View source for Quantum heat engines and refrigerators

A quantum [[heat engine]] is a device that generates power from the heat flow between  hot and cold reservoirs. 
The operation mechanism of the engine can be described by the laws of [[quantum mechanics]]. 
The first realization of a quantum heat engine was pointed out by [[Henry Evelyn Derek Scovil|Scovil]] and Schulz-DuBois in 1959,<ref name="scovil59">{{cite journal|last1=Scovil|first1=H. E. D.|last2=Schulz-DuBois|first2=E. O.|title=Three-Level Masers as Heat Engines|journal=Physical Review Letters|volume=2|issue=6|year=1959|pages=262–263|issn=0031-9007|doi=10.1103/PhysRevLett.2.262|bibcode=1959PhRvL...2..262S}}</ref> showing the connection of efficiency of the [[Nicolas Léonard Sadi Carnot|Carnot]] engine and the 3-level [[maser]].
Quantum [[refrigerator]]s share the structure of quantum heat engines with the purpose of pumping heat from a cold to a hot bath consuming power
first suggested by Geusic, Schulz-DuBois, De Grasse and Scovil.<ref name="GeusicBois1959">{{cite journal|last1=Geusic|first1=J. E.|last2=Bois|first2=E. O. Schulz-Du|last3=De Grasse|first3=R. W.|last4=Scovil|first4=H. E. D.|title=Three Level Spin Refrigeration and Maser Action at 1500 mc/sec|journal=Journal of Applied Physics|volume=30|issue=7|year=1959|pages=1113–1114|issn=0021-8979|doi=10.1063/1.1776991|bibcode=1959JAP....30.1113G}}</ref> When the power is supplied by a laser the process is termed [[optical pumping]] or [[laser cooling]],  suggested by [[David J. Wineland|Wineland]] and [[Theodor W. Hänsch|Hänsch]].<ref name="wineland">D. J. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975)</ref><ref name="HänschSchawlow1975">{{cite journal|last1=Hänsch|first1=T.W.|last2=Schawlow|first2=A.L.|title=Cooling of gases by laser radiation|journal=Optics Communications|volume=13|issue=1|year=1975|pages=68–69|issn=0030-4018|doi=10.1016/0030-4018(75)90159-5|doi-access=free|bibcode=1975OptCo..13...68H}}</ref><ref name="LetokhovMinogin1976">{{cite journal|last1=Letokhov|first1=V.S.|last2=Minogin|first2=V.G.|last3=Pavlik|first3=B.D.|title=Cooling and trapping of atoms and molecules by a resonant laser field|journal=Optics Communications|volume=19|issue=1|year=1976|pages=72–75|issn=0030-4018|doi=10.1016/0030-4018(76)90388-6|bibcode=1976OptCo..19...72L}}</ref>
Surprisingly [[heat engine]]s and [[refrigerator]]s can operate up to the scale of a single particle thus justifying the need for a quantum theory termed [[quantum thermodynamics]].<ref name="alicki1">{{cite journal|last1=Alicki|first1=R|title=The quantum open system as a model of the heat engine|journal=Journal of Physics A: Mathematical and General|volume=12|issue=5|year=1979|pages=L103–L107|issn=0305-4470|doi=10.1088/0305-4470/12/5/007|bibcode=1979JPhA...12L.103A}}</ref>

==The 3-level amplifier as a quantum heat engine==

[[File:Three-level-amp.pdf|thumb|The three level amplifier. Levels 1 and 3 are coupled to the hot reservoir. Levels 1 and 2 are coupled to the cold reservoir. Power is obtained when there is population inversion between levels 3 and 2.]]

The three-level-amplifier is the template of a quantum device. It operates by employing a hot and cold bath
to maintain [[population inversion]] between two energy levels which is used to amplify light by [[stimulated emission]]<ref>Yariv, Amnon (1989). ''Quantum Electronics'', 3rd ed., Wiley. {{ISBN|0-471-60997-8}}</ref>
The ground state level ('''1-g''') and the excited level ('''3-h''') are coupled to a hot bath of temperature <math>T_\text{h}</math>.
The energy gap is  <math>\hbar \omega_\text{h} = E_3-E_1 </math>. When the population on the levels equilibrate 
::<math >\frac{N_\text{h}}{N_\text{g}}=e^{-\frac{\hbar \omega_\text{h}}{k_\text{B} T_\text{h}}} </math>
where <math>\hbar=\frac{h}{2 \pi } </math> is the [[Planck constant]] and <math>k_\text{B}</math> is the [[Boltzmann constant]].
The cold bath of temperature <math>T_\text{c} </math> couples the ground ('''1-g''') to an intermediate level ('''2-c''') 
with energy gap <math> E_2-E_1=\hbar \omega_\text{c} </math>.
When levels '''2-c''' and '''1-g''' equilibrate then
::<math> \frac{N_\text{c}}{N_\text{g}}=e^{-\frac{\hbar \omega_\text{c}}{k_\text{B} T_\text{c}}} </math>.
The device operates as an [[amplifier]] when  levels ('''3-h''') and ('''2-c''') are coupled to an external field of frequency <math>\nu</math>.
For optimal resonance conditions  <math>\nu=\omega_\text{h}-\omega_\text{c}</math>. 
The efficiency of the amplifier in converting heat to power is the ratio of work output to heat input:
::<math> \eta=\frac{\hbar \nu}{\hbar \omega_\text{h}} =1-\frac{\omega_\text{c}}{\omega_\text{h}}</math>.
Amplification of the field is possible only for positive gain (population inversion)
<math>G =N_\text{h} - N_\text{c} \ge 0 </math>. 
This is equivalent to <math>\frac{\hbar \omega_\text{c}}{k_\text{B} T_\text{c}} \ge \frac{\hbar \omega_\text{h}}{k_\text{B} T_\text{h}}</math>.
Inserting this expression into the efficiency formula  leads to:
::<math>\eta =1-\frac{\omega_\text{c}}{\omega_\text{h}} \le 1- \frac{T_\text{c}}{T_\text{h}} = \eta_\text{c}</math>
where <math>\eta_\text{c} </math> is the [[Carnot cycle]] [[efficiency]]. 
Equality is obtained under a zero gain condition <math>G =0 </math>.
The relation between the quantum amplifier and the [[Nicolas Léonard Sadi Carnot|Carnot]] efficiency was first pointed out by Scovil and Schultz-DuBois:<ref name="scovil59" />

Reversing the operation driving heat from the cold bath to the hot bath by consuming power constitutes a [[refrigerator]].
The efficiency of the refrigerator defined as the [[coefficient of performance]] (COP) for the reversed device is:
::<math> \epsilon = \frac{\omega_\text{c}}{\nu} \le \frac{T_\text{c}}{T_\text{h}-T_\text{c}}</math>

==Types==

Quantum devices can operate either continuously or by a reciprocating cycle. 
Continuous devices include [[solar cell]]s converting solar radiation to electrical power, [[thermoelectric]]  where the output is current and [[lasers]] where the output power is coherent light.
The primary example of a continuous refrigerator is [[optical pumping]] and [[laser cooling]].<ref name="NareviciusBannerman2009">{{cite journal|last1=Narevicius|first1=Edvardas|last2=Bannerman|first2=S Travis|last3=Raizen|first3=Mark G|title=Single-photon molecular cooling|journal=New Journal of Physics|volume=11|issue=5|year=2009|pages=055046|issn=1367-2630|doi=10.1088/1367-2630/11/5/055046|doi-access=free|bibcode=2009NJPh...11e5046N|arxiv=0808.1383}}</ref><ref name="am1">{{cite journal|last1=Kosloff|first1=Ronnie|last2=Levy|first2=Amikam|title=Quantum Heat Engines and Refrigerators: Continuous Devices|journal=Annual Review of Physical Chemistry|volume=65|issue=1|year=2014|pages=365–393|issn=0066-426X|doi=10.1146/annurev-physchem-040513-103724|pmid=24689798|arxiv=1310.0683|bibcode=2014ARPC...65..365K|s2cid=25266545}}</ref> 
Similarly to classical reciprocating engines, quantum heat engines also have a cycle that is divided into different strokes. A stroke is time segment in which a certain operation takes place (e.g. thermalization, or work extraction). Two adjacent strokes do not commute with each other. The most common reciprocating heat machines are the four-stroke machine, and the two-stroke machine. Reciprocating devices have been suggested operating either by the [[Carnot cycle]]<ref name="geva2">{{cite journal|last1=Geva|first1=Eitan|last2=Kosloff|first2=Ronnie|title=A quantum-mechanical heat engine operating in finite time. A model consisting of spin-1/2 systems as the working fluid|journal=The Journal of Chemical Physics|volume=96|issue=4|year=1992|pages=3054–3067|issn=0021-9606|doi=10.1063/1.461951|bibcode=1992JChPh..96.3054G}}</ref><ref name="bender">{{cite journal|last1=Bender|first1=Carl M|last2=Brody|first2=Dorje C|author-link2=Dorje C. Brody|last3=Meister|first3=Bernhard K|title=Quantum mechanical Carnot engine|journal=Journal of Physics A: Mathematical and General|volume=33|issue=24|year=2000|pages=4427–4436|issn=0305-4470|doi=10.1088/0305-4470/33/24/302|arxiv=quant-ph/0007002|bibcode=2000JPhA...33.4427B|s2cid=5335}}</ref> or the  [[Otto cycle]].<ref name="tova">{{cite journal|last1=Feldmann|first1=Tova|last2=Kosloff|first2=Ronnie|title=Performance of discrete heat engines and heat pumps in finite time|journal=Physical Review E|volume=61|issue=5|year=2000|pages=4774–4790|issn=1063-651X|doi=10.1103/PhysRevE.61.4774|pmid=11031518|bibcode=2000PhRvE..61.4774F|arxiv=physics/0003007|s2cid=2277942}}</ref>

In both types the quantum description allows to obtain equation of motion for the working medium
and the heat flow from the reservoirs.

