Open two-level quantum battery

Task id: battery/TwoLevel-v0. Model: qenvs.models.qubit_battery.QubitBattery. Environment: qenvs.envs.BatteryCharging (shared with every model that defines a battery Hamiltonian).

The smallest quantum battery: one qubit whose excited state stores one quantum of energy \(\omega_0\). A classical drive charges it, a thermal bath at finite temperature discharges (and partially charges) it, and pure dephasing destroys the coherence that makes the stored energy extractable as work (ergotropy).

Physics

Rotating-frame Hamiltonian (\(\hbar = 1\)):

\[ H(t) = \frac{\Delta}{2}\,\sigma_z + \frac{u(t)}{2}\,\sigma_x , \qquad H_B = \omega_0\,|1\rangle\langle 1| . \]

\(H_B\) is the battery Hamiltonian used for the stored energy \(E = \operatorname{tr}(\rho H_B)\) and the ergotropy \(\mathcal{E} = \operatorname{tr}(\rho H_B) - \min_U \operatorname{tr}(U\rho U^\dagger H_B)\) (Allahverdyan, Balian and Nieuwenhuizen 2004). For a qubit with eigenvalues \(r_\pm\) of \(\rho\) (\(r_+ \ge r_-\)) this is \(\mathcal{E} = \omega_0 (p_1 - r_-)\).

Lindblad master equation \(\dot\rho = -i[H(t), \rho] + \sum_k \mathcal{D}[L_k]\rho\) with the thermal qubit dissipator of Breuer and Petruccione (2002, Sec. 3.4) at bath Bose occupation \(n_{\mathrm{th}}\) and coupling rate \(\gamma_1\), plus pure dephasing:

\[ L_\downarrow = \sqrt{\gamma_1 (n_{\mathrm{th}} + 1)}\;|0\rangle\langle 1| ,\qquad L_\uparrow = \sqrt{\gamma_1 n_{\mathrm{th}}}\;|1\rangle\langle 0| ,\qquad L_\phi = \sqrt{\gamma_\phi / 2}\;\sigma_z . \]

Without drive the populations obey the rate equation \(\dot p_1 = \gamma_1 n_{\mathrm{th}}\,p_0 - \gamma_1 (n_{\mathrm{th}} + 1)\,p_1\), so \(p_1(t)\) relaxes at rate \(\gamma_1 (2 n_{\mathrm{th}} + 1)\) to the Gibbs value

\[ p_1^{\mathrm{th}} = \frac{n_{\mathrm{th}}}{2 n_{\mathrm{th}} + 1}, \qquad \frac{p_1^{\mathrm{th}}}{p_0^{\mathrm{th}}} = \frac{n_{\mathrm{th}}}{n_{\mathrm{th}} + 1} = e^{-\beta \omega_0}. \]

Coherences decay at \(\gamma_1 (2 n_{\mathrm{th}} + 1)/2 + \gamma_\phi\). With \(n_{\mathrm{th}} = 0\) the model reduces exactly to TwoLevel with the same \(\gamma_1\) and \(\gamma_\phi\).

Closed, resonant, constant drive \(u\) from \(|0\rangle\): \(p_1(t) = \sin^2(u t / 2)\), hence \(E(t) = \omega_0 \sin^2(u t / 2)\), a \(\pi\)-pulse (\(\int u\,dt = \pi\)) stores \(E = \mathcal{E} = \omega_0\), and the instantaneous charging power is bounded by \(|dE/dt| \le \omega_0 u / 2\), with equality at \(u t = \pi/2\). Detuned: \(p_1(t) = (u/\Omega)^2 \sin^2(\Omega t/2)\), \(\Omega = \sqrt{u^2 + \Delta^2}\).

Basis and index conventions

Index	State	\(H_B\) energy
0	ground \(\lvert 0\rangle\)	0
1	excited \(\lvert 1\rangle\)	\(\omega_0\)

\(\sigma_z = \lvert 0\rangle\langle 0\rvert - \lvert 1\rangle\langle 1\rvert = \mathrm{diag}(1, -1)\), so \(\Delta > 0\) lowers the excited level in the rotating frame (same convention as TwoLevel). The emission operator is built explicitly as dq.basis(2, 0) @ dq.basis(2, 1).dag(); dynamiqs.sigmam() follows the QuTiP spin convention (index 0 = up) and would raise instead of lower here.

