Physically constrained quantum clock-driven dynamics
Dario Cilluffo Lea Lautenbacher Giovanni Spaventa Susana F. Huelga Martin B. Plenio


§ ABSTRACT 

Thermal machines are physical systems specifically designed to make thermal energy available for practical use through state transformations in a cyclic process. This concept relies on the presence of an additional element equipped with a clock, controlling which interaction Hamiltonian between the system and the reservoirs must act at a certain time and that remains unaffected during this process. In the domain of quantum dynamics, there is substantial evidence to suggest that fulfilling this final condition is, in fact, impossible, except in ideal and far-from-reality cases. In this study we start from one such idealized condition and proceed to relax the primary approximations to make the model more realistic and less ideal. The main result is a fully quantum description of the engine-clock dynamics within a realistic quantum framework. Furthermore, this approach offers the possibility to address the deeper and more fundamental challenge of defining meaningful time operators in the realm of quantum mechanics from a different standpoint.




Corresponding author:dario.cilluffo@uni-ulm.de
Institute of Theoretical Physics & IQST, Ulm University, Albert-Einstein-Allee 11 89081, Ulm, Germany


Institute of Theoretical Physics & IQST, Ulm University, Albert-Einstein-Allee 11 89081, Ulm, Germany


Institute of Theoretical Physics & IQST, Ulm University, Albert-Einstein-Allee 11 89081, Ulm, Germany


Institute of Theoretical Physics & IQST, Ulm University, Albert-Einstein-Allee 11 89081, Ulm, Germany


Institute of Theoretical Physics & IQST, Ulm University, Albert-Einstein-Allee 11 89081, Ulm, Germany



Physically constrained quantum clock-driven dynamics
    Martin B. Plenio
    September 9, 2024
====================================================





§ INTRODUCTION




An engine can generally be defined as an open system coupled to many portions of the surrounding environment, which act as reservoirs. The primary aim of an engine is generating power in the form of mechanical work <cit.>. 



The realization of thermodynamic cycles, whether classical or quantum, relies on time-dependent Hamiltonians, which often incur significant energy costs that are difficult to fully capture in theoretical models. This problem can be addressed by embedding the system within an expanded framework that includes an additional "clock" system to keep track of time. The engine and clock together evolve under a time-independent Hamiltonian in a larger Hilbert space <cit.>. While this approach increases the dimensionality and complexity of the system due to the expanded Hilbert space, it compensates by making the overall dynamics autonomous, removing the need for external time-dependent driving.
This framework has offered valuable insights into how the intrinsic physics of the time-keeping system <cit.> and the process of time sampling <cit.> affect the dynamics of generic quantum machines, with notable implications for quantum computing protocols <cit.>. Additionally, this perspective has become foundational in the resource theory of thermodynamics <cit.>.

In the most general framework, a quantum system that can either access or generate a time signal and use it to control the evolution of another system, while remaining robust to perturbations (such as back reactions from the engine), can serve as a clock <cit.>.
A minimal model including effectively all these features has been recently proposed in <cit.>: 
the time-dependent Hamiltonian coupling system and environment is modelled as a set of time-independent operators acting sequentially during the evolution time.
Each operator is correlated to the position of an external particle (the clock) freely moving under a Hamiltonian that is assumed to be linear in the momentum. 
The control mechanism is provided by an effective coupling between the particle and the engine that is assumed to commute with the free Hamiltonian of the engine (covariant operation).
As pointed out in <cit.>, this model is inherently nonphysical because of the particular free evolution assumed for the clock (pure translation), that requires an unbounded clock Hamiltonian.
This argument can be seen as another facet of the celebrated Pauli objection to the existence of a time operator in standard quantum mechanics <cit.>. Namely, the equation of motion of a self-adjoint time operator T̂ reads

    Ṫ̂̇ = i [Ĥ,T̂] = 1 ⇒  [T̂,Ĥ] = i 1 ,

thus if such operators were to exist, they would need to be unitarily equivalent to x̂ and p̂, implying that their spectra are continuous and unbounded from below. This situation would result in unstable and nonphysical descriptions of interacting quantum theories. In essence, the commutation relation involving T̂ and the Hamiltonian operator, as outlined in (<ref>), lacks exact physical solutions within the framework of standard non-relativistic quantum mechanics.


