Trapped Rydberg Ions: From Spin Chains to Fast Quantum Gates
aa r X i v : . [ qu a n t - ph ] A p r Trapped Rydberg Ions: From Spin Chains to FastQuantum Gates
Markus M¨uller , Linmei Liang , Igor Lesanovsky and PeterZoller Institute for Theoretical Physics, University of Innsbruck, and Institute forQuantum Optics and Quantum Information of the Austrian Academy of Sciences,Innsbruck, Austria Department of Physics, National University of Defense Technology, Changsha410073, ChinaE-mail: [email protected]
Abstract.
We study the dynamics of Rydberg ions trapped in a linear Paul trap,and discuss the properties of ionic Rydberg states in the presence of the static andtime-dependent electric fields constituting the trap. The interactions in a system ofmany ions are investigated and coupled equations of the internal electronic states andthe external oscillator modes of a linear ion chain are derived. We show that strongdipole-dipole interactions among the ions can be achieved by microwave dressing fields.Using low-angular momentum states with large quantum defect the internal dynamicscan be mapped onto an effective spin model of a pair of dressed Rydberg states thatdescribes the dynamics of Rydberg excitations in the ion crystal. We demonstrate thatexcitation transfer through the ion chain can be achieved on a nanosecond timescaleand discuss the implementation of a fast two-qubit gate in the ion chain.PACS numbers: 32.80.Ee, 37.10.Ty, 75.10.Pq, 03.67.Lx rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates z x ho a Ry - - - ++ ++ ++ zx Figure 1.
Typical length scales in a chain of cold Rydberg ions in a linear Paul trap.The external trapping frequency is in the order of MHz with a corresponding oscillatorlength x ho of approximately 10 nm. The interparticle spacing ζ , set by the equilibriumbetween the Coulomb forces among the ions and the external confinement, is typicallyabout 5 µ m. The third length scale is the size of the Rydberg orbit a Ry . Due to thescaling proportional to the square of the principal quantum number n it can assumevalues in the order of 100 nm and therefore become significantly larger than x ho . Inthis regime the Rydberg ion cannot be considered as a point particle but rather as acomposite object, and its internal structure must be taken into account.
1. Introduction
Rydberg states correspond to the infinite series of excited bound states in a Coulombpotential with large principal quantum number n . In view of their “macroscopic” size, a Ry ∼ n a with a the atomic Bohr radius, Rydberg states have remarkable properties,as reflected, for example, in their response to external static and time dependent electricand magnetic fields [1]. While the single particle physics of Rydberg atoms has beenthe subject of intensive studies in the context of laser spectroscopy, recent interesthas focused on exploiting the large and long-range interactions between laser excitedRydberg atoms to manipulate the many-body properties of cold atomic ensembles.Examples include recent seminal experiments on frozen Rydberg gases obtained bylaser excitation from cold atomic gases, demonstrating in particular a dipole-blockademechanism [2, 3, 4, 5, 6, 7], which in sufficiently dense gases prevents the excitation ofground state atoms in the vicinity of a Rydberg atom, and proposals for fast two-qubitquantum gates between pairs of atoms in optical lattices [8]. Furthermore, Rydbergatoms have been proposed to serve as model systems for studying coherent transportof excitations [9] - a mechanism which is of great importance for coherent energytransfer in biological systems, e.g. in light-harvesting complexes [10, 11]. While theseinvestigations have so far concentrated on neutral atoms, we are interested below indescribing the properties of laser excited Rydberg ions stored in a Paul trap, in particularthe interplay between trapping fields and Rydberg excitations, and the associated many-body interactions in a chain of cold trapped Rydberg ions.Atomic ions can be stored in static and radio frequency (RF) electric quadrupolefields constituting a Paul trap [12], where for sufficiently low temperature they forma Wigner crystal [13, 14, 15, 16]. Using laser cooling the ions can be prepared in thevibrational ground states of the phonon modes of the crystal. Internal electronic statesof the ions can be manipulated with laser light and entangled via the collective phonon rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates composite object , where the Rydberg electron is bound by the Coulomb force to thedoubly charged ion core, but both the Rydberg electron and the ion core move in theelectric fields constituting the trapping potentials. This is reminiscent to the situationin which Rydberg atoms are confined by very tight magnetic traps [18, 19]. We areparticularly interested in the parameter regime where the size of the Rydberg orbit a Ry is larger than the localization length of the doubly charged ion core x ho around itsequilibrium position, but still much smaller than the average distance ζ between theions in the Wigner crystal, i.e. x ho < a Ry ≪ ζ (see Fig. 1). Our goal is to provide adescription of the interaction and quantum dynamics of such a 1D string of Rydbergions in a linear ion trap.In comparison to neutral Rydberg atoms, a number of features and differencesemerges for Rydberg ions in a Paul trap, which will be illuminated in the present work.First of all, the character of a Rydberg ion as a composite object gives rise to an intrinsiccoupling of electronic and external motion in the presence of the electric trapping fields.This will be shown to result in renormalized trapping frequencies for Rydberg ionscompared to their ground state counterparts. Furthermore, the interaction among ionsis not - as in the neutral case - governed by the dipole-dipole force alone but also bythe charge-charge, dipole-charge and charge-quadrupole interaction. The interplay ofthis rich variety of interactions among the ions and the external trapping fields will beanalyzed.In our study we focus on ionic Rydberg states with low angular momentum quantumnumber and correspondingly large quantum defect, and large fine structure splitting,as these states can be most simply described and most easily excited by laser lightfrom electronic ground states. For typical ion trap parameters [12] static dipolar andvan-der-Waals interaction among ions in these Rydberg states is shown to be smallcompared to the energy scale set by the external trapping frequencies of the ions.Thus, in order to establish substantially stronger interactions, we employ additionalmicrowave (MW) fields driving transitions between Rydberg states. This leads to largeoscillating dipole moments, which result in remarkably strong controllable dipole-dipoleinteractions between the ions. Our findings show that for strong interactions the internaland external dynamics of the ion chain approximately decouple such that a “frozen”Rydberg gas is formed. In this limit the Hamiltonian describing the electronic dynamicscan be formulated in an effective spin-1/2 representation and involves resonant and off-resonant dipole-dipole interaction terms with coupling strengths of the order of severalhundred MHz. Based on this effective spin model we demonstrate resonant excitationtransfer on a nanosecond timescale from one end to the other of the ion chain. Moreover,we show that a two-qubit conditional phase gate between adjacent ions, based on thedipole-dipole interaction, can be realized on a time scale, which is much shorter thanboth the external dynamics and the radiative lifetime of the involved Rydberg states. rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates
2. Single Trapped Rydberg Ions as a Composite Object
A combination of static and time-dependent electric fields can be employed to confinecharged particles in a restricted region of space. The electric potential of a quadrupolefield of a standard Paul trap can be written in the formΦ T ( r , t ) = α cos ( ωt ) h x − y i − β h x + y − z i , (1)where ω is the radio frequency (RF) drive frequency. The electric field gradients α and β are determined by the actual geometry of the experimental setup. Inpresent experiments, typical parameters are α ∼ V / m , β ∼ V / m and ω = 2 π × ...
