Coherent Double Circuit Configurations

• The paper presents a systematic analysis of coherent double circuit configurations, detailing quantum, matter-wave, and combinatorial models with robust transport and interference properties.
• It employs Y-junction quantum circuit models and non-Hermitian Hamiltonians to demonstrate isolated eigenstates, superradiant transitions, and resilience against disorder.
• The study generalizes the concept via combinatorial frameworks, showing that cyclic group actions yield schurian, separable configurations that underpin efficient routing and switching.

Coherent double circuit configurations refer to systematically structured systems—across physical and combinatorial domains—composed of two mutually coherent, loop-like subsystems or "circuits." These configurations are characterized by explicit connectivity patterns (such as Y-junctions in quantum or matter-wave circuits, or disjoint fibers in algebraic combinatorics), supporting coherent transport, switching, and interference. They serve as model systems for the analysis of wave propagation, transmission phenomena, and group-theoretic symmetries in both quantum device engineering and algebraic combinatorics.

1. Quantum Double Circuit Models and Spectral Structure

The double-branch (Y-junction) quantum circuit is the prototypical physical realization of a coherent double circuit configuration. The system consists of a central node coupled to two discrete, finite-length branches. The relevant Hamiltonian for the closed (isolated) system is:

$$H_{\text{closed}} = v \sum_{n=1}^{N-1} (|1,n\rangle\langle 1,n+1| + |2,n\rangle\langle 2,n+1| + \text{h.c.}) + v(t_1|0\rangle\langle 1,1| + t_2|0\rangle\langle 2,1| + \text{h.c.})$$

with $N$ sites per branch, uniform intra-branch hopping $v$, and vertex couplings $t_1, t_2$ to the central site $|0\rangle$.

The eigenvalue spectrum decomposes into:

• Bulk (extended) states with Bloch-like character residing inside the energy band $|E| < 2v$.
• Two isolated eigenstates for $Q = t_1^2 + t_2^2 > 2$ with $|E_{\text{loc}}| > 2v$, exponentially localized at the central node and decaying with localization length $\xi = 1/\ln(Q-1)$. In the large-$N$ limit, $E_\text{loc} = \pm (Q/\sqrt{Q-1})v$ (Ziletti et al., 2011).

This decoupling of localized and extended states underlies the circuit's robust transport properties and coherent switching potential.

2. Coherent Transport, Transmission, and Open-circuit Dynamics

The open system is constructed by coupling semi-infinite leads (continuum channels) to the terminal sites of each branch with coupling strength $\gamma$, resulting in the effective non-Hermitian Hamiltonian:

$$H_{\text{eff}} = H_{\text{closed}} - \frac{i}{2}\gamma W, \quad W = |1,N\rangle\langle 1,N| + |2,N\rangle\langle 2,N|$$

Eigenvalues become $E_r - (i/2)\Gamma_r$, the imaginary parts encoding decay widths.

The exact transmission amplitude between branches 1 and 2 at resonance with an isolated state $E = E_{\text{loc}}$ is

$$T_{12}(E_{\text{loc}}) = 4\frac{|t_1|^2 |t_2|^2}{Q^2}$$

when end effects are negligible—the Hauser–Feshbach form (Ziletti et al., 2011). Transmission via the isolated state dominates coherent routing between the two arms.

A super-radiant transition occurs as $\gamma$ increases, characterized by a redistribution of widths: a few “super-radiant” states acquire widths $\sim\gamma$, while the rest are “trapped” with narrowed widths $\sim1/\gamma$. For $M=2$, the threshold is $\gamma_{sr} \approx 2v$.

3. Integrated Coherent Matter-wave Double Circuits

In atom-optical analogs, double-circuit configurations consist of two waveguiding loops (“rings”), coherently connected by a beamsplitter/switch region realized by “painted potentials.” This geometry enables the launching, splitting, and recombining of a Bose-Einstein condensate (BEC).

• Geometry: Two closed loops (radius $R\approx18\,\mu$m), linked via a Y-junction (arm separation $d\approx3.7\,\mu$m).
• Potential Engineering: Time-averaged optical dipole potentials, painted by a rapidly scanned red-detuned laser beam, generate precisely shaped guides, bends, and switchable Y-junctions.
• Operation: A single BEC wavepacket is injected, coherently split at the Y-junction (with adjustable power ratios via trap-depth modulation), optionally switched dynamically, and recombined for phase-sensitive interference.
• Readout: The relative phase between the rings is extracted from interference visibility after time-of-flight expansion. The Sagnac phase from rotation (for interferometry) is $\Delta\phi_\text{Sagnac} = \frac{4\pi mA}{h}\Omega$ (Ryu et al., 2014).

Optimization of bend radii, transverse mode matching, and junction smoothness are essential for minimizing transverse excitation and maintaining phase coherence.

4. Combinatorial Double-Circuit (Coherent Configuration) Perspective

In algebraic combinatorics, double-circuit configurations are formalized as the simplest nontrivial cases of quasiregular coherent configurations with two fibers. Let $G_1 = \mathbb{Z}/m\mathbb{Z}$ and $G_2 = \mathbb{Z}/n\mathbb{Z}$. One considers disjoint sets $\Omega_1 \cong G_1$, $\Omega_2 \cong G_2$, and sets $\Omega = \Omega_1 \cup \Omega_2$.

• Basic Relations: For $g \in G_1$, $R^1_g = \{(x, x+g): x \in \Omega_1\}$; for $h \in G_2$, $R^2_h = \{(y, y+h): y \in \Omega_2\}$; no nontrivial cross-fiber relations.
• Intersection Numbers: Nonzero only within a single fiber; for $R = R^i_g$, $S = R^i_h$, $T = R^i_{g+h}$, $p_{R, S}^T = 1$; all other $p$ vanish.
• Group Action: The configuration is schurian—arising from the orbitals of $C_m \times C_n$—and separable; these properties are guaranteed precisely when each component group has a distributive lattice of normal subgroups, which holds for finite cyclic groups (Hirasaka et al., 2018).

This structure provides an archetype for more elaborate “linked-quotient” coherent configurations and illustrates the direct interplay between group actions and combinatorial partitionings.

5. Disorder, Robustness, and Generalizations

The robustness of double-circuit coherence against disorder has been quantified in quantum models. For on-site disorder ($\epsilon_{a,n}$ uniformly in $[-W/2, W/2]$), bulk states localize with length $\sim(v/W)^2$. The isolated eigenstates extend robustly up to a critical disorder $W_{\text{cr}}$ satisfying:

• Near threshold $Q \rightarrow 2^+$: $W_{\text{cr}}/v \simeq (Q-2)^2/2$,
• Strong coupling $Q \gg 1$: $W_{\text{cr}}/v \simeq k\sqrt{Q}$, $k \approx 1.48$ for $M=2$.

Numerical diagonalization confirms the persistence of coherent, isolated states deep into the disordered regime (Ziletti et al., 2011).

In matter wave circuits, transverse and longitudinal mode engineering suppresses decoherence from imperfections. Techniques such as painting offset couplings, adopting clothoid transitions for bends, and dynamic trap-depth adjustment offer fine-tuned control over loss and mode-mismatch (Ryu et al., 2014).

6. Implementation Parameters and Practical Realizations

Characteristic system and experimental parameters for coherent double-circuit matter-wave implementations include:

Parameter	Value/Description	Source
Guide radius ($R$)	$18.6\,\mu$m	(Ryu et al., 2014)
Arm separation at Y-junction ($d$)	$3.7\,\mu$m	(Ryu et al., 2014)
Painting beam power	$30{-}50\,$mW/guide	(Ryu et al., 2014)
Transverse freq. ($\omega_r/2\pi$)	$1705\,$Hz	(Ryu et al., 2014)
TOF for interference	$2{-}3\,$ms	(Ryu et al., 2014)
BEC atom number ($N$)	$4\times10^3$	(Ryu et al., 2014)

These values have been demonstrated experimentally, establishing the feasibility of stable, high-visibility interference and arbitrary routing in double-circuit topologies. The same principles specify couplings, disorder thresholds, and transmission metrics in engineered quantum circuits (Ziletti et al., 2011).

7. Mathematical Generalizations and Structural Theorems

Double-circuit coherent configurations serve as the base case for a broad class of schurian and separable quasiregular coherent configurations, whenever all components correspond to cyclic groups or, more generally, to groups with distributive normal subgroup lattices. The lattice-theoretic criterion ensures that partial isomorphisms of the combinatorial structure can always be extended to global automorphisms, enforcing uniqueness up to isomorphism (separability) and realization by group orbitals (schurity) (Hirasaka et al., 2018).

This theory supports the explicit combinatorial modeling of disconnected or weakly coupled physical systems and underpins the analysis of symmetries, invariants, and transmission channels in coherent transport networks.