Abstract
Symmetry breaking phase transitions play an important role in nature. When a system traverses such a transition at a finite rate, its causally disconnected regions choose the new broken symmetry state independently. Where such local choices are incompatible, topological defects can form. The Kibble–Zurek mechanism predicts the defect densities to follow a power law that scales with the rate of the transition. Owing to its ubiquitous nature, this theory finds application in a wide field of systems ranging from cosmology to condensed matter. Here we present the successful creation of defects in ion Coulomb crystals by a controlled quench of the confining potential, and observe an enhanced power law scaling in accordance with numerical simulations and recent predictions. This simple system with welldefined critical exponents opens up ways to investigate the physics of nonequilibrium dynamics from the classical to the quantum regime.
Introduction
Phase transitions are ubiquitous in nature, from the freezing of water to the emergence of superfluidity, superconductivity and ferromagnetism, as well as symmetry breaking in cosmology^{1,2}. In particular, the universality of secondorder phase transitions makes their dynamics largely independent from their microscopic nature. This has led to a comprehensive effort to study them in a wide range of systems^{1,2,3,4,5,6,7,8,9,10}.
These studies test a theory of secondorder phase transition dynamics, the Kibble–Zurek mechanism (KZM), that predicts the scaling of defect formation with quench rate. In previous experiments, this scaling was studied primarily in homogeneous systems, such as in fluxoid formation in superconducting loops^{5} and ferroelectric domains in multiferroics^{6}. A more complete overview on KZM experiments can be found in Supplementary Note 1.
The main difficulty with observing the KZM scaling is the weak fractional power law dependence of defect densities that requires varying the control parameter over a large range to obtain a high sensitivity to the scaling exponent. This means that one has to change the quench time over orders of magnitude to gain a reliable signal. A way to enhance the scaling of defect density on quench rate that relies on causality (central to KZM) was suggested^{3} for investigation of systems in harmonic traps. Here, symmetry breaking begins in the centre of the trap, and the transition front propagates with a speed set by the quench rate and the local gradient of the potential. Where the front is faster than the relevant speed of sound, different parts of the system cannot communicate and they break symmetry independently as in the case of a homogeneous transition^{11,12}. However, further from the trap centre, the phase front slows, and as it becomes slower than the speed of sound defect production ceases (as the outlying parts of the system inherit broken symmetry from their neighbours). Thus, in harmonic traps the quench rate controls both the density of defects in the homogeneous region and the size of the fraction of the trap in which they can be produced. This leads to steeper scalings^{3,4,13}.
Here, we use chains of ions confined in harmonic traps that present a promising platform for quantum simulation and information processing^{14,15,16,17,18} as a wellcontrolled system to study the dynamics of symmetry breaking and the formation of topological defects. This system offers excellent experimental control over the quench rate of the trapping potential and offers the possibility of studying defect formation in a setting in which the sensitivity is improved due to the enhanced scaling that occurs in inhomogeneous systems^{3,4}. In contrast to previous experiments, no assumptions regarding the values of the critical exponents are necessary, as they can be derived using the Ginzburg–Landau model^{4}. In this study, we present the creation of stable topological defects in ion Coulomb crystals and measure the scaling of defect creation with respect to the controlled quench parameter as compared to the predictions of the KZM theory.
Results
Scaling of defect creation in a harmonic ion trap
Our experiment tests a theory—the KZM^{1,2}—that employs equilibrium scaling behaviour to predict their nonequilibrium consequences. Critical slowing down, that is, the divergence of the relaxation time of the system, implies that every secondorder phase transition traversed at a finite rate will inevitably depart from equilibrium near the critical point, where a discrete symmetry is chosen. In a trapped ion chain, as the triaxial confining potential is weakened in the transverse direction, the chain buckles and the ions form a ‘zigzag’ (see Fig. 1). This breaking of the (axial) symmetry is a secondorder phase transition described by the Ginzburg–Landau theory^{19}. It can lead to the formation of topological defects (for example, places where a ‘zig’ is followed by another ‘zig’ rather than a ‘zag’), as highlighted in Fig. 1 by red circles. Depending on the speed at which the transverse potential is quenched, the system chooses a final configuration of either a pure ‘zigzag’ or kinks dividing the two different orientations with a certain probability. The appearance of such kinks, that show solitonlike behaviour^{20}, is a result of the incompatible local choices of symmetry breaking.
