Measurement-induced operation of two-ion quantum heat machines
aa r X i v : . [ qu a n t - ph ] M a r Measurement-induced operation of two-ion quantum heat machines
Suman Chand ∗ and Asoka Biswas Department of Physics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India (Dated: September 27, 2018)We show how one can implement a quantum heat machine by using two interacting trapped ions,in presence of a thermal bath. The electronic states of the ions act like a working substance, while thevibrational mode is modelled as the cold bath. The heat exchange with the cold bath is mimicked bythe projective measurement of the electronic states. We show how such measurement in a suitablebasis can lead to either a quantum heat engine or a refrigerator, that undergoes a quantum Ottocycle. The local magnetic field is adiabatically changed during the heat cycle. The performance ofthe heat machine depends upon the interaction strength between the ions, the magnetic fields, andthe measurement cost. In our model, the coupling to the hot and the cold baths are never switchedoff in an alternative fashion during the heat cycle, unlike other existing proposals of quantum heatengines. This makes our proposal experimentally realizable using current tapped-ion technology.
I. INTRODUCTION
In recent years, the study of quantum thermodynamics[1, 2] has attracted a lot of attention to understand thefundamental relation between quantum mechanics andthermodynamics [3]. In this context, the concept of quan-tum heat engines (QHEs) was first introduced by Scoviland Schulz-Dubois using three-level masers [4]. Sincethen, a significant amount of effort has been devotedin studies of several quantum heat machines, includingdifferent heat engines [5–32], namely, Carnot [5, 20, 24],Otto [7, 8, 11–16, 18, 19, 22, 23, 25, 29], Brayton [30, 31],Diesel [31] and Stirling [32] and also the refrigerator [32–40].QHEs have been proposed using different working sub-stances, e.g., two-level systems [3, 6–19], multi-level sys-tems [20, 21], and harmonic oscillators [3, 19, 22, 23].Several proposals have been made to implement such en-gines in cavity QED [24], single ion [25], optomechanicalsystems [26], quantum dots [27], and cold bosons [28].In a standard heat engine, a working substance ex-tracts heat from a hot bath at an equilibrium temper-ature T H , does a certain amount of work W , and thenreleases the rest of the energy to the cold bath at an equi-librium temperature T L ( < T H ). The ideal Carnot enginesets an upper limit for the efficiency of such an engineat η C = 1 − T L /T H . Several authors have investigatedthe performance of the QHEs to determine whether thequantum nature of the associated heat baths providesany advantage over their classical counterparts [5, 41].For example, QHEs can operate with an efficiency be-yond the classical Carnot bound η C without violatingthe second law of thermodynamics by using quantum co-herent heat reservoirs [5, 29] or the squeezed heat bath[7].Entanglement [42–44] represents a nonclassical nonlo-cal correlation between two or more quantum systemsthat does not have any classical counterpart. It is quite ∗ [email protected] interesting to investigate how the entanglement in theworking substance affects the basic quantum thermody-namical quantities, namely work and heat. In fact, an en-tangled system is more efficient in extracting work thanthe system without such nonclassical properties [6, 7, 9–19, 30]. In this context, different types of interactionbetween the subsystems of the working substance havebeen employed, namely, Heisenberg XXX [9, 12–14] andXX interaction [10, 11, 15, 17, 30], Dzyaloshinski-Moriyainteractions [10], and squeezing [13], to show that theengine efficiency can be a function of the entanglementprevailing in the system. In this paper, we demonstratehow a quantum heat machine can be implemented usingtwo ions. The thermal environment works as a hot bath,while the common vibrational mode of the ion is madeto work like a cold bath. We explore the effect of thecoupling between the electronic states