Squeezing-Fueled Quantum Otto Engine via Measurement-Induced Cooling: The Two-Qubit Quantum Rabi Model

Asoka Biswas; S R Rathnakaran

REVIEW 4 major objections 4 minor 76 references

Squeezing-Fueled Quantum Otto Engine via Measurement-Induced Cooling: The Two-Qubit Quantum Rabi Model

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that a two-qubit quantum Otto engine, run on a single hot bath with measurement-induced cooling and cavity squeezing as fuel, operates above the standard quantum Otto efficiency bound while delivering more power.

desk verdict The paper's own analytic formulas imply Otto efficiency exactly; the above-Otto claim rests on excluding resource costs the paper later admits, so the abstract overstates the result. read the letter →

arxiv 2608.02521 v1 pith:G7LRVSBC submitted 2026-08-03 quant-ph

S R Rathnakaran , Asoka Biswas This is my paper · ORCID

classification quant-ph

keywords quantumOttoenginetwo-qubitRabimodelcavitysqueezingmeasurement-inducedcoolingnon-MarkovianbathhierarchicalequationsofmotionfuelQEDthermodynamic

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

The paper proposes a quantum Otto engine whose working medium is two qubits in a cavity (two-qubit quantum Rabi model), with a single non-Markovian hot bath, a projective measurement on the cavity replacing the cold bath, and a squeezing drive on the cavity playing the role of quantum fuel. The central aim is to show that cavity squeezing systematically improves both power output and efficiency, driving the engine above the standard quantum Otto limit while the limit-cycle efficiency asymptotically approaches that limit from above. The authors derive a perturbative analytic treatment of the squeezing, obtaining closed-form expressions for work and heat, and support the claim with numerical power-efficiency curves and phase-dependent contour maps. A sympathetic reader would care because the architecture points to an experimentally accessible cavity-QED route for quantum heat engines that convert squeezed vacuum and measurement information into work.

What carries the argument

The carrying machinery is the two-qubit quantum Rabi Hamiltonian in sector A plus the single-mode squeezing perturbation V = (μω/2)(e^{iφ}a^2 + e^{-iφ}(a†)^2). The principal analytic object is the squeezing-renormalised relaxation kernel Γ_eff = Γ_0(1 + 4rα²cosφ), where r is the squeezing parameter, α = g/ω the displacement amplitude, and φ the squeezing phase; it makes the heat absorbed during the hot isochore phase-dependent, so the squeezing phase acts as a thermal valve. The cold stroke is implemented by projective measurements Π_n(α) = D(α)|n><n|D†(α) on the cavity, with the engine post-selecting the outcome n* that minimises the qubit energy, in place of a cold thermal reservoir.

What would settle it

Evaluate the paper's own formulas: substituting W1 from Eq. (9), W2 from Eq. (30), and Q_H from Eq. (21) into η = -W/Q_H yields η = 1 - B_L/B_H exactly, so a numerical scan that instead multiplies the reported power by the post-selection probability p_{n*} and adds W_sq = ω sinh²r plus W_meas to the denominator would settle whether any efficiency above the Otto bound actually survives.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that a multi-resource architecture—two qubits coupled to a squeezed cavity, thermalised by a single non-Markovian hot bath, and cooled by projecting the cavity onto displaced Fock states—can outperform a conventional two-bath Otto engine. The two-qubit Rabi Hamiltonian splits by parity into an active sector A and an inert sector B; starting in sector A keeps the cycle analytically tractable. Weak squeezing turns the conditional cavity states into displaced-squeezed states, generates qubit coherence through a squeezing source term, and renormalises the population relaxation kernel to Γ_eff = Γ_0(1+4rα²cosφ). The paper claims this makes hot-bath heat

Load-bearing premise

The engine is assumed to run on the post-selected measurement branch that maximizes cooling, and the costs of preparing the squeezed cavity and of performing the measurement are excluded from the efficiency denominator.

Editorial extensions

If this is right

If correct, the same cavity-QED setup can serve as a quantum heat engine powered by a squeezed vacuum, with the squeezing phase as a controllable knob for heat flow and for trading power against efficiency.
The measurement stroke would let a single hot bath do the job of two baths, provided the post-selected outcome n* occurs with high probability; the engine's viability hinges on that branch being typical rather than rare.
In the limit-cycle regime, the cumulative efficiency would approach the Otto bound 1 - B_L/B_H from above rather than violate it, meaning the benefit is a finite-time enhancement paid for by non-thermal resources.
Reversing the cycle with a suitable field-ratio and measurement-protocol change would yield a measurement-assisted quantum refrigerator whose coefficient of performance is tuned by the squeezing phase.

