Measurement of the total spin angular momentum <Fz> of alkali-metal atoms

Atsushi Hatakeyama; Jeffery W. Martin; Kei Ishii; Runa Yasuda; Wolfgang Klassen

REVIEW 4 major objections 4 minor 41 references

Measurement of the total spin angular momentum <Fz> of alkali-metal atoms

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A paper establishes that integrated absorption coefficients of σ+ and σ− probe light directly yield the total spin projection ⟨Fz⟩ of cesium atoms in each ground hyperfine level, and uses this to estimate wall spin-relaxation probabilities.

desk verdict A genuinely new direct measurement of <Fz> with clean algebra but no error bars and an uncalibrated method; worth a serious referee, not yet established. read the letter →

arxiv 2504.19713 v1 pith:ZF3PR22D submitted 2025-04-28 physics.atom-ph

Runa Yasuda , Kei Ishii , Wolfgang Klassen , Jeffery W. Martin , Atsushi Hatakeyama This is my paper · ORCID

classification physics.atom-ph

keywords totalspinangularmomentumopticalpumpingabsorptionmonitoringcesiumD2linepolarizationanti-relaxationcoatingwallrelaxationspin-flowbalance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a direct optical method for measuring the expectation value ⟨Fz⟩ of the total spin angular momentum of alkali-metal atoms in the electronic ground state. The authors show that, for room-temperature cesium vapor with resolved ground hyperfine levels, the integrated absorption coefficients of σ+ and σ− probe light over the Doppler-broadened D2 lines are enough to determine ⟨Fz⟩ for each hyperfine level, without needing to resolve individual Zeeman sublevels. Using this method, they measure the largest ⟨Fz⟩ = 2.5 (polarization P = 0.63) in an anti-relaxation-coated cell. A simple spin-flow balance then yields single-collision wall spin-relaxation probabilities of about 1 for an uncoated cell and 4 × 10⁻³ for a coated cell. The method matters because ⟨Fz⟩ quantifies the angular-momentum reservoir available in spin-polarized vapor, which is central to optical pumping, spin-exchange, and magnetometry applications.

What carries the argument

The load-bearing object is the relation between integrated absorption coefficients and magnetic-sublevel populations, derived from electric-dipole transition strengths computed with Wigner 3j and 6j symbols. Summing the transition strengths over all excited hyperfine levels yields the linear forms in Eqs. (7) and (8), which make the integrated absorption coefficient $A^{{±}}$_F depend on ⟨Fz⟩_F through a slope of ∓420ℏ (for F = 3) or ±420ℏ (for F = 4). Dividing the difference A⁺_F − A⁻_F by the thermal-equilibrium sum Ā⁺_F + Ā⁻_F cancels the unknown atom number and cell-geometry factors, leaving ⟨Fz⟩_F. The same data also give the hyperfine populations n_{F=3} and n_{F=4}. A second mechanism, the spin-flow balance R_absℏ = C_FL R_absℏ + pΓ⟨Fz⟩N_totalℏ, connects the measured ⟨Fz⟩ to the wall collision rate Γ and the single-collision relaxation probability p.

What would settle it

A direct mechanical measurement of the angular momentum stored in the vapor, using a torsion pendulum based on the Einstein–de Haas effect, would give an independent value of ⟨Fz⟩ for the same cell and pumping conditions; if that value disagrees with the absorption-derived ⟨Fz⟩ beyond the stated 10–20% uncertainties, the absorption-integration method would be shown to be contaminated by systematic effects such as radiation trapping or velocity-changing collisions.

Watch

Extended reading notes

Core claim

The central claim is that the difference between the integrated absorption coefficients for σ+ and σ− probe light, each summed over all excited-state hyperfine levels reachable from a given ground hyperfine level, is proportional to ⟨Fz⟩ for that ground level. Specifically, Eqs. (14) and (15) give ⟨Fz⟩_{F=3} and ⟨Fz⟩_{F=4} as simple ratios of (A⁺_F − A⁻_F) to (Ā⁺_F + Ā⁻_F), with numerical prefactors −7/2 and +9/2. The total ⟨Fz⟩ is the sum of the two hyperfine contributions. The authors validate the low-probe-power regime and negligible excited-state population, and they measure the dependences on pump power, pump polarization, and repump power in both uncoated and coated cells. The highest polarization, P = 0.63 with ⟨Fz⟩ = 2.5, occurs in the coated cell using F = 4 → F′ = 3 pumping and F = 3 → F′ = 3 repumping. The wall spin-relaxation probability per collision, extracted from a steady-state spin-flow balance, is close to 1 for the uncoated glass surface and about 4 × 10⁻³ for the paraffin coating.

