Wigner-Lindblad Equations for Quantum Friction

Wigner-Lindblad equations for quantum friction

Denys I. Bondar,∗,† Renan Cabrera,† Andre Campos,† Shaul Mukamel,‡ and Herschel A. Rabitz†

†Princeton University, Princeton, NJ 08544, USA ‡University of California, Irvine, Irvine, CA 92697, USA

Abstract Introduction

Dissipative forces are ubiquitous and thus con- Realistic models of quantum systems must in- stitute an essential part of realistic physical clude dissipative interactions with an environ- theories. However, quantization of dissipation ment, which may be of various nature rang- has remained an open challenge for nearly a ing from a vacuum to a generic thermal bath. century. We construct a quantum counter- Nevertheless, construction of physically consis- part of classical friction, a velocity-dependent tent quantum models of dissipative forces has force acting against the direction of motion. In been a long standing problem since the birth of particular, a translationary invariant Lindblad quantum mechanics (see, e.g.,1–4). A common equation is derived satisfying the appropriate framework for describing open quantum sys- dynamical relations for the coordinate and mo- tems is to represent the state of the system by a mentum (i.e., the Ehrenfest equations). Nu- density matrix, whose evolution is governed by merical simulations establish that the model ap- the Lindblad equation.5,6 In this Letter, we con- proximately equilibrates. These ﬁndings signif- struct a model of quantum friction, whose clas- icantly advance a long search for a universally sical counterpart is a velocity-dependent force valid Lindblad model of quantum friction and acting against particle’s motion. open opportunities for exploring novel dissipation phenomena. Current paradigm New paradigm of ODM

Unitary model of Observaons Math formalism system + bath (Operaonal Dynamical) (Modeling) x

Trace bath out arXiv:1412.1892v3 [quant-ph] 20 Apr 2016 initial p Introduce approximaons

x Master equaon Master equaon

Lindbladian ? Observaons ? Lindbladian ! Observaons ! final p Figure 1: Currently paradigm for deriving master equations governing open system dynamics vs proposed novel approach of Operational Dy- namic Modeling (ODM).

1 (a) (b)

Figure 2: The initial (a) and final (b) Wigner (a) (b) functions for the harmonic oscillator evolving according to the model (1) governing the Ohmic dissipation [with eq (20)], γ = 0.07 a.u., L = 3 Figure 3: (a) The final Wigner function for a.u., and D = 0]. The circular solid lines depict the harmonic oscillator evolving according to the level set of the Hamiltonian H = (p2 +x2)/2 the model (1) governing the Ohmic dissipation a.u. (a) The Wigner function of the ground [with eq (20), γ = 0.07 a.u., L = 3 a.u., and state displaced along the momentum axis. The D = 0.0143 a.u.]. The circular solid lines depict reached steady state (b) is not a Gaussian dis- the level set of the Hamiltonian H = (p2 +x2)/2 tribution. a.u. The initial Wigner function is shown in figure 2(a). Note that the steady state approaches By employing the phase space representation the thermal Boltzmann state with kT = 1.166 of quantum mechanics,7–9 where an observable a.u. depicted in (b). O = O(x, p) is assumed to be a real-valued function of coordinate x and momentum p, and system’s state is represented by the Wigner function W = W (x, p), we derive the Lindblad- Wigner equation d i W = − (H?W − W?H) dt ~ + D[W ] + D0[W ], (1) H =p2/(2m) + U(x), (2) 2γ 1 D[W ] = A?W?A∗ − W?A∗ ?A ~ 2 1 − A∗ ?A?W , (3) 2 (a) (b) 2D 1 D0[W ] = x ? W ? x − W ? x ? x ~2 2 Figure 4: (a) The Wigner function of the 1 ∂2W − x ? x ? W = D , (4) Schrödingercat state at time t = 0. (b) The 2 ∂p2 Wigner function at later time t = 2 a.u. af- ←−−→ ←−−→! i ∂ ∂ ∂ ∂ ter evolving according to the model (1) gov- ? = exp ~ − , (5) 2 ∂x ∂p ∂p ∂x erning the Ohmic dissipation (system’s parameters are defined in figure 3.). As time pro- which guarantees completely positive dynamics gresses, the Wigner function’s negativity van- of the density matrix underlying the Wigner ishes and the state approaches the Boltzmann function W for an arbitrary operator A. In equilibrium shown in figure 3(b). standard derivations, one finds a family of relaxation operators A by assuming a weak cou- pling to a bath and expanding the dynamics

2 perturbatively. Here we adopt a different strat- egy: We require that the first moments of W satisfy the Ehrenfest equations d 1 hxi = hpi, (6) dt m d hpi = −hU 0(x)i − 2γh sign (p)f(|p|)i, (7) dt (a) characterizing motion of a particle of mass m interacting with an environment induced velocity-dependent friction. The conventional derivations of master equations (see figure 1) do not guaranteed satisfaction of these relations. Using the Operational Dynamical Mod- eling (ODM) algorithm to be described below, we construct an operator A that satisfies the (b) constraints (6) and (7)

s A = Lf |p| + ~ exp(−i sign (p)x/L). 2L (8)

(c) The classical limit of the Lindblad-Wigner equation (1) with eq (8) recovers the appropriate Fokker-Planck equation,10 ∂ D[W ] = 2γ [ sign (p)f(|p|)W ] + O ( ) . (9) ∂p ~ The Ehrenfest relations for the second moments may also be obtained from eq (1): (d) d hp2i = −2hpU 0(x)i dt − 2γ f(|p|) 2|p| − ~ + 2D, (10) L d 1 hxpi = hp2i − hxU 0(x)i dt m (e) − 2γh sign (p)f(|p|)xi, (11) d 2 γ L f 0(|p|)2 hx2i = hxpi + ~ . (12) dt m 2 f(|p|)

