Phase-Shift Noise (PSN) Overview

• Phase-Shift Noise (PSN) is a phenomenon characterized by stochastic phase perturbations and measurement errors observed in optical sensing, communication, and metrology applications.
• PSN mechanisms convert phase fluctuations into observable uncertainties such as timing jitter, amplitude noise, or channel-estimation errors, impacting overall system performance.
• Mitigation strategies include differential measurement, phase-tracking algorithms, and transduction techniques that cancel common-mode noise or convert phase errors into manageable signals.

Phase-Shift Noise (PSN) is not a universally standardized term. In the arXiv literature, it refers variously to uncertainty in extracting a measured phase lag, random phase diffusion on transmitted states, hardware-induced deviation between an intended phase shift and the realized one, or phase fluctuations that reappear as timing jitter, amplitude noise, channel-estimation error, or common-mode cavity detuning. The common structure is a stochastic perturbation of phase, or of a quantity inferred from phase, together with a domain-specific transduction mechanism and a mitigation strategy. In some papers the acronym is explicit, while in others closely related notions are discussed under “phase noise,” “phase-shift noise,” “phase error,” or “phase-noise spectroscopy” (Cheema et al., 2012, Li et al., 2023, Khanzadi et al., 2014).

1. Terminological scope and domain-specific meanings

The literature uses closely related terminology rather than a single canonical definition. In microcavity biosensing, the relevant object is the noise or error in measuring the phase lag of a modulated optical signal in phase-shift cavity ring-down spectroscopy (PS-CRDS) (Cheema et al., 2012). In coherent communications, PSN appears as Gaussian phase fluctuations or phase diffusion acting on BPSK coherent states or on oscillator phase processes (Jarzyna et al., 2015, Notarnicola et al., 2023, Khanzadi et al., 2014). In reconfigurable intelligent surfaces, PSN is the random deviation $\tilde{\theta}_n$ between the intended RIS phase $\overline{\theta}_n$ and the realized phase $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$ (Li et al., 2023). In Rydberg-EIT spectroscopy, phase fluctuations on incident light are converted into transmitted intensity noise, and the paper explicitly frames the effect as phase-noise spectroscopy / phase-shift noise (He et al., 2021).

Domain	Meaning of PSN or related term	Representative paper
Microcavity sensing	Phase-shift measurement error in PS-CRDS	(Cheema et al., 2012)
Optical / quantum communication	Gaussian phase fluctuations or phase diffusion	(Jarzyna et al., 2015, Notarnicola et al., 2023)
Active RIS	Random phase error of RIS elements	(Li et al., 2023)
Rydberg-EIT spectroscopy	Phase noise converted to amplitude noise	(He et al., 2021)
Digital timing / oscillators	Phase noise and jitter as timing fluctuation	(Calosso et al., 2016)

This diversity produces two recurrent misconceptions. First, “phase-based” readout is not noise-free; it changes which noise sources dominate. In PS-CRDS, for example, the critical uncertainty becomes phase detection precision rather than absolute transmitted intensity (Cheema et al., 2012). Second, lexical overlap with “phase shift” does not imply PSN. Lattice-QCD work on $\pi\pi$ scattering uses “phase shift” in Lüscher analysis, but the object there is a scattering observable rather than a noise process (Bulava et al., 2010).

2. Measurement noise in phase-shift readout and cavity metrology

A direct metrological use of PSN appears in PS-CRDS with biotinylated microtoroid resonators. The cavity converts a sinusoidally intensity-modulated input into an output with phase lag $\phi$ and reduced modulation depth, with the central lifetime relation

$\tan \phi = -\omega \tau .$

The phase lag is then used to infer the ring-down time $\tau$ and hence the cavity quality factor $Q$ (Cheema et al., 2012). The operational advantage is that the readout is much less sensitive to laser intensity fluctuations than conventional resonance-dip tracking, because the inferred quantity is the phase delay rather than the absolute transmitted intensity. The paper nevertheless identifies phase error as the limiting uncertainty and reports an error signal with mean $\pm 0.2662^\circ$, variance $0.1076^\circ$, mode $\overline{\theta}_n$0, and SNR between $\overline{\theta}_n$1 dB and $\overline{\theta}_n$2 dB. It further states that lock-in amplification could reduce the phase error to $\overline{\theta}_n$3, with projected SNR $\overline{\theta}_n$4–$\overline{\theta}_n$5 dB (Cheema et al., 2012).