Reading between the lines

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

Combining the paper's analytic expressions—W1 from Eq. (9), W2 from Eq. (30), and Q_H from Eq. (21)—gives η = -W/Q_H = 1 - B_L/B_H identically, independent of r, φ, τ_h, and bath parameters; if that is right, the above-Otto efficiency shown in the numerics is an artifact of the post-selected definition rather than a consequence of the derivation.
The reported power P = -W/(2τ+τ_h) omits the probability p_{n*} of actually obtaining the cooling outcome; the cycle-averaged power would be p_{n*} times the reported value, and if p_{n*} is not near unity the engine is not deterministic.
The paper's own Appendix D shows that including the squeezing-preparation cost W_sq = ω sinh²r and measurement cost W_meas bounds the resource-inclusive efficiency by a generalized Carnot-type bound, so the headline gain is best read as a paid-for conversion of non-passive free energy and information.
A natural testable extension is to measure the hot-stroke heat Q_H as a function of φ for fixed r: the predicted linear-in-r correction δQ_H ∝ -r cos φ gives a direct experimental signature of the claimed thermal-valve effect, distinct from any post-selection accounting.

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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

The thing to know: the headline claim does not survive contact with the paper's own equations. Combining Eqs. (9), (21), and (30) gives η = 1 − B_L/B_H identically. The squeezing strength r, the phase φ, the stroke time, and all bath parameters cancel. So the above-Otto efficiencies shown in Figs. 2–4 cannot be consequences of the analytic model as written; they must come from a different, unreconciled definition of work or heat. The paper never addresses that mismatch. This is the central problem.

What is genuinely new: the specific architecture—two-qubit Rabi model working medium, single non-Markovian HEOM hot bath, projective cavity-measurement cold stroke, and intramode cavity squeezing as fuel—is a combination I have not seen before. The perturbative expressions for Γ_eff, δQ_H, and the coherence source term S_sq are new for this model. The multi-cycle limit-cycle analysis with trace-distance convergence is a sensible idea, and the data availability statement is fine. Appendix D is also candid: it defines a resource-inclusive efficiency η_tot and concedes that the super-Otto operation is “paid for” by squeezing and measurement costs. That honesty makes the abstract harder to excuse.

Soft spots, in proportion: the post-selection issue is real and load-bearing. The cold stroke picks the measurement outcome n* that minimizes qubit energy, but the probability p_{n*} of that outcome is never reported. If p_{n*} is not near unity, the true cycle-averaged power is p_{n*} times the claimed value, and the engine cannot be treated as deterministic. That alone could invalidate the power claims. Second, the efficiency denominator excludes W_sq and W_meas; Appendix D’s own Eq. (D7) shows that including them keeps η_tot below the generalized bound. So the “above Otto” language is at best an operational figure of merit, not a fundamental advantage. Third, several perturbative coefficients (κ_0, f(ω, φ, τ)) are asserted without derivation, and HEOM convergence is stated but not shown with convergence plots. These are minor by comparison but still worth tightening.

Who is this for? Researchers working on measurement-assisted quantum engines and on the thermodynamic bookkeeping of quantum resources. It is a useful cautionary example and a starting point for a corrected analysis. But the paper should not be published with the current abstract and efficiency claims. The fix is clear: report conditional performance with the success probability, include resource costs in the headline efficiency, and reconcile the analytic and numerical definitions of work and heat.

My recommendation: send it to peer review, not desk reject. The model and derivations are substantive enough to warrant referee time, but the authors need to do major revision to make the claims match the math. If I were the editor, I would make acceptance conditional on resolving the central inconsistency.

Referee Report

4 major / 4 minor

Summary. The paper proposes a four-stroke quantum Otto engine built from two qubits in a cavity (the two-qubit quantum Rabi model), with a single non-Markovian hot bath treated via HEOM, a projective measurement on the cavity replacing the cold bath, and a squeezing drive as a purported fuel. It derives perturbative expressions for stroke work and hot heat, and reports numerical HEOM simulations showing power–efficiency curves, phase–squeezing contour maps, and multi-cycle convergence. The headline claim is that cavity squeezing raises both power and efficiency, driving the efficiency above the standard Otto limit and asymptotically converging to it from above in the limit-cycle regime.

