---
title: Displaced Rotating-Wave Model
url: https://www.emergentmind.com/topics/displaced-rotating-wave-model
type: topic
---

# Displaced Rotating-Wave Model

Searching arXiv for the cited papers and related usage of displaced rotating-wave methods.
The displaced rotating-wave model, as developed in displaced rotating-wave approximation (DRWA/GRWA) constructions, introduces a displacement before an RWA-type truncation so that counter-rotating effects are absorbed into a transformed basis rather than discarded ab initio. In the cited literature, this strategy appears in the isotropic and anisotropic quantum Rabi model, in inhomogeneously coupled multi-qubit Tavis–Cummings-type systems beyond the RWA, and in strongly detuned driven oscillators, where it yields Jaynes–Cummings-like effective Hamiltonians, block-tridiagonal parity-resolved problems, or drive-adapted residual fluctuation theories [1607.08459][1403.5893][2202.13172]. This suggests that the term is best understood as a method class centered on spin- or drive-conditioned displacement, with ordinary RWA recovered only as a limiting reduction.

## 1. Conceptual definition and operator structure

In the single-qubit Rabi setting, the displaced rotating-wave construction begins from a spin-dependent displacement,
\[
U_{\text{disp}}=\exp\!\left[\beta\,\sigma_x\,(a^\dagger-a)\right],
\]
with dimensionless displacement parameter \(\beta\). The transformed Hamiltonian contains renormalized longitudinal and transverse qubit terms generated by hyperbolic functions of \(a^\dagger-a\), and the practical approximation consists of retaining the energy-conserving contributions of those terms in the displaced basis [1607.08459].

In the multi-qubit beyond-RWA Tavis–Cummings setting, an equivalent displacement is conditioned on the \(\sigma_j^z\) configuration after a spin rotation,
\[
U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].
\]
For each spin configuration \(s=(s_1,\dots,s_N)\), the cavity is displaced by
\[
\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,
\]
and the non-orthogonality of the resulting displaced Fock states generates controlled inter-block couplings in a truncated analytic spectrum calculation [1403.5893].

In the strongly detuned driven-oscillator setting, the same logic is reformulated in a drive-adapted operator basis. Ladder operators \(b,b^\dagger\) are defined with the drive frequency \(\omega\), not the bare oscillator frequency \(\omega_0\), and a displacement \(D(\beta)\) is then chosen to eliminate linear terms in the \(\omega\)-rotating frame. The resulting description is explicitly presented as the spirit of a “Displaced Rotating-Wave Model” for off-resonant driving [2202.13172].

| Setting | Core transformation | Reduced description |
|---|---|---|
| Isotropic/anisotropic Rabi | \(U_{\text{disp}}=\exp[\beta\sigma_x(a^\dagger-a)]\) | JC-like effective Hamiltonian |
| GSRWA extension | \(U=S(\lambda)e^{\beta\sigma_x(a^\dagger-a)}\) | Displaced-squeezed JC-like blocks |
| Multi-qubit beyond RWA | \(U=\exp[(a^\dagger-a)\sum_j(g_j/\omega_c)\sigma_j^z]\) | Parity-resolved displaced blocks |
| Driven oscillator | Drive-adapted \(b\)-basis plus \(D(\beta)\) | Residual fluctuation theory around off-resonant steady state |

A recurrent point is that displacement does not merely reparametrize the oscillator; it changes which terms are regarded as perturbative. That distinction is central to all three formulations.

## 2. Single-qubit DRWA for isotropic and anisotropic Rabi models

The isotropic Rabi Hamiltonian is
\[
H_{\text{Rabi}}=\omega\,a^\dagger a+\frac{\Delta}{2}\,\sigma_z+g\,\sigma_x\,(a+a^\dagger),
\]
while the anisotropic Rabi Hamiltonian is
\[
H_{\text{aniso}}=\frac{\Delta}{2}\,\sigma_z+\omega\,a^\dagger a+g_r(\sigma_+a+\sigma_-a^\dagger)+g_{cr}(\sigma_+a^\dagger+\sigma_-a).
\]
The same model is reparametrized by
\[
\alpha=\frac{g_1+g_2}{2},\qquad \gamma=\frac{g_2-g_1}{2},
\]
so that
\[
H=\frac{\Delta}{2}\,\sigma_z+\omega\,a^\dagger a+\alpha(a^\dagger+a)\sigma_x+i\gamma\,\sigma_y(a^\dagger-a).
\]
The isotropic limit is \(\gamma=0\), equivalently \(g_1=g_2\) [1607.08459].

