Chapter 3: Engineering Implementation and Finite Complexity Readout

Source Theory: euler-gls-info/17-time-crystals-null-modular-z2-holonomy.md, §5; Appendix D

Introduction

The previous two chapters established the theoretical framework for time crystals:

• Chapter 01: Floquet-QCA and period doubling mechanism
• Chapter 02: Z₂ holonomy and topological invariants

Now we face practical questions: How to observe and measure time crystals in experiments?

This chapter will answer:

1. Which quantum platforms are suitable for realizing time crystals?
2. How to discriminate time crystal signals within finite measurement steps?
3. How many samples are needed for reliable discrimination (sample complexity)?
4. How to handle noise and dissipation?

Core Tool: DPSS windowed readout technique (review Chapter 20)

Everyday Analogy:

• Time Crystal Signal: Weak periodic “heartbeat”
• Noise: Background noise
• DPSS Windowing: High-sensitivity “stethoscope”
• Sample Complexity: How long to listen to confirm heartbeat exists

1. Experimental Platform Overview

1.1 Four Candidate Platforms

Time crystals can be realized on various quantum platforms, each with advantages and disadvantages:

TRL (Technology Readiness Level): Technology maturity level, 1-9 scale, higher numbers indicate greater maturity.

Mermaid Platform Comparison

graph TD
    A["Time Crystal<br/>Experimental Platforms"] --> B1["Cold Atoms"]
    A --> B2["Superconducting Qubits"]
    A --> B3["Ion Trap"]
    A --> B4["Solid-State Spin"]

    B1 --> C1["Coherence Time<br/>Second Scale"]
    B2 --> C2["Gate Speed<br/>Nanosecond Scale"]
    B3 --> C3["Fidelity<br/>99.9%"]
    B4 --> C4["Temperature<br/>Room Temperature"]

    D["DPSS Readout"] -.->|"Common Requirement"| B1
    D -.-> B2
    D -.-> B3
    D -.-> B4

    style A fill:#e1f5ff
    style B1 fill:#ffe1e1
    style B2 fill:#f5e1ff
    style B3 fill:#fff4e1
    style B4 fill:#e1ffe1
    style D fill:#ffe1f5

1.2 Platform Selection Criteria

Key Metrics:

(1) Floquet Band Gap ${\Delta }_{\mathrm{F}}$:

• Larger gap, stronger time crystal signal
• More robust to noise

(2) Coherence Time $T_2$:

• Need $T_2\gg N\cdot T$ ($N$ is number of measurement periods)
• Limits maximum measurable $N$

(3) Measurement Fidelity $F_{\mathrm{meas}}$:

• Readout error directly affects signal-to-noise ratio
• Need $F_{\mathrm{meas}}>95\%$

(4) Scalability:

• Number of lattice sites (system size)
• Parallel measurement capability

Selection Matrix:

2. Cold Atom Optical Lattice Realization

2.1 System Composition

Lattice: Periodic potential wells formed by optical standing waves

Atoms: Alkali metal atoms (e.g., ${}^{87}$Rb, ${}^{40}$K)

Spin States: Hyperfine levels $|F,m_F⟩$ simulate spin-1/2

Floquet Drive:

• Method 1: Periodic modulation of lattice depth
• Method 2: Raman pulse driving spin flips
• Method 3: Lattice oscillation (shaking)

Mermaid Cold Atom System

graph TD
    A["Laser Beams"] --> B["Optical Lattice<br/>Periodic Potential Wells"]
    B --> C["Atom Array<br/>Spin-1/2"]

    D["Floquet Drive"] --> E["Modulation Schemes"]
    E --> E1["Lattice Depth<br/>V(t) = V_0[1+A cos(omega t)]"]
    E --> E2["Raman Pulses<br/>Periodic Spin Flips"]
    E --> E3["Lattice Oscillation<br/>Position Modulation"]

    F["Measurement"] --> G["Single-Site Imaging<br/>Fluorescence Detection"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5

2.2 Specific Experimental Scheme

Step 1: Prepare Initial State $$|{\psi }_0⟩=|\uparrow \downarrow \uparrow \downarrow \cdots ⟩$$ (antiferromagnetic order)

Step 2: Floquet Drive Period $T=1$ ms, drive $N=100$ periods.