of the two ions onthe efficiency of the heat engine. We discuss the suitablestrategy such that the same system can also perform likea refrigerator.In a standard classical heat engine, the working sub-stance interacts with the hot bath and the cold bath inan alternative fashion. This assumes the ability to selec-tively switch off or switch on the coupling with the bathduring certain strokes of the heat cycle. In all the exist-ing proposals, as mentioned above, primary efforts havebeen made to directly map such classical heat strokesinto quantum heat engines. However, such a ‘reciprocat-ing’ cycle may not be feasible in quantum regime, as theworking substance experiences an always-on interactionwith the bath [45, 46]. In our model, we show that it israther possible to switch between the two baths, as re-quired in heat cycles, in presence of such an always-oninteraction. In this context, we propose use of the pro-jective measurement of the electronic states of the ionsin suitable basis, that leads to an effective heat exchangewith the cold bath. Further, suitable choice of projectedstates can lead to either a heat engine or a refrigeratorcycle. In view of the above, the two-ion system, as wedescribe next, poses as an experimentally feasible modelto implement a quantum heat machine.The paper is organized as follows. In Sec. II, we de-scribe our two-ion model and discuss how the quantumheat machines can be implemented in such system. Weconclude the paper in Sec. III. II. IMPLEMENTATION OF THE CYCLES OFQUANTUM HEAT MACHINESA. Model
We consider two trapped two-level ions with the lowestlying electronic states |±i as the relevant energy levels.These internal states of the ions interact with a commonvibrational mode a . The Hamiltonian that describes thissystem can be written as (in unit of Planck’s constant ~ = 1) H = H S + H ph + H int , (1)where H S = J (cid:16) σ (1)+ σ (2) − + σ (1) − σ (2)+ (cid:17) + B (cid:16) σ (1) z + σ (2) z (cid:17) ,H ph = ωa † a ,H int = k (cid:16) a † σ (1) − + σ (1)+ a (cid:17) + k (cid:16) a † σ (2) − + σ (2)+ a (cid:17) . (2)Here H S represents the unperturbed Hamiltonian of twoions, which interact with each other with the correspond-ing coupling constant J (as in the Heisenberg XX model), H ph is the energy of the vibrational mode with frequency ω , and H int defines the interaction between the inter-nal and the vibrational degrees of freedom of the ion.The interaction strength between the electronic transi-tions of the i th ion and the vibrational mode is given by k i , ( i ∈ , B is appliedalong the quantization axis. The cases J >
J < {| ++ i , | + −i , |− + i , |−−i} , the Hamiltonian H S can bewritten in the following matrix form: H S = B J J − B . (3)The eigenvalues of the above Hamiltonian H S are givenby E = − B, E = 2 B, E = − J, E = + J , (4) with the respective eigenstates | E i = = | − −i , | E i = = | + + i , | E i = 1 √ − = 1 √ |− + i − | + −i ) , | E i = 1 √ = 1 √ |− + i + | + −i ) . (5)Further, within the Lamb-Dicke limit, it is assumedthat the ionic vibration is confined to its two lowestlying energy levels, while the higher excited states arenot populated. As a result, the vibrational mode canbe considered as a two-level cold bath (with the rele-vant phonon-number states | i and | i ) with an averagephonon number ¯ n ph ≪
1. For example, one can achieve¯ n ph ≈ .
02 in a single Be ion, that can be cooled usingstandard ion trapping technique [47]. We emphasize thata finite-level system can act as a bath, as coupling to suchbath often leads to decoherence of the system (see, e.g.,[48]). Here the system S continuously interacts with thiseffective cold bath through the Hamiltonian H int , whilethe thermal environment at an equilibrium temperature T H interacts with both the system S and the vibrationalmode. B. Implementation of different strokes
In the following, we focus on the quantum Otto cycle,that consists of four strokes: two isochoric strokes andtwo adiabatic strokes. Here we show how to implementthese strokes with the system S and the two baths asidentified above.