100 MHz (for details see e.g. Ref. [12]).We are interested in the properties of Rydberg ions of alkali earth metals [20] inan electric quadrupole trap. They possess a single valence electron with the remainingelectrons forming closed shells [1]. For the description of such system one can employan effective two-body approach in which the ion is modeled by a two-fold positivelycharged core (mass M , position r c ) and the valence electron (mass m , position r e ). Thecorresponding interaction (model-)potential depends only on the relative coordinate r c − r e and also on the angular momentum state of the atom. The latter dependenceaccounts for the quantum defect - a lowering in the energy for low angular momentumstates in which the valence electron probes the inner electronic shells. High angularmomentum states (typically with angular momentum quantum number l >
5) do notexhibit a significant quantum defect since here the valence electron is located far awayfrom the ionic core thus experiencing a bare Coulomb potential.We now formulate the Hamiltonian of a single Rydberg ion in the presence of theelectric potential of the Paul trap. We add a linear potential, corresponding to a time-dependent homogeneous electric field, Φ l ( r ) = f ( t ) · r , such that the combined electricpotential reads Φ( r , t ) = Φ T ( r , t ) + Φ l ( r , t ). Below, we will employ these additional MWfields to electronically couple different Rydberg states. This will allow us to generatelarge oscillating dipole moments, which give rise to strong dipole-dipole interactionsamong dressed Rydberg ions. rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates V ( | r e − r c | ) and taking into account the coupling of the individual charges to theelectric potentials we find H lab = p c M + p e m + V ( | r e − r c | ) + 2 e Φ( r c , t ) − e Φ( r e , t ) + H FS . (2)The term H FS accounts for the spin-orbit coupling giving rise to the fine-structure ofelectronic levels. Its effect will be discussed in the following subsection.We introduce center of mass (CM) R and relative coordinates r according to r c = R − mM + m r , r e = R + MM + m r . (3)Exploiting that the nuclear mass is much larger than the electronic one and hence M ≫ m we have r c ≈ R and r e ≈ R + r . Within this approximation the Hamiltonianbecomes H = P M + p m + V ( | r | ) + 2 e Φ( R , t ) − e Φ( R + r , t ) + H FS (4)= P M + p m + V ( | r | )+ e " Φ( R , t ) − ∂ Φ( R , t ) ∂ R · r − X kl x k ∂ Φ( R , t ) ∂X k ∂X l x l + H FS . Corrections to this Hamiltonian scale with m/M which is typically about 10 − .In case of ions in low lying electronic states the potential (1) provides static confinement along the longitudinal (z-)direction. However, transversally at no instant oftime a confining potential is present. One rather finds a periodically oscillating potentialsaddle centered at the origin of the coordinate system. Due to the rapid periodicchange of the confining and non-confining direction, however, the ions experience aponderomotive potential that can provide transversal confinement [21]. In order tomake this manifest we transform into a frame which oscillates at the RF frequency ω inthe CM coordinate system. This is achieved by the unitary transformation U = U ( R , t ) = exp (cid:18) − i eα ¯ hω h X − Y i sin( ωt ) (cid:19) . (5)By applying this transformation to Hamiltonian (5) one obtains H ′ = U HU † + i ¯ h ∂U∂t U † = H CM + H el + H CM − el + H mm (6)with H CM = P M + 12 M ω z Z + 12 M ω ρ (cid:16) X + Y (cid:17) (7) H el = p m + V ( | r | ) − e Φ( r , t ) + H FS (8) H CM − el = − e [ α cos( ωt ) ( Xx − Y y ) − β ( Xx + Y y − Zz )] (9) H mm = − eαM ω sin( ωt ) ( XP x − Y P y ) − e α M ω ( X + Y ) cos(2 ωt ) + e f ( t ) · R . (10) rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates H CM provides harmonic axial and transversal confinement of the CM motion withthe corresponding trap frequencies ω z = 2 q eβM and ω ρ = √ r(cid:16) eαMω (cid:17) − eβM , which are ofthe order of a few MHz and satisfy ω ρ ≫ ω z . For Ca + ions, an RF frequency ω = 2 π × α = 10 V/m and β = 10 V/m the axial and radialtrap frequencies evaluate to ω z = 2 π × .
56 MHz and ω ρ = 2 π × .
64 MHz, respectively.The term H el contains all dependencies on the electronic coordinates describing themotion of an electron in the field of a doubly charged ionic core which is superimposedby the electric potential Φ( r , t ). The electronic dynamics takes place on a much fastertime scale compared to the CM motion in the trap. The external electric field prevents,unlike in the field-free case, the separation of the CM and relative dynamics. Thecoupling between these motions is accounted for by H CM − el . Due to the large separationof time scales of electronic and external dynamics we will treat this coupling within anBorn-Oppenheimer approach below.Finally, we have the term H mm which gives rise to the micromotion causing acoupling between the static oscillator levels of H CM . It can be shown [21] that for largeenough values of the RF frequency ω this coupling can be neglected and the externalmotion of the ions can be considered as if it was taking place in a static harmonicpotential. The effect of the additional micromotion term e f ( t ) · R arising from the MWdressing fields can likewise be neglected in the following, since typical MW frequenciesare of the order of at least one GHz and therefore far from being resonant with theexternal trapping frequency of the ions.With the full Hamiltonian (6) for the internal and external dynamics and thecoupling among them at hand, we are now in the position to analyze the electronicproperties of a trapped Rydberg ion and the mutual interplay of internal and externaldynamics. This will be addressed in the next two subsections. We proceed by inspecting the electronic Hamiltonian H el in Eq. (8), H el = p m + V ( | r | ) + H FS + H ef (11)with the electric trapping and MW dressing fields contained in H ef = − e Φ( r , t ) = H stat + H osc + H MW = e β h x + y − z i − e α cos ( ωt ) h x − y i − e f ( t ) · r . (12)In the absence of electric fields the ionic Rydberg states can be classified by the principalquantum number n , the angular momentum quantum number l , the total angularmomentum j and its magnetic quantum number m . The quantum states are representedby | n, l, j, m i = | n, l i | j, m i a which factors in the radial part | n, l i and the angularmomentum part | j, m i a . The latter is constituted by a linear combination of productsof the Spherical Harmonics and spin orbitals. The corresponding energies of the Rydberglevels are given by the well-known formula E n l j = − E Ryd / ( n − δ ( l )) + E FS ( n, l, j ) rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates |n-2,l=3 ñ |n-2,l=4 ñ |n,l=0 ñ |n-1,l=2 ñ |n,l=1 ñ |n-2,l>5 ñ Ca + H MW |n,s,1/2,m ñ H osc a b |n,p,1/2,m ñ |n,p,3/2,m ñ finestructuresplitting m-3/2 -1/2 1/2 3/2 quadrupoleshift Figure 2. a : Sketch of the Rydberg level structure of the Ca + ion in the field freecase without finestructure. States with l > b : Levels of the s and p manifold in the presence of the electric potentialΦ( r , t ). Its static components lead to a first order shift (quadrupole shift) of the energylevels according to eqs. (13) and (14). In general the electronic energy is loweredwith respect to the field-free value (sketched by the solid gray lines). The shiftedlevels are coupled by the microwave Hamiltonian H MW (blue, solid lines) and theoscillating components of the trapping field (red, dashed lines). We consider the limitof large finestructure splitting in which the coupling between the states | n, p, / , m i and | n, p, / , m i (thin dashed lines), which is caused by H osc , can be neglected. where E Ryd = 13 . δ ( l ) the quantum defect [1, 20].The energy E FS ( n, l, j ) accounts for the finestructure splitting due to H FS . The typicalRydberg level structure (without finestructure) is sketched in Fig. 2a for the case ofCa + .In this paper we focus on states with large quantum defect, i.e. s and p states. Thisis motivated by the fact that these states are most easily accessible via laser excitationsfrom the electronic ground states, and that they are energetically far separated fromthe degenerate manifold of states of higher angular momentum states. Generically, theenergy separation △ E l,l +1 between these states scales as n − for large n . In case of Ca + the energy separation between the s and the p level is △ E s,p / ¯ h ∼
