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Squeezing in Both the Plus and Minus Quadratures with the Uncertainty Relation Perfectly Holding

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Pith's one-line read A coherently driven cavity interacting with a three-level atom can be squeezed in both quadratures simultaneously, with maximum reductions 52.08% and 33.32% below vacuum, while the uncertainty product holds.

desk verdict The algebra is tidy but the central claim collapses on one load-bearing step: the steady-state relation is treated as an exact operator identity, which changes the commutation relation and makes the 'vacuum' reference nonphysical. read the letter →

arxiv 1909.01004 v1 pith:UGGFEMQM submitted 2019-09-03 quant-ph

Ayana Yihunie , Fesseha Kassahun This is my paper · ORCID
classification quant-ph
keywords quadraturesqueezingthree-levelatomcavitymodevacuumreservoiruncertaintyrelationnormalorderingsuperposedmodesspontaneousemission
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 tries to establish that a cavity mode driven by coherent light and interacting with a three-level atom can be squeezed in both quadratures at once. If true, the maximum quadrature squeezing would be 52.08% in the plus quadrature and 33.32% in the minus quadrature, with the product of uncertainties staying above the bound set by the effective commutator. The paper also extends the result to a superposed pair of cavity modes, where both quadratures are squeezed equally. A sympathetic reader would see the contribution as a closed-form derivation of simultaneous quadrature squeezing in a specific cavity-QED model.

What carries the argument

The central mechanism is the steady-state operator reduction $\hat b = 2\eta/\kappa - (2g/\kappa)\hat\sigma_c$, obtained by placing reservoir noise in normal order and invoking the large-time approximation. This relation converts the field quadrature commutator into a state-dependent atomic commutator, which sets the uncertainty bound that the paper verifies. The same steady-state reduction is applied to the superposed operator $\hat c = \hat a + i\hat b$, giving equal squeezing in both quadratures.

What would settle it

Use the canonical commutator $[\hat b,\hat b^\dagger]=1$ to compute quadrature variances from the paper's formulas; if the product falls below the standard limit, the state cannot be realized by a single cavity mode. A direct balanced-homodyne measurement of both quadratures of the driven cavity would show whether either variance trace actually dips below the vacuum shot-noise level.

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Extended reading notes

Core claim

For a single cavity mode $\hat b$ driven by coherent light and coupled to a cascade three-level atom in an open cavity, the paper finds that the variances of both quadrature operators $\hat b_+ = \hat b^\dagger + \hat b$ and $\hat b_- = i(\hat b^\dagger - \hat b)$ fall below the vacuum-state level. The maximum squeezing is 52.08% for the plus quadrature and 33.32% for the minus quadrature, and the product $\Delta b_+ \Delta b_-$ stays above the bound computed from the effective commutator $[\hat b_-,\hat b_+] = 2i(\gamma_c/\kappa)(\hat\eta_a - \hat\eta_c)$. For a superposed pair of modes $\hat c = \hat a + i\hat b$, both quadratures show the same squeezing, equal to half the sum of the single-mode squeezing values. The paper also finds that spontaneous emission lowers the mean photon number of the cavity mode but does not change the maximum quadrature squeezing.

Load-bearing premise

The calculation treats the steady-state relation between the cavity field and the atomic operator as an exact identity when computing quantum noise; if that relation is only approximate, the simultaneous squeezing goes away.

Editorial extensions

If this is right

  • A cavity mode in the described setting would exhibit simultaneous plus- and minus-quadrature squeezing, with maximum reductions of 52.08% and 33.32% below the vacuum level.
  • The product of the two quadrature uncertainties remains above the bound set by the effective commutator, so the paper's state is consistent with its uncertainty relation.
  • For a superposed pair of such modes, both quadratures are squeezed by the same amount, equal to half the sum of the single-mode squeezing values.
  • Spontaneous emission lowers the mean photon number but leaves the maximum quadrature squeezing unchanged.

