More bang for your buck: Towards super-adiabatic quantum engines
The reversible nature of thermodynamical cycles is an idealisation based on the assumption of perfect quasi-static dynamics. As a consequence of this assumption, ideal engines operate at the maximum efficiency but have zero power. Realistic engines, on the other hand, operate in finite-time and are intrinsically irreversible, implying friction effects at short cycle times. The engineering goal is to find the maximum efficiency allowed at the maximum possible power Calborn (); Esposito (). In the current technological age, with our ability to manipulate devices at the nanoscale and beyond, one must understand the consequences of engines which operate at the quantum mechanical level. In this domain one cannot avoid the emergence of both quantum and thermal fluctuations jrev (); campisi () which drastically alter the energetics of the cycle sekimoto (). Very recently, it has been shown that the Hamiltonian of a quantum system may be manipulated in such a way as to mimic an adiabatic process via a non-adiabatic shortcut DR03 (); Berry09 (); Chen10 (); STAreview (). A surge of experimental progress has demonstrated several of these proposals in the laboratory Schaff1 (); Schaff2 (); hfqd (). In this paper we show that, by utilising shortcuts to adiabaticity STAreview () in a quantum engine cycle, one can engineer a thermodynamical cycle working at finite power and zero friction. Our findings are elucidated using a harmonic oscillator undergoing a quantum Otto cycle.
Thermodynamics is the study of heat and its interconversion to mechanical work. It successfully describes the “equilibrium” properties of macroscopic systems ranging from refrigerators to black holes refrig (). The technological revolution potentially embodied by the incorporation of information technology is motivating the consideration and realisation of quantum devices going all the way down to the micro- and nano-scale nanotech (). This has forced us to revise our interpretation of thermodynamics so as to include ab initio both quantum and thermal fluctuations, which become prominent at those working scales. In fact, far from equilibrium, such fluctuations become dominant and cannot be neglected. In turn, thermodynamical quantities such as work and heat become inherently stochastic and should be reformulated as such.
Recently discovered work fluctuation theorems (FTs) and the corresponding framework, which is known to hold both in quantum and classical systems, are extremely useful for the task of setting up a quantum apparatus for thermodynamics tasaki (). FTs demonstrate that information on the equilibrium state of the systems is encoded in the associated probability distributions, and can be interpreted as refined statements of the second law applicable at the micro- and nano-scale helping us understand the origin of microscopic irreversibility in systems undergoing finite-time transformations jrev (); campisi (). Here we address the important related question of whether or not FTs can help us shed light on limitations or possible advantages of a quantum device operating in finite-time. A convenient platform to look for a quantitative answer is provided by quantum engine cycles, for which the thermodynamic laws must be recast appropriately mahlerbook (); scovil (); alicki (); ion_engine (). In fact, although its working principles might well be quantum mechanical, the efficiency of a reversible engine would always be limited by the second law, which makes the quantum version of cycles remarkably similar to their classical counterparts.
The assumed reversibility (quasi-stationarity) of an engine cycle, which implies its zero-power nature due to its infinitely long cycle-time, crashes with the reality of any practical machine, either quantum or classical. This is due to the finite-time operation which exposes it to the effects of friction-induced losses. While Ref. kosloff_lube () proposes the use of systematic noise to suppress frictional losses in the expansion and compression stages of a quantum Otto cycle, here we devise an innovative way to run a finite-time (and power) quantum cycle based on the use of shortcuts to adiabaticity DR03 (); Berry09 (); Chen10 (); STAreview (). As we will show, such Hamiltonian-engineering techniques, which have recently found considerable interest in the quantum community, allow us to drive the expansion and compression stages of a cycle, which are prone to frictional effects, virtually without any loss affecting the performance of a quantum engine within the finite-time of its cycle. Drawing inspiration from a recent ion-trap proposal ion_engine (), we will provide an example of a finite-time, fully frictionless quantum Otto cycle where the working medium is a quantum harmonic oscillator.
I Quantum Otto Cycle
In an Otto engine, a working medium (coupled alternatively to two baths at different temperatures ) undergoes a four-stroke cycle. In its quantum version, the state of the working medium is described by a density operator that is changed by the Hamiltonian . Here, is an adjustable work parameter, typical of the specific setting used to physically implement , whose value determines the equilibrium configuration of the system. As illustrated in Fig. 1, the cycle steps are as follows:
1) An adiabatic expansion performed by the change of the work parameter , where is the time at which this step ends. As a result of this transformation, work is extracted from the medium due to the change in its internal energy.
2) A cold isochore where heat is transferred from the working medium to the cold bath. This is associated with a heat flow from the medium to the cold reservoir it is in contact with.
3) An adiabatic compression performed by the reverse change of the work parameter and during which work is done on the medium.
4) A hot isochore during which heat is taken from the hot reservoir by the working medium.
If the engine is run in a finite-time, i.e. if we abandon the usual quasi-static assumption, friction is inevitably generated along the expansion and compression steps. We will elucidate the nature of this friction later on. In addition, one may (realistically) assume imperfect heat conduction during the isochores. Under such conditions, the work done by/on the engine across the cycle, and the heat exchanged by the working medium with the baths, become stochastic quantities with associated probability distributions . The efficiency of the engine is then defined as the ratio between the average total work per cycle and the average heat received from the hot bath, according to the equation
where is the average value of the quantity during step of the cycle. The power of the engine is defined accordingly as
where is the time needed by the step. In what follows we consider the case where friction only occurs along the adiabatic transformations. In particular, we neglect fluctuations in the heat flow, so that , and we assume that the thermalisation process associated with the isochoric transformations occur in a timescale that is much quicker than any other one in the dynamics of the cycle. Under these assumptions, the power of the engine can be approximately written as
Ii Finite-time thermodynamics
Before we quantify the efficiency of the engine, it is necessary to properly define the probability distribution of work of which is the first moment. We consider a dynamical system described by . The system is prepared by allowing it to equilibrate with a heat reservoir at inverse temperature with . The initial state of the system is the Gibbs state
with the associated partition function. At , the system-reservoir coupling is removed and a protocol is performed on the system taking the work parameter from its initial value to at a later time . For instance, the process could be either the expansion or the compression step of the Otto cycle, represented by a change of the parameter of the system Hamiltonian , where is the eigenstate with eigenvalue . As discussed in Ref. lutz (), in this context the very concept of work should be reformulated in a way to account for both the statistics of the initial state of the system and the inherently non-deterministic nature of quantum mechanics and measurements. The explicit expression for the work distribution function reads lutz ()
Here, letting , is the transition probability under the evolution operator associated with , and is the occupation probability of the initial mode which for a Gibbs ensemble reads . The moment of the distribution is . The average work extracted or done up to time is found for .
For finite systems, the statistical nature of work requires the second law of thermodynamics to be revised to the form , with the change in free energy and the equality holding for a quasi-static isothermal process (the inequality holding strictly for all quasi-static processes performed without the coupling to a thermal reservoir). For all non-ideal processes, the deficit between the average work and the variation in free energy can be accounted for by the introduction of the average irreversible work as . We anticipate that the behavior of will be contrasted, later on in this paper, with the average work performed onto or made by the system when a shortcut to adiabaticity is implemented. For such quantity, , in the absence of a heat bath. For a closed quantum system, the incoming heat flow is null and the entropy change due to the irreversibility of the process is
which can be recast as lutz2 () with the relative entropy between two density matrices and vedral (), the time-evolving state, and the corresponding equilibrium reference state at the initial temperature . Here, quantifies the degree of friction caused by the finite-time protocol on the expansion or compression stage of the engine cycle at hand. When a bath is reconnected this friction is manifested by dissipation into the bath and hence the decrease in the overall efficiency of the motor. For simplicity and for the point of demonstration we allow only this form of irreversibility in our engine cycle although in principle the same analysis can be done for fluctuating heat flows hexchange1 (); hexchange2 ().
Iii Friction-free finite-time engine
Recently there has been a significant amount of work devoted to the design of so-called super-adiabatic protocols, i.e. shortcuts to states which are usually reached by slow adiabatic processes DR03 (); Berry09 (); STAreview (). A typical approach for shortcuts to adiabaticity is to use ad hoc dynamical invariants to engineer a Hamiltonian model that connects a specific eigenstate of a model from an initial to a final configuration determined by a dynamical process. Here we will rely on an approach based on engineered non-adiabatic dynamics achieved using self-similar transformations Chen10 (); delcampo11 ().
