Quantum Battery: Ergotropy & Coherent Charging

Charging Drive

Cells N 25

Rabi rate Ω₀ 1.20

Dephasing γ (1/T₂) 0.15

Battery State

Stored energy E

0.00 ħω

Ergotropy W

0.00 ħω

Charging power P

0.00 ħω/s

Ω_eff

1.20

Extractable vs trapped energy

W = 0%passive = 0%

How it works

A quantum battery stores energy in the excited-state population of two-level cells, but not all of that energy can be pulled back out with a unitary operation. The extractable part is the ergotropy (Allahverdyan, Balian & Nieuwenhuizen, 2004): the maximum work obtainable by any cyclic unitary acting on the cell's state.

W = Tr(ρH) − Tr(ρ_passive H)
ρ_passive: same eigenvalues as ρ, reordered so the
           largest population sits on the lowest energy level

For a single driven cell on the Bloch sphere (Bloch vector s = (x, y, z), energy axis z, |z|,|s| ≤ 1), this reduces to a closed form used directly here:

Stored energy   E_cell = (1 + z) / 2         [units of ħω]
Ergotropy       W_cell = (z + |s|) / 2       |s| = √(x²+y²+z²)
Passive energy  E_cell − W_cell = (1 − |s|) / 2  ≥ 0

Passive energy vanishes only for a pure state (|s| = 1). Each cell is charged by a resonant drive and loses purity to pure dephasing at rate γ = 1/T₂, obeying the standard optical Bloch equations:

dx/dt = −γx
dy/dt = Ω_eff·z − γy
dz/dt = −Ω_eff·y

Charging power follows analytically from the same equations: P = N·(dE/dt) = −N·Ω_eff·y/2.

N — number of identical battery cells (grid on the left of the scene).
Ω₀ / γ — bare Rabi charging rate and dephasing rate; higher γ decoheres the state faster, opening a gap between E and W.
Individual vs Collective — individual charging uses Ω_eff = Ω₀ per cell; collective (Dicke-enhanced) charging couples all N cells to one mode, giving Ω_eff = Ω₀√N — the origin of the super-extensive charging-power advantage reported for collective quantum batteries (Ferraro et al., PRL 2018; Campaioli et al., 2017).

Cube color/height = excited population Left bar = stored energy E Right bar = ergotropy W

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