[Paper Review] Reverse Quantum Speed Limit: How Slow Quantum Battery can Discharge?

Brij Mohan, Arun Kumar Pati|arXiv (Cornell University)|Jun 25, 2020

Advanced Thermodynamics and Statistical Mechanics1 references4 citations

TL;DR

This paper introduces the reverse quantum speed limit (RQSL), a geometric lower bound on the time required for quantum evolution, derived from the Fubini-Study metric and gauge-invariant path lengths in projective Hilbert space. It demonstrates that quantum batteries can be designed to discharge slowly, with RQSL providing a tight lower bound on average discharging power, validated in two-level systems and cavity-QED models where the bound saturates.

ABSTRACT

We introduce the notion of reverse quantum speed limit for arbitrary quantum evolution which answers a fundamental question: ``how slow a quantum system can evolve in time?" Using the geometrical approach to quantum mechanics, the reverse speed limit follows from the fact that the gauge invariant length of the reference section is always greater than the Fubini-Study distance on the projective Hilbert space of the quantum system. We illustrate the reverse speed limit for two-level quantum systems with an external driving Hamiltonian and show that our results hold well. We find several examples where our bound is tight. We also find one practical application of the reverse speed limit in discharging process of quantum batteries which answers the question: ``how slow quantum batteries can discharge?" Our result provides a lower bound on the average discharging power of quantum batteries.

Motivation & Objective

• To establish a fundamental lower bound on quantum evolution time, answering 'how slow can a quantum system evolve?'
• To develop a geometric framework for reverse quantum speed limit (RQSL) using Fubini-Study distance and gauge-invariant path lengths
• To apply RQSL to quantum batteries, specifically addressing the minimal discharging time and maximal operational duration
• To identify physical systems where the RQSL bound is tight, ensuring practical relevance
• To provide a geometric foundation for optimizing quantum battery design and stability

Proposed method

• Formalize the reverse quantum speed limit using the geometric structure of the projective Hilbert space and the Fubini-Study metric
• Define the reference curve length and horizontal curve length in the principal bundle framework, showing the former always exceeds the latter
• Use the difference between reference and horizontal curve lengths to derive the curvature-dependent RQSL
• Apply the RQSL to two-level quantum systems driven by time-dependent Hamiltonians to verify bounds
• Model a quantum battery using N two-level atoms in a cavity with Jaynes-Cummings interaction, simulating discharging via time-evolved states
• Demonstrate saturation of the RQSL bound in the cavity-QED model, where reference, horizontal, and geodesic lengths are equal

Experimental results

Research questions

• RQ1 What is the minimal evolution speed for a closed quantum system, and what geometric constraints enforce it?
• RQ2 How can the reverse quantum speed limit be derived from the intrinsic geometry of quantum state space?
• RQ3 In what physical systems does the reverse quantum speed limit bound saturate, indicating optimal slow evolution?
• RQ4 How does the RQSL constrain the discharging time and average power of quantum batteries?
• RQ5 Can the RQSL be used to design quantum batteries with prolonged operational lifetimes?

Key findings

• The reverse quantum speed limit arises from the geometric inequality that the gauge-invariant length of the reference curve exceeds the Fubini-Study distance on the projective Hilbert space.
• For two-level systems under external driving, the RQSL bound is valid and can be saturated in specific dynamical regimes.
• In a cavity-QED model of quantum battery, the RQSL bound is saturated when the reference curve length, horizontal curve length, and geodesic distance are all equal to $ S_0/2 $.
• The average discharging power of quantum batteries is bounded from below by the RQSL, providing a new figure of merit for battery design.
• The RQSL is tight in the parallel discharging model of a multi-atom quantum battery, indicating fixed, maximal discharging time.
• The RQSL provides a geometric explanation for slow evolution, rooted in the curvature of the quantum state space, analogous to but distinct from the standard quantum speed limit.