RST theoretical background: The question is should two photons—created together in a single SPDC event—remain phase-aligned indefinitely, and whether a Hong–Ou–Mandel (HOM) interferometer could reveal a finite “coherence window” even for perfectly identical radiation units. In the RS2 framework the answer is straightforward: the photons begin with the strongest alignment that their discrete creation event allows, but they are not created with perfectly identical scalar phases, nor does any principle of the theory require them to maintain exact relational alignment once they separate. The HOM experiment becomes sensitive to the consequences of this fact. Crucially, this does not imply that either photon changes internally. It follows entirely from how RS2 distinguishes between two different kinds of phase.
In RS and RS2, a photon is a discrete unit of scalar motion. Its familiar sinusoidal electromagnetic (EM) waveform in extension space is only the projection of a deeper scalar rotation occurring in the natural reference system (NRS). This intrinsic scalar rotation is uniform, fixed, and unchanging throughout the photon’s existence. It never “drifts” or evolves unless the photon is absorbed and re-emitted as a new unit. Larson treats the photon’s internal motion as a one-dimensional scalar vibration, inherently uniform and not subject to internal change once emitted. Nehru’s bi-rotation model and Peret’s scalar-rotation interpretation differ in mechanism, but they likewise regard the photon’s internal scalar cycle as fixed and quantized. In all versions of the theory, nothing inside a photon “drifts”; only the relative comparison between two independently created photons can change over time.
However, this internal rotation possesses a scalar phase—the phase of the 3D scalar cycle—and this scalar phase exists only in the NRS. It is not the same thing as the electromagnetic phase of the sinusoid observed in space. These two phases are related only through projection, not identity. The scalar phase determines whether two independent photons can still produce two-photon interference; the EM phase is what interferometers manipulate. This distinction is essential, because one phase is intrinsic and inaccessible, while the other is externally controllable.
The electromagnetic phase in extension space is familiar to optics: it can be shifted by path length, dispersion, modulators, EOMs, birefringence, chirp, phase plates, and so on. It is straightforward to force two photons to meet at a beamsplitter with precisely matched EM phase. But this accessible EM phase is not the scalar phase that RS2 treats as fundamental. Adjusting it does not and cannot alter the intrinsic scalar rotation. The scalar phase cannot be directly manipulated from within extension space.
When an SPDC pair is created, both photons originate from the same discrete scalar transition. They inherit the same scalar rotation rate, and they begin with scalar phases aligned as closely as the discrete emission process permits. Importantly, RS2 does not require that the initial scalar phases of two distinct radiation units be perfectly identical. The emission event creates two separate units, each with its own scalar rotation; it does not produce two “halves” of a single continuous cycle. Thus the pair begins with extremely close—but not exact—scalar-phase agreement, and the theory imposes no mechanism that would enforce exact relational alignment thereafter.
This subtle point eliminates any need to invoke “ongoing drift” inside the photons. Each photon continues its scalar rotation perfectly. Nothing internal changes. The only fact that matters is that their scalar phases were not identical at the moment of emission. As the photons separate and follow their own perfectly uniform scalar cycles, the tiny initial mismatch produces increasingly noticeable consequences as the delay in the HOM interferometer grows. The relational comparison changes even though each photon remains internally flawless.
A helpful analogy is two perfect metronomes, both ticking at the same frequency and both started nearly simultaneously, but placed on separate tables with no coupling. Neither metronome ever changes its rate. Both are ideal oscillators. Yet because they are independent systems, the tiny initial mismatch in start time is not corrected by any shared mechanism. Over time, the relative phase becomes more noticeable, not because either unit drifts in itself, but because independence means the initial difference is not held to zero. This is exactly the RS2 situation: two perfect rotations, but no requirement that their relational alignment be preserved indefinitely.
