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formula
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\begin{array}{rlr}\hat{H}&=&\hbar\omega\left(\hat{n}+\frac{1}{2}\right)+\hbar\omega_{\mathrm{det}}\left(\hat{n}_{\mathrm{det}}+\frac{1}{2}\right)+\\&&\hbar g\,\hat{n}\:\hat{n}_{\mathrm{\mathrm{det}}}+\hat{H}_{\mathrm{drive}+{\mathrm{\mathrm{decay}}}}\,.\end{array}
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\omega
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\hat{n}
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\omega_{\mathrm{det}}
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\hat{n}_{\mathrm{det}}
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\left\langle\hat{n}_{\mathrm{det}}\right\rangle\gg 1
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\bar{X}(t)
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\left\langle\hat{n}\right\rangle(t)
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\Gamma/\kappa
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\dot{N}_{\mathrm{in}}
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\propto\left(\Gamma\tau_{\mathrm{avg}}\right)^{-1/2}
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X_{\mathrm{thr}}
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\kappa\tau_{\mathrm{avg}}=2
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\hat{\rho}
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\begin{array}{rlr}\dot{\hat{\rho}}&=&-i\sqrt{{\frac{\dot{N}_{\mathrm{in}}\kappa}{2}}}\left[\hat{a}+\hat{a}^{\dagger},\hat{\rho}\right]+\kappa\left(\hat{a}\hat{\rho}\hat{a}^{\dagger}-\frac{1}{2}\hat{n}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{n}\right)\\&&-2\Gamma\left[\hat{n},\left[\hat{n},\hat{\rho}\right]\right]-\sqrt{{4\Gamma}}\left(\hat{n}\hat{\rho}+\hat{\rho}\hat{n}-2\hat{\rho}\left\langle\hat{n}\right\rangle(t)\right)\xi(t).\,\,\,\,\,\end{array}
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\kappa
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\kappa_{\mathrm{det}}\gg\kappa
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\hat{n}=\hat{a}^{\dagger}\hat{a}
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\Gamma\equiv g^2\left\langle\hat{n}_{\mathrm{det}}\right\rangle/(4\kappa_{\mathrm{det}})
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1/\Gamma
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X(t)\equiv\langle\hat{n}\rangle(t)+\frac{1}{4}\sqrt{{\frac{1}{\Gamma}}}\xi(t).
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\left\langle\xi\right\rangle=0
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\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})
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\tau_{\mathrm{avg}}
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\tau_{\mathrm{avg}}\ll\kappa^{-1}
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\tau_{\mathrm{dark}}
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\dot{N}_{\mathrm{in}}^{-1}\gg\tau_{\mathrm{dark}}\gg\kappa^{-1}
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\dot{N}_{\mathrm{det}}
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\eta
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\eta\equiv\left.\frac{d\dot{N}_{\mathrm{det}}}{d\dot{N}_{\mathrm{in}}}\right|_{{\mathrm{\dot{N}}}_{\mathrm{in}}=0}.
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O(10^4)
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10^2/\kappa
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\Gamma/\kappa\ll 1
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\Gamma/\kappa\gg 1
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\Gamma/\kappa=0.6
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\dot{N}_{\mathrm{in}}=0
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\dot{N}_{\mathrm{in}}\sim\tau_{\mathrm{dark}}^{-1}
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\Gamma
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X_{\mathrm{thr}}=0.5
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\max(\eta)
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\Gamma/\kappa=4
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\alpha(t)
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\dot{\alpha}(t)=\left(-i\,\delta\omega(t)-\frac{\kappa}{2}\right)\alpha(t)+\sqrt{{\frac{\kappa}{2}}}\alpha_{\mathrm{\mathrm{L}}}^{\mathrm{in}}.
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\alpha_{\mathrm{L}}^{\mathrm{in}}
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\delta\omega(t)\equiv gn_{\mathrm{det}}(t)
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n_{\mathrm{det}}\gg 1
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\left\langle\delta\omega(t)\delta\omega(0)\right\rangle-\left\langle\delta\omega\right\rangle^2=g^2\bar{n}_{\mathrm{det}} e^{-\kappa_{\mathrm{det}}|t|/2}\,.
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\frac{\alpha(t)}{\sqrt{{\kappa_L}}\alpha_{\mathrm{L}}^{\mathrm{in}}}=\int_{-\infty}^t dt^{\prime}\exp\left[-i\int_{t^{\prime}}^t\delta\omega(t^{\prime\prime})dt^{\prime\prime}-\frac{\kappa}{2}(t-t^{\prime})\right].
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\delta\omega(t)
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\kappa_{\mathrm{det}}|t-t^{\prime}|\gg 1
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\left\langle|\alpha|^2\right\rangle
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\langle\exp[-iY]\rangle=\exp[-i\langle Y\rangle-\frac{1}{2}\mathrm{Var}Y]
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\langle|a_{\mathrm{\mathrm{R}}}^{\mathrm{out}}|^2\rangle=\frac{\kappa}{2}\langle|\alpha|^2\rangle=\left\langle\mathcal{T}\right\rangle|\alpha_{\mathrm{L}}^{\mathrm{in}}|^2,
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\langle\mathcal{T}\rangle=\left(1+4\frac{\Gamma}{\kappa}\right)^{-1}.
