017). A Bogoliubov-operator treatment later produced compact transcendental functions whose zeros reproduce the regular spectrum and whose explicit pole structure identifies two kinds of exceptional eigenvalues (Xie et al., 2018).

For the anisotropic closed model,

$a^\dagger,a$0

with anisotropy ratio $a^\dagger,a$1, the spectrum supports both first-order and continuous quantum phase transitions (Xie et al., 2019). The first-order transition is detected by a crossing between the ground state and first excited state, with critical coupling

$a^\dagger,a$2

At the special Stark couplings $a^\dagger,a$3, the model becomes analytically tractable by mapping to an effective oscillator, and all low-lying levels close at the critical coupling, indicating continuous quantum phase transitions (Xie et al., 2019).

The closed QRSM also exhibits convention-dependent collapse formulas. In the convention with Stark term $a^\dagger,a$4, the spectra collapse phenomenon occurs at $a^\dagger,a$5, and at the critical coupling infinite discrete spectra accumulate into a finite energy from below (Xie et al., 2018). In the convention with Stark term $a^\dagger,a$6, the corresponding critical structure appears at $a^\dagger,a$7 (Xie et al., 2019, Ye et al., 3 Jul 2026). This suggests that apparent differences in collapse thresholds across papers are largely notational rather than physical.

A recurrent misconception is to identify these closed-system crossings and collapses with established dissipative phase transitions. The closed anisotropic Rabi–Stark literature explicitly does not provide a master equation, Liouvillian spectrum, dissipative gap, or steady-state phase diagram, so closed-system quantum phase transitions should not be conflated with proven dissipative criticality (Xie et al., 2019).

3. Open-system formulations and dressed-basis dissipation

The open QRSM and open AQRSM are typically formulated by coupling the interacting light–matter system to independent cavity and qubit thermal baths. In the open anisotropic study, the total Hamiltonian is

$a^\dagger,a$8

with bath Hamiltonian

$a^\dagger,a$9

and system–bath couplings

$\Delta$0

The baths are thermal, not only zero-temperature loss channels, and no separate pure dephasing channel or non-Markovian reservoir is included in that formulation (Zhang et al., 31 Aug 2025).

The corresponding isotropic open QRSM adopts the same structural choice: one bosonic thermal bath coupled through $\Delta$1 and one qubit bath coupled through $\Delta$2, with Ohmic spectral densities and dressed transition operators built from matrix elements of $\Delta$3 and $\Delta$4 between eigenstates $\Delta$5 of the full interacting Hamiltonian (Ye et al., 3 Jul 2026). In the Otto-cycle literature, the dressed-basis Markovian master equation is written with transition projectors $\Delta$6, rates $\Delta$7, and thermal occupations $\Delta$8, with Ohmic spectral function

$\Delta$9

and the steady state is the Gibbs or canonical state in the dressed basis (Xu et al., 2024).

The reason for this dressed-basis treatment is explicit in the open QRSM and AQRSM literature: in ultrastrong coupling and deep-strong coupling the qubit and cavity cannot be treated as weakly interacting subsystems with independent bare decay channels, naive bare Lindblad operators such as $g$0 and $g$1 can predict unphysical effects such as photon emission from the interacting ground state, and dissipation must therefore be expressed in the eigenbasis of the full interacting Hamiltonian (Zhang et al., 31 Aug 2025, Ye et al., 3 Jul 2026).

Not all open treatments are equally complete. A minimal dissipative bridge appears in the multiphoton Rabi–Stark study, where openness is modeled only by bosonic loss through

$g$2

with no qubit relaxation, no qubit dephasing term, no thermal bath, and no dissipative spectral theory (Cong et al., 2019). This suggests a hierarchy in the literature: minimal Lindblad-loss tests, dressed weak-coupling thermal master equations, and, outside the Stark setting itself, non-Markovian stochastic-Schrödinger treatments for the driven dissipative Rabi model that may serve as methodological precursors rather than direct Stark-model solutions (Henriet et al., 2014).

4. Nonclassical light, correlations, and multiphoton processes

The open anisotropic quantum Rabi–Stark model has been used to study steady-state photon correlations and quadrature squeezing under dissipation and thermal environments. Using the quantum dressed master equation, the anisotropic study analyzes second- and higher-order correlation functions and shows that the nonlinear Stark coupling significantly modulates photon statistics, inducing distinct and tunable photon antibunching and bunching effects (Zhang et al., 31 Aug 2025). In that work, successive transition signatures of correlation functions provide a potential experimental probe for predicting quantum phase transitions, and the Stark coupling acts as a direct control parameter for photon squeezing (Zhang et al., 31 Aug 2025).

