Charger-mediated energy transfer in exactly-solvable models for quantum batteries
Gian Marcello Andolina, Donato Farina, Andrea Mari, Vittorio Pellegrini, Vittorio Giovannetti, Marco Polini
CCharger-mediated energy transfer in exactly-solvable models for quantum batteries
Gian Marcello Andolina,
1, 2, ∗ Donato Farina,
1, 2
Andrea Mari, Vittorio Pellegrini, Vittorio Giovannetti, and Marco Polini Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy NEST, Scuola Normale Superiore, I-56126 Pisa, Italy (Dated: November 27, 2018)We present a systematic analysis and classification of several models of quantum batteries involv-ing different combinations of two-level systems and quantum harmonic oscillators. In particular,we study energy-transfer processes from a given quantum system, termed “charger”, to anotherone, i.e. the proper “battery”. In this setting, we analyze different figures of merit, including thecharging time, the maximum energy transfer, and the average charging power. The role of couplingHamiltonians which do not preserve the number of local excitations in the charger-battery systemis clarified by properly accounting them in the global energy balance of the model.
I. INTRODUCTION
Currently there is worldwide interest in exploitingquantum phenomena such as superposition, quantum co-herence, and entanglement for future technologies inthe realms of communication, computation, simulation,and sensing/metrology. On a seemingly disconnettedpath, the possibility to use quantum resources to achievesuperior performances in the manipulation of energy iscurrently being intensively studied .In this context, a number of researchers has been work-ing on “quantum batteries” , i.e. quantum me-chanical systems for storing energy where genuine quan-tum effects can be used to obtain more efficient and fastercharging processes with respect to classical analog sys-tems. From an abstract point of view, the fact thatquantum coherent processes can be faster then classicaloperations is a known fact emerging from quantum infor-mation theory and, specifically, from the concept of quan-tum speed limits . The idea of exploiting quantumcoherence for efficiently charging (or discharging) quan-tum batteries has been studied in a fully abstract fash-ion , and, more recently, by expoiting concretemodels that can be implemented in laboratories .In this Article we follow the same research line but,differently from previous attempts , we focus onlyon minimal models of quantum batteries, which can besolved exactly. The simplicity of our toy models allowsus, on the one hand, to avoid all subtle approximationsand formal technicalities needed to handle more sophis-ticated models such as those studied in Refs. 15 and16, and, on the other hand, to identify general features,which are independent of the details of the specific ex-perimental implementation.To this end, we model a quantum battery as either atwo-level system (TLS) or a quantum harmonic oscilla-tor (QHO), the same simplified picture being also usedfor the charging system—see Fig. 1a) and b). The basicidea here is that TLSs and QHOs can be viewed as ele-mentary building blocks of more complex quantum bat-teries. Also, the models considered in this work can beexperimentally implemented. Indeed, TLSs and QHOs are ubiquitous in atomic and condensed matter physics.They are elementary building blocks of cavity QED ar-chitectures and in systems of trapped ions , ultra-cold atoms , superconducting circuits , and semi-conductor quantum dots .For the three charger-battery combinations illustratedin Figs. 1b) and by means of a unitary Hamiltonian in-teraction, we study energy transfer processes from thecharger initialized in an arbitrary state to the quantumbattery initialized in the ground state. We are particu-larly interested in understanding the relevance of quan-tum coherence for improving the efficiency of the charg-ing process and in clarifying the role of coupling termsthat do not commute with the local Hamiltonians of themodel. Among the main results of this Article, we em-phasize the following ones: i) when a TLS-based quantumbattery is charged via a QHO it is convenient to preparethe charger in a Fock state which, for sufficiently largeenergies, can be safely replaced by a coherent state giv-ing approximately equal performances; ii) in the previoussituation, we observe that the charging time is inverselyproportional to the square root of the charger energy.In our treatment, we focus on average energies(i.e. Hamiltonian expectation values) without taking intoaccount statistical fluctuations. For this reason, the pre-sented approach is applicable also in contexts other thanthat of quantum batteries, such as that of heat transportprocesses .Our Article is organized as follows. A general theoryof energy transfer and different models of quantum bat-teries are presented in Sect. II for the special case wherethe coupling Hamiltonian between the charger and thebattery preserve the local energy of the system. Thisanalysis is then extended to non-commuting Hamilto-nians (i.e. going beyond energy-preserving protocols) inSect. III. A brief summary and our main conclusions arefinally reported in Sect. IV. Useful technical details canbe found in Appendix A. a r X i v : . [ qu a n t - ph ] N ov t ⌧ H H B H A H B H A H B H A (a) g A ! gg A ! B ! B ! A ! B ! (b)(1)(2)(3) FIG. 1. (Color online) Panel (a) shows the time-dependentinteraction protocol that allows energy flow between thecharger, described by the Hamiltonian H A , and the bat-tery, described by the Hamiltonian H B . At time t < τ thetwo systems A and B do not interact and cannot exchangeenergy, their dynamics being governed by the Hamiltonian H = H A + H B . In the time interval 0 < t < τ the Hamilto-nian H is switched on and the two systems interact. Finally,the interaction is switched off at time τ , and the energy E B ( τ )stored in the battery B is a conserved quantity. Panel (b)illustrates cartoons of the three charger-battery toy modelsintroduced and studied in this Article. Sub-panel (1): energytransfer is studied between two qubits; sub-panel (2): energytransfer is studied between a quantum harmonic oscillator anda qubit; sub-panel (3): energy transfer is studied between twoquantum harmonic oscillators. II. ENERGY TRANSFER IN THECHARGER-QUANTUM BATTERY SETUP
In this Section we introduce a general theoreticalframework to address the charging process of a quan-tum battery schematically represented in Fig. 1. Weconsider two quantum systems, A and B, where A is the“charger”, initially containing some input energy, whileB is the proper “quantum battery”, initially preparedin the ground state. We denote by ρ A ( t ) and ρ B ( t ) thedensity matrices representing their respective quantum states and with H A and H B the corresponding time-independent local Hamiltonians. We can therefore iden-tify with E A ( t ) = tr[ H A ρ A ( t )] the energy of the chargerand with E B ( t ) = tr[ H B ρ B ( t )] the energy of the quan-tum battery. We assume that at time t = 0 the chargeris initialized in an arbitrary state while the battery is inits ground state, i.e. ρ AB (0) = ρ A (0) ⊗ | (cid:105)(cid:104) | B , (1)such that E A (0) > E B (0) = 0. We model thecharging process as the physical operation of letting Aand B interact for a finite amount of time τ , as in Fig. 1a).More precisely, we assume the following global Hamilto-nian H ( t ) = H + λ ( t ) H , (2)where H = H A + H B , H is some given interactionHamiltonian, and λ ( t ) is a dimensionless coupling con-stant, equal to 1 for t ∈ [0 , τ ] and 0 elsewhere. Physically,the “on/off” coupling constant λ ( t ) is the only classicalparameter which can be externally controlled. (This canbe implemented using a quantum clock, see e.g. Ref. 41).This implies that the total energy E ( t ) = tr[ H ( t ) ρ AB ( t )]is constant at all times with the exception of the switch-ing times, i.e. t = 0 and t = τ , where some non-zero en-ergy can be exchanged, representing the thermodynamicwork cost of switching on and off the interaction. Suchcost can be quantified as the total energy change at bothswitching points, i.e. δE sw ( τ ) ≡ [ E ( τ + ) − E ( τ − )] + [ E (0 + ) − E (0 − )]= tr {H [ ρ AB (0) − ρ AB ( τ )] } , (3)where ρ AB ( τ ) = e − i ( H + H ) τ ρ AB (0) e i ( H + H ) τ ( (cid:126) = 1throughout this Article).We first consider the case in which the interactionHamiltonian commutes with the sum of the local terms,[ H , H ] = 0 , (4)ensuring δE sw ( τ ) = 0 for every initial state. From aphysical point of view, this choice corresponds to energy-preserving protocols in which all the energy stored inthe quantum battery B at the end of the charging pro-cess originates, without any thermodynamic ambiguity,from the charger A. In this case, the performances ofthe charger-battery setup can be studied in terms of the(mean) energy stored in the battery and the correspond-ing average storing power, defined respectively as E s ( τ ) ≡ E B ( τ ) = tr[ H B ρ B ( τ )] , (5) P s ( τ ) ≡ E s ( τ ) /τ . (6)Upon optimization with respect to the charging time τ ,we can extract from these functionals a collection of fig-ures of merit which quantify the “quality” of a givencharging protocol from different perspectives. Specifi-cally, we define the maximum (mean) energy that can bestored in the quantum battery E s ≡ max τ [ E s ( τ )] ≡ E ( τ ) , (7)the maximum power,˜ P s ≡ max τ [ P s ( τ )] , (8)and their corresponding optimal charging times τ ≡ min E ( τ )= E s ( τ ) , ˜ τ ≡ min P (˜ τ )= ˜ P s ( τ ) . (9)Finally, we also introduce the charging power at maxi-mum energy, P s ≡ E s /τ = E s ( τ ) /τ , (10)which, due to the fact that τ and ˜ τ may not necessarilycoincide, will in general be smaller than ˜ P s .For non-commuting interactions [ H , H ] (cid:54) = 0 morecaution should be used when defining the figures of meritfor a given charging protocol. Indeed, in this case, thefinal energy of the quantum battery will not come onlyfrom the charger A but also from the classical modula-tion of the coupling constant λ ( t ) and, for this reason,the “quality” of the protocol has some degree of arbi-trariness depending on which of the two energy fluxes isactually desired. The analysis of this particular situationis postposed to Sect. III. In the next Section, instead,we study E s ( τ ) and P s ( τ ) for three alternative modelsof the charger-battery setting that fulfill the commuta-tivity identity (4) and admit full analytical treatment,looking for the presence of advantages associated withthe quantum structure of the system dynamics. As auseful tool for this analysis, we compare the optimalcharging times (9) to the quantum speed limit (QSL)time τ QSL18–21 that defines the minimum temporal in-terval needed to let a quantum system to evolve be-tween two orthogonal states under the action of its (time-independent) Hamiltonian H , i.e. τ QSL = π {(cid:104)H(cid:105) , (cid:104) δ H(cid:105)} , (11)with (cid:104)H(cid:105) indicating the gap between the mean value andthe ground-state energy of H , evaluated on the systeminput state, and (cid:104) δ H(cid:105) being instead the correspondingsquare root of the variance of H . A. Energy transfer between two TLSs
We begin by studying the simplest, yet non-trivial,case of a charger-battery setting which we will use asreference for the following study. Here, the chargerand quantum battery are two resonant TLSs (alsonamed qubits throughout this Article), coupled via anenergy-preserving interaction that merely shifts excita-tion quanta between the two qubits. Accordingly, wewrite the system Hamiltonian (2) in terms of the follow- ing components: H A = ω (cid:16) σ (A) z + 1 (cid:17) , (12) H B = ω (cid:16) σ (B) z + 1 (cid:17) , H = g (cid:16) σ (A) − σ (B)+ + σ (A)+ σ (B) − (cid:17) , where ω is the level spacing of each TLS, σ (S) z are Paulimatrices acting on the S = A,B subspaces, σ (S)+ , σ (S) − arespin ladder operators acting on the same subspaces, and g is the coupling strength. In this case, energy transfer isoccurring through the well-known Rabi oscillations, seeFig. 2. Indeed, exploiting the fact that Eq. (4) holds,one can easily show that, assuming the charger A to beinitialized in the excited state | (cid:105) A and the qubit B in theground state | (cid:105) B , the evolved system can be expressedas | Ψ( t ) (cid:105) AB = e − iω t (cid:2) cos( gt ) | (cid:105) A | (cid:105) B (13) − i sin( gt ) | (cid:105) A | (cid:105) B (cid:3) , yielding E s ( τ ) = ω sin ( gτ ) , P s ( τ ) = ω sin ( gτ ) τ , (14)for the quantities (5) and (6). The maximum energyis hence provided by E s = ω and is achieved at time¯ τ = π/ (2 g ) (the corresponding power at maximum en-ergy transfer (10) being P s = 2 gω /π ). The maximumpower instead is ˜ P s ≈ . gω and is achieved at time˜ τ ≈ . /g (result obtained by simple numerical inspec-tion of the function y = sin ( x ) /x , which has maximumvalue ˜ y ≈ .