A closely related but more explicitly differential strategy appears in optical ring-cavity phase-shift measurement. Two beams separated by exactly one cavity free spectral range are simultaneously resonant, so cavity-length fluctuations enter both PDH error signals as nearly common-mode noise. The differential combination suppresses the cavity contribution and leaves the desired differential phase shift. Experimentally, the differential error signal produces about $\overline{\theta}_n$6 dB reduction in cavity noise up to approximately half the cavity linewidth, $\overline{\theta}_n$7 kHz (Wang et al., 2020). In the simulated spin-squeezing measurement, the uncanceled probe signal corresponds to $\overline{\theta}_n$8 mrad phase sensitivity, whereas the canceled differential signal reaches $\overline{\theta}_n$9 mrad, a factor of $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$0 improvement (Wang et al., 2020). The residual floor is attributed primarily to detection electronics and laser relative intensity noise rather than to cavity-length fluctuations once the differential cancellation is applied.

These two cavity-based examples illustrate a common principle: the dominant PSN is often not removed by raw optical power, but by choosing a measurement variable whose nuisance phase contribution is either common-mode or weakly coupled to the desired observable.

3. Communication-theoretic PSN: phase diffusion, oscillator processes, and robust receivers

In communication systems, PSN is typically modeled as a random phase process acting on symbols or oscillators. For collective BPSK with Hadamard words, the phase noise model is an independent Gaussian fluctuation on each symbol,

$\theta_n=\overline{\theta}_n+\tilde{\theta}_n$1

with phases fluctuating around a reference phase fixed over the whole word (Jarzyna et al., 2015). The key result is that imperfect Hadamard interference converts the ideal erasure channel into a generalized erasure channel with uniform noise, but the optimized low-power mutual information still retains the Holevo-type nonlinear scaling

$\theta_n=\overline{\theta}_n+\tilde{\theta}_n$2

for $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$3 (Jarzyna et al., 2015). PSN therefore reduces the prefactor through $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$4 without destroying the superadditive scaling, provided the phase reference is stable over the whole word.

For binary coherent-state discrimination under phase diffusion, the noisy BPSK states become mixed states through a Gaussian phase-randomization channel. The hybrid near-optimum receiver (HYNORE) combines a weak-field homodyne-like branch with conditional displacement and photon counting, and the paper introduces the maximum tolerable phase noise $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$5 as a robustness metric (Notarnicola et al., 2023). The reported conclusion is that HYNORE is more robust than the displacement photon-number-resolving receiver (DPNR), with $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$6 for all energies considered, and that it beats the standard quantum limit in particular regimes, especially at high signal energy and also in some low-energy / low-noise regimes (Notarnicola et al., 2023). This does not make HYNORE universally optimal; the comparison is explicitly against DPNR and SQL, not against the Helstrom bound.

At the waveform level, 3MSK for DFT-s-OFDM makes robustness to phase noise a design objective by constraining the symbol-to-symbol phase transitions to

$\theta_n=\overline{\theta}_n+\tilde{\theta}_n$7

using the three baseband frequencies $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$8 and $\theta_n=\overline{\theta}_n+\tilde{\theta}_n$9 (Renfors et al., 2021). The receiver then performs recursive phase tracking inside a trellis-based detector without pilot or reference signals. Under PN in a TDL-E channel at $\pi\pi$0 GHz with $\pi\pi$1 kHz subcarrier spacing, 3MSK with $\pi\pi$2 degrades by only about $\pi\pi$3 dB at the $\pi\pi$4 BER point, whereas QPSK degrades by about $\pi\pi$5 dB; the paper states that 3MSK $\pi\pi$6 is about $\pi\pi$7 dB better than QPSK in the PN-impaired case (Renfors et al., 2021).

At the receiver-algorithm level, two additional models are prominent. One is oscillator PN with colored increments in single-carrier transmission: $\pi\pi$8 where the increment process has autocorrelation $\pi\pi$9 (Khanzadi et al., 2014). A soft-input MAP estimator reaches the Bayesian Cramér–Rao bound in the data-aided case, while a modified soft-input extended Kalman smoother based on an AR approximation of colored increments approaches MAP performance at sufficient model order; the paper reports that a first-order AR model can match MAP in the data-aided and $\phi$0 pilot-density cases, while $\phi$1 is needed for $\phi$2 pilots (Khanzadi et al., 2014). The second is LTE downlink PN modeled as a discrete-time Wiener process,

$\phi$3

which produces common phase error (CPE) and inter-carrier interference (ICI) in OFDM (Wang et al., 2017). Because LTE downlink has sparse reference symbols, the proposed iterative detection and estimation scheme uses PN-aware 2D MMSE channel estimation followed by residual PN and channel updates; five iterations are reported to be sufficient (Wang et al., 2017).