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\eta=2\langle\mathcal{T}\rangle
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\Gamma/\kappa=1/2
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2\pi\cdot 100\mathrm{MHz}
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2\pi\cdot 5\mathrm{GHz}
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1\mathrm{MHz}
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100\mathrm{MHz}
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40\%
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a_k\in\mathcal{C}
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k\in\mathbb{Z}
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\mathcal{C}
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f_{\mathrm{c}}
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\theta
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O(t)=\sqrt{{2x(t)}}cos\left(2\pi f_{\mathrm{c}} t+\theta\right)
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x(t)=JI(t)=JA\left(\mu+\sum_{k=-\infty}^\infty a_k q(t-k{T_\mathrm{s}})\right),
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\mu
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T_\mathrm{s}
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x(t)\geq 0
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t\in\mathbb{R}
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Q(\omega)=\intop_{-\infty}^\infty q(t)e^{-j\omega t} dt=0,\;|\omega|\geq 2\pi B,
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Q(\omega)
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\mathcal{C}=\left\{0, 1\right\}
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P_{\mathrm{opt}}=\frac{1}{T_\mathrm{s}}\intop_0^{T_\mathrm{s}}\mathbb{E}\left\{x(t)\right\} dt,
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\mathbb{E}\left\{\cdot\right\}
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\begin{array}{rl}P_{\mathrm{opt}}&=\frac{1}{T_\mathrm{s}}\intop_0^{T_\mathrm{s}} JA\left(\mu+\mathbb{E}\left\{a_k\right\}\sum_{k=-\infty}^\infty q(t-k{T_\mathrm{s}})\right) dt\\&=JA\left(\mu+\mathbb{E}\left\{a_k\right\}\overline{q}\right),\end{array}
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\overline{q}=\frac{1}{T_\mathrm{s}}\intop_{-\infty}^\infty q(t)dt=\frac{Q(0)}{T_\mathrm{s}}.
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P_\mathrm{max}=\max x(t)=JA\left(\mu+\max\sum_{k=-\infty}^\infty a_k q(t-k{T_\mathrm{s}})\right)
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\ldots, a_{-1}, a_0, a_1, a_2,\ldots
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y(t)=Rh(t)\otimes x(t)+n(t),
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\otimes
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h(t)=H(0)\delta(t)
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R=J=1
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H(0)=1
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N_0/2
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r(t)=y(t)\otimes g(t),
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G(\omega)=\left\{\begin{array}{cc}G(0)&|\omega|<2\pi B\\0&|\omega|\geq 2\pi B\end{array}.\right.
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G(\omega)=\zeta Q^*(\omega)
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\left(\cdot\right)^*
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\zeta
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\mathcal{C}\subset\mathbb{R}^+
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\mu=0
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\mathcal{C}\subset\mathbb{R}
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q(k{T_\mathrm{s}})=\left\{\begin{array}{ll}q(0),&k=0,\\0,&k\neq 0.\end{array}\right.
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{\operatorname{\mathrm{sinc}}}(x)=\sin(\pi x)/(\pi x)
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0\le\alpha\le 1
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B=(1+\alpha)/(2{T_\mathrm{s}})
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\mu>0
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in Data Studio
PH FORMULA CORPUS V1
PH_FORMULA_CORPUS_V1 is a large-scale formula corpus containing 160 million normalized mathematical expressions.
Each formula has been carefully normalized to ensure concise, consistent, and simplified representations.
🗓️ Timeline
- ✅ July 2025 – Released the formula corpus
- 🔜 Aug 2025 – Upcoming release of the synthetic datasets
- ⏳ Sep 2025 – Scheduled release of a model achieving commercial-grade qualit PHOCR
🔧 Normalization Process
- Parse formulas using the KaTeX syntax tree with customized macro definitions.
- Traverse the tree with a tailored algorithm to normalize the expression of each node.
Additional normalization steps include:
- Standardizing
\operatornameand\operatorname*usage - Unifying variations of
\textxxand\mathxxcommands - Normalizing spacing across all elements
📘 Dataset Usage
def test_parquet():
import random
import pyarrow.parquet as pq
s
input_file = "/path/to/formula_corpus_v1.parquet"
table = pq.read_table(input_file)
print(f"Total rows: {len(table)}")
num_rows = min(100, len(table))
indices = random.sample(range(len(table)), num_rows)
formulas = table.column("formula")
for idx in indices:
print(formulas[idx].as_py())
📚 Citation
Thank you for using PH_FORMULA_CORPUS_V1.
This dataset is developed and maintained by Puhuilab.
If you use this dataset in your research or applications, please remember to give appropriate credit.
To cite PH_FORMULA_CORPUS_V1 in academic work, please use the following BibTeX entry:
@misc{phformulacorpusv12025,
title={PHFormulaCorpusV1: A Large-Scale Normalized LaTeX Formula Dataset},
author={Puhui Lab},
year={2025},
url={https://huggingface.co/datasets/puhuilab/ph_formula_corpus_v1}
}
💡 Acknowledgment
The dataset follows the licenses of its original sources:
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