---
title: "Read photon blockade through Rabl’s level diagram"
description: "Read the setup, displaced levels, sideband spectrum and photon correlations in Rabl 2011 v2, then recalculate a bounded weak-drive curve."
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-optomechanics/reading-photon-blockade/"
last_modified: "2026-10-28"
---

[Quantum optomechanics](https://sandboxphysics.com/stories/quantum-optomechanics/)

L4 · Chapter 9

# Read photon blockade through Rabl’s level diagram

Sandbox Physics · 2026-10-28

Open Rabl’s The photon blockade effect in optomechanical systems, arXiv:1102.0278v2. Figure 2 explains why the first and second photons meet different resonances. Figure 3 shows single-photon sidebands. Figure 4 tests two-photon statistics. Following that order separates a frequency shift from antibunching and prepares a reproducible, limited theoretical comparison.

## Before you begin

Single-photon coupling from Chapter 3 and state distinctions from Chapter 8. Basic complex linear algebra is needed for the numerical step. Fock states have definite photon or phonon number. Detuning and linewidth conventions are defined below.

- [How strong light enhances a weak interaction](https://sandboxphysics.com/stories/quantum-optomechanics/linearization-and-sidebands/)
- [Where backaction goes when you watch one quadrature](https://sandboxphysics.com/stories/quantum-optomechanics/backaction-evasion-and-mechanical-states/)

- Convert the paper’s amplitude linewidth into energy linewidth.
- Derive the one-photon resonance from displaced oscillator energies.
- Check the weak-drive high-Q correlation against cutoff and analytic references.

## Figure 1 identifies the detected field

A weak coherent laser drives a cavity coupled to a mechanical mode. Split the outgoing field between two photon counters and compare coincidences with mean flux. The observable belongs to a specified output field. If a reflection channel includes the coherent incident field, its interference must enter the calculation.

Use bare detuning $\Delta_0=\omega_L-\omega_c$ and cavity energy linewidth $\kappa$. The paper’s Eq. 3 uses amplitude decay rate $\kappa_R$, so $\kappa=2\kappa_R$. It also introduces displaced detuning $\Delta=\Delta_0+g_0^2/\Omega_m$. Check which detuning appears on each axis.

Source locations: Rabl

## Figure 2 shifts the spring equilibrium for each photon number

Let $n=a^\dagger a$. Turn off the drive and fix photon number. Complete the square in the full mechanical Hamiltonian, defining $U=g_0^2/\Omega_m$. Each photon sector contains a ladder spaced by $\Omega_m$, shifted downward by $Un^2$.

Adding the first photon without changing the displaced phonon number costs $-\Delta_0-U$. Its resonance is $\Delta_0=-U$. At that setting, adding the second costs $-2U$ relative to the same laser and is detuned. Phonon-changing transitions add other routes, so this simple energy difference cannot predict the full correlation curve.

$$
\begin{aligned}\frac{H_n}\hbar&=\Omega_m\left(b^\dagger-\frac{g_0n}{\Omega_m}\right)\left(b-\frac{g_0n}{\Omega_m}\right)-\Delta_0 n-Un^2,\\ E_{n,m}/\hbar&=m\Omega_m-\Delta_0 n-Un^2.\end{aligned}
$$

Reversing the mechanical coordinate changes displacement sign and leaves the energy shift unchanged. The full single-photon interaction is retained.

$n=0$ $n=1$ $n=2$

Original displaced-oscillator ladder in the laser frame at the one-photon resonance. Each photon-number column has equally spaced phonon levels, shifted by a quadratic photon-number energy. The lowest zero- and one-photon levels coincide; the two-photon level is detuned.

Source locations: Aspelmeyer

## Figure 3 first asks whether one photon can enter

The horizontal axis is bare detuning and the vertical axis is normalized cavity excitation $S$. The zero-phonon line lies at $\Delta_0=-U$, with other lines spaced by mechanical frequency. In the zero-temperature high-Q weak-drive limit, overlaps between the ground state and displaced mechanical states give Poisson-like weights.

The published Figure 3 includes finite mechanical Q and corresponding broadening. Label the infinite-Q expression below separately. It explains positions and weights, but does not reproduce each point of that finite-width curve. This excitation spectrum uses the leading one-photon response and does not yet test photon–photon correlations.

$$
S(\Delta_0)=\sum_{m=0}^{\infty}\frac{e^{-s^2}s^{2m}}{m!}\frac{(\kappa/2)^2}{(\kappa/2)^2+(\Delta_0+U-m\Omega_m)^2},\qquad s=\frac{g_0}{\Omega_m}
$$

This is the zero-temperature high-Q limit of Eq. 8 in the energy-linewidth convention. Normalization uses the uncoupled cavity photon number on resonance.

## Figure 4 includes the second photon

Equal-time correlation $g^{(2)}(0)$ compares photon pairs with squared mean number. Its numerator uses $n(n-1)$ because a photon cannot pair with itself. A coherent reference gives one, and values below one indicate antibunching. The reference is specifically coherent light; general displaced squeezed Gaussian fields can have different statistics.

In weak drive, solve the stationary response in zero-, one- and two-photon sectors, each with a mechanical Fock ladder. Two-photon probability and squared one-photon probability both begin at fourth order in drive. Taking their leading ratio avoids time integration of extremely rare coincidences.

Increase the mechanical cutoff until both spectrum and correlation stabilize. Two photon sectors suffice for this leading weak-drive correlation, not arbitrary drive. Mechanical damping is negligible during optical response; the initial mechanical equilibrium is at zero temperature. Record the order in which weak drive and high Q are taken.

$$
g^{(2)}(0)=\frac{\langle a^{\dagger2}a^2\rangle}{\langle a^\dagger a\rangle^2}=\frac{\langle n(n-1)\rangle}{\langle n\rangle^2}
$$

A pure leakage channel proportional to the cavity field has this correlation, provided no extra coherent field is mixed into the detected output.

## Work one point to a reproducible value

Choose Figure 4a parameters $g_0/\Omega_m=0.5$ and paper amplitude linewidth $\kappa_R/\Omega_m=0.15$. Energy linewidth is therefore $0.30\Omega_m$. Squaring gives $U/\Omega_m=0.25$. Select the zero-phonon resonance $\Delta_0/\Omega_m=-0.25$ at zero temperature, high Q and weak drive.

The direct Fock response gives $S=0.783227$ and $g^{(2)}(0)=0.328740$. An independently constructed Eq. 11 coefficient sum checks the same high-Q curve. Increase mechanical cutoff from forty to sixty and compare again. Agreement and truncation differences are numerical diagnostics, separate from experimental error bars.

Equation 13 retains a few channels near the zero-phonon line. It explains the first antibunching dip. Across a broader detuning scan, additional phonon resonances cause deviations, so plot the approximation separately from the full sum.

## Compare finite coupling with the linearized approximation

The gray uncoupled cavity answers what happens when optomechanical coupling is removed. Testing linearization requires finite single-photon coupling. Find the radiation-pressure shifted mean field, then solve the covariance of the two optical and two mechanical quadratures around it. The offline benchmark saves this calculation with its bath and stable branch.

Write $a=\alpha+d$ and rotate the optical phase so $\alpha$ is positive real. Define $N=\langle d^\dagger d\rangle$ and $M=\langle dd\rangle$. Gaussian fourth moments reduce to sums of pairings of second moments. The resulting expression retains the magnitude and phase of squeezing; a Gaussian field can itself antibunch.

A controlled bright working point uses $\alpha=100$, $g_0/\Omega_m=0.0002$, $\kappa/\Omega_m=0.3$, $\gamma/\Omega_m=10^{-4}$, $\Delta_{\rm eff}/\Omega_m=-1$ and bath occupation one. It gives $N=0.00052932$ and correlation $1.000000143$. Increasing the coherent background while reducing single-photon coupling at fixed enhanced coupling suppresses the omitted term.

At the blockade point, choose $g_0/\Omega_m=0.5$, bare detuning $-0.25\Omega_m$, zero bath temperature and drive $\mathcal E/\Omega_m=0.00015$. A formal Gaussian calculation with $\gamma/\Omega_m=10^{-4}$ gives $g^{(2)}_{\rm G}(0)=2.40275$. The weak-drive infinite-Q Fock limit gives $0.328740$. The bath limits are labelled separately. Here the stable, physical Gaussian covariance is an extrapolation beyond useful linearization: fluctuations overwhelm the tiny coherent background, so stability cannot justify dropping the nonlinear interaction.

$$
g^{(2)}_{\mathrm G}(0)=\frac{\alpha^4+4\alpha^2N+2N^2+|M|^2+2\alpha^2\operatorname{Re}M}{(\alpha^2+N)^2}
$$

The anomalous moment uses the same phase as the coherent amplitude. Gaussian pair contractions give the fourth moment. The offline file includes full covariance, stability diagnostics and finite-bath sensitivity.

Source locations: Aspelmeyer · Lemonde

## Use the same reading procedure on the next paper

Record version, drive limit, bath model and loss definitions. Identify bare or shifted detuning and the vertical normalization. Find an independent zero-coupling check, change photon and phonon cutoffs or time step, and only then compare a selected source curve.

This slice recalculates a bounded weak-drive, zero-temperature high-Q model and its theoretical reference. Finite temperature, finite Q, strongly driven steady states and non-Gaussian mechanical preparation need their own solvers and observations. Photon antibunching can motivate a mechanical-state question, whose answer requires a mechanical observable.

## Predict, change, compare, explain

### [Recalculate the first antibunching dip](https://sandboxphysics.com/labs/optomechanical-photon-blockade/)

1. Choose the photon-blockade preset. Predict the zero-phonon resonance from single-photon coupling before comparing with the scan.
2. Broaden the cavity at fixed coupling. Compare excitation and equal-time correlation and explain why a spectral peak alone does not establish antibunching.
3. Export the weak-drive scan with energy-linewidth convention and mechanical cutoff. Compare the high-Q benchmark with Eq. 11 and plot Eq. 13 separately.

## Exercises with explanations

1. Double single-photon coupling. How far does the zero-phonon line shift?

   Read the explanation

   The shift scales quadratically, so its red shift becomes four times larger. Linewidth and sideband resolution still need separate checks.

2. Does photon correlation below one establish mechanical Wigner negativity?

   Read the explanation

   No. It establishes nonclassical statistics of the optical field. Mechanical Wigner negativity requires mechanical tomography or an equivalent dedicated observable.

[Previous: Where backaction goes when you watch one quadrature](https://sandboxphysics.com/stories/quantum-optomechanics/backaction-evasion-and-mechanical-states/)

## Read the sources

Derivations, worked examples and explanatory figures are authored for the stated models. Synthetic records and model references are labelled separately from experimental results. The source locations below identify the reviewed theory and the limited numerical comparison.

1. [Lemonde, Didier & Clerk, Antibunching and unconventional photon blockade with Gaussian squeezed states, arXiv:1410.6510v1 (2014)](https://arxiv.org/html/1410.6510v1)

   Section II, Eqs. 2–4: displaced Gaussian photon moments and amplitude squeezing. The course uses the general moment identity; its finite-coupling optomechanical covariance is a separate calculation.

2. [Aspelmeyer, Kippenberg & Marquardt, Cavity Optomechanics, arXiv:1303.0733v1 (2013); RMP 86, 1391 (2014)](https://arxiv.org/html/1303.0733v1)

   Pinned arXiv v1: II.1–II.2 cavity linewidth and mechanical spectra; III.1–III.3 radiation pressure and linearization; V.2 and VII.1 dynamical backaction and cooling; VI.1–VI.2 measurement and single-quadrature readout; X.6 nonlinear quantum optomechanics. Equations and examples here are derived in the course convention. This is a review, not a new experimental dataset.

3. [Rabl, The photon blockade effect in optomechanical systems, arXiv:1102.0278v2 (2011)](https://arxiv.org/html/1102.0278v2)

   Pinned v2, April 15, 2011: Eqs. 1, 3–6 and 11–13; Figs. 1–4. The theory predicts photon antibunching. The local weak-drive two-photon Fock calculation takes zero temperature and the high-Q limit; its selected comparison is labelled separately from finite-Q paper curves. Rabl uses the amplitude linewidth; our energy linewidth is twice it.