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advanced32 min studyLesson 480 in Physics

Relate intracavity dynamics to the measured output

Declare port phases and linewidths, derive reflection, and preserve vacuum noise while connecting cavity population to external photon flux.

Before you begin
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The question for today

What you’ll understand

  • Use a consistent sign convention for the Langevin and boundary equations.
  • Distinguish dimensionless cavity occupation from a traveling photon flux.
  • Compute resonance reflection and critical coupling with internal loss.

A detector outside a cavity measures the outgoing field, not the intracavity photon number directly. The outgoing amplitude contains both prompt incident-field reflection and radiation leaking from the resonator, so interference can cancel the output even when the cavity stores energy. Input–output theory relates those amplitudes while accounting for damping and the vacuum noise required by quantum commutators. We will use kappa as the photon-number decay rate and state the input–output phase convention explicitly.

Normalize a traveling field as a flux amplitude

The cavity annihilation operator is dimensionless and its number operator counts stored photons. A traveling input operator instead has a delta-function time commutator and units of inverse square-root second. Its normally ordered squared amplitude is photon flux. Multiplication by square root of a port decay rate converts that incoming flux amplitude into a cavity drive. Confusing these normalizations produces an incorrect steady photon number or a leaking flux missing a decay-rate factor.

Couple the cavity to an external measured port with energy decay kappa external and to an unobserved internal-loss bath with decay kappa internal. Their sum is the total photon-number decay. In a frame rotating at the drive frequency, define detuning as drive minus cavity frequency. Choose a plus sign for the incoming drive and a minus sign for emitted cavity amplitude in the boundary relation. Other port phase choices are allowed if both equations are transformed consistently.

a^˙=(iΔ−κ/2)a^+κex b^in+κi ξ^in,b^out=b^in−κex a^,κ=κex+κi\dot{\hat a}=(i\Delta-\kappa/2)\hat a+\sqrt{\kappa_{\mathrm{ex}}}\,\hat b_{\mathrm{in}}+\sqrt{\kappa_i}\,\hat\xi_{\mathrm{in}},\quad \hat b_{\mathrm{out}}=\hat b_{\mathrm{in}}-\sqrt{\kappa_{\mathrm{ex}}}\,\hat a,\quad \kappa=\kappa_{\mathrm{ex}}+\kappa_i
All κ rates and Δ=ωdrive−ωc are in s⁻¹; a is dimensionless while bin,bout,ξin have units s⁻¹ᐟ². This sign convention uses independent Markov input ports and a linear single cavity mode.↔ Scroll wide equations horizontally.

Solve the mean and then combine amplitudes

For a coherent incident beam, let beta be its mean input amplitude and take the internal bath to have zero mean. The steady cavity amplitude is obtained by dividing the drive by total amplitude damping minus i detuning. In a linear cavity with vacuum baths this gives a coherent intracavity state and occupation equal to the squared amplitude. Its output amplitude is the incident amplitude minus the leaking cavity amplitude, rather than a sum of their separately squared intensities.

On resonance the reflection coefficient is internal loss minus external coupling divided by their sum. A lossless one-sided cavity reflects all incident power with a phase reversal on resonance, despite nonzero stored energy. At equal external and internal rates the coherent reflected amplitude vanishes: incident energy is dissipated through the internal channel. If a second physical output mirror replaces internal absorption, that energy exits another port and must be included explicitly rather than labeled absorbed.

αss=κex βκ/2−iΔ,nˉ=∣αss∣2,r(Δ)=1−κexκ/2−iΔ,r(0)=κi−κexκi+κex\alpha_{\mathrm{ss}}=\frac{\sqrt{\kappa_{\mathrm{ex}}}\,\beta}{\kappa/2-i\Delta},\qquad \bar n=|\alpha_{\mathrm{ss}}|^2,\qquad r(\Delta)=1-\frac{\kappa_{\mathrm{ex}}}{\kappa/2-i\Delta},\qquad r(0)=\frac{\kappa_i-\kappa_{\mathrm{ex}}}{\kappa_i+\kappa_{\mathrm{ex}}}
|β|² is incident photon flux in s⁻¹, n̄ dimensionless cavity occupation, and r an amplitude reflection coefficient. These expressions assume a linear empty cavity, coherent stationary drive, and vacuum internal noise; reflected power fraction is |r|².↔ Scroll wide equations horizontally.

