Hermitian Squeezing–Dicke Model

• The Hermitian Squeezing–Dicke Model is a class of unitary collective-spin Hamiltonians that generate squeezing and enable Dicke state preparation through tailored adiabatic passages.
• It employs one-axis twisting dynamics (H(t)=χJz²+β(t)Jz+Ω(t)Jx) to create sequential avoided crossings that facilitate robust state transitions with near-perfect fidelity.
• Extensions to spin–boson and periodically driven systems demonstrate its versatility in achieving near-Heisenberg scaling on platforms such as optical cavities, trapped ions, and superconducting circuits.

Searching arXiv for the cited core paper and closely related uses of the term. The Hermitian Squeezing–Dicke Model denotes a class of Hermitian many-body constructions that combine Dicke-state structure with squeezing-generating interactions, but the term is not used uniformly across the literature. In one usage, it refers to a purely collective-spin model based on one-axis twisting and coherent control in the Dicke basis, with Hamiltonian $H(t)=\chi J_z^2+\beta(t)J_z+\Omega(t)J_x$, designed for rapid adiabatic passage to Dicke states and extreme spin-squeezed superpositions (Carrasco et al., 2023). In other usages, closely related Hermitian models appear as linear-to-quadratic collective-spin interpolations for Dicke-state preparation (Opatrný et al., 2015), generalized Dicke or gauge-invariant Dicke Hamiltonians whose ground states are squeezed (Shapiro et al., 2019, San et al., 2024, Hayashida et al., 2020), periodically driven Dicke models that realize effective two-axis countertwisting (Reilly et al., 2023), and even Hermitian light–matter systems whose bosonic Bogoliubov description acquires effective non-Hermitian features through squeezing (Wang et al., 25 Jun 2026). Across these variants, the common element is that squeezing is generated by Hermitian dynamics—unitary spin nonlinearities, Hermitian atom–photon couplings, or gauge-invariant quadratic terms—rather than by explicitly non-Hermitian Hamiltonians.

1. Definition and scope

In the collective-spin formulation emphasized by “Dicke State Generation and Extreme Spin Squeezing via Rapid Adiabatic Passage” (Carrasco et al., 2023), the system consists of $N$ identical spin-$1/2$ atoms with collective spin $J=N/2$, collective operators $J_x,J_y,J_z$, and Dicke states $|J,m\rangle$ defined by

$$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$

For even $N$, $m=0$ is a unique Dicke state; for odd $N$, $N$0 are degenerate (Carrasco et al., 2023). In that setting, the Hermitian Squeezing–Dicke Model is explicitly spin-only: $N$1 with all terms Hermitian and acting within the Dicke manifold (Carrasco et al., 2023).

A distinct but related usage appears in counterdiabatic Dicke-state preparation, where the baseline Hermitian interpolation is

$N$2

with the target Dicke state $N$3 as the unique ground state of the final quadratic Hamiltonian (Opatrný et al., 2015). Here the “Squeezing–Dicke” designation refers to a linear-to-quadratic Hermitian control problem rather than to rapid adiabatic passage in the Dicke basis.

The phrase also appears in broader Dicke-model contexts. In the generalized Dicke model with unequal rotating-wave and counter-rotating-wave couplings,

$N$4

the Hamiltonian is Hermitian for real $N$5, and the focus is photon-condensate squeezing near the superradiant phase transition rather than collective-spin RAP control (Shapiro et al., 2019). Likewise, the gauge-invariant Dicke model supports photon condensation into a nonclassical squeezed state with $N$6 but $N$7 (San et al., 2024).

These usages are related by structure rather than by a single canonical definition. A plausible implication is that “Hermitian Squeezing–Dicke Model” is best understood as an umbrella term for Hermitian Dicke-state or Dicke-Hamiltonian frameworks in which squeezing is intrinsic to the Hamiltonian dynamics or equilibrium state.

2. Collective-spin Hermitian model in the Dicke basis

The spin-only model of (Carrasco et al., 2023) is built from one-axis twisting,

$N$8

which is Hermitian and diagonal in the Dicke basis. With a linear control detuning $N$9, the Dicke-basis energies are

$1/2$0

The quadratic ladder in $1/2$1 is the key structural ingredient of the protocol, because it creates a sequence of avoided crossings that can be traversed adiabatically (Carrasco et al., 2023).

Collective rotations are implemented by

$1/2$2

and small-angle rotations around $1/2$3 or $1/2$4 coherently mix neighboring Dicke states (Carrasco et al., 2023). Expanding

$1/2$5

the amplitudes obey

$1/2$6

with nearest-neighbor couplings

$1/2$7

Thus $1/2$8 couples only adjacent Dicke states, while $1/2$9 fixes the instantaneous diabatic energies (Carrasco et al., 2023).

This formulation sharply distinguishes Dicke states from the Dicke superradiance Hamiltonian. In the RAP protocol, “Dicke states” means the symmetric spin eigenstates $J=N/2$0, whereas the spin–boson Hamiltonian

$J=N/2$1

is not used (Carrasco et al., 2023). That distinction matters because the Hermitian Squeezing–Dicke Model of (Carrasco et al., 2023) is entirely collective-spin and contains no bosonic mode.

3. Rapid adiabatic passage, target states, and metrological quantities

The rapid adiabatic passage protocol engineers sequential avoided crossings between neighboring $J=N/2$2 levels by chirping

$J=N/2$3

while using a transverse coupling $J=N/2$4, typically turned on and off smoothly with a Blackman shape (Carrasco et al., 2023). Adjacent diabatic levels $J=N/2$5 and $J=N/2$6 cross at

$J=N/2$7

with constant spacing

$J=N/2$8

The relevant Landau–Zener parameter is

$J=N/2$9

and the diabatic transition probability is

$J_x,J_y,J_z$0

A sufficient adiabaticity condition is $J_x,J_y,J_z$1 throughout the RAP sequence (Carrasco et al., 2023). The minimum adiabatic gap at the $J_x,J_y,J_z$2 crossing is

$J_x,J_y,J_z$3

The principal target for even $J_x,J_y,J_z$4 is $J_x,J_y,J_z$5. The preparation protocol initializes the system in the coherent spin state $J_x,J_y,J_z$6, ramps $J_x,J_y,J_z$7 before the first crossing, chirps $J_x,J_y,J_z$8 until the last crossing between $J_x,J_y,J_z$9 and $|J,m\rangle$0, and then turns $|J,m\rangle$1 off smoothly just after $|J,m\rangle$2, yielding $|J,m\rangle$3 with fidelity $|J,m\rangle$4 in the simulations (Carrasco et al., 2023). More generally, to target $|J,m\rangle$5, the sweep ends so that the last avoided crossing is between $|J,m\rangle$6 and $|J,m\rangle$7 (Carrasco et al., 2023).

The same framework generates extreme spin-squeezed states (ESS) satisfying

$|J,m\rangle$8

These are produced by running the $|J,m\rangle$9 RAP sequence but abruptly quenching $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$0 off slightly before the final crossing completes, leaving controlled population in $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$1 in addition to $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$2 (Carrasco et al., 2023). The reported overlaps are $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$3 with the ideal ESS and $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$4 (Carrasco et al., 2023).

For metrology, the key quantity is the quantum Fisher information

$$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$5

For Dicke states,

$$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$6

Hence

$$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$7

so $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$8 achieves near-Heisenberg scaling along $$J^2|J,m\rangle = J(J+1)|J,m\rangle,\qquad J_z|J,m\rangle = m|J,m\rangle .$$9 or $N$0 (Carrasco et al., 2023). For finite-contrast ESS, the Wineland parameter

$N$1

can satisfy $N$2, so Ramsey sensitivity beats the standard quantum limit (Carrasco et al., 2023).

4. Robustness, scaling, and implementation

The RAP scheme is reported to be robust to variations of the driving field and timing parameters. Simulations show fidelities $N$3 for broad ranges of $N$4, and multiplicative amplitude noise at the $N$5 level still gives final fidelity $N$6 (Carrasco et al., 2023). With $N$7 and $N$8, fidelities $N$9 are reported both for $m=0$0 and for ESS with $m=0$1 (Carrasco et al., 2023).

