REVIEW 3 major objections 5 minor 21 references
The small effect of graviton-induced decoherence
T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read Different quantum states of gravitational radiation imprint distinct, tiny decoherence signatures on a mechanical oscillator, captured by state-dependent Lindblad master equations.
desk verdict Solid open-systems derivation of graviton-bath master equations, but the squeezed 'dark sector' headline overstates: the protection is interaction-picture-only and Table I's squeezed row is not the correct Schrödinger-picture equation. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing machinery is the tidal interaction $H_I = -\frac{m}{4}\ddot{h}_{xx}(t,0)x^2$, which couples the mechanical oscillator quadratically to the graviton field and therefore generates two-phonon processes. In mode language the matter part of the interaction is $(b e^{-i\omega_m t}+b^\dagger e^{i\omega_m t})^2$, so every Lindblad jump operator is quadratic in $b$ and $b^\dagger$. The bath enters through its one- and two-point correlation functions; the gravitational spectral density $J^{(g)}(\Omega)\propto\Omega^5$ produces the vacuum rate $\gamma_V$ and, for a squeezed bath, the anomalous correlators combine into a rank-one Kossakowski matrix with the dressed dark states $|D_\pm\rangle$ satisfying $L_S|D_\pm\rangle=0$. The divergent Lamb-shift Hamiltonian is absorbed by a counterterm that fixes the mechanical frequency to its physical value.
What would settle it
Take the exact $\Omega^5$ gravitational spectral density, compute the second-order reduced dynamics without extending the one-sided integral to infinity, and compare the resulting decay of the $\rho_{01}$ coherence with the exponential rates in Table I; if non-Markovian corrections are comparable to $\gamma_V$ for realistic $\omega_m$, the exact Lindblad rates and the exact dressed-dark-sector protection would fail.
Extended reading notes
Core claim
The paper's central claim, stated on its own terms, is that after the Born-Markov and secular approximations and an explicit counterterm renormalization of the ultraviolet-divergent Lamb shift, the oscillator's reduced dynamics is exactly Lindblad for each bath state, with jump operators quadratic in the phonon operators. The universal tidal interaction $H_I = -\frac{m}{4}\ddot{h}_{xx}(t,0)x^2$ is quadratic in the displacement $x$, so one graviton exchanges with two mechanical phonons. In vacuum $L_V=b^2$, making $|0\rangle$ and $|1\rangle$ dark and preserving the coherence of the qubit they span; a coherent bath has the same dissipator plus a coherent-shift Hamiltonian that reproduces the classical geodesic-deviation equation; number and thermal baths add the excitation jump $b^{\dagger 2}$, coupling $\rho_{01}$ to an infinite ladder of coherences; and a squeezed bath yields the single jump operator $L_S=\cosh r(2\omega_m)b^2 - e^{i\phi(2\omega_m)}\sinh r(2\omega_m)b^{\dagger 2}$, whose kernel contains one dressed dark state in each parity sector, so the mutual coherence of $|D_+\rangle$ and $|D_-\rangle$ is exactly preserved by the dissipator. Quantitatively, all rates are multiples of the small vacuum rate $\gamma_V = \frac{32}{15}t_P^2\omega_m^3$.
Load-bearing premise
The load-bearing premise is that the graviton bath correlation functions decay fast enough for the Markov approximation, which extends the one-sided time integral to infinity, to be valid; because the gravitational spectral density behaves as $\Omega^5$, its time-domain correlations are power laws rather than exponentials, so this is not automatic.
Editorial extensions
If this is right
- Vacuum graviton baths protect the coherence of the $\{|0\rangle, |1\rangle\}$ qubit for all times, and the same even-odd parity decomposition of the Hilbert space holds for every bath state considered.
- A coherent graviton bath does not amplify decoherence relative to vacuum; its only extra effect is a coherent drive identical to the tidal action of a classical gravitational wave, so realistic strain values like $h_+=10^{-21}$ map directly to coherent-state amplitudes.
- Number and thermal graviton baths destroy the exact protection of $\rho_{01}$ and couple it to an infinite ladder of higher coherences, but for a CMB-like occupation number the coherence lifetime is of order $10^{60}\,\mathrm{s}$, so the effect is negligible in practice.
