Long-range interactions assisted shortcuts to adiabaticity and battery charging in open quantum critical systems

This paper demonstrates that long-range interactions serve as a valuable resource for optimizing shortcuts to adiabaticity and enhancing quantum battery charging in open critical systems by enabling algebraically decaying control protocols and reducing operational costs compared to short-range interactions.

The Gist

Imagine you are trying to drive a car from point A to point B as smoothly as possible. In the world of quantum physics, "driving" a system (like a collection of atoms) from one state to another without causing a crash (excitations or errors) is incredibly difficult, especially when you have to speed through a "traffic jam" known as a quantum critical point.

Usually, to avoid a crash, you have to drive very slowly (adiabatically). But in the quantum world, being too slow is often not an option because the environment (heat, noise) will mess things up. So, scientists use a technique called Shortcuts to Adiabaticity (STA). Think of STA as a "magic GPS" that tells the car exactly how to steer and accelerate to reach the destination instantly without hitting any bumps.

This paper explores what happens when you add long-range interactions to this mix. In a normal quantum system, particles only talk to their immediate neighbors (like people in a line whispering to the person next to them). In this study, the authors look at a system where particles can "whisper" across the entire room, even to people far away.

Here is the breakdown of their findings using simple analogies:

1. The Problem: The "Infinite Reach" Trap

In standard quantum systems with only short-range interactions (neighbors only), trying to use the "magic GPS" (STA) right at the critical traffic jam requires a very strange control: you would need to connect the steering wheel to a lever at the very end of the car, no matter how long the car is. This is like needing a control wire that stretches to infinity. It's theoretically possible but practically impossible to build.

2. The Solution: Long-Range Interactions as a "Super-Connector"

The authors studied a specific model (the Kitaev chain) where particles have long-range interactions. They found that when these long-range connections are present, the "magic GPS" doesn't need infinite wires.

• The Analogy: Instead of needing a wire that stretches to infinity, the control signal fades away gradually, like a radio signal getting weaker the further you go. The strength of the connection drops off in a predictable, smooth way (algebraically) rather than requiring an impossible, infinite reach.
• The Result: This makes the "shortcut" much easier to build and implement in real life.

3. Two Different Roads (Two Critical Points)

The system they studied has two different "traffic jams" (critical points) where things get tricky.

• Road A (The Good One): At one critical point, having long-range connections is a huge advantage. It actually makes the "traffic" less dense, allowing the system to move faster and smoother. The control signals needed are weaker and easier to manage.
• Road B (The Neutral One): At the other critical point, the long-range connections don't help much more than short-range ones. The physics behaves differently here, and the "long-range advantage" disappears.

4. Charging a Quantum Battery

The authors also applied this to quantum batteries. Imagine a battery that stores energy in quantum states. Usually, if you try to charge it quickly, you lose energy to heat (dissipation).

• The Trick: They proposed a modified "shortcut" method to charge this battery. Instead of just moving the system smoothly, they deliberately flipped the population of energy states (like filling the top shelf of a pantry before the bottom shelf).
• The Benefit: They found that using long-range interactions helps the battery store more usable energy (called ergotropy). It's like having a better charging cable that lets you pack more power into the battery before the heat kills the charge.

5. Heat and Cost

Every time you force a system to move fast, it generates heat (cost).

• The Finding: In the "good" scenario (Road A), using long-range interactions actually reduces the heat generated during the process. It's a more energy-efficient way to drive the system through the critical point.
• Temperature Matters: These benefits are most visible when the system is cold. If the system is too hot (high temperature), the random thermal noise drowns out the benefits of the long-range connections, making the system behave like a normal, messy one.

Summary

The paper claims that long-range interactions are a valuable tool for controlling quantum systems.

1. They make "shortcuts" (STA) physically possible by removing the need for impossible, infinite-range controls.
2. They reduce the energy cost (heat) of moving the system.
3. They can help charge quantum batteries more efficiently by storing more usable energy.

The authors suggest these findings are relevant for building future quantum technologies, such as quantum computers and quantum engines, and that these setups could potentially be tested in current experimental labs using ion traps or quantum simulators.

