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Precision and Work Fluctuations in Gaussian Battery Charging

Nicolai Friis, Marcus Huber

arXiv:1708.00749v2quant-ph

TL;DR

The paper asks how reliably batteries can be charged when operations are restricted to practically implementable Gaussian unitaries. It derives optimal protocols for arbitrary and Gaussian operations, finding that Gaussian charging is less effective than the unrestricted optimum but achieves asymptotically vanishing relative charge variance and work fluctuations at large input energies.

  • Problem

    Arbitrary operations can be difficult to realize in infinite-dimensional oscillator systems, motivating study of charging reliability under practical Gaussian-unitary control.

    Source: §1 · p. 2

  • Method

    The paper derives optimal charging protocols for arbitrary unitaries and for Gaussian unitaries, assessing battery precision and work fluctuations in thermal oscillator batteries.

    Source: §1 · pp. 2–3

  • Results

    Gaussian unitaries perform significantly less well than is possible in principle, yet optimal Gaussian charging makes relative charge variance and work fluctuations asymptotically vanish as input energy grows.

    Source: §1 · p. 3, §5 · p. 14

  • Takeaways & Limitations

    Gaussian operations provide a practical trade-off between performance, reliability, and implementability for battery charging.

    Source: §5 · p. 14

    The analysis mostly focuses on individual batteries; the roles of correlations and Gaussian-operation restrictions on charging power remain open questions.

    Source: §5 · p. 14

Abstract

from arXiv · show

One of the most fundamental tasks in quantum thermodynamics is extracting energy from one system and subsequently storing this energy in an appropriate battery. Both of these steps, work extraction and charging, can be viewed as cyclic Hamiltonian processes acting on individual quantum systems. Interestingly, so-called passive states exist, whose energy cannot be lowered by unitary operations, but it is safe to assume that the energy of any not fully charged battery may be increased unitarily. However, unitaries raising the average energy by the same amount may differ in qualities such as their precision, fluctuations, and charging power. Moreover, some unitaries may be extremely difficult to realize in practice. It is hence of crucial importance to understand the qualities that can be expected from practically implementable transformations. Here, we consider the limitations on charging batteries when restricting to the feasibly realizable family of Gaussian unitaries. We derive optimal protocols for general unitary operations as well as for the restriction to easier implementable Gaussian unitaries. We find that practical Gaussian battery charging, while performing significantly less well than is possible in principle, still offers asymptotically vanishing relative charge variances and fluctuations.

1 Introduction

The paper examines how practical control restrictions shape the precision and work fluctuations of quantum battery charging. It studies thermal harmonic-oscillator batteries charged by a fixed energy increase, comparing arbitrary and Gaussian unitaries.

1 Introduction

  • Gaussian unitaries are comparatively simple to implement in continuous-variable systems because they require driving Hamiltonians at most quadratic in creation and annihilation operators.The paper contrasts this practical family with arbitrary global operations that can be difficult to realize in large or infinite-dimensional systems.Source: §1 · p. 2

2 Charging a quantum battery

  • Batteries begin in thermal states at ambient temperature, treated as empty because they contain no extractable work, and a unitary raises their average energy by a fixed ∆E.For noninteracting modes, the initial thermal state is a product state.Source: §1 · p. 2, §2 · p. 3
  • Charging precision is quantified by final battery energy variance V, while charging-process fluctuations are quantified by the average squared deviation (∆W)2 of energy transitions from the average supply.The two quantities capture distinct concerns: variation in the stored charge and variation in energy transfer during charging.Source: §1 · p. 2, Figure 1 · p. 2, §2 · p. 3

1 Introduction

  • The study seeks optimal protocols for arbitrary unitaries and Gaussian unitaries, finding that Gaussian charging performs worse in principle but can asymptotically make relative charge variance and work fluctuations vanish.The asymptotic result applies to large energy inputs and uses combinations of displacements and single-mode squeezing.Source: §1 · p. 3

3 Fundamental Limits for Battery Charging

The section establishes unrestricted benchmarks for charging precision and fluctuations, then examines how those limits behave for single- and multi-mode batteries. At finite temperature, minimizing final-energy variance and charging fluctuations requires different protocols, which can involve temperature- and energy-specific operations.

3 Fundamental Limits for Battery Charging

  • At finite temperatures, protocols minimizing final-energy variance and charging fluctuations do not coincide and may require temperature- and energy-specific operations in particular subspaces.These unrestricted results serve as benchmarks for the precision and fluctuations achievable with Gaussian unitaries.Source: §3 · p. 4

3.A Fundamental limits for zero temperature

  • For an infinite-dimensional battery charged from its ground state, energy variance can grow arbitrarily large at any fixed energy input, while minimum standard-deviation change vanishes at integer multiples of ω.The minimum is obtained by a superposition of the two energy eigenstates nearest the available energy; its maximum standard-deviation change is ω/2.Source: 3.A Fundamental limits for zero… · p. 4

3.B Fundamental precision limits for arbitrary temperatures

  • For thermal batteries, an optimal-precision protocol reorders thermal-state weights around the target energy, then adjusts the mean energy; higher initial temperatures can yield decreasing final variances.The adjustment uses two-level rotations ordered by increasing added squared deviation per unit energy change.Source: 3.B Fundamental precision limits for… · p. 5

3.C Fundamental precision limits for multi-mode batteries

  • For different mode frequencies, correlated charging can marginally improve precision, but correlations are not the key ingredient; equal-frequency batteries can avoid such correlations.The optimal protocol's energy-adjustment step can use rotations between joint energy eigenstates that correlate the modes.Source: 3.C Fundamental precision limits for… · p. 6
  • With multiple modes, the optimal joint variance is never higher than charging only one mode, and local energy splitting can exploit individual-mode variance minima.The best split depends on temperature, energy input, and mode frequencies.Source: 3.C Fundamental precision limits for… · pp. 6–7

3.D Fundamental fluctuation limits for arbitrary temperatures

For arbitrary-temperature batteries, the optimal fluctuation protocol uses level shifts and can attain vanishing fluctuations for integer energy inputs; multiple modes can further reduce fluctuations. The following Gaussian-state preliminaries specify the state representation used in the next section.

3.D Fundamental fluctuation limits for arbitrary temperatures

  • For an input energy of one oscillator unit, shifting thermal-state weights upward and sending the last weight to the ground state makes fluctuations vanish as the shift range tends to infinity.For inputs below one unit, the protocol instead shifts from a selected higher level and adds the remaining energy with a two-level rotation; larger inputs combine this with integer shifts.Source: 3.D Fundamental fluctuation limits for… · p. 7
  • For multiple modes, local charging can yield vanishing fluctuations at energies mω_A + nω_B and can reduce fluctuations between those values by splitting energy across modes.The exact optimum depends on frequencies, temperatures, and input energy, requiring case-by-case analysis.Source: 3.D Fundamental fluctuation limits for… · p. 8

4.A Preliminaries: Phase space and Gaussian states

  • Gaussian states are defined by multivariate Gaussian Wigner functions and are fully determined by their first-moment vector and covariance matrix.Gaussian unitaries act on first moments by affine maps, with displacements and real symplectic matrices.Source: 4.A Preliminaries: Phase space and… · pp. 8–9

4.B Charging precision for single-mode Gaussian unitaries

The section evaluates energy variance for single-mode Gaussian charging, including displacement protocols. Pure displacements provide a comparison case, but are neither the minimum-variance nor the maximum-variance Gaussian protocol.

4.B Charging precision for single-mode Gaussian unitaries

  • At zero temperature, displacement of the vacuum creates a coherent state, providing the ground-state case considered in the charging comparison.Source: 4.B Charging precision for single-mode… · p. 9
  • For a displaced thermal state, the energy variance is obtained from its first moments and covariance, with displacement changing the first moments while leaving the thermal covariance unchanged.The stated variance expression includes a term proportional to the squared first-moment norm and the initial thermal variance.Source: 4.B Charging precision for single-mode… · p. 10
  • Pure displacements are neither optimal nor worst-case for Gaussian charging, while their energy standard deviation grows asymptotically with the square root of the energy increase.Figure 5 compares their performance with the Gaussian variance extrema and the arbitrary-unitary lower bound for a ground-state battery.Source: 4.B Charging precision for single-mode… · p. 10, Figure 5 · p. 10

4.C Optimal and worst-case Gaussian charging precision

For fixed energy input and temperature, the extremal variance among single-mode Gaussian protocols can be found by optimizing over squeezing. The worst case is pure squeezing, whereas the minimum generally combines squeezing and displacement.

