Quantum battery charging bound tested for up to five qubits

Anass Jad of Abdelmalek Essaadi University and Palacký University, with colleague Abderrahim El Allati of Abdelmalek Essaadi University and Université Grenoble Alpes, established a quantum speed limit for charging the N-qubit Dicke battery. Their work demonstrates that charging time is fundamentally limited by a single rate, ΓN = 2λ √(n/N), combining light-matter coupling, charger amplitude and system size. This means increasing the number of qubits can paradoxically slow the process. The researchers found the bound is “geometric,” reducing a complex optimization problem to a function of the angle swept by the collective spin on the Bloch sphere, and that only the coherent component of the energy driving the battery delivers extractable work. This distinction allows for a speed limit on work extraction, something “no instantaneous-power argument can supply.” Dicke Battery Ergotropy Limit for N-Qubits This rate combines these factors, revealing a complex interplay between system size and charging efficiency, and was demonstrated through theoretical analysis of the battery’s ergotropy, its maximum useful work. The work establishes a quantum speed limit for how quickly this type of battery can be charged, a bound previously lacking analytical clarity for systems beyond a few qubits.

The team proved that the charging time, τ*(ε), to reach a target fraction ε of the maximum extractable work adheres to this limit in the classical-field limit for any light-matter coupling λ, register size N greater than or equal to two, and target fraction ε between zero and one. The supporting mathematical framework for these calculations is detailed elsewhere, but the core result is a quantifiable constraint on charging speed. Numerical integration of the full Dicke Hamiltonian, with a dynamical charger and a mixed battery state, finds the bound respected at every admitted charging time for two to five qubits, and approached most closely in the limit of small target work. While the capacity of the Dicke battery grows linearly with the number of qubits (N), the normalized charging rate decreases with the square root of N, resulting in a total power that scales with the square root of N. This means the collective advantage of the Dicke battery resides in its power output, not in a faster normalized charging speed.

Geometric Quantum Speed Limit for Charging Time This limit isn’t simply a matter of energy transfer, but a fundamental bound on how quickly extractable work can be established within the battery, a distinction rooted in the nature of the energy delivered. This simplification allows for a direct calculation of the minimum charging time, defined as the required rotation angle divided by the maximum angular velocity, a concept analogous to the Mandelstam-Tamm bound but applied to ergotropy. The ergotropy itself, though calculated through a complex optimization, reduces to a fixed function, ε = sin2(θ/2), of the polar angle θ swept by the collective Bloch vector, further demonstrating the geometric nature of the limit. The charger’s vacuum contribution delivers energy but does not contribute to extractable work, resulting in a short-time coefficient of 4/N, independent of charger occupation. The composite rate ΓN = 2λ √(n/N) governs the amplitude of the classical drive, which in turn limits the angular velocity of the collective Bloch vector, establishing a direct link between the drive and the achievable charging speed.

The team’s work connects to a broader problem in light-matter physics: determining the fastest rate at which a collective spin system can be rotated to a target polarization given a fixed-amplitude coherent drive, offering insights beyond the specific context of quantum batteries. The analysis confirms the derived bound, providing a fundamental limit on the speed of ergotropy charging in this system, and establishing a framework for understanding the interplay between energy transfer, work extraction and quantum speed limits.

Collective Spin Dynamics and Angular Velocity The charging time for an N-qubit Dicke battery is fundamentally linked to the rate at which its collective spin rotates on the Bloch sphere, a connection revealed by new theoretical work. The analysis establishes a bound independent of the initial charger occupation, a result not achievable through arguments based solely on instantaneous power transfer. The drive, mediated by a collective dipole coupling of strength λ, limits the rate at which this angle can grow, establishing a maximum angular velocity of 2ΓN. This means the minimum charging time is fundamentally determined by the geometry of the spin’s evolution, not simply the energy transferred. The researchers found this bound mirrors the structure of the well-known Mandelstam-Tamm relation, but with the energy uncertainty replaced by the rate ΓN. While increasing the number of qubits (N) and the charger strength (n) would intuitively suggest faster charging, the square root relationship means the overall rate decreases as N increases, lengthening the required charging time. The analysis also confirms that the chord length provides a weaker estimate than the arc length, coinciding only as the target extractable work, ε, approaches zero. The paper states, echoing the structure of the Mandelstam-Tamm relation.