Technical Summary

Technical Summary: Long-range interactions assisted shortcuts to adiabaticity and battery charging in open quantum critical systems

Problem Statement
Quantum systems driven out of equilibrium generally suffer from non-adiabatic excitations, which are detrimental to tasks such as quantum annealing, state preparation, and the operation of quantum engines and computers. While Shortcuts to Adiabaticity (STA) have been rigorously studied in closed systems to suppress these excitations, their application in open quantum systems—where dissipation is ubiquitous—remains less explored. A specific challenge in many-body open quantum critical systems is the complexity of the control protocols required. In systems with short-range interactions, implementing STA often demands counterdiabatic (CD) control terms involving interactions between infinitely distant spins at critical points, rendering them practically unimplementable. Furthermore, the role of long-range interactions (LRIs) in reducing the cost of STA and enhancing the performance of quantum technologies, such as quantum batteries, in the presence of dissipation has not been fully established.

Methodology
The authors investigate these issues using a long-range Kitaev (LRK) chain comprising spinless fermions on a one-dimensional lattice. The system is driven through quantum critical points by a time-dependent chemical potential $\mu(t)$ and coupled to a Fermionic bath. The interaction strength decays algebraically with distance as $1/|i-j|^\alpha$, where $\alpha$ is a tunable parameter.

The study employs a master equation approach (Lindblad type) to model the open system dynamics. The authors design a control protocol comprising two components:

1. Unitary Control: A counterdiabatic Hamiltonian ($H_{CD}$) is derived to ensure the system follows the instantaneous eigenstates of the bare Hamiltonian, suppressing non-adiabatic transitions.
2. Non-unitary Control: Time-dependent Lindblad operators and transition rates are engineered to constrain the system's trajectory to a specific target state (either thermal equilibrium or a population-inverted state) despite the presence of dissipation.

The analysis focuses on two distinct critical points ($\mu = -1$ and $\mu = +1$) and compares the behavior of the system in the long-range regime ($1 < \alpha < 2$) versus the short-range regime ($\alpha > 2$). The authors analytically derive the spatial decay of the CD coupling strengths and the transition rates, verifying these with numerical integration. Additionally, the protocol is adapted to model a quantum battery by engineering a population-inverted trajectory to maximize ergotropy (extractable work).

Key Contributions and Results

• Algebraic Decay of Control Terms: A primary finding is that in the long-range regime ($1 < \alpha < 2$) near the critical point $\mu = -1$, the required CD coupling strength $h_m$ between sites separated by distance $m$ decays algebraically ($h_m \sim m^{-(2-\alpha)}$ at criticality and $m^{-\alpha}$ away from it). This stands in sharp contrast to short-range interactions (or the $\alpha \to \infty$ limit), where STA at criticality typically requires infinite-range interactions (constant $h_m$). This algebraic decay suggests that LRIs can make exact STA protocols practically realizable by avoiding the need for infinite-range control.
• Regime-Dependent Advantages: The benefits of LRIs are not universal across all critical points. Near $\mu = -1$, the energy gap $E_k(\alpha)$ increases as the interaction range increases (decreasing $\alpha$), leading to reduced non-adiabatic excitations and lower control costs. Conversely, near $\mu = +1$, the energy gap increases with decreasing interaction range (increasing $\alpha$), negating the advantages of LRIs in that specific regime.
• Dissipative Control and Heat Current: In the presence of dissipation, the study shows that LRIs can reduce the energetic cost of STA. Specifically, in the low-temperature regime near $\mu = -1$, the heat current generated during the STA protocol is suppressed for smaller $\alpha$ (longer interaction ranges). In the high-temperature regime, thermal fluctuations dominate, washing out the signatures of criticality and the dependence on $\alpha$.
• Quantum Battery Charging: The authors propose a modified STA protocol to charge a quantum battery by driving the system along a population-inverted trajectory. They demonstrate that for $\mu \approx -1$, longer interaction ranges (smaller $\alpha$) enhance the ergotropy of the battery. This enhancement is attributed to the specific dispersion relations in the long-range regime. However, this advantage is absent near $\mu = +1$.
• Multi-body Interactions: The authors note that while the control operators are local in momentum space, their real-space representation involves multi-body interaction terms even away from criticality, a consequence of the need to control entropy in open systems.

Significance
The paper establishes long-range interactions as a valuable resource for quantum control in open many-body systems. By demonstrating that LRIs can transform the required control fields from infinite-range (unimplementable) to algebraically decaying (potentially implementable), the work addresses a major barrier to practical STA implementation. Furthermore, the results suggest that tuning the range of interactions can optimize the performance of quantum technologies, specifically by reducing the thermodynamic cost of control and enhancing the energy storage capacity (ergotropy) of quantum batteries in dissipative environments. The authors conclude that these protocols are potentially implementable in current experimental platforms, such as ion traps and quantum simulators.