Open Quantum Rabi Model Overview

• The open quantum Rabi model is a light–matter system that combines a bosonic mode and a two-level system with full counter-rotating interactions and various dissipative couplings.
• It investigates nonequilibrium phenomena, including dissipative phase transitions, bistability, and universal scaling through both Markovian and non-Markovian environments.
• Its rich symmetry structure—featuring Z2 parity conservation and significant counter-rotating effects—guides experimental realizations in circuit-QED and trapped-ion platforms.

The open quantum Rabi model comprises formulations in which a single bosonic mode and a two-level system are coupled through the full counter-rotating light–matter interaction while the mode, the qubit, or both are also coupled to dissipative or structured environments. Its coherent core is typically written as $$H_{\mathrm{QRM}}=\omega_c a^\dagger a+\frac{\omega_q}{2}\sigma_z+g\,\sigma_x(a+a^\dagger)$$, or in an equivalent basis-rotated form, and its open-system extensions include Markovian oscillator damping, parity-preserving two-photon relaxation, thermal reservoirs treated in the dressed eigenbasis, and non-Markovian Ohmic baths. Across these variants, the central issues are the fate of the model’s $Z_2$ symmetry, the dynamical role of counter-rotating terms beyond the rotating-wave approximation, the structure of nonequilibrium steady states and dissipative phase transitions, and the use of the Rabi system itself as a critical environment for auxiliary probes (Hwang et al., 2017, Malekakhlagh et al., 2018, Liu et al., 2022, Hu et al., 2021).

1. Canonical formulations

The cited literature formulates the open quantum Rabi model in several distinct ways rather than through a single universal generator. In the Markovian cavity-loss setting, the coherent Hamiltonian is the standard Rabi Hamiltonian together with oscillator damping at rate $\kappa$, as in $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$, with $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$ in one common convention. A circuit-QED realization uses the equivalent rotating-frame Hamiltonian $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$, again with photonic loss as the dominant dissipative channel. Other formulations replace one-photon leakage by even-photon relaxation, $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$, or retain explicit thermal and non-Markovian environments instead of a Lindbladian closure (Hwang et al., 2017, Ning et al., 19 Jun 2025, Malekakhlagh et al., 2018, Liu et al., 2022, Pirozzi et al., 17 Mar 2026, Henriet et al., 2014).

A useful way to organize these variants is to separate them by both bath structure and system–bath coupling operator. Global thermalization studies treat the full interacting QRM as an effective multilevel system and derive dressed-state master equations for either two individual heat baths or one common heat bath. Non-Markovian approaches instead keep the environment explicitly, for example through an Ohmic spectral density $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$ or through a combined spectral density $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$ that mixes a discrete cavity mode with an Ohmic continuum. The resulting terminology is therefore model-specific: “open quantum Rabi model” may mean a Lindblad problem, a global thermal master equation, or a Hamiltonian system-plus-bath construction.

Formulation	Representative dynamics	Distinctive feature
Markovian cavity-loss OQRM	$\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$	Dissipative phase transition and nonequilibrium steady state
Two-photon-loss OQRM	$Z_2$0	Exact preservation of parity by an even-photon jump operator
Thermal-bath OQRM	Global master equation in the dressed QRM eigenbasis	Thermalization depends on bath topology
Non-Markovian OQRM	Explicit bath Hamiltonian with frequency-dependent $Z_2$1	Bath-induced criticality and memory effects

2. Symmetry, counter-rotating structure, and parity sectors

The decisive structural difference between the Rabi model and the Jaynes–Cummings limit is the presence of counter-rotating terms. In the closed QRM the conserved symmetry is not excitation number but the discrete parity operator $Z_2$2. Open-system extensions inherit or modify this structure according to the bath coupling. In the two-photon-relaxation model, $Z_2$3, so the Lindbladian preserves the same $Z_2$4 symmetry as the coherent Hamiltonian. This allows a number-parity basis $Z_2$5, $Z_2$6, in which both the Hamiltonian and the phenomenological non-Hermitian effective description become block diagonal, and the closed-model recursion relations extend to complex eigenfrequencies by the substitution $Z_2$7 (Malekakhlagh et al., 2018).