The team verified the bound numerically at finite charger occupation, finding no violations and observing the closest approach in the limit of small target work.

Coherent Drive Impact on Work Extraction This finding reveals that the charging time is about efficiently converting energy into a form that can be used, and the analysis demonstrates that the familiar square-root estimate for charging time is actually a chord length representing a weaker bound than the true arc length when a large fraction of maximum work is required.

The team’s calculations show that the ergotropy, a measure of quickly extractable work, collapses onto a single angle representing the battery’s state, simplifying the complex optimization problem typically associated with determining extractable work. This means the initial coefficient determining charging speed is independent of how occupied the charger is with photons. The analysis further clarifies the role of the drive’s components, revealing that the incoherent portion deposits energy without contributing to the battery’s ability to perform work, and the team found that the mean energy stored is governed by the second moment ⟨(a + = 4n + 1, while the passive-state energy is governed by the quadrature variance, which is unity. This detail highlights that maximizing energy storage does not necessarily equate to maximizing the amount of work the battery can ultimately deliver, and the researchers found that the squared first moment ⟨a + = 4n represents the extractable work. The underlying dynamics resemble semiclassical spin rotation.

The team’s claim focuses specifically on work extraction, not the rotation itself, and the team’s calculations suggest that the ergotropy is defined by an optimization over unitaries, yet it collapses onto the single angle ε = sin2(θ/2), with the charger vacuum depositing energy but no extractable work. This reduction is not part of the rotation problem and is what allows a kinematic bound on the angle to become a thermodynamic bound on the charging time. Numerical Validation with 2-5 Qubit Systems Calculations confirm the predicted charging time bound even with finite charger occupation, extending validation to systems of up to five qubits. This approach allowed for a direct comparison between the analytical bound derived in the classical-field limit and numerical results obtained at varying system sizes and charger occupations. Extrapolating the results to infinitesimally short time intervals further solidified the agreement, demonstrating that the numerical data converged to the theoretical value of 4/N, where N represents the number of qubits. This consistency is notable because it confirms the cancellation identified in Eq. (10) within the source material, establishing that the vacuum quadrature deposits energy into the battery without contributing to extractable work at leading order. A key finding from the numerical verification is the exclusion of a value, A = 4(n + 1/4)/(Nn), which would be observed if the charger vacuum did contribute extractable work. Light-Matter Coupling Rate: Γ_N Definition The derivation of this limit relies on the classical-field approximation, where the charger’s quadrature is approximated by a c-number, 2 n cosωct, simplifying the complex quantum dynamics. Numerical verification excluded a coefficient of A = 4(n + 1/4)/(Nn), a value that would have altered the established charging time bound, and the factor 1/N arises from the cancellation within the equations governing excitation amplitude, which remains independent of the number of qubits while ergotropy scales with N. This bound has practical implications for circuit-QED implementations, establishing the necessary coupling strength to achieve charging within a target time, and for parameters of ε = 0.8, ns, n = 10, N = 2 and ω0/2π = 5GHz, the required coupling falls within the capabilities of current architectures. This limit is a design target for future quantum batteries, not a practical obstacle. Further research will focus on a rigorous finite-n statement, potentially replacing ΓN with a fluctuation-corrected rate, and extending the derivation to encompass arbitrary pulse-shaping protocols, open-system dynamics and models like the Tavis-Cummings and Jaynes-Cummings models. An experimental test using superconducting circuits, where ΓN is directly controlled by coupling and drive, appears feasible given current technology, and the data supporting these findings are available upon reasonable request from the authors. Source: https://iopscience.iop.org/article/10.1088/2058-9565/aebdc0 More like thisQuantum PhysicsQuantum operations guide shows how physics limits information processingQuantum PhysicsAnderson orthogonality catastrophe is seen in quantum dot experimentsQuantum Research NewsPhysicists demonstrate a spin-state memory for quantum informationPhysicsResidual coherence dominates entanglement in neutrino oscillation, study findsStay currentSee today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: