Quantum spin models with electrons in Penning traps
aa r X i v : . [ qu a n t - ph ] J un Quantum spin models with electrons in Penning traps
G. Ciaramicoli, I. Marzoli, and P. Tombesi
Dipartimento di Fisica, Universit`a degli Studi di Camerino, 62032 Camerino, Italy (Dated: October 28, 2018)
Abstract
We propose a scheme to engineer an effective spin Hamiltonian starting from a system of elec-trons confined in micro-Penning traps. By means of appropriate sequences of electromagneticpulses, alternated to periods of free evolution, we control the shape and strength of the spin-spininteraction. Moreover, we can modify the effective magnetic field experienced by the particle spin.This procedure enables us to reproduce notable quantum spin systems, such as Ising and XY models. Thanks to its scalability, our scheme can be applied to a fairly large number of trappedparticles within the reach of near future technology. PACS numbers: 03.65.-w, 03.67.Ac, 03.67.Lx, 75.10.Jm . INTRODUCTION Single electrons confined in Penning traps may represent a valid, experimentally viablesystem for the implementation of a quantum processor [1, 2, 3]. Our proposals have beenencouraged by the astonishing results obtained in high precision experiments with a singleelectron [4, 5, 6, 7, 8] and by the advances in trapping technology, from micro-traps [9] toscalable open planar Penning traps [10, 11]. In this spirit, it has also been put forward howto realize a quantum information channel, based on interacting spin chains, by means oftrapped electrons [12, 13].In this paper we focus on a linear array of electrons, each one confined in a micro-Penningtrap. Our aim is to prove that, from the same physical system, we can derive a variety ofinteracting spin models. In particular, we show how to design and control the relevantterms in the effective spin Hamiltonian. As a result a system of trapped electrons can beexploited to study the dynamics of a wide range of quantum spin models. We recall thatthese models are very important for the understanding of the rich phenomenology observedin several quantum many-body systems, such as quantum magnets and high temperaturesuperconductors. Moreover quantum spin systems are able to exhibit quantum phase tran-sitions [14]. To this end, it is critical to control and vary system parameters like the appliedmagnetic field and the spin-spin coupling strength. Our method to shape the effective spin-spin interaction employs sequences of electromagnetic pulses alternated to periods of freeevolution. This technique is similar to the refocusing schemes used in nuclear magneticresonance (NMR) experiments and relies, from the theoretical point of view, on the averageHamiltonian theory [15]. We point out that a system of trapped electrons presents severaladvantages over NMR implementations. The most important ones are scalability and thepossibility to independently adjust the values of relevant quantities, like the spin precessionfrequencies and the spin-spin coupling. Indeed their values depend on external parameterssuch as the magnetic field gradient, the inter-particle distance and the voltage applied tothe trap electrodes [3, 12].Also other systems, like linear or planar arrays of trapped ions, enjoy some of theseproperties and, therefore, it has been proposed to use them as a quantum simulator forinteracting spin chains [16, 17, 18]. However working with trapped electrons we naturallyhave a system of spin one-half particles, without the need for artificially creating an effec-2ive two-level system. Another difference between trapped ions and electrons relies in thetypical resonance frequencies. Ions are controlled by means of a sophisticated laser setup,while trapped electrons are manipulated by microwave or radio-frequency fields. In thisrespect trapped electrons can benefit from the same technology already developed for NMRspectroscopy.In this paper we consider both the case of electrons with the same spin precession fre-quency [12] and the case of electrons with different spin precession frequencies [3]. Thefrequency addressability, which is necessary to manipulate specific particles in the array, isobtained with the insertion of a magnetic field gradient. However this condition is requiredonly to modify the interaction range and topology. In the case of electrons with the samespin precession frequency, we prove that, by flipping the spin state twice, we can effectivelyreduce or even cancel the spin dynamics due to the external uniform magnetic field. Thisway the effective spin system is subjected to a weaker magnetic field, whose intensity can godown to zero, without affecting the overall trap stability. The same resonant electromagneticfield, used to flip the spin, is able to produce coherent superpositions of the two spin statesby adjusting its phase and duration. These operations are the building blocks of specificpulse sequences that allow to engineer the effective spin Hamiltonian. By iterating suchpulse sequences, we obtain various interesting spin Hamiltonians such as the Ising model orthe XY model. In addition, if we want to customize the interaction range and coordinationnumber, we should apply similar sequences of pulses