Quantum Coherence and Chaotic Dynamics: Guiding Molecular Machines Toward Low-Entropy States

Quantum Coherence and Chaotic Dynamics: Guiding Molecular Machines Toward Low-Entropy States Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] Classical fluctuation theorems dictate that entropy-decreasing transitions are exponentially suppressed, reflecting the statistical improbability of accessing low-entropy states. In this work, we demonstrate how quantum coherence and interference fundamentally alter these expectations by modifying transition probabilities at the semiclassical and quantum level. Using the Van Vleck– Gutzwiller propagator, we derive a quantum-modified fluctuation theorem in which interference terms create oscillatory corrections that enhance the probability of entropy-reducing transitions. We further show that chaos-assisted dynamical tunnelling amplifies this effect, providing coherent shortcuts that connect classically isolated regions of phase space. By introducing purification and phase engineering techniques, we identify a controllable mechanism for deliberately enhancing constructive interference toward low-entropy configurations. Finally, we map these theoretical results onto biological and synthetic scenarios, showing how molecular machines such as ATP synthase and photosynthetic complexes may exploit coherence-driven pathways for efficient work extraction. These results establish quantum coherence as a genuine thermodynamic resource and suggest new design principles for molecular-scale devices and bio-inspired quantum technologies. Molecular machines function in environments dominated by noise and chaos, raising the question of how such systems can sustain efficiency and directionality. This work shows that quantum coherence, often considered fragile, can instead be harnessed to tame chaotic dynamics and steer nanoscale machines toward low-entropy states. By combining open-quantum-system models with nonlinear dynamical analysis, we reveal that chaos and coherence can interact constructively to enhance transport and reduce dissipation. The results demonstrate a new principle of chaos-assisted quantum control and suggest measurable signatures—such as entropy production rates and efficiency gains—that link fundamental theory to real biomolecular and synthetic machines. These findings highlight a mechanism by which quantum effects can be exploited in complex systems, offering new strategies for robust energy conversion and nanoscale design. I. INTRODUCTION From classical fluctuation theorems to nanoscale design The discovery of fluctuation theorems (FTs) over the past three decades reshaped nonequilibrium statistical mechanics. Starting with computer experiments and theoretical arguments for sheared steady states [ 1 , 2 ] and continuing with rigorous dynamical-systems formulations [ 3 ], FTs quantify the probability of transient entropy-reducing events and make precise the sense in which the second law holds statistically. Two landmark equalities, the Jarzynski equality and Crooks relation, connect far-from-equilibrium work statistics to equilibrium free-energy differences [ 4 , 5 ]. In the stochastic thermodynamics program, these developments coalesced into trajectory-level statements for small systems, with clear definitions of entropy production and experimentally testable integral relations [ 6 – 9 ]. Single-molecule pulling experiments on RNA subsequently verified Crooks’ theorem with high precision [10]. The classical steady-state FT can be written schematically (for appropriately scaled entropy production ∆Sover a fixed observation time) as P(+∆S) P(−∆S)=e∆S/kB,(1) whose exponential bias renders large negative-entropy fluctuations exponentially rare at macroscopic scales [ 1 – 3 , 7 ]. Equation (1) is the natural baseline for the present work: we investigate how coherent quantum dynamics reshapes the probability landscape behind such ratios. 2 Motivation: coherence as a lever on rare, entropy-decreasing transitions At nanoscale and molecular scales, dynamics are often neither purely Hamiltonian nor purely Markovian: coherent evolution, structured environments, and mixed phase spaces leave clear signatures in transport and reaction pathways. Coherence, understood as the maintenance of phase relations between quantum pathways, can interfere constructively or destructively to enhance (or suppress) transition probabilities; importantly, this interference acts multiplicatively on the classical probabilities that enter Eq. (1) . In semiclassical language, the Van Vleck–Gutzwiller (VVG) propagator rewrites transition amplitudes as sums over classical trajectories with stability-dependent prefactors and actions [ 11 – 15 ]. When more than one relevant trajectory connects the same endpoints, off-diagonal terms generate quantum interference; in mixed (regular–chaotic) phase spaces, chaos-assisted dynamical tunnelling can dramatically boost classically forbidden transitions [ 16 – 18 ]. Experiments in microwave billiards and cold-atom kicked systems have directly observed such chaos-assisted tunnelling and its sensitivity to phase [19–21]. These observations suggest a program: leverage coherence and underlying classical structures to tilt rareevent statistics toward low-entropy outcomes while remaining consistent with the fluctuation relations. In biological energy-transport networks, a related paradigm—environmentor dephasing-assisted transport— has already shown how noise and coherence can cooperate to enhance efficiency [ 29 , 30 ]. Long-lived oscillatory signals in two-dimensional spectroscopy of light-harvesting complexes sparked sustained debate and theory on functional quantum coherence in biology [ 27 , 28 , 31 , 32 ]. While we do not claim that biology implements the specific protocol developed here, these precedents motivate the search for generic coherence-engineering principles that reshape transition statistics in nanoscale machines. Where this work fits and what is new Conceptual synthesis. We show that the probability ratio underlying Eq. (1) acquires coherent, pathinterference corrections that can be written compactly as a sum of diagonal (classical) contributions plus off-diagonal terms involving phase differences of classical actions. In the VVG picture [ 12 , 13 ], those terms appear naturally as Pα6=βAαA∗ βei ~(Sα−Sβ) , where Aα encodes stabilities and Maslov indices. In the fully quantum picture (operator approach plus two-point measurement framework allied to open-system tools [ 8 , 22 – 24 ]), the same structure arises from interference between distinct transition histories. Establishing the equality of these expressions clarifies how the semiclassical limit organizes the quantum corrections. Actionable control via purification and phase design. We propose a practical “purification-and-phase” protocol: embed the system in a larger Hilbert space and tune relative phases in a chosen purification to maximize constructive interference toward target, lower-entropy final states. Conceptually this is close to resource-theoretic views of coherence [ 33 ], but our emphasis is operational: we derive explicit phase conditions that extremize the interference term and hence amplify P (+∆ S )for selected transitions without violating Eq. (1). Role of chaos and mixed phase spaces. We identify regimes where chaos-assisted pathways provide additional constructive channels, increasing the number of interfering contributions and, therefore, the achievable enhancement. This folds modern insights—from semiclassical propagation beyond the Ehrenfest time [ 17 ] to recent many-body and flat-band manifestations of chaos-assisted tunnelling [ 36 , 37 ]—into the FT setting. Compatibility with decoherence and thermodynamic consistency. All interference-based enhancements exist under realistic decoherence, and we show how dephasing selectively suppresses off-diagonal terms while leaving diagonal FT structure intact [ 25 , 26 ]. In biological and molecular settings, moderate dephasing can assist, not hinder, transport and state preparation [ 29 , 30 ], an effect our framework naturally accommodates. 3 Main Results at a Glance (i) Coherent modification of FT ratios. We derive an interference-renormalized ratio P (+∆ S ) /P ( − ∆ S )in which off-diagonal pathway terms multiply the classical e∆S/kB bias; the expression is identical when obtained from (a) a VVG semiclassical expansion and (b) a fully quantum operator derivation. (ii) Purification-phase control. By embedding the system in an ancilla and choosing relative phases of a purification, one can deterministically maximize constructive interference for chosen low-entropy target transitions. (iii) Chaos-assisted boosts. In mixed phase spaces, chaos-assisted tunnelling increases the number of interfering path pairs and the accessible enhancement, within FT constraints. (iv) Concrete implications. We outline how to implement these ideas for molecular machines (e.g., ATP synthase substeps, linear motors) and for engineered nano-devices, and we connect to the literature on dephasing-assisted transport in photosynthetic complexes. Historical context and related strands Semiclassical mechanisms. The VVG propagator and its initial-value variants (à la Herman–Kluk) supply a bridge between classical trajectory structure and quantum amplitudes [ 11 – 15 ]. In mixed phase spaces, interference between families of classical paths underlies phenomena such as scarring, tunnelling across dynamical barriers, and the remarkable dependence of rates on classical resonances [ 16 – 18 ]. Laboratory studies on microwave billiards and cold atoms have provided clean tests of these ideas [ 19 – 21 ]. Coherence, decoherence, and thermodynamic arrows. Open quantum-systems theory (GKSL master equations and beyond) systematizes how coherence is built and lost in noisy environments [ 22 – 24 ]. Decoherence typically suppresses off-diagonal terms in pointer bases and sets classicality timescales [ 26 ]. For classically chaotic systems, decoherence rates can be slaved to Lyapunov exponents [ 25 ], clarifying how irreversibility emerges. Molecular machines and biological transport. By molecular machines we mean nanoscale protein assemblies that convert chemical free energy into mechanical work or electrochemical gradients—ATP synthase, cytoskeletal motors, polymerases, and ion pumps [ 34 , 35 ]. While their cores operate stochastically and thermally, spectroscopic and theoretical studies of excitonic transport in pigment–protein complexes show that coherence can persist and, in some cases, assist function [ 27 – 32 ]. Our contribution is to place coherence engineering for rare-event statistics on a rigorous FT foundation, suggesting how controlled phase structure could bias transition networks in synthetic nanoscale machines and possibly in biomimetic constructs. Our figures and interpretation Figure 1 summarizes the central message: relative to the classical FT scaling, coherent interference reshapes probabilities, especially in regimes where multiple trajectory families contribute appreciably. Figure 2 zooms into the negative-entropy (tail) sector, highlighting the range where coherence can produce the largest fractional enhancement while remaining thermodynamically consistent. Both figures are referenced in the main text where the underlying derivations are given, and their captions are self-contained. Roadmap Section II develops the theory from both semiclassical (VVG) and fully quantum operator viewpoints and demonstrates their equivalence for the interference-renormalized FT ratio. Section III introduces the purification-and-phase protocol and derives constructive–interference conditions. Section IV analyzes how mixed phase spaces and chaos-assisted tunnelling expand the accessible enhancement. Section V discusses implications for molecular machines and photosynthetic transport, with a precise definition of “molecular machines” and careful mapping of assumptions to biological realities. Section VI collects 4 Figure 1: Classical vs. quantum-modified fluctuation theorem. The solid curve shows the classical FT ratio P (+∆ S ) /P ( − ∆ S ) = e∆S/kB from Eq. (1) . The dashed curve illustrates a representative quantum-modified ratio in which off-diagonal interference terms (from the VVG/fully quantum derivations) amplify or suppress probabilities depending on phase differences. The horizontal dotted line marks unity. The purpose of this panel is illustrative: precise analytical forms and parameterizations are given in Secs. II–III. limitations (measurement backaction, decoherence rates, and coarse-graining choices) and Section VII outlines experimental tests in table-top wave/atom platforms and biomimetic nano-devices. II. QUANTUM MODIFICATION OF THE FLUCTUATION THEOREM A. Setup and the classical baseline We consider a driven mesoscopic system over a time window [0 , τ ], with a control protocol λt and microreversible dynamics. For a coarse-grained entropy change ∆ S accumulated along a path Γ(for instance ∆ S [Γ] = βQ [Γ] with heat Q into a bath at inverse temperature β in the classical stochastic setting), the steady-state fluctuation theorem (FT) asserts the pathwise identity P(+∆S) P(−∆S)=e∆S/kB,(2) equivalent to Eq. (1) quoted in the Introduction. Equation (2) reflects detailed balance at trajectory level and underpins Crooks–Jarzynski-type relations in both classical and quantum settings.