In order to employ eq (1), the following free parameters must be specified i) f(p) ≥ 0 – the Figure 5: Quantum (solid red lines) [eq (1)] vs velocity dependence of the dissipative force, ii) classical (dashed blue lines) [eq (9)] dissipative γ ≥ 0 – a friction coefficient, iii) L > 0 – a dynamic of a harmonic oscillator. Parameters length-scale constant defining the dynamics of for both systems are identical (parameters are second-order moments (10)-(12), and iv) D ≥ 0 specified in figure 3.) Time-evolution of the first is a dephasing constant chosen such that dy- [(a), (b)] and second [(c), (d)] order moments. (e) The total energy variation. 3 namics equilibrates to the Boltzmann state with This state of the field is unsatisfactory because some temperature. non-Lindblad master equations are known to Equation (5) defines the Moyal prod- lead to negative probabilities,26,27 whereas the uct,7–9 which is a result of mapping the non- violation of the Ehrenfest equations lead to commutative matrix product in the Hilbert unphysical artifacts.28 space into the phase space. As a result, the A comparative review29 of major quantum dissipator (3) is obtained by Wigner trans- dissipation theories further revealed that no forming the Lindblad equation for the density existing model is simultaneously i) complete matrix, thus the Wigner function’s marginals positive, ii) translationally invariant, and iii) stay positive throughout the entire evolution. asymptotically approaching thermal equilib- The dissipator (4) describes dephasing, the loss rium. The present model (1) exactly obeys the of quantum coherence;1,11,12 whereas, the dis- first two properties, whereas the latter can be sipator (3) causes amplitude damping. The satisfied approximately (this can be achieved usage of sign and modulus functions in eqs (8) exactly in the free particle case). Further- and (7) are necessary to ensure that the fric- more, our simulations confirm that the dy- tion force acts against the particle’s motion. namics of our model does not cause uncontrol- The dissipator (3) is translationally invariant lable spreading of the wave packet even at zero (more precisely, Galilei-covariant): a spatial temperature, and approaches thermal equilib- displacement W (x, p) → W (x + c, p) implies rium at higher temperatures, thereby overcom- D[W ](x, p) → D[W ](x + c, p). Numerical sim- ing computational and physical inconsistencies ulations establish that the long-time dynamics plaguing other dissipative theories. The model govern by eq (1) rigorously does not equilibrate; (1) is obtained as a unique consequence of the however, the dynamics can be said to approx- Ehrenfest constrains (6) and (7) and the re- imately equilibrate. Namely, one can find a quirements for the dynamics to be Lindblad, value of the dephasing constant D in eq (4) translationary invariant, and state independent such that the steady state closely resembles the [i.e., A = A(x, p) in eq (3) does not depend on Boltzmann state with temperature T . In this the Wigner function]. sense, we numerically define the temperature A difficulty of constructing physical models dependence of D = D(T ). of quantized friction lies in fundamental limitations of the current paradigm for modeling open system dynamics (see figure 1). First, the com- Comparison with other the- bined system plus bath are assumed to evolve ories unitarily; second, the environmental degrees of freedom are traced out by making a number of Current quantum friction models can be approximations. This procedure neither guar- roughly divided into the two categories: i) Lind- antees that the resultant master equation can blad models not obeying the Ehrenfest relations reproduce the observations characterizing phe- have been proposed in.13–15 The fundamental nomenon of interest, nor that the equations to reason for the Ehrenfest equation violation is have a desired mathematical structure. the ubiquitous usage of A [eq (3)], which is It is noteworthy that the limitations of the taken to be linear with respect to the coordi- current paradigm persist even if no approxima- nate and momentum [see the comment after tion is necessary to trace the bath out. For ex- eq (14)]. ii) Non-Lindblad models3,16–22 obey- ample, the Hu-Paz-Zhang master equation17–19 ing the Ehrenfest relations that preserve state’s for a harmonic oscillator interacting with a lin- positivity for sufficiently high temperatures. ear passive heat bath of oscillators is exact un- Contrary to the claims, the master equations der the assumption that the bath is initially in23–25 belong to the same category. In par- at equilibrium and not coupled to the oscilla- ticular, the model in23 produces uncontrollable tor. The obtained non-Lindblad master equa- heating26,27 greatly spreading the wave packet. tion preserves the density matrix’s positivity

