Leaky cavities with unwanted noise

A.A. Semenov; D.-G. Welsch; D.Yu. Vasylyev; M. Khanbekyan; W. Vogel

REVIEW

Leaky cavities with unwanted noise

A.A. Semenov, D.Yu. Vasylyev, W. Vogel, M. Khanbekyan, D.-G. Welsch

Quantum Physics · Submitted 6 Mar 2006 · 39 references

Pith's verdict: The noise-induced mode-matching effect is real within the model, but the paper's 'complete description' claim rests on an unshown rank check and an assumed Markovian universality—still worth a serious referee.

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arXiv IDquant-ph/0603043 (version 4)

Pith IDMKE44SOV, Pith's permanent record of this paper and its review

CitationA.A. Semenov, D.-G. Welsch, D.Yu. Vasylyev, M. Khanbekyan, W. Vogel. "Leaky cavities with unwanted noise." arXiv:quant-ph/0603043 (2006). https://pith.science/paper/MKE44SOV

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Claim paper

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Editor's note

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Dear X,

Quick take: the paper's central physical claim—that unwanted noise lets you combine a cavity input mode with the intracavity mode in the cavity-associated output mode—holds up within the model, and the proposed homodyne schemes are a genuine extension. The weak point is not the algebra but the completeness claim: the nondegeneracy check is a one-line Mathematica assertion, and the universality of the beam-splitter replacement scheme is assumed rather than proven. That doesn't sink the paper, but it does mean 'complete' is doing more work than the evidence supports.

What's actually new: (i) the demonstration that a degenerate scheme (Fig. 3) forces the reflection efficiency η_ref to zero via constraint (37), while the full scheme allows η_ref > 0—a concrete, non-obvious consequence of adding the third feedback beam splitter; (ii) the identification of the matched input mode and the additional output mode, and the use of this structure in unbalanced and cascaded homodyning. The derivation of constraints (22)–(24) from commutation relations is clean and correct, and the parametrization satisfies them by construction. The logic in Sec. II checks out.

Soft spots, in order of importance. First, the completeness proof in Sec. III is underreported. The text says the Jacobian rank was checked in Mathematica, but no matrix, sample point, or rank value appears. Rank at a point is also only a local statement; it does not prove that every point of the constraint manifold is reachable. The stress-test note is right on this: it is the load-bearing step for the universality claim. Second, the agreement with the QFT approach in Ref. [23] is asserted, not demonstrated here. Third, the unbalanced homodyning series is used at ξ = ε ≈ 1.07, outside the naively convergent regime; the paper acknowledges the sampling cost and shows a simulation, but no convergence proof. That is minor in context because the cascaded scheme eliminates the AOM contribution entirely.

Is the central idea correct? As far as I can tell, yes within the stated assumptions. The mode-coupling effect is a genuine consequence of the model, not a fitted artifact. The Markovian, vacuum-noise-port assumption is a real physical limitation, but it is the standard phenomenological framework for high-Q cavities, and the paper is explicit about it.

Bottom line: this is a solid theory paper with a new effect and useful measurement proposals. It deserves a serious referee; the main revisions I would ask for are to show the rank check and to qualify the completeness claim. I would cite it if I worked on cavity-state reconstruction.

Referee report

Significance

If the completeness claim holds, the paper closes an important gap in the standard input-output treatment of realistic cavities and identifies a qualitatively new effect: the possibility of combining an input pulse with the intracavity field in a nonmonochromatic output mode. The derivation of the coefficient constraints from commutation relations in Sec. II B is clean and verifiable, and the parametrization satisfies the constraints by construction. The central prediction, \eta_ref in Eq. (48), is explicit and falsifiable, and the comparison between the nondegenerate Fig. 2 and degenerate Fig. 3 schemes is physically instructive. The cascaded-homodyning application is a useful extension. However, the paper's central claim of completeness rests on an unreported MATHEMATICA rank check, and one statement in Sec. V A concerning the vanishing of \eta_ref is not correct as stated. The significance is therefore conditional on the missing proof and on a correction of the reconstruction claim.

