aa r X i v : . [ qu a n t - ph ] D ec Quantum advantage of two-level batteries in self-discharging process
Alan C. Santos ∗ Departamento de F´ısica, Universidade Federal de S˜ao Carlos,Rodovia Washington Lu´ıs, km 235 - SP-310, 13565-905 S˜ao Carlos, SP, Brazil (Dated: December 23, 2020)Devices that use quantum advantages for storing energy in the degree of freedom of quantum systems havedrawn attention due to their properties of working as quantum batteries. However, one can identify a number ofproblems that need to be adequately solved before a real manufacturing process of these devices. In particular,it is important paying attention to the ability of quantum batteries in storing energy when no consumption centeris connected to them. In this paper, by considering quantum batteries disconnected from external chargingfields and consumption center, we study the decoherence e ff ects that lead to charge leakage to the surroundingenvironment. We identify this phenomena as a self-discharging of QBs, in analogy to the inherent decay ofthe stored charge of conventional classical batteries in a open-circuit configuration. The quantum advantageconcerning the classical counterpart is highlighted for single- and multi-cell quantum batteries. I. INTRODUCTION
Over the last few years, one has been paid some attentionto the development of quantum devices able to store energy tobe used for later processing. The result after years of studiesis a number of works discussing the performance of quantumbattery (QB) concerning its charging power [1–10] and workextraction [11–15]. One can highlight as a significant advanceto this field the work developed by Allahverdyan, Balian andNieuwenhuizen, who addressed the question of how much en-ergy can be extracted from quantum system by unitary oper-ations [16]. Based on quantum thermodynamics theory, theyshowed that the this extractable maximum energy is given by E = X D i = X D n = ̺ n ǫ i (cid:16) |h ̺ n | ǫ i i| − δ ni (cid:17) . (1)also known as ergotropy , with D being the dimension of theHilbert space and we write H = P D n = ǫ n | ǫ n i h ǫ n | and ρ = P D n = ̺ n | ̺ n i h ̺ n | , so that ̺ ≥ ̺ ≥ · · ·≥ ̺ D and ǫ ≤ ǫ ≤ · · ·≤ ǫ D .However, it is important to keep in mind we are still farfrom a real development of practical QBs due to a numberof di ff erent reasons. In fact, if we understand that the manu-facturing process of these devices is not only justified by itshigh charging power, we raise a number of question that havenot been adequately addressed yet. As one of these questions,we highlight here the phenomenon known as self-discharging (SD) of batteries [17–21]. This process leads to the loss ofcharge due to inherent characteristic of the system used asworking fluid for storing energy and it happens regardlesswhether the battery is connected to some consumption hub.Although we have a number of proposals of two-level QBs indi ff erent systems, such as spin systems [22], quantum cavi-ties [2, 13, 23, 24], among others [3, 25, 26], the SD mecha-nism for this kind of QB is yet an open question.Based on studies of SD in commercial batteries, in this pa-per we introduce a strategy to study SD processes in QBs.The key point is defining how to put a QB as an open-circuit,which is intuitively done by the absence of external charging ∗ ac [email protected] r C B L F (on) A (on) (a) Closed circuit
C B r L F (off) A (off) (b) Open circuit
Figure 1. Sketch of (1a) an classical open circuit composed by acharger C, the battery B, an internal resistance r and an external cir-cuit L (a lamp), where the resistance r is considered to describe theohmic self-discharging of a classical battery [19]. The arrow showsthe equivalent of the classical circuit in a quantum approach. An ex-ternal field F is used as charger and an auxiliary qubit A is used asconsumption center (the “lamp”). (1b) Classical open-circuit config-uration used to study SD processes and its quantum counterpart aswe are proposing here to do similar studies of SD in QBs. or extracting energy fields, and interaction with auxiliary sys-tems used as consumption hubs. We apply our approach tosingle- and multi-cell QBs, where we identify quantum ad-vantage in SD of two-level storing energy devices.