This issue has deep historical roots within the genesis of quantum theory <cit.>.
Over the years, various strategies have emerged to confront this challenge, each presenting distinctive viewpoints. These strategies  include the proposition of non-self-adjoint time operators <cit.>, as well as explorations within the framework of relativistic quantum field theory <cit.>. Remarkably, an alternative perspective has advocated for the retention of unbounded operators <cit.>.
Other attempts in the literature, as documented in <cit.>, often rely on reasonable finite-dimensional approximations of T̂.
In this study, we extend the ideal model presented in <cit.> to encompass a realistic quantum framework, addressing complexities previously unaccounted for. Specifically, our approach involves utilizing a quantum field as our clock-system and tracking time by monitoring the motion of a coherent pulse as it propagates along a linear trajectory.
Under the same fundamental assumptions as in <cit.> regarding the coupling Hamiltonian between the system and the engine, we demonstrate that this coupling directly influences the dispersion law of the wave-packet, resulting in a significant deterioration of the clock's performance.
This result paves the way to address the challenge of approximating the solution of (<ref>) by means of infinite-dimensional operators that form an integral part of the model.











The paper is structured as follows: in Sec. <ref> we define the global model of engine and clock and describe the dynamics of the system using the framework of quantum collision models (QCM) <cit.>. Subsequently, in Sec. <ref> we introduce a model for the interaction between the control system and the engine. This model, designed for situations characterized by minor degradation, serves to restrict deviations from the desired dynamics we aim to implement. Sec. <ref> explicitly addresses the issue of degradation and in Sec. <ref> we return to the problem of time operators to quantitatively assess how the unavoidable degradation of physical clocks affects the precise definition of time operators in QM. 



§ DEFINITION OF ENGINE AND CLOCK


Following the reasoning in <cit.> we consider a bipartite Hilbert space ℋ = ℋ_E ⊗ℋ_C, where E identifies a generic open system and its surrounding environment, and C an auxiliary system denoted as clock.
The role of the clock is controlling the evolution of the system without any external control.
The total Hamiltonian reads

    Ĥ_ tot = Ĥ_E + Ĥ_C + V̂_EC  ,

where H_E and H_C are the free Hamiltonians of engine and clock, respectively. The identities over the complementary Hilbert spaces are omitted. In <cit.> the interaction between engine and clock takes the form

    V̂_EC = ∫_ℝ d x  V̂_E (x)⊗|x⟩_C⟨x|  ,

where |x⟩_C⟨x| projects over the different position states of the clock

and V̂_E∈ℋ_E. We choose as a clock a one-dimensional bosonic field with free Hamiltonian 

    Ĥ_C = ∫ d k  ω_k â_k^†â_k   .

The initial state of the clock is a pulse described by 

    |ψ⟩^(0)_C = ∫ dx  ξ_x_0(x)  𝒳(x)  |vac⟩≡ψ̂^†_C (x_0) |vac⟩ ,

where ξ_x_0(x) is the pulse envelope function centered on x_0, with velocity c=1 and 𝒳 is an operator acting on the ladder operators of the field x (e.g. displacement or squeezing). Within the quasimonochromatic approximation <cit.>, i.e. the spectral width is much smaller than the average frequency of the wave-packet. 
In the following, we will assume a linear dispersion relation ω_k = v_g k in the free Hamiltonian (<ref>). These assumptions correspond to taking as our clock a single-photon pulse traveling in a vacuum or a particle with a very narrow distribution in momentum. 
For simplicity, we assume a gaussian envelope with frequency bandwidth Ω

    ξ(x) = (Ω^2/2π)^1/4 e^-Ω^2 (x-x_0)^2/4 ,

with 𝒳 = ℐ, where ℐ is the identity map, i.e. a single-particle pulse.

We can distinguish between two different times only when the pulse has moved by an amount that is larger than its width (in <ref> there's an example of this process). 
This leads naturally to a first discretisation of the interaction defined by a time window of size W and discretising V̂_E(s) on them.
In light of this intuition we make the following replacements in (<ref>):

    V̂_E (s)    →∑_i V̂_E(s_i) Θ_W(s-s_i) , 
    
    |s _C s |    →1/W∫_𝒟_s dx  xx ,

where Θ(s-s_i) is the rectangle function of size W centered on the time s_i, which can be chosen among the times within the ith window. We denote with 𝒟_q the integration range corresponding to a time window centered on the value q.
Let's assume that s_i corresponds to the centre of the window.  
In the second equation the measure of position is defined, i.e. we integrate the number of excitations per length over the region corresponding to s.
More detailed information about W and the discrete potential V̂(s_i) will come in subsequent sections. 