4. Hardware-induced phase errors, cancellation architectures, and invariance mechanisms

In active RIS-aided systems, PSN is explicitly a hardware impairment. Each RIS element realizes

$\phi$4

with $\phi$5 modeled as either von Mises or uniform, and the coherent-combining loss summarized by

$\phi$6

The LMMSE channel estimator and the data-phase SINR both contain $\phi$7, and the paper identifies a non-zero N-MSE floor at high transmit power due to PSN. It further reports that ergodic capacity saturates at high transmit power because PSN and imperfect CSI scale together, even though increasing the number of RIS elements remains highly beneficial (Li et al., 2023). In this setting PSN is not merely additive noise; it simultaneously reduces coherent signal gain and increases effective interference terms.

A different architecture avoids explicit phase tracking altogether. In passive optical phase noise cancellation over fiber, optical phase perturbations $\phi$8 are mapped into an RF beat note carrying $\phi$9, processed open-loop, and then re-embedded into the outgoing optical carrier (Hu et al., 2020). The residual remote phase-noise PSD follows

$\tan \phi = -\omega \tau .$0

so the scheme preserves the conventional delay-limited suppression and bandwidth of approximately $\tan \phi = -\omega \tau .$1 (Hu et al., 2020). For the 145 km fiber spool, this corresponds to about $\tan \phi = -\omega \tau .$2 Hz, and the stabilized link reaches Allan deviation $\tan \phi = -\omega \tau .$3 at $\tan \phi = -\omega \tau .$4 s and $\tan \phi = -\omega \tau .$5 at $\tan \phi = -\omega \tau .$6 s, with no servo bumps (Hu et al., 2020). The paper explicitly states that the method does not beat the propagation-delay bound; its advantage is faster response and phase recovery in an open-loop design.

Direction-Shift Keying (DSK) provides a stronger invariance result. In the mmWave distributed-antenna model, the received signal contains a random carrier-frequency-offset / phase-noise term $\tan \phi = -\omega \tau .$7, but the optimal detector uses ratios

$\tan \phi = -\omega \tau .$8

so the phase-noise factor cancels exactly (Chraiti et al., 1 Sep 2025). The paper then distinguishes channel coherence time (CCT), which scales like $\tan \phi = -\omega \tau .$9, from direction coherence time (DCT), which scales like $\tau$0, and explicitly states that phase noise does not appear in the final DCT expression (Chraiti et al., 1 Sep 2025). The simulations show that DSK remains essentially unchanged across five decades of phase-noise standard deviation, whereas SSK degrades severely. A plausible implication is that some systems can be made PSN-resilient not by estimating phase more accurately, but by basing detection on observables, such as TDoA / DoA signatures, from which the phase-noise factor cancels algebraically.

5. Phase-noise transduction, spectroscopy, and device metrology

Several works analyze PSN through conversion into another observable. In Rydberg-EIT noise spectroscopy, phase noise on the incident optical fields is converted into amplitude noise on the transmitted probe because the complex susceptibility near EIT resonance is strongly dispersive and absorptive (He et al., 2021). The conversion is highly sensitive to two-photon detuning: it is strongly suppressed at two-photon resonance and enhanced away from it. The paper reports that near resonance the probe intensity fluctuations are more than $\tau$1 dB higher than at $\tau$2 MHz detuning around $\tau$3 MHz analyzing frequency, and that modulation-enhanced PNS extends the analyzable frequency range from roughly $\tau$4 kHz to $\tau$5 MHz, with SNR improvement about a factor of $\tau$6 relative to unmodulated PNS (He et al., 2021). In this setting the atoms act as a detuning-dependent phase-noise transducer.

High-speed InGaAs P-I-N photodiodes exhibit an analogous conversion from optical power fluctuations to microwave phase fluctuations. The AM-to-PM coefficient is defined as

$\tau$7

and the predicted single-sideband phase noise due to laser RIN is

$\tau$8

(Taylor et al., 2011). The reported $\tau$9 values range roughly from $Q$0 to $Q$1 rad depending on photodiode and photocurrent. PD1 and PD2 exhibit nulls where $Q$2 goes to zero, with one prominent example around $Q$3 mA, but the null positions can shift by about $Q$4 mA with temperature, bias, or optical mode shape (Taylor et al., 2011). This makes PSN-like phase instability a consequence of device nonlinearity and saturation rather than of an external phase process alone.

Measurement infrastructure itself can also be designed around phase-noise transduction. For DACs and DDSs, a carrier-suppressed scheme reduces the carrier by about $Q$5–$Q$6 dB and amplifies the modulation index seen by the analyzer by $Q$7, where $Q$8 is the residual carrier amplitude fraction (Calosso et al., 2019). The AD9144 demonstration measured noise over $Q$9 decades of frequency, and the reported flicker noise followed the exact $\pm 0.2662^\circ$0 law with maximum discrepancy of $\pm 0.2662^\circ$1 dB over $\pm 0.2662^\circ$2 decades (Calosso et al., 2019). The paper explicitly notes that the method allows AN to be measured with a PN analyzer and vice versa through controlled AM/PM cross-conversion.

Digital-electronics work supplies a general framework for interpreting such phenomena. Phase noise and jitter are related by

$\pm 0.2662^\circ$3

and the paper distinguishes phase-type and time-type processes (Calosso et al., 2016). It reports flicker noise between $\pm 0.2662^\circ$4 and $\pm 0.2662^\circ$5 dBrad$\pm 0.2662^\circ$6/Hz at $\pm 0.2662^\circ$7 Hz offset and white noise down to $\pm 0.2662^\circ$8 dBrad$\pm 0.2662^\circ$9/Hz in favorable cases (Calosso et al., 2016). This provides a device-level language for PSN-like timing fluctuations in comparators, dividers, PLLs, and distribution chains.

6. Event-like phase shifts, detection theory, and acronym ambiguity

Not all phase-related uncertainty is modeled as a stationary noise process. In change-point analysis of instantaneous phase, the object of interest is an abrupt phase-shift event in a noisy time series,

$0.1076^\circ$0

with $0.1076^\circ$1 the Heaviside step function (Marshall et al., 2014). Two non-parametric detectors are proposed: the cumulative summation statistic $0.1076^\circ$2 and the phase-derivative statistic $0.1076^\circ$3. The paper states that CUSUM has higher power for identifying single shift events, whereas the PD estimator has better temporal resolution for multiple ones (Marshall et al., 2014). In weakly coupled Rössler attractors, CUSUM achieves mACC $0.1076^\circ$4 and AUROC $0.1076^\circ$5, compared with PD mACC $0.1076^\circ$6 and AUROC $0.1076^\circ$7, while in EEG beta-band recordings the PD method detects $0.1076^\circ$8 phase shift events with an average ISI $0.1076^\circ$9 ms, corresponding to about $\overline{\theta}_n$00 shifts per second (Marshall et al., 2014). This literature treats phase shifts as meaningful events that may be obscured or mimicked by noise, rather than as nuisance phase diffusion.

A final point of terminology is that the acronym “PSN” is itself overloaded. In explicit-substitution calculi, PSN denotes Preservation of Strong Normalization, formalized as $\overline{\theta}_n$01, and has no connection to phase or noise (0909.5045). Likewise, scattering “phase shift” in lattice QCD refers to the $\overline{\theta}_n$02 phase shift extracted from finite-volume energies using Lüscher’s method, not to a stochastic phase perturbation (Bulava et al., 2010). For this reason, technical usage of “PSN” should always be interpreted in its local disciplinary context.

Taken together, the literature shows that Phase-Shift Noise is best understood not as a single universal model, but as a family of phase-perturbation phenomena characterized by three elements: a stochastic phase mechanism, a transduction pathway that makes the perturbation observable, and a mitigation or inference strategy tailored to that pathway. Depending on context, the dominant tools are differential cavity readout, stochastic estimation in colored oscillator models, structured modulation and pilot-free tracking, algebraic cancellation in detector metrics, or conversion of phase noise into amplitude noise for spectroscopy and metrology (Cheema et al., 2012, Khanzadi et al., 2014, Chraiti et al., 1 Sep 2025).