Let us consider a quantum harmonic oscillator with time-dependent frequency as the working medium of the engine cycle Chen10 (). The Hamiltonian model that we consider is thus , where and are the position and momentum operators of an oscillator of mass . Inspired by the scheme put forward in ion_engine (), we will use the tunable harmonic frequency to implement the compression and expansion steps of the Otto cycle. In line with the experimental proposal for the realisation of a microscopic Otto motor put forward in ion_engine (), the frequency of the harmonic trap embodies the volume of the chamber into which the working medium is placed, while the corresponding pressure is defined in terms of the change of energy per unit frequency.
Needless to say, in the compression or expansion stage of the Otto cycle, the frequency of the trap will have to be varied, so that takes here the role of the work parameter introduced when discussing FTs. We now suppose to subject the working medium to a change in the work parameter occurring in a time and corresponding to, say, one of the friction-prone steps of the Otto cycle. Our goal is to design an appropriate shortcut to adiabaticity to arrange for a fast, frictionless evolution between the equilibrium configuration of the working medium at and that at . In order to do this, we remind that the wavefunction of an initial eigenstate of is known to follow the self-similar evolution Chen10 ()
where , is the energy of the eigenstate being considered at , and the scaling factor is the solution of the Ermakov equation
with the initial conditions and . The shortcut to adiabaticity that we seek is then found by inverting the Ermakov equation and complementing the previous set of boundary conditions with , and with and . Instances of solutions to this problem can be found as illustrated in the Appendix, where we give the explicit form of the scaling factor such that the finite-time dynamics that takes the initial state to the final one actually mimics the wanted adiabatic evolution (albeit for any , is in general different from the eigenstate of ). The choice of a harmonic oscillator is not a unique example as similar self-similar dynamics can be induced in a large family of many-body systems delcampo11 () and other trapping potentials, such as a quantum piston DB12 ().
Let us consider the fluctuations induced in the expansion and compression stages of the Otto cycle when the above shortcut to adiabaticity is implemented. Let us consider a driving Hamiltonian with instantaneous eigenstates and eigenvalues . In the adiabatic limit, the corresponding transition probabilities tend to for all . The average work simplifies then to . On the other hand, in a shortcut to adiabaticity, only the weaker condition holds. For the time-dependent harmonic oscillator, it follows that
In the adiabatic limit and .
Figure 2(a) shows that the average work along a shortcut to an adiabatic expansion in comparison with in the corresponding adiabatic processes (the behaviour observed during a shortcut to a compression is mirrored in time). It is very important to stress that is the work done on either adiabat until the reconnection with the bath, i.e. just prior to the isochoric heating or cooling stage. The standard deviation of the work distribution is displayed in Fig. 2(b). In turn, this provides a further characterisation of the work fluctuations along the shortcut through the width of . It is interesting to notice that upon completion of the stroke, the non-equilibrium deviation of both the average work and the standard deviation from the adiabatic trajectory disappear.
We shall now analyse the non-equilibrium deviation with respect to the adiabatic work . Note that this expression is equivalent to the deviation of the mean energy of the motor along the super-adiabats from its (instantaneous) adiabatic expression. For an isothermal reversible process and . Differently, for the adiabatic dynamics associated to stages 1 and 3 of the Otto cycle, conservation of the population in is satisfied provided that , as it is the case for a large-class of self-similar processes (here, is introduced by noticing that the physical adiabatic state at time is characterised by the occupation probabilities ) Chen10 (); delcampo11 (); DB12 (). As a result, the reference state is not the physical instantaneous equilibrium state resulting from the adiabatic dynamics, and we find
From this result, it is clear that, in general, . However, it is straightforward to check that, at the final time of the process , we have , which implies and, in turn, the frictionless nature of the process [cf. Fig. 2(c)]. The time-evolution of the different contribution to , i.e. and , are displayed in Figure 2(d). This result is remarkable in the context of the quantum Otto cycle: If the baths are reconnected at just the right time after both the compression and expansion stages, then the efficiency of an ideal reversible engine can be reached in finite-time, therefore implementing a perfectly frictionless finite-time cycle. As we have built our engine so that friction is the only source of irreversibility, the super-adiabatic engine clearly reaches the maximum efficiency of an ideal quasi-static engine in a finite-time only.
