Vanishing efficiency of a speeded-up ion-in-Paul-trap Otto engine^(a)

One of the challenges to realize quantum technologies and devices that outperform classical counterparts, is to achieve an exhaustive control over the state and dynamics of quantum systems. To this effect, shortcuts to adiabaticity (STA) [1,2] stand as a useful toolbox, mimicking the final result of a slow adiabatic evolution in shorter times, which avoids the drawbacks of long-time processes, such as decoherence. An open question is to determine their energy cost.

As devices based on quantum properties, such as quantum computers, engines and refrigerators are being proposed, it is important to understand their energy flows and costs. A useful model for an engine, exactly solvable but complex enough to represent friction and heat leaks [3], considers a quantum harmonic oscillator as the working medium of an Otto engine [4]. During the cycle, the oscillator undergoes two power strokes between two frequencies and two thermalizations. Compared to the macroscopic Otto engine, the frequency plays the role of an inverse volume, and the harmonic potential the role of the piston [3]. This quantum engine could be implemented by an ion (primary system) in a Paul trap (control system), our model hereafter. Analyzing specific elementary operations is important to reach or test more general conclusions. In particular, expansions and compressions, apart from being strokes of the Otto engine, have been key to develop STA in theory [5] and experiments [6,7]. While the results of our calculations are model dependent, they also give hints on what to expect in a broad domain of systems as discussed in the last section.

In a standard analysis the (angular) frequency $\omega(t)$ is assumed to be classical, with definite values, whereas the particle is treated quantally. This hybrid classical-quantum scenario is justified by the different scales involved and is a general feature, well discussed in foundational work on quantum mechanics [8], when driving microscopic systems. A related and widespread feature, again a consequence of different scales, is the negligible effect of the particle on the classical control, whereas the classical control determines the quantum dynamics. As a consequence, we can design useful STA protocols that are independent of the particle state and its dynamics.

The basic performance criteria of a thermal device are power output and efficiency. Any thermal device that operates in finite time incurs in rate-dependent losses that diminish efficiency, as opposite to a device operating reversibly with no power output [9]. At the microscopic level, increasing the rate of a given transformation usually increases quantum friction [10,11], i.e., undesired excitations at final time which imply a waste of energy. STA, however, suppress quantum friction in the power strokes [12–14]. It may therefore seem that STA enhance the power output arbitrarily without affecting efficiency, enabling a "perpetual-motion machine of the third kind" [15]. While there is widespread agreement that some kind of "cost" inherent in the STA process precludes these machines, many different "costs" have been put forward [3,13,14,16–25], which are not necessarily in conflict if regarded as different aspects of the system energy or its interactions [2].

Here, we refine the viewpoint originally proposed in [26] with more accurate work definitions, perfecting as well the model in [27] which is applied to a new set of operations in Paul traps (in [27] we only considered transport and the Hamiltonian was not as detailed as here). We identify the cost with the exclusive work for the total system, which is essentially the energy consumption to set the driving protocol, i.e., the classical parameters of the Hamiltonian for the primary system. Several examples dealing with STA demonstrate that this perspective is crucial to reach sensible conclusions [26,27]. Beyond STA see, e.g., [15,28,29] as examples of the need to account for all the energy flows.

This work focuses on the compression/expansion strokes of the quantum harmonic Otto cycle, without explicitly modeling the heat baths nor their interaction with the working medium. A comprehensive review of the quantum harmonic Otto cycle may be found in [3], see, e.g., [30,31] for some recent results.

We point out two factors that lead to different definitions for the work done or required by a transformation of a microscopic system. The first one corresponds to the definition of work in externally driven systems [32]. Suppose a simple setting described by the Hamiltonian $H(y,t) = H_0(y)+\lambda(t)y$ with externally driving potential $\lambda(t)y$ along some coordinate y. The exclusive work definition evaluates only the change of the internal energy defined by the "unperturbed" H₀. It considers this difference as work injected to the unperturbed system by the action of $\lambda(t)$ [33] and corresponds to the standard expression of force times displacement. Instead, an inclusive definition evaluates work as the change of the energy represented by the total Hamiltonian, including the external influence [34].