Plugging Eqs. (<ref>) and (<ref>) into Eq.(<ref>), the interaction term can be expressed as

    V̂_EC→   ∫_ℝ ds ∑_i V̂_e(s_i) Θ_W(s-s_i) ⊗1/W∫_𝒟_s dx  xx
       =
    1/W∑_i   V̂_e(s_i) ⊗∫_ℝ ds Θ_W(s-s_i) ∫_𝒟_s dx  xx
       =
    1/W∑_i   V̂_e(s_i) ⊗∫_𝒟_i dx  xx ,

where 𝒟_i is now the region including s_i (following the previous assumption it is [s_i-W/2,s_i+W/2]).

In the interaction picture with respect to the free Hamiltonian of the clock and the engine, we have

    V̂_EC^I(t)  =
    1/W∑_i   V̂^I_E(s_i) ⊗∫_𝒟_i dx  x-tx-t ,

with V̂^I_E(s_i)=e^i Ĥ_E tV̂_E(s_i) e^-i Ĥ_E t (we omit the time dependence)
and the total propagator is given by

    𝒰̂_t,0 = e^-i (Ĥ_E + Ĥ_C) t 𝒯exp{-i ∫_0^t dτV̂^I_EC(τ) } ,

where 𝒯 is the time ordering operator. 





We now discretise the time axis into shorter intervals t = n Δ t with n∈ℕ and the time step Δ t≪ W. Thus the propagator in (<ref>) is decomposed as

    𝒰̂_t,0   = ∏_n 𝒰̂_n 
       = ∏_n e^-i (Ĥ_E + Ĥ_C) Δ t 𝒯exp{-i ∫_t_n-1^t_n dτV̂^I_EC(τ) } .

In the limit of Δ t → 0
we can approximate the propagator (<ref>) through the first order of the Magnus expansion <cit.>:




    ℋ_n^(0)   = 1/Δ t1/W∑_i   V̂_e^I(s_i) ⊗∫_𝒟_i dx  ∫_t_n-1^t_n dτx-τx-τ
       =
    
    
    
    
    
    
    1/W∑_i  V̂^I_e(s_i) ⊗∫_t_i-W/2-t_n^t_i+W/2-t_n dx  δn̂_x
       =
    1/W∑_i  V̂^I_e(s_i) ⊗N̂_W(i+1/2) -n ,

where δn̂_α:=1/Δ t∫_t_n-1^t_n dτα-τα-τ and N̂_W(i+1/2) -n := ∫_t_i-W/2-t_n^t_i+W/2-t_n dx  δn̂_x. 
Thus we obtain the discrete-time propagator

    𝒰̂_n ≃ e^-i (Ĥ_e + Ĥ_c) Δ t exp{-i Δ t /W∑_i  V̂^I_e(s_i) ⊗N̂_W(i+1/2) -n} .

Note that in our approximation we used the operator in Eq. (<ref>) instead of the field operator ψ̂^†_c (s) ψ̂_c (s) to avoid biasing by the shape of the pulse.

None of the previous assumptions prevent pulse broadening, as it depends on the clock-engine interaction Hamiltonian. Note that unlike the ideal scenario proposed in <cit.>, we cannot decompose the propagator in Eq.(<ref>) into two separate operators for the clock and the engine. Consequently, the engine and the clock exhibit correlations during their evolution. This back-reaction affects the clock's state and provides the physical basis for degradation. 






§ CLOCK DEGRADATION AND DISPERSION LAW


The free evolution of the clock ladder operators reads 

    e^i Ĥ_C  tâ_x e^-i Ĥ_C t = â_x- t ,

i.e. the free Hamiltonian of the clock only translates the state of the field. 
It becomes evident that when the clock evolves exclusively under this Hamiltonian with the assumption of a linear dispersion, it might behave as an ideal clock in the sense of <cit.>. Consequently, we can explore the potential for degradation by examining the emergence of nonlinearity in the dispersion law. Starting from (<ref>) we have, in Fourier space

