At a glance
The state of a laser-driven two-level atom is one arrow on the Bloch sphere. This sim lets you set the laser's strength and its detuning from resonance, and shows how the arrow's path and the excited-state population Pe respond. The same picture describes a qubit, an atomic clock and a spin in an NMR magnet.
What you'll learn
- How an atom's state becomes an arrow on a sphere, and what the laser does to that arrow.
- How the laser strength Ω and the detuning δ together set the arrow's turning axis and turning rate.
- How to read Pe(t): how high it can get and how often it repeats.
- How drive strength and tuning compare as ways to excite the atom.
Where every run starts: the arrow on |g⟩, the laser axis (pink) in the equator.
What's on screen
- State arrow (gold): the atom's state. South pole |g⟩, north pole |e⟩.
- Laser axis (pink): the line the laser turns the arrow about.
- Trail (teal): the arrow tip's path over the last 1¼ turns.
- Strip-chart: Pe against time, last 10 µs. The dashed red line is the cap (section 05).
- Energy levels: |g⟩, |e⟩ and the laser's photon energy ħωL as a pink arrow. A gap between its tip and |e⟩ is ħδ, drawn magnified ×10⁸.
- Live readouts: Pe, the effective Rabi rate and the max population cap. The Guided Inquiry reveals them step by step.
See itRun the opening settings. Watch for: the gold arrow turning about the pink axis, with the teal trail following its tip.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Laser strength Ω/2π | How hard the laser drives the atom (the Rabi frequency Ω, in MHz). Moving it restarts the run from |g⟩. | 0.2–2 MHz opens at 0.5 | |
| Detuning δ/2π | How far the laser frequency sits from the atom's resonance, δ = ωL − ω₀. The ⌖ δ=0 button inside it snaps back to resonance. Moving it restarts the run. | −3 to +3 MHz opens at 0 | |
| View azimuth | Spins the camera around the sphere. The physics does not change. | 0°–360° opens at 35° | |
| View elevation | Tilts the camera up or down. The physics does not change. | −80° to +80° opens at 20° | |
| Live readouts | The excited population Pe, the effective Rabi rate Ω′/2π, and the max population cap. | panel | |
| ▶ Play / ⏸ Pause | Runs the atom, or freezes it so you can read Pe at one instant. | top bar | |
| Speed | Plays the run faster or slower on screen. The physics does not change. At 1× one µs of run time takes about one second. | 0.25×–4× opens at 1× | |
| ↻ Reset | Puts Ω/2π back to 0.5 MHz and δ/2π to 0, and restarts the run. The two view sliders keep their values. | top bar |
Fixed: a closed two-level atom with no decay, in the rotating-wave approximation. Every run starts in |g⟩.
The laser turns the arrow about an axis
A pure state is a unit arrow b = (bx, by, bz). Its height gives the population, Pe = (1 + bz)/2, so Pe = 0 at the south pole |g⟩ and 1 at the north pole |e⟩. In the frame that rotates with the laser, and keeping only the near-resonant part of the field, the drive is constant:
Ω is the drive strength and δ = ωL − ω₀ the detuning: the two sliders.
A constant drive turns the arrow rigidly about a fixed axis n̂ at angular rate Ω′, starting from |g⟩. Section 04 gives n̂ and Ω′.
One rigid rotation, always from the ground state.
On resonance (δ = 0), n̂ lies along x in the equator and Ω′ = Ω. The arrow then runs along a great circle through both poles: it reaches |e⟩ and returns once per turn. At Ω/2π = 0.5 MHz a turn takes 1/(0.5 MHz) = 2 µs, with Pe = 1 first at 1 µs. Changing Ω changes the turning rate and nothing else.
Key idea. Driving the atom means turning its arrow. Pe is only the height of the tip, which is what the strip-chart plots.
See itStay on resonance and set Ω/2π to 2 MHz. Watch for: the chart's peaks still touching 1, now four times as often as at 0.5 MHz.
Detuning tips the axis
Off resonance, the axis leaves the equator and the turning rate grows:
The axis n̂ and its tilt θ above the equator (negative means below).
Ω′ is never smaller than Ω.
- Tilt. θ = 0 at δ = 0. For δ > 0 the +x end of the axis dips below the equator, and for δ < 0 it rises above. θ depends only on the ratio δ/Ω.
- Path. The arrow still starts at |g⟩, which is now off the axis, so it circles the axis on a circle through |g⟩ of radius Ω/Ω′. The more tilted the axis, the smaller the circle.
- Reach. That circle passes through |e⟩ only if |e⟩ and |g⟩ are equally far from the axis, which needs the axis in the equator. For δ ≠ 0 the arrow never reaches |e⟩.
See itDetune by δ/2π = 1 MHz at Ω/2π = 0.5 MHz. Watch for: the pink axis tipping, and the teal trail shrinking to a small circle near |g⟩.
