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Towards an experimental implementation of entanglement harvesting in superconducting circuits: effect of detector gap variation on entanglement harvesting

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A detector model built from a superconducting tunable coupler shows that the coupling-induced drop in qubit energy gap suppresses spacelike entanglement harvesting but leaves genuine harvesting intact for detectors in causal contact.

desk verdict A genuinely useful circuit-to-detector mapping and parameter scan for entanglement harvesting in superconducting circuits, but the two-level/adiabatic bridge is explicitly deferred just where it is needed most. read the letter →

arxiv 2505.01516 v1 pith:COGD4PDV submitted 2025-05-02 quant-ph gr-qchep-th

Adam Teixidó-Bonfill , Xi Dai , Adrian Lupascu , Eduardo Martín-Martínez This is my paper · ORCID
classification quant-phgr-qchep-th
keywords entanglementharvestingUnruh-DeWittdetectorsuperconductingcircuitsvariableenergygapderivativecouplingvacuumtunablecouplerfluxqubit
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 builds a middle-ground detector model—called VGSD, for variable-gap, spatial-derivative coupling—that sits between the idealized Unruh-DeWitt detector and a concrete superconducting circuit built from a tunable coupler and a flux qubit. It argues that in that device the qubit's energy gap cannot be held constant: as the coupling is switched on, the gap drops approximately linearly with the coupling strength, $\Delta\Omega/2\pi \approx 5.2\lambda$ GHz for the fitted parameters. Using this model, the paper finds that increasing the (negative) gap variation reduces the entanglement acquired by spacelike detectors and, at the strongest explored coupling, eliminates spacelike harvesting entirely. For detectors in causal contact, entanglement still appears, and the field-correlation part $M_+$ dominates the communication part $M_-$, so the entanglement counts as genuine harvesting. The upshot is that a near-future superconducting-circuit experiment could demonstrate genuine entanglement harvesting, with detector placement in or near lightlike contact the more forgiving route.

What carries the argument

The object that carries the calculation is the VGSD interaction Hamiltonian, $\hat H_I(t)=\hbar c\sum_\nu\lambda_\nu\chi_\nu(t)\hat\mu_\nu(t)\partial_x\hat\phi_C(t,x_\nu)$, in which the monopole phase accumulates the time-dependent gap, $\varphi_\nu(t)=\int_0^t \Omega_\nu(t')\,dt'$, and the field is coupled through its spatial derivative with an exponential cutoff $C(\omega)=e^{-\omega/2\Omega_{\rm cut}}$. The gap-variation effect enters entirely through $\chi_\varphi(t)=e^{i\varphi(t)}\chi(t)$, which replaces the constant-phase $\chi(t)$ of a standard UDW detector. The cutoff makes the vacuum correlators closed-form, $J(t)=\Omega_{\rm cut}^2/(1+i\Omega_{\rm cut}t)^2$, which keeps the negativity integrals tractable. The communication-versus-harvesting split is carried by decomposing the off-diagonal amplitude into $M=M_+ + M_-$, where $M_+$ comes from the field anticommutator (pre-existing correlations) and $M_-$ from the commutator (communication).

What would settle it

Take two TC+FQ detectors in a fixed spacelike configuration, ramp the coupling to $\lambda=-1$ (so $\Delta\Omega/2\pi\approx-5.2$ GHz), and measure the two-qubit negativity after the interaction: the model predicts zero, so any appreciable non-zero negativity would falsify the suppression claim. A complementary check is to measure the qubit population after fast switching; detectable population in excited levels beyond the first would show the two-level adiabatic assumption that the entire model rests on is violated.

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Extended reading notes

Core claim

The paper's central claim is that the entanglement-harvesting behavior of a realistic superconducting implementation is governed by a simple linear relation between the detector energy gap and its coupling: $\Omega(t)=\Omega_0+\Delta\Omega\,\chi(t)$, with the fitted device giving $\Delta\Omega/2\pi\approx 5.2\lambda$ GHz. In the scenarios explored numerically, this coupling-induced gap reduction makes spacelike harvesting harder: the negativity $\mathcal{N}$ shrinks, the usable range of interaction durations and switching shapes narrows, and at $\lambda=-1$ (spin-boson coupling $\alpha\approx0.32$) strict spacelike harvesting disappears. The same gap variation does not block harvesting for detectors that can signal each other; there the negativity can even increase, and the estimator $|M_+|/(|M_+|+|M_-|)$ shows the entanglement is predominantly drawn from pre-existing field correlations rather than from communication mediated by the field. Because the field lives on a 1+1D transmission line, the negativity acquired near the light cone does not keep decaying with detector separation—it plateaus. The paper presents this as evidence that genuine entanglement harvesting can be realized in existing or near-future superconducting circuits, provided experiments account for the gap variation in their choice of switching and detector placement.

Load-bearing premise

The entire harvesting calculation assumes the tunable-coupler flux-qubit circuit remains in its two-level qubit subspace during switching—no leakage to higher energy levels—under adiabatic conditions that the paper notes are subtle and not yet proven; if that assumption fails, the VGSD detector no longer describes the device.

Editorial extensions

If this is right

  • Spacelike harvesting in this device is best at weak or moderate coupling; pushing to ultra-strong coupling does not help because the accompanying gap reduction suppresses the signal.
  • Detectors placed at or near lightlike contact remain a viable route to genuine harvesting, since the field-correlation contribution dominates even when communication is possible.
  • Because negativity near the light cone plateaus with distance in 1+1D, detectors can be spaced farther apart without sacrificing entanglement, which relaxes fabrication constraints.
  • Compact switching functions (cosine ramps or isosceles trapezoid) with delay $t_\Delta = t_d - T$ at fixed $t_d$ are the practical way to maximize spacelike harvesting with a fixed device geometry.
  • Improved device designs that decouple the gap from the coupling would restore and enhance spacelike harvesting, so gap variation is a concrete design target.

Reading between the lines

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

  • The same variable-phase detector could be used to revisit other relativistic quantum information protocols, such as quantum energy teleportation or communication capacity, where the chirp-like phase would act as a tunable resonance condition.
  • The model predicts a sharp, testable scaling law: for detectors kept at fixed distance from the light cone ($t_\Delta-t_d$ constant), measured negativity should saturate rather than decay as separation grows; observing continued decay would indicate the 1+1D idealization fails.
  • The gap variation is in effect a frequency chirp, so a natural (unexplored) control is to drive the qubit with a compensating chirp to restore a constant effective gap; this could rescue spacelike harvesting without redesigning the device.
  • The predicted disappearance of spacelike harvesting at $\alpha\approx0.32$ could be tested as a threshold: ramp the coupling at fixed geometry and watch the two-qubit negativity vanish, which would provide a calibration of the model against the real device.
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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 / 5 minor

Summary. The paper proposes a variable-gap, spatial-derivative-coupled detector (VGSD) as an intermediate model between ideal Unruh-DeWitt detectors and a specific superconducting circuit implementation: a flux qubit with a tunable coupler (TC+FQ) coupled to a transmission line. Starting from the circuit Hamiltonian (15), the authors apply a sequence of approximations (two-level truncation, adiabatic free evolution, transversal coupling, linear gap dependence) to arrive at the interaction-picture Hamiltonian (31). They then compute, to second order in the coupling λ, the two-detector state and negativity for vacuum-field entanglement harvesting, using analytic expressions for the field integrals (Appendices B-C) and numerically exploring six parameter scenarios (Table I) with different switching functions. The main results are that increasing the (negative) gap variation reduces spacelike harvesting, can eventually eliminate it at the strongest couplings, and does not prevent genuine harvesting (as estimated by the M+/(M−+M+) ratio) for causally connected detectors, with some enhancement in causal contact. The paper also highlights a 1+1D effect: the harvested negativity does not decay asymptotically along the lightcone as the detector separation grows.

Significance. If the device-mapping assumptions hold, this is a useful and fairly concrete step toward experimental entanglement harvesting in circuit QED. The paper's strengths include a careful derivation of the detector state and negativity, closed-form expressions for the field correlators with the exponential cutoff, a transparent decomposition of the entanglement into communication and genuine-harvesting parts following [58], and a systematic parameter scan with experimentally motivated values. The 1+1D lightcone plateau is a genuinely interesting observation. However, the experimental-prediction status of the results rests on an unproven two-level/adiabatic reduction at ultra-strong coupling, and the perturbative truncation at O(λ²) is not controlled at the largest coupling considered (λ=−1). These issues do not invalidate the VGSD model calculations, but they currently limit the paper's central claim that the plots describe the TC+FQ device.

major comments (3)
  1. [§III A 1–2 and Eq. (31)] The reduction from the TC+FQ Hamiltonian (15) to the two-level variable-gap detector (31) is load-bearing for the paper's experimental claim, but it is explicitly assumed rather than justified. The text states that the conditions for the adiabatic/two-level approximation in the ultra-strong regime "are subtle and will be explored elsewhere" and "we will operate under the assumption that they hold", with the cited experimental validation [63,68,69] valid only for weaker couplings. Yet Table I includes scenarios with γ=0.14–0.22 (α=0.1–0.32), and Figures 17–21 use switching durations T=0.13–0.71 ns, which are comparable to or only a few times the qubit period 1/Ω0≈0.137 ns. A few oscillation cycles do not by themselves guarantee adiabaticity, and no leakage estimate is provided. If population leaves the qubit subspace during switching, the phase φ(t) in Eq. (32) and the effective coupling in Eq. (31) do not describe the device, and the harvesting curves in Figures 14–22 lose their experimental meaning. Please provide a leakage estimate (e.g., a numerical solution of the multilevel Schrödinger equation for the proposed fβ(t) schedules) or explicitly restrict the experimental claims to the weak- and moderate-coupling scenarios where the reduction is safer.
  2. [§IV, Eqs. (48)–(52), and Table I, scenario 5] The final state and the negativity are computed to leading order in λ, with neglected terms of O(λ⁴). In scenario 5 of Table I, λ=−1, so λ²=1 and the uncomputed O(λ⁴) corrections are nominally the same order as the reported leading contribution. The plots of N/λ² in Figures 11 and 21 are therefore not quantitatively controlled for that scenario, and the statement that spacelike harvesting is eliminated at the strongest coupling depends on a cancellation that is not demonstrated. Please estimate the size of the next-order corrections (for instance, by computing the fourth-order contribution for a simplified switching profile) or soften the quantitative claims at λ=−1. This does not necessarily destroy the qualitative trend, but the present manuscript does not support quantitative statements at this coupling.
  3. [§VII vs §VI D and Figure 21] The concluding paragraph says that "increasing the gap variation reduces the entanglement acquired by spacelike detectors but does not completely cancel it", whereas Section VI D states that the larger negative ΔΩ "mak[es] spacelike harvesting impossible for strong couplings", and Figure 21 shows zero spacelike negativity for scenario 5. These statements are in direct tension. Please reconcile them by specifying the parameter range over which the "does not completely cancel" claim is intended, since the abstract and introduction frame the paper around a reduction that is partial, while the body shows a complete cancellation in one explored case.
minor comments (5)
  1. [§VI A and Eqs. (30), (66)] The stated fixed parameter ΔΩ/(2π)≈5.2·λ GHz is slightly inconsistent with the chain ΔΩ/(2π)≈−23·γ GHz (Eq. (30)) and λ≈−4.53·γ (Eq. (66)), which gives ΔΩ/(2π)≈5.08·λ GHz. Please check the conversion and use a consistent value throughout.
  2. [Figures 14–22] The figure labels abbreviate the gap variation as "Ωv" in several panels (e.g., rows of Figures 14–16), which is confusing because v is also the speed of light in the transmission line. The label should be ΔΩ/(2π) or a clearly defined symbol.
  3. [§II A] The notation for the cutoff function is used consistently in the main text, but the variable "x−" appears in Eq. (54) before its definition as a difference of positions; please define it explicitly at first use, alongside t−.
  4. [§III A 3 and Eq. (28)] The statement that neglecting γz and γid "does not affect entanglement harvesting at leading order" is supported by three conditions, but the third condition says "the field is initially prepared in states diagonal in the Fock basis", which is not the same as the vacuum/zero-mean Gaussian states later used in Eq. (45). Please clarify whether the assumed field states satisfy this condition.
  5. [Appendix C] There is a typo in the heading "1. Symmetric switching functions equal up to a time shift and equal detectors": it should read "equal up to a time shift" consistently, and the paragraph beginning "The follwing change of variables" is missing a "l". These are presentation issues only.