The second factor corresponds to the role of the control. Gibbs already stated that the force that induces a given transformation on a system is often affected by the configuration of an external body [35], here the "control system". The energy needed to manipulate this body should be taken into account when discussing efficiencies, and more so in the context of quantum technologies with a macroscopic control system and a microscopic "primary system".

Let us combine these factors to propose two useful definitions: Total work is the exclusive total energy consumed by the control plus primary system; Microscopic work is an inclusive definition for the primary system that disregards the energy cost to set the control parameters. The microscopic work is the definition found in most studies on quantum thermal machines. It is not invariant under gauge transformations that shift in time the zero energy point [32], and thus, according to Cohen-Tannoudji et al., it is not a physical quantity [36]. The gauge, however, may be fixed by the experimental setting to make energy differences physically meaningful [32]. Using the proper gauge the microscopic work may constitute an indicator of the total work if it is proportional to it. This proportionality is not guaranteed, but it indeed occurs, as shown below, for a specific regime in our model.

We consider a one-dimensional quantum harmonic Otto engine whose working medium is a single ion of mass m and electric charge Q. The Hamiltonian for the working medium in the absence of the longitudinal harmonic potential reads $H_{S,0}(x)=\frac{p^2}{2m}$, where p is the momentum of the ion. The ion is trapped in a harmonic potential generated by a segmented linear Paul trap, the control system. Following [27], we model it as a circuit formed by a controllable power source that generates an electromotive force (efm) $\mathcal{E}(t)$ and a low-pass electronic filter formed by a resistor (resistance R) and a capacitor (capacitance C), with Hamiltonian [37] $H_C(q,t) = q^2 / 2C - \mathcal{E}(t) q$, where q is the charge in the capacitor. Through the modified Hamilton equation $\dot p_q = - \frac{\partial H_C}{\partial q} - \frac{\partial \mathcal{F}}{\partial \dot q}$, which accounts for friction through Rayleigh's dissipation function [37] $\mathcal{F} = R \dot q^2 /2$, we get the dynamics of the control system,

$$\begin{equation} \mathcal{E} (t)= \frac q C + R \dot q. \end{equation} \tag{ 1 }$$

The interaction between the control and the ion is $H_{SC}(x,q,t) = Q \phi(x) q / C$, where x is the coordinate of the ion and $\phi(x)$ is a (dimensionless) electrostatic potential that depends on the geometry of the trap [38] given by $\phi(x)=a e^{-x^2/2b^2}$ with a and b constant [27]. The potential is approximately harmonic near the origin with angular frequency $\omega(t)$ determined by $\partial^2 H_{SC} / \partial x^2 |_{x=0}= m \omega^2(t)$ and related to q by

$$\begin{equation} q = - \frac {b^2 m C} { a Q } \omega^2(t). \end{equation} \tag{ 2 }$$

Upon Taylor expansion around x = 0, the interaction becomes $H_{SC} = m \omega^2(t)x^2 / 2 - b^2 m \omega^2(t)$. The last term, usually ignored since it does not affect the ion dynamics, fixes the gauge, see fig. 1.

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The global Hamiltonian is quadratic,

$$\begin{equation} H(x,q,t)=\frac{p^2}{2m} + \frac m 2 \omega^2(t)x^2 - b^2 m \omega^2(t) + \frac 1 {2C} q^2 - \mathcal{E}(t) q. \end{equation} \tag{ 3 }$$

It governs the evolution of two interacting degrees of freedom, one of them macroscopic and classical, the charge in the capacitor, and the other microscopic and quantum. The overwhelming difference in scale enables us to make a clear separation and treat the dynamics of the capacitor as effectively independent of the ion dynamics, whereas the dynamics of the quantum system is governed by $H_S=H_{S,0}+H_{SC}$, where $\omega(t)$ is treated as an external parameter, whose evolution is designed in what follows using invariant-based inverse engineering.