After the displacement transform, the Hamiltonian becomes
\[
\begin{aligned}
H_1={}&\omega\,a^\dagger a+\omega\beta^2-2\beta\omega\alpha+(\alpha-\omega\beta)(a^\dagger+a)\sigma_x \\
&+\frac{\Delta}{2}\Big\{\sigma_z\cosh\!\big[2\beta(a^\dagger-a)\big]-i\sigma_y\sinh\!\big[2\beta(a^\dagger-a)\big]\Big\} \\
&+\gamma(a^\dagger-a)\Big\{-\sigma_z\sinh\!\big[2\beta(a^\dagger-a)\big]+i\sigma_y\cosh\!\big[2\beta(a^\dagger-a)\big]\Big\}.
\end{aligned}
\]
This explicitly shows that the original counter-rotating structure induces both longitudinal \(\sigma_z\) dressing and transverse \(\sigma_y\) dressing, while the choice \(\beta\simeq \alpha/\omega\) partially cancels the linear \(\sigma_x\) term [1607.08459].

The DRWA/GRWA step is then to discard highly off-resonant terms generated by the hyperbolic functions and retain only energy-conserving couplings. In that approximation the Hamiltonian acquires a JC-like form,
\[
H_{\text{eff}}^{\text{GRWA}}\simeq \omega\,a^\dagger a+\frac{\Delta_{\text{eff}}}{2}\sigma_z+g_{\text{eff}}(\sigma_+a+\sigma_-a^\dagger)+E_0.
\]
The effective qubit splitting and coupling are renormalized by the displacement. In practice, this description accurately captures spectra at moderate \(g/\omega\), but the same source states that it gives incorrect ground-state energy at ultrastrong coupling \(g/\omega\gtrsim 0.5\), particularly for large positive detuning \(\Delta/\omega\gtrsim 1\), because virtual excitations induced by counter-rotating terms are not well approximated by a displacement alone [1607.08459].

A frequent misconception is that the displaced scheme is just the ordinary RWA in shifted coordinates. The transformed Hamiltonian above shows otherwise: the hyperbolic operator structure carries nontrivial qubit dressing, and the truncation is performed only after that dressing has been introduced.

## 3. Generalized squeezing extension of the displaced model

The generalized squeezing rotating-wave approximation (GSRWA) augments displacement by a squeezing transform,
\[
U=V\,U_{\text{disp}},\qquad U_{\text{disp}}=e^{\beta\sigma_x(a^\dagger-a)},\qquad V=S(\lambda)=e^{\lambda(a^2-a^{\dagger 2})}.
\]
The Bogoliubov identities are
\[
V a V^\dagger=a\cosh(2\lambda)+a^\dagger\sinh(2\lambda),\qquad
V a^\dagger V^\dagger=a\sinh(2\lambda)+a^\dagger\cosh(2\lambda).
\]
Its stated motivation is that counter-rotating terms produce virtual excitations and two-photon processes \(a^2\) and \(a^{\dagger 2}\) that cannot be captured by displacement alone [1607.08459].

After squeezing, and after neglecting explicit \(a^2\) and \(a^{\dagger 2}\) terms, the transformed Hamiltonian \(H'=V H_1 V^\dagger\equiv H_0'+H_1'\) is written in terms of
\[
\eta_0=\omega\sinh^2(2\lambda)+\omega\beta^2-2\beta\alpha,\quad
\eta_1=\omega\big(\cosh^2(2\lambda)+\sinh^2(2\lambda)\big),
\]
\[
\eta_2=\cosh(2\lambda)+\sinh(2\lambda),\quad
\eta=\omega\big(\cosh(2\lambda)-\sinh(2\lambda)\big).
\]
The squeezing therefore renormalizes the oscillator frequency, the linear coupling, and the arguments of the hyperbolic functions [1607.08459].