II. SINGLE-CELL QBS
The study of SD in classical batteries is done in the follow-ing way: after charging the battery or capacitor, we let the sys-tem evolves in a open-circuit configuration, where inevitablysome amount of energy is lost, leading then to the SD process.As illustrated in Fig. 1, a similar approach can also be appliedto QBs. The dynamics of a QB can be written as˙ ρ ( t ) = H [ ρ ( t )] + R [ ρ ( t )] , (2)where H [ ρ ( t )] describes the unitary dynamics of the systemand / or coherent interaction with consumption hub and R [ ρ ( t )]encodes all losses processes due to the coupling with the en-vironment. In particular, let us also assume that H [ ρ ( t )] = i ~ [ H + H int + H ( t ) , ρ ( t )] , (3)with H int is a Hamiltonian used to deal with internal interac-tions between the cells of the battery H ( t ) being an arbitraryfield used to inject or extract energy from the QB. Then, wesay that the battery is in a open-circuit configuration when H ( t ) =
0, situation in which one can adequately study the SDof QBs using a similar strategy as done for classical batteries.As a first case, let us consider the most elementary caseof a QB composed by a single two-level system with internalHamiltonian H = ~ ωσ + σ − , σ − = σ + † = | g i h e | , driven by themaster equation˙ ρ ( t ) = i ~ [ H , ρ ( t )] + R rel [ ρ ( t )] , (4)which describes the relaxation processes leading to energydissipation, with R rel [ ρ ( t )] = Γ (cid:0) σ − ρ ( t ) σ + − { σ + σ − , ρ ( t ) } (cid:1) , (5)being σ z = | e i h e | − | g i h g | . In particular, we assume this pro-cess in our study because it is the most common decoherenceprocesses in a large of physical systems, e.g. in nuclear spinsystems [27–29], cavity quantum electrodynamics [30] andsuperconducting qubits [31–33]. Then, by writing the densitymatrix in the QB basis | e i and | g i as˙ ρ ( t ) = ̺ e ( t ) | e i h e | + ̺ g ( t ) | g i h g | + (cid:16) ̺ eg ( t ) | e i h g | + h.c. (cid:17) , (6)the solution of the dynamics can be found as ̺ e ( t ) = ̺ e (0) e − Γ t , ̺ eg ( t ) = ̺ eg (0) e − i ω t e − Γ t . (7)From this general result for the dynamics considered here,we can study di ff erent situations. In particular, we are in-terested in the decay performance when we use quantum re-sources (coherence) to store energy in the QB. A. Discharging from full charge state
As a first discussion, let us assume the case of a batteryinitially in a full charged state ̺ eg (0) = ̺ e (0) =
1. In thiscase, the available energy as given by the ergotropy E ( t ) andthe internal energy U ( t ) read (with E max = ~ ω ) E ( t ) = Θ ( τ c − t ) (cid:16) e − t Γ − (cid:17) E max , U ( t ) = E max e − t Γ , (8)where Θ ( x ) is the Heaviside theta (0 if x < x ≥
0) and τ c is the population crossing time. This crossing is inevitabledue to the decay where the instantaneous population in excited ̺ e ( t ) decreases from 1 to 0 and the population in ground state ̺ g ( t ) goes through the contrary direction. Then, ̺ e ( t ) > ̺ g ( t )for t < τ c , ̺ e ( t ) = ̺ g ( t ) for t = τ c , and ̺ e ( t ) < ̺ g ( t ) for t > τ c , con-sequently, the basis ordering requested by the Eq. (1) changes in time during the dynamics. From a simple calculation byimposing ̺ e ( t ) = ̺ g ( t ) we can show that τ c = / Γ .The decay of both quantities E ( t ) and U ( t ) is well-describedby the monotonic decreasing exponential e − t /τ d , with τ d = / Γ being a characteristic decay time scale. This kind of decayprocess is similar to the ohmic behavior of SD for conven-tional classical batteries [17, 18], where we also get a descrip-tion in terms of a single decay time scale. Then we under-stand this result as intuitively expected due to the absence ofany quantumness in the battery during the process (quantumstate coherence, for example). It is also worth highlightingthat the ergotropy sudden death arising to the population or-dering during the system evolution. With this example weargue that, although we have some quantum advantage in thecharging process of the a two-level system [25], the fact ofa quantum system to be charged in the full charge state can-not be taken as quantum concerning SD processes, since itsbehavior is equivalent to conventional classical batteries. B. Exploring quantum advantage