==Quantum reciprocating heat engine and refrigerator==
{{see also|Reciprocating heat engine}}
Quantum versions of most of the common [[thermodynamic cycle]]s have been studied, for example the [[Carnot cycle]],<ref name="geva2" /><ref name="bender" /><ref name="QuanLiu2007">{{cite journal|last1=Quan|first1=H. T.|last2=Liu|first2=Yu-xi|last3=Sun|first3=C. P.|last4=Nori|first4=Franco|title=Quantum thermodynamic cycles and quantum heat engines|journal=Physical Review E|volume=76|issue=3|pages=031105|year=2007|issn=1539-3755|doi=10.1103/PhysRevE.76.031105|pmid=17930197|bibcode=2007PhRvE..76c1105Q|arxiv=quant-ph/0611275|s2cid=3009953}}</ref> [[Stirling cycle]]<ref name="WuChen1998">{{cite journal|last1=Wu|first1=F.|last2=Chen|first2=L.|last3=Sun|first3=F.|last4=Wu|first4=C.|last5=Zhu|first5=Yonghong|title=Performance and optimization criteria for forward and reverse quantum Stirling cycles|journal=Energy Conversion and Management|volume=39|issue=8|year=1998|pages=733–739|issn=0196-8904|doi=10.1016/S0196-8904(97)10037-1|bibcode=1998ECM....39..733W }}</ref> and [[Otto cycle]].<ref name="tova" /><ref name="Kieu2006">{{cite journal|last1=Kieu|first1=T. D.|title=Quantum heat engines, the second law and Maxwell's daemon|journal=The European Physical Journal D|volume=39|issue=1|year=2006|pages=115–128|issn=1434-6060|doi=10.1140/epjd/e2006-00075-5|bibcode=2006EPJD...39..115K|arxiv=quant-ph/0311157|s2cid=119382163}}</ref>

The Otto cycle  can serve as a template for other reciprocating cycles. 
[[File:Q-otto-cycle.pdf|thumb|Quantum [[Otto cycle]] shown in the Entropy <math>\Omega</math> plane
where the energy entropy and the [[Von Neumann entropy]] are displayed. <math>\Omega</math> is the 
internal frequency of the device and is controlled externally. It mimics the inverse volume in the [[Otto cycle]].
The red and blue lines are the hot and cold isochores. The cycle represents a heat pump.]]
It is composed of the following four segments:

*Segment  <math> A \rightarrow B </math>   ''isomagnetic'' or ''[[isochoric process]]'', partial equilibration with the cold bath under constant Hamiltonian. The dynamics of the working medium is characterized by the  propagator <math> { U}/ </math>.
*Segment <math> B \rightarrow C </math>  ''[[magnetization]]''  or  [[adiabatic compression]], the external field  changes expanding the gap between energy levels of the Hamiltonian. The dynamics is characterized by the propagator <math> { U}_\text{ch} </math>.
*Segment <math>C \rightarrow D </math> ''isomagnetic'', or ''isochoric process'' partial equilibration with the hot bath described by the propagator <math> U_\text{h} </math>.
*Segment <math>D \rightarrow A</math> ''[[demagnetization]]'' or  [[adiabatic expansion]] reducing the energy gaps in the Hamiltonian, characterized by the  propagator <math> U_\text{hc} </math>.

The propagator of the four stroke cycle 
becomes <math> U_\text{global}</math>,  which is the ordered product of the segment propagators:
::<math>
{U}_\text{global}~~=~~        { U}_\text{hc} { U}_\text{h} {U}_\text{ch} {U}_\text{c}      
</math>
The propagators are linear operators defined  on a vector space which completely determines the state of the working medium.
Common to all thermodynamic cycles the consecutive segment propagators do not commute <math>[{\ U}_i,{ U}_j] \ne 0</math>.
Commuting propagators will lead to zero power.

In a reciprocating quantum heat engine the working medium is a quantum system such as spin systems<ref name="FeldmannKosloff2003">{{cite journal|last1=Feldmann|first1=Tova|last2=Kosloff|first2=Ronnie|title=Quantum four-stroke heat engine: Thermodynamic observables in a model with intrinsic friction|journal=Physical Review E|volume=68|issue=1|pages=016101|year=2003|issn=1063-651X|doi=10.1103/PhysRevE.68.016101|pmid=12935194|bibcode=2003PhRvE..68a6101F|arxiv=quant-ph/0303046|s2cid=23777311}}</ref> or an harmonic oscillator.<ref name="yair">{{cite journal|last1=Rezek|first1=Yair|last2=Kosloff|first2=Ronnie|title=Irreversible performance of a quantum harmonic heat engine|journal=New Journal of Physics|volume=8|issue=5|year=2006|pages=83|issn=1367-2630|doi=10.1088/1367-2630/8/5/083|doi-access=free|bibcode=2006NJPh....8...83R|arxiv=quant-ph/0601006}}</ref> For maximum power the cycle time should be optimized. 
There are two basic timescales in the reciprocating refrigerator the cycle time <math> \tau_\text{cyc} </math> and the internal
timescale <math> 2 \pi /\omega </math>. In general when <math> \tau_\text{cyc} \gg 2\pi/\omega </math> the 
engine operates in quasi-adiabatic conditions. The only quantum effect can be found at low temperatures
where the unit of energy of the device becomes <math> \hbar \omega </math> instead of <math>k_\text{B} T</math>.
The efficiency at this limit is <math> \eta = 1 -\frac{\omega_\text{c}}{\omega_\text{h}}</math>, always smaller than the [[Carnot efficiency]] <math>\eta_\text{c}</math>. At high temperature and for the harmonic working medium the efficiency at maximum power becomes <math> \eta = 1-\sqrt{\frac{T_\text{c}}{T_\text{h}}} </math> which is the [[endoreversible thermodynamics]] result.<ref name="yair"/>

