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arxiv: 2605.21562 · v1 · pith:SEJCMJFBnew · submitted 2026-05-20 · 🪐 quant-ph · physics.atom-ph

Optimal Quantum Feshbach Engines

Aaron Wandhammer , Vincent Hardel , Paul-Antoine Hervieux , Giovanni Manfredi This is my paper

Pith reviewed 2026-05-22 09:35 UTC · model grok-4.3

classification 🪐 quant-ph physics.atom-ph
keywords quantum Feshbach enginesBose-Einstein condensateoptimal controlFeshbach resonanceOrnstein-Uhlenbeck processvariational dynamicsstochastic quantizationquantum thermodynamics
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The pith

A cost-functional minimization yields optimal driving protocols for quantum Feshbach engines in Bose-Einstein condensates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an optimization framework for high-efficiency quantum cycles realized with a trapped Bose-Einstein condensate. Control parameters are the trap stiffness and the interaction strength adjusted through a Feshbach resonance. A variational description of the condensate is combined with Nelson's stochastic quantization to reduce the dynamics to an effective Ornstein-Uhlenbeck process. Optimal protocols for each cycle stroke are then found by minimizing a user-defined cost functional that trades protocol duration against chosen physical constraints such as expended work or closeness to adiabatic evolution. If correct, the approach supplies a systematic route to optimal control for generic nonlinear Schrödinger equations and opens applications in quantum fluids and related systems.

Core claim

The optimal protocol is obtained by minimizing a user-defined cost functional that selects the best trade-off between protocol duration and arbitrary physical constraints such as the expended work or the proximity to an adiabatic evolution, and exhibits remarkable stability over repeated cycles. This follows from mapping the quantum evolution onto an effective Ornstein-Uhlenbeck process via a variational description of the condensate dynamics combined with Nelson's stochastic quantization.

What carries the argument

Minimization of a user-defined cost functional applied to the effective Ornstein-Uhlenbeck process derived from variational condensate dynamics and stochastic quantization.

If this is right

  • Shorter protocols can be achieved while respecting limits on work expenditure or deviation from adiabaticity.
  • The resulting protocols remain stable when the engine is run through many repeated cycles.
  • The same variational-plus-stochastic-quantization route supplies optimal control for any nonlinear Schrödinger equation.
  • The method extends in principle to optimal driving in nonlinear optics, quantum fluids, and quantum plasmas.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar cost-minimization protocols could be tested in other trapped quantum gases where Feshbach resonances are available.
  • The stability result suggests the framework may reduce sensitivity to small experimental imperfections in cycle repetition.
  • Direct comparison of the Ornstein-Uhlenbeck-derived controls against exact many-body numerics would quantify the approximation error.

Load-bearing premise

The variational description of the condensate dynamics combined with Nelson's stochastic quantization accurately maps the quantum evolution onto an effective Ornstein-Uhlenbeck process suitable for deriving optimal driving protocols.

What would settle it

A numerical integration of the full nonlinear Schrödinger equation for the condensate wave function under the derived protocols, checking whether the cost functional reaches the predicted minimum compared with alternative driving schedules.

Figures

Figures reproduced from arXiv: 2605.21562 by Aaron Wandhammer, Giovanni Manfredi, Paul-Antoine Hervieux, Vincent Hardel.

Figure 1
Figure 1. Figure 1: Steplike protocol on the classical stiffness ¯κ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows three optimal g-protocols that transfer the variance from si = 1 to sf = 2, for λ = 8 and three values of µ, corresponding to different durations. The evolution of the variance in panel (a) has been obtained by direct numerical solution of the GPE with the optimal protocol g(t) shown in panel (b). In order to highlight the remarkable stability of the solutions, the simulations have been extended beyo… view at source ↗
Figure 5
Figure 5. Figure 5: Total power (a) and efficiency (b), both [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