4 and satisfies the Ehrenfest relations. However, we shall now establish, no Lindblad dynamics the Hu-Paz-Zhang propagation of states ini- with an A linear in x and p satisfies the Ehren- tially uncorrelated with the environment leads fest equations (6) and (7). Indeed, substituting to instantaneous infinite spreading of the wave A = ax + bp into eq (14) leads to packet,30 which could be fixed by modulating ∗ ∗ ∗ ∗ the friction coefficient during the evolution.31 (a b − ab )x = 0, (a b − ab )p = 4ip, (15) Further revealed problems have lead to the conclusion that the model is of very limited physi- where a contradiction becomes evident. This cal utility.32 conclusion holds in the case of Lindblad mod- To overcome these fundamental limitations, els with several such A operators. Our model a new paradigm of ODM33 has been recently (8) circumvents this no-go result due to its new put forth, enabling the generation of models di- non-linear dependence on x and p. rectly from observed data (see figure 1). To de- The action of the dissipative force is expected rive master equations, ODM needs two inputs: to be translationary invariant. One observes di- observed data recast in the form of Ehrenfest rectly from the definition of the Moyal product relations and a specified mathematical struc- (5) that if ture of the equation of motion. As an outcome, A(x, p) = g(p) exp(iCx),C∗ = C, (16) ODM guarantees that the resulting equations of motion have the desired physical structure to then the dissipator D[W ] (3) is translationary reproduce the supplied dynamical observations. invariant. Formally, the dissipator D[W ] with This formalism has provided new interpreta- A given by eq (16) obeys tion of the Wigner function,34 unveiled conceptual inconstancies in finite-dimensional quan- ∂ ∂ ∂2 ∂2 ∂2 D[W ] = F p, , , , , , ··· W, tum mechanics,35 formulated dynamical models ∂x ∂p ∂x2 ∂x∂p ∂p2 in topologically nontrivial spaces,36 advanced (17) the study of quantum-classical hybrids,37 and lead to development of efficient numerical tech- where ··· denotes higher order derivatives, and niques.12,38 the function F explicitly does not depend on x. Additionally, eq (6) is satisfied for any real valued g(p). Therefore, substituting the expansion Derivation ∞ X n We begin by identifying the amplitude dump- g(p) = gn(p)~ (18) ing dissipator D[W ] [eq (3)], thus the dephasing n=0 0 coefficient D is set to zero to ignore D [W ] [eq into eqs (16) and (14), we recursively finding (4)]. Substituting eq (1) into eqs (6) and (7) all terms in expansion (18) (see the supporting and then dropping the averaging, which is jus- information). Finally, the resultant series can tified by A being state independent, we obtain be then summed up, thereby leading to the ex- equations for an unknown function A = A(x, p), act solution (8) where L is a positive length- defining the Lindblad dissipator (3), scale constant to balance the dimensionality. ∂A ∂A∗ The value of L dictates the time-dynamics of A∗ ? − ?A = 0, (13) second-order moments [eqs (10)-(12)]. ∂p ∂p A simple steady state solution is found by ad- ∂A ∂A∗ A∗ ? − ?A = −4i sign (p)f(|p|). (14) ditionally requiring the spatial homogeneity ∂x ∂x ∂W dW The Lindblad models for Omic friction, = 0, = 0,U = 0, D = 0 f(p) = |p|, has been widely studied (see, ∂x dt e.g.,13,14,23,24,39,40), where A was found to be =⇒ W = const/f(|p|). (19) a linear combination of x and p. However, as