The RWA-like reduction is performed by retaining number-conserving terms in the oscillator basis,
\[
\cosh\!\big[2\beta\eta(a^\dagger-a)\big]\approx G(a^\dagger a),\qquad
\sinh\!\big[2\beta\eta(a^\dagger-a)\big]\approx R(a^\dagger a)a^\dagger-aR(a^\dagger a),
\]
\[
(a^\dagger-a)\sinh\!\big[2\beta\eta(a^\dagger-a)\big]\approx F(a^\dagger a),\qquad
(a^\dagger-a)\cosh\!\big[2\beta\eta(a^\dagger-a)\big]\approx T(a^\dagger a)a^\dagger-aT(a^\dagger a).
\]
The corresponding matrix elements are given in terms of associated Laguerre polynomials,
\[
G_{n,n}=\chi\,L_n\!\big(4\beta^2\eta^2\big),\qquad
R_{n+1,n}=\frac{2\beta\eta\,\chi}{n+1}\,L_n^1\!\big(4\beta^2\eta^2\big),
\]
with \(\chi=\exp(-2\beta^2\eta^2)\), and with \(F_{n,n}\), \(T_{n+1,n}\), and \(G_{n+2,n}\) defined accordingly. Higher-order multiphoton terms \(O(\beta^2\eta^2)\) and \(O(\beta^3\eta^3)\) are neglected within this approximation [1607.08459].

The effective GSRWA Hamiltonian is then
\[
\begin{aligned}
H_{\text{GSRWA}}={}&\eta_0+\eta_1 a^\dagger a+\sigma_z\left[\frac{\Delta}{2}G(a^\dagger a)-\gamma\eta F(a^\dagger a)\right] \\
&+\left\{\eta_2(\alpha-\omega\beta)+\frac{\Delta}{2}R(a^\dagger a)-\gamma\eta T(a^\dagger a)\right\}a^\dagger\sigma_-+\text{h.c.},
\end{aligned}
\]
or, equivalently,
\[
H_{\text{GSRWA}}\simeq \omega_{\text{eff}}\,a^\dagger a+\frac{\Delta_{\text{eff}}(n)}{2}\sigma_z+g_{\text{eff}}(n)(\sigma_+a+\sigma_-a^\dagger)+E_0,
\]
with
\[
\omega_{\text{eff}}\equiv \eta_1,\quad
\Delta_{\text{eff}}(n)\equiv \Delta G_{n,n}-2\gamma\eta F_{n,n},\quad
g_{\text{eff}}(n)\equiv \eta_2(\alpha-\omega\beta)+\frac{\Delta}{2}R_{n+1,n}-\gamma\eta T_{n+1,n}.
\]
Each excitation manifold \(\{|n,+z\rangle,\ |n+1,-z\rangle\}\) reduces to a \(2\times 2\) block that can be diagonalized analytically, as in the standard RWA [1607.08459].

The connection back to DRWA is explicit. When \(\lambda\to 0\),
\[
H_{\text{GSRWA}}(\beta,\lambda)\xrightarrow{\lambda\to 0}H_{\text{DRWA/GRWA}}(\beta),
\]
with \(\eta_0\to \omega\beta^2-2\beta\alpha\), \(\eta_1\to\omega\), \(\eta_2\to 1\), and \(\eta\to\omega\). This is the precise sense in which GSRWA is a generalization rather than a separate approximation scheme [1607.08459].

## 4. Variational parameters, observables, and cat-like ground states

For the isotropic case \((\gamma=0)\), the effective ground state is \(|G\rangle=|0\rangle\otimes|-z\rangle\), and the GSRWA ground-state energy is
\[
E_0^{\text{GSRWA}}(\beta,\lambda)=\omega\sinh^2(2\lambda)+\omega\beta^2-2\beta\alpha-\frac{\Delta}{2}\exp\!\big[-2\beta^2 e^{-4\lambda}\big].
\]
The stationarity conditions are
\[
\frac{\partial E_0}{\partial \beta}=0,\qquad \frac{\partial E_0}{\partial \lambda}=0,
\]
with the explicit equations given in the source. For small \(\beta,\lambda\lesssim 1\), the approximate solutions are
\[
\beta\simeq \frac{\alpha}{\omega+\Delta e^{-4\lambda}\exp\!\Big[-\frac{2\alpha^2}{(\omega+\Delta)^2}e^{-4\lambda}\Big]},
\qquad
\lambda\simeq \frac{\Delta}{2\omega}\,\frac{\alpha^2}{(\omega+\Delta)^2}.
\]
These formulas show that \(\lambda\) grows with \(\Delta\) and \(\alpha\), while \(\beta\) is reduced by squeezing through the factor \(e^{-4\lambda}\). In the deep-strong-coupling limit \(g/\omega\gg 1\), the optimum approaches \(\beta\simeq \alpha/\omega\) and \(\lambda\simeq 0\), so GSRWA collapses to GRWA [1607.08459].

For the anisotropic case \((\gamma\neq 0)\), the ground-state energy becomes
\[
E_0^{\text{GSRWA}}(\beta,\lambda)=\omega\sinh^2(2\lambda)+\omega\beta^2-2\beta\alpha-\Big(\frac{\Delta}{2}+2\gamma\beta e^{-4\lambda}\Big)\exp\!\big[-2\beta^2 e^{-4\lambda}\big],
\]
and \(\beta,\lambda\) are obtained numerically from the stationarity equations. The same source states that \(\lambda\) increases with \(\Delta/\omega\) and \(g/\omega\) up to ultrastrong coupling, then decreases in deep-strong coupling, while \(\beta\) grows roughly linearly with \(g/\omega\) in deep-strong coupling. For \(\gamma<0\), a level crossing occurs with the first excited manifold, and beyond the crossing the variational minimization must be performed on the lower eigenvalue of that excited block rather than on \(E_0^{\text{GSRWA}}\) itself [1607.08459].

For the isotropic ground state, the comparison baselines are
\[
E_0^{\text{GVM}}(\beta)=\omega\beta^2-2\beta\alpha-\frac{\Delta}{2}e^{-2\beta^2},
\qquad
E_0^{\text{GRWA}}=-\frac{\alpha^2}{\omega}-\frac{\Delta}{2}e^{-2\alpha^2/\omega^2}.
\]
The stated conclusion is that the squeezing terms substantially lower the energy and match numerics up to \(g/\omega\approx 1\), notably at positive detuning \(\Delta/\omega\gtrsim 1\), where GRWA and GVM deviate [1607.08459].

The mean photon number in the isotropic case is given analytically as
\[
\langle n\rangle^{\text{GSRWA}}=\omega\sinh^2(2\lambda)+\omega\beta^2.
\]
The baseline expressions are
\[
\langle n\rangle^{\text{GVM}}\simeq \omega\beta^2\simeq \frac{\omega\alpha^2}{\big[\omega+\Delta e^{-2\alpha^2/(\omega+\Delta)^2}\big]^2},
\qquad
\langle n\rangle^{\text{GRWA}}=\omega\beta^2=\alpha^2/\omega.
\]
The source emphasizes that the squeezing contribution corrects the \(\Delta\)-independence of GRWA and improves agreement with numerics across detuning and coupling [1607.08459].

The lab-frame ground state is a Schrödinger-cat-like entangled state,
\[
|\Psi_0\rangle=\frac{1}{\sqrt{2}}\Big[e^{-\beta(a^\dagger-a)}|0\rangle_s\otimes|+x\rangle
+e^{+\beta(a^\dagger-a)}|0\rangle_s\otimes|-x\rangle\Big],
\]
where \(|0\rangle_s=V^\dagger|0\rangle\) is the squeezed vacuum. Equivalently,
\[
|\Psi_0\rangle\approx \frac{1}{\sqrt{2}}\Big(|-\beta,\lambda\rangle\otimes|+x\rangle+|\beta,\lambda\rangle\otimes|-x\rangle\Big),
\]
with \(|\alpha,\lambda\rangle\equiv D(\alpha)S(\lambda)|0\rangle\). The effective displacement seen by the qubit is
\[
\beta'=\beta\,\eta_2=\beta\big(\cosh(2\lambda)+\sinh(2\lambda)\big),
\]
so that the overlap of the two oscillator branches is \(e^{-2{\beta'}^2}\), and the reduced qubit density matrix has eigenvalues
\[
p_\pm=\frac{1\pm e^{-2{\beta'}^2}}{2},
\]
leading to entanglement entropy
\[
S=-p_+\log_2 p_+-p_-\log_2 p_-.
\]
The paper’s specific claim is that this displaced-squeezed cat provides a better description than a purely displaced cat in the ultrastrong regime and at large \(\Delta\) [1607.08459].

## 5. Multi-qubit displaced-basis extensions beyond the RWA

For \(N\) inhomogeneously coupled qubits, the beyond-RWA Tavis–Cummings-type Hamiltonian is analyzed in a rotated spin basis,
\[
H=\omega_c\,a^\dagger a+\sum_{j=1}^{N}\left(-\frac{\omega_j}{2}\,\hat{\sigma}_j^x+g_j\,(a^\dagger+a)\,\hat{\sigma}_j^z\right),
\]
with \(\hbar=1\). The model conserves a global \(Z_2\) parity,
\[
\Pi=\left(\prod_{j=1}^{N}\hat{\sigma}_j^x\right)e^{i\pi a^\dagger a},\qquad [H,\Pi]=0,
\]
and the Hilbert space decomposes into even and odd parity sectors [1403.5893].

For each spin configuration, displaced oscillator states \(|n\rangle_{A(\beta)}\) are defined by
\[
A(\beta)=\hat{D}^\dagger(\beta)a\hat{D}(\beta)=a+\beta,
\qquad
\hat{D}(\beta)=e^{\beta(a^\dagger-a)}.
\]
They are orthonormal within a given displacement but non-orthogonal across different displacements; the explicit overlap formula is given in the source. This non-orthogonality is not a technical nuisance but the mechanism that generates effective inter-block couplings in the truncated theory [1403.5893].

For \(N=2\), the four spin product states generate four displacements,
\[
\beta_1=\frac{g_1+g_2}{\omega_c},\quad
\beta_2=\frac{g_2-g_1}{\omega_c},\quad
\beta_3=-\beta_2,\quad
\beta_4=-\beta_1.
\]
Parity symmetry reduces the dynamical equations to two coupled infinite systems with diagonal terms \(\varepsilon_{im}=\omega_c(m-\beta_i^2)\) and overlap-mediated off-diagonal terms \(\Omega_{mn}^\kappa\) and \(W_{mn}^\kappa\), related by \(\Omega_{mn}^\kappa=W_{nm}^\kappa\). In general \(N\), one obtains \(2^{N-1}\) coupled equations of the same type, with a primitive block size \(2^{N-1}\times 2^{N-1}\) [1403.5893].

The truncation hierarchy is explicit. In zeroth order, inter-block couplings between different displaced oscillator numbers are neglected, and for \(N=2\) one obtains a \(2\times 2\) determinant with eigenvalues
\[
E_m^{\kappa\pm}=m\omega_c-\frac{\omega_c}{2}(\beta_1^2+\beta_2^2)\pm \theta_m^\kappa,
\qquad
\theta_m^\kappa=\sqrt{(\Omega_m^\kappa)^2+\frac{\omega_c^2}{4}(\beta_1^2-\beta_2^2)^2}.
\]
In first order, nearest-block couplings between \(m\) and \(m+1\) are retained, yielding a \(4\times 4\) determinant and a quartic equation. The analytic roots are given in the source in Ferrari form, together with the rule for discarding the pseudo-solutions that arise from double counting neighboring blocks [1403.5893].

The two-qubit model also exhibits exact and quasi-exact structures. For identical qubits and homogeneous couplings \((g_1=g_2\equiv g,\ \omega_1=\omega_2)\), the singlet states
\[
|\psi_{\text{singlet}(m)}\rangle=\frac{1}{\sqrt{2}}\big(|10\rangle-|01\rangle\big)\otimes|m\rangle
\]
are exact eigenstates with
\[
E_m=m\omega_c.
\]
For homogeneous coupling with symmetric or asymmetric detuning, there is a constant exact eigenvalue
\[
E=\omega_c
\]
under the conditions \(\omega_1+\omega_2=2\omega_c\) or \(\omega_1-\omega_2=2\omega_c\), with the corresponding exact eigenstates given explicitly in the source [1403.5893].

These exact lines are linked to the paper’s discussion of hidden symmetry and integrability. Using the fidelity
\[
F(\lambda,\delta)=|\langle\psi(\lambda)|\psi(\lambda+\delta)\rangle|^2,
\]
the authors identify nontrivial same-parity level crossings. The stated conclusion is that the homogeneous coupled two-qubit model with \(\omega_1=\omega_2\) or \(\omega_1\pm\omega_2=2\omega_c\) is integrable in Braak’s sense on those parameter manifolds [1403.5893].

The displaced-basis formalism also yields analytic dynamics. For an initial state \(|10\rangle|m\rangle_{A_2}\), the population probability \(P_{10}(m,t)\) is expressed as a superposition of oscillatory components determined by \(\theta_m^\kappa\), and homogeneous coupling reduces that expression to two-frequency combinations. With an initial coherent field \(|z\rangle\), the total probability
\[
P_{10}(z,t)=\sum_{m=0}^\infty p(m)\,P_{10}(m,t),
\qquad
p(m)=e^{-|z|^2}\frac{|z|^{2m}}{m!},
\]
acquires analytic collapse–revival envelopes. The same formalism produces a closed expression for the single-qubit inversion and an X-form reduced density matrix for Bell-state initial conditions, from which the concurrence
\[
C(t)=\left|\sum_{m=0}^\infty\sum_{\kappa=\pm}\frac{p(m)\big(1+\kappa(-1)^m\big)}{2}\,q_{2m}^{\kappa+}\right|
\]
is obtained [1403.5893].

In the “displaced rotating-wave model” perspective used by that work, the main point is that displacement enables tractable analytics beyond the RWA in inhomogeneous multi-qubit systems, while parity symmetry controls the allowed block structure and the hierarchy of truncations.

## 6. Off-resonant driven oscillators, validity regimes, and limitations

For the driven harmonic oscillator,
\[
H(t)=\frac{p^2}{2m}+\frac{1}{2}m\omega_0^2x^2-F_0\cos(\omega t)\,x,
\]
the exact steady-state amplitude is
\[
X=\frac{F_0}{m(\omega_0^2-\omega^2)}.
\]
If one works in the standard bare-\(\omega_0\) ladder-operator basis and applies the ordinary RWA in the \(\omega\)-rotating frame, the predicted amplitude becomes
\[
x_{\rm RWA}(t)=-\frac{F_0/m}{2\omega_0\Delta}\cos(\omega t),
\qquad \Delta=\omega-\omega_0,
\]
so that
\[
\frac{x(t)}{x_{\rm RWA}(t)}=\frac{2\omega_0}{\omega_0+\omega}.
\]
The cited paper states that the standard quantum description, using creation and annihilation operators built from the natural frequency \(\omega_0\), necessarily leads to incorrect results when combined with the RWA for off-resonant driving [2202.13172].

