Amplification Fraction in Frequency Metrology

• Amplification fraction is the proportion of system output directly from amplification, influencing noise behavior and resource efficiency.
• Dual-frequency systems exploit correlated noise cancellation to effectively halve the amplifier count while improving signal stability.
• In weak value amplification protocols, a small amplification fraction concentrates Fisher information, optimizing precision in quantum metrology.

Amplification fraction is a foundational concept in diverse fields including frequency metrology, quantum measurement, signal processing, and nonlinear dynamics. At its core, the term refers to the portion of a system's output—or, more specifically, of the measured signal or information—that results directly from an amplification process, relative to the input or, in some contexts, compared to a conventional method of signal readout or measurement. The precise definition and quantitative role of the amplification fraction vary depending on the physical context, the nature of noise or loss mechanisms, and the architectural choices in the amplifying system.

1. Dual-Frequency Amplification and Noise Correlation

In ultra-stable frequency metrology, particularly for weak microwave signals from cryogenic sapphire oscillators, the amplification fraction is critically impacted by the architecture of the readout system (Creedon et al., 2011). Conventional systems employ two independent amplifier chains (individually amplified, IA), with each signal amplified separately before mixing to produce a beat note. Each amplifier introduces both flicker (1/f) phase noise and thermal (white) noise, contributing independently to the overall phase noise spectral density: $$S_\phi^\mathrm{(IA)} = \frac{2\alpha}{f} + \frac{2k_B T_0 N}{P_\mathrm{inp}}$$ where α is the flicker noise coefficient, $k_B$ the Boltzmann constant, $T_0$ the ambient temperature, $N$ the amplifier’s noise figure, and $P_\mathrm{inp}$ the input power.

By contrast, the dual-frequency (DF) readout system combines both signals before amplification. The shared amplifier imposes nearly identical (correlated) flicker noise on both signals. When the outputs are mixed, this correlated flicker noise cancels, leaving only the thermal noise: $$S_\phi^\mathrm{(DF)} = \frac{2k_B T_0 N}{P_\mathrm{inp}}$$ Thus, the DF system's amplification fraction for flicker noise contribution drops to zero at the output, a result that dramatically improves frequency stability when the oscillator instability approaches the $10^{-16}$ regime. For white noise–limited operation, both IA and DF systems show identical performance, but the DF architecture achieves this with half the number of amplifiers, maximizing resource efficiency (the effective amplification fraction per amplifier is doubled in terms of usable output information).

2. Quantification of Amplification Fraction in Weak Value Amplification Protocols

In quantum metrology, especially within the paradigm of weak value amplification (WVA) (Jordan et al., 2013, Alves et al., 2014), the amplification fraction refers to the proportion of measured events—conditioned by post-selection—that actually contribute to the extracted signal or Fisher information.

Consider a system preselected in state $|i\rangle$, weakly coupled to a meter, and postselected onto $|f\rangle$. The postselection probability is $\gamma = |\langle f|i\rangle|^2$, often much less than one. Despite this, the amplified shift in the meter’s pointer, characterized by the weak value $A_w = \langle f|A|i\rangle/\langle f|i\rangle$, leads to the concentration of (nearly) all Fisher information into this small subensemble: $I' = (A_w)^2 \gamma N / \sigma^2$ where $N$ is the total trial number and $\sigma$ is the meter width. In idealized situations, $(A_w)^2\gamma \simeq 1$, ensuring that the information extracted with the small amplification fraction $\gamma$ matches that of the standard (unamplified) procedure. Photon recycling protocols can increase the effective Fisher information by a factor $1/\gamma$ if the unused photons are recirculated.

The amplification fraction is thus not just a statistical cost but represents the efficiency of information transfer per detected ensemble in the context of metrological bounds.

3. Suppression of Correlated Noise and Resource Efficiency

The implementation of a shared-amplifier architecture directly leverages the amplification fraction to suppress dominant noise sources. When correlated noise processes—here, flicker phase noise—are transformed into common-mode signals, proper system design can ensure their complete removal at the measurement output. The relevant model equations are:

• Individual amplifier (IA): $S_\phi^\mathrm{(IA)}$, doubling both flicker and white noise per amplifier.
• Dual-frequency (DF): $S_\phi^\mathrm{(DF)}$, eliminating flicker, only doubling the white noise.

This change translates to an effective amplification fraction for flicker noise terms at the output: $$\text{Amplification Fraction}^{(\text{flicker, DF})} = 0$$

$$\text{Amplification Fraction}^{(\text{flicker, IA})} = 1$$

thus, for identical input conditions, the DF system effectively doubles the value per amplifier invested, particularly as instability of the source approaches the regime where flicker noise would dominate without such cancellation.

4. Thermal Noise as Limiting Factor and Practical Measurement Floor

In both architectures, the ultimate noise floor is dictated by the white (thermal) phase noise, given by: $$S_\phi^\mathrm{(thermal)} = \frac{k_B T_0 N}{P_\mathrm{inp}}$$ leading, in the time domain, to fractional frequency instability

$$\sigma_T(\tau) = \frac{2}{\pi f_0}\sqrt{\frac{k_B T_0 N}{P_\mathrm{inp}}}\,\tau^{-3/2}$$

where $f_0$ is the microwave frequency and $\tau$ the integration time. When input powers are low (below –80 dBm), both DF and IA systems converge to the same white noise–limited floor, but the DF's ability to operate without flicker noise in the output regardless of instability level renders the amplification fraction with respect to correlated noise processes near-optimal.

5. Scaling, System Design, and Future Ultra-Stable Oscillators

The amplification fraction, in the sense of information-to-resource ratio, directly influences system scalability. As the intrinsic noise floor of frequency sources is lowered (e.g., in next-generation sapphire oscillators), conventional readout architectures become flicker-noise-limited even for extremely small $\alpha$, limiting further advances in measured fractional instability. The DF system, with its halved amplifier requirements and immunity to flicker noise, becomes essential for practical scaling to parts-in-$10^{16}$ or better. The performance metrics, resource usage, and the utility of the amplification fraction as a system-level figure of merit are summarized in the table below.

Architecture	Number of Amplifiers	Flicker Noise at Output	White Noise at Output	Performance at Low Power
IA	2	$2\alpha/f$	$2k_B T_0 N / P_\mathrm{inp}$	Equal to DF, both thermal-limited
DF	1	0 (correlated, canceled)	$2k_B T_0 N / P_\mathrm{inp}$	Equal to IA, both thermal-limited

6. Significance in Metrology and Systemic Advancements

The effective control of the amplification fraction, via suppression of correlated (flicker) noise and optimal allocation of amplification resources, is a key enabler for ultra-high-resolution frequency measurement and for advancing towards measurement systems capable of assessing fractional frequency stabilities down to parts in $10^{16}$. The DF scheme's robust design, eliminating the contribution of the flicker noise term even at extremely low power while operating at maximal resource efficiency, is poised to become foundational in the next generation of frequency metrology systems.

7. Outlook and Broader Context

Although the precise definition of amplification fraction in this context applies most directly to the ratio of noise contributions (or equivalent resource efficiency), the general principle extends to quantum metrological protocols, networked oscillators, and any system where shared amplification and noise correlation properties can be engineered. The critical insight is that maximizing the effective amplification fraction—minimizing dissipative resource use per unit of high-fidelity signal—will be a central design consideration in all ultra-low-noise measurement systems as physical and technological grounds for improvement narrow to those determined by fundamental device noise and architecture-intrinsic efficiency.