Thermodynamic Cost for Classical Counterdiabatic Driving

TL;DR

This paper investigates the thermodynamic cost of classical counterdiabatic driving, establishing a universal trade-off between speed and cost, and introduces new cost definitions relevant when excess work vanishes, with applications to harmonic oscillators.

Contribution

It provides a new complementary definition of thermodynamic cost for classical systems and demonstrates the classical counterpart of quantum results on counterdiabatic driving.

Findings

A universal speed-cost trade-off for classical counterdiabatic driving.

New cost measure relevant when average excess work is zero.

Application to a time-dependent harmonic oscillator.

Abstract

Motivated by the recent growing interest about the thermodynamic cost of Shortcuts to Adiabaticity (STA), we consider the cost of driving a classical system by the so-called Counterdiabatic Driving (CD). To do so, we proceed in three steps: first we review a general definition recently put forward in the literature for the thermodynamic cost of driving a Hamiltonian system; then we provide a new complementary definition of cost, which is of particular relevance for cases where the average excess work vanishes; finally, we apply our general framework to the case of CD. Interestingly, we find that in such case our results are the exact classical counterparts of those reported in Funo et al. PRL, 118(10):100602, 2017. In particular we show that a universal trade-off between speed and cost for CD also exists in the classical case. To illustrate our points we consider the example of a time-dependent harmonic oscillator subject to different strategies of adiabatic control.

Tables

Table 1. Table 1 : Comparison between the thermodynamic costs based on the first two moments of average excess work for the protocols ( 34 ) and ( 35 ).

Protocol
${\omega }_{\mathrm{lin}}^2$	$1.27\times {10}^{-4}$	$2.75\times {10}^{-2}$
${\omega }_{\sin }^2$	$4.78\times {10}^{-4}$	$5.84\times {10}^{-2}$

Equations

$$\mathbf{X}:=\frac{\partial H_0}{\partial \mathbf{\lambda }}\mathrm{and}\Delta \mathbf{X}:=\mathbf{X}-{⟨\mathbf{X}⟩}_{\lambda }.$$

$${⟨\mathcal{P}⟩}_{\mathbf{\lambda }}:=\dot{\mathbf{\lambda }}\cdot {⟨\mathbf{X}⟩}_{\mathbf{\lambda }}\mathrm{and}{⟨{\mathcal{P}}_{\mathrm{ex}}⟩}_{\mathbf{\Lambda }}:=\dot{\mathbf{\lambda }}\cdot {⟨\Delta \mathbf{X}⟩}_{\mathbf{\Lambda }},$$

$${⟨W_{\mathrm{ex}}(t)⟩}_{\mathbf{\Lambda }}$$

$${⟨W_{\mathrm{ex}}⟩}_{\tau }^{(1)}:={\tau }^{-1}{\int }_0^{\tau }{⟨W_{\mathrm{ex}}(t)⟩}_{\mathbf{\Lambda }}dt.$$

$${⟨{{\mathcal{P}}_{\mathrm{ex}}}^2⟩}_{\mathbf{\Lambda }}={\dot{\lambda }}^i{\dot{\lambda }}^j{⟨\Delta X_i\Delta X_j⟩}_{\mathbf{\Lambda }}$$

$${⟨W_{\mathrm{ex}}^2(t)⟩}_{\mathbf{\Lambda }}$$

$${⟨W_{\mathrm{ex}}⟩}_{\tau }^{(2)}:={\tau }^{-1}{\int }_0^{\tau }\sqrt{{⟨W_{\mathrm{ex}}^2(t)⟩}_{\mathbf{\Lambda }}}dt.$$

$$\Omega (E,\mathbf{\lambda }):=\int d\mathbf{z}\Theta [E(\mathbf{\lambda })-H_0(\mathbf{z},\mathbf{\lambda })]$$

$$H_1(z,\mathbf{\lambda })=\dot{\mathbf{\lambda }}\cdot \mathbf{\xi }={\dot{\lambda }}^i{\xi }_i(z,\mathbf{\lambda }),$$

