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advanced32 min studyLesson 476 in Physics

Solve a driven atom with relaxation

Track population, coherence, and saturation with explicit decay-rate and Rabi-frequency conventions.

Before you begin
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The question for today

What you’ll understand

  • Declare population and transverse coherence decay rates.
  • Derive the steady excited population and its saturation limit.
  • Distinguish a mixed dissipative steady state from pure Bloch-sphere geometry.

A continuously driven atom both exchanges amplitude with a field and loses excitation to its surroundings. Pure-state Rabi oscillations therefore cannot describe its long-time behavior. The optical Bloch equations couple the population difference to two coherence components and include separate longitudinal and transverse relaxation rates. Their steady solution explains why an ordinary coherently driven two-level atom approaches half excitation rather than complete inversion, and why increasing intensity broadens its resonant response.

Choose rates and a rotating-frame Hamiltonian

Use the population difference w as excited population minus ground population, and u and v as the expectations of the usual Pauli x and y operators. A real drive has Hamiltonian coupling hbar Omega divided by two. The detuning is drive frequency minus atomic frequency, giving a negative detuning term in the rotating-frame Hamiltonian. Defining these signs before writing the equations makes phase conventions and steady coherences reproducible.

Let Gamma be the excited-population decay rate and gamma two the transverse coherence decay rate. For radiative decay plus independent pure dephasing, gamma two equals half Gamma plus the dephasing rate. Population and amplitude decay rates therefore must not be given the same symbol without qualification. The Markov master-equation description assumes a suitable weak, memoryless environment and the rotating-wave drive approximation. Strong spectral memory or additional atomic levels requires a broader model.

Hrotℏ=−Δ2σz+Ω2σx,w=ρee−ρgg,γ2=Γ2+γϕ\frac{H_{\mathrm{rot}}}{\hbar}=-\frac\Delta2\sigma_z+\frac\Omega2\sigma_x,\qquad w=\rho_{ee}-\rho_{gg},\qquad \gamma_2=\frac\Gamma2+\gamma_\phi
Ω, Δ, Γ, γ₂, and γφ are in s⁻¹, with Δ=ωdrive−ωatom. σz has eigenvalue +1 on |e⟩. Ω is the population Rabi angular frequency on resonance without damping.↔ Scroll wide equations horizontally.

Coherence and population form one coupled system

Hamiltonian precession rotates the Bloch components, while damping contracts the transverse components toward zero and relaxes the population difference toward minus one. The drive converts population difference into coherence and coherence back into population. This coupling is why an incoherent two-rate picture misses transient Rabi oscillations even when it can approximate a steady population. It also makes clear that a finite excited population need not imply a pure superposition of ground and excited states.

Set the time derivatives to zero and solve the coherence equations in terms of w. Substitution into the population equation gives a saturation parameter proportional to drive amplitude squared and with a Lorentzian detuning dependence. The excited population is one half times that parameter divided by one plus the parameter. At weak drive it is proportional to intensity. At arbitrarily large steady drive it approaches one half because coherent excitation and stimulated de-excitation balance while decay still favors the ground.

u˙=Δv−γ2u,v˙=−Δu−Ωw−γ2v,w˙=Ωv−Γ(w+1),ρee=1+w2\dot u=\Delta v-\gamma_2u,\qquad \dot v=-\Delta u-\Omega w-\gamma_2v,\qquad \dot w=\Omega v-\Gamma(w+1),\qquad \rho_{ee}=\frac{1+w}{2}
u=⟨σx⟩ and v=⟨σy⟩ are dimensionless. These equations use the displayed rotating Hamiltonian and include population decay to |g⟩ plus transverse pure dephasing; the atomic density matrix can be mixed.↔ Scroll wide equations horizontally.

Saturation changes rates and linewidths

The fluorescence photon rate for a closed radiative transition is Gamma times excited population, so it approaches Gamma divided by two at strong steady drive. Detection captures only a fraction set by collection and efficiency. With no pure dephasing, the resonant saturation parameter is twice Omega squared divided by Gamma squared, and the population resonance broadens with the square root of one plus that parameter. The linewidth statement must name this steady population response and use the same angular-frequency convention.

