Zeno Friction and Anti-Friction in Quantum Collision Models

# Zeno Friction and Anti-Friction in Quantum Collision Models

###### Abstract

We have analyzed the friction induced on a quantum system as it moves over and interacts with a surface which is itself composed quantum systems. Specifically, we model the interactions between the system and the surface with a Collision Model. We show that under some natural assumption (that nothing happens in no time, and that things happen at a finite rate) the magnitude of the friction induced by this interaction decreases as for large enough velocities, . Specifically, we predict this phenomena occurs in the Zeno regime, where the system’s coupling to each element of the surface is so short that not much evolution happens within each interaction. In order to investigate friction at low velocities and with velocity dependent couplings strengths, we motivate and develop what we call one dimensional convex collision models. Within these models we were then able to computed an analytic expression for the friction-velocity profile in a generic scenario. We are thus able to determine exactly the conditions under which the usual friction-velocity profiles arise within these models. Finally, through physically motivated examples, we demonstrate the possibility of anti-friction in which the system is accelerated by its interaction with the surface. We associate this phenomena with active material and inverted populations.

## I An idea for this or a follow-up paper

We have one system traveling along a surface. As it travels along the surface it encounters one fresh ancilla after another.

Now what if we also want to consider the scenario in which two surfaces are sliding past another, given that this is an important realistic case? Actually, results for this case should be obtainable by a very simple modification of our setup:

Now our system is no longer a lone cowboy, now it is part of a surface itself. And now as our system travels along the surface it doesn’t encounter a sequence of fresh ancillas. It encounters ancillas in the state that was the final state of the ancilla that it interacted with!

The two surfaces could be composed of systems of a different nature or they could be of the same nature. Either case should be interesting.

If our systems are classical and all initially in the same uncorrelated state then in effect, the sliding motion of two surfaces becomes equivalent to having two systems (one from each surface, opposite another) whose coupling strength is time dependent. So we have two systems interacting through an interaction Hamiltonian with a periodically time-dependent prefactor. I think that kind of bi-partite system is interesting in itself especially in the quantum case.

Now if we have two surfaces sliding past another quantumly then things can get much richer because initially unentangled systems that make up the two surfaces can all get entangled. This entanglement may be far reaching and could have interesting and possibly useful consequences for quantum tech. [Dan: I can look into this a bit over winter break, but I think it would require enough modifications to the formalismt that we should only consider it as a new paper/project.]

## Ii Introduction

In the literature, the term “quantum friction” has historically been used to refer to two very different scenarios. The first scenario relates to certain necessary inefficiencies of heat engines which arises when operating them too quickly [Edu: I didn’t know that was called quantum friction. Is it really? it is quantum thermo literature? we need references of this since I only knew about the Casimir-related case].[Dan: In classical thermodynamics it is just called friction, in quantum thermo they say things like “frictionless quantum engines” and probably in a few texts they say “quantum friction”. Overall the connection of this work to this sort of friction is tenuous. I would be fine removing it from the paper and focusing more on relationship to quantum friction.] In the second case “quantum friction” refers to a Casimir-type force directed against an object’s velocity which arises from the object’s interaction with a nearby surface. This interaction is mediated by the surrounding quantum field. We will now briefly review these two scenarios outlining the scenario we consider in this paper [Edu: We should add the many references that are considered important in this field. You can get them from the paper with Pablo].[Dan: I will reread that introduction and find more references.]

It is a standard result of classical thermodynamics that the efficiency of a heat engine is bounded by the Carnot efficiency. However, this bound is only reached by reversible heat engines which must operate quasi-statically, taking an infinite amount of time on each cycle. That is, any classical heat engine with maximum efficiency must have zero power. In other words, when operating a heat engine at any finite speed there will necessarily be inefficiencies TextbookChapter. These inefficiencies, viewed as a sort of penalty for going too fast, are broadly referred to as friction. The same results do not hold for quantum heat engines. Using a techniques called shortcut to adiabaticity STA, which implement adiabatic transformations in finite time, one can apply “quantum lubrication” QuantumLube and produce a friction-less quantum engines.

