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Multimode advantage in continuous variable quantum battery

We provide an architecture for a multimode quantum battery (QB) based on the framework of continuous variable (CV) systems. We examine the performance of the battery by using a generic class of multimode initial states whose parameters can be tuned to produce separable as well as entangled states and that can be charged locally as well as globally by Gaussian unitary operations. Analytical calculations show that a separable state is equally advantageous as an entangled one for two- and three-mode batteries when taking the figures of merit as the second moments of the change in energy. In order to produce a stable quantum battery consisting of an arbitrary number of modes, we derive compact analytical forms of the energy fluctuations and prove that for a multimode separable Gaussian initial state, fluctuations decrease as the number of modes increases, thereby obtaining a scaling analysis. Moreover, we demonstrate that local displacement as a charger is better to minimize the fluctuations in energy than that involving the squeezing unitary operation.


I. INTRODUCTION
Small-scale energy-storing devices like batteries are necessary due to the growing trend of miniaturizing electronic devices, especially at the molecular and subatomic levels. Batteries typically store chemical energy and convert it into electrical energy through wellknown reduction-oxidation reactions. Although these batteries play a crucial role in our daily life, starting from medical instruments to household appliances, navigation, and also in transportation systems, they have several significant constraints at the microscopic level. The principles of quantum mechanics have been shown to modify the laws of thermodynamics [1,2] and improve the performance of a variety of devices, including that of quantum thermal machines [3,4].
A quantum battery (QB) is a device that can store energy, and from which the energy can be extracted via unitary operations. It typically consists of a collection of d-dimensional quantum systems governed by a Hamiltonian having a non-degenerate energy spectrum [5]. In the last few years, a series of QB models, charged via global and local unitary operators, have been proposed, which include many-body quantum batteries [6][7][8][9][10][11], interacting spin-chains [12], Dicke quantum battery [13,14], quantum battery based on superconducting qubits [15] and batteries formulated with the Sachdev-Ye-Kitaev (SYK) [16] and also the Lipkin-Meshkov-Glick (LMG) models [17][18][19]. Numerous investigations have also been conducted to analyze the detrimental impact on the performance of the QB by taking into account the inevitable interaction of the system with the environment [20][21][22][23][24]. Going beyond these distinguishable systems, the modeling of energystoring devices with indistinguishable particles in the ultra-cold atomic setup in the test bed of bosons and fermions has also been addressed in recent times [25]. Despite the tremendous progress in the field of quantum batteries, it is still unclear how non-classicality, like quantum entanglement [26] or quantum coherence [27] affects the performance of quantum thermal machines.
In particular, global entangling operations are found to be more beneficial than the local operations, which are further supported by a very recent work [28], although it may not generate entanglement between the subsystems [29][30][31].
All the aforementioned QBs are designed on systems having discrete and finite degrees of freedom. Interesting studies on QBs in infinite dimensional systems are limited. Variables like the position and momentum of a particle possess a continuous spectrum and are used to characterize continuous variable (CV) systems. Prominent examples include the harmonic oscillator, through which many physical systems having an important role in quantum optics [32] have been realized and which also provide significant benefits in the fields of quantum error correction [33,34], quantum teleportation [35], quantum computation [36] and quantum cryptography [37]. Apart from all these theoretical proposals, tremendous progress on the experimental front has been reported while employing CV systems to execute quantum information tasks, such as dense coding [38,39], teleportation [40,41], quantum cloning [42] and quantum key distribution [43]. As a thermal machine, harmonic oscillators are also used as a charger to deposit energy in a two-level system [44,45], while non-Gaussian charging is shown to be optimal [46] to charge a QB made of infinite-dimensional bosonic systems under the constraints of low environmental temperature and in the presence of strong squeezing [47]. All these proposals for CV batteries involve few or a single mode.
