Displaced Rotating-Wave Model

• Displaced Rotating-Wave Model is a quantum method that applies spin- or drive-conditioned displacements to include counter-rotating effects within an effective energy-conserving framework.
• It underpins techniques such as the generalized squeezing RWA and multi-qubit displaced-basis approaches to produce Jaynes–Cummings-like effective Hamiltonians and tractable spectral analyses.
• The model enhances accuracy in regimes like ultrastrong coupling and off-resonant driving by optimizing variational parameters and absorbing virtual excitation contributions.

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 (Zhang, 2016, Mao et al., 2014, Košata et al., 2022). 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 (Zhang, 2016).

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 (Mao et al., 2014).

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 $\beta$0 is then chosen to eliminate linear terms in the $\beta$1-rotating frame. The resulting description is explicitly presented as the spirit of a “Displaced Rotating-Wave Model” for off-resonant driving (Košata et al., 2022).

Setting	Core transformation	Reduced description
Isotropic/anisotropic Rabi	$\beta$2	JC-like effective Hamiltonian
GSRWA extension	$\beta$3	Displaced-squeezed JC-like blocks
Multi-qubit beyond RWA	$\beta$4	Parity-resolved displaced blocks
Driven oscillator	Drive-adapted $\beta$5-basis plus $\beta$6	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

$\beta$7

while the anisotropic Rabi Hamiltonian is

$\beta$8

The same model is reparametrized by

$\beta$9

so that

$a^\dagger-a$0

The isotropic limit is $a^\dagger-a$1, equivalently $a^\dagger-a$2 (Zhang, 2016).

After the displacement transform, the Hamiltonian becomes

$a^\dagger-a$3

This explicitly shows that the original counter-rotating structure induces both longitudinal $a^\dagger-a$4 dressing and transverse $a^\dagger-a$5 dressing, while the choice $a^\dagger-a$6 partially cancels the linear $a^\dagger-a$7 term (Zhang, 2016).

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,

$a^\dagger-a$8

The effective qubit splitting and coupling are renormalized by the displacement. In practice, this description accurately captures spectra at moderate $a^\dagger-a$9, but the same source states that it gives incorrect ground-state energy at ultrastrong coupling $\sigma_j^z$0, particularly for large positive detuning $\sigma_j^z$1, because virtual excitations induced by counter-rotating terms are not well approximated by a displacement alone (Zhang, 2016).

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,

$\sigma_j^z$2

The Bogoliubov identities are

$\sigma_j^z$3

Its stated motivation is that counter-rotating terms produce virtual excitations and two-photon processes $\sigma_j^z$4 and $\sigma_j^z$5 that cannot be captured by displacement alone (Zhang, 2016).

After squeezing, and after neglecting explicit $\sigma_j^z$6 and $\sigma_j^z$7 terms, the transformed Hamiltonian $\sigma_j^z$8 is written in terms of

$\sigma_j^z$9

$$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$0

The squeezing therefore renormalizes the oscillator frequency, the linear coupling, and the arguments of the hyperbolic functions (Zhang, 2016).

The RWA-like reduction is performed by retaining number-conserving terms in the oscillator basis,

$$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$1

$$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$2

The corresponding matrix elements are given in terms of associated Laguerre polynomials,

$$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$3

with $$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$4, and with $$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$5, $$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$6, and $$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$7 defined accordingly. Higher-order multiphoton terms $$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$8 and $$U=\exp\!\Big[(a^\dagger-a)\sum_{j=1}^{N}\frac{g_j}{\omega_c}\,\sigma_j^z\Big].$$9 are neglected within this approximation (Zhang, 2016).

The effective GSRWA Hamiltonian is then

$s=(s_1,\dots,s_N)$0

or, equivalently,

$s=(s_1,\dots,s_N)$1

with

$s=(s_1,\dots,s_N)$2

Each excitation manifold $s=(s_1,\dots,s_N)$3 reduces to a $s=(s_1,\dots,s_N)$4 block that can be diagonalized analytically, as in the standard RWA (Zhang, 2016).

The connection back to DRWA is explicit. When $s=(s_1,\dots,s_N)$5,

$s=(s_1,\dots,s_N)$6

with $s=(s_1,\dots,s_N)$7, $s=(s_1,\dots,s_N)$8, $s=(s_1,\dots,s_N)$9, and $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$0. This is the precise sense in which GSRWA is a generalization rather than a separate approximation scheme (Zhang, 2016).

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

For the isotropic case $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$1, the effective ground state is $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$2, and the GSRWA ground-state energy is

$\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$3

The stationarity conditions are

$\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$4

with the explicit equations given in the source. For small $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$5, the approximate solutions are

$\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$6

These formulas show that $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$7 grows with $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$8 and $\beta(s)=\frac{1}{\omega_c}\sum_{j=1}^N s_j\,g_j,$9, while $b,b^\dagger$0 is reduced by squeezing through the factor $b,b^\dagger$1. In the deep-strong-coupling limit $b,b^\dagger$2, the optimum approaches $b,b^\dagger$3 and $b,b^\dagger$4, so GSRWA collapses to GRWA (Zhang, 2016).

For the anisotropic case $b,b^\dagger$5, the ground-state energy becomes

$b,b^\dagger$6

and $b,b^\dagger$7 are obtained numerically from the stationarity equations. The same source states that $b,b^\dagger$8 increases with $b,b^\dagger$9 and $\omega$0 up to ultrastrong coupling, then decreases in deep-strong coupling, while $\omega$1 grows roughly linearly with $\omega$2 in deep-strong coupling. For $\omega$3, 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 $\omega$4 itself (Zhang, 2016).