Damping alone cannot preserve a quantum mode

If the cavity operator were merely multiplied by an exponentially decaying factor, its equal-time commutator would decay as well, contradicting the bosonic algebra. The bath inputs supply exactly the noise contribution needed to preserve that commutator. Vacuum noise can therefore be essential even though its normally ordered incoming photon flux is zero. Omitting it may accidentally preserve a classical mean-field solution while making the quantum fluctuations inconsistent.

With the drive removed, stored photon number decays at kappa and mean amplitude at kappa divided by two. Under drive, however, the output flux includes interference with the incident beam and is not generally kappa external times stored occupation. The wave lab can display addition of two coherent amplitudes with a relative phase, but its controls do not set cavity linewidth, detuning, ports, or vacuum noise. It illustrates the boundary interference arithmetic and leaves the actual cavity dynamics to these equations.

[b^in(t),b^in†(t′)]=δ(t−t′),⟨n^(t)⟩=⟨n^(0)⟩e−κt,⟨a^(t)⟩∝e−κt/2[\hat b_{\mathrm{in}}(t),\hat b_{\mathrm{in}}^\dagger(t^{\prime})]=\delta(t-t^{\prime}),\qquad \langle\hat n(t)\rangle=\langle\hat n(0)\rangle e^{-\kappa t},\qquad \langle\hat a(t)\rangle\propto e^{-\kappa t/2}
The decay expressions are for an undriven linear cavity with vacuum inputs, in its rotating frame. Markov temporal fields have white-noise commutators; their integrated observables and finite detector bandwidths must be used for physical measurements.↔ Scroll wide equations horizontally.

Work it through

Balance reflected and internally lost photons

A resonant empty cavity has κex=3.0×10⁶ s⁻¹ and κi=1.0×10⁶ s⁻¹. A coherent input supplies 1.0×10⁶ photons/s. Find intracavity occupation, reflected flux, and internal-loss flux in steady state.

  1. Total photon decay is κ=4.0×10⁶ s⁻¹. On resonance n̄=4κex|β|²/κ²=4×3×1/16=0.75 photons.

  2. The reflection amplitude is (1−3)/(1+3)=−1/2, so reflected power fraction is one quarter. Reflected photon flux is 2.5×10⁵ s⁻¹, including prompt-leakage interference.

  3. Internal loss removes photons at κi n̄=7.5×10⁵ s⁻¹. Adding that to reflected flux returns the full input 1.0×10⁶ s⁻¹, as a stationary energy balance requires.

The result

The cavity stores mean occupation 0.75, reflects 2.5×10⁵ photons/s, and loses 7.5×10⁵ photons/s internally. Leakage intensity alone would give the wrong measured reflection.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Combine prompt and cavity amplitudes

For independent runs Reset and set BOTH phase 180°, amplitude ratio 1; then Reset and set phase 0°, ratio 1. Predict zero and doubled summed amplitudes. The fixed-frequency sine model illustrates coherent boundary interference; it contains no cavity, port rates, linewidth, or quantum bath noise.

Combine prompt and cavity amplitudesTwo waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Current readings: Sum at x = 0: 0.866; Resultant amplitude: 1.732; Phase difference: 60°. All parameters are adjustable with the labeled controls below.−3.14−1.5701.573.14−3.4−1.701.73.4x · radiansAmplitude
SumFirst componentSecond component
Sum at x = 0
0.866
Resultant amplitude
1.732
Phase difference
60°
t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). 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 forOutput detection combines amplitudes before squaring, so zero reflection can coexist with stored cavity energy.

Pause & reason

Check your understanding

Under a consistent single-port input–output convention, why can resonant reflection vanish at κex=κi?

Your turn

Try it yourself

A resonant cavity has κex=κi=2.0×10⁶ s⁻¹ and input flux 1.0×10⁶ photons/s. Find stored occupation and reflected flux. Then double both rates at fixed input flux and determine the new occupation and photon-number ringdown time.

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

Give me a hint

At critical coupling reflection vanishes; stored occupation scales inversely with a common scaling of both rates.

Compare with the solution

Initially κ=4.0×10⁶ s⁻¹ and n̄=4κexΦ/κ²=0.5, with zero coherent reflected flux in the ideal model. After doubling both rates κ=8.0×10⁶ s⁻¹ and n̄=0.25. The undriven photon-number ringdown time is 1/κ=125 ns. Critical coupling persists, while the field-amplitude ringdown time is twice the photon-number time.

Keep the idea

Take it with you

  • Traveling flux amplitudes and dimensionless cavity operators have different units.
  • Port phases and amplitude-decay conventions must remain consistent between equations.
  • Vacuum inputs preserve quantum commutators even when their normal-ordered flux vanishes.

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