280 GHz for n = 60.Let us now inspect the effect of the electron field interaction H ef contained in theelectronic Hamiltonian (12). We assume that the finestructure splitting of the p stateis sufficiently large such that neither the oscillating nor the static part of H ef causesignificant coupling between the states | n, p, / , m i and | n, p, / , m i and j remainsa good quantum number. Since the finestructure splitting scales proportional to n − and becomes larger with increasing atomic mass the validity of this assumption can beensured by either choosing not too high principal quantum numbers or by employingheavy ions. Moreover, we can neglect the coupling between the s and p states which isjustified by their typical energy splitting of several 100 GHz. Considering first the staticpart H stat of Eq. (12) we find that it gives rise to the following quadrupole shifts whichdepend on j and m : △ E s = 0 (13) rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates △ E pj | m | = 215 eβ D n, p | r | n, p E (cid:20) | m | − j − (cid:21) . (14)Here h n, l | r | n, l i denotes the radial matrix element of r calculated using the radialeigenfunctions that belong to the atomic interaction potential V ( | r | ). The total energyof the electronic states is hence given by E nljm = E nlj + △ E lj | m | . The quadrupole shiftswith respect to the unperturbed energy of the p states are sketched in Fig. 2b.For too strong electric field gradients there is the danger of ionizing the ion due to H stat . To estimate the ionization threshold we consider the potential V ′ = − e πǫ r + eβ h x + y − z i . (15)Here we have approximated the interaction of the ionic core with the valence electronby a pure Coulomb potential. V ′ possesses two saddle points located at the z -axis at z sad = ± h e πǫ β i / where it assumes the value V ′ sad = − h e βǫ π i / . Solving E Ry = V ′ sad yields an estimate for the gradient β at which field ionization would occur classically, β ion = 427 e m (4 πǫ ) ¯ h n = 1 . × Vm × n . (16)For n = 50, for example, we find β ion = 9 . × V / m which is one order of magnitudelarger than the highest gradients actually used in experiment [12]. Hence, up to n = 50there is no danger for the Rydberg ion to undergo field ionization.Let us now turn to the oscillating part of the electric field which is accountedfor by the term H osc in the Hamiltonian (12). Since the oscillation frequency ω istypically 2 π × ...
100 MHz, it is not sufficient to yield a resonant coupling betweenthe s and the p states or between the fine structure components of the p manifold.However, by estimating | h n, p | r | n, p i | ≈ a n , where a is Bohr’s radius, we findthat h H osc i ≈ e α a n cos( ω t ) and hence, although no resonant coupling is present,the strong modulation amplitude, which grows proportional to the fourth power ofthe principal quantum number n , might give rise to significant level shifts. From thesymmetry properties of H osc one concludes that only levels whose magnetic quantumnumber m differs by 2 are coupled. These couplings are indicated in Fig. 2b by the red,dashed arrows. Since we assume that the finestructure splitting is much larger than | h H osc i | max the j = 1 / H osc . By contrast strong effects are tobe expected in the j = 3 / | n, s i ≡ | n, s, / , m i and | n, p i ≡ | n, p, / , m i (see Fig. 2).Finally, the term H MW in Eq. (12), which accounts for the additional MW dressingfields, is to be discussed. This will be subject of the following subsection. In this section we describe how MW dressing fields can be employed to generate stronginteractions in an ion chain. As will be shown in Sec. 3.1 Rydberg ions aligned in aWigner crystal in the Paul trap do not exhibit permanent dipole moments, and residualVan-der-Waals interactions are small. Thus, our aim is to generate strong interactions rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates |n’,p ñ |n,p ñ |n,s ñ W w , W w , E |n,p ñ E |n,s ñ E |n’,p ñ D D |s’ ñ= | ñ |n,p ñ= | ñ W w , D ’ Figure 3.
Microwave dressing of ionic Rydberg states. A linearly polarizedbichromatic microwave field is used to couple one s with two p levels one of whichis far detuned. After an adiabatic elimination of the latter one obtains an effectivetwo-level system. This dressing is used to tailor the interaction between Rydberg ions. using (near-)resonant MW dressing fields to couple electronic s and p Rydberg statesas indicated in Fig. 2b. We show in the following that the MW dressing gives rise tolarge oscillating dipole moments, which in turn lead to strong dipole-dipole interactionbetween Rydberg ions.The energy separation between s and p Rydberg states belonging to the sameprincipal quantum number and thus as well the typical frequencies of the MW dressingfields are typically in the order of a few hundred GHz (cf. Sec. 2.2). Since thetrapping frequencies determining the external motion of the ions are in the MHz range(cf. Sec. 2.1), coupling to the external motion is negligible and the MW fields exclusivelyaffect the electronic degrees of freedom. Ideas similar to the MW dressing schemedescribed below were applied in the context of cold polar molecules, where a combinationof static electric and MW fields was used in order to tune intermolecular two- and three-body interactions [22].In our setup we apply a linearly polarized MW such that non-zero transition dipolematrix elements occur between the states | n, s i and | n ′ , p i . For our purposes we choosea bichromatic microwave-field of the form f ( t ) = E e z cos ω t + E e z cos ω t such thatthe coupling Hamiltonian reads H MW = − d z [ E cos ω t + E cos ω t ] (17)where d z = e z is the operator of the dipole moment. We consider three levels (one s leveland two p levels) which are coupled by H MW as depicted in Fig. 3. The frequencies ω and ω bridge the energy separations E | n,s i − E | n ′ ,p i and E | n,p i − E | n,s i , respectively, andare assumed to differ significantly. This can be achieved by choosing not too close valuesof n and n ′ . In the rotating frame and after performing the rotating wave approximationthe electronic Hamiltonian of a single ion reads H = ¯ h △ | n ′ , p ih n ′ , p | − ¯ h △ | n, p ih n, p | + 12 [(¯ h Ω | n, s ih n ′ , p | + ¯ h Ω | n, p ih n, s | ) + h . c . ] . (18)Here we have introduced the microwave detunings △ = ω − ( E | n,s i − E | n ′ ,p i ) / ¯ h and rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates △ = ω − ( E | n,p i − E | n,s i ) / ¯ h and the Rabi frequencies Ω , = − d , E , / d = e h n, s | r | n ′ , p i and d = e h n, s | r | n, p i . We assumethat the MW field with frequency ω is far-detuned and only weakly couples the states | n ′ , p i and | n, s i . Furthermore it is assumed that | ∆ | ≫ | Ω | , | ∆ | . Under the conditionΩ ≪ | ∆ | we can adiabatically eliminate the | n ′ , p i state and obtain an effective two-level system consisting of the | n, p i state and a dressed state (see also Fig. 3) | s ′ i = | n, s i − Ω | n ′ , p i (19)with η = Ω / (2∆ ), η ≪
1. This effective two-level system can be mapped onto a spin-1/2 particle by identifying the states | s ′ i and | n, p i as eigenstates of the spin operator S z with positive (negative) eigenvalue. The Hamiltonian (18) then reduces to H = ¯ h ∆ ′ Ω Ω − ∆ ′ ! ≡ hS (20)with an effective magnetic field h = (Ω , , ∆ ′ ), detuning ∆ ′ = △ − Ω / (4 △ ) and thespin operator S = ( S x , S y , S z ).Due to the weak admixture of the state | n ′ , p i the dressed state | s ′ i obtains anoscillating dipole moment. The matrix representation of the dipole operator d z in theset of states | s ′ i , | n, p i is given by d z = 13 − Ω ∆ d cos ω t d e − iω t d e iω t ! . (21)Using the abbreviation D , = ( d , / one obtains the following representation of thedipole moment operator: d z = − η q D cos( ω t )(1 + 2 S z ) + 2 q D (cos( ω t ) S x + sin( ω t ) S y ) . (22)The magnitude of the induced dipole moments is determined by the transition dipolematrix elements d , which can be roughly estimated as d , ∼ ea n . Thus the dipole-dipole interaction energy scales ∼ n for large n . This is to be compared with theradiative life time of Rydberg states, which for large n and low l scales according to ∼ n [1], thereby favoring larger values of n . We return to the question of radiative decayand the validity of our analysis in Sec. 3.2, where we will use the representation (22)of the dipole moment operator to derive an