We develop an optimization framework for high-efficiency quantum cycles implemented with a trapped Bose-Einstein condensate, whose control parameters are the trap stiffness and the interaction strength tuned via a Feshbach resonance. Optimal driving protocols for each stroke of the cycle are obtained from a variational description of the condensate dynamics combined with Nelson's stochastic quantization, which maps the quantum evolution onto an effective Ornstein-Uhlenbeck process. The optimal protocol is obtained by minimizing a user-defined cost functional that selects the best trade-off between protocol duration and arbitrary physical constraints (such as the expended work or the proximity to an adiabatic evolution), and exhibits remarkable stability over repeated cycles. The method also provides a systematic route to optimal control for generic nonlinear Schr\"odinger equations, paving the way to optimal control strategies in fields as diverse as nonlinear optics, quantum fluids, and quantum plasmas.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops an optimization framework for quantum thermodynamic cycles realized with a trapped Bose-Einstein condensate, with control parameters given by the trap stiffness and the Feshbach-tuned interaction strength. A variational reduction of the condensate dynamics is combined with Nelson's stochastic quantization to map the evolution onto an effective Ornstein-Uhlenbeck process; optimal protocols for each stroke are then obtained by minimizing a user-defined cost functional that trades protocol duration against physical constraints such as expended work or deviation from adiabaticity. The resulting protocols are asserted to exhibit remarkable stability over repeated cycles, and the approach is presented as a general method applicable to other nonlinear Schrödinger equations.

Significance. If the variational-to-stochastic mapping remains faithful under time-dependent controls, the framework would supply a concrete, computationally tractable route to optimal driving in interacting quantum fluids. The explicit incorporation of a tunable cost functional and the reported cycle-to-cycle stability would constitute a useful advance over purely adiabatic or heuristic protocols, provided the underlying effective process accurately reflects the quantum dynamics.

major comments (2)
  1. [Derivation of the effective OU process] The central mapping from the variational Gross-Pitaevskii dynamics to an effective Ornstein-Uhlenbeck process via Nelson's stochastic quantization is load-bearing for all optimality claims. The manuscript must demonstrate that the resulting drift and diffusion coefficients remain consistent with the underlying nonlinear quantum evolution when the trap frequency and scattering length are varied rapidly; otherwise the derived protocols are optimal only for the reduced stochastic model, not for the physical condensate.
  2. [Numerical results on cycle stability] The stability over repeated cycles is asserted without quantitative benchmarks. Direct comparison of the optimized protocols against the exact many-body or full Gross-Pitaevskii evolution (e.g., via fidelity or work variance after N cycles) is required to substantiate that the stability is not an artifact of the variational ansatz.
minor comments (2)
  1. [Optimization procedure] The precise functional form of the cost functional (including any weighting parameters between duration and constraint terms) should be stated explicitly, together with the numerical values or ranges used in the reported optimizations.
  2. [Variational ansatz] Notation for the variational parameters (width, phase, etc.) and their stochastic interpretation should be introduced once and used consistently; occasional shifts between deterministic and stochastic descriptions obscure the mapping.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment in turn below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Derivation of the effective OU process] The central mapping from the variational Gross-Pitaevskii dynamics to an effective Ornstein-Uhlenbeck process via Nelson's stochastic quantization is load-bearing for all optimality claims. The manuscript must demonstrate that the resulting drift and diffusion coefficients remain consistent with the underlying nonlinear quantum evolution when the trap frequency and scattering length are varied rapidly; otherwise the derived protocols are optimal only for the reduced stochastic model, not for the physical condensate.

    Authors: We agree that explicit verification of the drift and diffusion coefficients under rapid time-dependent controls is necessary to support the optimality claims for the physical system. The mapping is constructed by first reducing the Gross-Pitaevskii dynamics to a variational ansatz and then applying Nelson's stochastic quantization to obtain the effective Ornstein-Uhlenbeck process. In the revised manuscript we will add a dedicated subsection that numerically compares the time-dependent drift and diffusion terms obtained from the stochastic mapping against direct integration of the variational equations for representative rapid changes in trap frequency and scattering length. These comparisons confirm consistency within the parameter regime explored, and the added material will clarify the domain of validity of the reduced model. revision: yes

  2. Referee: [Numerical results on cycle stability] The stability over repeated cycles is asserted without quantitative benchmarks. Direct comparison of the optimized protocols against the exact many-body or full Gross-Pitaevskii evolution (e.g., via fidelity or work variance after N cycles) is required to substantiate that the stability is not an artifact of the variational ansatz.

    Authors: The cycle stability is demonstrated and quantified within the variational framework that underlies the entire optimization procedure. We acknowledge that benchmarks against full many-body or exact Gross-Pitaevskii evolution would provide additional reassurance. Such simulations, however, remain computationally prohibitive for the interaction strengths and cycle numbers considered here. In the revision we will include explicit quantitative measures (work variance and fidelity after 100 cycles) computed within the variational model, together with a more detailed discussion of the known accuracy of the chosen ansatz for the dynamics of trapped condensates. This will make the scope and limitations of the reported stability clearer. revision: partial

standing simulated objections not resolved
  • Direct numerical comparison of optimized protocols against exact many-body or full Gross-Pitaevskii evolution over multiple cycles, owing to prohibitive computational cost.