The ceiling and the rate of Pe
Reading the height of the tip off the turning motion gives the strip-chart's curve:
One factor sets how high, the other how often.
The dashed cap line on the chart.
- Ceiling. The first factor is the cap, the square of the circle's radius Ω/Ω′.
- Rate. The sin² factor repeats every 2π/Ω′, which is 1/(Ω′/2π) in time. Pe first peaks at half of that.
- Opposite trends. As |δ| grows, Ω′ grows: the cap falls and the rate rises.
Ω/2π
δ/2π
These sliders move the sim too.
| Detuning, at Ω/2π = 0.5 MHz | Ω′/2π | one turn takes | cap |
|---|---|---|---|
| 0 | 0.500 MHz | 2.000 µs | 1.000 |
| 0.5 MHz | 0.707 MHz | 1.414 µs | 0.500 |
| 1 MHz | 1.118 MHz | 0.894 µs | 0.200 |
| 2 MHz | 2.062 MHz | 0.485 µs | 0.059 |
See itRun δ = 0, then δ/2π = 1 MHz, each for two peaks. Watch for: the second run topping out at 0.20 instead of 1, with its peaks closer together.
A stronger drive widens the resonance
At fixed Ω the cap depends only on δ: Pemax = 1/(1 + (δ/Ω)²). It is a peak of height 1 and half-width Ω, falling to ½ at |δ| = Ω.
- Strength widens the peak and leaves its top at 1. At δ/2π = 1 MHz the cap is 0.20 for Ω/2π = 0.5 MHz and 0.80 for 2 MHz. At δ = 0 it is 1 for both.
- A narrow peak is selective. Keeping the cap above 0.99 needs |δ| ≤ Ω/√99 ≈ 0.1 Ω, which is 50 kHz at Ω/2π = 0.5 MHz. A wider peak forgives more detuning but responds less sharply to a small frequency error. Spectroscopists call this widening by a stronger probe power broadening.
See itHold δ/2π = 1 MHz and compare Ω/2π = 0.5 MHz with 2 MHz. Watch for: the cap readout, and the dashed line on the chart, rising from 0.200 to 0.800.
Where it matters
- Qubit gates. Any two-level system, such as a superconducting or trapped-ion qubit, obeys these equations. A resonant drive lasting π/Ω turns |g⟩ into |e⟩, and a detuned one falls short. The sim keeps the drive on, so Pe first peaks at π/Ω and the arrow keeps turning.
- Atomic clocks and spectroscopy. A laser is tuned to an atomic resonance. The narrower the response to δ, the better a small drift shows, and a strong probe broadens the line it measures.
- NMR and MRI. A spin driven by a radio-frequency field follows the same rotation. It tips fully only on resonance, which MRI uses to excite a chosen slice.
Go deeper: the rotating frame and the closed-form solution
The three components the sim evaluates every frame. No numerical integration is involved.
- The sphere is drawn in the frame turning at the laser frequency. The part of the field that nearly matches the atom becomes the constant drive above. The rest oscillates at about twice the transition frequency and averages out: the rotating-wave approximation, excellent when Ω and δ are MHz and the transition is near 400 THz.
- bz is the population difference; bx and by carry the phase relative to the laser.
- Nothing decays here, so the oscillation never dies down. A real atom also decays and dephases.
Summary
In one line. The laser turns the atom's Bloch arrow about a fixed axis at rate Ω′; detuning tips that axis, which lowers the ceiling on Pe and raises the rate.
Key points
- Pe = (1 + bz)/2 is the height of the arrow's tip.
- The axis is (Ω, 0, −δ)/Ω′ and the turning rate is Ω′ = √(Ω² + δ²).
- Pe peaks at Ω²/(Ω² + δ²) once every 1/(Ω′/2π).
- Only δ = 0 reaches |e⟩. A stronger drive widens the resonance, not its top.
Common mix-ups
- "A stronger laser excites the atom more." On resonance the flop is already complete, so Ω sets only the rate. Off resonance it raises the cap, which still reaches 1 only at δ = 0.
- "Detuned, the atom still reaches |e⟩; it just takes longer." It never does, and the oscillation gets faster, not slower.
- "Red and blue detuning behave differently." Pe depends on δ², so ±δ give the same cap and rate. Only the direction of the tilt, and the sign of bx, flip.
Things to try on your own
- At Ω/2π = 0.5 MHz, find the largest δ/2π, in steps of 0.05 MHz, with the cap at least 0.9. Repeat at 1 MHz and compare with Ω/3.
- Choose Ω and δ so the cap is about 0.5 and one turn takes about 1 µs.
- Set δ/2π = +1 MHz, then −1 MHz. Compare the cap, the rate and the lean of the trail.