    Ĥ_C + V̂_EC =    ∫ dk v_g k 1_E â^†_kâ_k   +  ∫ dk dk' ℱ[V̂_E](k-k')  ⊗â^†_kâ_k' ,

where ℱ[V̂_E](k) = ∫ dx V̂_E(x) e^- i k x denotes the Fourier transform of the engine Hamiltonian.
Even in the case in which ℱ[V̂_E]_E ∝ f(k,t) δ(k-k'), the interaction term still introduces a complex and non-trivial deviation from linearity into the dispersion relation. One of the consequences of the non-linearity of the dispersion relation is wave-packet broadening <cit.> [We are not considering Airy wavepackets <cit.>, which are well-known for being the only nonspreading-wavepacket solutions to Schrödinger's equation. Nonetheless, it's important to note that Airy wavepackets exhibit acceleration, making them unsuitable as candidates for position-based timekeeping.
]. When the wave-packet width becomes comparable to W, from the point of view of the engine, we are not able to select which of the transformations is the one we must implement to reproduce the target dynamics: in other words, we do not know what time it is. We interpret this phenomenon as a manifestation of clock degradation arising from the interaction with the engine. This interaction leads to the emergence of an effective mass ∝ (∂^2_k ω_k)^-1, even when the free clock field is originally massless. On the other hand, the coupling (<ref>) is a translation of (<ref>) for quantum fields, which in turn finds its origins in the broader concept of the ancilla-clock system initially introduced in <cit.>. Therefore, for clocks falling within this broad class, our findings provide direct evidence of the intrinsic connection between degradation and the effective mass of the system operating as a clock.
Directly solving the clock's dynamics under (<ref>) in order to quantify the effects of degradation is unpractical. In the next section we take advantage of our model to introduce a time operator formalism that will enable us to achieve a comprehensive characterization of clock degradation.



§ TIME OPERATORS


 Inspired by the functional form of the Hamiltonian in Eq.(<ref>), we define a continuous clock time operator 

    T̂ = 1/c∫ dx   x  â_x^†â_x ,

where c is the speed of light in the vacuum. It is interesting to note that given any continuous dispersion relation ω(k) for the clock system, we can find an expression for the corresponding (<ref>), as shown by the following

    Given the time operator defined above, and given a continuous dispersion relation ω(k), the commutator with Ĥ_C reads
    
    [T̂,Ĥ_C] = i Λ̂

    where 
    
    Λ̂=1/c∫ dk ω'(k)  â_k^†â_k ,
 
    with ω'(k) = dω(k)/d k. 


    See Appendix <ref>.

The theorem above implies the Heisenberg equation of motion

    Ṫ̂̇_H(t) = i [T̂_H(t),Ĥ_C] = ∫ dk  ω'(k)â_k^†â_k = Λ̂ ,

which is generally far off from the ideal case of (<ref>), and the clock states will in general spread, a phenomenon that causes superpositions of different time states and reduces the resourcefulness of the clock system. Having introduced a time operator, we are in the position of defining an associated quantifier of degradation by means of the variance of the operator T̂:

    Given a time operator T̂ and an initial clock state |ψ_C⟩, the degradation D(t) of the clock is defined as 
    
    D(t) = √(Var(T̂)) = (⟨T̂^2⟩ - ⟨T̂⟩ ^2)^1/2 .


Given a dispersion relation and its corresponding modified Heisenberg algebra [T̂,Ĥ_C]=iΛ̂, we find that the time dependency of D(t) can be extracted: it is linear for any (continuous) choice of dispersion relation ω(k), and proportional to the variance of the operator Λ̂:

Given a dispersion relation ω(k), and its corresponding modified commutator [T̂,Ĥ_C]=iΛ̂, the clock degradation obeys
    
    D(t) = √(Var (Λ̂)) t



    See Appendix <ref>.

The theorem above suggests that it might always be possible to completely eliminate the problem of degradation at all future times by preparing initial clock states such that Var(Λ̂)=0, i.e. eigenstates of Λ̂. Interestingly enough, under fairly general assumptions this possibility is ruled out, unless the dispersion relation is linear, as shown by the following

    Given a single-particle clock state |ψ_C⟩, whose support in k-space is defined by a compact interval Ω∈ℝ with non-zero length, and given a continuous and injective dispersion relation ω(k):Ω→ℝ, one has D(t)=0 ∀ t if and only if ω(k) is linear in the whole support Ω.


