Abstract
Phase transitions have been a research focus in manybody physics over past decades. Cold ions, under strong Coulomb repulsion, provide a repealing paradigm of exploring phase transitions in stable confinement by electromagnetic field. We demonstrate various conformations of up to sixteen lasercooled ^{40}Ca^{+} ion crystals in a homebuilt surfaceelectrode trap, where besides the usually mentioned structural phase transition from the linear to the zigzag, two additional phase transitions to more complicated twodimensional configurations are identified. The experimental observation agrees well with the numerical simulation. Heating due to micromotion of the ions is analysed by comparison of the numerical simulation with the experimental observation. Our investigation implies very rich and complicated manybody behaviour in the trappedion systems and provides effective mechanism for further exploring quantum phase transitions and quantum information processing with ultracold trapped ions.
Introduction
The lasercooled ions confined in radiofrequency (rf) traps and Penning traps can condense into crystalline states and form different ordered configurations under control of the confining potentials^{1,2,3,4,5,6,7,8,9,10,11,12}. The variation of the configurations corresponds to structural phase transitions of the ion crystals, such as the lineartozigzag phase transition predicted in theory^{13,14} and later experimentally verified^{15,16,17}. Recent observations have also found that rapid implementation of the lineartozigzag phase transition leads to formation of defects in the ion crystal chain, obeying the inhomogeneous KibbleZurek mechanism, a topological phase transition relevant to the early universe^{18,19,20,21,22,23,24}. Besides, if those ion crystals are lasercooled down to ultracold states, it was predicted theoretically that the structural phase transition of the ion crystals from the linear to the zigzag can be mapped into an Ising model in a transverse field^{25,26,27}. Different from in the classical regime, the lineartozigzag phase transition occurring in quantum regime is temperature dependent^{28}. So the trapped ion crystals provide an experimental toolbox to explore the variation from a classical phase transition to a quantum counterpart^{29}.
In the present work, we demonstrate the control of the ion crystals in our homemade surfaceelectrode trap (SET). The SETs have recently attracted much attention due to relatively simple fabrication as well as the possibility of trapping and shuttling short linear ion crystals, the latter of which is the prerequisite of a scalable quantum information processing^{30,31,32,33,34,35}. Even in the case of a few ions, the confined ion crystals in a single layer lattice structure^{36,37}, the controllable geometric structures of the ions and the flexible architecture of electrodes^{31,32} make the SETs very promising for quantum simulation, such as for the spinspin coupling models^{26,27} in condensed matter physics and for fundamental feature in thermodynamics^{38,39}. Particularly, the variable twodimensional geometry of qubits in SETs is essential to measurementbased quantum computing^{40,41}, error correcting codes^{42,43} and quantum annealing^{44,45}.
Our present work intends to explore structural phase transitions with lasercooled ^{40}Ca^{+} ion crystals. The various conformations of the ions reflect typical manybody behaviour and also form the prerequisite of trappedion quantum information processing, from which we may know the distribution of the future qubits under our control and the influence from the rf heating. Due to strong asymmetry in our SET with the potential well in the yaxis much steeper than in both x and zaxes, the ion crystals are distributed only in the xz surface (further justified later). As such, we define the anisotropy as , rather than ^{13,14,15,16,17}, to characterize different ion crystal conformations, where is the trap frequency in x (y, z)axis. With up to sixteen lasercooled ^{40}Ca^{+} ion crystals, we will demonstrate different configurations in variation with α and explore secondorder phase transitions never experimentally identified before. By comparing simulated results with the experimental observation, we will also investigate the rf heating along different directions using the Langevin thermostat molecular dynamics (MD) method.
Results
Experimental setup and trapping potentials
Our SET is a 500 μm scale planar trap with five electrodes^{46,47,48,49,50} as shown in Fig. 1(a). The electrodes are made of copper on a vacuumcompatible printed circuit board substrate. The electrodes labeled as EC and ME represent the end electrodes and middle electrodes, respectively and SE represents four control electrodes. There are three horizontal electrodes as central electrodes, two of which, i.e., the RF electrodes, are applied by rf voltages and the middle one AE applied by a dc voltage works as a compensation electrode.
