An Introduction to Trapped Ions, Scalability and ...

Chapter 9

An Introduction to Trapped Ions, Scalability and Quantum Metrology Alastair Sinclair

Abstract This article presents an introductory overview of three separate experimental aspects of ion trapping. It begins by discussing the various conventional approaches to confining charged particles, along with standard experimental techniques for laser cooling, coherent spectroscopy and quantum state preparation. Ion heating, a potential obstacle to experiments in quantum coherence is also discussed. For trapped ions to continue to advance in the field of quantum information, scalable trapping arrays are considered an essential technological component. Examples of the various approaches which have been pursued are outlined. A specific case study of a microtrap developed at NPL is presented, to exemplify the considerations needed in creating an operational device. A significant application of trapped ions is in quantum metrology, and more specifically in optical atomic clocks. The operational principle of a single-ion clock is described, and candidate species are highlighted. Advanced techniques for quantum state preparation and readout can now be used to enable frequency comparisons with unprecedented precision. This suggests that trapped ions will offer new levels of measurement sensitivity, the impact of which could range across optical atomic clocks, fundamental physics and navigation.

1 Introduction Laser-cooled trapped ions have been a highly useful experimental system for over 30 years. Trapping individual atomic particles provides a convenient means of preparing isolated, confined quantum systems with which to investigate the predictions of quantum mechanics. A particularly good example of this is an experiment that demonstrated Young’s interference using two 198 Hg+ ions [1]. Through state preparation and polarisation-sensitive detection, the experiment showed either the wave A. Sinclair (B) National Physical Laboratory, Middlesex, Hampton Road, Teddington TW11 0LW, UK e-mail: [email protected] E. Andersson and P. Öhberg (eds.), Quantum Information and Coherence, Scottish Graduate Series, DOI: 10.1007/978-3-319-04063-9_9, © Springer International Publishing Switzerland 2014

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or particle nature of the photons emitted by the laser-cooled ions. Over the past 20 years, single trapped ions have also played an important part as the atomic reference in the development of optical atomic clocks [2], with fractional uncertainties continually decreasing over time. Clocks based on individual Al+ ions have demonstrated unprecedented precision in frequency comparisons [3], and are an indication that a redefinition of the SI second is likely to occur in the not-too-distant future. Linear strings of trapped ions have played a significant role in the rapid development of experimental research into quantum information processing. Many different technological approaches are being pursued [4], however entangled states of trapped ions are arguably at the forefront of this research, with many landmark experimental demonstrations [5]. Quantum gates and algorithms, deterministic quantum teleportation, and multi-qubit entangled states are just some of the indicators of this success. These developments are now impacting on quantum simulation and on quantum metrology. This article is structured into three distinct parts. Section 2 gives an introduction to radiofrequency ion traps, and the basic experimental techniques required to operate them. This is balanced with a brief description of some theoretical aspects necessary to understand ion confinement and associated ion-laser interactions. These foundations are illustrated by experimental data. Section 3 discusses the scaling-up of trapped ion systems, for the purpose of enabling entangled states of many more particles, in particular for application to quantum information. It presents a brief survey of some of the technical approaches being pursued; a specific case study illustrates the many practical considerations required in developing a scalable trap system. Section 4 gives an introduction to the use of trapped ions for quantum metrology. This covers single ion clocks, recently developed techniques to perform highly sophisticated spectroscopy, and some examples of the types of measurements that are enabled as a result. None of these topics are covered exhaustively; where appropriate, reference is made to the relevant literature, including several review articles that provide significantly more detail.

2 Charged Particles: Trapping, Cooling, and Coherent Interactions This section gives a brief overview of the theory associated with Paul traps [6], where dynamic potentials are used to confine ions. Some examples of specific traps as used in actual experiments are presented. The routine techniques for creating and cooling ions, and detecting their electronic state, are also described. The theoretical basis for coherent interactions between a two-level trapped ion and a laser is introduced. Spectroscopy of a single ion is used throughout to illustrate the main features. A full description of the relevant theory is described in extensive detail in the review article by Leibfried et al. [7].

9 An Introduction to Trapped Ions, Scalability and Quantum Metrology

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2.1 Radiofrequency Ion Traps The desired ideal condition for confinement of charged particles is an harmonic potential. The most popular designs of traps are those that result in a dynamic confining electric potential Φ(x, y, z, t) which for any instant in time is close to a quadrupolar shape at the trap centre. In general, the potential can be written as a sum of static and time varying components, with amplitudes UDC and U0 respectively, Φ(x, y, z, t) =

1 1 UDC (αx 2 + βy2 + γz2 ) + U0 cos(Ωt)(α x 2 + β y2 + γ z2 ). (1) 2 2

The time varying component oscillates at angular frequency Ω, and since the potential must satisfy Laplace’s equation ∇ 2 Φ = 0, then the geometric coefficients are constrained as follows: α+β+γ =0

α + β + γ = 0

(2)

These constraints show that it is not feasible to create a 3D potential minimum, thus a charged particle may only be trapped in a dynamical fashion. Two options for the geometrical factors to satisfy the conditions of (2) imposed on the potential are as follows: (3) α=β=γ=0 α + β = −γ − (α + β) = γ > 0

α = −β .

(4)

The first option (3) yields a potential that offers 3D confinement in a pure oscillating field, i.e. there is no static component, and is realised with a cylindrically symmetric Paul trap (see Fig. 1a). Nowadays, a variant of this type of trap (see Fig. 1b) is typically used to store a single ion, and is applied to the development of optical frequency standards [2]. The second option (4) creates a potential where there is dynamical confinement in 2D, with a static potential providing confinement in the third dimension. This is the configuration of a linear Paul trap, (see Fig. 1c) which is capable of storing linear ion strings, and is applied to quantum information science with trapped ions [5]. It is important to analyse the equation of motion for an ion trapped in a potential of either configuration [7]. This enables one to understand the influence of experimental parameters on achieving a stable, bound motion of the ion. In one dimension, the equation of motion for an ion of mass m and charge +e is x¨ = −

e e ∂Φ = − [UDC α + U0 cos(Ωt)α ]x. m ∂x m

Through using the substitutions

(5)

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ξ=

Ωt , 2

ax =

4eUDC α , mΩ 2

qx =

2eURF α mΩ 2

(6)

the equation of motion (5) is transformed to the Mathieu equation d2x + [ax − 2qx cos(2ξ)]x = 0 dξ 2

(7)

which is a differential equation with periodic coefficients. Note that the parameters ax and qx contain the physical quantities specific to the experimental system. Stable solutions to (7) are required to ensure that the ions motion is bound to the trapping potential, and are of the general form x(ξ) = Aeiβx ξ

∞ n=−∞

C2n ei2nξ + Be−iβx ξ

∞

C2n e−i2nξ

(8)

n=−∞

where A and B are constants, and the C2n satisfy a specific recursion relation obtained by using (8) in (7). The exponent βx and coefficients C2n are functions of the parameters ax and qx only. The extent of the stability region for the Mathieu equation, i.e. that region of parameter space where stable solutions exist, correspond to those pairs of (ax , qx ) yielding 0 ≤ βx ≤ 1. The same is true for the other two dimensions y and z, and stable trapping of an ion is only feasible when the ions motion is bounded in all three dimensions. The type of trap (e.g. cylindrically symmetric or linear, see examples in Fig. 1), determines the exact shape of the stability region. From an experimental perspective, any deviations from the ideal electrode geometry also influence the shape and extent of the stability region. The electrode geometry configurations shown in Fig. 1a, b are cylindrically symmetric, and provide rf confinement in 3D according to Eq. (3). In general, a small dc potential may be applied to the endcap electrodes to change the shape of the potential. This geometry modifies the parameter relations to α = β = −γ/2 and α = β = −γ /2, so that the a and q parameters of the Mathieu equation are related as (9) az = −2ax = −2ay , qz = −2qx = −2qy In the geometry of Fig. 1a, α = α , β = β and γ = γ , and to a good approximation α = 2/(r02 + 2z02 ). For a specific size of trap (with r0 , z0 ), these relationships then enable one to calculate the appropriate set of operating parameters (U0 , Ω) to ensure stable confinement of a specific ion species (charge e, mass m). It is usual for these cylindrically symmetric traps to be operated with the parameter qz ∼ 0.4, and az ∼ 0. The full detail of the stability region is illustrated in the review by Leibfried et al. [7]. The electrode geometry presented in Fig. 1c is that of a linear trap, which provides confinement by a rf field in 2D and a static field in the third dimension, according to Eq. (4). From these, the q-parameters of the Mathieu equation are determined to be


