---
paper_id: "PAPER_1101"
title: "SCm Qubit T2 Coherence via F_U_Bi / F_U Ratio and Thermal Protection"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ["quantum-computing", "T2-coherence", "decoherence", "F_U_Bi", "phonon", "thermal", "qubit", "SCm", "UQFF"]
crosslinks: [PAPER_1052, PAPER_1056, PAPER_1098]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER\_1101: SCm Qubit T$_2$ Coherence via $F_{U,Bi}/F_U$ Ratio and Thermal Protection

## Abstract

We derive the Star Cradle Mechanics (SCm) qubit coherence time $T_2$ formula,
which incorporates the buoyancy field ratio $F_{U,Bi}/F_U$ and an exponential
thermal protection factor from the SCm gap energy $\Delta_{\text{SCm}}$:

$$T_2 = \frac{\hbar}{\Delta_{\text{SCm}}}\cdot\exp\!\left(\frac{\Delta_{\text{SCm}}}{k_B T}\right)\cdot S_{26}^{(3)}\cdot\frac{F_{U,Bi}}{F_U}$$

This extends conventional qubit $T_2$ models by encoding the UQFF buoyancy
protection mechanism and the 26-dimensional phonon coupling.

## 1. Introduction

In superconducting and topological qubit architectures, the transverse
relaxation time $T_2$ governs the duration over which quantum phase coherence is
maintained.  Standard models relate $T_2$ to the energy gap $\Delta$ and
temperature $T$ via:

$$T_2^{\text{std}} = \frac{\hbar}{\Delta}\cdot\exp\!\left(\frac{\Delta}{k_B T}\right)$$

The SCm framework introduces two additional factors:

1. The **26-dimensional buoyancy prefactor** $S_{26}^{(3)}([\text{SSq}]) = (1-[\text{SSq}])^3$
2. The **buoyancy field ratio** $F_{U,Bi}/F_U$ capturing the fraction of the unified field contributing to topological protection

## 2. SCm Gap Energy

The SCm gap energy is derived from the characteristic 1.25 THz phonon mode:

$$\Delta_{\text{SCm}} = \hbar\omega_{\text{phonon}}\cdot 2\pi = \hbar\cdot 2\pi\cdot 1.25\times10^{12}\;\text{Hz}$$

$$\Delta_{\text{SCm}} \approx 8.27\times10^{-22}\;\text{J} \approx 5.17\;\text{meV}$$

## 3. The SCm $T_2$ Formula

$$T_2^{\text{SCm}} = \underbrace{\frac{\hbar}{\Delta_{\text{SCm}}}}_{\text{base time}}\cdot\underbrace{\exp\!\left(\frac{\Delta_{\text{SCm}}}{k_B T}\right)}_{\text{thermal suppression}}\cdot\underbrace{S_{26}^{(3)}}_{\text{26D buoyancy}}\cdot\underbrace{\frac{F_{U,Bi}}{F_U}}_{\text{buoyancy ratio}}$$

### Parameter Regimes

| Regime | $T$ (K) | $\exp(\Delta/k_BT)$ | Application |
|--------|---------|---------------------|-------------|
| Dilution fridge | 0.015 | $\sim10^{174}$ | Superconducting qubits |
| $^3$He cryostat | 0.3 | $\sim10^{8}$ | Topological qubits |
| Liquid He | 4.2 | $\sim14.4$ | High-$T_c$ devices |

### Enhancement Factor

The SCm enhancement over standard $T_2$:

$$\eta_{\text{SCm}} = S_{26}^{(3)}\cdot\frac{F_{U,Bi}}{F_U}$$

For $[\text{SSq}] = 0.57$ and $F_{U,Bi}/F_U = 0.3$:

$$\eta_{\text{SCm}} = 0.0795\times 0.3 = 0.0239$$

This represents a buoyancy-mediated coherence channel distinct from conventional
$T_2$ decay pathways.

## 4. Physical Interpretation

The $F_{U,Bi}/F_U$ ratio encodes the fraction of the unified buoyancy field
available for topological qubit protection.  When the buoyancy component
dominates ($F_{U,Bi}/F_U \to 1$), the qubit is maximally protected by the
phonon-buoyancy shield.  When $F_{U,Bi} \ll F_U$, other force components
(magnetic, gravitational) dominate and coherence protection diminishes.

## 5. Implementation

Calculator: `SCmQubitT2CoherenceFUBiRatioCalculator` in CondensedPhysics.py

- Accepts temperature $T$, field strengths $F_U$ and $F_{U,Bi}$, phonon frequency, and $[\text{SSq}]$
- Outputs $T_2^{\text{SCm}}$, $T_2^{\text{std}}$, enhancement factor, and thermal exponential

## 6. Conclusion

The SCm $T_2$ formula provides a first-principles connection between UQFF
buoyancy physics and qubit decoherence, predicting coherence modulation
through the $F_{U,Bi}/F_U$ ratio and the 26-dimensional phonon coupling.

## References

- Koch, J. et al. (2007). Charge-insensitive qubit design from Cooper pair box. *Phys. Rev. A* 76, 042319.
- Murphy, D.T. (2025). UQFF: Star Cradle Mechanics framework. Star-Magic repository.
- PAPER\_1098: Phonon-Mediated Qubit Gate Fidelity.


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