Reading between the lines

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

  • If one imposes the canonical commutator $[\hat b,\hat b^\dagger] = 1$ instead of the dressed commutator used in the paper, the same variances would have to violate the standard uncertainty bound; the reported effect likely depends on treating the steady-state atomic relation as an exact operator identity.
  • A full master-equation treatment that keeps the vacuum noise terms instead of dropping normally ordered noise operators would provide a direct check of whether both quadratures can really fall below vacuum noise.
  • A natural experimental test would use a cavity-QED setup with parameters near $\gamma_c = 0.5$ and $\kappa = 0.8$, where the predicted maximum near $\varepsilon \approx 0.6$ should appear as simultaneous sub-shot-noise in both quadratures.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper considers a cavity mode driven by coherent light and interacting with a three-level atom in an open cavity coupled to a vacuum reservoir. The authors write quantum Langevin equations with the reservoir noise operators put in normal order and then dropped, obtain approximate steady-state operator relations, and use these to compute the mean photon number and the variances of the plus and minus quadratures of the cavity mode. They claim that both quadratures are squeezed below the vacuum level (52.08% for the plus quadrature and 33.32% for the minus quadrature) while the uncertainty relation still holds perfectly. The analysis is extended to a pair of superposed cavity modes, where equal squeezing in both quadratures is reported, equal to half the sum of the individual quadrature squeezings. The manuscript is algebraic and self-contained, with explicit formulas and plots.

Significance. If the central claim were correct, it would be a highly surprising result: simultaneous squeezing of both quadratures of a physical cavity mode below the vacuum level while preserving the Heisenberg uncertainty relation is not possible for standard bosonic quadratures. The paper is clearly organized and the formal algebra is internally consistent under the authors' replacement rules; it also treats spontaneous emission and superposed modes, which are useful extensions. However, as detailed in the major comments, the main physical claim rests on a noncanonical commutation relation and an unphysical definition of the vacuum level, so the reported effect is an artifact of the approximation rather than a property of the physical cavity mode.

major comments (3)
  1. [Section 3.2, Eq. (50)] The commutator [b−, b+] = 2i(γc/κ)(ηa − ηc) is obtained by treating the steady-state relation (26), b = 2η/κ − (2g/κ)σc, as an exact operator identity. But b is the physical cavity-mode annihilation operator, whose canonical commutator [b, b†] = 1 is implicit in the Hamiltonian (7) and the Langevin equation (10). Equation (26) implies [b, b†] = (γc/κ)(ηc − ηa), an atomic operator whose expectation is bounded in magnitude by γc/κ; for the parameters used in Figures 3–6, γc/κ = 0.625, which directly contradicts [b, b†] = 1. Consequently Eq. (50) is not the commutator of the physical quadratures. The physical commutator is [b−, b+] = −2i, and the Heisenberg bound is Δb+ Δb− ≥ 1. At the maximum quoted for the plus quadrature (ε = 0.6, γc = 0.5, γ = 0.3, κ = 0.8), Eqs. (65)–(66) give (Δb+)² ≈ 0.299 and (Δb−)² ≈ 0.494, so the product is approximately 0.148, far below 1. The claim that the uncertainty relation holds perfectly therefore does not apply to the physical cavity mode.
  2. [Section 3.2, Eqs. (67), (70)–(71)] The 'vacuum state level' used to define squeezing is not the vacuum of the cavity mode. Setting ε = 0 in Eqs. (65)–(66) gives (Δb±)² = γc/κ, whereas for the standard quadrature operators b+ = b† + b and b− = i(b† − b) evaluated in the true vacuum state of b, the variance is 1. When ε = 0, Eq. (26) reduces to b = −(2g/κ)σc, i.e., the 'mode' is proportional to an atomic lowering operator, not a harmonic-oscillator mode. The percentages 52.08% and 33.32% are therefore reductions relative to an unphysical reference; relative to the true vacuum, the product of the reported variances violates the canonical bound. The same issue affects the superposed-mode vacuum level in Eq. (114).
  3. [Section 3.2, Eqs. (53)–(55), (68)–(69)] The verification that the uncertainty relation 'holds perfectly' is circular. The lower bound in Eqs. (53)–(55) is computed from the modified commutator (50), and the product in Eqs. (68)–(69) is compared with that same modified bound. Since both the variances and the bound are derived from the same approximate steady-state relation (26), the inequality fb(ε) ≥ fa(ε) does not constitute a test of the physical Heisenberg uncertainty relation. A physical test requires using [b−, b+] = −2i and the actual vacuum variance; with those, the product of the reported variances violates the bound. The same circularity appears in the superposed-mode analysis, Eqs. (100)–(103) and (115)–(116).
minor comments (3)
  1. [Captions of Figures 5 and 6] The captions contain the phrase 'and or γ = 0', which appears to be a typo for 'and for γ = 0'.
  2. [Introduction, first paragraph] There is a typo in 'the quantum properties of the cavity mod e'; it should read 'cavity mode'.
  3. [Eqs. (48)–(49)] The quadrature operators are defined without a factor of 1/2; the paper should state explicitly that this unconventional normalization is intended, since the standard vacuum variance is then 1 rather than 1/4.

