Measurement Induced Synthesis of Coherent Quantum Batteries

Quantum coherence represented by a superposition of energy eigenstates is, together with energy, an important resource for quantum technology and thermodynamics. Energy and quantum coherence however, can be complementary. The increase of energy can reduce quantum coherence and vice versa. Recently, it was realized that steady-state quantum coherence could be autonomously harnessed from a cold environment. We propose a conditional synthesis of N independent two-level systems (TLS) with partial quantum coherence obtained from an environment to one coherent system using a measurement able to increase both energy and coherence simultaneously. The measurement process acts here as a Maxwell demon synthesizing the coherent energy of individual TLS to one large coherent quantum battery. The measurement process described by POVM elements is diagonal in energy representation and, therefore, it does not project on states with quantum coherence at all. We discuss various strategies and their efficiency to reach large coherent energy of the battery. After numerical optimization and proof-of-principle tests, it opens way to feasible repeat-until-success synthesis of coherent quantum batteries from steady-state autonomous coherence.

average amount of energy from a quantum system 2,17 , increasing the potential usefulness of the thermal reservoirs 18,19 or increase in the power output of thermal machines 20,21 , supported by an experimental demonstration 22 .
To synthesize a higher dimensional quantum coherent battery from the independent TLSs, Maxwell's demon principle can be advantageously used. It is a conditional procedure based on the measurement of a system 23 . For increasing the energy only, it would be sufficient to measure TLSs individually in the energy representation and keep only excited systems in a battery, whereas the de-excited systems will be brought once more into interaction with a thermal bath and reused until all of them have maximal energy (the so-called repeat-until-success strategy). Efficient energy measurement is, therefore, a thermodynamic resource 7,8,[24][25][26][27][28][29] . However, for keeping coherence, the energy measurement is counterproductive since it causes loss of all system coherence. Therefore, to synthesize two-level systems with a partial coherence into a large coherent system requires a different measurement strategy. However, we still demand it to use only measurement elements diagonal in the TLS energy basis. This fact ensures that the measurement does not directly project on a superposition of energy eigenstates and that it is not a source of energy on its own 30,31 .
In this paper, we propose such strategy which is coherently synthesizing excited two-level systems to a larger coherent system with the possibility to enhance both energy and coherence on two and more TLS by the collective quantum measurement with elements diagonal in the energy basis. Such synthesizing unifies the battery build-up and charging process. As we find out the outcome of the measurement, we acquire information about the battery state and, based on it, we decide on the next step of the protocol, see Fig. 1a. If the TLSs did not synthesize as required, we let them interact with the thermal bath surrounding again to provide once more residual coherent energy, and repeat it until the success. In this way, we can asymptotically approach deterministic charging at a cost of the number of TLS interacting with the bath.
Our main result shows the possibility of elementary coherent energy synthesizing, i.e., the increase of both the initial energy and coherence 32 (both small but nonzero) employing quantum measurement on the system. This result is quantitatively presented in Fig. 1b. Such energy and coherence increasing measurement can be extended and optimized further, as we subsequently propose, to reach maximal coherent energy of the battery. Although this charging stage is, in a single protocol run, conditional with a limited probability of success p s , it can be repeated until it succeeds and TLS are synthesized and coherently charged with the overall success probability arbitrarily close to one. Our work represents a bottom-to-top approach and analyzes only the basic recycling strategy. We generalize the protocol to higher number N of TLS, although sticking with the general idea of global measurement on these copies, and open the road for future numerical optimization in this direction, as well. Top-to-bottom numerical approach based purely on numerical optimization requires the insight presented here anyway.
As our focus is on coherent energy synthesis, we consider all other incoherent energy operations, e.g., energy increase without coherence change, to be without any cost. The information, obtained in our measurement-based  13 and enters the process, assumed to be almost discharged with low initial energy E 0 and coherence C 0 (red loops connecting the levels). Such bath enables, due to the structure 13 of the system-bath interaction Hamiltonian, the formation of coherence in energy basis of a generic two-level system, independently of its initial state.
Step 2) represents the conditional stage of the charging operation. If the outcome of this step is successful (see step 3) with the single run probability of success p s , the battery is made more coherent and simultaneously charged, i.e., has higher energy > E E f 0 and increased coherence > C C f 0 as well, both with respect to the eigenbasis of Hamiltonian (2). If the charging fails, with the single run probability = − p p (1 ) f s in step 2), the battery is completely discharged and incoherent. In principle it can be recycled by bringing it in contact with the cold coherence-inducing bath 13 again and the protocol is repeated until success (RUS). If the charging does not successfully occur in R recycling runs, the total probability of failure, , is exponentially converging to zero with the number of repetitions, see the inset for a typical behavior. (b) Parametric 3D plot of the relative entropy of coherence difference ΔC, Eq. (6), on x-axis, the normalized average energy difference ΔE/E, Eq. (5), on z-axis and single run probability of success p s on y-axis. It demonstrates the basic step in the presented protocol (a): increase in the energy and coherence of the battery (after the charging process) with sufficient probability of success.
protocol, is physical, thus the measurement result has to be stored in some classical physical register (memory) which, after its filling, has to be refreshed with nonzero incoherent energy cost, dictated by Landauer's principle 29 . We do not explicitly consider such necessary incoherent energy resources in this paper, as at this stage we are primarily interested in the effect of coherent energy synthesis, and not so much in classical and incoherent resources allowing its functionality.

coherently charging a pair of two-Level Systems
In the following subsection, we describe the system and the variables of interest. The next subsections introduce the main idea of the paper, i.e., the conditional charging procedure applied to a pair of TLS in pure initial states, showing also which projective measurement is optimal concerning our goal. The following subsection generalizes the results for two classes of partially coherent states, namely, for partially mixed states due to dephasing and spontaneous emission of energy from TLS. energy and coherence. As the system, we consider two independent copies of two-level systems (TLS), denoted 1 and 2, with a non-zero energy gap E between the states. The preferred basis of the pair of TLSs, which we will uniquely refer to throughout the paper, will be given by the TLS energy eigenstates labeling the ground and excited state, respectively, | 〉 | 〉 g e { , } j j , for both subsystems = j 1, 2. The Hamiltonians defining these states read where E is the energy gap of each TLS, yielding the Hamiltonian of the total system From an energetic point of view, the Hamiltonian (2) represents a four-level system with the following energy eigenstates: | 〉 g g 1 2 ground state with energy −E, doubly degenerate first excited states | 〉 | 〉 e g g e { , } 1 2 1 2 with zero energy, and | 〉 e e 1 2 with energy +E. All the energy considerations will be made relative to this total Hamiltonian (2). The average energy and the variance of energy of any quantum state of our system is determined in a standard way as To quantify coherence of the state, we use the relative entropy of coherence (denoted simply "coherence" from now on) defined as 32 diag where ρ ρ ρ = −ˆŜ( ) Tr( ln ) is von Neumann entropy and ρŜ( ) diag is the entropy of the diagonal version of the state with respect to the energy basis. Thus, the latter one effectively amounts to the Shannon entropy of state ρˆ concerning energy eigenbasis. The maximum value of coherence C depends on the dimension D of the system Hilbert space. For any system ≤ ≤ C D 0 l n holds 32 , thus, e.g., for a single TLS it is upper bounded by = C ln2. If the single basic step of the protocol increases both coherence and energy, this result would constitute a prerequisite to use such coherent energy for an iterative procedure on many TLSs.
The choice of the relative entropy of coherence, Eq. (4), as our measure is definitely not unique. As another possibility, we may use as well the l 1 -norm based measure 32 . Although both are proper coherence measures, we prefer relative entropy of coherence. Such preference is motivated by the recognized connection of C ρ( ) with thermodynamic quantities 2 and the notion of von Neumann entropy, naturally connecting this measure and thermodynamics in much broader sense.
As we are interested in changes of the battery energy and coherence, we focus on the behavior of the energy difference between the final energy after the charging and the initial energy before the charging process, respectively. Similarly, we focus on the change of the coherence Synthesizing the 2-TLS coherent battery made from pure states. Before we analyze more realistic cases of TLS in mixed states, we discuss the ideal case showing the main idea of our approach. We consider charging of the battery consisting of a pair of TLS in the following initial state j j j where the Hamiltonians defining the respective energy eigenstates are in Eq. (1). This type of state can be, in principle, a steady state of TLS, originating from a certain type of interaction with a thermal bath 13 . This bath can increase the TLS energy and coherence with respect to its ground state g. Such case will serve below as an optimization target for harnessing of coherent energy from the environment. The corresponding initial state of the TLS pair is As it is well known, projective measurement applied to the system can change the state closer to a desired one. We want to choose such measurement that will charge the battery, i.e., increase its energy but also coherence, if possible. Moreover, we want to achieve this goal without using projectors on states with any coherence with respect to the energy eigenbasis of the Hamiltonian (2). If we project on the superposition of energy eigenstates, coherence can be induced by the measurement itself. Therefore, only projectors diagonal in the measurement basis are allowed. Namely, we choose the pair Application of such measurement results in two conditional post-measurement states s is the probability of success of the charging procedure, applied to a pair of TLS, serving as a new normalization factor. Equation (11) represents the final state after successful round of the battery charging process. In the case of the success, we fuse the initially independent pair of TLS into a larger superposition, thus creating and coherently charging the battery simultaneously. The failure of the protocol results in reducing the initial energy and coherence to zero, by bringing the TLS pair to the energetic ground state. Thus, these pairs (failed to be charged) can be used once more to harness a partial coherence from the environment. If such a strategy is repeated, the probability of failure after R independent repetitions (meaning that the battery will be not charged even in a single event) decreases as with p the probability of single TLS excitation, Eq. (7). As illustrated in the inset of Fig. 1a, such event becomes almost impossible at an exponential rate. The state (11) determines the final energy E f and its variance 〈Δ 〉 E f 2 with respect to the Hamiltonian (2), yielding It can be compared to the initial average energy and the initial energy variance of the state (8) The dependence of both on the excitation probability p, c.f. Eq. (7), are shown in Fig. 2a 2 , c.f. Eqs. (13) and (14), hence concentrating the energy distribution of the final battery. Together with the fact, that the average value of energy is increased by the protocol, > E E f 0 , one can understand the situation as an increase in the quality of coherent energy. Simultaneously, the projective measurement does not distinguish the states | 〉 | 〉 | 〉 e e e g g e , , 1 2 and therefore it allows not only to keep the initial coherence, but even to increase it. Having characterized the initial and final states |Ψ〉 i , |Ψ 〉 f , we can determine their coherence C 0 , C f , respectively, Eq. (4), in a straightforward way. The initial coherence of the state |Ψ〉 i is (due to TLS independence) additive, and reads , not solvable analytically. Therefore, it is more informative to focus on the relation between the final and initial coherence and deal with p as with an implicit parameter. Such an approach yields approximate functional relation valid for every p This result provides immediately = → C lim ln2 . This represents the value of the initial/ final coherence where the curves C f and C 0 cross each other. As such,  C 0 represents the maximum value of the coherence attainable, if coherence increase is required.
We would like to stress that our protocol basically decomposes any initial state Eq. (8) into component in the one dimensional | 〉 g g 1 2 subspace and its orthogonal complement from three dimensional subspace spanned by  (8), either has a non-zero component in the above mentioned 3D subspace (for > p 0), or does not have it (for = p 0) in which case there is no increase of the energy and coherence after the successful measurement, causing the discontinuity of the values at = p 0. This fact is taken into account in all our derivations and is stressed by the condition ≠ p 0 in Eq. (11) and the following.
Synthesizing the 2-TLS coherent battery made from mixed states. Whereas the previous subsection has introduced the main idea of our measurement-based charging process, the following lines will describe the effect of partial coherence of the TLS initial state. Such states are characterized, e.g., by the property ρ < Tr( ) 1 2 and typically result from dephasing or spontaneous emission processes acting on the TLS. In the following lines, we will study these two archetypal processes individually.
We start with the pure dephasing process acting individually on each TLS because it is usually the fastest process, which reduces coherence after it is harnessed from the environment. In this case, the pure initial state (7) will be modified as p e e p p e g g e p g g j www.nature.com/scientificreports www.nature.com/scientificreports/ where ε ≤ < 0 1 . The total initial state of the system reads ρ ρ ρ = ⊗ˆî 1 2 , Eq. (18). The charging procedure is considered identical to the one in the previous subsection characterized by a pair of projectors = −ˆˆP P P { , 1 } 0 1 0 , Eq. (9), but applied to the state ρˆi, resulting in the successfully charged state where p s is the same value of the success probability, Eq. (11), independent of ε. We stress, that the partially coherent state, ε < 1, does not change the results for the average energy E f , Eq. (3), obtained in the previous subsection, cf. Eq. (13) and Fig. 2a. This is intuitively the case, because E 0 as well as E f , depend only on the diagonal terms appearing in the state (18). Hence, to recall, the initial energy of the battery is i 0 On the other hand ε < 1 strongly influences the values of the initial coherence Eq. (19). Therefore, the mixedness of the initial state also affects the relation between C f and C 0 , cf. Eq. (17), and their difference Plot of ΔC, Eq. (21), is shown in Fig. 2b for several values of parameter ε. Following the same logic as in the previous subsection, we characterize the upper bound on the initial coherence value C 0 that can be still increased. Equation (21) yields the following result  The initial energy of the battery now reads As in the previous lines it is more useful to show the approximate result for coherence change ΔC, Eq. (6), than the individual summands, yielding 0 a good approximation for η ≤ ≤ 0 1 /3. The interesting fact that ΔC increases with η stems from the different speed at which C 0 and C f decrease with η. This opens the gap ΔC between them and simultaneously increases the lower-value crossing point, = C C f 0 , in which the charging protocol stops increasing the coherence. This value reads (if such point of coincidence exists), see Fig. 3b, Comparison of Eqs. (27) and (22) reveals one striking difference. Although both processes (dephasing and spontaneous emission) of the initial states decrease the initial value of coherence C 0 in a similar way, the effect of charging yields results with qualitatively opposite behavior. Whereas lowering ε, Eq. (18), quickly decreases  C 0 (Fig. 2b), the increase of η, Eq. (23), leads to the increase of  C 0 (Fig. 3b).  applied to the pure initial state |Ψ〉 i , Eq. (8). Clearly, the case = a 1, = b 0 represents the projector used at the beginning of this section. On the other hand, = a 1, = b 1 is clearly an optimal choice for maximization of the final energy, as it results in removing all the populations except those of the state | 〉 e e 1 2 , preparing the analogy of the successful state (11) with maximum energy, but certainly zero coherence in the energy eigenbasis. Hence, Eq. (28) represent a generalization of the projections to the set of POVM's, but is still compatible with the previous important choice of measurement elements that are diagonal in the eigenbasis of the Hamiltonian (2).
Due to the more complicated structure of the analytical results obtained using the elements of (28), we restrict ourselves to a graphical representation of the results, see Fig. 4. They present values of the coherence optimized for each fixed excitation probability p, Eq. (7), where the optimization is done over a certain range of a, b parameters, see below. The energy plots show, for each fixed p, the post-measurement value corresponding to the optimal a (or b) parameters determined by optimization of the coherence. Figure 4 shows the results for a general class of POVM, cf. Eq. (28), and its restricted analogue with = a 1, ≤ ≤ b 0 1 . We have chosen these two classes, because they represent POVM yielding two qualitatively different results, and we compare them both with the projector case = a 1, = b 0, see Fig. 4. The results for the more general class indicate a substantial coherence increase over the whole range of p, reaching the maximum attainable value ≈ C a b ( , ) ln4 f for a pair of TLS, but this is paid by a smaller increase in the energy compared to the simple projection, Eq. (9) and Fig. 4. For the latter POVM class, we found the increase of coherence in the lower p region (up to = p 1/2) reaching Fig. 4b, and higher increase in energy compared to the simple projection, Eq. (9), whereas POVM defined in Eq. (28) gives an increase in energy with respect to the initial value E 0 only in the range of ≤ ≤ p 0 1 /2. This increase is significantly lower compared to the projector-based protocol, Fig. 4a. The optimization process for both classes of POVMs was made with respect to the final post measurement coherence C f , Eq. (4). Therefore, the general class, Eq. (28), was optimized over a and b for each fixed p, resulting into C a b ( , ) f . Figure 4a shows the corresponding final energy  (1 1 )( whereas the relationship between the optimal value a and the excitation probability p can be well approximated by the polynomial dependence ≈ − − a p p 1 3/2 8 2 4 in the interval ≤ ≤ p 0 2 /5. To summarize this subsection, we found that due to the presence of additional free parameters a, b, Eq. (28), which allows for subsequent optimization, POVM protocol offers more resulting options compared to the simple projector based protocol. We have presented two different types of results, out of a number of options. POVM allows both for increase of C a b ( , )

Synthesizing N-tLS coherent Battery Made from pure States
This section generalizes the results described for a pair of TLS in the previous section. Here, a larger number of systems is subjected to the charging protocol. Namely, we consider a battery given by ≥ N 2 noninteracting copies of TLS, characterized by the total Hamiltonian where Ĥ j are respective Hamiltonians of the j-th TLS, given in Eq. (1). In the following we restrict our attention only to the pure (ε = 1) initial states of the system, cf. Eq. (18)), noting that generalization can be done numerically in a straightforward way.
The total initial state of the N TLS's battery is therefore, cf. Eq. (7), j N 1, , labels the copies of TLS. Subsequently, the initial energy of the system is given by a straightforward (due to its additivity) generalization of Eq. (14), yielding  f (green line), for the general type of POVM, shows substantial increase over the whole range of p (e.g., green arrow for = . p 0 2 in (b)), compensated by smaller increase in the energy compared to the simple projection, Eq. (9) (e.g., green arrow for = .
p 0 2 in (a)). The results for restricted type of POVM (orange lines) show improvement of the coherence values (e.g., orange arrow for = .
p 0 2 in (b)) over the interval of < p 1/2, and simultaneous increase of the post measurement energy values (e.g., orange arrow for = . p 0 2 in (a)) on the same interval of p, with respect to the simple projector case, Eq. (9). www.nature.com/scientificreports www.nature.com/scientificreports/ The charging process is analogous to previous subsections and consists of application of a pair of global pro- (2) increases with the number N of TLS constituting the battery, whereas the energy gain is limited by . Notably, this last property is positive due to the fact that the higher N we use, the less we are restricted by the choice of parameters for which we observe the positive effect of the charging protocol, i.e., the energy E f N ( ) and coherence C f N ( ) increase, over the respective initial values. Moreover, it follows from Eqs. (37) and (38) that for fixed initial state excitation probability p, Eq. (7), we obtain At this point we want to mention another possible charging protocol employing N TLS. Its detailed analysis is beyond the scope of this paper. It does not rely on global measurement on all TLS, but instead, on sequential pairwise application of the protocol introduced in the previous section, to the initial state (31)  At this point, we would like to stress that such (simplified) protocol can be applied on all types of states (pure, dephased, after spontaneous emission, etc.) mentioned in the previous sections, due to its pairwise structure and it can be combined (in certain stages) with the repeat-until-success strategy introduced in the first section.

conclusions and outlook
We have presented a protocol for the repeat-until-success charging of the quantum battery by means of quantum measurement application. The building blocks of our battery are factorized copies of identical two-level systems (TLS) with nonzero, although possibly very small, population of the excited state, and some small residual coherence that can be achieved by a suitable interaction with a sufficiently cold thermal bath 13 . A pair of TLS subsequently undergoes a global, unity-resolving measurement with two possible outcomes, each represented by an operator diagonal in the TLS energy basis. www.nature.com/scientificreports www.nature.com/scientificreports/ If the protocol succeeds, we synthesize the initially independent pair of TLS into a system with higher coherent energy, thus creating and charging the battery simultaneously. The failure of the protocol results in reducing the initial energy and coherence to zero. These two possibilities represent the outcomes of the inherently conditional protocol, however with the possibility of increasing the success probability arbitrarily close to one (using repeat-until-success strategy), turning the protocol effectively into a deterministic one. Our results show that the partial purity and/or previous spontaneous emission of the energy of the initial state decreases the positive effect of the charging, but does not prevent the charging in principle. The variance of energy is decreased in this type of charging protocol.
The results of the projector-based charging protocol were generalized to the case of using N-TLS, either in a global or pairwise approach. The final value of the coherent energy can be increased even more, due to optimization, if the measurement consists of POVM elements. Such protocol is superior in the value of coherence and energy with respect to the results of the projective measurement. The results should stimulate proof-of-principle experimental verification of such energy synthesizing using controlled quantum systems. To verify the observability of simultaneous energy and coherence increase, experimental tests can be implemented using photons [33][34][35][36] , employing such optical toolbox also with atoms 37-39 , or solid-state systems 40,41 . The implementation of measurement strategy for another experimental platforms is under development. This optical experimental tests would stimulate quantum thermodynamic analysis 2,42 of such synthesis, already used to analyze energy manipulations 27,43,44 .
We have verified that many paths are open to synthesize energy in coherent quantum battery for further optimization. This optimization is a demanding top-to-bottom task, therefore, it has to be solved numerically.