to selected subsets of spins in the ar-ray. The resulting spin system can exhibit a nearest neighbor (NN) as well as a long rangeinteraction. The number of pulses in each sequence is relatively small and, most notably,does not depend on the number of spins in the array, thus making our procedure scalable.The paper is organized as follows. In Sec. II we briefly present the system of trappedelectrons and review the derivation of the effective spin-spin interaction. In Sec. III wedescribe how to prepare and manipulate, with an additional oscillating magnetic field, thespin state of each electron in the array. In Sec. IV we show how to engineer the spinHamiltonian by applying appropriate electromagnetic pulse sequences, that allow to controlthe strength and the range of the interaction. The capability of our technique to reproducea given Hamiltonian is analyzed in Sec. V. Finally in Sec. VI we summarize our results anddiscuss future perspectives. The more technical details, concerning the design of the pulsesequences and the estimate of fidelity, are reported, respectively, in Appendices A and B.3 I. ARRAY OF TRAPPED ELECTRONS
Let us consider a system of N electrons confined in an array of micro-Penning traps inthe presence of linear magnetic gradients. The Hamiltonian of the system can be written as H = N X i =1 H NCi + N X i>j H Ci,j , (1)where H NCi = ( p i − e A i ) m e + eV i − ge ¯ h m e σ i · B i (2)represents the single electron dynamics inside a trap and H Ci,j = e πǫ | r i − r j | (3)describes the Coulomb interaction between electrons i and j . In Eqs. (2) and (3) m e , e , g , and σ i are, respectively, the electron mass, charge, gyromagnetic factor, and Pauli spinoperators. We assume that the micro-traps are aligned along the x axis and that x i, is theposition of the center of the i -th trap. The electrostatic potential V i ( x i , y i , z i ) ≡ V z i − [( x i − x i, ) + y i ] / ℓ (4)is the usual quadrupole potential of a Penning trap, where V is the applied potential dif-ference between the trap electrodes and ℓ is a characteristic trap length. The magnetic field[3] B i ≡ − b x i − x i, ) i + y i j ] + ( B ,i + bz i ) k (5)is the sum of the trapping magnetic field B ,i k , providing the radial confinement, with alocal linear magnetic gradient b around the i -th trap. The associated vector potential A i ≡
12 ( B ,i + bz i )[ − y i i + ( x i − x i, ) j ] (6)preserves the cylindrical symmetry of the unperturbed trapping field.Following the approach described in [3, 4, 12] the Hamiltonian, Eq. (2), of a single electroncan be written as H NCi ≃ − ¯ hω m,i a † m,i a m,i + ¯ hω c,i a † c,i a c,i + ¯ hω z a † z,i a z,i + ¯ h ω s,i σ zi + g ε ¯ hω z (cid:16) a z,i + a † z,i (cid:17) σ zi − g ε ¯ hω z s ω z ω c,i (cid:16) σ (+) i a c,i + σ ( − ) i a † c,i (cid:17) (7)4here the annihilation operators a m,i , a c,i , a z,i refer, respectively, to the magnetron, cyclotronand axial oscillators of the i -th electron and σ ( ± ) i ≡ ( σ xi ± iσ yi ) /
2. The frequencies of the dif-ferent electron motions are ω m,i ≃ ω z / (2 ω c,i ), ω c,i ≃ ( | e | B ,i /m e ) − ω m,i , ω z = q eV / ( m e ℓ )and ω s,i ≡ g | e | B ,i / (2 m e ). The Hamiltonian (7) has been obtained under the assumptions ω m,i ≪ ω z ≪ ω c,i and b | z i | /B ,i ≪
1. We also assume that the cyclotron motion is in theground state and the amplitude of the magnetron motion is sufficiently small (axialization)[19]. The dimensionless parameter ε ≡ | e | bm e ω z s ¯ h m e ω z (8)represents the coupling, due to the magnetic gradient, between internal and external degreesof freedom of the particle.Similarly, if the oscillation amplitude of the electrons is much smaller than the inter-trapdistance, the part of the Hamiltonian describing the Coulomb interaction can be written as[12] H Ci,j ≃ ¯ hξ i,j ( a z,i + a † z,i )( a z,j + a † z,j ) − ¯ hξ i,j ω z ω c,i (cid:16) a c,i a † c,j + a † c,i a c,j (cid:17) , (9)where ξ i,j ≡ e / (8 πǫ m e ω z d i,j ) with d i,j being the distance between the i -th and j -th particle.Now we apply to the system Hamiltonian the unitary transformation [20] S = N X i =1 g ε " σ zi ( a † z,i − a z,i ) + ω z ω a,i s ω z ω c,i (cid:16) σ ( − ) i a † c,i − σ (+) i a c,i (cid:17) , (10)with ω a,i ≡ ω s,i − ω c,i . This transformation formally removes, to the first order in ε , theinteraction between the internal and the external degrees of freedom in Hamiltonian (7)and, at the same time, introduces a coupling between the spin motions of different electrons.Consequently the spin part of the system Hamiltonian can be recast as [12] H s ≃ N X i =1 ¯ h ω s,i σ zi + ¯ h N X i>j (cid:16) J zi,j σ zi σ zj − J xyi,j σ xi σ xj − J xyi,j σ yi σ yj (cid:17) , (11)where J zi,j = (cid:18) g (cid:19) ξ i,j ε = (cid:18) g (cid:19) ¯ he πε m e b ω z d i,j , (12) J xyi,j = (cid:18) g (cid:19) ξ i,j ε ω z ω a,i ω c,i = (cid:18) g (cid:19) ¯ he πε m e b ω a,i ω c,i d i,j . (13)5he effective spin Hamiltonian (11) exhibits a long range interaction between all the particlesin the chain. The coupling strength decreases with the third power of the distance betweenparticles, i.e. with a dipole-like behavior. Moreover, J zi,j and J xyi,j depend, respectively, on theaxial frequency and the cyclotron and anomaly frequencies. Since the trapping frequenciesform a well defined hierarchy, the coupling in the longitudinal and transverse direction canbe utterly different. For example, for typical experimental values of the cyclotron and axialfrequencies, such as ω c / π ≃