Let us address a final important point. The efficiency in Eq. (1) of an Otto cycle diminishes explicitly with the breakdown of adiabaticity ion_engine (). In contrast, the super-adiabatic engine put forward in this proposal does achieve the maximum possible value . It should be noted, quite strikingly, that if unlimited resources are available, there is no fundamental lower-bound on the running time of the adiabats . However, it is worth taking a pragmatic approach here and attempt at the quantification of the energy costs associated with the running of our super-adiabatic engine. To this end, we have considered the time-averaged dissipated work for , ensuring for all . The cut-off time was taken to be the maximum running time along the shortcut of the super-adiabat before the trap is inverted. Indeed, when this occurs, the adiabatic eigenenergies are not well defined, implying the break-down of our formalism. The explicit expression for are reported in the Appendix. Figure 3 shows that the cost of running the super-adiabatic engine exhibits a neat power-law behaviour for a wide range of parameters.
An explicit upper bound for the power of an engine run can be calculated using the fundamental limitations set by quantum speed limit, as shown in the Appendix.
We have demonstrated the possibility to perform a fully frictionless quantum cycle working in a finite-time only. Our proposal exploits the idea of shortcuts to adiabaticity, which allowed us to bypass the detrimental effects of friction on the compression and expansion stages in an important thermodynamical cycle such as the Otto cycle. We believe that our study embodies only one example of the potential brought about by the fascinating combination of shortcuts to adiabaticity and the framework for out-of-equilibrium dynamics of a quantum system. The possibilities to achieve maximum efficiency of a quantum engine with virtually no friction, yet within a finite operating time, is a tantalizing result with the potential to revolutionise the design of micro- and nano-scale motors operating at the verge of quantum mechanics.
Acknowledgements.We greatly thanks G. De Chiara, B. Damski, S. Deffner, R. Dorner, C. Jarzynski, E. Passemar and N. Sinitsyn for discussions and comments on the manuscript. AdC is supported by the U.S. Department of Energy through the LANL/LDRD Program and a LANL J. Robert Oppenheimer fellowship; JG acknowledges funding from IRCSET through a Marie Curie International Mobility fellowship; MP thanks the UK EPSRC for a Career Acceleration Fellowship and a grant under the “New Directions for EPSRC Research Leaders” initiative (EP/G004579/1).
Driving protocol of the super-adiabats
In this part of the Appendix we illustrate the formal procedure for the determination of the scaling factor used in the super-adiabatic steps of our proposal. The simplest interpolation of the actual solution of the Ermakov equation in the main Letter with the boundary conditions stated in the main text is found to be the polynomial
with . This solution guarantees that, in the adiabatic limit, and , while more generally the eigenstates of the initial oscillator will evolve according to the scaling law in Eq. (6) of the main Letter. The modulation of is the responsible for the speed-up of the transformations performed along the super-adiabats. In turn, the implementation of such modulation is the price to pay for the achievement of such advantage. The explicit form can be extracted from the Ermakov equation, , using Eq. (10) for , see Fig. (4).
for sufficiently small values of , can become purely imaginary (requiring the inversion of the trap into an expelling barrier). The condition provides the cut-off time used in Fig. 3 of the main Letter.
Nonequilibrium work fluctuations
We next present the derivation of Eq. (9) in the main Letter. Consider as reference states the fictitious equilibrium state and the adiabatic one . Then, in the adiabatic limit the average work can be rewritten as
with the instantaneous partition function. For a general nonequilibrium process, the average work reads instead
As a result, nonequilibrium deviations from the mean adiabatic work take the form
It is worth considering an an alternative approach, where is used as a reference state and the dynamics is restricted to the class of self-similar processes Chen10 (); delcampo11 (); DB12 (), for which conservation of the population in the mode as a function of time is satisfied provided that
as it is the case for the adiabatic dynamics associated to the shortcuts discussed here. Under such condition the partition of the instantaneous equilibrium state remains constant . Using , the average work in the adiabatic limit reads
where we have introduced the von Neuman entropy of an arbitrary state . More generally,
This leads to the following compact expression for nonequilibrium work deviations form the adiabatic path,
The two expressions for , Eqs. (13) and (17), agree for self-similar processes and vanish at the end of the stroke (either 1 or 3 in Fig. 1 of the main Letter) both for a shortcut and in the adiabatic limit.
Upper bound to power through the quantum speed limit
The quantum speed limit for a driven quantum system DL11 () allows us to derive an upper bound for the power of the engine. For simplicity, we can consider a equal-time shortcuts along the two super-adiabats so that . Then, it follows that
where with respect to the ground state energy, , and the angle in Hilbert space between initial and target states is
in terms of the fidelity . In a super-adiabatic engine, equals
where is the inverse temperature of the cold bath during stage .
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