Fortschr. Phys. 48 (2000) 9±±11, 811±±821 Ion Trap Quantum Computer with Bichromatic Light Anders Sùrensen*) and Klaus Mùlmery ) Institute of Physics and Astronomy, University of Aarhus DK-8000  Arhus C, Denmark Abstract In the ion trap quantum computer, quantum bits are represented by the internal state of the trapped ions, and the collective vibrations of the ions are exploited for the coherent manipulation of the state of several bits. We have developed a method to perform gates on different ions by illuminating the ions with bichromatic light. Our gates are independent of the motional state of the ions and they may be applied to ions in any state or, e.g., thermal mixture of vibrational states. Moreover, our proposal allows the production of multiparticle entangled states even in situations without experimental access to the individual ions in the trap. Maximally entangled states of any number of ions are produced by simply illuminating all ions with the same beams of bichromatic light. 1. Introduction At sufficiently low temperatures trapped ions freeze into a crystal where they are located such that the Coulomb repulsion among the ions equilibrates the confining force from the trapping potential. The vibrations of the ions are strongly coupled due to the Coulomb interaction, and in the harmonic oscillator approximation the vibrational Hamiltonian may be diagonalized to form a set of collective vibrational modes, like the phonon modes in an ordinary crystal. An electromagnetic field tuned to resonance with a vibrational frequency may be used to selectively excite a single collective mode. Instead of applying an electromagnetic field with the vibrational frequency, one may also excite a vibration by tuning a laser to one of the upper or lower sidebands of the ions, i.e., by choosing the frequency of the laser equal to the resonance frequency of an internal transition in an ion plus or minus a vibrational frequency. The laser is then on resonance with an excitation of the internal transition and a simultaneous change in the vibrational motion. This coupling of internal and external degrees of freedom has been extensively used for precise control of the quantum state of trapped ions [1, 2], and in 1995 Cirac and Zoller proposed that the ion trap can be used for quantum computation [3]. In the ion trap quantum computer each qubit is represented by the internal states of an ion, and by using long lived states, like for instance hyperfine structure states, the qubits may be very effectively shielded from the surroundings. Single qubit rotation and two qubit gates are achieved by focusing a laser on each ion and by exploiting the collective vibrations for interaction between the ions. In the original proposal [3] the system is restricted to the joint motional ground state of the ions. By tuning a laser to a sideband, a vibration is excited if a particular ion is in a certain internal state. Upon subsequent laser irradiation of another ion, the internal state of this ion is changed only if the vibrational motion is excited. At the end of the gate the vibrational excitation is removed and additional gates may subsequently be implemented. *) E-mail: anderss@ifa.au.dk ) E-mail: moelmer@ifa.au.dk y 812 A. Sùrensen and K. Mùlmer, Ion Trap QC with Bichromatic Light For various technical reasons the vibrations of the ions are subject to heating, and this makes cooling to the ground state a very difficult task. For two ions, cooling to the ground state followed by internal state entanglement of the ions has only recently been accomplished [4]. We have developed alternative implementations of quantum gates [5] that are both insensitive to the initial vibrational state and robust against changes in the vibrational motion (heating) occurring during operation, as long as the ions are in the Lamb-Dicke regime, i.e., their spatial excursions are restricted to a small fraction of the wavelength of the exciting radiation. Our gate may also be useful in situations without experimental access to the individual ions [6], where it efficiently produces a maximally entangled state of all the trapped