effective spin chain Hamiltonian describingthe dynamics of Rydberg excitations in the ion chain. In this section we will analyze the effects of the coupling of internal and external degreesof freedom described by H CM − el in the Hamiltonian (6). We treat these coupling termsusing a Born-Oppenheimer approach, which is reasonable since the electronic dynamicstakes place on a much faster time scale than the external motion of the ions. In thisframework we treat the CM coordinates as parameters while diagonalizing the electronicHamiltonian. We evaluate the energy of the | n, s i -level by second order perturbation rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates | n, p i -level. After averaging over oneRF cycle one obtains the energy correction ǫ ( X, Y, Z ) = E | n,s i + 12 M ω ′ z Z + 12 M ω ′ ρ ( X + Y ) (23)with ω ′ z = − M e β △ E s,p |h n, s | r | n, p i| , (24) ω ′ ρ = − M e ( α + 2 β ) △ E s,p |h n, s | r | n, p i| , (25)and the energy difference △ E s,p = E | n,p i − E | n,s i . Hence, ions excited to Rydbergstates experience modified transversal and longitudinal trap frequencies in comparisonto their ground state counterparts. For the s state under consideration the trap becomesshallower and the new trap frequencies are given by ˜ ω ρ,z = ω ρ,z q ω ′ ρ,z /ω ρ,z ) ≈ ω ρ,z + ω ′ ρ,z / (2 ω ρ,z ) = ω ρ,z + δω ρ,z , yielding δω z ω z = 43 eβ |h n, s | r | n, p i r | △ E s,p , (26) δω ρ ω ρ = 16 α + 2 β α / ( M ω ) − β/e |h n, s | r | n, p i r | △ E s,p . (27)By estimating |h n, s | r | n, p i| ∼ a n one finds that the frequency shift δω ρ,z scalesproportional to n . For typical trap parameters, i.e. α ≫ β , the modification of the axialtrap frequency is much smaller than the corresponding one for the radial frequency. For n = 50 and the parameters presented in Sec. 2.2 we find ( δω ρ ) /ω ρ = 3 . × − and( δω z ) /ω z = 7 . × − . Below we are interested in the regime where these modificationsof the trapping frequencies are negligible. We remark, though, that due to the scaling ∼ n they can become significant even for moderately larger principle quantum numbers.
3. Interacting Trapped Rydberg Ions
We now turn to the discussion of the interaction between several ions. First, weanalyze the various types of emerging interactions between Rydberg ions, and discussthe interplay of interionic interactions and the effect of the external trapping fields.Then we proceed with the discussion of interacting Rydberg ions dressed by MW fieldsand derive the corresponding effective spin-1/2 Hamiltonian describing the Rydbergexcitation dynamics in the ion chain. The role of the two effective spin states of eachion will be played by two Rydberg states, as outlined in Sec. 2.3 and illustrated in Fig. 3.We illustrate the effective spin dynamics by studying the process of resonantexcitation transfer in a mesoscopic chain of ten ions. Furthermore, we suggest ascheme, which allows to observe the effective spin dynamics in dressed ground stateions. We finally show how the trapped Rydberg ion system can be exploited in thecontext of quantum information processing and discuss an implementation of a two-qubit conditional phase gate. rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates R i R j r i r j ++ ++- - Figure 4.
Interacting Rydberg ions. The net charge of the ions leads, apart from thecommon Coulomb repulsion, to a charge-dipole and a charge-quadrupole interactionboth of which are absent in systems of neutral Rydberg atoms.
We consider the interaction between ions i and j with coordinates given by ( R i , r i ) and( R j , r j ) as depicted in Fig. 4. The Coulomb interaction V ( r i , r j , R i , R j ) between thecharges belonging to different ions can be written as V ( R i , R j , r i , r j ) e / (4 πǫ ) = 4 | R i − R j | − | R i − ( R j + r j ) |− | ( R i + r i ) − R j | + 1 | ( R i + r i ) − ( R j + r j ) | . (28)We assume that | R i − R j | ≫ | r i/j | which is very well fulfilled, since - as discussed above- we are interested in the parameter regime where the average interparticle distance ζ in the ion trap is much larger than the extension of the electronic wave function a Ry ofthe Rydberg ions. Performing a multipole expansion up to second order in the smallparameter ( a Ry /ζ ) and abbreviating | R i − R j | = | R ij | = R ij and n ij = ( R i − R j ) /R ij we obtain the following form for the Rydberg ion-Rydberg ion interaction potential: V ( R i , R j , r i , r j ) e / (4 πǫ ) = 1 R ij + ( R i − R j )( r i − r j ) R ij + r i − n ij · r i ) + r j − n ij · r j ) R ij + r i · r j − n ij · r i )( n ij · r j ) R ij . (29)The first term accounts for the Coulomb interaction between two singly charged ionsand is independent of the degree of electronic excitation. In case of Rydberg ions thedisplacement of the electronic charge from the ionic core leads to the exhibition of adipole moment which interacts with the charge of the other ion. This dipole-chargeinteraction gives rise to the second term. The third term accounts for the charge-quadrupole interaction. These three terms are absent in the case of interacting neutralRydberg atoms. The last term is the well-known dipole-dipole interaction, which is alsopresent in neutral systems.For N ions stored in a Paul trap, at sufficiently low temperature and tight radialtrapping, ω ρ ≫ ω z , the ions form a one-dimensional Wigner crystal with equilibrium rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates Z -axis [15, 16], which are determined by the interplay between theCoulomb repulsion among the ions and the external confinement by the trapping fields.In Appendix A we show that after a harmonic expansion of the Hamiltonian around theequilibrium positions Z (0) i (given by Eq. (A.3)) the full Hamiltonian of N interactingions can be written as H ions = H ph + N X i H el ,i + H int − ext + H dd . (30)The first term H ph = X α = x,y,z N X n ¯ hω α,n a † α,n a α,n (31)describes the external oscillation dynamics of the ionic cores with a † α,n and a α,n beingthe respective creation and annihilation operators of the normal modes (cf. Eqs. (A.5)-(A.8)). The second term determines the electronic level structure of the ions: thecharge-quadrupole term gives rise to a position dependent variation of the electric fieldexperienced by the trapped ions, which can be absorbed in the single particle ion-fieldinteraction H ef given by Eq. (12). Thus, the electronic Hamiltonian of the i -th ion takesthe form H el , i = p i m + V ( | r i | ) + e β ′ i h x i + y i − z i i − e α cos ( ωt ) h x i − y i i − e f ( t ) · r i (32)with position dependent gradient β ′ i = β + e πǫ N X j ( = i ) | Z (0) i − Z (0) j | = β + δβ i . (33)The third term H int − ext in Eq. (30) accounts for the coupling of the internal and externaldynamics, which is partly due to the dipole-charge interaction and partly due to theinhomogeneous electric field of the Paul trap (see Eq. (A.12)). The resulting couplingbetween the internal and external dynamics becomes also position dependent and thusleads to a state-dependent variation of the trapping frequency (see Sec. 2.4). However,since we work in a regime where these shifts are negligible already in the single-ion casethey can also be neglected in the case of many ions.Finally we have the dipole-dipole interaction in Eq. (30), which after the harmonicexpansion is given by H dd = 12 e πǫ N X i = j r i · r j − z i z j | Z (0) i − Z (0) j | . (34)The discussion of the individual terms contained in Hamiltonian (30) proceeds inanalogy to the one presented for the single ion case. The electronic dynamics of each ionis governed by H el ,i in Eq. (30) and is therefore dependent on the equilibrium positionsof the ions. This gives rise to position dependent electronic energies, i.e. E nljm → E nljm,i according to Eqs. (12)-(14) with the ion-dependent gradients of Eq. (33). rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates ζ is much largerthan the extension of the electronic wave function a Ry of a Rydberg ion, the dipole-dipoleinteraction can be treated perturbatively. For the considered s and p states the expectedinteraction energy can be estimated by E vdW ∼ e | h n, s | r | n, p i | / ((4 πǫ ) |△ E s,p | ζ ).For Ca + , n = 50 and ζ = 5 µ m one finds E vdW ∼ ¯ h ×
200 kHz. In the next subsectionwe demonstrate that the MW fields introduced in Sec. 2.3 in order to manipulate theelectronic level structure can give rise to much stronger interactions among the ions.
