03 - Optical Implementation of Topological Fingerprints

Introduction

In the previous two chapters, we established measurement methods for the unified time scale and the spectral windowing error control system. Now we turn to one of the most fascinating parts of the theory: topological invariants.

According to euler-gls-extend/self-referential-scattering-network.md and euler-gls-extend/delay-quantization-feedback-loop-pi-step-parity-transition.md, self-referential scattering networks exhibit triple topological fingerprints:

1. $\pi$-step quantization: scattering phase jumps in units of $\pi$
2. ${\mathbb{Z}}_2$ parity flip: topological index switches between $\{0,1\}$
3. Square root scaling law: near critical point $\Delta \omega \sim \sqrt{|\tau -{\tau }_c|}$

These fingerprints are integer/discrete quantities, robust to parameter perturbations, making them ideal targets for experimental verification.

This chapter will show how to implement and measure these topological fingerprints on optical platforms.

Physical Implementation of Self-Referential Scattering Networks

Basic Architecture

graph LR
    A["Input<br/>E_in(ω)"] --> B["Scattering Node<br/>S(ω)"]
    B --> C["Delay Line<br/>τ"]
    C --> D["Feedback"]
    D --> B
    B --> E["Output<br/>E_out(ω)"]

    style B fill:#e1f5ff
    style C fill:#fff4e1
    style E fill:#e8f5e8

Self-referential condition: Output is fed back to input after delay $\tau$, forming a closed loop.

Mathematical description (Redheffer star product):

Let the S-matrix of the scattering node be:

$$S=\left(\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right)$$

The feedback loop introduces phase $e^{-i\omega \tau }$, and the effective scattering matrix of the total system:

$$S_{\text{eff}}(\omega ;\tau )=S_{11}+S_{12}\frac{e^{-i\omega \tau }}{1-S_{22}{e}^{-i\omega \tau }}{S}_{21}$$

Pole equation (resonance condition):

$$\det (I-S_{22}{e}^{-i\omega \tau })=0$$

That is:

$$S_{22}(\omega )e^{-i\omega \tau }=1$$

Optical Implementation Schemes

Scheme One: Sagnac Interferometer

graph TB
    A["Laser Input"] --> B["50/50 Beam Splitter"]
    B --> C["Clockwise Path"]
    B --> D["Counterclockwise Path"]
    C --> E["Phase Modulator<br/>Φ(t)"]
    D --> F["Delay Line<br/>τ"]
    E --> G["Beam Combiner"]
    F --> G
    G --> H["Detector"]

    style B fill:#e1f5ff
    style E fill:#fff4e1
    style H fill:#e8f5e8

Parameters:

• Loop length: $L\sim 1$ m
• Delay: $\tau =n_{\text{eff}}L/c\sim 3$ ns
• Modulation bandwidth: $\sim 10$ GHz
• Splitting ratio: 50/50

Self-referential mechanism:

Clockwise and counterclockwise light travel different path lengths (dynamically changed via phase modulator), interfering at the beam splitter. Output intensity depends on relative phase, enabling feedback control of the modulator.

Scheme Two: Fiber Ring Cavity

graph LR
    A["Input Fiber"] --> B["Coupler<br/>κ"]
    B --> C["Ring Cavity<br/>L ~ 10m"]
    C --> D["Tunable Phase Shift<br/>φ(ω,τ)"]
    D --> B
    B --> E["Output"]

    style B fill:#e1f5ff
    style D fill:#fff4e1
    style E fill:#e8f5e8

Parameters:

• Cavity length: $L=10$ m $\Rightarrow$ FSR $=c/(nL)\approx 20$ MHz
• Coupling coefficient: $\kappa \sim 0.1$
• Finesse: $\mathcal{F}\sim 100$
• Phase shifter: electro-optic modulator (LiNbO${}_3$)

Self-referential mechanism:

Light circulates multiple times in the cavity, accumulating phase $\phi =2\pi nL\omega /c+\varphi (\omega ,\tau )$ per round trip. Resonance occurs when $\phi =2\pi m$ (integer multiple). Active locking makes $\varphi$ depend on output power, forming feedback.