1. Ignition stroke
In this isochoric process (1 →
2, see Fig. 1), the ionsinteracts with the hot bath and get thermalized to anequilibrium temperature T H . To estimate the heat ex-changed by the system with the bath during this stroke,we start by rewriting the Hamiltonian H in the joint ba-sis {| + + 1 i , | + + 0 i , | + − i , |− + 1 i , |− − i , | + − i , FIG. 1. Schematic diagram of a quantum Otto cycle usingtwo trapped ions. The solid red (dashed blue) lines refer tothe adiabatic (isochoric) processes. The insets display the rel-evant electronic states and the states of the vibrational mode. |− + 0 i , |− − i} in the following matrix form: H = B + ω B k k k ω J k J ω − B + ω k k
00 0 0 0 k J
00 0 0 0 k J − B , (6)where we have assumed that coupling between the elec-tronic states and the vibrational mode are the same forboth the ions, i.e., k = k = k . The eigenstates | U n i may be written in terms of the joint basis as | U n i = a n | + + 1 i + a n | + + 0 i + a n | + − i + a n |− + 1 i + a n |− − i + a n | + − i + a n |− + 0 i + a n |− − i , n ∈ [1 , . (7)The interaction with the thermal bath leads to the fol-lowing mixed state: ρ ( H )1 = X n =1 p n | U n ih U n | , p n = exp ( − U n /k B T H ) P n =1 exp ( − U n /k B T H ) , (8)where p n is the occupation probability of the n th eigen-state | U n i (with corresponding eigenvalue U n ) of the totalHamiltonian H . Note that this state is achieved at thesteady state irrespective of the initial preparation of theions.The reduced density matrix of the system S can beobtained by taking the partial trace over the vibrational states as ρ ( H ) S = 1 P H X n =1 e − U n /K B T H (cid:2) | ++ i h ++ | ( a n + a n )+ | + −i h + −| ( a n + a n ) + |− + i h− + | ( a n + a n )+ |−−i h−−| ( a n + a n )+( a n a n + a n a n )( | ++ i h + −| + | + −i h ++ | )+( a n a n + a n a n )( | ++ i h− + | + |− + i h ++ | )+( a n a n + a n a n )( | ++ i h−−| + |−−i h ++ | )+( a n a n + a n a n )( | + −i h− + | + |− + i h + −| )+( a n a n + a n a n )( | + −i h−−| + |−−i h + −| )+( a n a n + a n a n )( |− + i h−−| + |−−i h− + | )](9)where P H = P n =1 exp[ − U n /k B T H ] is the normalizationconstant.This can be rewritten in terms of the energy eigenstates | E i i of the system Hamiltonian H S through the inversetransformation of the Eq. (5) | ++ i = | E i , | + −i = 1 √ | E i − | E i ) , |− + i = 1 √ | E i + | E i ) , |−−i = | E i . (10)Using above equation, we can get the occupation proba-bility P i of the i th eigenstate | E i i ( i ∈ [1 , P ( T H ) = X n =1 e − U n /K B T H (cid:0) a n + a n (cid:1) ,P ( T H ) = X n =1 e − U n /K B T H (cid:0) a n + a n (cid:1) ,P , ( T H ) = X n =1 e − U n /K B T H h(cid:16) a n ∓ a n a n (cid:17) + (cid:16) a n ∓ a n a n (cid:17) + (cid:16) a n ∓ a n a n (cid:17) + (cid:16) a n ∓ a n a n (cid:17)i . (11)The average energy of the system under considerationcan be written as U = P i =1 E i P i . Here the change in the E i s corresponds to the heat exchange, while the changein the probabilities refer to the certain work done duringthe cycle [49]. Based on the initial preparation of the ion,if the initial probability for being in the i th eigenstate is P i ( T L ), then the heat exchanged with the hot bath bythe system S during this stroke is given by Q H = X i =1 E Hi { P i ( T H ) − P i ( T L ) } . (12)Note that in this process, the magnetic field is kept fixedat B = B H , such that the eigenvalues E Hi of the systemHamiltonian H S also remain constant and therefore nowork is done. Due to the change in the occupation prob-abilities, only the heat is exchanged during this cycle.