Significance. Some ingredients are genuinely valuable: a symmetry-reduced two-qubit Rabi model with solvable structure, an explicit attempt to quantify the squeezing and measurement costs in Appendix D, and a HEOM-based numerical treatment of non-Markovian effects. However, the central claim fails: the analytic work and heat formulas imply the standard Otto efficiency exactly, independent of squeezing; the numerical 'super-Otto' results rely on an efficiency definition that omits exactly the resource costs the paper itself identifies, and on post-selecting a single measurement outcome without branch averaging. The resource-inclusive efficiency defined in Eq. (D7) does not beat the generalized bound, as the authors concede. Thus the manuscript as submitted does not establish the advertised result. The main value is a structured framework and an honest resource-cost appendix, not the claimed demonstration.

major comments (4)

[§II (Eqs. (9), (21), (30))] Combining W1 = −(B_H−B_L), Q_H = B_H(1−e^{−Φ})(1−tanh β_H B_H), and W2 = (B_H−B_L)[tanh β_H B_H + e^{−Φ}(1−tanh β_H B_H)] gives W = W1+W2 = −(B_H−B_L)(1−tanh β_H B_H)(1−e^{−Φ}), so η = −W/Q_H = 1−B_L/B_H identically. All dependence on r, φ, τ_h, and bath parameters cancels. This directly contradicts the abstract's central claim that squeezing drives the efficiency above the standard Otto limit. The numerical curves in Figs. 2–4 that show η above this value must be using a different work/heat definition or different initial conditions than Eqs. (9), (21), and (30); the manuscript does not reconcile the analytic and numerical results.
[§II.A, Eq. (7) and Eq. (20); §III.B] The analytic calculation assumes the expansion starts from the pure state |Ψ⟩=|−−⟩⊗|ξ_{−−}^{(n)}⟩, so P(0)=−1. In the multi-cycle limit-cycle analysis of §III.B, the initial state at vertex A is the post-measurement state from the previous cycle. Unless the cold stroke always projects onto a branch that exactly reproduces a pure |−−⟩ state with unit probability, the analytic formulas for Q_H and W do not describe the limit-cycle engine, and the claimed saturation to η_Otto in Fig. 5 cannot be inferred from them. A limit-cycle calculation must use the actual fixed point of the map, not a single-cycle pure-state ansatz.
[§II.A.4 (Eqs. (31)–(32)) and §III] The cold stroke post-selects the outcome n* that minimizes qubit energy and reports Q_C, W, and P without multiplying by the outcome probability p_{n*}. Work and power are not linear in the post-selection survival probability; if p_{n*} is not close to unity, the true cycle-averaged power is substantially lower and the engine is not deterministic. The manuscript does not report p_{n*} or provide a properly averaged treatment of all measurement branches. This is load-bearing for the numerical power–efficiency claims.
[§III.A, Appendix D, Conclusion] The efficiency η=−W/Q_H used in Figs. 2–4 excludes the squeezing preparation cost W_sq=ω sinh²r and the measurement cost W_meas (Eqs. (D3), (D6)). Appendix D defines η_tot = −W/(Q_H+W_sq+W_meas) and states it does not exceed the generalized bound; the Conclusion explicitly says the operational gain is 'supplied and paid for' by squeezing and measurement resources. As written, the headline above-Otto result is an artifact of the resource-excluded efficiency definition. The paper should either report η_tot as the main efficiency or clearly frame the result as an operational gain with externally paid resources; it cannot claim 'above the standard quantum Otto limit' while omitting those costs.

minor comments (4)

[§II.A, after Eq. (7)] The text says 'Note that ⟨E_q^A⟩=B_L' but Eq. (7) gives ⟨E_q^A⟩=−B_L for the state |−−⟩. Please correct this typo.
[Appendix C, Eqs. (C21), (C37)] The transient correction δΓ^(2)(t)=−κ_0 μ²ω²e^{−Γ_eff t} with κ_0≈α²+1/2, and the squeezing overlap correction R_sq(τ)≈1+r f(ω,φ,τ), involve coefficients that are not derived from a complete calculation. As written, the claim of fully analytic expressions is overstated; a derivation or a clearly stated numerical extraction procedure is needed.
[§III.B and Abstract] The abstract says the efficiency remains above the Otto bound throughout and asymptotically converges to it from above, while §III.B states the cumulative efficiency 'saturates to the Otto bound'. Please clarify whether the cumulative efficiency and the single-cycle efficiency plotted in Figs. 2–4 are different quantities and how the two statements are consistent.
[Appendix D.2] The phrase 'ideal projective measurements is unbounded upwards' is unclear and likely means 'the resource cost of ideal projective measurements can diverge'. Please rephrase.

