Physica B 460 (2015) 105–113
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Physica B journal homepage: www.elsevier.com/locate/physb
Ion Coulomb crystals Michael Drewsen Aarhus University, Department of Physics and Astronomy, Ny Munkegade 120, DK8000 Aarhus C, Denmark
art ic l e i nf o
a b s t r a c t
Available online 24 November 2014
The following text will give a brief introduction to the physics of the spatially ordered structures, socalled Coulomb crystals, that appear when conﬁned ions are cooled to sufﬁciently low temperatures. It will as well brieﬂy comment on the very diverse scientiﬁc applications of such crystals, which have emerged the past two decades. While this document lacks ﬁgures, it includes a substantial number of references in which more detailed information can be found. It is the hope that the text will stimulate the reader to dig deeper into one or more of the discussed subjects and inspire her/him to think about new potential applications. & 2014 Elsevier B.V. All rights reserved.
Keywords: Coulomb crystals Ion trapping Ion cooling
1. Introduction Coulomb crystals constitute a special class of spatial ordered structures of matter. While usual solid matter ranging from simple molecules over clusters to solids with long range ordered structures form through the quantum delocalized nature of electrons and hence are governed by the laws of quantum mechanics, Coulomb crystals are exclusively based on classical electromagnetic interactions between conﬁned charged particles of the same sign of charge, and hence relying only on the classical laws of physics. The ﬁrst elaborated considerations of Coulomb crystals was made by Wigner in the 1930s , where he was interested in understanding the change in the electronic structure and properties of matter when the density of electrons become so low that despite the rather large DeBroglie wavelength of electrons, their individual wavefunctions does not overlap. Under such conditions Wigner suggested the formation of regularly ordered structures with each electron taking up a particular position in space. Wigner never managed to realize such crystals of electrons, but around 1990, Coulomb crystallization of two-dimensional electron gases was demonstrated within GaAs/GaAlAs quantum well structures . Prior to these experiments, Wigner crystals had been produced on the surface of super ﬂuid helium [3,4]. Three-dimensional Coulomb crystals of electrons have though still not been realized. When considering Coulomb crystals made up of electrons, such structures are also often referred to as Wigner-crystals. Coulomb crystals can as well be composed of positively charged particles such as atomic ions at various charge stages. For decades people within the astrophysical community have been considerably interested in such objects. This is due to the fact that E-mail address: [email protected]
http://dx.doi.org/10.1016/j.physb.2014.11.050 0921-4526/& 2014 Elsevier B.V. All rights reserved.
when a star dies, it will eventually turn into one of two very exotic stellar objects named White Dwarfs and Neuron Stars, which to various extents are assumed to incorporate Coulomb crystallized matters . In case the mass of the original star is below a certain critical value after the nuclear burning processes have terminated, the remaining parts of the star will contract under the inﬂuence of gravity until the point where all the electrons form a degenerated gas which will be able to produce an equalizing quantum pressure. Such a remnant star with a radius of typically a few hundreds of kilometers will mainly consist of fully stripped oxygen and carbon atoms embedded in the degenerated electron gas. When eventually the so-called White Dwarf cools down by thermal radiation to a temperature, which becomes lower than the very high Fermienergy of such a dense electron gas (typically equivalent to millions of kelvins), quantum mechanics prohibits collisional energy exchanges between the ions and electrons. The ions can though still exchange energy through their Coulomb interactions within the conﬁning potential created by the degenerated electron gas, and eventually at sufﬁciently low temperatures, Coulomb crystals of the fully stripped ions will be formed. In the case of initially more massive stars, the quantum pressure of the degenerated electron gas cannot counteract the gravitational contracting forces, and eventually most of the material of the star will be turned into a degenerated gas of neutrons which then provide a quantum pressure that can counteract a gravitational collapse. This situation leads to a so-called neutron star of the size of only few tens of kilometers in diameters. While nuclear burning in White Dwarfs typically stops with the formation of mainly oxygen and carbon, in Neutron Stars fusion processes continues until the formation of iron after which no further energy can be released through fusion. While the main part of neutron stars are made up of neutrons, it is believed that their crusts will