Now, let us explore quantum coherence of a two-level QBto study the same process. As an immediate consequence ofconsidering quantum coherence, we remark the impossibilityof starting the system in a full charged state. Hence, in orderto have a fair comparison between classical and quantum per-formance we will assume that the system starts with ergotropy E (0) = E / = E max /
2. Under this choice, we can compare thegain of an state with maximal coherence | ψ i = ( | e i + | g i ) / √ ρ cl = (3 / | e i h e | + (1 / | g i h g | . It is straightfor-ward conclude that the ergotropy for both states is E / . Then,the ergotropy for the classical and quantum version, respec-tively, give E cl ( t ) = E / Θ (cid:16) τ (1 / − t (cid:17) (cid:16) e − t Γ − (cid:17) ~ ω, (9a) E qu ( t ) = E / e − t Γ / (cid:16) − e t Γ / √ − e t Γ / + e − t Γ (cid:17) , (9b)where τ (1 / = / / Γ . Then, it is immediate to concludesome substantial di ff erence between the two initial states. Dif-ferent from the classical case, the existence of quantum co-herence leads us to a SD phenomena that cannot be explainedby an ohmic process, a characteristic of commercial classicalsupercapacitors [19–21]. In the Fig. 2a we show the time evo-lution of the above quantities, highlighting the half-life time τ / (concerning the initial amount of stored energy) for eachkind of battery. Then, one can see the advantage of storingenergy in quantum information concerning classical one.For completeness, we also consider the portion of en-ergy stored in the system as ergotropy from quantity ε ( t ) = E ( t ) / U ( t ), being U ( t ) = tr( ρ ( t ) H ) the internal energy of thesystem. We understand the parameter ε ( t ) in the followingway: the energy conservation law says that we need to spenda total amount of energy U ( t ) to store an total E ( t ) of ergotropyin the QB, so that some portion of energy cannot be extractedfrom the QB by unitary operations. Thus, it gives us a way to (a) Ergotropy E ( t ) / E / (b) ε ( t ) Figure 2. (2a) Ergotropy (as a multiple of E / ) for the initial classi-cal and quantum states. Vertical dashed lines denotes the respectivehalf-life time for classical and quantum states given, respectively, by τ cl1 / Γ = / ≈ .
36 and τ qu1 / Γ = + √ / ≈ .
65. (2b)E ffi ciency coe ffi cient ε ( t ) for the same process. quantify the percentage of U ( t ) that is wasted in the process.In fact, in case ε ( t ) = U ( t ) = E ( t ), while ε ( t ) < U ( t ) > E ( t ) and hence there is an amount of ‘dead en-ergy’ that cannot be later extracted by unitary operations, be-ing ε ( t ) = III. N-CELL QBS
In general we also are interested in exploring quantum ef-fects in multi-cell QBs as, for example, entanglement. Byconsidering a linear chain of N cells, we let the system evolv-ing under a decay process described by the master equationconsidered as˙ ρ ( t ) = i ~ [ H ( N ) + H linint , ρ ( t )] + R rel [ ρ ( t )] + R icl [ ρ ( t )] , (10)where H ( N ) = P Nn = ~ ωσ + n σ − n , H int is the interaction Hamilto-nian between cells of the battery given as H linint = P n J ~ σ − n σ + n + , R rel [ ρ ( t )] being independent relaxation, as in Eq. (5) for eachcell, and the second term reads R icl [ ρ ( t )] = X n ± Γ icl (cid:16) σ − n ρ ( t ) σ + n + − { σ + n + σ − n , ρ ( t ) } (cid:17) . (11)The additional term R icl [ ρ ( t )] describes the intracell leak-age, in which a given cell n of the battery would emit anexcitation and then this excitation is temporarily stored inits nearest-neighbor cell. Therefore, the sum P n ± needs tobe computed over the two nearest-neighbor of the n -th cell,namely, the terms with { n , ( n + } and { n , ( n − } . It isworth mentioning that the internal energy U we are interesteddoes not take into account the Hamiltonian H linint , since the ex-istence of interactions will promote positive / negative contri-butions in U and it eventually