72 at ˜ x ≈ . B. Energy transfer between a QHO and a TLSbattery
We now focus on the case in which the charger A isdescribed by a QHO while the quantum battery B is stilldescribed by a TLS. The relevant Hamiltonians are H A = ω a † a , (15) H B = ω (cid:16) σ ( B ) z + 1 (cid:17) , H = g (cid:16) a † σ ( B ) − + aσ ( B )+ (cid:17) , where a † ( a ) is the creation (destruction) bosonic oper-ator acting on A, and where again ω and g are respec-tively the characteristic frequency of both systems andthe coupling strength parameter of the model. The modeldescribed by the total Hamiltonian H = H A + H B + H is the so-called Jaynes-Cumming model , which can beseen as the building block of much more complicatedmany-body models such as the Tavis-Cummings andDicke models . π/ π π/ πgτ . E s ( τ ) / ω (a) π/ π π/ πgτ . P s ( τ ) / ( g ω ) (b) FIG. 2. (Color online) Panel (a) displays the stored energy E s ( τ ) (in units of ω ) as a function of gτ , for the case of twocoupled qubits. Panel (b) shows the average charging power P s ( τ ) (in units of gω ) as a function of gτ . We clearly seethat the quantum battery is charged by the charging qubitvia Rabi oscillations. We now note that from the commutativity relation (4)the operator K = a † a + σ z /
2, which counts the totalnumber of excitations, commutes with the full Hamilto-nian H and is therefore a constant of the motion. Wecan hence solve the dynamics by restricting the anal-ysis to subspaces with a given number n of excitationsspanned by the vectors | n (cid:105) A | (cid:105) B and | n − (cid:105) A | (cid:105) B , whereHamiltonian simplifies to the one described in the pre-vious Section—see Eq. (12)—with appropriate renor-malized parameters. Here the eigenvectors of H are |± , n (cid:105) = ( | n (cid:105) A | (cid:105) B ± | n − (cid:105) A | (cid:105) B ) / √ ω ± ,n = nω ± √ ng . Thereforeif we start from the initial quantum state | n (cid:105) A | (cid:105) B , itstemporal evolution is given by | Ψ n ( t ) (cid:105) AB = e − inω t (cid:2) cos( √ ngt ) | n (cid:105) A | (cid:105) B − i sin( √ ngt ) | n − (cid:105) A | (cid:105) B (cid:3) . (16) Consider next the case of a generic input of the form (1)where we fix the initial energy to the value E A (0) = E in and hence the average number of excitations to K = E in /ω . Expanding ρ AB (0) on the Fock basis | n (cid:105) A | (cid:105) B ,from Eq. (16) we can calculate the mean stored energyand the average charging power: E s ( τ ) = ω (cid:88) n p ( K ) n sin ( √ ngτ ) , (17)and P s ( τ ) = ω (cid:80) n p ( K ) n sin ( √ ngτ ) τ , (18)where p ( K ) n is the diagonal part of ρ A (0) in the Fock basis,subject to the constraint of yielding the selected initialenergy, i.e. (cid:80) n np ( K ) n = K .Let us first study the case of an initial state of the Focktype. In this case, p ( K ) K = 1 and p ( K ) n (cid:54) = K = 0, and Eqs. (17)and (18) become E Fs ( τ ) = ω sin ( √ Kgτ ) , (19) P Fs ( τ ) = ω sin ( √ Kgτ ) τ , (20)where “F” denotes that the initial state of the charger isa Fock state. The maximum of Eq. (19) is E Fs = ω andis achieved for the first time at¯ τ = π/ (2 √ Kg ) . (21)At this special time the battery gets completely charged,resulting in a final state of the AB system that exactlyfactorizes, i.e. | K − (cid:105) A | (cid:105) B . Due to the properties of thefunction sin ( x ) /x —Sect. II A—the maximum value ofthe power (20) is instead provided by ˜ P Fs ≈ . gω √ K and is achieved at time˜ τ ≈ . / ( √ Kg ) , (22)which, apart from a multiplicative constant, exhibits thesame 1 / √ K scaling of Eq. (21). Compared with the twoqubits model of the previous Section, Eq. (14), in thepresent case there is still a transfer of only one quan-tum of energy from A to B but in a time window thatis reduced by a factor 1 / √ K . Thus we can say that,from the initial number K of excitations in the system,only one is eventually transferred from the charger to thequantum battery, with the other K − / √ K improvement reported in Eq. (21)which, despite the lack of collective behavior stemmingfrom the mutual interactions between K qubit batteriescoupled to a single common photonic mode, mimics asimilar scaling observed in Ref. 15. Such advantage canalso be connected with the QSL bound (11) confirmingan argument of Ref. 14. Indeed, by direct evaluation, wehave (cid:104)H(cid:105) = Kω and (cid:104) δ H(cid:105) = (cid:113) (cid:104) δ H A (cid:105) + Kg ≥ √ Kg , π/ π π/ π √ Kgτ / E s ( τ ) / ω (a) π/ π π/ π √ Kgτ / P s ( τ ) / ( √ K g ω ) (b) π/ π π/ π √ Kgτ / E s ( τ ) / ω (c) π/ π π/ π √ Kgτ / P s ( τ ) / ( √ K g ω ) (d) FIG. 3. (Color online) Panel (a) displays the stored energy E s ( τ ) (in units of ω ) as a function of √ Kgτ , for the case ofa qubit charged by a QHO. The initial number of excitation is K = 3. Different curves refer to results obtained for threedifferent choices of the initial state of the charger: Fock state (blue solid line), coherent state (red dashed line), and Gibbs state(dash-dotted green line). Panel (b) shows the average charging power P s ( τ ) (in units of √ Kgω ) as a function of √ Kgτ . Theinitial number of excitation is K = 3. Color coding as in panel (a). Panel (c) same as in panel (a) but for K = 20. The blackdotted line represents the energy of a Gibbs state of the qubit, with temperature equal to that of the initial Gibbs state of theQHO. Note that, for long times (not shown), the system has revivals, due the unitarity of the time evolution. Panel (d) showsthe average charging power corresponding to panel (c). This figure clearly shows the “optimality” of the Fock state, which isthe best choice for maximizing the energy and power. Note that, for K (cid:29) E s ( τ ) and P s ( τ ) calculated for an initial Fock state. which, for K big enough, gives τ QSL (cid:39) π/ (2 √ Kg ) repro-ducing the scaling of Eq. (21).Consider next the case where A is initialized in ageneric (not necessarily Fock) input state. From a closeinspection of Eqs. (17) and (18) it turns out that, forfixed K , the values of E Fs and ˜ P Fs are bigger than thecorresponding quantities one can obtain with any otherinput state of A having the same expectation value ofthe input energy of the selected Fock state. Indeed, fromEq. (17) we have E s ( τ ) ≤ ω = E Fs , while from Eq. (18) we obtain P s ( τ ) = ω g (cid:88) n √ n p ( K ) n (cid:20) sin ( g √ nτ ) g √ nτ (cid:21) ≤ ω g max x (cid:20) sin ( x ) x (cid:21) (cid:88) n √ n p ( K ) n ≤ ω g √ K max x (cid:20) sin ( x ) x (cid:21) = ˜ P Fs , (23)where in the second inequality we used the concavity ofthe function √ x to write (cid:80) n p ( K ) n √ n ≤ √ K . These rela-tions are also evident in Fig. 3 where we plot the storedenergy E Fs ( τ ) and the average charging power P Fs ( τ ) ofthe Fock input case, together with the corresponding val-ues of E s ( τ ) and P s ( τ ) obtained for different choices ofthe input state of A (namely the case of a coherent in-put and the one of a thermal distribution, characterizedby a Poissonian distribution p n = e − K K/n ! and a Gibbsdistribution p n = [ K/ ( K + 1)] n / ( K + 1), respectively).According to the above analysis, for fixed input meanenergy of the charger A, Fock states provide optimal per-formances with respect to all our figures of merit. A Fockstate, however, is not always easy to be prepared exper-imentally for an arbitrary number of photons K . Onemay therefore be interested in replacing it with a moreaffordable coherent state |√ K (cid:105) having the same energy.Luckily, from our previous formulas (see also Fig. 3) it isevident that for K (cid:29)
1, a coherent state and a Fock stateproduce almost indistinguishable results. More generally,this fact is valid for every initial state with a sufficientlypeaked energy distribution { p ( K ) n } n , i.e. a state such that (cid:104) ( a † a ) (cid:105) (cid:28) (cid:104) a † a (cid:105) . Such weak dependence on the specificinitial state is clearly crucial for the purpose of validatingexperimentally the 1 / √ K scaling of the optimal chargingtimes reported in Eqs. (21) and (22).Finally, we note that the role of quantum coherenceis not crucial in the charging step of a quantum bat-tery. Indeed, Fock states, which provide optimal perfor-mances, have no coherence in the basis of the eigenstatesof the Hamiltonian of the charger. Furthermore, becauseof Eq. (23), any coherent combination of Fock states isnot optimal. The role of entanglement is more much sub-tle and is thoroughly discussed in Ref. 47. C. Energy transfer between two QHOs
We now study the case in which both A and B areQHOs with a quadratic Hamiltonian H characterized bythe following terms: H A = ω a † a , (24) H B = ω b † b , H = g ( a † b + ab † ) . The operator H A + H B + H can be diagonalized in termsof the “normal” bosonic operators, γ ± = ( a ± b ) / √ ω ± = ω ± g which,to guarantee overall stability, are taken positive by as-suming | g | ≤ ω .As usual, we fix the initial mean energy of the chargerA ( E A (0) = E in ) and define the average number of ex-citations, K = E in /ω . In order to calculate the storedenergy (5) we find then useful to adopt the Heisenbergrepresentation writing E s ( τ ) = tr[ ρ AB (0) H B ( τ )], with H B ( τ ) ≡ e i H τ H B e − i H τ . Expressing hence a and b asfunctions of the normal operators γ ± and using that the latter evolve simply as γ ± ( t ) = e − iω ± t γ ± , we obtain H B ( τ ) = ω (cid:40) a † a + b † b (25) − (cid:20) e − i gτ a † a − b † b + b † a − a † b ) + H . c . (cid:21) (cid:41) . This considerably simplifies the calculation of E s ( τ ) sincethe initial state contains no excitations on B, yielding E s ( τ ) = Kω sin ( gτ ) , (26) P s ( τ ) = Kω sin ( gτ ) τ , (27)the formulas applying irrespectively from the details ofthe initial state (a direct consequence of the quadraticform of the Hamiltonian, for which the dynamics of thefirst and second moments—e.g. (cid:104) a (cid:105) , (cid:104) b (cid:105) , (cid:104) a † a (cid:105) , etc—is in-dependent of higher-order ones).Equations (26) and (27) have exactly the same depen-dence on time of Eq. (14) for the case of two TLSs model.Hence, the optimal charging times of the two models co-incide, i.e. ¯ τ = π/ (2 g ) and ˜ τ ≈ . /g , and exhibit nospeedup in K . Nonetheless, due to the higher storingcapability of the QHO battery which has now an un-bounded energy spectrum, in the present case the valuesfor the associated maximal stored energy and maximalpower (i.e. E s = Kω and ˜ P s ≈ . gKω ) show a linearincrease in K that was absent in the model of Sect. II A. Itis also worth stressing that the K improvement for ˜ P s re-ported here has a completely different origin with respectto the √ K power improvement observed in Sect. II B. In-deed, due to the absence of an unbounded energy spec-trum for the battery of the QHO-TLS model, the √ K im-provement of the previous Section is just a consequence ofthe speedup in the charging time (22) which, as alreadynoticed, is instead absent in the present model. The valueof ¯ τ = π/ (2 g ) obtained here, can finally be comparedwith the QSL time of Eq. (11). An analogous calculationof Sect. II B gives τ QSL (cid:39) π/ (2 √ Kg ) in the large K limit,revealing that, at variance with the QHO-TLS case, theobserved ¯ τ does not saturate the QSL bound. This isdue to the fact that, before reaching a state of maximalcharging for B, the system has to travel between a finitenumber of orthogonal states. While the bound can beapplied for each of this transition, we should take intoaccount that we have to travel through many orthogo-nal states. This simple example shows that the predic-tions of a quantum advantage based on a speed limitargument are not always correct independently of thespecific model. III. THEORY OF ENERGY TRANSFER IN THENON-COMMUTING CASE