This STA technique rests on the parameterization of a quadratic invariant in terms of a scaling factor $\rho(t)$ that determines the state width and satisfies the Ermakov equation [1,5]. The evolution of the control parameter is computed from the Ermakov equation as $\omega^2(t)= \omega_0^2 / \rho^4 - \ddot{\rho} / \rho$, where the dots represent time derivatives, and $\rho(t)$ is designed to satisfy the boundary conditions $\rho(0)=1$, $\rho(t_f)= (\omega_0 / \omega_f)^{1/2}$, $\dot{\rho}(t_b)=0$ and $\ddot{\rho}(t_b)=0$, with $t_b=0,t_f$, so that initial eigenstates of H_S(0) evolve according to

$$\begin{equation} \psi_n (x,t) = e^{\frac{i m \dot{\rho}}{2 \hbar \rho} x^2} \frac1{\sqrt{\rho}} \phi_n\left(\frac x {\rho} \right)\!, \end{equation} \tag{ 4 }$$

and become at t_f eigenstates of H_S(t_f). To interpolate we use the ansatz $\rho(t)=\sum_{i=0}^5 \rho_i (t/t_f)^i$, with the $\rho_i$ fixed by the boundary conditions. We will also use the notation $\omega_0 \equiv \omega(0)$ and $\omega_f \equiv \omega(t_f)$.

The expectation value of H_S(t) for the dynamical modes (4) is

$$\begin{equation} \epsilon_n (t)= \frac{(2n + 1) \hbar}{4 \omega_0} \left( \dot{\rho}^2 + \omega^2(t) \rho^2 + \frac{\omega_0^2}{\rho^2} \right) - b^2 m \omega^2(t), \end{equation} \tag{ 5 }$$

see fig. 1, and compare the insets that depict $\widetilde{\epsilon}_0=\epsilon_0+ b^2 m \omega^2(t)$ and $\epsilon_0$. The gauge term dominates the evolution of $\epsilon_n(t)$ over the vibrational energy of the ion and, moreover, increases the energy of the ion during expansions and decreases it during compressions.

In our inverse engineering protocol, once the desired $\omega(t)$ has been set, the dynamics of the capacitor charge q and the electromotive force $\mathcal{E}$ that we have to implement are found from eqs. (2) and (1).

Work is commonly computed by integrating over time the instantaneous power. The total work considers the evolution of the exclusive instantaneous power $\mathcal P$ of the composite unperturbed system driven by $H_0=H+ \mathcal{E}(t) q$. To calculate this total power including the effect of the primary system (backaction) we consider first a fully classical approximation. Later we shall substitute the variables dealing with the quadratic microscopic system by quantum expectation values. For systems described by a Rayleigh dissipation function, the modified Hamilton equations imply that the (exclusive) total power contributes to change the energy of the unperturbed system and to overcome the dissipation, see the appendix. In our model,

$$\begin{equation} \mathcal P = \mathcal{E}(t) \dot{q}= \mathrm{d}H_0/\mathrm{d}t + R \dot q^2. \end{equation} \tag{ 6 }$$

We can separate it as $\mathcal P = \mathcal P_C + \mathcal P_S$, the power needed to generate the dynamics on the Paul trap (without the ion) and the power to overcome the backaction by the ion,

$$\begin{eqnarray} \mathcal P_C &= \frac{\partial H_{0,C}}{\partial t} + R \dot q^2 = (R \dot q + q/C ) \dot q, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \mathcal P_S &= \frac{\partial H_S}{\partial t} = \left(1- \frac {x^2} {2 b^2}\right)\frac{aQ}C \dot q, \end{eqnarray} \tag{ 8 }$$

where $H_{0,C}=q^2 / 2C$. In $\mathcal P_S$ the first term is due to the gauge term. As $q\gg Q$ we expect $|\mathcal P_S| \ll |\mathcal P_C|$, and $\mathcal P \approx \mathcal P_C = \mathcal E(t) \dot q$, see eq. (1), which holds in all calculations. In fact the dominance of $\mathcal P_C$ is needed to set state-independent shortcut protocols.

The possibility to "regenerate" (store and reuse) the energy that flows out of the total system during negative-power time segments is accounted for by a factor $-1\le \mu \le1$ [26] multiplying the negative power in the total integrated work,

$$\begin{equation} W_T = \int_0^t \mathcal P_{C_+} \mathrm{d}t' + \mu \int_0^t \mathcal P_{C_-} \mathrm{d}t'. \end{equation} \tag{ 9 }$$

Here $\mathcal P_{C_{\pm}} = \Theta(\pm \mathcal P_C) \mathcal P_C$, and Θ is the Heaviside function. In the Paul trap both signs need consumption, $\mu = -1$.

