π-Step Quantization Mechanism

Argument Principle, Spectral Flow, and Rigorous Proof of Delay Quantization Steps

Introduction

In previous chapter, we saw poles of closed-loop scattering matrix move with delay parameter $\tau$. When poles cross real axis, system’s phase response jumps.

But why π? Why not other values? Is magnitude of this transition precisely predictable?

This chapter gives rigorous mathematical proof, revealing necessity of π-step. We will use Argument Principle in complex analysis and Spectral Flow theory in topology, establishing quantitative relationship from pole trajectories to phase transitions.

Argument Principle: Topological Counting in Complex Plane

Statement of Argument Principle

Let $f(z)$ be meromorphic function on complex plane (holomorphic except for finite poles), with $N_0$ zeros and $N_{\infty }$ poles (counting multiplicity) inside closed contour $\gamma$.

Then along $\gamma$ going around once, argument change of function is:

$${\Delta }_{\gamma }\arg f=\frac{1}{2\pi }{\oint }_{\gamma }d(\arg f)=N_0-N_{\infty }$$

In more intuitive language: Net winding number of phase $\arg f$ of $f(z)$ along contour, equals number of zeros minus number of poles inside contour.

Geometric Intuition

Imagine $f(z)$ is a mapping from complex plane to complex plane. When $z$ goes around $\gamma$ once, $f(z)$ draws a closed curve in image space.

• If $\gamma$ contains a zero, $f(z)$ winds around origin once (positive direction);
• If $\gamma$ contains a pole, $f(z)$ winds around origin once (negative direction).

Argument principle is essentially a topological invariant: Winding number is independent of continuous deformation of contour, only depends on number and type of singularities inside.

Logarithmic Derivative Integral

Argument principle has equivalent integral form:

$$\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f^{\prime }(z)}{f(z)}dz=N_0-N_{\infty }$$

Proof: Let $f(z)=(z-z_0{)}^{m}g(z)$, where $g(z_0)\ne 0$. Then:

$$\frac{f^{\prime }(z)}{f(z)}=\frac{m}{z-z_0}+\frac{g^{\prime }(z)}{g(z)}$$

Second term integrates to zero along small circle (because $g$ holomorphic), first term integrates to $2\pi im$.

Summing over all zeros and poles gives argument principle.

Scattering Phase and Determinant Argument

From Scattering Matrix to Total Phase

For scattering matrix $S(\omega ;\tau )$, define total phase:

$$\phi (\omega ;\tau )=\arg \det S(\omega ;\tau )$$

This is a real-valued function, defined modulo $2\pi$. To study phase changes, we need to choose a continuous branch.

At fixed frequency $\omega ={\omega }_{\ast }$, treat $\phi$ as function of $\tau$:

$$\phi (\tau )\equiv \phi ({\omega }_{\ast };\tau )$$

When $\tau$ changes from ${\tau }_1$ to ${\tau }_2$, phase change is:

$$\Delta \phi =\phi ({\tau }_2)-\phi ({\tau }_1)$$

Since phase defined modulo $2\pi$, we need to track phase along continuous path, ensuring no artificial $2\pi$ jumps.

Determinant of Closed-Loop Scattering

For Schur complement form:

$$S^{\circlearrowleft }(\omega ;\tau )=S_{ee}+S_{ei}\mathcal{C}(\omega ;\tau )[I-S_{ii}\mathcal{C}{]}^{-1}{S}_{ie}$$

where $\mathcal{C}(\omega ;\tau )=R(\omega )e^{i\omega \tau }$.

Using matrix determinant identity:

$$\det (A+UCV)=\det (A)\det (I+CV{A}^{-1}U)$$

Get:

$$\det S^{\circlearrowleft }=\det S_{ee}\cdot \det \left[I+S_{ee}^{-1}{S}_{ei}\mathcal{C}(I-S_{ii}\mathcal{C}{)}^{-1}{S}_{ie}\right]$$

Further using $\det (I+AB)=\det (I+BA)$:

$$\det S^{\circlearrowleft }=\det S_{ee}\cdot \frac{\det (I-S_{ii}\mathcal{C}+S_{ie}X{S}_{ei}\mathcal{C})}{\det (I-S_{ii}\mathcal{C})}$$

(where $X=S_{ee}^{-1}$ and above can be simplified)

Most crucial observation:

Zeros and poles of determinant determined by zeros of denominator $\det (I-S_{ii}\mathcal{C})$.

Pole Equation and Spectral Flow

Let $S_{ii}(\omega )$ and $R(\omega )$ be smooth near real frequency. Poles satisfy:

$$\det [I-R(\omega )e^{i\omega \tau }]=0$$

Extend $\omega$ to complex plane, $\omega ={\omega }_r+i{\omega }_i$.

For fixed $\tau$, pole trajectory $\omega (\tau )$ is family of curves satisfying above equation.

Definition of Spectral Flow: When $\tau$ changes from ${\tau }_1$ to ${\tau }_2$, sum of poles crossing real axis (from upper half-plane to lower half-plane, or reverse) is called spectral flow:

$$\mathrm{Sf}({\tau }_1\to {\tau }_2)=\#\{\text{downward crossings}\}-\#\{\text{upward crossings}\}$$

This is a signed topological count.

Proof of π-Step Theorem

Theorem Statement

Theorem (π-Step Transition)

Under assumptions:

1. Scattering matrix $S(\omega ;\tau )$ analytic in real frequency and parameter space;
2. At delay $\tau ={\tau }_c$, exactly one pole ${\omega }_c\in \mathbb{R}$ located on real axis;
3. This pole simple (i.e., ${\partial }_{\tau }\Im \omega ({\tau }_c)\ne 0$), crossing direction unique;
4. In small neighborhood of $({\omega }_c,{\tau }_c)$, no other poles or zeros cross real axis.

Then at fixed frequency ${\omega }_{\ast }={\omega }_c$, phase transition is:

$$\Delta \phi =\underset{ϵ\to 0^{+}}{\lim }[\phi ({\tau }_c+ϵ)-\phi ({\tau }_c-ϵ)]=\pm \pi$$

And sign determined by pole crossing direction:

• If pole from upper half-plane to lower half-plane (downward crossing): $\Delta \phi =-\pi$
• If pole from lower half-plane to upper half-plane (upward crossing): $\Delta \phi =+\pi$

Proof Strategy

We decompose problem into three steps:

1. Local Factorization: Near $({\omega }_c,{\tau }_c)$, factorize $\det S$ as product of pole factor and smooth factor;
2. Argument Change of Pole Factor: Calculate phase jump caused by single pole crossing real axis;
3. Continuity of Smooth Factor: Prove smooth factor’s phase contribution continuous, produces no jump.

Step 1: Local Factorization

In neighborhood $U$ of $({\omega }_c,{\tau }_c)$, exist holomorphic functions $z(\tau )$ and $g(\omega ;\tau )$ such that:

$$\det S(\omega ;\tau )=(\omega -z(\tau ){)}^m\cdot g(\omega ;\tau )$$

where:

• $z({\tau }_c)={\omega }_c$ (pole exactly at ${\omega }_c$ when $\tau ={\tau }_c$)
• $m=+1$ corresponds to zero, $m=-1$ corresponds to pole
• $g({\omega }_c;{\tau }_c)\ne 0$ (smooth factor non-zero)

This factorization comes from combination of residue theorem and implicit function theorem.

Since we consider poles (resonances), usually $m=-1$.