In the thermodynamic limit, near secondorder phase transitions, the relaxation time τ and the healing length ξ (which set the time and the length scales on which the order parameter returns to its equilibrium value) diverge as τ=τ_{0}/ε^{νz} and ξ=ξ_{0}ε^{v} as the dimensionless distance ε to the critical point approaches zero. Here, τ_{0} and ξ_{0} depend on the microphysics, while the critical exponents v and z define the universality class of the transition. The distance from the critical value is given by the dimensionless parameter , where λ_{c} is the critical value of the control parameter λ at which the phase transition occurs. The change of ε with time takes the system from the symmetric (ε<0) to the symmetrybroken (ε>0) phase. In what follows, we take ε=t/τ_{Q}, where τ_{Q} is the timescale on which the phase transition is crossed. The ratio of ξ and τ yields the speed of sound at which the information about the choice of broken symmetry can propagate. The size of domains that can coordinate symmetry breaking is set at the instant when, because of critical slowing down, the state of the system ceases to follow changes imposed by the timedependent control parameter ε. This happens when the relaxation time τ matches the timescale of the changes imposed on the system. This time, , determines the size of the domains , that is, the typical separation between topological defects.
In del Campo et al.^{4}, Coulomb crystals trapped in a harmonic axially confining potential were proposed to measure the scaling of defect formation to test KZM. In such crystals, the charge density along the ion chain is larger in the centre and reduces towards the edges. Within the local density approximation, this inhomogeneity is inherited by the critical control parameter λ_{c}. In our case, this control parameter is the transverse secular frequency ν_{t} describing the radial confinement. The spatial dependence of its critical value is given by . Here, ω(x) is the characteristic frequency of the axial Coulomb coupling along the trap axis x. It is given in terms of the mass m and charge Q of the ion, and the axial interion spacing a(x), which reaches its minimum value at the centre of the ion chain, that is, at x=0. Laser cooling provides a dissipative force acting on the ions described by the friction parameter η (for a definition of η and its experimental parameters, see Methods). In the underdamped regime accessible in our experiment, the characteristic response time is fixed by the inverse of ω(0) and the sound velocity becomes independent of η (and x). As a result, the system exhibits meanfield critical exponents v=1/2 and z=1 (see del Campo et al.^{4}).
We consider a quench of the transverse secular frequency of amplitude δ in a timescale τ_{Q}, that is, . During the quench, the phase transition is first crossed at the centre of the chain. The velocity of the front υ_{F} with which the critical point propagates depends on the quench rate and varies along the chain. The dynamics of an inhomogeneous phase transition is governed by the interplay of υ_{F} and the sound velocity υ. For extremely fast quenches, the inequality υ_{F}>υ holds everywhere in the system, and the paradigmatic KZM for homogeneous systems is recovered. The sharpness of this inequality was studied in detail (see Supplementary Note 2). In this regime the density of kinks d scales with the quench rate as , in agreement with the KZM. At lower quench rates, υ_{F}>υ is only satisfied in a fraction of the system, which depends on the quench rate. This results in an effective system size , where defects can be produced with densities predicted by KZM^{3,4}. Outside this spatial region, the critical front is slower than υ, and defects will not form^{11}. As a result of this interplay between propagating phase front and finite speed of sound, the total number of kinks in the inhomogeneous KZM is substantially reduced and exhibits a more pronounced dependence on the quench rate, that is, , see Fig. 2.