The geometric lab always stays on a pure-state Bloch sphere. It can show that changing relative phase at fixed mixing leaves basis populations unchanged while rotating transverse coherence. It cannot display contraction into the sphere, radiative relaxation, saturation, or a mixed steady state. A half-excited steady atom is therefore not automatically the equatorial pure state shown by the lab. Its purity also depends on the magnitude of its coherences, which the open-system equations determine.

s=Ω2γ2Γ(Δ2+γ22),ρeess=s2(1+s),Rem=Γρeess≤Γ2s=\frac{\Omega^2\gamma_2}{\Gamma(\Delta^2+\gamma_2^2)},\qquad \rho_{ee}^{\mathrm{ss}}=\frac{s}{2(1+s)},\qquad R_{\mathrm{em}}=\Gamma\rho_{ee}^{\mathrm{ss}}\le\frac\Gamma2
s and ρee are dimensionless; Rem is in photons/s for a closed radiative transition. The steady solution assumes constant drive and rates. For γφ=0, s(0)=2Ω²/Γ² and the population-response FWHM is Γ√(1+s(0)) in angular-frequency units.↔ Scroll wide equations horizontally.

Work it through

Compute a radiatively broadened steady state

A closed two-level atom is driven on resonance with Γ=2.0×10⁶ s⁻¹, no pure dephasing, and Ω=1.0×10⁶ s⁻¹. Find γ₂, s, steady excited population, and emitted photon rate.

  1. Without pure dephasing the coherence rate is Γ/2=1.0×10⁶ s⁻¹. It is not the population decay rate.

  2. On resonance s=Ω²/(Γγ₂)=0.5. The excited population is 0.5/[2(1.5)]=1/6.

  3. The photon emission rate is Γ/6≈3.33×10⁵ s⁻¹. This is an emission rate into all allowed radiative directions, not the detected count rate of a finite collection system.

The result

γ₂=1.0×10⁶ s⁻¹, s=0.5, steady excitation is 1/6, and emission is about 3.33×10⁵ photons/s. A steady coherent drive does not fully invert this closed two-level atom.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Separate populations from coherence phase

For independent runs Reset and set BOTH mixing angle 90°, relative phase 0°; then Reset and set mixing 90°, phase 90°. Predict equal basis probabilities one half but different transverse Bloch directions. This pure two-state geometry has no damping or mixed-state saturation and does not solve the optical Bloch equations.

Separate populations from coherence phaseA normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Current readings: Probability |0⟩: 71.1%; Probability |1⟩: 28.9%; Bloch (x, y, z): 0.78, 0.45, 0.42. All parameters are adjustable with the labeled controls below.xyz|0⟩|1⟩Basis probabilitiesP(0)71.1%P(1)28.9%
Bloch vector / P(0)P(1)Phase latitude
Probability |0⟩
71.1%
Probability |1⟩
28.9%
Bloch (x, y, z)
0.78, 0.45, 0.42
t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forMatching populations does not establish matching coherence or purity.

Pause & reason

Check your understanding

For a closed coherently driven two-level atom under steady strong resonant drive, what limit follows from the optical Bloch model?

Your turn

Try it yourself

At fixed detuning and decay rates a drive gives s=1. Double its field amplitude. Find the new saturation parameter and steady excited population, and compare with the original population.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Rabi frequency follows field amplitude, while s follows its square.

Compare with the solution

Doubling field amplitude doubles Ω and quadruples s to 4. The new excited population is 4/[2(5)]=0.4. Originally it was 1/[2(2)]=0.25. The population does not quadruple because saturation changes the denominator. The result assumes the same closed transition and that stronger driving still respects the two-level and rotating-wave approximations.

Keep the idea

Take it with you

  • Population decay Γ and coherence decay γ₂ are different rates.
  • A steady closed coherently driven two-level atom saturates at half excitation.
  • Pure Bloch-sphere geometry cannot represent general dissipative steady states.

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