Load-bearing premise

The measurement assumes that, after keeping the probe power low and the excited-state population negligible, the integrated absorption coefficients are distorted by no other systematic process and therefore exactly encode the ground-state sublevel populations; the authors state that independent calibration by a torsion-pendulum measurement is planned but has not yet been performed.

Editorial extensions

If this is right

The absorption-integration method extends directly to other alkali-metal atoms with well-resolved ground hyperfine levels, providing a way to measure ⟨Fz⟩ and ⟨Sz⟩ without optically resolving Zeeman sublevels.
The demonstrated coated-cell polarization P = 0.63, with the extracted per-collision relaxation probability 4 × 10⁻³, quantifies how effectively anti-relaxation coatings preserve angular momentum in alkali vapor reservoirs.
Because the method yields ⟨Fz⟩ and hyperfine populations separately, it can track how spin is distributed between the two ground hyperfine levels during optical pumping with and without repumping.
The spin-flow balance gives a steady-state route to wall spin-relaxation probabilities, complementing time-resolved relaxation-in-the-dark measurements.
The low probe-power requirements (below 10 µW in the uncoated cell and below 50 nW in the coated cell) make the method applicable in regimes where other monitoring approaches distort the population.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the linear form of Eqs. (14) and (15) suggests the method could be adapted to time-resolved measurements, tracking ⟨Fz⟩ dynamics during pulsed pumping or during wall-collision relaxation, using a fast probe scan.
If the planned torsion-pendulum measurement disagrees with this optical method, the most likely cause would be a systematic distortion of the integrated absorption coefficients by radiation trapping, velocity-changing collisions, or an unpumped background; that comparison would directly identify which correction is needed.
The method's requirement of resolved ground hyperfine levels but not resolved excited levels means it could be applied to alkali vapors with buffer gas as long as the ground hyperfine splitting remains visible, potentially covering a wider parameter space than conventional birefringence monitoring.
The spin-flow model could be extended to extract not just the per-collision relaxation probability but also the angular-momentum transfer to the cell walls, which is relevant to surface spin physics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Quick take: this paper reports a new way to measure total spin projection in alkali vapor without assuming a spin-temperature distribution. The central algebra in Sec. II is sound: the integrated absorption ratio with sigma+ and sigma- probe light yields _F via Eqs. (14)-(15), and the global factor cancels cleanly. The thermal-equilibrium 9:7 absorption ratio check and the probe-power independence test provide real support. The authors show sensible dependencies on pump power, polarization, and repump power, and the headline result of = 2.5 (P = 0.63) in a coated cell is concrete.

The soft spots are proportionate. First, no error bars appear anywhere, despite eight repeated runs per condition. The paper says the data are averaged but never reports the scatter, so the numerical claims have no stated uncertainty. Second, the method has no independent calibration. The authors say this themselves in Sec. IV.B: they plan a torsion-pendulum Einstein-de Haas measurement but have not performed it. That honesty is to their credit, but it means the central numbers rest on internal consistency checks, not an external standard. Third, the spin-flow model in Sec. IV.C is the weakest section. The extracted wall-collision probability p = 1.6 for the uncoated cell is unphysical, and attributing it to 10-20% parameter uncertainty plus omitted processes like radiation trapping is a sign that the model inputs, or the measured feeding them, are not yet quantitatively consistent. To be fair, that inconsistency does not directly invalidate the central measurement; the spin-flow analysis is a secondary application with several hand-chosen inputs, including CFL from an assumed linear population distribution and Gamma from cell dimensions. But it should be recast as illustrative until the independent check is done.

The citation pattern looks fine. They cite prior work measuring Sz or reconstructing sublevel populations without extracting Fz, and their novelty claim holds up. This is an incremental but real subfield contribution, not a breakthrough.

I would send this to a serious referee. The method deserves scrutiny, and the authors are doing honest work. The referee should press for error bars, for an explanation of the p = 1.6 result, and for a clear accounting of systematic effects (radiation trapping, beam overlap, velocity-changing collisions) that could bias the integrated absorption ratio. If those are addressed, this becomes a solid methods paper.

Recommendation: engage; accept for review.