$$\delta z=\delta \mathbf{\lambda }\cdot \{z,\mathbf{\xi }\}.$$

$$⟨\dots {⟩}_{E,\mathbf{\lambda }}:=\frac{1}{{\partial }_E\Omega }\int dz\delta (E-H_0)\dots ,$$

$${⟨\mathbf{\xi }⟩}_{E,\mathbf{\lambda }}=0,$$

$${⟨f(z)⟩}_{\mathbf{\Lambda }}:=\int d{z}_0\rho (z_0){⟨f(z)⟩}_{E,\mathbf{\lambda }}.$$

$${⟨W_{\mathrm{ex}}(t)⟩}_{\mathbf{\Lambda }}:={⟨W(t)-W_{\mathrm{ad}}(t)⟩}_{\mathbf{\Lambda }},$$

$$W(t)=H_{\mathrm{CD}}(z_t,{\mathbf{\lambda }}_t)-H_0(z_0,{\mathbf{\lambda }}_0)$$

$$W_{\mathrm{ad}}(t)=H_0(z_t,{\mathbf{\lambda }}_t)-H_0(z_0,{\mathbf{\lambda }}_0)$$

$${⟨W_{\mathrm{ex}}(t)⟩}_{\mathbf{\Lambda }}={⟨H_1(z_t,{\mathbf{\lambda }}_t)⟩}_{\mathbf{\Lambda }}={\dot{\lambda }}^i{⟨{\xi }_i⟩}_{\mathbf{\Lambda }}.$$

$${⟨W_{\mathrm{ex}}(t)⟩}_{\mathbf{\Lambda }}$$

$${⟨W_{\mathrm{ex}}^2(t)⟩}_{\mathbf{\Lambda }}$$

$$\delta (\Delta W{)}^2:=\mathrm{Var}[W(t)]-\mathrm{Var}[W_{\mathrm{ad}}(t)].$$

$$\delta (\Delta W{)}^2$$

$$={⟨H_{\mathrm{CD}}(z_t,{\mathbf{\lambda }}_t{)}^2-H_0(z_t,{\mathbf{\lambda }}_t{)}^2⟩}_{\mathbf{\Lambda }}$$

$$={⟨H_1(z_t,{\mathbf{\lambda }}_t{)}^2⟩}_{\mathbf{\Lambda }}$$

$$={\dot{\lambda }}^i{\dot{\lambda }}^j{⟨{\xi }_i{\xi }_j⟩}_{\mathbf{\Lambda }},$$

$$\delta (\Delta W{)}^2=\int d{z}_0\rho (z_0){⟨{\xi }_i{\xi }_j⟩}_{E,\mathbf{\lambda }}{\dot{\lambda }}^i{\dot{\lambda }}^j,$$

$$g_{ij}^E:={⟨{\xi }_i{\xi }_j⟩}_{E,\mathbf{\lambda }}$$

$$g_{ij}^{(n)}=⟨n(t)|{\hat{\xi }}_i^{†}{\hat{\xi }}_j|n(t)⟩.$$

$$l(\rho (z_0),\rho (z_{\tau })):={\int }_0^{\tau }\sqrt{\delta (\Delta W{)}^2}dt.$$

$${⟨W_{\mathrm{ex}}⟩}_{\tau }^{(2)}=\frac{1}{\tau }l(\rho (z_0),\rho (z_{\tau }))={⟨\delta \Delta W⟩}_{\tau },$$

$$\delta (\Delta W{)}^2\le ⟨H_{\mathrm{CD}}^2⟩-{⟨H_{\mathrm{CD}}⟩}^2=\mathrm{Var}[H_{\mathrm{CD}}]$$

$$\tau \ge \frac{l(\rho (z_0),\rho (z_{\tau }))}{{⟨\mathrm{Std}[{\mathrm{H}}_{\mathrm{CD}}]⟩}_{\tau }},$$

