Geometric Criteria for Complete Mode Conversion in Detuned Systems via Piecewise-Coherent Modulation

Abstract

Static phase detuning fundamentally constrains coherent state transfer in asymmetric classical and quantum systems. We introduce a Bloch-sphere formulation for piecewise-coherent modulation that recasts coupled-mode dynamics as geometric trajectories, transforming algebraic control into path optimization. The approach reveals a cone of inaccessibility at the target pole and yields exact geodesic criteria for complete mode conversion in detuned systems. Leveraging this framework, we break time-reversal symmetry to realize a magnet-free optical isolator with near-unity contrast. Furthermore, for detuning larger than coupling between modes, we develop a recursive multi-step protocol enabling deterministic transfer for arbitrary detunings and derive a universal geometric lower bound on the required number of coupling-switching events.

Figures (4)

• Figure 1: (a) Schematic of the directional coupler modulator (DCM) structure. First, we study the energy transfer between the supermodes in a single directional coupler modulator (DCM-1) and later, build a system with a second DCM-2 to study non-reciprocal optical transmission, [b] representation of directional coupler modes on a Bloch sphere with the curve showing the state evolution for $\Omega t_t = 0.29\pi$ and $\Omega t_2 = 0.71\pi$ leading to complete transfer of energy from $A_1$ to $A_2$, [c] permissible values of $\phi$ and $\Delta/\kappa$ for complete exchange of energy between the supermodes of the directional coupler, [d] and [e] optical energy transferred to $A_2$ supermode as a function of $\Omega_1t_1$ and $\Omega_2t_2$ for two point $P_1,P_2$ respectively as shown in [c] showcasing possible range of energy transfer between the supermodes.
• Figure 2: State evolution of supermodes on Bloch sphere in under [a] non-zero $\Delta$ with a static coupling, and [b] non-zero $\Delta$ with the precession vector changes at $t_1$ leading to a complete transfer of power to $A_2$ state at time $t_2$. The red cone depicts the cone of inaccessibility.
• Figure 3: State evolution of supermodes on Bloch sphere in [a] forward direction, and [b] reverse direction.
• Figure 4: (a) Variation of number of coherent modulation steps required for complete state tranefer from the north to south pole, (b) complete conversion for $\Delta/\kappa=2$ using $N=2$, and (c) inability of the two step piecewise protocal for complete transfer.