Friction as a consistent quantum-mechanical concept
aa r X i v : . [ qu a n t - ph ] O c t Friction as a consistent quantum-mechanical concept
Dmitry V. Zhdanov, ∗ Denys I. Bondar, † and Tamar Seideman ‡ University of Bristol, Bristol BS8 1QU, UK Tulane University, New Orleans, Louisiana 70118, USA Northwestern University, Evanston, Illinois 60208, USA
A quantum analog of friction (understood as a completely positive, Markovian, translation-invariant and phenomenological model of dissipation) is known to be in odds with the detailedbalance in the thermodynamic limit. We show that this is not the case for quantum systems withinternal (e.g. spin) states non-adiabatically coupled to translational dynamics. For such systems,a quantum master equation is derived which phenomenologically accounts for the frictional effectof a uniform zero temperature environment. A simple analytical example is provided. Conjecturesregarding the finite temperature case are also formulated. The results are important for efficientsimulations of complex molecular dynamics and quantum reservoir engineering applications.
I. INTRODUCTION
When dealing with complex dissipative environments,modern quantum scientists substantially rely on intu-ition, alike ancient craftsmen taming the elements tobuild mills or sailing vessels. Specifically, there is a vastexperimental evidence that quantum effects play a piv-otal role even in such complex and manifestly dissipativeprocesses as photosynthesis [1–3]. However, it not clearhow coherent quantum dynamics is induced and guidedby dissipative interactions. Despite several recent con-ceptual breakthroughs in the areas of quantum reservoirengineering and topologically protected phases of mat-ter [6], the analysis of coherent dynamics in real-wouldopen systems is impeded by a prohibitively complex mi-croscopic modelling of nonperturbative system-bath in-teractions.In classical mechanics, the similar curse of dimensional-ity is escaped from by introducing friction, a phenomeno-logical non-conservative force resisting the relative mo-tion of objects and converting their kinetic energies intoheat [7]. Despite being very simple, the concept of fric-tion is proven useful even in explaining some quantum-level dynamics involving strong system-bath couplings,such as in the case of simple chemical reactions in liq-uid solutions [8]. However, there is no unique way toquantize non-conservative forces [7]. As result, exist-ing phenomenological quantum dissipative models, be itquantum optical master equation [9], F¨oster, or Redfieldmodels [10, 11], depend on the system Hamiltonian in acomplex way. This dependence cannot be simplified tofew friction and diffusion coefficients, especially, in thelow temperature regime. Furthermore, in order to en-sure that the surrounding physical bath stays unchanged, ∗ [email protected] † [email protected] ‡ [email protected] For instance, it was proven that quantum information process-ing can be fully dissipation-driven [4]. Furthermore, dissipationenables optical and mechanical nonreciprocal couplings – a keyingredient for implementing a quantum analog of sailing vessel[5]. quantum dissipation model must be nontrivially read-justed each time the Hamiltonian is altered (e.g., by anexternal field or as a result of chemical reactions). Fur-thermore, the discussed models describe the relaxationdynamics in terms of interstate transition rates. Thispicture is not natural when dealing with dynamics ofessentially semiclassical vibration wavepackets in manyphotochemical reactions.To mitigate the above complications, in this paper, wedevelop a phenomenological quantum dissipation modelpossessing favorable features of classical friction. Forbrevity, we will refer to this model as “quantum fric-tion” . We restrict our analysis to the simplest case ofa homogeneous environment at a zero temperature andmaster equations linear with respect to the system den-sity matrix. It is explained in detail elsewhere that thecorresponding quantum friction model must satisfy thefollowing four criteria [13, 14]:i Markovianity,ii positivity,iii translation invariance, andiv asymptotic approach to the canonical equilibriumstate.Specifically, the Markovianity ensures that the quan-tum friction is memoryless. Positivity guarantees thatthe model is quantum-mechanically consistent, i.e., thatany initial positive density matrix remain positive at alltimes. Translation invariance makes quantum frictioncoordinate-independent, as the corresponding classicalfriction is a velocity-only dependent force.The problem of finding a phenomenological modelobeying these four criteria has a long history and hasbeen the subject of many controversies over the years(see, e.g., Refs. [15–17]). As early as in 1976, Lind-blad demonstrated that this problem has no resolution We want to disambiguate our broad notion of “quantum friction”from its narrow meaning of the force acting on atoms flying nearsurfaces [12]. for a harmonic oscillator [18]. Subsequent failed searchesforced Kohen et.al. to conjecture in their 1997 review [19]that the incompatibility of all four criteria is a genericproperty, though with the sagacious comment: “exceptin special cases”, which were not known at that time.Later, free quantum Brownian motion was identified byVacchini [20] to be such a special case. We have proventhat no other exceptions exist among quantum systemswith N ext translation degrees of freedom [14].In this paper, we report a new class of “special cases”with all four criteria satisfied at zero temperature. Theseare the systems with internal degrees of freedom, such asspin. The explicit forms of quantum friction dissipatorsare derived for such systems.The paper is organized as follows. We start with for-malizing the four-criterion problem in Sec. II. A construc-tive proof of existence of its solution at zero temperaturefor systems with internal degrees of freedom is presentedin Section III. The corresponding quantum friction dissi-pators are also derived and analyzed in detail. The ob-tained results are illustrated on the simplest analyticallytractable example of a two-dimensional harmonic oscilla-tor in Sec. IV. In the following section V we discuss theconjectures regarding the existence of quantum frictionat finite temperatures. The paper concludes with a briefsummary and outlook. II. FORMULATION OF THE PROBLEM
The object of our analysis will be the quantum systemswith e N int > | e i i coupled to N ext =0 trans-lation degrees of freedom. (Hereafter, the tilde ˜ marksthe quantities associated with the internal states.) Anexample of such a system is a molecule with the follow-ing vibronic model Hamiltonian:ˆ H = N ext X k =1 µ k ˆ p k + V (ˆ x ) | {z } vibrational part + f N int X i =1 e E i | e i ih e i | | {z } spin \ electronic part + X i = j g i,j | e i ih e j | | {z } spin \ electronic couplings + f N int X i =1 δV i (ˆ x ) | e i ih e i | | {z } vibronic couplings . (1)Here the sets of nuclear coordinates ˆ x = { ˆ x , ..., ˆ x N ext } andmomenta ˆ p = { ˆ p , ..., ˆ p N ext } represent the translation de-grees of freedom and “internal” states | e i i correspond todifferent electronic and/or spin states of the molecule.The symbols g i.j , V i (ˆ x ) and e E i denote the coupling con-stants, coupling operators and internal states eigenener-gies, respectively. Assume that the molecule is immersedinto a homogeneous environment (such as gaseous me-dia or uniform solvent) at zero temperature. Our goalis to construct a phenomenological model of the dissipa-tive back-action of such environment on the system whichsatisfies the criteria i – iv of quantum friction. The question which dissipative processes simultane-ously satisfy criteria i and ii was resolved by Lindblad[21]. The answer is a master equation now bearing hisname: ∂∂t ˆ ρ = L [ˆ ρ ] , L = L + L rel , (2)where the superoperator L [ ⊙ ]= i ~ [ ⊙ , ˆ H ] accounts forunitary evolution of the system isolated from environ-ment and the dissipation term L rel describing the system-environment couplings is L rel = N lbd X k =1 L lbdˆ A k , (3a) L lbdˆ L [ˆ ρ ] def = ˆ L ˆ ρ ˆ L † − ( ˆ L † ˆ L ˆ ρ +ˆ ρ ˆ L † ˆ L ) . (3b)The goal of this work is to identify the conditions underwhich the dissipator L rel defined by Eq. (3a) additionallyobeys the criteria iii and iv.The translation invariance criterion iii with respect tothe n -th coordinate is formally defined as[ˆ p n , L rel [ˆ ρ ]]= L rel [[ˆ p n , ˆ ρ ]] . (3c)A general recipe on imposing this criterion on the Lind-blad term (3a) was found by Holevo [22, 23] and furtheranalyzed in applications by Vacchini [24, 25]. Specifically,to satisfy Eq. (3c) it is necessary and sufficient that theLindblad operators ˆ A k in Eq. (3a) take the formsˆ A k def = e − i κ k ˆ x ˆ f k (ˆ p ) . (3d)Here ˆ f k ( p ) ∈ C f N int × f N int and κ k ∈ R N ext are arbitrary pa-rameters and matrix-valued functions. They generallyshould be treated as a phenomenological quantities andcan be deduced from empirical fits to time evolution ofhigher-order averages, e.g., h ˆ x i , h ˆ p i . In the cases whenthe dissipative term L rel describes random collisions withlight environmental particles, the values of ~ κ k and f k ( p )can be associated with the characteristic momentum ex-change in a collision event and the scattering amplitudein the momentum space [14].It is worth to show why the dissipative dynamics satis-fying criteria i – iii deserves the name quantum friction.The average positions and momenta of a wavepacket,evolving according to Eqs. (3), satisfy equations ddt h ˆ p n i = i ~ h [ ˆ H, ˆ p n ] i + h ˆ F fr n i , (4a) ddt h ˆ x n i = i ~ h [ ˆ H, ˆ x n ] i + h ˆ G fr n i , (4b)where F fr (ˆ p )= − P k ~ κ k f † k (ˆ p ) f k (ˆ p ) (5)defines the position-independent force h ˆ F fr i which, undera proper choice of operators f † k (ˆ p ), acts in the oppositedirection to momenta h ˆ p i similarly to the conventionallydefined classical friction. Note, however, that this forceis paired with term h ˆ G fr i in Eq. (4b) where G fr n (ˆ p )= 12 i ~ X k (cid:16) f † k (ˆ p ) ∂f k (ˆ p ) ∂ ˆ p n − ∂f † k (ˆ p ) ∂ ˆ p n f k (ˆ p ) (cid:17) . (6)The presence of term h ˆ G fr i can be physically attributedto changed effective masses of moving particles “dressed”by the environment. This term is of the same natureas the “position diffusion”, a well-known peculiarity ofquantum Brownian motion [26].Recall that the classical frictional forces are accompa-nied by thermal fluctuations satisfying the fluctuation-dissipation theorem. Similarly, criteria i – iii are notsufficient to define thermodynamically consistent quan-tum friction, and the additional criterion iv is requiredto ensure detailed balance at the thermal equilibrium: L rel [ˆ ρ th T ]=0 . (7)Here ˆ ρ th T = ˆ ρ | t →∞ ∝ e − ˆ Hk B T is the stationary Gibbs systemstate corresponding to temperature T . It is worth stress-ing that Eq. (7) implies non-vanishing thermal fluctua-tions even at the zero bath temperature.The criteria i, ii and iv are fulfilled, e.g., by quantumoptical master equation [9]. A variety of other modelsare known where some three out of four criteria i – ivare satisfied (see Ref. [19] for detailed review). Howeverexcept for free Brownian motion [20], no model is knownwhich satisfies all the Eqs. (3) and (7). Non-existence ofsuch model for the case e N int =1 was rigorously proven byus recently [14]. In particular, we have shown that thecriteria i, ii and iii are compatible only with the followingweaker variant of the condition (7):Tr[ˆ ρ th T ′ L rel [ˆ ρ th T ]]=0 for any T ′ , (8)which we will refer to as the relaxed thermalization (RT)condition.The condition (8) always holds when Eq. (7) is satis-fied. In fact, it can be shown that Eq. (8) only guaran-tees that the steady state of the model coincides with thetrue equilibrium ˆ ρ th T up to the first order in system-bathcouplings. Thus, Eq. (8) is expected to reasonably wellapproximate the exact thermalization criterion (7) whenthe bath-induced decay and decoherence times are largecompared to all the characteristic dynamical timescalesof the system [14]. III. RECONCILING THE FOUR-CRITERIONCLASH AT ZERO TEMPERATURE