Counter-rotating terms are not a perturbative detail. In the broader Rabi-family literature they explicitly remove excitation-number conservation and reduce a $Z_2$8 structure to a discrete $Z_2$9 one. The closed one-dimensional Rabi-Hubbard model makes this especially clear: counter-rotating on-site and hopping processes destroy the fixed-filling structure that supports Mott phases in the Jaynes–Cummings–Hubbard limit, leaving instead a parity-breaking coherent transition and a photon-number instability. That equilibrium result is not itself an open-system statement, but it fixes the underlying symmetry logic that carries into lossy cavity and circuit-QED arrays when the rotating-wave approximation fails (Flottat et al., 2016).

In the Markovian single-mode open QRM, the steady-state order parameter is likewise parity breaking. The normal phase has $\kappa$0, whereas the superradiant phase has two symmetry-related branches with nonzero $\kappa$1. In parity-preserving dissipators this symmetry remains exact at the level of the generator; in other settings, parity can survive as the organizing principle for the coherent core while the bath changes the spectral and steady-state structure.

3. Dissipative phase transitions and multicritical steady-state structure

The central phase-transition result for the Markovian open QRM is that a single oscillator mode coupled to a single qubit can exhibit a second-order dissipative phase transition in the finite-component thermodynamic limit $\kappa$2, with $\kappa$3 held finite. The critical coupling is shifted by damping to $\kappa$4. Below threshold the unique stable steady state is the normal phase, $\kappa$5; above threshold two stable superradiant steady states appear, with

$\kappa$6

The asymptotic decay rate closes linearly, $\kappa$7, the oscillator population diverges as $\kappa$8, the critical quadrature variance scales with the same exponent, and the finite-$\kappa$9 critical scaling is $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$0. The corresponding scaling function coincides with that of the open Dicke model, so the open QRM belongs to the same nonequilibrium universality class as the open Dicke model rather than the closed QRM (Hwang et al., 2017).

Anisotropy between rotating and counter-rotating couplings greatly enlarges the steady-state landscape. In the anisotropic open quantum Rabi model,

$$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$1

with oscillator damping $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$2, the phase diagram contains not only normal and superradiant phases but also a bistable region where both are stable. The superradiant phase exists only in a finite anisotropy window, the normal phase can re-enter at large coupling, and first-order and second-order dissipative phase transitions meet at multicritical points. On the ordinary second-order line, both the asymptotic decay rate and the oscillator fluctuation number scale with exponent $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$3; on the first-order boundaries, the normal-side exponents remain $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$4 while the superradiant-side exponents become $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$5; and along the tangent approach to the tricritical point the exponents become $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$6. Finite-$$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$7 Wigner functions distinguish the phases by Gaussian, two-lobed, and trimodal structures, the last being the phase-space signature of bistability (Lyu et al., 2023).

A further refinement incorporates the diamagnetic $$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$8 term required by the Thomas–Reich–Kuhn sum rule,

$$\dot\rho=-i[H_{\rm Rabi},\rho]+\kappa\mathcal D[a]\rho$$9

with $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$0, $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$1, and cavity loss $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$2. In this TRK-consistent open model, the isotropic line $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$3 obeys the no-go theorem and does not support a superradiant transition, whereas anisotropy $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$4 restores dissipative superradiant criticality. The steady-state phase diagram becomes asymmetric in $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$5, includes isolated bistable regions, and has fewer tricritical points than the $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$6-free case. Near ordinary critical lines the photon-number fluctuation exponent is $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$7, but when the two critical-line branches merge it becomes $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$8. The paper emphasizes that this anomalous scaling is controlled by the intersection of the two critical branches rather than by tricriticality itself (Wang et al., 14 Jan 2026).