Taking logarithm:

$$\log \det S(\omega ;\tau )=m\log (\omega -z(\tau ))+\log g(\omega ;\tau )$$

Taking imaginary part (i.e., argument):

$$\arg \det S(\omega ;\tau )=m\arg (\omega -z(\tau ))+\arg g(\omega ;\tau )$$

Step 2: Argument Change of Pole Factor

Fix $\omega ={\omega }_c$, define auxiliary function:

$$h(\tau )={\omega }_c-z(\tau )$$

Linear expansion near ${\tau }_c$:

$$z(\tau )\approx z({\tau }_c)+(\tau -{\tau }_c)z^{\prime }({\tau }_c)={\omega }_c+(\tau -{\tau }_c)z^{\prime }({\tau }_c)$$

So:

$$h(\tau )\approx -(\tau -{\tau }_c)z^{\prime }({\tau }_c)$$

By assumption, $z^{\prime }({\tau }_c)$ has non-zero imaginary part:

$$z^{\prime }({\tau }_c)=a+ib,b\ne 0$$

Then:

$$h(\tau )=-(\tau -{\tau }_c)(a+ib)$$

For $\tau >{\tau }_c$ and $\tau <{\tau }_c$:

$$h({\tau }_c+ϵ)\approx -ϵ(a+ib)$$

$$h({\tau }_c-ϵ)\approx ϵ(a+ib)$$

Calculating argument:

$$\arg h({\tau }_c+ϵ)=\arg [-(a+ib)]=\pi +\arg (a+ib)$$

$$\arg h({\tau }_c-ϵ)=\arg (a+ib)$$

Transition:

$$\Delta \arg h=\arg h({\tau }_c+ϵ)-\arg h({\tau }_c-ϵ)=\pi$$

(If $b>0$; if $b<0$ then $-\pi$)

Step 3: Continuity of Smooth Factor

Smooth factor $g({\omega }_c;\tau )$ non-zero near ${\tau }_c$, can choose continuous logarithm branch:

$$\log g({\omega }_c;\tau )=\log |g({\omega }_c;\tau )|+i\arg g({\omega }_c;\tau )$$

Since $g$ non-zero and smooth, $\arg g$ is continuous function of $\tau$, produces no jump at ${\tau }_c$.

Step 4: Total Phase Transition

Combining Steps 2 and 3:

$$\phi ({\omega }_c;\tau )=m\arg ({\omega }_c-z(\tau ))+\arg g({\omega }_c;\tau )$$

At $\tau ={\tau }_c$:

$$\Delta \phi =m\cdot \Delta \arg h+\underset{=0}{\underbrace{\Delta \arg g}}=m\cdot (\pm \pi )$$

For pole $m=-1$:

$$\Delta \phi =-(\pm \pi )=\mp \pi$$

Sign depends on sign of imaginary part of $z^{\prime }({\tau }_c)$, i.e., direction of pole crossing real axis.

Conclusion: Phase transition magnitude exactly $\pm \pi$, QED.

Calculation of Delay Quantization Steps

Implicit Equation for Step Positions

Condition for pole crossing real axis: There exist real frequency ${\omega }_c\in \mathbb{R}$ and delay ${\tau }_c$ such that:

$$\det [I-R({\omega }_c)e^{i{\omega }_c{\tau }_c}]=0$$

and $\Im {\omega }_c=0$.

For simple case (single eigenvalue $\lambda (\omega )$):

$$\lambda ({\omega }_c)e^{i{\omega }_c{\tau }_c}=1$$

Writing as:

$$|\lambda ({\omega }_c)|=1,{\omega }_c{\tau }_c+\arg \lambda ({\omega }_c)=2\pi n$$

First condition requires $\lambda$ has modulus 1 on real axis (lossless); second condition gives:

$${\tau }_c=\frac{2\pi n-\arg \lambda ({\omega }_c)}{{\omega }_c}$$

This is implicit equation, because $\lambda ({\omega }_c)$ itself also depends on frequency.

Explicit Approximation: Slow-Variation Approximation

If in small frequency window, $\lambda (\omega )$ approximately constant: $\lambda (\omega )\approx {\lambda }_0{e}^{i{\varphi }_0}$, then:

$${\tau }_k=\frac{2\pi k-{\varphi }_0}{{\omega }_{\ast }}$$

This gives family of equally spaced steps, spacing:

$$\Delta \tau =\frac{2\pi }{{\omega }_{\ast }}$$

Exactly “one optical period” round-trip time!