    See Appendix <ref>.   

As a corollary to the theorem above, we can consider the limit case in which our initial clock state is extremely well localised in a spatial window of width W. Since good clock states must be sufficiently localised in position, 
they must necessarily be sufficiently delocalised in momentum and therefore the support Ω in the theorem above becomes the whole real line as W→ 0, forcing the dispersion relation to be linear everywhere. 

We can now investigate what happens to (<ref>) in the case of a linear dispersion relation ω(k)=ck. Following the results above one has

    Λ̂=∫ dk  â_k^†â_k=N̂_C ,

where N̂_C counts the excitations of the field. Remarkably, this is the unique case in which we can construct resourceful clock pulses that have Var(Λ̂)=0 but are not eigenstates of the clock Hamiltonian (rendering their resourcefulness as clock states void, due to their stationarity).
The initial state of the clock (<ref>) has a fixed number of excitations N, and furthermore such number of excitations is exactly conserved during the dynamics due to the fact that [N̂_C,Ĥ]=0. Thus, within our assumptions, when starting from a clock state with well-defined particle number N̂_C|ψ_C⟩=N|ψ_C⟩, we can always define a rescaled time operator by introducing the projector ℙ_N onto the N-particles sector 

    τ̂ = 1/Nℙ̂_NT̂ℙ̂_N

such that

    τ̇̂̇ = 1 ,

that corresponds to the original Pauli relation[The rescaling procedure outlined here has a classical counterpart. The angular velocity of the arm is the same in all the classical watches and is achieved by rescaling the tangential speed of the arm's tip with the length of the arm.]. 
This definition allows us to conclude that, as long as the joint clock-engine dynamics does not couple sectors with different number of particles, the Pauli relation can be obeyed exactly if the effective dispersion relation of the clock after being coupled to the engine can be maintained linear. 
In other words, as expected, we found that any wave-packet with a fixed total number of excitations traveling through a medium with linear dispersion law (i.e. any massless wavepacket) behaves as an ideal clock, in the sense of <cit.>.
However, as pointed out in the previous section, this is fundamentally impossible, due to the fact that the interaction with the engine will inevitably correct the bare dispersion relation of the clock, generating a finite effective mass



and thus introducing unavoidable degradation of any clock state that is sufficiently localised in position. This trade-off between degradation and resourcefulness of the clock states is a consequence of the Heisenberg uncertainty relation for the clock's position and momentum, and can then be translated into a lower-bound

    D(t) ≥| ⟨Λ̂⟩ |/2 ΔĤ_C,

where ΔĤ_C is the variance of Ĥ_C, which follows directly from the general Robertson-Schrödinger uncertainty relation associated to the pair of noncommuting observables T̂ and Ĥ_C <cit.>.




Since Λ̂ is a constant of motion either in the presence of inherent nonlinearity of the clock's dispersion relation or of an engine, we can explicitly compute |Λ|. Indeed only the initial state of the clock matters and we have control on that. Let the initial state of the clock, in the frequancy domain, be

    |ψ⟩_0 = ∫ dk ξ̃(k) â^†_k |0⟩ ,

where the ξ̃(k) is the spectral density of the pulse and we are considering a single-particle pulse.
Then we can calculate the expectation value of Λ at any time as

    Λ̂   = 1/c∫ dk dk' dk”ξ̃(k) ξ̃(k”) ω'(k') â_kâ^†_k'â_k'â^†_k”_0
    
       
    = 1/c∫ dk dk' dk”ξ̃(k) ξ̃(k”) ω'(k') δ_k k'δ_k' k”
       = 1/c∫ dk|ξ̃(k)|^2 ω'(k)

Using the chain rule we find

    Λ̂   = 1/c∫ dk|ξ̃(k)|^2 ω'(k) = - 1/c∫ dk d/dk(|ξ̃(k)|^2) ω(k)

where we exploited the fact that the initial state of the clock is defined with a finite frequency bandwidth Ω (ξ̃ decays exponentially elsewhere). This also means that we can rescale k → k - k_0, where k_0 is the central momentum of the pulse, and plug into the previous expression the expansion of ω up to the second order in k.  