Step 3: Measure Local Spin $$⟨{\sigma }_x^z{⟩}_n=⟨{\psi }_n|{\sigma }_x^z|{\psi }_n⟩$$

Step 4: DPSS Windowing Analysis Apply DPSS windowing to sequence $\{a_n{\}}_{n=0}^{N-1}$ ($a_n=⟨{\sigma }_x^z{⟩}_n$).

Expected Results:

• Main frequency at $\omega =\pi$ (normalized frequency)
• Energy peak significantly above noise floor

Parameter Estimates (Source Theory §5.3):

2.3 Advantages and Disadvantages of Cold Atoms

Advantages: ✅ Long coherence time (second scale) → Can measure large $N$ ✅ Single-site resolved imaging → Precise local measurement ✅ Tunable interactions → Flexible control of ${\Delta }_{\mathrm{F}}$ ✅ Low temperature environment → Low noise

Disadvantages: ❌ Complex preparation (vacuum system, laser cooling) ❌ Temperature sensitive (need $\mu$K level) ❌ Destructive measurement (atoms lost after fluorescence) ❌ Slow cycle rate (minutes per experiment)

3. Superconducting Qubit Realization

3.1 System Composition

Qubits: Josephson junctions

Coupling: Capacitive or inductive coupling

Floquet Drive: Microwave pulse sequences

Measurement: Dispersive readout

Mermaid Superconducting System

graph TD
    A["Superconducting Chip"] --> B["Qubit Array<br/>Josephson Junctions"]
    B --> C["Couplers<br/>Tunable Coupling"]

    D["Microwave Drive"] --> E["Floquet Pulses"]
    E --> E1["X Gate<br/>Spin Flip"]
    E --> E2["Z Gate<br/>Phase Accumulation"]
    E --> E3["Two-Qubit Gates<br/>Entanglement Generation"]

    F["Readout"] --> G["Dispersive Measurement<br/>Non-Destructive"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5

3.2 Specific Experimental Scheme

Step 1: Prepare Initial State Prepare via single-qubit gate sequences: $$|{\psi }_0⟩=|01010101\cdots ⟩$$

Step 2: Floquet Drive Period $T=1\mu$s, drive $N=1000$ periods.

Step 3: Projective Measurement $$P_z=⟨{\sigma }_x^z{⟩}_n$$

Step 4: Repeat Measurement Repeat $M=10^4$ times, statistical average.

Parameter Estimates:

3.3 Advantages and Disadvantages of Superconducting Qubits

Advantages: ✅ Fast manipulation (nanosecond gates) → High efficiency ✅ Programmable architecture → Flexible control sequences ✅ Non-destructive measurement → Repeatable readout ✅ Mature technology → Industrialization potential

Disadvantages: ❌ Short coherence time ($\mu$s scale) → Limits $N$ ❌ Crosstalk noise → Affects multi-qubit systems ❌ Low temperature environment (mK level) → Complex equipment ❌ Relatively small gap → Signal-to-noise ratio challenge

4. DPSS Windowed Readout Scheme

4.1 Problem Setup

Measurement Sequence: $$a_n=\mathrm{tr}({\rho }_0{U}_F^{†n}O{U}_F^n),n=0,1,\dots ,N-1$$

Ideal Time Crystal Signal ($m=2$): $$s_n=s_0(-1{)}^n$$

Actual Measurement (with noise): $$a_n=s_n+{\eta }_n$$

where ${\eta }_n$ is noise, assumed:

• Zero mean: $\mathbb{E}[{\eta }_n]=0$
• Finite variance: $\mathbb{E}[{\eta }_n^2]={\sigma }_{\eta }^2$
• Finite correlation length: $\mathbb{E}[{\eta }_n{\eta }_m]\approx 0$ (when $|n-m|$ large)

Mermaid Measurement Model

graph LR
    A["Ideal Signal<br/>s_n = s_0(-1)^n"] --> C["Actual Measurement<br/>a_n"]
    B["Noise<br/>eta_n"] --> C

    C --> D["DPSS Windowing<br/>w_n * a_n"]
    D --> E["Fourier Transform<br/>hat a(omega)"]

    E --> F["Main Frequency Detection<br/>omega = pi"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5

4.2 DPSS Window Function

DPSS Definition (Review Chapter 20 Section 02): Discrete prolate spheroidal sequences (DPSS) are solutions to the following optimization problem:

$$\underset{\{w_n\}}{\max }\frac{{\sum }_{|k|\le W}|{\hat{w}}_k{|}^2}{{\sum }_k|{\hat{w}}_k{|}^2}$$

subject to constraint ${\sum }_n|w_n{|}^2=1$.

