Spin Squeezing and the Tavis–Cummings Model in Biquaternionic Form

Introduction

A single spin-1/2 is the smallest quantum system with an orientation, and its noise is the smallest quantum noise: a coherent spin state has fluctuations $\Delta J_\perp = \tfrac{\hbar}{2}\sqrt{N}$ distributed isotropically in the plane transverse to the mean spin. This is the standard quantum limit. When $N$ spin-1/2 particles are coupled collectively — to one another or to a common field — the noise can be redistributed: one transverse component can be squeezed below the standard quantum limit at the expense of the conjugate component. This is spin squeezing, and it is the resource of quantum metrology.

The model in which the collective coupling is cleanest is the Tavis–Cummings model: $N$ two-level systems (the spins) coupled with a common strength $g$ to a single bosonic mode. It is the $N$-atom generalisation of the Jaynes–Cummings model, and it exhibits the collective enhancement $g\sqrt{N}$ that underlies superradiance and the collective Rabi splitting. It is a driven multi-spin-1/2 system, and it is the subject of this article, in the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ of the read-list articles.

The article establishes:

1. The collective spin. The operators $\tilde{J}_\alpha = \sum_{i=1}^N\tilde{S}_\alpha^{(i)}$ are Hermitian elements of $\mathbb{M}_+$ on the $N$-spin space; the symmetric states are the Dicke states, and the collective coherent state is the highest-weight state of a rotor.
2. The Tavis–Cummings Hamiltonian and its collective coupling. In the rotating-wave approximation the interaction is $g(a^\dagger \tilde{J}_- + a\tilde{J}_+)$, and in the single-excitation sector the collective coupling is $\hbar g\sqrt{N}$, giving the collective Rabi splitting $2\hbar g\sqrt{N}$. Both statements are verified below.
3. Spin squeezing. The Wineland parameter $\xi_R^2 = N(\Delta J_\perp)^2/|\langle\mathbf{J}\rangle|^2$ defines the noise relative to the standard quantum limit; the one-axis twisting Hamiltonian $\frac{\chi}{\hbar}\tilde{J}_3^2$ drives $\xi_R^2$ below $1$. The exact diagonalisation in the Dicke basis gives $\xi^2_{\min}$ decreasing with $N$, consistent with the standard $N^{-2/3}$ scaling.
4. The algebra's reading. The collective observables are elements of $\mathbb{M}_+$, the mean spin and the covariance are trace pairings, the squeezing ellipsoid is the image of the coherent-state noise ball under the non-uniform rotor that implements the twist — a shear, not a rigid rotation — and the Tavis–Cummings interaction is an $\mathbb{M}_+$ coupling between the spin and the mode.

The notation is that of the read-list articles: $\tilde{S}_k = \tfrac{\hbar}{2}ie_k$, $\tilde\Pi_\pm(\hat\mu) = \tfrac12(e_0\pm i\hat\mu)$, $\tilde{\rho} = \tfrac12(e_0+i\mathbf{r})$, the trace formula $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$, and the isomorphism $\Phi$ with $ie_k\mapsto\sigma_k$.

The companion articles supply the pieces: - Companion article Angular Momentum and Spin in Biquaternionic Form, for the collective spin, the ladder operators and the Dicke states. - Companion article Quantum Mechanics in Biquaternionic Form, for the bosonic mode and its coupling to a two-level system. - Companion article The Harmonic Oscillator in Biquaternionic Form, for the mode algebra of the cavity field.

The Collective Spin

Collective Operators

For $N$ spin-1/2 particles label the spin operators by particle and component,

$$ \tilde{S}_\alpha^{(i)} = \frac{\hbar}{2}ie_\alpha^{(i)}, \qquad \alpha = 1,2,3, $$

with $[\tilde{S}_\alpha^{(i)},\tilde{S}_\beta^{(j)}] = i\hbar\,\delta_{ij}\,\epsilon_{\alpha\beta\gamma}\tilde{S}_\gamma^{(i)}$. The collective spin is the sum over the particles,

$$ \tilde{J}_\alpha = \sum_{i=1}^N \tilde{S}_\alpha^{(i)}, $$

so that $\tilde{J}_\alpha$ is again an element of $\mathbb{M}_+$ and the collective algebra is the same $su(2)$ algebra with $j = N/2$. The raising and lowering combinations $\tilde{J}_\pm = \tilde{J}_1\pm i\tilde{J}_2$ shift the collective magnetic quantum number by $\pm1$. These are the standard collective operators of the Dicke model; the representation theory of the resulting total angular momentum $j = N/2$ is treated in the companion article Angular Momentum and Spin in Biquaternionic Form, and here the emphasis is on the spin-1/2 constituents and their collective noise.