    Λ̂   = - 1/c∫ dk d/dk(|ξ̃(k)|^2) (c k +  k^2/2 M)
    
       =  - N   - ∫ dk |ξ̃(k)|^2 k/Mc = - N   -  k/Mc

where N is the number of excitations of the pulse. We put N=1 and plug this result into the uncertainty relation, thus

    ΔT̂ΔĤ_C ≥1/2|Λ̂| = 1/2+  k /M c ,

which introduces a positive finite correction to the ideal minimum uncertainty. This aligns with a scenario where the clock's performance deviates from ideal behavior.








§ CONCLUSIONS


Our study digs into the realm of driven systems, and has revealed the fundamental challenge of achieving the precise time keeping condition in realistic scenarios, which has been shown possible except in idealized and far-from-reality situations. 
Building upon one such ideal scenario, our study relaxes primary approximations, striving to create a more realistic and pragmatic model of control systems. 
The main achievement of our research lies in the development of a comprehensive quantum description of engine-clock dynamics within a truly realistic quantum framework. 
While a connection between the presence of mass and the possibility of keeping time has been explored also experimentally <cit.>, we focused on the characterization of the clock's degradation phenomenon. Our formalism has proven effective in capturing its complexity. Notably, it enables us to establish a clear connection between the degradation of a quantum clock and its mass, even at low energies.
It's worth noting that we make an implicit but reasonable assumption that the markers on the timeline, allowing us to track the clock's motion, are uniformly spaced and considerably wider than the initial pulse width. Consequently, this degradation will inevitably lead to the clock's diminished performance over an extended period. As for the possibility of implementing a protocol involving adaptive adjustments to the time window widths, this falls outside the scope of our current work. It's important to notice that such a procedure would likely necessitate the involvement of an external agent, which contradicts our goal of achieving autonomous evolution for the machine. For the same reasons, we also exclude packets that exhibit acceleration, such as Airy's <cit.>.
Finally we point out that we operated under the assumption of linear susceptibility within the propagation medium, a premise in models similar to the one outlined in <cit.>. However, it is important to note that nonlinear effects are well-documented for their capacity to induce phenomena that can counteract dispersion effects, and in some cases, even lead to their complete suppression. For instance, solitons are a prime example of such nonlinear effects. This perspective opens up intriguing avenues for the exploration of moving-particle clocks within the domain of nonlinear quantum mechanics.




§ ACKNOWLEDGEMENTS






§ PROOFS OF THE STATEMENTS IN SECTION VI


For the sake of clarity, in this appendix we will focus exclusively on the dynamics of the clock setting aside the explicit goal of implementing a particular transformation. Thus we consider V(t) a continuous variable as well as in <cit.>, i.e. we work in the limit of W→ 0.
Therefore the clock Hamiltonian keeps the form in (<ref>) while the total Hamiltonian is turned into






    Ĥ = Ĥ_E⊗1_C + 1_E⊗Ĥ_C + ∫ dx V̂_E(x) ⊗â_x^†â_x .

When considering the time operator

    T̂ = 1/c∫ dx   x â_x^†â_x ,

where c is a reference speed, we are interested in the commutator [T̂_H(t),Ĥ_C], where T̂_H(t) is the Heisenberg picture representation of T̂, which will give us the equation of motion we are looking for. In order to compute the commutator with Ĥ_C, we make use of the following

    
    T̂ = i/clim_ϵ→ 01/ϵ∫ dk( â_k^†â_k+ϵ - â^†_k â_k )



We start by expressing â_x,â_x^† as anti-transforms of â_x,â_x^†, and we make use of the fact that, in the sense of distributions ∫ dx x e^-ikx = 2π i δ'(k) where δ'(x) is such that ∫ dx δ'(x)f(x) = - f'(0). Putting this together we can write

    T̂ = 1/2π∫ dx ∫ dk ∫ dk'   x e^i (k'-k) xk^†k'
        = 1/2π∫ dk ∫ dk' (∫ dx  x e^i (k'-k) x) k^†k'
         =1/2π∫ dk ∫ dk' (-2π i δ'(k'-k) ) k^†k'
       = -i ∫ dk k^†∫ dk' δ'(k'-k) k' = i ∫ dk k^†[ ∂/∂ k'k']_k'=k .
 
Finally, by using the definition of derivative

    ∂/∂ kâ_k  = lim_ϵ→ 01/ϵ( â_kϵ - â_k ) ,

we obtain the result.