A HOM interferometer is sensitive to the projected compatibility of the scalar rotations, not to the EM overlap alone. At very small path delays, the scalar phases remain close enough that the projections into extension space interfere, producing the characteristic dip in coincidences. As the delay increases, the consequences of the small initial phase difference eventually exceed the window in which coherent projection is possible. Nothing about either photon changes; the coincidence rate returns to baseline because the scalar rotations were never perfectly matched and are not coupled.
This naturally produces a finite “Frame Width of Coupling”: the interval over which the underlying scalar phases remain compatible enough for interference to appear in projection. Because scalar motion is discrete, RS2 allows that the HOM dip could exhibit a slightly flatter central region and sharper edges than a purely convolutional joint-temporal-amplitude model would predict, especially as experimental jitter decreases. The feature need not form an exact rectangle, but RS2 does not forbid a compact-support-like shape at high precision. Reducing pump-timing jitter, path drift, and detector uncertainty could allow any intrinsic structure to become increasingly visible if it exists.
Experimental motivation from a QM perspective:
Standard treatments of SPDC usually assume that the joint temporal amplitude (JTA) of the biphoton is smooth—typically Gaussian or sinc-derived—because that is the form produced by idealized Gaussian pumps and approximate phasematching models. In practice, however, quantum optics does not require this smoothness. Several well-established theoretical models predict that real SPDC sources can exhibit sharply bounded or partially truncated spectral structure, which in the temporal domain becomes a finite interval in which the JTA is approximately constant, bounded by sharp edges. These include:
- Finite phasematching bandwidths with abrupt edges in periodically poled crystals (Fejer et al., 1992; Dixon et al., 2014).
- Rectangular or step-shaped pump spectra obtained via pulse shaping (Silberberg et al., 2000).
- Engineered quasi-phase-matching gratings designed to impose spectral discontinuities (Quesada & Sipe 2014; Ansari et al., 2018).
- Time-bin style temporal gating, which creates effectively compact temporal support (Franson 1989; Thew et al., 2014).
- Group-velocity-matched SPDC where the JSA becomes highly sensitive to phasematching boundaries (Mosley et al., 2008).
- Hard spectral filtering, which mathematically enforces a truncated JSA.
When any of these effects are present—in isolation or combined—the corresponding temporal amplitude can include a region where the overlap integral remains constant for a finite delay range. This produces a flat or quasi-flat region in the HOM interference dip. Such plateaus are not exotic: they are a direct and model-independent prediction of quantum mechanics whenever the biphoton JTA possesses compact-support features.
What has prevented their observation is not theoretical difficulty but experimental precision. These structures could be only a few femtoseconds wide, and traditional HOM interferometers lacked the timing stability and delay resolution to probe that regime. With modern <5 fs stabilized delay lines and high-resolution scans, this fraction of parameter space has finally become experimentally accessible.
The motivating question is therefore straightforward:
Do real SPDC sources—when examined at the highest temporal resolution currently achievable—exhibit the compact-support structure that several quantum-optical models already allow?
A positive result would reveal sharper spectral boundaries or tighter pump–crystal interactions than typically assumed, refining how we model real biphoton sources and updating the approximations used in quantum-optical theory. A null result would place the strongest constraints to date on the abruptness of SPDC biphoton structure and justify continued use of smooth-JTA approximations.
Either outcome is valuable. The experiment does not challenge quantum mechanics; it explores a theoretically permitted but experimentally unexplored region of two-photon coherence using tools precise enough to test it for the first time.
Measuring the Frame Width of Coupling (FWC) in a Hong–Ou–Mandel (HOM) Experiment
1) What we are measuring
A Hong–Ou–Mandel (HOM) interferometer measures how two photons interfere when their arrival times differ by only a few femtoseconds.
The
Frame Width of Coupling (FWC) is defined as the delay interval τ over which the photons still interfere (coherent two-photon behavior).
Outside that range, they behave independently and the coincidence rate returns to baseline.
<figure 1> Diagram of a basic HOM interferometer with SPDC source and delay line.
2) How the measurement works
When two indistinguishable photons reach a 50:50 beamsplitter at nearly the same moment, they “bunch” and exit the same port.