The time between crossings is $m=0$2. For $m=0$3, $m=0$4 can be engineered to be weakly dependent on $m=0$5 in cavity implementations, and the paper shows that $m=0$6 can be scaled at least $m=0$7 without loss of fidelity, so the total RAP time $m=0$8 can be independent of $m=0$9 (Carrasco et al., 2023). With more aggressive schedules such as piecewise-linear $N$0, the time can decrease $N$1 for moderate $N$2, while in the regime $N$3 a fast regime with total time $N$4 still exists (Carrasco et al., 2023).

The model is compatible with several hardware platforms. The explicitly listed implementations are optical cavities with alkaline-earth(-like) atoms, trapped ions, Rydberg arrays, and superconducting circuits (Carrasco et al., 2023). In cavity QED, $N$5 is related to cavity cooperativity and detunings, and for $N$6 can be engineered to be weakly dependent on $N$7 (Carrasco et al., 2023). Trapped ions and Rydberg arrays approximate OAT through finite-range Ising interactions in the symmetric subspace, and superconducting qubit ensembles can realize tunable nonlinear collective couplings (Carrasco et al., 2023).

The main nonidealities are photon scattering, inhomogeneous coupling, dephasing, and finite-range interaction nonuniformities (Carrasco et al., 2023). The paper notes that RAP robustness helps, and that turning off $N$8 after state creation avoids unwanted evolution and preserves fidelity (Carrasco et al., 2023). This suggests that, within the collective-spin usage of the term, Hermiticity is associated not merely with formal self-adjointness but with fully unitary squeezing dynamics prior to decoherence.

5. Relation to other Hermitian Dicke-state preparation schemes

A separate Hermitian route to Dicke-state preparation is the counterdiabatic interpolation of (Opatrný et al., 2015). There the system is driven from a linear coherent Hamiltonian $N$9 to the quadratic Dicke Hamiltonian

$N$00

using

$N$01

with

$N$02

The exact transitionless-driving term $N$03 is Hermitian by construction, and experimentally accessible Hermitian approximations are built from operator monomials $N$04 such as $N$05 and higher-order generalizations (Opatrný et al., 2015).

For $N$06 and $N$07, the simulations show a clear hierarchy: no compensation gives final fidelity $N$08, while adding $N$09 yields $N$10 and squeezing $N$11 dB for the $N$12 Dicke target (Opatrný et al., 2015). This scheme shares with (Carrasco et al., 2023) the goal of preparing $N$13 through Hermitian dynamics, but differs in mechanism: it interpolates between ground states rather than traversing sequential avoided crossings in the Dicke ladder.

Another relevant comparison is with steady-state preparation in an open generalized Dicke model (Masson et al., 2018). The Hamiltonian

$N$14

is Hermitian, but the target Dicke state $N$15 is stabilized only together with cavity loss $N$16 (Masson et al., 2018). In the large-$N$17 regime the steady state approaches the strongly spin-squeezed Dicke state $N$18, quantified by the Dicke squeezing parameter

$N$19

for the ideal $N$20 Dicke state (Masson et al., 2018). This is not a purely unitary Hermitian Squeezing–Dicke Model in the sense of (Carrasco et al., 2023), but it demonstrates that Hermitian coherent couplings can still be the organizing structure even when dissipation selects the attractor.

6. Extensions to spin–boson Dicke models and broader interpretations

Several later works extend the Hermitian squeezing–Dicke idea to genuine spin–boson Dicke Hamiltonians. In the periodically driven Dicke model,

$N$21

driving at the parametric resonance $N$22 yields the effective two-axis countertwisting Hamiltonian

$N$23

under a rotating-wave approximation (Reilly et al., 2023). The quantum Fisher information then reaches $N$24, with peak time

$N$25

which is faster than one-axis twisting (Reilly et al., 2023). This is a Hermitian squeezing construction, but it is no longer Dicke-basis RAP; it is a periodically driven Dicke-type model whose effective dynamics mimics TACT.

