Achieving Identical Stored Energy in Cascaded Collisional Quantum Battery Charging: Analytical Result

The gist: An adaptive-angle measurement protocol for cascaded collision models in quantum battery charging achieves complete suppression of stored-energy fluctuations for excited-state chargers and substantial suppression for superposition-state chargers.

Introduction to Quantum Batteries

Quantum batteries exploit quantum effects to transfer, store and release energy [1]. They may serve as energy-supply units compatible with quantum architectures such as quantum computers, quantum sensors and quantum simulators [2, 3]. These devices provide a framework for quantum thermodynamics to uncover new principles for energy manipulation and examine the associated quantum advantages [4]. Various models have been extensively studied to assess performance including stored energy, charging power and ergotropy [13–17].

Model of the System

The simplest yet nontrivial configuration investigated is a two-qubit charger–battery system where both are individual two-level qubits [20, 26, 27]. The total system Hamiltonian consists of components describing the charger and battery evolution [4]. The charging process involves a unitary evolution with interaction referred to as a one-step (discrete) collision between the charger and the battery [22].

Projective Measurement and Conditional States

After a collision of duration τ, i.e., at t = τ, the charger is projectively measured in a basis spanned by specific states [5]. This measurement leads to two conditional pure states for the battery:

1. With probability pk0, the battery assumes state φ0i = a00i + a11i/√pk0 = √1 − e0i + √e exp(iα)1i, where e = a12/pk0 and α = arg(a1) − arg(a0) [9].

2. With probability pk1, the battery assumes state φ1i = b00i + b11i/√pk1 = √1 − e'0i + √e' exp(iα')1i, where e' = b12/pk1 and α' = arg(b1) − arg(b0) [9].

Adaptive Measurement Protocol

To eliminate trajectory-dependent energy disparities, the measurement basis vectors on the charger are dynamically tuned by choosing a suitable projective basis such that the stored energies of the two branches are identical, i.e., E = hφ0Hbφ0i = hφ1Hbφ1i [11]. This condition is equivalent to a12 pk0 = b12 pk1 [12], which reduces to b02 a12 = a02 b12 [8]. This constraint on θ after the n-th collision is given by a complex expression involving coefficients from the unitary evolution [13]. This scheme is termed an adaptive-angle measurement protocol, where θ is denoted as the adaptive angle θa [13].

Stored Energies in the Battery

For cascaded collision models, where chargers are noninteracting and initially uncorrelated, each charger collides with the battery exactly once [22]. The stored energy after the (n + 1)-th collision is given by en+1 = a12 + b12 [17], which can be expressed in terms of previous states as an algebraic identity rather than a physical superposition of the two post-selected branches [18]. When each fresh charger qubit is prepared in the excited state, i.e., ea = 1, the stored energy becomes en+1 = en + g2omega2S2τ (1 − en) [19]. By mathematical induction, all trajectory branches after the n-th collision possess strictly equal battery energies [20].

Results and Discussion

In simulations with 10 6 stochastic quantum trajectories, complete suppression of stored-energy fluctuations is achieved when the charger qubit is in the excited state, as Eq. (19) shows that en+1 is independent of the phase αn in φb(nτ)i [20]. This leads to four possible descendant branches with identical stored energies after two collisions, given by En+2 = en+2ωb = [(1 − g2omega2S2τ)en+1 + g2omega2S2τ]ωb [18]. For superposition-state chargers (ea < 1), distinct battery energies develop among trajectory branches after the n-th collision, implying that stored-energy fluctuations exist [3]. However, the adaptive-angle protocol is superior to fixed-angle protocols in suppressing these fluctuations when ea < 1, as shown by the ratio η = hσ(En)iA/hσ(En)iF decreasing continuously from 0.5840 to 0 as ea increases [3].

Conclusions

This work establishes an adaptive-angle measurement paradigm for collisional quantum battery charging, in which post-collision measurement bases for chargers are dynamically tuned to suppress trajectorydependent energy disparities [22]. For each “fresh” charger prepared in the excited state, stored-energy fluctuations are completely eliminated across different quantum trajectories [20]. For superposition-state chargers, substantial reduction of stored-energy fluctuations is achieved relative to the minimal-fluctuation fixed-measurement basis [3]. The proposed protocol maintains its performance across a broad range of detunings and provides an effective strategy for achieving high-uniformity quantum battery charging.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 12674301) [Acknowledgments].

1. Adaptive measurement protocol for charger states enables complete suppression of stored-energy fluctuations when using excited-state chargers, as shown by Eq. (19): When each fresh charger qubit is prepared in the excited state, i.e., ea = 1, Eq. (18) reduces to en+1 = en + g squared / Ω squared S squared τ.

2. The adaptive-angle measurement protocol achieves complete elimination of stored-energy fluctuations for excited-state chargers, as demonstrated by the result: By mathematical induction, all trajectory branches after the ˜n-th collision possess strictly equal battery energies.

3. For superposition-state chargers (where ea < 1), the adaptive scheme provides a substantial reduction of stored-energy fluctuations compared to fixed measurements, as quantified by the ratio η: The endpoint ea = 1 corresponds to the charger qubit in the excited state φai = 1i, i.e., the case of complete suppression of stored-energy fluctuations discussed above.

4. The system can be used for high-uniformity quantum battery charging, which is a key requirement for reliable devices, by employing this protocol to ensure nearly identical stored energy among distinct realizations.