FR10-A · The phase a spectrum misses

Joint Spectral Phase & Photon Purity

Can identical joint intensities hide different photon purities? Joint intensity keeps each amplitude’s size and discards its phase. Those phases decide whether the two possible herald outcomes leave the same signal mode. Equal intensities can therefore hide either a pure signal photon or a mixed one.

Turn one pair amplitude through a full circle. The four intensities stay equal, but the heralded photon changes from one coherent mode to an equal mixture.

Heralded purity $0.637$

Effective mode number $1.57$

Ideal HOM coincidence $0.182$

Largest Schmidt weight $0.761$

Same intensity, different coherence

Turn the hidden phase

Drag the gold clock. All four circles keep the same brightness; their arrows reveal what intensity discards.

Joint phase

\theta

0.65\pi

What the diagram represents

The model has two orthonormal collected spectral bins on each arm and a normalized pure, low-gain photon-pair state. A herald detector ignores the idler bin. The phase diagram represents joint-state amplitudes, not an apparatus or propagation. HOM predictions assume independent identical heralded states, one photon per input, a balanced lossless splitter and matched polarization. Multipair emission, loss and detector noise are omitted.

Each circle is one joint signal–idler frequency-bin amplitude. Its area shows the same probability in all four cells; the arrow shows phase. This joint phase is not a local phase plate acting on only one photon.

Focus the diagram and use the left or right arrow key to turn the selected phase.

Model assumptions

Ideal model predictions

Intensity stays still. Purity changes.

Joint intensity and reduced-state coherence

\rho_s=FF^\dagger

A complete phase turn

$\mathcal P=\operatorname{Tr}(\rho_s^2),\qquad C_0=\frac{1-\mathcal P}{2}$

Read the coefficient matrix

$F(\theta)=\frac12\begin{pmatrix}1&1\\1&e^{i\theta}\end{pmatrix},\qquad |F_{jk}|^2=\frac14$

Rows name signal bins; columns name idler bins. Ignoring the idler outcome adds the two possible signal states as a mixture. The two eigenweights are the probabilities of the orthogonal Schmidt modes.

Start with the essentials

Focus question: Can identical joint intensities hide different photon purities?
One-sentence intuition: Equal joint intensities do not determine whether different herald outcomes leave the same signal mode.

Core mathematical model

The phase hidden from intensity

F(\theta)=\frac12\begin{pmatrix}1&1\\1&e^{i\theta}\end{pmatrix},\qquad |F_{jk}|^2=\frac14

Rows label two orthonormal signal bins, and columns label two orthonormal idler bins. Each cell has the same probability for every phase. Only the lower-right amplitude changes direction in the complex plane.

Ignoring the idler frequency

\rho_s=FF^\dagger=\frac14\begin{pmatrix}2&1+e^{-i\theta}\\1+e^{i\theta}&2\end{pmatrix}

Each possible idler outcome contributes one column of the coefficient matrix. Because the outcome is unread, their density matrices add. At zero phase the columns describe the same signal state; at half a turn they describe orthogonal signal states.

From Schmidt weights to interference

\begin{aligned}\lambda_\pm&=\frac{1\pm|\cos(\theta/2)|}{2}\\\mathcal P&=\lambda_+^2+\lambda_-^2=\frac{3+\cos\theta}{4}\\K&=\frac1{\mathcal P},\qquad C_0=\frac{1-\mathcal P}{2}\end{aligned}

The weights are the squared singular values of the pair amplitude. For independent identical mixed single photons at a balanced splitter, zero-delay coincidence depends on their state overlap, which here equals purity.

Common difficulties

A joint phase differs from a local phase

Typical misconceptionA phase plate on either photon can turn just one joint amplitude.

Better mental modelA local signal phase rotates a whole row, and a local idler phase rotates a whole column. Such local unitary changes preserve Schmidt weights. Rotating only one cell changes the joint state.

A mixed photon still carries exactly one photon

Typical misconceptionLow purity means there is sometimes no photon or more than one.

Better mental modelThis model always conditions on one generated pair. Purity describes uncertainty about the signal mode after the idler frequency is ignored. Vacuum, multipair emission and loss need a larger model.

Run the experiment

01
Start with one signal mode
Choose One shared mode. Predict whether the idler frequency can tell the two signal preparations apart.
What to observe: All arrows align. Both herald outcomes leave the same signal state, the largest Schmidt weight is one, and ideal zero-delay HOM coincidence vanishes.
02
Keep the intensity and lose purity
Choose Two equal modes and compare the four intensity cells with the previous preset.
What to observe: Every intensity remains one quarter. The two signal states are now orthogonal; both Schmidt weights are one half. Purity is one half and ideal HOM coincidence is one quarter.
03
Find what changes under phase reversal
Compare A hidden quarter turn with Reverse the phase. Then drag the gold clock through a full turn.
What to observe: Opposite phases have the same purity and HOM coincidence, but the imaginary coherence changes sign. A phase-sensitive mode projection can distinguish them.

Joint Spectral Phase & Photon Purity

Intensity stays still. Purity changes.

Joint intensity and reduced-state coherence

A complete phase turn