2. Expansion stroke
During this adiabatic cycle (2 →
3, see Fig. 1),the magnetic field is modified from B H to B L , suchthat the occupation probabilities of the four eigenstates {| E i i , i = 1 , , , } remain unchanged. Consequentlythere is no heat exchange between the system andheat bath. However, the corresponding eigenvalues E Hi change to the values E L , = ∓ B L and E L , = ∓ J . Thisamounts to the following work done by the system S dur-ing this cycle: W = X i =1 P i ( T H ) (cid:0) E Li − E Hi (cid:1) . (13)
3. Exhaust stroke
In an usual Otto engine, this stroke is associated withcooling of the system through heat release to the coldbath. In the present case, in this stroke (3 →
4, see Fig.1), the system exchanges heat Q L with the cold bathand the system Hamiltonian changes from H S ( B H ) to H S ( B L ). To estimate the Q L , we start with the followingstate of the ions, that is adiabatically evolved thermalstate, as attained at the end of the expansion stroke: ρ ( L )1 = U † I ρ ( H )1 U I , (14)where U I = T exp (cid:20) − i Z τ dt ′ H ( t ′ ) (cid:21) ,H ( t ) = H S ( t ) + H ph + H int ,H S (0) = J (cid:16) σ (1)+ σ (2) − + σ (1) − σ (2)+ (cid:17) + B H (cid:16) σ (1) z + σ (2) z (cid:17) ,H S ( τ ) = J (cid:16) σ (1)+ σ (2) − + σ (1) − σ (2)+ (cid:17) + B L (cid:16) σ (1) z + σ (2) z (cid:17) . (15)Here T stands for time-ordering and τ is the finite timefor the adiabatic change of the magnetic field from B H and B L during the expansion stroke. The state ρ ( L )1 canbe written in the joint basis of the electronic states andthe vibrational mode as ρ ( L )1 = X m,n =1 ρ ( mn )1 | m i h n | , | m i , | n i = | + + 1 i , | + + 0 i , |± ∓ i , |− − i , |± ∓ i , |− − i . (16)Note that, at thermal equilibrium, the system S is en-tangled with the vibrational mode. To this end, we pro-pose a projective measurement of the state of the systemS, thereby disentangling S from the vibrational mode.Further, if the two-ion system is measured in the groundstate, the occupation probabilities of the higher excitedstates reduce to zero. This mimics the release of heat tothe cold bath, as is usually required in an exhaust strokeof an Otto engine. Clearly, such a measurement-induced heat exchange depends upon the choice of the projectedstate. In fact, as shown later in this Section, the systemcan work as a quantum heat engine or a quantum refrig-erator, depending upon the measurement basis. In thefollowing, we choose the system eigenstates | E i i as themeasurement basis.Generally speaking, upon projection onto the eigen-state | E i i of the system, the density matrix ρ ( L )1 getsfactorized and can be written as ρ ( L )1 | meas = | E i ih E i | X k,l =0 r ( i ) kl | k i h l | , (17)where | k i , k ∈ , r ( i ) kl are the relevant density matrix elements be-tween the states | k i , | l i , corresponding to the projectiononto | E i i . In this way, the system exchanges heat withthe cold bath and thereafter gets decoupled from the coldbath.Through this heat-exchange process, the probabilitydistribution { P i } of the eigenstates also changes, whilemaintaining the corresponding eigenvalues identical. Thelocal magnetic field B L is kept constant during thisstroke. If the final occupation probability for the i theigenstate becomes P i ( T L ), the heat exchange betweenthe system and the cold bath can be calculated as Q L = X i =1 E Li [ P i ( T L ) − P i ( T H )] . (18)Note that, as in the ignition stroke, no work is done dur-ing this stroke as well. The measurement process, asdescribed above, is apparently probabilistic and relies onthe result of the measurement. A reasonable alternativeoption for decoupling the system from the bath couldbe to use the non-selective measurement, as described in[50]. J Q H , Q L , W Q H Q L W J η ( a ) ( b )FIG. 2. Variation of (a) heat exchanged Q H (dotted blue)and Q L (solid red), with the hot and the cold bath, respec-tively and the net work done (dot-dashed magenta) W and(b) the efficiency η as a function of the coupling constant J .The others parameters for the cycle are B H = 10 , B L = 5 , k =0 . , ω = 1 , k B T H = 3 .
5. Here the measurement is done in thebasis | E i . The physically acceptable parameter region forthe engine to operate is J ≤ B L = 10.