In finitesized systems, τ and ξ, that diverge to infinity in an infinite system, ‘round off’ near the critical point. However, sufficiently far away from criticality the system will still behave like an infinite system^{21}. This is the regime of relevance to the KZM, according to which the phase of the broken symmetry is chosen at the freezeout time , away from the critical point. We have carried out numerical simulations with 30–100 ions to check how the defect scaling depends on the system size (the simulation for our experimental system with 30 ions is shown in Fig. 2). We consistently observe a linear KZM scaling indicating homogeneous and inhomogeneous regimes. For parameters such that a quench leads to only one or no defects, a steeper scaling is observed, see Fig. 2. For this condition, a doubling of the Kibble–Zurek scaling has been reported previously in homogeneous systems in a series of works^{5,22,23}. In the inhomogeneous case, when the kink density d becomes approximately equal to the probability to observe a single kink, this suggests a scaling of . Further details of the inhomogeneous regime can be found in Supplementary Note 2.
Creation of topological defects in ion Coulomb crystals
In ion Coulomb crystals, two types of metastable defect configurations that are robust against thermal fluctuations have been identified in numerical simulations, that is, odd and extended kink solitons^{4,20}. Experimentally, spontaneous kink formation has been observed during the process of crystallization induced by laser cooling^{24,25}. In our experiment, we laser cool about 30 ^{172}Yb^{+} ions into a linear crystal in a threedimensional harmonic trap^{26} and create kinks via a controlled quench of the radial potential, see Methods section. We induce a slight ellipticity in the radial trapping frequencies and to confine the ions to a plane, and sweep both frequencies linearly over the critical value. In the final zigzag configuration, 1.3. The phase transition is governed by the lower transverse frequency . The structure of the Coulomb crystals is then detected and analyzed by imaging the fluorescence of the ions onto a camera. Figure 3 shows the experimentally obtained stable configurations of both types of kinks.
For quench times, t=2τ_{Q} in the order of the oscillation period T=1/ν_{x} of the axial secular motion of the ions, kink formation in the twodimensional ‘zigzag’ structure is observed as a break in the periodic order of the position of the ions. Depending on the final radial interion spacing, odd or extended kink configurations are stable. The statistics described in this work is based on measurements of extended kinks, as in this configuration, the energy barrier for a defect, that is, the Peierls–Nabarro potential^{27}, is larger and gives an inherently higher stability compared with odd kinks. As extended kinks are a result of initially created odd kinks during the radial quench and no more than one defect per quench is created in our experimental regime, both final configurations show the same statistics. This allows us to use extended kinks to test KZM scaling. In Supplementary Movies 1–3, we show the experimental creation of odd and extended kinks, as well as a simulation of the timeresolved transformation of a odd kink into an extended kink.
To obtain the statistics of kink formation, the radial quench cycle is repeated up to 4,400 times for each quench time and the density of defects d is defined as the average number of kinks per quench cycle. We measure the kink density as a function of the quench time τ_{Q} at a fixed axial frequency ν_{x}/2π=24.6±0.5 kHz.
The kink densities obtained in our experiment range from 0.01 to 0.24, that is, from one kink, for every 100 quenches, to another kink for every 4–5 quenches. The ultrahigh vacuum with a pressure of 5 × 10^{−9} Pa assures a low rate of spontaneous kink formation due to melting and recrystallization after background gas collisions. We experimentally determined the rate as one kink in 67 s of observation time in the ‘zigzag’ configuration. This allows the detection of kinks created in a controlled quench down to a minimum probability of 6 × 10^{−4}, corresponding to ln(d)=−7.4. Collisions also limit the lifetime of our kinks that were experimentally determined to be 1.6 s (see Supplementary Fig. S1).
Kink dynamics simulations and analysis of observed scaling
In finite systems, losses can occur not only when pairs of kinks annihilate but also when kinks move out of the ‘zigzag’ region. The Peierls–Nabarro potential of the initially created odd kinks builds up during the quench from the centre and decreases to zero towards the outer linear parts of the crystal. The mobility of kinks is determined by the depth of this potential in comparison with their kinetic energy, which is gradually dissipated by the phononic coupling to the lasercooled crystal at a rate quantified by the friction parameter η. Once a kink is able to change sites, it experiences an outward acceleration due to the inhomogeneity of the potential, enhancing the losses.