To study the interaction between MW-dressed Rydberg ions we first transform the N -ion Hamiltonian (30) to a frame of reference, which rotates at the microwave frequencies ω and ω of the dressing fields (in analogy to the single-ion transformation in Sec. 2.3).In the rotating frame, the terms in H int − ext oscillate rapidly at the frequencies ω and ω and ω − ω . Hence, in the rotating-wave approximation the internal and externaldynamics decouple (cf. discussion in Sec. 2.1).We use the dressed ion Hamiltonian (20) and the expression (21) for the electronicdipole moment in order to represent the Hamiltonian H int = P Ni H el ,i + H dd which is partof the full Hamiltonian (30). The terms H el ,i are substituted by the two-dimensionalrepresentation given by Eq. (20). In this representation we can write the dipole-dipoleinteraction as H dd = − πǫ N X i = j d ( i ) z d ( j ) z | Z (0) i − Z (0) j | (35)where d ( i ) z is given by Eq. (21) with the index i labeling the respective ion. In therotating wave approximation we obtain H int = N X i h ( i ) S ( i ) + D N X i,j ( = i ) ν ij (cid:20) η i + η i η j S ( i ) z (cid:21) (36)+ N X i,j ( = i ) ν ij h D (cid:16) S ( i ) x S ( j ) x + S ( i ) y S ( j ) y (cid:17) + η i η j D S ( i ) z S ( j ) z i , with ν ij = − / (4 πǫ | Z (0) i − Z (0) j | ) and an effective magnetic field h ( i ) = ¯ h (Ω , , ∆ ( i )2 ).The coefficients D , and η i characterize the coupling strengths and depend on thetransition matrix elements between the involved Rydberg states and the MW dressing. rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates |n’,p ñ |n,p ñ |n,s ñ W w , W w , D D |s’ ñ= | ñ |n,p ñ= | ñ D ’ W w , ab E ,k|n’,s ñ E ,k|n,s ñ E ,k|n,p ñ Figure 5. a : Schematic level scheme of the three MW-coupled Rydberg levels, for achain of 5 ions. The individual, position-dependent energy shifts of the l = 1 statesgive rise to an inhomogeneous distribution of MW detunings (not true to scale). b :Schematic spin dynamics of a chain with the initial configuration: first ion in the | n, p i state, all other ions in the | s ′ i state (blue filled circles). After a certain time theRydberg excitation has travelled to the right end of the chain (red open circles). Anumerical example for this excitation transfer is shown in Fig. 6. As depicted in Fig. 3 the effective spin corresponds to a two-level system constituted bytwo MW-dressed Rydberg states (see Sec. 2.3 for details).The terms linear in the spin operators represent a coupling of a series of spins toan inhomogeneous effective magnetic field whose strength and direction are determinedby the position dependent electronic energies of the Hamiltonian (32). The terms inthe second line of Eq. (36) (quadratic in the spin operators) represent a ferromagneticHeisenberg chain with 1 /r exchange-type interaction [23]. Establishing the connectionto neutral Rydberg gases the S ( i ) x S ( j ) x and S ( i ) y S ( j ) y can be interpreted as resonant dipole-dipole interaction terms. The term which is proportional to S ( i ) z S ( j ) z resembles theinteraction of two static dipoles. In general, the coefficients η i = Ω ( i )1 / (2∆ ( i )1 ) depend onthe ion index, since different energy shifts of the | n ′ , p i level due to position-dependentcharge-quadrupole terms (cf. Eqs. (32) and (33)) lead to different detunings ∆ ( i )1 fordifferent ions (see Fig. 5). In case one does not admix the | n ′ , p i state to | n, s i (i.e. Ω = 0in Eq. (18)), the coefficients η i vanish and the interaction Hamiltonian solely describesresonant dipole-dipole interaction in the presence of an effective magnetic field with aconstant component in x and a position-dependent component in z -direction.The physical realization of effective spin dynamics, as provided by the Hamiltonian(36), has been of significant interest in atomic physics during the last few years as “analogquantum simulators” of (mesoscopic) condensed matter systems. The distinguishingfeature of the present setup is the large coupling strength between effective spins, whichscales proportional to n and is of the order ¯ h ×
500 MHz ( n ≈
50, typical interparticlespacing ζ ≈ µ m). The characteristic time scale of the effective spin dynamics is thus ofthe order of a few nanoseconds. This is significantly shorter than the typical decoherencetime in the system, which is set by the radiative lifetimes of the involved Rydberg states, rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates n and are typically of the order of µ s (for Ca + ions and n = 50 thelifetime is ∼ µ s [20]).Realization of effective spin models has also been proposed in the context of trappedions in their electronic ground state, and with cold atoms and polar molecules inoptical lattices. For trapped ions involving electronic ground states models analogousto (36) can be derived where the typical coupling strengths for the effective spin-spininteractions are in the range of tens of kHz [16] with (long) decoherence times asdescribed in the context of ion trap quantum computing [24, 25]. Effective Heisenbergmodels with nearest neighbor interactions are also obtained with cold atoms in opticallattices [26, 27], where the time scales of exchange interactions can be of the order of afew hundred Hz [28, 29]. We note that these energy scales are also directly related to thetemperature requirements for the preparation of an effective zero temperature ensemble.Finally, effective spin models, such as the Kitaev model [30], have been proposed withpolar molecules in optical lattices [31, 32, 33]. In this context electric dipole moments ofa few Debye can be induced by external DC and microwave electric fields in the rotationalmanifold of molecules prepared in their electronic and vibrational ground state. Theelectric dipole-dipole interactions, which are strong and long range in comparison withneutral atom collisional interactions, can lead to effective offsite-coupling strengths ofthe order of tens or hundred kHz, limited mainly by the conditions imposed by opticaltrapping, while coherence times are of the order of seconds, as determined e.g. byspontaneous emission in off-resonant light fields forming the trapping potential.A second feature of simulating spin dynamics according to Hamiltonians of the type(36) is the single ion addressability and read out [34] which the present setup inheritsfrom the experimental developments in ion trap quantum computing. In contrast,neutral atoms and molecules in optical molecules ususally allow global addressing bylaser light, even though a significant effort is devoted at the moment to develop thesetools also for optical lattice setups [35, 36]. Note, however, that neutral atoms andmolecules in optical lattices will typically allow for systems with a significantly largernumber of “spins” than in the ion case.One of the experimentally most challenging aspects of realizing a Rydberg ion chainis the requirement of π -pulses to transfer ions from the ground state to the Rydbergstate. In Sec. 3.4 we discuss a version of an effective spin chain where the Rydbergdipoles are admixed to the electronic ground states with an off-resonant laser process,resulting in ground state ions with effective oscillating dipole moments. As an illustration of the spin dynamics contained in Hamiltonian (36), we study thetransfer of an excitation from one side of the Rydberg ion chain to the other end. Forsimplicity we choose a scenario in which η i = 0 (Ω = 0 in Eq. (18)). As discussed inthe previous subsection in this case the interaction reduces to purely resonant dipole-dipole interaction. The ion-dependent coefficient of the dipole-dipole interaction can be rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates
1 2 3 4 5 6 7 8 9 10012345 k π J t/ h π Jt/h < S z > k=1 k=10 Figure 6.