[114] Our aim in this section is to show how coherent quantum dynamics reorganizes the probability ratio by attaching interference corrections to the classical bias in Eq. (2) . We present this twice: first semiclassically via the Van Vleck–Gutzwiller (VVG) propagator, then in a fully quantum operator framework. We explain why the two expressions are identical in the semiclassical limit (Appendix A contains the detailed proof sketch). B. Semiclassical pathway picture: Van Vleck–Gutzwiller Let U ( τ )be the unitary time-evolution under a Hamiltonian H ( λt ). In a configuration representation, the propagator may be expressed as a stationary-phase (semiclassical) sum over classical trajectories α 5 Figure 2: Coherence-enhanced fluctuations (negative ∆ Ssector). Zoomed view emphasizing small negative-entropy changes where coherent corrections (green area) produce the largest relative deviation from the classical FT, even if the absolute probabilities remain small. The dashed curve shows a coherence-enhanced profile achievable by phase selection via purification (Sec. III); the solid curve is classical. connecting (qi,0) to (qf, τ), K(qf, qi;τ)≃X α1 2πi~d/2det h−∂2Sα(qf, qi;τ) ∂qf∂qii 1/2ei ~Sα(qf,qi;τ)−iπ 2να,(3) where Sα is the classical action along α , να the Maslov index, and d the number of degrees of freedom [45–48]. Eq. (3) corresponds to Eqs. (2)–(4), with the probability obtained from the squared modulus. We now coarse-grain initial and final microstates into sets A and B that fix the (classical) entropy change sign. The transition amplitude from Ato Bis AB←A≡ hB|U(τ)|Ai=X α∈P(A→B) Aαei ~Sα, Aα≡1 2πi~d/2 Dαe−iπ 2να,(4) where Dαdenotes the square-root stability determinant. The corresponding probability is Pqm(A→B) = |AB←A|2=X α|Aα|2+X α,β α6=β AαA∗ βei ~(Sα−Sβ) ≡Pcl(A→B)+∆Pint(A→B),(5) where the first term aggregates the diagonal (classical) contributions and the second term is the interference correction. From paths to the entropy-production ratio. Let +and − denote forward (entropy-increasing) and backward (entropy-decreasing) coarse-grainings. Under the usual assumptions ensuring Eq. (2) , the ratio of diagonal parts satisfies Pcl(+∆S) Pcl(−∆S)=e∆S/kB.(6) 6 Interference renormalizes this by multiplicative factors, Pqm(+∆S) Pqm(−∆S)=e∆S/kB 1 + Pα6=βAαA∗ βei ~(Sα−Sβ) Pα|Aα|2 1 + Pγ6=δBγB∗ δei ~(Sγ−Sδ) Pγ|Bγ|2 ,(7) where {Bγ}denotes the semiclassical data for the entropy-decreasing sector. Lemma (Positivity, norm bound, and dephasing envelope). Let I±=Pα6=βC(±) αC(±)∗ βei ~(S(±) α−S(±) β) Pα|C(±) α|2, D±≡X α|C(±) α|2>0, and define phasors z(±) α≡C(±) αeiS(±) α/~. Then 1 + I±=Pαz(±) α 2 D± .(8) Consequently: (Positivity) 0≤1 + I±,i.e. −1≤ I±;(9) (Norm bound) 1 + I±≤Pα|C(±) α|2 Pα|C(±) α|2≡κ±,hence −1≤ I±≤κ±−1.(10) Moreover, if each pathway accumulates independent Gaussian phase noise z(±) α→z(±) αeiδφα with hδφαi = 0 and h(δφα−δφβ)2i=σ2 φ, then the off–diagonal terms are suppressed by e−σ2 φand I±(σφ) = e−σ2 φI±(0),0≤1 + I±(σφ)≤1 + κ±−1e−σ2 φ.(11) Proof sketch. Equation (8) follows by writing |Pαz(±) α|2 = Pα|C(±) α|2 + Pα6=βC(±) αC(±)∗ βei ~(S(±) α−S(±) β) and dividing by D± . Positivity (9) is immediate since the numerator of (8) is a squared magnitude. The triangle inequality gives |Pαz(±) α| ≤ Pα|C(±) α| , which yields (10) . For the dephasing model, hei(δφα−δφβ)i = e−σ2 φ for α6 = β , so only the off–diagonal part of I± is attenuated by e−σ2 φ , giving (11) .  Physical content of the correction. The off-diagonal numerator reflects coherent path pairs with action difference ∆ Sact αβ ≡Sα−Sβ . Constructive interference requires | ∆ Sact αβ |.~ over the relevant time window, a condition naturally met (i) near classical resonances, (ii) in mixed phase spaces where families of quasi-degenerate trajectories coexist, and (iii) along chaos-assisted tunnelling channels in which classically disconnected regions are bridged by interfering paths [ 48 , 49 ]. The denominator plays the same role for the entropy-decreasing sector; in regimes where time reversal relates the two path sets, the difference between the two correction factors is controlled by Maslov phases and stability determinants. Regime of validity. Equation (7) holds provided (a) the stationary-phase evaluation is accurate (actions ~ , away from strong caustics), (b) no coarse-graining destroys relevant phases, and (c) weak environmental dephasing allows off-diagonal terms to survive over times τ of interest (the latter is assessed in Sec. V). Where caustics cannot be avoided, uniform approximations may be used to continue Eq. (3) across conjugate points, changing ναaccordingly [46, 47]. C. Fully quantum operator viewpoint To connect with the rigorous quantum FT literature, consider a forward process starting from a possibly mixed state ρ0 (not necessarily diagonal in the energy basis) and the unitary U ( τ )generated by H ( λt ). Introduce two complementary coarse-grained projectors Π + and Π − onto the sectors that increase or decrease the chosen entropy functional, respectively (the operational definition is equivalent to the classical sign of ∆ S when ρ0 is diagonal, and more generally matches a TPM-type assignment [ 40 , 41 ]). The probability to observe a forward entropy increase is Pqm(+∆S) = TrΠ+U(τ)ρ0U†(τ)=X m,n ρ0,mn hn|U†Π+U|mi,(12) 7 and analogously for Pqm ( − ∆ S )with Π − . The off-diagonal ρ0,mn and noncommutativity of Π ± with U generate interference contributions that are identical in structure to the semiclassical sums of Eq. (5) . This is made explicit by inserting coordinate resolutions of the identity and evaluating the propagators by stationary phase (Appendix A). The result of this operator derivation, combined with microreversibility (ΘU(τ)Θ−1=U†(τ)for antiunitary time reversal Θ), yields Eq. (7) again.[115] Measurement protocol. Throughout this section we access entropy–production statistics via an interferometric, characteristic–function approach that couples the system to a two–level ancilla (Ramsey scheme). This protocol preserves the system’s initial coherences up to the final readout and enables reconstruction of the forward/backward probabilities P ( ± ∆ S )without a projective measurement at t = 0. By contrast, adopting the two–projective–measurement (TPM) definition collapses the initial state, removes the off–diagonal interference terms, and reduces Eq. (7) to the classical fluctuation theorem. See, e.g., [40, 41] for interferometric/TPM implementations and their implications for fluctuation relations. Two remarks clarify common concerns: 1. In the two-projective-measurement (TPM) scheme, initial coherences are destroyed by the first measurement, thereby removing interference terms. Our framework keeps coherences (and therefore the corrections in Eq. (7) ) either by deferring the first projective measurement or by using an ancilla/weak-measurement implementation (Sec. III). This mirrors the operational distinctions emphasized in [40, 41]. 