- A squeezed graviton bath makes the bare coherence $\rho_{01}$ initially decay, then re-cohere and saturate, while the mutual coherence of the dressed states $|D_+\rangle$ and $|D_-\rangle$ is exactly preserved; exploiting this protection in the laboratory frame requires a phase reference to the squeezing phase.
Reading between the lines
- Because $\gamma_V\propto\omega_m^3$, the paper's predictions imply that pushing the mechanical frequency far above the MHz benchmark is the most direct lever for making graviton-induced decoherence less astronomically small, a consequence the authors leave implicit.
- A natural extension the paper does not perform is to replace the Markov approximation with the exact power-law correlations of the $\Omega^5$ spectral density; time-dependent rates could spoil the exactness of the dressed dark sector.
- Laboratory analogues with engineered squeezed reservoirs, for example squeezed light driving a two-phonon optomechanical jump, could test the qualitative recoherence signature of $\rho_{01}$ without invoking real gravitons.
- The paper's parity-sector decomposition suggests that any parity-preserving bath will decohere even and odd number ladders independently; mixing the sectors would require a non-parity-preserving interaction or initial state.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives reduced dynamics of a harmonic oscillator coupled through the tidal (quadratic-in-displacement) interaction to quantized gravitational waves in multi-mode vacuum, coherent, number, thermal, and squeezed states. It obtains Lindblad master equations with explicit rates and jump operators, renormalizes the ultraviolet-divergent Lamb shift, identifies even/odd parity sectors, and analyzes coherence protection in the vacuum and squeezed cases, including a claimed dressed dark sector for squeezed graviton baths. The supplemental material contains detailed derivations of the matter and gravity Hamiltonians, bath correlation functions, secular approximation, renormalization, and the resulting coherence hierarchies.
Significance. The paper is systematic and largely self-contained: the decoherence rates and jump operators are computed from the microscopic Hamiltonian rather than fitted, and the analysis covers several physically motivated graviton states in one framework. The distinction between vacuum, coherent, number, thermal, and squeezed gravitational environments is conceptually useful, and the quantitative estimates (e.g., decoherence timescales of order 10^60 s for CMB thermal gravitons) make the smallness of the effects concrete. The main novelty, however, is the squeezed-bath dark sector, and that claim is not supported in the form presented: the Schrödinger-picture master equation used for the squeezed case is not the correct transform of the derived interaction-picture equation, so the claimed coherence protection is only a property of the interaction-picture dissipator at fixed time, not of the physical reduced dynamics. The vacuum and number/thermal results appear sound and would survive the correction.
major comments (3)
- [Table I, Eq. (18), Supplemental Eqs. (S202) and (S250)] The Schrödinger-picture squeezed master equation is not the unitary transform of the interaction-picture equation derived in the paper. From \dot\rho_I = \gamma_V D[L_S]\rho_I one obtains \dot\rho_S = -i[H_m,\rho_S]/\hbar + \gamma_V D[L_S(t)]\rho_S with L_S(t)=\cosh r\,e^{-2i\omega_m t}b^2 - e^{i\phi}\sinh r\,e^{2i\omega_m t}b^{\dagger 2}, and D[L_S(t)]\neq D[L_S(0)] because [H_m,L_S] is not proportional to L_S. Consequently Eq. (35) defines instantaneous dark states of the dissipator only; the full reduced dynamics does not preserve the dressed qubit of Eq. (36), and the abstract's claim of a coherence-protected squeezed sector is not established for the physical dynamics. The caveat in Section 5 that the protection is exact for the squeezed dissipator alone acknowledges part of this, but Table I still lists a time-independent L_S together with the free Hamiltonian term, which is not a correct master equation in any picture.
- [Section 5, Eqs. (33)-(34), Fig. 2 (right)] The numerical squeezed-coherence curves are obtained by solving the interaction-picture dissipator, as stated in the text, whereas Eqs. (33)-(34) and the surrounding discussion present R_{01,S}(t) as coherence ratios of the density matrix. Since the correct Schrödinger-picture dissipator contains e^{\pm 4i\omega_m t} cross terms, the plotted recoherence is an interaction-picture effect and should not be presented as a property of the lab-frame coherence without explicitly reporting the time-dependent basis or the transformation. This is a consequence of the previous comment and should be fixed together with it.