For the isotropic ground state, the comparison baselines are

$\omega$5

The stated conclusion is that the squeezing terms substantially lower the energy and match numerics up to $\omega$6, notably at positive detuning $\omega$7, where GRWA and GVM deviate (Zhang, 2016).

The mean photon number in the isotropic case is given analytically as

$\omega$8

The baseline expressions are

$\omega$9

The source emphasizes that the squeezing contribution corrects the $\omega_0$0-independence of GRWA and improves agreement with numerics across detuning and coupling (Zhang, 2016).

The lab-frame ground state is a Schrödinger-cat-like entangled state,

$\omega_0$1

where $\omega_0$2 is the squeezed vacuum. Equivalently,

$\omega_0$3

with $\omega_0$4. The effective displacement seen by the qubit is

$\omega_0$5

so that the overlap of the two oscillator branches is $\omega_0$6, and the reduced qubit density matrix has eigenvalues

$\omega_0$7

leading to entanglement entropy

$\omega_0$8

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 $\omega_0$9 (Zhang, 2016).

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

For $\beta$00 inhomogeneously coupled qubits, the beyond-RWA Tavis–Cummings-type Hamiltonian is analyzed in a rotated spin basis,

$\beta$01

with $\beta$02. The model conserves a global $\beta$03 parity,

$\beta$04

and the Hilbert space decomposes into even and odd parity sectors (Mao et al., 2014).

For each spin configuration, displaced oscillator states $\beta$05 are defined by

$\beta$06

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 (Mao et al., 2014).

For $\beta$07, the four spin product states generate four displacements,

$\beta$08

Parity symmetry reduces the dynamical equations to two coupled infinite systems with diagonal terms $\beta$09 and overlap-mediated off-diagonal terms $\beta$10 and $\beta$11, related by $\beta$12. In general $\beta$13, one obtains $\beta$14 coupled equations of the same type, with a primitive block size $\beta$15 (Mao et al., 2014).

The truncation hierarchy is explicit. In zeroth order, inter-block couplings between different displaced oscillator numbers are neglected, and for $\beta$16 one obtains a $\beta$17 determinant with eigenvalues

$\beta$18

In first order, nearest-block couplings between $\beta$19 and $\beta$20 are retained, yielding a $\beta$21 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 (Mao et al., 2014).

The two-qubit model also exhibits exact and quasi-exact structures. For identical qubits and homogeneous couplings $\beta$22, the singlet states

$\beta$23

are exact eigenstates with

$\beta$24

For homogeneous coupling with symmetric or asymmetric detuning, there is a constant exact eigenvalue

$\beta$25

under the conditions $\beta$26 or $\beta$27, with the corresponding exact eigenstates given explicitly in the source (Mao et al., 2014).

These exact lines are linked to the paper’s discussion of hidden symmetry and integrability. Using the fidelity

$\beta$28

the authors identify nontrivial same-parity level crossings. The stated conclusion is that the homogeneous coupled two-qubit model with $\beta$29 or $\beta$30 is integrable in Braak’s sense on those parameter manifolds (Mao et al., 2014).

The displaced-basis formalism also yields analytic dynamics. For an initial state $\beta$31, the population probability $\beta$32 is expressed as a superposition of oscillatory components determined by $\beta$33, and homogeneous coupling reduces that expression to two-frequency combinations. With an initial coherent field $\beta$34, the total probability

$\beta$35

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

$\beta$36

is obtained (Mao et al., 2014).

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,

$\beta$37

the exact steady-state amplitude is

$\beta$38

If one works in the standard bare-$\beta$39 ladder-operator basis and applies the ordinary RWA in the $\beta$40-rotating frame, the predicted amplitude becomes

$\beta$41

so that

$\beta$42

The cited paper states that the standard quantum description, using creation and annihilation operators built from the natural frequency $\beta$43, necessarily leads to incorrect results when combined with the RWA for off-resonant driving (Košata et al., 2022).

The proposed fix is to define ladder operators with the drive frequency $\beta$44,

$\beta$45

so that in this basis the mismatch $\beta$46 appears as squeezing terms $\beta$47. In the $\beta$48-rotating frame, the stationary solution is

$\beta$49

and it reproduces the exact response amplitude. If one then performs a displacement by $\beta$50, 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 (Košata et al., 2022).

The same method is extended to the driven Duffing oscillator,

$\beta$51

In the drive-adapted $\beta$52-basis, the stationary mean-field equation is

$\beta$53

with

$\beta$54

The discriminant

$\beta$55

controls bistability, and the source states that the phase boundary in the $\beta$56 plane is tracked to within the numerical precision $\beta$57 relative discrepancy at small $\beta$58 in the $\beta$59-basis, whereas the bare RWA shows visible shifts (Košata et al., 2022).

Across the cited literature, the validity regimes are sharply differentiated rather than uniform. For the Rabi problem, displacement only is recommended when $\beta$60 or in the deep-strong limit $\beta$61, where squeezing is negligible, while GSRWA is recommended for ultrastrong coupling $\beta$62 and/or large positive detuning $\beta$63 (Zhang, 2016). For inhomogeneous multi-qubit systems, the displaced-basis truncations are accurate when $\beta$64 and $\beta$65, and can be extended into deeper coupling by including first-order and higher-order inter-block couplings (Mao et al., 2014). 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 $\beta$66 (Košata et al., 2022).

The main limitations are equally explicit. GSRWA neglects explicit $\beta$67 and $\beta$68 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 (Zhang, 2016). The multi-qubit displaced-basis approach relies on truncation order and on overlap-mediated couplings between non-orthogonal displaced states (Mao et al., 2014). The drive-adapted Duffing treatment remains an RWA and mean-field theory, so higher harmonics still generate small deviations in the deeply nonlinear regime (Košata et al., 2022).

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.