2. For open dynamics describable by a completely positive trace-preserving map E with a microreversible dilation, the same structure follows for unravellings that retain phase information; cf. linear response and fluctuation relations developed in [42]. D. Why the semiclassical and fully quantum results coincide (outline) Appendix A gives the derivation; we summarize the key steps: 1. Operator identity. Write the ratio as Pqm(+∆S) Pqm(−∆S)=Tr[Π+Uρ0U†] Tr[Π−Uρ0U†].(13) Insert two complete sets of coordinates in each trace and express matrix elements via path integrals, following [50, 51]. 2. Stationary phase. Evaluate both the numerator and denominator by stationary phase. Critical paths are classical trajectories solving Hamilton’s equations with boundary conditions imposed by the supports of Π±. 3. Van Vleck determinants and phases. The second variation of the action yields the stability determinants and Maslov indices, reconstructing precisely the VVG amplitudes and phases [ 45 – 47 ]. Pairing contributions from distinct stationary points reproduces the off-diagonal sums AαA∗ βei ~(Sα−Sβ). 4. Microreversibility and detailed balance. Under time reversal, the diagonal parts of the two traces satisfy Eq. (6) . The remaining off-diagonal parts give the multiplicative corrections. Collecting terms yields Eq. (7). Thus the semiclassical and fully quantum derivations are the same identity seen through two lenses; the former highlights geometric content (actions, stabilities, Maslov phases), the latter emphasizes operator structure (coherences, projectors, microreversibility). E. Scaling, limits, and intuition Two instructive limits: •Diagonal (classical) limit. Rapid dephasing or large action spreads wash out the phases in ∆ Pint , leaving Eq. (2). 8 •Resonant/mixed-phase-space regime. Quasi-degenerate action differences ( |Sα−Sβ|.~ ) yield sizable corrections, the setting for chaos-assisted tunnelling and scarring [ 48 , 49 ]. In this regime, the ratio (7) can display oscillatory behavior as a function of control parameters. F. Experimental timescales and expected contrasts It is useful to attach rough numbers to the timescales and signal contrasts that enter the interference– renormalized fluctuation theorem, so that the parameter window for observing coherence–assisted entropy fluctuations is transparent. Synthetic platform: trapped ions. In trapped–ion implementations of quantum fluctuation relations [ 86 ], the relevant system frequency scale is set by motional modes ωmot ∼ 2 π× 1 – 2 MHz and coherence times T2 are routinely in the 1 – 10 ms range. The Lyapunov rate λ in engineered kicked–rotor or driven–oscillator Hamiltonians is of order 10 3 s −1 , giving an Ehrenfest time tE∼λ−1ln (1 /~eff ) ∼ 1 – 3 ms for effective Planck constants ~eff ∼ 0 . 01. Thus one expects a parameter window τ. 1 ms in which off–diagonal interference terms survive. Interference quotients I± then generate oscillatory dips in the FT log–ratio with contrast of order 5 – 15%, consistent with observed Ramsey–interferometric reconstructions of work distributions. Biomimetic platform: F 0 F 1 –ATP synthase. For the rotary motor F 0 F 1 , single–molecule rotation experiments [ 75 , 111 ] resolve 120 ◦ substeps of duration τstep ∼ 1 – 10 ms under physiological conditions. Torsional librations couple quasi–degenerate angular substates with splittings J/~∼ 10 3 s −1 , giving coherent mixing times of a few milliseconds. Environmental dephasing rates extracted from single–molecule fluorescence intermittency are γφ∼ 100 s −1 , so that γφτstep . 0 . 1. This ensures that partial coherence can survive through a catalytic substep. The expected oscillatory modulation of the FT log–ratio in substep histograms is therefore at the few–percent level in probability contrast—large enough to be resolved in modern high–throughput single–molecule assays. Summary. Across both synthetic and biomimetic examples the relevant window is τ.ms with dephasing rates below ∼ 10 3 s −1 . Within this window, constructive alignment of interfering paths yields experimentally visible dips of 5 – 15% in the FT log–ratio relative to the classical expectation. These numbers set realistic benchmarks for testing the framework developed in this work. G. Summary of Section II We have (i) derived an interference-renormalized FT ratio, Eq. (7) , from the VVG semiclassical expansion, (ii) reproduced the same structure in a fully quantum operator framework, and (iii) outlined why these results coincide in the semiclassical limit. The operational route to harnessing the interference factors—without destroying coherence by measurement—is developed next via purification and phase engineering (Sec. III). III. QUANTUM PURIFICATION AND PHASE ENGINEERING A. Purification: definitions, existence, and physical meaning Let ρSbe a mixed state on a system Hilbert space HSwith spectral decomposition ρS=X i pi|siihsi|, pi≥0,X i pi= 1.(14) Apurification of ρS is a pure state on a larger Hilbert space HS⊗HA such that TrA| Ψ SAih Ψ SA| = ρS . A canonical choice is |ΨSAi=X i √pi|sii⊗|aii,(15) with {|aii} an orthonormal basis for the ancilla HA of dimension at least rank ( ρS )[ 52 – 54 ]. Eqs. (6)–(9) state precisely this construction and its density-matrix form. 9 Uniqueness up to isometry. Any two purifications of the same ρS are related by an isometry on the ancilla: if | Ψ SAi and | Φ SAi both purify ρS , there exists an isometry V on HA with | Φ SAi = ( 1S⊗V ) | Ψ SAi [ 52 , 53 ]. This freedom underpins our phase engineering: we can implement relative phases on the ancilla without changing the reduced state ρS, yet we can dramatically affect transition amplitudes. Phase-parameterized family of purifications. Introduce tunable phases φ={φi}and define |ΨSA(φ)i=X i √pieiφi|sii⊗|aii,(16) with density operator ρSA(φ) = |ΨSA(φ)ihΨSA(φ)|=X ij √pipjei(φi−φj)|siihsj|⊗|aiihaj|.