Dicke / Tavis-Cummings battery

qenvs.models.DickeBattery, task battery/Dicke-v0 (environment BatteryCharging).

A collective quantum battery after Ferraro et al. (2018): $N$ identical two-level systems (the cells) are charged by a single cavity mode (the charger) that initially holds $n$ photons. All cells couple to the cavity with the same, time-dependent strength $g(t)$, which is the control. Charging $N$ cells collectively is faster than charging them one by one: the charging time shrinks with $N$ (collective advantage).

Physics

With $\hbar = 1$, collective spin operators $J_\pm, J_z$ ($j = N/2$) and cavity operators $a, a^\dagger$:

\[ H_{\mathrm{TC}}(t) = \omega_0 \left(J_z + \tfrac{N}{2}\right) + \omega_c\, a^\dagger a + g(t)\, \left(a^\dagger J_- + a J_+\right) \qquad (\texttt{rwa=True}) \]

\[ H_{\mathrm{Dicke}}(t) = \omega_0 \left(J_z + \tfrac{N}{2}\right) + \omega_c\, a^\dagger a + g(t)\, \left(a + a^\dagger\right)\left(J_+ + J_-\right) \qquad (\texttt{rwa=False}) \]

The two differ by the counter-rotating terms $a J_-$ and $a^\dagger J_+$. The Tavis-Cummings (TC) Hamiltonian conserves the excitation number $\hat N_{\mathrm{exc}} = a^\dagger a + J_z + N/2$; the Dicke Hamiltonian only conserves its parity and creates photons out of the vacuum. There is no diamagnetic $A^2$ term: this is the bare Dicke Hamiltonian as used by Ferraro et al. (2018).

The stored energy is measured by the battery Hamiltonian

\[ H_B = \omega_0 \left(J_z + \tfrac{N}{2}\right), \qquad 0 \le \langle H_B \rangle \le N\omega_0 , \]

which is $0$ for all cells in the ground state and $N\omega_0$ when fully charged. Energy and ergotropy are properties of the cells alone: the model's battery_reduced(state) traces out the cavity (dq.ptrace(state, 0)) and returns $H_B$ on the $(N+1)$-dimensional Dicke space, and BatteryCharging evaluates both quantities on that reduced state. For closed dynamics the global state stays pure, so an ergotropy computed on the full cells+cavity state with $H_B \otimes 1$ would equal the energy identically; on the reduced cell state it is strictly below the energy whenever the cells are entangled with the cavity (see Reward).

Optional collective dissipators (Lindblad jump operators):

rate	jump operator	meaning
`kappa`	$\sqrt{\kappa}\, a$	cavity photon loss
`gamma`	$\sqrt{\gamma}\, J_-$	collective (superradiant) emission
`gamma_phi`	$\sqrt{\gamma_\phi}\, J_z$	collective dephasing

For $N = 1$ the dephasing jump $\sqrt{\gamma_\phi}\,\sigma_z/2$ damps the coherence at rate $\gamma_\phi / 2$, i.e. DickeBattery(gamma_phi=x) corresponds to TwoLevel(gamma_phi=x/2).

Basis and index conventions

Permutation-symmetric (Dicke) subspace $|j, m\rangle$, $j = N/2$, $m = -j, \dots, +j$, stored at indices $0, \dots, N$: index 0 = all cells ground, index $N$ = all cells excited (library rule: higher index = more excited).
$J_+$ is built from $\langle j, m+1 | J_+ | j, m \rangle = \sqrt{j(j+1) - m(m+1)}$, $J_- = J_+^\dagger$, $J_z = \mathrm{diag}(-j, \dots, j)$. For $N = 1$ this gives $J_+ = |1\rangle\langle 0|$, $J_z = \sigma_z / 2$ (collective_spin_matrices(1)).
Cavity Fock space $|0\rangle, \dots, |n_{\max}\rangle$ with $a|n\rangle = \sqrt{n}\,|n-1\rangle$ (boson_matrices(n_max)).
Tensor order: battery (first) $\otimes$ cavity (second), $\dim = (N+1)(n_{\max}+1)$; the qarrays carry dims = (N + 1, n_max + 1), so dq.ptrace(state, 0) is the cell state.
Initial state: all cells ground, cavity in the Fock state $|n\rangle$, $|{-j}\rangle \otimes |n\rangle$ (ket for closed models, density matrix when any rate is positive).
Note that QuTiP's jmat orders $m$ from $+j$ down to $-j$, the opposite of this library; the QuTiP cross-checks permute the spin index accordingly.