ions. This part of our proposal has recently been used to create a 4-particle entangled state [7]. In section 2 we present our original proposal, and in section 3 we present a more recent proposal which allows a significant speed-up of the gate. In sec. 4 we analyse the preparation of multi-particle entangled states. In sec. 5 we confront our proposal with the five plus two requirements for quantum computing, suggested by David DiVincenzo in the introductory article of this issue of Fortschritte der Physik [8], and we briefly present the effect of various errors on the fidelity of the gate. 2. Slow Two-colour Gate Like in the original ion trap scheme [3], we address individual ions with single laser beams, but quantum logic gates involving two ions are performed through off-resonant laser pulses. To perform our gate on two ions we illuminate both ions with bichromatic light detuned by the same amount above and below resonance. This laser setting couples the states jggni $ fjegn  1i; jgen  1ig $ jeeni, where the first (second) letter denotes the internal state e or g of the first (second) ion and n is the quantum number for the relevant vibrational mode of the trap, see Fig. 1. We choose the detuning from the sideband so large that the intermediate states jegn  1i and jgen  1i are not populated in the process. As we shall show below, the internal state transition is then insensitive to the vibrational quantum number n, and it may be applied even with ions which exchange vibrational energy with a surrounding reservoir. If we tune the lasers sufficiently close to the sidebands, we can neglect all other vibrational modes and concentrate on one collective degree of vibrational excitation of the ions. In this case our system can be described by the following Hamiltonian ( h ˆ 1) H ˆ H0 ‡ Hint H0 ˆ n ay a ‡ 1=2† ‡ weg Hint ˆ P Wi; j s ‡i ei hi 2 i; j P s zi =2 1† i a‡ay † wj t† ‡ h:c:† ; ν |een> ω=ωeg −δ |gen+1> |gen> |gen-1> |egn+1> |egn> |egn-1> ω eg |ggn> |egn> ω=ωeg +δ |een+1> |een> |een-1> |gen> |ggn+1> |ggn> |ggn-1> Fig. 1: Energy level diagram for two ions with quantized vibrational motion illuminated with bichromatic light. The one photon transitions indicated in the figure are not resonant, d 6ˆ n, and the only resonant transitions are from jggni to jeeni and from jegni to jgeni. Various transition paths involving intermediate states with vibrational number n differing by unity are identified. 813 Fortschr. Phys. 48 (2000) 9±±11 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0 2000 4000 6000 8000 10000 12000 14000 νt Fig. 2: Rabi oscillations between jggi and jeei. The figure shows the time evolution of the internal atomic density matrix elements qgg; gg (full line), qee; ee (long dashed line) and Im qgg; ee † (short dashed line). The magnitude of Re qgg; ee † is below 0.03 and is not shown. In the initial state, the ions are in the internal ground state and a coherent vibrational state with mean excitation n ˆ 2. Parameters are d ˆ 0:90n, W ˆ 0:10n, and h ˆ 0:10: where n is the frequency of the vibration, ay and a are the ladder operators of the quantized oscillator, and weg is the energy difference between the internal states e and g. Pauli matrices si represent the internal degrees of freedom for the i'th ion, and wj and Wi;j are the frequency and Rabi frequency of the j'th laser addressing the i'th ion. The exponential y eihi a‡a † represents the spatial dependence of the plane wave laser field and the recoil of the ions upon absorption of a photon (eikx with x replaced by ladder operators x / a ‡ ay ). We tune the lasers close to the center-of-mass vibrational mode where all ions participate equally in the vibration, so that the coupling of the recoil to the vibration is identical for all ions, i.e., hi ˆ h for all i. We consider an ion trap operating in the Lamb-Dicke limit, that is, the Lamb-Dicke parameter h is much smaller than unity for all ions, and the ions are p cooled to a regime with vibrational quantum numbers n ensuring that h n ‡ 1 is well below unity. Note that this may still allow n-values well above unity. We wish to perform an operation on the