100 GHz and ω z / π ≃
100 MHz, the ratio J xyi,j /J zi,j is lessthan 10 − . Therefore, for practical purposes J xyi,j is often negligible with respect to J zi,j . Inparticular, this is true when the difference between the spin frequencies of different particlesis much larger than their xy spin-spin coupling strength. In this case the spin Hamiltonianreduces to H s ≃ N X i =1 ¯ h ω s,i σ zi + ¯ h N X i>j J zi,j σ zi σ zj . (14)In Hamiltonian Eq. (14) we have used the rotating wave approximation (RWA) to neglectthe interactions between spins along the x and y directions, since they give rapidly rotatingterms. The Hamiltonian (14) is, therefore, similar to the nuclear spin Hamiltonian of themolecules used to perform NMR experiments [15]. However in NMR systems the spinfrequency differentiation and the spin-spin couplings are determined by the chemical natureof the molecules, whereas in our system they depend on the value of the applied fields, thatare under control of the experimenter. III. SPIN STATE MANIPULATION
In this section we describe how to prepare and manipulate the spin state with an externaloscillatory field. Let us consider a magnetic field b p ( t ) oscillating in the xy plane withfrequency ω and phase θ such that b p ( t ) = b p [ i cos( ωt + θ ) + j sin( ωt + θ )] . (15)If we add this field to the system, the spin Hamiltonian, Eq. (11), becomes (here and in therest of the paper we set ¯ h = 1) H ≃ N X j =1 ω s,j σ zj + χ N X j =1 [ σ (+) j e − i ( ωt + θ ) + σ ( − ) j e i ( ωt + θ ) ] , (16)6ith χ ≡ g | e | b p / (2 m e ). In deriving the Hamiltonian (16), we assumed that the interactionbetween the electrons and the oscillating magnetic field is much stronger than the spin-spincoupling. Hence, the terms in Eq. (11) proportional to J zi,j and J xyi,j can be neglected. In thecase of a system with spin frequency differentiation the field (15), applied for an appropriatetime t with frequency ω = ω s,j , affects only the spin states of the resonant j -th electron | ↓i j → e i ( ω s,j / t cos (cid:18) χt (cid:19) | ↓i j − ie − i ( ω s,j / t − iθ sin (cid:18) χt (cid:19) | ↑i j , (17) | ↑i j → e − i ( ω s,j / t cos (cid:18) χt (cid:19) | ↑i j − ie i ( ω s,j / t + iθ sin (cid:18) χt (cid:19) | ↓i j . (18)Without spin frequency differentiation, the single qubit addressing with microwave radiationis, of course, no longer possible. Therefore when all the spins have the same precessionfrequency ω s , a single resonant pulse suffices to produce the evolution of Eqs. (17) and (18)for each particle in the array. From Eqs. (17) and (18) we see that by changing durationand phase of the applied pulse we can prepare and manipulate at will the spin states of thetrapped electrons. In particular, if we apply a pulse for a time ¯ t = π/χ with θ = 0, we canflip the spin state of each particle | ↓i j → − ie − i ( ω s,j / t | ↑i j , (19) | ↑i j → − ie i ( ω s,j / t | ↓i j . (20)We define this transformation as F ≡ N O j =1 {− i [ σ (+) j e − i ( ω s,j / t + σ ( − ) j e i ( ω s,j / t ] } . (21)It is not difficult to verify that the inverse transformation F − is obtained with a pulse ofthe same duration ¯ t but with phase π .If we move to the interaction picture (IP) with respect to the Hamiltonian P Ni =1 ( ω s,i / σ zi ,the system evolution is given by Eqs. (17) and (18) with ω s,j = 0. Consequently, the spinflip operation, Eqs. (19) and (20), turns into | ↓i j → − i | ↑i j , (22) | ↑i j → − i | ↓i j . (23)The above transformations correspond to the application of the operator − iσ xj . In a similarway a pulse applied for the time ¯ t with θ = π/ − iσ yj . Furthermore, always working in IP, if the pulse is applied for atime ¯ t/
2, we can obtain the pseudo-Hadamard operations G x ≡ N O j =1 ( − iσ xj ) √ θ = 0 , (24) G † x ≡ N O j =1 ( iσ xj ) √ θ = π, (25) G y ≡ N O j =1 ( − iσ yj ) √ θ = π , (26) G † y ≡ N O j =1 ( iσ yj ) √ θ = − π . (27)The coherent superposition of the spin states | ↑i , | ↓i for each particle can be achieved witha single multi-frequency pulse. Hence, an appropriate choice of the frequency, duration andphase of the pulses allows for performing, apart from irrelevant phase factors, single qubitoperations on each spin of the array. IV. ENGINEERING THE SPIN HAMILTONIAN
In this section we show that, by using the additional magnetic field (15), we can alsoadjust and control the form of the effective spin Hamiltonian, starting from the modelsgiven by Eqs. (11) and (14). This is achieved by applying to the system specific sequencesof pulses alternated to periods of free evolution. Our approach is inspired to the refocusingschemes used in NMR experiments [15]. Similarly to this technique a key point is the choiceof the different time scales. Spin operations, operated by means of pulses, should be virtuallyinstantaneous with respect to the free evolution of the system. Therefore, the pulse durationshould be much shorter than the free evolution time.
A. Tuning of the effective magnetic field
The spin Hamiltonian, Eq. (11), in the case of spins with the same precession frequencycan be recast as H s ≃ H + H c , (28)8here H ≡ N X i =1 ω s σ zi , (29) H c ≡ N X i>j (cid:16) J zi,j σ zi σ zj − J xyi,j σ xi σ xj − J xyi,j σ yi σ yj (cid:17) . (30)In the following we shall prove that, by sending resonant pulses of the kind of Eq. (15),it is possible to reduce or even cancel the effects on the spin dynamics of the Hamiltonianterm H . This result corresponds to an effective modulation of the external magnetic field,without affecting the trapping stability of the whole set up.In particular, by applying a sequence consisting of a pulse producing the spin flip trans-formation F , Eq. (21), followed by a period of free evolution t and by a pulse producing theinverse transformation F − , we can change the sign of the Hamiltonian term H F − e − iH s t F = exp[ − i ( − H + H c ) t ] . (31)To prove Eq. (31) we use the identity F − e − iH s t F = exp[ − i ( F − H F + F − H c F ) t ] . (32)Now we have F − H F = N X j =1 [ σ (+) j e − i ( ω s / t + σ ( − ) j e i ( ω s / t ] (cid:18) ω s σ zj (cid:19) [ σ (+) j e − i ( ω s / t + σ ( − ) j e i ( ω s / t ]= − N X j =1 ω s σ zj = − H . (33)The identity F − H c F = H c follows from the commutation relation [ H c , F ] = 0, which can beverified with some algebra. Moreover we observe that [ H , H c ] = 0, because the interactionHamiltonian preserves the total magnetization P Ni =1 σ zi . From this last consideration andfrom Eq. (31) we find F − e − iH s t F e − iH s t = exp[ − iH eff ( t + t )] , (34)with H eff ≡ t − t t + t H + H c . (35)The left hand side of relation (34) represents a sequence consisting of a period t of freeevolution, a pulse producing the transformation F , a period t of free evolution and a pulse9roducing F − . From the right hand side of Eq. (34), we see that this sequence is equivalentto the system evolution for the total time t + t according to the Hamiltonian H eff . Hence,we can obtain an effective reduction, by a factor ( t − t ) / ( t + t ), of the Hamiltonian term H . This result can be viewed as a decrease of the magnitude of the uniform magnetic fieldas far as the electron spin dynamics is concerned. Notice that for t = t we can completelysuppress the dynamical effects due to the term H . B. Design and control of the spin-spin coupling
Let us now consider a system with spin frequency differentiation. If we add another fieldconsisting of a superposition of terms resonant with the spin frequencies b s ( t ) = N X k =1 b s [ i cos( ω s,k t ) + j sin( ω s,k t )] , (36)the spin Hamiltonian of Eq. (14) becomes in IP with respect to P Ni =1 ( ω s,i / σ zi H IP ≃ H z + H bs , (37)with H z ≡ N X i>j J zi,j σ zi σ zj , (38) H bs ≡ η N X i =1 (cid:16) σ (+) i + σ ( − ) i (cid:17) = η N X i =1 σ xi , (39)where η ≡ g | e | b s / (4 m e ). Hence, the application of the oscillating field (36) gives rise to aneffective static transverse magnetic field, whose strength can be controlled and modified,since it depends on the field amplitude b s . This tool may turn out useful in reproducingquantum models like Ising system of spins. In this case the parameter η should be comparableto the coupling strength J zi,j between the spins.Moreover, we can engineer the spin-spin coupling, that is introduce an effective spin-spininteraction along the x and y axes. This is achieved by means of sequences of pulses, of thekind given in Eqs. (24), (25), (26), and (27), affecting all the spins in the array. Indeed, itcan be easily proved that G x e − iH z t G † x = e − iG x H z G † x t = exp − i N X j>k J zj,k σ yj σ yk t , (40) G y e − iH z t G † y = e − iG y H z G † y t = exp − i N X j>k J zj,k σ xj σ xk t . (41)10ence a sequence of two specific pulses, alternated to a period of free evolution under theHamiltonian H z , effectively modifies the direction of the spin-spin coupling. Now, if wecombine the three operations (39), (40), and (41) we have that for t , t , t ≪ π/J zi,i +1 , π/ηe − i ( H z + H bs ) t ( G y e − iH z t G † y )( G x e − iH z t G † x ) ≃ e − iH eff ( t + t + t ) , (42)with H eff = τ η N X i =1 σ xi + τ N X i>j J zi,j σ zi σ zj + τ N X i>j J zi,j σ xi σ xj + τ N X i>j J zi,j σ yi σ yj , (43)where τ i = t i / ( t + t + t ). In deriving relation (42) we used the approximate identity [21] e − iA t e − iA t . . . e − iA n t n ≃ e − i ( τ A + τ A + ... + τ n A n ) t (44)with t = P ni =1 t i and τ i = t i /t , which is valid, to first order in t , for t i much shorterthan the typical time scale of the dynamics due to the Hamiltonian A i . However, moreelaborate sequences of pulses (see Appendix B) give approximations to higher orders in t [22]. A recursive application of the sequence (42) determines an effective evolution under theHamiltonian H eff . We point out that, in this Hamiltonian, Eq. (43), we can independentlycontrol and change the values of the parameters τ i ’s and η , since they depend, respectively,on the free evolution times t i ’s and on the pulse amplitude b s . Consequently we obtain anHamiltonian H eff with a variable relative strength of the spin-spin coupling in the x , y , and z directions and a tunable transverse magnetic field. Notice that we can also set τ i = 0for any desired i or η = 0. This is achieved by simply choosing t i = 0 or switching off theexternal field b s ( t ). In this way various interesting quantum spin models can be derivedfrom Hamiltonian (42). For example for τ = τ = 0 we obtain the Ising model, whereas for τ = 0 ( τ = 0) we obtain the XY model in its usual (rotated) basis. C. How to modify the interaction range and topology