Circularity Check

3 steps flagged · score 8.0 of 10

The claimed above-Otto efficiency is built into the resource-excluded definition of η and is absent from the paper's own analytic equations, which give exactly η = 1 − B_L/B_H.

self definitional [Section III.A (definition of η and P); Appendix D (η_tot, Eq. D7)]
"The output power is defined as P = −W/(2τ + τ_h), where W = W1 + W2 is the total work per cycle and 2τ + τ_h is the full cycle duration, while the efficiency is given by η = −W/QH, with QH denoting the heat absorbed from the hot bath. ... ηtot = −W/(QH + Wsq + Wmeas)."

The 'above Otto' result is produced by choosing η = −W/QH, so that the squeezing-preparation work Wsq and the measurement cost Wmeas are excluded from the denominator. Appendix D shows that when these costs are included, ηtot does not exceed the generalized bound. The claimed enhancement is therefore an artifact of the efficiency definition, not a consequence of the model dynamics.
other [Section II.A, Eqs. (9), (21), (30)]
"W1 = −(BH − BL). (9) ... QH = BH (1 − e^{−Φ(th)})(1 − tanh(βH BH)). (21) ... W2 = (BH − BL)[tanh(βH BH) + e^{−Φ(th)}(1 − tanh(βH BH))]. (30)"

Substituting (9), (21), and (30) gives W = W1 + W2 = −(BH − BL)(1 − tanh βH BH)(1 − e^{−Φ}), and therefore η = −W/QH = (BH − BL)/BH = 1 − BL/BH identically. The squeezing strength r, phase φ, stroke time τ_h, and bath parameters all cancel. The analytic model thus predicts the standard Otto efficiency, not the claimed above-Otto enhancement; any super-Otto value in the numerics must come from a different definition of work or heat, so the headline prediction is an external input rather than a derived result.

1 more flagged steps

self definitional [Section II.A.4, Eqs. (31)–(32); Section III.A power definition]
"The maximum cooling is achieved for n = n∗, for which the E(q)n becomes minimum. The energy extracted from the qubit subsystem, therefore, is QC = E(q)n∗ − ⟨EqD⟩. (32) ... The probability of each outcome is pn = Tr(Π̂n ρ̂globalD)."

The engine output is computed only for the post-selected branch n∗; the outcome probability p_{n∗} and all discarded branches are omitted from W and P. Hence P = −W/(2τ+τ_h) is a conditional, post-selected quantity, not the cycle-averaged engine output. If p_{n∗} is not unity, the true mean power is p_{n∗} times the reported value. This selection is an input assumption that creates the apparent gain rather than a consequence of the thermodynamic model.

full rationale

The central claim—that cavity squeezing drives the Otto efficiency above the standard quantum Otto limit—is not supported by the paper's own analytic derivation. Combining Eqs. (9), (21), and (30) gives exactly η = 1 − B_L/B_H, independent of r, φ, τ_h, and the bath parameters; the above-Otto effect appears only in the numerical discussion and in the resource-excluded definition of η. The paper itself concedes in Appendix D that the resource-inclusive efficiency η_tot = −W/(Q_H + W_sq + W_meas) obeys the generalized bound, and the Conclusion states that the enhancement is 'supplied and paid for' by squeezing and measurement resources. Additionally, the cold stroke is post-selected on the outcome n∗ that minimizes the qubit energy, with no averaging over p_{n∗} in the reported work or power. Together these choices make the claimed super-Otto behavior a definitional artifact: the figure of merit is constructed to omit the fuel and measurement costs, and the analytic equations—where the costs are consistently excluded—yield no enhancement at all. No load-bearing self-citation chain was identified; the problem is not citation-based but definitional and internal inconsistency.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The engine's headline performance depends on hand-chosen dissipative parameters (λ, γ), a fixed ultrastrong coupling α=0.5, finite stroke times, and a post-selected measurement whose success probability is never evaluated. The perturbative derivation introduces at least two unspecified/ad hoc quantities (κ_0 and R_sq). The central efficiency claim is further shaped by the bookkeeping choice of excluding the preparation cost of the squeezed state and the measurement cost. These are the free choices the reader 'pays for'.

free parameters (10)

bath reorganization energies λ_c, λ_q = 0.1, 0.1
Hand-chosen in Appendix B; set the qubit and cavity dissipation strengths and thereby Γ0, Q_H and W.
Drude cutoff frequencies γ_c, γ_q = 0.1, 0.1
Hand-chosen; set bath memory time and the dephasing rate Γ_φ.
HEOM truncation (max depth, N_k) = 4, 4
Hand-chosen truncation orders; convergence only asserted, affecting numerical accuracy of all figures.
cavity-qubit displacement α = 0.5
Coupling g = ωα chosen for Figs. 2-5; enters Γ_eff, δQ_H, and the crossing overlap.
cavity frequency ω = 5
Sets energy scale; chosen.
stroke durations τ, τ_h = 1, 1
Finite-time stroke durations; chosen; the non-adiabaticity depends on them.
hot bath temperature T = 1 (and 0.5, 2 in Fig. 3)
Bath temperature; chosen per simulation.
field ratio B_H/B_L scan = B_L=0.5, B_H up to 10 B_L
Determines the Otto bound η_Otto and the power-efficiency curve.
coefficient κ_0 in δΓ^(2)(t) = α² + 1/2 (≈0.75)
Ad hoc constant in Eq. (C21) for the second-order transient rate; not derived from the Hamiltonian.
function f(ω,φ,τ) in R_sq(τ) = unspecified (computed numerically)
First-order squeezing correction to the Franck-Condon overlap (Eq. C37); only stated to exist, not given.

assumptions (6)

domain assumption Two-qubit QRM Hamiltonian with symmetric constraints (J_x=J_y=J/2, J_z=0, g_1=g_2=g/2, B_1=B_2=B/2) and initial condition in sector A fully describes the engine.
The paper confines dynamics to sector A and ignores sector B leakage (Section II). If the B sector is populated, the working medium has extra levels and the derived W and Q_H fail.
domain assumption HEOM with Drude-Lorentz baths, hierarchy depth 4 and N_k=4, is numerically exact for the parameters used.
Appendix B states the hierarchy and parameters but only asserts convergence ('Convergence is verified'), providing no error analysis.
domain assumption The hot bath couples to the qubits only through σ_x^A and to the cavity through (a+a†), so population dynamics decouple from coherences and obey a rate equation (C5-C8).
This structure yields the closed-form Q_H and W2; any additional bath coupling would change the relaxation kernel Φ.
ad hoc to paper The cold stroke is an ideal projective measurement Π_n(α) = D(α)|n⟩⟨n|D†(α)⊗I with post-selection on the outcome n* that minimizes qubit energy.
Post-selection on n* is essential to the cycle; the probability p_{n*} is not modeled, so the reported output is a conditional rather than an average quantity.
ad hoc to paper The work in the unitary strokes is computed from the qubit-only Hamiltonian B(t)σ_z^A (Eqs. 6 and 29), excluding the cavity energy change due to squeezing and measurement.
This bookkeeping choice makes η = -W/Q_H equal to the Otto bound; including cavity energy would change W and the reported efficiency.
ad hoc to paper The transient relaxation correction requires a coefficient κ_0 ≈ α²+1/2 (Eq. C21) and a squeezing overlap correction R_sq(τ) ≈ 1 + r f(ω,φ,τ) (Eq. C37), which are not derived from a complete calculation.
κ_0 and f are inserted to close the perturbative expansion; f is 'computed numerically' and left unspecified.

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Cite this review

Pith. "Pith review of Squeezing-Fueled Quantum Otto Engine via Measurement-Induced Cooling: The Two-Qubit Quantum Rabi Model." pith.science (2026). https://pith.science/paper/G7LRVSBC

@misc{pith2026260802521,
  author       = {Pith},
  title        = {Pith review of: Squeezing-Fueled Quantum Otto Engine via Measurement-Induced Cooling: The Two-Qubit Quantum Rabi Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7LRVSBC}},
  note         = {Machine review of arXiv:2608.02521}
}

read the original abstract

We investigate a quantum Otto engine (QOE) constructed from the two-qubit quantum Rabi model, operating within a cavity quantum electrodynamics (QED) architecture. The engine operates with two qubits as the working substance and a single non-Markovian hot thermal bath, modeled via the hierarchical equations of motion (HEOM) formalism. In place of a conventional cold thermal reservoir, the cooling stroke is realized through a projective measurement protocol on the cavity mode, which acts as an ancillary subsystem and effectively mimics a cold bath for the qubit working medium via measurement back-action. A squeezing drive applied to the cavity mode serves as a quantum fuel. We demonstrate that cavity squeezing systematically enhances both the power output and operational efficiency of the engine - the work extracted per unit of heat drawn from the hot bath-driving it above the standard quantum Otto limit. In the limit-cycle regime, the efficiency, while remaining above the Otto bound throughout, asymptotically converges to it from above. This identifies squeezing as a controllable quantum resource for thermodynamic optimization. Our results reveal that the interplay between qubit-cavity coupling, measurement-induced cooling, and non-equilibrium squeezing gives rise to a multi-resource thermodynamic architecture with performance characteristics inaccessible to conventional two-bath quantum Otto engines, thereby providing a concrete route toward experimentally realizable quantum heat engines in cavity QED platforms.

Figures

Figures reproduced from arXiv: 2608.02521 by the authors.

**Figure 2.** FIG. 2: Parametric plot showing the variation of Power [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: Output power [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: Thermodynamic contour maps of the quantum [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5: Multi–cycle performance of the squeezed ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references

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The modulation follows a linear schedule:B(t) =B L + (BH −B L)t/τ

Expansion StrokeA→B In the first stage, the qubit frequency is non- adiabatically ramped from a lower valueB L to a higher valueB H over a finite time durationτ. The modulation follows a linear schedule:B(t) =B L + (BH −B L)t/τ. After this transformation, the evolved state of the sys- tem is given by ˆρ global B = ˆUexp(τ)ˆρglobal A ˆU † exp(τ), where ˆUe...
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The resulting effective dynamics enables analytical treatment via Braak’s solu- tion to the asymmetric QRM [12, 55]. Upon imposing the symmetric constraints,J x =J y = J/2, Jz = 0,g 1 =g 2 =g/2, andB 1(t) =B 2(t) =B(t)/2, this configuration allows the system to be described in terms of symmetric and antisymmetric spin subspaces. Defining ⃗S=⃗ σ(1) +⃗ σ(2)...
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To capture the non-perturbative system–bath correlations beyond the Born–Markov approximation, we employ the hierarchical equations of motion (HEOM) formalism [56]

Hot Isochoric StrokeB→C During the hot isochoric stroke, the qubit frequency is held fixed atB H while the system interacts with a thermal reservoir at temperatureTfor a finite timet h, partially thermalising the joint qubit–cavity state. To capture the non-perturbative system–bath correlations beyond the Born–Markov approximation, we employ the hierarchi...
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Compression StrokeC→D The global state atCis given by Eq. (11). The unitary dynamics during this compression stroke are given by: ˆUcom(τ) = ˆUexp(t−τ).(23) As in the case of expansion stage, since [H eff ,ˆσA z ] = 0 holds also during the compression stroke, the unitary ˆUcom(τ) maps each qubit sector to itself: ˆUcom(τ) |m⟩⟨m′| ⊗Ωcav m,m′ ˆU † com(τ) = ...
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We cool the qubits via a measurement-based stroke that projects the cavity mode onto displaced Fock states

Cold Stroke via Projective MeasurementD→A ′ After the unitary return stroke C→D, the system re- sides in a global state ˆρglobal D . We cool the qubits via a measurement-based stroke that projects the cavity mode onto displaced Fock states. We define the necessary pro- jector as Π n(α) =D(α)|n⟩⟨n|D †(α)⊗I q1q2 . For eachn, the cavity is projected, and the...
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The relevant Hamil- tonian ˆHA can be diagonalized with Glauber’s displace- ment operator ˆD(α) = exp(αa † −α ∗a)

Exact diagonalization when r = 0 Block A: This subspace consists of states with total spinS= 1, and the coupling to the qubits modifies the cavity mode through displacement. The relevant Hamil- tonian ˆHA can be diagonalized with Glauber’s displace- ment operator ˆD(α) = exp(αa † −α ∗a). The eigenstates and corresponding energies are: |ψ±± n ⟩(0) =|±±⟩⊗ ˆ...
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Perturbative approach forr̸= 0 a. Perturbative Energy Corrections in Block A To understand the influence of squeezing on the eigenenergies of the system, we consider the effective Hamiltonian ˆHeff = ˆHA + ˆV ,(A3) whereH A denotes the unperturbed Hamiltonian of the two-qubit Rabi model andVis the squeezing term treated as a perturbation, as given in Eq. ...
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Derivation of the relaxation kernelΦ(t)and the effective rateΓ eff During the hot isochoric stroke, the populations of the two relevant qubit sectors evolve under bath-induced transitions. Since the population dynamics decouples from coherences in this subspace, it can be written in rate-equation form: ˙p++(t) =−Γ ↓(t)p ++(t) + Γ↑(t)p −−(t),(C5) 12 withp ...

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

The modulation follows a linear schedule:B(t) =B L + (BH −B L)t/τ

Expansion StrokeA→B In the first stage, the qubit frequency is non- adiabatically ramped from a lower valueB L to a higher valueB H over a finite time durationτ. The modulation follows a linear schedule:B(t) =B L + (BH −B L)t/τ. After this transformation, the evolved state of the sys- tem is given by ˆρ global B = ˆUexp(τ)ˆρglobal A ˆU † exp(τ), where ˆUe...

[2] [2]

The resulting effective dynamics enables analytical treatment via Braak’s solu- tion to the asymmetric QRM [12, 55]. Upon imposing the symmetric constraints,J x =J y = J/2, Jz = 0,g 1 =g 2 =g/2, andB 1(t) =B 2(t) =B(t)/2, this configuration allows the system to be described in terms of symmetric and antisymmetric spin subspaces. Defining ⃗S=⃗ σ(1) +⃗ σ(2)...

[3] [3]

To capture the non-perturbative system–bath correlations beyond the Born–Markov approximation, we employ the hierarchical equations of motion (HEOM) formalism [56]

Hot Isochoric StrokeB→C During the hot isochoric stroke, the qubit frequency is held fixed atB H while the system interacts with a thermal reservoir at temperatureTfor a finite timet h, partially thermalising the joint qubit–cavity state. To capture the non-perturbative system–bath correlations beyond the Born–Markov approximation, we employ the hierarchi...

[4] [4]

Compression StrokeC→D The global state atCis given by Eq. (11). The unitary dynamics during this compression stroke are given by: ˆUcom(τ) = ˆUexp(t−τ).(23) As in the case of expansion stage, since [H eff ,ˆσA z ] = 0 holds also during the compression stroke, the unitary ˆUcom(τ) maps each qubit sector to itself: ˆUcom(τ) |m⟩⟨m′| ⊗Ωcav m,m′ ˆU † com(τ) = ...

[5] [5]

We cool the qubits via a measurement-based stroke that projects the cavity mode onto displaced Fock states

Cold Stroke via Projective MeasurementD→A ′ After the unitary return stroke C→D, the system re- sides in a global state ˆρglobal D . We cool the qubits via a measurement-based stroke that projects the cavity mode onto displaced Fock states. We define the necessary pro- jector as Π n(α) =D(α)|n⟩⟨n|D †(α)⊗I q1q2 . For eachn, the cavity is projected, and the...

[6] [6]

The relevant Hamil- tonian ˆHA can be diagonalized with Glauber’s displace- ment operator ˆD(α) = exp(αa † −α ∗a)

Exact diagonalization when r = 0 Block A: This subspace consists of states with total spinS= 1, and the coupling to the qubits modifies the cavity mode through displacement. The relevant Hamil- tonian ˆHA can be diagonalized with Glauber’s displace- ment operator ˆD(α) = exp(αa † −α ∗a). The eigenstates and corresponding energies are: |ψ±± n ⟩(0) =|±±⟩⊗ ˆ...

[7] [7]

Perturbative approach forr̸= 0 a. Perturbative Energy Corrections in Block A To understand the influence of squeezing on the eigenenergies of the system, we consider the effective Hamiltonian ˆHeff = ˆHA + ˆV ,(A3) whereH A denotes the unperturbed Hamiltonian of the two-qubit Rabi model andVis the squeezing term treated as a perturbation, as given in Eq. ...

[8] [8]

Derivation of the relaxation kernelΦ(t)and the effective rateΓ eff During the hot isochoric stroke, the populations of the two relevant qubit sectors evolve under bath-induced transitions. Since the population dynamics decouples from coherences in this subspace, it can be written in rate-equation form: ˙p++(t) =−Γ ↓(t)p ++(t) + Γ↑(t)p −−(t),(C5) 12 withp ...