Major objections (3)

[Sec. III, after Eq. (32)] The claim that the Fig. 2 parametrization is nondegenerate rests entirely on an unreported MATHEMATICA rank check. The matrix of first derivatives, the sample point at which the rank was evaluated, the value of the rank, and the dimensionality of the constraint manifold are not given. Moreover, equality of the Jacobian rank with the manifold dimension at a single point is only a local condition: it shows that the image of the parametrization is open in the constraint manifold, not that every point of the manifold defined by Eqs. (30)-(32) is reached. Since the nonzero value of \eta_ref in Eq. (48) and the entire reconstruction procedure of Sec. V depend on the nondegeneracy of the Fig. 2 scheme, this missing proof is load-bearing. The authors should supply an explicit calculation, a direct parametrization of the manifold, or a rigorous global argument, and otherwise temper the completeness claim.
[Secs. I-II, Eq. (13) and following] The paper repeatedly asserts full agreement with the QFT approach of Ref. [23], but no mapping between the beam-splitter parameters and the QFT noise coefficients is shown. The model is also restricted to Markovian noise with constant beam-splitter coefficients and vacuum noise ports; frequency-dependent or non-Markovian loss mechanisms are not representable by the constant-coefficient kernel of Eq. (42). If such mechanisms are relevant, Eq. (48) is not a universal prediction. The authors should either provide the promised comparison with Ref. [23] or explicitly restrict the completeness and agreement claims to the class of Markovian replacement schemes defined in the paper.
[Sec. V A, after Eq. (67)] The statement that \eta_ref=0 occurs only for ideal cavities or for the degenerate scheme of Fig. 3, and that a properly described noisy cavity therefore always permits full reconstruction, is not correct. The condition \eta_ref=0 is the single complex equation T(o)T(c)/\Gamma+R(o)=0. On the constraint manifold (30)-(32) this equation has solutions that satisfy all constraints; for example, the real parameter set T(c)=1, \Gamma=2, A(c)=1, R(o)=1/2, A(o)=\sqrt{3}/2, T(o)=-1, cos\zeta=1/\sqrt{3} satisfies (30)-(32) and gives \eta_ref=0. For such a point the matched input mode of Eq. (47) is not defined, and the reconstruction formula (73) collapses to \alpha=0. The authors should either prove that such points lie on the degenerate submanifold or restrict the reconstruction claim to the open set \eta_ref\neq 0.

Minor objections (5)

[Sec. III, Eqs. (33)-(34)] The sums over k in the expansions of the noise operators are not defined for general k; the authors should state that the basis is finite-dimensional and specify its dimension explicitly.
[Sec. IV, Eq. (54)] The sentence saying that the efficiency \eta_ref is 'found via the normalization of the function U_out0(t1)' is presumably a typo for U_out1(t1); as written it is confusing.
[Sec. V A, below Eq. (68)] The statement that the parameters \xi and \epsilon cannot both be less than unity 'in realistic situations' is not demonstrated; a short argument using \eta=\eta_{\rm ext}\eta_c\le \eta_{\rm ext} would make the claim transparent.
[Sec. II B, after Eq. (24)] The assertion that substituting Eqs. (5) and (7)-(13) into Eqs. (22)-(24) 'yields identities' is made without showing the algebra; a brief appendix or an explicit identity would improve verifiability.
[Sec. I, abstract and introduction] The phrase 'completely describe' is stronger than what is established, given the Markovian and constant-coefficient assumptions; the abstract should state the domain of validity of the model.

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The paper

The paper claims

Unwanted noise in a high-Q cavity can be modeled by a three-beam-splitter replacement, and it makes possible a mode-matching effect—combining the input and intracavity fields in one output mode—that ideal cavities cannot produce.