It is widely known that the interaction of neutral (but polarizable) objects in the presence of a quantum field can induce forces on them. For instance, the Casimir forces and Van der Waals forces are examples of this phenomenon. In these scenarios the neutral objects are treated as boundary conditions against which the quantum field theory equations are solved. That is, they often ignore the internal structure of the macroscopic objects that set the boundary conditions111We do note that in PhysRevA.98.032507 the friction induced on an atom moving relativistically over a surface is computed, treating the atom as an Unruh-DeWitt detectors, i.e. with internal structure. However, we note that in this work the surface is still treated as a boundary condition.. When considering moving objects (for instance an atom moving over an infinite plate) the induced forces may have some component directed against the system’s motion. This motion resisting force has been called “quantum friction.” [Achim: Casimir and related forces are conservative rather than dissipative. Do people really call them friction?] [Dan: Yes, see Edu’s paper Arxiv 1807.01727 for examples.] [Edu: Yes and we should add references to that field. There are many to cite that can be seen in the intro to that paper. It is a field that has been going on for many years.]

In this paper we propose a new method of characterizing the emergence of quantum friction [Edu: I think we should steer away from calling what we do quantum friction. Rather is a quantum model for friction that doesn’t consider the quntum field aspect of it which is so important in what people call ‘quantum friction’. I would strongly suggest that we don’t call it that, which is an established thing and can backfire since what we do is definitely not quantum friction with a field fluctuations (the Casismir-like approach that has gotten the name).][Dan: I agree, we are already introducing a new term, “Zeno Friction” for the high speed phenomena. That’s enough new terms/usage of terms for one paper.] which combines aspects of these two approaches described above. As in the Casimir force notion of quantum friction, we consider an object moving over and interacting with a surface. However, our approach simplifies the scenario by not considering the presence of any quantum fields mediating the interaction between moving particle and surface. Rather, we model the effective interaction of the moving particle with every constituent of the surface through a Collision Model. Unlike previous works, this simplification allows us to treat both the object and the macroscopic surface as quantum systems. Specifically, we take the this surface to be composed of many quantum systems which the traveling system interacts with directly. As in the thermodynamics approach, we characterize our notion of friction through the cost paid by an external agent controlling the motion of the system above the surface.

## Iii Friction in Collision Models

Consider a quantum system, , pulled by an external agent at a fixed speed over a line of ancillary systems, as in Fig. 1. Suppose that these ancillas are separated by a distance and that the system only interacts with the nearest ancilla at any given moment such that the system interacts with a new ancilla, , every .

During the system’s interaction with each ancilla, they are free to exchange energy between their internal states. However since an external agent is coupling and decoupling the systems, one does not in general expect to conserve systems’ total internal energy. That is, in general the agent must do some work to maintaining the system’s fixed trajectory. This work222Note that the work associated with each interaction is defined in terms of the expected energy changes of the system and ancilla. Generally, the work required to perform a quantum process is associated with a distribution of work costs RevModPhys.83.771. In general, these distributions can have variances comparable to their averages. An analysis of the quantum fluctuations of this work cost (and of the friction we define from it) would be interesting but is beyond the scope of this paper. [Edu: Is this something a referee could raise? I know I would if I think about it. If the deviation is comparable to the average the whole analysis may not be correct.][Dan: I was hoping to raise this point in preparation for a referee picking at the issue. One defense is “We do note however, that the interactions we consider are applied repeatedly such that fluctuations of the individual interactions will tend to cancel out and make the total work cost relatively more certain.”] must satisfy,

 δES+δEA+δW=0, (1)

where and are the expected changes in the system and ancilla’s internal energies due to the interaction.

This situation is reminiscent of an experiment commonly done in freshman physics labs, see Fig. 2. The students attach a force meter to a set of weights and use it to pull across the table at a fixed speed. The force registered by the force meter is (equal and opposite to) frictional force, , of the weights on the table. The energy cost333[Dan: New smaller footnote here which refers to the full argument in the appendix.] Note that we are associating the agent’s entire energy cost with a frictional force, not just the part of this work that is ultimately converted to heat. We discuss the reasons for and consequences of this choice at length in appendix A, but basically our choice here means that we consider a modern car’s regenerative braking system a source of friction. of maintaining this fixed speed over a distance is,

 ΔW=−fΔx. (2)

Similarly, in the scenario described above, we can identify the energy cost of maintaining a fixed speed per distance travelled as the friction (time-averaged over a single interaction),

 f \coloneqq−δWδx=δESδx+δEAδx=1v(δESδt+δEAδt). (3)

Note that in our sign convention if both the system and ancilla gain energy due to their interaction the friction is positive.