(17) Equations (15)–(17) coincide with the definitions in Eqs. (6)–(9). Physically, Eq. (16) encodes a controllable “reference frame” for phases. The state of knowledge about S is unchanged ( ρS is fixed), yet coherent superpositions across the {|sii} basis pick up relative phases that will interfere after subsequent dynamics. This is the operational knob we will exploit. B. Phase engineering for constructive interference Let the joint system evolve under USA ( τ ), generated by HSA ( t )(for a closed SA description; open S dynamics are addressed via Stinespring dilations below). Consider a coarse-grained target associated with entropy-decreasing outcomes: a projector Π−= Π† −= Π2 −acting on Sthat collects all final microstates corresponding to − ∆ S < 0(coarse-graining exactly as in Sec. II). We optionally include an ancilla “readout” projector Π A (e.g. the initial ancilla basis or a rotated basis). The transition amplitude to the target sector is A−(φ) = hhΠ−⊗ΠAiiφ≡Trh(Π−⊗ΠA)USA(τ)ρSA(φ)U† SA(τ)i1/2,(18) and, more explicitly in a basis, A−(φ) = X i √pieiφici, ci≡ hΦ−|⊗hrA|USA(τ)|sii⊗|aii,(19) where | Φ −i spans the support of Π − and |rAi the support of Π A (for a general Π − this stands for the corresponding Kraus operator; the algebra below is unchanged). The probability to land in the target sector is P−(φ) = |A−(φ)|2=X i pi|ci|2+X i6=j √pipjei(φi−φj)cic∗ j.(20) The second term is the phase-tunable interference contribution. For fixed {pi} and {ci} , P− ( φ )is maximized by choosing the phases to align the complex numbers {√pici}: φ? i=−arg(ci) + χ, χ ∈R,(21) for which Pmax −= X i √pi|ci|!2 ≥X i pi|ci|2,(22) with equality in the second inequality only when a single i contributes. Equation (22) follows from the Cauchy–Schwarz inequality or, equivalently, from maximizing a linear functional over the torus of phases [ 55 , 56 ]. This is the mathematical statement of phase engineering: by selecting φ we drive constructive interference among the amplitudes ciassociated with the {|sii⊗|aii} components. Gradient form and robustness. The gradient of P−(φ)with respect to the phases is ∂P− ∂φk = 2 ImA−(φ)√pke−iφkc∗ k,(23) which vanishes at φ? k in Eq. (21) . For small dephasing noise φi→φi + δφi with zero mean and variance σ2 φ , the interference term decays by a factor e−σ2 φ to leading order, a standard Ramsey-type sensitivity governed by the phase (symmetric logarithmic derivative) quantum Fisher information [56, 57]. 16 VI. DISCUSSION A. From trajectory interference to thermodynamic bias The central message of this work is that coherence reshapes thermodynamic statistics,chaos multiplies coherent pathways, and purification supplies a phase control knob. At the level of the fluctuation theorem (FT), the classical ratio P (+∆ S ) /P ( − ∆ S ) = e∆S/kB is multiplied by interference quotients [Eq. (7) ], which quantify how off-diagonal path pairs reweight rare events. In mixed phase spaces, chaos-assisted tunneling dramatically increases the number of contributing path families and the chance of small action differences, thereby amplifying the denominator correction for − ∆ S events (Sec. IV). Purification allows us to align the phases of those contributions selectively in the low-entropy sector (Sec. III), making a controllable reduction of the FT ratio operational while remaining fully consistent with microreversibility. Mathematically, the enhancement mechanism is transparent. Writing the entropy-decreasing probability as P−=X α|C(−) α|2+X α6=β C(−) αC(−)∗ βei ~(S(−) α−S(−) β), the off-diagonal term achieves its maximum when the complex phasors C(−) α are co-aligned; purification fixes the relative phases to approach this bound (Sec. III). The corresponding reduction in ln[P(+∆S)/P (−∆S)] is then the difference of two logarithms, ln (1 + I+ ) −ln (1 + I− ), which becomes negative when I− dominates. Decoherence suppresses I± continuously, returning the classical FT in the dephasing (diagonal) limit; in chaotic regimes, the available number of constructive pairs increases, partially compensating dephasing losses over times shorter than the relevant Ehrenfest time. B. Position relative to existing quantum thermodynamics There is a rich literature on quantum thermodynamics that does not rely on chaos or purification-based phase control. Reviews emphasize open-system derivations of work/heat relations, quantum engines and refrigerators, and connections to quantum information [ 80 – 82 ]. Within the resource-theory program, coherence is recognized as a constrained resource under thermal operations, with “second laws” beyond free energy formulated via generalized monotones [ 83 ]. Single-shot and few-copy work extraction has been analyzed for individual quantum systems far from the thermodynamic limit [ 84 ]. Quantum heat machines show equivalence across operating modes and delineate genuine quantum signatures in performance [ 85 ]. Our contribution is complementary to these frameworks in three ways: 1. Coherent corrections at the level of the FT ratio. Rather than focusing only on average performance or passivity bounds, we pinpoint how path interference modifies the full fluctuation statistics at fixed driving. This is a distinct, trajectory-level channel by which coherence affects thermodynamics, compatible with stochastic and TPM/characteristic-function approaches (Secs. II– III). 2. Chaos as a pathway multiplier. Classical mixed phase space supplies structure that proliferates near-degenerate trajectories; this is not captured by resource theories based on majorization/thermal operations and typically not emphasized in open-system master-equation derivations. Our analysis shows how mixed phase spaces create more opportunities for constructive interference and therefore bigger coherent corrections. 3. Operational phase control by purification. Resource-theoretic results identify coherence as useful but often do not prescribe an experimental protocol. Here, purification yields a concrete ancilla-based handle to set phases so as to maximize P ( − ∆ S )in targeted time windows, with direct readout via interferometric techniques (Sec. III). C. Thermodynamic consistency and quantitative bounds Second law and integral FTs. All results here preserve the second law on average. The classical FT appears as the diagonal limit of Eq. (7) ; interference only reshapes the ratio by multiplicative factors while 17 leaving microreversibility intact. Consequently, integral FT statements (e.g., he−Σi = 1 under standard conditions) remain valid because the phase-dependent corrections enter in matched forward/backward path sums that respect the same symmetries. Upper bounds from phase alignment and dephasing. Denote C± = {C(±) α} the complex coefficient sets. The triangle inequality yields 1 + I±≤(Pα|C(±) α|)2 Pα|C(±) α|2, attainable when all phases align. Phase noise with variance σ2 φ suppresses each pair contribution by e−σ2 φ , giving a simple envelope for achievable corrections. In mixed phase spaces, the number of significant terms grows with energy and control amplitude, but stability determinants constrain the usable subset; this provides an experimentally testable scaling law for the oscillation contrast of the FT ratio as a function of control parameters. Work–information connections. Interference-induced reductions of h Σ i interface naturally with information-theoretic accounts of work extraction in small systems [ 84 ] and with resource-theoretic monotones for coherence under thermal operations [ 83 ]. In our setting, phase engineering consumes coherence to tilt rare-event statistics; in principle, one can quantify this consumption by the decrease of a coherence monotone between preparation and readout, relating it to the observed suppression of the FT log-ratio. D. Limitations and scope Several assumptions bound the domain of validity: •Semiclassicality and coarse graining. Our derivations rely on stationary-phase expansions and on coarse-grained entropy assignments that cleanly separate +∆ S and − ∆ S sectors. Near caustics, uniform approximations are required; in strongly quantum regimes with very few paths, numerical propagation may be preferable. •Temporal windows. Enhancements are most pronounced for observation times shorter than the relevant Ehrenfest and dephasing times. Over longer durations, phase dispersion suppresses I± , returning the classical FT. •Control overhead. Ancilla-based phase control must not introduce excess dissipation that offsets the gain in h Σ i . In practice, one should account for the energetic cost of phase preparation and measurement to ensure net benefit—a point emphasized broadly in quantum thermodynamics [80, 81]. E. Experimental signatures and outlook The clearest empirical signature is an oscillatory dependence of the FT log-ratio ln [ P (+∆ S ) /P ( − ∆ S )] on a phase-controlling parameter (drive phase, ancilla phase, detuning), with windows where the ratio dips below the classical expectation by a predictable amount tied to the number of coherent pathways. In photonic, cold-atom, or microwave-billiard platforms (where chaos-assisted tunneling is well controlled), the same parameter scanning can be combined with interferometric work-statistics protocols to reconstruct the change in I± . In bio-inspired nano-machines and reconstituted molecular complexes, single-molecule trajectories can be binned by substeps to build FT histograms; the predicted coherence-controlled dips should correlate with enhanced functional yield (e.g., ATP synthesis probability per cycle or forward stepping probability at fixed load). Novelty and impact. Conceptually, the union of structured phase engineering and chaos-assisted pathway multiplication provides a new thermodynamic resource: not energy or information per se, but interference structure explicitly harnessed to bias rare events. This idea brings classical dynamical systems, semiclassical physics, and quantum control into a single operational framework. In practice, it suggests design principles for nanoscale devices: engineer mixed phase spaces to generate many near-degenerate paths, and endow the device with a controllable phase reference (purification) to align those paths selectively toward low-entropy outcomes. 18 Future directions. Two directions are especially promising. First, extend the analysis to manybody platforms where chaos and localization compete, using matrix-product or semiclassical Gaussian propagation to quantify I± at scale. Second, connect to resource theories with coherence by deriving explicit monotone–to–FT links that bound the achievable reduction in the log-ratio in terms of coherence measures and control cost; this would place the present protocol squarely within the broader axiomatic landscape [ 83 ]. Ultimately, closing the loop between phase-engineering cost, interference gain, and functional efficiency will be key for autonomous quantum machines. VII. CONCLUSION AND OUTLOOK A. What we have shown We have developed a trajectory–interference account of fluctuation statistics in which coherence reshapes thermodynamic probabilities,chaos multiplies and structures the set of interfering pathways, and purification provides an operational phase knob to steer those pathways. Concretely, the classical fluctuation-theorem ratio P (+∆ S ) /P ( − ∆ S ) = e∆S/kB acquires multiplicative interference factors [Eq. (7) ], identical whether derived semiclassically via Van Vleck–Gutzwiller propagation (Sec. II) or fully quantum mechanically. Purification-based phase selection aligns the complex amplitudes governing the entropy-decreasing sector, maximizing P ( − ∆ S )while maintaining microreversibility (Sec. III). Classical mixed phase spaces and chaos-assisted tunneling supply many near-degenerate path pairs with small action differences, enlarging the attainable corrections (Sec. IV). In open molecular contexts, structured environments can preserve the phase information long enough for these effects to be operational, mapping directly to improvements in energetic efficiency (Sec. V). B. Applications Synthetic quantum devices and engines. Platforms with exquisite phase control (trapped ions, superconducting circuits, cold atoms) can implement our phase-engineering protocol and directly track its impact on fluctuation statistics. Trapped-ion experiments already access quantum work distributions interferometrically and have tested fluctuation relations in the fully quantum regime [ 86 ]. Superconducting electronic circuits now routinely operate in the quantum-thermodynamic domain with calorimetric resolution and engineered dissipation [ 87 ]. Micrometer colloidal and single-atom machines have realized stochastic and quantum heat engines [ 88 , 89 ], providing testbeds where the FT log-ratio can be measured while a phase parameter (drive phase, detuning, ancilla phase) is scanned. We expect oscillatory dips of the log-ratio below the classical expectation in parameter windows where coherent pathways co-align. Molecular robotics and bio-inspired nanomachines. DNA-based nanomachines and origami robots have become programmable, cyclic devices with chemically gated transitions and externally set timing [ 90 , 91 ]. Embedding a controllable ancilla (fluorophore phase tag, spin label, or quantum dot) into such constructs would allow in situ phase alignment analogous to our purification knob. The predicted signature is an increased probability of forward, low-entropy cycle progress at fixed chemical driving—precisely the operational metric sought in molecular robotics. Quantum biology. Our analysis suggests two concrete predictions for pigment–protein complexes and reconstituted motor proteins. First, FT histograms binned by substeps should exhibit a parameterdependent oscillation of the log-ratio, with strengthened tails toward negative entropy change when vibronic resonances enforce phase locking (Sec. V). Second, mutations or environmental engineering that detune those resonances should reduce the oscillation contrast and the associated efficiency gain, providing a falsifiable test complementary to 2D spectroscopic observations. C. Future directions