The backaction term is in fact the inclusive microscopic work. The quantum version of eq. (8) takes the same form with expectation values, $H_S\to\langle H_S\rangle$, $x^2\to\langle x^2\rangle$. Defining work at the quantum level is not straightforward [39] and, furthermore, the relation between the system energy and inclusive work holds only if the gauge is appropriately fixed according to the experiment [32,40]. We evaluate the microscopic work for the shortcut process as the difference between final and initial energies determined by H_S. The contribution from each mode is $\langle W_m \rangle = \sum_n p_n^0 \int_0^{t_f} \langle \mathcal P_S \rangle_n \mathrm{d}t$, where $\langle \mathcal P_S \rangle_n = \mathrm{d}\epsilon_n/\mathrm{d}t=(2n+1)\hbar \rho^2\omega\dot\omega/(2\omega_0) - 2 b^2 m \omega \dot \omega$. Assuming an initial thermal distribution with $p_n^0 = e^{-\beta \epsilon_n(0)}/Z$, inverse temperature β, and $Z=\sum_n e^{-\beta \epsilon_n(0)}$, we get

$$\begin{equation} \langle W_m \rangle = \frac{\hbar}2 (\omega_f - \omega_0) \coth \left( \frac{\beta \hbar \omega_0} 2 \right) - b^2 m (\omega_f^2- \omega_0^2).\enskip \end{equation} \tag{ 10 }$$

Importantly, for any realistic β, $\langle W_m \rangle$ is negative when $\omega_0 > \omega_f$ and positive when $\omega_0 < \omega_f$. Owing to the gauge term, the microscopic work in each power stroke behaves oppositely to what it is commonly expected when ignoring the physical gauge, see again the inset of fig. 1.

The total work in eq. (9) comes from the two terms of $\mathcal P_C$ in eq. (7), an Ohmic dissipation $R\dot{q}^2$ and the change in the potential energy of the capacitor. Using the time constant of the circuit (RC) and $q/\dot{q}=\omega/ (2 \dot \omega)$ we identify different regimes, dominated by the dissipative term or by the capacitor term. When $\omega/ (2 \dot \omega) \ll RC$, $\mathcal P_C$ is dominated by dissipation in the resistor, and, compare eqs. (7) and (8) and note the dominance of the gauge (first term) in eq. (8),

$$\begin{equation} \mathcal P_C / \dot q \approx RC \, \mathcal P_S / (a Q). \end{equation} \tag{ 11 }$$

Alternatively, when $\omega/ (2 \dot \omega) \gg RC$, the instantaneous microscopic power per unit charge is proportional to the control power input per unit charge,

$$\begin{equation} \mathcal P_C / q \approx \mathcal P_S / (a Q). \end{equation} \tag{ 12 }$$

Figure 2 shows the total work consumed by the power strokes in different regimes, for final times that yield a monotonic $\omega(t)$. For an expansion stroke dominated by the capacitor, the consumption decreases first as we reduce t_f (up to $t_f\approx 0.21\ \mu \text{s}$ for the green dash-dotted line). For faster protocols, $\omega/ (2 \dot \omega) \gg RC$ becomes unattainable and the total work is dominated by dissipation. We check numerically that for both power strokes, as $t_f\rightarrow0$, the total work scales as $1/t_f^{\,\,\,5}$. This scaling agrees with [26,27] and contrasts with Landauer's estimate of the energy dissipation as being proportional to the "velocity of the process" when studying the cost of computation [41].

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The microscopic power output of the engine is calculated along a cycle time τ, adding compression and expansion terms, $P = {(\langle W_m^{\rm comp} \rangle + \langle W_m^{\rm exp} \rangle)}/ \tau$. STA protocols increase this quantity by reducing τ and keeping the microscopic work output of adiabatic processes.