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still have fully stripped iron ions Fe26 þ present, which even under the extreme high density conditions still is believed to form ion Coulomb crystals! Due to the required conditions for formation of Coulomb crystals of ions, as will be revisited in the next chapter, it was ﬁrst with the development of laser cooling techniques in the late 1970s and early 1980s that it became possible to form ion Coulomb crystals in laboratories. However, since these ﬁrst demonstrations by the groups of Dr. Wineland, NIST, Boulder, USA , and Walther, Max-Planck Institute for Quantum Optics, Garching, Germany , the number of groups and the breath of scientiﬁc investigations based on ion Coulomb crystals created in laboratory have went through an amazing development. The rest of this contribution is organized as follows: In Section 2 a discussion of the condition for Coulomb crystallization will be provided, and in Section 3 how these requirements can be met in the case of positively charged ions will be presented. Structural properties of Coulomb crystals of various dimensionalities will be the focus of Section 4 with examples in the form of crystals of atomic ion species. In Section 5, some of the dynamical properties of Coulomb crystals will be brieﬂy discussed. Many applications of ion Coulomb crystals in a large variety of scientiﬁc investigations will be the theme of Section 6. Section 7 will ﬁnally conclude the presentation.
2. Conditions for Coulomb crystallization The condition for Coulomb crystallization is often formulated in terms of the so-called plasma coupling parameter, Γ, which is deﬁned in the following way:
Q2 . 4π ϵ 0akBT
Here Q is the charge of the particle involved, a is the Wigner–Seitz radius, corresponding to the particle density, and T is the temperature of the particles. Essentially, Γ is just a measure of the ratio of the inter-particle Coulomb energy to their kinetic energy:
ECoul . Ekin
For inﬁnitely large three-dimensional identical charged particle systems, often also referred to as one-component plasmas (OCPs), it can be shown that many of the thermodynamical properties scale with the value of Γ . With respect to Coulomb crystallization, it has been proven by several groups (see, e.g., [9,10]) that the criterion is given by
Γ ≥ 175
or, as can be shown by introducing the local single ion oscillation frequency, or similarly the plasma frequency ωplas :
nQ2 ϵ 0M
a ≥ 104 ×
where Δrexc is the root-mean-square single particle excursion around its equilibrium position due to its kinetic energy. This inequality is very similar to the Lindemann criterion  for usual solids with respect to the motion of the atomic nuclei. For valence electrons in usual solids, even at the absolute zerotemperature, the quantum mechanical excursion of the individual
h2ϵ 0 Q2M
where M is the mass of the charged particles. For exotic stellar objects like White Dwarfs with interior ions being mainly C6 þ and O8 þ , and neutron star crusts with Fe26 þ ions, the critical Wigner–Seitz radii for quantum melting become a ∼ 10−12 m and a ∼ 10−14 m , respectively. Surprisingly, despite the high density of such objects, quantum melting is not a problem, and classical Coulomb crystallization happens already at temperature in the million-kelvin range ! In contrast to short-range interacting particles leading to closed packed crystal lattices, the ground state of a inﬁnite Coulomb crystal is a body-centered cubic (bcc) structure . Hexagonal closed packed (hcp) and face centered cubic (fcc) structures correspond, however, only to slightly excited states . In the case of effective harmonic conﬁnement potentials, it is straight forward through the application of Poisson's equation to prove that ﬁnite size one-component systems will also have constant particle density at low temperatures. For such ﬁnite systems, one ﬁnds that the condition for reaching Coulomb crystallization, deﬁned as the temperature at which there is a maximum in the speciﬁc heat, is ﬁrst reached at larger values of Γ, or equivalently at lower temperatures. For instance, for isotropic conﬁned systems ranging from ∼100 to ∼10000 particles, the corresponding critical values of Γ varies from ∼500 to ∼200 . So far, criterions for crystallization under various anisotropic conﬁnement conditions have not been studied in detail, but a variation in the values of Γ can be expected even for the same number of ions due to changes in the surface effects.