    Given the time operator defined above, and given a continuous dispersion relation ω(k), the commutator with Ĥ_C reads
    
    [T̂,Ĥ_C] = i Λ̂

    where Λ̂=1/c∫ dk ω'(k) â_k^†â_k .


The quantity we need to compute is

    [T̂,Ĥ_C] =  i/clim_ϵ→ 01/ϵ∫ dk ∫ dk'  ω_k'[ â_k^†â_k+ϵ - â^†_k â_k ,  â_k'^†â_k'] .

By making use of the relations 

    [k^† a_k+ϵ, a^†_k'k'] = k^†k'δ(k+ϵ-k') - k'^† a_k+ϵδ(k-k')
         [k^†k, a^†_k'k'] = ( k^†k' - k'^†k) δ(k-k') ,

we get

    [T̂,Ĥ_C] =  i/clim_ϵ→ 01/ϵ(∫ dk  ω_k+ϵk^†k+ϵ - ∫ dk ω_kkk+ϵ) 
        = i/clim_ϵ→ 01/ϵ∫ dk   (ω_k+ϵ - ω_k )kk+ϵ .

Now, if ω(k) ≡ω_k is a continuous function of k we can write 

    lim_ϵ→ 0(ω_k+ϵ - ω_k /ϵkk+ϵ) 
        =lim_ϵ→ 0ω_k+ϵ - ω_k /ϵ×lim_ϵ→ 0kk+ϵ = ω'(k)kk ,

and therefore

    [T̂,Ĥ_C] = i/c∫ dk ω'(k)k^†k


From the theorem above, we can easily drawn some initial conclusions, as exemplified in the following two corollaries:

    Given the time operator defined above, and given any continuous dispersion relation 
    ω(k), 
    the resulting modified commutator
    
    [T̂,Ĥ_C] = i Λ̂

    is such that 
    [Λ̂,Ĥ_C]=0, 
    i.e. 
    Λ̂ 
    is a constant of motion. In particular, the Heisenberg picture representation of 
    Λ̂ 
    reads
    
    Λ̂_H(t) = e^iĤ_CtΛ̂ e^-iĤ_Ct = Λ .




    Given the time operator defined above, and given a linear dispersion relation ω(k)=ck, the commutator with Ĥ_C reads
    
    [T̂,Ĥ_C] = i N̂

    where N̂=∫ dk â_k^†â_k is the number operator.


As a consequence, note that the Heisenberg equations of motion for this time operator read

    Ṫ̂̇_H = i [Ĥ_C,T̂_H] = i[Ĥ_C,T̂] = Λ̂ .

We are now in the position to define the concept of a clock's degradation, i.e. the spreading of a clock state under its own Hamiltonian dynamics.


    Given a time operator T̂ and an initial clock state |ψ_C⟩, the degradation D(t) of the clock is defined as 
    
    D(t) = √(Var(T̂)) = (⟨T̂^2⟩ - ⟨T̂⟩ ^2)^1/2 .



Given the modified Heisenberg algebra [T̂,Ĥ_C]=iΛ̂, we can characterize the resulting degradation of a clock state. We find that the time dependency of D_0(t) can be extracted: it is linear for any (continous) choice of dispersion relation ω(k), and proportional to the variance of the operator Λ̂.


Given a dispersion relation ω(k), and its corresponding modified commutator [T̂,Ĥ_C]=i Λ̂, the clock degradation obeys
    
    D(t) = √(Var (Λ̂)) t



    First, we exploit the freedom of writing expectation values in the Schrödinger or Heisenberg picture as follows
    
    ⟨T̂^2⟩ = ⟨ψ_C(t)|T̂^2 |ψ_C(t)⟩ = ⟨ψ_C|T̂^2_H(t) |ψ_C⟩ .

Then, we use the Heisenberg equation of motion

    Ṫ̂̇_H = Λ̂_H(t) = Λ̂

implying the solution T̂_H(t) = Λ̂ t, which we can plug in the expression for D(t) and get

    D_0(t) = √(⟨ψ_c|Λ̂^2 t^2|ψ_c⟩ - ⟨ψ_c|Λ̂ t |ψ_c⟩^2) .

By extracting the time dependency we get the result.