Referee Report

4 major / 4 minor

Summary. The paper presents a technique for measuring the total spin angular momentum projection ⟨Fz⟩ of cesium atoms in the ground state, using integrated absorption coefficients of σ+ and σ− probe light on the D2 transitions. The authors derive closed-form expressions (Eqs. (14)–(16)) that relate the difference of integrated absorption coefficients for the two probe polarizations to ⟨Fz⟩ for each hyperfine level, with the unknown overall factor canceled by normalizing to thermal-equilibrium absorption. They validate the probe power regime and the thermal-equilibrium ratio, then apply the method to uncoated and anti-relaxation-coated cells under various optical pumping conditions, reporting a maximum ⟨Fz⟩ = 2.5 (P = 0.63). Finally, they introduce a spin-flow balance model (Eq. (21)) to estimate single-collision wall spin-relaxation probabilities p, obtaining p ≈ 1.6 for the uncoated cell and p ≈ 4 × 10⁻³ for the coated cell.

Significance. If the method is valid, it provides a relatively simple absorption-based route to ⟨Fz⟩ for alkali-metal vapors, a quantity that is often inferred indirectly. The derivation in Sec. II is transparent and the algebraic steps leading to Eqs. (14)–(16) are correct given the stated assumptions. The internal consistency checks—the thermal-equilibrium 9:7 absorption ratio and the probe-power independence—are good and give confidence that the probe itself does not perturb the system in the chosen regime. The paper also makes concrete, falsifiable predictions (e.g., the dependence on pump power, polarization, and repump power) that agree qualitatively with the data. However, the central quantitative claims currently lack an independent calibration: the paper itself states that the method's validity should ultimately be examined with a torsion-pendulum measurement. The absence of uncertainty estimates and the unphysical p = 1.6 in the spin-flow analysis further limit the strength of the conclusions.

major comments (4)

[Sec. IV (all figures and Table II)] No error bars or uncertainty estimates are reported anywhere, despite the statement in Sec. III that 'all experiments are performed eight times, and the data are averaged.' The quoted values of ⟨Fz⟩ and p are point estimates; for example, the difference between p = 1.6 for the uncoated cell and the physical upper bound of 1 is central to the spin-flow discussion, but without propagated uncertainties it is impossible to judge whether this is a statistically significant discrepancy. The authors should report the standard deviation of the eight repeated measurements and estimate systematic contributions from probe-power calibration, polarization purity, and beam-overlap geometry.
[Sec. IV.C, Eq. (21) and Table II] The spin-flow balance yields p = 1.6 for the uncoated cell, which exceeds the physical maximum of 1 for a probability. The statement that 'an uncertainty of 10–20%' and omitted processes make this consistent with 1 is not quantitatively justified: a 10–20% uncertainty cannot shift 1.6 to ≤ 1. This indicates either missing spin-loss channels (radiation trapping, spin-exchange collisions, or hyperfine-changing collisions) or a mismatch between the measured ⟨Fz⟩ and the model's assumptions. The derived wall relaxation probability for uncoated surfaces is therefore not quantitatively supported; the model needs to be extended or the claim softened.
[Sec. II, Eqs. (4)–(6), and Sec. III] The derivation assumes a uniform spatial distribution and factorizes the velocity distribution f(v) from the sublevel populations n_{F,m_F}. Under optical pumping, however, the pump beam (8–9 mm diameter) is smaller than the probe beam (12 mm diameter) and intersects it at a 5° angle, so the probed volume contains atoms outside the pumped region and a nonuniform polarization profile. The measured integrated absorption coefficients are averages over this profile, so the ⟨Fz⟩ extracted from Eqs. (14)–(15) is a beam-averaged quantity, whereas the spin-flow model in Eq. (21) treats the whole cell as uniformly polarized. This mismatch could bias both the quoted polarization values and the derived p; the authors should use matched beam sizes or explicitly model the overlap.
[Sec. IV.B] The paper states that 'the validity of the method should ultimately be examined using an alternative experimental approach' and that a torsion-pendulum measurement is planned but not yet performed. Given that the central quantitative claims rest on an uncalibrated absorption model, the results are conditional. The abstract and conclusions should either state this limitation explicitly or the authors should provide an independent cross-check (e.g., Faraday rotation or spin-noise spectroscopy) to support the method before presenting ⟨Fz⟩ = 2.5 as the headline result.

minor comments (4)