For shorter cycle times the working medium cannot follow adiabatically the change in the external parameter.
This leads to friction-like phenomena. Extra power is required to drive the system faster.
The signature of such dynamics is the development of coherence causing extra dissipation.
Surprisingly the dynamics leading to friction is quantized meaning that frictionless solutions to the [[adiabatic expansion]]/compression
can be found in finite time.<ref name="CampoGoold2014">{{cite journal|last1=Campo|first1=A. del|last2=Goold|first2=J.|last3=Paternostro|first3=M.|title=More bang for your buck: Super-adiabatic quantum engines|journal=Scientific Reports|volume=4|issue=1|pages=6208|year=2014|issn=2045-2322|doi=10.1038/srep06208|pmid=25163421|pmc=4147366|doi-access=free|bibcode=2014NatSR...4E6208C}}</ref>
<ref name="BeauJaramillo2016">{{cite journal|last1=Beau|first1=Mathieu|last2=Jaramillo|first2=Juan|last3=del Campo|first3=Adolfo|title=Scaling-Up Quantum Heat Engines Efficiently via Shortcuts to Adiabaticity|journal=Entropy|volume=18|issue=5|year=2016|pages=168|issn=1099-4300|doi=10.3390/e18050168|doi-access=free|bibcode=2016Entrp..18..168B|arxiv=1603.06019}}</ref> As a result, optimization has to be carried out only with respect to the time allocated
to heat transport. In this regime the quantum feature of coherence degrades the performance.
Optimal frictionless performance is obtained when the coherence can be cancelled.

The shortest cycle times <math> \tau_\text{cyc} \ll 2\pi/\omega </math>, sometimes termed sudden cycles,<ref name="FeldmannKosloff2012">{{cite journal|last1=Feldmann|first1=Tova|last2=Kosloff|first2=Ronnie|title=Short time cycles of purely quantum refrigerators|journal=Physical Review E|volume=85|issue=5|pages=051114|year=2012|issn=1539-3755|doi=10.1103/PhysRevE.85.051114|pmid=23004710|bibcode=2012PhRvE..85e1114F|arxiv=1204.4059|s2cid=31174895}}</ref> have universal features. In this case coherence contributes to the cycles power.

A [[two-stroke engine]] quantum cycle equivalent to the Otto cycle based on two [[qubits]] has been proposed.
The first qubit has frequency <math>\omega_\text{h}</math> and the second <math>\omega_\text{c}</math>. The cycle is composed
of a first stroke of partial equilibration of the two qubits with the hot and cold bath in parallel.
The second power stroke is composed of a partial or full swap between the qubits.
The swap operation is generated by a unitary transformation which preserves the [[entropy]] 
as a result it is a pure power stroke.<ref name="AllahverdyanHovhannisyan2010">{{cite journal|last1=Allahverdyan|first1=Armen E.|last2=Hovhannisyan|first2=Karen|last3=Mahler|first3=Guenter|title=Optimal refrigerator|journal=Physical Review E|volume=81|issue=5|pages=051129|year=2010|issn=1539-3755|doi=10.1103/PhysRevE.81.051129|pmid=20866207|bibcode=2010PhRvE..81e1129A|arxiv=1007.4307|s2cid=12750223}}</ref><ref name="UzdinKosloff2014">{{cite journal|last1=Uzdin|first1=Raam|last2=Kosloff|first2=Ronnie|title=The multilevel four-stroke swap engine and its environment|journal=New Journal of Physics|volume=16|issue=9|year=2014|pages=095003|issn=1367-2630|doi=10.1088/1367-2630/16/9/095003|doi-access=free|bibcode=2014NJPh...16i5003U|arxiv=1404.6182}}</ref>

The quantum Otto cycle refrigerators shares the same cycle with [[magnetic refrigeration]].<ref name="ShirronMcCammon2014">{{cite journal|last1=Shirron|first1=Peter J.|last2=McCammon|first2=Dan|title=Salt pill design and fabrication for adiabatic demagnetization refrigerators|journal=Cryogenics|volume=62|year=2014|pages=163–171|issn=0011-2275|doi=10.1016/j.cryogenics.2014.03.022|bibcode=2014Cryo...62..163S}}</ref>

