QCL Subband Eigenstates & Population-Inversion Dynamics (2D)

Quantum-Well Design

Well width L 5.5 nm

Bias per period 260 mV

Lattice temperature 77 K

Injection current density 800 A/cm²

Laser State

BELOW THRESHOLD

Photon energy (FD solver)

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Emission wavelength

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Population inversion ΔN(t)

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Lower-level lifetime τ₁

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FD grid vs analytic E₁

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Time to steady state

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How it works

Same physics as the 3D cascade view, computed two genuinely different ways instead of animated as a moving staircase. Top panel: a finite-difference discretization of the 1D Schrödinger equation turns the quantum well into an N×N tridiagonal Hamiltonian matrix, diagonalized on-device with the classical Jacobi eigenvalue algorithm (real numerical linear algebra, not the closed-form particle-in-a-box formula) to obtain E₁, E₂ and the actual probability-density curves |ψ₁(x)|², |ψ₂(x)|²:

H·ψ = E·ψ,  H tridiagonal: Hᵢᵢ=2t, Hᵢ,ᵢ₊₁=−t,  t=ħ²/(2m*Δx²)
solved via cyclic Jacobi rotations until off-diagonal norm → 0

Bottom panel: the two-level rate equations are integrated forward in time with RK4 instead of solved algebraically for their instantaneous steady state, so you see the actual transient — N₂(t) and N₁(t) climbing from zero and settling — every time you move a slider:

dN₂/dt = R_pump·η(V) − N₂/τ₂
dN₁/dt = N₂/τ₂ − N₁/τ₁
ΔN(t) = N₂(t) − N₁(t)

Well width — rebuilds the Hamiltonian; the solver re-diagonalizes and the wavefunction panel redraws with new E₁, E₂ and photon energy.
Bias per period — sets injection efficiency η(V), a Gaussian resonance peaking at the design alignment, which scales the pump term feeding N₂.
Lattice temperature — lengthens τ₁ (LO-phonon backfilling), visibly slowing how fast the lower level drains once you push T up.
Injection current density — sets the pump rate; watch N₂(t) rise and re-settle to a new plateau when you change it.

The "FD grid vs analytic E₁" readout cross-checks the numerical eigensolver against the textbook infinite-well formula E_n = n²π²ħ²/(2m*L²) live, confirming the two independent computations agree to within numerical grid error.

|ψ₂(x)|² — upper subband |ψ₁(x)|² — lower subband N₂(t), N₁(t) traces ΔN(t), threshold line