3. Ion Coulomb crystals in laboratories By standard ion trapping techniques, either in the form of Penning traps or radiofrequency (rf) traps also named Paul traps , the highest practical reachable particle density of singly charged ions is ∼1015 m 3, corresponding to a∼10 μm. With this value of a, inequality (3) and the deﬁnition of Γ in Eq. (1) lead to the following requirement in terms of temperature to obtain Coulomb crystallization:
T ≤ TCC ∼ 10 mK
Consequently, before the advent of laser cooling techniques [15,16] it was impossible to reach the condition for ion Coulomb crystallization, since until then all other cooling techniques were limited by the temperature of liquid helium, and hence typically to temperatures in the range of a few kelvins. On the other hand, for standard laser Doppler cooling the minimum temperature is effectively given by 
by the condition:
Δrexc/a < 0.1
electrons makes it impossible to fulﬁll inequality (5), and delocalizing electrons leading to speciﬁc energy band structures emerge as a consequence. More generally, one can derive a rather simple approximate condition for this ‘quantum melting’ to happen at absolute zero temperature (unpublished):
=Γcool ∼ 0.1 − 1 mK, kB
where Γcool is the spectral width of the cooling transition. This temperature is clearly below TCC , and even low enough to create crystals of the smallest ﬁnite systems. The production of ion Coulomb crystals through laser cooling of ions were pioneered by the groups of David Wineland (NIST, Boulder, Colorado, USA) and Prof. Walther (MPQ, Garching, Germany) in the 1980s, and the past decades an ever increasing
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number of groups world wide have taken part in the further development, exploration and application of these exotic crystalline structures. An interesting issue concerning Coulomb crystallization in both Penning and rf traps is the fact that in both cases the crystallized ions in addition to their random thermal motion perform synchronized driven motion, either in the form of rotation (Penning traps) or quadrupolar oscillations (rf traps). Interestingly, the corresponding averaged kinetic energy can easily exceeds the thermal energy by 104–106 orders of magnitude!
4. Structural properties. (1D, 2D and 3D structures) While the lowest energy state of inﬁnite constant density Coulomb crystals is a bcc structure [18,9], the structures of ﬁnite size crystals depend strongly on the number of ions as well as the shape of the trapping potential. Even in the case of trapping conditions leading to constant density (i.e., three-dimensional harmonic trapping potentials), three-dimensional crystals can have many different features. For the isotropic conﬁnement case, where the conﬁning force can be written in terms of a single force constant κ:
Ftrap(r) = − κr ,
the over-all structure of the Coulomb crystals must be spherical due to spherical symmetry. However, what is not obvious, is the fact that for crystals (or more correctly clusters) containing up to a few thousand ions, the total potential energy is minimized by having the ions organized in concentric shells with a speciﬁc “magic” numbers of ions  as the temperature goes to zero. Within each shell the ions furthermore form closed two-dimensional structures, which to a high degree is like a hexagonal lattice, the minimum energy state for an inﬁnite two-dimensional planar crystal . Already with more that a few ions present, it is essentially impossible to predict the ground state conﬁguration, and Monte Carlo or Molecular Dynamics (MD) simulations have to be introduced to ﬁnd those . However, since it is well known from cluster physics that the number of metastable conﬁgurations increases roughly exponentially with the number of particles forming the cluster, it soon becomes extremely difﬁcult to ensure that a particular simulation have reached the minimum energetic state. The uncertainty in the predicted magic numbers of ions from MD simulations within each shell is though so small that for a given spherical shell structure, the number of ions can be determined with an uncertainty much smaller than the square-root of the total number of ions . Hence, comparing MD simulations with projection images of crystals recorded in the