==Continuous quantum engines==

Continuous quantum engines are the quantum analogues of [[turbine]]s. The work output mechanism is coupling to an external periodic field, typically the electromagnetic field. Thus the [[heat engine]] is a model for a [[laser]].<ref name="am1"/> 
The models differ by the choice of their working substance
and heat source and sink. Externally driven two-level,<ref name="Gelbwaser-KlimovskyAlicki2013">{{cite journal|last1=Gelbwaser-Klimovsky|first1=D.|last2=Alicki|first2=R.|last3=Kurizki|first3=G.|title=Minimal universal quantum heat machine|journal=Physical Review E|volume=87|issue=1|pages=012140|year=2013|issn=1539-3755|doi=10.1103/PhysRevE.87.012140|pmid=23410316|bibcode=2013PhRvE..87a2140G|arxiv=1209.1190|s2cid=18826566}}</ref> three level<ref name="GevaKosloff1996">{{cite journal|last1=Geva|first1=Eitan|last2=Kosloff|first2=Ronnie|title=The quantum heat engine and heat pump: An irreversible thermodynamic analysis of the three-level amplifier|journal=The Journal of Chemical Physics|volume=104|issue=19|year=1996|pages=7681–7699|issn=0021-9606|doi=10.1063/1.471453|bibcode=1996JChPh.104.7681G}}</ref> four-level<ref name="ScullyChapin2011">{{cite journal|last1=Scully|first1=M. O.|last2=Chapin|first2=K. R.|last3=Dorfman|first3=K. E.|last4=Kim|first4=M. B.|last5=Svidzinsky|first5=A.|title=Quantum heat engine power can be increased by noise-induced coherence|journal=Proceedings of the National Academy of Sciences|volume=108|issue=37|year=2011|pages=15097–15100|issn=0027-8424|doi=10.1073/pnas.1110234108|pmid=21876187|pmc=3174605|doi-access=free|bibcode=2011PNAS..10815097S}}</ref><ref name="HarbolaRahav2012">{{cite journal|last1=Harbola|first1=Upendra|last2=Rahav|first2=Saar|last3=Mukamel|first3=Shaul|title=Quantum heat engines: A thermodynamic analysis of power and efficiency|journal=EPL (Europhysics Letters)|volume=99|issue=5|year=2012|pages=50005|issn=0295-5075|doi=10.1209/0295-5075/99/50005|bibcode=2012EL.....9950005H|s2cid=13833767 |url=https://epljournal.edpsciences.org/10.1209/0295-5075/99/50005/pdf }}</ref> and coupled harmonic oscillators<ref name="Kosloff1984">{{cite journal|last1=Kosloff|first1=Ronnie|title=A quantum mechanical open system as a model of a heat engine|journal=The Journal of Chemical Physics|volume=80|issue=4|year=1984|pages=1625–1631|issn=0021-9606|doi=10.1063/1.446862|bibcode=1984JChPh..80.1625K}}</ref> have been studied.

The periodic driving splits the energy level structure of the working medium. This splitting allows the two level engine to couple
selectively to the hot and cold baths and produce power. On the other hand, ignoring this splitting in the derivation of the equation of motion  will violate the [[second law of thermodynamics]].<ref name="SzczygielskiGelbwaser-Klimovsky2013">{{cite journal|last1=Szczygielski|first1=Krzysztof|last2=Gelbwaser-Klimovsky|first2=David|last3=Alicki|first3=Robert|title=Markovian master equation and thermodynamics of a two-level system in a strong laser field|journal=Physical Review E|volume=87|issue=1|pages=012120|year=2013|issn=1539-3755|doi=10.1103/PhysRevE.87.012120|pmid=23410296|bibcode=2013PhRvE..87a2120S|arxiv=1211.5665|s2cid=25511420}}</ref>

Non thermal fuels have been considered for quantum heat engines. The idea is to increase the energy content of the hot bath without
increasing its entropy. This can be achieved by employing coherence<ref name="Scully2003">{{cite journal|last1=Scully|first1=M. O.|title=Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence|journal=Science|volume=299|issue=5608|year=2003|pages=862–864|issn=0036-8075|doi=10.1126/science.1078955|pmid=12511655|bibcode=2003Sci...299..862S|s2cid=120884236|doi-access=free}}</ref> or a squeezed thermal bath.<ref name="RoßnagelAbah2014">{{cite journal|last1=Roßnagel|first1=J.|last2=Abah|first2=O.|last3=Schmidt-Kaler|first3=F.|last4=Singer|first4=K.|last5=Lutz|first5=E.|title=Nanoscale Heat Engine Beyond the Carnot Limit|journal=Physical Review Letters|volume=112|issue=3|pages=030602|year=2014|issn=0031-9007|doi=10.1103/PhysRevLett.112.030602|pmid=24484127|bibcode=2014PhRvL.112c0602R|arxiv=1308.5935|s2cid=1826585}}</ref> These devices do not violate the second law of thermodynamics.

==Equivalence of reciprocating and continuous heat machines in the quantum regime==

Two-stroke, Four-stroke, and continuous machine are very different from each other. However it was shown<ref name="UzdinLevy2015">{{cite journal|last1=Uzdin|first1=Raam|last2=Levy|first2=Amikam|last3=Kosloff|first3=Ronnie|title=Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures|journal=Physical Review X|volume=5|issue=3|pages=031044|year=2015|issn=2160-3308|doi=10.1103/PhysRevX.5.031044|doi-access=free|bibcode=2015PhRvX...5c1044U|arxiv=1502.06592}}</ref> that there is a quantum regime where all these machines become thermodynamically equivalent to each other. While the intra cycle dynamics in the equivalence regime is very different in different engine types, when the cycle is completed they all turn out to provide the same amount of work and consume the same amount of heat (hence they share the same efficiency as well). This equivalence is associated with a coherent work extraction mechanism and has no classical analogue. These quantum features have been demonstrated experimentally.<ref name="Klatzow2019">{{cite journal|last1=Klatzow|first1=James|last2=Becker|first2=Jonas N |last3=Ledingham|first3=Patrick M|last4=Weinzetl|first4=Christian|last5=Kaczmarek|first5=Krzysztof T|last6=Saunders|first6=Dylan J|last7=Nunn|first7=Joshua|last8=Walmsley|first8=Ian A|last9=Uzdin|first9=Raam|last10=Poem|first10=Eilon|title=Experimental demonstration of quantum effects in the operation of microscopic heat engines|journal=Physical Review Letters|volume=122|issue=11|pages=110601|year=2019|issn=1079-7114|doi=10.1103/PhysRevLett.122.110601|pmid=30951320|bibcode=2019PhRvL.122k0601K|arxiv=1710.08716|s2cid=206318713}}</ref>