The Dicke States

The symmetric subspace is spanned by the Dicke states $|j = N/2,\,m\rangle$, $m = -N/2,\dots,N/2$, in which $j+m$ particles are up and $j-m$ are down, in the fully symmetric combination. They satisfy

$$ \tilde{J}_3|j,m\rangle = \hbar\,m\,|j,m\rangle, \qquad \tilde{J}_\pm|j,m\rangle = \hbar\sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle . $$

The subspace has dimension $N+1$, while the full $N$-spin space has dimension $2^N$; the non-symmetric states carry total angular momentum $j < N/2$ and are not reached by the collective operators. The collective dynamics generated by $\tilde{J}_\alpha$ — and in particular the Tavis–Cummings interaction and the twisting Hamiltonians below — stays within the symmetric subspace if the initial state is symmetric.

The Coherent Spin State

The collective coherent state is the product of $N$ identical spin-1/2 coherent states,

$$ |\theta,\phi\rangle = \bigotimes_{i=1}^N \left(\cos\frac{\theta}{2}|{+}\rangle_i + \sin\frac{\theta}{2}e^{i\phi}|{-}\rangle_i\right), $$

which equals the highest-weight state of the collective spin rotated to the direction $(\theta,\phi)$. Its mean spin is $\langle\mathbf{J}\rangle = \tfrac{N\hbar}{2}\hat{n}(\theta,\phi)$, and its transverse variance is isotropic,

$$ \langle J_\perp^2\rangle = \frac{N\hbar^2}{4}, \qquad \Delta J_\perp = \frac{\hbar\sqrt{N}}{2}. $$

This is the standard quantum limit: the noise is the quantum noise of $N$ independent spins, and it is the reference against which squeezing is measured.

The Bloch Ball of the Collective Spin

The collective state space, restricted to the symmetric subspace, is the spin-$N/2$ state space, whose Bloch-ball analogue has dimension $N$ (see the companion article Angular Momentum and Spin in Biquaternionic Form). For $N=1$ it reduces to the spin-1/2 Bloch sphere of this subcategory. The mean spin $\langle\mathbf{J}\rangle$ is a vector of length at most $\tfrac{N\hbar}{2}$; a coherent state sits on the boundary, and a state with reduced mean spin sits inside. Squeezing is a statement about the covariance of $\mathbf{J}$, and it is therefore not visible in the mean spin alone.

The Tavis–Cummings Model

The Hamiltonian

The Tavis–Cummings model couples $N$ two-level systems to a single bosonic mode with the Hamiltonian

$$ \tilde{H} = \hbar\omega_c\,a^\dagger a + \omega_a\,\tilde{J}_3 + g\left(a^\dagger \tilde{J}_- + a\,\tilde{J}_+\right), $$

where $a,a^\dagger$ are the mode operators, $\omega_c$ the mode frequency, $\omega_a$ the level splitting, and $g$ the single-spin coupling. The interaction conserves the total excitation number

$$ \tilde{N}_{\mathrm{exc}} = a^\dagger a + \frac{\tilde{J}_3}{\hbar} + \frac{N}{2}, $$

which counts the photons plus the number of excited spins; this is the collective version of the Jaynes–Cummings excitation number. The model is integrable in each excitation sector.

The Rotating-Wave Approximation

The interaction kept is the rotating-wave term: the mode annihilates a photon while a spin is raised ($a\tilde{J}_+$), and creates one while a spin is lowered ($a^\dagger\tilde{J}_-$). The neglected counter-rotating terms $a^\dagger\tilde{J}_+$ and $a\tilde{J}_-$ oscillate at the sum frequency $\omega_a+\omega_c$ in the interaction picture and average to zero when $g\ll\omega_a,\omega_c$; this is the rotating-wave approximation, the same approximation as in the single-spin Rabi problem. In the algebra both terms are elements of $\mathbb{M}_+$; the approximation selects the slowly varying combination.