Circularity Check

0 steps flagged

No circularity detected; derivation relies on external mapping and user-defined cost

full rationale

The paper obtains optimal driving protocols by combining a variational description of condensate dynamics with Nelson's stochastic quantization to produce an effective Ornstein-Uhlenbeck process, then minimizing an explicitly user-defined cost functional that trades off duration against arbitrary constraints. This structure introduces an external optimization objective and a stochastic mapping rather than deriving results that reduce by the paper's own equations to self-fitted parameters or self-citations. No load-bearing step equates a prediction to its input by construction, and the abstract presents the method as a systematic route for generic nonlinear Schrödinger equations without evidence of ansatz smuggling or uniqueness imported from prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the variational description for BEC dynamics and the accuracy of Nelson's stochastic quantization mapping to an Ornstein-Uhlenbeck process; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Nelson's stochastic quantization maps the quantum evolution onto an effective Ornstein-Uhlenbeck process
    This mapping is invoked to enable the variational optimization of the condensate dynamics.

pith-pipeline@v0.9.0 · 5671 in / 1289 out tokens · 34500 ms · 2026-05-22T09:35:22.738629+00:00 · methodology

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Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    J. I. Cirac and P. Zoller, Goals and opportunities in quan- tum simulation, Nature Physics8, 264 (2012)

    work page 2012

  2. [2]

    Deffner and S

    S. Deffner and S. Campbell, Quantum Thermodynam- ics: An introduction to the thermodynamics of quantum information (2019), version Number: 1

    work page 2019

  3. [3]

    Torrontegui, S

    E. Torrontegui, S. Ib´ a˜ nez, S. Mart´ ınez-Garaot, M. Mod- ugno, A. Del Campo, D. Gu´ ery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, Shortcuts to Adiabaticity, inAdvances In Atomic, Molecular, and Optical Physics, Vol. 62 (Elsevier, 2013) pp. 117–169

    work page 2013

  4. [4]

    Gu´ ery-Odelin, A

    D. Gu´ ery-Odelin, A. Ruschhaupt, A. Kiely, E. Tor- rontegui, S. Mart´ ınez-Garaot, and J. G. Muga, Short- cuts to adiabaticity: Concepts, methods, and applica- tions, Rev. Mod. Phys.91, 045001 (2019)

    work page 2019

  5. [5]

    J. Li, K. Sun, and X. Chen, Shortcut to adiabatic control of soliton matter waves by tunable interaction, Scientific Reports6, 38258 (2016)

    work page 2016

  6. [6]

    Torrontegui, S

    E. Torrontegui, S. Ib´ a˜ nez, X. Chen, A. Ruschhaupt, D. Gu´ ery-Odelin, and J. G. Muga, Fast atomic trans- port without vibrational heating, Physical Review A83, 013415 (2011)

    work page 2011

  7. [7]

    Del Campo, Shortcuts to Adiabaticity by Counter- diabatic Driving, Physical Review Letters111, 100502 (2013)

    A. Del Campo, Shortcuts to Adiabaticity by Counter- diabatic Driving, Physical Review Letters111, 100502 (2013)

    work page 2013

  8. [8]

    M. V. Berry, Transitionless quantum driving, Journal of Physics A: Mathematical and Theoretical42, 365303 (2009). 7

    work page 2009

  9. [9]

    Masuda and K

    S. Masuda and K. Nakamura, Fast-forward of adiabatic dynamics in quantum mechanics, Proceedings of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences466, 1135 (2009)

    work page 2009

  10. [10]

    C. A. Plata, A. Prados, E. Trizac, and D. Gu´ ery- Odelin, Taming the Time Evolution in Overdamped Systems: Shortcuts Elaborated from Fast-Forward and Time-Reversed Protocols, Physical Review Letters127, 190605 (2021)

    work page 2021

  11. [11]

    Raynal, T

    D. Raynal, T. De Guillebon, D. Gu´ ery-Odelin, E. Trizac, J.-S. Lauret, and L. Rondin, Shortcuts to Equilibrium with a Levitated Particle in the Underdamped Regime, Physical Review Letters131, 087101 (2023)

    work page 2023

  12. [12]

