Quantum Battery: Ergotropy Phase Portrait

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

This is a 2D-native view of the same quantum battery: instead of a 3D Bloch sphere, the coherence components (x, y) of the ensemble-averaged Bloch vector are plotted directly as a point moving in their own natural plane — the equatorial phase portrait of the optical Bloch equations. This plane is not a camera projection of a sphere; x and y are the two real dynamical variables that already live in a plane by definition, with the population z shown separately as a vertical gauge and color.

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 (Bloch vector s = (x, y, z), |s| ≤ 1):

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

Each cell is charged by a resonant drive and loses purity to pure dephasing at rate γ = 1/T₂:

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

The phase-portrait point (x, y) traces a decaying spiral around the origin — the radius of that spiral is exactly √(x²+y²), the transverse part of |s|, so its shrinkage toward the origin is a direct, literal picture of dephasing destroying extractable ergotropy. Charging power follows analytically: P = N·(dE/dt) = −N·Ω_eff·y/2.

N — number of identical battery cells (grid, top-left of the canvas).
Ω₀ / γ — bare Rabi charging rate and dephasing rate; higher γ shrinks the spiral 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.

Grid cell color = excited population Spiral = coherence (x,y) phase portrait Bars = stored energy E vs ergotropy W