The Single-Excitation Sector and the Collective Rabi Splitting

Restrict to the manifold with one excitation, spanned by the states $|1,\text{all down}\rangle$ (one photon, no excited spin) and $|0,\text{one excited}\rangle$ (no photon, one spin excited, in the symmetric combination). On resonance $\omega_a = \omega_c = \omega$ the two states are degenerate at energy $\hbar\omega(1-N/2)$, and the interaction couples them with the matrix element

$$ \langle 1,\text{all down}|g\,a\,\tilde{J}_+^{\vphantom{}^{*}}|0,\text{one excited}\rangle = \hbar g\sqrt{N}, $$

because $\tilde{J}_+|j,-j\rangle = \hbar\sqrt{N}\,|j,-j+1\rangle$ with $j = N/2$. The two eigenvalues are therefore separated by

$$ \boxed{\;\Delta E = 2\hbar g\sqrt{N}\;} $$

the collective Rabi splitting. The square-root enhancement is the signature of the collective coupling: it is the same $\sqrt{N}$ that appears in superradiance and in the collective Lamb shift. The result was verified numerically: for $N = 1,4,16,100$ the eigenvalue splitting of the $2\times2$ block equals $2\hbar g\sqrt{N}$ exactly.

The eigenstates of the block are the polaritons, the symmetric and antisymmetric combinations of "photon" and "excited spin", which are the collective analogues of the dressed states of the single-spin Rabi problem.

The Holstein–Primakoff Linearisation

Small Excitations

Near the fully polarised state $|j,\,j\rangle$ (all spins up) the collective operators can be represented by a single bosonic mode,

$$ \tilde{J}_3 = \hbar\left(\frac{N}{2} - b^\dagger b\right) + O(1/N), \qquad \tilde{J}_+ \approx \hbar\sqrt{N}\,b^\dagger, \qquad \tilde{J}_- \approx \hbar\sqrt{N}\,b, $$

the Holstein–Primakoff transformation truncated at leading order. Substituting into the Tavis–Cummings Hamiltonian gives a quadratic two-mode problem,

$$ \tilde{H} \approx \hbar\omega_c\,a^\dagger a + \hbar\omega_a\left(\frac{N}{2}-b^\dagger b\right) + \hbar g\sqrt{N}\left(a^\dagger b + a b^\dagger\right), $$

whose normal modes are obtained by diagonalising the $2\times2$ coupling. On resonance the normal-mode splitting is again $2\hbar g\sqrt{N}$; the linearised theory is the large-$N$ limit of the exact single-excitation result, and it is the standard route to the collective normal modes.

Superradiance and the $\sqrt{N}$ Enhancement

The same collective enhancement governs the coupling of the spins to a radiating field. The emission amplitude of a collective state carries the factor $\sqrt{N}$, so the coupling to a cavity mode is $g\sqrt{N}$ and the radiated intensity carries $N$: compared with the same number of independently radiating dipoles, whose amplitudes add incoherently, the emission is cooperatively enhanced — superradiance. For the Dicke state $|j,0\rangle$ the enhancement of the rate is quadratic in $N$. In the algebra these are statements about the collective raising and lowering operators $\tilde{J}_\pm$ and their matrix elements in the Dicke states; the enhancement factors are the $\sqrt{N}$ of the collective algebra.

Spin Squeezing

The Standard Quantum Limit

For a coherent spin state the transverse variance is $(\Delta J_\perp)^2 = \tfrac{\hbar^2 N}{4}$ and the mean spin is $|\langle\mathbf{J}\rangle| = \tfrac{\hbar N}{2}$. The Wineland squeezing parameter is

$$ \xi_R^2 = \frac{N\,(\Delta J_\perp)^2_{\min}}{|\langle\mathbf{J}\rangle|^2}, $$

where $(\Delta J_\perp)^2_{\min}$ is the smallest variance in the plane perpendicular to the mean spin. The numerator is $N$ times the squeezed variance, the denominator the squared mean spin; for a coherent state $\xi_R^2 = 1$, a state with $\xi_R^2 < 1$ is spin squeezed, and the metrological phase sensitivity is improved by the factor $\xi_R$ below the standard quantum limit.

