---
title: Hermitian Squeezing–Dicke Model
url: https://www.emergentmind.com/topics/hermitian-squeezing-dicke-model
type: topic
---

# Hermitian Squeezing–Dicke Model

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 [2306.03190]. In other usages, closely related Hermitian models appear as linear-to-quadratic collective-spin interpolations for Dicke-state preparation [1510.07261], generalized Dicke or gauge-invariant Dicke Hamiltonians whose ground states are squeezed [1909.10252], [2409.02701], [2009.02630], periodically driven Dicke models that realize effective two-axis countertwisting [2310.07694], and even Hermitian light–matter systems whose bosonic Bogoliubov description acquires effective non-Hermitian features through squeezing [2606.26770]. 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” [2306.03190], 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 \(N\), \(m=\pm 1/2\) are degenerate [2306.03190]. In that setting, the Hermitian Squeezing–Dicke Model is explicitly spin-only:
\[
H(t)=\chi J_z^2+\beta(t)J_z+\Omega(t)J_x ,
\]
with all terms Hermitian and acting within the Dicke manifold [2306.03190].

A distinct but related usage appears in counterdiabatic Dicke-state preparation, where the baseline Hermitian interpolation is
\[
H_0(t)=A_c(t)H_c+A_n(t)H_n,\qquad H_n=\chi(J_z-nI)^2 ,
\]
with the target Dicke state \(|J_z=n\rangle\) as the unique ground state of the final quadratic Hamiltonian [1510.07261]. 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,
\[
H=\omega a^\dagger a+\frac{\epsilon}{2}S_z+\frac{g}{\sqrt{N}}(aS^+ + a^\dagger S^-)+\frac{J}{\sqrt{N}}(aS^- + a^\dagger S^+) ,
\]
the Hamiltonian is Hermitian for real \(\omega,\epsilon,g,J\), and the focus is photon-condensate squeezing near the superradiant phase transition rather than collective-spin RAP control [1909.10252]. Likewise, the gauge-invariant Dicke model supports photon condensation into a nonclassical squeezed state with \(\eta=\langle \hat a\rangle=0\) but \(\bar n=\langle \hat a^\dagger \hat a\rangle\neq 0\) [2409.02701].

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 [2306.03190] is built from one-axis twisting,
\[
H_{\text{OAT}}=\chi J_z^2 ,
\]
which is Hermitian and diagonal in the Dicke basis. With a linear control detuning \(\beta(t)\), the Dicke-basis energies are
\[
E_m(t)=\chi m^2+\beta(t)m .
\]
The quadratic ladder in \(m\) is the key structural ingredient of the protocol, because it creates a sequence of avoided crossings that can be traversed adiabatically [2306.03190].

Collective rotations are implemented by
\[
R_n(\theta)=e^{-i\theta J_n},\qquad n\in\{x,y,z\},
\]
and small-angle rotations around \(x\) or \(y\) coherently mix neighboring Dicke states [2306.03190]. Expanding
\[
|\psi(t)\rangle=\sum_m a_m(t)|J,m\rangle ,
\]
the amplitudes obey
\[
i\dot a_m(t)=E_m(t)a_m(t)+\frac{\Omega(t)}{2}\left(\zeta_+(m)a_{m+1}(t)+\zeta_-(m)a_{m-1}(t)\right),
\]
with nearest-neighbor couplings
\[
\zeta_\pm(m)=\sqrt{(J\mp m)(J\pm m+1)} .
\]
Thus \(\Omega(t)J_x\) couples only adjacent Dicke states, while \(\chi J_z^2+\beta(t)J_z\) fixes the instantaneous diabatic energies [2306.03190].

