Wigner-Lindblad equations for quantum friction Denys I. Bondar,∗,y Renan Cabrera,y Andre Campos,y Shaul Mukamel,z and Herschel A. Rabitzy yPrinceton University, Princeton, NJ 08544, USA zUniversity of California, Irvine, Irvine, CA 92697, USA E-mail: [email protected] 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 dissipa- tion 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 mas- ter 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 fig- ure 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) Schr¨odingercat 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 param- eters 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 re- laxation 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(jpj)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 rela- tions. 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 jpj + ~ exp(−i sign (p)x=L): 2L (8) (c) The classical limit of the Lindblad-Wigner equation (1) with eq (8) recovers the appropri- ate Fokker-Planck equation,10 @ D[W ] = 2γ [ sign (p)f(jpj)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(jpj) 2jpj − ~ + 2D; (10) L d 1 hxpi = hp2i − hxU 0(x)i dt m (e) − 2γh sign (p)f(jpj)xi; (11) d 2 γ L f 0(jpj)2 hx2i = hxpi + ~ : (12) dt m 2 f(jpj) In order to employ eq (1), the following free pa- rameters 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 limita- tions 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.