Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

For quadratic potentials, this paper establishes that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model is reproduced at any temperature by a classical, non-Markovian stochastic process in phase space.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 17:40 UTC pith:VPPE6DPS

load-bearing objection A useful QBM Monte Carlo recipe built on a standard equivalence, but the exactness claim has an unproven initial-correlation step and the printed noise formulas contain a dimensionally wrong factor. the 4 major comments →

arxiv 2512.08641 v3 pith:VPPE6DPS submitted 2025-12-09 quant-ph

Quantum Brownian Motion as a Classical Stochastic Process in Phase Space

Dmitriy Kondaurov , Evgeny Polyakov This is my paper
classification quant-ph MSC 81S3081S4082C31
keywords quantum Brownian motionCaldeira-Leggett modelWigner functiongeneralized Langevin equationquantum fluctuation-dissipation theoremnon-Markovian stochastic processMonte Carlo simulationdecoherence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that, for a particle coupled to a harmonic-oscillator bath through a quadratic potential, the full quantum dynamics is exactly represented by an ensemble of classical phase-space trajectories obeying a generalized Langevin equation; the only quantum input is the noise statistics, taken from the bath's thermal Wigner distribution via the quantum fluctuation-dissipation theorem. Starting from the correlated thermal equilibrium state of particle plus bath, arbitrary preparations of the particle—including Schrödinger-cat superpositions—enter as weighted averages over trajectory ensembles. This turns a difficult non-Markovian open-quantum-system problem into a Monte Carlo simulation of classical stochastic trajectories, valid from high temperature down to zero temperature. For smooth nonquadratic potentials, the off-diagonal width of the reduced density matrix, λ = ℏ/sqrt(⟨p²⟩_eq), shrinks as the bath cutoff grows, giving a controlled small parameter for approximation. If correct, the method provides an exact simulator for driven-dissipative quantum protocols and exposes failures of high-temperature master equations at low temperature.

Core claim

The central claim is an identity: the Wigner-transformed propagator of the quantum Caldeira-Leggett model, starting from a correlated thermal state, equals the propagator of a classical non-Markovian process whose noise correlation is the symmetrized quantum noise spectrum, i.e., the quantum fluctuation-dissipation theorem. The proof goes through the Wigner function, which for a quadratic Hamiltonian obeys the classical Liouville equation; eliminating bath variables yields the generalized Langevin equation with noise ξ_W(t) built from bath Wigner-sampled initial conditions. For quadratic potentials this is exact at any temperature and for any particle preparation represented by a Wigner prep

What carries the argument

The load-bearing object is the Wigner function of the total particle-plus-bath system, whose evolution for the quadratic Caldeira-Leggett Hamiltonian is the classical Liouville equation. Eliminating the bath variables from the classical trajectories yields the generalized Langevin equation with colored noise drawn from the thermal Wigner functions of the bath oscillators; the ensemble-averaged noise obeys the quantum fluctuation-dissipation theorem, including zero-point fluctuations. Preparations and interventions are converted from operators to phase-space preparation functions λ(r₀,p₀|r̄,p̄), which act as trajectory weights. The natural small parameter for nonquadratic potentials is the eq

Load-bearing premise

The reduction of the correlated thermal equilibrium state to a long evolution from a factorized preparation at t = −T is assumed, not proven: the paper's appendix justifies it in one sentence, and real-time evolution does not literally reproduce the imaginary-time structure of e^(−βH); if this relaxation step fails, the exactness claim for correlated initial states is unsupported.

What would settle it

Take a finite bath with a few oscillator modes where the exact correlated thermal state e^(−βH)/Z can be computed, run the paper's protocol (factorized state at t = −T, evolve to t = 0), and compare the resulting reduced Wigner function elementwise to the exact state. Any nonvanishing difference as T → ∞ for fixed β and cutoff would falsify the initialization claim; alternatively, a mismatch between the GLE noise correlation used and the symmetrized quantum noise of the exact state would falsify the mapping.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For quadratic potentials, observables of the open quantum particle, including non-Markovian memory effects, can be computed exactly at all temperatures by averaging classical trajectories.
  • The Monte Carlo scheme reproduces exact analytical results at zero temperature and captures fast early-time thermalization and decoherence driven by vacuum fluctuations—effects missed by high-temperature white-noise and master-equation treatments.
  • Arbitrary sequences of measurements and manipulations are representable as trajectory weights, so the same classical trajectory engine simulates driven-dissipative protocols such as cat-state decoherence.
  • For smooth nonquadratic potentials, the error of the classical-stochastic approximation is controlled by λ/L, justifying extension to anharmonic systems when the bath cutoff is large.

Where Pith is reading between the lines

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

  • The paper's Monte Carlo scheme uses rejection sampling with |λ| and complex weights; its practical efficiency for strongly nonclassical states with large negative Wigner regions may degrade, since no sign-problem analysis is given.
  • If the negative-time equilibration assertion fails, the exactness claim starting from the correlated thermal state would be limited to factorized or otherwise uncorrelated initial states; this is testable by comparing the protocol's state at t = 0 with e^(−βH)/Z for a finite bath.
  • The identified small parameter suggests a systematic perturbation theory in λ/L—for example, gradient-expansion corrections to the generalized Langevin equation—that would extend exactness to weakly anharmonic potentials; the paper leaves this as an outlook.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper claims that the exact reduced dynamics of a Caldeira-Leggett Brownian particle in a quadratic potential can be represented, at any temperature, as an ensemble of classical non-Markovian Langevin trajectories with noise obeying the quantum fluctuation-dissipation theorem. It further claims that arbitrary particle preparations and interventions can be handled via Wigner-function weights, and that the scheme applies to anharmonic potentials through a small-parameter expansion controlled by the bath-induced coherence length. The manuscript validates the method against the analytical results of Grabert et al. and Breuer-Petruccione. However, the presentation contains a load-bearing normalization error in the quantum FDT and an unjustified reduction of the correlated thermal equilibrium state to a negative-time evolution from a factorized state; the latter is not merely a technical gap but is inconsistent with unitary evolution for pure preparations. The numerical implementation appears to use a different noise correlator than the displayed theory, so the internal consistency of the validation is unclear. The central exactness claim for correlated equilibrium states is therefore unsecured.

Significance. If the mapping held as stated, it would be a practically valuable all-temperature stochastic representation of non-Markovian open quantum dynamics, including zero-point driven decoherence and arbitrary Wigner-weight preparations. The benchmarks against known exact solutions are a genuine strength and give the reader useful non-trivially falsifiable checks. However, because of the FDT normalization contradiction and the unsupported correlated-initial-state step, the paper's main theorem—exactness starting from the correlated thermal state—is not established. The factorized-initial-state version of the mapping is essentially standard, so the novelty and impact hinge on the correlated-initial-state construction. With those gaps fixed or explicitly narrowed, the Monte Carlo scheme could be a useful contribution to the literature.

major comments (4)
  1. [Eqs. (28), (29), (33), (34), (37)] The quantum FDT is written with a spurious factor of ω. For the spectral density J(ω) defined in Eq. (9), the symmetrized noise correlator for the GLE is C(t-s) = (ℏ/π) ∫_0^∞ dω J(ω) coth(ℏω/2kBT) cos[ω(t-s)]. Equations (29) and (33) contain an additional factor of ω, making them dimensionally inconsistent. Concretely, the T=0 Ohmic result should be Eq. (30), but the extra-ω formula gives a different ε-dependence. Moreover, the numerical synthesis in Eqs. (34) and (37) has no ω factor and therefore generates the correct correlator, not Eq. (33). Thus the displayed theory and the numerical implementation are mutually inconsistent; the validation in Sec. V cannot confirm Eqs. (28)–(33) as written.
  2. [Sec. III.A, Eq. (24), App. A.3, Sec. IV.B] The reduction of the correlated equilibrium state ρ_SB^β to a negative-time evolution from a factorized state is not established. If W(r,p;-T) in Eq. (24) corresponds to a pure particle state, unitary evolution from the factorized initial state cannot converge to the mixed Gibbs state Eq. (13); the global state remains on a unitary orbit. Appendix A.3 asserts in one sentence that boundary terms disappear, but no mixing, ergodicity, or initial-slip control is given. This gap is load-bearing because Eq. (14) and the Sec. IV.B initialization protocol both rely on sampling the correlated equilibrium state. Please either provide a rigorous derivation for the reduced dynamics (e.g., via the imaginary-time/Matsubara influence functional) or narrow the theorem to factorized initial preparations and mark the correlated case as a separate, non-exact assumption.
  3. [Eq. (28)] Equation (28) includes an imaginary antisymmetric term −i sin[ω(t−s)] in the two-time correlation. A real classical stochastic force ξ_W constructed from sampled initial oscillator coordinates can only have a real symmetric two-time correlation, namely Eq. (29). Presenting Eq. (28) as the noise correlation of the 'classical stochastic process' conflates the non-symmetrized operator correlation with the stochastic-process correlation. The manuscript should clarify that Eq. (28) is the operator correlation and that Eq. (29) is the correlation actually realized by the real classical noise ξ_W.
  4. [Sec. VI.A, Eq. (49)] Equation (49) claims that ⟨p²⟩_eq diverges in the wide-band limit for an Ohmic bath. For the exponential-cutoff spectral density J(ω)=γmω e^{-εω} used throughout the paper, J(ω)/ω = mγ e^{-εω}, so the integral remains finite as the upper cutoff is removed. The divergence only occurs for a hard-cutoff spectral density with J∝ω over the entire range. This should be corrected or reformulated, and it directly affects the statement that the coherence length λ vanishes for the family of baths considered in the paper.
minor comments (5)
  1. [Eq. (22)] The Wigner transform is missing the normalization factor 1/(2πℏ) (or an equivalent convention statement). As written it is inconsistent with the standard Weyl correspondence used in Eq. (25).
  2. [Sec. II.A] The text states 'we consider free Brownian motion with V(x)=0', but the Hamiltonian in Eq. (1), the numerical equation (38), and Sec. VI all allow or use a potential V(x). Please clarify which potentials are used in the analytical and numerical results.
  3. [Introduction] The citation '[3?–6]' is malformed; the question mark should be removed and the intended references checked.
  4. [Sec. IV.B] Initializing all trajectories with x(−T_eq)=0, p(−T_eq)=0 corresponds to a delta-function Wigner function, which is not a valid quantum particle preparation. If this is merely a numerical equilibration device, this should be stated explicitly; otherwise the sampling procedure is not a sampling of the factorized Wigner state described in Sec. III.A.
  5. [Eq. (32)] The symbol p^{(j)}_S(t) is not defined; it should presumably be p^{(j)}(t).

Circularity Check

0 steps flagged

No significant circularity: the GLE mapping is derived from the quadratic Wigner/Liouville structure and influence-functional identities, and is benchmarked against external analytical results rather than fitted.

full rationale

The central derivation is not circular. For the quadratic CL Hamiltonian, the Wigner function obeys the classical Liouville equation (Eq. 23), so total-system trajectories are exact; eliminating the bath gives the GLE (Eq. 26) with noise statistics set by the quantum thermal Wigner distributions (Eqs. 27-29). Appendix A.2 proves equality of the classical and quantum propagators for factorized initial states by evaluating the q-integral of the influence functional (Eqs. A10-A15), and no parameter is fitted to make the target results come out. The numerical benchmarks are checked against Grabert et al. Eq. (42) and Breuer-Petruccione Eq. (46), which are external analytical results. The Wigner preparation/weight scheme is a standard representation, not a fit. The anharmonic O(lambda/L) claim is asserted rather than derived, and the correlated-initial-state step rests on App. A.3's one-sentence negative-time 'trick' ('Directing T->infinity, terms connecting dynamics with initial state at t=-T disappear...'), but that is an unproved relaxation/ergodicity assumption about the physical initialization, not a structural circularity: the map itself does not reduce to its input by definition, and no equation is equivalent to its own output by construction. I therefore find no significant circularity; the flagged items are rigor/correctness concerns, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities (no new fields, particles, or forces). It introduces a technique (signed Wigner preparation weights) and a heuristic small parameter (lambda/L), neither of which is an invented entity with a falsifiable handle. The free parameters listed are physical inputs (gamma, epsilon, T, m, hbar, sigma0), not fitted values. The load-bearing assumptions are the equilibration/imaginary-time replacement and the uniform validity of the classical GLE with quantum noise; the anharmonic small-parameter claim is an ad hoc assertion.

axioms (4)
  • domain assumption A factorized product state at t=-T evolved to t=0 converges to the correlated thermal equilibrium state rho_SB^beta as T->infinity; equivalently, negative real-time propagation can replace imaginary-time (Matsubara) propagation for initial correlations.
    Sec. III.A step 1 and Appendix A.3 ('Directing T->infinity, terms connecting dynamics with initial state at t=-T disappear'). Requires relaxation/ergodicity in the thermodynamic limit; not proven. Load-bearing for the exactness-from-correlated-equilibrium claim.
  • domain assumption Bath initial conditions sampled from free-oscillator thermal Wigner functions produce noise whose symmetrized statistics equals the quantum FDT even after system-bath correlations have developed.
    Sec. III.B, Eqs. (27)-(29). Standard for linear coupling, but it is exactly the identity that converts the operator GLE into a classical process with real colored noise; it holds only for symmetrized (single-time) quantities, not for time-ordered multi-time correlators.
  • ad hoc to paper For smooth anharmonic potentials the GLE-with-quantum-noise dynamics is accurate to O(lambda/L) with lambda = hbar / sqrt(<p^2>_eq), uniformly in time.
    Sec. VI.B, Eq. (51). Asserted without derivation or numerical demonstration. The lambda->0 argument (Eqs. 49-50) is shown only for the free-particle case; for confined potentials <p^2>_eq is dominated by zero-point motion and does not diverge, and short-time coherences generated by a preparation are not obviously controlled by the equilibrium lambda.
  • standard math For a quadratic Hamiltonian, Wigner-function evolution is governed by the classical Liouville equation, so the total-system quantum state is a classical ensemble of phase-space trajectories.
    Sec. III.A, Eq. (23). Standard result; requires the Hamiltonian to be at most quadratic, which the paper states.

pith-pipeline@v1.3.0-alltime-deepseek · 13754 in / 39009 out tokens · 397085 ms · 2026-08-03T17:40:05.275295+00:00 · methodology

0 comments
read the original abstract

We establish that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model, with at most quadratic external potential, can be mapped, at any temperature, onto a classical, non-Markovian stochastic process in phase space. Starting from a correlated thermal equilibrium state between the particle and bath, we demonstrate that this correspondence is exact for quadratic potentials under arbitrary quantum state preparations of the particle itself. Our approach allows to consider arbitrary initial quantum states - including highly non-classical superpositions - which are incorporated via their Wigner functions, which serve as statistical weights for trajectory ensembles. Furthermore, the formalism naturally accommodates external manipulations and measurements modeled by preparation functions acting at arbitrary times, enabling the simulation of complex driven-dissipative quantum protocols. For more general, smooth potentials, we identify a natural small parameter: the density matrix becomes strongly quasidiagonal in the coordinate representation, with its off-diagonal width shrinking as the bath's spectral cutoff increases, suggesting a controlled parameter for a possible approximation.

Figures

Figures reproduced from arXiv: 2512.08641 by Dmitriy Kondaurov, Evgeny Polyakov.

Figure 1
Figure 1. Figure 1: FIG. 1. Time evolution of the coordinate dispersion [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Momentum dispersion [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum Brownian transport in a correlated Gaussian force

    cond-mat.stat-mech 2026-07 reject novelty 4.0

    A no-dissipation Caldeira-Leggett model is claimed to give bath-particle mean-square displacement ~t^5 and velocity ~t^3, but the derivation is internally inconsistent.

Reference graph

Works this paper leans on

50 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Its formal solution, substituting the particle’s trajectoryx(t), is xi(t) =x i(0) cos(ωit) + pi(0) miωi sin(ωit) + ci miωi Z t 0 sin[ωi(t−τ)]x(τ)dτ

    Reduced Equation of Motion The classical equations of motion derived from Hamil- ton’s equations are m¨x(t) = X i cixi(t)−x(t) X i c2 i miω2 i ,(2) mi ¨xi(t) +m iω2 i xi(t) =c ix(t).(3) Equation (3) for the bath oscillators is a linear inho- mogeneous equation. Its formal solution, substituting the particle’s trajectoryx(t), is xi(t) =x i(0) cos(ωit) + pi...

  2. [2]

    The corresponding memory kernel isM(t) = (2mγ/π)ε/(ε 2 +t 2)

    Spectral Density and Thermodynamic Limit In the thermodynamic limit of a continuous bath, the system properties are encoded in the spectral density J(ω) = π 2 X i c2 i miωi δ(ω−ω i).(9) 3 The memory kernel is then expressed as M(t) = 2 π Z ∞ 0 dω J(ω) ω cos(ωt).(10) For an ohmic bath with an exponential cutoff on Λ (cor- responding to a time scaleε= Λ −1)...

  3. [3]

    Fluctuation-Dissipation Theorem The stochastic forceξ(t) derives from the uncertain initial conditions{x i(0), pi(0)}of the bath. Assuming these are sampled from a Gibbs distribution at tempera- tureT, i.e.,ρ B ∝exp(−H B/kBT), the noise is Gaussian with zero mean and its autocorrelation function is given by the classical fluctuation-dissipation theorem (F...

  4. [4]

    Joint State and Preparation The total system’s quantum state is described by a density matrixρ SB . We consider the physically relevant scenario where, at an initial timet= 0, the total sys- tem is prepared starting from a joint thermal equilibrium state at inverse temperatureβ= 1/(k BT): ρβ SB =Z −1e−βH ,(13) whereHis the full Hamiltonian in Eq. (1) andZ...

  5. [5]

    Consider a sequence of times 0< t 1 < t2 <· · ·< tN

    Reduced Quantum Dynamics with Intermediate Interventions The formalism of preparation operators naturally ex- tends to describe a sequence of interventions—such as measurements, unitary kicks, or state preparations— applied to the particle at specific intermediate times. Consider a sequence of times 0< t 1 < t2 <· · ·< tN . At each timet k, an interventio...

  6. [6]

    (13), an initial preparation is performed att= 0: ρSB (0+) = X j0 O(0) j0 ρβ SB O(0) j0 † .(16)

    Initial Preparation: Starting from the joint thermal stateρ β SB from Eq. (13), an initial preparation is performed att= 0: ρSB (0+) = X j0 O(0) j0 ρβ SB O(0) j0 † .(16)

  7. [7]

    Unitary Evolution: The state evolves unitarily un- der the full Hamiltonian until the first intervention: ρSB (t− 1 ) =U(t 1)ρ SB (0+)U †(t1),whereU(t) =e −iHt/ℏ . (17)

  8. [8]

    Intervention att k: At timet k, the intervention op- erators are applied: ρSB (t+ k ) = X jk O(k) jk ρSB (t− k ) O(k) jk † .(18) The state then continues its unitary evolution until the next intervention or the final time

  9. [9]

    The observable 4 ˆAacts solely on the particle’s Hilbert space

    Final State and Observables: After the last inter- vention att N , the state evolves to the any time t > tN : ρSB (t) =U(t−t N )ρ SB (t+ N )U †(t−t N ).(19) The physically relevant object is the reduced den- sity matrix of the particle att, obtained by tracing out the bath: ρS(t) = TrB ρSB (t) .(20) The expectation value of any particle observable ˆA at a...

  10. [10]

    Crucially, evolving the system from this factorized state at t=−Ttot= 0 under the full interacting Hamilto- nian allows the particle and the bath to fully equi- librate

    At a distant past timet=−T(withT→ ∞), we assume a factorized, positive initial Wigner func- tion: W SB (−T) =W(r, p;−T)× Y i W β i (ri, pi).(24) Here,Wis an arbitrary (positive) Wigner function for the Brownian particle, andW β i is the thermal Wigner function for thei-th bath oscillator at in- verse temperatureβ= 1/(k BT), which for a har- monic oscillat...

  11. [11]

    An ensemble of initial conditions (r(j), p(j),{r (j) i , p(j) i }) is sampled from the dis- tributionW SB (−T)

  12. [12]

    For each samplej, the classical equations of motion (Hamilton’s equations derived from the fullH) are solved numerically fromt=−Tto the desired final time

  13. [13]

    Expectation values of system observables at any timetare computed by averaging the correspond- ing Weyl symbolO W (r, p) over this ensemble of classical trajectories: ⟨ ˆO(t)⟩ ≡Tr h ˆOρS(t) i = lim N→∞ 1 N NX j=1 OW r(j)(t), p(j)(t) . (25) B. The Reduced Stochastic Process for the Brownian Particle The Hamiltonian structure of the CL model allows the bath...

  14. [14]

    Begin simulations at a sufficiently distant past time tstart =−T eq, withT eq chosen to be several times longer than the bath’s memory timeτ M and the system’s relaxation timeγ −1

  15. [15]

    Initialize the particle with arbitrary coordinates and momentum (e.g.,x(−T eq) = 0, p(−Teq) = 0)

  16. [16]

    (38) fromt start tot=

    Evolve the system under Eq. (38) fromt start tot=

  17. [17]

    The state att= 0 is then statistically indistinguishable from the true thermal equilibrium stateρ β SB

    This equilibration period allows the system-bath correlations to fully develop. The state att= 0 is then statistically indistinguishable from the true thermal equilibrium stateρ β SB

  18. [18]

    Store the particle’s phase-space coordinates (x(j)(0−), p(j)(0−)) at the end of this equilibration run. This initialization procedure implicitly samples from the correct Wigner distributionW SB (r, p,{ri, pi},0) corre- sponding toρ β SB , circumventing the need for its explicit construction. C. Monte Carlo Sampling of Preparations and W eighted Averages T...

  19. [19]

    For each trajectoryj, the coordinates (¯r (j),¯p(j)) = (x(j)(0−), p(j)(0−)) serve as the input to the prepa- ration function

  20. [20]

    A new set of post-preparation coordinates (r(j) i , p(j) i ) is sampled. In this work, we employ a simple rejection sampling method: candidate points (r, p) are drawn uniformly from a region encompassing the essential support of the target distribution (e.g., within several standard devia- tions of its mean), and accepted with a probability proportional t...

  21. [21]

    Each trajectory is assigned a complex weight w(j) =λ(r (j) 0 , p(j) 0 |¯r(j),¯p(j)).(39)

  22. [22]

    (38), generating its future evolution

    The trajectory is then propagated forward from the new initial condition (r (j) 0 , p(j) 0 ) att= 0 + under Eq. (38), generating its future evolution. For a sequence of interventions at times{t k}, this reweighting/resampling procedure is applied at each in- tervention time using the corresponding preparation function. The expectation value of a particle ...

  23. [23]

    Exact Quantum Dynamics (Our method): The col- ored noise correlation uses the full quantum FDT, Eq. (33)

  24. [24]

    High-Temperature White Noise: The noise cor- relation uses the classical FDT,⟨ξ(t)ξ(0)⟩ ∝ 2mγkBT δ(t), corresponding to the standard Markovian limit

  25. [25]

    High-Temperature Master Equation (ME): Dy- namics from the common Caldeira-Leggett master equation [7, 12]: ˙ρ=− i ℏ [H0, ρ]− iγ 2ℏ [x,{p, ρ}]− mγkBT ℏ2 [x,[x, ρ]].(45) The parameters areγ=π/2,ε= 0.01, andT= 1 (in units whereℏ= 1). Crucially, for these parameters, kBT≈1 isnotmuch greater than the effective cutoff en- ergyℏ/ε≈100, violating the condition (...

    2021

  26. [26]

    (A3) withγ(t) = 1 m M(t)

    Classic non-Markovian stochastic process For generalized Langevin equation [22] m¨x+ Z t ti dsM(t−s) ˙x(s) =ξ(t) (A1) with colored noise⟨ξ(s)ξ(s+t)⟩=σ(t) propagator (with the normalized prefactor omitted) is written as K(x f pf tf |xipiti) = Z xf xi Dx Z Dzδ( ˙xi− pi m )δ( ˙xf − pf m )eiS[x,z] , (A2) where effective action is S[x, z] = Z tf ti dsz(s)[¨x(s...

  27. [27]

    Calculate explicitly R Dq, slicing [0, t] by intervalsτ= t/N: [t 0, t1, ..., tN ], soq(t k) =q k

    Quantum influence functional with factorized initial condition For non-correlated state we have ρSB 0 =ρ 0 ⊗ρ B β (A4) J(r f qf tf |riqiti) = Z rf ri Dr Z qf qi Dqe i ℏ A[r,q] (A5) A[r, q] =− Z tf ti dsmq(s)[¨r(s) + Z s ti duγ(s−u) ˙r(u)]+ +m( ˙rf qf −˙riqi) + i 2ℏ Z tf ti ds Z tf ti duq(s)N(s−u)q(u) (A6) Go to Wigner representationW(rpt) = 1 2πℏ R dqρ(rq...

  28. [28]

    Replacing integration in imaginary time with a negative time for correlated initial states To comparisong with [17] we show, as it taken into account correlation between particle and environment in t= 0. Notice thatρ SB β is stationary state of full Hamil- tionian, so we can identically rewrite it as U(0,−T)ρ SB β U(−T,0) (A16) Because of preparation acts...

  29. [29]

    Sur la th´ eorie du mouvement brownien,

    P. Langevin, “Sur la th´ eorie du mouvement brownien,” C. R. Acad. Sci. Paris146, 530–533 (1908)

    1908

  30. [30]

    Reaction-rate theory: fifty years after Kramers,

    P. H¨ anggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Rev. Mod. Phys.62, 251–341 (1990)

    1990

  31. [31]

    Quantum tunneling in the presence of an arbitrary linear dissipation mechanism,

    A. J. Leggett, “Quantum tunneling in the presence of an arbitrary linear dissipation mechanism,” Phys. Rev. B30, 1208–1218 (1984)

    1984

  32. [32]

    Quantum Brownian Motion in Bose-Einstein Condensates,

    C. Charalambous, “Quantum Brownian Motion in Bose-Einstein Condensates,” Ph.D. thesis, Universitat Polit` ecnica de Catalunya (2020)

    2020

  33. [33]

    Surface diffusion within the Caldeira–Leggett formalism,

    E. E. Torres-Miyares, G. Rojas-Lorenzo, J. Rubayo- Soneira, and S. Miret-Artes, “Surface diffusion within the Caldeira–Leggett formalism,” Phys. Chem. Chem. Phys. 24, 15871–15890 (2022)

    2022

  34. [34]

    Applicabil- ity of the Caldeira–Leggett Model to Vibrational Spec- troscopy in Solution,

    F. Gottwald, S. D. Ivanov, and O. K¨ uhn, “Applicabil- ity of the Caldeira–Leggett Model to Vibrational Spec- troscopy in Solution,” J. Phys. Chem. Lett.6, 2722–2727 (2015)

    2015

  35. [35]

    Quantum tunnelling in a dissipative system,

    A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Ann. Phys. (N.Y.)149, 374–456 (1983)

    1983

  36. [36]

    Path integral ap- proach to quantum Brownian motion,

    A. O. Caldeira and A. J. Leggett, “Path integral ap- proach to quantum Brownian motion,” Physica A121, 587–616 (1983)

    1983

  37. [37]

    Weiss,Quantum Dissipative Systems, 4th ed., Series in Modern Condensed Matter Physics Vol

    U. Weiss,Quantum Dissipative Systems, 4th ed., Series in Modern Condensed Matter Physics Vol. 13 (World Sci- entific, Singapore, 2012)

    2012

  38. [38]

    C. W. Gardiner and P. Zoller,Quantum Noise(Springer, Berlin, 2004)

    2004

  39. [39]

    On the Generators of Quantum Dynamical Semigroups,

    G. Lindblad, “On the Generators of Quantum Dynamical Semigroups,” Commun. Math. Phys.48, 119–130 (1976)

    1976

  40. [40]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2002)

    2002

  41. [41]

    Completely Positive Quantum Dissipa- tion,

    B. Vacchini, “Completely Positive Quantum Dissipa- tion,” arXiv:quant-ph/0002094 (2000)

    Pith/arXiv arXiv 2000

  42. [42]

    Test Particle in a Quantum Gas,

    B. Vacchini, “Test Particle in a Quantum Gas,” Phys. Rev. A63, 033803 (2001)

    2001

  43. [43]

    R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals(McGraw-Hill, New York, 1965)

    1965

  44. [44]

    The Theory of a Gen- eral Quantum System Interacting with a Linear Dissipa- tive System,

    R. P. Feynman and F. L. Vernon, “The Theory of a Gen- eral Quantum System Interacting with a Linear Dissipa- tive System,” Ann. Phys. (N.Y.)24, 118–173 (1963)

    1963

  45. [45]

    Quantum Brownian Motion: The Functional Integral Approach,

    H. Grabert, P. Schramm, and G.-L. Ingold, “Quantum Brownian Motion: The Functional Integral Approach,” Phys. Rep.168, 115–207 (1988)

    1988

  46. [46]

    Low-Temperature and Long-Time Anomalies of a Damped Quantum Particle,

    P. Schramm and H. Grabert, “Low-Temperature and Long-Time Anomalies of a Damped Quantum Particle,” J. Stat. Phys.49, 767–810 (1987)

    1987

  47. [47]

    Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise,

    B. L. Hu, J. P. Paz, and Y. Zhang, “Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise,” Phys. Rev. D45, 2843–2861 (1992)

    1992

  48. [48]

    Quantum description of Einstein’s Brownian motion,

    F. Petruccione and B. Vacchini, “Quantum description of Einstein’s Brownian motion,” Phys. Rev. E71, 046134 (2005)

    2005

  49. [49]

    Dissipation in the Caldeira-Leggett model,

    L. Ferialdi, “Dissipation in the Caldeira-Leggett model,” Phys. Rev. A95, 052109 (2017)

    2017

  50. [50]

    Path Integral Solution for Nonlinear Gener- alized Langevin Equations,

    P. H¨ anggi, “Path Integral Solution for Nonlinear Gener- alized Langevin Equations,” inPath Integrals from meV to MeV: Tutzing ’92, edited by H. Grabert, A. Inomata, L. Schulmann, and U. Weiss (World Scientific, Singapore, 1993), pp. 289–301

    1993