Now suppose that instead of being pulled by an external agent, the system moves over the ancillas freely, carried by its inertia. Since the system is still coupling and decoupling to the ancillas, there is still a energy cost to be paid by whatever is controlling the system’s motion. It is natural to expect that this cost is paid out of the system’s kinetic energy, such that (if the friction is positive) it slows down and ultimately stops. Note that in order to apply our formalism to this case it is necessary to check that the velocity does not change much during each interaction.

As we will see in later, in certain scenarios, the friction may be negative, resulting in anti-friction. That is, the systems’ internal energies may be lowered by their interaction. Paralleling the above argument, the energy lost by the systems will go into whatever is coupling and decoupling them. That is, the energy may either go to the system’s kinetic energy or to the external agent pulling the system. When a system experiences anti-friction it pushes forward, either speeding up or pressing into whatever is holding it back.

We will now analyze the friction discussed above using the framework of Collision Models. In the interaction (note, we start counting from ) the system and ancilla states are updated as,

 ρS(nδt) →TrA(U(δt)(ρS(nδt)⊗ρA(0))U(δt)†) (4) =ΦS(δt)[ρS(nδt)], ρA(0) →TrS(U(δt)(ρS(nδt)⊗ρA(0))U(δt)†) (5) =ΦA,n(δt)[ρA(0)],

where is some unitary operator on the joint system, , describing their interaction. Note while the system’s update map, , is independent of , the ancilla’s update map, , can depend on the interaction number, , through the system’s current state, . Further note that always depends on linearly.

From these update formulas, we can compute the expected change in the system’s internal energy as,

 δES,n=TrS(^HS(ΦS(δt)−\openoneS)[ρS(nδt)]), (6)

where is the local Hamiltonian of and is the identity channel on . Likewise we can compute the expected change in the ancilla’s internal energy as,

 δEA,n=TrA(^HA(ΦA,% n(δt)−\openoneA)[ρA(0)]), (7)

where is the local Hamiltonian of and is the identity channel on . From these we can identify the average friction during the interaction,

 fn=δES,nδx+δEA,nδx=1v(δES,nδt+δEA,nδt). (8)

As we will now see, under some natural assumptions, this collisional model of friction yields bizarre phenomenology at high speeds.

### iii.1 Collisional Friction in the Zeno Regime

It is often natural to expect that nothing can happen in no time and that when things do happen they happen at a finite rate. We can capture these intuitions by making some regularity assumptions about the update maps’ behaviors around . Specifically, we could assume that

 ΦS(δt)→\openoneSandΦ%A,n(δt)→\openoneAasδt→0, (9)

and that,

 Φ′S(0)andΦ′A,n(0)exist (10)

where the primes indicate a derivative with respects to . For instance, these assumptions hold if the unitary matrix, , in (4) and (5) describing the interaction between and are generated by a Hamiltonian, , which is independent of (and therefore of ). That is, .

Given these regularity assumptions, it follows that the friction decays as as . Specifically taking the limit (or equivalently ) in (8) we find,

 fn =1vTrS(^HSΦ′% S(0)[ρS(nδt)]) (11) +1vTrA(^HAΦ′% A,n(0)[ρA(0)])+o(v−1),

for large . Note that we are using small-o notation here since we have not assumed and are second differentiable at . This means we see less friction as we go faster. This goes against a common intuition that friction is a penalty for going fast; In Zeno friction, we see no friction.

As a concrete example, suppose that the unitary matrix governing the interaction is given by

 U(δt)=exp(−i^Hδt/ℏ)  where  ^H=^HS+^HA+^HSA (12)

with independent of the systems’ relative velocity, . In this case we can easily compute and (as in Layden:2015b). From Layden:2015b we have

 Φ′S(0)[ρS] =−iℏ[^HS+TrA(^HSAρA(0)),ρS] (13) =−iℏTrA([^HS+^HSA,ρS⊗ρA(0)]).

