A unified framework for anomalous boson bunching

Technical Summary: A Unified Framework for Anomalous Boson Bunching

Problem Statement
The paper addresses a paradox in quantum optics known as "anomalous boson bunching." Standard physical intuition and the Hong–Ou–Mandel (HOM) effect suggest that the probability of indistinguishable bosons bunching (occupying the same output modes) is maximized when the particles are fully indistinguishable. Conversely, introducing distinguishability (e.g., via time delays or polarization) should monotonically decrease this probability.

However, recent experimental and theoretical findings have demonstrated "anomalous" scenarios where partially distinguishable photons exhibit a higher probability of bunching into a subset of output modes than fully indistinguishable photons. This phenomenon violates the "Multimode Bunching Conjecture" (Conjecture P1), which posits that bunching probability is maximized for fully indistinguishable inputs. This conjecture was linked to the Bapat–Sunder conjecture regarding matrix permanents, which was recently disproven by Drury. While the existence of anomalous bunching is established, the underlying mechanism and the full scope of the phenomenon remain poorly understood. Specifically, it is unclear how distinguishability can enhance bunching and whether this effect is limited to specific multimode configurations or extends to other regimes.

Methodology and Theoretical Framework
The authors develop a unified theoretical framework based on the concept of total indistinguishability.

1. Reformulation of Indistinguishability: The paper argues that boson bunching is not governed solely by the internal indistinguishability of input photons (e.g., polarization, frequency) but by the total indistinguishability of the postselected output state.

2. Mathematical Formalism:

   • The system considers $n$ single bosons injected into an $m$-mode linear interferometer described by a unitary $U$.
   • The multimode bunching probability $P_\kappa$ (for a subset of output modes $\kappa$) is traditionally expressed as $P_\kappa = \text{perm}(H \odot S_{int})$, where $H$ is a matrix derived from the interferometer and $\kappa$, and $S_{int}$ is the internal distinguishability matrix (Gram matrix of internal states).
   • The authors decompose the spatial contribution by defining a normalized spatial distinguishability matrix $S_\kappa$ derived from the postselected spatial states restricted to $\kappa$.
   • They define the total distinguishability matrix as the Hadamard product: $S_{tot} = S_\kappa \odot S_{int}$.
   • The bunching probability is rewritten as $P_\kappa = P^{cl}_\kappa \cdot \text{perm}(S_{tot})$, where $P^{cl}_\kappa$ is the classical bunching probability. Here, $\text{perm}(S_{tot})/n!$ represents the weight of the permutationally symmetric component of the total state (internal + spatial).

3. Core Proposition: The authors propose that the bunching probability is maximized when the total indistinguishability is maximized. Crucially, because $S_{tot}$ depends on both the internal state and the specific interferometer/output subset, maximizing total indistinguishability does not necessarily require the internal states to be identical ($S_{int} \neq E$). This allows for situations where adding internal distinguishability alters the phases in the Hadamard product such that $\text{perm}(S_\kappa \odot S_{int}) > \text{perm}(S_\kappa)$, leading to anomalous bunching.

Key Contributions and Results
The paper identifies and analyzes four distinct "facets" of anomalous bunching, all arising from violations of the Bapat–Sunder inequality ($\text{perm}(A \odot B) \leq \text{perm}(A)$ for positive semidefinite matrices $A, B$):

1. Enhanced Multimode Bunching (Original Scenario):

   • The authors revisit the original counterexamples where partially distinguishable photons (e.g., with time delays) bunch more strongly into a subset of modes than fully indistinguishable photons.
   • They demonstrate that this occurs because the specific internal distinguishability, when combined with the spatial distinguishability $S_\kappa$ of the interferometer, increases the total indistinguishability $\text{perm}(S_{tot})$ beyond the case where $S_{int}$ is the identity matrix.

2. Enhanced Single-Mode Bunching:

   • Contrary to the standard HOM intuition, the authors show that adding a second independent internal degree of freedom (e.g., adding time delays to photons that already differ in polarization) can increase the single-mode bunching probability.
   • This effect is independent of the specific interferometer and arises purely from the interplay of two internal distinguishability matrices ($S_A \odot S_B$) violating the Bapat–Sunder inequality.
   • They provide an explicit counterexample involving seven photons with manipulated polarization and time-bin states, though they note the experimental probability is extremely low ($\sim 10^{-4}$).

3. Quantum Enhancement in Multimode vs. Single-Mode:

   • The paper investigates the "quantum enhancement factor" (ratio of quantum to classical bunching probability).
   • It is generally expected that single-mode bunching provides the maximal quantum enhancement ($n!$ for indistinguishable particles).
   • The authors show that for partially distinguishable particles, a multimode bunching event can exhibit a larger quantum enhancement factor than a single-mode bunching event with the same input state. This implies that the "quantumness" of the interference is not strictly bounded by the single-mode case when distinguishability is present.

4. Spatial Anomalous Bunching in Nested Interferometers:

   • The authors demonstrate that anomalous bunching can occur purely through spatial degrees of freedom, without internal distinguishability.
   • In a setup with multiple copies of an interferometer, sending photons in a superposition across these copies (creating spatial distinguishability) can result in higher bunching probabilities than sending all photons to a single interferometer. This effectively uses spatial superposition to mimic the role of internal states in the original anomaly.

Significance and Claims
The paper claims to resolve the apparent tension between the existence of anomalous bunching and the fundamental link between indistinguishability and bosonic interference. By shifting the focus from "internal indistinguishability" to "total indistinguishability" (including spatial postselection), the authors restore a direct, intuitive connection: bunching is always maximized when total indistinguishability is maximized.

The significance of this framework is threefold:

1. Unification: It provides a single mathematical structure (Eq. 18) that explains all known instances of anomalous bunching, including those that violate the Multimode Bunching Conjecture.
2. Clarification: It clarifies that anomalous bunching is not a failure of quantum mechanics but a consequence of the non-monotonic behavior of matrix permanents under Hadamard products.
3. Discovery: It reveals previously unexplored facets of the phenomenon, specifically that anomalous effects can occur in single-mode bunching and can be driven purely by spatial degrees of freedom.

The authors remain modest regarding practical applications, noting that while these effects offer new insights into the role of distinguishability in multiphoton interference, the specific configurations required to observe them (particularly the single-mode enhancement) are currently challenging to realize experimentally due to the extremely low probabilities involved. The work primarily serves as a theoretical clarification and a classification of the conditions under which the Multimode Bunching Conjecture fails.