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# Catalysis of heat-to-work conversion in quantum machines

Contributed by L. Davidovich, September 22, 2017 (sent for review June 26, 2017; reviewed by Adolfo del Campo and Muhammad Suhail Zubairy)

## Significance

The traditional (19th century) rules of thermodynamics were conceived for engines that convert heat into work. Recently, these rules have been scrutinized, assuming that the engines have quantum properties, but we still have no complete answer to the question: Are these rules then the same as the traditional ones? Here, we subject a “piston”—an oscillator that extracts work from the engine—to energy “pumping” that renders this oscillator quantum and nonlinear. We show that even weak pumping may strongly catalyze the heat-to-work conversion rate. This catalysis, analogous to its chemical-reaction counterpart, is a manifestation of “quantumness” in heat engines, yet it adheres to the traditional laws of thermodynamics.

## Abstract

We propose a hitherto-unexplored concept in quantum thermodynamics: catalysis of heat-to-work conversion by quantum nonlinear pumping of the piston mode which extracts work from the machine. This concept is analogous to chemical reaction catalysis: Small energy investment by the catalyst (pump) may yield a large increase in heat-to-work conversion. Since it is powered by thermal baths, the catalyzed machine adheres to the Carnot bound, but may strongly enhance its efficiency and power compared with its noncatalyzed counterparts. This enhancement stems from the increased ability of the squeezed piston to store work. Remarkably, the fraction of piston energy that is convertible into work may then approach unity. The present machine and its counterparts powered by squeezed baths share a common feature: Neither is a genuine heat engine. However, a squeezed pump that catalyzes heat-to-work conversion by small investment of work is much more advantageous than a squeezed bath that simply transduces part of the work invested in its squeezing into work performed by the machine.

The intimate rapport of thermodynamics with the theory of open quantum systems and its applications to quantum heat engines has been long and fruitful. The landmarks of this rapport have been Einstein’s theory of spontaneous and stimulated emission (1), the determination of maser efficiency (2⇓–4), and its extension to the micromaser (5). Among the diverse proposals for quantum heat engines (6⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–35), intriguing suggestions have been made to boost the Carnot efficiency through bath preparation in nonthermal [population-inverted (29), phase-coherent (phaseonium) (30), or squeezed (31, 35)] states.

However, quantum machines fueled by such nonthermal baths adhere to rules that differ from those of quantum heat engines (32, 33, 36) (*Discussion*). Here, instead, we restrict ourselves to machines fueled by thermal baths, but introduce the concept of catalysis known from the theory of chemical reaction (37), whereby a small amount of catalyst (here, a weak pump) strongly enhances the reaction rate (here, the heat-to-work conversion).

We illustrate this concept for the minimal model (18, 20, 34) of a fully quantized heat machine wherein a two-level system (TLS) acts as the working fluid (WF) that simultaneously interacts with hot and cold baths and is dispersively (off-resonantly) coupled to a piston mode that undergoes amplification and extracts work. This model is here extended by subjecting the quantized piston mode to nonlinear (quadratic) pumping. Our motivation for considering this scheme is that nonlinearly pumped parametric amplifiers may produce squeezed output (38⇓⇓⇓–42). We wish to find out whether this property may catalyze the machine performance. To this end, we investigate work extraction by combining quantum-optical amplification and dissipation theory (38⇓⇓–41) with thermodynamics (43).

Our main insight is that the quadratic pumping (5) of the piston mode provides a powerful handle on the performance of the machine, which is determined by the piston state nonpassivity (43⇓⇓⇓⇓⇓⇓–50): the capacity of the piston state to store work. In analogy to the potential energy stored in a classical (mechanical) device or the charging energy of a battery, nonpassivity [also known as ergotropy (48)] is a unique measure of work extractable from a quantum state. We find that under quadratic pumping, the piston mode evolves into a thermal-squeezed state that strongly enhances its work capacity (nonpassivity) compared with its linearly pumped or unpumped counterparts. The resulting catalysis effects are that the output power and efficiency of heat-to-work conversion are drastically enhanced, and the piston “charging efficiency” (i.e., the fraction of piston energy convertible to work) may approach unity. On the other hand, since the machine is fueled by thermal baths, the Carnot efficiency bound remains valid upon subtracting the work invested by the pump, so that the machine abides by the first and second laws of thermodynamics (51).