Typically, accelerating the thermodynamic cycle increases dissipation, diminishing work output and/or rising the energy required to perform the cycle, reducing efficiency. For a typical engine the input energy is the heat absorbed from the hot bath, $\langle {\mathcal{Q}} \rangle = \hbar \omega_0 / 2 [\coth(\beta_1\hbar\omega_0/2) -\coth(\beta_2\hbar\omega_f/2)]$, with $\beta_{1,2}=1/{k_B T_{1,2}}$ and k_B the Boltzmann constant. Since the shortcut does not affect the heat absorbed, it may seem that the efficiency is not affected. Our STA engine, however, is not a typical engine, as the expansion of the piston (in our case the trap expansion) is externally driven rather than being a consequence of the push by the hot working medium. The compression is similarly externally driven with a cost. Thus, the energy used to generate the external driving in eq. (9) constitutes an extra energy demand. We thus redefine the efficiency as microscopic work output divided by the cost¹

$$\begin{equation} \eta = \frac{\langle W_m^{\rm comp} \rangle + \langle W_m^{\rm exp} \rangle}{\langle {\mathcal{Q}} \rangle + W_T^{\rm exp} + W_T^{\rm comp}}, \end{equation} \tag{ 13 }$$

which is negligible $(\eta \approx 0)$ for the protocol durations in which STA are of interest due to the scale difference between the working medium and the control. For slower and slower processes (see fig. 3), the dissipation in the control decreases and the efficiency increases thereof, together with a diminishing microscopic power output. If $\mu=-1$, as considered so far, the efficiency increase saturates at the reversible work to charge and discharge the capacitor. For a $\mu=1$ setting (full regeneration), there is a crossover to a linear growth with t_f around the final time for which $\dot \omega / \omega^2 >1$ during the process, since the dissipated energy diminishes as $\sim t_f^{-1}$, consistently with Landauer's prediction for long times. Finally it reaches the conventional efficiency of the reversible Otto cycle $\eta=(\langle W_m^{\rm comp} \rangle + \langle W_m^{\rm exp} \rangle )/\langle {\mathcal{Q}}\rangle$, as the total work becomes negligible compared with the heat input.

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To calculate the engine performance criteria we use the microscopic work in the numerators. Along the cycle the gauge contributions to the microscopic work in each power stroke exactly cancel each other, making engine microscopic power and efficiency truly physical quantities whether we use or do not use the "physical gauge" provided by the experiment. This effect is similar to what happens with the Jarzynski inequality [42], also given in terms of inclusive work.

The idea of finding the optimal time path for the motion of the piston has been present in the field of finite-time thermodynamics for long [43]. However, already Andresen et al. pointed out that such trajectory may be optimal for some performance criterion but detrimental for others [9], as we see here in a clear and extreme way.

We have studied the energetic cost of performing fast STA expansions/ compressions for an ion in a Paul trap. In the context of an Otto cycle, the inclusion of this cost in the "debit" represented by the denominator of the efficiency implies a vanishing energetic efficiency. The result mirrors the paradigm that slow reversible engines provide maximal efficiencies and zero power. Here the opposite result holds: for the fast STA engine the efficiency vanishes, but microscopic power is enhanced. These findings beg for a more speculative extrapolation and discussion of potential implications in a broader context, beyond the specific model. We shall thus indulge now into arguments that intend to be reasonable rather than based on a formal generic analysis, and suggestive of further research.

A first, preliminary observation is that STA processes on a microscopic system are, in standard applications and from the point of view of the microscopic system, externally driven processes which involve a semiclassical Hamiltonian that depends on time-dependent classical parameters, i.e., with a given value at any time. (Applications of STA to optical devices are exceptional since, in the effective Hamiltonian, the position along the optical device plays the role of time, see, e.g., [44] and [2]). The different STA techniques design the time dependence of these classical parameters to take the control-dependent quantum system to the same results of slow adiabatic driving. The semiclassical nature of the Hamiltonian implies a macroscopic object or apparatus that determines the small system dynamics but, at the same time, is quite insensitive to it. These are the conditions in our model and we expect this scenario to be, if not universal, broadly applicable. Even if friction for the microscopic, primary object is evaded by the STA process, changing the classical parameters with the necessary speed involves an energy cost which, to be sure, is system dependent, but will lead to increased macroscopic dissipation for faster speeds. Power input will be required to change inertias of the control parameters and fight dissipation losses. Smart designs for specific systems might minimize the role of inertia and benefit from regeneration mechanisms $(\mu\approx 1)$, but never to the point of achieving perpetual cyclic motion of the classical parameters for free. This is quite a fundamental feature that can hardly be ignored when doing energy accounting. A cornerstone of finite-time thermodynamics is that processes without dissipation do not occur in nature if performed in finite time. In particular, the scale difference between macroscopic energy costs, even if small, and the microscopic energies involved makes the prospect for an energetically efficient cycle quite challenging. It also challenges scalability, i.e., the idea that a device that combines many STA quantum engines may outperform a classical engine.