We performed numerical simulations to investigate the dynamics that can lead to losses of kinks before they can be detected experimentally. As a result, we have identified two regimes: (1) for very slow quenches, losses occur due to the fact that defects escape into the linear region before the confining potential has built up sufficiently, and (2) for fast quenches, an enhanced amount of kinetic energy is introduced into the system that allows defects to leave the ‘zigzag’ region even after the quench has completed.
To separate the effect of these losses from the density of kinks predicted by KZM, the time evolution of the number of defects for each quench cycle is determined in our simulations. We compared the absolute density of defects created during one cycle with the density of defects that remained after the ion motion has been damped out by the optical cooling forces (depending on the value of η, that is, at 350–500 μs after the radial quench). After this time, a constant number of defects is observed in the simulations, defining the density of stable kinks that are detectable in our experiment.
Figure 4a shows the simulation results for the created and experimentally detectable kink densities for different values of η. There is a region with negligible losses from ≈−2 to ≈−2.5, where the experimental detection allows us to directly observe the density of created kinks. This is the region we use to compare the scaling of the measured kink density with the KZM predictions. The most important aspect regarding KZM tests, seen in Fig. 4a, is the independence of the density of created kinks (filled symbols) from the friction coefficient η. This proves that the phase transition happens in the underdamped regime, with welldefined dynamical critical exponent, z=1, see del Campo et al.^{4}
In Fig. 4b, the experimentally measured densities are shown, compared with simulated results for various values of η. A power law fit to the experimental data in the region from ln[1/(ν_{x}τ_{Q})]=−1.9 to −2.6 yields a scaling of σ=2.7±0.3. This is in good agreement with the value of σ=2.63±0.13 obtained in a fit to the numerical data for the amount of kinks created (filled symbols in Fig. 4a) from ln[1/(ν_{x}τ_{Q})]=−2 to −3.2. The matching amount of losses at fast quenches is consistent with the assumed friction coefficient in the simulations. The data were taken over months with a repeatability indicated by the error bars.
For quench times ln[1/(ν_{x}τ_{Q})]<−1.7, the probability to produce more than one kink per ion chain becomes negligible. This corresponds to the regime in which the scaling coefficient is doubled to σ=8/3≈2.67, which is in agreement with the observed values in both experiment and simulations.
Discussion
We have demonstrated kink creation in the linearto‘zigzag’ structural phase transition in ion Coulomb crystals and studied its dependence on the quench rate of the trapping potential in a regime in which a maximum of one defect is created. In this regime, we have observed a doubling of the inhomogeneous scaling. The observed defect creation in our harmonic ion trap shows the influence of inhomogeneities in phase transitions, where the role of causality is enhanced and the formation of defects is suppressed by orders of magnitude compared with homogeneous systems.
The experimentally accessible regime could be extended by enlarging the size of the crystal and by increasing the friction due to laser cooling. In this way, future experiments with Coulomb crystals may be able to measure the scaling in the other two regimes shown in Fig. 2. Accessing the inhomogeneous KZM regime requires a larger system size in which it is easier to create more than one kink per chain. In order to avoid longrange interactions between defects, one could exploit the properties of the more localized odd kinks. Calculations of the PeierlsNabarro potential of the different types of kinks will allow identification of optimum ion numbers, trap parameters to reduce losses and enable stable trapping of multiple kinks^{28,29}. Experimental limitations in the system size can be overcome by using, for example, cryogenic ion traps that allow storage of large Coulomb crystals with long lifetimes^{30}. The homogeneous KZM scaling could be tested with ions trapped in ring trap geometries^{31} or linear ion traps with anharmonic axial confinement^{32}.
Our results open up ways to explore the various properties of discrete nonlinear stable excitations, which have attracted considerable theoretical and experimental effort^{33}. In such systems, the often counterintuitive nonlinear physics of solitons, breathers and their thermal fluctuations becomes accessible. Furthermore, to reach the onset of quantum effects in the phase transition, temperatures of a few tens of microkelvin, lowheating rates on the order of a hertz and possibly the use of a lighter ion species are required^{34,35}. By implementing recently demonstrated advanced lasercooling methods to prepare chains of ions^{36,37,38} in the groundstate and ion traps with subhertz heating rates^{39}, this work could be extended to study phase transitions in the quantum regime^{40}.