From this we can compute the first term in (11) to be,

 1vTrS(^HSΦ′S% (0)[ρS(nδt)]) (14) =1v−iℏTrSA(^HS[^HS+^HSA,ρS⊗ρA(0)]) =1v−iℏTrSA([^HS,^HS+^HSA]ρS⊗ρA(0)) =1v−iℏ⟨[^HS,^HSA]⟩n

where we have used the identity and defined as the expectation value taken with respects to the joint state at , that is . The second term in (11) can be computed by the same method to be . Thus in total the friction is

 fn =1v⟨iℏ[^H% SA,^HS+^HA]⟩n+O(v−2). (15)

Note that as expected the presence of friction is directly related to the non-conservation of the systems’ local energies under the interaction Hamiltonian.

This phenomena of decreasing friction at higher velocities is not the sort of velocity dependence that we are used to seeing in our everyday encounters with friction; typically the amount of friction either increases or stays constant at increasing speeds. One is lead to wonder: at what speeds do we expect to start seeing Zeno friction?

To estimate the speeds associates with the Zeno friction, let us consider a particle travelling through the air at a speed interacting with nitrogen molecules via a Van der Waals interaction (with energy scale ) as it crosses their Van der Waals radius (). The perturbative expansion underlying (11) and (15) requires that the amount of evolution happening in each interaction is small, . Taking the duration of the interaction to be the crossing time, , we find this requires,

 v≫2rEℏ=43km/s=1.5×10−4c. (16)

It is interesting to note that this critical speed is approximately halfway (geometrically) between walking speed and the speed of light [Edu: I smiled reading this explanation. Feels like (good) lecture material to say in class but not sure about the paper.][Dan: I like it still.]. While speeds much larger than this are not common in everyday life, some particles from space enter the atmosphere at speeds of . [Edu: What do you mean? muons can be much faster than that even () are you thinking of something in particular?][Dan: I think I was looking for a neutral particle so that it would be appropriate to treat as a Van Der Walls interaction. I don’t remember where this number came from.]

An important caveat to our prediction of Zeno Friction at high velocities is that interaction must obey the regularity assumptions, (9) and (10). These can be naturally negated by taking the coupling strength between and to increase with their relative velocity. For example, if is generated by then as ; That is, something happens in no time. Such velocity dependent couplings could arise naturally if the systems’ couple to each others external/kinetic degrees of freedom.

[Edu: The discussion is good but I feel we are lacking a selling point in the argumentation at this point: a reader can take this as “Oh! they are not able to explain friction. Dissappointing.” However we should perhaps emphasize that there are regime wehere we expect this to dominate and that this is a new kind of friciton that no one else has anticipated before no?][Dan: You’re right, I have rephrased the discussion away from “why don’t we see Zeno friction in everyday scenarios” to “Zeno friction is a new phenomena in a new regime”. Also there is a new line of text below promising we will get to an explaination of everyday friction.]

Barring this possibility, we expect to see Zeno friction at high velocities. That is, we predict that for velocity independent couplings the amount of friction will begin decreasing at high enough speeds.

In order to explore friction at low velocities (outside of the Zeno regime) and the possibility of velocity dependent couplings we will now particularize to a simplified class of collision models. Using these models we be able to reproduce the common friction-velocity profiles we experience everyday. We will also explore scenarios within this model exhibiting anti-friction.

## Iv One-dimensional convex collision models

### iv.1 Motivation and Definition

One of the most widely used collision models PhysRevLett.113.100603; PhysRevA.75.052110; PhysRevE.97.022111; 1367-2630-16-9-095003; PhysRevA.76.062307 is the partial swap interaction first described in Scarani2002. It consider a system, S, interacting with an ancilla, , via the partial swap Hamiltonian, , where is the unitary matrix which swaps the states of and as . Note that is self-adjoint, , as well as unitary such that . For example, if and are qubits then is the isotropic spin coupling.

Evolution under the partial swap Hamiltonian for a time is described by the partial swap unitary,

 U(t) =exp(−i^Hswt/ℏ) (17) =cos(Jt)^\openoneSA−isin(Jt)Usw, (18)

where is the identity operator on the joint system . Evolving by this unitary from an initially uncorrelated state, the reduced state of the system is,

 ρS(t) =TrA(U(t)(ρS(0)⊗ρA% (0))U(t)†) (19) =cos(Jt)2 ρS(0)+sin(Jt)2 ρA(0) −icos(Jt)sin(Jt) TrA([Usw,ρS(0)⊗ρA(0)]).

A similar expression holds for the reduced state of the ancilla. The cross terms in these expressions vanish if and are diagonal in the same444“Same” here meaning that and are diagonal in the same basis. basis yielding,

 ρS(t) =cos(Jt)2 ρS(0)+sin(Jt)2 ρA(0), (20) ρA(t) =cos(Jt)2 ρA(0)+sin(Jt)2 ρS(0). (21)

That is, the system and ancilla oscillate between their own initial states and the other’s initial state at a rate . Note that each system evolves within a one-dimensional space as a convex combination of two fixed endpoint. We invite you to imagine this evolution as the information about the system’s initial condition is being passed from S to A and back in the same way that a harmonic oscillator passes its energy between its position and momentum.

More realistically one might expect that during this interaction the ancilla is connected to a larger environment into which it leaks some information about the system’s initial condition at some rate, . Using our harmonic oscillator analogy, one can imagine that the information is dissipated into the environment while in system A in the same way that the energy of a damped oscillator is dissipated while it is stored as momentum. Motivated by this analogy, one can model the effect of A’s environment by taking

 ρS(t) =ϕS(t) ρS(0)+(1−ϕS(t)) ρA(0), (22) ρA(t) =ϕA(t)ρA(0)+(1−ϕA(t)) ρS(0), (23)

with

 ϕS(t) =e−2γAt(cos(ωt)+γAωsin(ωt))2, (24) ϕA(t) =1−e−2γAtJ2ω2sin(ωt)2, (25)

and is the damped oscillation rate. Note that if the oscillation is over-damped. Figure 3 a,b) shows the evolution of the two systems coupled this way when under damped and critically damped.

Alternatively, one could imagine that instead of swapping their initial states back and forth, the systems interact by repeatedly entangle and then disentangle. For example, the joint system could evolve as,

 ρSA(t)=ϕ(t)ρS(0)⊗ρA(0)+(1−ϕ(t))|ψ⟩⟨ψ|, (26)

where is a maximally entangled state and describes the systems’ evolution. In this case the dynamics of the systems’ reduced states are,

 ρS(t) =ϕ(t) ρS(0)+(1−ϕ(t)) ^\openoneS% /DS, (27) ρA(t) =ϕ(t)ρA(0)+(1−ϕ(t)) ^\openoneA% /DA, (28)

where and are the dimensions of the system and ancilla respectively. Again note that each system evolves within a one-dimensional space as a convex combination of two fixed endpoint.

We can capture the common elements of these examples in the following definition. In a one-dimensional convex collision model the interaction updates the system and ancilla states as,

 ρS(nδt) →ϕS(δt)ρS(nδt)+(1−ϕS(δt))ρS,⊙, (29) ρA(0) →ϕA,n(δt)ρA(0)+(1−ϕ%A,n(δt))ρA,⊙,n, (30)

for some and and some target states and . As in a generic collision model, we allow the details of the ancilla’s update to depend on the interaction number, , via a linear dependence on the current state of the system, .

While in general both and can depend on , we will now assume for simplicity that is independent of . Note that this is the case in all of our motivational examples.

### iv.2 Friction in one-dimensional convex collision models

We will now calculate the average friction during the interaction, , for a generic one-dimensional convex collision model.

First we note that the system’s update equation, (29), can be easily solved yielding,

 ρS(nδt)=ϕS(δt)nρS(0)+(1−ϕS(δt)n)ρS,⊙. (31)

Next, we note that there is a natural interpolation scheme between the discrete time steps, , given by,

 ρS(t) =e−ΓtρS(0)+(1−e−Γt) ρ%S,⊙, (32)

where

 Γ\coloneqq−1δtLn(ϕS(δt)). (33)

See Fig 3 c) for an illustration of such an interpolation scheme. Note that the interpolation scheme exactly matches the system state at the end of every interaction. Also note that if (such that the system reaches its target state after just one interaction and stays there) then . Thus in this case the interpolation scheme predicts system reaches its target state just after and stays there. [Dan: Added a comment to help people interpret the divergences in which appear later.]

Note that since the dependence on of the ancilla’s target state is assumed to come from a linear dependence on , it must be of the form,

 ρA,⊙,n=ϕS(δt)nρA,⊙,0+(1−ϕS(δt)n)ρA,⊙,∞. (34)

Note that we can fit this with the same interpolation scheme as the system’s state.

Next, from equation (31) we can compute the system’s internal energy at as,

 ES(nδt)=ϕS(δt)nES(0)+(1−ϕS(δt)n)ES,⊙, (35)

where is the system’s initial energy and is the energy of the system’s target state. From this we can compute the change in the system’s energy during the interaction,

 δES,n =(1−ϕS(δt))ϕS(δt)n(ES,⊙−ES(0)). (36)

Note that this is just a geometric sequence with a common ratio and normalized to have a sum of .

Similarly we can calculate that after the interaction the energy of the ancilla is,

 EA,n=ϕA(δt)EA(0)+(1−ϕA(δt))EA,⊙,n. (37)

where is the energy of the ancilla’s initial state and is the energy of the ancilla’s target state. The change in the ancilla’s energy due to this interaction is,

 δEA,n=(1−ϕA(δt)) (EA,⊙,% n−EA(0)). (38)

From these we find that the friction averaged over the interaction is,

 fn =(1−ϕS(δt)) ϕS(δt)n ES,⊙−ES(0)δx (39) +(1−ϕA(δt)) EA,⊙,n−EA(0)δx.

That is, in the interaction the system takes an (ever diminishing) step towards its target state while the ancilla takes it first (and only) step towards its target state.

We will now separate this friction into a permanent/transient parts which remains/vanish as . Specifically, we find the permanent friction to be,

 f∞=(1−ϕA(δt)) EA,⊙,∞−EA(0)δx. (40)

Note that at late times the system has always reached its target state so the only energy cost is moving each ancilla one step towards its target state at . Thus the permanent friction depends only on the dynamics of the ancillas.

The transient part of the friction is defined as,

 ftransient,n \coloneqqfn−f∞ (41) =(1−ϕS(δt)) ϕS(δt)n ES,⊙−ES(0)δx +(1−ϕA(δt)) EA,⊙,n−EA,⊙,∞δx.

The transient friction is associated with the system approaching its target state, , and with the ancilla’s target state approaching its final target state, . As discussed above (in equations (32) and (34)) these both happen exponentially at a rate . We can factor this exponential decay out of both these terms we find

 ftransient,n=ftr exp(−Γnδt)=ftr ϕS(δt)n (42)

where

 ftr\coloneqqftransient,0 =(1−ϕS(δt))ES,⊙−E%S(0)δx (43) +(1−ϕA(δt))EA,⊙,0−EA,⊙,∞δx.

Thus is fully captured by the quantities, , and as,

 fn=f∞+ftr exp(−Γnδt). (44)

Note that while we are characterizing the friction in terms of the interpolation parameter, , our analysis only ever evaluated the states and energies of the systems at times, , where the interpolation scheme is exact. The above equation can equivalently be interpreted as saying the transient friction decays geometrically by a factor of each interaction. The benefit of using the interpolation scheme is that it allows for fair comparisons of this decay for systems with different (or equivalently travelling at different speeds). We will now make some general comments about each of these quantities.

First we note the the permanent friction, , and the transient friction, can both be either positive or negative depending on the energies of the system and ancilla’s initial and target states. Specifically, we expect to see anti-friction when the energy of the system and ancilla’s target state is lower than their initial state. As we will discuss later, such situations arise naturally from states with inverted populations.

Next we note that , , and can all depend on the systems’ relative velocity through their dependence on .