4. Non-Markovian baths, bath-induced criticality, and nonequilibrium universality

A distinct branch of the subject does not begin from a Lindbladian at all. In a driven and dissipative Rabi model treated by stochastic Schrödinger equations, the cavity mode is coupled to a spin through the full Rabi interaction, driven by $$H_{\rm Rabi}=\omega_0 a^\dagger a+\frac{\Omega}{2}\sigma_z-\lambda(a+a^\dagger)\sigma_x$$9, and embedded in a combined environment with spectral density

$$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$0

Here the Ohmic bath acts non-Markovianly on the spin, while cavity leakage is introduced separately in the Markov approximation through an imaginary cavity frequency part $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$1. The resulting reduced-spin dynamics captures beyond-RWA effects such as the Bloch–Siegert shift, non-Markovian damping, and Kondo-type correlations, with a renormalized tunneling $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$2 and an effective detuning $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$3 (Henriet et al., 2014).

The non-Markovian open quantum Rabi model with an Ohmic bath attached to the oscillator coordinate changes the critical theory even more radically. In that model,

$$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$4

with

$$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$5

the bath induces a Berezinskii–Kosterlitz–Thouless transition rather than the isolated QRM criticality. The problem maps onto an effective spin-boson model with low-frequency dissipation parameter $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$6, a delocalized phase, and a localized phase where tunneling renormalizes to zero in the $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$7 limit. The order parameter is

$$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$8

the quoted critical coupling for the main parameter set is $$H_{\rm Rabi}= \frac{\Omega}{2}\sigma_y+\delta a^\dagger a+\eta \sigma_x(a+a^\dagger)$$9, and the relaxation time diverges with the BKT essential singularity

$$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$0

In driven passages through the transition, the freeze-out time is determined by a Lambert-$$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$1 expression rather than a power law, and the excitation energy at freeze-out follows $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$2. The paper also notes an internal inconsistency between the text and figure caption for the excitation-probability exponent, whereas the near-$$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$3 excitation-energy scaling is the stable conclusion (Pirozzi et al., 17 Mar 2026).

These non-Markovian results sharpen a basic conceptual divide. In Markovian open QRMs, dissipation typically modifies or broadens an otherwise coherent critical point. In the Ohmic non-Markovian OQRM, the bath defines the universality class itself. This distinction is not cosmetic: it changes both the equilibrium critical theory and the nonequilibrium Kibble–Zurek scaling structure.

5. Thermalization, parity-resolved relaxation, and analytical open-system solutions

Open-QRM dynamics depends strongly on how the bath couples to the system. In the two-photon-relaxation problem, the jump operator $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$4 preserves parity and therefore separates the dynamics into even and odd sectors. This has direct dynamical consequences. At $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$5, an even initial state relaxes to zero photons, whereas an odd initial state relaxes to one photon because pair loss cannot remove the last odd photon. For $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$6, even- and odd-parity initial states continue to display qualitatively distinct transient and steady-state behaviors, and in the odd sector the dressed mode connected to $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$7 remains nearly dark under two-photon loss up to about $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$8. The paper stresses that this parity-sensitive behavior is not an ultrastrong-coupling peculiarity but already appears at weak coupling (Malekakhlagh et al., 2018).

Thermalization studies in the dressed QRM eigenbasis show that the reservoir topology matters as much as the Hamiltonian. For the open QRM coupled to two individual heat baths, one through $$\dot{\rho}=-i[H_s,\rho]+2\kappa_{c2}\mathcal D[a^2]\rho$$9 and one through $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$0, the dressed-state global master equation thermalizes to the Gibbs state of $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$1 only if the two baths have the same temperature or if one of the two couplings vanishes. By contrast, when both subsystems couple to one common heat bath, thermalization always occurs. In the thermalized regime, the mixed-state entanglement between qubit and bosonic mode is quantified by the logarithmic negativity $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$2: it vanishes at high temperature, approaches the ground-state entanglement at low temperature, and reaches its largest values approximately around $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$3 in the resonant case (Liu et al., 2022).