Shannon Number: $$N_0=2NW$$

where $N$ is number of samples, $W$ is normalized bandwidth.

Principal DPSS Sequence $w_n^{(0)}$ corresponds to largest eigenvalue ${\lambda }_0\approx 1$.

Mermaid DPSS Properties

graph TD
    A["DPSS Window Function<br/>w_n^(0)"] --> B["Time Domain Localization<br/>Support on [0,N-1]"]
    A --> C["Frequency Domain Localization<br/>Concentrated in [-W,W]"]

    D["Shannon Number<br/>N_0 = 2NW"] -.->|"Degrees of Freedom"| A

    E["Main Leakage<br/>1-lambda_0 << 1"] -.->|"Energy Concentration"| C

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1

4.3 Windowed Phase Accumulation

Windowed Fourier Spectrum (Source Theory §5.2): $$\hat{a}(\omega )=\sum _{n=0}^{N-1}{w}_n^{(0)}{a}_n{\mathrm{e}}^{-\mathrm{i}\omega n}$$

Main Frequency Detection ($m=2$ time crystal): $$\hat{a}(\pi )=\sum _{n=0}^{N-1}{w}_n^{(0)}{a}_n(-1{)}^n$$

Ideal Signal Contribution: $$\hat{s}(\pi )=s_0\sum _n{w}_n^{(0)}|(-1{)}^n{|}^2=s_0\sum _n|w_n^{(0)}{|}^2=s_0$$

Noise Variance (Source Theory §5.3): $$\mathrm{Var}(\hat{a}(\pi ))={\sigma }_{\eta }^2\sum _n|w_n^{(0)}{|}^2={\sigma }_{\eta }^2$$

(assuming window function normalized ${\sum }_n|w_n^{(0)}{|}^2=1$)

4.4 Signal-to-Noise Ratio and Discrimination Criterion

Signal-to-Noise Ratio: $$\mathrm{SNR}=\frac{|\mathbb{E}[\hat{a}(\pi )]{|}^2}{\mathrm{Var}(\hat{a}(\pi ))}=\frac{|s_0{|}^2}{{\sigma }_{\eta }^2}$$

Discrimination Criterion: Set threshold $\gamma$, discrimination rule:

$$\{\begin{array}{ll}|\hat{a}(\pi )|>\gamma & \Rightarrow \text{Time crystal exists}\\ |\hat{a}(\pi )|\le \gamma & \Rightarrow \text{No time crystal}\end{array}$$

Error Probability (Chebyshev inequality): $$P(\text{error})\le \frac{{\sigma }_{\eta }^2}{{\gamma }^2}$$

Choose $\gamma =c{\sigma }_{\eta }$ ($c>1$), then: $$P(\text{error})\le \frac{1}{c^2}$$

5. Sample Complexity Theory

5.1 Theorem Statement

Theorem 5.1 (Source Theory §5.3):

Under the following conditions:

(1) Floquet Band Gap: ${\Delta }_{\mathrm{F}}>0$

(2) Bounded Noise: $\{{\eta }_n\}$ zero mean, finite correlation length, variance ${\sigma }_{\eta }^2$ bounded

(3) DPSS Windowing: Use DPSS basis sequence $w_n^{(0)}$ with appropriate bandwidth $W$

Then to discriminate whether period $2T$ time crystal signal exists with error probability at most $\epsilon$, required complexity steps $N$ satisfy:

$$\boxed{N\ge C{\Delta }_{\mathrm{F}}^{-2}\log (1/\epsilon )}$$

where $C$ is a constant (depends on system details).

Mermaid Theorem Structure

graph TD
    A["Theorem 5.1<br/>Sample Complexity"] --> B["Condition 1<br/>Gap Delta_F > 0"]
    A --> C["Condition 2<br/>Noise Finite Variance"]
    A --> D["Condition 3<br/>DPSS Windowing"]

    B --> E["Conclusion"]
    C --> E
    D --> E

    E --> F["N >= C * Delta_F^-2 * log(1/epsilon)"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#ffcccc
    style F fill:#aaffaa

5.2 Proof Outline

Step 1: Relationship Between Signal Amplitude and Band Gap

Floquet band gap ${\Delta }_{\mathrm{F}}$ controls amplitude and dissipation time of time crystal signal:

$$|s_0|\sim {\Delta }_{\mathrm{F}}\cdot \exp (-\gamma /{\Delta }_{\mathrm{F}})$$

where $\gamma$ is noise strength.

Step 2: DPSS Energy Concentration

DPSS window function is nearly ideal band-limited near frequency band $[-W,W]$, observation at main frequency $\omega =\pi$ is mainly sensitive to time crystal signal, noise is suppressed.

Step 3: Large Deviation Estimate

Using Chebyshev inequality or Chernoff bound, require:

$$\frac{|s_0{|}^2}{{\sigma }_{\eta }^2/N}\ge c_0\log (1/\epsilon )$$

i.e., signal-to-noise ratio sufficiently large.

Step 4: Solve for $N$

From $|s_0|\sim {\Delta }_{\mathrm{F}}$, get:

$$N\ge C\frac{{\sigma }_{\eta }^2}{{\Delta }_{\mathrm{F}}^2}\log (1/\epsilon )$$

After normalization, this is the theorem conclusion.

Detailed calculations see Source Theory Appendix D.

5.3 Complexity Analysis

Band Gap Dependence:

• Larger gap ${\Delta }_{\mathrm{F}}$, stronger signal, fewer samples needed
• $N\propto {\Delta }_{\mathrm{F}}^{-2}$ (inverse square)

Error Requirement:

• Smaller error tolerance $\epsilon$, more samples needed
• $N\propto \log (1/\epsilon )$ (logarithmic growth, mild)

Practical Estimates:

Mermaid Complexity Trends

graph LR
    A["Gap Increases<br/>Delta_F up"] --> B["Signal Strengthens<br/>s_0 up"]
    B --> C["Sample Demand Decreases<br/>N down"]

    D["Error Requirement Strengthens<br/>epsilon down"] --> E["Sample Demand Increases<br/>N up"]

    F["N ~ Delta_F^-2 * log(1/epsilon)"]

    style A fill:#e1ffe1
    style B fill:#e1f5ff
    style C fill:#aaffaa
    style D fill:#ffcccc
    style E fill:#ffaaaa
    style F fill:#ffe1f5

6. Noise Robustness and Error Control

6.1 Noise Source Classification

Noise in Experiments:

(1) Quantum Projection Noise:

• Inherent randomness of measurement
• Variance $\sim 1/\sqrt{M}$ ($M$ is number of repetitions)

(2) Technical Noise:

• Laser intensity fluctuations (cold atoms)
• Microwave power fluctuations (superconducting)
• Magnetic field noise

(3) Dissipation and Decoherence:

• Spin relaxation ($T_1$ process)
• Phase decoherence ($T_2$ process)

(4) Measurement Error:

• Readout fidelity $<100\%$
• Crosstalk

Mermaid Noise Sources

graph TD
    A["Noise Sources"] --> B1["Quantum Projection<br/>Inherent Randomness"]
    A --> B2["Technical Noise<br/>Environmental Fluctuations"]
    A --> B3["Dissipation<br/>T1, T2"]
    A --> B4["Measurement Error<br/>Readout Errors"]

    B1 --> C["Total Variance<br/>sigma_eta^2"]
    B2 --> C
    B3 --> C
    B4 --> C

    C --> D["Affects SNR<br/>Signal-to-Noise Ratio Decreases"]

    style A fill:#e1f5ff
    style B1 fill:#ffe1e1
    style B2 fill:#f5e1ff
    style B3 fill:#fff4e1
    style B4 fill:#e1ffe1
    style C fill:#ffe1f5
    style D fill:#ffcccc

6.2 Error Mitigation Strategies

Strategy 1: Increase Repetition Count $${\sigma }_{\mathrm{eff}}^2=\frac{{\sigma }_{\eta }^2}{M}$$

Requires $M$ independent repetitions.

Strategy 2: Optimize Window Function Bandwidth

Choose $W\approx 0.1$ (normalized), so DPSS concentrates near main frequency $\omega =\pi$, suppressing out-of-band noise.

Strategy 3: Dynamical Decoupling

Insert decoupling pulse sequences in Floquet drive gaps to extend $T_2$.

Strategy 4: Post-Processing Filtering

Apply low-pass filtering to measurement sequence $\{a_n\}$ to remove high-frequency noise.