lead to an unfair comparisonwhen we consider di ff erent kind of geometry beyond the lin-ear one (as we shall consider soon). For this reason, onlycontributions of the QB energy basis is taken into account, (a) ε ( t ) for classical state (b) ε ( t ) for quantum state Figure 3. E ffi ciency coe ffi cient ε ( t ) for classical and quantum statesfor some values of N . Here we consider N = J = Γ and Γ icl = Γ / we mean, U = tr( H ( N ) ρ ). We analyze the dynamics in samescenario as before by taking the classical state of N cells as ρ cl ( N ) = ⊗ Nn = ρ cl and the quantum state as | ψ ( N ) i = ⊗ Nn = | ψ i .Because each cell starts in a configuration with ergotropy E / ,then the initial amount of ergotropy in the QB as function of N is E / ( N ) = N /
2. In Fig. 3a we show the e ffi ciency pa-rameter ε ( t ) for the classical state, normalized by the initialamount of ergotropy in each case, and the Fig. 3b shows thesame quantity for the quantum version.As in the single-qubit case, here we can see quantum ad-vantage for N -cells quantum battery (at least up to N = ffi ciency in storing energyas ergotropy, as computed by the parameter ε (0), it shows aclear quantum advantage scaling with N . In fact, the value ε qu (0) = ffi ciency of the quantum version forany N . On the other hand, the quantity ε cl (0) ≈ .
667 says wehave a total of ‘dead energy’ given by ∼ .
3% per cell, henceit is easy to see that the total amount of worst energy scalesas ∼ . ~ ω N . It is worth mentioning that due to the initialstate considered here, no quantum advantage is obtained fromentanglement, what does our proposal useful due to the verysimple state we use as resource to store energy. A. Exploring network e ff ects It is reasonable assumption to think that the kind of in-teractions between the cells of an QB would develop somerole in its charging process. For this reason, it is worth ana-lyzing how the system geometry a ff ects the SD. In this sce-nario, we write the interaction Hamiltonian in a general wayas H int = P { n , m } J ~ σ − n σ + m , in which the sum P { n , m } is done overall direct interactions between the QB cells. In addition, weassume that connection between each pair of cells induces acollective decay channel each other, so that the master equa-tion is modified as R icl [ ρ ( t )] = X { n , m } Γ icl (cid:0) σ − n ρ ( t ) σ + m − { σ + m σ − n , ρ ( t ) } (cid:1) . (12)This kind of dynamics is usually found in light mediatedinteraction in atomic systems [34–36] and superconductingqubits [32, 33], for example. To illustrate our results, weconsider four kind of geometries for a five-cell QB. They are (a) Internal energy U ( t ) / U (b) E ffi ciency coe ffi cient ε ( t ) Figure 4. (4a) Internal energy as a multiple of the initial internalenergy U = h ψ ( N ) | H ( N ) | ψ ( N ) i = N ~ ω/
2. (4b) E ffi ciency coe ffi -cient ε ( t ) for the (main) quantum and (inset) classical states. Herewe assume N = J = Γ and Γ icl = Γ / named here as central, linear, circular and symmetric network,as shown in the legends of the Fig. 4 (from top to bottom, re-spectively).It is intuitive to imagine that the most intracell decay, theslower the energy leakage from the system into the surround-ing environment. On the other hand, consider the result shownin Fig. 4a, in which we compute the instantaneous internal en-ergy U ( t ) = tr( ρ ( t ) H ( N )) for the quantum state, being ρ ( t ) thethe solution of the Eq. (10). This behavior suggests a comple-mentary discussion, being the topology most relevant to thecollective energy decay than the number of intracell channels.As example, we can mention the linear and central geome-tries, in which they have same number of connections but theyhave di ff erent property of storing internal energy (excitation).In addition, and most important here, the e ffi ciency in stor-ing energy as ergotropy is strongly a ff ected by the geometry.As highlight, by using a central geometry we guarantee moree ffi ciency than the other cases, as it is seen in Fig. 4b. In ad-dition to keep energy for more time, the capacity of storingergotropy is also enhanced relatively to the other geometries.In conclusion, we can see that the connectivity between eachcell of the QB can be used as resource to design QBs withdi ff erent SD characteristics. If we use classical states to usedstore ergotropy, the connectivity does not matter as we can seein Fig. 4b (inset).In order to give more details about the quantumness of thebattery, we also consider the dynamics of the entanglementand coherence of the system during the dynamics. As mea-sure of entanglement, we use the pairwise concurrence be-tween two cells and we take into account an average over allconnections of the battery. Mathematically this reads C ( t ) = (1 / N { n , m } ) X { n , m } C nm ( t ) , (13)where N { n , m } denotes the number of total connections in thebattery, for example, one has N { n , m } = N { n , m } = C nm ( t ) is obtained bytaking the reduced density matrix of the n -th and m -th cells.This quantity is computed from definition of concurrence byHill-Wootters that reads C ( ˆ ρ ) = max { , λ − λ − λ − λ } ,where λ , · · · , λ are the eigenvalues in decreasing order ofthe matrix ˆ R = ( ˆ ρ / ˆ˜ ρ ˆ ρ / ) / , where ˆ˜ ρ = ( ˆ σ y ⊗ ˆ σ y ) ˆ ρ ∗ ( ˆ σ y ⊗ ˆ σ y ), (a) Coherence (b) Concurrence Figure 5. Coherence and entanglement dynamics for the system as-sumed in Fig 4. Here N = J = Γ and Γ icl = Γ / with ˆ ρ ∗ being the complex conjugate of ˆ ρ written in the highlyentangled Bell basis [37]. On the other hand, the coher-ence is computed from its conventional definition based onthe l -norm normalized coherence [38, 39] defined as C ( t ) = (1 / C max ) P i , j , i | ρ i j ( t ) | , with ρ i j ( t ) being the matrix elements ofthe system state. This measure of coherence is enough for ourdiscussion, since our reference basis is well defined by theempty and charged states of the battery [9].We identify some amount of coherence and entanglementthat arises during the decay dynamics starting from the clas-sical, but they are very small quantities (of order of 10 − ) andwe neglected them. On the other hand, these quantities arevery significant for the case where initial state of the QB hasinitial coherence, therefore we show these quantities in Fig. 5.The behavior for coherence and entanglement leads us to con-clude that the resistance of a QB against decay process is as-sociated with the amount of coherence in the system. In fact,while the entanglement can increase during the dynamics dueto collective decay of the system, the e ffi ciency follows anequivalent decreasing process as followed by the coherence.Moreover, to the best of our knowledgment, it is not trivialto conclude whether entanglement develops some role in thekind of process we are considering in this manuscript. Wemean, from the dynamics for the central geometry we can no-tice that the collective decay induces both entanglement birthsand death. However, the e ffi ciency cannot be perfectly ex-plained by the entanglement behavior. This result reinforcesa recent discussion raised in Refs. [9, 22], where it has beenprovided some evidences that entanglement is not the mainresource for QBs. IV. CONCLUSION
In this paper we have characterized the SD process of two-level QBs, whose the main decoherence process leads to en-ergy losses from the system to some surrounding environment.The quantum advantage is first explored for single-cell de-vices, where the usage of quantum coherence to store energyleads to a slower decay concerning the process in which en-ergy is initially stored in classical states. When we increasethe system, both coherence and geometry of the intracell con-nections seem to be good sources to design robust QB againstthese undesired e ff ects that discharges our QB, even whetherit is not coupled to some consumption hub. This study opensperspectives for new advances in QBs, since keeping energystored in QB for long times after the charging process is a keytask for the development of realistic energy storing quantumdevices. ACKNOWLEDGMENTS
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