==Heat engines and open quantum systems==

The elementary example operates under quasi equilibrium conditions. Its main quantum feature is the discrete energy level structure.
More realistic devices operate out of equilibrium possessing friction heat leaks and finite heat flow.
[[Quantum thermodynamics]] supplies a dynamical theory required for systems out of equilibrium such as heat engines, thus,
inserting dynamics into thermodynamics.
The theory of [[open quantum system]]s constitutes the basic theory. For heat engines a reduced description of the dynamics
of the working substance is sought, tracing out the hot and cold baths.
The starting point is the general Hamiltonian of the combined systems:
::<math>
H = H_\text{s}s + H_\text{c} + H_\text{h}\text{h} + H_\text{sc}+H_\text{sh}
</math>
and the system Hamiltonian <math> H_\text{s}(t) </math>  is time dependent.
A reduced description leads to the equation of motion of the system:
::<math>
\frac{d}{dt} \rho = -\frac{i}{\hbar}[H_\text{s},\rho] + L_\text{h} (\rho)+L_\text{c}(\rho)
</math>
where <math>\rho</math> is the density operator describing the state of the working medium and <math>L_\text{h/c}</math> is the generator of dissipative dynamics
which includes the heat transport terms from the baths.
Using this construction, the total change in energy of the sub-system becomes:
::<math>
\frac{d}{dt} E = \left\langle \frac{\partial H_\text{s}}{\partial t} \right\rangle + \langle L_\text{h} (H_\text{s}) \rangle+\langle L_\text{c} (H_\text{s}) \rangle
</math>
leading to the dynamical version of the [[first law of thermodynamics]]:<ref name="alicki1" />
* The power <math> P =\left\langle \frac{\partial H}{\partial t} \right\rangle </math>  
* Heat currents <math> J_\text{h} = \langle L_\text{h} (H_\text{s}) \rangle </math> and <math> J_\text{c} = \langle L_\text{c} (H_\text{s}) \rangle </math>.

The rate of [[entropy production]] becomes:
::<math>
\frac{d S}{dt} = -\frac{J_\text{h}}{T_\text{h}}-\frac{J_\text{c}}{T_\text{c}} \ge 0
</math>

The global structure of [[quantum mechanics]] is reflected in the derivation of the reduced description.
A derivation which is consistent with the laws of thermodynamics is based on
the weak coupling limit.
A thermodynamical idealization assumes that the  system and the baths are uncorrelated, meaning that the total state 
of the combined system becomes a tensor product at all times:
::<math>
 \rho =  \rho_\text{s} \otimes  \rho_\text{h} \otimes \rho_\text{c} ~.
</math>
Under these conditions  the  dynamical equations of motion become: 
<math>
\frac{d}{dt} \rho_\text{s} = {L}  \rho_\text{s}~,
</math>
where <math>{L}</math> is the Liouville superoperator described in terms of the system's Hilbert space,
where the reservoirs are  described implicitly.
Within the formalism of quantum open system, <math> L </math> can take the form of the 
Gorini-Kossakowski-Sudarshan-Lindblad (GKS-L) [[Andrey Markov|Markov]]ian generator or also known just as [[Lindblad equation]]
.<ref name="Kosloff2013">{{cite journal|last1=Kosloff|first1=Ronnie|title=Quantum Thermodynamics: A Dynamical Viewpoint|journal=Entropy|volume=15|issue=12|year=2013|pages=2100–2128|issn=1099-4300|doi=10.3390/e15062100|doi-access=free|bibcode=2013Entrp..15.2100K|arxiv=1305.2268}}</ref> Theories beyond the weak coupling regime have been proposed.<ref>{{cite journal |last1=Ludovico |first1=M. F. |last2=Lim |first2=J. S. |last3=Moskalets |first3=M. |last4=Arrachea |first4=L. |last5=Sánchez |first5=D. |title=Dynamical energy transfer in ac-driven quantum systems |journal=Physical Review B |date=21 April 2014 |volume=89 |issue=16 |page=161306 |doi=10.1103/PhysRevB.89.161306 |bibcode=2014PhRvB..89p1306L |hdl=10261/116187 |s2cid=119265583 |hdl-access=free }}</ref>
<ref name="GallegoRiera2014">{{cite journal|last1=Gallego|first1=R|last2=Riera|first2=A|last3=Eisert|first3=J|title=Thermal machines beyond the weak coupling regime|journal=New Journal of Physics|volume=16|issue=12|year=2014|pages=125009|issn=1367-2630|doi=10.1088/1367-2630/16/12/125009|doi-access=free|bibcode=2014NJPh...16l5009G|arxiv=1310.8349}}</ref><ref name="EspositoOchoa2015">{{cite journal|last1=Esposito|first1=Massimiliano|last2=Ochoa|first2=Maicol A.|last3=Galperin|first3=Michael|title=Quantum Thermodynamics: A Nonequilibrium Green's Function Approach|journal=Physical Review Letters|volume=114|issue=8|pages=080602|year=2015|issn=0031-9007|doi=10.1103/PhysRevLett.114.080602|pmid=25768745|bibcode=2015PhRvL.114h0602E|arxiv=1411.1800|s2cid=11498686}}</ref>