Strategy 5: Quantum Error Correction Codes

Encode logical time crystal states in multi-qubit systems, tolerate single-qubit errors.

6.3 Weak Non-Unitary Perturbation

Non-Unitary Floquet Evolution (Source Theory §3.5 Corollary G):

Actual Floquet operator $U_F$ may not be completely unitary (dissipation), define non-unitary deviation:

$${\Delta }_{\mathrm{nonU}}(E)=|U_F^{†}(E)U_F(E)-\mathbb{I}{|}_1$$

Robustness Condition:

If satisfied: $${\int }_{\mathcal{I}}{\Delta }_{\mathrm{nonU}}(E)\mathrm{d}E\le \epsilon$$

and error budget ${\mathcal{E}}_h\le {\delta }_{\ast }-\epsilon$ (review Chapter 21 Section 04), then parity label ${\nu }_{\mathrm{chain}}$ unchanged.

Physical Meaning: As long as dissipation is “sufficiently small” (in integral sense), time crystal phase remains stable.

7. Experimental Parameter Design Examples

7.1 Cold Atom Scheme

Goal: Discriminate period-doubling time crystal with error rate $\epsilon =10^{-2}$.

System Parameters:

• Number of lattice sites: $10\times 10=100$
• Floquet period: $T=1$ ms
• Band gap: ${\Delta }_{\mathrm{F}}=2\pi \times 100$ Hz
• Coherence time: $T_2=5$ s

DPSS Parameters:

• Bandwidth: $W=0.05$ (normalized)
• Shannon number: $N_0=2NW$

Sample Requirement: $$N\ge C{\Delta }_{\mathrm{F}}^{-2}\log (1/\epsilon )\sim 10^4$$

Total Measurement Time: $$t_{\mathrm{total}}=N\cdot T=10^4\times 1\mathrm{ms}=10\mathrm{s}$$

✅ Satisfies $t_{\mathrm{total}}<T_2$

Repetition Count: $$M=100\text{ independent experiments}$$

Total Experiment Time: $$T_{\exp }=M\cdot t_{\mathrm{total}}\sim 1000\mathrm{s}\approx 17\min$$

Mermaid Cold Atom Parameter Flow

graph TD
    A["Goal<br/>epsilon = 0.01"] --> B["Band Gap<br/>Delta_F = 2pi × 100 Hz"]
    B --> C["Number of Samples<br/>N ~ 10^4"]

    C --> D["Total Time<br/>t = N × T = 10 s"]

    E["Coherence Time<br/>T_2 = 5 s"] --> F{" t < T_2 ? "}

    D --> F

    F -->|"No"| G["Need Optimization<br/>Increase Delta_F"]
    F -->|"Yes"| H["Scheme Feasible<br/>Repeat M=100 times"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffcccc
    style G fill:#ffaaaa
    style H fill:#aaffaa

7.2 Superconducting Qubit Scheme

Goal: Discriminate period-doubling time crystal with error rate $\epsilon =10^{-2}$.

System Parameters:

• Number of qubits: 20
• Floquet period: $T=10\mu$s
• Band gap: ${\Delta }_{\mathrm{F}}=2\pi \times 1$ MHz
• Coherence time: $T_2=50\mu$s

Sample Requirement: $$N\ge C{\Delta }_{\mathrm{F}}^{-2}\log (1/\epsilon )\sim 500$$

Total Measurement Time: $$t_{\mathrm{total}}=N\cdot T=500\times 10\mu \mathrm{s}=5\mathrm{ms}$$

✅ Satisfies $t_{\mathrm{total}}\ll T_2$

Repetition Count: $$M=10^4\text{ times}$$

Total Experiment Time: $$T_{\exp }=M\cdot t_{\mathrm{total}}\sim 50\mathrm{s}$$

Advantages:

• Fast cycling (ms scale)
• Can repeat $M$ many times to improve statistics

8. Connection with Chapter 20 Experimental Schemes

8.1 Unified Time Scale Measurement

Chapter 20 Section 01 (Unified Time Scale) gives triple equivalence: $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

In Floquet Time Crystals: $${\kappa }_F(\omega )=\frac{1}{2\pi }\mathrm{tr}{Q}_F(\omega ),Q_F=-\mathrm{i}{U}_F^{†}{\partial }_{\omega }{U}_F$$