4.C Optimal and worst-case Gaussian charging precision

IV.C.1 Worst-case Gaussian charging precision

For Gaussian charging, pure single-mode squeezing gives the worst precision, while combining squeezing and displacement minimizes the variance. The optimal Gaussian strategies also yield sublinear growth in variance and fluctuations at large input energy.

IV.C.1 Worst-case Gaussian charging precision

  • For any temperature and input energy, pure single-mode squeezing gives the maximal standard-deviation increase among Gaussian unitaries.The increase is linear in input energy asymptotically.Source: IV.C.1 Worst-case Gaussian charging… · p. 11

IV.C.2 Optimal Gaussian charging precision

  • Figure 6 plots standard-deviation change against input energy, comparing worst-case and optimal Gaussian charging with optimal non-Gaussian values across temperatures.The curves cover temperatures T = 0 to 10 in steps of 1.Source: Figure 6 · p. 11
  • At large input energy, the optimal Gaussian precision strategy has V−/∆E → 0, so its variance grows sublinearly.It combines squeezing and displacement; most energy asymptotically goes into displacement.Source: IV.C.2 Optimal Gaussian charging… · p. 11

4.D Charging precision for multi-mode Gaussian unitaries

  • In multi-mode precision charging, optimal energy allocation depends on mode number, temperatures, and frequencies; equal-frequency batteries optimally split energy evenly.For pure displacements, by contrast, equal-frequency modes have variance independent of the energy split.Source: 4.D Charging precision for multi-mode… · p. 12

4.E Charging fluctuations for single-mode Gaussian unitaries

  • At large input energy, Gaussian work fluctuations grow sublinearly; minimizing them invests almost all energy into displacement, while the worst case favors squeezing.The optimal and worst fluctuation strategies use nonzero displacement, and their extremal parameters can be found numerically.Source: 4.E Charging fluctuations for… · p. 13
  • For multi-mode Gaussian fluctuations, allocating energy across batteries can be useful because the optimal protocol gives sublinear fluctuation growth.The allocation depends on the modes’ frequencies.Source: 4.E Charging fluctuations for… · p. 13

5 Conclusion

The paper derives fundamental bounds and optimal protocols for charging precision and work fluctuations, then identifies optimal and worst-case Gaussian operations. Gaussian performance falls short of optimal non-Gaussian protocols, but optimal Gaussian charging has asymptotically vanishing relative variance and fluctuations.

5 Conclusion

  • For finite energy input, precision and work fluctuations are unbounded above in principle, while optimal protocols provide lower bounds at given energies and temperatures.The protocols require precise interventions in particular subspaces of an infinite-dimensional Hilbert space, making them difficult to realize.Source: §5 · p. 14
  • For single-mode batteries, pure squeezing is least favorable for precision, while squeezing-displacement combinations attain the Gaussian precision and fluctuation extrema.For multiple modes, access to multiple batteries is generally useful, while correlations help only indirectly.Source: §5 · p. 14
  • Gaussian operations perform less well than optimal non-Gaussian protocols, yet optimal Gaussian charging has asymptotically vanishing relative variance and fluctuations.The authors characterize this as a trade-off between performance, reliability, and practicality.Source: §5 · p. 14
  • The paper does not investigate fundamental and practical limits for finite-dimensional battery charging in depth.Some results, including optimal precision charging, translate to finite-dimensional systems, but others apply only to the infinite-dimensional case.Source: §5 · p. 14

Appendix

The optimal precision protocol first rearranges thermal-state probabilities to minimize squared distance from the target energy, then uses two-level rotations to reach that energy with minimal additional increase in squared distance.

