# Frictional lubricity enhanced by quantum mechanics

^{a}International School for Advanced Studies (SISSA), I-34136 Trieste, Italy;^{b}Consiglio Nazionale delle Ricerche, Istituto Officina dei Materiali (CNR-IOM), Democritos National Simulation Center, I-34136 Trieste, Italy;^{c}International Centre for Theoretical Physics (ICTP), I-34151 Trieste, Italy

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Contributed by Erio Tosatti, February 14, 2018 (sent for review January 22, 2018; reviewed by Aleksandr Volokitin and Vladan Vuletic)

## Significance

Sliding friction, a corner of physics still under debate after centuries, is currently studied in nanosystems such as optical lattice emulators, where parameters are easy to control but where applicability of classical mechanics is no longer guaranteed. The quantum effects on friction being largely unexplored, we show here that even a basic theoretical model predicts important novelties. For a particle dragged over large barriers, whose classical stick–slip friction would be large, Landau–Zener tunneling through the barrier can avert stick–slip, with an increase of lubricity. This and related forms of quantum lubricity should be observable for cold ions sliding in optical lattices.

## Abstract

The quantum motion of nuclei, generally ignored in the physics of sliding friction, can affect in an important manner the frictional dissipation of a light particle forced to slide in an optical lattice. The density matrix-calculated evolution of the quantum version of the basic Prandtl–Tomlinson model, describing the dragging by an external force of a point particle in a periodic potential, shows that purely classical friction predictions can be very wrong. The strongest quantum effect occurs not for weak but for strong periodic potentials, where barriers are high but energy levels in each well are discrete, and resonant Rabi or Landau–Zener tunneling to states in the nearest well can preempt classical stick–slip with nonnegligible efficiency, depending on the forcing speed. The resulting permeation of otherwise unsurmountable barriers is predicted to cause quantum lubricity, a phenomenon which we expect should be observable in the recently implemented sliding cold ion experiments.

Friction is, among all basic physical phenomena, the one in most need of fundamental work. In particular, the main current understanding of friction, largely based on mesoscale and nanoscale developments, is essentially classical (1). Quantum effects in sliding friction, despite some early and laudable work (2⇓–4) including experimental suggestions (5), have not been discussed very thoroughly so far. In most cases, in fact, the forced motion of atoms, molecules, and solids is considered, and simulated, just classically. The quantum effects that may arise at low temperatures, connected with either quantum freezing of the phonons or a slight quantum smearing of classical energy barriers, are not generally deemed to be dramatic and have received very little attention. At the theoretical level, in particular, quantum frictional phenomena were not pursued after and beyond those described by the seminal path-integral Monte Carlo study in the commensurate Frenkel–Kontorova model (2, 3). Possible reasons for this neglect are the scarcity of well-defined experimental frictional realizations where quantum effects might dominate and, symmetrically, on the theory side, the lack of easily implementable quantum dynamical simulation approaches. Cold ions in optical lattices (6) offer brand new opportunities to explore the physics of sliding friction, including quantum aspects. Already at the classical level, and following theoretical suggestions (7), recent experimental work on cold ion chains demonstrated important phenomena such as thermolubricity (8), the Aubry transition (9⇓–11), and multiple frictional slips (12). The tunability of the perfectly periodic optical potential that controls the motion of atoms or ions should make it possible to access regimes where quantum frictional effects can emerge.

Here we show, hopefully anticipating experiment, that a first, massive quantum effect will appear already in the simplest sliding problem, that of a single particle forced by a spring to move in a periodic potential: a quantum version of the renowned Prandtl–Tomlison model, and a prototypical system that should also be realizable experimentally by a cold ion dragged by a time-dependent confining potential. As we will show, the main quantum effect amounts to a force-induced Landau–Zener (LZ) tunneling, of course well-known and studied in many different contexts (13⇓⇓–16) outside of sliding friction. The effect of LZ tunneling on friction is striking because it shows up preferentially for strong optical potentials and high barriers, where classical friction is large, while resonant tunneling between levels in nearby potential wells can cause it to drop—a phenomenon that we may refer to as quantum lubricity.