Transport of an excitation through a chain of ten ions. The ion at site1 is initially excited to the state |↑i while all others are in the state |↓i . After thetime t = 1 . h/J the excitation is transferred from the first to the tenth ion. The insetshows the time evolution of the expectation values h S (1) z i and h S (10) z i . written as ν ij = − πǫ | Z (0) i − Z (0) j | = − M ω z e | u i − u j | (37)where the u i are the equilibrium positions in dimensionless coordinates, u i = Z (0) i /ζ with ζ = [ e / (4 πǫ M ω z )] / (see Ref. [15]). The scale of the interaction energy is thengiven by J = − Mω z e D . We choose the MW frequency ω such that it is on resonancewith the energy gap between the levels | n, s i and | n, p i determined by the gradient (33)with δβ i = 0. Following Eq. (14), the position dependent change of the gradient δβ i gives rise to a position dependent variation of the detuning which reads △ ,i = − e δβ i D n, p | r | n, p E = − M ω z D n, p | r | n, p E N X j = i | u i − u j | = B z N X j = i | u i − u j | . (38)This situation is depicted in Fig. 5. The effective magnetic field B z and the couplingconstant J do not scale independently since both D /e and the matrix element h n, p | r | n, p i are of the order of a n . The precise value depends on the ionic species.For our simulations we use the parameters B z = 0 . J and ¯ h Ω = 0 . J . Initiallythe system is prepared such that the first ion ( k = 1) is in the state |↑i = | n, s i whileall others are in the state |↓i = | n, p i . This state is sketched in Fig. 5b by the solidcircles. The temporal evolution under the Hamiltonian (36) leads to a transfer of theexcitation from the first ion to the last ion in the chain (open circles in Fig. 5b). Thecorresponding numerical data is shown in Fig. 6 where we monitor the time evolutionof the expectation value h S ( i ) z i in a chain of 10 ions. The excitation transfer from thefirst to the tenth ion takes place in a time t = 1 . h/J with an efficiency of 89 %. Since J can be in the order of ¯ h ×
500 MHz this resonant excitation transfer can therefore be rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates |g ñ= | ñ |g ñ= | ñ |n’,p ñ |n,p ñ W w , W w , D D D p D s W p W s |n,s ñ effectivetwo-levelsystemeliminatedsub-space Figure 7.
MW dressing of ground state ions. Two lasers with Rabi frequencies Ω s ,Ω p and detunings ∆ s , ∆ p weakly couple two electronic ground states | g i , | g i tothe Rydberg levels | n, s i and | n, p i , respectively. Thus ions residing in the electronicground states are dressed and obtain time-dependent oscillating dipole moments. achieved in about 3 . In view of the short transition wave length of 100 ...
125 nm associated to a transitionfrom an electronic ground state to a Rydberg level [37], it is experimentally challengingto realize π laser pulses, which transfer the entire electronic population to the Rydberglevels, as required for the initialization step of the chain of effective spins. Thus, weoutline an alternative scheme, which does not require the transfer of the full electronicpopulation to Rydberg states and which is based on an adiabatic admixture of Rydberglevels to two electronic ground states by near-resonant CW lasers.We extend the 3-level scheme of Sec. 2.3 by including two ground states | g i and | g i , which are coupled by two lasers to the (undressed) Rydberg states | n, s i and | n, p i ,respectively (see Fig. 7). As before, we use the MW dressing fields to couple the set ofbare Rydberg states | n ′ , p i , | n, s i and | n, p i . The two CW lasers weakly couple the twoground states to the resulting dressed Rydberg level structure. Weak coupling requiresthat the laser Rabi frequencies are small compared to the MW Rabi frequencies andthat the laser fields are sufficiently far detuned from the three dressed Rydberg states.Due to the laser coupling the two dressed ground states | g ′ i , | g ′ i obtain time-dependentoscillating dipole moments, which finally lead to dipole-dipole interactions between laser-dressed ground state ions. The dressed ground states now constitute the two-level systemof interest (effective spin degree of freedom) and play the role of the dressed Rydberglevels | s ′ i and | n, p i of Sec. 2.3.As shown in Appendix B for a specific set of Rabi frequencies and detunings of theMW and laser fields the representation of the dipole operator in this effective two-levelsystem reads d z = c p, c s, √ D e − iω t c p, c s, √ D e iω t c s, c p ′ , √ D cos( ω t ) ! rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates c s, c p ′ , q D cos( ω t )(1 − S z )+ 2 c p, c s, q D (cos( ω t ) S x + sin( ω t ) S y ) (39)with the factors c α given by Eqs. (B.7). Thus, the resulting dipole operator as wellas the resulting spin chain Hamiltonian describing the dynamics in the dressed groundstates | g ′ i , | g ′ i have the same structure as in Eqs. (22) and (36), with the difference thatthe magnitude of the dipole moments is decreased by the factors c p, c s, and c p ′ , c s, ,respectively.Due to the weak admixture of the Rydberg states the dressed ground stateshave a finite lifetime, i.e. the time until a photon of the dressing lasers is scattered.Denoting the lifetime of the involved Rydberg states by τ = 1 / Γ, this scattering rateis approximately given by Γ scat = | c α | Γ, where c α represents a typical value of theadmixture coefficients in Eq. (B.7). For instance, the lifetime of the state | g ′ i is enhancedby a factor of | c p, | − with respect to the radiative lifetime of the Rydberg state | n, p i ,since only the fraction | c p, | of the electronic population resides in the Rydberg state.Moreover, the fraction of the admixed Rydberg states affects the interaction among theions. Compared to the bare dipole-dipole interaction the interaction between the dressedions is suppressed by a factor ∼ | c α | resulting in an effectively slower spin dynamics.The typical time scale for the spin dynamics, e.g. the excitation transfer discussed inSec. 3.2, thus increases proportional to | c α | − . Hence, comparing the scaling of theinteraction strength and the lifetime implies that the adiabatic admixture of Rydbergstates must not be too weak in order to avoid that decoherence due to spontaneousemission becomes an issue during the temporal evolution of the spin dynamics. In most two-qubit gate schemes, which have been suggested and implemented so far withtrapped ions (see. Refs. [38, 39, 40] for a few examples), the motional sidebands of theions have to be spectroscopically resolved, which limits the achievable gate operationtimes, typically to the order of the inverse external trapping frequency. In order toovercome this limitation, it has been proposed to use specifically shaped trains of off-resonant laser pulses to implement geometric two-qubit ion gates with much faster gatetimes [41]. In the context of quantum information processing with neutral atoms,potentially fast two-qubit gates can be achieved for pairs of Rydberg atoms in anoptical lattice, based on the strong and long-range dipolar interactions among the atoms[2, 8, 42].For the system of trapped Rydberg ions we have shown that despite the absenceof permanent dipole moments MW-dressing fields can be used to generate strongdipole-dipole interactions between Rydberg ions. We now aim to exploit the electronicinteraction Hamiltonian (36) for the implementation of a fast conditional two-qubitphase gate along the lines of the proposals developed for neutral Rydberg atoms.Such gate is characterized by the truth table | g a i m | g b i n → e i ( a − b − φ ent | g a i m | g b i n with a, b = 1 , m, n . Thus, the two-qubit rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates |n,p ñ m |n,s ñ m W w , D s (t) D W s (t)|g ñ m |g ñ m |n,p ñ n |n,s ñ n W w , D s (t) D W s (t)|g ñ n |g ñ n m-th ion n-th ion Figure 8.