Finally, we note that the magnitude of the permanent and transient friction are both bounded as,

 |f∞| ≤|EA,⊙,∞−EA(0)|δx, (45) |ftr| ≤|ES,⊙−ES(0)|δx+|EA,⊙,0−EA,⊙,∞|δx, (46)

and therefore so is the total friction. Note that bounds are velocity independent, such that this model cannot predict for all . However, as we will see we can predict this velocity profile in the low velocity regime.

As we discussed in Sec III.1, the friction at high velocities depends on how the systems’ update maps behave for small . For instance, suppose our regularity assumptions, (9) and (10), are satisfied such that we can expand and around as,

 ϕS(δt) =1−δtϕS,1+O(δt2), (47) ϕA(δt) =1−δtϕA,1+O(δt2), (48)

then for large velocities we can expand the friction parameters as,

 f∞(v) =EA,⊙,∞−EA(0)vϕ%A,1+O(v−2), (49) ftr(v) =ES,⊙−ES(0)vϕS,1+EA,⊙,0−EA,⊙,∞vϕA,1+O(v−2), Γ(v) =ϕS,1+O(v−1).

Note that as expected the magnitude of the friction goes as for large , that is we see Zeno Friction.

If we do not meet these regularity assumptions then at high velocities we will not see Zeno friction. For instance, if and then

 f∞(v→∞) =EA,⊙,∞−EA(0)δxFA, (50) ftr(v→∞) =ES,⊙−ES(0)δxFS+EA,⊙,0−EA,⊙,∞δxFA Γ(v) =Ln(FS)vδx+O(1),

for large . That is, the permanent and transient friction both approaches a constant at high speeds, although the decay rate becomes large, so the transient friction will vanish quickly.

The friction at low velocities depends on the how the system’s interact for long times. For instance, if and then

 f∞(v→0) =EA,⊙,∞−EA(0)δxFA, (51) ftr(v→0) =ES,⊙−ES(0)δxFS+EA,⊙,0−EA,⊙,∞δxFA Γ(v→0) =0.

That is, the permanent and transient friction both approaches a constant at zero speeds. Note that since the decay rate goes to zero, so the transient friction will vanish very slowly.

If instead and decay polynomially to for large as,

 ϕS(δt) =1−δt−pϕS,p (52) ϕA(δt) =1−δt−pϕA,p, (53)

for some then we find for small velocities,

 f∞(v) =EA,⊙,∞−EA(0)δxp+1ϕA,pvp (54) ftr(v) =ES,t−ES(0)δxp+1ϕS,pvp+EA,t,0−EA,⊙,∞δxp+1ϕA,pvp Γ(v) =ϕS,−pδxp+1vp+1.

Thus we can recover any scaling behavior for small velocities by picking an appropriate exponential, .

## V Examples

We will now consider several example scenarios.

### v.1 Damped Partial Swap Interaction

Consider a spin qubit, S, moving at a speed relative to a line of spin qubit ancillas. Suppose that the ancillas are separated by a distance and that the system interacts only with the nearest ancilla such that it meets a new ancilla, A, every . Suppose that the system and ancillas are initially in thermal states,

 ρS(0)=(1+aS(0)σz)/2,^HS=ℏωSσz/2, (55) ρA(0)=(1+aA(0)σz)/2,^HA=ℏωAσz/2, (56)

with respects to their local Hamiltonians. Note that for either system, corresponds to the ground state () with the system’s temperature increasing as increases. At the system is at infinite temperature, maximally mixed (). For the state has an inverted population ().

Suppose the system couples to each ancilla via the isotropic spin coupling, . As discussed in Section IV.1 this coupling induces a partial swap interaction between the systems and corresponds to the one-dimensional convex collision model, (29), with

 ϕS(δt) =cos(Jδt)2,ρS,⊙=ρA(0), (57) ϕA(δt) =cos(Jδt)2,ρA,⊙,n=ρS(nδt). (58)

As discussed in Section IV.1 this situation can be modified to include each ancilla dissipating information into its environment at a rate by instead taking and to be,

 ϕS(δt) =e−2γAδt(cos(ωδt)+γAωsin(ωδt))2, (59) ϕA(δt) =1−e−2γAδtJ2ω2sin(ωδt)2, (60)

where is the damped oscillation rate. Note that if the oscillation is over damped, , then is imaginary. The identities, and are useful in this case.