laboratories constitutes a reliable method to obtain information on the number of ions as well as the ion density in experiments. Since the ions will always have a ﬁnite thermal energy in the laboratories, it is indeed possible also to observe the formation of energetically excited metastable conﬁgurations which may even have extended long range ordered structures with bcc or fcc character at ion numbers much smaller than ∼10000, expected from theoretical investigations to be the minimum number in order to have such structures as the ground states . Sometimes crystals composed of domains of both fcc and bcc structures can even appear . In another special trapping situation, where the ions are conﬁned by an isotropic harmonic potential in two dimensions while being freely to move along the third perpendicular direction, concentric cylindrically shell structures represent the minimum
energy states. In this situation, the number of shells depends rather on the linear density of particles along the unconﬁned axis than on the total number of ions. In the laboratories this speciﬁc situation can effectively be realized by choosing a ring shaped two-dimensional rf quadrupole trap as has been pioneered by Prof. Walther, MPQ , or by letting the ion species of interest (the low mass species) be part of a two-species crystal in a standard linear rf quadrupole trap . In the ﬁrst case the ring shaped conﬁnement will lead to an effective periodic boundary condition for the ions along the ring, which will not deviate substantially from the inﬁnitely long 2D conﬁnement situation, when the circumference is much larger than the typical ion–ion distance and diameter of the crystal. In the two-species case, the heavier ion species will feel an effective conﬁning force which is weaker than for the lighter mass one in the plane with rf ﬁeld conﬁnement, and hence the lighter mass ions will segregate within a cylindrical volume along the rf ﬁeld free axis . Empirically, it has been shown that the shell structure of this inner component of the two-species crystal to a very high degree mimics the inﬁnite case due to the effective electrical shielding by the surrounding heavier species . In an extreme case a string of 43 equidistantly spaced 24Mg + ions was situated within a crystal of ∼300040Ca + ions . In the more general case of harmonic conﬁnement with an isotropic potential in two dimensions, but a different force constant in the third one (κz ≠ κx = κy) the equipotential surfaces are described by concentric spheroids. While a weakly interacting gas of ions in thermal equilibrium would have a spatial distribution with contour surfaces of the density corresponding to the equipotential surfaces of the conﬁning potential, this is not true for Coulomb crystallized ion ensembles [24–26]. In cases where the number of ions in a Coulomb crystal is so large that the extension in all three dimensions exceeds signiﬁcantly the spacing of the individual ions, it can be expected that the shape of the crystal should be very similar to the case of a conﬁned cold charged ﬂuid with a constant charge density. For this latter situation it is indeed possible to ﬁnd an analytic expression for the outer spheroidal shape of such a ﬂuid in a potential with spheroidal equipotential surfaces . However, due to the Coulomb repulsion, the shapes deviate from the equipotential surfaces of the trapping potential. In a series of experiments, it has been veriﬁed that the outer contour of Coulomb crystals indeed follow the same shape as the cold charged liquid model, except at extreme conﬁnement cases leading to the formation of real 2D (pancake) or 1D (string) structures [24,25]. For the number of ions reaching up to a few thousand, the ions will as in the isotropic conﬁning case furthermore be organized in concentric spheroidal shells [28,25]. The 2D planar structures [20,29,30] and 1D string structures [20,31,32] constitute special cases, where the liquid model does not sufﬁce to describe the shape. Here, different approaches including MD simulations have to be used to obtain more detailed structural information. Both in theory and by experiments it has been shown that under speciﬁc trapping conditions, Coulomb crystals can have negative Poisson coefﬁcients, meaning the crystal when