## The Model and Basic Assumptions

In our illustration of catalysis for a quantum heat-powered engine, the WF is composed of a TLS, S, which is dissipatively coupled to two thermal baths all the time. S is off-resonantly coupled to a pumped harmonic oscillator, dubbed a piston, P, which can collect and store the extracted work. The cold and hot baths, denoted by C and H, respectively, are “spectrally nonoverlapping,” as detailed below. P is not coupled to its own bath to avoid energy dissipation, which would disturb the thermodynamical balance of heat and work in the total system.

The Hamiltonian has the form (**1** is

The key feature we consider in Eq. **1** is the coupling of the quantized piston to an external pumping Hamiltonian*SI Text*), which oscillates at frequency

A cavity-based nonlinear parametric amplifier (5, 38) coupled to two heat baths with different temperatures and spectra can realize the present model (Fig. 1*A*). The intracavity WF of the machine may be an atomic gas (56), an optomechanical setup (57), or a collection of superconducting flux qubits (53⇓–55). The SP coupling in Eq. **2** is experimentally realizable by a flux qubit which is dispersively coupled to high-Q (phonon) mode of a nanomechanical cavity (cantilever) that acts as the P mode (53, 58). Alternatively, P can be a field mode of a coplanar resonator whose quantized electromagnetic field quadrature

## Outline of the Dynamical Analysis

The dynamics of such pumped quantum open systems, consistent with the laws of thermodynamics, is given in terms of the Floquet expansion (18, 20, 34, 43) of the Lindblad (Markovian) equations (59), which involves the bath response at the

The Lindblad master equation for the piston mode *SI Text*).

Work extraction requires *SI Text*), with small ratio *SI Text*) can be simplified.

The corresponding FP equation for the quantized P may be solved analytically (60) for an initial Gaussian state, *SI Text*) in the form of a 2D Gaussian with maximal and minimal widths **11**) along the respective orthogonal axes *SI Text*). The width *B*), causing squeezing.

## Work Extraction

To evaluate work extraction by P at the steady state of S, we take into account the pumping in the energy balance according to the first law of thermodynamics (10, 43, 51).

## Work Efficiency Bound for Pumped Piston

For a given

For Gaussian states (used here), the passive state **6** and using Eq. **7**, we find**8** is the net rate of extractable work converted from heat. The first term **8**, reflects the rise with time of the temperature

We note the following fundamental difference between the present machine and a usual heat engine. The usual power (or rate of work) is given by *SI Text*) wherein the machine is approximately a heat engine. It therefore must abide by the second law and the ensuing Carnot bound. However, as we show, its performance may be strongly catalyzed by the pump squeezing, a surprising and hitherto-unexplored effect.

To obtain better insight into the catalytic nature of nonlinear pumping in this setup, we compute the thermodynamic engine efficiency, which is defined as the ratio of the net work (or power) output to the heat input supplied by H to *SI Text*) for any Gaussian states in terms of *SI Text*) taken with respect to the initial state of P. Only thermal states can be considered as “natural” initial states. For such states, *SI Text*), well defined as long as the same pump beam is used for the parametric amplifier and for the preparation of the initial coherent state.

Keeping those observations in mind, we derive in *SI Text* the expressions of the passivity increase and *SI Text*)*SI Text*),*SI Text*). The number of passive quanta can be expressed in terms of the widths **10** can be rewritten as *SI Text*),*B*) that corresponds to setting *SI Text*). The above result remains valid for arbitrary initial Gaussian states, although the general expressions (detailed in *SI Text*) are more involved.

By contrast, for linear pumping or in the absence of any pumping (**6**) or the power (in Eq. **8**) becomes small only in the semiclassical limit (when *SI Text*)*SI Text*). Without any pump, the efficiency is reduced to**13** and **14**) is very small (*SI Text*).

To maximize the efficiency in Eqs. **13** and **14**, **14**], linearly pumped [**13**], and nonlinearly pumped [**10**, **12**] situations reveals that quadratic pumping may dramatically enhance the maximal work capacity, as shown in Fig. 2*A* for a small initial piston charging *SI Text*). However, the passive energy remains equal to *SI Text*).