Computing the average friction during the interaction we find, where,

 f∞(v) =0 (61) ftr(v) =(ℏωS(1−e−2γAδx/v(cos(ωδxv)+γ% Aωsin(ωδxv))2)−ℏωAJ2ω2e−2γAδx/vsin(ωδxv)2)aA(0)−aS(0)δx Γ(v)

Note that the friction is entirely transient. This is because at late times the system has reached its target state, that is to the the ancilla’s initial state. Thus at late times the partial swap interaction does not effect the reduced state of either system.

For large velocities we can expand the friction as a series in to find,

 ftr(v) =ℏJ2δxv2(ωS−ωA)(aA(0)−aS(0))+O(v−3) Γ(v) =J2δxv+O(v−2). (62)

We note that these expansions are hold independently of if the evolution is over-, under-, or critically damped. Note that as predicted in Sec III.1, the magnitude of the friction goes to zero as the velocity increases.

Note that the transient friction at large velocities can be either positive or negative depending on the system’s energy gaps and their initial polarizations. In this example we see anti-friction at high velocities when the system with the higher energy gap also has a higher population number.

Taking the limit of small velocities we find,

 ftr(v→0) =ℏωSδx(aA(0)−a%S(0)), (63) Γ(v→0) ={2γA,γA≤J2γA−2√γ2A−J2,γA>J. (64)

Note that the transient friction as can be either positive or negative depending on the systems’ initial polarizations. We see anti-friction when the system has a higher population number than the ancilla. This is the case for instance if the ancilla are all in their ground state.

At intermediate velocities the transient friction can oscillate and change signs as shown in Fig 4.

Note that in this and all following figures we have picked our dimensionful quantities along the lines of the Van der Waals interaction example discussed above, see (16). For reference, a force of nN acting on a nitrogen atom with mass amu results in an acceleration of . From an initial speed of km/s, this force stops the atom in ps over a distance of nm. Travelling this distance, the atom would cross Van der Walls radii.

We can modify this scenario to avoid Zeno friction (i.e., to have friction at large velocities) by having a velocity dependent coupling. For example we could take the coupling strength to be velocity dependent as for some . Calculating the friction in this case yield the same result as before (see (61)) but with . Note that for large velocities the dynamics is always underdamped. Likewise for small velocities the dynamics is always over damped.

For large velocities, we can expand the transient friction and decay rate as,

 ftr(v) =ℏsin(kδx)2δx(ωS−ωA)(aA(0)−aS(0))+O(v−1) (65) Γ(v) =vδxLog(sec(kδx)2)+O(1). (66)

Note that as anticipated in this case the friction does not decay as for large . However, note that since the friction’s decay rate is proportional to , the friction at high velocities decays quickly. In this case we see anti-friction at high velocities when the system with the higher energy gap also has a higher population number.

For small velocities, we can expand the the transient friction and decay rate as,

 ftr(v) =k2vℏωSγA(aA(0)−aS(0))+O(v2) (67) Γ(v) =k2v2γA+O(v3). (68)

Note that in this regime we recover the usual friction dependence . Moreover note that in this regime the friction’s decay rate is very small, meaning that while the friction is entirely transient it will last a relatively long time. We see anti-friction when the system has a higher population number than the ancilla. This is the case for instance if the ancilla are all in their ground state.

Figure 5 shows the behavior of the friction at intermediate velocities. Note that the dynamics can be critically damped at intermediate velocities.

### v.2 Entangle-disentangle Interaction

Next let us consider the second motivating example described in Sec. IV.1 in which the system and ancilla repeatedly entangle and disentangle with each other. As discussed above this dynamics can be described by the one-dimensional convex collision model with,

 ϕS(δt) =ϵ+(1−ϵ)cos(Jδt)2,ρS,⊙=^\openoneS/DS, (69) ϕA(δt) =ϵ+(1−ϵ)cos(Jδt)2,ρA,⊙,n=^\openoneA/DA (70)

for some and some oscillation rate