stretched along one direction, it will lead to a lateral expansion . Another interesting situation is the one where two ion species with the same charge-to-mass ratio is simultaneously crystallized in, e.g., a linear rf trap. In this situation both ion species feel the same effective potential and a priori could be expected to mix. On a gross scale this is also true according to MD simulations, and for large systems one indeed ﬁnds a simple cubic lattice (as in CsCl salt) as the minimum energy state . For smaller ﬁnite systems of a few thousand ions, common shell structures like for a single species is formed, however, with the common shells consisting of slightly displaced sub-shells of the two species . It was also
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found by MD simulations that the degree of mixing is very sensitive to the exact values of the charge-to-mass ratios of the two ion species. A difference of only one per mille is essentially enough to observe at least partial segregation of the two species . As we will discuss brieﬂy in Section 6, such segregation effects may be studied in details by experimenting with Highly Charged Ion (HCI), and may turn out to have relevance for the composition of Coulomb crystals within White Dwarfs, where 16O8 + and 12C 6 + ions, which have nearly the same charge-to-mass ratio, are present. Coulomb crystals with non-constant particle densities can as well be created and studied through the application of higher order pole rf traps, like hexapole and octopole traps. Recently, Coulomb crystallization of ions in particular octupole traps have been considered both theoretically [35–37] and experimentally [38,39]. In such higher order traps the ground state conﬁguration of the ions will never be a single string, but rather ring/tube-like structures  or multi-stringlike structures  depending on the trapping conditions. For few ions where near hexagonal lattice-like pancake structures appears for 2D systems under rf quadrupole conﬁnement, in higher order traps, rings of ions are instead formed. These new types of structure may, for instance, ﬁnd applications within the ﬁeld of quantum simulations (see Section 6).
5. Dynamical properties Like usual solids, Coulomb crystals have a series of vibrational modes (phonon modes). In general, if a Coulomb crystal consists of N ions then such a crystal will have 3N normal modes with corresponding distinct eigenmode frequencies . For smaller crystals, the individual modes can be found either through direct computations  or through the analysis of the motions of ions in MD simulations after the introduction of a motional excitation . For larger spheroidal crystals which may in real experiments actually be found to be a series of isomeric conﬁguration with each their particular mode pattern and spectrum, a simpler way to gain information on mode structures and frequencies is once again to approximate the real granular Coulomb crystals by a cold charged liquids with the same shape and size, and ﬁnd such object's eigenfrequencies and eigenmodes. As has been shown by Prof. Dan Dubin, UCSD, USA, the modes of such cylindrical symmetric objects can be characterized by two quantum numbers (l,m), where m describes the azimuthal part of the eigenmode, and l the longitudinal part with l-1 indicating the number of nodal planes along the symmetry axis . As long as the characteristic length scale for the (l,m)-modes exceeds signiﬁcantly the ion–ion spacing within the Coulomb crystals, these modes may provide a good description of the real crystal modes. Experiments conducted both in Penning  and linear rf traps  have in fact proved that the modes like the (l, m)-modes can be excited in ion Coulomb crystals. For highly anisotropic trapping conditions leading to either 1D string structures or 2D pancake-like structures a sub-set of 1D  and 2D  modes can be found more easily. As we shall discuss in the next chapter, various modes of such dimensionally restricted crystals have found applications within quantum information processing as well as quantum simulation.
6. Applications As should be clear from this section, the applications of Coulomb crystals in carrying out research beyond the studies of the basic properties of such crystals are extremely broad.
