Delocalization and heat transport in multidimensional trapped ion systems

We study the connection between heat transport properties of systems coupled to different thermal baths in two separate regions and their underlying nonequilibrium dynamics. We consider classical systems of interacting particles that may exhibit a certain degree of delocalization and whose effective dimensionality can be modified through the controlled variation of a global trapping potential. We focus on Coulomb crystals of trapped ions, which offer a versatile playground to shed light on the role that spatial constraints play on heat transport. We use a three-dimensional model to simulate the trapped ion system and show in a numerically rigorous manner to what extent heat transport properties could be feasibly tuned across the structural phase transitions among the linear, planar zigzag, and helical configurations. By solving the classical Langevin equations of motion, we analyze the steady state spatial distributions of the particles, the temperature profiles, and total heat flux through the various structural phase transitions that the system may experience. The results evidence a clear correlation between the degree of delocalization of the internal ions and the emergence of a nonzero gradient in the temperature profiles. The signatures of the phase transitions in the total heat flux as well as the optimal spatial configuration for heat transport are shown.

Figure 1

Spatial probability density $P\left(\mathbf{q}\right)$ (6) of the trapped ion system obtained from a single stochastic realization with $t={\tau }_{ss}=5\times {10}^{-2}\mathrm{s}$ and $\sigma =2\mu \mathrm{m}$. The axial frequency is set to ${\omega }_x/2\pi =50\mathrm{kHz}$, and the radial ones are ${\omega }_z=13{\omega }_x$ and ${\omega }_y=n_y{\omega }_x$. The labels (a) (13,13), (b) (10,13), and (c) (7,13) correspond to the anisotropy parameters $(n_y,n_z)$. The positions $\mathbf{q}$ with values of $P\left(\mathbf{q}\right)$ below 5% of its maximum value are not depicted. The total extension of the system along the axial direction is reduced from approximately $420\mu \mathrm{m}$ in the linear chain to $402\mu \mathrm{m}$ in the complete zigzag configuration. The vesta software was used for the visualization of the spatial distributions [53].

Figure 2

The same as Fig. 1 for fixed axial frequency ${\omega }_x/2\pi =50\mathrm{kHz}$, equal radial frequencies ${\omega }_y={\omega }_z$, and with anisotropy parameters (a) (13,13), (b) (9,9), and (c) (7,7). In this case, the positions $\mathbf{q}$ with values of $P\left(\mathbf{q}\right)$ below 6.5% of its maximum value are not depicted. The total extension of the system along the axial direction is reduced from approximately $420\mu \mathrm{m}$ in the linear chain to $402\mu \mathrm{m}$ in the complete helical configuration.

Figure 3

The same as Fig. 1, for fixed axial frequency ${\omega }_x/2\pi =50\mathrm{kHz}$, radial frequencies ${\omega }_y=9{\omega }_x$ and ${\omega }_z=n_z{\omega }_x$ and with anisotropy parameters (a) (9,10), (b) (9,9), and (c) (9,8). The total extension of the system along the axial direction is approximately $410\mu \mathrm{m}$ in the three configurations. Note that, in the lowest panel, the planar zigzag configuration is confined on the $xz$ plane whereas, in the upper one, the planar zigzag configuration is on the $xy$ plane.

Figure 4

Steady axial distributions ${\mathrm{\Theta }}_i\left(x\right)$ (7) for the same eight ions in (a) the linear (L) configuration given by $(n_y,n_z)=(13,13)$, (b) the zigzag (Z) configuration given by $(n_y,n_z)=(9,8)$, and (c) the helical (H) configuration given by $(n_y,n_z)=(7,7)$. We consider $c_x=2000$ cells with size $\mathrm{\Delta }=0.22\mu \mathrm{m}$ along the $x$ direction, set the time values $t={\tau }_{ss}=4\times {10}^{-2}\mathrm{s}$, and average over more than 500 stochastic trajectories. The peaks on both sides (black line) correspond to the $N_L=3$ and $N_R=3$ extreme ions that are connected to the laser reservoirs and evidence the strong spatial confinement of such ions in the spatial regions where the beams are focused. Such regions are depicted by the colored areas. The blue and red lines correspond to the same two internal ions. For a better comparison, the intensity of the six peaks of the extreme ions (black line) has been divided by a factor of 2 in the middle panel and by a factor of 11 in the lower one.

