Semiclassical Backreaction: A Qualitative Assessment

Fabio van Dissel; George Zahariade

REVIEW 3 major objections 5 minor 76 references

Semiclassical Backreaction: A Qualitative Assessment

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In a two-oscillator toy model with biquadratic coupling, both the mean-field and truncated Wigner approximations extend the time for which a classical background description matches full quantum evolution, and which method works better is…

desk verdict A clean, honest benchmark of mean-field versus truncated Wigner backreaction in a two-oscillator toy model, with the caveat that the unstable-regime comparison may be measuring dephasing of a delocalized wavefunction rather than true semiclassical validity. read the letter →

arxiv 2411.19825 v2 pith:YW5SHMHT submitted 2024-11-29 hep-th gr-qcquant-ph

Fabio van Dissel , George Zahariade This is my paper · ORCID

classification hep-thgr-qcquant-ph

keywords semiclassicalbackreactionquantumbreaktimemean-fieldapproximationtruncatedWignerparametricresonanceentanglemententropycoupledharmonicoscillatorsMathieuinstabilitybands

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 asks when two standard semiclassical recipes for quantum backreaction can be trusted: replacing quantum variables in the classical equations by their expectation values (the mean-field method) versus evolving classical equations from many random initial conditions and averaging (the truncated Wigner method). The authors compare both against an exact numerical quantum evolution of the background variable in a toy system of two biquadratically coupled harmonic oscillators. They find that both methods generically track the quantum result longer than the classical solution alone, so both capture at least some backreaction. The better method depends on classical stability: far from the Mathieu instability bands the mean-field method wins, often by orders of magnitude, while inside the bands both methods fail sooner and truncated Wigner does only slightly better while capturing dissipative amplitude decay. The authors interpret the failures through growing entanglement between the two oscillators, which destroys the classicality of the background.

What carries the argument

The load-bearing object is the classical stability analysis of the background: linearizing the $y$ oscillator around $x = x_0 \cos t$ gives the Mathieu equation $\delta\ddot{y} + [(\Omega/\omega)^2 + \lambda x_0^2 \cos^2 t]\,\delta y = 0$, whose instability bands are centered at $\lambda x_0^2 = 2(n^2 - (\Omega/\omega)^2)$. The paper uses these bands to organize the entire parameter scan: quantum and semiclassical break times drop near the bands because the interaction rapidly entangles $x$ and $y$. The two semiclassical schemes are the mean-field equations, in which $x$ couples to $|z|^2 = \langle \hat{y}^2 \rangle$, and the truncated Wigner ensemble, in which $y$ and $p_y$ are sampled from the initial Wigner distribution and each pair is evolved classically before averaging over realizations. The comparison is made quantitative by the integrated relative error $\delta(t)$ and by the threshold $0.05$ that defines the break times.

What would settle it

Compute the reduced density matrix of $x$ in an unstable parameter region such as $\lambda = 0.1$, $x_0 = 5$, $\Omega/\omega = 0.4$, and test whether the $x$ probability density becomes bimodal or its purity collapses before the truncated Wigner break time; if the background becomes multi-modal before the claimed break time, then the benchmark $\langle \hat{x}(t) \rangle$ itself is not a classical trajectory and the break-time comparison would need to be re-evaluated.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is an assessment rather than a theorem: in the parameter range studied, the semiclassical break times $t_{\mathrm{MF}}$ and $t_{\mathrm{TW}}$ both exceed the quantum break time $t_q$, meaning each method extends the time over which $x(t)$ is a good classical stand-in by a measurable amount. Generically $t_{\mathrm{MF}} > t_{\mathrm{TW}}$ for parameters lying away from the classical instability bands of the Mathieu equation, sometimes by orders of magnitude, while inside the bands $t_{\mathrm{TW}} > t_{\mathrm{MF}}$ but only marginally. For classically stable parameters and weak collective coupling, the mean-field method reproduces the full quantum evolution essentially exactly at the perturbative level, including the frequency shift $\lambda \omega/4\Omega$, whereas truncated Wigner eventually dissipates the amplitude because its random phases decorrelate. The paper reads these patterns through the lens of entanglement: instability bands generate strong entanglement, and strong entanglement shortens every break time and reverses the ranking of the two methods.