$$H_0(p,q,\omega (t))=\frac{p^2}{2}+{\omega }^2(t)\frac{q^2}{2}.$$

Full text

I Introduction

Shortcuts to Adiabaticity (STA) is the design of nonadiabatic processes that reproduce in a finite time the same final state that would result from an adiabatic, infinitely slow, protocol. Due to their importance in developing techniques for nanoengineering, such strategies have called the attention of the scientific community during the last years both at the theoretical and at the experimental level Demirplak and Rice (2003); Berry (2009); Ibáñez et al. (2012); Jarzynski (2013); del Campo (2013); Torrontegui et al. (2013); Couvert et al. (2008); Chen et al. (2010); Schaff et al. (2011); Bason et al. (2012); An et al. (2016); Martínez et al. (2016). Although the main practical motivation for STA is related to their quantum applications, there is a growing interest in understanding their classical counterparts, both because the classical scheme can be useful for new quantum strategies and because STA can be exploited for the design of protocols that can speed up the convergence in Jarzynski’s equality Jarzynski (2013); Deng et al. (2013); Deffner et al. (2014); Jarzynski et al. (2017); Okuyama and Takahashi (2017); Acconcia and Bonança (2015); Acconcia et al. (2015).

One of the most noteworthy techniques for STA is Counterdiabatic Driving (CD). In CD one uses an auxiliary Hamiltonian appropriately tailored so that the dynamics generated by the original Hamiltonian plus the auxiliary term preserves exactly the adiabatic invariant of the system del Campo (2013); Jarzynski (2013); Deng et al. (2013); Deffner et al. (2014); Jarzynski et al. (2017); Okuyama and Takahashi (2017); Sels and Polkovnikov (2017). Moreover, since the auxiliary term vanishes at the beginning and at the end of the CD, one can show that the work distribution at the end of the protocol is the same as that of the bare adiabatic process Deng et al. (2013). Therefore it would seem that such driving effectively boosts the adiabatic dynamics at a finite rate without any extra cost and that the duration of the CD process can be pushed to zero with no a priori bound Del Campo et al. (2014); Beau et al. (2016), besides those set by quantum mechanics Bhattacharyya (1983). This seems to be suspicious from a thermodynamic perspective. Indeed, recently there has been a surge of interest in trying to quantify the real cost of driving systems by CD and more in general by STA Zheng et al. (2016); Deffner and Campbell (2017); Funo et al. (2017); Campbell and Deffner (2017); Abah and Lutz (2016); Torrontegui et al. (2017). In particular, in Funo et al. (2017) a sensible definition of the thermodynamic cost for CD in the quantum case has been given, based on a universal trade-off between the speed and the excess of work fluctuations at any intermediate time during the protocol, and it has been shown that such cost can be further related to the geometric structure of the Hilbert space.

In this work we revisit such results for classical systems and we set them into the more general perspective of the thermodynamic cost for driving a Hamiltonian system out of equilibrium put forward in Sivak and Crooks (2012). Remarkably, we obtain that the most natural extension of the thermodynamic cost introduced in Sivak and Crooks (2012) when applied to CD provides the exact classical counterpart of the analysis in Funo et al. (2017), including the existence of a universal bound on the speed of CD. Moreover, as a by-product we obtain the classical equivalent of the geometric tensor proposed by Provost and Vallee Funo et al. (2017); Provost and Vallee (1980). We conclude with two examples, the first one illustrating the bound for CD and the second one focusing on the importance of considering the excess of work fluctuations also for STA different from CD.

II Thermodynamic cost of driving

In this section we briefly review the definition of the thermodynamic cost for driving a Hamiltonian system out of equilibrium given in Sivak and Crooks (2012) (see also Zulkowski et al. (2012, 2013); Sivak and Crooks (2016)) in terms of the average excess work and then we extend it to consider fluctuations of the excess work, in order to be able to deal with cases where the average excess work vanishes, as it happens e.g. in CD.

Let us consider a driven Hamiltonian system with Hamiltonian $H_{0}({\bm{z}},{\bm{\lambda}}(t))$, where ${\bm{z}}=({\bm{p}},{\bm{q}})$ denotes a point in phase space and ${\bm{\lambda}}(t)$ represents a set of time-dependent parameters that can be externally moved according to a predetermined protocol. One can define the conjugate forces and their deviations from the equilibrium values as

[TABLE]

Here the notation is the same as in Sivak and Crooks (2012) (up to a sign in the first definition) and $\left<\ldots\right>_{{\bm{\lambda}}}$ stands for the equilibrium average at fixed values of the control parameters ${\bm{\lambda}}(t)$. Accordingly, the (instantaneous) average power and the average power excess are

[TABLE]

where the subscript $\bf\Lambda$ denotes the average over the ensemble following the dynamics. Using the standard definition of work Jarzynski (1997), the average excess work can be calculated to be Sivak and Crooks (2012)

[TABLE]

and thus

[TABLE]

represents a good general definition for the thermodynamic cost of performing the nonequilibrium protocol on the system.

However, differently from Sivak and Crooks (2012), here we want to consider not only the possibility of arbitrarily moving the parameters ${\bm{\lambda}}$, but also that of using auxiliary fields to control the evolution of the system over a predetermined path, as it happens in the case of STA. As we will see, in such case the average excess work vanishes. Therefore it is important to generalize the above construction and define a thermodynamic cost which applies to the case of vanishing average excess work. This is the task of the remainder of this section. The most natural generalization of the cost (3) is provided by considering the fluctuations of the excess work. From the definitions above it follows that

[TABLE]

and accordingly one can compute

[TABLE]

together with the associated thermodynamic cost

[TABLE]

This definition is completely general and applies to any system for which the combined action of the driving protocol and of the external fields makes the average excess work vanish at any moment of time. In the following we consider one such case, i.e. CD.

III The Cost of CD

Let us now apply the above general discussion to the case of STA and in particular to CD. The strategy of CD is to add an auxiliary term $H_{1}({\bm{z}},{\bm{\lambda}}(t))$ to the bare Hamiltonian of the system, in such a way that the dynamics under the total Hamiltonian function $H_{\rm{CD}}=H_{0}+H_{1}$ preserves the adiabatic invariant

[TABLE]

exactly for all $t\in[0,\tau]$, with $\tau$ being the duration of the protocol Jarzynski (2013); Okuyama and Takahashi (2017). From now on we consider only systems with one degree of freedom, as it is usual in the analysis of classical STA. In such case, it has been shown that $H_{1}(z,{\bm{\lambda}})$ can be written as

[TABLE]

where $\bm{\xi}$ is the generator that converts displacements in the space of parameters ${\bm{\lambda}}\rightarrow{\bm{\lambda}}+\delta{\bm{\lambda}}$ into displacements in the phase space $z\rightarrow z+\delta z$ according to Jarzynski (2013); Deffner et al. (2014); Okuyama and Takahashi (2017)

[TABLE]

The invariance of (8) may be restated in terms of the adiabatic energy shell $\mathcal{E}(t):=\{z|H_{0}(z,{\bm{\lambda}}(t))=E(t)\}$ as follows: the CD guarantees that points starting on the same energy shell remain on the corresponding adiabatic energy shell during the evolution Jarzynski et al. (2017). Considering this fact, it is natural to introduce a microcanonical measure associated with the dynamics under the CD control Jarzynski (2013)

[TABLE]

where $E$ is the instantaneous energy of the adiabatic energy shell $\mathcal{E}(t)$. It follows that Jarzynski (2013)

[TABLE]

and that, for any initial distribution of points over the phase space $\rho(z_{0})$ that evolve under $H_{\rm{CD}}$, the mean value of an observable $f(z)$ at time $t$ is given by

[TABLE]

Using (13), a natural definition of the instantaneous average of excess work is

[TABLE]

where

[TABLE]

is the work done by the CD up to time $t$ and

[TABLE]

is the corresponding work done by the bare adiabatic protocol. Note that in (14) we use the same notation as in (3) because they coincide, as we now show. From the definitions (14)–(16), one immediately arrives at

[TABLE]

On the other side, starting from (3) we have

[TABLE]

where in the second equality we used (12) and $\left<\partial H_{0}/\partial{\bm{\lambda}}\right>_{{\bm{\lambda}}}=\left<\partial H_{0}/\partial{\bm{\lambda}}\right>_{\bf\Lambda}$, which follows from observing that the dynamics with the CD forces each point to follow the equilibrium trajectory of the bare system.