==The quantum [[absorption refrigerator]]==

The [[absorption refrigerator]]  is of unique importance in setting an autonomous quantum device.
Such a device requires no external power and operates without external intervention in scheduling the operations
.<ref name="PalaoKosloff2001">{{cite journal|last1=Palao|first1=José P.|last2=Kosloff|first2=Ronnie|last3=Gordon|first3=Jeffrey M.|title=Quantum thermodynamic cooling cycle|journal=Physical Review E|volume=64|issue=5|pages=056130|year=2001|issn=1063-651X|doi=10.1103/PhysRevE.64.056130|pmid=11736037|bibcode=2001PhRvE..64e6130P|arxiv=quant-ph/0106048|s2cid=8201978}}</ref><ref name="LindenPopescu2010">{{cite journal|last1=Linden|first1=Noah|last2=Popescu|first2=Sandu|last3=Skrzypczyk|first3=Paul|title=How Small Can Thermal Machines Be? The Smallest Possible Refrigerator|journal=Physical Review Letters|volume=105|issue=13|pages=130401|year=2010|issn=0031-9007|doi=10.1103/PhysRevLett.105.130401|pmid=21230755|bibcode=2010PhRvL.105m0401L|arxiv=0908.2076|s2cid=2707740}}</ref><ref name="LevyKosloff2012">{{cite journal|last1=Levy|first1=Amikam|last2=Kosloff|first2=Ronnie|title=Quantum Absorption Refrigerator|journal=Physical Review Letters|volume=108|issue=7|pages=070604|year=2012|issn=0031-9007|doi=10.1103/PhysRevLett.108.070604|pmid=22401189|bibcode=2012PhRvL.108g0604L|arxiv=1109.0728|s2cid=6981288}}</ref> The basic construct includes three baths; a power bath, a hot bath and a cold bath.
The tricycle model is the template for the absorption refrigerator. 
[[File:Tricyle-1.png|thumb|Quantum tricycle absorption refrigerator. The device is composed from three baths where <math> T_\text{d} \ge T_\text{h} \ge T_\text{c} </math>. Heat flows from the power reservoir and cold bath to the hot bath.]]

The tricycle engine has a generic structure.
The basic model consists of three thermal baths: A hot bath with temperature <math>T_\text{h}</math>, 
a cold bath with temperature <math> T_\text{c} </math>
and a work bath with temperature <math>T_\text{d}</math>.

Each bath is connected to the engine via a frequency filter which can be modeled by three oscillators:
::<math>
 H_0 = \hbar \omega_\text{h}  a^{\dagger}  a +\hbar \omega_\text{c}  b^{\dagger}  b + \hbar \omega_\text{d}  c^{\dagger}  c~~,
</math>
where <math>\omega_\text{h}</math>, <math>\omega_\text{c}</math> and <math> \omega_\text{d}</math>
are the filter frequencies on resonance <math>\omega_\text{d}=\omega_\text{h}-\omega_\text{c} </math>.

The device operates as a refrigerator by removing an excitation from the cold bath as well as from the work bath
and generating an excitation in the hot bath. The term <math> a^{\dagger}  b  c </math>  in  the Hamiltonian is non linear
and crucial for an engine or a refrigerator.
::<math>
H_I = \hbar \epsilon \left(  a  b^{\dagger}  c^{\dagger} +  a^{\dagger}  b  c \right)~~,
</math>
where <math>\epsilon </math> is the coupling strength.

The first-law of thermodynamics represents the energy balance of heat currents originating from the three baths and collimating on the system:
::<math>
\frac{dE_\text{s}}{dt}= { J}_\text{h} + { J}_\text{c} +{J}_\text{d} ~~.
</math>
At steady state no heat is accumulated in the tricycle, thus <math>\frac{dE_\text{s}}{dt}= 0 </math>. 
In addition, in steady state the entropy is only generated in the baths, leading to the [[second law of thermodynamics]]:
::<math>
\frac{d}{dt}\Delta { S}_\text{u}~=~-\frac{{J}_\text{h}}{T_\text{h}} - \frac{{ J}_\text{c}}{T_\text{c}} -\frac{{ J}_\text{d} }{T_\text{d}}~\ge~0~~.
</math>
This version of the second-law is a generalisation of the statement of [[Clausius theorem]]; 
heat does not flow spontaneously from cold to hot bodies.
When the temperature <math>T_\text{d} \rightarrow \infty </math>, no entropy is generated in the power bath. 
An energy current with no accompanying [[entropy production]]  is equivalent to generating pure power:
<math> { P}={J}_\text{d} </math>, where <math>{ P}</math> is the power output.