- [Section 4, after Eq. (15); Supplemental Eq. (S245)] The Markov approximation is introduced by extending the upper integration limit to infinity with the statement that the bath correlation functions decay sufficiently fast, but for the gravitational spectral density J^{(g)}(\Omega)\propto\Omega^5 the time-domain correlation decays as a power law (\sim \tau^{-6}) rather than exponentially. The paper should state explicitly why the Markov limit is justified here, for example by estimating the correlation time \tau_c\sim\omega_m^{-1} relative to the extremely long relaxation time \gamma_V^{-1}, or by estimating the size of the leading non-Markovian correction. I do not regard the power-law tail as automatically invalidating the rates, but the argument is load-bearing for every entry in Table I and should be made explicit.
minor comments (5)
- [Abstract and Section 1] The phrase 'two-phonon rather single-phonon processes' is missing a 'than' and should read 'two-phonon rather than single-phonon processes'.
- [Section 2, text before Eq. (9)] There is a duplicated definite article: 'the the resulting tidal interaction Hamiltonian reads' should be 'the resulting tidal interaction Hamiltonian reads'.
- [Section 5, paragraph after Eq. (39)] The sentence 'It is important to emphasize that this protection is exact for the squeezed dissipator alone' should appear at the first introduction of the dressed dark sector, not only at the end of the section, and the abstract and Table I should carry the same qualification.
- [Supplemental Eq. (S194)] The expectation values for a single multi-mode squeezed vacuum state contain redundant delta-function factors (e.g., \delta^{(3)}(k'-k_0)\delta^{(3)}(k''-k_0)\delta^{(3)}(k'-k'')), which make the expressions harder to read; a clearer notation or an explanatory sentence would help.
- [Section 4, Eq. (16)] The passage from Eq. (15) to Eq. (16) suppresses the first-order term in Eq. (13) by assuming a vanishing one-point function, but the coherent case is then treated via Footnote 1; it would be clearer to move that discussion into the main derivation so that the status of the one-point function is explicit at the point where it is dropped.
Circularity Check
No significant circularity: all rates, jump operators, and dark states are computed from the microscopic Hamiltonian and stated bath states rather than fitted or imported.
full rationale
Derivation chain: Sec. 2 builds the matter Hamiltonian and quantized TT graviton field; Eq. (9) derives the tidal interaction from minimal coupling, and Eq. (10) gives its interaction-picture form. Eq. (16) is the Born-Markov master equation with the bath correlator D^(g) defined by Eq. (17). Sec. 5 and Supplement D evaluate the correlators for each bath state, giving the rates and jump operators in Table I. In particular, γ_V = (32/15)t_P^2 ω_m^3 follows from the spectral density J^(g)(Ω) ∝ Ω^5 in Eqs. (S244)–(S246); no rate is fitted to the target decoherence predictions. The coherent-shift Hamiltonian H_CS is defined from the one-point function in Eq. (19), and its equality with the classical tidal Hamiltonian in Eq. (26) follows algebraically from G_k^λ = (m/4)x_ZPF^2 G_{k,xx}^λ ω_k^2 together with the definition of h^CL_xx in Eq. (25); this is a consistency check, not an input. The squeezed dark states are obtained as the kernel of L_S in Eq. (35), where L_S is the rank-one diagonalization of the computed Kossakowski matrix (Supplement F.3, Eqs. (S249)–(S250)); the states are explicitly constructed by recurrence rather than assumed. Ref. [10] is cited for the vacuum two-phonon result, but the same result is re-derived in Supplement F.2 (Eqs. (S304)–(S307)), so that self-citation is not load-bearing. The paper itself qualifies the protected sector: 'this protection is exact for the squeezed dissipator alone. The free oscillator Hamiltonian does not leave the dressed dark sector invariant,' which places any concern about the physical claim in the domain of interpretation or correctness rather than circularity. The Markov approximation is explicitly stated as an assumption; even if the Ω^5 spectral density makes its validity questionable, that is a correctness risk, not a circular step. No load-bearing step reduces, by the paper's own equations or by self-citation, to its own inputs.
Assumptions & free parameters
assumptions (7)
- domain assumption Linearized quantum gravity: the gravitational field is a free massless spin-2 field quantized in the TT gauge, with no graviton self-interactions or back-reaction.
- domain assumption The matter system is a single harmonic oscillator representing the relative displacement of two particles, or of an object versus the trap minimum, with the center-of-mass motion decoupled.
- domain assumption Born approximation: the total initial state factorizes and the graviton bath remains approximately stationary.
- domain assumption Markov approximation: bath correlation functions decay fast enough to extend the one-sided integral to infinity, yielding a delta-function resonance.
- standard math Secular approximation: rapidly oscillating matter terms at e^(+-2i omega_m t) and e^(+-4i omega_m t) are discarded.
- ad hoc to paper Renormalization condition: the counterterms H_CT cancel the entire UV-divergent Lamb-shift Hamiltonian, fixing the mechanical frequency omega_m as the physical frequency and removing the induced Kerr term.
- domain assumption For number and squeezed baths, the occupation and squeezing profiles are assumed isotropic and polarization-independent.
Cite this review
Pith. "Pith review of The small effect of graviton-induced decoherence." pith.science (2026). https://pith.science/paper/424XOCTO
@misc{pith2026260807257,
author = {Pith},
title = {Pith review of: The small effect of graviton-induced decoherence},
year = {2026},
howpublished = {\url{https://pith.science/paper/424XOCTO}},
note = {Machine review of arXiv:2608.07257}
}
read the original abstract
We derive and investigate the reduced dynamics of a nonrelativistic quantum matter system interacting with quantized gravitational waves. Because gravitational radiation couples tidally to the mass quadrupole moment, we emphasize that the leading interaction is quadratic in the mechanical displacement. Rather than treating it as a complication to be removed by linearization, we retain this full quadratic dependence, which gives rise to two-phonon rather single-phonon processes typical in dipolar interactions. We clarify its relation to the effective linear coupling that often appears in optomechanical literature, but which conceals the delicate quantum signatures of the full quadratic coupling. Using the Born-Markov and secular approximations, we microscopically derive Lindblad master equations for the matter dynamics induced by graviton baths in multi-mode vacuum, coherent, number, thermal, and squeezed states. We treat the ultraviolet-divergent Lamb-shift Hamiltonian by an explicit renormalization procedure, with the mechanical frequency identified as the physical frequency. Remarkably, owing to the nature of the interaction, the matter Hilbert space decomposes into even- and odd-parity sectors for all bath states considered. For a coherent graviton bath, the decoherence is identical to that of the vacuum, while the bath one-point function produces a coherent-shift Hamiltonian that exactly reproduces the tidal interaction with a classical gravitational wave. For a vacuum graviton bath, we recover the coherence protection between the two lowest mechanical number states, while number and thermal graviton baths remove this protection and enhance decoherence, although the effect remains small for realistic parameters. By contrast, a squeezed graviton bath enlarges the coupled coherence structure, and gives rise to a novel, coherence-protected sector spanned by dressed dark states.
Figures
Reference graph
Works this paper leans on
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[14]
It is, by definition, a mixture of pure states and it can only be defined via the density operator
Single and multi-mode thermal state A thermal state cannot, of course, be defined in terms of a ket state. It is, by definition, a mixture of pure states and it can only be defined via the density operator. Let us directly consider the multi-mode thermal state. The multi-mode thermal state is fundamentally the Gibbs state of the complete free gravitationa...
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The energy-momentum tensor for a point particle We describe the internal mode as an effective point particle of invariant rest massm. Its covariant energy-momentum tensor in the main text was written as T µν(x) =c 2 Z dτ pµ(τ)p ν(τ)p −pλpλ δ(4)(x−z(τ)),(S131) wherez µ(τ) is the worldline,τis proper time,u µ =dz µ/dτ, andp µ =mu µ. We use the normalization...
