Edge-Resolved Two-Boson Interference

• The paper demonstrates how resolved two-boson interference exploits spatial, modal, or temporal detection to reveal exchange symmetry effects, such as Hong–Ou–Mandel suppression.
• It details various platforms—from beam splitters to lattice quantum walks and SSH models—that illustrate the unique interplay of interaction, dimerization, and boundary effects in two-body systems.
• The study further illustrates resonant Fano–Feshbach conversion and mode/time-resolved interference, linking theoretical predictions with practical observables across diverse quantum experiments.

Searching arXiv for relevant papers on two-boson interference, edge effects, and SSH/doublon physics. arXiv search query: "two boson interference edge SSH doublon Hong Ou Mandel" Edge-resolved two-boson interference denotes a class of two-particle interference phenomena in which the bosonic exchange symmetry is read out with spatial, modal, temporal, or boundary resolution. In the literature, the phrase covers several closely related settings: output-channel-resolved Hong–Ou–Mandel suppression at a beam splitter, spatially resolved correlation matrices in interacting quantum walks, and boundary-localized two-body states in dimerized lattices whose two-particle sector is mapped onto a higher-dimensional single-particle problem (Tichy, 2013, Lahini et al., 2011, Liberto et al., 2016, Gorlach et al., 2018). Across these settings, the central observables are coincidence probabilities, second-order correlation functions, and real-space or mode-space intensity distributions, all of which expose how two indistinguishable bosons interfere when multiple many-body alternatives lead to the same detected outcome.

1. Exchange interference and resolved detection channels

The canonical starting point is the two-input, two-output beam splitter. With one boson in each input mode,

$$\ket{\chi_{\text{ini}}^{(2)}}=\hat a_1^\dagger \hat a_2^\dagger \ket{\text{vac}},$$

the beam splitter acts as

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$

with reflection probability $R$ and transmission probability $T=1-R$ (Tichy, 2013). For distinguishable particles, the coincidence event $(1,1)$ is obtained by adding probabilities,

$P_{\text{D}(1,1)} = T^2 + R^2,$

whereas for identical bosons the two many-body alternatives interfere destructively in the coincidence channel. At a balanced beam splitter, the bosonic output is

$$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$

so that

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$

This is the Hong–Ou–Mandel effect in its resolved-output form (Tichy, 2013).

A crucial distinction in this literature is between coincidence suppression and same-port occupation. One review emphasizes that the vanishing of the coincidence event for perfect bosons is caused by destructive interference between the two many-particle paths, whereas the preference for double occupation is also a kinematic property of bosons; for the $(2,0)$ event there is only one contributing process, so its probability is not sensitive to a relative phase between alternatives in the same way (Tichy, 2013). This matters when “edge-resolved” language is used loosely for output-channel-resolved counting: the decisive signature is usually the redistribution of pair counts across resolved channels, especially the suppression of split-channel coincidences.

The same interference logic appears in other geometries. In the two-particle two-slit experiment, the joint state for identical particles is

$$\Psi(x,y)=N_\Psi\big(\psi(x)\phi(y)\pm \psi(y)\phi(x)\big), \qquad N_\Psi=\big(2\pm 2|\langle \psi|\phi\rangle|^2\big)^{-1/2},$$

and the observed coincidence pattern combines ordinary path interference with exchange terms (Sancho, 2014). The paper identifies two regimes: when the final overlap is near unity, bosons and distinguishable particles can have almost the same coincidence patterns, whereas fermions remain clearly different; when the mode distributions are not similar, distinguishable, bosonic, and fermionic patterns separate clearly (Sancho, 2014). This provides a caution against equating large overlap with a necessarily strong visible bosonic deviation in every resolved detection pattern.

2. Spatially resolved two-boson correlations in lattice quantum walks

In the interacting quantum-walk setting, the relevant notion of resolution is the full two-particle correlation matrix rather than boundary localization. For two bosons on a one-dimensional lattice, the dynamics is governed by the Bose–Hubbard Hamiltonian

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$0

with single-particle density

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$1

and two-particle correlation function

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$2

(Lahini et al., 2011). A fluctuation correlator,

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$3

is used for comparison to classical light in the nonlinear photonic experiment (Lahini et al., 2011).

The central result is that two-boson quantum walks on a 1D lattice develop strongly spatially structured two-particle correlations shaped by the interplay of bosonic Hanbury Brown–Twiss interference and on-site interactions in the Bose–Hubbard model (Lahini et al., 2011). The correlation pattern depends on the interaction strength $$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$4, but not on the sign of $$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$5: attractive and repulsive interactions produce the same spatial correlation structure, even though the interacting spectrum is shifted in energy up or down (Lahini et al., 2011).

