Hong–Ou–Mandel (HOM) Effect: Why the Beam Splitter Needs a $\pi$ Phase Shift

1. Introduction

In the Hong–Ou–Mandel (HOM) effect, two indistinguishable photons incident on a beam splitter interfere in such a way that they always exit the same output port (bunching). A key requirement is that reflection from the bottom side introduces a $\pi$ phase shift, while reflection from the top does not.

2. The Correct Beam Splitter Transformation

A 50:50 beam splitter transforms input modes ($\hat{a}_1, \hat{a}_2$) into output modes ($\hat{a}_3, \hat{a}_4$):

$$ \begin{cases} \hat{a}_1 \rightarrow \frac{1}{\sqrt{2}}(\hat{a}_3 + \hat{a}_4) \\ \hat{a}_2 \rightarrow \frac{1}{\sqrt{2}}(\hat{a}_3 - \hat{a}_4) \end{cases} $$

Matrix form:

$$ U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$

3. Proof of Unitarity

A matrix $U$ is unitary if $U^\dagger U = I$. Verification:

$$ U^\dagger U = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = I $$

⇒ $U$ is unitary.

4. Without $\pi$ Phase Shift

Incorrect transformation matrix:

$$ U_{\text{wrong}} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} $$

Fails unitarity check:

$$ U_{\text{wrong}}^\dagger U_{\text{wrong}} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \neq I $$

Key Results

Case	Matrix	Unitary?	HOM Effect?
Correct	$\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}$	✅ Yes	✅ Bunching
No π shift	$\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & 1\end{pmatrix}$	❌ No	❌ No bunching

Conclusion

The $\pi$ phase shift ensures unitarity: $U^\dagger U = I$, required for destructive interference in the $|1,1\rangle$ output state. Essential for observing the HOM dip in coincidence measurements.

HOM Effect Phase Shift and Unitarity

Hong–Ou–Mandel (HOM) Effect: Why the Beam Splitter Needs a \(\pi\) Phase Shift

1. Introduction

2. The Correct Beam Splitter Transformation

3. Proof of Unitarity

4. Without \(\pi\) Phase Shift

Key Results

Conclusion