Physical Meaning of Steps

Delay quantization steps ${\tau }_k$ correspond to: Each time feedback loop’s round-trip phase $\omega \tau$ increases by $2\pi$, system “winds” one full circle in phase space, pole trajectory completes one “longitudinal mode” jump.

Analogy:

• Fabry-Perot cavity: Longitudinal mode spacing $\Delta \nu =c/(2L)$;
• Optical microring: Free Spectral Range (FSR) $\Delta \omega =2\pi /{\tau }_{\mathrm{round}}$;
• Self-referential scattering network: Step spacing $\Delta \tau =2\pi /\omega$.

This is same physical phenomenon (discrete spectrum from periodic boundary conditions) manifesting in different parameter spaces!

Group Delay Double-Peak Merger and Square-Root Scaling

Definition of Group Delay

Group delay matrix:

$$Q(\omega ;\tau )=-iS(\omega ;\tau {)}^{†}\frac{\partial S}{\partial \omega }$$

Its trace gives total group delay:

$${\tau }_g(\omega ;\tau )=\mathrm{tr}Q(\omega ;\tau )=-\Im \left[\mathrm{tr}\left(S^{-1}\frac{\partial S}{\partial \omega }\right)\right]$$

In Schur complement form, can write as:

$${\tau }_g={\tau }_g^{(0)}+{\tau }_g^{(\mathrm{fb})}$$

where second term is feedback contribution.

Origin of Double-Peak Structure

Near π-step $\tau \approx {\tau }_c$, pole trajectory ${\omega }_{\mathrm{pole}}(\tau )$ approaches real axis.

When scanning frequency $\omega$, group delay shows Lorentzian resonance peak near pole:

$${\tau }_g(\omega )\sim \frac{\Gamma }{(\omega -{\omega }_{\mathrm{pole}}{)}^2+{\Gamma }^2}$$

where $\Gamma =|\Im {\omega }_{\mathrm{pole}}|$ is resonance width.

When $\tau$ approaches ${\tau }_c$, two poles (from $n$ and $n+1$ longitudinal modes) simultaneously approach real axis, producing two resonance peaks.

Peak positions:

$${\omega }_{\pm }(\tau )\approx {\omega }_c\pm \delta \omega (\tau )$$

where $\delta \omega$ is peak spacing.

Derivation of Square-Root Scaling Law

Near $\tau ={\tau }_c$, pole trajectory can use local Puiseux expansion (branch expansion):

$${\omega }_{\mathrm{pole}}(\tau )\approx {\omega }_c+A(\tau -{\tau }_c{)}^{1/2}+B(\tau -{\tau }_c)+\cdots$$

This is because pole crossing real axis corresponds to branch point in complex frequency plane, similar to behavior of $\sqrt{z}$ at origin.

Substituting into pole equation, expanding to leading order, can rigorously derive:

$$A^2\sim \frac{{\partial }_{\tau }\lambda }{{\partial }_{\omega }\lambda }$$

(Specific coefficient depends on local form of $\lambda (\omega )$)

Therefore, scaling law for peak spacing:

$$\Delta \omega (\tau )=\delta \omega (\tau )\sim C\sqrt{|\tau -{\tau }_c|}$$

where $C$ is constant determined by system parameters.

Experimental Fingerprint

Square-root scaling is unique fingerprint of π-step:

1. Far from step ($|\tau -{\tau }_c|\gg \Delta \tau$): Group delay single peak, large peak width;
2. Approaching step: Single peak splits into double peak, peak spacing shrinks as $\sqrt{|\tau -{\tau }_c|}$;
3. Exactly at step: Double peaks merge into extremely sharp peak (theoretically width tends to zero);
4. Crossing step: Peak disappears or flips (phase transition $\pi$).