Load-bearing premise

The comparison assumes that the expectation value $\langle \hat{x}(t) \rangle$ from the full quantum simulation is a faithful description of the classical background, which holds only while the $x$ wavefunction stays localized and approximately monomodal; the authors check this for some parameter choices but do not establish it rigorously, especially in the strongly entangled unstable regimes where the qualitative conclusions matter most.

Editorial extensions

If this is right

In the two-oscillator model, both semiclassical methods give $t_{\mathrm{MF}}, t_{\mathrm{TW}} > t_q$ generically, so even approximate backreaction is better than no backreaction for describing the background.
Far from instability bands, the mean-field method is generally more accurate than truncated Wigner, sometimes by orders of magnitude, and for weak collective coupling it reproduces the full quantum evolution including the frequency shift.
Inside instability bands, both methods fail sooner, but truncated Wigner performs marginally better and is the only one of the two that captures the dissipative decay of the background amplitude.
Break-time scaling with occupation number $N$ changes from linear to logarithmic when the system is classically unstable, showing that classical instability spoils the standard quantum break-time heuristic.
In the cosmological setting that motivates the model, the results indicate that truncated Wigner is the appropriate semiclassical method for parametric-resonance-driven preheating, while a mean-field approach should be more suitable for the stable analogue, reheating.

Reading between the lines

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

A practical decision rule is implicit in the results: scan the classical stability bands of the background first, then choose the mean-field method far from resonance and truncated Wigner near resonance; this could save considerable computational cost in applications.
The failure mechanism suggests that entanglement entropy, not just the break-time error, could serve as a predictive diagnostic: break times appear to shrink as entanglement grows, and tracking both quantities together might give an early warning of semiclassical breakdown.
The three-mode extension in the paper hints that in field theory a single unstable resonant mode may be enough to shorten global semiclassical break times, but whether many stable modes can wash out that effect remains an open question that could be tested by adding more modes numerically.
The authors' 'amusing side project' of inverting the logic---designing interactions that classicalize a quantum state---is a concrete testable extension: one could search for Hamiltonians under which a highly excited number state evolves into a coherent state, which would clarify the boundaries of the backreaction picture.

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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,

The paper does something genuinely useful: it takes two backreaction schemes that are routinely used in preheating and semiclassical gravity—mean-field and truncated Wigner—and checks them against an exact solution of the two-oscillator Schrödinger equation. The central qualitative map, that mean-field wins far from classical instability bands and truncated Wigner wins marginally inside them, is new and clearly presented. The analytic weak-coupling check in Appendix A is a real strength: mean-field exactly reproduces the frequency shift, and the argument for why TW fails in that regime is convincing. The numerics are well documented, with convergence tests and a stated threshold for break times. I also appreciate the candor in Section V, where the authors explicitly flag that their benchmark assumes ⟨x̂⟩ remains a faithful classical variable.

That flag is also the paper's main soft spot, and it lands exactly where the conclusions are sharpest. In the unstable cases, the entanglement entropies in Table I are large (S_e ≈ 1.2–1.9 at t=25). When the x marginal is that mixed, the L1 distance from ⟨x̂⟩ may reward whichever method best tracks the centroid of a delocalized, possibly multimodal wavefunction. The paper checks Gaussianity only for some parameter choices and leaves a rigorous study for later. So the claim that TW captures dissipative amplitude decay inside instability bands is not yet established as a statement about classical background accuracy; it might be measuring dephasing. This does not destroy the paper, but it does mean the title's promise—regime of validity—is only fully delivered in the stable regime, where the benchmark is secure.

Minor points: the heuristic formulas (21)-(22) are schematic and not used to generate figures, so they add little. The replication data is referenced but not linked in the preprint, which is annoying. Self-citations are not a problem here; the cited works are relevant and the comparison is not fitted.

Who should read this: anyone using mean-field or TW for backreaction in cosmology or field theory, and people working on quantum break times. It deserves a serious referee—the question matters, the toy model is well chosen, and the stable-regime result is solid. My recommendation: send it to peer review, but require the authors to either strengthen the monomodality check inside the instability bands or soften the interpretation there.

Best, [Your name]

Referee Report

3 major / 5 minor

Summary. The paper examines two semiclassical backreaction schemes, the mean-field (MF) approximation and the stochastic truncated Wigner (TW) method, in a toy model of two bi-quadratically coupled harmonic oscillators. The authors compare the background expectation value ⟨x̂(t)⟩ obtained from a fully quantum Schrödinger evolution with the predictions of the two semiclassical methods, defining semiclassical break times via an L1 error functional. They map break times over parameter space and relate their qualitative findings to the stability structure of the Mathieu equation and to the entanglement entropy between the two oscillators. The central claims are that both methods generically extend the validity of a classical description beyond the purely classical solution, that MF is more accurate far from classical instability bands, and that TW performs somewhat better inside the instability bands while capturing dissipative amplitude decay.