Circularity Check

1 steps flagged · score 4.0 of 10

The model's purely-transverse VGSD Hamiltonian leans on an unpublished self-citation ([70]) for the step that drops longitudinal coupling; the harvesting computations themselves are independent, so this is partial, not total, circularity.

  1. self citation load bearing [Section III A 3, after Eq. (27); support for dropping γz and γid before Eq. (31)]
    "In this paper we are going to assume that the longitudinal coupling γz and the term γid are zero. For fε such that the qubit transition frequency is minimal, the longitudinal coupling γz was shown to be negligible in [44]."

    The only support cited for the step that lets the TC+FQ circuit be replaced by the purely transverse VGSD Hamiltonian of Eq. (31) is reference [70], listed in the bibliography as 'A. Teixidó-Bonfill and E. Martín-Martínez, Work in preparation'—an unpublished citation whose authors are two of the present authors. The claim that dropping γz and γid leaves leading-order entanglement harvesting unchanged is not derived in this paper and has no published, machine-checked, or otherwise external corroboration. Since Eqs. (31), (37), and all subsequent negativity and M± computations rely on this purely transverse two-level model, the device-to-detector bridge depends on an in-house, unverifiable citation rather than on an independently established result.

full rationale

The paper's target quantities (negativity N and the M± split) are computed from the VGSD Hamiltonian via a Dyson expansion; they are not fit to the results they are claimed to predict. The fixed circuit parameters (Ω0, ΔΩ from the linear fit, Z0, v, Ωcut, td) trace to previously published experimental characterizations or to [44]'s numerical diagonalization based on measured critical currents and capacitances; those are external inputs, even though some co-authors overlap. The W± decomposition used to identify genuine harvesting is taken from [58,59], both published, and the extension to derivative coupling is cited to [59]; this is a legitimate use of prior independent work, not a circular reduction. The explicitly deferred two-level/adiabatic justification in §III A 1–2 is a validity gap (leakage could invalidate the detector model), not a circularity, and is acknowledged by the authors. The one circularity-adjacent flaw is the load-bearing, unpublished self-citation [70] used to justify dropping γz and γid at leading order; without that step, Eq. (31) and all subsequent harvesting plots are not guaranteed to describe the TC+FQ device. Because the core numerical exploration remains independent of that lemma, the overall circularity is partial (score 4), not total.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the reduction of a specific tunable-coupler circuit to a two-level variable-gap detector and on a second-order perturbative calculation. The strongest inputs (ΔΩ relation, Ω0, circuit parameters) come from fits to experimental data in [44], while the harvesting/communication split relies on the W± decomposition from [58,59]. No new physical entities are introduced.