5 Therefore, if f(p) is chosen to be an inversely tions (10)-(12): proportional to a thermal equilibrium state, the D free particle dynamics of the model (8) equili- = f(|p|) 2|p| − ~ brates without the need for the dephasing dis- γ L sipator D0[W ] (4). mω 2 f 0(|p|)2 − L , (21) Both quantum corrections in eqs (10) and ~ 2 f(|p|) st (12), i.e., the terms proportional to ~, are po- sition independent, thereby reconfirming the where h· · · ist denotes averaging over a steady translational invariance. According to eq (12), state. the quantum correction to Ohmic dissipation The steady state for the model with D = (with f(p) = |p|) is singular, therefore the func- 0.0143 (a.u.) is shown in figure 3. For a suf- tion f(p) should be regularized. For example, ficiently high value of D, the ring in figure 2(b) we employ the following smoothing in numeri- is washed out and the Wigner function of the cal simulations (see, e.g., figures 2-5) steady state looks like a gaussian [figure 3(a)], which well approximates the Boltzmann equi- p f(p) = p2/ p2 + 2, 2 = 0.5. (20) librium for some temperature [figure 3(b)]. The larger the dephasing coefficient D, the more ac- The Ehrenfest relations (6), (7), (10)-(12) have curate the equilibration dynamics. Addition- been verified numerically for diferent values of ally, we have also verified that the approxi- parameters and initial conditions. mate equilibration dynamics occurs in the case Equation (12) establishes that the steady of anharmonic oscillators. Figure 4 establishes state need not coincide with thermal equilib- that the evolution generated by the model (1) rium. As an example, consider a harmonic os- washes out the quantum interference initially cillator. The equilibrium state is characterized present in the Schrödingercat state, while the by the identity hxpi = 0, which contradicts the dynamics equilibrates. steady state condition dhx2i/dt = 0 because the Despite a simple look of the Wigner func- quantum correction in eq (12) is strictly posi- tions in figures 2 and 3, full time-dependent tive. Furthermore, if the steady state is posi- dynamics are rich in quantum features. Fig- tive, then its Wigner function should be more ure 5 compares quantum dissipative dynamics, pronounced in the second and fourth quadrants governed by the Lindblad-Wigner equation (1), of the phase space (where xp < 0) to com- with the corresponding classical Fokker-Planck pensate for the quantum correction [this small evolution (9). Even though both quantum and asymmetry as can be noticed in figures 2(b) and classical master equations satisfy the same first 3(a)]. order Ehrenfest theorems (6)-(7), time evolu- Figure 2 shows the initial state with aver- tion of the expectation values of the coordi- age momentum p = 3 a.u. (arbitrary units, nate [figure 5(a)] and momentum [figure 5(b)] ~ = m = 1, are employed in the simula- exhibit quantitative differences. Since the op- tions) reaches the steady state with a circularly tical polarizability is proportional to hxi, the shaped Wigner function. The latter has a char- predicted quantum corrections may be observed acteristic asymmetry, as discussed above. The via non-linear spectroscopy.41 The correction to reached state does not resemble the Boltzmann the second order Ehrenfest theorems (10)-(12), thermal equilibrium. enforcing the Heisenberg uncertainty priciple, To allow the dynamics to equilibrate ap- qualitative change open system dynamics [fig- proximately, we include the dephasing dissi- ures 5(c) and 5(d)]. As a result, the expectation pator D0[W ] [eq (4)] with a non-vanishing D. value of energy in classical dissipative dynam- In the case of a harmonic oscillator [U(x) = ics monotonically decreases; whereas, energy re- mω2x2/2], the fluctuation-dissipation theorem vives in quantum case at short time scales [fig- follows from the second order Ehrenfest rela- ure 5(e)].

6 Outlook based derivation overcomes all these fundamental weaknesses by deriving Lindblad equations In order to describe quantum dissipative dy- enforced to be compatible with the Ehrenfest namics, emerging in many areas of physics, equations. This formalism opens new horizons there is a need for a Lindblad model satisfying in quantum non-equilibrium statistical mechan- the Ehrenfest relations (6) and (7), with long- ics. time dynamics converging to an equilibrium state. Currently, the lack of such model has Acknowledgement D.I.B., R.C., H.A.R. re- been substituted by a multitude of dissipative spectively acknowledge financial support from theories. Using ODM (figure 1), we have found NSF CHE 1058644, DOE DE-FG02-02-ER- the translationally invariant Wigner-Lindblad 15344 and ARO-MURI W911NF-11-1-0268. model (1) exactly obeying the Ehrenfest equa- D.I.B. was also supported by 2016 AFOSR tions (6) and (7). Furthermore, according Young Investigator Research Program. A.G.C to numerical simulations, our model not only was supported by the Fulbright foundation. shows that a state with non-vanishing mean S. M. gratefully acknowledges the support of velocity [figure 2(a)] approximately approaches NSF CHE-1361516 and the Chemical Sciences, the Boltzmann equilibrium (figure 3), but also Geosciences, and Biosciences division, Office of exhibits pronounce quantum corrections (figure Basic Energy Sciences, Office of Science, U.S. Department of Energy. We want to thank 5) even in the case of a harmonic oscillator. 29 The following Ehrenfest relation ubiquitously David Tannor for drawing our attention to arises in molecular dynamics41 and Dmitry Zhdanov for numerous insightful discussions. Z t Supporting Information Available: d 2 hpi(t) = −mω hxi(t) − dτγ(t − τ)hpi(τ), Maple code for symbolic derivation of the main dt −∞ (22) results of the paper. where the time dependent dissipation coefficient γ(t) is connected with the spectral den- References sity of the bath, which characterizes the nature (1) Gardiner, C.; Zoller, P. Quantum noise: of dissipative dynamics. Such a generalization a handbook of Markovian and non- of the developed model requires the applica- Markovian quantum stochastic methods tion of ODM to non-Markovian dynamics. In with applications to quantum optics; this case, the Lindblad-Wigner equation (1) will Springer, 2004. have to be replaced by a corresponding time- 42 convolutionless master equation (see, e.g., ), (2) Razavy, M. Classical and quantum dissi- thus leading to a time dependent extension of pative systems; World Scientific, 2005. the relaxation operator (8). The presented derivation of the master equa- (3) Bolivar, A. O. The dynamical- tion directly from time evolution of expecta- quantization approach to open quan- tion values embodied in Ehrenfest relations, is tum systems. Annals Phys. 2012, 327, a long-sought alternative to the current cum- 705–732. bersome paradigm for obtaining equations of motions (see figure 1). A master equation is (4) Caldeira, A. O. An Introduction to Macro- typically obtained by performing a number of scopic Quantum Phenomena and Quan- approximations after the bath is traced out of tum Dissipation; Cambridge University a combined system-bath model. Such a deriva- Press, 2014. tion usually leads to either a non-Lindblad (5) Kossakowski, A. On quantum statistical master equation or a model incapable of re- mechanics of non-Hamiltonian systems. producing observations. The presented ODM- Rep. Math. Phys. 1972, 3, 247 – 274.