Figure 5

Steady radial distributions ${\mathrm{\Theta }}_i\left(y\right)$ [(a)–(c)] and ${\mathrm{\Theta }}_i\left(z\right)$ [(d)–(f)] (7) of the two internal ions (blue and red lines) whose axial distributions ${\mathrm{\Theta }}_i\left(x\right)$ are depicted in Fig. 4 for the linear (L), zigzag (Z), and helical (H) configurations considered in such figure. We consider $c_y=c_z=2000$ cells with size $\mathrm{\Delta }=0.024\mu \mathrm{m}$ along both radial directions and set the time values and perform the stochastic average as in Fig. 4.

Figure 6

The number of ions arranged in the (a) linear, the (b) zigzag, and (c) the helical configuration, obtained from the numerical simulation of the dynamics in the interval $[1\times {10}^{-2},2\times {10}^{-2}]\mathrm{s}$ of the steady state. A diagram (d) of the anisotropy map that shows the values of $(n_y,n_z)$ leading to the linear string along the axial axis (gray square area at high values of both $n_y$ and $n_z$), the zigzag configuration on the $xz$ plane (below the diagonal $n_y=n_z$), the zigzag configuration on the $xy$ plane (above the diagonal $n_y=n_z$), and the helical configuration (along the diagonal $n_y=n_z$). In panel (d), the values of anisotropy parameters $(n_y,n_z)$ corresponding to the spatial probability densities depicted in Figs. 1, 2, 3 are shown (blue circles). The zigzag-linear transition (ZL line), helical-linear transition (HL line), and zigzag-helical-zigzag transition (ZHZ line) considered to perform the analysis of the transport properties through the various structural phase transitions are also indicated.

Figure 7

(a) Steady state temperature profiles $T\left(x\right)$ (8) across the axial direction for the complete linear (L), zigzag (Z), and helical (H) configurations. The colored areas are the regions where the two heat reservoirs are acting. The labels indicate the values of the anisotropy parameters $(n_y,n_z)$. The ion spatial distributions corresponding the different configurations are depicted in Figs. 1 (L and Z) and 2 (L and H). (b) Temperature gradients across the region occupied by the internal ions, corresponding to the temperature profiles shown in (a), and obtained for intervals of the $x$ coordinate symmetrically arranged around the center of the system and with increasing size ${\mathrm{\Delta }}_T$.

Figure 8

(a) Steady state temperature profiles $T\left(x\right)$ (8) across the axial direction for the linear (L) and the zigzag (Z) configurations depicted in Fig. 1. The colored areas are the regions where the two heat reservoirs are acting. (b) The component of the steady total heat flux $\mathbf{J}$ (18) across the axial direction for anisotropy parameters selected along the ZL line depicted in Fig. 6, corresponding to the zigzag-linear structural phase transition occurring for $n_z=13$ and different values of $n_y$. The dashed lines indicate the anisotropy parameters whose temperatures profiles are shown in (a). The results were obtained from numerical simulations of the dynamics in the interval $[4\times {10}^{-2},8\times {10}^{-2}]\mathrm{s}$ of the steady state and the average over more than 1000 stochastic trajectories. The error bars in the total heat flux provide a measure of the fluctuations around such an average and are given by the corresponding standard deviations. We consider $c_x=500$ cells with size $\mathrm{\Delta }=0.85\mu \mathrm{m}$ along the axial direction to get the temperature profiles.

Figure 9

The same as Fig. 8 for (a) the linear (L) and the helical (H) configurations depicted in Fig. 2 and for (b) anisotropy parameters selected along the HL line depicted in Fig. 6, corresponding to the helical-linear structural phase transition occurring for different values of $n_y=n_z$.

Figure 10

The same as Fig. 8 for (a) the zigzag (Z) and the helical (H) configurations depicted in Fig. 3 and for (b) anisotropy parameters selected along the ZHZ line depicted in Fig. 6, corresponding to the zigzag-helical-zigzag structural phase transition occurring for $n_y=9$ and different values of $n_z$.