FR12 · Single-photon optomechanics

Optomechanical Photon Blockade

Can one photon move a mirror enough to keep a second photon out? Yes, if a single photon shifts the cavity resonance by more than its width. Radiation pressure changes the mechanical equilibrium, so a second photon at the first photon’s frequency can miss resonance. Phonon-assisted paths can let pairs through again.

Tune a weak laser across a cavity with a movable mirror. Compare the nonlinear photon-pair response with an uncoupled cavity, and watch phonon sidebands open extra resonances.

Model assumptions

A zero-temperature, weak-drive quantum calculation retains photon sectors through two photons and 40 mechanical Fock states. Mechanical relaxation is neglected during the optical response. Frequencies use the mechanical angular frequency; cavity linewidth is the energy decay rate. The geometry and displacement are schematic. Curves are theoretical predictions, and photon antibunching does not establish a nonclassical mechanical state.

A MOVABLE MIRROR / A CHANGED RESONANCE

Schematic geometry and enlarged mirror motion. Drag the mirror to tune the cavity. 01 laser · 02 cavity · 03 splitter · 04/05 counters. Light ribbons show the steady field envelope; they are not detected photon events.

Photon-pair correlation

g^{(2)}(0)=0.329

Normalized cavity excitation

S=0.783

Leading weak-drive occupation

\bar n=1.76\times10^{-4}

Single-photon energy shift

U/\Omega_m=0.250

At this resonance, photon pairs are suppressed.

The mechanical ladder leaves sidebands

Cyan: nonlinear quantum response. Gray: uncoupled coherent cavity at the same bare detuning and linewidth. Gold marks the selected frequency.

$\Delta_0/\Omega_m\ \longrightarrow\qquad S$

A bright peak need not admit pairs

Coral: equal-time pair correlation from the Fock calculation. The coherent reference is one. The vertical axis is logarithmic.

$\Delta_0/\Omega_m\ \longrightarrow\qquad g^{(2)}(0)$

Why the first and second photons differ

Radiation pressure shifts the mechanical equilibrium by an amount proportional to photon number. Completing the square in the oscillator Hamiltonian lowers the energy by photon number squared. Mechanical excitations add extra routes into the cavity.

E_{n,m}/\hbar=-\Delta_0n-(g_0^2/\Omega_m)n^2+\Omega_m m

This optical antibunching test does not reconstruct the mechanical state. A nonclassical mechanical claim needs a separate mechanical witness.

Rabl (2011), Eqs. 1, 3, 8, 11–13 and Fig. 4(a)

Study quantum optomechanics from moving mirrors to single photons

Start with the essentials

Focus question: Can one photon move a mirror enough to keep a second photon out?
One-sentence intuition: The radiation-pressure energy shift grows as photon number squared, so the first and second photons generally need different resonance frequencies.

Core mathematical model

The displaced mechanical ladder

\frac{E_{n,m}}{\hbar}=-\Delta_0 n-\frac{g_0^2}{\Omega_m}n^2+\Omega_m m

The integers count photons and phonons in the displaced oscillator basis. Detuning is laser frequency minus bare cavity frequency.

Counting photon pairs relative to coherent light

g^{(2)}(0)=\frac{\langle a^{\dagger2}a^2\rangle}{\langle a^\dagger a\rangle^2}

The coherent reference is one. Values below one mean fewer coincident photon pairs at the same mean intensity.

Common difficulties

Which system is nonclassical?

Typical misconceptionPhoton antibunching proves that the mirror is in a mechanical cat state.

Better mental modelAntibunching is a statement about the optical field. A mechanical nonclassicality claim requires an appropriate mechanical observable or state reconstruction.

Run the experiment

01
Compare the first two resonances
Compare One photon fits and Pairs find a resonance. Follow the gold marker on the correlation curve.
What to observe: A small change in laser frequency opens the two-photon path. The excitation spectrum alone does not measure pair suppression.
02
Resolve the mechanical sideband
Choose The sideband opens, then broaden the cavity linewidth.
What to observe: A photon can enter with a mechanical excitation. Broadening the optical resonance merges these distinct paths.
03
Remove the interaction
Choose An uncoupled cavity, or reduce the single-photon coupling to zero.
What to observe: The excitation becomes a Lorentzian and the pair correlation returns to the coherent value one.

Optomechanical Photon Blockade

The mechanical ladder leaves sidebands

A bright peak need not admit pairs