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Interaction Hamiltonian from minimal coupling: from TT to FNC Let us start with the universal linearized-gravity coupling, SI = 1 2 Z d4x hµνT µν.(S133) For a point particle, T µν(t,x) = pµpν p0 δ(3)(⃗ x−⃗ q(t)),(S134) where we have denotedp µ =mu µ. Evaluating the spatial integral and taking the nonrelativistic limitu µ = (1,˙xi), gives the Lagrangian in...
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Quantization modes With suppressed TT superscript, usingx 0 =ct, the action (S88) becomes S(g) = Z dt L(g), L (g)(t) = Z d3xL (g).L (g) = c2 64πG h ˙hij ˙hij −c 2∂khij∂khij i .(S91) The momentum canonically conjugate toh ij is therefore Πij ≡ ∂L(g) ∂ ˙hij = c2 32πG ˙hij.(S92) In terms of the canonical variables, the action reads S(g) = Z dt d3x h Πij ˙hij...
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Polarization tensor After fixing the residual gauge freedom, we found that (Eq. (S86)), e′ µν =e TT µν .(S115) Suppressing the prime (and the TT superscript),e µν denotes a tensor satisfying ηµνeµν = 0, k µeµν = 0, u µeµν = 0.(S116) For a fixed propagation direction, the space of symmetric tensors satisfying these conditions is two-dimensional. We therefo...
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Interaction Hamiltonian from minimal coupling: from FNC to TT Let us start with the universal linearized-gravity coupling, SI = 1 2 Z d4x hµνT µν.(S138) For a point particle, T µν(t,x) = pµpν p0 δ(3)(⃗ x−⃗ q(t)),(S139) where we have denotedp µ =mu µ. We now take the nonrelativistic limit and go to the FNC coordinates, keeping factors ofv i ≡˙xi andcexplic...
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Interaction Hamiltonian from the general-relativistic point-particle Lagrangian Let us start with the reparametrization-invariant action of a free relativistic point particle S=−mc Z ds, ds 2 =g µν dxµdxν.(S146) 30 We choose our coordinates asx µ = (ct, xi) and parametrize the curve by a monotonic affine parameter, which we take to beλ=t. Switching to pro...
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V acuum state The vacuum state is defined by gk′,λ′ |0⟩= 0.(S153) Using the above, together with its Hermitian conjugate, we quickly note that the one-point correlation functions vanish, ⟨0|g k′,λ′ |0⟩= 0,(S154a) ⟨0|g † k′,λ′ |0⟩= 0.(S154b) Employing, in addition, the commutation relations (S152), for the two-point correlation functions we immediately fin...
Show all 21 references
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Single multi-mode number state For an orthonormal single multi-mode number state|n k0,λ0 ⟩, the annihilation operator acts as gk′,λ′ |nk0,λ0 ⟩= √nk0,λ0 δ(3)(k0 −k ′)δ λ′ λ0 |nk0,λ0 −1⟩.(S160) This, together with its Hermitian conjugate and the commutation relations (S152), giv...
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Multi-mode number state For an orthonormal multi-mode number state|n ⟩ ≡N (k0,λ0) |nk0,λ0 ⟩, the annihilation operator acts as gk′,λ′ |n⟩= √nk′,λ′ | · · ·nk′,λ′ −1· · · ⟩.(S163) This, together with its Hermitian conjugate and the commutation relations (S152), give the two-poin...
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[10]
Single multi-mode coherent state The coherent state can be defined in multiple ways. For a single multi-mode coherent state we write |αk0,λ0 ⟩=D k0,λ0 |0⟩,(S170) where the displacement operator is given by Dk0,λ0 =e αk0 ,λ0 g† k0 ,λ0 −α∗ k0 ,λ0 gk0 ,λ0 .(S171) Note the unitari...
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Multi-mode coherent state For a multi-mode coherent state|α ⟩ ≡N (k0,λ0) |αk0,λ0 ⟩, the annihilation operator acts as gk′,λ′ |α⟩=α k′,λ′ |α⟩.(S181) Using this, its Hermitian conjugate, and the commutation relations (S152), we obtain ⟨α|g † k′,λ′gk′′,λ′′ |α⟩=α ∗ k′,λ′ αk′′,λ′′,...