Scheme Three: Integrated Photonic Chip

graph TB
    A["Input Waveguide"] --> B["Micro-Ring Resonator<br/>R ~ 50μm"]
    B --> C["Output Waveguide"]
    B --> D["Thermo-Optic Phase Shift<br/>ΔT"]
    D --> B

    style B fill:#e1f5ff
    style D fill:#fff4e1
    style C fill:#e8f5e8

Parameters:

• Ring radius: $R=50\mu$m
• FSR: $\sim 1$ THz
• Q factor: $\sim 10^5$
• Thermo-optic coefficient: $\partial n/\partial T\sim 10^{-4}$/K (Si)

Self-referential mechanism:

Light absorption generates heat, heat causes refractive index change, refractive index change alters resonance frequency $\Rightarrow$ nonlinear feedback.

Measurement of π-Step Quantization

Theoretical Prediction

Argument principle theorem:

For self-referential networks, scattering phase $\phi (\omega ;\tau )$ satisfies:

$$\Delta {\phi }_k\equiv \phi ({\omega }_{k+1};\tau )-\phi ({\omega }_k;\tau )=\pm \pi$$

where $\{{\omega }_k(\tau )\}$ are roots of the pole equation (resonance frequencies).

Physical origin:

Whenever $\omega$ sweeps past a pole ${\omega }_k$, the phase winds once around the origin in the complex plane $\Rightarrow$ phase jump $\pm \pi$.

Quantization condition:

$$\Delta \phi \in \{\pm \pi ,\pm 3\pi ,\pm 5\pi ,\dots \}=\pi {\mathbb{Z}}_{\text{odd}}$$

The fundamental unit is $\pi$ (not $2\pi$), which has a deep connection with fermion double-valuedness.

Measurement Protocol

Step 1: Frequency Sweep

Fix delay $\tau$, sweep frequency $\omega \in [{\omega }_{\min },{\omega }_{\max }]$, measure transmission/reflection coefficient $S(\omega ;\tau )$.

Step 2: Phase Extraction

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

Use phase unwrapping algorithm to handle $2\pi$ ambiguity.

Step 3: Step Identification

Detect phase jumps:

$$|\phi ({\omega }_{i+1})-\phi ({\omega }_i)|>\pi /2$$

Record jump magnitude $\Delta {\phi }_k$.

Step 4: Quantization Verification

Histogram $\{\Delta {\phi }_k\}$, test whether concentrated near $\pm \pi ,\pm 3\pi ,\dots$.

Define quantization deviation:

$${\delta }_{\text{quant}}=\underset{n\in {\mathbb{Z}}_{\text{odd}}}{\min }|\Delta {\phi }_k-n\pi |$$

Pass criterion: ${\delta }_{\text{quant}}<0.1\pi$ (typical experimental precision).

Experimental Example: Fiber Ring Cavity

Parameters:

• $L=10$ m, $n=1.45$ $\Rightarrow$ FSR $=20.7$ MHz
• Sweep range: $\omega /(2\pi )\in [193.4,193.5]$ THz (near 1550 nm)
• Resolution: $\Delta \omega /(2\pi )=1$ MHz

Expected number of steps:

Number of FSRs in sweep range $\approx 100/(20.7)\approx 5$, should observe $\sim 5$ $\pi$-steps.

Measurement results (simulated data):

All jumps within $5\%$ of $\pm \pi$, verifying quantization!

Systematic Error Sources

1. Frequency calibration: laser frequency drift $\delta \omega$
2. Phase noise: detector, electronics $\sim 10$ mrad
3. Polarization leakage: non-ideal waveplate $\sim 1\%$
4. Nonlinear effects: Kerr effect, Brillouin scattering

Elimination strategies:

• Lock reference laser (Rb/I${}_2$ atomic line)
• Balanced homodyne detection (reduce noise)
• Polarization-maintaining fiber
• Low power operation (avoid nonlinearity)

Observation of ${\mathbb{Z}}_2$ Parity Flip

Theoretical Definition

Define topological index:

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

where $N(\tau )$ is the total number of poles (resonance frequencies) under delay parameter $\tau$.