6.1. Non-linear dynamics The fact that the interaction between conﬁned ions by the nature of the Coulomb interaction is non-linear, made already around 1990 the smallest non-trivial ion systems consisting of two ions an interesting platform for the study of chaotic behavior in rf traps [44–47]. The appearance of bifurcation in the motional state of the ions as well as the existence of instability of two-ion crystals within the usual parameter range for single ion stable motion was studied . Later, experiments with larger (∼1000 ions) systems furthermore showed that when the plasma frequency of such crystals reached half the rf drive frequency, parametric excitation would lead to crystal melting . In some cases the parametric resonance could be switched off by induced crystal rotation , which led to lower particle densities and consequently lower plasma frequencies. 6.2. Statistical physics and thermodynamics Coulomb crystals in the form of ion-strings and so-called zigzag conﬁgurations have recently also found applications within statistical physics . For instance, several groups have recently investigated the formation of defects [51,52] (often referred to as kinks) in zig-zag conﬁgurations when formed through fast changes in the trapping parameters leading to a change in the equilibrium conﬁgurations from string to zig-zag structures. Through these experiments, universal scaling laws for the number of defects have been shown to follow predictions of the Kibble–Zurek model in the special inhomogeneous case [53,54]. From a thermodynamics perspective, ion Coulomb crystals constitute as well an interesting “playground”. Recently, e.g., the thermal diffusion of a single impurity ion in a chain of ions has been investigated experimentally , and models to study heat transport in such systems have been proposed . Also recently, it has been proposed to use strings of trapped ions in combination with periodic corrugated potentials in the form of induced dipole forces to study fundamental issues related to friction  as well as quantum phase transitions  in connection with a Coulombversion of the Frenkel–Kontorova model [59,57]. 6.3. Solid-state physics As mentioned already, Coulomb crystals constitute a special class of solids, which make them natural objects for studies of various structural formations and dynamics. In the future, the combination of trapping of ions in harmonic traps with periodic corrugated potentials in the form of standing wave dipole potentials, should make it possible not only to pin speciﬁc metastable structures by addressing selected lattice planes, but as well make it feasible to make controlled changes in the lattice structure, e.g., from a bcc to a fcc structure and vice versa, through only changes in the over-all trap potential . While such experiments have not yet been carried out, ﬁrst experiments of optical trapping and pinning of few ions within the nodal/antinodal planes of an optical standing wave ﬁeld has recently been demonstrated [61–63]. 6.4. Plasma physics By nature Coulomb crystals are the solid-state representation of a one-component plasma (OCP) [8,64], and hence are in this respect an interesting special state of such plasma. By performing experiments in Penning as well as rf traps it becomes indeed possible to study OCPs under both magnetized and non-magnetized conditions [40,65]. By introducing different ion species in the same trap it becomes furthermore possible to study multi-species OCPs in or in the vicinity of Coulomb crystallization.
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6.5. Nuclear fusion
6.9. Cavity QED related experiments
When laser-cooled ions collide with hotter impurity ions within a magnetized plasma in a Penning trap, the correlated collision dynamics resembles to a high degree the processes governing thermal nuclear fusion in dense stellar interiors [66–68]. Hence, in the future, Penning trap experiments may actually be used to model nuclear fusion in hot dense plasmas.
Ion Coulomb crystals are as well extremely interesting objects for the study of cavity QED effects. For a single atomic ion, lasercooled to its quantum mechanically motional ground state of an harmonic trap potential with oscillation frequencies of 0.1–1 MHz, the resulting ground state wavefunction will have a spatial extension of only a few nanometers, and hence much smaller than the wavelength of light in the optical or near optical range (NIR–UV range). Consequently, it is possible to position a single ion very precisely with respect to the nodes and antinodes of a standing wave cavity ﬁeld [80,75,81], and hence control the effective coupling between the ion and the cavity ﬁeld . In the recent past, there has been a series experiments demonstrating and exploiting the strong spatial localization even with ions cooled only to the Doppler limit [80,75,61–63,82,83] Larger Coulomb crystals are as well interesting objects when the goal is the investigation and exploitation of the collective effect in the coupling of atoms to cavity ﬁelds. Recently, collective strong coupling, where the rate of exchange of a collective electronic excitation of ions in a Coulomb crystal with a photon in the cavity ﬁeld exceeds both the excited state decay rate as well as the decay rate of photons out of the cavity, was demonstrated . This is an interesting regime, since it enables the study of the fundamental quantum dynamics of the combined photon-ensemble system without decoherence, as well as open for, e.g., the realization of a quantum memory for light  and novel photon counters . With the latter prospects in mind, recently, electromagnetically induced transparency (EIT) and an all-optical switch was demonstrated .