In this section, we prove the existence of quantum fric-tion simultaneously satisfying criteria i – iv for systemscharacterized by the Hamiltonian (1) with e N int >
1. Ourmethodology is to first figure out the necessary require-ments to satisfy the criteria i – iii and RT criterion (8) for e N int > L rel satisfying criteria i – iii needs to have the form definedby Eqs. (3). In the case T =0, substitution of Eqs. (3)allows to cast Eq. (8) into N lbd independent extremalconditions J k = Tr[ˆ ρ th0 L lbdˆ A k [ˆ ρ th0 ]] → max ( k =1 , ..., N lbd ) . (9)Specifically, Eq. (9) can be obtained from Eq. (8) andequalities Tr[ L lbdˆ A k [ˆ ρ ]]=0 which guarantee that J k ≤ k . Hence, Eq. (8) can be satisfied only when all J k take their maximal values J k =0.Let | n i and E n be the system Hamiltonian eigenstatesand eigenvalues, respectively. Each of these states can berepresented in the form | n i = f N int X i =1 | e i i | Ψ n,i i , (10)where | Ψ n,i i are the vibrational parts of the eigenstates.Note that | Ψ n,i i are neither normalized nor orthogonal.Below we will deal with their momentum wavefunctionsΨ n,i ( p ) and also with the operators Ψ n,i (ˆ p ) obtained viathe substitution p → ˆ p . In addition, we will use the no-tation | ϕ k,n i = ˆ A k | n i . The expressions for J k in Eq. (9)can now be rewritten as J k = |h | ϕ k, i| −h ϕ k, | ϕ k, i . (11)Here we accounted for the fact that the thermodynamicequilibrium at T =0 corresponds to the ground stateˆ ρ th0 = | ih | . The Cauchy-Schwarz inequality requires that |h | ϕ k, i| ≤h ϕ k, | ϕ k, ih | i = h ϕ k, | ϕ k, i , (12)where the equality holds iff | i ∝ | ϕ k, i . Hence, it followsfrom Eqs. (9) and (11) that J k =0 iff ˆ A k | i = α k | i for all k ( α k ∈ C ) . (13)Eqs. (13) always can be resolved with respect to ˆ A k .The general solution can be written in terms of opera-tors ˆ f k (ˆ p ) introduced in Eq. (3d) asˆ f k (ˆ p )= ˆ f ,k (ˆ p )+ α k f N int X i =1 Ψ ,i (ˆ p − ~ κ k )Ψ ,i (ˆ p ) | e i ih e i | , (14a)where ˆ f ,k (ˆ p ) is the position-independent operator satis-fying the equation ˆ f ,k (ˆ p ) | i =0 . (14b)For instance, the general solution for ˆ f ,k in the case of atwo-level internal subsystem ( e N int =2) can be representedasˆ f ,k (ˆ p )= X i,j =0 ( − i − j G − i,k (ˆ p )Ψ ∗ ,i (ˆ p )Ψ ,j (ˆ p ) | g − i ih g − j | , (15)where G ,k ( p ) ∈ C and G ,k ( p ) ∈ C are arbitrary functions.One can notice that the operators (15) always con-tain off-diagonal terms between different internal states | e i i . Hence, the population of internal states is conservedwhen ˆ f ,k =0 for all k . In the case when | e i i represent theelectronic states of a molecule, the latter model can rep-resent instantaneous events (e.g., the direct collisions oflight particles with the molecule’s nuclei) not involvingelectrons.Equations (14) answer the question on when criteriai – iii can be satisfied together with the relaxed ther-malization condition (8). Let us now turn to the condi-tions required to satisfy strict thermalization condition iv(Eq. (7)). The substitution of Eqs. (14) allows to rewritecondition (7) as N lbd X k =1 α k | i = N lbd X k =1 f k (ˆ p ) † f k (ˆ p ) | i = N lbd X k =1 α k A † k (ˆ p ) | i . It is easier to analyze this relation after multiplying bothits sides by operator P f N int j =0 | e ih e j | Ψ ,j (ˆ p ), which gives ∀ p : N lbd X k =1 α k f N int X j =1 (cid:16) | Ψ ,j ( p ) | −| Ψ ,j ( p − ~ κ k ) | (cid:17) =0 . (16)As discussed in Ref. [14], the effect of the part of Lindbladoperator A k (ˆ p ) proportional to α k can be associated withan instantaneous inelastic collision with massless particle,such as photon, having momentum ~ κ k . In light of thisinterpretation, the exact thermalization condition (16)requires invariance of the momentum distribution withrespect to entire sequence of such collisions described byoperator L rel . Apart from exceptional cases, the condi-tion (16) can be satisfied iff either α k =0 or κ k =0 for each k . Note, however, that the terms L lbdˆ A k corresponding to κ k =0 do not contribute to friction ˆ F fr in Eq. (4a). Inother