Clearly, from (12) and (13), the above equations imply that on average the CD does no extra work with respect to the bare adiabatic protocol, as in the quantum case Funo et al. (2017).

Since the average excess work vanishes, to study the cost of CD we now focus on the excess of work fluctuations according to (6). Repeating the steps in (18), we arrive at the expression

[TABLE]

This increase in the work fluctuations is the price to be paid in the case of CD and it has a thermodynamic cost quantified by (7). Indeed, in the following we show that this definition of the thermodynamic cost is exactly the classical counterpart of the one proposed in Funo et al. (2017).

Following Funo et al. (2017), we now define the difference in the instantaneous work fluctuations as

[TABLE]

where ‘Var’ stands for the variance. Expanding each term on the right hand side and then using (17), we arrive at the following result

[TABLE]

that is, the instantaneous modifications in the work fluctuations due to the CD coincide with the second moment of the excess work (cf. eq. (19)). Moreover, from (21) and (13), we obtain

[TABLE]

and from comparison with Funo et al. (2017) we conclude that

[TABLE]

is the classical equivalent of the quantum geometric tensor

[TABLE]

This metric induces a natural distance between the initial and final statistical states of the system under the CD dynamics, $\rho(z_{0})$ and $\rho(z_{\tau})$, given by

[TABLE]

Notice that such distance is well defined because $\delta(\Delta W)^{2}\geq 0$.

IV A classical Speed Limit for CD

In this section we use the results in the previous section to show that there exists a universal trade-off between speed and cost for CD in the classical case.

From (7), (19), (21) and (25) it follows that

[TABLE]

where $\delta\Delta W:=\sqrt{\delta(\Delta W)^{2}}$. Therefore we have expressed the thermodynamic cost (7) in terms of the statistical length between the initial and final distribution in phase space. Furthermore, considering that $\delta(\Delta W)^{2}=\left<H_{\rm CD}^{2}\right>_{\bf\Lambda}-\left<H_{0}^{2}\right>_{\bf\Lambda}$ (cf. equation (21)), and noting that $\left<H_{0}^{2}\right>_{\bf\Lambda}\geq\left<H_{0}\right>_{\bf\Lambda}^{2}=\left<H_{\rm CD}\right>_{\bf\Lambda}^{2}$, we obtain the following inequality

[TABLE]

and hence, using (25), we get

[TABLE]

where ‘Std’ is the standard deviation. This result provides the same bound on the duration of the process as the one obtained in Funo et al. (2017) (although in Funo et al. (2017) it is further reduced with the use of the Bures distance in the Hilbert space). The remarkable point about our derivation of such relation is the fact that we showed explicitly that it does not depend on any quantum property. We conclude that this bound depends solely on the nature of the counterdiabatic control and it sets a limit on the duration of a CD exactly in the same way as its quantum counterpart.

V Examples

In this section we illustrate the above results using a paradigmatic example, namely the driven harmonic oscillator, with bare Hamiltonian

[TABLE]

We analyze first the CD, showing that the bound (28) forbids drivings at any speed, and then we consider a different type of driving, which is not CD but still realizes a STA for this system. We use this latter case to argue for the need of taking into account the cost provided by the average excess of work fluctuations also in cases different from CD.