During the preparation of this manuscript, we became aware of similar results presented in another paper^{41}.
Methods
Atomic system and ion trap technology
We lasercool ^{172}Yb^{+} ions on the ^{2}S_{1/2}–^{2}P_{1/2} transition at a wavelength of 370 nm to temperatures of a few mK, slightly above the Doppler cooling limit^{42} of T=0.5 mK for this transition. Two repump lasers at wavelengths 935 and 638 nm deplete the populations of higher lying metastable states that are coupled to the cooling cycle via radiative decay or collisions. A single coil on top of the vacuum chamber produces a quantization field of about B=0.2 mT at the ion position to optimize the repumping efficiency of the 935 nm laser. A detailed description of the experimental setup can be found in Pyka et al.^{26}
Laser cooling the ions via a single travelling wave leads to a friction coefficient of the optical cooling force given by at Δ=−Γ/2 (see Metcalf and van der Straten^{42}), where s is the saturation of the cooling transition, k is the wave vector of the cooling laser and Δ is the detuning of the laser with respect to the resonance of the transition with line width Γ. The transverse beam profile is focused down to horizontal and vertical waist sizes ω_{x}=8.8 mm and ω_{y}=80 μm. For a laser power of 630 μW, this results in an estimated experimental friction coefficient of η_{exp}=4 × 10^{−21} kg s^{−1} along a single axis. The simulations that have been carried out apply a friction parameter η_{sim} to each axis and show the most agreement with the experiment for η_{sim}=(2.5…3.0) × 10^{−21} kg s^{−1}.
The ion trap is a threedimensional segmented linear radio frequency (rf) Paul trap, which was designed for highprecision spectroscopy on linear ion crystals. It offers full control to compensate stray fields in three dimensions and, in particular, has low axial micromotion over a large range of several hundreds of micrometres^{26}. The ions are trapped in a loading segment via photoionization and then shuttled to a spectroscopy segment that is protected from the atomic beam. Avoiding contamination of the electrodes makes it possible to perform measurements with highly reproducible experimental parameters. The axial and radial secular frequencies in this segment were measured repeatedly, and a maximum deviation of σ_{ax}=0.5 kHz and σ_{rad}=1 kHz over several weeks was observed. For a linear crystal of about 30 ions with length l~300 μm, the maximum axial rf field component along the crystal is as low as 500 V m^{−1}, whereas the radial field does not exceed 200 V m^{−1} corresponding to a total micromotion amplitude of only x_{mm}=12 nm.
The ion crystal is imaged with a selfbuilt lens system optimized for minimum aberrations onto an electron multiplying CCD camera with a magnification of M=28 and an experimentally estimated resolution of about 1.5 μm over the full ion chain length of 300 μm. At the given axial trap frequency, about 30 ions can be imaged in the zigzag configuration.
Radial quench cycle
The rf voltage driving the radial confinement is amplitude modulated by a control signal using an rf mixer. Its actual amplitude on the ion trap electrodes is monitored by a short antenna inside the helical resonator (U_{mon}), the impedance of which matches the amplified rf voltage U_{rf} to the trap capacitance^{26}. By observing parametric heating of the ions, we can measure the radial as well as axial secular frequencies directly as a function of the trap voltages with a relative precision of 10^{−3}.
To drive quenches of the radial confinement, a linear function, shown schematically in Supplementary Fig. S2, is applied as a control signal to the rf mixer. Owing to the characteristics of the mixer, the measured slope of the ramp deviates from the ideal slope, and therefore is fitted to the monitored signal U_{mon} (see Supplementary Fig. S3) to obtain the effective quench time used in the analysis as shown in Fig. 4b. The characteristic time constant of the resonant circuit of 7 μs limits the fastest possible quench time, but its effect on the ramps used in this work is negligible.
The ramp has also been tested for linearity in trap frequency by verifying that a linear fit to the ramp near the point of the linear zigzag transition deviates by <2% from an ideal linear function between the frequencies measured at the extreme values of the ramp, see Supplementary Fig. S4.
