Nonequilibrium quantum thermodynamics in Coulomb crystals

We present an in-depth study of the nonequilibrium statistics of the irreversible work produced during sudden quenches in proximity to the structural linear-zigzag transition of ion Coulomb crystals in 1+1 dimensions. By employing both an analytical approach based on a harmonic expansion and numerical simulations, we show the divergence of the average irreversible work in proximity to the transition. We show that the nonanalytic behavior of the work fluctuations can be characterized in terms of the critical exponents of the quantum Ising chain. Due to the technological advancements in trapped-ion experiments, our results can be readily verified.

Figure 1

(a) Irreversible work $W_{\text{IRR}}$ and (b) variance ${\sigma }^2$ for small quenches with amplitude $\mathrm{\Delta }\omega =0.01$ within the same phase (circles in the zigzag phase and squares in the linear phase) at zero temperature as a function of the initial frequency squared ${\omega }^2$. The vertical dotted line indicates the critical frequency. Also shown is the contribution of the soft mode to (c) the irreversible work $W_{\text{IRR}}$ and (d) the variance ${\sigma }^2$ as a function of $|{\omega }^2-h_1|$. The straight lines are the approximate scaling expressions (8) and (9). Various system sizes $L$ have been considered here: $L=60$ (purple), 72 (blue), 90 (cyan), 108 (green), 120 (yellow), 132 (orange), and 144 (red).

Figure 2

Irreversible work $W_{\text{IRR}}$ at $T=0$ for small quenches in ${\omega }^2$, considering and increasing number $L$ of ions. Here $L=60$ (red), 72 (orange), 90 (yellow), 108 (green), 120 (cyan), 132 (blue), and 144 (purple). The left inset shows the location ${\omega }_{☆}^2$ of the maximum of $W_{\text{IRR}}$ in the control parameter ${\omega }^2$, as a function of the system size $L$. Extrapolation to the thermodynamical limit of ${\omega }_{☆}^2$, performed via fitting (gray line), shows that it is compatible with the critical point ${\omega }_C^2$ calculated via the DMRG algorithm (shaded area). The right inset shows the value of the maximum of $W_{\text{IRR}}$ as a function of the ion chain length $L$, showing an accurate linearity.

Figure 3

Collapse of the irreversible work of the full anharmonic model according to the scaling ansatz defined in Eq. (12). The color coding for the symbols is as in Fig. 2.