It is interesting to note that, under fairly general assumptions, linear dispersion relations are the only ones that guarantee the existence of non-degrading clock states, as shown by the following


    Given a single-particle clock state |ψ_C⟩, whose support in k-space is defined by a compact interval Ω∈ℝ with non-zero length, and given a continuous and injective dispersion relation ω(k):Ω→ℝ, one has D(t)=0 ∀ t if and only if ω(k) is linear in the whole support Ω


     Let us suppose that D(t)=0 identically. Then Var(Λ̂)=0 on |ψ_C⟩, which means that |ψ_C⟩ is an eigenstate of Λ̂. Furthermore, since ω(k) is an injective function, |ψ_C⟩ cannot be an eigenstate of Ĥ_C if Ω has non-vanishing length. However, since [Λ̂,Ĥ_C]=0, the only possibility is that Λ̂ is degenerate on Ω, i.e. the function ω'(k) is constant on Ω. Therefore ω(k) is linear on Ω.

     Conversely, let us suppose that ω(k) is linear. Then Λ̂ = N̂ and, since N̂|ψ_C⟩=|ψ_C⟩ we have Var(N̂)=0 and therefore D(t)=0.




§ ON THE FORM OF THE INTERACTION


The choice of the window W and the interaction is crucial for the scenario we want to describe. We operate under the assumption that the interaction in (<ref>) does not allow energy exchange between the engine and the clock. Thus apart from translation of the clock states, the pulse may suffer degradation, i.e. pulse broadening in time. The most critical parameter for this timekeeping approach based on the position is then the width of the wave-packet. 

One possibility can be choosing W in order to include all the pulse envelope (W ≫Ω^-1 for the Gaussian pulse (<ref>), as depicted in <ref>)
and

    V̂^I_e(s_i)=
    v̂_i   , for i  even ;
    1 , for i  odd .

This choice guarantees that a specific Hamiltonian v_i acts on the engine at a specified time window, at the price of a phase shift due to the identity. 

The presence of intermittent dead windows naturally introduces the concept of a “period" in our system, similar to mechanical clocks: the interval between two successive ticks corresponds to a phase of the evolution during which the oscillating component disengages from the surrounding mechanism (<ref>).
More precisely, for each Δ t, the potential v_i undergoes modulation proportional to the radiation flux over the ith time window. 
In this scenario the width of the pulse in terms of Δ t is crucial for the speed at which one can induce a certain V̂_e(t) on the engine.






Let us consider W(i+1/2) - n' ∈ [2m W, (2m+1)W], with m ∈ℕ. In this case the pulse encompasses two time windows. According to (<ref>) and considering the action of N̂_α on the state of the clock, the generator reads

    1/W∑_i     V̂^I_e(s_i) ⊗N̂_W(i+1/2) -n'
        =α_n'V̂^I_e(s_2m) + β_n'V̂^I_e(s_2m+1)
        =α_n'v̂_2m + β_n'1 ,

where we define α_n' = 1/WN̂_W(2m+1/2) -n'_c^1/2 and β_n' = 1/WN̂_W(2m+1+1/2) -n'^1/2_c. Thus in this case

    𝒰̂_n'   = exp{-i (α_n'v̂_2m + β_n'1)Δ t}
       = exp{-i α_n'v̂_2mΔ t}exp{-iβ_n'1Δ t}
       = exp{-i α_n'v̂_2mΔ t} e^-iβ_n'Δ t ,

and the global phase factor can be neglected.
At each n' the potential v̂_2m is weighted by the factor α_n' that ranges between 0 (the pulse is crossing another window) and a maximum that depends on the normalisation of the pulse (when the pulse is exactly centered on the 2mth time window).
This scheme allows us to approximate the action of the target time-dependent generator V_e(t) on a mesh of time windows of size W. Nevertheless, a notable limitation is that each generator V_e(s_i) demands an effective time frame of 2W for execution. This, in conjunction with the modulation, imposes significant constraints on the accuracy of the achieved transformation. Notably, as W increases, our discrete series offers a closer approximation to the continuous generator. Such trade-offs, that arise from the need to bridge the gap between continuous and discrete representations, are a common challenge in the realm of approximating continuous-time systems.
Up to this point, we have not directly accounted for any clock degradation process. In the upcoming section, we explore the impact of clock degradation on the wave-packet dynamics, examining how the Hamiltonian in (<ref>) affects also its shape. Understanding the underlying reasons for this degradation is crucial, given our reliance on the position of the particle in our time-keeping protocol.

