Significance. If the central claims hold, the paper would provide a valuable quantitative benchmark for semiclassical backreaction methods used widely in cosmology and quantum field theory, where fully quantum computations are usually unavailable. The paper has clear strengths: Appendix A gives a clean analytic demonstration that MF reproduces the weak-coupling frequency shift while TW necessarily dephases; Appendix C documents convergence tests and energy conservation for the Schrödinger solver; and the break-time definition is explicit and used consistently across parameter scans. The work also advances a falsifiable heuristic linking method performance to entanglement and classical instability. However, the significance of the unstable-regime conclusions is conditional on the validity of ⟨x̂(t)⟩ as a proxy for a classical background, and the authors explicitly acknowledge in Sec. V that this validity is questionable exactly in the regimes where their sharpest conclusions are drawn.

major comments (3)

[Sec. V] The central comparison in Sec. IV uses Eq. (16), which measures the L1 distance from the full quantum expectation value ⟨x̂(t)⟩. The manuscript itself states in Sec. V that this quantity is a valid benchmark only while ψ(x,y) is well localized and approximately monomodal in x, and that only a partial Gaussianity check was performed. In the unstable-regime cases of Table I (e.g., λ=0.1, x0=5, Ω/ω=0.4), the entanglement entropy at t=25 is S_e≈1.86 while the reported MF break time is t_MF=9 and the TW break time is t_TW=17; such a large entropy indicates a highly mixed x marginal, so ⟨x̂⟩ may cease to represent any single classical background before t_TW. Consequently, the claim that TW outperforms MF inside instability bands is not yet established as a statement about the validity of a classical description of x; the L1 metric may instead reward whichever method best tracks the centroid of a delocalized wavefunction. I recommend computing a quantitative localization diagnostic for the x marginal (e.g., the variance or inverse participation ratio of the position distribution, or a time-dependent Gaussianity measure) and reporting whether the benchmark remains classically meaningful up to the claimed break times.
[Sec. IV, Fig. 1] The interpretation that TW captures dissipative amplitude decay in unstable cases rests on the same benchmark. The decay of ⟨x̂(t)⟩ in the right panels of Fig. 1 can originate either from genuine dissipation of a localized background or from phase randomization across a broad or multimodal x distribution. Without monitoring the x probability density, the apparent agreement of x_TW(t) with ⟨x̂(t)⟩ does not demonstrate that TW gives the correct classical background trajectory. Please add explicit evidence on localization of the x marginal (for example, snapshots of the x probability density at representative times or a time-dependent Gaussianity measure) before making the dissipation claim.
[Table I] The entanglement entropy S_e in Table I is evaluated at the fixed fiducial time t=25 for all parameter sets, while the reported break times for the unstable cases are much shorter (e.g., t_MF=9 and t_TW=17 for the first row). This makes the stated causal ordering 'unstable modes result in larger entanglement, which in turn reduces the break times' difficult to assess, because S_e at t=25 is evaluated after one or both semiclassical methods have already broken down. Reporting S_e at the respective break times, or plotting S_e(t) for each row, would provide direct support for the proposed link between entanglement growth and break-time reduction.

minor comments (5)

[Sec. IV / Fig. 2-4 captions] The text states that the instability bands shown in the figures are the shifted bands with n^2 → n^2(1+λω/4Ω)^2, but the captions refer only to Eq. (18). Please state the shift explicitly in the captions to avoid ambiguity.
[Sec. III.A.2] The TW variant fixes the x initial conditions while sampling y and p_y; this is a nontrivial modification of the standard TW prescription. The text says it 'corresponds to the assumption of classicality for the x degree of freedom,' but a brief discussion of the range of validity or prior use of this variant would help.
[Eqs. (21)-(22)] The heuristic formulas for t_MF and t_TW involve α, β, 'Entanglement', and 'Diffusion', none of which are defined quantitatively. Since the text itself describes them as schematic, a sentence indicating how (if at all) these quantities could be extracted from the numerics would improve reproducibility.
[Sec. IV, threshold choice] The paper asserts that qualitative conclusions are insensitive to the 0.05 threshold, but no threshold-scan result is shown. A supplementary plot varying the threshold for a prototypical parameter point would make this assertion verifiable.
[Appendix A.3] Typographical issues: the heading 'Trunctated Wigner' should be 'Truncated Wigner'; Sec. I contains 'ocurrences' and 'an a posteriori'; footnote 7 contains 'bonna fide' instead of 'bona fide'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the semiclassical methods are benchmarked against an independent Schrödinger-equation solution.

full rationale

The central claim, that mean-field and truncated-Wigner methods extend the useful classical description of x beyond the quantum break time, is benchmarked against a fully quantum mechanical solution of the Schrödinger equation. The break-time definition in Eq. (16) compares each semiclassical x(t) and the classical solution x_cl(t) to the independently computed expectation value ⟨x̂(t)⟩; no parameter of either semiclassical method is fitted to that benchmark, and neither method's equations are derived from the quantity being predicted. The mean-field equations (10)-(11) and the truncated-Wigner sampling of Eq. (12) followed by classical evolution of Eqs. (7)-(8) are standard implementations, and no uniqueness theorem or ansatz is imported from the authors' prior work to force the comparison. The self-citations to Refs. [43,44] are background formalism citations for the mean-field method, not load-bearing evidence for the numerical results, and the perturbative analysis in Appendix A follows Ref. [66] as an independent consistency check. The heuristic formulas (21)-(22) are explicitly schematic and are not used to generate any figure. The paper's own stated limitation, that ⟨x̂(t)⟩ is only a faithful classical-background proxy while the x wavefunction remains localized and approximately monomodal, is a validity caveat rather than a circularity: it concerns whether the chosen benchmark is physically interpretable in the unstable regime, which is a correctness risk, not an identification of the prediction with the input. Because the comparisons are self-contained against an external numerical benchmark and no central step reduces to a fit or to a self-citation chain, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the representativeness of the two-oscillator model, the validity of the initial state, and the use of the quantum expectation value of x as a classical benchmark; these are domain assumptions the authors partly flag. No new physical entities are introduced, and no physical parameters are fitted to data. Hand-chosen numerical conventions are listed separately.

free parameters (2)

break-time threshold = 0.05
Chosen by hand in the definition of break times via Eq. (16). The authors state qualitative conclusions are insensitive to this value.
entanglement entropy truncation m_max = 28
Maximum occupation number for the y oscillator when computing S_e in Table I. The state is checked to be contained to a few parts in 10^2 in this truncated space.

assumptions (6)

domain assumption The two-oscillator model is a faithful minisuperspace reduction of the two-field Lagrangian (1).
Sec. II-III map the field theory to a homogeneous mode plus one Fourier mode; Sec. V admits extension to many modes is unclear.
domain assumption The initial wavefunction (6), a displaced coherent state for x and zeroth-order adiabatic vacuum for y, is the correct semiclassical state to order lambda.
Sec. III.A.1 states the expression is approximate to first order in lambda; higher-order corrections may introduce non-Gaussianity and entanglement.
domain assumption The quantum expectation value of x is a valid classical variable for assessing method accuracy.
Section V explicitly lists this as a main limitation; it holds only while the x wavefunction is localized and monomodal.
domain assumption Classical parametric instability of the Mathieu equation (17) controls the growth of quantum entanglement and hence break times.
Sec. IV and Discussion use the instability bands as the organizing feature and rely on Refs. [50-52] for the entanglement-Lyapunov connection.
standard math Standard properties of Glauber coherent states and Wick factorization for the initial state.
Sec. II uses coherent states and factorization of correlators to define the classical limit.
standard math The instability bands of the Mathieu equation are given by lambda x0^2 = 2(n^2 - (Omega/omega)^2).
Sec. IV uses the standard Floquet/Mathieu stability diagram.

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Pith. "Pith review of Semiclassical Backreaction: A Qualitative Assessment." pith.science (2026). https://pith.science/paper/YW5SHMHT

@misc{pith2026241119825,
  author       = {Pith},
  title        = {Pith review of: Semiclassical Backreaction: A Qualitative Assessment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YW5SHMHT}},
  note         = {Machine review of arXiv:2411.19825}
}

read the original abstract

The backreaction of quantum degrees of freedom on classical backgrounds is a poorly understood topic in theoretical physics. Most often it is treated within the semiclassical approximation with the help of various ad hoc prescriptions accounting for the effect of quantum excitations on the dynamics of the background. We focus on two popular ones: (i) the mean-field approximation whereby quantum degrees of freedom couple to the classical background via their quantum expectation values; (ii) the (stochastic) Truncated Wigner method whereby the fully coupled system is evolved using classical equations of motion for various randomly sampled initial conditions of the quantum degree of freedom, and a statistical average is performed a posteriori. We evaluate the performance of each method in a simple toy model against a fully quantum mechanical treatment, and identify its regime of validity. We interpret the results in terms of quantum entanglement and loss of classicality of the background.

Figures

Figures reproduced from arXiv: 2411.19825 by the authors.

**Figure 1.** Some typical evolutions of ⟨xˆ(t)⟩ (black dotted line), xMF (t) (blue line), xTW (t) (red line) and xcl(t) (green line). We illustrate the importance of classical instability by choosing parameters in such a way that we are either in a completely stable regime (left panels, λ = 0.01, x0 = 5) or in a regime where the dynamics (at least intermittently) move through an instability band (right panels, λ = 0.1, x0 = 10).… view at source ↗

**Figure 2.** Quantum, and semiclassical break times for the MF and TW methods (as defined via Eq. (16) [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Same plot as Fig. 2 with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Same plot as Fig. 2 with [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Plot showing the regions in the (λ, x0) parameter plane where tMF > tTW (blue dots) and tMF < tTW (red dots) for different values of the ratio Ω/ω: 0.4 (left), 0.8 center, and 1.2 (right). The shaded region represent the instability bands. A heuristic picture emerges: …

**Figure 6.** Figure 6: The three dimensional version of Fig. 5. We plot the difference in break times [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: The break times tq (green), tMF (blue) and tTW (red) plotted against N = x 2 0/2 (a proxy for ℏ −1 ) for different values of the collective coupling λN and for Ω/ω = 0.8. The leftmost plot corresponds to λN = 0.15, and to systems that remain stable all the way through …

**Figure 8.** Figure 8: Plot of the convergence parameter as defined in Eq. (C4) as a function of time, for the operators [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]

**Figure 9.** Figure 9: The degree of conservation of the expectation value of [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]

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Works this paper leans on

76 extracted references · 51 canonical work pages

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Quantum Mechanics As discussed at the beginning of the section, the Schrödinger picture state att = 0 is prepared in such a way that its wavefunction can be written (to first-order inλ) as ψ0(x, y) = N0e−(x−x0)2/2e−((Ω/ω)2+λx2 0) 1/2 y2/2, (6) where N0 is a normalization factor. This corresponds to the tensor product of a coherent state with a large occup...

work page
[2]

(8) As mentioned in Sec

Semiclassical approximations The starting point of any semiclassical method is the classical equations of motion stemming from Hamiltonian (5), ¨x + 1 + λy2 x = 0, (7) ¨y + h (Ω/ω)2 + λx2 i y = 0. (8) As mentioned in Sec. II and at the beginning of Sec. III, we have prepared our quantum system in such a way that its dynamics yield ⟨ˆx(t)⟩ = x0 cos t and ⟨...

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[3]

Mean Field approximation While keepingx classical we promotey to a quantum operatorˆy. Working in the Heisenberg picture we can expand it as ˆy(t) = z(t)ˆa† + z∗(t)ˆa, where the creation and annihilation operators are defined with respect to the initial adiabatic ground state such thatˆa |0⟩ = 0 (see Refs. [43, 44] for details). The dynamics of the comple...

work page
[4]

correctness

Truncated Wigner method: 12 The Wigner function of our initial state (6) is given by W0(x, y, px, py) = 1 π Z dx′dy′ψ∗ 0(x + x′, y+ y′)ψ0(x − x′, y− y′)e2i(pxx′+pyy′) = N 2 0 e−(x−x0)2 e−p2 xe−((Ω/ω)2+λx2 0) 1/2 y2 e−((Ω/ω)2+λx2 0) −1/2 p2 y , (12) and can be interpreted as a joint probability distribution for the initial phase space variables of our syst...

work page
[5]

decoheres

Measure of correctness and semiclassical break time Each of the two semiclassical methods described above gives a value forx(t). We would like to know at what point in time this value deviates significantly from the true value,⟨ˆx(t)⟩, obtained by evolving the wavefunction with the Schrödinger equation. For this purpose, we first need to introduce an appr...

work page 2021
[6]

Quantum Mechanics In this derivation, we closely follow Ref. [66]. We start by writing the quantum state of the coupled system as |ψ⟩ = ∞X n,m=0 cn,m(t)fn(t) |n⟩x |m⟩y , (A1) 26 where |n⟩x and |m⟩y are eigenstates of the free Hamiltonians forx and y in Eq. (5) respectively. The functions fn(t) are introduced for future convenience and correspond to the ex...

