Computation of equilibrium strategies can optimize the cavity detuning parameters for ergotropy protection in open quantum batteries.

Game-theoretic equilibrium strategies (specifically Nash equilibrium or Pareto-optimal solutions) applied to the optimization of cavity detuning parameters (Δ = ω_cavity - ω_qubit) in Jaynes-Cummings or Tavis-Cummings open quantum battery models will demonstrably preserve ergotropy (maximum extractable work) at levels ≥15% higher than unoptimized or heuristically-tuned detuning parameters under identical decoherence conditions (T1, T2 relaxation times in the range 1–100 μs), across at least three distinct environmental noise models (Markovian, non-Markovian, and structured reservoir).

Phase 1 — Numerical Simulation (Weeks 1–4): Implement open quantum battery model using QuTiP (Python) with Lindblad master equation solver. Define ergotropy functional E(ρ(t)) = Tr[Hρ(t)] - min_{U unitary} Tr[HUρ(t)U†]. Parameterize cavity detuning Δ as the control variable. Implement game-theoretic framework with two players: Player 1 (optimizer, maximizes ergotropy), Player 2 (environment, maximizes decoherence impact). Compute Nash equilibrium via iterated best-response or linear programming for zero-sum formulation.

Phase 2 — Benchmarking (Weeks 5–6): Compare equilibrium-optimized Δ* against: (a) Δ=0 (resonance), (b) random Δ sampling, (c) gradient descent optimization, (d) analytical dispersive limit approximation.

Phase 3 — Robustness Testing (Weeks 7–8): Vary N_qubits ∈ {1,2,4,8}, noise models ∈ {Markovian, non-Markovian (HEOM), structured bath}, temperature T ∈ {10, 50, 100 mK}, and coupling g ∈ {1, 10, 100} MHz.

Phase 4 — Statistical Validation (Week 9): Run 500 Monte Carlo trajectories per configuration. Compute confidence intervals, effect sizes (Cohen's d), and p-values.

# Experimental Validation Package — Core Implementation Sketch
# Ergotropy Protection via Game-Theoretic Cavity Detuning Optimization

import numpy as np
import qutip as qt
import nashpy as nash
from scipy.optimize import minimize
from itertools import product

# ── 1. SYSTEM PARAMETERS ──────────────────────────────────────────────────────
class QuantumBatteryConfig:
    omega_c = 2 * np.pi * 5.0e9   # cavity frequency (Hz)
    omega_q = 2 * np.pi * 5.0e9   # qubit frequency (Hz)
    g       = 2 * np.pi * 100e6   # coupling strength (Hz)
    kappa   = 2 * np.pi * 1e6     # cavity decay rate (Hz)
    gamma1  = 2 * np.pi * 0.1e6   # qubit relaxation rate (Hz)
    gamma_phi = 2 * np.pi * 0.05e6 # dephasing rate (Hz)
    N_fock  = 10                   # Fock space truncation
    N_qubits = 1                   # number of qubits

# ── 2. HAMILTONIAN CONSTRUCTION ───────────────────────────────────────────────
def build_hamiltonian(config, delta):
    """Jaynes-Cummings Hamiltonian with detuning delta = omega_c - omega_q"""
    a   = qt.tensor(qt.destroy(config.N_fock), qt.qeye(2))
    sm  = qt.tensor(qt.qeye(config.N_fock), qt.sigmam())
    sz  = qt.tensor(qt.qeye(config.N_fock), qt.sigmaz())
    
    H_cavity = (config.omega_c + delta) * a.dag() * a
    H_qubit  = 0.5 * config.omega_q * sz
    H_int    = config.g * (a.dag() * sm + a * sm.dag())
    return H_cavity + H_qubit + H_int

# ── 3. LINDBLAD COLLAPSE OPERATORS ────────────────────────────────────────────
def build_collapse_ops(config):
    a  = qt.tensor(qt.destroy(config.N_fock), qt.qeye(2))
    sm = qt.tensor(qt.qeye(config.N_fock), qt.sigmam())
    sz = qt.tensor(qt.qeye(config.N_fock), qt.sigmaz())
    return [
        np.sqrt(config.kappa)    * a,
        np.sqrt(config.gamma1)   * sm,
        np.sqrt(config.gamma_phi)* sz
    ]

# ── 4. ERGOTROPY CALCULATOR ───────────────────────────────────────────────────
def compute_ergotropy(rho, H):
    """
    E = Tr[H*rho] - min_{U} Tr[H * U*rho*U†]
    Minimum achieved by passive state: eigenvalues of rho sorted
    descending paired with eigenvalues of H sorted ascending.
    """
    rho_evals, rho_evecs = np.linalg.eigh(rho.full())
    H_evals, _           = np.linalg.eigh(H.full())
    
    # Sort: rho descending, H ascending
    rho_evals_sorted = np.sort(rho_evals)[::-1]
    H_evals_sorted   = np.sort(H_evals)
    
    E_current = np.real(np.trace(H.full() @ rho.full()))
    E_passive = np.real(np.dot(rho_evals_sorted, H_evals_sorted))
    return max(0.0, E_current - E_passive)

# ── 5. MASTER EQUATION SOLVER ─────────────────────────────────────────────────
def solve_open_battery(config, delta, t_max_factor=10):
    H   = build_hamiltonian(config, delta)
    c_ops = build_collapse_ops(config)
    
    # Initial state: qubit excited, cavity vacuum
    psi0 = qt.tensor(qt.basis(config.N_fock, 0), qt.basis(2, 0))
    rho0 = psi0 * psi0.dag()
    
    t_max = t_max_factor / config.gamma1
    tlist = np.linspace(0, t_max, 100)
    
    result = qt.mesolve(H, rho0, tlist, c_ops, [], 
                        options=qt.Options(rtol=1e-8, atol=1e-10))
    
    ergotropies = [compute_ergotropy(rho, H) for rho in result.states]
    return tlist, ergotropies