Assumptions and validity

Rotating-wave approximation (rwa=True, the default) is justified for $g \ll \omega_0$ near resonance ($\omega_c \approx \omega_0$). The check rwa_weak_coupling shows that at $g = 0.01\,\omega_0$ the TC and Dicke energies agree to $6 \times 10^{-4}$ for $t \in [0, \pi/(g\sqrt{N})]$, while at $g = 0.5\,\omega_0$ they differ by order 1 (counter_rotating_terms). The default task bound action_bound = 0.5 therefore lets an agent leave the RWA regime; set a smaller bound if you need the TC model to be physically faithful.
Symmetric subspace. Working in the $j = N/2$ sector is exact as long as the Hamiltonian and every dissipator are permutation invariant, which holds for the collective operators used here. Local (per-cell) decay or dephasing couples the symmetric sector to lower-$j$ sectors and is not supported in v0; a permutation-invariant (PIQS-style, Shammah et al. 2018) solver is planned.
Fock cutoff (read this before using rwa=False). For TC the excitation number is conserved, so $n_{\max} = n$ is exact (the default). For the Dicke model the counter-rotating terms create photons, and the default $n_{\max} = n + 4$ is a convenience, not a convergence guarantee. The cutoff that is needed grows with $N$ and with the coupling. Measured for the Ferraro protocol ($n = N$), comparing the stored energy with a reference at a much larger cutoff:

coupling	$N$	default $n_{\max} = n + 4$	converged return (cutoff needed)
$g = 0.1\,\omega_0$, 20 steps of `dt = 0.2`	2, 3, 4	max energy error $8.5\times10^{-6}$, $1.1\times10^{-4}$, $5.2\times10^{-4}$	adequate
$g = 0.1\,\omega_0$	6, 8	$4.2\times10^{-3}$, $4.3\times10^{-2}$	raise $n_{\max}$
$g = 0.5\,\omega_0$ (task default bound), 20 steps	2	return $1.017$	$1.0005$ ($n + 10$)
$g = 0.5\,\omega_0$	3	$1.754$ (15 percent off)	$1.530$ ($n + 24$)
$g = 0.5\,\omega_0$	6	$2.608$	$2.604$ ($n + 56$; $n + 24$ still gives $2.752$)

The $N = 6$ row shows that a return close to the converged value at the default cutoff can be accidental, and that a fixed "$n + 24$" rule does not generalise. The reliable criterion is the top_fock observable (population of the highest kept Fock level): raise n_max until its maximum over the episode is below about $10^{-4}$ and the return stops changing. In the table, top_fock reaches $0.07$ to $0.10$ at the default cutoff and $1.3\times10^{-4}$ for $N = 6$ at $n + 56$. The check cutoff_convergence covers the weak-coupling case $N = 2$, $g = 0.1\,\omega_0$ only. * Energies and rates are in units of $\omega_0$ (= 1 by default), times in $1/\omega_0$.

Parameters

field	default	meaning
`n_cells`	3	number of cells $N$
`n_photons`	`None` ($= N$)	initial cavity Fock number $n$ (Ferraro protocol: $n = N$)
`n_max`	`None` ($n$ if `rwa` else $n + 4$)	Fock cutoff (see above for `rwa=False`)
`omega0`	1.0	cell frequency, energy unit of $H_B$
`omega_c`	1.0	cavity frequency; detuning $\omega_c - \omega_0$
`rwa`	`True`	Tavis-Cummings (`True`) or Dicke (`False`)
`kappa`, `gamma`, `gamma_phi`	0.0	collective dissipation rates (any $> 0$ switches to Lindblad dynamics)

Observables reported in info: energy ($\langle H_B\rangle$ of the reduced cell state), ergotropy (of the reduced cell state with respect to $H_B$), n_photons ($\langle a^\dagger a\rangle$), excitation ($\langle \hat N_{\mathrm{exc}}\rangle$), top_fock (population of the Fock level $|n_{\max}\rangle$, a truncation monitor), plus the generic power and purity (of the full cells+cavity state) of BatteryCharging.

Control semantics

One control channel, the coupling $g(t)$, piecewise constant over each control period dt and clipped to $[-g_{\max}, g_{\max}]$ with $g_{\max} =$ action_bound (task default 0.5). Physically $g \ge 0$. Replacing $g \to -g$ for the whole protocol is a gauge symmetry: $U = (-1)^{a^\dagger a}$ maps $a \to -a$ and leaves every observable unchanged (also for the Dicke term, which is linear in $a + a^\dagger$), so negative actions are allowed. A sign change within a protocol is physical and can reverse charging: for $N = n = 1$ the steps $[+g, -g]$ with $g\,\mathrm{dt} = \pi/4$ return the cell to the ground state while $[+g, +g]$ charges it fully (tested). Setting $g = 0$ decouples the cells and holds the stored charge (closed system), which makes "charge, then switch off" the natural optimal strategy.

Task kwargs (qenvs.make("battery/Dicke-v0", ...)): n_cells, n_photons, n_max, omega0, omega_c, rwa, kappa, gamma, gamma_phi, horizon (20), dt (0.2), action_bound (0.5), reward ("energy").

Reward

Per-step gain of the stored quantity $Q$, so the undiscounted return equals $Q(\text{final}) - Q(\text{initial})$ with $Q = \langle H_B\rangle$ (reward="energy") or the ergotropy of the reduced cell state $\rho_B = \mathrm{tr}_{\mathrm{cav}}\,\rho$ with respect to $H_B$ (reward="ergotropy"). The return is bounded by $N\omega_0$. For closed dynamics the ergotropy is strictly below the energy whenever the cells and the cavity are entangled, so the two rewards are different objectives: for $N = 2$, constant $g = 0.5$ and $t = 1$ the energy is $0.784\,\omega_0$ but the ergotropy is only $0.277\,\omega_0$ (while the global state is pure, purity = 1). The two coincide only when the reduced cell state is pure, e.g. at full charge.

For Tavis-Cummings charging from a Fock state the reduced cell state stays diagonal in the Dicke basis (the excitation number is conserved), so its ergotropy is exactly zero until the cell populations become inverted. reward="ergotropy" is then a sparse reward with an exactly zero gradient on that plateau (for example $N = 2$, constant $g = 0.3$ for 6 steps of dt = 0.2 gives return 0). Use reward="energy" for early training or gradient-based optimisation and evaluate the ergotropy afterwards.

Closed-form landmarks (resonant TC, constant $g$):

$N = n = 1$ (Jaynes-Cummings): $P_e(t) = \sin^2(g t)$. At $g t = \pi/4$ the reduced cell state is $\mathrm{diag}(1/2, 1/2)$: energy $\omega_0/2$, ergotropy $0$. At $g t = \pi/2$ the cell is fully excited: ergotropy = energy = $\omega_0$ (reduced_ergotropy).
$N = n = 2$: in the two-excitation manifold the coupling matrix has eigenfrequencies $0$ and $\pm\sqrt{6}\,g$, so $E(t)/\omega_0 = \tfrac{2}{3}\sin^2(\sqrt{6}\,g t) + \tfrac{4}{9}\,(1 - \cos\sqrt{6}\,g t)^2$, with the first maximum $E_{\max} = \tfrac{8}{9}\,N\omega_0$ at $t^* = \pi/(\sqrt{6}\,g)$ (two_cells_closed_form).