In equilibrium Dicke physics, the ground state near the superradiant critical point is intrinsically squeezed. For the isotropic Hermitian Dicke model with counter-rotating terms,

$N$26

the critical coupling is

$N$27

and the ground state is analytically a two-mode squeezed vacuum in the photon–atom basis (Hayashida et al., 2020). In the equal-frequency case,

$N$28

so the squeezed quadrature variance vanishes at $N$29, giving perfect intrinsic squeezing at the superradiant phase transition (Hayashida et al., 2020).

The gauge-invariant Dicke model sharpens this distinction between coherence and squeezing. Its Hermitian Hamiltonian,

$N$30

enforces gauge constraints through the diamagnetic contribution, and the ground state has $N$31 but $N$32 (San et al., 2024). In its quadratic limit,

$N$33

so the field condenses into a squeezed vacuum rather than a coherent state (San et al., 2024).

Two further developments broaden the meaning of the term. “Dicke materials as a resource for quantum squeezing” studies an effective Hermitian Dicke model in solids,

$N$34

with critical coupling $N$35, perfect two-mode squeezing at the superradiant critical point, and perturbative stability against finite temperature, dilute disorder, and modest local interactions (Sharma et al., 23 Mar 2026). By contrast, “Non-Hermiticity of an anomalous superradiant phase” starts from the fully Hermitian Hamiltonian

$N$36

and shows that its bosonic Bogoliubov–de Gennes matrix is non-Hermitian because anomalous terms mix creation and annihilation operators, leading to an effective $N$37-symmetric dynamical matrix and a complex excitation spectrum in the anomalous superradiant phase (Wang et al., 25 Jun 2026). This indicates that, in some contexts, a Hermitian Squeezing–Dicke Model is interesting precisely because Hermitian squeezing can generate effective non-Hermitian physics without dissipation.

7. Conceptual distinctions, misconceptions, and current significance

A persistent source of confusion is the word “Dicke.” In (Carrasco et al., 2023), Dicke states are the symmetric spin states $N$38 and the model is spin-only. In (Shapiro et al., 2019, Hayashida et al., 2020, San et al., 2024, Sharma et al., 23 Mar 2026), and (Wang et al., 25 Jun 2026), the reference is instead to the Dicke spin–boson Hamiltonian and its generalizations. These are related traditions, but not interchangeable.

A second misconception is to equate all Dicke-related squeezing with one-axis twisting. The RAP model of (Carrasco et al., 2023) indeed uses $N$39, but measurement-induced Dicke-state preparation by heterodyne QND measurement is structurally different: its Hermitian part is the dispersive interaction $N$40, while squeezing and Dicke projection arise from conditional measurement back-action rather than from a unitary nonlinear spin Hamiltonian (Vanderbruggen et al., 2010). That work explicitly states that no mapping to OAT or TAT is implied (Vanderbruggen et al., 2010).

A third distinction concerns squeezing criteria. For finite-contrast spin states, Wineland-type metrological squeezing is natural, as in the ESS analysis of (Carrasco et al., 2023). For Dicke-class states built from two non-orthogonal spinors, the relevant measure in (Akhilesh et al., 2019) is instead the coordinate-independent Kitagawa–Ueda parameter

$N$41

with squeezing when $N$42. That paper does not introduce any Hamiltonian and therefore does not, strictly speaking, define a Hermitian Squeezing–Dicke Model (Akhilesh et al., 2019). It provides structural criteria for when Dicke-class states are squeezed, not a dynamical realization.

Taken together, the literature supports a precise but plural understanding. In its narrow sense, the Hermitian Squeezing–Dicke Model is the collective-spin Hamiltonian $N$43 used for rapid adiabatic passage to Dicke and extreme spin-squeezed states (Carrasco et al., 2023). In a broader encyclopedic sense, it names a family of Hermitian Dicke-related frameworks—spin-only, spin–boson, gauge-invariant, periodically driven, counterdiabatic, and material realizations—in which squeezing is generated, stabilized, or revealed by Hermitian interactions rather than by explicitly non-Hermitian dynamics.