The proposed fix is to define ladder operators with the drive frequency \(\omega\),
\[
\hat{x}=\sqrt{\frac{\hbar}{2m\omega}}\,(b+b^\dagger),
\qquad
\hat{p}=i\sqrt{\frac{\hbar m\omega}{2}}\,(b^\dagger-b),
\]
so that in this basis the mismatch \(\omega_0\neq \omega\) appears as squeezing terms \(b^2+(b^\dagger)^2\). In the \(\omega\)-rotating frame, the stationary solution is
\[
\beta\equiv \langle\tilde b\rangle=\frac{2F_b\omega}{\omega_0^2-\omega^2},
\]
and it reproduces the exact response amplitude. If one then performs a displacement by \(D(\beta)\), the linear terms vanish and the residual dynamics is governed by detuning, squeezing, and nonlinear corrections. This is explicitly identified as the essence of a displaced rotating-wave model for off-resonant driving [2202.13172].

The same method is extended to the driven Duffing oscillator,
\[
H_D=\frac{p^2}{2m}+\frac{1}{2}m\omega_0^2x^2+\frac{\alpha}{4}x^4-F_0\cos(\omega t)\,x.
\]
In the drive-adapted \(b\)-basis, the stationary mean-field equation is
\[
0=\frac{\omega_0^2-\omega^2}{2\omega}r_b-\frac{F_b}{\hbar}+\frac{3\alpha\hbar}{4m^2\omega^2}(r_b+r_b^3)
\equiv c_b r_b^3+(A_b+c_b)r_b-\frac{F_b}{\hbar},
\]
with
\[
A_b=\frac{\omega_0^2-\omega^2}{2\omega},\qquad c_b=\frac{3\alpha\hbar}{4m^2\omega^2}.
\]
The discriminant
\[
D_b=\left(\frac{q_b}{2}\right)^2+\left(\frac{p_b}{3}\right)^3
\]
controls bistability, and the source states that the phase boundary in the \((\omega,F_0)\) plane is tracked to within the numerical precision \((\sim 10^{-5}\) relative discrepancy at small \(F_0)\) in the \(b\)-basis, whereas the bare RWA shows visible shifts [2202.13172].

Across the cited literature, the validity regimes are sharply differentiated rather than uniform. For the Rabi problem, displacement only is recommended when \(g/\omega\ll 1\) or in the deep-strong limit \(g/\omega\gg 1\), where squeezing is negligible, while GSRWA is recommended for ultrastrong coupling \(0.3\lesssim g/\omega\lesssim 1\) and/or large positive detuning \(\Delta/\omega\gtrsim 1\) [1607.08459]. For inhomogeneous multi-qubit systems, the displaced-basis truncations are accurate when \(\omega_j\ll \omega_c\) and \(g_j/\omega_c\sim 0.1\), and can be extended into deeper coupling by including first-order and higher-order inter-block couplings [1403.5893]. For the driven oscillator, the drive-adapted construction is exact for the linear problem and remains perturbatively controlled for residual fluctuations when the dropped oscillatory terms are small relative to \(2\omega\) [2202.13172].

The main limitations are equally explicit. GSRWA neglects explicit \(a^2\) and \(a^{\dagger 2}\) terms after squeezing and truncates higher-order multiphoton processes through the Laguerre-polynomial reduction; excited-state spectra beyond the lowest blocks require care in regimes with strong multiphoton dressing [1607.08459]. The multi-qubit displaced-basis approach relies on truncation order and on overlap-mediated couplings between non-orthogonal displaced states [1403.5893]. The drive-adapted Duffing treatment remains an RWA and mean-field theory, so higher harmonics still generate small deviations in the deeply nonlinear regime [2202.13172].

A common misconception is that all failures of the RWA should be attributed directly to omitted counter-rotating terms. The driven-oscillator analysis instead shows that off-resonant errors can arise from using ladder operators tied to the wrong frequency, while the Rabi-model analysis shows that even after displacement, virtual excitations and two-photon processes may require squeezing for quantitative accuracy. Taken together, these results define the displaced rotating-wave model not as a single approximation of fixed scope, but as a hierarchy in which displacement selects the appropriate reference frame and subsequent RWA-type truncations are judged by how completely that frame captures virtual, off-resonant, or configuration-dependent dressing.

Source: https://www.emergentmind.com/topics/displaced-rotating-wave-model