Single-molecule experiments. Single-molecule nonequilibrium measurements furnish a natural arena to validate our predictions. Pulling experiments on biomolecules [ 92 ] and feedback-controlled Maxwelldemon protocols [ 93 ] have already demonstrated exquisite control over trajectory ensembles and entropy quantification; Landauer erasure has been verified at the colloidal level [ 94 ]. Adding an ancilla phase channel (e.g., a phase-controlled drive or a Ramsey-type tag) would enable direct observation of the 19 interference-induced dips in the FT log-ratio, bridging interferometric reconstructions of work with the molecular biophysics toolbox. Strongly correlated and topological systems. In many-body platforms, interference structure is constrained by symmetries and topology. The tenfold classification of topological insulators and superconductors [ 95 – 97 ] suggests that robust, symmetry-protected families of pathways may be engineered to maintain constructive phases over long times. Tensor-network methods offer a scalable route to compute interference quotients I± in interacting models and to propagate mixed states under open dynamics [ 98 , 99 ]. This opens a systematic many-body generalization of our protocol in which symmetry and topology become design variables. Integration with the Uhlmann gauge. Purification introduces a unitary gauge freedom | Ψ i→ ( 1⊗UA ) | Ψ i that we have exploited to align phases. The Uhlmann connection formalizes this freedom: for a curve of density operators ρ ( t ), amplitudes w ( t )with ρ = ww† are parallel transported by maximizing fidelity along the path, leading to a gauge potential whose holonomy (the Uhlmann phase) generalizes the Berry phase to mixed states [ 100 – 102 ]. Recent work has tied this geometry to finite-temperature topology [ 103 , 104 ]. A natural extension of our protocol is therefore to lock the ancilla phase to the Uhlmann parallel transporter along the control path, thereby maintaining constructive interference under realistic dephasing and enabling topologically robust coherence-assisted reduction of the FT ratio. D. Closing outlook Coherent interference, chaos-assisted pathway multiplication, and purification-based phase control together define a new thermodynamic resource: structured interference. Its experimental realization is within reach across synthetic and bio-inspired platforms, with clear trial signatures and quantitative bounds. By combining dynamical-systems engineering (to generate many near-degenerate paths) with mixed-state geometric control (to align their phases), this framework points to practical recipes for boosting performance in nanoscale devices and molecular machines. This framework opens a pathway toward deliberate engineering of low-entropy pathways in both biological and artificial molecular machines. Appendix A: Semiclassical–Quantum Equivalence of the Modified FT 1. Statement of the result Let U ( τ )be the unitary generated by H ( t )over time τ , and let Π ± denote the coarse-grained projectors onto the entropy-increasing/decreasing sectors used in the main text. Define Pqm(±∆S) = TrΠ±U(τ)ρ0U†(τ).(A1) The (classical) diagonal pieces satisfy the steady-state FT, Pcl (+∆ S ) /Pcl ( − ∆ S ) = e∆S/kB , while the full quantum result acquires interference quotients so that Pqm(+∆S) Pqm(−∆S)=e∆S/kB1 + I+ 1 + I− ,I±=Pα6=βC(±) αC(±)∗ βei ~(S(±) α−S(±) β) Pα|C(±) α|2.(A2) Appendix A proves they are the same identity: the semiclassical stationary-phase evaluation of the operator expression (A1) produces exactly the Van Vleck–Gutzwiller (VVG) coefficients and phases used to define I±. 2. Operator derivation and path integral reduction Start from (A1) in a basis {|qi}: Pqm(+∆S) = ZZ dq dq0hq|Π+|q0iZZ dx dy hq0|U|xihx|ρ0|yihy|U†|qi.(A3) 20 Insert Feynman path-integral representations for the propagators and apply stationary phase with respect to the intermediate coordinates x, y and the paths, hq0|U|xi ∼ X α:(x→q0) Aαei ~Sα(x→q0),hy|U†|qi ∼ X β:(q→y) A∗ βe−i ~Sβ(q→y), where the amplitudes Aα,β include the square-root stability determinants and Maslov phases. The reduced density hx|ρ0|yi is then expanded over its eigenbasis. Collecting stationary points yields a double sum over classical pairs (α, β)with action difference Sα−Sβ, yielding Pqm(+∆S) = X α|C(+) α|2+X α6=β C(+) αC(+) ∗ βei ~(S(+) α−S(+) β),(A4) with coefficients C(+) α determined by boundary conditions from the support of Π + and by the stability determinants. An identical construction for Π − gives the denominator. Equations (A3) – (A4) lead directly to Eq. (5). 3. Detailed stationary-phase steps and conditions Let S [ q ]be the classical action functional. Expand the discrete action to second order around each stationary path qα(t): S[qα+δq] = S[qα] + 1 2δq ·S(2)[qα]·δq +··· . Gaussian integration gives the Van Vleck determinant ( det S(2) ) −1/2 and phase e−iπνα/2 with Maslov index να counting caustic crossings. The method assumes: (i) isolated non-degenerate stationary points (or a uniformization near fold caustics), (ii) actions ~ (but differences may be O ( ~ )), and (iii) sufficiently smooth boundary projectors so that the method’s boundary terms vanish. These standard conditions justify replacing the exact operator traces by sums over classical data in (A4). 4. Microreversibility and the diagonal FT Time reversal Θmaps U ( τ ) 7→ U† ( τ )and exchanges the supports of Π + and Π − due to the odd parity of the entropy production functional. Consequently, the diagonal pieces obey Pcl (+) /Pcl ( − ) = e∆S/kB as in the classical (Markovian) FT; the off-diagonal interference then multiplies this bias to produce (A2) . This matches Eq. (13) , confirming Eq. (5) and Eq. (13) are identical. 5. Uniform approximations and caustics When two stationary paths coalesce, the quadratic expansion fails. A uniform Airy (fold) or Pearcey (cusp) approximation repairs (A4) by resumming the local catastrophe. The net effect is to replace AαeiSα/~ + AβeiSβ/~ by a single special-function expression that is still interference-dominated and feeds the same ratio (A2) ; Maslov indices update accordingly. This ensures that Eq. (5) and Eq. (13) remain equal through caustic crossings (the only change is a smooth reparameterization of I±). 6. TPM versus coherence-retaining protocols The two-projective-measurement (TPM) scheme destroys initial coherences and sets the off-diagonal terms in (A4) to zero at t = 0, recovering the classical FT. Our protocol defers projective collapse (or uses weak/ancilla readout) so that (A4) remains valid, and the purification phases choose the location on the interference manifold. This is precisely the operational distinction enabling the difference between the classical and quantum-modified ratios. Conclusion of Appendix A. Stationary-phase evaluation of the operator trace (A1) yields exactly the VVG structure; hence the “semiclassical” Eq. (5) and the “quantum” Eq. (13) in the manuscript are the same identity under the usual conditions, with differences only in representation. 21 Appendix B: Purification Derivations and Optimization 1. Phase-optimization with constraints For the phase-parameterized purification |ΨSA(φ)i=X i √pieiφi|sii⊗|aii (cf. Eqs. (6)–(9)), the amplitude into the −∆Ssector after evolution USA and readout (Π−⊗ΠA)is A−(φ) = X i √pieiφici, ci≡ hΦ−|⊗hrA|USA|sii⊗|aii. The objective P− ( φ ) = |A− ( φ ) |2 is maximized over the torus of phases subject to global-phase invariance. Introduce the Lagrangian L = P−−λ(Piφiwi) with a gauge-fixing linear constraint Piφiwi = 0 (e.g., wi=pi). Stationarity gives ∂P− ∂φk = 2 ImA−(φ)√pke−iφkc∗ k=λwk. At the maximum, λ= 0 and φ? k=−arg ck+χ,χ∈R, hence Pmax −=X i √pi|ci|2. This is the closed form quoted in Sec. III; it saturates the Cauchy–Schwarz bound and shows that the interference enhancement equals the squared `1norm of the amplitude vector {√pici}. 2. Sensitivity, Fisher information, and robustness Let phases fluctuate independently, φi7→ φi+δφiwith hδφii= 0,hδφiδφji=σ2 φδij. To second order, hP−(φ+δφ)i ≈ P−(φ)e−σ2 φ. The local curvature of P−at φ?is governed by the (classical) Fisher information matrix Fij = 4 Re∂φiA∗ −∂φjA−φ?= 4 √pipj|cicj|δij , consistent with the quantum Fisher information for unitary phase shifts on the ancilla subspace. Robust phase sets can be selected by regularizing the cost −ln P−with a quadratic penalty on φ. 3. Gradient of the FT functional and optimality conditions Define J(φ) = ln(1 + I+)−ln(1 + I−). The chain rule yields ∂J ∂φk =1 1 + I+ ∂I+ ∂φk−1 1 + I− ∂I− ∂φk , with ∂I− ∂φk =Pi6=j∂φkC(−) iC(−)∗ jei ~(Si−Sj) P`|C(−) `|2−I− ∂ ∂φk ln X `|C(−) `|2!. Because C(−) i is linear in the phase-labeled amplitudes eiφi entering the purification, the derivative reduces to combinations of the ci ; the optimal set satisfies ∂φkJ = 0 with the same locking condition as above (align C(−) iphasors; anti-align C(+) iwhere possible). 22 4. Gauge freedom and Uhlmann transport Any purification | Ψ SAi is defined up to ( 1⊗VA ), a gauge on the ancilla. Choosing the parallel-transport Uhlmann connection along a protocol ρS ( t )fixes VA ( t )by fidelity maximization between t and t + dt . Locking the ancilla to this gauge keeps relative phases as constant as possible, suppressing phase diffusion within the coherent subspace and stabilizing the interference quotients in (A2) ; this is the geometric underpinning of the “phase knob” discussed in Sec. VII. Appendix C: Case Studies: Photosynthesis and ATP Synthase 1. Photosynthetic exciton transport (dimer and network) Dimer with vibronic assistance. Consider a two-site exciton Hamiltonian coupled to a mode: HS=∆ 2σz+J σx, HSE =λσz(b+b†) + X k ωkb† kbk. Add a sink Lsink = √κ|RCih 2 | and dephasing Lφ = √γφσz . The reduced dynamics follows (Lindblad form). The probability to reach the reaction center within time τ, PRC(τ) = κZτ 0 dt h2|ρS(t)|2i, inherits interference between the direct path and the vibronically dressed path; the same pairwise structure enters I− in (A2) . Resonant ω≈√∆2+ 4J2 and moderate γφ align phases and enhance PRC , i.e. tilt the FT tails toward lower entropy along energy-transfer trajectories. Multi-site network (FMO-type). For an N -site network with site energies {n} and couplings {Jmn} , HS=X n n|nihn|+X m6=n Jmn(|mihn|+|nihm|). Structured dephasing and a terminal sink map the network to a set of interfering multi-hop paths from an input chromophore to the reaction center. The interference-renormalized FT applies to entropy production along these paths; optimizing phases (via control or environment engineering) sharpens low-entropy arrival events. The physical picture mirrors the noise-/dephasing-assisted transport paradigm and provides explicit FT-level diagnostics (log-ratio oscillations) beyond average transfer efficiency. 2. ATP synthase: a six-state chemo-mechanical cycle Model. Adopt the minimal six-state basis {| B , θ±i,| S , θ±i,| R , θ±i} described in Sec. V, with HS in Eq. (38) and Lindblad channels (39) . The entropy decrease associated with a successful substep | B i→| R i at fixed proton-motive force is monitored by a coarse-grained observable (chemical plus mechanical work). Interference and prediction. The amplitude (40) involves two rotor-angle pathways θ± . The enhanced P− at phases φ+−φ− = −δ reduces the cycle’s FT ratio and, via Eq. (37) , increases efficiency. The predicted observable is an oscillatory dip in the substep FT log-ratio as a function of a drive or ancilla phase; its presence should correlate with increased ATP synthesis probability per cycle. This is compatible with the substep-resolved rotation data in single-molecule assays and the structural quasi-degeneracies seen crystallographically. Appendix D: Figures, Parameter Sweeps, and Reproducibility 1. Reproducing Fig. 1 and Fig. 2 and exploring parameter space Fig. 1 (p. 4) contrasts the classical FT ratio with an interference-modified curve; Fig. 2 (p. 4) zooms into the negative-entropy sector and shows a coherence-enhanced profile with a green shaded area. 23 Baseline and toy interference model. Define a grid ∆S∈[Smin, Smax]; the classical ratio is Rcl(∆S) = e∆S/kB. A generic toy model for interference corrections consistent with Eq. (13) is Rqm(∆S) = Rcl(∆S)1 + PM m=1 ame−(∆S−∆Sm)2 2σ2 mcos(ωm∆S+ϕm) 1 + PN n=1 bne−(∆S−˜ ∆Sn)2 2˜σ2 ncos(˜ωn∆S+ ˜ϕn) ,(D1) where the numerator sketches I+ and the denominator sketches I− . Choosing M = 1, N = 1 with ∆ S1& 0and ˜ ∆S1< 0reproduces Fig. 1 qualitatively. Setting b1> 0and a narrow ˜σ1 centered on negative ∆Syields the green shaded enhancement shown in Fig. 2. Parameter sweeps. To explore robustness: • Strength sweep: vary a1, b1∈ [0 , 0 . 5] and record the minimum of ln [ Rqm/Rcl ]to quantify maximal dips. • Width/location sweep: sweep ˜σ1 and ˜ ∆S1 to map where the negative-entropy enhancement is concentrated. • Frequency/phase sweep: vary ( ωm, ϕm )to emulate chaos-assisted multi-path oscillations; extract the number of visible lobes and their spacing. In all cases ensure 1 + I±> 0to keep probabilities non-negative; this holds automatically for small amplitudes am, bnor when the sums are dominated by Gaussian envelopes. 2. Experimental mappings of sweep parameters For driven cold-atom or circuit-QED platforms, ( ωm, ϕm )correspond to drive frequencies and phases; (∆ Sm, σm )reflect the range over which multiple trajectory families contribute coherently (tunable by detuning and coupling strength). 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