By fitting $\Delta \omega$ vs $\tau$ data, can precisely determine step position ${\tau }_c$ and scaling constant $C$.

Connection with Scale Identity

Phase Slope and Scale Density

Scale identity:

$$\kappa (\omega ;\tau )=\frac{1}{\pi }\frac{\partial \phi }{\partial \omega }=\frac{1}{2\pi }\mathrm{tr}Q(\omega ;\tau )$$

At fixed $\tau$, integrating over frequency:

$${\int }_{{\omega }_1}^{{\omega }_2}\kappa (\omega ;\tau )d\omega =\frac{1}{\pi }[\phi ({\omega }_2;\tau )-\phi ({\omega }_1;\tau )]$$

This is net phase change in frequency window (normalized to $\pi$ units).

Transition of Frequency Window Integral

Now fix frequency window $[{\omega }_c-\delta ,{\omega }_c+\delta ]$, let delay $\tau$ cross step ${\tau }_c$.

Define integral:

$$I(\tau )={\int }_{{\omega }_c-\delta }^{{\omega }_c+\delta }\kappa (\omega ;\tau )d\omega$$

Proposition: When $\tau$ crosses ${\tau }_c$, $I(\tau )$ jumps one unit:

$$\Delta I=I({\tau }_c+0)-I({\tau }_c-0)=\pm 1$$

Proof: When $\tau <{\tau }_c$, frequency window contains one pole ($n$-longitudinal mode); When $\tau >{\tau }_c$, this pole has left window, replaced by $(n+1)$-longitudinal mode.

By argument principle, phase difference at window boundaries changes by $\pm \pi$, so normalized integral changes by $\pm 1$.

This completely matches π-step theorem:

$$\Delta \phi =\pm \pi \iff \Delta I=\pm 1$$

Perspective of Unified Time Scale

Scale density $\kappa (\omega ;\tau )$ can be understood as “physical time density per unit frequency”.

Frequency window integral $I(\tau )$ is “total accumulated time in this frequency window”.

π-step corresponds to: When delay parameter crosses quantization step, system’s “effective time” in this frequency window suddenly increases or decreases by one unit.

This is a quantized transition of time, formally analogous to energy level transitions in quantum mechanics—except here what transitions is “time scale”, not energy!

Cumulative Effect of Multiple Steps

Spectral Flow Counting and Integer Invariant

When delay $\tau$ increases from ${\tau }_0$ to $\tau$, may cross multiple steps ${\tau }_1,{\tau }_2,\dots ,{\tau }_K$.

Define spectral flow count:

$$N(\tau )=\sum _{k:{\tau }_k<\tau }\Delta n_k$$

where $\Delta n_k=\Delta {\phi }_k/\pi =\pm 1$ is transition direction of $k$-th step.

This is an integer topological invariant, recording “net number of steps” system has experienced.

Z₂ Reduction and Parity

Although $N(\tau )$ is integer, in many physical problems, only its parity is essential:

$$\nu (\tau )=N(\tau )\mathrm{mod}2\in \{0,1\}$$

This is a Z₂ topological index.

In next chapter, we will discuss in detail why Z₂ parity is more “fundamental” than integer counting, and its deep connection with fermion statistics.

Numerical Verification and Experimental Calibration

Numerical Simulation Scheme

To verify π-step theory, can perform following numerical experiments:

1. Choose Model: Take single-channel feedback model or simple matrix model;
2. Parameter Scan: Fix frequency ${\omega }_{\ast }$, scan delay $\tau \in [{\tau }_{\min },{\tau }_{\max }]$;
3. Phase Calculation: For each $\tau$, calculate $\phi ({\omega }_{\ast };\tau )=\arg S_{\mathrm{tot}}({\omega }_{\ast };\tau )$;
4. Phase Unwrapping: Use phase unwrapping algorithm to remove artificial $2\pi$ jumps;
5. Step Identification: Identify jumps of magnitude $\pm \pi$ on $\phi (\tau )$ curve;
6. Scaling Law Fitting: Near each step, scan frequency, extract group delay double-peak spacing $\Delta \omega (\tau )$, fit $\Delta \omega \sim \sqrt{|\tau -{\tau }_c|}$.