mutual state of two ions selected freely within the string of ions, and we assume the same Rabi frequency W for all lasers. With the choice of detunings described above, the only energy conserving transitions are between the states jggni and jeeni and between jegni and jgeni, where we specify only the states of the ~ for the transition two ions of interest and of the collective vibration. The Rabi frequency W between jggni and jeeni, via intermediate states m, can be determined in second order perturbation theory, ~ W 2 !2 ˆ P heenj H int jmi hmj Hint jggni 2 ; Eggn ‡ wj Em m 2† where the laser energy wj is the energy of the laser exciting the intermediate state jmi. If we restrict the sum to jegn  1i and jgen  1i, we get ~ˆ W Wh†2 ; n d 3† where d ˆ w1 weg is the detuning of the blue detuned laser. For the transition between jegni and jgeni we find the same effective Rabi frequency. The remarkable feature in Eq. (3) is that it contains no dependence on the vibrational quantum number n. This is due to interference between the paths indicated in Fig. 1. If we 814 A. Sùrensen and K. Mùlmer, Ion Trap QC with Bichromatic Light take a path involving jn p ‡ 1i, we have a factor of n ‡ 1 appearing in the numerator p ( n ‡ 1 from raising and n ‡ 1 from lowering the vibrational quantum number). In paths involving jn 1i we obtain a factor of n. Due to the opposite detunings, the denominators in Eq. (2) have opposite signs and the n dependence disappears when the two terms are subtracted. The coherent evolution of the internal atomic state is thus insensitive to the vibrational quantum numbers, and it may be observed with ions in any superposition or mixture of vibrational states. To confirm the validity of our pertubative analysis we have performed a direct numerical integration of the ScroÈdinger equation with the Hamiltonian (1) to all orders in h. We have considered a situation, where both ions are initially in the internal ground state. For the vibrational state, we have investigated a number of different states, including Fock, coherent and thermal states, all yielding qualitatively similar results. The outcome of the computation for a coherent state of vibrational motion can be seen in Fig. 2, where we show the evolution of relevant terms of the atomic internal state density matrix qij; kl ˆ Trn q jkli hijj†, where i; j; k; l ˆ e or g, and where Tr n denotes the partial trace over the unobserved vibrational degrees of freedom. The figure clearly shows that we have Rabi oscillations between the atomic states jggi and jeei, and the values of the off diagonal element qgg; ee confirm that we have a coherent evolution of the internal atomic state which is not entangled with the vibrational motion. Superimposed on the sinusoidal curves are small oscillations with a high frequency due to off resonant couplings of the type jggni ! jegn ‡ 1i, jggni ! jgen 1i, and jggni ! jegni. The magnitude of these oscillations and the deviation from ideal transfer between jggi and jeei can be suppressed by decreasing W. No particularly demanding assumptions have been made for the experimental parameters. With a vibrational frequency n=2p ˆ 200 kHz, the transition shown in Fig. 2, require Rabi frequencies W=2p of modest 20 kHz, and the evolution from jggi to jeei is accomplished in 5 ms. This time scale is, however, longer than the time scale in the original ion trap proposal [3], due to our use of off-resonant interactions instead of resonant couplings. To be relevant for real computational tasks, it is necessary that our evolution is robust against decoherence effects on this longer time scale. An important source of decoherence is heating of the vibrational motion, and it is a major asset of our proposal that it is insensitive to this interaction with the environment. To validate that the internal state evolution is in fact stable againstpheating, we intro duce a thermal reservoir described by relaxation operators c1 ˆ G 1 ‡ nth † a and 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0 5000 10000 νt 15000 0 5000 10000 15000 νt Fig. 3: Rabi oscillations in a heating trap. The left panel shows the result of a single Monte Carlo realization with a total of 39 jumps occurring at times indicated by the arrows. The right panel is an average over 10 realizations. The curves represent qgg; gg (full line), qee; ee (long dashed) and Im qgg; ee † (short dashed). Parameters are d ˆ 0:90n, W ˆ 0:05n, h ˆ 0:1, G ˆ 2  10 4 n, and nth ˆ 2. Fortschr. Phys. 48 (2000) 9±±11 815 p y c2 ˆ Gnth a , where G characterizes the strength of the interaction and nth is the mean vibrational number in thermal equilibrium. We analyse the dynamics of the system using Monte Carlo wavefunctions [9], which evolve with a non-Hermitian Hamiltonian interrupted by jumps at random times. The result of the computation can be seen in Fig. 3, where we show (left) the result of a single Monte Carlo realization with quantum jumps indicated by arrows and (right) the average over 10 realizations. In the figure we have chosen nth ˆ 2. This rather low value could represent a heating mechanism counteracted by laser cooling on a particular ion reserved for this purpose. In the simulations 34 vibrational quanta are exchanged with the reservoir on average, and we wish to emphasize that with the proposed scheme, the gate is almost unaffected even though the duration of the gate is much longer than the coherence time of the channel used to communicate between the qubits. 3. Fast Two-color Gate Recently, Milburn [10] has proposed a realization of a multi-bit quantum gate in the ion trap, which also operates when the ions are vibrationally excited: Adjusting the phases of laser fields resonant with side band transitions, one may couple internal state operators to different quadrature components, e.g., position and momentum X and P, of the collective oscillatory motion. In Ref. [10] it is proposed to use the two Hamiltonians H1 ˆ l1 Jz X and H2 ˆ l2 Jz P, P expressed in terms of the collective spin operators Ji ˆ jik i ˆ x; y; z†, where the sum is k over the ions irradiated by the lasers, and where jik is the pseudo-spin operator for the atom k, which may be defined from the generic representation of two level systems by Pauli spin matrices jik ˆ s ik =2. l1 and l2 denote the strengths of the Hamiltonians, variable in time. By alternating application of the Hamiltonians H1 and H2 we obtain the exact propagator eiH2 t eiH1 t e iH2 t e iH1 t 2 2 ˆ eil1 l2 Jz t 4† because the commutator of the oscillator position and momentum is a constant. The interaction contained in Jz2 between the ions has been established via the vibrational motion, but after the gate this motion is returned to the initial state and is not in any way entangled with the internal dynamics. Note that if H1 and H2 involve different internal state operators e.g. jz and jy for two different ions, it is the product of these operators that appears in Eq. (4), and any pairwise quantum computing gate operation can be constructed, e.g., the C-NOT operation. We shall now show that our bichromatic scheme utilizes the same mechanism, and that the analysis of Milburn points to a much more rapid application than considered in the previous section. In the Lamb-Dicke limit with lasers detuned by d our bichromatic interaction Hamiltonian in the interaction picture becomes Hint ˆ 2WJx cos dt† ‡ p sin n p 2 hWJy ‰x cos n d† t ‡ sin n ‡ d† t†Š ; d† t ‡ cos n ‡ d† t† 5† where we have introduced the dimensionless position and momentum operators x ˆ p12 a ‡ ay † and p ˆ pi2 ay a†. Choosing not too strong laser intensities W  d and tuning close to the sideband n d  d we may neglect the Jx term and the terms oscillating at n ‡ d in Eq. (5), and our interaction is a special case of the Hamiltonian Hint ˆ Jy f t† x ‡ Jy g t† p : 6† 816 A. Sùrensen and K. Mùlmer, Ion Trap QC with Bichromatic Light p ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ ÿÿÿÿÿÿ þþþþþþ -A(t) (G(t),-F(t)) þþþ ÿÿÿ ÿÿÿ þþþ ÿÿÿ þþþ A(t) ÿÿÿ þþþ ÿÿÿ þþþ ÿÿÿ þþþ ÿÿÿ þþþ ÿÿÿ þþþ x Fig. 4: The paths traversed in phase space and the function A t† in Milburns proposal (rectangular) and in our proposal (circle). With a change from Jz to Jy the interaction considered by Milburn [10] may also be put in this