7 (6) Lindblad, G. On the generators of quan- (19) Hu, B. L.; Paz, J. P.; Zhang, Y. Quantum tum dynamical semigroups. Commun. Brownian motion in a general environ- Math. Phys. 1976, 48, 119–130. ment: Exact master equation with non- local dissipation and colored noise. Phys. (7) Curtright, T.; Fairlie, D.; Zachos, C. Fea- Rev. D 1992, 45, 2843–2861. tures of time-independent Wigner functions. Phys. Rev. D 1998, 58, 025002. (20) Bolivar, A. O. Quantum Tunneling at Zero Temperature in the Strong Fric- (8) Curtright, T. L.; Zachos, C. K. Quantum tion Regime. Phys. Rev. Lett. 2005, 94, Mechanics in Phase Space. Asia Pacific 026807. Physics Newsletter 2012, 01, 37. (21) Cleary, L.; Coffey, W. T.; Dowling, W. J.; (9) Curtright, T.; Fairlie, D. B.; Zachos, C. K. Kalmykov, Y. P.; Titov, S. V. Phase space A Concise Treatise on Quantum Mechan- master equations for quantum Brownian ics in Phase Space; World Scientific, 2013. motion in a periodic potential: compari- (10) Gardiner, C. Stochastic methods; son of various kinetic models. J. Phys. A Springer, 1985. 2011, 44, 475001.

(11) Zurek, W. H. Decoherence, einselection, (22) Bolivar, A. O. Non-Markovian quantum and the quantum origins of the classical. Brownian motion: a non-Hamiltonian ap- Rev. Mod. Phys. 2003, 75, 715–775. proach. arXiv:1503.07951 2015,

(12) Cabrera, R.; Bondar, D. I.; Jacobs, K.; (23) Diosi, L. On High-Temperature Marko- Rabitz, H. A. Eﬃcient method to gener- vian Equation for Quantum Brownian ate time evolution of the Wigner function Motion. EPL 1993, 22, 1. for open quantum systems. Phys. Rev. A (24) Vacchini, B. Completely Positive Quan- 2015, 92, 042122. tum Dissipation. Phys. Rev. Lett. 2000, (13) Berman, M.; Kosloﬀ, R.; Tal-Ezer, H. So- 84, 1374–1377. lution of the time-dependent Liouville-von (25) Petruccione, F.; Breuer, H.-P. The theory Neumann equation: dissipative evolution. of open quantum systems; Oxford Univ. J. Phys. A 1992, 25, 1283. Press, 2002.