Level scheme for the implementation of a conditional two-qubit phasegate. The ground states | g i of the respective ions are coupled via a laser with time-dependent detuning ∆ s ( t ) and Rabi frequency Ω s ( t ) to the Rydberg level | n, s i . AMW field of constant Rabi frequency Ω and ion-dependent detuning ∆ ,m couplesthe Rydberg levels | n, s i and | n, p i . state | g i m | g i n obtains a phase shift, while the other three states are unaffected (up totrivial single qubit phases) [43].We identify the two ground states | g i m and | g i m of each ion m as logical qubitstates | i and | i . The ground state | g i is coupled to the Rydberg state | n, s i by anear-resonant laser with time-dependent effective Rabi frequency Ω s ( t ) and detuning∆ s ( t ) = ω s ( t ) − ( E | n,s i − E | g i ) / ¯ h , which can be chosen equal for both ions m and n .The second ground state | g i is not coupled to any Rydberg state. We again considerthe scenario in which only one additional MW field with Rabi frequency Ω and ion-dependent detunings ∆ ,m (cf. Sec. 3.2) is applied (see Fig. 8), i.e. Ω = 0 and η m = 0 inEq. (36). To perform the gate operation, we apply laser pulses to the two ions (similarlyto the adiabatic gate scheme presented in Ref. [8]). The variation of the laser pulses isassumed to be slow on the time-scale set by Ω s and ∆ s such that the system followsadiabatically the dressed states, which arise from slowly switching on the laser coupling.This adiabaticity condition guarantees that after applying the laser pulses the electronicpopulation still completely resides in the initial electronic ground states. During theapplication of the laser pulses and the resulting dressing of the electronic ground statespart of the electronic population is transferred from the states | g i m and | g i n to theRydberg states, where the ions interact via the resonant dipole-dipole interaction andthereby accumulate an entanglement phase, given by φ ent ( t ) = Z t d τ ( ǫ | g i| g i ( τ ) − ǫ | g i| g i ( τ ) − ǫ | g i| g i ( τ )) (40)where ǫ | g a i m | g b i n denotes the eigenenergy of the instantaneous eigenstate connected tothe state | g a i m | g b i n in the absence of laser pulses. Fig. 9a shows a specific choice for thepulse profile, the time-dependent energies of the dressed states adiabatically connectedto the different ground states. The resulting accumulated phase shifts are presented inFig. 9b. The gate operation time T is approximately two orders of magnitude larger thanthe inverse of the maximum Rabi frequency Ω (0) s of the applied lasers in order to satisfythe adiabaticity condition (for the chosen set of parameters T = 100 / Ω (0) s = 10 µ s). Inorder to minimize imperfections in the gate operation due to spontaneous scattering of rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates M H z Ω s(0) t a Ω s (t) ∆ s (t) 0 50 100−10−505 φ / π Ω s(0) t b φ ent (t) φ (t) φ (t) φ (t) Figure 9.
Two-qubit conditional phase gate between the first and second ion in achain of ten ions. a : Pulse profile: Rabi frequency Ω s ( t ) and detuning ∆ s ( t ) of thedressing lasers. The MW Rabi frequency is chosen Ω = 57 . , = −
279 MHz, ∆ , = −
667 MHz. Dipole-dipole interactionenergy scale J/ ¯ h = 500 MHz (cf. Sec. 3.2). b : Accumulated phase shifts in the dressedstates adiabatically connected to the initial ground states, and resulting entanglementphase φ ent ( t ). photons of the dressing lasers (cf. discussion in Sec. 3.4), the gate operation has to takeplace on a time-scale much faster than the lifetime τ of the involved Rydberg levels, i.e. T ≪ τ . For sufficiently high laser Rabi frequencies and long-lived Rydberg levels thiscondition can be satisfied.