form, with f t† and g t† alternating between zero and non-vanishing constants. The exact propagator for the Hamiltonian (6) may be represented by the ansatz U t† ˆ e iA t† Jy2 e iF t† Jy x and the SchroÈdinger equation i F t† ˆ „t 0 f t 0 † dt 0 ; d dt e iG t† Jy p ; 7† U t† ˆ HU t† then leads to the expressions G t† ˆ „t 0 g t0 † dt 0 ; and A t† ˆ „t 0 F t 0 † g t 0 † dt 0 : 8† In the xp phase space the operator U performs translations x; p† ! x ‡ Jy G t†; p Jy F t†† entangled with the internal state of the ions. This is illustrated in Fig. 4, where p the circular trajectory corresponds to our harmonic interaction, f t† ‡ ig t† ˆ 2 hW exp i n d† t†, and the rectangular trajectory corresponds to Milburns pulsed interaction. Within the present formulation, the trick devised by Milburn is to use functions f t† and g t† such that F t† and G t† both vanish after a period t. At 2 this instant the propagator reduces to U t† ˆ e iA t† Jy , i.e., we are left with an internal state evolution which is independent of the external vibrational state, and the vibrational motion is returned to its original state, be it the origin as in Fig. 4 or any other point in phase space. In Fig. 5 we presents two different methods of achieving F t†  G t†  0. In (a) we operate in a slow regime where the analysis of sec. 2 is valid. This regime may also be described by Eq. (7) with small values of the functions F t† and G t†. (The non-zero values of F t† and G t† are responsible for the small fast oscillations in the figure). In the slow gate the internal state is only weakly entangled with the vibrational motion and the gate ~ has acquired the desired value, irrespective of the current may be stopped when A t†  Wt value of F t† and G t†. A different regime is obtained if we only require F t† and G t† to vanish at the end of the gate. If we chose n d† t ˆ K2p, where K is any integer, G t† ˆ F t† ˆ 0 at the end of the gate, but the internal state may be strongly entangled with the vibrational motion in the course of the gate. This allows faster gate operation. See Fig. 5 (b), where n d† t ˆ 2
2p at the time nt  250, where the maximally entangled state p12 jggi i jeei† is created.pFor  a general value of K the maximally entangled state may be created after a time t ˆ phWK . With the same Rabi frequency W and K ˆ 1 our gate is even faster than the original ion trap proposal [3]. 817 Fortschr. Phys. 48 (2000) 9±±11 4. Multi-particle Entanglement Suppose that there are only two ions in the trap and we illuminate both ions with a beam of bichromatic light. If the beam has the same intensity in both colours, this is exactly the situation depicted in Fig. 1, and the internal state performs coherent Rabi oscilation between jggi and jeei. If the ions are initially prepared in the ground state jggi, an EPR-state p1 jggi i jeei† may be generated. Incidentally, it turns out that we may also generate 2 maximally entangled states 1 jYi ˆ p eifg jgg . . . gi ‡ eife jee . . . ei† ; 2 9† for any number of ions, by simply illuminating all ions with the bichromatic light. The phases fg and fe have values specified below. The states in Eq. (9) have several very interesting applications both in fundamental physics and technology. They are SchroÈdinger cat superpositions of states of mesoscopic separation, and they are ideal for spectroscopic investigations. In current frequency standards each atom or ion evolves individually, hence the improvement in frequency uncertainty, as the number of particles (N) is increased, is of statistical nature, i.e., the relative uncertainty behaves like p1N . In current frequency standards, the duration of the measurements is much shorter than the decoherence time of the internal atomic state, and by binding the ions together as in Eq. (9) we are sensitive to the phase evolution of the excited state jee . . . ei which is proportional to N, and consequently the frequency uncertainty is proportional to N1 resulting in an improved sensitivity [11]. In experiments with a duration exceeding the time scale of internal atomic decoherence tdec , however, the shorter coherence time tdec =N of the entangled