free parameters (2)
  • ΔΩ (gap variation coefficient) = ΔΩ/(2π) ≈ 5.2·λ GHz (equivalently -23 GeV·γ from Fig. 5)
    Linear best fit to numerically computed qubit frequency vs transversal coupling γx (Fig. 5), based on Hamiltonian parameters fitted to experiment in [44]. Sets the gap variation in every scenario (Table I).
  • Ω0 (free qubit gap) = Ω0/(2π) ≈ 7.3 GHz
    The qubit transition frequency at zero coupling, extracted from the same numerical diagonalization with [44] parameters; an input, but a fitted value.
assumptions (8)
  • domain assumption Two-level truncation of the TC+FQ circuit (only the two lowest energy eigenstates are kept).
    Section III A 1: justified when higher transition frequencies are large compared to qubit frequency; the authors state the exact conditions at ultra-strong coupling will be explored elsewhere.
  • domain assumption Adiabatic theorem governs the free qubit evolution while the gap is tuned, preventing transitions out of the qubit subspace.
    Section III A 2: assumed for fβ tuned slowly; geometric phases set to zero by phase convention; validity at ultra-strong coupling deferred.
  • domain assumption Longitudinal coupling and identity term are zero (γz=γid=0), leaving pure transversal coupling.
    Section III A 3: said negligible at the symmetry point in [44] and not affecting leading-order harvesting under the stated sufficient conditions.
  • domain assumption Energy gap depends linearly on the switching function: Ω(t)=Ω0+ΔΩ χ(t).
    Section III A 4, Eq. (29): from linear best fit to numerically diagonalized Hamiltonian; approximate for the explored fβ range.
  • domain assumption Neglect of the renormalization term Φ₋²/(2ℓ0Δx) in the continuum limit.
    Section II C: term diverges as Δx→0; authors expect it does not change conclusions, with analysis deferred to future work.
  • domain assumption Exponential UV cutoff at 50 GHz models the transmission line attenuation.
    Section II A, Eq. (9): value matched to experiment in [63]; equivalent to a spatial smearing of the detector.
  • domain assumption Perturbative expansion in λ truncated at O(λ²) is valid for all scenarios, including λ=-1.
    Section IV, Eqs. (40)-(52): no estimate of O(λ⁴) terms; at λ=-1, λ²=1 so omitted terms are formally of the same order as the leading term.
  • domain assumption The M+/M- decomposition (from W± field commutator/anticommutator) separates genuine harvesting from communication.
    Section V: uses results of [58,59]; the estimator |M+|/(|M+|+|M-|) is interpreted as the fraction of genuinely harvested entanglement.

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Cite this review

Pith. "Pith review of Towards an experimental implementation of entanglement harvesting in superconducting circuits: effect of detector gap variation on entanglement harvesting." pith.science (2026). https://pith.science/paper/COGD4PDV

@misc{pith2026250501516,
  author       = {Pith},
  title        = {Pith review of: Towards an experimental implementation of entanglement harvesting in superconducting circuits: effect of detector gap variation on entanglement harvesting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/COGD4PDV}},
  note         = {Machine review of arXiv:2505.01516}
}
read the original abstract

Motivated by the prospect of experimental implementations of entanglement harvesting in superconducting circuits, we propose a model of variable-gap particle detector that aims to bridge some of the gaps between Unruh-DeWitt (UDW) models and realistic implementations. Using parameters tailored to potential experimental setups, we investigate entanglement harvesting in both spacelike-separated and causally connected scenarios. Our findings reveal that while variations in the energy gap reduce the ability to harvest entanglement for spacelike-separated detectors, detectors in causal contact can still become entangled through their interaction with the field. Notably, our analysis shows that (due to the derivative coupling nature of the model) even for causally connected detectors, the entanglement primarily originates from the field's correlations. This demonstrates the potential for genuine entanglement harvesting in the lab and opens the door to near-future entanglement harvesting experiments in superconducting circuits.

Figures

Figures reproduced from arXiv: 2505.01516 by the authors.

Figure 1
Figure 1. FIG. 1. Lumped circuit model of a resistance-less transmis [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Circuit model of a flux qubit. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Circuit model of the tunable coupler connected to the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Lumped circuit model of a transmission line coupled [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The blue dots show the superconducting qubit tran [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Gaussian switching shape, with 5 standard deviations shown. (b) Cosine ramps switching shape with [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Plots for the scenario 1 of Table I: [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Analogous to Figure 7, but for scenario 2 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Analogous to Figure 7, but for scenario 3 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Analogous to Figure 7, but for scenario 4 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Analogous to Figure 7, but for scenario 5 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Analogous to Figure 7, but for scenario 6 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Spacetime diagrams with gray rectangles represent [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 21
Figure 21. Figure 21: However, negativity in causal contact grows [PITH_FULL_IMAGE:figures/full_fig_p017_21.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Negativity acquired by spacelike detectors. To ensure the negativity is the largest, we explore the boundary where [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Negativity acquired by spacelike detectors. The plots are analogous to Figure 14, with the only difference being that [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Negativity acquired by spacelike detectors. The plots are analogous to Figure 14, with the exceptions of fixing [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Scenario 1 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Analogous to Figure 17, but for scenario 2 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p022_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Analogous to Figure 17, but for scenario 3 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p023_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Analogous to Figure 17, but for scenario 4 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Analogous to Figure 17, but for scenario 5 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p024_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Analogous to Figure 17, but for scenario 6 of Table I, with [PITH_FULL_IMAGE:figures/full_fig_p024_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23 [PITH_FULL_IMAGE:figures/full_fig_p029_23.png]

