Ergotopy transport in a one dimensional spin chain

TL;DR

This work analyzes how extractable work, quantified as ergotropy, can be shuttled along a one-dimensional XX spin chain with tunable exchange couplings, bridging uniform and perfect-state-transfer (PST) regimes. It demonstrates a quantum advantage when ergotropy is encoded in quantum coherences, enabling more efficient transport in non-PST chains, while population-encoded ergotropy faces finite-length transport limits; PST couplings enable perfect shuttling and exhibit robustness to disorder for coherent ergotropy. The authors derive exact transition-amplitude expressions in the PST and uniform limits, characterize ergotropy dynamics under both encodings, and study the thermodynamic cost of turning on the chain interactions via quantum work statistics, revealing smaller fluctuations for PST. Overall, the results highlight key differences between ergotropy transport and conventional information or energy transfer and have implications for the design of quantum batteries and energy transport in quantum networks.

Abstract

We examine the transport of useful energy, i.e. extractable work as quantified by the ergotropy, along a spin chain with tuneable exchange couplings between the sites. We focus on, and interpolate between, the two physically relevant limits of uniform interaction strengths and engineered couplings which achieve perfect state transfer (PST). By modelling the individual constituents as quantum batteries, we consider how the manner in which the extractable work appears in the initial state of the first site impacts the chain's ability to transport ergotropy to the final site. For non-PST couplings, we establish that there is a clear quantum advantage when the ergotropy is initially endowed in quantum coherences and demonstrate that this ergotropy is more efficiently transferred. For extractable work encoded in a population inverted state, we show that this considerably limits the length of chain over which any ergotropy can be faithfully transported. For PST couplings, we consider the robustness to disorder and again demonstrate a quantum advantage for coherently endowed ergotropy. Finally, we examine the work probability distribution associated with quenching on the interactions which provides insight into the work cost in switching on the couplings. We show that PST couplings lead to smaller fluctuations in this work cost, indicating that they are more stable.

Figures (6)

• Figure 1: $\varepsilon_{max}$ of the last qubit in the chain, as a function of $N$, evaluated at the first reflection time, Eq. \ref{['eq:ReflectTime']}. The inset of the top panel is the same calculation but over a time interval $Jt\!\in\!(0,1000) \gg 2N$. The dashed curves correspond to initial states given by Eq. \ref{['pureinitial']} and the solid curves for Eq. \ref{['mixedinitial']}. Each pair of curves is initialized with the same $\varepsilon_{in}$ = $\{0.5,\,1,\,2\}$ corresponding to $\theta = \{\frac{\pi}{3},\,\frac{\pi}{2},\,\pi\}$ and $q=\{0.625,\,0.75,\,1\}$, with colours {green, blue, red} respectively and can be determined from Eqs. \ref{['eq:Ergcoh']} and \ref{['eq:Ergmix']}. Parameters: $J = 1,$$B = 1$. Panels correspond to a different values of $\alpha$: (a) 0, (b) 0.1, and (c) 0.5.
• Figure 2: $\varepsilon_{max}$ endowed in coherences (dashed curves) and populations (solid curves) of the last qubit in the chain evaluated at $Jt$ given by Eq. \ref{['eq:ReflectTime']}, as a function of $\varepsilon_{in}$. We show three system sizes, $N=\{10,25,100\}$. Parameters: $\alpha = 0, J=1, B=1$. Inset:The value of $\varepsilon_{in}$ that corresponds to the peak value of $\varepsilon_{max}$, as a function of chain size $N$. The corresponding value is marked with a star for the system sizes shown in the main panel.
• Figure 3: $\varepsilon_{\eta}$, calculated at $Jt$ given by Eq. \ref{['eq:ReflectTime']} as a function of $\varepsilon_{in}$ endowed in coherences (a) and populations (b) and $N$. Parameters: $\alpha = 0, J=1, B=1$ for both panels.
• Figure 4: (a) $\varepsilon_{max}$ averaged over $1000$ realisations of disorder amplitude $\Delta$ evaluated at the first reflection time, Eq. \ref{['eq:ReflectTime']} with $\alpha\!=\!1$. The curves from top to bottom in both panels correspond to $N=\{5,25,50\}$. The upper, blue and lower, red curves correspond to coherent and mixed states, respectively. The shaded region denotes one standard deviation from the mean. (b) Figure of merit from Eq. \ref{['eq:gamma']} as a function of $\Delta$ for various $N$. This is computed from the data in (a).
• Figure 5: (a) Work distribution, $P(W)$, resulting from suddenly quenching on the interactions in the chain interpolating between uniform (leftmost, blue, $\alpha\!=\!0$) and PST (rightmost, red, $\alpha\!=\!1$) regimes. (b) and (c) Show the two limits together with the probability density functions, Eq. \ref{['eq:pdfCA']} and Eq. \ref{['eq:pdfXX']}, shown by the continuous black lines. For panels (a-c) we assume the first site is in a maximally charged state, i.e. $|1\rangle$ corresponding to $\theta\!=\!\pi$ and $q\!=\!1$. (d-g) Work distributions corresponding to states with the same initial ergotropy present in the populations (d,e) or the coherences (f,g) of the first site. The top-row corresponds to PST couplings, $\alpha\!=\!1$ and the bottom row for uniform couplings $\alpha\!=\!0$. Panels (d) and (e) correspond to an active mixed initial state with $q\!=\!0.92$. Panels (f) and (g) correspond to a pure, coherent initial state with $\theta\!=\!\frac{3\pi}{4}$. All panels assume $J\! = \!1$, $B\!=\!1$, and $N\!=\!50$.