==Quantum refrigerators and the [[third law of thermodynamics]]==

There are seemingly two independent formulations of the [[third law of thermodynamics]] both originally were stated by [[Walther Nernst]]. The first formulation is known as the [[Nernst heat theorem]], and can be phrased as:

*The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.

The second formulation is  dynamical, known as the ''unattainability principle'':<ref name="Landsberg1956">{{cite journal|last1=Landsberg|first1=P. T.|title=Foundations of Thermodynamics|journal=Reviews of Modern Physics|volume=28|issue=4|year=1956|pages=363–392|issn=0034-6861|doi=10.1103/RevModPhys.28.363|bibcode=1956RvMP...28..363L}}</ref>

*It is impossible by any procedure, no matter how idealized, to reduce any assembly to [[absolute zero]] temperature in a finite number of operations.

At steady state the [[second law of thermodynamics]] implies that the total [[entropy production]] is non-negative.
When the cold bath approaches the absolute zero temperature,
it is necessary to eliminate the [[entropy production]] divergence at the cold side
when <math>T_\text{c} \rightarrow 0 </math>, therefore
::<math>
\dot S_\text{c} \propto - T_\text{c}^{\alpha}~~~,~~~~\alpha \geq 0~~.
</math>
For <math>\alpha=0</math> the fulfillment of the [[second law of thermodynamics|second law]] depends on the [[entropy production]] of the other baths, 
which should compensate for the negative [[entropy production]] of the cold bath. 
The first formulation of the third law modifies this restriction. 
Instead of <math>\alpha \geq 0</math> the third law imposes <math>\alpha > 0 </math>, 
guaranteeing that at  absolute zero the entropy production at the cold bath is zero: <math>\dot S_\text{c} = 0</math>. 
This requirement leads to the scaling condition of the heat current <math>{ J}_\text{c} \propto T_\text{c}^{\alpha+1}</math>.

The second formulation, known as the unattainability principle can be rephrased as;<ref name="LevyAlicki2012">{{cite journal|last1=Levy|first1=Amikam|last2=Alicki|first2=Robert|last3=Kosloff|first3=Ronnie|title=Quantum refrigerators and the third law of thermodynamics|journal=Physical Review E|volume=85|issue=6|pages=061126|year=2012|issn=1539-3755|doi=10.1103/PhysRevE.85.061126|pmid=23005070|bibcode=2012PhRvE..85f1126L|arxiv=1205.1347|s2cid=24251763}}</ref>
*No refrigerator can cool a system to [[absolute zero]] temperature at finite time.

The dynamics of the cooling process is governed by the equation
::<math>
 { J}_\text{c}(T_\text{c}(t)) = -c_V(T_\text{c}(t))\frac{dT_\text{c}(t)}{dt}~~.
</math>
where <math>c_V(T_\text{c})</math> is the heat capacity of the bath. Taking <math>{ J}_\text{c} \propto T_\text{c}^{\alpha+1}</math> and <math>c_V \sim T_\text{c}^{\eta} </math> with <math>{\eta} \geq 0 </math>, we can quantify this formulation by evaluating the characteristic exponent <math>\zeta</math> of the cooling process,
::<math>
 \frac{dT_\text{c}(t)}{dt} \propto -T_\text{c}^{\zeta}, ~~~~~ T_\text{c}\rightarrow 0, ~~~~~ {\zeta=\alpha-\eta+1}
</math>
This equation introduce the relation between the characteristic exponents  <math>\zeta</math> and <math>\alpha</math>. When <math>\zeta < 0</math> then the bath is cooled to zero temperature in a finite time, which implies a violation of the third law. It is apparent from the last equation, that the unattainability  principle is more restrictive than the [[Nernst heat theorem]].

== References ==

{{Reflist}}

==Further reading==
Deffner, Sebastian and Campbell, Steve. "Quantum Thermodynamics: An introduction to the thermodynamics of quantum information", (Morgan & Claypool Publishers, 2019).<ref>{{Cite book|title=Quantum Thermodynamics|doi=10.1088/2053-2571/ab21c6|year=2019|last1=Deffner|first1=Sebastian|isbn=978-1-64327-658-8|s2cid=195791624}}</ref>

F. Binder, L. A. Correa, C. Gogolin, J. Anders, G. Adesso (eds.) "Thermodynamics in the Quantum Regime. Fundamental Aspects and New Directions." (Springer 2018)

Gemmer, Jochen, M. Michel, and Günter Mahler. "Quantum thermodynamics. Emergence of thermodynamic behavior within composite quantum systems. 2." (2009).

Petruccione, Francesco, and Heinz-Peter Breuer. The theory of open quantum systems. Oxford university press, 2002.

== External links ==
*{{cite web | url=http://phys.org/news/2014-01-nanoscale-standard-efficiency-limit.html | title=Nanoscale heat engine exceeds standard efficiency limit | website=phys.org}}

<!--- Categories --->
[[Category:Quantum mechanics]]
[[Category:Heat pumps]]
[[Category:Thermodynamics]]