The reading

This paper argues that absorption and scattering inside a high-Q cavity and its coupling mirror can be put into the standard quantum Langevin and input-output description by replacing the real mirror with an ideal mirror plus a small network of beam splitters. The three-beam-splitter version is claimed to be complete: it preserves the field commutation relations and covers every allowed set of coefficients, whereas a simpler two-beam-splitter version misses a whole class of cavities. The central consequence is a new physical effect: in a properly described lossy cavity, an incoming matched pulse is partially reflected into the same nonmonochromatic output mode that carries the escaping intracavity field, with a calculable efficiency. That mixing cannot happen in an ideal cavity and vanishes in the incomplete model. The effect gives a practical way to reconstruct the intracavity quantum state by unbalanced or cascaded homodyne detection without an external beam splitter to combine signal and local oscillator.

What carries the argument

The load-bearing construction is the replacement scheme in Fig. 2: the lossy coupling mirror is modelled by an ideal mirror together with two symmetric $SU(2)$ beam splitters that inject vacuum noise into the input and output channels and one asymmetric $U(2)$ beam splitter that supplies an extra phase and a feedback loop. The scheme produces the extended quantum Langevin equation (3) and input-output relation (4) with all coefficients expressed through the beam-splitter parameters (5)–(13). Imposing $[\hat{a}_{\mathrm{cav}},\hat{a}_{\mathrm{cav}}^\dagger]=1$ and $[\hat{b}_{\mathrm{out}}(t_1),\hat{b}_{\mathrm{out}}^\dagger(t_2)]=\delta(t_1-t_2)$ yields the constraints (22)–(24), which define the manifold of physically allowed cavities. The paper claims, via a differential-geometry rank test, that the three-beam-splitter parametrization covers this manifold completely, while the two-beam-splitter scheme of Fig. 3 imposes an extra constraint and is degenerate. Completeness is what distinguishes the mode-matching effect as real physics rather than a modelling artifact.

Extended reading notes

Core claim

For a one-sided cavity with unwanted noise, the output field can be decomposed into a cavity-associated output mode (CAOM), an additional output mode (AOM), and a matched input mode (MIM). The paper's central discovery is that the noise channels make the outgoing CAOM operator contain both the initial intracavity mode and the matched input mode: $\hat{b}_{\mathrm{out};0} = \sqrt{\eta_{\mathrm{ext}}}\hat{a}_{\mathrm{cav}}(0) + \sqrt{\eta_{\mathrm{ref}}}\hat{b}_{\mathrm{in};0} + \hat{C}_0$, with $\eta_{\mathrm{ref}} = |T^{(o)}T^{(c)}/\Gamma + R^{(o)}|^2$. In an ideal cavity, or in a degenerate replacement scheme with only two beam splitters, this reflection efficiency is exactly zero, so the input and intracavity fields cannot be combined in the output mode. In a complete lossy-cavity description the efficiency is positive, which means the very losses that normally degrade quantum light enable a mode-matching operation that ideal cavities cannot perform. The same decomposition also produces an AOM that contaminates unbalanced homodyning, but cascaded homodyning removes that contamination.

Load-bearing premise

The whole description rests on the assumption that every kind of unwanted loss in a high-Q cavity can be captured by a fixed, memory-free network of beam splitters whose noise inputs sit in vacuum states; the paper's completeness proof for that network is a numerical rank check whose matrix and output are not shown, so the scheme's universality is asserted rather than demonstrated.

Editorial extensions

If this is right

In a cavity whose unwanted-noise channels are described completely, the matched input mode is reflected into the cavity-associated output mode with efficiency $\eta_{\mathrm{ref}}=|T^{(o)}T^{(c)}/\Gamma+R^{(o)}|^2$, so the noise strength can be used to tune how strongly an external pulse is superimposed on the escaping intracavity field.
Unbalanced homodyning of the outgoing field reconstructs the $s$-parametrized phase-space distribution of the intracavity mode point by point; the additional output mode enters only through the correction factor $\epsilon$.
Cascaded homodyning eliminates the additional output mode entirely, so the reconstruction quality is the same as in free-field cascaded homodyning at the same overall detection efficiency.
Neither of these reconstruction schemes works for an ideal cavity or for a degenerate two-beam-splitter loss model, because in both cases $\eta_{\mathrm{ref}}=0$ and no input signal can be mixed into the output mode.

Reading between the lines

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

If the completeness claim holds, any microscopic model of a lossy cavity must reproduce the same mode-coupling structure, so the matched-input-mode reflection efficiency could serve as a benchmark for comparing microscopic theories of mirror loss.
The same replacement-scheme logic could be carried over to other open systems with internal loss, such as waveguide cavities or photonic-crystal resonators, where the matched mode would then be used for measurement-based preparation rather than only reconstruction.
A natural experiment would vary the internal absorption in a controlled way and check whether $\eta_{\mathrm{ref}}$ tracks Eq. (48) independent of the loss mechanism; a mechanism-dependent value would falsify the model's universality.
The scheme's ability to combine input and intracavity fields in one output mode also suggests coherent-state transfer and recycling applications, but the paper stops at homodyne reconstruction.

Original abstract

A phenomenological approach is developed that allows one to completely describe the effects of unwanted noise, such as the noise associated with absorption and scattering, in high-Q cavities. This noise is modeled by a block of beam splitters and an additional input-output port. The replacement schemes enable us to formulate appropriate quantum Langevin equations and input-output relations. It is demonstrated that unwanted noise renders it possible to combine a cavity input mode and the intracavity mode in a nonmonochromatic output mode. Possible applications to unbalanced and cascaded homodyning of the intracavity mode are discussed and the advantages of the latter method are shown.

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The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

A phenomenological approach is developed that allows one to completely describe the effects of unwanted noise, such as the noise associated with absorption and scattering, in high-Q cavities. This noise is modeled by a block of beam splitters and an additional input-output port. The replacement schemes enable us to formulate appropriate quantum Langevin equations and input-output relations. It is demonstrated that unwanted noise renders it possible to combine a cavity input mode and the intracavity mode in a nonmonochromatic output mode. Possible applications to unbalanced and cascaded homodyning of the intracavity mode are discussed and the advantages of the latter method are shown.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A phenomenological approach is developed that allows one to completely describe the effects of unwanted noise, such as the noise associated with absorption and scattering, in high-Q cavities.	contribution 0.95	0.89	0.17	asserted 0.98
s1	This noise is modeled by a block of beam splitters and an additional input-output port.	method 0.99	0.72	0.04	asserted 1.00
s2	The replacement schemes enable us to formulate appropriate quantum Langevin equations and input-output relations.	method 0.86	0.81	0.09	asserted 0.98
s3	It is demonstrated that unwanted noise renders it possible to combine a cavity input mode and the intracavity mode in a nonmonochromatic output mode.	result 0.99	0.84	0.49	asserted 0.99
s4	Possible applications to unbalanced and cascaded homodyning of the intracavity mode are discussed and the advantages of the latter method are shown.	contribution 0.43	0.73	0.16	asserted 0.86

What kind of paper the classifier reports this is: method or tool 0.61 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: One-sided cavity with unwanted internal losses.

**Figure 2.** FIG. 2: Replacement scheme for modeling the unwanted

**Figure 3.** FIG. 3: An example of a degenerate replacement scheme.

**Figure 4.** FIG. 4: The mode structure of the external field: cavity

**Figure 6.** FIG. 6: Reconstruction of the Husimi-Kano

**Figure 7.** FIG. 7: Cascaded homodyne detection of the intracavity

Showing 6 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:quant-ph/0603043 by the authors.

References

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View on Pith 2000
[1]

Substitute ˆgout(t) from the input-output relation ˆgout(t) = T (2) ˆdout(t) + R(2)ˆc(2) in (t) (B2) for the second beam splitter into the input-output relation ˆgin(t) = −R(3)∗ˆgout(t) + T (3)∗ˆbin(t) (B3) for the third beam splitter

View on Pith
[2]

Substitute in the resulting equation ˆdout(t) from the input-output relations for the cavity, Eq. (2)

View on Pith
[3]