This produces the well-known
HOM dip in the coincidence rate.
The dip shape comes from the overlap of the two-photon joint temporal amplitude A(t):
- If A(t) is smooth (Gaussian/sinc-like), the dip is smooth.
- If A(t) effectively has a finite temporal support—a window where the overlap exists—then the dip develops a flat bottom with sharper edges.
Importantly, several quantum-optics models already allow such compact-support features in A(t), so a plateau is fully compatible with standard QM.
To model this possibility, we use:
- a normal coherence envelope of width σc
- multiplied by a rectangular window of width TFWC
This produces a plateau whose width directly equals T
FWC.
Modern delay stabilization (< 5 fs RMS) makes this experimentally resolvable.
<figure 2> Two-Photon Overlap in HOM (Standard View).
3) Standard quantum-optics interpretation
In ordinary quantum optics, everything is explained by the joint temporal amplitude A(t), which is shaped by:
- pump pulse shape
- crystal phase-matching
- spectral filtering
- dispersion
A finite FWC is simply a feature of the source’s A(t).
It does
not imply new physics or modify quantum mechanics in any way.
Several established theoretical models permit compact-support or truncated temporal structure, including:
- engineered quasi-phase-matching gratings
- rectangular or sharply shaped pump spectra
- group-velocity–matched SPDC
- hard spectral filtering
- finite phasematching bandwidths with abrupt edges
This experiment asks a basic question that has rarely been measured directly:
Is the HOM dip always smooth, or can it show a finite plateau?
Either outcome is useful:
- Smooth dip → coherence decays continuously (typical SPDC behavior).
- Plateau → A(t) contains a compact-support feature, still fully compatible with QM.
4) RST (Reciprocal System) interpretation
In RST/RS2:
- A photon is not a wave packet but a unit of radiation—a quantum of scalar motion.
- This scalar motion has conjugate components in space–time and the reciprocal region (“time–space”).
- In RS2, the familiar oscillation of radiation is the projection of a deeper scalar rotation in the underlying scalar dimension.
- Two photons created together begin with matched scalar phases because they originate as adjacent radiation units.
Adjacency remains, but exact scalar-phase alignment need not persist.
Small differences in effective scalar motion gradually produce a phase offset.
Interference in space–time is therefore observed only while their reciprocal phases remain aligned.
Thus, in the RS/RST interpretation:
TFWC measures the duration over which the scalar phases of the two radiation units remain aligned in the time–space region.
Once that alignment is lost, interference ends—even though reciprocal adjacency remains.
RST regards the HOM dip as the projection into ordinary 3D “extension space” of the underlying scalar motion.
This provides structural context but does not change the experiment’s operational meaning.
<figure 3> Time-Space Adjacency and Scalar-Phase Alignment (RST View).
5) What the scan actually measures
As the delay τ is scanned:
Inside |τ| < T
FWC/2 → coherent overlap → suppressed coincidences
Outside → no coherent overlap → coincidences return to baseline
Thus, the dip directly measures the temporal width of two-photon coherence.
6) Practical notes
- Delay resolution: ~10 fs steps, interleaved for effective ~5 fs
- Counts per point: ≳ 5,000
- Scan range: ±2 ps to establish baseline
- Stability: < 5 fs RMS via interferometric lock
- Pump-power sweeps: check that multi-pair events do not artificially flatten the dip
7) Interpretation of results
Smooth, rounded dip:
Continuous coherence envelope; standard SPDC behavior.
Flat-bottom dip with sharp edges:
Compact-support feature in the joint temporal amplitude.
In QM, this is source structure.
In RS/RST, it measures the finite duration of scalar-phase alignment in the reciprocal domain.
Summary
The HOM interferometer provides direct access to the temporal extent of two-photon interference.
- The Frame Width of Coupling TFWC quantifies this interval.
- Quantum-optics view: TFWC refines the shape of A(t).
- RST/RS2 view: TFWC measures how long the scalar phases of adjacent radiation units remain aligned in the reciprocal region.