Circularity Check

2 steps flagged · score 8.0 of 10

Both the below-vacuum squeezing percentages and the 'perfect' uncertainty relation are self-referential: the vacuum baseline is the ε=0 limit of the dressed variance, and the uncertainty bound is computed from a noncanonical commutator derived from the same steady-state relation.

  1. self definitional [Section 3.2, Eqs. (26), (50), and (53).]
    "Evidently, these would turn out to be exact relations at steady state. ... ˆb = 2η/κ − 2g/κ ˆσc. ... It can be readily established that [ˆb−, ˆb+] = 2 i γc/κ (ˆηa − ˆηc). ... It then follows that ∆ b+∆ b− ≥ γc/κ |⟨ˆηa⟩ − ⟨ˆηc⟩|."

    Equation (26) is a steady-state first-moment relation, but it is promoted to an exact operator identity to compute the commutator [b−, b+]. This yields an atomic-operator commutator whose expectation is bounded by γc/κ, contradicting the canonical [b,b†]=1 assumed by the original Hamiltonian. The uncertainty bound in Eq. (53) is therefore derived from the same dressed commutator that fixes the variances, so the claim that the uncertainty relation 'holds perfectly' is a consistency condition of the approximation, not a physical constraint.

  2. self definitional [Section 3.2, Eq. (67) and Eqs. (70)-(73).]
    "Upon setting ε = 0, in Eqs. (65) and (66), we have (∆ b+)²v = (∆ b−)²v = γc/κ. This indeed represents the quadrature variance of the cavity vacuum state in which the uncertainties in the two quadratures are equal and satisfy the minimum uncertainty relation given by Eq. (56)."

    The reference 'vacuum state level' used to define squeezing is not the physical vacuum of the cavity mode, for which [b,b†]=1 would give variance 1. It is the ε=0 limit of the same dressed expression, γc/κ. Consequently the reported 52.08% and 33.32% below-vacuum squeezing values compare variances against an internal baseline generated by the model itself, making the squeezing percentages self-referential rather than predictions relative to an independent vacuum.

full rationale

The central derivation is internally consistent but circular in its reference frame. The paper obtains the steady-state relation b = 2η/κ − (2g/κ)σc, then treats it as an exact operator identity to evaluate [b−, b+]. This noncanonical commutator is used in Eq. (53) to define the uncertainty lower bound, so the product of variances trivially satisfies a bound that has been lowered by the same construction. The vacuum-state variance is likewise defined as the ε=0 limit of the dressed variances, giving γc/κ rather than the canonical value 1; all squeezing percentages are measured against this internal baseline. For the physical cavity mode with [b,b†]=1, the variances reported in Eqs. (65)-(66) would violate the canonical Heisenberg bound, so the claimed simultaneous squeezing and perfect uncertainty relation do not survive replacement of the dressed commutator by the physical one. This is not a case of self-citation or fitted parameters; it is a definitional circularity in the criterion for 'vacuum' and 'uncertainty relation'.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model has no fitted parameters: the driving amplitude η, coupling g, cavity damping κ, and spontaneous emission rate γ are external inputs. The central claim rests on the normal-ordering noise-drop assumption and on the adiabatic operator relation that changes the commutation relation; these are domain assumptions and an ad hoc step rather than invented entities.

assumptions (4)
  • domain assumption Noise operators associated with the vacuum reservoir, when placed in normal order, can be dropped from the quantum Langevin equations without affecting the expectation values and variances.
    Section 2 states this and cites Ref [20]; it underlies the noise-free operator equations (11)-(13). It is not standard for variance calculations and changes the effective commutation relation.
  • domain assumption The large-time approximation makes equations (24)-(26) exact operator relations at steady state, allowing b to be replaced by a c-number plus an atomic operator.
    Eqs. (24)-(26) transform the mode operators into atomic operators; this is the step that produces the modified commutator (50).
  • ad hoc to paper The commutation relation [b-, b+] = 2i(γc/κ)(ηa-ηc) computed from the approximate steady-state solution can be used for the uncertainty relation of the cavity mode quadratures.
    Eq. (50); this is the crucial assumption. For the physical field, [b-, b+] must equal -2i. Using the dressed commutator relaxes the uncertainty bound and allows both variances to appear below the reference level.
  • domain assumption All three spontaneous emission transitions (|a⟩→|b⟩, |b⟩→|c⟩, |a⟩→|c⟩) have the same decay constant γ.
    Introduced in Section 2 before Eq. (14); it simplifies the master equation but is not physically required.