6.6. Geoscience Another not immediate connection of Coulomb crystals to other branches of science is the observed stick-slip motion of the orientation of ion Coulomb crystals in Penning traps with respect to the phase of a rotating dipolar or quadrupolar perturbation. However, it turns out that the probability distributions of the amplitude of the slips are well described by power law relations, similar to those found in experiments on avalanches and slips in granular systems, as well as in models of earthquakes . 6.7. Exotic stellar objects Besides being of fundamental plasma physics interest, OCPs in the form of fully ionized atoms are expected to play an important role for the cooling mechanisms of White Dwarfs . Since such stellar objects are supposed to consist mostly of 16O8 + and 12C 6 + ions, which have nearly identical charge-to-mass ratios, it would be interesting to carry out experiments with two species with nearly the same charge-to-mass ratio. By performing experiments with Highly Charged Ions (HCIs) (e.g. 27 Al3 +, 36 Ar 4 + , 45Sc 5 + , 54 Fe6 + , …, 180Hf 20 + , 189Os21 + , …) cooled sympathetically through Coulomb interactions with simultaneously trapped and lasercooled 9Be+ ions, one can explore the effect of tiny differences in the charge-to-mass ratio on the segregation of the species. In such experiments, the fact that different species will crystallize at different temperatures (see Eq. (1)) may play an important role. In previous two-species MD simulations of ions in a linear rf trap, it was found that the degree of mixing is very sensitive to the exact values of the charge-to-mass ratios . However, much more detailed work has to be carried out in order to provide results of high enough quality to be relevant for the investigations of the cooling of White Dwarfs. 6.8. Test of quantum mechanics Ion Coulomb crystals in the form of few ion systems are quite unique objects for studying fundamental consequences of quantum mechanics. One of the ﬁrst things to be studied was the concept of quantum jumps, i.e., the observation of the abrupt jumps in the quantum state of a single quantum system upon measurements during induced dynamics. In a series of experiments with single atomic ions, this concept was veriﬁed by driving narrow transitions between the ground and a metastable excited electronic state [70–72]. The fact that spontaneous decay is in fact also a very fundamental quantum phenomenon has beautifully been demonstrated by altering the boundary conditions for the light emission of a single ion, either by introducing a single mirror [73,74] or an optical cavity  to establish single photon selfinterference. The interference of single photons in scattering processes involving two ions have furthermore led to the observation of super-/sub-radiant states  as well as Young's double slit like results . Experiments focused on the investigation of the so-called quantum Zeno effect has additionally been carried out with ion Coulomb crystals [78,79].
6.10. Quantum information processing Due to the existence of common normal modes of vibration for the particles constituting a Coulomb crystal, quantum correlated states can be created between the internal states of ions in such crystals through the applications of spatially localized laser beams addressing narrow line transitions in the individual ions . By detuning an excitation laser frequency up (blue) or down (red) by a common mode frequency with respect to the transition frequency, deterministic entanglement can be created between the internal states of an ion and the common motional state . By carefully addressing individual ions with appropriate laser pulses, it was shown by Cirac and Zoller in 1995  that universal quantum gate operations can be performed between any two qubits represented by the internal states |g 〉 ≡ |0〉 and |e〉 ≡ |1〉 of the ions. Hence, an ion Coulomb crystal in the form of a string of N ions, can be used as a quantum register of N qubits in a general purpose quantum computer. Since this ﬁrst proposal for quantum computing with cold ions several other schemes have been proposed, often including the so-called Mølmer–Sørensen gates , and an impressive series of important results have been obtained experimentally regarding multi-particle entanglement [92,93] and quantum gate operations [94–96]. 6.11. Quantum simulations The possibility of performing universal gates between any pairs of ion qubit states in a N qubit system as brieﬂy discussed above makes it in principle also possible to emulate the quantum dynamics of any Hamilton operator acting on a quantum system in a Hilbert space of dimension 2N . While such emulations can in principle be carried out on classical computers, the dimension of the Hilbert space increases exponentially with the number of qubits involved, and eventually becomes intractable. On the other hand using quantum gates to represent the dynamics, this