The radial electric potential is produced by the rf voltage with amplitude V_{RF} ~ 640 V (0Peak) and frequency MHz. The axially dc electric potential is produced by a voltage V_{EC} = 40 V applied on the endcap electrodes. Depending on the parameters above, the rf potential null is above the trap surface by about 910 μm (See Supplementary Information), which does not generally coincide with the dc potential minimum. As a result, throughout the experimental implementation, we keep the two minima overlapping by adjusting the compensation voltage, which can effectively reduce the rf heating.
For convenience of description, we label the SET electrodes from 1 to 13, with the electrodes No. 1 to No. 11 applied by dc voltages and the other two, i.e., No. 12 and No. 13, by rf voltages, as plotted in Fig. 1a. With dc voltages V_{i} applied on the ith electrode, static potential is generated as^{51}
where , , and are labeled in the inset of Fig. 1a. For the two rf electrodes, we have (i = 12, 13) with the rf frequency Ω_{rf}. So the total effective trapping potential energy in the SET at time t is given by,
where Q is the ion charge. For clarity and simplicity, we employ the pseudopotential approximation in part of our treatments below, where the pseudopotential energy is expressed as
with m the ion mass. So for N ions confined in the SET, the total potential energy is given by
Experimental observation and numerical simulation
Our experiment starts from the loading of ^{40}Ca^{+} ions by twostage photoionization^{52} using a 423 nm laser for the 4S_{0}4P_{1} transition of the calcium atoms, followed by the second excitation by a 380 nm light emitting diode. The trapped ions are Doppler cooled by a gratingstabilized 397 nm laser, with assistance of a gratingstabilized 866 nm laser for D_{3/2}state repumping^{50}. We detect the 397 nm laserinduced fluorescence by an electronmultiplying CCD (EMCCD) (PhotonMax512, Princeton Instruments) along the yaxis. As a result, we cannot experimentally identify the ion crystals distributed along the yaxis. But our numerical simulation (See Methods and Supplementary Information) clearly identifies that, for the ions initially confined as a line in zaxis with , the ion crystals change to twodimensional configurations in xz plane with the increase of α. Under the condition of α < 0.7, the ion crystals distribute for less than 2 μm along yaxis, 0 to 40 μm in xaxis and 50 to 150 μm in zaxis. As a result, there is no threedimensional conformation of the ion crystals under current trapping condition.
For our purpose, we confine three to sixteen lasercooled ions, for each of which we gradually raise or lower the trapping potentials and try to avoid hysteresis (or nonlinearity)^{53,54,55,56} in the observation of configuration changes of the ion crystals. In our operations, the voltage V_{EC} remains unchanged, but the voltage V_{ME} is applied on the middle electrodes decreased from 20 V to −30 V, which increases but decreases , as shown in Fig. 1(b). So the ion crystal configuration changes with the increase of α. Meanwhile, the compensation voltage V_{AE} is adjusted to reduce the ions’ heating to the best, i.e., the dc potential minimum overlapping with the rf potential null. We plot the ion crystals involving ten ions as an example in Fig. 2 which images the change of the ion crystal spatial distributions for a wide range of the applied voltages on the middle electrodes. With the increase of α, the lower trapping frequency in xaxis leads to more serious rf heating (due to the ions more distant from the rf potential null) and the resolution blurring of the individual ions in our observation (see discussion later about heating). Some blurring cases, e.g. for , are also due to nonequilibrium states in the process of the structural phase transition. Nevertheless, considering the center of each ion, we may still identify the configurations of the ion crystals in those cases, which can be justified by the pseudopotential approximation. As shown in Fig. 2, the observed configurations of the ion crystals are in good agreement with the simulated results under pseudopotential approximation.