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Alternatively, it can be derived using the Bogoliubov transformations
Single multi-mode squeezed vacuum state A single multi-mode squeezed vacuum state is defined as |ξk0,λ0 ⟩=S k0,λ0 |0⟩,(S188) where the squeezing operator is given by Sk0,λ0 =e 1 2 ξ∗ k0 ,λ0 g2 k0 ,λ0 − 1 2 ξk0 ,λ0 g†2 k0 ,λ0 .(S189) Note the unitarity of the squeezing operator...
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Multi-mode squeezed vacuum state For a multi-mode squeezed vacuum state|ξ ⟩ ≡N (k0,λ0) |ξk0,λ0 ⟩, one finds the (simpler) similarity transformation (same mode), S† k′,λ′ gk′,λ′ Sk′,λ′ =g k′,λ′ uk′,λ′ +g † k′λ′ vk′,λ′,(S198) whereu k′,λ′ andv k′,λ′ are defined in (S193). Using ...
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Rotating-wave approximation (R W A) for the interaction Hamiltonian For completeness, we note how the interaction Hamiltonian would look if one performed a rotating-wave approximation directly at the Hamiltonian level. This approximation keeps only the near-resonant terms in t...
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Secular approximation for the matter operators The evaluation requires the matter part of interaction-picture interaction Hamiltonian at the two timestandt−τ. These are H (m) II (t) =b 2e−2iωmt +b †2e2iωmt +b †b+bb †,(S224) and H (m) II (t−τ) =b 2e−2iωm(t−τ) +b †2e2iωm(t−τ) +b...
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− 6 5 Λ5 − 8 3 Λ3ω2 m −32Λω 4 m + 32ω5 m log Λ + 2ωm Λ−2ω m # , χΛ 2 =αℏ
Curing the divergences: the Lamb-shift Hamiltonian and renormalization For a stationary two-point bath correlator, we obtained HLS =ℏ X Ω∈{0,±2ωm} S(Ω)A†(Ω)A(Ω) =ℏS(2ω m)b †2b2 +ℏS(−2ω m)b 2b†2 +ℏS(0) (2b †b+ 1) 2 (S251) In the case of vacuum, we have S(ω) = 1 ℏ2 P Z ∞ 0 dΩ J(...
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Coherent bath For the multi-mode coherent graviton bath, we found that the nonvanishing one-point function generates the additional coherent-shift Hamiltonian (S187), while the dissipative contribution remains identical to that of the vacuum. The expectation value of the metri...
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The standard harmonic-oscillator ladder relations are b|q⟩= √q|q−1⟩, b† |q⟩= p q+ 1|q+ 1⟩
Number and thermal baths Coherences.—Let us, for convenience, rewrite the common-structure master equation (ρ≡ρ (m)(t)) ˙ρ=−iω m[b†b, ρ] +γ↓D[b2]ρ+γ ↑D[b†2]ρ(S281) For notational convenience, we usenas a state-dependent shorthand, defined by n= ( n(2ωm),for the number-state ca...
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Reducing to the vacuum result.—Let us turn our attention to the vacuum, in which case the evolution of the coherences should confirm our previous result
Using the master equation (S287), for a genericρ 01, we have ˙ρ01 = 2 √ 3γ ↓ρ23 −4γ ↑ρ01.(S299) The next equation governs the evolution of theρ 23, which is given by ˙ρ23 =γ ↓ 4 √ 15ρ 45 −4ρ 23 +γ ↑ 2 √ 3ρ 01 −16ρ 23 .(S300) The next equation governs the evolution of theρ 45, ...
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Squeezed bath Let us, for convenience, rewrite the master equation (ρ≡ρ (m)(t),r≡r(2ω m), ϕ≡ϕ(2ω m)), ˙ρ=−iω m[b†b, ρ] +γV D[LS ]ρ, L S = coshr b2 −e iϕ sinhr b†,2 (S325) It is useful to go back to the non-diagonal Kossakowski matrix version of the equation, with N≡sinh 2 r, M...
Reviewed August 10, 2026 · model on record in the stance chip above.
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