In the isotropic open QRSM, quadrature squeezing is characterized by

$g$3

with squeezing criterion $g$4 (Ye et al., 3 Jul 2026). Because parity implies $g$5 in the low-temperature state, the squeezing factor reduces to

$g$6

and it can be decomposed into contributions from two-photon processes $g$7 through

$g$8

The low-excitation regime is governed mainly by the constructive $g$9 channel $r$0 and the destructive $r$1 channel $r$2 (Ye et al., 3 Jul 2026).

The open-squeezing study reports that positive Stark coupling enhances squeezing, negative Stark coupling suppresses it, the standard dissipative QRM at $r$3 has weak squeezing with $r$4, and near $r$5 the squeezing can reach approximately $r$6; away from the critical regime, positive $r$7 can lower the squeezing factor to roughly $r$8 (Ye et al., 3 Jul 2026). The paper attributes this to the Stark-controlled redistribution of low-lying amplitudes: positive $r$9 increases $U$0 and decreases $U$1, strengthening the constructive even-photon channel and weakening the destructive odd-photon channel, whereas negative $U$2 does the reverse (Ye et al., 3 Jul 2026).

A related closed-system dynamical line of work shows that the full Rabi–Stark Hamiltonian supports selective odd-$U$3-photon interactions. In that formulation the effective one-photon detunings are occupation dependent,

$U$4

so the Stark term produces Fock-selective resonances; the full model then generates selective multiphoton transitions, especially odd-$U$5, with strengths scaling as $U$6 (Cong et al., 2019). The same work shows that a simple bosonic-loss Lindblad channel degrades but does not immediately destroy selective three-photon oscillations: for $U$7, $U$8, $U$9, and initial state $r=1$0, the peak population of $r=1$1 is about $r=1$2 at $r=1$3 and about $r=1$4 at $r=1$5 (Cong et al., 2019).

5. Quantum criticality in the open model

Open-system work on the QRSM treats squeezing and correlation observables as signatures of the same first-order and second-order quantum phase transitions known from the closed Hamiltonian. For positive Stark coupling, the closed QRSM has a first-order quantum phase transition at

$r=1$6

and the open steady state at low temperature retains a sharp optical signature: the squeezing factor exhibits a sudden jump from a squeezed regime to a normal regime when $r=1$7 crosses the corresponding location (Ye et al., 3 Jul 2026). The physical mechanism is ground-state parity switching, which abruptly redistributes weight from even-excitation configurations favorable to $r=1$8 squeezing toward odd-excitation content and the destructive $r=1$9 channel (Ye et al., 3 Jul 2026).

Near the second-order superradiant transition at $U=0$0, the relevant effective size is

$U=0$1

or $U=0$2 near $U=0$3, and one quadrature becomes critical while the canonically conjugate quadrature becomes nearly perfectly squeezed (Ye et al., 3 Jul 2026). At the critical point, the squeezing factor obeys

$U=0$4

with fitted slopes approaching $U=0$5, so $U=0$6; using $U=0$7 from the ground-state problem and the scaling relation $U=0$8 gives $U=0$9 (Ye et al., 3 Jul 2026). The finite-size scaling ansatz

$r=0$0

produces data collapse in the quantum-critical regime (Ye et al., 3 Jul 2026).

Thermal fluctuations eventually destroy this critical behavior. Let $r=0$1 denote the gap between the two lowest levels at the critical point; the open-squeezing work states that

$r=0$2

and the scaling regime persists only while $r=0$3 is not violated. Once $r=0$4, thermal population of low-lying excited states spoils the near-ground-state critical behavior (Ye et al., 3 Jul 2026). This criterion is one of the clearest experimentally oriented results in the present open-model literature.

The open anisotropic work reaches a parallel conclusion through correlation functions rather than squeezing: successive transition signatures in steady-state photon correlations provide a potential experimental probe for predicting quantum phase transitions (Zhang et al., 31 Aug 2025). This suggests a broader pattern in open Rabi–Stark physics: criticality is often inferred optically, through steady-state nonclassical observables, rather than through direct spectroscopy of the closed Hamiltonian.

6. Thermodynamic, stability, and methodological extensions

A thermodynamic variant of the open AQRSM appears when the anisotropic Rabi–Stark Hamiltonian is used as the working medium of a quantum Otto engine alternately coupled to hot and cold reservoirs (Xu et al., 2024). During the isochoric strokes the model evolves under a dressed-state Markovian master equation, and the first law is written as

$r=0$5

The cycle supports heat-engine, refrigerator, heater, and accelerator regimes depending on the signs of $r=0$6, $r=0$7, and $r=0$8, and finite-time repeated cycling converges to a limit cycle monitored by a fidelity $r=0$9, with the reported result that for $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$00, $$H_{\text{AQRSM}}=\omega_0 a^\dagger a+\left(\frac{\Delta}{2}+U a^\dagger a\right)\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-) +gr(a\sigma_- + a^\dagger \sigma_+),$$01 (Xu et al., 2024). Around first-order critical points the engine is comparatively robust, whereas near the continuous quantum phase transition the ideal-cycle work, efficiency, and peak power are