By contrast, linear pumping generates an energy contribution which is independent of thermal energy, so that the passive and nonpassive contributions remain additive *SI Text*). Consequently, the ergotropy increase generated by heat input is then very limited (Fig. 2, Fig. S1, and *SI Text*). Note that, for any pumping, the fundamental requirement is

Importantly, the charging efficiency, i.e., the ratio between maximum useful work and the total energy stored in the piston, is enhanced*C* illustrates that quadratic pumping may drastically enhance both work extraction and charging efficiency in the quantized P mode, compared with its unpumped counterpart.

## Discussion

Here we set out to explore: Does the fact that a quantum machine is fueled by a heat bath imply that the machine conforms to the traditional rules of thermal (heat) engines? Conversely, does the quantumness of parts of a thermal machine endow it with unique resources? To answer these questions, we have derived the efficiency of a heat-fueled machine whose quantized piston is subject to quadratic pumping. It reveals the possibility of strong catalysis of heat-to-work conversion.

It is instructive to compare the present machine with machines powered by certain nonthermal baths, such as a squeezed-thermal or coherently displaced thermal bath, which render the WF steady state nonpassive (31⇓–33). The Carnot bound may nominally be surpassed in such machines at the expense of work supplied by the bath, but the comparison of their efficiency bound with the Carnot bound of heat machines is inappropriate, because this is imposed by the second law only on heat imparted by the bath. Such nonthermal machines do not adhere to the rules of a heat engine, since they receive both work and heat from external sources (32). Namely, the ability to attain super-Carnot efficiency is an effect of work transferred from the nonthermal bath to the WF (Fig. 3*A*).

By contrast, in the present setup (Fig. 3*B*), work supplied by the pump to the piston, thereby squeezing it and rendering it nonpassive, is a catalyst: It allows for strongly enhanced heat-to-work conversion efficiency. The Carnot bound does limit this heat-to-work conversion efficiency because the work contribution from the pumping or piston state preparation is subtracted, the only net energy input being the hot bath.

Another important difference between the two kinds of machines is that in our scheme, the work invested is recovered in the internal energy of P, whereas in a machine where a squeezed thermal bath is used, most of the work invested in squeezing the bath is lost in the bath since only a small part of it is transferred to the WF (36).

Our scheme is also convenient from an experimental point of view since it is much easier to squeeze a single-mode harmonic oscillator (piston) than a bath. A micromaser fed by two-atom clusters (16, 64⇓⇓–67) prepared in nearly equal superposition of doubly excited and doubly unexcited states may also strongly squeeze a cavity-field piston coupled to two heat baths (33). Cyclic cavity-mirror shaking is another squeezing mechanism (68). The nonpassivity of the output may be verified by homodyning the piston with a local oscillator (5).

In the present work, we consider a WF comprising a single TLS or a dilute sample thereof, with less than one TLS per cubic wavelength, such that collective effects are negligible (69). It would be worth investigating further potential beneficial collective effects (70) in presence of multiple TLSs.

To conclude, the hitherto-unexplored heat-to-work conversion catalysis has been shown to arise from the ability of pump-induced nonlinear (squeezed) piston dynamics to increase and sustain its nonpassivity and thereby its capacity to convert heat to work. Thus, squeezing may provide a uniquely advantageous resource to thermal machines.

## Acknowledgments

L.D. and C.L.L. thank Nicim Zagury for valuable discussions. G.K. was supported by the Israel Science Foundation and Alternative Energy Research Initiative. C.L.L. was supported by the College of Agriculture Engineering and Science of the University of KwaZulu-Natal. L.D. and C.L.L. were supported by the Brazilian Agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico, Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, and the National Institute of Science and Technology for Quantum Information.

## Footnotes

↵

^{1}A.G. and C.L.L. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: ldavid{at}if.ufrj.br.

Author contributions: L.D. and G.K. designed research; A.G., C.L.L., L.D., and G.K. performed research; and A.G., C.L.L., L.D., and G.K. wrote the paper.

Reviewers: A.D., University of Massachusetts Boston; and M.S.Z., Texas A&M University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1711381114/-/DCSupplemental.

Published under the PNAS license.

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