Quantum Batteries as Work Sources for Phase-Locked Parametric Amplification

Technical Summary: Quantum Batteries as Work Sources for Phase-Locked Parametric Amplification

Problem Statement
Quantum batteries (QBs) have been proposed as finite quantum systems capable of storing energy and performing work, with recent hardware proposals suggesting their use to replace continuously supplied microwave drives in superconducting quantum technologies. A specific application under investigation is the replacement of the continuous pump tone in quantum-limited parametric amplifiers (JPAs) with a locally precharged bosonic quantum battery. While removing continuous pump lines could reduce cryogenic heat loads and wiring overhead, it remains unclear whether a finite quantum system can fulfill the dual role of the pump: acting as both an energy source and a phase reference. In standard classical models, the pump is an undepleted, ideal phase reference; however, quantizing the pump reveals a potential resource distinction between stored energy and phase coherence. This paper investigates whether a finite bosonic battery can generate a phase-locked amplifier field or if it merely generates signal-idler pairs without the necessary phase locking.

Methodology
The authors model a nondegenerate parametric amplifier where the pump is treated as a finite quantum system (mode $c$) rather than a classical tone, interacting with signal ($a$) and idler ($b$) modes via a closed trilinear Hamiltonian:
$$H_I = i\hbar g (c a^\dagger b^\dagger - c^\dagger a b)$$
To isolate the role of phase coherence, the study compares pump batteries with identical photon-number distributions but varying phase properties:

1. Coherent states: Possessing both energy and phase coherence.
2. Phase-randomized coherent states: Having the same Poissonian photon-number distribution as coherent states but zero first-order phase coherence ($\langle c \rangle = 0$).
3. Fock states: Possessing sharply defined energy but no phase reference.
4. Displaced-squeezed states: Used to analyze the trade-off between coherent displacement and number-noise reduction.

The study progresses from a closed-system analysis to an open-system model incorporating leakage through signal and idler ports. Crucially, the emitted fields are collected into auxiliary temporal modes ($A$ and $B$) using a cascaded-systems formalism. This allows for the measurement of genuine temporal output modes rather than relying on equal-time proxies. The performance is evaluated using two primary metrics:

• Collected Energy: The total number of collected signal-idler pairs, $n_f = (N_A + N_B)/2$.
• Phase-Sensitive Interference: The interference observable $I^{(f)}_{\min}$ and the normalized pair coherence $C_f$. A value of $I^{(f)}_{\min} < 1$ (below the vacuum level) certifies the generation of a phase-locked, EPR-squeezed field.

Key Contributions and Results

1. Resource Distinction: The paper establishes that stored pump energy and phase coherence are distinct resources. In both closed and open systems, phase-randomized pumps (and Fock pumps) generate signal-idler pair numbers comparable to coherent pumps with the same mean energy. However, these incoherent pumps fail to produce anomalous two-mode coherence ($\langle ab \rangle = 0$) and cannot generate a phase-locked amplifier field.
2. Temporal-Mode Certification: In the open-system model with matched collector bandwidths ($\gamma_f/\lambda \approx 2$), the distinction becomes operationally clear. A coherent pump produces a sub-vacuum interference dip ($I^{(f)}_{\min} \approx 0.553$) and finite phase locking ($C_f > 0$). In contrast, a phase-randomized pump with nearly identical collected energy ($n_f \approx 0.469$) yields $I^{(f)}_{\min} \approx 1.94$ and $C_f = 0$, remaining above the vacuum reference.
3. Role of Squeezing: The analysis of displaced-squeezed pump batteries reveals that weak amplitude squeezing can slightly improve the interference dip (e.g., reducing $I^{(f)}_{\min}$ from 0.553 to 0.547) by reducing finite-pump number fluctuations while preserving the coherent displacement. However, strong squeezing transfers energy away from the coherent component, degrading the phase reference and reversing the gain. Thus, squeezing assists but cannot replace the need for phase coherence.
4. Bandwidth Dependence: The resource separation is robust across collector bandwidths. While the depth of the interference dip varies with bandwidth, only the phase-coherent pump remains below the vacuum interference reference, whereas phase-randomized pumps consistently fail the phase-sensitive test.

Significance and Claims
The paper claims that the operational signature of a functional quantum battery for parametric amplification is not merely the ability to store and transfer energy, but the ability to store phase-coherent energy.

• Operational Requirement: Replacing a continuous microwave pump tone with a quantum battery requires the battery to possess a phase reference capable of locking the amplifier output. Stored energy alone is insufficient to generate the phase-locked field required for quantum-limited amplification.
• Hardware Implication: In the context of superconducting readout chains, this distinction is critical for estimating the benefits of removing pump lines. The authors provide a heat-budget estimate suggesting that if a phase-coherent battery could replace the continuous pump, it could reduce the active and passive heat load per qubit, potentially increasing the cooling-limited qubit capacity by a factor of roughly 1.7 (from $\sim 5.8 \times 10^3$ to $\sim 9.8 \times 10^3$).
• Modesty of Claims: The authors explicitly state that they do not claim every control line can be removed. Instead, they identify the specific resource (phase coherence) required for a precharged on-chip mode to take over the work-source role of a microwave pump. They acknowledge that practical implementation faces challenges such as recharge overhead, residual calibration needs, and battery lifetime, which would reduce the theoretical gains.

In summary, the work certifies that phase-locked parametric amplification distinguishes between energy storage and phase-coherent work, demonstrating that a quantum battery must provide phase coherence, potentially assisted by number-noise reduction, to function effectively as a pump source.