    Hardel, G

    V. Hardel, G. Manfredi, P.-A. Hervieux, and R. Goerlich, Shortcuts to adiabaticity in harmonic traps: A quantum- classical analog, Physical Review E110, 054138 (2024)

    work page 2024

  13. [13]

    Nelson, Derivation of the Schr¨ odinger Equation from Newtonian Mechanics, Physical Review150, 1079 (1966)

    E. Nelson, Derivation of the Schr¨ odinger Equation from Newtonian Mechanics, Physical Review150, 1079 (1966)

    work page 1966

  14. [14]

    Castin, Bose-Einstein condensates in atomic gases: simple theoretical results, ArXiv 10.48550/arxiv.con- mat/0105058 (2001)

    Y. Castin, Bose-Einstein condensates in atomic gases: simple theoretical results, ArXiv 10.48550/arxiv.con- mat/0105058 (2001)

    work page doi:10.48550/arxiv.con- 2001

  15. [15]

    Dalfovo, S

    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics71, 463 (1999)

    work page 1999

  16. [16]

    Kagan, E

    Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Evolu- tion and Global Collapse of Trapped Bose Condensates under Variations of the Scattering Length, Physical Re- view Letters79, 2604 (1997)

    work page 1997

  17. [17]

    Roati, M

    G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Mod- ugno, A. Simoni, M. Inguscio, and G. Modugno, K39 Bose-Einstein Condensate with Tunable Interactions, Physical Review Letters99, 010403 (2007)

    work page 2007

  18. [18]

    D’Errico, M

    C. D’Errico, M. Zaccanti, M. Fattori, G. Roati, M. Ingus- cio, G. Modugno, and A. Simoni, Feshbach resonances in ultracold 39K, New Journal of Physics9, 223 (2007)

    work page 2007

  19. [19]

    C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fes- hbach resonances in ultracold gases, Reviews of Modern Physics82, 1225 (2010)

    work page 2010

  20. [20]

    Del Campo, Frictionless quantum quenches in ultra- cold gases: A quantum-dynamical microscope, Physical Review A84, 031606 (2011)

    A. Del Campo, Frictionless quantum quenches in ultra- cold gases: A quantum-dynamical microscope, Physical Review A84, 031606 (2011)

    work page 2011

  21. [21]

    Schaff, X.-L

    J.-F. Schaff, X.-L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, Shortcut to adiabaticity for an interacting Bose-Einstein condensate, Europhysics Letters93, 23001 (2011)

    work page 2011

  22. [22]

    Brandner, M

    K. Brandner, M. Bauer, M. T. Schmid, and U. Seifert, Coherence-enhanced efficiency of feedback-driven quan- tum engines, New Journal of Physics17, 065006 (2015)

    work page 2015

  23. [23]

    M. Beau, J. Jaramillo, and A. Del Campo, Scaling-up quantum heat engines efficiently via shortcuts to adia- baticity, Entropy18, 10.3390/e18050168 (2016)

    work page doi:10.3390/e18050168 2016

  24. [24]

    F. Wu, L. Chen, F. Sun, C. Wu, F. Guo, and Q. Li, Quan- tum degeneracy effect on performance of irreversible otto cycle with ideal bose gas, Energy Conversion and Man- agement47, 3008 (2006)

    work page 2006

  25. [25]

    J. Li, T. Fogarty, S. Campbell, X. Chen, and T. Busch, An efficient nonlinear Feshbach engine, New Journal of Physics20, 015005 (2018)

    work page 2018

  26. [26]

    Keller, T

    T. Keller, T. Fogarty, J. Li, and T. Busch, Feshbach en- gine in the Thomas-Fermi regime, Physical Review Re- search2, 033335 (2020)

    work page 2020

  27. [27]

    V. V. Nautiyal, R. S. Watson, and K. V. Kheruntsyan, A finite-time quantum Otto engine with tunnel coupled one-dimensional Bose gases, New Journal of Physics26, 063033 (2024)

    work page 2024

  28. [28]

    J. A. Estrada, F. Mayo, A. J. Roncaglia, and P. D. Mininni, Quantum engines with interacting Bose- Einstein condensates, Phys. Rev. A109, 012202 (2024)

    work page 2024

  29. [29]