words, despite the underlying dissipative processformally satisfies criteria i – iii of quantum friction, itdoes not involve direct momentum transfer between thesystem and the bath. Hence, it cannot be physically in-terpreted as a frictional process and, thus, is out of scopefor our programme.The remaining possibility to obey condition (16) bysetting α k =0 and κ k =0 in Eqs. (14) leads to exactly ther-malizable dissipator L rel satisfying Eq. (7). This findingthat all four criteria i – iv can be simultaneously satisfiedis the central result of this work. For example, in the caseof the two-level internal subspace considered above, thisis achieved by setting α k =0 in Eq. (14a) and choosingarbitrary functions G ,k ( p ), G ,k ( p ) and vectors κ k inEq. (15). It is worth to stress, however, that unlike clas-sical friction, this solution leads to non-vanishing term h G fr i in Eq. (4b).What makes quantum friction possible in the case e N int >
1? Our formal results admit the following physicalinterpretation. The very notion of friction is implicitly attached to the classical concept of bath. When con-sidering classical dynamics, it is sufficient to treat thebath as an infinite heat tank at a constant temperature.However, this model fails to account for proximity effectsresponsible for spatial and/or temporal system-bath cor-relations. These correlations turned out to be crucialfor quantum thermalization [14]: Without them, micro-scopic perpetual motion would be possible. Internal de-grees of freedom enable quantum friction by serving asan ancilla subsystem to phenomenologically mimic prox-imity effects on translation degrees of freedom. We haveseen that the thermalizability criterion iv can be fulfilledeven if this ancilla subsystem consists of just two quan-tum states. However, a very essence of proximity effectsimplies that this mechanism can work only when the ex-ternal and ancilla internal degrees of freedom are couplednon-adiabatically. This implies that the system groundstate | i is such that | Ψ ,i i 6∝ | Ψ ,j = i i and | Ψ ,i i 6 =0 forall i and j . If this condition is satisfied, quantum frictionsimultaneously thermalizes both the external degrees offreedom and the ancilla subsystem. Otherwise, neitherof these degrees of freedom can be thermalized, as can beseen, e.g., from Eq. (15) in the case of | Ψ , i =0. IV. AN ILLUSTRATION: THE DAMPEDHARMONIC OSCILLATOR
Let us illustrate the conclusions of the previous sectionby quantizing the familiar classical model of a dampedharmonic oscillator¨ x +2 γ ( ˙ x ) ˙ x + ω x =0 , (17)where ω and γ = γ ( ˙ x ) are the oscillator’s frequency andfrictional damping rate, respectively. Recall that accord-ing to celebrated Lindblad’s result [18], the model (17)cannot be quantized within criteria i – iv imposed onthe friction term under the assumption that an oscillat-ing body is a structureless point particle. Here we as-sume that the oscillating body is a prealigned diatomicmolecule AB of mass M with one internal harmonic de-gree of freedom: a stretching vibrational mode x char-acterized by the reduced effective mass µ and potentialenergy ˆ V vib ∝ ˆ x .Consider the dynamics of such a molecule in a har-monic dipole trap, as shown in Fig. 1, assuming that thelong-term thermalization dynamics is guided by an effec-tive friction force (e.g., due to radiation decay), whichstirs the system into the thermodynamic equilibriumwith an environment. Assuming that atom A is a pri-mary contributor to the induced dipole moment, the sys-tem Hamiltonian can be written in the form:ˆ H = ˆ p M + ˆ p µ + ˆ V dipole + ˆ V vib , where x in an external center-of-mass molecular coor-dinate and ˆ V dipole ≃ Mω (ˆ x − µm ˆ x ) is a laser-induced Figure 1. The ball-and-spring model of a quantum dampedharmonic oscillator (Eq. (17)) with an additional internalstructure represented by the molecular stretching mode x (See Sec. IV for details). trapping potential. It is convenient to rewrite the Hamil-tonian (IV) in the normal mode representation:ˆ H = X l =1 ~ ω l ˆ a ′ l † ˆ a ′ l . (18)Here ˆ a ′ l is the annihilation operator for l -th normal mode:ˆ a ′ l = 1 √ ~ p β ′ l ˆ x ′ l + i q β ′ l ˆ p ′ l ! ( β ′ l =( ~ ω l ) − ) , (19)and the operators ˆ x ′ l and ˆ p ′ l of normal coordinates andmomenta are defined asˆ x ′ = √ m cos( ϕ )ˆ x −√ m sin( ϕ )ˆ x , ˆ x ′ = √ m sin( ϕ )ˆ x + √ m cos( ϕ )ˆ x , ˆ p ′ = cos( ϕ ) √ m ˆ p − sin( ϕ ) √ m ˆ p , ˆ p ′ = sin( ϕ ) √ m ˆ p + cos( ϕ ) √ m ˆ p . (20)We are interested in the situation when the external andinternal motions are coupled, i.e., when ϕ =0 , π .Let us derive the phenomenological frictional dissipa-tor for the case of cold environment T =0. Accordingto the conclusions of Sec. III, the dissipator of interestshould be of the form (3) withˆ A k = e − iκ k ˆ x f k (ˆ p , ˆ p , ˆ x ) (21)and satisfy Eq. (13).It is helpful to introduce the operatorsˆ a = a (ˆ x , ˆ p , ˆ p )= √ (cid:16) ˆ a ′ p β ′ sin( ϕ )+ˆ a ′ p β ′ cos( ϕ ) (cid:17) , ˆ a = a (ˆ x , ˆ p , ˆ p )= √ (cid:16) ˆ a ′ p β ′ cos( ϕ ) − ˆ a ′ p β ′ sin( ϕ ) (cid:17) , (22)which satisfy ˆ a , | i =0 . (23)Here | i is the system’s ground state and also the equi-librium Gibbs state for T = 0. It can be written inmomentum representation as h p , p | i∝ e − P l =1 12 β ′ l p ′ l . (24) The general solution of the equation ˆ A k | i = α k | i compliant with the translation invariance condition (3c)with respect to x isˆ A k = e − iκ k ˆ x ˆ G ′ k ˆ a + α k e − i ~ κ k √ m ˆ a . (25)Here ˆ G ′ k = G ′ k (ˆ p , ˆ p , ˆ x ) is an arbitrary operator indepen-dent of ˆ x . Eq. (25) is nothing but the specialization ofthe general solution (14). It obeys the exact thermaliza-tion condition (7) if α k =0 for all k and satisfies only RTcondition (8) otherwise.As expected, the dissipator L rel corresponding to solu-tion (25) with α k =0 vanish when the internal and exter-nal degrees of freedom are dynamically decoupled (e.g.,when e N int =1 or ϕ =0). However, the solutions of bothEqs. (14a) and (25) with α k =0 do exist even in the lat-ter case. A simple computation shows that these so-lutions reduce to f k (ˆ p ) ∝ e ˆ p κkm ω obtained in Ref. [14].The latter solution (and, more generally, the last termin Eq. (25)) indicate that the system-environment cor-relations are minimized (for any given κ ) when the as-sociated effective force h G fr k i depends exponentially onmomenta ˆ p . Interestingly that unlike typical classicalfriction forces anti-aligned with momenta, the last termin Eq. (25) has a long exponentially vanishing “endother-mic” tail representing a force aligned with p . For a fur-ther discussion on the physics of these tails see Ref. [14].In contrast, the solution (25) with α k =0 represents anarbitrary nonlinear friction force acting on the externaldegree of freedom x . Importantly, this force exists dueto the dissipative coupling between the external and in-ternal degrees of freedom. The origin for this couplingcan be illustrated by the classical ball-and-spring modelof a diatomic molecule where each ball is subjected to anindependent non-linear friction force. In this model, thetotal effective friction force applied to the system’s cen-ter of mass depends on the relative velocity of the balls,whereas the decay of the internal oscillation depends onthe center-of-mass velocity. The hallmark of the quan-tum friction is inability to cancel these interdependenciesout even for the linear friction F fr1 ∝ p . V. DISCUSSION ON THE FINITETEMPERATURE CASE
So far, our analysis has been restricted to interactionsof quantum systems with baths cooled down to the zerotemperature. An existence of quantum friction forcesat finite temperatures is an open question beyond thescope of this work. Nevertheless, we would like to brieflydiscuss insights that might help to find the answer.First, note that in the case T =0 the exactly thermal-izable frictional dissipator (defined by Eqs. (3) and (14)with α k =0) has the property that each individual term L lbdˆ A k independently satisfies the condition (7). However,the analogous property cannot hold at finite tempera-tures for any Lindblad operator ˆ A k of the form (3d): ∀ T =0 : L lbdˆ A k [ˆ ρ th T ] =0 . (26)The proof of inequality (26) is given in Appendix A. Thisresult can be intuitively understood in the simplest case N ext =1 as follows. The most general form of the opera-tor ˆ f k in Eq. (3d) corresponding to a single translationaldegree of freedom isˆ f k = N ext X i,j =1 X k | e i ih e j | c i,j,k ˆ p k , (27)where c i,j,k are complex coefficients to be determined.Now, imagine that we truncated the translational basisto K states. It is obviously impossible to satisfy the ex-act thermalization condition in this approximation. In-deed, in order to turn the inequality (26) into the equal-ity while satisfying the condition (7), ( e N int K ) − e N K − c i,j,k (here we excluded the complex scaling factor).As discussed in Ref. [14], the terms L lbdˆ A k can be re-garded as Markovian approximations for relaxation pro-cesses similar to ones involved in the Doppler cooling.Importantly, each term L lbdˆ A k represents an independentfrictional process in such an interpretation. At the sametime, the inequality (26) shows that these processes can-not be independent since each of them drives the systemout of thermal equilibrium. This contradiction showsthat if the friction-like Markovian dissipator L rel ex-ists for finite temperatures, it must consist of severalinterdependent terms of the form L lbdˆ A k , and hence hasa non-trivial physical interpretation, as in non-TI case[27]. Alternatively, the contradiction may signify that nofriction-like quantum process exists at nonzero temper-atures. Nevertheless, if the latter conjecture is correct,the conventional interpretation of friction in the classicallimit would require revisiting.At the same time, the RT condition ∀ T ′ : Tr[ˆ ρ th T ′ L lbdˆ A k [ˆ ρ th T ]]=0 (28)can be seamlessly satisfied. Indeed, Eq. (28) sets only e N int K − e N K − L rel for the harmonic oscil-lator example of Sec. IV satisfying criteria i – iii and RTcondition (28) at a finite temperature is given by Eqs. (3)withˆ A k ∝ e − iκ k ( ˆ x ′ ϕ )+ˆ x ′ ϕ ) ) √ m e κ k ( β ′ λ p ′ ϕ )+ β ′ λ p ′ ϕ ) ) √ m = e − iκ k ˆ x e κ k √ m ( β ′ λ ˆ p ′ sin( ϕ )+ β ′ λ ˆ p ′ cos( ϕ ) ) , (29)where λ l = tanh( ~ ω l k B T ). VI. CONCLUSION
The concept of friction (defined as the phenomenologi-cal dissipative model satisfying criteria i – iv) can be con-sistently extended into quantum mechanics for systemswith internal degrees of freedom in the case of a zero-temperature environment. This finding complementsthe previous no-thermalization-without-correlations re-sult [14] implying that such dissipators are absent forstructureless particles. We proved and illustrated on theanalytically tractable example that the internal degreesof freedom enable the quantum friction by serving asan ancilla subsystem to harvest the required correlationsand mimic system-bath quantum proximity effects. In-formally, this implies that in order to be thermodynam-ically consistent, quantum friction must dissipate heatboth into an environment and inside the system itself.For this to be true, external and ancilla degrees of free-dom need to be non-adiabatically coupled.Quantum friction can be used as a simple phenomeno-logical relaxation model to simulate the non-equilibriumdynamics of complex molecular systems strongly coupledto an homogeneous environment (e.g., a molecule in asolvent). Such a model is guaranteed to be consistent inthe thermodynamic limit and may allow for substantialmemory and time savings in numerical studies of funda-mental photoinduced processes, such as photoisomeriza-tion, light-induced charge and energy transfer in organicmaterials.The existence of friction-like quantum dissipators atfinite temperatures remains an intriguing open question.The affirmative or negative answers would challenge themicroscopic or semiclassical theories of friction. For com-putational applications permitting approximate thermal-ization, the relaxed thermalization workaround (8) maybe used in place of taxing microscopic models [28, 29], ifsystem-bath couplings are weak.