Single-Period Time Increment: $$\Delta {\tau }_F={\int }_{{\Omega }_F}{w}_F(\omega ){\kappa }_F(\omega )\mathrm{d}\omega$$

8.2 PSWF/DPSS Windowing Technique

Chapter 20 Section 02 (Spectral Windowing) established DPSS theory, directly applied in this chapter:

• Shannon number $N_0=2NW$
• Main leakage upper bound $1-{\lambda }_0\le \cdots$
• Triple error decomposition

Specialization in Time Crystals:

• Main frequency fixed at $\omega =\pi$ (period doubling)
• Bandwidth $W$ chosen to optimize signal-to-noise ratio

8.3 Topological Fingerprint Measurement

Chapter 20 Section 03 (Topological Fingerprints) discussed:

• π-step ladder
• Z₂ parity flip
• Square-root scaling law

Manifestation in Time Crystals:

• Z₂ holonomy ${\mathrm{hol}}_{{\mathbb{Z}}_2}({\Gamma }_F)$ is topological fingerprint
• Windowed parity threshold criterion (Chapter 21 Section 04 Theorem G)

Mermaid Chapter Connections

graph TD
    A["Chapter 20: Experimental Schemes"] --> B1["Section 01: Unified Time Scale<br/>kappa(omega)"]
    A --> B2["Section 02: DPSS Windowing<br/>Shannon Number N_0"]
    A --> B3["Section 03: Topological Fingerprints<br/>Z_2 Parity"]

    C["Chapter 22: Time Crystals"] --> D1["Floquet Time Scale<br/>kappa_F(omega)"]
    C --> D2["DPSS Readout<br/>Main Frequency omega=pi"]
    C --> D3["Z_2 Holonomy<br/>hol(Gamma_F)"]

    B1 -.->|"Application"| D1
    B2 -.->|"Application"| D2
    B3 -.->|"Application"| D3

    style A fill:#e1f5ff
    style B1 fill:#ffe1e1
    style B2 fill:#f5e1ff
    style B3 fill:#fff4e1
    style C fill:#e1ffe1
    style D1 fill:#ffe1f5
    style D2 fill:#f5e1ff
    style D3 fill:#fff4e1

9. Chapter Summary

9.1 Core Content Review

Experimental Platforms:

• Cold atoms: Long $T_2$, precise control
• Superconducting qubits: Fast gates, programmable
• Ion traps: Ultra-long coherence, high fidelity
• Solid-state spin: Room temperature, integration

DPSS Windowed Readout: $$\hat{a}(\omega )=\sum _{n=0}^{N-1}{w}_n^{(0)}{a}_n{\mathrm{e}}^{-\mathrm{i}\omega n}$$

Sample Complexity (Theorem 5.1): $$\boxed{N\ge C{\Delta }_{\mathrm{F}}^{-2}\log (1/\epsilon )}$$

Noise Robustness:

• Increase repetition $M$ to reduce variance
• Optimize window bandwidth $W$ to suppress noise
• Dynamical decoupling to extend $T_2$

Experimental Parameter Examples:

• Cold atoms: $N\sim 10^4$, total time 10 s
• Superconducting qubits: $N\sim 500$, total time 5 ms

9.2 Key Insights

1. Finite Complexity is Practical Constraint: Theoretically predicted time crystals must be discriminated within finite measurement steps $N$, sample complexity gives realizability criterion.

2. Band Gap is Core Parameter: Larger ${\Delta }_{\mathrm{F}}$, stronger signal, smaller $N$ requirement. Primary goal of experimental design is to maximize band gap.

3. DPSS is Optimal Windowing: Under given $N$ and $W$, DPSS maximizes frequency domain energy concentration, minimizes worst-case error.

4. Cross-Platform Unified Framework: Cold atoms, superconducting, ion traps, though physically different, all follow same DPSS windowing theory and sample complexity theorem.

5. Deep Connection with Scattering Theory: Time crystal readout is essentially frequency domain scattering measurement, unified time scale ${\kappa }_F(\omega )$ runs throughout.

9.3 Preview of Next Chapter

Next chapter (04-time-crystal-summary.md) will:

• Synthesize entire chapter theory (00-03)
• Discuss open problems and future directions
• Role of time crystals as “unified time scale phase lockers”
• Complementary relationship with FRB observations and δ-ring scattering

End of Chapter

Source Theory: euler-gls-info/17-time-crystals-null-modular-z2-holonomy.md, §5; Appendix D