Experimental Measurement Protocol

On optical or microwave platform:

1. Equipment: Tunable delay line + vector network analyzer (or optical interferometer);
2. Measurement: Scan two-dimensional parameter space $(\omega ,\tau )$, record complex scattering coefficients $S(\omega ;\tau )$;
3. Data Processing:
   • Extract phase $\phi (\omega ;\tau )=\arg S(\omega ;\tau )$;
   • Calculate group delay ${\tau }_g(\omega ;\tau )=-\partial \phi /\partial \omega$;
4. Feature Identification:
   • Plot phase contour map on $(\omega ,\tau )$ plane, identify “phase cliffs” (π-steps);
   • Near steps, observe group delay double-peak merger;
5. Quantitative Verification:
   • Measure step spacing $\Delta \tau$, compare with theoretical prediction $2\pi /\omega$;
   • Fit square-root scaling law, extract system parameters.

Chapter Summary

Core Theorems

π-Step Theorem: Under assumption of simple pole crossing real axis, local transition of closed-loop scattering phase exactly $\pm \pi$.

Delay Quantization: Step positions determined by implicit equation $\omega \tau +\arg \lambda (\omega )=2\pi n$, under slow-variation approximation, steps equally spaced, spacing equals round-trip time of one optical period.

Square-Root Scaling Law: Group delay double-peak spacing $\Delta \omega \sim \sqrt{|\tau -{\tau }_c|}$, this is local behavior from branch point, can serve as experimental fingerprint of π-step.

Connection with Scale Identity: Integral of scale density in frequency window $\int \kappa d\omega$ jumps one unit at step, equivalent to phase transition $\Delta \phi =\pm \pi$.

Physical Picture

π-step is not “accidental behavior” of system, but topological necessity: Pole crosses real axis, argument principle guarantees phase exactly winds half circle. This is unified manifestation of complex analysis geometry and physical causality.

Why is π Special?

Mathematically, $\pi$ is natural measure of “half circle”; physically, π-step corresponds to “half resonance”—system at critical point between resonance and anti-resonance.

More deeply, distinction between π vs $2\pi$ reflects topological divide of single-valuedness vs double-valuedness:

• Ordinary functions: Go around once return to original value (single-valued)
• Self-referential feedback: Go around once flip sign (double-valued)

This is exactly theme of next chapter’s Z₂ parity transition!

Thought Questions

1. Generalization of Argument Principle: If contour contains multiple poles crossing real axis simultaneously, is total phase transition equal to algebraic sum of contributions from each pole?

2. Non-Simple Poles: If pole is double (i.e., $m=2$), is phase transition $2\pi$? Try analyzing from local factor $(\omega -z(\tau ){)}^{-2}$.

3. Pole Merging: If two poles simultaneously cross real axis and positions coincide, what happens? (Hint: This corresponds to “special point” or “singularity in parameter space”)

4. Experimental Noise: In actual measurements, phase data contains noise. How to robustly identify π-steps? (Hint: Use integer property of frequency window integral)

5. High-Dimensional Generalization: If there are two tunable parameters $({\tau }_1,{\tau }_2)$, does π-step generalize to “phase lines” in parameter plane? Can these lines form topological network?

Preview of Next Chapter

After proving necessity of π-step, next chapter explores deeper topological structure:

Z₂ Parity Transition and Topological Index

We will:

• Construct topological parity index $\nu (\tau )=N(\tau )\mathrm{mod}2$
• Prove its flipping rule under evolution $\nu ({\tau }_k+0)=\nu ({\tau }_k-0)\oplus 1$
• Establish connection with fundamental group, Null-Modular double cover
• Explain why parity is more “fundamental” than integer

Let us continue deeper into topological mysteries of self-referential scattering!