The Squeezing Ellipse

The covariance of the collective spin is the symmetric $3\times3$ matrix

$$ \mathrm{Cov}(J_\alpha,J_\beta) = \tfrac{1}{2}\langle \tilde{J}_\alpha\tilde{J}_\beta + \tilde{J}_\beta\tilde{J}_\alpha\rangle - \langle\tilde{J}_\alpha\rangle\langle\tilde{J}_\beta\rangle , $$

whose eigenvalues are the variances along the principal axes. In the plane perpendicular to the mean spin the covariance reduces to a $2\times2$ matrix $\begin{pmatrix}\langle\Delta J_a^2\rangle & C\\ C & \langle\Delta J_b^2\rangle\end{pmatrix}$, and the squeezing ellipse is the error ellipse of the two quadratures. The smaller eigenvalue is the squeezed variance; the larger is the anti-squeezed one. The product of the two principal variances obeys the uncertainty relation, so squeezing is a redistribution and never a reduction of the total noise.

One-Axis Twisting

The simplest generator of squeezing is the one-axis twisting Hamiltonian

$$ \tilde{H}_{OAT} = \frac{\chi}{\hbar}\,\tilde{J}_3^2 , $$

which is diagonal in the Dicke basis and therefore exactly solvable. Starting from the coherent state along $+x$, the evolution

$$ |\psi(\chi t)\rangle = e^{-i\chi t\tilde{J}_3^2/\hbar^2}|\theta=\tfrac{\pi}{2},\phi=0\rangle $$

shears the noise ellipse: the variance in the $y$–$z$ plane develops an off-diagonal covariance, the smaller principal variance drops below $\hbar^2N/4$, and $\xi_R^2$ falls below $1$. The mean spin also decreases, since the twisting rotates different Dicke components at different rates; the competition between the shrinking variance and the shrinking mean spin fixes an optimal twisting time.

Numerical Illustration

Computing the exact evolution in the Dicke basis and minimising $\xi_R^2$ over $\chi t$ gives, for the pure one-axis twisting model:

The optimal time decreases and the noise decreases with $N$, consistent with the standard one-axis-twisting scaling $\xi^2_{\min}\sim N^{-2/3}$: the computed optimal times $0.228,0.153,0.101,0.066$ follow $N^{-2/3}$ with a coefficient between $0.9$ and $1.1$, and the minimum variances give $\xi^2_{\min}N^{2/3} = 1.42,\,1.46,\,1.44,\,1.39$ — the standard scaling with a prefactor of order unity. At $\chi t = 0$ the parameter is $\xi^2 = 1$ to machine precision, as it must be for a coherent state.

Two-Axis Twisting

The two-axis countertwisting Hamiltonian $\frac{\chi}{\hbar}(\tilde{J}_+^2 + \tilde{J}_-^2)$ — or equivalently a squeezing generator with two orthogonal axes — reaches the Heisenberg scaling $\xi^2\propto N^{-1}$ in the ideal case, faster than one-axis twisting. It is harder to realise physically because it requires a nonlinear coupling with two non-commuting axes, but it is the standard benchmark for the best achievable squeezing from a nonlinear collective Hamiltonian. The framework describes it in the same language: a Hermitian element of $\mathbb{M}_+$ that is quadratic in the collective spin.

The Biquaternion Reading

Collective Observables in $\mathbb{M}_+$

Each $\tilde{J}_\alpha$ is a sum of Hermitian elements $ie_\alpha^{(i)}$ and is Hermitian; the mean spin is the trace pairing

$$ \langle \tilde{J}_\alpha\rangle = \mathrm{Tr}\!\left(\tilde{\rho}\,\tilde{J}_\alpha\right), \qquad \tilde{\rho} = \text{the collective state}, $$

and the covariance is the symmetrised trace pairing of $\tilde{J}_\alpha\tilde{J}_\beta$. Squeezing is thus a statement about the trace pairings of the collective $\mathbb{M}_+$ elements and their products: it is the anisotropy of a quadratic pairing, not of a linear one. The mean spin is a vector, the covariance is a symmetric tensor, and the physics of squeezing lives entirely in the tensor.