The spin Hamiltonian (38) can be written as N X i>j J zi,j σ zi σ zj ≡ H z + H z + . . . , (45)where H zn ≡ N − n X i =1 J zi,i + n σ zi σ zi + n (46)11epresents the coupling between the n -th nearest neighbor spins. In our system the inter-action between spins has a dipole-like nature, i.e. it decreases with the third power of theinter-particle distance. Consequently only the first few terms H zn at the right hand side ofEq. (45) play a significant role. In the following we are going to outline a procedure to in-dependently control and modify, in a relatively simple way, the strength and the sign of therelevant H zn terms. In other words we can design the interaction topology by enhancing orsuppressing the coupling between the n -th nearest neighbors. This is achieved by iterativelyapplying to the system appropriate sequences consisting of pulses alternated to periods offree evolution. In our scheme each pulse affects simultaneously a specific subset of spins inthe array. Notice that if a particular pulse sequence S modifies the spin Hamiltonian H z , wecan extend the same kind of coupling to the other directions by simply performing, accordingto Eqs. (40) and (41), the sequences G y SG † y and G x SG † x . Therefore, we restrict ourselvesto the transformations affecting the spin-spin coupling along the z direction. Furthermore,we are going to prove that the number of pulses in each sequence does not depend on thenumber of spins in the array, thus making our technique scalable with the system size.As an example, we describe how to suppress the second nearest neighbor interaction bymaking use of three different transformations, defined as σ xo , σ xc and σ xc . Each of themcan be performed by a single multi-frequency pulse. The transformation σ xo consists in thesimultaneous application of σ x to all the spins in the odd sites of the array. The transfor-mations σ xc and σ xc flip, instead, alternated couples of neighboring spins. In particular, σ xc affects the spin couples { , } , { , } , . . . , whereas σ xc affects the couples { , } , { , } , . . . .We prove in Appendix A that the sequence σ xc e − iH z t σ xo e − iH z t σ xc e − iH z t (47)corresponds to the system evolution for a time t under the effective Hamiltonian H eff ≃ H z + H z , where the coupling of each spin with its second nearest neighbors has been removed.Indeed, since the term H z is small, the above sequence well approximates an effective NNHamiltonian. The transformations σ xo , σ xc and σ xc are the building blocks to construct othersequences which realize different kinds of spin Hamiltonians. For example, as described inAppendix A, we can easily invert the sign of H z , thus switching from an anti-ferromagnetic toa ferromagnetic interaction, or make the coupling strength between first and second nearestneighbors equal. This last case corresponds to an effective change in the array topology,12 FIG. 1: Schematic drawing of a linear chain of N spins (upper part). When the first and secondnearest neighbor coupling strengths are made equal, the linear chain becomes equivalent to a planararray (lower part). since the number of nearest neighbors passes from two (linear chain) to four (see Fig. 1).In order to affect the coupling between neighboring spins of order higher than three, weshould apply simultaneously the transformation σ x to selected subsets of three or more spinsalong the chain. For example, as we show in Appendix A, with a seven pulse sequence wecan suppress the interaction between a spin and all its neighbors from the second up to thesixth nearest neighbors. In such a way we improve our approximation of a NN interactingspin chain. V. FIDELITY
In this section we discuss the performances of the scheme, based on the detailed analysisreported in Appendix B. We emphasize that our treatment focuses on the limitations due tothe mapping of the system of trapped electrons into the desired target system of interactingspins. Hence, most experimental imperfections are not considered here.As described in the previous section, to derive the effective spin Hamiltonian we makeuse of the approximate identity (44) or of more sophisticated approximations [22]. As aconsequence we introduce an error [21]