To characterize the configuration changes, we employ the centertocenter distance Δx (Δz) of two outermost ions in axis, which are found to be very sensitive to the potential change. By measuring Δx and Δz in each image of the ion crystal configuration, we define and find some abrupt raising in the curves of W with respect to α, implying the structural phase transitions. As shown below, W can be considered as an order parameter, which changes from zero to different nonzero values corresponding to different structural phases. Figure 3 exemplifies the cases of 10 and 13 ions with three such phase transitions in the change of the ion crystal configurations. With respect to the cusplike phase transition in the thermodynamical limit, finite numbers of the ions only show the abrupt raising with definite slopes around the critical points of the phase transitions, in which the slopes vary for different numbers of the ions. The first phase transition, occurring at α < 0.15, is for the lineartozigzag phase transition which has been investigated previously in different iontrap systems^{13,14,15}. But the second one has never been reported experimentally before, which happens at 0.2 < α < 0.3, corresponding to the phase transition from the zigzag to the ellipse encircling a single ion or an ion string. The third phase transition represents the configuration change to a more complicated case, e.g., the concentric ellipses. Such a case, however, with α > 0.5, occurs in a much lower depth trap in xaxis, in which the ion crystal melting has handicapped our exact measurement. So the third phase transition in Fig. 3 is only theoretically predicted. We will come back to this point later by treating the rf heating in the case of α > 0.5.
Power laws
To give a more complete impression on this topic, we list in Table 1 different structural phase transitions occurring for different numbers of the ions, where the fewion cases (N ≤ 5) are omitted due to the same as the wellknown results in previous publications^{15,17}. Although our SET is different from the rf linear traps or Penning traps in the potential or the potential symmetry, there is no fundamental difference in ion crystal configurations if the ion number N is less than 6, where the only phase transition is from the linear to the zigzag. For more than five ions involved in the trapped ion crystals, however, there are more complicated configurations and thereby more phase transitions. This can be understood from Table 1 that more ions involved lead to more complicated configurations. Particularly, more phase transitions occur gradually with more ions involved, in which the same phase transition might occur in the case of a smaller α.
Table 1 also presents the possibility of experimentally observing the third phase transition with α < 0.5 if fourteen or more ions are involved in the ion crystals. However, our experiments with fourteen to sixteen ions show serious melting before reaching the critical point of the third phase transition. To understand the experimental difficulty, we have to consider the influence from the rf heating, as discussed later.
The previous studies have shown the scaling behaviour at the critical point of the phase transition from the linear to the zigzag^{13,14,15,17}. Here we assume the similar scaling behaviour in other phase transitions by defining and , where and are, respectively, the critical anisotropic parameters for the first and second phase transitions, N is the ion number and and are the corresponding constants determined by the fitting. The third phase transition is not considered here due to lack of enough experimental data. We have compared the experimental data with the simulation values at the critical points of the two phase transitions in Fig. 4, where the curves are plotted by numerical simulation based on the definitions of the scaling behaviour given above and the experimental data are averaged from the observed data within the abruptly raising regimes of the curves in Fig. 3. We label in Fig. 4 the deviation from the average values of the measurements by error bars, which are determined by the mean square root. We find that the numerical values (i.e., the curves) fit the experimental data within the range of the statistical error.
Discussion
Our experimental values above for the first phase transition are in very good agreement with the previous theoretical results^{13}, even better than the results in^{15}, as listed in Table 2. This might be due to the fact that the powerlaw expression intrinsically depends on the number of the ions involved^{13,14,15}. We are working on 6–16 ions, more than considered in^{15} and thereby obtain the parameter values closer to that in Schiffer’s calculation (involving 10–500 ions) in^{13}. Moreover, despite nongenerality, the expression of the power law implies the onset of a secondorder phase transition. Besides, the powerlaw expression is also useful for understanding the relevance of the phase transitions to the values of α in Table 1. Rewriting the powerlaw expression as , for the positive constant c and the negative constant β, we surely have smaller values of with more ions involved.