The Twisting as a Rotor

One-axis twisting is conjugation by a non-uniform rotor. In the Dicke basis the evolution operator $\exp(-i\chi t\tilde{J}_3^2/\hbar^2)$ acts on each $|j,m\rangle$ with a phase depending on $m^2$; written on the spinor it is a diagonal rotor whose angle is proportional to $m^2$. It is not a rigid rotation of the collective Bloch ball — rigid rotations are generated by the linear $\tilde{J}_\alpha$ and are the unit quaternions of $\mathbb{H}_{\mathbb{B}}$ — but a shear of the ball, generated by the quadratic $\tilde{J}_3^2$. The distinction is the algebraic content of the difference between a rotation (which preserves the noise ellipse) and a squeezing (which shears it). The squeezing ellipsoid is the image of the coherent-state noise ball under this shear.

The Tavis–Cummings Coupling

The interaction term $g(a^\dagger\tilde{J}_- + a\tilde{J}_+)$ is an $\mathbb{M}_+$ coupling between the spin sector and the mode: $a^\dagger\tilde{J}_-$ and its adjoint are conjugate elements, and their sum is Hermitian. Its matrix element in the single-excitation sector, $\hbar g\sqrt{N}$, is the trace pairing evaluated between the two collective idempotents, and the collective Rabi splitting $2\hbar g\sqrt{N}$ is the associated spectral gap. The algebra thus places the collective coupling on the same footing as the single-spin Rabi coupling of the magnetic-resonance problem, with the collective operator replacing the single spin.

What the Algebra Adds and What It Does Not

Standard physics, transcribed. The collective spin algebra, the Dicke states, the Tavis–Cummings Hamiltonian, the $\sqrt{N}$ collective coupling, the Holstein–Primakoff linearisation, superradiance, the Wineland parameter, one- and two-axis twisting are all standard.

What the algebra organises.

• Collective observables are elements of one algebra. The $N$-spin collective operators are sums of $\mathbb{M}_+$ elements and remain in $\mathbb{M}_+$; the collective dynamics is described by the same trace pairings as the single-spin dynamics.
• Squeezing is a tensor property. The mean spin is a linear trace pairing and the covariance a quadratic one; squeezing is the anisotropy of the quadratic pairing, and the algebra's trace formula expresses both.
• Rotation versus shear. Rigid rotations of the collective state are rotors in $\mathbb{H}_{\mathbb{B}}$; squeezing is generated by the quadratic $\tilde{J}_3^2$ and is a shear, not a rotation. The algebraic distinction matches the physical one.
• The collective coupling is a trace pairing. The $\hbar g\sqrt{N}$ matrix element and the $2\hbar g\sqrt{N}$ splitting are the same spectral structure as in the single-spin case, with the collective operator in place of the spin.

What the algebra does not supply. The number of spins $N$, the coupling $g$, the frequencies and the initial state are external data. The algebra supplies the collective structure and the pairings; it does not select the Hamiltonian.

Open Questions

1. The informational reading. Spin squeezing is often described as the creation of multipartite entanglement among the spins, and the squeezing parameter bounds a metrological entanglement measure. That reading belongs to the informational subcategory; whether the collective $\mathbb{M}_+$ algebra has a natural entanglement structure is left there.

2. The exact Tavis–Cummings dynamics at finite $N$. The model is integrable in each excitation sector; the finite-$N$ dynamics of the collective spin, including the revivals and the collapse, is standard. Does the algebraic formulation streamline the finite-$N$ treatment?

3. Dissipation and superradiance. The cooperative decay that gives superradiance requires an open system; the framework is unitary, and the open-system extension is the same gap noted for the Rabi problem.

4. The best squeezing. Two-axis countertwisting reaches the Heisenberg scaling only in the ideal case; the optimal control problem for squeezing is a question about the reachable set of the collective algebra, and it is not addressed here.

5. Empirical contact. As everywhere in the series, the reformulation reproduces the standard predictions; whether it implies a measurable deviation is open.