[Sec. III] There is a typo: 'exhibites' should be 'exhibits' in the description of the probe beam profile.
[Sec. IV.B] There is a typo: 'The resulrs found' should be 'The results found' in the paragraph about future modeling.
[Eq. (17)] The definition P = ⟨Fz⟩/4 is clear for Cs, but the maximum value of ⟨Fz⟩ = 4 corresponds to complete population of the F = 4, m_F = 4 state; this should be stated explicitly so that the normalization is not confused with the maximum possible spin projection of a single atom.
[Sec. II, Eqs. (11)–(12)] The notation $\bar{A}^{\pm}_F$ is used for thermal-equilibrium absorption coefficients but is not explicitly defined before its first use; a brief definition would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Fz extraction inverts measured absorption coefficients via known Clebsch-Gordan strengths, with no fitted parameter or self-citation chain forcing the result.

full rationale

The derivation from transmittance to Fz is self-contained. Equations (9)-(15) express the measured integrated absorption coefficients A+F and A-F as sums over ground-state sublevel populations with fixed transition strengths from Table I, then algebraically invert the ratios to obtain <Fz>F. The thermal-equilibrium references Abar+F and Abar-F are also measured with the pump off and are not fitted parameters, and the common factor (L/V)aNtotal cancels exactly in the ratio. Thus the central quantity is neither defined in terms of itself nor obtained from a fitted parameter. The spin-flow section is also not circular: p is solved from Eq. (21), Rabs = CFL Rabs + pGamma<Fz>Ntotal, using independently estimated Rabs, Gamma, Ntotal and the measured <Fz>. The phrase in Sec. IV.C that the linear sublevel distribution 'ensures that the observed <Fz>F=4 is reproduced' describes how the spontaneous-emission factor CFL is modeled; it does not make p an input or rename the measured Fz as a prediction. The paper explicitly states that the method's validity should ultimately be checked with an alternative torsion-pendulum measurement, and that the unphysical p=1.6 for the uncoated cell is attributed to 10-20% parameter uncertainty and omitted processes such as radiation trapping. Those are external-validation and modeling limitations, not circularity. Self-citations appear for coated-cell preparation and the planned torsion pendulum, but none is load-bearing for the central extraction, and no uniqueness theorem or ansatz is smuggled in via self-citation.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central Fz measurement depends on no adjustable parameters: all proportionality constants cancel in Eqs. (14)-(15), and the ground-hyperfine absorption ratio 9:7 validates the transition strengths. The spin-flow estimate of the wall-collision probability p is the only place where model inputs (CFL, Gamma, Ntotal) enter; these are estimated from geometry and from an assumed linear sublevel distribution rather than measured independently.

free parameters (3)

CFL (average spin emission factor per absorbed photon) = 0.26 (uncoated), 0.24 (coated)
Estimated in Sec. IV.C by assuming a linear population distribution among F=4 magnetic sublevels chosen to reproduce the measured Fz_{F=4}. This hand-chosen input enters Eq. (21) and affects the derived wall-collision probability p.
Gamma (wall collision rate) = 1.5e4 s^-1 (uncoated), 1.6e4 s^-1 (coated)
Estimated geometrically from cell dimensions and mean atomic speed; used in Eq. (21) to convert p to a relaxation time. It is a model input, not an independently measured quantity.
Atomic density Ntotal = 3.4e10 cm^-3
Derived from transmittance spectra via Eq. (22) and checked against the Cs vapor pressure. Used in the spin-flow normalization for p.

assumptions (7)

domain assumption Ground-state hyperfine levels (F=3,4) are well resolved within the Doppler-broadened D2 line, while excited-state hyperfine levels are unresolved, so the sums over F' in Eq. (6) are valid.
Stated in Sec. II and supported by Fig. 1; the ground-state splitting is 9.2 GHz versus a 378 MHz Doppler width.
domain assumption Excited-state populations are negligible, so the ground-state populations sum to 1.
Stated in Sec. II and argued in Sec. IV.A from the small decrease of nF4+nF3 at high pump power (Fig. 5).
domain assumption The probe laser is in the low-intensity limit (Beer's law, no optical pumping by the probe).
Checked experimentally in Fig. 4 for the uncoated cell; the probe power is kept below 10 uW (uncoated) and 50 nW (coated).
domain assumption The atomic spatial distribution is uniform over the probe region.
Stated in Sec. II. In the coated cell the pump and probe beams differ in diameter, so polarization is likely nonuniform, but the extraction treats the sample as homogeneous.
standard math The velocity distribution f(v) is the room-temperature Maxwell distribution and integrates to unity over the scanned Doppler line.
Used in Eqs. (4)-(6); the integrated absorption coefficient over all velocities gives the total sublevel population.
ad hoc to paper Eq. (21) spin-flow balance: spin input from absorbed pump photons equals spin loss via spontaneous emission and wall collisions, with radiation trapping and other channels neglected.
Introduced in Sec. IV.C; the authors acknowledge neglected processes such as radiation trapping.
ad hoc to paper CFL is evaluated assuming a linear population distribution among F=4 magnetic sublevels.
Stated in Sec. IV.C; the assumed distribution is chosen to reproduce the observed Fz_{F=4}.