Closed models (gamma1 = gamma_phi = 0) carry kets and are integrated with sesolve; open models carry density matrices and use mesolve. The observation is the flattened real and imaginary parts of the state plus \(t / T\): 5 numbers closed, 9 open.

Assumptions and validity

Rotating-wave approximation: \(H(t)\) is the rotating-frame Hamiltonian; the drive frequency is absorbed into \(\Delta\) and the counter-rotating term is dropped. Valid for \(|u|, |\Delta| \ll \omega_0\). Note that the task default action_bound = 2.0 (in units of \(\omega_0\)) and the pi-pulse example sit outside this regime: in these units the results are the formal RWA solution, and Bloch-Siegert shifts and counter-rotating effects are not modelled. Lower action_bound (or raise omega0) when physical fidelity to a lab qubit matters.
Markovian (Lindblad) bath with flat spectral density around \(\omega_0\); the dissipator is the one of the undriven qubit and is applied unchanged while the drive is on (no dressed-state / Floquet master equation). Valid for \(\gamma_1, \gamma_\phi \ll \omega_0\) and weak drive.
Dephasing is modelled as a white-noise \(\sigma_z\) channel.
initial='thermal' is the Gibbs state of \(H_B\) at the bath occupation, i.e. the battery starts in equilibrium with its bath (zero ergotropy).

Parameters and units

Energies in units of \(\omega_0\) (default 1), time in units of \(1/\omega_0\), rates in units of \(\omega_0\), \(\hbar = 1\).

Field	Default	Meaning
`omega0`	1.0	battery level splitting, energy unit of \(H_B\)
`detuning`	0.0	\(\Delta\), drive-qubit detuning
`gamma1`	0.0	bath coupling rate; emission \(\gamma_1 (n_{\mathrm{th}}+1)\), absorption \(\gamma_1 n_{\mathrm{th}}\)
`n_th`	0.0	bath Bose occupation at \(\omega_0\) (0 = zero temperature)
`gamma_phi`	0.0	pure dephasing rate
`initial`	`'ground'`	`'ground'` (\(\lvert 0\rangle\)) or `'thermal'` (Gibbs state, needs `gamma1 > 0` or `gamma_phi > 0` when `n_th > 0`)

Negative rates, unknown initial values, and a mixed thermal state in a closed model raise ValueError at construction.

Control semantics and bounds

One control channel: the action \(u\) multiplies \(\sigma_x / 2\), so \(u\) is the Rabi frequency. Actions are held constant for one control period dt (piecewise-constant pulse) and clipped to \([-\text{action\_bound}, \text{action\_bound}]\) (default 2.0). The ODE integrator (adaptive Tsit5 by default) takes as many internal steps as it needs inside each dt.

Reward

reward='energy' (default): per-step gain of stored energy \(r_k = E(t_{k+1}) - E(t_k)\). reward='ergotropy': per-step gain of ergotropy. The undiscounted return telescopes to \(Q(T) - Q(0)\). info reports energy, ergotropy, p1, average power \(E(t)/t\) and purity every step.

import jax.numpy as jnp
import qenvs

env = qenvs.make("battery/TwoLevel-v0", horizon=10, dt=0.2, gamma1=0.05, n_th=0.2)
actions = jnp.full((env.horizon, env.action_dim), jnp.pi / (env.horizon * env.dt))
state, rewards = env.rollout(actions)  # pi-pulse; sum(rewards) = E_final - E_initial

Validation

All checks live in qenvs.validation.qubit_battery.CHECKS, run in tests/test_qubit_battery.py and are printed by qenvs validate. Measured errors below are from one run on CPU with x64 enabled (dynamiqs 0.3.6, jax 0.7.2, qutip 5.3.1); every check was also run in float32 and passed. The three closed-system analytic checks integrate with Tsit5(rtol=atol=1e-8) so the error reflects the model rather than the integrator; the default Tsit5() settings are validated by solver_convergence.

Check	Reference	Tolerance	Measured error (x64)
`thermal_steady_state`	zero drive from \(\lvert 0\rangle\), \(\gamma_1 = 0.8\), \(n_{\mathrm{th}} = 0.6\), \(\gamma_\phi = 0.3\), \(t = 30\): \(p_1 \to n_{\mathrm{th}}/(2 n_{\mathrm{th}}+1)\)	1e-5	8.0e-7
`thermal_state_stationary`	Gibbs initial state, zero drive, \(t \le 10\): \(\max_t \lVert \rho(t) - \rho(0) \rVert_{\max}\)	1e-6	7.8e-15
`pi_pulse_charging`	closed, \(\omega_0 = 1.3\), \(u t = \pi\): \(E = \mathcal{E} = \omega_0\)	1e-6	4.7e-9
`ergotropy_le_energy`	20 random open steps: \(0 \le \mathcal{E} \le E\) (max violation)	1e-7	0
`charging_power_bound`	closed, \(u = 1.4\), \(\omega_0 = 0.9\): \(\lvert \max_t \lvert dE/dt \rvert - \omega_0 u/2 \rvert\), 4th-order finite differences on 201 points	1e-4	9.1e-8
`env_return_equals_gain`	random open rollout: \(\lvert \sum_k r_k - (Q_T - Q_0) \rvert\) for energy and ergotropy	1e-6	0
`qutip_cross_check`	QuTiP 5 `mesolve`, same \(H(t)\) (12 random piecewise-constant steps, \(\Delta = 0.3\)) and jump operators (\(\gamma_1 = 0.4\), \(n_{\mathrm{th}} = 0.6\), \(\gamma_\phi = 0.15\)): \(\max \lvert \rho_{\mathrm{dq}} - \rho_{\mathrm{qt}} \rvert\)	1e-6	6.6e-8
`density_matrix_constraints`	random open rollout: hermiticity, \(\lvert \operatorname{tr}\rho - 1 \rvert\), negativity	1e-6	1.1e-16
`t1_decay_matches_two_level`	\(n_{\mathrm{th}} = 0\), from \(\lvert 1\rangle\): \(p_1 = e^{-\gamma_1 t}\) and equal to `TwoLevel`	1e-6	3.4e-7
`detuned_rabi`	closed, \(u = 1.2\), \(\Delta = 0.9\): \(p_1 = (u/\Omega)^2 \sin^2(\Omega t/2)\)	1e-6	5.4e-9
`thermal_state_passive`	Gibbs state: trace 1, \(E = \omega_0 p_1^{\mathrm{th}}\), \(\mathcal{E} = 0\)	1e-6	0
`solver_convergence`	open constant-drive rollout: default `Tsit5()` vs exact `Expm()`	1e-6	2.9e-8

Discrimination: the checks were run against deliberately broken models (control sign flipped, control halved or doubled, emission and absorption swapped, dq.sigmam() used for emission, dephasing rate doubled, detuning sign flipped or doubled, wrong Gibbs population, absorption channel dropped). Every mutant failed at least one check; the QuTiP cross-check alone catches the control-sign, detuning-sign and dephasing-rate errors, which population-only checks cannot see.

References

A. E. Allahverdyan, R. Balian and Th. M. Nieuwenhuizen, "Maximal work extraction from finite quantum systems", EPL 67, 565 (2004). Ergotropy.
R. Alicki and M. Fannes, "Entanglement boost for extractable work from ensembles of quantum batteries", Phys. Rev. E 87, 042123 (2013). Quantum batteries, passive states.
H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002), Sec. 3.4. Thermal qubit master equation, Bose occupation and the \(n_{\mathrm{th}} + 1\) / \(n_{\mathrm{th}}\) rates.
I. I. Rabi, "Space quantization in a gyrating magnetic field", Phys. Rev. 51, 652 (1937). Rabi formula used for the closed-system checks.

Known limitations

The bath dissipator is the undriven one; for strong drives the physical dissipator becomes drive-dependent (Floquet or dressed-state master equation), which this model does not include.
initial='thermal' with n_th > 0 requires an open model, since a mixed initial state needs the density-matrix integrator; a closed unitary charge from a thermal state is therefore not available in this version.
The environment casts actions to float32 before building \(H(t)\); the QuTiP cross-check uses the same rounded amplitudes.
qutip_cross_check needs the qenvs[dev] extra. Without QuTiP it is reported as skipped (status SKIP) with a "not run" detail rather than raising, so qenvs validate still runs; a skipped check never counts as passed.
Only one qubit: no collective effects. See battery/Dicke-v0 for \(N\)-qubit charging.