This formulation sharply distinguishes Dicke states from the Dicke superradiance Hamiltonian. In the RAP protocol, “Dicke states” means the symmetric spin eigenstates \(|J,m\rangle\), whereas the spin–boson Hamiltonian
\[
H_{\text{Dicke}}=\omega a^\dagger a+\omega_0 J_z+g(a+a^\dagger)J_x
\]
is not used [2306.03190]. That distinction matters because the Hermitian Squeezing–Dicke Model of [2306.03190] 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 \(|J,m\rangle\) levels by chirping
\[
\beta(t)=\alpha t\,u(-t)
\]
while using a transverse coupling \(\Omega(t)\), typically turned on and off smoothly with a Blackman shape [2306.03190]. Adjacent diabatic levels \(|J,m\rangle\) and \(|J,m-1\rangle\) cross at
\[
t_{m,m-1}=\frac{\chi(1-2m)}{\alpha},
\]
with constant spacing
\[
\tau=\frac{2\chi}{\alpha}.
\]

The relevant Landau–Zener parameter is
\[
\Gamma_m=\frac{a_m^2}{\alpha},\qquad
a_m=\frac{\Omega(t)}{2}\sqrt{(J+m)(J-m+1)},
\]
and the diabatic transition probability is
\[
P_{\text{LZ}}=e^{-2\pi \Gamma_m}.
\]
A sufficient adiabaticity condition is \(\Gamma_m\gg 1\) throughout the RAP sequence [2306.03190]. The minimum adiabatic gap at the \(m\leftrightarrow m-1\) crossing is
\[
\Delta_{\min}(m)\approx \Omega\,\zeta_-(m)=\Omega\sqrt{(J+m)(J-m+1)} .
\]

The principal target for even \(N\) is \(|J,0\rangle\). The preparation protocol initializes the system in the coherent spin state \(|J,J\rangle\), ramps \(\Omega(t)\) before the first crossing, chirps \(\beta(t)\) until the last crossing between \(|J,1\rangle\) and \(|J,0\rangle\), and then turns \(\Omega(t)\) off smoothly just after \(t=0\), yielding \(|J,0\rangle\) with fidelity \(>0.999\) in the simulations [2306.03190]. More generally, to target \(|J,n\rangle\), the sweep ends so that the last avoided crossing is between \(|J,n+1\rangle\) and \(|J,n\rangle\) [2306.03190].

The same framework generates extreme spin-squeezed states (ESS) satisfying
\[
(\chi J_z^2-\Omega_{\text{ESS}}J_x)|\Psi_{\text{ESS}}\rangle=\lambda |\Psi_{\text{ESS}}\rangle .
\]
These are produced by running the \(|J,0\rangle\) RAP sequence but abruptly quenching \(\Omega(t)\) off slightly before the final crossing completes, leaving controlled population in \(|J,\pm 1\rangle\) in addition to \(|J,0\rangle\) [2306.03190]. The reported overlaps are \(\gtrsim 0.999\) with the ideal ESS and \(\langle J_x\rangle=J/2\) [2306.03190].

For metrology, the key quantity is the quantum Fisher information
\[
F_Q[|\psi\rangle;J_n]=4\,\mathrm{Var}_\psi(J_n).
\]
For Dicke states,
\[
F_Q[|J,m\rangle;J_z]=0,\qquad
F_Q[|J,m\rangle;J_{x,y}]=N^2/2-2m^2+N .
\]
Hence
\[
F_Q[|J,0\rangle;J_{x,y}]=N^2/2+N,
\qquad
\Delta\phi_{\min}\ge \frac{1}{\sqrt{F_Q}}\approx \frac{\sqrt{2}}{N}\quad (N\gg 1),
\]
so \(|J,0\rangle\) achieves near-Heisenberg scaling along \(x\) or \(y\) [2306.03190]. For finite-contrast ESS, the Wineland parameter
\[
\xi_R^2=\frac{N\,\Delta J_\perp^2}{\langle J\rangle^2}
\]
can satisfy \(\xi_R^2\ll 1\), so Ramsey sensitivity beats the standard quantum limit [2306.03190].

## 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 \(>0.999\) for broad ranges of \(\{\alpha,\Omega(t),t_{\text{on}},t_{\text{off}},t_2\}\), and multiplicative amplitude noise at the \(5\%\) level still gives final fidelity \(\approx 0.997\) [2306.03190]. With \(\Omega_{\max}\approx 0.88\chi\) and \(\alpha=0.1\chi^2\), fidelities \(\ge 0.999\) are reported both for \(|J,0\rangle\) and for ESS with \(\langle J_x\rangle=J/2\) [2306.03190].