V.1 With CD

The auxiliary Hamiltonian (9) in this case reads Deffner et al. (2014)

[TABLE]

For a general protocol $\omega(t)$ we have that the adiabatic energy shell is described by the ellipse

[TABLE]

where in the last equality we used the adiabatic invariant for the harmonic oscillator Goldstein et al. (2014). In this case there is only one parameter $\omega$ and thus there is only one metric component in (23), which can be calculated at any time $t$ using the energy shell (31), to obtain

[TABLE]

Here $\omega:=\omega(t)$, $X:=1-{1}/{\omega^{2}}$, $F\left(X\right)$ and $K\left(X\right)$ are the complete elliptic integral of the second and first kind respectively, and we are using $\omega_{0}=1$ throughout the rest of the paper. Considering an initial canonical distribution with $\beta=1$, together with the protocol

[TABLE]

(cf. Funo et al. (2017)), where $s:=t/\tau$, we can compute numerically the instantaneous excess of work fluctuations (22) and the thermodynamic cost (26) for different values of $\tau$. The results are displayed in Fig. 1.

As expected, the shorter the protocol the larger the instantaneous excess of work fluctuations; this affects the thermodynamic cost of the CD, which diverges in the limit $\tau\rightarrow 0$, in agreement with (28).

V.2 With STA from linear response theory

In Acconcia and Bonança (2015), the authors found a whole class of degenerate protocols which realize a STA for the harmonic oscillator (29) under weak driving, obtained from linear response theory (see also Acconcia et al. (2015)). The relevance of such protocols here is the fact that, unlike CD, they have $\left<W_{\rm ex}(t)\right>_{\bf\Lambda}\neq 0$ for $0<t<\tau$, while they all have $\left<W_{\rm ex}(\tau)\right>_{\bf\Lambda}=0$ for appropriate values of $\tau$. Thus we can use these examples to illustrate the importance of comparing the excess of work fluctuations during the protocol with the mean excess work, by showing that there are further cases (beside CD) in which the former term cannot be neglected. We remark that this is something one would intuitively expect for small systems. We consider here in particular two protocols: the linear one

[TABLE]

and the sine protocol

[TABLE]

(cf. Acconcia and Bonança (2015)). Taking a canonical initial distribution with $\beta=1$ as in the CD case (cf. Sec. V.1) and using values of $\tau$ for which such protocols effectively provide a STA ($\tau=\pi$ for the linear case and $\tau\approx 1.2$ for the sine protocol), we can compute numerically both the thermodynamic cost stemming from the average excess work at any intermediate time – equation (4) – and its analogue stemming from the excess of work fluctuations, equation (7). In Table 1 we show the results. In both cases $\left<W_{\rm ex}\right>_{\tau}^{(2)}$ is greater than $\left<W_{\rm ex}\right>_{\tau}^{(1)}$ by two orders of magnitude. Therefore we consider that estimates of the thermodynamic cost of a protocol based on the average work excess alone cannot be fully precise in such cases.

This aspect has been somehow overlooked in the literature and further research in this direction is required in all cases where similar controls are proposed.

VI Conclusions

In this work we have formulated a general framework to analyze the thermodynamic cost of driving a Hamiltonian system by using the combined action of moving its parameters and possibly also adding extra fields. Interestingly, in the particular case of CD our results provide the classical counterpart of the findings in Funo et al. (2017). Besides, the metric (23) is completely new in the classical case (see also the related discussion in Kolodrubetz et al. (2013)). For instance, we remark that this tensor is different from the one proposed in Sivak and Crooks (2012), because there the metric quantifies the average excess power, which vanishes in the case of CD. In addition, we showed that there exists a speed limit (28) for CD stemming from (23), which provides a bound on the speed at which CD can be performed, exactly in the same way as in the quantum case. However here the analysis is completely classical. Based on this argument, we argued that (23) is a relevant object for the analysis of the thermodynamic cost of CD, and that more in general one should consider both the cost based on the average excess work and that based on the average excess of work fluctuations according to the definitions (4) and (7) above. Although we worked here only with systems with one degree of freedom and the extension of all the results to interacting many-body systems is not direct (see e.g. the discussion in Weinberg et al. (2017)), we expect that these tools will provide useful complementary information for assessing the thermodynamic cost of general driving protocols for classical systems.

VII Acknowledgements

The authors would like to thank A del Campo, K Funo, and M Ueda for their valuable comments. AB is funded by a DGAPA–UNAM postdoctoral fellowship. DT acknowledges financial support from Conacyt (CVU 442828).