Orientational Melting in a Mesoscopic System of Charged Particles

Lucia Duca, Naoto Mizukami, Elia Perego, Massimo Inguscio, and Carlo Sias

Phys. Rev. Lett. 131, 083602 – Published 25 August 2023

Abstract

A mesoscopic system of a few particles can undergo changes of configuration that resemble phase transitions but with a nonuniversal behavior. A notable example is orientational melting, in which localized particles with long-range repulsive interactions forming a two-dimensional crystal become delocalized in common closed trajectories. Here we report the observation of orientational melting occurring in a two-dimensional crystal of up to 15 ions. We measure density-density correlations to quantitatively characterize the occurrence of melting, and use a Monte Carlo simulation to extract the angular kinetic energy of the ions. By adding a pinning impurity, we demonstrate the nonuniversality of orientational melting and create novel configurations in which localized and delocalized particles coexist. Our system realizes an experimental testbed for studying changes of configurations in two-dimensional mesoscopic systems, and our results pave the way for the study of quantum phenomena in ensembles of delocalized ions.

Received 26 October 2022
Revised 4 March 2023
Accepted 17 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.083602

Physics Subject Headings (PhySH)

Atom & ion trapping & guiding Crystallization Thermodynamics

Trapped ions

Atomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Lucia Duca ^1,2, Naoto Mizukami^1,2,3, Elia Perego^1,2,†, Massimo Inguscio^2,4,5, and Carlo Sias ^1,2,4,*

¹Istituto Nazionale di Ricerca Metrologica (INRiM), 10135 Torino, Italy
²European Laboratory for Nonlinear Spectroscopy (LENS), 50019 Sesto Fiorentino, Italy
³Politecnico di Torino, 10129 Torino, Italy
⁴Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), 50019 Sesto Fiorentino, Italy
⁵Department of Engineering, Campus Bio-Medico University of Rome, 00128 Rome, Italy

^*To whom inquiries should be addressed. c.sias@inrim.it
^†Present address: Department of Physics, University of California, Berkeley, California, USA.

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Vol. 131, Iss. 8 — 25 August 2023

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Images

Figure 1
Sketch of the physical system. (a) The ion trap is composed by 4 rf (gray, only two shown) and 4 dc (yellow) electrodes. The inset shows a picture of a two-dimensional crystal of 7 ${}^{138}{Ba}^{+}$ ions in a potential with trap frequencies $(ω_{x}, ω_{y}, ω_{z}) = 2 π \times (400, 121, 97) kHz$ . (b) A reduction of the $ω_{y} / ω_{z}$ ratio towards unity corresponds to a decrease of the height $V_{B}$ of the potential barrier associated to the rigid rotation of the crystal. When $V_{B}$ is reduced, e.g., when the angular periodic potential changes from deep (dashed dark blue) to shallow (solid light blue), the particles spatial distribution spreads as illustrated in the figure by the two light and dark blue shaded thermal distributions and their corresponding sketches in (c). When the barrier is further lowered, the particles undergo orientational melting.
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Figure 2
Accessing orientational melting by changing the particles’ confining potential. Left: stability diagram of the ion trap calculated for ${}^{138}{Ba}^{+}$ , expressed as a function of $a_{y}$ and $q_{y}$ [31]. The yellow curve corresponds to the condition $ω_{y} / ω_{z} = 1$ , as expected from the simulation of our trap, and fits inside the stability region (blue area). The location of this curve is confirmed by experimental data (yellow dots) obtained from fitting the ions’ spatial distribution and corresponding to a radially symmetric crystal [31]. Right: images of 5 (i) and 7 (ii)–(iv) ${}^{138}{Ba}^{+}$ ions at $q_{y} = - 0.182$ and different values of $a_{y}$ . The images illustrate how crystallization and ellipticity change as a function of $a_{y}$ across the melting transition. The images are taken at $ω_{y} / ω_{z} = (3.9, 1.2, 1.1, 0.9)$ , $ω_{y} = 2 π \times (246, 121, 107, 91) kHz$ , and $q_{y} = - 0.182$ , top to bottom.
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Figure 3
Characterization of orientational melting for 4 and 7 ions. (a) Amplitude $C$ of the angular density-density correlation function $g (θ_{N T})$ calculated along the elliptic trajectory (inset). The onset of melting is clearly different for 4 (blue data) and 7 (red data) ions, as shown in the raw images in (b) taken in the regions (i) and (ii). The dashed black (red) line corresponds to the correlation amplitude expected for a crystal of 4 (7) ions at the best fitting temperature of $E_{T 4} / k_{B} = 102 mK$ ( $E_{T 7} / k_{B} = 96 mK$ ), as calculated from a Monte Carlo simulation. The blue (red) shaded area for 4 (7) ions represents a change of $\pm 10 mK$ from the best fitting curve. (c) Increase of the angular spread $σ$ as the melting transition is approached (see inset). $σ$ is obtained by fitting the density distribution [31], and the values are normalized by the angular separation $θ_{N T}$ between the ions. The gray central area corresponds to trap conditions for which no density modulation is visible, see images (iii) in (b). The dashed black and red lines correspond to the spread of the simulated density profiles for 4 and 7 ions at a temperature $E_{T 4} / k_{B}$ and $E_{T 7} / k_{B}$ , respectively. The error bars in (a),(c) indicate the standard deviation of the mean over 3 to 10 images.
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Figure 4
Orientational melting in the presence of a pinning impurity. (a) Single snapshot images of a crystal of 6 to 15 ions in the presence of an impurity ion of a different isotope. The data are taken at the onset of melting $ω_{y} / ω_{z} = 1.18$ and $q_{y} = - 0.182$ , corresponding to $(ω_{x}, ω_{y}, ω_{z}) = 2 π \times (401, 116, 98) kHz$ . The impurity appears as a dark ion (red circle). The upper (lower) images correspond to the impurity located in the outer (inner) shell. The impurity ion suppresses melting in the hosting shell (see text and Ref. [31]). (b) We quantify the level of localization for different $N$ by calculating the ratio between the correlations’ amplitudes obtained with the dark ion in the inner ( $C_{In}$ ) and outer shell ( $C_{Out}$ ) (green dots). The error bars of the data correspond to the standard deviation of the mean, which is calculated on 4 to 18 images depending on $N$ . The dashed line is a guide to the eye. The plot shows that $N = 7$ and $N = 14$ correspond to magic numbers for which the melting transition is disfavored. (c) Angular kinetic energy of the outer ring for $N = 7$ , 10, 14. The temperature is extracted by comparing the measured correlations $C_{In}$ and their standard deviations with a Monte Carlo simulation (see text).
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