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Reference graph

Works this paper leans on

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    Two-level approximation To reduce the tunable coupler + flux qubit circuit into a qubit, we need to truncate the energy levels, keeping only the two lowest-energy eigenvectors of ˆHtc+fq. Ne- glecting higher energy levels can be justified when their transition frequencies are large compared to the qubit transition frequency. Then, if the coupling is switc...

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    Taking the adiabatic approximation on the free qubit evolution Changing the parameterfβ over time can induce tran- sitions in the qubit and even cause the TC+FQ circuit of Figure 3 to leave the qubit subspace. These transitions occur even if the circuit is not connected to the trans- mission line, and are a consequence of the fact that the energy eigensta...

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    (22) shows that the coupling occurs through ˆ γ5

    Transversal coupling approximation The interaction Hamiltonian obtained in Eq. (22) shows that the coupling occurs through ˆ γ5. Consider ˆγqb 5 to be the restriction of ˆγ5 onto the qubit subspace. Then, ˆγqb 5 can be expressed in as a linear combination of identity and Pauli operators, ˆγqb 5 =γxˆσx +γyˆσy +γzˆσz +γidˆ11, (27) where ˆσz =|0fβ⟩⟨0fβ|−| 1f...

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    The exact form of this dependence is also obtained using the numerical diagonalization of the Hamiltonian of the tunable coupler + flux qubit

    Approximated linear dependence of the gap on the instantaneous transversal coupling strength The energy gap of the qubit ℏΩ varies with fβ, which can be rewritten as a dependence of Ω on the instanta- neous transversal coupling strength γx. The exact form of this dependence is also obtained using the numerical diagonalization of the Hamiltonian of the tun...

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    Pozas-Kerstjens and E

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    The adiabatic approximation of the free qubit evo- lution

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    The transversal coupling assumption, γid =γz = 0

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    The resulting simplified model resembles a qubit parti- cle detector with variable gap and spatial derivative cou- pling

    The linear approximation of the dependence of Ω on γx. The resulting simplified model resembles a qubit parti- cle detector with variable gap and spatial derivative cou- pling. This detector model has the following interaction picture interaction Hamiltonian, ˆHint(t) =−φ0 ℓ0 ...

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    (54) into Eq

    Integrating over the field modes last Substituting Eq. (54) into Eq. (49) leads to Lνν = λ2 2π Z ∞ 0 dωω C(ω)2|gχφν(ω)|2, Lab = λ2 2π Z ∞ 0 dωω C(ω)2 cos(ωtd)gχφa(ω)gχφb(ω)∗, M =−λ2 2π Z ∞ 0 dωω C(ω)2 cos(ωtd) × Z dt dt′e−iω|t−t′|χφa(t)χφb(t′). (55) Here, we used the Fourier t...

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    (55) and a quite helpful way of evaluating these integrals is to integrate over ω before performing the time integrals

    Integrating over the field modes first An alternative to Eq. (55) and a quite helpful way of evaluating these integrals is to integrate over ω before performing the time integrals. Doing this,Wvac xx′ becomes Wvac xx′(t−,x−) = 1 4πc2 J t− +|x−| c +J t−−|x−| c , J (t) = Z ∞ 0 d...