Data analysis and statistics
For each quench time, up to 4,400 pictures are taken in several series of measurements over 3 months. Accounting for this, the kink density for each quench time is calculated as the mean of the densities from the individual series for that time weighted by the number of pictures per series. The first and the biggest contribution to the error bars is the scatter of the individual series at a given quench time, calculated as sample s.d. The second and the independent contribution accounts for statistical uncertainty of the data, assuming a binomial distribution.
Numerical simulations
For Doppler cooled ions, the energy distribution of the ion chain obeys the Fokker–Planck distribution^{43}. Hence, the dynamics of the jth ion can be described by the Langevin equation:
where is the ponderomotive trapping potential, is the Coulomb potential energy, η is the friction coefficient due to laser cooling and ε_{j}(t) is the stochastic force. The amplitude of the stochastic force, ε_{j}(t), is related to the temperature T and the friction coefficient η by the fluctuationdissipation relation ‹ε_{αj}(t)ε_{βk}(t′)›=2η k_{B}Tδ_{αβ}δ_{ij}δ(t−t^{′}), with α, β=x, y, z. Collisions with background particles are not included in this model.
To simulate the quench from a lineartozigzag chain, the radial trapping frequencies, and , decrease linearly with time at rates corresponding to the experimentally measured values. The stochastic differential equation (1) was numerically integrated between 1,500 and 3,000 times per quench time τ_{Q} in order to achieve a low statistical uncertainty.
Additional information
How to cite this article: Pyka, K. et al. Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals. Nat. Commun. 4:2291 doi: 10.1038/ncomms3291 (2013).
References
 1
Kibble, T. W. B. Topology of cosmic domains and strings. J. Phys. A: Math. Gen. 9, 1387–1398 (1976).
 2
Zurek, W. H. Cosmological experiments in superfluid helium? Nature 317, 505–508 (1985).
 3
Zurek, W. H. Causality in condensates: gray solitons as relics of BEC formation. Phys. Rev. Lett. 102, 105702 (2009).
 4
del Campo, A., de Chiara, G., Morigi, G., Plenio, M. B. & Retzker, A. Structural defects in ion chains by quenching the external potential: the inhomogeneous KibbleZurek mechanism. Phys. Rev. Lett. 105, 075701 (2010).
 5
Monaco, R., Mygind, J., Rivers, R. J. & Koshelets, V. P. Spontaneous fluxoid formation in superconducting loops. Phys. Rev. B 80, 180501(R) (2009).
 6
Griffin, S. M. et al. Scaling behavior and beyond in the hexagonal manganites. Phys. Rev X 2, 041022 (2012).
 7
Bowick, M. J., Chandar, L., Schiff, E. A. & Srivastava, A. M. The cosmological Kibble mechanism in the laboratory: string formation in liquid crystals. Science 263, 943–945 (1994).
 8
Chuang, I., Durrer, R., Turok, N. & Yurke, B. Cosmology in the laboratory: defect dynamics in liquid crystals. Science 251, 1336–1342 (1991).
 9
Weiler, C. N. et al. Spontaneous vortices in the formation of BoseEinstein condensates. Nature 455, 948–951 (2008).
 10
Sadler, L. E., Higbie, J. M., Leslie, S. R., Vengalattore, M. & StamperKurn, D. M. Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate. Nature 443, 312–315 (2006).
 11
Dziarmaga, J., Laguna, P. & Zurek, W. H. Symmetry breaking with a slant: topological defects after an inhomogeneous quench. Phys. Rev. Lett. 82, 4749–4752 (1999).
 12
Kibble, T. W. B. & Volovik, G. E. On phase ordering behind the propagating front of a secondorder transition. JETP Lett. 65, 102–107 (1997).
 13
del Campo, Retzker, A. & Plenio, M. B. The inhomogeneous Kibble–Zurek mechanism: vortex nucleation during BoseEinstein condensation. New J. Phys. 13, 083022 (2011).
 14
Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012).
 15
Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590–593 (2010).
 16
Friedenauer, A., Schmitz, H., Glueckert, J., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nat. Phys. 4, 757–761 (2008).