# ── 6. GAME-THEORETIC PAYOFF MATRIX ──────────────────────────────────────────
def build_payoff_matrix(config, delta_range, noise_levels):
    """
    Player 1 (optimizer): chooses delta
    Player 2 (environment): chooses noise amplification factor
    Payoff: ergotropy at t = 1/gamma (Player 1 maximizes, Player 2 minimizes)
    """
    n_delta = len(delta_range)
    n_noise = len(noise_levels)
    A = np.zeros((n_delta, n_noise))  # Player 1 payoff matrix
    
    for i, delta in enumerate(delta_range):
        for j, noise_factor in enumerate(noise_levels):
            cfg_modified = QuantumBatteryConfig()
            cfg_modified.gamma1   *= noise_factor
            cfg_modified.gamma_phi *= noise_factor
            cfg_modified.kappa    *= noise_factor
            
            tlist, ergs = solve_open_battery(cfg_modified, delta)
            # Ergotropy at t = 1/gamma (index ~10 out of 100)
            A[i, j] = ergs[10]
    
    return A

# ── 7. NASH EQUILIBRIUM COMPUTATION ──────────────────────────────────────────
def find_nash_equilibrium(A):
    """Zero-sum game: B = -A for Player 2"""
    B = -A
    game = nash.Game(A, B)
    equilibria = list(game.support_enumeration())
    
    if not equilibria:
        raise ValueError("No Nash equilibrium found — check payoff matrix")
    
    # Return first equilibrium (mixed strategy)
    sigma1, sigma2 = equilibria[0]
    delta_idx = np.argmax(sigma1)  # Pure strategy approximation
    return delta_idx, sigma1, sigma2

# ── 8. MAIN VALIDATION PIPELINE ──────────────────────────────────────────────
def run_validation(config):
    delta_range  = np.linspace(-10*config.g, 10*config.g, 20)
    noise_levels = np.linspace(0.5, 2.0, 10)
    
    print("Building payoff matrix...")
    A = build_payoff_matrix(config, delta_range, noise_levels)
    
    print("Computing Nash equilibrium...")
    delta_idx, sigma1, sigma2 = find_nash_equilibrium(A)
    delta_star = delta_range[delta_idx]
    print(f"Equilibrium detuning: Δ* = {delta_star/(2*np.pi*1e6):.2f} MHz")
    
    # Benchmark comparisons
    strategies = {
        'Nash_Optimized': delta_star,
        'Resonance':      0.0,
        'Random':         np.random.choice(delta_range),
        'Dispersive':     10 * config.g,  # dispersive limit
    }
    
    results = {}
    for name, delta in strategies.items():
        tlist, ergs = solve_open_battery(config, delta)
        results[name] = {
            'tlist': tlist,
            'ergotropies': ergs,
            'E_at_decoherence': ergs[10],
            'time_averaged_E': np.mean(ergs)
        }
        print(f"{name}: E(t=1/γ) = {ergs[10]:.4f}")
    
    return results, delta_star, A

# ── 9. STATISTICAL VALIDATION ─────────────────────────────────────────────────
def monte_carlo_validation(config, delta_star, n_trajectories=500):
    from scipy.stats import mannwhitneyu
    
    ergs_optimized = []
    ergs_baseline  = []
    
    for _ in range(n_trajectories):
        # Add parameter noise (±5% variation)
        cfg_noisy = QuantumBatteryConfig()
        cfg_noisy.g      *= (1 + 0.05 * np.random.randn())
        cfg_noisy.kappa  *= (1 + 0.05 * np.random.randn())
        cfg_noisy.gamma1 *= (1 + 0.05 * np.random.randn())
        
        _, ergs_opt = solve_open_battery(cfg_noisy, delta_star)
        _, ergs_res = solve_open_battery(cfg_noisy, 0.0)  # resonance baseline
        
        ergs_optimized.append(ergs_opt[10])
        ergs_baseline.append(ergs_res[10])
    
    stat, p_value = mannwhitneyu(ergs_optimized, ergs_baseline, 
                                  alternative='greater')
    
    # Cohen's d
    d = (np.mean(ergs_optimized) - np.mean(ergs_baseline)) / \
        np.sqrt((np.std(ergs_optimized)**2 + np.std(ergs_baseline)**2) / 2)
    
    print(f"Mann-Whitney U p-value: {p_value:.4e}")
    print(f"Cohen's d: {d:.3f}")
    print(f"Mean ergotropy improvement: "
          f"{100*(np.mean(ergs_optimized)-np.mean(ergs_baseline))/np.mean(ergs_baseline):.1f}%")
    
    return p_value, d, ergs_optimized, ergs_baseline

# ── 10. ENTRY POINT ───────────────────────────────────────────────────────────
if __name__ == "__main__":
    config = QuantumBatteryConfig()
    results, delta_star, payoff_matrix = run_validation(config)
    p_val, cohens_d, ergs_opt, ergs_base = monte_carlo_validation(
        config, delta_star, n_trajectories=500
    )
    
    # SUCCESS CHECK
    improvement = (np.mean(ergs_opt) - np.mean(ergs_base)) / np.mean(ergs_base)
    success = (p_val < 0.01) and (cohens_d >= 0.8) and (improvement >= 0.15)
    print(f"\nVALIDATION {'PASSED' if success else 'FAILED'}")
    print(f"Improvement: {100*improvement:.1f}% | p={p_val:.3e} | d={cohens_d:.2f}")