The time between crossings is \(\tau=2\chi/\alpha\). For \(N\lesssim 10^3\), \(\chi\) can be engineered to be weakly dependent on \(N\) in cavity implementations, and the paper shows that \(\alpha\) can be scaled at least \(\propto N\) without loss of fidelity, so the total RAP time \(N\tau\approx N\chi/\alpha\) can be independent of \(N\) [2306.03190]. With more aggressive schedules such as piecewise-linear \(\beta(t)\), the time can decrease \(\propto \log(N)/N\) for moderate \(N\), while in the regime \(\chi\propto 1/N\) a fast regime with total time \(\propto \log(N)\) still exists [2306.03190].

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 [2306.03190]. In cavity QED, \(\chi\) is related to cavity cooperativity and detunings, and for \(N\lesssim 10^3\) can be engineered to be weakly dependent on \(N\) [2306.03190]. 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 [2306.03190].

The main nonidealities are photon scattering, inhomogeneous coupling, dephasing, and finite-range interaction nonuniformities [2306.03190]. The paper notes that RAP robustness helps, and that turning off \(\chi\) after state creation avoids unwanted evolution and preserves fidelity [2306.03190]. 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 [1510.07261]. There the system is driven from a linear coherent Hamiltonian \(H_c=\vec c\cdot \vec J\) to the quadratic Dicke Hamiltonian
\[
H_n=\chi (J_z-nI)^2 ,
\]
using
\[
H_0(t)=A_c(t)H_c+A_n(t)H_n
\]
with
\[
A_c(t)=\omega_{\max}\cos^3\!\left(\frac{\pi t}{2T}\right),\qquad
A_n(t)=\chi_{\max}\sin^3\!\left(\frac{\pi t}{2T}\right) .
\]
The exact transitionless-driving term \(H_B(t)\) is Hermitian by construction, and experimentally accessible Hermitian approximations are built from operator monomials \(L_k\) such as \(J_zJ_y+J_yJ_z\) and higher-order generalizations [1510.07261].

For \(N=30\) and \(\chi_{\max}T=2\), the simulations show a clear hierarchy: no compensation gives final fidelity \(F\approx 0.19\), while adding \(L_1+L_2+L_3+L_4\) yields \(F\approx 1-2\times 10^{-7}\) and squeezing \(\approx -65\) dB for the \(n=0\) Dicke target [1510.07261]. This scheme shares with [2306.03190] the goal of preparing \(|J_z=n\rangle\) 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 [1810.11176]. The Hamiltonian
\[
H=\omega_0 S_z+\omega a^\dagger a+\frac{\lambda}{\sqrt N}(a+a^\dagger)(S_++S_-)+\frac{U}{N}S_z a^\dagger a
\]
is Hermitian, but the target Dicke state \(|N/2,0\rangle\) is stabilized only together with cavity loss \(\kappa D[a]\rho\) [1810.11176]. In the large-\(U\) regime the steady state approaches the strongly spin-squeezed Dicke state \(|N/2,0\rangle\), quantified by the Dicke squeezing parameter
\[
\xi_D = N \frac{(\Delta S_z)^2 + 1/4}{\langle S_x^2 + S_y^2 \rangle},
\qquad
\xi_D=\frac{1}{N+2}
\]
for the ideal \(m=0\) Dicke state [1810.11176]. This is not a purely unitary Hermitian Squeezing–Dicke Model in the sense of [2306.03190], 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,
\[
H=\hbar \Delta J_z+\hbar \chi \cos(\omega t) J_x^2 ,
\]
driving at the parametric resonance \(\omega=2\Delta\) yields the effective two-axis countertwisting Hamiltonian
\[
H_{\text{TACT}}=\hbar \frac{\chi}{4}(J_x^2-J_y^2)
=\frac{\hbar \chi}{8}(J_+^2+J_-^2)
\]
under a rotating-wave approximation [2310.07694]. The quantum Fisher information then reaches \(F_Q\approx 0.65 N^2\), with peak time
\[
t_{\text{peak}}\approx \frac{\ln(N^2)+4}{N|\chi|},
\]
which is faster than one-axis twisting [2310.07694]. 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,
\[
\frac{H}{\hbar}=\omega_a a^\dagger a+\omega_b\left(S_z+\frac{N}{2}\right)+\frac{2g}{\sqrt N}(a+a^\dagger)S_x ,
\]
the critical coupling is
\[
g_c=\frac{1}{2}\sqrt{\omega_a\omega_b},
\]
and the ground state is analytically a two-mode squeezed vacuum in the photon–atom basis [2009.02630]. In the equal-frequency case,
\[
e^{2r_-}=\sqrt{\frac{\omega_a-2g}{\omega_a}},
\]
so the squeezed quadrature variance vanishes at \(g\to g_c^-\), giving perfect intrinsic squeezing at the superradiant phase transition [2009.02630].