Appendix A: Proof of the inequality (26)
The proof is by contradiction. Suppose that the in-equality (26) can be turned into equality. This assump-tion implies that h n | L lbdˆ A k [ˆ ρ th θ ] | l i =0 for all n an l or, moreexplicitly, that ∀ n, l : R n,l − S n,l =0 , (A1)where R n,l = X m γ m ( θ ) h n | ϕ k,m ih ϕ k,m | l i ,S n,l = h ϕ k,n | ϕ k,l i γ l ( θ )+ γ n ( θ )2 . Here γ i ( θ )= h i | ˆ ρ th θ | i i = e − Ei/θ P n e − En/θ is a positive statisticalweight of the i -th eigenstate.The equalities (A1) imply that P n,l | R n,l | − P n,l | S n,l | =0. After some algebra,this relation reduces to X m,n |h ϕ k,m | ϕ k,n i| (cid:20) γ m ( θ ) − γ n ( θ )2 (cid:21) =0 . (A2)Since the term in square brackets is positive for all m = n ,we can conclude that the necessary condition for ther-malization is that h ϕ k,m | ϕ k,n i =0, or h m | ˆ A † k ˆ A k | n i =0 forall m = n . The latter implies that ∀ m : ˆ A † k ˆ A k | m i = ˆ f † k ˆ f k | m i = c k,m | m i , (A3)where c k,m are certain nonnegative constants. The non-negative Hermitian operator ˆ A † k ˆ A k can be expanded asˆ A † k ˆ A k = N − X i,j =0 | e i ih e j | F i,j (ˆ p ) , (A4)where F i,j ( p ) are some complex-valued functions, suchthat F i,j ( p )= F ∗ j,i ( p ). Using Eq. (A4), the equality (A3)can be rewritten in the matrix form X j F i,j ( p )Ψ m,j ( p )= c m Ψ m,i ( p ) . (A5)Note that here we can treat p as c-numbers. The e N int × e N int matrix F has at most e N int distinct eigenvalues λ k . Each eigenstate | n i is associated with one of theseeigenvalues λ ( n ). Let us choose the set of e N int indices r k , such that | r k i is associated with λ k . Then, each ofremaining eigenstates should be representable as a linearcombination of | r k i ,Ψ m,i ( p )= X k : λ ( r k )= λ ( m ) c m,r k ( p )Ψ r k ,i ( p ) , (A6) with p -dependent coefficients c r k ,m ( p ). Furthermore, λ ( n ) = λ ( m ) ⇒ X i Ψ ∗ n,i ( p )Ψ m,i ( p )=0 for all p . (A7)The basis states | n i can generally satisfy constraints (A6)and (A7) only in two cases: 1) some of the internal statesare decoupled from the rest (i.e., the dynamic space splitsinto isolated subspaces) and 2) all λ k are equal. Here weare not interested in case 1) and assume that the dynam-ics of all the external and internal degrees of freedom ismixed by couplings. Case 2) implies thatˆ A † k ˆ A k = λ ≥ , (A8)where λ is some nonnegative constant. Without loss ofgenerality, it is sufficient to consider two cases: λ =0 and λ =1.The case λ =0 implies that ∀ m : ˆ A k | m i =0, which canbe satisfied only if ˆ A k =0. Hence, we can exclude the case λ =0 from consideration.Consider now λ =1. In this case, the equality L lbdˆ A k [ˆ ρ th T ]=0 takes the formˆ A k ˆ ρ th T ˆ A † k − ˆ ρ th T =0 . (A9)It is easy to show that in the case of non-degenerate γ k ( θ )the above equality can be satisfied iff ˆ A k =1, i.e., when L lbdˆ A k =0. This completes the proof. ACKNOWLEDGMENTS
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