For the initial state with both bosons on the same site,

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$6

increasing $$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$7 causes the particles to propagate increasingly as a bound pair, and the density becomes more localized (Lahini et al., 2011). For the adjacent-site initial state,

$$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$8

the noninteracting limit exhibits HBT bunching, while strong interactions produce a fermion-like anti-bunching pattern: the bosons avoid being on the same side and the correlation matrix resembles that of two noninteracting fermions started in the same configuration (Lahini et al., 2011). Importantly, in this adjacent-site case the density $$\hat a_j^\dagger \rightarrow \hat U \hat a_j^\dagger \hat U^{-1} = i\sqrt{R}\,\hat b_j^\dagger + \sqrt{T}\,\hat b_{3-j}^\dagger ,$$9 changes only weakly with $R$0, whereas the two-particle correlations are highly sensitive to interaction strength (Lahini et al., 2011).

The spectral explanation relies on the separation of the two-particle spectrum into a main scattering band and a bound-pair miniband. Writing the eigenfunction as

$R$1

the bound-pair miniband contains states with large weight at $R$2, and in the strong interaction limit

$R$3

(Lahini et al., 2011). If the bosons start on the same site, the initial state overlaps strongly with the bound-pair miniband; if they start on different sites, the state mainly overlaps with scattering states. In the strong-$R$4 regime, the scattering sector becomes hard-core, and the Jordan–Wigner mapping explains why the two-particle correlator $R$5 becomes identical to that of noninteracting fermions (Lahini et al., 2011).

A common misconception is to read these correlation maps as boundary physics. The paper explicitly states that “At all stages the particles are far from the lattice boundaries,” so the reported structures are spatially resolved bulk correlations, not edge-state or boundary-resolved phenomena (Lahini et al., 2011).

3. Boundary-localized two-body states in the SSH model

A stricter use of edge-resolved two-boson interference appears in dimerized Su–Schrieffer–Heeger lattices with interactions, where open boundaries generate genuinely edge-localized two-body states. In this setting the two-body Hilbert space has its own band structure and boundary physics (Liberto et al., 2016). The interacting SSH Hamiltonian is

$R$6

with

$R$7

and

$R$8

(Liberto et al., 2016). For open boundaries, two terminations are distinguished: D1 starts and ends with a strong bond $R$9, while D2 starts and ends with a weak bond $T=1-R$0 (Liberto et al., 2016).

The standard mapping from two particles in one dimension onto one particle in a two-dimensional lattice is central. The first-quantized wavefunction $T=1-R$1 is treated as a single-particle wavefunction on a 2D square lattice, with the diagonal $T=1-R$2 representing double occupancy, that is, the closed interaction channel (Liberto et al., 2016). In this representation, edge localization in the two-body problem appears as localization near boundaries or corners of the 2D array.

The two-body spectrum contains three type I scattering continua and several interaction-generated bound-state bands. In the fully dimerized limit $T=1-R$3, each cell hosts three strong-link two-body eigenstates $T=1-R$4, $T=1-R$5, and $T=1-R$6, with energies

$T=1-R$7

which broaden into narrow two-body bands for finite $T=1-R$8 (Liberto et al., 2016). An additional out-of-cell bound state $T=1-R$9, centered predominantly on neighboring cells, appears around energies $(1,1)$0 and is generated by a virtual-process-induced effective nearest-neighbor interaction mediated mainly by the $(1,1)$1 channel (Liberto et al., 2016).

Under open boundary conditions, edge-bound states arise from the interplay of interactions, SSH dimerization, and boundary termination. A central result is that edge-bound states are present not only in D2, where single-particle edge states exist, but also in D1, which does not support single-particle edge states (Liberto et al., 2016). The identified states include $(1,1)$2, $(1,1)$3-$(1,1)$4, $(1,1)$5, and $(1,1)$6 (Liberto et al., 2016). This is one of the clearest instances in which two-body boundary physics is not reducible to the single-particle problem.

The same paper argues that, for large values of the interactions, there is a breaking of the standard bulk-boundary correspondence (Liberto et al., 2016). In the strong-interaction limit, the doublons behave like an SSH particle with effective hoppings

$(1,1)$7

but the boundary induces an on-site energy shift

$(1,1)$8

which breaks chiral symmetry (Liberto et al., 2016). The consequence is that the naive single-particle bulk-boundary correspondence does not directly govern the interacting two-body problem.