In this paper, we go beyond this restriction and explore the question -does the battery's efficiency increase if it is constructed with several modes? If so, is multimode entanglement required for this improvement? We report here that although the second answer is negative, the first one is affirmative. Precisely, we present a design of a QB consisting of an arbitrary number of non-interacting Gaussian systems, that are charged via local Gaussian operations [48] namely, squeezing and displacement, and global squeezers. To analyze the performance, the second moments in the energy change, known as charg- Schematic diagram depicting the functioning of a multimode continuous variable quantum battery. Initially, it is uncharged and has a minimum amount of energy in each mode which is shown as filled in light (blue) color. The phase space description of each mode is taken to be the same and is shown as light (green) ellipses. The local charging operations are considered to be either displacement (D c ) or squeezing (Ŝ c ). The dark (blue) color represents the increment in the energy of the charged battery from the dotted line. The charged modes are either squeezed or displaced which are depicted as dark (green) ellipses in phase space.
ing precision [46] -determining the stability of the battery, and the work fluctuation -quantifying how accurately the battery can be charged, are computed after optimizing the parameters involved in the charging process and the initial state of the battery. Specifically, we choose a class of generic states comprising two-and three-modes with varying degrees of entanglement, as the initial states of the QB, thereby representing both separable and entangled states of different entanglement strengths. We highlight that when charging is carried out using global and local operations, and the energy to be stored is fixed to a given amount, fully separable and entangled states provide an identical level of stability in the QB. Our results demonstrate that the modeling proposed here definitely lowers the experimental cost of the CV battery. By charging the system with local squeezing and displacement operations, we also demonstrate the multimode advantage of the stability of the CV battery. In particular, our analyses, supplemented with numerical data, show that by considering initial separable states with an arbitrary number of modes, the fluctuation during discharging decreases sharply with the increase of modes, when squeezing unitaries are used as the charging operation. Although the work fluctuation does not vary with the number of modes in the case of local displacement, the fact that the charging precision de-creases as the number of modes grows, once more emphasizes the constructive role played by multiple modes present in the battery. The minimum values attained by both the figures of merit are lower in the case of local displacement as charging operations than that provided through squeezing. Such modal superiority has a twofold implication: firstly, it boosts the stability of the battery against fluctuations in energy, and secondly, storing more energy in the battery is possible, since chargin a single mode is restricted by experimental limitations (especially in the case of squeezing). Therefore, a higher number of modes directly implies a more perfectly functioning CV quantum battery.
The paper is organized as follows: In Sec. II, we introduce the basic notions of CV quantum battery and its charging operations along with the figures of merit. The importance of considering the entangled state as the initial state and the role of global as well as local charging are reported in Sec. III for effectively small system size. Motivated by the results obtained in Sec. III, a scaling analysis is performed by considering a fully separable initial state consisting of an arbitrary number of modes, and the multimodal advantage is reported in Sec. IV. Finally, in Sec. V, we summarize our results.

II. CONTINUOUS VARIABLE QUANTUM BATTERY AND ITS FIGURES OF MERIT
A device consisting of an arbitrary number of quantum systems, in which energy can be stored and extracted whenever required via quantum operations, is called a quantum battery. The initial battery state can either be a product state of decoupled subsystems or a multisite entangled state. The battery HamiltonianĤ B can be used to describe the energy of the system. In order to store (charging) or extract (discharging) work from the system, the battery is evolved by a charging HamiltonianĤ C which must be non-commuting with respect toĤ B . Like the battery Hamiltonian, the charging operation can either be local, addressing individual subsystems, or an interacting global one, which acts on the entire system. The choice of the battery Hamiltonian, and the charging Hamiltonian, depends on the system of interest [7,8,10,12,20,23].

Design of the CV Battery
Instead of working in finite dimensions, we propose a quantum battery built of infinite dimensional systems, specifically continuous variable (CV) systems. In particular, we choose the initial state of the battery as an N -mode Gaussian state which can be described by its first and second moments in the phase space. The underlying Hamiltonian describing the energy of the system isĤ B = ∑ N j=1 ω jNj whereN j =â † jâ j denotes the number operator. In a more accomplished way, we consider the system consisting of N harmonic oscillators, each of frequency ω j of the mode j. In order to probe the role of quantum correlations present in the system, we choose the initial state to be both correlated and uncorrelated, which, e.g. can be prepared through the application of beam splitters on an N -mode product state with alternate squeezing r, −r, ... in consecutive modes [49]. Hence, the class of correlated states can be characterized by N − 1 parameters τ j which represents the transmittivity of the beam splitter j. Notice that most of the well-known entangled states can be constructed in this way as we will describe later.
Charging. The goal of the charging operation is to increase the energy of the system through unitary evolution which also accounts for the work stored in the system in the case of closed dynamics. This evolution changes the state of the system to another state which, in turn, creates an energy difference ∆E U C , where U C denotes the charging unitary. By constructing the battery Hamiltonian, we choose the initial state and the charging unitary in such a way that the total change in energy becomes equal to ∆E U C . We analyze the stability of the battery during charging and discharging. We also provide the most efficient modal distribution of the energy, ∆E U j C , in an arbitrary mode j, such that ∑ j ∆E U j C = ∆E U C . As charging operations, local squeezing and displacement unitaries act on each mode while global squeezing operations are performed on the entire system. Both pictures address the question of whether entanglement in the initial state or entangling charging operations can help to achieve optimal figures of merit.
• Squeezing operation. The squeezing operator . ., acts on the mode j of the N -mode battery, where, ζ j = δ j e iθ j is the squeezing parameter, with δ j and θ j respectively being the squeezing degree and the squeezing angle. Here, i = √ −1 and * denotes complex conjugation. The total local charging operator for the N -mode battery is given bŷ S c =Ŝ 1 (ζ 1 ) ⊗ . . . ⊗Ŝ N (ζ N ). We refer to θ j as the phase of the charger.
• Displacement operation. If the battery is charged via local displacements, the charging operator on jâ j is the local displacement operator characterized by α j = |α j |e iφ j with φ j being the phase of the charging unitary.
Performance indicators. Instead of studying the usual work output of the battery, which is typically studied in spin systems [50], we are interested in quantifying the stability of the quantum battery (QB). Precisely, the charging process can be designed in such a way, that at the time of discharge, it can produce the same amount of energy that is stored in the battery. This is quantified by a parameter known as the charging precision [46], which is defined for an N -mode system as where V(ρ k ) stands for the variance, given by withĤ B being the battery Hamiltonian. Here, ρ k s are the states under consideration with k = 0, 1 corresponding to the initial and final states of the battery respectively, and ρ 1 = U C ρ 0 U † C with U C being the charging unitary. If the charging strength is γ j (δ j , |α j |) and its corresponding phase is ν j (θ j , φ j ) for each mode of the battery, evidently, we have where {s} is the collection of initial state parameters e.g. the initial squeezing r, and ω j s are the parameters of the battery Hamiltonian. For the mode j, we can invert the expression for a fixed energy increment ∆E U j C to obtain γ j , i.e., Using Eq. (4), we can obtain the charging precision as a function of the increased energy of the different modes, i.e., By fixing the initial state parameters, {s} and the total energy increment our task is to minimise Eq. (5), i.e., ∆σ (N) , over the set of the charging phases {ν j }, and the energy distribution, The minimum value is denoted as ∆σ min . Although a general battery Hamiltonian includes arbitrary ω j s, for ease of calculation, we take them all to be equal to unity. However, the conclusions remain the same even under this assumption, since the overall behavior is qualitatively similar when one considers arbitrary ω j s. In particular, the difference in taking arbitrary ω j s would be reflected in the optimal energy distribution {∆E U j C } of the charging process and the optimal charging phases {ν j }, although the effect of the number of modes on the figures of merit remains unaffected.
Another indicator of the performance of the battery is the energy fluctuation during the charging process. Towards defining it, let us first rewrite ∆E U C as To store ∆E U C amount of energy in the battery by the corresponding charging process U C , the deviation of stored energy for an N -mode system is quantified by a quantity called work fluctuation [46], which is defined as where and the expectation value is taken with respect to the are the energies of the uncharged and charged states of the battery respectively, which give ∆E We will use ∆W (N ) as another indicator to check the performance of the QB during the charging process. Note that the optimization of the work fluctuation ∆W (N ) proceeds along the same line as that of ∆σ (N ) as elucidated through Eqs (3) -(5).
For a well-functioning battery, we are required to minimize the energy fluctuation, to make the battery stable during the charging and discharging processes which, in turn, implies the minimization of both ∆σ (N ) and ∆W (N ) , thereby achieving ∆σ (N ) min and ∆W (N ) min . To this end, our objective is to increase the energy of the battery by an amount ∆E U C , which is distributed equally between different modes, and check whether it is helpful to increase the number of modes of the QB. Such a study can also reveal the role of entanglement in achieving an efficient quantum battery in this setup.

III. NECESSITY OF ENTANGLEMENT FOR MULTIMODE BATTERIES
In order to analyze the performance of the battery consisting of N -modes, it is natural to investigate whether entanglement is necessary for the proper functioning of a multimode quantum battery. Note that both entangled and product states can be prepared from the ground state of the battery Hamiltonian by global and local operations respectively, as discussed in the previous section. Specifically, we compare the trends of ∆σ (N ) and ∆W (N ) obtained from two-and three-mode entangled states with those of squeezed separable states as initial states.

A. Two-mode CV battery: Entangled vs. Product states
To design the initial state of a two-mode battery, we use a beam splitter BS 12 with transmittivity τ where 0 ≤ τ ≤ 1. Two squeezed vacuum modes, one squeezed in the position quadrature with squeezing strength r and the other squeezed in the momentum quadrature with the same strength, are impinged on the beam splitter, and the output state is entangled, depending on τ. Though the transmissivity (i.e., the state parameter) can take its value up to 1, the entanglement content of the output state is symmetric with τ = 0.5. It can be shown that the entanglement of the output state increases with τ up to τ = 0.5 which leads to a well-known two-mode squeezed vacuum (TMSV) state while τ = 0 corresponds to a separable state. A generic state belonging to this one-parameter family with varying entanglement content may be conveniently represented in the phase space picture by the following displacement vector and covariance matrix as d (2) Note that initially, the energy of the uncharged battery is E (2) 0 = 2 sinh 2 r. Such a choice of the initial state can help us to probe both entangled states having different entanglement content and the product state as uncharged QBs. Squeezing and displacement operations on each mode as well as the global two-mode squeezing operator are applied to store energy in the battery. In each case, our aim is to determine the optimal value of τ, and, therefore, the entanglement, for which charging in both the modes yields the minimum value of ∆σ (2) and ∆W (2) .

Charging with local squeezing
The charging operator for the two-mode battery consists of local squeezing operators acting on the individual modes, given byŜ c =Ŝ 1 (ζ 1 ) ⊗Ŝ 2 (ζ 2 ), where the local squeezing operatorsŜ j (ζ j ) have been defined in Sec. II 1. Upon charging, the increase in the total energy of the battery is When the battery is charged in both the modes, we can represent the increase in energy of the two modes. We can similarly calculate the second moments of the energy and hence can obtain the charging precision ∆σ (2) as given in Eq. (1). The charging precision is initially a function of r, τ, δ j s, and θ j s, although using Eq. (13), we can write it as ∆σ (2) = f (r, ∆EŜj c , θ j , τ) for a fixed squeezing strength of the initial state, r. We distribute the total energy ∆EŜ c as k∆EŜ c and (1 − k)∆EŜ c in the two modes and find that the charging precision is minimum at k = 1/2 after optimizing over k, θ j , and τ. Now, while finding the optimum entanglement content of the battery, i.e., the optimal value of τ for a given ∆EŜ c , which occurs when ∆EŜ1 min turns out to be independent of τ with optimal values of θ j , such that θ 1 + θ 2 = π, for a fixed r, as depicted in Fig. 2(a). Our results indicate that to obtain a stable QB, the unentangled state is as good as the entangled one as an initial state. This implies that entanglement may be a sufficient condition for a good battery but is not necessary. This is true irrespective of the initial squeezing strength (initial energy of the uncharged battery) and the energy provided to the battery via the charging process. Thus for two-mode quantum batteries, a twomode separable state is an optimum choice.

Charging through entangling squeezer
Let us investigate the reverse scenario where the initial state is a two-mode separable state (2-S MSV) while the global squeezing operation is applied in both the modes together to charge the battery and study the role of entanglement generation in the battery. Mathematically, the two-mode product state, in the phase space, is expressed as where Ξ j denotes the covariance matrix corresponding to a single-mode squeezed vacuum (SMSV ), In the phase space, the symplectic matrix representation of the global squeezing operation [51] reads aŝ where A = cos θ sinh δ, B = sin θ sinh δ, and C = cosh δ, with δ and θ being the squeezing strength and the squeezing angle respectively. Here, the change in energy, ∆EŜ c is a function of δ, θ, r, i.e., ∆EŜ c = f (δ, θ, r).
min (z-axis) is shown against τ 1 (x-axis) and τ 2 (y-axis) for squeezing and displacement chargers of the three-mode initial states respectively. The energy specifications in this case correspond to r = 0.5, ∆ED c (∆EŜ c ) = 15 (dark blue) and r = 1, ∆ED c (∆EŜ c ) = 24 (light orange). All the axes are dimensionless.
In this case, storing energy in a single mode is not possible due to the global charging operation, although we can quantify how much energy is incorporated in each mode. We find that ∆EŜ c = 2 sinh 2 δ cosh 2r and ∆EŜ1 c = ∆EŜ2 c = ∆EŜ c /2 i.e., the total energy increment is divided between two modes equally. We proceed to check the behavior of ∆σ (2) with the variation of total energy ∆EŜ c to find if there is any advantage due to quantum correlations developed while storing the energy. By optimizing ∆σ (2) with respect to the charging parameter, θ to obtain ∆σ (2) min , we observe that the minimum value exceeds the value obtained in the previous scenario where the initial state is an entangled one, charged via local operations. It implies that the stability of the QB obtained via local operations is higher than that via global ones. Furthermore, the configuration for optimal charging precision again corresponds to the squeezing angle θ = π/2. Therefore, entangling operations cannot enhance the stability of the battery than that obtained from the separable initial states with local chargers.

Charging via displacement
Instead of squeezing unitary, let us find whether a change of unitary can lead to a more stable QB or not. In particular, the battery is charged via local displacements applied individually in each mode. The charging operator in this case readŝ whereD j (α j ) are the local displacement operators as described in Sec. II 1. The energy gained by the battery upon charging has a much simpler form than the one in the case of local squeezer and depends only on the parameters |α j |.
Again, similar to the previous case, we can calculate ∆σ (2) = f (r, |α j |, φ j , τ) and since α j = ∆EDj c (from Eq. (18)), we obtain ∆σ (2) in terms of r, ∆EDj c , φ j s and τ. By fixing a value of r, we again find that ∆ED1 c = ∆ED2 c = ∆ED c /2, leads to minimal fluctuation i.e., ∆σ (2) min for any value of τ with optimal phase configuration. Moreover, the optimal charging precision again turns out to be independent of τ (see Fig. 2(b)). Thus for every value of τ, we obtain a particular set of (φ 1 , φ 2 ) (e.g. for τ = 0, φ 1 = π/2 and φ 2 = 0) which yields the same minimum value of ∆σ (2) . We can safely conclude that in terms of the charging precision of the QB, entangled and product initial states perform equally well. From the analysis of the local charging operations, we arrive at the following observation.

Observation 1.
For a two-mode battery charged in both the modes through local squeezing and displacement unitaries, entanglement plays no role in the optimal performance as quantified by the charging precision, ∆σ (2) min .
Remark 1. Notice that, the minimum value of ∆σ (2) is independent of τ and hence the entanglement of the initial state although it always occurs at θ 1 = θ 2 = π/2 in case of squeezing charger. However, this is not the case for the local displacement operations, i.e., ∆σ Remark 2. By setting τ = 0, we find that the optimal value of ∆σ (2) min is independent of θ j when the battery is charged via squeezing operations. This implies that for a two-mode separable battery, the optimal stability only depends on the squeezing degree of the charger.

B. Entangled three-mode battery
Let us take the initial state of the QB as a three-mode entangled state, which in general can be prepared by a tritter [35,52] composed of two beam splitters. A class of such three-mode entangled states can be parameterized by the transmission coefficients of two beam splitters BS 12 and BS 23 which act on mode pairs (1, 2) and (2, 3) respectively. The first and third modes are squeezed in the momentum quadrature while the second mode is position squeezed, all of the modes having degree r. The resulting state can be characterized by [49] and Ξ where Note that by setting τ 1 = 1/3 and τ 2 = 1/2, we can prepare the Basset-Hound state [52][53][54].
The energy of such an uncharged battery is E (3) 0 = 3 sinh 2 r. Again local displacement and squeezing unitaries are applied individually on the three modes to charge the battery.

Charging with local squeezing and displacement
When local squeezing unitaries are applied to three modes, each with a degree δ j and angle θ j (j = 1, 2, 3), the increase in the energies of the individual battery modes is given by ∆EŜ2 c = sinh δ 2 cosh δ 2 cos θ 2 sinh 2r(1 − 2τ 1 τ 2 ) + sinh δ 2 cosh 2r , with ∆EŜ c = ∑ 3 j=1 ∆EŜj c . In this case, the charging precision can be shown to be optimized when ∆EŜj c = ∆EŜ c /3 in all the modes. Considering that we distribute the energy ∆EŜ c as k 1 ∆EŜ c , k 2 ∆EŜ c and (1 − k 1 − k 2 )∆EŜ c in the three modes such that k 1 + k 2 ≤ 1, we observe that ∆σ (3) min occurs at k 1 = k 2 = 1/3. The further analysis proceeds in a manner similar to the twomode case, where we invert Eqs. (22) - (24) to obtain δ j in terms of r, ∆EŜj c s, θ j s and τ j s and hence we write ∆σ (3) = f (r, ∆EŜj c , θ j , τ j ) using Eq. (1). In this scenario, where charging is performed equally in all three modes, the minimum charging precision is obtained, for any initial squeezing r and ∆EŜ c when θ j = π/2 and is independent of τ 1 and τ 2 (see Fig. 2 (c)). This implies that the entanglement is only a sufficient requirement for optimal performance since the biseparable configuration (obtained with τ 2 = 0), as well as the fully separable state (obtained by setting τ 1 = 0), perform equally well. Furthermore, when the battery comprises fully separable modes, ∆σ (3) min is independent of the squeezing angles θ j , thereby providing more freedom during the charging process. Thus, we can easily work with a fully separable state. For example, with r = 1 and min ≈ 2.51. Thus for a three-mode battery as well, entanglement is only a sufficient condition for optimum performance when charging is done via local squeezing operations. In case of charging via local displacements, having parameters α j = |α j |e iφ j , then the increase in energy of the battery is given as, where ∆EDj c is the energy change in mode j. Again, we can estimate ∆σ (3) in terms of the displacement parameters, the initial squeezing strength, and τ j and . Given an initial squeezing strength r and an energy change ∆ED c , the charging precision attains minimum at some set of values of τ j s with some optimal choice of φ j s depending on entanglement content when all the modes are equally charged (see Fig. 2 (d)). Specifically, at τ 1 = 0 (fully separable), we find that the optimal choice of phases, which yields the minimum charging precision ∆σ (3) min , is φ 1 = π, φ 2 = φ 3 = π/2. Furthermore, when τ 1 = 0, ∆σ (3) min is independent of τ 2 , implying that we can dispense with the beam splitters altogether and hence we prove that the fully separable squeezed state is the optimal initial state for building a stable QB. Therefore, in the three-mode case, we obtain a condition to build a stable battery that is truly independent of entanglement, i.e., for both the chargers, the results obtained above can be consolidated as follows.
Observation 2. When a three-mode battery is charged via local squeezing and displacement operations equally in all the modes, the separable state is the most efficient choice.

IV. N -MODE SEPARABLE BATTERY
Let us argue that the unentangled state is the appropriate choice as an initial state which is influenced by the results obtained in the preceding section. Analyzing our aforementioned observations, we first notice the following: 1. Neither an entangled state with local charging nor a separable state with global charging leads to a stable battery as designed here.
2. The phase parameters involved in the charging process (i.e., squeezing or displacement angles) play a crucial role in the optimization of the charging precision.
3. Moreover, it turns out that equal charging in all the modes corresponds to the best charging configuration.
From these observations, it is natural to take an Nmode separable state with all the modes having equal initial squeezing (r), as the initial state of an N -mode battery and consider the charging operations as local.
The initial state in this case can be represented by the displacement vector and covariance matrix as The initial average energy of this system is while the average of the squared energy takes the form as The corresponding second moment of the energy, turns out to be V(ρ 0 ) = N cosh 2 r sinh 2 r. By applying local squeezing and displacement unitaries in each mode, the variation of the figures of merit of the battery against the number of modes can be obtained. Before presenting the results, let us state a lemma, which will be frequently applied.
For any multivariate function f (x 1 , x 2 , . . . x n ) which is symmetric to permutations of its arguments, the global extremum occurs at x 1 = x 2 = · · · = x n .
Proof. Since f is a symmetric function of its arguments, we have f (x 1 , x 2 , . . . x n ) = f (x 2 , x 1 , . . . x n ) = . . . and all possible permutations of x j . Suppose that f min(max) = f (x 0 1 , x 0 2 , . . . x 0 n ). Then all possible permutations of x 0 j as the argument of f would still yield f max(min) . Since the extremum is a global one, this implies that x 0 1 = x 0 2 = . . . x 0 n . A direct consequence of this lemma is that the figures of merit considered here, being symmetric functions of the increment in mode energies ∆E U j C , can be minimized by distributing the entire energy ∆E U C equally in each mode i.e., by choosing ∆E This has been extensively verified for all the models under consideration.

A. Charging with local squeezing
The charging of an N -mode battery by applying local squeezing in each mode is performed with the modal squeezing parameter, ζ j = δ j e iθ j . Hence the total charging operator becomesŜ c =Ŝ 1 (ζ 1 ) ⊗Ŝ 2 (ζ 2 ) ⊗ ... ⊗ S N (ζ N ). Upon charging, the total energy is given by where NŜ j c denotes the average energy of the mode j after charging and ∆EŜj c is the corresponding energy increment in that mode, which is given as ∆EŜj c = sinh δ j cosh δ j cos θ j sinh 2r + sinh δ j cosh 2r .
Hence, the total increase in energy after charging reads as On the other hand, the average of the squared energy is given by with N 2 S j c being the mean of squared energy of the mode j, which takes the form as 3 cosh 4δ j + 4 sinh 2δ j sinh 2r(3 sinh 2δ j cos 2θ j sinh 2r − 4 cos θ j ) + 12 sinh 4δ j cos θ j sinh 4r − 16 cosh 2δ j cosh 2r + (9 cosh 4δ j + 3) cosh 4r + 1 .
Having the expressions for the averages of the energy and the squared energy, one can find the standard deviation in the energy before and after the charging process. Thereafter, from Eqs. (1) and (2), we can find the charging precision ∆σ In order to calculate the work fluctuation, we first need to find {Ĥ B ,Ĥ B } ρ 0 with respect to the initial state of the battery ρ 0 . Using some operator algebra, we finally obtain the expectation value of the above-mentioned anticommutator as and using Eq. (7), we can obtain the expression of ∆W Advantage of multiple modes in the battery for charging with local squeezing Notice first that the separable N -mode state we consider as the initial state of the battery is symmetric with respect to the permutation of the constituent modes. As a result, both ∆σ (N ) and ∆W (N ) are symmetric functions of the energy increment in mode j, ∆E Hence, considering Lemma 1, the minimum for both the figures of merit are obtained when each mode has its energy raised by ∆E U j C = ∆E U C /N , which is the optimal charging configuration. An interesting point to note is that for a fixed initial squeezing r of the battery modes, an equal energy increment in each mode, implies that all the modes have equal charging parameters. In other words, we have δ 1 = δ 2 = · · · = δ N = δ and θ 1 = θ 2 = · · · = θ N = θ. This simplifies our analysis considerably since we only need to optimize over a single parameter i.e., θ, in this charging scenario. is depicted against the number of modes N in Fig. 3 (a). Clearly, ∆σ (N ) min decreases with N . Precisely, from Fig. 3 (a), we can see that ∆σ Work fluctuation: In case of ∆W (N ) , unlike the charging precision, we note that ∆W (N ) is phase dependent in the optimal charging configuration. The minimum of work fluctuation, ∆W (N ) min , is achieved when all the squeezing angles vanish. In Fig. 3 (b), we plot ∆W (N ) min with N while keeping ∆E (N ) S c fixed at some values. We demonstrate that with an increase in the number of modes, the optimal work fluctuation decreases, and hence the efficiency of the battery enhances. We can also see that ∆W (N ) min can be approximated by a function proportional to N −0.49 which matches exactly with the charging precision.

B. Local displacement as charging operation
Let us now move to a situation where charging is achieved via local displacements in each mode. Such a study can provide a platform to make a comparison between the two charging operations, namely squeezing and displacement. In this case, the entire charging unitary can be represented asD = ⊗ N j=1D j (α j ), where α j = |α j |e iφ j as defined before. In this case, the change in energy of the battery upon charging is given by where ∆EDj c = |α j | 2 is the energy gained by the battery mode, j. In order to estimate the second moment of the energy in the charged battery, we need to calculate the total energy and its squared version are given by [55] ND j c =sinh 2 r + |α j | 2 (38) in terms of the photon number. The mean energy and its second power reduce to The second expression can lead to the variance in the final energy, V(ρ 1 ) (N ) through Eq. (2), thereby obtaining the charging precision using Eq. (1) as ∆σ (N ) = f (r, |α j |, φ j ). Noting from Eq. (37) that |α j | = ∆EDj c , it is straightforward to calculate ∆σ (N ) = f (r, ∆EDj c , φ j ).
The expression for ∆W (N ) can also be computed by noting that, for charging via displacement, we have from which we can obtain the work fluctuation using Eq. (7).

Multimodal advantage for charging with local displacement
Akin to the case of charging via local squeezing, here also an equal charging in all the modes corresponds to the optimal charging configuration and yields the best figures of merit according to Lemma 1. Hence, for a fixed initial squeezing r of the battery modes, optimal charging configuration implies that |α j | = |α| and φ j = φ for charging via local displacement. In this situation, we find that the minima of both ∆σ (N ) as well as ∆W (N ) , is determined by the phase factors of the charging unitaries, i.e., by θ and φ.
Charging precision: At φ = π/2, we obtain that the least charging precision steadily decreases with N for a fixed r and ∆E (N ) D c . Interestingly, the scaling with N is found to 1 N 3/2 (see Fig. 3 (c)). Observation 4. The above discussion demonstrates that displacement unitary can provide better stability of the quantum battery in terms of the charging precision with the increase of the number of modes in comparison with the squeezing one.
Work fluctuation: On the other hand, the work fluctuation attains its least value when we fix φ = π/2. However, our numerical studies show that ∆W (N ) is independent of the number of modes, and in the optimal charging configuration, it remains the same for all N as shown in Fig. 3 (d) at a fixed value of the initial squeezing r and the energy change ∆E Thus an increase in the number of modes has no effect on the work fluctuation for charging via local displacement operations, although the multimode advantage is evident from the behavior of ∆σ (N ) .

V. CONCLUSION
In recent years, different designs of quantum batteries (QB) have been proposed either to increase the storage capacity and stabilize their energy extraction or to take care of their implementations in certain physical systems. Till now, the majority of works on quantum batteries deal with discrete spin systems, although continuous variable (CV) quantum systems can be a potential candidate for energy storage devices. In the implementation of quantum information processing tasks, such systems are demonstrated to have some advantages over finite dimensional systems, including the stability of QBs. In CV systems, the fluctuation in the extraction (storage) of energy from (in) the QB during discharging (charging) can also be quantified.
We considered pure multimode Gaussian states as the initial state of the CV quantum battery. To charge the battery, local squeezing and displacement operations as well as entangling squeezing operations were carried out. We established that increasing the number of modes contributes positively to the stability of the system by conducting the whole analysis in the phase space description and taking into account the second moments in the change of energy. One of the counterintuitive results we found was that, after optimizing the relevant parameters, entanglement between modes is not a prerequisite for creating a stable quantum battery. In sharp contrast, multimodal entanglement is required for quantum advantage in quantum communication protocols including cryptography and metrological problems. However, our results immediately imply that such a QB is economically less expensive to create than a QB requiring entanglement since separable multimode states turn out to be the optimal ones with local unitary operations, especially displacements.
Furthermore, we established that while the strengths of the squeezing and displacement operations of the charger determine the energy to be stored in the battery, the behavior of considered figures of merit, namely the optimal charging precision and the work fluctuation, is purely dictated by the corresponding phases. It demonstrates that, in contrast to standard CV protocols like dense coding, teleportation, and quantum illumination, which primarily rely on the magnitude of the squeezing or displacement present in the resource state, QB requires other features for good stability. This possibly indicates that quantum communication or quantum sensing protocols do not require the same resources as thermodynamic tasks like storing energy in batteries. Moreover, the modal advantage in the stability of the battery provides the theoretical groundwork for developing effective thermal storage devices in the future. the means and variances ofN j , which will be necessary for our analytical calculations. Note that there exists a mapping between the two parameterizations of d k and Ξ kl . In the phase space formalism, the evolution operator of the state can be represented by a symplectic matrix, S in terms of which we can derive the first and second moments of the evolved state as (A8)