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Fig. 1 Illustrations of some typical electrode geometries that are used for trapping ions. a A cylindrically symmetric 3D Paul trap trap with electrodes of hyperbolic form, where the rf is applied to the ring. b A cylindrically symmetric 3D Paul trap, known as a “ring” trap. The rf potential is applied to the “ring”, while the “endcaps” are at ground. c A linear Paul trap, often known as a “rod” trap, since the electrodes are made of metallic rods. The rf potential is applied to diagonally opposite rod electrodes for dynamic confinement in 2D (the radial plane). A static potential is applied to the endcaps for confinement in the third dimension

qx = −qy , qz = 0

(10)

and again it is typical for linear traps to be operated with qx ∼ 0.4 to ensure the ion trap is operating within the stability region specific to the linear trap [7]. Following further analysis of solutions to the Mathieu equation [7], the lowest order approximation to the ions trajectory is determined in the limit |ax |, qx2 1. The result is a solution of the form ri (t) = r0i cos(ωi t + φi )[1 +

qi cos(Ωt)] 2

(11)

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where i = x, y, z. This illustrates the origin of the two components to the trapped ions motion; the term r0i cos(ωi t + φi ) results in harmonic motion at a frequency Ω ωi = βi , where βi = 2

ai +

qi2 2

(12)

which is sometimes referred to as the secular motion. Superimposed on top of this is secondary motion at the rf drive frequency . The amplitude of this is smaller than that of the harmonic motion at ωi , and as such it is termed “micromotion”. If this faster motion has a small enough amplitude, it can be neglected and the motion of the ion is well approximated by an harmonic oscillator of frequency ωi . This is known as the “pseudopotential” approximation. In constructing an ion trap, the aim is to create a good approximation to a harmonic potential, containing minimal anharmonic components. The ideal harmonic potential is created using electrodes of hyperbolic form as in Fig. 1a. Ion confinement, as characterised by motional frequency, is proportional to the amplitude of the potential’s harmonic component. Anharmonic terms cause the ions motion to be nonlinear, with motional frequencies dependent on oscillation amplitude. The most desirable situation is to maximise the ions motional frequency while minimising anharmonicities. The trap efficiency parameter, εtrap , is a direct measure for the effectiveness of a trap structure in achieving these aims. It quantifies the harmonic component of the potential, relative to that created by ideal hyperbolic electrodes (εtrap = 1) with the same ion-electrode distance [8]. The first practical versions of a cylindrical Paul trap consisted of hyperbolic electrodes, following the original vision of Paul [6], to create a perfect harmonic potential. However, these traps had a comparatively large ion-electrode distance and extremely limited optical access, and are not suited to confinement of single ions in the Lamb-Dicke regime. One specific example, used at NPL (see Fig. 1a) [9], had an ion-electrode distance r0 = 5 mm. With rf amplitude U0 = 320V at frequency Ω/2π = 1.78 MHz, a motional frequency of ωr /2π ∼ 100 kHz was achieved for 88 Sr + ions, which is very low in comparison to most traps used today. For storing single ions, the ring trap illustrated earlier in Fig. 1b is much more appropriate. For comparison, one such trap operated at NPL [10] had an ion-electrode distance r0 = 500 μm. With a rf amplitude U0 = 450V at frequency Ω/2π = 14 MHz, 88 Sr + yielded motional frequencies of (ωr , ωz )/2π = (0.72, 1.16) MHz. Higher motional frequencies can be achieved, as demonstrated by the group at Innsbruck; with 40 Ca+ , (ωr , ωz )/2π = (2.1, 4.5) MHz was reported in a ring trap used for quantum state engineering [11]. Tight confinement of single ions in the Lamb-Dicke regime, together with much greater optical access, can be achieved in an endcap trap [12, 13], as illustrated in Fig. 2. In this design of trap, the “ring” has effectively been split in half and pulled back over the endcap electrodes. In such a trap constructed for 88 Sr + at NPL, the ion-electrode distance z0 = 280 μm, and with U0 = 390 V at Ω/2π = 15.9 MHz, motional frequencies of (ωr , ωz )/2π = (1.95, 3.96) MHz were


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Fig. 2 Example of cylindrically-symmetric endcap trap. The rf is applied to the inner electrodes of the concentric arrangement, and this geometry affords good optical access

obtained. This type of endcap trap is now used in the single ion optical clock systems at NPL [14, 15]. Linear traps, where the electrodes are made from metallic rods [16, 17], were the original conventional form for storing ion strings. In order to be rigid and still permit optical access, these typically have an ion-electrode distance r0 = 1.0 mm [16], however motional frequencies in the radial plane of 1 MHz ω/2π 2 MHz are still achievable. One of the most successful linear trap designs of “macroscopic” metallic electrodes is the “blade” trap developed by the Innsbruck group [18]. This had a reduced ion-electrode distance r0 = 0.8 mm, and could achieve motional frequencies (ωr , ωz )/2π = (5, 1.2) MHz with 40 Ca+ , however this necessitated an rf amplitude at the ∼kV level.