Summary

For $N$ spin-1/2 particles the collective operators $\tilde{J}_\alpha = \sum_i\tilde{S}_\alpha^{(i)}$ are Hermitian elements of $\mathbb{M}_+$ obeying the same $su(2)$ algebra with $j = N/2$. The symmetric states are the Dicke states $|j,m\rangle$, and the collective coherent state is the product of $N$ identical coherent states with mean spin $\tfrac{\hbar N}{2}\hat{n}$ and transverse variance $\tfrac{\hbar^2N}{4}$ — the standard quantum limit.

The Tavis–Cummings model,

$$ \tilde{H} = \hbar\omega_c\,a^\dagger a + \omega_a\,\tilde{J}_3 + g\left(a^\dagger\tilde{J}_- + a\tilde{J}_+\right), $$

conserves the excitation number $a^\dagger a + \tilde{J}_3/\hbar + N/2$. In its single-excitation sector the collective coupling is $\hbar g\sqrt{N}$ and the collective Rabi splitting is

$$ \Delta E = 2\hbar g\sqrt{N}, $$

verified numerically for $N = 1,4,16,100$. The Holstein–Primakoff linearisation reproduces this at large $N$, and the same $\sqrt{N}$ governs superradiance.

Spin squeezing is measured by the Wineland parameter

$$ \xi_R^2 = \frac{N\,(\Delta J_\perp)^2_{\min}}{|\langle\mathbf{J}\rangle|^2}, $$

equal to $1$ for a coherent state and below $1$ for a squeezed state. One-axis twisting, $\tilde{H}_{OAT} = \frac{\chi}{\hbar}\tilde{J}_3^2$, is diagonal in the Dicke basis and exactly solvable; minimising $\xi_R^2$ over the twisting time gives $0.354,0.230,0.143,0.087$ for $N = 8,16,32,64$, consistent with the standard $N^{-2/3}$ scaling and with $\xi_R^2 = 1$ at $\chi t = 0$. Two-axis countertwisting reaches the ideal Heisenberg scaling $\xi^2\propto N^{-1}$.

In the algebra the mean spin is a linear trace pairing $\mathrm{Tr}(\tilde{\rho}\tilde{J}_\alpha)$ and the covariance a quadratic one; squeezing is the anisotropy of the quadratic pairing. Rigid rotations of the collective state are rotors in $\mathbb{H}_{\mathbb{B}}$, while the twisting generator $\tilde{J}_3^2$ is a shear; the Tavis–Cummings coupling $g(a^\dagger\tilde{J}_- + a\tilde{J}_+)$ is an $\mathbb{M}_+$ coupling whose single-excitation gap is the collective Rabi splitting.

Summary of Notation

Further Reading

• M. Tavis and F. W. Cummings, "Exact Solution for an $N$-Molecule–Radiation-Field Hamiltonian," Physical Review 170 (1968) 379–384, for the model.
• R. H. Dicke, "Coherence in Spontaneous Radiation Processes," Physical Review 93 (1954) 99–110, for the collective spin states and superradiance.
• M. Kitagawa and M. Ueda, "Squeezed Spin States," Physical Review A 47 (1993) 5138–5143, for one- and two-axis twisting and the squeezing parameter.
• D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, "Spin Squeezing and Reduced Quantum Noise in Spectroscopy," Physical Review A 46 (1992) R6797–R6800, for the metrological squeezing parameter.
• J. Ma, X. Wang, C. P. Sun, and F. Nori, "Quantum Spin Squeezing," Physics Reports 509 (2011) 89–165, for a comprehensive review of spin squeezing.
• A. S. Sørensen and K. Mølmer, "Entanglement and Extreme Spin Squeezing," Physical Review Letters 86 (2001) 4431–4434, for the relation between squeezing and collective entanglement.
• B. M. Garraway, "The Dicke Model in Quantum Optics: Dicke Model Revisited," Philosophical Transactions of the Royal Society A 369 (2011) 1137–1155, for the collective coupling and the normal-mode structure.
• T. Holstein and H. Primakoff, "Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet," Physical Review 58 (1940) 1098–1113, for the bosonic representation of collective spins.
• C. Gross, "Spin Squeezing, Entanglement and Quantum Metrology with Bose–Einstein Condensates," Journal of Physics B 45 (2012) 103001, for the experimental realisation and the algebraic reading of the squeezing Hamiltonians.