A complementary analytical treatment considers a weakly coupled open QRM with a thermal bath acting only on the bosonic mode through a local Lindblad equation. Using the Bargmann-space holomorphic formalism, the open dispersive Jaynes–Cummings model is solved exactly and the open QRM is solved perturbatively up to second order in $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$4. The notable steady-state result is that, within this local-Lindblad setting,

$J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$5

whereas the Jaynes–Cummings value is $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$6. Thus the stationary qubit populations in the open QRM depend on both qubit and boson frequencies, even at weak coupling, because the counter-rotating terms change the selected zeroth-order steady state (Vaaranta et al., 2024).

Criticality can also be detected through reduced dynamics rather than through a steady state. If an auxiliary two-level atom is dispersively coupled to the cavity field, the QRM acts as an effective environment for the probe, and the probe coherence is governed by the Loschmidt echo

$J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$7

In the short-time limit,

$J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$8

with $J(\omega)=\alpha\omega\Theta(\omega_c-\omega)$9. Near the QRM critical point, the photon-number variance becomes sharply enhanced and the auxiliary atom undergoes sudden decoherence, so criticality appears directly as a loss of probe coherence (Hu et al., 2021).

6. Experimental realizations and closed-system baselines

Open-QRM phenomenology has moved from proposal to experiment. A circuit-QED realization engineered the effective Hamiltonian

$$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$0

in a rotating Floquet frame using a frequency-tunable Xmon qubit coupled to a lossy resonator, with dominant resonator decay described by $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$1. The reported operating regime had $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$2, $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$3 to $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$4, $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$5, $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$6, and $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$7. At $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$8, steady-state photon-number distributions showed clear population transfer to higher photon numbers as the coupling increased: $$J(\omega)=\pi g^2\delta(\omega-\omega_0)+2\pi\alpha\omega e^{-\omega/\omega_c}$$9 rose from $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$0 at $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$1 to $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$2 at $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$3. The mean photon number increased and saturated without the persistent oscillations characteristic of a closed QRM, directly demonstrating a finite nonequilibrium steady state generated by competition between coherent Rabi dynamics and photon leakage (Ning et al., 19 Jun 2025).

Trapped ions remain the most flexible proposed platform for critical open-QRM studies. One two-ion scheme identifies $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$4, $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$5, and $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$6, while sympathetic cooling of the common mode supplies the oscillator damping required for the open-QRM master equation. The anisotropic open-QRM multicritical proposal uses a mixed-species $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$7 pair, with independently tunable red and blue Raman sidebands generating $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$8 and $\dot\rho=-i[H,\rho]+\kappa\mathcal D[a]\rho$9, and sympathetic cooling implementing $Z_2$00 (Hwang et al., 2017, Lyu et al., 2023).

Interpretation of open-QRM experiments and steady states remains anchored to closed-system reference results. In the closed single-mode QRM, fidelity-susceptibility scaling at the critical point $Z_2$01 in the $Z_2$02 limit yields $Z_2$03 and $Z_2$04. The closed ground-state wavefunction also admits a complementary description in terms of quadpolarons and bipolarons, with a crossover scale $Z_2$05 and a hidden strong-coupling scaling $Z_2$06. For many-body extensions, the closed one-dimensional Rabi-Hubbard model shows that counter-rotating terms remove generic Mott phases, leave a $Z_2$07 coherent transition, and drive a photon-number instability as $Z_2$08. This suggests that open cavity and circuit-QED arrays inherit a parity-breaking coherent backbone that dissipation regularizes and reshapes rather than creating from scratch (Wei et al., 2017, Ying et al., 2015, Flottat et al., 2016).

Taken together, these results define the open quantum Rabi model as a family of symmetry-sensitive, strongly nonperturbative open light–matter problems rather than as a single canonical master equation. Markovian one-photon loss, parity-preserving two-photon relaxation, global thermal reservoirs, and structured non-Markovian baths produce sharply different steady states, relaxation channels, and universality classes. What remains common is that counter-rotating coupling is indispensable, parity is usually the central organizing symmetry, and finite-component systems can nevertheless display genuine criticality, bistability, and universal scaling once the relevant thermodynamic or bath-controlled limit is identified.