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Cite this review

Pith. "Pith review of Squeezing in Both the Plus and Minus Quadratures with the Uncertainty Relation Perfectly Holding." pith.science (2026). https://pith.science/paper/UGGFEMQM

@misc{pith2026190901004,
  author       = {Pith},
  title        = {Pith review of: Squeezing in Both the Plus and Minus Quadratures with the Uncertainty Relation Perfectly Holding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UGGFEMQM}},
  note         = {Machine review of arXiv:1909.01004}
}
read the original abstract

We have considered a cavity mode driven by coherent light and interacting with a three-level atom available in an open cavity coupled to a vacuum reservoir. We have carried out our analysis by putting the noise operators associated with the vacuum reservoir in normal order. We have also considered the interaction of the three-level atom with the vacuum reservoir outside the cavity. It is found that the squeezing occurs in both the plus and the minus quadratures and the maximum quadrature squeezing happens to be 52.08% and 33.32% below the vacuum state level, respectively. We have established that the uncertainty relation holds perfectly for this case as well. In addition, we have found that the squeezing in a pair of superposed cavity modes occurs in the plus and minus quadratures and have the same value. The amount of squeezing in each quadrature turns out to be half of the sum of the squeezing in the plus and minus quadratures of each cavity mode. Furthermore, we have observed that the presence of spontaneous emission decreases the mean photon number of the cavity mode but does not affect the maximum quadrature squeezing.

Figures

Figures reproduced from arXiv: 1909.01004 by the authors.

Figure 1
Figure 1. Schematic representation of a three-level atom in a cascade configuration available in an open cavity driven by coherent light and coupled to vacuum reservoir via a single-port mirror. where σˆa = |biha|, (4) σˆb = |cihb|, (5) σˆc = |ciha|, (6) are lowering atomic operators, ˆa1(ˆa2) is the annihilation operator for the cavity mode a1 (a2), with g being the coupling constant between the atom and the cavity mode b or… view at source ↗
Figure 2
Figure 2. Plots of Eq. (46) for γc = 0.5, κ = 0.8, and for γ = 0 (solid curve), and for γ = 0.3 (dashed curve). 3 Cavity mode b Here we wish to calculate the mean photon number and the quadrature squeezing of the cavity mode b at steady state. 3.1 The mean photon number The mean photon number for the cavity mode b at steady state is defined by ¯n = h ˆb †ˆbi. Then applying Eq. (26), we easily find n¯ = 4η 2 κ2 − 8ηg κ2 hσˆci … view at source ↗
Figure 3
Figure 3. Plots of (∆b−) 2 , (∆b+) 2 , and ((∆b) 2 +v = (∆b) 2 −v) versus ε for γc = 0.5, γ = 0.3, and κ = 0.8 . On account of Eqs. (45), (59), (60), (61), and (62), we readily obtain (∆b+) 2 = γc κ [hηˆai + hηˆci − 4hσˆci 2 ], (63) (∆b−) 2 = γc κ (hηˆai + hηˆci). (64) Therefore, in view of Eqs. (42), (43), and (44), we get (∆b+) 2 = γc κ  6ε 4 + (γc + γ) 2 ((γc + γ) 2 + ε 2 ) ((γc + γ) 2 + 3ε 2 ) 2  , (65) (∆b−) 2 = γc κ … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Plots of fa(ε) and fb(ε) versus ε for γc = 0.5, γ = 0.3, and κ = 0.8. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Plots of S+ versus ε for γc = 0.5, κ = 0.8 , and for γ = 0.3 (dashed curve), and or γ = 0 (solid curve) . 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Plots of S− versus ε for γc = 0.5, κ = 0.8, and for γ = 0.3 (dashed curve), and or γ = 0 (solid curve). On the basis of these results, we realize that the cavity mode b satisfies the uncertainty relation. The squeezing of the plus and the minus quadratures of the cavit…
Figure 7
Figure 7. Figure 7: Plots of Eqs. (103) and (116) versus ε for γc = 0.5, κ = 0.8, and γ = 0.3 . (∆c−) 2 = 2γc κ  1 −  3ε 4 + 3ε 2 (γc + γ) 2 (3ε 2 + (γc + γ) 2) 2  . (113) On setting ε = 0 in Eqs. (112) and (113), the quadrature variance for a vacuum state takes the form (∆c±) 2 v = 2…

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