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    Therefore, W− is not affected by the amount of pre-existing entanglement in the field, while W + is

    The expectation value of [ ˆϕ(x), ˆϕ(x′)] does not de- pend on the field state, while the expectation value of{ˆϕ(x), ˆϕ(x′)} does. Therefore, W− is not affected by the amount of pre-existing entanglement in the field, while W + is

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    Furthermore, even non-perturbatively, detec- tors cannot communicate by coupling to commut- ing field observables in the interaction picture (see, e.g

    W + does not participate in communication at lead- ing order, which is instead mediated by W− [73– 76]. Furthermore, even non-perturbatively, detec- tors cannot communicate by coupling to commut- ing field observables in the interaction picture (see, e.g. Appendix of [59]). Si...

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    The causal propagator, and thus W−, vanishes outside the light cone

    The commutator [ ˆϕ(x), ˆϕ(x′)] is proportional to the difference between the retarded and advanced Green’s functions (the classical causal propaga- tor [77]). The causal propagator, and thus W−, vanishes outside the light cone. On the other hand, the field anticommutator has ...

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    Its shape is X(s) =e−s2 , (70) plotted in Figure 6(a)

    Gaussian Switching Shape The Gaussian switching is common in the literature of entanglement harvesting and we explore it to ease com- parisons with established results. Its shape is X(s) =e−s2 , (70) plotted in Figure 6(a)

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    Its shape is X(s) =    1 |s|≤ Sf 2, 1 2 + 1 2 cos π 2|s|−Sf 1−Sf Sf 2 <|s|< 1 2, 0 1 2≤|s|, (71) with an example plotted in Figure 6(b)

    Cosine Ramps Switching Shape The cosine ramps switching function lasts for a finite time and has a continuous derivative. Its shape is X(s) =    1 |s|≤ Sf 2, 1 2 + 1 2 cos π 2|s|−Sf 1−Sf Sf 2 <|s|< 1 2, 0 1 2≤|s|, (71) with an example plotted in Figure 6(b). Sf is the por...

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    Its shape is X(s) =    1 |s|≤ Sf 2, 1−2|s| 1−Sf Sf 2 <|s|< 1 2, 0 1 2≤|s|, (72) with an example plotted in Figure 6(c)

    Isosceles Trapezoid Switching Shape The isosceles trapezoid switching function lasts a finite time and is continuous. Its shape is X(s) =    1 |s|≤ Sf 2, 1−2|s| 1−Sf Sf 2 <|s|< 1 2, 0 1 2≤|s|, (72) with an example plotted in Figure 6(c). Sf is the portion of time that the...

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    Place the detectors at a fixed distance, i.e. with a constant td, and pick their switching delay to be t∆ =td−T . The reason for this choice stems from detectors being spacelike as long as |t∆|≤ td−T . For larger|t∆|, detectors would be in lightlike con- tact, and for even lar...

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    la Caixa

    Place two simultaneously switched detectors, t∆ = 0, as close as possible while keeping their in- teractions spacelike. This is achieved by allowingtd to take the value td =T . This choice is illustrated in a spacetime diagram in subfigure 13(b) and the acquired negativity is ...

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    The diagonal solid lines indicate full lightlike contact

    isosceles trapezoid, T = 0.16 ns,Sf = 0.2. The diagonal solid lines indicate full lightlike contact. Outside the dashed lines, either there is no light contact (compact switchings) or only the 5σ tails of the Gaussian can interact. To aid visualization, the values of negativit...

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    Moreover, we assume that the energy gap of both detectors depends linearly on their switching functions, as in Eq

    Symmetric switching functions equal up to a time shift and equal detectors We can further simplify the equations Lµν andM for the type of χν(t) explored in this article, which satisfy χν(t) =χ(t−tν), (C1) 27 with χ(t) = χ(−t), and tν controlling the time at which the detector ...

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    (C16) Then, performing the change of variables in Eq

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    −81.5 2 .94 18 .6 −81.5 196 . 6.18 24 .8 2.94 6 .18 46 .1 −32.4 18.6 24 .8 −32.4 87 .7   (pF)−1. The figure shows that the dependency Ω(γx) is approx- imately linear, which implies that Ω(t)≈ Ω0 + ∆Ωχ(t). (29) Here, we choseγx(t) =γχ(t), withχ(t) a switching func- tion whi...

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