 17
Monroe, C., Meekhof, D. M., King, B. E., Itano, W. M. & Wineland, D. J. Demonstration of a fundamental quantum logic gate. Phys. Rev. Lett. 75, 4714–4717 (1995).
 18
Häffner, H. et al. Scalable multiparticle entanglement of trapped ions. Nature 438, 643–646 (2005).
 19
Fishman, S., de Chiara, G., Calarco, T. & Morigi, G. Structural phase transitions in low dimensional ion crystals. Phys. Rev. B 77, 064111 (2008).
 20
Landa, H., Marcovitch, S., Retzker, A., Plenio, M. B. & Reznik, B. Quantum coherence of discrete kink solitons in ion traps. Phys. Rev. Lett. 104, 043004 (2010).
 21
Zurek, W. H., Bettencourt, L. M. A., Dziamarga, J. & Antunes, N. D. Shards of broken symmetry: topological defects as traces of the phase transition dynamics. Acta. Phys. Polon. B 31, 2937–2962 (2000).
 22
Saito, H., Kawaguchi, Y. & Ueda, M. KibbleZurek mechanism in a quenched ferromagnetic BoseEinstein condensate. Phys. Rev. A 76, 043613 (2007).
 23
Dziarmaga, J., Meisner, J. & Zurek, W. H. Winding up of the wavefunction phase by an insulatortosuperfluid transition in a ring of coupled BoseEinstein condensates. Phys. Rev. Lett. 101, 115701 (2008).
 24
Schneider, C., Porras, D. & Schaetz, T. Experimental quantum simulations of manybody physics with trapped ions. Rep. Prog. Phys. 75, 024401 (2012).
 25
Mielenz, M. et al. Trapping of topologicalstructural defects in Coulomb crystals. Phys. Rev. Lett. 110, 133004 (2013).
 26
Pyka, K., Herschbach, N., Keller, J. & Mehlstäubler, T. E. A highprecision segmented Paul trap with minimized micromotion for an optical multipleion clock. Appl. Phys. B doi: 10.1007/s0034001355805 (2013).
 27
Braun, O. M. & Kivshar, Y. S. The FrenkelKontorova Model: Concepts, Methods, and Applications SpringerVerlag (2004).
 28
Partner, H. L. et al. Dynamics of topological defects in ion Coulomb crystals. Preprint at http://arxiv.org/abs/1305.6773 (2013).
 29
Landa, H., Brox, J., Mielenz, M., Schaetz, T. & Reznik, B. Structure, dynamics and bifurcations of discrete solitons in trapped ion crystals. Preprint at http://arxiv.org/abs/1305.6754 (2013).
 30
Poitzsch, M. E., Bergquist, J. C., Itano, W. M. & Wineland, D. J. Cryogenic linear ion trap for accurate spectroscopy. Rev. Sci. Inst. 67, 129–134 (1996).
 31
Birkl, G., Kassner, S. & Walther, H. Multipleshell structures of lasercooled ^{24}Mg^{+} ions in a quadrupole storage ring. Nature 357, 310–313 (1992).
 32
Lin, G. D. et al. Largescale quantum computation in an anharmonic linear ion trap. Europhys. Lett. 86, 60004 (2009).
 33
Sato, M., Hubbard, B. E. & Sievers, A. J. Nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev. Mod. Phys. 78, 137–157 (2006).
 34
Silvi, P., De Chiara, G., Calarco, T., Morigi, G. & Montangero, S. Full characterization of the quantum linearzigzag transition in atomic chains. Preprint at http://arxiv.org/abs/1301.3386 (2013).
 35
Shimshoni, E., Morigi, G. & Fishman, S. Quantum structural phase transition in chains of interacting atoms. Phys. Rev. A 83, 032308 (2011).
 36
Lin, Y. et al. Sympathetic electromagneticallyinducedtransparency laser cooling of motional modes in an ion chain. Phys. Rev. Lett. 110, 153002 (2013).
 37
Morigi, G., Eschner, J. & Keitel, C. H. Ground state laser cooling using electromagnetically induced transparency. Phys. Rev. Lett. 85, 4458–4461 (2000).