## Model and Methods

Our model, sketched in Fig. 1, consists of a single quantum particle of mass M in the one-dimensional periodic potential created, for instance, by an optical lattice, of strength

We can understand the basic mechanism leading to quantum frictional dissipation by considering the instantaneous eigenstates of *A* for a reduced Hilbert space with four states per well. Denoted by *E*_{n}(*t*) and *E*_{n} and

At the anticrossing at **3**), which as we shall see is proportional to the frictional dissipation, is negligible. In that low-velocity case, a quantum particle is transmitted adiabatically without friction. This is therefore a regime, which one might designate of quantum superlubricity, where friction may vanish nonanalytically as in Eq. **3** in the limit of zero speed (see *Inset* in Fig. 3)—totally unlike the classical case, where friction vanishes linearly with v (viscous friction). Quantum superlubricity should be realized at sufficiently low temperatures, only thermally destroyed in favor of viscous lubricity as soon as temperature T is large enough to upset the LZ physics behind the mechanism. This, however, is not expected to occur until T becomes considerably larger than the tunneling gap

Moving on to larger speeds

To calculate the quantum frictional dissipation rate, we describe the particle motion by means of a weak coupling Born–Markov quantum master equation (QME), based on a time-evolving density matrix

## Results

Fig. 4 shows, for an arbitrary but convenient choice of parameters, the time-dependent population probability of the first three instantaneous eigenstates,

The mechanism just described predicts an advancement of the average position of the particle as well as a corresponding onset of dissipated power that are very different from those of ordinary Langevin frictional dynamics (18), which, with all parameters except ℏ are the same as in the quantum case, would describe the classical forced sliding of the same particle. Fig. 5 compares the average particle position versus time in the quantum and classical cases. The “quantum slips” occur rather suddenly, on an LZ tunneling time scale (22)

Because it occurs at a lower spring loading, the resonant barrier permeation strongly reduces the overall mechanical friction work exerted by the pulling spring. Fig. 3 shows the amount of energy absorbed by the bath (friction) at the end of each period as a function of velocity. In the classical case, for time scales much shorter than the characteristic thermal hopping of the barrier, friction grows logarithmically with speed, due to thermally activated slip, as is well known for stick–slip at finite temperature (1, 23⇓–25)

The quantum dissipation rate is by comparison, within the present parameter choice, smaller by a factor **3**) of transition from the *Inset*. We should note that Eq. **7** is approximate first of all because it does not include higher excited states; moreover, it is only valid when velocity is low enough that the cooling rate

This conceptually simple form of quantum lubricity might, in some variant, be within experimental reach for cold ions in optical lattices. The parameters used in our simulations assume a particle with the mass *M* of ^{171}Yb, and a lattice spacing

## Conclusions

In summary, comparison of classical stick–slip with quantum friction for a particle sliding in a periodic potential foreshadows major differences. A classical particle slides from a potential well to the next by overcoming the full potential barrier, whereas a quantum particle can permeate the barrier by LZ resonant tunneling to a discrete level in the nearby well, a process suddenly and narrowly available at a well-defined position of the harmonic trap, leading to discontinuous forward jump, as shown in Fig. 5. This quantum slip preempts the classical slip, giving rise to quantum lubricity. The potential energy accumulated by the particle during sticking, and frictionally dissipated after the quantum slip, is just the amount sufficient to reach the resonant condition with the excited state in the next well. Conversely, the classical potential energy increase necessary for classical slip is close to the top of the barrier, with a correspondingly larger amount of dissipated energy during and after the slip. In addition to this quantum lubricity effect, a regime of quantum superlubricity is in principle expected at sufficiently low temperatures, where the friction decay with velocity decreasing to zero should be nonanalytical, with all derivatives equal to zero. It will be of interest in the future to pursue these quantum novelties in more detail, as soon as experimental realizations for single and many-particle systems will emerge.

## Acknowledgments

Research was supported by the EU FP7 under European Research Council (ERC) Advanced Grant 320796 MODPHYSFRICT, and in part by European Cooperation in Science & Technology (COST) Action MP1303.

## Footnotes

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^{1}To whom correspondence should be addressed. Email: tosatti{at}sissa.it.

Author contributions: F.P., G.E.S., and E.T. designed research; T.Z., F.P., and G.E.S. performed research; T.Z., F.P., G.E.S., and E.T. analyzed data; and T.Z., F.P., G.E.S., and E.T. wrote the paper.

Reviewers: A.V., Samara State Technical University; and V.V., Massachusetts Institute of Technology.

The authors declare no conflict of interest.

Published under the PNAS license.

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