With the Ferraro protocol ($n = N$ photons) and a constant coupling, the first energy peak reaches $N\omega_0$ only for $N = 1$; for $N = 2, 4, 8$ it reaches $0.89, 0.86, 0.88$ of $N\omega_0$, because the cells and the cavity do not fully swap excitations for $N > 1$. Later peaks were not searched; time-dependent couplings (the point of the task) can do better.

import jax.numpy as jnp
import qenvs

env = qenvs.make("battery/Dicke-v0", n_cells=3, horizon=20, dt=0.2)
# constant coupling g = 0.5 for the first 10 steps, then decouple and hold
actions = jnp.concatenate([jnp.full((10, 1), 0.5), jnp.zeros((10, 1))])
state, rewards = env.rollout(actions)
print(float(rewards.sum()))  # 2.58: stored energy in units of omega0 (max 3)

Validation

All numbers below are from qenvs.validation.dicke.CHECKS on this machine (dynamiqs 0.3.6, JAX 0.7.2, QuTiP 5.3.1, CPU). The analytic and cross-check integrations use a tight adaptive Tsit5 (rtol=1e-8, atol=1e-10) so that they measure physics rather than the default solver tolerance; the default solver is covered by solver_convergence (closed) and open_solver_convergence (Lindblad). Every check passes in float64 and in float32.