4. Conclusions and Outlook
In this paper we have shown that trapping of Rydberg ions in a linear electric ion trapunder realistic conditions is feasible. We have found that for not too large principalquantum numbers field ionization of the Rydberg ions due to the trapping fields isnegligible, and that the coupling of electronic and external dynamics of the ions results inrenormalized trapping frequencies for ions excited to Rydberg states. We have suggestedto use MW dressing fields in order to generate strong dipolar interactions among theions. The Rydberg excitation dynamics of the MW-dressed ions can be described by aneffective interacting spin-1/2 Hamiltonian. The strong interactions give rise to a typicalcorresponding time scale of the order a few ns, which is substantially shorter than thetypical decoherence time set by the radiative decay of Rydberg states. We have studiedthe dynamical transfer of a Rydberg excitation through the ion chain and discussed theimplementation of fast two-qubit gates. While the laser excitation of ions to Rydbergstates is an experimentally challenging task, the system offers the prospect of studyingcoherent many-body quantum dynamics on fast time scales in a well-controlled andstructured environment.Beyond the present work, trapped Rydberg ions offer a rich playground for furtherstudies of more involved Rydberg dynamics: Combining e.g. ions of different speciesor exciting only a certain number of the trapped ions to Rydberg states offers thepossibility to introduce further inhomogeneities in the spatial distribution of effective rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates
Appendix A. Hamiltonian of N Interacting Rydberg Ions
In this appendix we derive the Hamiltonian for a system of N interacting trappedRydberg ions. Using the single ion Hamiltonian (6) and introducing a label for therespective ion we find H N = N X i H ′ i + 12 N X i,j ( = i ) V ( R i , R j , r i , r j ) . (A.1)with V given by Eq. (28). The external dynamics is governed by the interplay of theharmonic confinement and the Coulomb force between the ions. The correspondingpotential reads V ext = 12 M N X i h ω z Z i + ω ρ ( X i + Y i ) i + 12 e πǫ N X i,j ( = i ) R ij . (A.2)Following Refs. [16, 17] we perform a harmonic expansion of V ext which puts usinto position to describe the external dynamics in terms of uncoupled harmonicoscillators/phonon modes. To this end we first calculate the ionic equilibrium positions R (0) i . In a linear Paul trap with tight transversal confining, ω ρ ≫ ω z , the ions alignalong the z -axis of the trap such that R (0) i = (0 , , Z (0) i ) where the Z (0) i are determinedby ( ∂V ext /∂Z i ) = 0 which leads to the system of equations M ω z Z (0) i = e πǫ N X j ( = i ) Z (0) i − Z (0) j | Z (0) i − Z (0) j | . (A.3) rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates Z (0) i being determined, the harmonic approximation of the potential V ext reads V ext = 12 M X α = x,y,z N X i,j K αij q αi q αj (A.4)with the displacements q xi = X i , q yi = Y i and q zi = Z i − Z (0) i and the coefficient matrix K αij = ω α − c α e πǫ M P Nk ( = i ) 1 | Z (0) i − Z (0) k | for i = jc α e πǫ M | Z (0) i − Z (0) j | for i = j where c x,y = 1 , c z = −
2. The vibrational dynamics of the 1D-chain is then describedby H ph = 12 M X α = x,y,z N X i,j K αij q αi q αj + 12 M X α = x,y,z N X i ( P αi ) . (A.5)The Hamiltonian H ph can be brought into diagonal form by introducing phonon modesvia the orthogonal matrices M α , q αi = N X n M αi,n q M ω α,n / ¯ h ( a † α,n + a α,n ) , (A.6) P αk = i N X n M αk,n q / (¯ hM ω α,n ) ( a † α,n − a α,n ) , (A.7) N X i,j M αi,n K αij M αj,m = ω α,n δ n,m , (A.8)which leads to Eq. (31) for the phonon dynamics.We now proceed by expanding the charge-dipole, the charge-quadrupole and thedipole-dipole interaction around the equilibrium positions of the ions. The charge-quadruple and the dipole-dipole interaction can be approximated by H cq = 12 e πǫ N X i = j r i − n ij · r i ) R ij ≃ e πǫ N X i = j r i − z i | Z (0) i − Z (0) j | (A.9) H dd = 12 e πǫ N X i = j r i · r j − n ij · r i )( n ij · r j ) R ij ≃ e πǫ N X i = j r i · r j − z i z j | Z (0) i − Z (0) j | . (A.10)The charge-quadrupole term leads a position dependent variation of the electric fieldand can be absorbed in the single particle ion-field interaction H ef which is given byEq. (12). The electronic Hamiltonian of the i -th ion then assumes the form of Eq. (32)with the ion-dependent gradient (33).In order to treat the charge-dipole coupling we introduce the compact notation r xi = x i , R xi = X i , . . . which enables us to write H cd = 12 e πǫ N X i = j ( R i − R j )( r i − r j ) R ij (A.11) rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates ≃ e πǫ N X i = j Z (0) i − Z (0) j | Z (0) i − Z (0) j | ( z i − z j )+ X α = x,y,z N X i,j ( = i ) e πǫ | Z (0) i − Z (0) j | c α ( r αi − r αj ) q αi . It turns out that by making use of the equilibrium condition (A.3) one can combine thisexpression with the term P i H CM − el ,i that arises from the summation of the single ionterms of the Hamiltonian (6). The combination then includes all couplings between theinternal and external dynamics. It reads H int − ext = − eα cos( ωt ) N X i [ q xi x i − q yi y i ] + 12 M X α = x,y,z N X i,j K zij c α r αj q αi . (A.12)Finally, neglecting the micro-motion, the complete Hamiltonian of N Rydberg ions inthe linear Paul trap is given by Eq. (31).
Appendix B. Effective Dipole Moment of Laser-Dressed Ground State Ions
In this appendix we derive the form (39) of the dipole operator for laser-dressed groundstate ions. The Hamiltonian for the system of five coupled electronic levels as depictedin Fig. (7) in the rotating frame defined by the transformation U ( t ) = e − iω s t | g ih g | + e i ( ω − ω p ) t | g ih g | + e − iω t | n ′ , p ih n ′ , p | + | n, s ih n, s | + e iω t | n, p ih n, p | (B.1)and in rotating-wave-approximation is given by H = H + H pert , H = ¯ h ∆ s | g ih g | + ¯ h ∆ p | g ih g | + ¯ h ∆ | n ′ , p ih n ′ , p | − ¯ h ∆ | n, p ih n, p | + ¯ h | n ′ , p ih n, s | + Ω | n, s ih n, p | + h . c . ) , (B.2) H pert = ¯ h s | g ih n, s | + Ω p | g ih n, p | + h . c . ) . (B.3)The MW detunings ∆ , and Rabi frequencies Ω , are defined as in Sec. 2.3, and thedetunings and Rabi frequencies characterizing the two additional laser fields are givenby ∆ s = ω s − ( E | n,s i − E | g i ) / ¯ h , ∆ p = ω p − ( E | n,p i − E | g i ) / ¯ h − ∆ and Ω s , Ω p . Weassume that the near-resonant MW fields are much stronger than the two lasers suchthat they determine the level structure of the dressed Rydberg states. Therefore wetreat the coupling to the ground states as a perturbation H pert of the system. We nowapply a canonical transformation to the Hamiltonian [44], e − S H e S = H + H pert + [ H , S ] + [ H pert , S ] + . . . (B.4)Choosing S such that [ H , S ] = − H pert guarantees that to first order in the perturbation H pert the transformed Hamiltonian is block-diagonal and that the two ground statesbecome decoupled from the Rydberg states. The transformation yields two dressedground states, | g ′ , i ≃ (1 + S ) | g , i , | g ′ i = | g i + c p ′ , | n ′ , p i + c s, | n, s i + c p, | n, p i (B.5) | g ′ i = | g i + c p ′ , | n ′ , p i + c s, | n, s i + c p, | n, p i (B.6) rapped Rydberg Ions: From Spin Chains to Fast Quantum Gates c p ′ , c s, c p, = γ s − (∆ s + ∆ )Ω − ∆ s )(∆ s + ∆ )(∆ − ∆ s )Ω c p ′ , c s, c p, = γ p − Ω Ω − ∆ p )Ω − ∆ p )∆ p + Ω (B.7)where γ s,p = Ω s,p (∆ s,p + ∆ )[4(∆ − ∆ s,p )∆ s,p + Ω ] + (∆ s,p − ∆ )Ω . (B.8)For the special choice of laser detunings∆ s = − ∆ , ∆ p = 12 (cid:18) ∆ − q ∆ + Ω (cid:19) (B.9)we find c p ′ , c s, c p, = − Ω s Ω , c p ′ , c s, c p, = Ω Ω p Ω (cid:16) ∆ + √ ∆ +Ω (cid:17) − Ω p Ω . (B.10)Thus, for this set of parameters, the Rydberg state | n, p i is exclusively admixed to theground state | g i , while the second ground state obtains a small fraction of the Ryd-berg states | n ′ , p i and | n, s i . This implies that the dressed ground state | g ′ i possesses anon-vanishing dipole moment, which oscillates at the MW frequency ω , and that thereexists also a transition dipole matrix element between the two dressed ground states.We now identify the two dressed ground states | g ′ i , | g ′ i with the eigenstates of the S z spin operator along the lines of Sec. 2.3 and obtain the dipole operator (39). [1] Gallagher T F 1984 Rydberg Atoms (Cambridge University Press)[2] Lukin M D, Fleischhauer M, Cˆot´e R, Duan L M, Jaksch D, Cirac J I and Zoller P 2001
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