(14) Gao, S. Dissipative Quantum Dynamics (26) Munro, W. J.; Gardiner, C. W. Non- with a Lindblad Functional. Phys. Rev. rotating-wave master equation. Phys. Rev. Lett. 1997, 79, 3101–3104. A 1996, 53, 2633–2640.

(15) Vacchini, B.; Hornberger, K. Quantum (27) Stenholm, S. Quantum theory of linear linear Boltzmann equation. Phys. Rep. friction. Brazilian J. Phys. 1997, 27, 214– 2009, 478, 71–120. 237.

(16) Caldeira, A. O.; Leggett, A. J. Path inte- (28) Wiseman, H.; Munro, W. Comment on gral approach to quantum Brownian mo- ”Dissipative Quantum Dynamics with a tion. Physica A 1983, 121, 587–616. Lindblad Functional”. Phys. Rev. Lett. (17) Ford, G. W.; Kac, M. On the quantum 1998, 80, 5702–5702. Langevin equation. J. Stat. Phys. 1987, (29) Kohen, D.; Marston, C. C.; Tannor, D. J. 46, 803–810. Phase space approach to theories of quan- (18) Ford, G. W.; Lewis, J. T.; tum dissipation. J. Chem. Phys. 1997, O’Connell, R. F. Quantum langevin 107, 5236–5253. equation. Phys. Rev. A 1988, 37, 4419.

8 (30) Ford, G. W.; O’Connell, R. F. Exact so- (40) Ferialdi, L. Exact Closed Master Equa- lution of the Hu-Paz-Zhang master equation for Gaussian Non-Markovian Dynam- tion. Phys. Rev. D 2001, 64, 105020. ics. Phys. Rev. Lett. 2016, 116, 120402.

(31) Fleming, C. H.; Roura, A.; Hu, B. L. (41) Mukamel, S. Principles of Nonlinear Opti- Exact analytical solutions to the master cal Spectroscopy; Oxford University Press, equation of quantum Brownian motion for 1999. a general environment. Ann. Phys. 2011, 326, 1207–1258. (42) Breuer, H.-P. Non-Markovian generalization of the Lindblad theory of open quan- (32) Ford, G. W.; O’Connell, R. F. Limitations tum systems. Phys. Rev. A 2007, 75, on the utility of exact master equations. 022103. Ann. Phys. 2005, 319, 348–363.

(33) Bondar, D. I.; Cabrera, R.; Lompay, R. R.; Ivanov, M. Y.; Rabitz, H. A. Operational Dynamic Modeling Transcending Quan- tum and Classical Mechanics. Phys. Rev. Lett. 2012, 109, 190403.

(34) Bondar, D. I.; Cabrera, R.; Zh- danov, D. V.; Rabitz, H. A. Wigner phase-space distribution as a wave function. Phys. Rev. A 2013, 88, 052108.

(35) Bondar, D. I.; Cabrera, R.; Rab- itz, H. A. Conceptual inconsistencies in ﬁnite-dimensional quantum and classical mechanics. Phys. Rev. A 2013, 88, 012116.

(36) Zhdanov, D. V.; Seideman, T. Wigner representation of the rotational dynamics of rigid tops. Phys. Rev. A 2015, 92, 012129.

(37) Radonjić, M.; Popović, D. B.; Prvanović, S.; Burić, N. Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction. Phys. Rev. A 2014, 89, 024104.

(38) Cabrera, R.; Bondar, D. I.; Cam- pos, A. G.; Rabitz, H. A. Relativistic decoherence: Eﬃcient numerical simulations of spin 1/2 relativistic open quantum systems. arXiv:1409.1247 2014,

(39) Lindblad, G. Brownian motion of a quantum harmonic oscillator. Rep. Math. Phys. 1976, 10, 393.