check	reference	tolerance	measured (float64)	measured (float32)
`operator_algebra`	$[J_+, J_-] = 2J_z$ for $N \le 6$; $[a, a^\dagger] = 1$ below the cutoff	1e-12	3.6e-15	3.6e-15
`jaynes_cummings_limit`	$N = 1$, $n \in \{1, 3\}$, resonant TC, $g = 0.5$: $P_e(t) = \sin^2(g\sqrt{n}\,t)$ (Jaynes-Cummings)	3e-6	1.1e-8	4.2e-7
`detuned_jaynes_cummings`	$N = 1$, $g = 0.5$, $(\omega_0, \omega_c, n) \in \{(1, 1.7, 1), (1, 0.3, 2), (1.3, 1.0, 1)\}$: $P_e = \frac{4g^2 n}{\Omega^2}\sin^2\frac{\Omega t}{2}$, $\Omega^2 = (\omega_c - \omega_0)^2 + 4g^2 n$, $\langle a^\dagger a\rangle = n - P_e$, and the coupling expectation $\langle V\rangle = (d/g)\,P_e$ with $d = \omega_c - \omega_0$ (odd in $d$, so it fixes the sign of the detuning)	3e-6	1.3e-8	7.2e-7
`two_cells_closed_form`	$N = n = 2$, $g = 0.5$: $t^* = \pi/(\sqrt{6}\,g)$ (grid of 601 points, parabolic refinement) and $E(t^*) = \tfrac{8}{9} N\omega_0$; error $= \max(	\Delta t^*	, 100\,	\Delta E_{\max}
`reduced_ergotropy`	$N = n = 1$: reduced cell state at $g t = \pi/4$ has $E = \omega_0/2$, ergotropy $0$; at $g t = \pi/2$ ergotropy $= E = \omega_0$	1e-6	3.8e-9	1.2e-7
`omega0_scaling`	resonant TC, $N = 2$, same random $g(t)$ with $\omega_0 = 1$ and $1.5$: energy and ergotropy scale exactly by $1.5$ (max abs residual over 10 env steps)	1e-6	1.3e-9	4.1e-8
`excitation_conservation`	closed TC, random piecewise-constant $g(t)$, $n_{\max} = 6 > n$: $\langle \hat N_{\mathrm{exc}}\rangle = n_{\mathrm{photons}}$ at every time, including $t = 0$ (renormalised state; norm drift reported in `detail`)	1e-6	1.3e-15	1.3e-15
`ergotropy_bounds`	$0 \le \mathcal{E}(\rho_B) \le \langle H_B\rangle \le N\omega_0$ along 15-step random rollouts (closed TC and open Dicke), plus $	\mathrm{tr}\rho - 1	$ for the open rollout; min gap $E - \mathcal{E}$ in `detail` ($1.7\times10^{-3}$ closed, $0.175$ open)	1e-6
`collective_advantage`	Ferraro et al. 2018: $t^(N)$ strictly decreasing, log-log slope of $t^$ vs $N$ within $-0.5 \pm 0.25$ for $N \in \{1, 2, 4, 8\}$, $g = 0.05$	0.25 (on the slope)	fitted exponent $-0.401$; $t^* = 31.42, 25.64, 18.37, 13.89$	$-0.401$
`rwa_weak_coupling`	Dicke ($n_{\max} = n + 6$) vs TC energy, $N = 2$, $g = 0.01\,\omega_0$, $t \in [0, \pi/(g\sqrt{N})]$	2e-3	5.8e-4	5.8e-4
`counter_rotating_terms`	same at $g = 0.5\,\omega_0$: energies must differ by $> 0.05$ (error = shortfall)	0	0 (difference 1.03)	0 (1.03)
`cutoff_convergence`	closed Dicke, $N = 2$, constant $g = 0.1\,\omega_0$, 20 steps of `dt = 0.2`: $\langle H_B\rangle(t)$ with $n_{\max} = 6$ (default) vs $12$; max `top_fock` population at $n_{\max} = 6$ in `detail` ($2.4\times10^{-5}$)	1e-3	8.5e-6	8.3e-6
`qutip_tavis_cummings`	open TC, $N = n = 3$, $\kappa = 0.1$, $\gamma = 0.05$, $\gamma_\phi = 0.03$, random $g(t)$: final $\rho$ vs QuTiP 5 `mesolve` (max abs element)	1e-5	5.8e-11	6.1e-8
`qutip_dicke`	open Dicke, $N = n = 3$, $n_{\max} = 7$, $\kappa = 0.1$, random $g(t)$: final $\rho$ vs QuTiP 5 `mesolve`	1e-5	5.8e-10	6.4e-8
`qutip_detuned`	open TC, $N = n = 2$, $\omega_0 = 0.8$, $\omega_c = 1.3$, $\kappa = 0.05$, $\gamma = 0.02$, random $g(t)$: final $\rho$ vs QuTiP 5 built with explicit $\omega_0$, $\omega_c$ (the coherences fix the sign of the detuning)	1e-5	4.0e-11	8.7e-8
`collective_dissipators`	$N = 1$, $g = 0$, no QuTiP: $\langle a^\dagger a\rangle = e^{-\kappa t}$ (cavity loss), $\langle H_B\rangle/\omega_0 = e^{-\gamma t}$ from $	e\rangle$ (emission), $2	\langle J_-\rangle	= e^{-\gamma_\phi t/2}$ from $(
`density_matrix_constraints`	open Dicke after random driving: Hermitian, trace 1, positive semidefinite	1e-6	1.1e-16	0
`solver_convergence`	default Tsit5 (env rollout) vs exact `Expm` propagator, closed Dicke, dim 32, 8 random steps	2e-5	4.4e-6	5.1e-6
`open_solver_convergence`	default Tsit5 vs `Tsit5(rtol=1e-10, atol=1e-12)`, open Dicke ($\kappa = 0.1$, $\gamma = 0.05$), dim 21, 8 random steps, final $\rho$	2e-5	5.4e-6	6.1e-6

Notes on the scaling check: the fitted exponent is $-0.40$ for $N \in \{1, 2, 4, 8\}$ and stays at about $-0.40$ when the fit is extended offline to $N = 16, 32, 48$ ($t^* = 10.60, 8.06, 6.87$). This is the finite-$N$ exponent of the TC model with $n = N$ photons at $g = 0.05\,\omega_0$ in this protocol (constant coupling, first energy peak). Ferraro et al. state an $N^{-1/2}$ law; whether this protocol approaches $-1/2$ at larger $N$ is not established here. The check therefore accepts a window $[-0.75, -0.25]$ rather than asserting $-0.5$. It is still discriminating: it fails if $t^*$ does not decrease with $N$ (no collective advantage).