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Pith. "Pith review of Measurement of the total spin angular momentum <Fz> of alkali-metal atoms." pith.science (2026). https://pith.science/paper/ZF3PR22D

@misc{pith2026250419713,
  author       = {Pith},
  title        = {Pith review of: Measurement of the total spin angular momentum <Fz> of alkali-metal atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZF3PR22D}},
  note         = {Machine review of arXiv:2504.19713}
}

read the original abstract

It is important to evaluate the total spin angular momentum of alkali-metal atoms if the atoms serve as a reservoir of angular momenta. We use an absorption-monitoring technique to measure <Fz>, i.e., the expectation values of the quantization (z) axis components of the total angular momentum of cesium (Cs) atoms in the electronic ground state in both uncoated and anti-relaxation-coated vacuum cells at room temperature. Cs atoms are polarized via optical pumping and probed using their D2 transitions. The probe laser frequency is varied across the Doppler-broadened D2 transition; the <Fz> values are derived using the integrated absorption coefficients. The largest <Fz> is 2.5 for the coated cell. We then use a simple model of spin flow through vapor cells to estimate the atomic spin relaxation probabilities after a single surface collision.

Figures

Figures reproduced from arXiv: 2504.19713 by the authors.

**Figure 2.** FIG. 2. Schematic diagram of the experimental setup. OC: opt [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Transmittance spectra for the uncoated cell with [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Dependence of the integrated absorption coefficients [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. Dependencies of the spin [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. Dependencies of the uncoated cell spin [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. Dependencies of the uncoated cell spin [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. Transmittance spectra for the coated cell with the [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: shows the pump power dependence of the spin z components. As in the uncoated cell, hFzi increases with increasing pump power. However, it becomes saturated around 40 µW (83 µW/cm2 ), which is much lower than in the uncoated cell, reflecting the strong hyperfine pumping…

**Figure 9.** Figure 9: FIG. 9. Dependencies of the spin [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10. Dependencies of the spin [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: FIG. 11. Spin flow during optical pumping through the cell. Th [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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T. G. Walker and W. Happer, Rev. Mod. Phys. 69, 629 (1997)

1997

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9 µ W (1.7 µ W/cm2)

2 mW (8.7 mW/cm 2), and the probe laser power is 1 . 9 µ W (1.7 µ W/cm2). B. Coated cell Polarization of alkali-metal atoms in uncoated vapor cells is randomize d by atomic colli- sions with the cell walls [7]. Spin relaxation times are thus generally dete rmined by deriving the mean travel times between the walls; this is 7 × 10−5 s in the uncoated cell ...

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J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, Phys. Rev. Lett. 89, 130801 (2002)

2002

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the relaxation in the dark

from Eqs. (9), (10), (11) and (12) as follows: nF =3 = ∑ mF =3 nF =3,m F =3 = 7 16 A+ F =3 + A− F =3 ¯A+ F =3 + ¯A− F =3 , (19) nF =4 = ∑ mF =4 nF =4,m F =4 = 9 16 A+ F =4 + A− F =4 ¯A+ F =4 + ¯A− F =4 , (20) where 7/16 and 9/16, coeﬃcients of relative populations of hyperﬁ ne levels in thermal equi- librium, are reasonably derived. III. EXPERIMENT AL SET...

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Appelt, A

S. Appelt, A. B.-A. Baranga, C. J. Erickson, M. V. Romalis , A. R. Young, and W. Happer, Phys. Rev. A 58, 1412 (1998)

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I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romali s, Nature 422, 596 (2003)

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Arimondo, M

E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977)

1977

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Gerginov, A

V. Gerginov, A. Derevianko, and C. E. Tanner, Phys. Rev. L ett. 91, 072501 (2003)

work page 2003