E ≡ k U − U ′ k ≡ max | ψ i : || ψ i| =1 | ( U − U ′ ) | ψ i| , (48)which measures the distance between the desired evolution U and the approximated evolu-tion U ′ . For instance, in our case the target unitary operator U = exp( − iH eff t ), with theeffective spin Hamiltonian H eff of the kind of Eq. (43), is approximated to the fourth orderin t by the sequence S , Eq. (B2) [22]. By using some algebra (see Appendix B), we can13ound the error associated to the application of a single sequence S with E S ≤ ( J z t ) f ( N ) , (49)where J z ≡ J zi,i +1 is the NN coupling strength and f ( N ) is, in good approximation, anincreasing linear function of the number N of electrons in the array. The exact form of f ( N )depends on the specific target spin Hamiltonian. From this result, we see that the error issmall whenever the time evolution is much shorter than the flipping time, i.e. t ≪ π/J z . Weprove in Appendix B that, if we iterate m times the sequence S the total error is E ≤ m E S .Therefore, if we want to simulate the system evolution for a given time T = mt , to keep theaccuracy high we should apply the same sequence S m times
E ≤ ( J z T ) f ( N ) m . (50)For a given simulation time T and coupling strength J z , the error E decreases with thenumber of iterations m and, therefore, with the total number of pulses.In appendix B we provide the explicit expression of f ( N ) for the XY and NN Isingmodels. Consequently we are able to estimate the upper bound of E in both cases. In ouranalysis, we also take into account the error introduced by the derivation of the effectivespin-spin coupling [12]. In particular, the error E c due to the canonical transformation (10)satisfies the relation E c ≤ N (cid:18) ¯ k + 12 (cid:19) ε , (51)where ¯ k is the mean axial oscillator excitation number. We consider an array of 50 electronswith inter-particle distance of 100 µ m, ω s / π = 100 GHz, ω z / π = 160 MHz, and a magneticgradient b ≃
200 T/m. With these parameters we obtain, according to Eq. (12), a NNcoupling constant J z = 10 Hz. We also assume that the spin frequencies of neighboringelectrons differ of about 2 MHz, each pulse has a duration of the order of µ s and the axialmotion is cooled to the ground state. In order to simulate the XY model (NN Ising model)for a time T = 1 s with fidelity of 99%, we need to iterate the specific sequence S about100 (50) times. In particular the simulation of the XY model requires about 3000 pulses,whereas the Ising model with NN coupling requires about 2000 pulses. Notice that the Isingmodel with dipole-like coupling requires no pulse sequence, since it is obtained, directly, byapplying the field (36). Therefore, in this case we only take into account, as a source oferror, the thermal excitation of the axial oscillator which, according to Eq. (51), is of theorder of 10 − . 14 I. CONCLUSIONS
In this paper we have proposed a scalable technique for easily controlling and adjustingthe effective Hamiltonian of a system of interacting spins. The underlying physical systemconsists of an array of trapped electrons in micro-Penning traps. The electron spin is pre-pared and manipulated with an external resonant magnetic field. These spin operations,applied to all the particles or subsets of them, are alternated to periods of free evolution in afashion similar to NMR refocusing schemes. To selectively address the electrons in the array,it is necessary to introduce a detuning between the characteristic spin frequencies by meansof a magnetic gradient. In particular, we have shown that, in the case of a system withoutspin frequency differentiation, a two pulse sequence permits to reduce or even cancel theeffect on the spin dynamics of the uniform magnetic field, without affecting the overall trapstability. This is potentially useful for the observation of quantum phase transitions [14],where it is important to modulate the ratio between the external magnetic field and the spin-spin coupling strength. In the case of a system with spin frequency differentiation, we haveproved that with a repeated application of appropriate pulse sequences we can modify andcontrol the interaction terms in the effective spin Hamiltonian. As a result a wide range ofspin Hamiltonians can be obtained, such as the Ising model and the XY models. Moreover,specific pulse sequences allow to control the sign and strength of the coupling between the k -th nearest neighbors for any significant value of k (first, second, . . . , nearest neighbors).As an example, we provide a prescription to obtain an Hamiltonian with substantially onlyNN coupling starting from a dipole-like interaction. In our scheme the number of pulses ineach sequence is relatively small and does not depend on the number of spins in the array.We derive an analytical formula to estimate the fidelity of our method for simulating theeffective spin Hamiltonian, as a function of the coupling strength, the simulation time andthe number of particles. Our estimates show that it is feasible to simulate the Ising, with NNcoupling, and the XY model with fidelity of 99% for a system of 50 electrons with a couplingstrength J z = 10 Hz. Of course, the evaluation of the performances of a real experimentwould require a closer analysis of all the possible sources of errors and decoherence. This is,however, beyond the scope of the present work.15 PPENDIX A
In this appendix, we are going to prove that spin flip operations, applied to subsets ofparticles in the array, result in an effective sign change in the interaction between neighborsof arbitrary order. The starting point is represented by the relation σ xi σ zi σ xi = − σ zi , (A1)which reverts the sign of the i -th spin operator. We define the following operators σ xo ≡ N/ O i =1 σ x i − , (A2) σ xc k ≡ O i ∈ c k σ xi σ xi +1 with c k = { k, k + 4 , k + 8 , . . . } for k = 1 , , (A3) σ x T k ≡ O i ∈T k σ xi σ xi +1 σ xi +2 with T k = { k, k + 6 , k + 12 , . . . } for k = 1 , , , (A4) σ x Q k ≡ O i ∈Q k σ xi σ xi +1 σ xi +2 σ xi +3 with Q k = { k, k + 8 , k + 16 , . . . } for k = 1 , , , , (A5)that affect simultaneously different subsets of spins. Moreover, we observe that( σ xo ) = ( σ xc k ) = ( σ x T k ) = ( σ x Q k ) = k , so that we can use the identity Ae B A = exp ( ABA ) , (A7)which holds true for any pair of operators A and B , whenever A = σ xo it follows that σ xo σ zi σ zj σ xo = ( − i + j σ zi σ zj , (A8)which amounts to a sign change in the interaction between spins with different parity. Con-sequently, given the Hamiltonian H z of Eq. (38), the transformation σ xo H z σ xo inverts thecoupling between neighbors of odd orders σ xo e − iH z t σ xo = e − iσ xo H z σ xo t = exp " − i N X k =1 ( − k H zk t . (A9)This property allows us to make equal in strength the coupling between first and secondnearest neighbors σ xo e − iH z t σ xo e − iH z t ≃ exp − iJ ′ N − X j =1 σ zj ( σ zj +1 + σ zj +2 ) t , (A10)16ith J ′ ≡ (2 / J zi,i +1 .With a three-pulse sequence σ xc e − iH z t σ xo e − iH z t σ xc = exp − i N/ X k =1 ( − k H z k t , (A11)we remove the coupling between odd order neighbors and alternatively change the sign ofthe coupling in H z between even order neighbors. To prove Eq. (A11) we use the identity σ xo = σ xc σ xc and the commutation relation [ σ xc H z σ xc , σ xc H z σ xc ] = 0 in order to obtain σ xc e − iH z t σ xo e − iH z t σ xc = σ xc e − iH z t σ xc σ xc e − iH z t σ xc = e − i ( σ xc H z σ xc + σ xc H z σ xc ) t . (A12)The transformation σ xc k H z σ xc k selectively changes the sign in H z to the operators σ zj and σ zj +1 with j ∈ c k , according to Eq. (A3). Consequently we have σ xc H z σ xc = N − X i =1 ( − i +1 J zi,i +1 σ zi σ zi +1 − N − X i =1 J zi,i +2 σ zi σ zi +2 + N − X i =1 ( − i J zi,i +3 σ zi σ zi +3 + . . . , (A13) σ xc H z σ xc = N − X i =1 ( − i J zi,i +1 σ zi σ zi +1 − N − X i =1 J zi,i +2 σ zi σ zi +2 + N − X i =1 ( − i +1 J zi,i +3 σ zi σ zi +3 + . . . . (A14)Hence, to demonstrate Eq. (A11), we sum Eq. (A13) and Eq. (A14) obtaining σ xc H z σ xc + σ xc H z σ xc = − H z + 2 H z + . . . = 2 N/ X k =1 ( − k H z k (A15)Notice that in the sum the coupling between nearest neighbors of odd orders cancels out.By combining the sequences (A9) and (A11), we can invert the sign of the coupling upto third nearest neighbors( σ xc e − iH z t σ xo e − iH z t σ xc )( σ xo e − iH z t σ xo ) ≃ exp [ − i ( − H z − H z − H z ) t ] , (A16)thus turning a ferromagnetic interaction into an anti-ferromagnetic one and viceversa. An-other consequence of Eq. (A11) is σ xc e − iH z t σ xo e − iH z t σ xc e − iH z t ≃ exp [ − i ( H z + H z ) t ] , (A17)where the coupling between second nearest neighbors has been removed.The approach described so far can be extended in order to cancel coupling terms of higherorder. For example, to remove both the second and third nearest neighbor couplings in H z we make use of the transformations defined in Eq. (A4). They simultaneously affect sets of17hree nearest neighbors in alternate succession. With arguments similar to those used forverifying Eq. (A11), we can demonstrate the following identity( σ x T e − iH z t σ x T )( σ x T e − iH z t σ x T )( σ x T e − iH z t σ x T ) = e − i ( H z − H z − H z − H z + ... ) t/ . (A18)By combining Eq. (A11) and Eq. (A18) we prove that the sequence( σ xc e − iH z t σ xo e − iH z t σ xc )( σ x T e − iH z t σ x T )( σ x T e − iH z t σ x T )( σ x T e − iH z t σ xT ) e − iH z t (A19)corresponds to the evolution for a time (4 / t under the Hamiltonian ( H z + H z + . . . ), wherethe coupling between second and third nearest neighbors has been removed. The implemen-tation of this sequence requires six pulses, since each couple of consecutive transformationsin Eq. (A19) is equivalent to a single transformation affecting simultaneously a specificsubset of spins in the array.It is worth to point out that with a seven-pulse sequence we can approximate the NNmodel in a very accurate way, i.e. we can suppress the interaction between a spin and allits neighbors from the second up to the sixth nearest neighbors. This is achieved by usingthe four transformations defined in Eq. (A5), that simultaneously affect alternated sets offour nearest neighbors. It can be proved that( σ x Q e − iH z t σ x Q )( σ x Q e − iH z t σ x Q )( σ x Q e − iH z t σ x Q )( σ x Q e − iH z t σ x Q ) = e − i ( H z − H z − H z − H z + H z + ... ) t . (A20)From Eq. (A11) and Eq. (A20) we have that the sequence( σ xc e − iH z t σ xo e − iH z t σ xc )( σ x Q e − iH z t σ x Q )( σ x Q e − iH z t σ x Q )( σ x Q e − iH z t σ x Q )( σ x Q e − iH z t σ x Q ) e − iH z t (A21)corresponds to the evolution for a time 2 t under the Hamiltonian ( H z + H z + . . . ), where thecoupling between nearest neighbors from the second up to the sixth order has been removed.The implementation of the above sequence requires just seven pulses, since each couple ofconsecutive transformations in Eq. (A21) is equivalent to a single transformation affectingsimultaneously a specific subset of spins in the array.18 PPENDIX B
The sequence of unitary operators to the left hand side of relation (44) approximates theevolution under the target Hamiltonian H = n X i =1 τ i A i (B1)to first order in t . However, more elaborate combinations of unitary operators provide betterapproximations. For example the sequence [22] S ≡ ¯ S S ¯ S S − ¯ S ¯ S ¯ S ¯ S S ¯ S S S S S ¯ S − S ¯ S S (B2)with S k ≡ e − i k t A e − i k t A . . . e − i k t n A n (B3)and ¯ S k ≡ e − i k t n A n . . . e − i k t A e − i k t A , (B4)approximates the unitary operator e − iHt , with t ≡ P ni =1 t i and τ i ≡ t i /t , to the fourth orderin t .The error introduced by approximating the unitary operator U with the unitary operator U ′ can be measured by the quantity [21] E ≡ k U − U ′ k ≡ max | ψ i : || ψ i| =1 | ( U − U ′ ) | ψ i| . (B5)Now if we want to approximate the evolution under the Hamiltonian Eq. (B1) for a time T = mt we can apply m times the sequence (B2). From the generalization of the inequality k AB − CD k ≤ k A − C k + k B − D k , (B6)verified for any unitary operator A , B , C , D , we have that the error E of our approximationsatisfies the inequality E ≤ P mi =1 E i , where E i is the error introduced by the i -th applicationof the sequence (B2). More specifically we evaluate the error E i ≡ k e − i ( P ni =1 τ i A i ) t − S k , (B7)when the operators A i ’s are typically of the kind P i>j J zi,j σ ki σ kj , with k = x, y, z . We explicitlyexpand each operator to the right hand side and find that their difference is proportional to19 , because the sequence S approximates the desired unitary evolution exp( − i P ni =1 τ i A i t ) tothe fourth order in t . Moreover, we make use of the inequality k αA + βB k ≤ | α |k A k + | β |k B k , (B8)which holds true for any pair of operators A and B and complex numbers α , β . Finally, weobserve that k C k = 1 if C is any product of Pauli operators σ ki . This approach lead us tothe following estimate for the error, defined in Eq. (B7), E i ≤ ( J z t ) f ( N ) , (B9)where J z ≡ J zi,i +1 is the nearest neighbor coupling strength and f ( N ) is, in good approxi-mation, an increasing linear function of the number N of electrons in the array, dependingon the specific form of the spin Hamiltonian.When we apply m times the sequence S , from the previous discussion it follows that E ≤ m ( J z t ) f ( N ). Now if we indicate with T = mt the total simulation time, we obtain E ≤ ( J z T ) f ( N ) m (B10)or, equivalently, m ≤ s ( J z T ) f ( N ) E . (B11)Equation (B11) gives an upper bound to the number of iterations required to mimic thedesired evolution with an error E . For example, to approximate the XY model in therotated basis, we choose A = η N X i =1 σ xi + N X i>j J zi,j σ zi σ zj , (B12) A = η N X i =1 σ xi + N X i>j J zi,j σ yi σ yj . (B13)In this case, the outlined approach gives for N > f ( N ) ≃ . N − .
85. The simulationof the NN Ising model is achieved with A = η N X i =1 σ xi + N X i>j J zi,j σ zi σ zj , (B14) A = η N X i =1 σ xi − N − X i =1 J zi,i +2 σ zi σ zi +2 , (B15)and, for N > f ( N ) ≃ . N − . CKNOWLEDGMENTS
This research was supported by the European Commission through the Specific TargetedResearch Project
QUELE , the Integrated Project FET/QIPC
SCALA , and the ResearchTraining Network
CONQUEST . [1] G. Ciaramicoli, I. Marzoli, and P. Tombesi, Phys. Rev. Lett. , 017901 (2003).[2] G. Ciaramicoli, I. Marzoli, and P. Tombesi, Phys. Rev. A , 032301 (2004).[3] G. Ciaramicoli, F. Galve, I. Marzoli, and P. Tombesi, Phys. Rev. A , 042323 (2005).[4] L. S. Brown and G. Gabrielse, Rev. Mod. Phys. , 233 (1986).[5] S. Peil and G. Gabrielse, Phys. Rev. Lett. , 1287 (1999).[6] B. D’Urso, B. Odom, and G. Gabrielse, Phys. Rev. Lett. , 043001 (2003).[7] B. D’Urso, R. Van Handel, B. Odom, D. Hanneke, and G. Gabrielse, Phys. Rev. Lett. ,113002 (2005).[8] B. Odom, D. Hanneke, B. D’Urso, and G. Gabrielse, Phys. Rev. Lett. , 030801 (2006).[9] M. Drndi´c, C. S. Lee, and R. M. Westervelt, Phys. Rev. B , 085321 (2001).[10] S. Stahl, F. Galve, J. Alonso, S. Djekic, W. Quint, T. Valenzuela, J. Verd`u, M. Vogel, and G.Werth, Eur. Phys. J. D , 139 (2005).[11] J. R. Castrej´on-Pita and R. C. Thompson, Phys. Rev. A , 013405 (2005).[12] G. Ciaramicoli, I. Marzoli, and P. Tombesi, Phys. Rev. A , 032348 (2007).[13] G. Gualdi, V. Kostak, I. Marzoli, and P. Tombesi, submitted for publication.[14] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, U.K.,1999).[15] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. , 1037 (2004).[16] E. Jan´e, G. Vidal, W. D¨ur, P. Zoller, and J. I. Cirac, Quantum Information and Computation , 15 (2003).[17] D. Porras and J. I. Cirac, Phys. Rev. Lett. , 207901 (2004).[18] X.-L. Deng, D. Porras, and J. I. Cirac, Phys. Rev. A , 063407 (2005).[19] H. F. Powell, D. M. Segal, and R. C. Thompson, Phys. Rev. Lett. , 093003 (2002).[20] F. Mintert and Ch. Wunderlich, Phys. Rev. Lett. , 257904 (2001).
21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cam-bridge University Press, Cambridge, U.K., 2000).[22] A. T. Sornborger and E. D. Stewart, Phys. Rev. A , 1956 (1999)., 1956 (1999).