Which check catches what (verified by mutating the model): a drift that uses $\omega_0$ for the cavity fails detuned_jaynes_cummings and qutip_detuned; a flipped detuning sign leaves every population unchanged and is caught by the $\langle V\rangle$ term of detuned_jaynes_cummings and by qutip_detuned (coherences); a reversed cavity-loss or emission operator or a wrong dephasing prefactor fails collective_dissipators (no QuTiP needed); an $H_B$ without $\omega_0$ fails detuned_jaynes_cummings, omega0_scaling and qutip_detuned; ergotropy evaluated on the full cells+cavity state fails reduced_ergotropy; a missing $+N/2$ offset fails excitation_conservation (and reduced_ergotropy, ergotropy_bounds if it is in $H_B$); a task factory that drops a kwarg fails test_task_factory_forwards_every_model_kwarg. The QuTiP cross-checks are the independent confirmation of the operator construction, the basis ordering, the jump-operator directions and the control sign: a reversed $J_-$, a factor 2 in $g$, or a mis-signed counter-rotating term changes the final density matrix at the $10^{-2}$ level. Without qutip installed the three QuTiP checks are reported as skipped (status SKIP in qenvs validate, skipped: true in --json) with an explanatory detail; they never count as passed.

Known limitations

Collective dissipation only. Local decay / dephasing (which breaks permutation symmetry) is not available; planned via a PIQS-style solver.
n_max for rwa=False must be converged by the user. The default $n + 4$ is adequate only for weak coupling and small $N$ ($|g| \lesssim 0.1\,\omega_0$, $N \le 4$); at the task default action_bound = 0.5 the needed cutoff grows with $N$ (see the table in Assumptions and validity). Raise n_max until top_fock stays below about $10^{-4}$.
The symmetric subspace is $N + 1$ dimensional, but the open-system Liouvillian is $((N+1)(n_{\max}+1))^2$ dimensional, so open-system episodes with large $N$ and large cutoffs are slow on CPU.
Only the coupling $g(t)$ is a control; cavity drive and detuning control are not part of -v0.
No thermal cavity or thermal cell bath in -v0 (initial cavity state is a pure Fock state).
The environment clips actions to $[-g_{\max}, g_{\max}]$ and casts them to float32.

References

D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, M. Polini, "High-Power Collective Charging of a Solid-State Quantum Battery", Phys. Rev. Lett. 120, 117702 (2018). Dicke battery, collective charging advantage.
M. Tavis, F. W. Cummings, "Exact Solution for an N-Molecule-Radiation-Field Hamiltonian", Phys. Rev. 170, 379 (1968).
R. H. Dicke, "Coherence in Spontaneous Radiation Processes", Phys. Rev. 93, 99 (1954).
G. M. Andolina, M. Keck, A. Mari, V. Giovannetti, M. Polini, "Quantum versus classical many-body batteries", Phys. Rev. B 99, 205437 (2019). Ergotropy and collective charging of Dicke batteries.
P. A. Erdman, G. M. Andolina, V. Giovannetti, F. Noé, "Reinforcement learning optimization of the charging of a Dicke quantum battery", Phys. Rev. Lett. 133, 243602 (2024), arXiv:2212.12397.
N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, F. Nori, "Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance", Phys. Rev. A 98, 063815 (2018). PIQS.
E. T. Jaynes, F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser", Proc. IEEE 51, 89 (1963). Single-cell limit used in jaynes_cummings_limit and detuned_jaynes_cummings.