A.1 Optimal precision charging protocol

  • The protocol seeks a unitary state with a target average energy and minimal final energy variance, starting from a thermal state.The initial state’s probabilities decrease with energy, and the target increase is ∆E.Source: §A.1 · p. 15

A.1.I Part I of the protocol

  • Part I assigns the largest probability weights to energy levels closest to the target, producing the unique orbit state with minimal squared distance from that target.Its mean energy may not equal the target, so the squared distance is not yet the final energy variance.Source: §A.1 · p. 15, A.1.I Part I of the protocol · p. 15

A.1.II Part II of the protocol

  • Part II uses two-level rotations to correct the mean energy while minimizing the increase in squared distance; once the target is reached, that distance equals the energy variance.The rotation angle is adjusted when a rotation can reach the target; otherwise the protocol continues with further level pairs.Source: A.1.II Part II of the protocol · pp. 15–17
  • Figure A.1 illustrates the rearrangement and correction for T = 3, with Part I overshooting the target mean energy before Part II rotations reach it.In the illustrated case, the target is closer to level 4 than level 5.Source: Figure A.1 · p. 16

A.2 Wigner representation of squared number operator

This appendix rewrites the squared number operator in phase-space form, then uses the Wigner function to extend the calculation from pure states to mixed and multimode states.

A.2 Wigner representation of squared number operator

  • The appendix rewrites the squared number operator using position and momentum operators before deriving its phase-space representation.The resulting representation is then inserted into the calculation of the squared number operator’s expectation.Source: §A.2 · p. 17
  • Linearity of the Wigner function extends the calculation from pure states to arbitrary single-mode mixed states and, for local number operators, to arbitrary multimode states.The derivation uses the Wigner function and the wave-function expansion in Hermite polynomials.Source: §A.2 · p. 18

A.3 Extremal Gaussian precision and fluctuations

This appendix establishes extremal Gaussian charging precision and fluctuations. For maximal variance, the unconstrained extremum is not physically realizable; the maximum under the energy constraint uses all input energy for squeezing.

A.3 Extremal Gaussian precision and fluctuations

  • For fixed energy input and temperature, the extremal values of Gaussian charging precision exist and are unique.The maximum of V+(r) occurs at a positive squeezing parameter; negative squeezing extrema are excluded from the allowed range.Source: §A.3 · p. 18, A.3.I Maximal variance · p. 18

A.3.I Maximal variance

  • The unconstrained local maximum of V+(r) is physically unrealizable because its squeezing energy exceeds the available input energy.The realizable maximum is attained at the largest squeezing parameter allowed by the global energy constraint.Source: A.3.I Maximal variance · p. 19
  • The worst Gaussian charging precision is achieved when all input energy is invested in single-mode squeezing.The unconstrained variance maximum would require more squeezing energy than the total input energy allows.Source: A.3.I Maximal variance · p. 19

A.3.II Minimal variance

The Gaussian protocol has a unique minimum of the charging variance for each energy input and initial temperature, and that minimum is physically realizable.

A.3.II Minimal variance

  • The Gaussian charging-variance minimum exists uniquely for every energy input and initial temperature.The minimizing squeezing parameter is the unique nonnegative solution of the extremal condition.Source: A.3.II Minimal variance · p. 19
  • Unlike the variance maximum, the minimum is physically realizable for all energy inputs and initial temperatures.At the minimum, the input energy splits between squeezing and displacement.Source: A.3.II Minimal variance · p. 19

A.3.III Maximal fluctuations

The appendix analyzes maximal work fluctuations during single-mode Gaussian charging at fixed input energy. It establishes a unique maximum for each valid parameter pair corresponding to fixed energy input and temperature.

A.3.III Maximal fluctuations

  • The maximum is located by transforming the squeezing parameter to u = e^-2r and solving fλ(u) = χ for a unique uχ.For every λ and χ, the relevant solution lies below the unique minimum uλ of fλ(u).Source: A.3.III Maximal fluctuations · p. 20
  • The maximal work-fluctuation extremum exists uniquely for fixed input energy and temperature.The maximization is formulated through W+(r), equivalent to the squared fluctuation up to terms independent of r.Source: A.3.III Maximal fluctuations · pp. 19–20

A.3.IV Minimal fluctuations

For fixed input energy, the appendix proves that minimal work fluctuations under single-mode Gaussian charging exist uniquely. The proof uses a monotonicity condition for the transformed minimization problem.

A.3.IV Minimal fluctuations

  • The minimization is recast using v = e^-2r, where hλ(v) decreases strictly from infinity to one.This monotonicity yields a unique vχ satisfying hλ(vχ) = χ for every χ and λ.Source: A.3.IV Minimal fluctuations · pp. 20–21
  • The minimum work-fluctuation extremum exists uniquely for fixed input energy and temperature.The second-derivative test confirms a minimum at the unique solution of the transformed extremal condition.Source: A.3.IV Minimal fluctuations · p. 21

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