In this Section we discuss how the process of energyexchange between a charger and a quantum battery ismodified when the condition [ H , H ] = 0 is not fulfilled.In this case δE sw ( τ ) (cid:54) = 0, meaning that the protocol de-scribed by Eq. (2) does not simply enable energy trans-fer from A to B, since some energy is externally injectedinto or extracted from the whole system, via the suddenquench of the interaction Hamiltonian. To characterizethe performances of these special charger-battery modelswe are hence forced to introduce a new functional E t ( τ )which, at variance with Eq. (6), accounts only for theprocess of energy transfer from A to B, while properlyneglecting the extra energy contributions induced by theexternal switching of H .Clearly, there is a certain degree of arbitrariness in giv-ing such definition. In this Article we offer the followingoperational definition of E t ( τ ):1) If δE sw ( τ ) <
0, some energy is extracted fromthe system A + B, which has a “credit” towardsthe external world. We can therefore safely statethat all the energy stored in B comes from A set-ting E t ( τ ) = E s ( τ );2) If δE sw ( τ ) >
0, some energy is injected into thesystem, which has a “debit” towards the externalworld. If the energy E A ( τ ) in A is sufficient to com-pensate this energy debit, i.e. if E A ( τ ) ≥ δE sw ( τ ),we state that the remaining energy in B is a trans-ferred energy, E t ( τ ) = E s ( τ ). Otherwise, if the en-ergy E A ( τ ) in A is not sufficient, we subtract fromthe energy in B the remaining amount needed topay the debit. Therefore, the transferred energy isgiven by E t ( τ ) = E s ( τ ) − [ δE sw ( τ ) − E A ( τ )].Summarizing, our definition of E t ( τ ) can then be ex-pressed as E t ( τ ) = E s ( τ ) − max (cid:8) , δE sw ( τ ) − E A ( τ ) (cid:9) . (28)With the help of the above quantity, in the remainingpart of this Section we study the efficiency of the twospecific cases of charger-battery models with non com-muting H and H . In the first case—Sect. III A—we re-lax the hypothesis that the two subsystems A and B arein resonance. In this case the charging protocol does notact on the system by controlling the coupling strength g between A and B. Rather, control occurs on the fre-quency of the subsystem A, which can be brought in res-onance with B or tuned away from it. In the secondcase—Sect. III B—we explicitly include into the Hamil-tonian terms that do not simply transfer excitations of H between the two subsystems. These terms can beneglected when the coupling constant is small, invokingthe so-called “rotating wave approximation” (RWA) .Hence, this beyond-RWA regime better describes the casein which the two subsystems A and B are strongly cou-pled. In what follows we present a simple model havinga critical point in the spectrum and we show that, nearthe critical point, both battery and charger are exter-nally charged via quenches and their energy increases asa power law in time. Although strong coupling can be thought of being an obvious choice to reduce the charg-ing time, since in this case ˜ τ ∼ /g , below we show thatthis regime is not optimal in the sense that it does notfit the ideal scenario of pure-energy-exchange between Aand B.For the sake of simplicity, both Sect. III A and III Bdeal with the case of two QHOs. A. The detuning protocol
So far we have analyzed a charging protocol in whichthe coupling between the two subsystems A and B isturned on and off. However, this protocol may be exper-imentally challenging. A more practical way to controlenergy exchange between the two subsystems A and Bconsists in manipulating the frequency of the charger A,an experimentally viable route with the technology de-scribed in Ref. 46. The new protocol goes as following.The two subsystems A and B are initially largely detunedand energy transfer is therefore strongly suppressed. Ina time window τ , the detuning is set to zero and the twosubsystems interact. Finally, the subsystem A is againlargely detuned from B and energy flow is again blocked.Formally, the system under study consists of two QHOswith a time-dependent Hamiltonian (2) with components H = [ ω + δω ] a † a + ω b † b + g ( a † b + ab † ) , H = − δωa † a . (29)The quantities ω and g and the operators a and b havethe same meaning as in Eq. (24) and δω is the detuningbetween the two subsystems. The latter is assumed tohave not a definite sign but to be large in modulus withrespect to the coupling, namely | δω/g | (cid:29) λ ( t ) of Eq. (2), H dictatesthe evolution at times t ∗ (cid:54)∈ [0 , τ ], while H = H + H generates the evolution at time t ∈ [0 , τ ]. Accordingly,at time t ∗ (cid:54)∈ [0 , τ ] the two subsystems are largely detuned,and energy exchange is suppressed. Using the well-knownSchrieffer-Wolff transformation , in this time window wecan effectively rewrite H as H eff0 = (cid:2) ω + δω + g /δω (cid:3) a † a + (cid:2) ω − g /δω (cid:3) b † b (30)up to corrections on the order of g /δω . This effectiveHamiltonian, which is valid at all times t ∗ (cid:54)∈ [0 , τ ] pro-vided | δω/g | (cid:29)