The gauge-invariant Dicke model sharpens this distinction between coherence and squeezing. Its Hermitian Hamiltonian,
\[
H_{GI}=\hat a^\dagger \hat a+\Delta\left\{\hat J_z\cosh[2f(\hat a-\hat a^\dagger)]+i\hat J_x\sinh[2f(\hat a-\hat a^\dagger)]\right\},
\]
enforces gauge constraints through the diamagnetic contribution, and the ground state has \(\eta=\langle \hat a\rangle=0\) but \(\bar n=\langle \hat a^\dagger \hat a\rangle>0\) [2409.02701]. In its quadratic limit,
\[
\tanh(2r)=\frac{2D}{\omega_c+2D},
\qquad
\bar n=\sinh^2 r ,
\]
so the field condenses into a squeezed vacuum rather than a coherent state [2409.02701].

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,
\[
H/\hbar=\omega_0 J_z+\omega a^\dagger a+\frac{g}{\sqrt N}(a+a^\dagger)J_x ,
\]
with critical coupling \(g_c=\sqrt{\omega\omega_0}/2\), perfect two-mode squeezing at the superradiant critical point, and perturbative stability against finite temperature, dilute disorder, and modest local interactions [2603.22416]. By contrast, “Non-Hermiticity of an anomalous superradiant phase” starts from the fully Hermitian Hamiltonian
\[
\hat H=\hat H_{\text{Dicke}}+\eta(\hat a^2+\hat a^{\dagger 2})+\frac{2\gamma}{N}(J_x^2-J_y^2)
\]
and shows that its bosonic Bogoliubov–de Gennes matrix is non-Hermitian because anomalous terms mix creation and annihilation operators, leading to an effective \(\mathcal{PT}\)-symmetric dynamical matrix and a complex excitation spectrum in the anomalous superradiant phase [2606.26770]. 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 [2306.03190], Dicke states are the symmetric spin states \(|J,m\rangle\) and the model is spin-only. In [1909.10252], [2009.02630], [2409.02701], [2603.22416], and [2606.26770], 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 [2306.03190] indeed uses \(H_{\text{OAT}}=\chi J_z^2\), but measurement-induced Dicke-state preparation by heterodyne QND measurement is structurally different: its Hermitian part is the dispersive interaction \(H_{\text{int}}=\hbar \chi J_z a_0^\dagger a_0\), while squeezing and Dicke projection arise from conditional measurement back-action rather than from a unitary nonlinear spin Hamiltonian [1003.0157]. That work explicitly states that no mapping to OAT or TAT is implied [1003.0157].

A third distinction concerns squeezing criteria. For finite-contrast spin states, Wineland-type metrological squeezing is natural, as in the ESS analysis of [2306.03190]. For Dicke-class states built from two non-orthogonal spinors, the relevant measure in [1901.02377] is instead the coordinate-independent Kitagawa–Ueda parameter
\[
\xi = \frac{2(\Delta \hat S_\perp)_{\min}}{\sqrt N},
\]
with squeezing when \(\xi<1\). That paper does not introduce any Hamiltonian and therefore does not, strictly speaking, define a Hermitian Squeezing–Dicke Model [1901.02377]. 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 \(H(t)=\chi J_z^2+\beta(t)J_z+\Omega(t)J_x\) used for rapid adiabatic passage to Dicke and extreme spin-squeezed states [2306.03190]. 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.

Source: https://www.emergentmind.com/topics/hermitian-squeezing-dicke-model