Skip to main content
Sandbox Physics
Cavity and waveguide quantum electrodynamics

L1 · Chapter 1

How an excitation moves from an atom into light

Sandbox Physics ·

An excited atom need not release its energy straight away. Coupled to a single cavity mode, it can pass the excitation back and forth. Connected to a long waveguide, the light can travel away. We first need to describe where the excitation is before asking what quantum state has been sent.

Keep just two possibilities

Prepare an excited atom beside an empty cavity. Keep one cavity mode, a pattern of electromagnetic oscillation that exchanges energy with the atom. With losses closed, there are only two possibilities: the atom is excited and the cavity is empty, or the atom is in its ground state and the cavity holds one photon.

Write these possibilities as ∣e,0⟩|e,0\rangle and ∣g,1⟩|g,1\rangle. The letter describes the atom and the number describes the cavity. Their complex amplitudes are cec_e and ccc_c. The probability of finding the atom excited is ∣ce∣2|c_e|^2. Because the measurement outcomes are exclusive, their probabilities add to one.

The angle of an amplitude arrow is not a direction in ordinary space. It determines whether alternative processes reinforce or cancel. Keeping only populations would discard information needed when light returns and is absorbed again.

∣ψ(t)⟩=ce(t)∣e,0⟩+cc(t)∣g,1⟩,∣ce∣2+∣cc∣2=1|\psi(t)\rangle=c_e(t)|e,0\rangle+c_c(t)|g,1\rangle,\qquad |c_e|^2+|c_c|^2=1
The single-excitation subspace reduces the problem to two complex numbers while preserving their relative phase.

An excitation oscillates between atom and cavity

Excitation probabilityPP
00.5101.573.14
Scaled timegtgt
  • Atom
  • Cavity
Original analytic reference for a resonant, lossless single-mode model. The sum of both curves is one.

Reading: Jaynes and Cummings · Sheremet et al.

Coupling turns the two amplitude arrows

Time is measured in seconds. The detuning Δ=ωa−ωc\Delta=\omega_a-\omega_c and coupling gg are angular frequencies in radians per second. A frame rotating at the cavity frequency removes their common rapid rotation. Choose the phase convention g>0g>0 and use the rotating-wave approximation, retaining terms that exchange one excitation. Coupling, detuning and decay rates must be small compared with the carrier frequencies; ultrastrong coupling requires the omitted terms.

A derivative gives the change of an amplitude per second. The imaginary factor on the right makes one amplitude turn in response to the other. An instant of constant population therefore need not be an instant of constant quantum state. Initially the atom amplitude is one and the cavity amplitude is zero.

c˙e=−iΔce−igcc,c˙c=−igce\dot c_e=-i\Delta c_e-igc_c,\qquad \dot c_c=-igc_e
A star means complex conjugation. It reflects an amplitude arrow across the real axis. The real part is its component along that axis.
d∣c∣2dt=2Re⁡(c∗c˙),ddt(∣ce∣2+∣cc∣2)=0\frac{d|c|^2}{dt}=2\operatorname{Re}(c^*\dot c),\qquad \frac d{dt}(|c_e|^2+|c_c|^2)=0
Insert the amplitude equations into the squared-magnitude derivatives. Exchange terms cancel, while pure detuning changes only phase, proving norm conservation.

Calculate the first complete transfer

On resonance, Δ=0\Delta=0. Differentiate the first equation and substitute the second. The atom amplitude obeys the same mathematical equation as a harmonic oscillator. The initial conditions select a cosine for the atom and a negative imaginary sine for the cavity. Their squared magnitudes exchange population.

Worked example: take g/(2π)=5 MHzg/(2\pi)=5\,\mathrm{MHz}. The first full transfer requires gt=π/2gt=\pi/2, giving t=50 nst=50\,\mathrm{ns}. The cavity population is then one. After the same interval again the excitation has returned to the atom. This result assumes a lossless single mode.

c¨e=−g2ce,Pe=cos⁡2(gt),Pc=sin⁡2(gt)\ddot c_e=-g^2c_e,\qquad P_e=\cos^2(gt),\qquad P_c=\sin^2(gt)
Population has half the period of an amplitude. State which quantity is oscillating before comparing a quoted frequency.

Detuning limits how much can be transferred

When the two frequencies differ, the atom amplitude acquires additional phase during the exchange. The arrows cannot keep adding in the direction that transfers population most efficiently. Oscillation continues, with a smaller maximum cavity population.

Solving the same equations gives angular frequency Ω=Δ2+4g2\Omega=\sqrt{\Delta^2+4g^2}. At Δ=2g\Delta=2g, the maximum cavity population is one half. Faster oscillations can accompany less transfer, so an earlier peak alone is not evidence of stronger coupling.

A waveguide introduces amplitudes at many positions. Each segment adds another possible location of the excitation. Extending two complex numbers into a chain of complex numbers prepares us to describe an open system.

Pc(t)=4g2Δ2+4g2sin⁡2 ⁣(Δ2+4g2 t2)P_c(t)=\frac{4g^2}{\Delta^2+4g^2}\sin^2\!\left(\frac{\sqrt{\Delta^2+4g^2}\,t}{2}\right)
Zero detuning restores complete exchange. Large detuning suppresses it.

Reading: Jaynes and Cummings · Sheremet et al.

Try the prediction

Pass an excitation into the cavity

  1. Watch the alternating populations and predict when they exchange on resonance.
  2. Change only detuning. Compare the maximum cavity population and the timing, then return to resonance.
  3. Explain the maximum transfer with the formula above and predict how loss would spoil the return.

Exercises and explanations

  1. If gg is halved, when does the first complete transfer occur?

    Read the explanation

    The time is inversely proportional to coupling, so fifty nanoseconds becomes one hundred nanoseconds. Coupling is an angular frequency; retain the factor of pi in the conversion.

  2. The atom population is one half. Does that determine which way the excitation moves next?

    Read the explanation

    No. The same populations can have different relative phases and therefore different subsequent exchange directions. The two complex amplitudes contain the missing information.

  3. Why are two amplitudes insufficient when the cavity is replaced by a long waveguide?

    Read the explanation

    Light can occupy many positions and modes. Their phases determine whether it can later return to the atom. The spatial amplitudes and their relative phases retain the information needed for a return.

Sources and calculation scope

  1. Jaynes and Cummings, Comparison of quantum and semiclassical radiation theories, Proc. IEEE 51, 89–109 (1963)

    Section II.B, pp. 93–95, Eqs. (35)–(36): single-mode exchange with detuning, checked in the scanned author archive at bayes.wustl.edu. Chapter 1 independently derives its two-amplitude rotating-wave, single-excitation limit.

  2. Sheremet et al., Waveguide quantum electrodynamics, Rev. Mod. Phys. 95, 015002 (2023)

    The theory overview and Appendix A place single-excitation propagation and input–output relations in their model context. Our two-amplitude examples and schematic figures are original teaching calculations.

FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy