Asymmetric Quantum Harmonic Otto Engine Under Hot Squeezed Thermal Reservoir

Kirandeep Kaur; Monika; Shishram Rebari; Varinder Singh

REVIEW 3 major objections 4 minor 41 references

Asymmetric Quantum Harmonic Otto Engine Under Hot Squeezed Thermal Reservoir

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Quantum Otto engine: sudden expansion caps efficiency at 1/2

desk verdict The algebra is clean but the central 1/2 efficiency bound for sudden expansion ignores the squeezing phase, which makes it not universal. read the letter →

arxiv 2411.16115 v2 pith:7B6KM5DO submitted 2024-11-25 quant-ph cond-mat.stat-mech

Monika , Kirandeep Kaur , Varinder Singh , Shishram Rebari This is my paper · ORCID

classification quant-phcond-mat.stat-mech

keywords quantumOttoenginesqueezedthermalreservoirtime-dependentharmonicoscillatorsuddenexpansionstrokecompressionefficiencyboundfrictionphasediagram

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 studies a quantum Otto engine whose working fluid is a harmonic oscillator, with one of the two work strokes driven suddenly and the other adiabatically. Its central claim is that the order of the sudden stroke matters decisively: if the expansion is sudden, the efficiency is capped at $1/2$ no matter how strongly the hot reservoir is squeezed, whereas if the compression is sudden, the efficiency can approach unity. The paper backs this with analytic upper bounds on efficiency and explicit formulas for the efficiency at maximum work, and shows that increasing squeezing grows the engine's operating region in the cycle's phase diagram at the expense of the refrigerator regime. The relevance is that squeezed thermal reservoirs are often proposed as a quantum resource to beat Carnot efficiency; this work identifies a separate stroke-dependent ceiling imposed by quantum friction.

What carries the argument

The central object is the time-dependent harmonic oscillator, whose nonadiabatic work strokes are characterized by adiabaticity parameters $\lambda_{AB}$ and $\lambda_{CD}$. In the sudden-switch limit these take the value $\lambda = (1+z^2)/(2z)$ with $z = \omega_c/\omega_h$, while in the adiabatic limit $\lambda = 1$. The analysis is carried out in terms of the compression ratio $z$, the inverse-temperature ratio $\tau = \beta_h/\beta_c$, and the squeezing parameter $r$ entering through $\cosh(2r)$. The key mechanism that produces the $1/2$ ceiling is the factorization of the sudden-expansion efficiency into two positive pieces, one of which is bounded by $1/2$; no equivalent factor appears in the sudden-compression case, whose efficiency bound approaches unity as $r$ grows.

What would settle it

A direct numerical integration of the oscillator's Lindblad master equation with a squeezed bath, without enforcing full relaxation to the ideal squeezed thermal state, would test whether the sudden-expansion efficiency can exceed $1/2$; observing an efficiency above $1/2$ in such a simulation, or in a trapped-ion realization of the sudden-expansion cycle, would falsify the claimed bound.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that for an asymmetric quantum harmonic Otto engine with a hot squeezed thermal reservoir, in the high-temperature limit the efficiency of the sudden-expansion configuration is bounded above by $1/2$, while the sudden-compression configuration can approach unit efficiency. The authors obtain closed-form expressions for the upper bound on efficiency, Eqs. (16) and (22), and for the efficiency at maximum work, Eqs. (17) and (23), both depending only on the Carnot efficiency $\eta_c$ and the squeezing parameter $r$. They attribute the sudden-expansion ceiling to quantum friction: the sudden frequency switch creates coherences that carry parasitic energy later dissipated as heat. They also compute the full phase diagram and find that squeezing enlarges the engine regime and shrinks the refrigerator regime, with only the engine mode surviving in the large-squeezing limit.

Load-bearing premise

The result assumes the hot isochore relaxes the oscillator completely to a squeezed thermal state with the reservoir's temperature and squeezing parameter; if relaxation is incomplete, or the squeezing phase participates in the energy balance, the efficiency bounds and optimal points no longer follow.

Editorial extensions

If this is right

For the sudden-expansion configuration, squeezing alone cannot push efficiency above $1/2$; only making the expansion more adiabatic would raise the ceiling.
The sudden-compression configuration is the promising route to high efficiency, approaching unity with strong squeezing.
Efficiency at maximum work depends only on $\eta_c$ and $r$, not on the oscillator frequencies, so the optimal work point $z^* = [\tau \,\mathrm{sech}(2r)]^{1/3}$ is universal in these variables.
Increasing squeezing expands the engine mode and contracts the refrigerator mode; in the $r \to \infty$ limit the Otto cycle is an engine for all $\tau$ and $z$ in the studied region.
The compression ratio $z = \sqrt{\tau \,\mathrm{sech}(2r)}$ marks the crossover where sudden expansion and sudden compression give equal work output.

Reading between the lines

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

The $1/2$ ceiling suggests a design rule: for any working fluid, sudden expansion that generates coherences caps efficiency, so the expansion stroke should be the slow controlled stroke whenever high efficiency is the goal.
If the squeezed reservoir does not fully imprint its phase on the oscillator, the efficiency bounds are likely to shift; testing that sensitivity would require dropping the full-relaxation assumption.
The universality of $z^*$ implies a calibration protocol: measure work at that ratio to infer the effective squeezing parameter of a reservoir without full state tomography.
The phase-diagram result that squeezing suppresses refrigeration could matter for quantum absorption refrigerators, where the same cycle is run in reverse.

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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

Quick take: the paper is cleanly written and the algebra is consistent, but the central result—the 1/2 efficiency cap for sudden expansion—does not survive a proper treatment of the squeezing phase. The energy expressions in Eqs. (2)–(4) treat the squeezed thermal state as if its quench dynamics were phase-independent. They are not. For a squeezed thermal state with phase φ, the variances in x and p are asymmetric unless cos φ = 0, so after a sudden frequency change the energy carries a term (1−z²) sinh(2r) cos φ. The paper’s λ_CD scaling picks out the special case φ = π/2 and silently discards all other phases.

What the paper does well: the optimization of efficiency and work for the two asymmetric configurations is carried through in closed form, the cubics and trigonometric solutions check out, and the phase diagrams in Figs. 5–6 are useful. The reciprocal-sum argument for the 1/2 bound is elegant within the assumed model.

The soft spot is load-bearing. For φ = π, the stress-test gives η → 1−z² near the positive-work threshold, so the bound is not universal. For φ = 0 the engine shuts down at large squeezing. The authors never specify the phase, so the claims are overgeneralized. Fixing this means either restricting the model to a stated phase or redoing the optimization with the full phase-dependent energy. The latter would change the cubics, the bounds, and the phase diagrams.

The reader’s conditional verdict is fair on the algebra, but the stress-test note lands. Secondary issues: heavy self-citation (Refs. [20],[23],[24],[27],[37],[38]) with little demarcation of what is new, and a few typos ("he compression ratio", "coth" formatting). These are minor.

This paper is for the quantum thermodynamics / quantum Otto engine community. It deserves a serious referee, because the question is real and the analytic machinery is mostly sound. But the referee should send it back for major revision on the phase issue rather than accepting the main result as stated. I would not cite the 1/2 bound or the phase diagram until the phase dependence is settled.

Referee Report

3 major / 4 minor

Summary. The paper studies a quantum Otto engine whose working fluid is a time-dependent harmonic oscillator, coupled to a hot squeezed thermal reservoir and a cold thermal reservoir. Asymmetry is introduced by making one adiabatic branch sudden and the other adiabatic: either sudden expansion with adiabatic compression, or sudden compression with adiabatic expansion. The authors derive high-temperature analytic expressions for work and efficiency in both configurations, optimize them to obtain upper bounds on efficiency and efficiency at maximum work, and construct the full phase diagram of the cycle. The main advertised results are that the sudden-expansion configuration has an efficiency upper bound of only 1/2, while the sudden-compression configuration can approach unity, and that increasing squeezing enlarges the engine regime at the expense of the refrigeration regime.

Significance. If the central claims were correct, the sharp asymmetry between sudden expansion and sudden compression would be a useful addition to the literature on finite-time quantum Otto engines with squeezed reservoirs. The paper is self-contained in its derivations and offers closed-form expressions, and the proof of the 1/2 bound is arithmetically correct within the phase-independent model assumed in Eqs. (1)-(6). However, the main claim rests on a sudden-quench energy formula that is not valid for a generic squeezed thermal state, because it omits the squeezing phase; the paper never specifies that phase. In addition, the analytic optimization solution for the sudden-expansion case is used outside its domain of validity. These issues affect the central efficiency bounds and the phase diagrams, so the present version requires substantial revision.

major comments (3)

[Section II, Eq. (4), and Section III A] The sudden-quench energy of a squeezed thermal state is not phase-independent. For a squeezed thermal state with squeezing parameter r and phase phi, after a sudden quench omega_h -> omega_c = z omega_h one has E_D = (omega_h/4) coth(beta_h omega_h/2)[(1+z^2) cosh(2r) + (1-z^2) sinh(2r) cos(phi)], using the standard convention in which <x^2> is proportional to cosh(2r) - sinh(2r) cos(phi) and <p^2> to cosh(2r) + sinh(2r) cos(phi). The phase term is absent from Eq. (4), which is valid only for cos(phi)=0. In the high-temperature limit, taking cos(phi)=-1 gives E_D = (1/(2 beta_h))(e^{-2r} + z^2 e^{2r}), leading to W = (1-z)/(2 z beta_h)[z(1+z)e^{2r} - 2 tau] and, for large r, Qh ~ e^{2r}/(2 beta_h) and eta ~ 1-z^2. For large r, any fixed z<1 (e.g. z=1/2) lies inside the engine window and gives eta~3/4, and eta approaches 1 as z approaches the lower edge of the engine window. This directly contradicts the abstract's claim that the sudden-expansion efficiency is at most 1/2. The squeezing phase must be specified, and the phase-dependent corrections must be included in the work, efficiency, and phase-diagram calculations.
[Appendix A 1 and Eq. (15)] The discriminant condition for the sudden-expansion cubic is stated incorrectly, and the trigonometric solution in Eq. (15) is used outside its domain. For Eq. (14), the discriminant is D = 108 tau^2 (2 tau - cosh(2r))(tau - cosh(2r))^2 cosh^3(2r), which is positive only when cosh(2r) < 2 tau. For typical parameters in Figure 2, e.g. tau = 0.2 and r = 0, or for any r with cosh(2r) > 2 tau, D < 0, the cubic has one real root, and the argument of cos^{-1} in Eq. (15) lies outside [-1,1]. The plotted curves in Figure 2 and the expression for eta_up^SE in Eq. (16) cover exactly those parameter regions, so Eq. (15) cannot be the solution used there. The authors should either provide the Cardano root for D < 0 or explicitly restrict Eqs. (15) and (16) to cosh(2r) < 2 tau and supply correct numerical or alternative analytic results elsewhere.
[Section IV, Eqs. (24)-(25), and Table I] The phase dependence also invalidates the sudden-expansion entries of Table I and Figures 5. Because Q_c^SE = E_A - E_D inherits the phase-dependent E_D discussed above, the engine/refrigerator boundary for sudden expansion changes with the squeezing phase. For the same counterexample as in the first comment (cos(phi)=-1, large r), the engine window is approximately sqrt(2 tau) e^{-r} < z < 1 with efficiency approaching 1-z^2, which is qualitatively different from the table's boundary z >= (sqrt(1 + 8 tau / cosh(2r)) - 1)/2. The conclusion that squeezing enlarges the engine mode at the expense of the refrigeration regime is therefore not established for a generic squeezed reservoir unless the phase is specified and included in the derivation.

minor comments (4)

[Section III A, after Eq. (12)] The text says "he compression ratio" where it should say "the compression ratio."
[Abstract and Section III A] Equation (11) is a strict inequality, so the abstract's phrase "is 1/2 only" should be softened to "is bounded above by 1/2 and approaches 1/2 only in the limit" to match the mathematics.
[Fig. 1 caption] The provided manuscript has garbled placeholder characters in the figure caption; this should be fixed in the production version.
[Section IV] The statement that for r -> infinity the Otto cycle has only the engine mode is asserted without derivation; a short limiting argument would make this claim easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central efficiency bounds and phase diagrams are derived algebraically from stated model energies, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained given its modeling assumptions. Equations (1)-(4) define the cycle energy values using the standard Husimi results for harmonic oscillator states, and Eqs. (5)-(9) are algebraic consequences for heat, work, and efficiency. The sudden-expansion upper bound η_SE < 1/2 follows directly from the positivity condition and the definition f1 = 1 - ω_c^2/ω_h^2, not from any quantity constructed to equal the claimed bound. The high-temperature work and efficiency expressions, the optimization conditions, and the cubic equations are all derived within the paper, and the trigonometric solutions are standard results reproduced in Appendix A. The paper does cite several works involving its own authors (Refs. [20], [23], [24], [27], [37], [38]), but these citations are contextual or provide mathematical tools; they do not supply the central 1/2 or unit-efficiency conclusions. No fitted parameter, no target value, and no uniqueness claim is imported to force the results. A possible physical objection about the phase dependence of a squeezed state under a sudden quench would be a correctness or modeling issue, not circularity, because the paper's equations do not reduce to their inputs by definition.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; model parameters (temperature ratio, frequency ratio, squeezing) are scanned rather than fitted. The main domain assumptions are ideal thermalization to the squeezed reservoir state and the high-temperature limit.

assumptions (4)

standard math Energy of a thermal harmonic oscillator is (omega/2)coth(beta omega/2), with a lambda factor after sudden frequency change.
Used in Eqs. (1)-(4), based on Husimi [30]; standard quantum mechanics.
domain assumption The hot squeezed reservoir drives the oscillator into a squeezed thermal state with energy (omega_h/2)coth(beta_h omega_h/2)cosh(2r).
Section II and Eq. (3); assumes ideal thermalization to the squeezed steady state.
domain assumption High-temperature approximation coth(beta_i omega_i/2) approximately 2/(beta_i omega_i).
Used to obtain closed-form optimization results in Section III; stated as the working regime.
standard math Sudden-switch adiabaticity parameter lambda = (omega_c^2 + omega_h^2)/(2 omega_c omega_h).
Used for sudden strokes; standard result for a sudden frequency quench, cited to [30].

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Pith. "Pith review of Asymmetric Quantum Harmonic Otto Engine Under Hot Squeezed Thermal Reservoir." pith.science (2026). https://pith.science/paper/7B6KM5DO

@misc{pith2026241116115,
  author       = {Pith},
  title        = {Pith review of: Asymmetric Quantum Harmonic Otto Engine Under Hot Squeezed Thermal Reservoir},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7B6KM5DO}},
  note         = {Machine review of arXiv:2411.16115}
}

read the original abstract

We study a quantum harmonic Otto engine under a hot squeezed thermal reservoir with asymmetry between the two adiabatic branches introduced by considering different speeds of the driving protocols. In the first configuration, the driving protocol for the expansion stroke is sudden-switch in nature and compression stroke is driven adiabatically, while the second configuration deals with the converse situation. In both cases, we obtain analytic expressions for the upper bound on efficiency and efficiency at optimal work output, which reveals a significant difference between the two configurations. Additionally, we find that the maximum achievable efficiency in sudden expansion case is 1/2 only while it approaches unity for the sudden compression stroke. Further, we study the effect of increasing degree of squeezing on the efficiency and work output of the engine and indicate the optimal operational regime for both configurations under consideration. Finally, by studying the full phase-diagram of the Otto cycle we observe that the operational region of the engine mode grows with increasing squeezing at the expense of refrigeration regime.

Figures

Figures reproduced from arXiv: 2411.16115 by the authors.

**Figure 2.** FIG. 2. Variation in upper bound on efficiency and efficiency [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4. Efficiency for sudden expansion (Eq.13) and com [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. Complete phase diagram of an Otto cycle as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Complete phase diagram of the Otto cycle as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 38 canonical work pages

[20]

Piccitto, M

G. Piccitto, M. Campisi, and D. Rossini, New Journal of Physics 24, 103023 (2022)

2022
[23]

Shastri and B

R. Shastri and B. P. Venkatesh, Physical Review E 106, 024123 (2022)

work page 2022
[24]

P. E. Harunari, F. S. Filho, C. E. Fiore, and A. Rosas, Physical Review Research 3, 023194 (2021)

work page 2021
[27]

Zhang, Physica A: Statistical Mechanics and its Ap- plications 559, 125083 (2020)

Y. Zhang, Physica A: Statistical Mechanics and its Ap- plications 559, 125083 (2020)

work page 2020
[37]

Alecce, F

A. Alecce, F. Galve, N. L. Gullo, L. Dell’Anna, F. Plas- tina, and R. Zambrini, New Journal of Physics17, 075007 (2015)

work page 2015
[38]

C ¸ akmak, F

S. C ¸ akmak, F. Altintas, and¨O. E. M¨ ustecaplıo˘ glu, Phys- ica Scripta 91, 075101 (2016)

work page 2016
[1]

Here, A = −3τ /2 cosh(2r), B = 0, C = τ [2τ − cosh(2r)] /2 cosh2(2r)

Sudden Expansion Case For the sudden expansion stroke, discriminant D of the equation, z3 − 3z2τ 2 cosh(2r) + τ [2τ − cosh(2r)] 2 cosh2(2r) = 0, (A4) is given by D = 108 τ 2 [2τ − cosh(2r)] [τ − cosh(2r)]2 cosh3(2r) > 0 . Here, A = −3τ /2 cosh(2r), B = 0, C = τ [2τ − cosh(2r)] /2 cosh2(2r)

work page
[2]

Here, A = 0, B = −3τ cosh(2r)/ cosh(2r) [2 cosh(2r) − τ ], C = 2τ 2/ cosh(2r) [2 cosh(2r) − τ ]

Sudden Compression Case In our case, the discriminant of cubic equation z3 − z [3τ cosh(2r)] cosh(2r) [2 cosh(2r) − τ ] + 2τ 2 cosh(2r) [2 cosh(2r) − τ ] = 0 (A5) will be D = 108 τ 3 [2 cosh(2r) − τ ] [τ − cosh(2r)]2 cosh2(2r)>0. Here, A = 0, B = −3τ cosh(2r)/ cosh(2r) [2 cosh(2r) − τ ], C = 2τ 2/ cosh(2r) [2 cosh(2r) − τ ]

work page

Show all 41 references

[3]

Y. A. Cengel, M. A. Boles, and M. Kano˘ glu, Thermo- dynamics: an engineering approach, Vol. 5 (McGraw-hill New York, 2011)

2011
[4]

Borgnakke and R

C. Borgnakke and R. E. Sonntag, Fundamentals of ther- modynamics (John Wiley & Sons, 2020)

2020
[5]

K. Kaur, V. Singh, J. Ghai, S. Jena, and ¨O. E. M¨ ustecaplıo˘ glu, Physica A: Statistical Mechanics and its Applications 576, 125892 (2021)

2021
[6]

K. Kaur, A. Jain, L. S. Singh, R. Singla, and S. Rebari, Journal of Non-Equilibrium Thermodynamics (2024)

2024
[7]

Singh and R

V. Singh and R. S. Johal, Physical Review E 98, 062132 (2018)

2018
[8]

Klaers, S

J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Physical Review X 7, 031044 (2017)

2017
[9]

Manzano, F

G. Manzano, F. Galve, R. Zambrini, and J. M. Parrondo, Physical review E 93, 052120 (2016)

2016
[10]

Kosloff and Y

R. Kosloff and Y. Rezek, Entropy 19, 136 (2017)

2017
[11]

Huang, T

X. Huang, T. Wang, and X. Yi, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 86, 051105 (2012)

2012
[12]

Manzano, Physical Review E 98, 042123 (2018)

G. Manzano, Physical Review E 98, 042123 (2018)

2018
[13]

R. J. de Assis, J. S. Sales, J. A. da Cunha, and N. G. de Almeida, Physical Review E 102, 052131 (2020)

2020
[14]

J. Wang, J. He, and Y. Ma, Physical Review E 100, 052126 (2019)

2019
[15]

Singh and R

V. Singh and R. S. Johal, Physical Review E 100, 012138 (2019)

2019
[16]

Schnabel, Physics Reports 684, 1 (2017)

R. Schnabel, Physics Reports 684, 1 (2017)

2017
[17]

Fearn and M

H. Fearn and M. Collett, Journal of Modern Optics 35, 553 (1988)

1988
[18]

Marian and T

P. Marian and T. A. Marian, Physical Review A 47, 4474 (1993)

1993
[19]

J. Son, P. Talkner, and J. Thingna, PRX Quantum 2, 040328 (2021)

2021
[21]

N. M. Myers, O. Abah, and S. Deffner, New journal of physics 23, 105001 (2021)

2021
[22]

K. Kaur, S. Rebari, and V. Singh, Journal of Non- Equilibrium Thermodynamics (2024)

2024
[25]

Singh, V

V. Singh, V. Shaghaghi, T. Pandit, C. Beetar, G. Be- nenti, and D. Rosa, arXiv preprint arXiv:2310.06512 (2023)

2023 arXiv
[26]

Singh and ¨O

V. Singh and ¨O. E. M¨ ustecaplıo˘ glu, Physical Review E 102, 062123 (2020)

2020
[28]

Roßnagel, O

J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, Physical review letters 112, 030602 (2014)

2014
[29]

Singh, S

V. Singh, S. Singh, O. Abah, and ¨O. E. M¨ ustecaplıo˘ glu, Physical Review E 106, 024137 (2022)

2022
[30]

V. V. Nautiyal, arXiv preprint arXiv:2411.13041 (2024)

2024 arXiv
[31]

V. V. Nautiyal, R. S. Watson, and K. V. Kheruntsyan, 9 New Journal of Physics (2024)

2024
[32]

Husimi, Progress of Theoretical Physics 9, 381 (1953)

K. Husimi, Progress of Theoretical Physics 9, 381 (1953)

1953
[33]

Rezek, Entropy 12, 1885 (2010)

Y. Rezek, Entropy 12, 1885 (2010)

2010
[34]

Plastina, A

F. Plastina, A. Alecce, T. J. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. Zambrini, Physical review letters 113, 260601 (2014)

2014
[35]

C ¸ akmak, F

S. C ¸ akmak, F. Altintas, A. Gen¸ cten, and ¨O. E. M¨ ustecaplıo˘ glu, The European Physical Journal D71, 1 (2017)

2017
[36]

C. L. Latune, I. Sinayskiy, and F. Petruccione, The Eu- ropean Physical Journal Special Topics 230, 841 (2021)

2021
[39]

Singh, Physical Review Research 2, 043187 (2020)

V. Singh, Physical Review Research 2, 043187 (2020)

2020
[40]

Singh, T

V. Singh, T. Pandit, and R. S. Johal, Physical Review E 101, 062121 (2020)

2020
[41]

Barnett and P

S. Barnett and P. M. Radmore, Methods in theoretical quantum optics, Vol. 15 (Oxford University Press, 2002)

2002

Pith tools

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

[1] [20]

Piccitto, M

G. Piccitto, M. Campisi, and D. Rossini, New Journal of Physics 24, 103023 (2022)

2022

[2] [23]

Shastri and B

R. Shastri and B. P. Venkatesh, Physical Review E 106, 024123 (2022)

work page 2022

[3] [24]

P. E. Harunari, F. S. Filho, C. E. Fiore, and A. Rosas, Physical Review Research 3, 023194 (2021)

work page 2021

[4] [27]

Zhang, Physica A: Statistical Mechanics and its Ap- plications 559, 125083 (2020)

Y. Zhang, Physica A: Statistical Mechanics and its Ap- plications 559, 125083 (2020)

work page 2020

[5] [37]

Alecce, F

A. Alecce, F. Galve, N. L. Gullo, L. Dell’Anna, F. Plas- tina, and R. Zambrini, New Journal of Physics17, 075007 (2015)

work page 2015

[6] [38]

C ¸ akmak, F

S. C ¸ akmak, F. Altintas, and¨O. E. M¨ ustecaplıo˘ glu, Phys- ica Scripta 91, 075101 (2016)

work page 2016

[7] [1]

Here, A = −3τ /2 cosh(2r), B = 0, C = τ [2τ − cosh(2r)] /2 cosh2(2r)

Sudden Expansion Case For the sudden expansion stroke, discriminant D of the equation, z3 − 3z2τ 2 cosh(2r) + τ [2τ − cosh(2r)] 2 cosh2(2r) = 0, (A4) is given by D = 108 τ 2 [2τ − cosh(2r)] [τ − cosh(2r)]2 cosh3(2r) > 0 . Here, A = −3τ /2 cosh(2r), B = 0, C = τ [2τ − cosh(2r)] /2 cosh2(2r)

work page

[8] [2]

Here, A = 0, B = −3τ cosh(2r)/ cosh(2r) [2 cosh(2r) − τ ], C = 2τ 2/ cosh(2r) [2 cosh(2r) − τ ]

Sudden Compression Case In our case, the discriminant of cubic equation z3 − z [3τ cosh(2r)] cosh(2r) [2 cosh(2r) − τ ] + 2τ 2 cosh(2r) [2 cosh(2r) − τ ] = 0 (A5) will be D = 108 τ 3 [2 cosh(2r) − τ ] [τ − cosh(2r)]2 cosh2(2r)>0. Here, A = 0, B = −3τ cosh(2r)/ cosh(2r) [2 cosh(2r) − τ ], C = 2τ 2/ cosh(2r) [2 cosh(2r) − τ ]

work page