coupling	\(N\)	default \(n_{\max} = n + 4\)	converged return (cutoff needed)
\(g = 0.1\,\omega_0\), 20 steps of `dt = 0.2`	2, 3, 4	max energy error \(8.5\times10^{-6}\), \(1.1\times10^{-4}\), \(5.2\times10^{-4}\)	adequate
\(g = 0.1\,\omega_0\)	6, 8	\(4.2\times10^{-3}\), \(4.3\times10^{-2}\)	raise \(n_{\max}\)
\(g = 0.5\,\omega_0\) (task default bound), 20 steps	2	return \(1.017\)	\(1.0005\) (\(n + 10\))
\(g = 0.5\,\omega_0\)	3	\(1.754\) (15 percent off)	\(1.530\) (\(n + 24\))
\(g = 0.5\,\omega_0\)	6	\(2.608\)	\(2.604\) (\(n + 56\); \(n + 24\) still gives \(2.752\))

rate	jump operator	meaning
`kappa`	\(\sqrt{\kappa}\, a\)	cavity photon loss
`gamma`	\(\sqrt{\gamma}\, J_-\)	collective (superradiant) emission
`gamma_phi`	\(\sqrt{\gamma_\phi}\, J_z\)	collective dephasing

field	default	meaning
`n_cells`	3	number of cells \(N\)
`n_photons`	`None` (\(= N\))	initial cavity Fock number \(n\) (Ferraro protocol: \(n = N\))
`n_max`	`None` (\(n\) if `rwa` else \(n + 4\))	Fock cutoff (see above for `rwa=False`)
`omega0`	1.0	cell frequency, energy unit of \(H_B\)
`omega_c`	1.0	cavity frequency; detuning \(\omega_c - \omega_0\)
`rwa`	`True`	Tavis-Cummings (`True`) or Dicke (`False`)
`kappa`, `gamma`, `gamma_phi`	0.0	collective dissipation rates (any \(> 0\) switches to Lindblad dynamics)