    Mishra, T

    C. Mishra, T. Busch, and T. Fogarty, Shortcuts to adi- abaticity in anisotropic Bose-Einstein condensates, New Journal of Physics27, 033009 (2025)

    work page 2025

  30. [30]

    C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Mod- ugno, D. S. Wiersma, and M. Inguscio, Effect of Optical Disorder and Single Defects on the Expansion of a Bose- Einstein Condensate in a One-Dimensional Waveguide, Physical Review Letters95, 170410 (2005)

    work page 2005

  31. [31]

    Manfredi and P.-A

    G. Manfredi and P.-A. Hervieux, Fidelity decay in trapped Bose-Einstein condensates, Phys. Rev. Lett. 100, 050405 (2008)

    work page 2008

  32. [32]

    F. Haas, G. Manfredi, P. K. Shukla, and P.-A. Hervieux, Breather mode in the many-electron dynamics of semi- conductor quantum wells, Phys. Rev. B80, 073301 (2009)

    work page 2009

  33. [33]

    Hurst, K

    J. Hurst, K. L´ evˆ eque-Simon, P.-A. Hervieux, G. Man- fredi, and F. Haas, High-harmonic generation in a quantum electron gas trapped in a nonparabolic and anisotropic well, Phys. Rev. B93, 205402 (2016)

    work page 2016

  34. [34]

    Bacciagaluppi, Nelsonian mechanics revisited, Foun- dations of Physics Letters12, 1 (1999)

    G. Bacciagaluppi, Nelsonian mechanics revisited, Foun- dations of Physics Letters12, 1 (1999)

    work page 1999

  35. [35]

    G. E. Uhlenbeck and L. S. Ornstein, On the Theory of the Brownian Motion, Physical Review36, 823 (1930)

    work page 1930

  36. [36]

    The initial condition is computed self-consistently and not assumed to be Gaus- sian

    We use a standard finite-difference scheme (Crank- Nicolson with split operator between the kinetic and po- tential parts of the Hamiltonian). The initial condition is computed self-consistently and not assumed to be Gaus- sian

    work page

  37. [37]

    Rosales-Cabara, G

    Y. Rosales-Cabara, G. Manfredi, G. Schnoering, P.-A. Hervieux, L. Mertz, and C. Genet, Optimal protocols and universal time-energy bound in Brownian thermo- dynamics, Physical Review Research2, 012012 (2020)

    work page 2020

  38. [38]

    For simplicity, we use the same symbol for a function of time such as ¯κ(t) and the corresponding function of the variance ¯κ(s) = ¯κ(t(s))

    work page

  39. [39]

    Other choices were explored in [12]

    work page

  40. [40]

    E. Q. Simmons, R. Sajjad, K. Keithley, H. Mas, J. L. Tanlimco, E. Nolasco-Martinez, Y. Bai, G. H. Fredrick- son, and D. M. Weld, Thermodynamic engine with a quantum degenerate working fluid, Phys. Rev. Res.5, L042009 (2023)

    work page 2023

  41. [41]

    R. S. Watson and K. V. Kheruntsyan, Quantum many- body thermal machines enabled by atom-atom correla- tions, SciPost Phys.18, 190 (2025)

    work page 2025

  42. [42]

    Campbell, I

    S. Campbell, I. D’Amico, M. A. Ciampini, J. Anders, N. Ares, S. Artini, A. Auff` eves, L. Bassman Oftelie, L. P. Bettmann, M. V. S. Bonan¸ ca, T. Busch, M. Campisi, M. F. Cavalcante, L. A. Correa, E. Cuestas, C. B. Dag, S. Dago, S. Deffner, A. Del Campo, A. Deutschmann- Olek, S. Donadi, E. Doucet, C. Elouard, K. Ensslin, P. Erker, N. Fabbri, F. Fedele, G....

    work page 2026

  43. [43]

    J. Koch, K. Menon, E. Cuestas, S. Barbosa, E. Lutz, T. Fogarty, T. Busch, and A. Widera, A quantum engine in the BEC–BCS crossover, Nature621, 723 (2023)

    work page 2023

  44. [44]

    Menon, T

    K. Menon, T. Busch, and T. Fogarty, Leveraging quan- tum statistics to enhance heat engines, Quantum Science and Technology10, 045039 (2025)

    work page 2025

  45. [45]

    H. R. Lewis, Classical and quantum systems with time- dependent harmonic-oscillator-type Hamiltonians, Phys. Rev. Lett.18, 510 (1967)

    work page 1967