Although our slow operations can be reasonably described under the pseudopotential approximation, a complete consideration of the timedependent potential in the SET is necessary for fully understanding the details in the configuration change of the ions, such as the ions’ heating due to the rf potential^{57,58}. As such, we simulate the dynamics of the system by solving the MD equations (See Methods). The heating effect due to the micromotion of the ions occurs in three dimensions of the SET, which is strongly relevant to the positions of the ions from the rf potential null. In the case of few tens of trapped ions with α < 0.7, since our simulation identifies a tiny distribution of the ions along the yaxis and we constantly keep the potential minimum at the rf potential null by adjusting V_{AE}, we may focus our investigation on the heating in x and zaxes during the configuration change. The temperature of the ions is assessed by the kinetic energy owned by the ions. For a comparison, we compute the energies from both the secular motion and the micromotion in the two dimensions. As shown in Fig. 5, the micromotion energies in both directions are proportional to the distance square, behaving as quadratic functions. In contrast, the secular motion energies are near constants along x and y axes. Besides, the overall temperature in zaxis is much less than in xaxis, implying negligible heating in zaxis compared to in xaxis. This reflects a fact that we have negligible rf potential along zaxis (See Supplementary Figure 2). With the increase of α, the ion crystals form the configurations with more components away from zaxis, which leads to a rapid increase of the rf heating. Meanwhile, the increase of α means stronger rf heating and weaker confinement in xaxis. This is why we cannot observe experimentally the third phase transition in Fig. 3 since the ion crystals turn to be seriously melting when α > 0.5 and then escape from the trap in the case of a bigger α. More details for quantitative estimate of the energies can be found in Supplementary Information.
On the other hand, the discrepancy between the experimental values and the simulated results also indicates the imperfection in our operations with respect to the ideal consideration. We estimate the imperfectioninduced errors within 4.7% and 7.5%, respectively, in the first and second phase transitions, including 0.15% error relevant to ±0.3 kHz deviation in measuring x and zaxial frequencies, 0.03% (0.06%) error due to ±53 Hz (±166 Hz) uncertainty of the dc potential in z(x) axis and 4.45% error from ±13.34 kHz uncertainty in the xaxial rf potential. There are some other unclear errors in the second phase transition.
Following on from this work, we expect to explore quantum mechanically structural phase transitions, which can be mapped into a quantum phase transition of Ising model subject to a transverse field^{25,27} and demonstrate temperature dependence^{28}, in future experiments by further cooling the ions down to the vibrational ground state. To this end, an improved SET with higher symmetric structure and deeper potential is expected. This new SET will also help implementing quantum computing tasks with ultracold ions confined and moved in a scalable fashion and error correcting codes and quantum algorithms accomplished under control. Particularly, the micromotioninduced heating might be effectively suppressed in the SET if the transverse motional modes and welldesigned strong laser pulses are employed^{59}.
Methods
The minimum energy analysis
Under the pseudopotential approximation, the trapping potential of our SET can be analytically expressed as in Eq. 3 and plotted in Fig. 6. With the pseudopotential, we have simulated the stable configurations of the ion crystals as in Fig. 2, where different trapping potentials induce different configurations and the position of each cooled ion can be solved by minimizing the total potential energy . The energy minimum analysis is carried out by the gradient descent method^{60}, i.e., a firstorder optimization algorithm for finding a local minimum of a function.
The numerical simulation involving the micromotion
A complete description of motion of the ion crystals requires involvement of the rf potential. In this case, the dynamics of the ions at a specific temperature T can be simulated by the MD method. For the jth ion, the Langevin equation is
where is the coordinate under the total energy potential , m is the mass of the jth ion and η is the friction coefficient induced by the laser cooling. is the stochastic force which obeys following ensemble average relations: and . Eq. (4) is numerically simulated using the values kgs^{−1} and mK by the Brownian dynamics^{18,19,61}.
Additional Information
How to cite this article: Yan, L. L. et al. Exploring structural phase transitions of ion crystals. Sci. Rep. 6, 21547; doi: 10.1038/srep21547 (2016).
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Acknowledgements
This work is supported by National Fundamental Research Program of China under Grant No. 2012CB922102 and by National Natural Science Foundation of China under Grants No. 11274352 and No. 11104325.
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M.F. and L.C. proposed the idea. W.W., L.C., F.Z. and S.J.G. performed the experiments. L.L.Y. and M.F. carried out the calculation. M.F. and L.L.Y. wrote the paper. M.F. and X.T. analysed the data. All authors discussed the results and commented on the manuscript.
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Yan, L., Wan, W., Chen, L. et al. Exploring structural phase transitions of ion crystals. Sci Rep 6, 21547 (2016). https://doi.org/10.1038/srep21547
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