check	reference	tolerance	measured (float64)	measured (float32)
`operator_algebra`	\([J_+, J_-] = 2J_z\) for \(N \le 6\); \([a, a^\dagger] = 1\) below the cutoff	1e-12	3.6e-15	3.6e-15
`jaynes_cummings_limit`	\(N = 1\), \(n \in \{1, 3\}\), resonant TC, \(g = 0.5\): \(P_e(t) = \sin^2(g\sqrt{n}\,t)\) (Jaynes-Cummings)	3e-6	1.1e-8	4.2e-7
`detuned_jaynes_cummings`	\(N = 1\), \(g = 0.5\), \((\omega_0, \omega_c, n) \in \{(1, 1.7, 1), (1, 0.3, 2), (1.3, 1.0, 1)\}\): \(P_e = \frac{4g^2 n}{\Omega^2}\sin^2\frac{\Omega t}{2}\), \(\Omega^2 = (\omega_c - \omega_0)^2 + 4g^2 n\), \(\langle a^\dagger a\rangle = n - P_e\), and the coupling expectation \(\langle V\rangle = (d/g)\,P_e\) with \(d = \omega_c - \omega_0\) (odd in \(d\), so it fixes the sign of the detuning)	3e-6	1.3e-8	7.2e-7
`two_cells_closed_form`	\(N = n = 2\), \(g = 0.5\): \(t^* = \pi/(\sqrt{6}\,g)\) (grid of 601 points, parabolic refinement) and \(E(t^*) = \tfrac{8}{9} N\omega_0\); error $= \max(	\Delta t^*	, 100\,	\Delta E_{\max}
`reduced_ergotropy`	\(N = n = 1\): reduced cell state at \(g t = \pi/4\) has \(E = \omega_0/2\), ergotropy \(0\); at \(g t = \pi/2\) ergotropy \(= E = \omega_0\)	1e-6	3.8e-9	1.2e-7
`omega0_scaling`	resonant TC, \(N = 2\), same random \(g(t)\) with \(\omega_0 = 1\) and \(1.5\): energy and ergotropy scale exactly by \(1.5\) (max abs residual over 10 env steps)	1e-6	1.3e-9	4.1e-8
`excitation_conservation`	closed TC, random piecewise-constant \(g(t)\), \(n_{\max} = 6 > n\): \(\langle \hat N_{\mathrm{exc}}\rangle = n_{\mathrm{photons}}\) at every time, including \(t = 0\) (renormalised state; norm drift reported in `detail`)	1e-6	1.3e-15	1.3e-15
`ergotropy_bounds`	\(0 \le \mathcal{E}(\rho_B) \le \langle H_B\rangle \le N\omega_0\) along 15-step random rollouts (closed TC and open Dicke), plus $	\mathrm{tr}\rho - 1	$ for the open rollout; min gap \(E - \mathcal{E}\) in `detail` (\(1.7\times10^{-3}\) closed, \(0.175\) open)	1e-6
`collective_advantage`	Ferraro et al. 2018: \(t^(N)\) strictly decreasing, log-log slope of \(t^\) vs \(N\) within \(-0.5 \pm 0.25\) for \(N \in \{1, 2, 4, 8\}\), \(g = 0.05\)	0.25 (on the slope)	fitted exponent \(-0.401\); \(t^* = 31.42, 25.64, 18.37, 13.89\)	\(-0.401\)
`rwa_weak_coupling`	Dicke (\(n_{\max} = n + 6\)) vs TC energy, \(N = 2\), \(g = 0.01\,\omega_0\), \(t \in [0, \pi/(g\sqrt{N})]\)	2e-3	5.8e-4	5.8e-4
`counter_rotating_terms`	same at \(g = 0.5\,\omega_0\): energies must differ by \(> 0.05\) (error = shortfall)	0	0 (difference 1.03)	0 (1.03)
`cutoff_convergence`	closed Dicke, \(N = 2\), constant \(g = 0.1\,\omega_0\), 20 steps of `dt = 0.2`: \(\langle H_B\rangle(t)\) with \(n_{\max} = 6\) (default) vs \(12\); max `top_fock` population at \(n_{\max} = 6\) in `detail` (\(2.4\times10^{-5}\))	1e-3	8.5e-6	8.3e-6
`qutip_tavis_cummings`	open TC, \(N = n = 3\), \(\kappa = 0.1\), \(\gamma = 0.05\), \(\gamma_\phi = 0.03\), random \(g(t)\): final \(\rho\) vs QuTiP 5 `mesolve` (max abs element)	1e-5	5.8e-11	6.1e-8
`qutip_dicke`	open Dicke, \(N = n = 3\), \(n_{\max} = 7\), \(\kappa = 0.1\), random \(g(t)\): final \(\rho\) vs QuTiP 5 `mesolve`	1e-5	5.8e-10	6.4e-8
`qutip_detuned`	open TC, \(N = n = 2\), \(\omega_0 = 0.8\), \(\omega_c = 1.3\), \(\kappa = 0.05\), \(\gamma = 0.02\), random \(g(t)\): final \(\rho\) vs QuTiP 5 built with explicit \(\omega_0\), \(\omega_c\) (the coherences fix the sign of the detuning)	1e-5	4.0e-11	8.7e-8
`collective_dissipators`	\(N = 1\), \(g = 0\), no QuTiP: \(\langle a^\dagger a\rangle = e^{-\kappa t}\) (cavity loss), \(\langle H_B\rangle/\omega_0 = e^{-\gamma t}\) from $	e\rangle$ (emission), $2	\langle J_-\rangle	= e^{-\gamma_\phi t/2}$ from $(
`density_matrix_constraints`	open Dicke after random driving: Hermitian, trace 1, positive semidefinite	1e-6	1.1e-16	0
`solver_convergence`	default Tsit5 (env rollout) vs exact `Expm` propagator, closed Dicke, dim 32, 8 random steps	2e-5	4.4e-6	5.1e-6
`open_solver_convergence`	default Tsit5 vs `Tsit5(rtol=1e-10, atol=1e-12)`, open Dicke (\(\kappa = 0.1\), \(\gamma = 0.05\)), dim 21, 8 random steps, final \(\rho\)	2e-5	5.4e-6	6.1e-6