We do not discard the possibility of systems and "sweet spots" where energy balances may be not so unfavorable. Also, a vanishing or small energy efficiency is not necessarily a problem depending on the aim of the STA process. If we are interested in fast adiabatic-like ion cooling by expansion, for example, the energy cost may be worth paying. As for quantum engines, it might be the case that the "quality" of the microscopic work achieved, i.e., the degrees of freedom put in motion when it functions, is worth the energy expense. A known example of a relatively inefficient but extremely useful device is the laser.

There is also room to explore alternatives to some of the premises applied so far. The microscopic/macroscopic divergence of scales might be absent in some systems. A quantum and macroscopic primary system is a route to explore [2], possibly at the price of making the STA protocol state dependent.

As already stated, the device studied here differs fundamentally from a typical engine. In a typical engine the heated or cooled down working medium moves the piston; in our STA quantum engine, the motion of the element that plays the role of the piston, the harmonic potential, is not at all a consequence of the dynamics of the ion, but an externally controlled evolution which consumes energy. Let us mention in this regard that quantum autonomous engines replicate the behavior of typical engines, because the controller is not driven but it evolves under the action of a time-independent Hamiltonian and the baths. In contrast to our STA engine, there is no need of power input to drive the controller. An experimental implementation was presented in [45]. Here the challenge is to apply STA to a system without time dependence in the Hamiltonian. The work on optics could be a reference about how to do that, this is quite an open and nontrivial issue.

We have also limited the analysis and discussion so far mostly to the power strokes but STA have also been proposed to accelerate the thermalization processes in Markovian [46,47] or non-Markovian regimes [48]. Corresponding energy consumptions should be studied.

We thank A. Polkovnikov for commenting on the paper. This work was supported by the Basque Country Government (Grant No. IT986-16) and PGC2018-101355-B-I00 (MCIU/AEI/FEDER,UE).

Here we prove that the expression in eq. (6) of the main text can be generalized to any system whose dissipation is proportional to the velocity squared, and thus describable by a Hamiltonian complemented by a Rayleigh dissipation function $\mathcal{F} = \frac 1 2 \gamma \dot q^2$. We do it for one dimension in a classical setting, but it can be extended to higher dimensions. We start by separating the total Hamiltonian into the unperturbed system and the external, time-dependent force term, $H(q,p_q,t)= H_0(q,p_q) - F(t) q$. The rate of change of the unperturbed Hamiltonian then reads

$$\begin{equation*} \frac{\mathrm{d}H_0}{\mathrm{d}t} = \frac{\mathrm{d}H}{\mathrm{d}t} + \dot F q + F \dot q. \end{equation*}$$

Rewriting the time derivative of the total Hamiltonian,

$$\begin{equation*} \frac{\mathrm{d}H}{\mathrm{d}t} = \frac{\partial H}{\partial t} + \frac{\partial H}{\partial q} \dot q + \frac{\partial H}{\partial p} \dot p, \end{equation*}$$

and then using modified Hamilton equations, $\dot q = \partial H / \partial p_q$ and $\dot p_q = - \partial H / \partial q - \partial \mathcal{F} / \partial \dot q$, we get

$$\begin{equation*} \frac{\mathrm{d}H_0}{\mathrm{d}t} = \frac{\partial H}{\partial t} - \dot q \frac{\partial \mathcal{F} }{ \partial \dot q} + \dot F q + F \dot q. \end{equation*}$$

Notice that the only explicitly time-dependent element of the Hamiltonian is the external force, and thus $\partial H / \partial t = - \dot F q$. Finally, performing the derivative of the Rayleigh function and reordering the terms, we find

$$\begin{equation*} \frac{\mathrm{d}H_0}{\mathrm{d}t} + \gamma \dot q^2 = F \dot q. \end{equation*}$$