2.2 Some Elementary Experimental Ion-Trapping Techniques The choice of ion species is influenced by the specific application. The various energy level structures available across the different species offer different advantages and disadvantages. For example, the availability of a high-Q optical transition with minimal sensitivity to environmental perturbations, is perhaps the primary consideration for a single ion optical clock. However, practical considerations such as availability of suitable lasers for cooling and spectroscopy can also influence the choice. The criteria for an ion with a suitable hyperfine qubit transition will of course be different, although sensitivity to external perturbations remains a key consideration. Ions such as Be+ , Mg+ , Ca+ , Sr + , Ba+ , Cd + , Hg+ , and Yb+ have been trapped and cooled by various research groups around the world. An interactive catalogue of candidate ions is maintained by the Monroe group at Maryland [19], and contains extensive data for the various species. The original method for creating the ions to be trapped was an inefficient technique based on electron bombardment of the atomic vapour. This required a relatively high atomic flux, and necessarily polluted the vacuum chamber and trap structure with

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stray charges. The former contributed to excessive ion heating rates in the trap, while the latter caused unpredictable time-varying electric fields in the vicinity of the trap. Nowadays, the standard method to create ions from a low-flux atomic vapour is using a photoionisation process. This uses a flux at least ∼1000 times weaker and creates no stray charge in the vacuum chamber, thus vastly reducing the two major drawbacks of the earlier method. Resonant photoionisation has been used in various species, for example, Ca+ [20–22], Sr + [23], and Yb+ [24]. An alternative approach using ultrafast pulses was demonstrated using Cd + [25]. Some basic features of Doppler cooling and spectroscopy of a single ion will now be illustrated using the example of 88 Sr + confined in an endcap trap [13]. The relevant energy levels for 88 Sr + are shown in Fig. 3; 40 Ca+ [11, 17, 21] has a similar structure with analogous transitions at different wavelengths. Doppler cooling is facilitated by the strong 2 S1/2 –2 P1/2 dipole transition at λ = 422 nm; decay to the metastable 2D 2 2 3/2 level necessitates a repumper on the D3/2 – P1/2 transition at λ = 1092 nm. Doppler cooling by a red-detuned laser beam is a balance between (1) the cooling force arising from velocity dependent absorption of photon momenta from the cooling laser, and (2) heating due to the random walk which results from the recoils of spontaneously emitted photons. The minimum equilibrium energy is given by kB Tmin = Γ /2, at a laser detuning of Δ = −Γ /2, where Γ is the transition linewidth determined by the excited state lifetime [7]. For optimised Doppler cooling, the cooling laser intensity needs to be well below saturation. If not, then it will power-broaden the transition significantly, increasing the effective Γ and the scattering rate. The heating rate due to the random walk from spontaneously scattered photons increases and the equilibrium Doppler-cooled energy will not be minimised. The photons scattered during laser cooling are detected, either with a photomultiplier or a CCD camera. As the cooling laser frequency is scanned from below resonance, fluorescence increases to a maximum at line centre; see Fig. 4. Above resonance the fluorescence effectively disappears as the ion becomes Doppler heated in the bluedetuned laser beam. This fluorescence signal is used to determine the presence and state of the trapped ion. The 2 S1/2 –2 D5/2 optical transition in 88 Sr + (natural linewidth 0.4 Hz) has been studied in the context of an optical clock reference transition [14]. In addition, it could be used as an optical qubit, since the analogous transition in 40 Ca+ has been used in this way by the Innsbruck group to great effect in trapped ion quantum information processing. Pulsed-probe spectroscopy [9], based on the standard electron shelving technique [27], is employed to characterize this transition. After a spectroscopy pulse, the cooling laser is used to determine whether the ion is in the ground 2 S1/2 state (where it will fluoresce) or the metastable 2 D5/2 state (where it will not fluoresce). The spectrum illustrated in Fig. 5 shows the transition is sufficiently narrow that sidebands arising from the ions motional modes are clearly resolved from the carrier. Optical spectroscopy on transitions such as this necessitates a laser with a narrow linewidth (typically