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problem can be solved , though the concatenation of many gate operations can become rather inefﬁcient. In some cases, like e.g. the study of magnetism and more generally spin–spin interactions at the quantum level, much more effective simulations can be carried out through emulating the two states of a magnetic dipole or spin by the two internal states of the ions, and apply appropriate light ﬁelds, which illuminate all the ions, to mimic magnetic ﬁelds [98–100,43]. By such quantum simulators one can simplify the number of operations needed considerable compared to “digital” quantum computing [101–103]. Whereas digital quantum processing procedure always will need some kind of individual addressing of all the ions, “analogue” quantum simulators have a larger potential to eventually be extended to twoand three-dimensional Coulomb crystals. 6.12. High precision spectroscopy The fact that ions within Coulomb crystals are both spatially very localized and motionally cold, makes such crystals ideal for highresolution spectroscopy. In particular, at such low temperatures where the quantized excitation of the normal modes is negligible, extremely narrow electronic transitions in the ions can be probed with ultra-high precision . Today, one of the most precise measurements of an optical transition in an atom has been performed in a 27 Al+ ion being sympathetically cooled  by a single laser-cooled 24Mg + ion to form a two-ion Coulomb crystal [106–108]. Since a large range of ion species, including ions of exotic isotopes, super-heavy element ions , highly charged ions [110,111] as well as molecular ions [112–114], can be effectively sympathetically crystallized through Coulomb interactions with laser-cooled ions [115,112,116], such reﬁned spectroscopy techniques may in the near future lead to a vast amount of new and improved spectroscopic data of importance for a variety of fundamental physics investigations ranging from nuclear to astrophysics. 6.13. Cold molecular ion research Sympathetic cooling of molecular ions through the Coulomb interaction with simultaneously trapped laser-cooled atomic ions has the past years opened for a broad range of molecular investigations of both physical and chemical interest [112,117–129]. By bringing the molecular ions into a crystalline state, it is indeed possible to exploit the speciﬁc properties of two-species Coulomb crystals to learn about the number of molecular ions present without being able to observe them directly, since the ﬂuorescence from the atomic ions will reveal the total size of the crystal [112,117,130]. By monitoring the growth in the number of dark ions in a reaction experiment, one can, e.g., infer reaction rates [112,117], and by recording the reduction in the size of the crystal through photodissociation of the molecular ions, one can, e.g., determine the internal state distribution of the molecular ions [120,121]. This latter method has been implemented to directly prove that though the translational motion of the molecular ions can be brought into the millikelvin range, the internal rovibrational temperature of the molecular ions is hardly changed due to the very distant (∼10 μm) Coulomb interactions . Indeed the cooling of the internal degrees of freedom of the molecules is so weak that they typically get in equilibrium with the radiation ﬁeld in the trapping region, which for a room temperature trap means ∼300 K [120,121]. The past years, several methods have though been devised to produce cold rovibrational molecular ions by the direct production of homonuclear molecular ions in speciﬁc internal states through Resonance Enhanced Multi-Photon Ionization , laser-induced rotational cooling [120,121] and buffer gas cooling either by laser-cooled atoms  or a cryogenic
helium gas . While the lowest internal temperatures have so far been obtained with a helium buffer gas (∼8 K) , buffer gas cooling by an ensemble of laser-cooled atoms has the prospect to bring the internal temperature into the microkelvin range. Combined with sympathetic sideband cooling to translational temperatures in the microkelvin regime, this will enable investigations of ion chemistry in a whole new ultracold regime.
7. Conclusion By this brief introduction to ion Coulomb crystals and their applications, I hope, I have managed to show how extremely versatile these objects are. Clearly, there is material enough for a whole book, however, the aim of the this work has primarily been to provide the reader with a text and references from which more detailed knowledge can be acquired elsewhere.
Remark It should be noted that the present text will in a slightly different format as well be published in the Proceedings for the International School of Physics (Enrico Fermi), Course CLXXXIX, 2013.
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