4. Interference between scattering and closed channels

Edge-resolved two-boson interference in the SSH context is not limited to static localization. It also includes resonant conversion between scattering and bound channels. In the two-body SSH model, the $(1,1)$9 bound-state band has energy near $P_{\text{D}(1,1)} = T^2 + R^2,$0, and when $P_{\text{D}(1,1)} = T^2 + R^2,$1 matches the energy of a scattering state, the bound state crosses the continuum and a Fano–Feshbach resonance occurs (Liberto et al., 2016). The most notable resonance is around

$P_{\text{D}(1,1)} = T^2 + R^2,$2

where the $P_{\text{D}(1,1)} = T^2 + R^2,$3 bound state becomes resonant with the higher type I continuum (Liberto et al., 2016).

The physical picture is a two-channel interference process. Two incoming scattering bosons can couple into the closed-channel bound state when their energy matches the bound-state energy, producing a strong enhancement of the diagonal population $P_{\text{D}(1,1)} = T^2 + R^2,$4, hybridization in the spectrum, and resonance peaks in the bound-state occupation (Liberto et al., 2016). The paper simulates two Gaussian wave packets colliding and uses the diagonal density

$P_{\text{D}(1,1)} = T^2 + R^2,$5

as a direct measure of the bound-state occupation; it exhibits a sharp resonance peak when the incident scattering energy matches the bound-state energy (Liberto et al., 2016).

A closely related driven-dissipative formulation is developed in a linear photonic simulator for the two-body Bose–Hubbard physics of an SSH chain (Gorlach et al., 2018). There the two-particle wavefunction

$P_{\text{D}(1,1)} = T^2 + R^2,$6

is mapped onto a single-particle problem in a 2D lattice, with the main diagonal $P_{\text{D}(1,1)} = T^2 + R^2,$7 carrying the interaction energy $P_{\text{D}(1,1)} = T^2 + R^2,$8 (Gorlach et al., 2018). Because the resonators are coherently pumped and lossy, the steady-state amplitudes satisfy

$P_{\text{D}(1,1)} = T^2 + R^2,$9

with $$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$0 to preserve bosonic symmetry (Gorlach et al., 2018).

This platform detects bulk doublon bands, doublon edge states, and Feshbach-resonance-like scattering features through spatially resolved transmission spectra. Pumping far from the diagonal mimics two bosons launched from separated positions, and the steady-state pattern can show two propagating “stripes,” a collision on the diagonal, and conversion into a localized doublon feature (Gorlach et al., 2018). The paper explicitly describes this as the photonic analogue of two-body interference near a boundary and of bound-pair formation during collision (Gorlach et al., 2018). When the pump is close to a corner, boundary structure, single-particle SSH edge modes, and doublon modes can hybridize, and the spectrum can show sharp edge-localized resonances (Gorlach et al., 2018).

5. Time-resolved and mode-resolved two-boson interference

Resolved two-boson interference is not only spatial. A distinct extension replaces spatial path alternatives by alternatives separated in time. In the partial-time-reversal duality of a beam splitter and a parametric amplifier, the beam-splitter Hamiltonian

$$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$1

is mapped to

$$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$2

and the beam splitter transmittance $$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$3 is mapped to the inverse gain $$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$4, with the 50:50 HOM point $$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$5 corresponding to gain $$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$6 (Cerf et al., 2020). In the Heisenberg picture,

$$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$7

while the parametric down-converter acts as

$$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$8

with $$\ket{\chi_{\textrm{fin},B}^{(2)}} =\frac{1}{\sqrt 2}\left(\ket{2,0}_{1,2}-\ket{0,2}_{1,2}\right),$$9 (Cerf et al., 2020).

The resolved signature is a dip in the joint detection probability for one photon in each output mode of the amplifier. The beam-splitter coincidence probability,

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$0

is mapped to

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$1

which vanishes at $$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$2 (Cerf et al., 2020). For a gain-2 amplifier fed with one photon in each input mode,

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$3

so the $$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$4 term disappears even though higher photon-pair terms remain (Cerf et al., 2020).

The mechanism is identified as timelike indistinguishability. The interfering alternatives are that the original pair passes through the amplifier without a stimulated event, or that the input pair is annihilated into the pump while a new pair is stimulated into existence (Cerf et al., 2020). Their amplitudes,

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$5

cancel exactly at $$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$6 (Cerf et al., 2020). The same paper further derives extended suppression laws: for integer gain $$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$7,

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$8

which is the partial-time-reversal dual of the extended HOM zero

$$P_{\text{B}(2,0)}=\frac12,\qquad P_{\text{B}(1,1)}=0,\qquad P_{\text{B}(0,2)}=\frac12 .$$9

(Cerf et al., 2020). In this sense, mode-resolved and time-resolved interference are exact analogues of spatial HOM suppression, but in an active, number-nonconserving Bogoliubov transformation.

6. Experimental platforms and diagnostic observables

Several experimental platforms realize different aspects of resolved two-boson interference. In the microwave domain, two independent, on-demand single photons from transmon-qubit–resonator sources were sent into a 50:50 beam splitter and measured using time-resolved heterodyne quadrature detection (Lang et al., 2013). The two sources were tuned to the same emission frequency,

$(2,0)$0

with cavity linewidths

$(2,0)$1

corresponding to a wavepacket duration on the order of $(2,0)$2 (Lang et al., 2013). The central observables were

$(2,0)$3

and

$(2,0)$4

At zero delay, the measured $(2,0)$5 around the overlap region, demonstrating HOM coalescence, while finite delays restored distinguishability (Lang et al., 2013). From moments up to fourth order, the output density matrix was reconstructed with fidelity about $(2,0)$6 to the ideal NOON state and negativity $(2,0)$7, confirming entanglement of the beam-splitter output modes (Lang et al., 2013).

In nonlinear photonic lattices, weak-interaction classical intensity-correlation fluctuations resemble the quantum two-boson correlation patterns, especially when interactions are not too strong (Lahini et al., 2011). The comparison is made through the classical fluctuation correlator $(2,0)$8. At stronger nonlinearity, however, the classical and quantum results diverge: the quantum system evolves toward fermionic-like anti-bunching, whereas the classical system becomes increasingly localized and does not reproduce the quantum fermionization effect (Lahini et al., 2011). This sharply delineates the weak-interaction regime in which classical-wave analogues remain faithful.

Cold-atom implementations are proposed as the cleanest route to the strong-interaction regime. The required capabilities are to prepare exactly two atoms at chosen lattice sites, let them tunnel freely, tune $(2,0)$9 for example using a Feshbach resonance, and image site occupations and correlations (Lahini et al., 2011). The predicted outcomes are configuration-dependent: if the atoms are placed on the same site, they propagate as a bound pair with localized density; if placed on different sites, the density changes only weakly with $$\Psi(x,y)=N_\Psi\big(\psi(x)\phi(y)\pm \psi(y)\phi(x)\big), \qquad N_\Psi=\big(2\pm 2|\langle \psi|\phi\rangle|^2\big)^{-1/2},$$0, but the two-particle correlation evolves strongly from bunching to fermion-like anti-bunching as $$\Psi(x,y)=N_\Psi\big(\psi(x)\phi(y)\pm \psi(y)\phi(x)\big), \qquad N_\Psi=\big(2\pm 2|\langle \psi|\phi\rangle|^2\big)^{-1/2},$$1 grows (Lahini et al., 2011).

A further optical analogue appears in a four-core microstructured fiber simulating two interacting bosons in a double-well potential (Longhi, 2011). There, propagation distance $$\Psi(x,y)=N_\Psi\big(\psi(x)\phi(y)\pm \psi(y)\phi(x)\big), \qquad N_\Psi=\big(2\pm 2|\langle \psi|\phi\rangle|^2\big)^{-1/2},$$2 plays the role of time, the transverse coordinates $$\Psi(x,y)=N_\Psi\big(\psi(x)\phi(y)\pm \psi(y)\phi(x)\big), \qquad N_\Psi=\big(2\pm 2|\langle \psi|\phi\rangle|^2\big)^{-1/2},$$3 represent the two-particle configuration space, and the four optical cores encode the four occupation sectors (Longhi, 2011). The observables

$$\Psi(x,y)=N_\Psi\big(\psi(x)\phi(y)\pm \psi(y)\phi(x)\big), \qquad N_\Psi=\big(2\pm 2|\langle \psi|\phi\rangle|^2\big)^{-1/2},$$4

resolve whether the system occupies same-well or opposite-well sectors (Longhi, 2011). The platform visualizes the crossover from independent Rabi oscillations to correlated pair tunneling and then to fragmented-pair tunneling in the fermionization limit (Longhi, 2011). This is not an SSH boundary problem, but it is a configuration-space-resolved interference experiment in which different tunneling pathways interfere through bosonic exchange symmetry.

Taken together, these experiments and proposals show that edge-resolved two-boson interference is not a single protocol but a family of resolved measurements. Depending on the platform, the “edge” may mean a physical lattice boundary, a corner in the 2D mapped two-body configuration space, a resolved output port, or a temporally distinguished mode in a Bogoliubov process. What remains invariant is the object being probed: a two-boson amplitude structure that is inaccessible to single-particle density alone and becomes visible only through resolved two-particle observables (Liberto et al., 2016, Gorlach et al., 2018, Cerf et al., 2020).