 38
Morigi, G. Cooling atomic motion with quantum interference. Phys. Rev. A 67, 033402 (2003).
 39
Poulsen, G., Miroshnychenko, Y. & Drewsen, M. Efficient groundstate cooling of an ion in a large roomtemperature linear Paul trap with a subHertz heating rate. Phys. Rev. A 86, 051402 (2012).
 40
Shimshoni, E., Morigi, G. & Fishman, S. Quantum zigzag transition in ion chains. Phys. Rev. Lett. 106, 010401 (2011).
 41
Ulm, S. et al. Observation of the Kibble–Zurek scaling law for defect formation in ion crystals. Nat. Commun. 4, 2290 (2013).
 42
Metcalf, H. J. & van der Straten, P. Laser Cooling and Trapping Springer (1999).
 43
Morigi, G. & Eschner, J. Doppler cooling of a Coulomb crystal. Phys. Rev. A 64, 063407 (2001).
Acknowledgements
We thank B. Damski and R. Rivers for suggestions and comments on the manuscript, L. Yi for his contributions to the detection software, E. Passemar for providing statistics codes and K. Thirumalai for assistance in the lab. This work was supported by NSF PHY1125915, the United States Department of Energy through the LANL/LDRD Program and a LANL J. Robert Oppenheimer fellowship (A.d.C.), the EU STREP PICC, the Alexander von Humboldt Foundation (M.B.P.), Career Integration Grant (CIG) no. 321798 (A.R.), by EPSRC (R.N.) and by DFG through QUEST. A.d.C. and W.H.Z. are grateful to KITP for hospitality.
Author information
Affiliations
Contributions
The experiment was initiated and led by T.E.M. based on the theory developed by A.d.C., M.B.P. and A.R. in del Campo et al.^{4} R.N. developed the numerical codes and carried out the numerical work with advice from M.B.P. and A.R. A.d.C., W.H.Z., R.N. and A.R. worked on the theory for the extended KZM scaling. T.E.M., K.P. and J.K. designed the experiment with input from R.N. T.B., H.L.P., J.K. and K.P. carried out the experiment. K.P., J.K., D.M.M. and K.K. performed data analysis and carried out numerical simulations. All authors contributed to the discussion of results and participated in manuscript preparation.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Figures, Notes and References
Supplementary Figures S1S5, Supplementary Notes 12 and Supplementary References (PDF 856 kb)
Supplementary Movie 1
Odd kink formation. Probabilistic creation of odd kinks is shown in real time by repeating a 70 μs rf ramp on a chain of 29 ions. The ramp back to the initial state is stretched to 0.5 s to illustrate the transition. Stable odd kinks are visible at times 00:05, 00:20, and 01:03, and an unstable kink forms and disappears at 00:30. (MOV 23838 kb)
Supplementary Movie 2
Extended kink formation. Probabilistic creation of extended kinks is shown in real time by repeating a 70 μs rf ramp on a chain of 30 ions. The ramp back to the initial state is stretched to 0.5 s to illustrate the transition. Stable extended kinks are visible at times 00:08, 00:17, 00:49, and 0:54, and an unstable kink forms and disappears at 00:33. (MOV 11443 kb)
Supplementary Movie 3
Extended kink forming from an odd kink during a simulated quench. First, an odd kink is created at the center of the ion chain. As the ramp further reduces the axial potential, the odd kink transforms into an extended kink. The quench time is = 100μs. (MOV 434 kb)
Rights and permissions
About this article
Cite this article
Pyka, K., Keller, J., Partner, H. et al. Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals. Nat Commun 4, 2291 (2013). https://doi.org/10.1038/ncomms3291
Received:
Accepted:
Published:
Further reading

Twodimensional supersolidity in a dipolar quantum gas
Nature (2021)

Experimentally testing quantum critical dynamics beyond the Kibble–Zurek mechanism
Communications Physics (2020)

Kibble–Zurek universality in a strongly interacting Fermi superfluid
Nature Physics (2019)

Dynamical equilibration across a quenched phase transition in a trapped quantum gas
Communications Physics (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.