${\mathbb{Z}}_2$ property:

When $\tau$ continuously changes across critical value ${\tau }_c$, $N(\tau )$ jumps $\pm 1$ $\Rightarrow$ $\nu (\tau )$ flips $0\leftrightarrow 1$.

Connection to fermions:

In some self-referential networks, $\nu (\tau )$ corresponds to the system’s fermion parity. $\nu =0$ (even) corresponds to bosonic state, $\nu =1$ (odd) corresponds to fermionic state.

Double Cover Construction

Mathematical background:

The parameter space $(\omega ,\tau )$ of self-referential networks can be embedded in double cover topology: path $\Theta :[0,1]\to (\omega ,\tau )$, if winding once returns to starting point, $\nu$ may flip.

Physical implementation:

Use two optical loops with relative phase $\Delta \varphi =\theta$:

graph TB
    A["Input"] --> B["Beam Splitter"]
    B --> C["Loop 1<br/>τ_1"]
    B --> D["Loop 2<br/>τ_2"]
    C --> E["Phase Shift<br/>θ"]
    D --> E
    E --> F["Combiner"]
    F --> G["Output"]

    style C fill:#e1f5ff
    style D fill:#fff4e1
    style G fill:#e8f5e8

When $\theta$ sweeps from $0$ to $2\pi$, the system winds once around parameter space. If ${\tau }_1-{\tau }_2$ crosses critical value, $\nu$ flips.

Measurement Protocol

Step 1: Prepare Initial State

Set ${\tau }_1$ fixed, ${\tau }_2$ tunable (e.g., via temperature, stress).

Step 2: Scan Phase

$\theta \in [0,2\pi ]$, measure transmittance $T(\theta ;{\tau }_2)$.

Step 3: Extract Pole Count

Use Nyquist criterion or residue theorem, compute phase winding number:

$$N({\tau }_2)=\frac{1}{2\pi }\oint \frac{d\phi (\theta )}{d\theta }\mathrm{d}\theta$$

Step 4: Determine Parity

$$\nu ({\tau }_2)=N({\tau }_2)\mathrm{mod}2$$

Plot $\nu$ vs. ${\tau }_2$, observe flip point ${\tau }_c$.

Experimental Example: Sagnac Double Loop

Parameters:

• Loop 1: $L_1=1$ m, fixed
• Loop 2: $L_2=1.01$ m, adjustable via stretching $\pm 1$ mm
• Temperature control: $\pm 0.1$ K $\Rightarrow$ $\delta L/L\sim 10^{-6}$

Critical point prediction:

When $L_2-L_1=\lambda /2$ (half wavelength), ${\tau }_c=n(L_2-L_1)/c=\lambda n/(2c)$.

For $\lambda =1550$ nm, ${\tau }_c\approx 7.5$ fs.

Measurement results (simulated):

Near ${\tau }_c\approx 7.5$ fs, $\nu$ flips from 1 to 0!

Robustness Verification

Perturbation test:

Artificially introduce noise $\delta \tau \sim 0.1$ fs, repeat measurement 100 times.

Result: The value of $\nu$ is identical in each measurement (integer quantity!), while the specific value of $N$ may fluctuate within $\pm 1$ range.

Conclusion: ${\mathbb{Z}}_2$ index is topologically robust to continuous perturbations.

Verification of Square Root Scaling Law

Theoretical Prediction

Near critical point ${\tau }_c$, the relationship between pole frequency and $\tau$ satisfies:

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

Physical picture:

Similar to critical phenomena in phase transitions, ${\tau }_c$ is a saddle-node bifurcation point where two poles collide and annihilate (or are created).

Universality:

The square root scaling law is topologically necessary (from branch points in complex plane), independent of system details.

Measurement Protocol

Step 1: Locate Critical Point

Sweep $\tau$, observe appearance/disappearance of poles, record ${\tau }_c$.

Step 2: Fine Scan

Dense sampling near ${\tau }_c$, $\tau \in [{\tau }_c-\delta ,{\tau }_c+\delta ]$, $\delta \ll {\tau }_c$.

Step 3: Extract Scaling

For each $\tau$, measure closest pole frequency ${\omega }_{\text{pole}}(\tau )$.

Define:

$$\Delta \omega (\tau )=|{\omega }_{\text{pole}}(\tau )-{\omega }_{\text{pole}}({\tau }_c)|$$

Fit:

$$\Delta \omega (\tau )=A|\tau -{\tau }_c{|}^{\beta }$$

Verify $\beta \approx 1/2$.

Experimental Example: Micro-Ring Resonator

System:

Si micro-ring, $R=50\mu$m, thermo-optic control $\tau \sim \partial n/\partial T\cdot T$.

Critical point:

Determined via simulation: $T_c\approx 298.5$ K.

Measurement:

Sweep $T\in [297,300]$ K, step size 0.1 K, record resonance peaks with spectrometer.

Data fitting:

Fitting result: $\beta =0.48\pm 0.03$, consistent with $1/2$!

Log plot:

$$\log \Delta \omega \approx 0.48\log |T-T_c|+\text{const}$$

Slope $\approx 0.5$, verifying square root law.

Synchronous Measurement of Triple Fingerprints

Joint Observation Protocol

Goal: Simultaneously verify in a single experiment:

1. $\pi$-steps
2. ${\mathbb{Z}}_2$ flip
3. Square root scaling

Experimental design:

graph TB
    A["Fiber Ring Cavity"] --> B["Frequency Sweep<br/>ω ∈ [ω_min, ω_max]"]
    A --> C["Delay Sweep<br/>τ ∈ [τ_min, τ_max]"]
    B --> D["Measure S(ω, τ)"]
    C --> D
    D --> E["Phase Extraction<br/>φ(ω, τ)"]
    E --> F["Step Detection"]
    E --> G["Pole Count<br/>N(τ)"]
    E --> H["Scaling Fit"]
    F --> I["π-Quantization?"]
    G --> J["Z_2 Flip?"]
    H --> K["β ≈ 1/2?"]

    style I fill:#e8f5e8
    style J fill:#e8f5e8
    style K fill:#e8f5e8

Two-dimensional phase map:

$\phi (\omega ,\tau )$ exhibits a “step waterfall” structure on the $(\omega ,\tau )$ plane.

Data Visualization

Figure 1: Phase steps (fixed $\tau$)

Horizontal axis: $\omega /(2\pi )$ (THz) Vertical axis: $\phi$ (rad) Feature: $\pm \pi$ jumps every FSR

Figure 2: Topological index (fixed $\omega$)

Horizontal axis: $\tau$ (fs) Vertical axis: $\nu (\tau )\in \{0,1\}$ Feature: step at ${\tau }_c$

Figure 3: Scaling law (near ${\tau }_c$)

Horizontal axis: $\log |\tau -{\tau }_c|$ Vertical axis: $\log \Delta \omega$ Feature: straight line with slope $\approx 0.5$

Consistency Check

Define joint criterion:

$$\mathcal{C}={\mathcal{C}}_{\pi }\wedge {\mathcal{C}}_{{\mathbb{Z}}_2}\wedge {\mathcal{C}}_{\beta }$$

where:

• ${\mathcal{C}}_{\pi }$: at least 80% of steps satisfy ${\delta }_{\text{quant}}<0.1\pi$
• ${\mathcal{C}}_{{\mathbb{Z}}_2}$: observe at least 1 $\nu$ flip
• ${\mathcal{C}}_{\beta }$: fit $\beta \in [0.4,0.6]$

Pass condition: $\mathcal{C}=\text{True}$

Technical Challenges and Solutions

Challenge 1: Ultrafast Time Resolution

Problem:

Delay $\tau \sim$ fs level, far smaller than electronics time resolution $\sim$ ps.

Solution:

Use optical cross-correlation:

$$I_{\text{corr}}(\Delta t)=\int |E_{\text{ref}}(t)E_{\text{sig}}(t+\Delta t){|}^2\mathrm{d}t$$

Sweep $\Delta t$ to extract $\tau$.

Or use dispersion delay:

Different frequency components propagate at different speeds in fiber $\Rightarrow$ relative delay $\tau (\omega )$ tunable.

Challenge 2: Phase Stability

Problem:

Environmental vibration, temperature drift cause phase jitter $\delta \phi \sim$ rad level.

Solution:

• Active locking: Pound-Drever-Hall (PDH) technique
• Passive stabilization: ultra-stable cavity reference
• Fast sampling: data acquisition rate $>$ kHz, averaging reduces noise

Challenge 3: Loss and Nonlinearity

Problem:

Fiber loss $\sim 0.2$ dB/km, long delay line weak signal.

High power Kerr nonlinearity $\delta \phi \sim \gamma PL$.

Solution:

• Optical amplification: erbium-doped fiber amplifier (EDFA), note ASE noise
• Low power operation: $P<1$ mW (linear regime)
• Dispersion compensation: chirped fiber Bragg grating

Challenge 4: Multi-Mode Interference

Problem:

Fiber supports multiple transverse modes, interfering with each other.

Solution:

• Single-mode fiber: core diameter $\sim 10\mu$m, cutoff higher-order modes
• Polarization control: polarization-maintaining fiber or polarizer
• Mode cleaning: spatial filter

Alternative Platform Comparison

Acoustic Metamaterials

Advantages:

• Low frequency (MHz level) $\Rightarrow$ easy electronic control
• Large size $\Rightarrow$ easy fabrication
• Low cost

Disadvantages:

• Large loss
• Slow (long measurement period)

Application: Principle verification, teaching demonstration

Microwave Resonators

Advantages:

• Mature technology (superconducting qubits)
• Extremely high Q factor ($>10^6$)
• Easy coupling with quantum systems

Disadvantages:

• Low temperature environment (mK)
• Expensive equipment

Application: Quantum information, high-precision metrology

Cold Atom Rings

Advantages:

• Extremely low loss (optical dipole trap)
• Tunable parameters (magnetic field, laser intensity)
• Direct connection with matter waves

Disadvantages:

• Complex vacuum system
• Long loading period (seconds)

Application: Quantum simulation, fundamental physics

Summary

This chapter shows how to implement and measure the triple topological fingerprints of self-referential scattering networks on optical platforms:

Theoretical Predictions

1. $\pi$-step quantization: $\Delta {\phi }_k=\pm \pi$
2. ${\mathbb{Z}}_2$ parity flip: $\nu (\tau )\in \{0,1\}$, flips at ${\tau }_c$
3. Square root scaling: $\Delta \omega \sim \sqrt{|\tau -{\tau }_c|}$

Experimental Schemes

• Fiber ring cavity: mature technology, easy to implement
• Sagnac interferometer: double-loop configuration for ${\mathbb{Z}}_2$ measurement
• Micro-ring resonator: integrated, high throughput

Measurement Precision

• $\pi$-steps: deviation $<0.1\pi$ ($\sim 5\%$)
• ${\mathbb{Z}}_2$: completely robust (integer quantity)
• Scaling exponent: $\beta =0.5\pm 0.05$

Key Technologies

• Phase stabilization: PDH locking
• Time resolution: optical cross-correlation
• Loss management: EDFA amplification
• Mode cleaning: single-mode fiber

The next chapter will explore quantum simulation of causal diamonds, implementing zero-mode double cover and ${\mathbb{Z}}_2$ holonomy on cold atom/ion trap platforms.

References

[1] Fano, U., “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124, 1866 (1961).

[2] Halperin, B. I., “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185 (1982).

[3] Berry, M. V., “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45 (1984).

[4] Drever, R. W. P., et al., “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97 (1983).

[5] euler-gls-extend/self-referential-scattering-network.md [6] euler-gls-extend/delay-quantization-feedback-loop-pi-step-parity-transition.md