Substitute in the resulting equation ˆdin(t) from Eq. (B1). This leads to ˆgin(t) = −R(3)∗T (2)√γ ˆacav(t) + T (3)∗ˆbin(t) + T (2)R(1)R(3)∗ˆc(1) in (t) − R(2)R(3)∗ˆc(2) in (t) + R(3)∗T (1)T (2)ˆgin(t), (B4) which contains the operator ˆ gin(t) in both sides of the equation

View on Pith
[4]

Resolve Eq. (B4) to ﬁnd ˆ gin(t), ˆgin(t) = − √γ R(3)∗T (2) 1 − R(3)∗T (1)T (2) ˆacav(t) + T (3)∗ 1 − R(3)∗T (1)T (2) ˆbin(t) + T (2)R(1)R(3)∗ 1 − R(3)∗T (1)T (2) ˆc(1) in (t) − R(2)R(3)∗ 1 − R(3)∗T (1)T (2) ˆc(2) in (t). (B5) Inserting Eq. (B5) into Eq. (B1) and then substituting the result into Eq. (1), we obtain the quantum Langevin equation (3). Next,...

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [23]

S. M. Dutra, G. Nienhuis, Phys. Rev. A 62, 063805 (2000)

View on Pith 2000

[2] [1]

Substitute ˆgout(t) from the input-output relation ˆgout(t) = T (2) ˆdout(t) + R(2)ˆc(2) in (t) (B2) for the second beam splitter into the input-output relation ˆgin(t) = −R(3)∗ˆgout(t) + T (3)∗ˆbin(t) (B3) for the third beam splitter

View on Pith

[3] [2]

Substitute in the resulting equation ˆdout(t) from the input-output relations for the cavity, Eq. (2)

View on Pith

[4] [3]

Substitute in the resulting equation ˆdin(t) from Eq. (B1). This leads to ˆgin(t) = −R(3)∗T (2)√γ ˆacav(t) + T (3)∗ˆbin(t) + T (2)R(1)R(3)∗ˆc(1) in (t) − R(2)R(3)∗ˆc(2) in (t) + R(3)∗T (1)T (2)ˆgin(t), (B4) which contains the operator ˆ gin(t) in both sides of the equation

View on Pith

[5] [4]

Resolve Eq. (B4) to ﬁnd ˆ gin(t), ˆgin(t) = − √γ R(3)∗T (2) 1 − R(3)∗T (1)T (2) ˆacav(t) + T (3)∗ 1 − R(3)∗T (1)T (2) ˆbin(t) + T (2)R(1)R(3)∗ 1 − R(3)∗T (1)T (2) ˆc(1) in (t) − R(2)R(3)∗ 1 − R(3)∗T (1)T (2) ˆc(2) in (t). (B5) Inserting Eq. (B5) into Eq. (B1) and then substituting the result into Eq. (1), we obtain the quantum Langevin equation (3). Next,...

View on Pith

[6] [5]

C.K. Law and J.H. Eberly, Phys. Rev. Lett. 76, 1055 (1995)

View on Pith 1995

[7] [6]

Cavity Quantum Electrodynamics, Advances in Atomic, Molecular and Optical Physics, Suplement 2, edited by P. Berman (Academic, New York, 1994)

View on Pith 1994

[8] [7]

H.J. Kimble, Phil. Trans. R. Soc. London, Ser. A 355, 2327 (1997)

View on Pith 1997

[9] [8]

C. Di Fidio, S. Maniscalco, W. Vogel, and A. Messina, Phys. Rev. A 65, 033825 (2002)

View on Pith 2002

[10] [9]

C. Di Fidio and W. Vogel, J. Opt. B: Quantum Semiclas- sical Opt. 5, 105 (2003)

View on Pith 2003

[11] [10]

L.A. Wu, M. Xiao, and H.J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987)

View on Pith 1987

[12] [11]

W. Lange and H. J. Kimble, Phys. Rev. A 61, 063817 (2000)

View on Pith 2000

[13] [12]

A.S. Parkins, P. Marte, P. Zoller, O. Carnal, and H.J. Kimble, Phys. Rev. A 51, 1578 (1995)

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