1, shows that the interaction between Aand B is effectively quenched and exchange of quanta be-tween the two subsystems is strongly suppressed. Thanksto this effective decoupling, we can define two effectivelocal Hamiltonians acting on A and B, i.e. H effA = (cid:2) ω + δω + g /δω (cid:3) a † a , H effB = (cid:2) ω − g /δω (cid:3) b † b , (31)which are approximate constants of the motion. Oncelocal Hamiltonians on A and B are defined, we can apply π/ π π/ πgτ − / / (a) π/ π π/ πgτ / (b) FIG. 4. (Color online)Figures of merit for the detuning proto-col described in Sect. III A. Panel (a) shows the stored energy E s ( τ ) (blue solid line) and the switching energy δE sw ( τ ) (reddashed line), in units of Kω , and as functions of gτ . Resultsin this panel have been obtained by setting δω = ω / g = ω /
10. Since δω >
0, there is no difference between thestored energy and the transferred energy, i.e. E t ( τ ) = E s ( τ ).Panel (b) shows the stored energy E s ( τ ) (blue solid line),the transferred energy E t ( τ ) (black dash-dotted line), andthe switching energy δE sw ( τ ) (red dashed line). Results inthis panel have been obtained by setting δω = − ω / g = ω /
10. Since δω <
0, some energy is injected into thesystem and E t ( τ ) ≤ E s ( τ ). the general analysis described in Sect. II to calculate allrelevant quantities. For simplicity we set K = 1.At times t ∈ [0 , τ ], the coupling parameter λ ( t ) is equalto one and due to the presence of H the two subsystemsare in resonance. In this time interval, H = H + H isidentical to that reported in Eq. (24) and, as long as weconsider a density matrix of the form (1) as input state forthe system, the dynamical evolution can be described asin Sect. II C. Hence, it is straightforward to calculate the stored energy, the average charging power, and δE sw ( τ ): E s ( τ ) = (cid:20) ω − g δω (cid:21) sin ( gτ ) , (32) P s ( τ ) = (cid:20) ω − g δω (cid:21) sin ( gτ ) τ ,δE sw ( τ ) = − δω sin ( gτ ) . In the case δω > δE sw ( τ ) < E t ( τ ) = E s ( τ ). In the case δω > B. Beyond the RWA
We now study the case of two QHOs with counter-rotating terms included in the interaction Hamiltonian,i.e. H A = ω a † a , (33) H B = ω b † b , H = g ( a + a † )( b + b † ) . In the limit g (cid:28) ω counter-rotating terms, i.e. terms ofthe form a † b † and ab , can be safely neglected and onerecovers Eq. (24).The full Hamiltonian H = H A + H B + H , whichdictates the dynamical evolution, has eigenvalues ω ± = (cid:112) ω ± gω . We therefore assume | g | /ω ≤ / a ( t ) = R aa ( t ) a + R ab ( t ) b + R aa † ( t ) a † + R ab † ( t ) b † ,b ( t ) = R ba ( t ) a + R bb ( t ) b + R ba † ( t ) a † + R bb † ( t ) b † , (34)where the quantities R ij ( t ) are calculated in Appendix A.By the same token, the local Hamiltonian for B getstransformed into H B ( t ) = ω (cid:2) R ∗ ba ( t ) a † + R ∗ bb ( t ) b † + R ∗ ba † ( t ) a + R ∗ bb † ( t ) b (cid:3) × (cid:2) R ba ( t ) a + R bb ( t ) b + R ba † ( t ) a † + R bb † ( t ) b † (cid:3) , (35)leading to the following expression for the stored energy E s ( τ ) ω = (cid:2) | R ba † ( τ ) | + | R bb † ( τ ) | (cid:3) + (cid:104) a † a (cid:105) A (cid:2) | R ba † ( τ ) | + | R ba ( τ ) | (cid:3) + (cid:2) (cid:104) aa (cid:105) A R ∗ ba † ( τ ) R ba ( τ ) + H . c . (cid:3) , (36)where for the sake of simplicity we have denoted the av-erage of an operator O evaluated on the initial state of π/ π π/ πgτ / / E s ( τ ) / ( K ω ) (a) π π π πω + τ E s ( τ ) / ( K ω ) (b) π/ π π/ πgτ − / / δ E s w ( τ ) / ( K ω ) (c) π π π πω − τ δ E s w ( τ ) / ( K ω ) (d) π/ π π/ πgτ / E t ( τ ) / ( K ω ) (e) π/ π π/ πω − τ / E t ( τ ) / ( K ω ) (f) FIG. 5. (Color online) Panel (a) displays the stored energy E s ( τ ) (in units of Kω ) as a function of gτ , for the case of twocoupled QHOs, evaluated by setting g = 0 . ω . Different curves refer to results obtained for three different choices of theinitial state of the charger: stored energy for an initial Fock or a thermal state evaluated by setting K = 3 (blue solid line);stored energy for an initial Fock or a thermal state evaluated by setting K = 100 (dark blue solid line); stored energy for aninitial coherent state evaluated by setting K = 3 (red dash-dotted line); stored energy for an initial coherent state evaluatedby setting K = 100 (dark red dash-dotted line). The same color code is used for all other panels. The stored energy showsoscillations similar to the RWA case (see Fig. 2), with counter-rotating terms causing only quantitative corrections. Panel (c)displays the switching energy δE sw ( τ ) (in units of Kω ) as a function of gτ , evaluated for g = 0 . ω . Panel (e) displays thetransferred energy E t ( τ ) (in units of Kω ) as a function of gτ , evaluated for g = 0 . ω . Panel (b) displays the stored energy E s ( τ ) (in units of Kω ) as a function of ω + τ and evaluated for g → ω /
2. Due to the vicinity to the critical point, the storedenergy increases as a power law. Panel (d) displays the switching energy δE sw ( τ ) (in units of Kω ) as a function of ω − τ ,evaluated for g → ω /
2. This quantity measures the energy that is externally injected. The power-law increase of this quantityis clear. Panel (f) displays the transferred energy E t ( τ ) (in units of Kω ) as a function of ω − τ , evaluated for g → ω /