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[7]

(10) and (11) and we wish to find a perturbative solution with the appropriate initial conditions

Mean-Field The MF equations of motion are Eqs. (10) and (11) and we wish to find a perturbative solution with the appropriate initial conditions. Previously, we saw that in the limit of small couplings, the main correction to the classical solutionx(t) = x0 cos t was a change in the frequency, proportional to λ. With this in mind, we perform a two-timing ...

work page
[8]

classical

T runctated Wigner It is significantly more difficult to derive an analytic estimate for the results of the TW method. However, here we will argue that after some time it will necessarily deviate from the weak collective coupling behavior of quantum mechanics. The best way to see this is to inspect one of the many classical realizations obtained in the TW...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Quantum Mechanics As discussed at the beginning of the section, the Schrödinger picture state att = 0 is prepared in such a way that its wavefunction can be written (to first-order inλ) as ψ0(x, y) = N0e−(x−x0)2/2e−((Ω/ω)2+λx2 0) 1/2 y2/2, (6) where N0 is a normalization factor. This corresponds to the tensor product of a coherent state with a large occup...

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[2] [2]

(8) As mentioned in Sec

Semiclassical approximations The starting point of any semiclassical method is the classical equations of motion stemming from Hamiltonian (5), ¨x + 1 + λy2 x = 0, (7) ¨y + h (Ω/ω)2 + λx2 i y = 0. (8) As mentioned in Sec. II and at the beginning of Sec. III, we have prepared our quantum system in such a way that its dynamics yield ⟨ˆx(t)⟩ = x0 cos t and ⟨...

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[3] [3]

Mean Field approximation While keepingx classical we promotey to a quantum operatorˆy. Working in the Heisenberg picture we can expand it as ˆy(t) = z(t)ˆa† + z∗(t)ˆa, where the creation and annihilation operators are defined with respect to the initial adiabatic ground state such thatˆa |0⟩ = 0 (see Refs. [43, 44] for details). The dynamics of the comple...

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[4] [4]

correctness

Truncated Wigner method: 12 The Wigner function of our initial state (6) is given by W0(x, y, px, py) = 1 π Z dx′dy′ψ∗ 0(x + x′, y+ y′)ψ0(x − x′, y− y′)e2i(pxx′+pyy′) = N 2 0 e−(x−x0)2 e−p2 xe−((Ω/ω)2+λx2 0) 1/2 y2 e−((Ω/ω)2+λx2 0) −1/2 p2 y , (12) and can be interpreted as a joint probability distribution for the initial phase space variables of our syst...

work page

[5] [5]

decoheres

Measure of correctness and semiclassical break time Each of the two semiclassical methods described above gives a value forx(t). We would like to know at what point in time this value deviates significantly from the true value,⟨ˆx(t)⟩, obtained by evolving the wavefunction with the Schrödinger equation. For this purpose, we first need to introduce an appr...

work page 2021

[6] [6]

Quantum Mechanics In this derivation, we closely follow Ref. [66]. We start by writing the quantum state of the coupled system as |ψ⟩ = ∞X n,m=0 cn,m(t)fn(t) |n⟩x |m⟩y , (A1) 26 where |n⟩x and |m⟩y are eigenstates of the free Hamiltonians forx and y in Eq. (5) respectively. The functions fn(t) are introduced for future convenience and correspond to the ex...

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[7] [7]

(10) and (11) and we wish to find a perturbative solution with the appropriate initial conditions

Mean-Field The MF equations of motion are Eqs. (10) and (11) and we wish to find a perturbative solution with the appropriate initial conditions. Previously, we saw that in the limit of small couplings, the main correction to the classical solutionx(t) = x0 cos t was a change in the frequency, proportional to λ. With this in mind, we perform a two-timing ...

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[8] [8]

classical

T runctated Wigner It is significantly more difficult to derive an analytic estimate for the results of the TW method. However, here we will argue that after some time it will necessarily deviate from the weak collective coupling behavior of quantum mechanics. The best way to see this is to inspect one of the many classical realizations obtained in the TW...

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