4. Compression Stroke
For this stroke (4 → B L to B H . The system remains in contact with the hot bath.During the expansion stroke, the occupation probabili-ties of the energy eigenstates | E i i remain unaltered. Theeigenvalues change from E Li to E Hi due to the change inthe magnetic field. This leads to the following work doneduring this stroke: W = X i =1 P i ( T L )( E Hi − E Li ) . (19)It must be borne in mind that after the compressionstroke ends, the heat machine goes into the next cycle,starting with the ignition stroke. During this stroke, thesystem gets thermalized to the state (8), irrespective ofits initial state and therefore the cycle continues in a sim-ilar fashion. Further, as long as the Lamb-Dicke limit ismaintained, the vibrational mode remains confined to itstwo lowest energy levels and can be reused as a cold bathduring the next cycle. J Q H , Q L , W -40-2002040 Q H Q L W J ε ( a ) ( b )FIG. 3. Variation of (a) heat exchanged Q H (dotted blue)and Q L (solid red), with the hot and the cold bath, respec-tively and the net work done W (dot-dashed magenta) and(b) the COP ε as a function of the coupling constant J . Theparameters are the same as in Fig. 2. Here the measurementis done on the state | E i . C. Efficiency and COP of the heat machine
In the following, we consider the measurement in dif-ferent eigenstates of H S . Case I: Projection in | E i state : This state is theground state as long as J < B [see Eq. (4)]. So theprojection of the state of the system S into | E i corre-sponds to cooling of the system. This heat can be therebyextracted from the system and transferred into the vibra-tioanal mode. In this case, we find that the heat releasedinto the cold bath is Q L <
0, while the heat absorbedby the system becomes Q H > W >
0. This situation clearly refers to executing a quantum heatengine. The efficiency of the heat engine is defined as J Q H , Q L , W -40-2002040 Q H Q L W J ε ( a ) ( b )FIG. 4. Variation of (a) heat exchanged Q H (dotted blue)and Q L (solid red), with the hot and the cold bath, respec-tively and the net work done W (dot-dashed magenta) and(b) the COP ε as a function of the coupling constant J . Theparameters are the same as in Fig. 2. Here the measurementis done on the state | E i . η = Work OutputHeat Input = Q H + Q L Q H . It is easy to see from theFig. 2(a) that the system S behaves like a heat enginefor the parameter regime J ≤ B L , i.e., as long as | E i remains the ground state. Note that in this regime, theefficiency increases for increasing values of J and becomesnear to unity. Beyond this regime, one attains a unphys-ical situation. In Fig. 2(b), we show how the efficiency η approaches unity with increase in J to its upper limit2 B L . Further in the limiting case of uncoupled spins(i.e., J = 0), the efficiency becomes η = 1 − B L B H , thatmatches with the results for a single-spin quantum Ottoengine [51]. Case II: Projection in | E i and | E i states : If the mea-surement is done in the other eigenstates, we obtain apossibility of the refrigerator action, in which the systemabsorbs heat from the cold bath ( Q L >
0) and releasesheat into the hot bath ( Q H < W < J fora given set of values of magnetic fields B L and B H . J Q H , Q L , W -20-10010 Q H Q L W J ε ( a ) ( b )FIG. 5. Variation of (a) heat exchanged Q H (dotted blue)and Q L (solid red), with the hot and the cold bath, respec-tively and the net work done W (dot-dashed magenta) and(b) the COP ε as a function of the coupling constant J . Theparameters are the same as in Fig. 2. Here the measurementis done in the state | E i . The physically acceptable parameterregion for the refrigerator to operate is J ≤ B L = 10. We here emphasize that the states | E i and | E i arethe eigenstates with positive eigenvalues and correspondto excited states. So projecting the system into theseeigenstates refers to heating of the system, as one wouldrequire in the stroke associated with the cold bath ina refrigeration cycle. The performance of a refrigeratoris quantified in terms of the coefficient of performance(COP) ε = Heat Input | Work Output | = Q L | Q H + Q L | . In Figs. 3(b) and4(b), we show the variation of ε with J . Clearly themeasurement in the | E i state leads to a much betterperformance as a refrigerator for a given value of J . Case III: Projection in | E i state : Quite interestingly,for a certain regime, J ≤ B L , the measurement in the | E i state leads to a refrigeration effect. This is because,in this parameter regime, | E i remains an excited state[see Eq. (4)] and the measurement in such a state leads toan effective heating of the system S (i.e., Q L > Q H and W remain negative [see Fig. 5(a)].This refers to a situation, in which the system behavesas a refrigerator. However, for larger values of J , thethe performance ε decreases with J , as shown in Fig.5(b). For J > B L , one reaches an unphysical regime, inwhich neither a heat engine nor a refrigerator action isachievable.We show in the parametric plots in Fig. 6 how theefficiency η and the COP ε vary with the work done bythe system or on the system, respectively. It is clear fromFig. 6(a) that both the work output and the efficiency ofthe heat engine are large for J = 2 B L , if one measuresthe system in the state | E i . Similarly, for the measure-ment in | E i [Fig. 6(b)] and | E i [Fig. 6(d)] states, therequired work to be done becomes less, while the coef-ficient of performance of the refrigerator increases, as J is increased. The measurement in | E i state is not a de-sirable choice for refrigeration, because to obtain a largeCOP, one would require a large amount work [Fig. 6(c)]. D. Effect of measurement
In the discussion above, we have not included the mea-surement cost. This cost would eventually restrict theperformance of a heat engine and a refrigerator. The costfor the projective measurement for one qubit is k B T ln 2[52], which is the same as the cost of classical measure-ment of one bit. This leads to modified definition of theefficiency [53] as η ′ = energy outputenergy input = Q in + Q out Q in + M , (20)where M is the cost of measurement. In the present caseof two qubits, M = 2 k B T ln 2.As an example, we show in Fig. 7 how the efficiency η ′ varies with J and the work W done by the system,when the system is measured in the state | E i . Clearly,the achievable efficiency becomes less than that in Fig.2(b), considering the effect of measurement cost. W η W -20 -15 -10 ε ( a ) ( b ) W -10 -9.8 -9.6 -9.4 ε W -10 -8 -6 -4 ε ( c ) ( d )FIG. 6. (a) (Color online) Parametric variation of the ef-ficiency η with the work done W by the system, when thesystem is measured in | E i state. Parametric variation of theCOP ε with the work done W on the system, when the systemis measured in (b) | E i (c) | E i , and (d) | E i states. In allthe plots, J is varied from 0 to 2 B L . The others parametersare the same as in Fig. 2. J η ′ W η ′ ( a ) ( b )FIG. 7. Variation of the efficiency η ′ (a) as a function of(a) coupling constant J and (b) work W done by the sys-tem, when the system in measured in | E i state. In (b), J ischanged from 0 to 2 B L . The others parameters are the sameas in Fig. 2. III. CONCLUSION
We have shown how two interacting trapped ions canbe employed to perform as a quantum Otto machine.These ions interact with a thermal bath at an equilibriumtemperature T H , while the common vibrational mode ofthe ions is chosen as the relevant cold bath. In order toperform the adiabatic stroke of the Otto cycle, we changethe local magnetic field adiabatically. A projective mea-surement of the electronic states during one of the iso-choric strokes leads to heat exchange with the cold bath.We find that by suitable choice of the projected state,one can effect either heat release to the cold bath or heatabsorption from the cold bath, thereby leading to a heatengine or a refrigerator operation. Particularly speaking,projection onto the ground state of the system Hamilto-nian results in a heat engine, while that onto the otherstates leads to refrigeration. The efficiency of the heatengine or the coefficient of performance of the refriger-ator depends on the magnetic fields and the interactionstrength between the two ions. We assess the perfor-mance of these heat machines, by including the measure- ment cost, as a function of the interaction strength. Weemphasize that our model is feasible with the currenttrapped-ion technology, as we do not need to switch offthe interaction with any of the baths during the heat cy-cle, and still can mimic all the heat strokes of a standardOtto cycle. [1] G. Gemma, M. Michel, and G. Mahler, Quantum Ther-modynamics (Springer, New York, 2004).[2] E. P. Gyftopoulos, and G. P. Beretta,
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