2. Onlya small amount of the corresponding stored energy seen in panel (b) can be counted as transferred energy, while the majorityof the energy is externally injected. (cid:104) O (cid:105) A = tr A [ Oρ A (0)]. We notice that (cid:104) a † a (cid:105) A = K is the mean value of excitations in thecharger at the beginning of the protocol and is pro-portional to the initial energy, so the first two lines inEq. (36) do not depend on the details of the initial state.On the contrary, for a coherent state as initial state ofA, we have (cid:104) aa (cid:105) A = α , while for both Fock and thermalstates of A (cid:104) aa (cid:105) A = 0. Hence, the third line in Eq. (36) is different from zero only in the case of a coherent state,while this quantity does not distinguish between a Fockand a thermal state.The switching energy δE sw ( τ ) can be calculated as fol-lowing. We first note that E (0) = 0. We thereforeneed to calculate only the interaction energy at time τ ,i.e. δE sw ( τ ) = − E ( τ ). With analogous steps to whatdescribed just above we find δE sw ( τ ) g = − (cid:110)(cid:2) R ab ( τ ) + R ∗ ab † ( τ ) (cid:3)(cid:2) R bb † ( τ ) + R ∗ bb ( τ ) (cid:3) + c . c . (cid:111) − (cid:104) a † a (cid:105) A (cid:110)(cid:2) R aa ( τ ) + R ∗ aa † ( τ ) (cid:3)(cid:2) R ba † ( τ ) + R ∗ ba ( τ ) (cid:3) + c . c . (cid:111) − (cid:110) (cid:104) aa (cid:105) A (cid:2) R aa ( τ ) + R ∗ aa † ( τ ) (cid:3)(cid:2) R ba ( τ ) + R ∗ ba † ( τ ) (cid:3) + H . c . (cid:111) . (37)The above considerations for Eq. (36) still hold and alsoEq. (37) can distinguish only between the coherent stateand the other two choices of initial states of the charger.It is useful to make a distinction between three situations.In the weak-coupling | g | /ω (cid:28) / | g | /ω (cid:46) /
2. In this case the counter-rotatingterms give quantitative corrections, see Figs. 5(a), (c),and (e), while the oscillating behavior of E s ( τ ) is stillpresent. Finally, the case | g | /ω → / R ij ( t ), which con-tains the function sin( ω ± t ) /ω ± . When ω ± → ω ± t ) /ω ± → t , which explains the power-law behav-ior.A comment on the strong-coupling and critical regimesis now in order. In the weak-coupling regime, counter-rotating terms can be neglected and rotating terms in H of the form ab † + a † b are the “best interaction Hamilto-nian” from the point of view of energy transfer, since,by definition, they just transfer excitations from A to Band viceversa. On the other hand, in the strong-couplingand critical regimes counter-rotating terms in H of theform a † b † + ab cannot be neglected and create/destroya pair of excitations in the two systems A and B. Now,the impact of these terms is detrimental from the pointof view of energy transfer. This is particularly clear inthe critical regime, where the both E s ( τ )—Fig. 5(b)—and the energy of the charger increase as power laws.This growing energy is externally injected in the systemvia the time-dependent modulation of the coupling con-stant λ ( t ) and only a small amount is exchanged betweenA and B. In summary, in the strong-coupling and crit-ical limits our results cannot be interpreted in terms ofpure energy exchange between the two subsystems. Thesimplest interpretation is, in contrast, in terms of twocoupled systems that are externally charged. IV. CONCLUSIONS
In this work we presented a systematic classificationand analysis of several simplified models of energy trans-fer for quantum batteries and of their associated chargingprocesses. Our approach, based only on different com-binations of two-level systems and harmonic oscillators,allowed us to derive exact results without the necessityof introducing any particular assumption or approxima-tion. The set of models considered in this work coversmany paradigmatic situations including the non trivialone when the interaction does not preserve the total num-ber of excitations.Some of the results obtained in this work for toy mod-els of quantum batteries are expected to hold in general.For example, the scaling of the charging time with aninverse power-law of the charger energy is expected to begeneral—see also Ref. 15. Moreover, we believe that thefact that quantum coherences in the basis of the eigen-states of the charger Hamiltonian are not a necessaryingredient in order to achieve optimal figures of merit isa general result, provided that no counter-rotating termsare at play. Finally, the fact that the strong-couplingregime is not suitable for studying the ideal scenario ofpure energy exchange between charger and battery is alsoexpected to hold true in more complicated models.Possible future outlooks and applications of our workcould be: theoretical or experimental implementations ofour models in specific systems and real devices, the devel-opment of a more detailed analysis taking into accountalso the presence of energy fluctuations, the extensionof the considered models to systems of arbitrary dimen-sion, the presence of loss or other noisy mechanisms, andcharging of the battery via an external classical field .Since it would be highly desirable for our quantum bat-tery to store energy for a relatively long time, a thoroughstudy of the role of dissipative effects during the storagestep should also be carried out and is left for future work.Within the general context of quantum enhanced tech-nologies, we hope that the simple yet exactly solvable1models of quantum batteries considered in this workcould represent a solid starting point stimulating newideas and further research lines. ACKNOWLEDGMENTS
We gratefully thank M. Campisi, F.M.D. Pellegrino,D. Ferraro, P.A. Erdman, V. Cavina, and M. Keck foruseful discussions.
Appendix A: Details on the calculation of Eq. (34)
In this Appendix we show the details of the calculationof Eq. (34). First of all, in order to find the time evolu-tion of the ladder operator it is useful to diagonalize theproblem. We define as A the vector made by the ladderoperators involved in the problem: A = aba † b † . (A1)In a similar way we denote as γ the vector of made bythe operators that diagonalize the Hamiltonian, i.e. γ = γ − γ + γ †− γ † + . (A2)Diagonalization of the Hamiltonian consists in findingthe transformation A = M γ , where M is a 4 × M is a straightforward textbook task . In theHeisenberg representation the vector evolved at time t , γ ( t ), is related via a diagonal matrix D to the vectoreingenmodes at the initial time γ ( t ) = Dγ : D = e − iω − t e − iω + t e iω − t
00 0 0 e iω + t . (A3)Our goal is to find A ( t ) = R ( t ) A , where R ( t ) is the ma-trix in Eq. (34). In order to find such transformation, weexpress A in terms of the eigenmodes γ , we evolve theeigenmodes, and then we express the eingemodes in termsof the initial ladder operators, using the inverse transfor-mation M − , i.e. A ( t ) = (cid:2) M D ( t ) M − (cid:3) A . Hence wefind: R aa ( t ) = 12 (cid:104) cos( ω − t ) + cos( ω + t ) (cid:105) − i (cid:104)(cid:0) ω + ω − ω (cid:1) sin( ω − t ) ω − + (cid:0) ω + ω ω (cid:1) sin( ω + t ) ω + (cid:105) , (A4) R ab ( t ) = 12 (cid:104) − cos( ω − t ) + cos( ω + t ) (cid:105) − i (cid:104) − (cid:0) ω + ω − ω (cid:1) sin( ω − t ) ω − + (cid:0) ω + ω ω (cid:1) sin( ω + t ) ω + (cid:105) ,R aa † ( t ) = ig (cid:104) sin( ω − t ) ω − − sin( ω + t )) ω + (cid:105) ,R ab † ( t ) = − ig (cid:104) sin( ω − t ) ω − + sin( ω + t )) ω + (cid:105) . From the fact that the Hamiltonian is symmetric withrespect to the exchange a ↔ b , we have: R ba ( t ) = R ab ( t ) , R bb ( t ) = R aa ( t ) , (A5) R ba † ( t ) = R ab † ( t ) , R bb † ( t ) = R aa † ( t ) . ∗ [email protected] M.F. Riedel, D. Binosi, R. Thew, and T. Calarco, Quan-tum Sci. and Tech. , 030501 (2017). A. Ac´ın, I. Bloch, H. Buhrman, T. Calarco, C. Eichler,J. Eisert, D. Esteve, N. Gisin, S.J. Glaser, F. Jelezko,S. Kuhr, M. Lewenstein, M.F. Riedel, P.O. Schmidt,R. Thew, A. Wallraff, I. Walmsley, and F.K. Wilhelm,arXiv:1712.03773. J. Goold, M. Huber, A. Riera, L. del Rio, and P.Skrzypczyk, J. Phys. A: Math. Theor. , 143001 (2016). S. Vinjanampathy and J. Anders, ContemporaryPhysics , 545 (2016). R. Alicki, and R. Kosloff, arXiv:1801.08314. P. Strasberg, G. Schaller, T.Brandes, and M.Esposito,Phys. Rev. X , 021003 (2017). M. Campisi, P. H¨anggi, and P. Talkner, Rev. Mod.Phys. , 1653 (2011). D. Gelbwaser-Klimovsky, W. Niedenzu and G. Kurizki,Advances In Atomic, Molecular, and Optical Physics, ,329 (2015). M. Horodecki and J. Oppenheim, Nature Comm. , 2059(2013). R. Alicki and M. Fannes, Phys. Rev. E , 042123 (2013). K.V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, andA. Ac´ın, Phys. Rev. Lett. , 240201 (2013). G.Watanabe, B. P. Venkatesh, P.Talkner, and A. delCampo, Phys. Rev. Lett. 118, 050601 (2017) F.C. Binder, S. Vinjanampathy, K. Modi, and J. Goold,New J. Phys. , 075015 (2015). F. Campaioli, F.A. Pollock, F.C. Binder, L. C´eleri, J.Goold, S. Vinjanampathy, and K. Modi, Phys. Rev.Lett. , 150601 (2017). D. Ferraro, M. Campisi, G.M. Andolina, V. Pellegrini, andM. Polini, Phys. Rev. Lett. , 117702 (2018). T.P. Le, J. Levinsen, K. Modi, M.M. Parish, and F.A.Pollock, Phys. Rev. A , 022106 (2018). I. Henao and R.M. Serra, Phys. Rev. E , 062105, (2018). S. Deffner and S. Campbell, J. Phys. A Math. Theor. 50,453001 (2017). V. Giovannetti, S. Lloyd, and L. Maccone, Europhys.Lett. , 615621 (2003). V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. A , 052109 (2003). V. Giovannetti, S. Lloyd, and L. Maccone, J. Opt. B:Quantum Semiclass. Opt. S807 (2004). S. Haroche and J. Raimond,
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