state may lead to the same resolution for that state and for an uncorrelated ensemble of atoms [12]. To describe the dynamics leading to the generation of states like Eq. (9), we note that states with Ne excited ions which are symmetric with respect to interchange of the ions may be described by the eigenstates jJMi ˆ jN=2; Ne N=2i of the Jz operator Jz jJMi ˆ M jJMi. Initially all ions are in the ground state jgi corresponding to the spin state jN=2; N=2i, and since the interaction in Eq. (6) only involves the Jy -operator which conserves the total angular momentum, the quantum state remains in the J ˆ N=2 subspace. As shown in section 3 the effective time evolution at times t ˆ K n2pd is described by the propagator U t† ˆ exp iA t† Jy2 † which distribute the population onto all jJ; Mi states, with M differing from N=2 by an even number. If N is even, population is transferred all the way to the state jN=2; N=2i, which is the second component of (9). We show in Fig. 6 the evolution of the populations of the two extremal states as we vary the effective stregth (a) 1 0.8 0.6 0.4 0.2 0 (b) 1 0.8 0.6 0.4 0.2 0 0 1000 2000 3000 4000 νt 0 200 400 600 νt Fig. 5: Time evolution of density matrix elements calculated using (7). (a) Pertubative regime (b) Fast gate. The first curve (counting from above at nt  1000 in (a) and nt  130 in (b)) represents qgg; gg , the second is the imaginary part of qgg; ee , the third is qee; ee , and the last curve is the real part of qgg; ee . In (a) the physical parameters are d ˆ 0:9n, h ˆ 0:1, and W ˆ 0:1n. In (b) the physical parameters are d ˆ 0:95n, h ˆ 0:1, and W ˆ 0:177n. The parameters in (b) are chosen such that a maximally entangled state p12 jggi ijeei† is formed at the time nt  250. 818 A. Sùrensen and K. Mùlmer, Ion Trap QC with Bichromatic Light 1 0.8 0.6 0.4 0.2 0 0 0.4 0.8 1.2 1.6 2 2.4 A(t) 2.8 3.2 3.6 4 Fig. 6: Evolution of the population of the joint ionic ground state jgg . . . gi ˆ jN=2; N=2i (curves starting from the value of unity at A t† ˆ 0), and the joint ionic excited state jee . . . ei ˆ jN=2; N=2i. Results are presented for different values of the number of ions: N ˆ 4 (solid curves), N ˆ 8 (dashed curves), and N ˆ 20 (dotted curves). At A t† ˆ p=2 the states are in a 50-50 superposition and the state (9) is obtained. of the interaction A t† for different (even) numbers of ions. At the instant when A t† ˆ p=2, both populations equal one half, and the state is precisely of the form (9), with fg ˆ p=4 and fe ˆ p=4. This result can be understood from the rotation propertiesP of angular momenta. The initial state can be expanded on eigenstates of Jy : jN=2; N=2i ˆ M cM jN=2; Miy . A similar expansion of jN=2; N=2i, may be obtained by a rotation around the y-axis by an angle p, jJJi ˆ 1†2J exp ipJy † jJ Ji (with the conventional choice of phases [13]). The rotation is easily performed on each P term in the My basis where the operator is diagonal, and it follows that jN=2; N=2i ˆ M cM eip N M† jN=2; Miy . Inserting the values for fg and fe mentioned above, pwe can therefore write the state (9) as p  P ip=4 = 2 ‡ 1†M eip=4 = 2† jN=2; Miy (N even). The net factors multiplying the M cM e initial amplitudes cM are unity for M even and i for M odd. The action of U t† in the Jy basis amounts to a multiplication of each amplitude cM by exp iA t† M 2 †, and for A t† ˆ p=2 this factor just attains the desired value of unity for M even and i for M odd. Mathematically, this analysis resembles the one for production of superpositions of states of mesoscopic distance by propagation of an optical field through an amplitude dispersive medium [14]. The pairwise interaction between the ions does not produce a coherent coupling of the two components in (9) if N is odd. In this case, however, states like (9) with fg ˆ p=8 and fe ˆ 7p=8 may be produced by applying an additional linear coupling H ˆ xJy [6]. 5. The Prospects of Ion Trap Quantum Computing 5.1. The five plus two requirements for quantum computation, proposed by Divincenzo [8] 1. A scalable physical system with well characterized qubits. The ion trap proposal is scalable by inclusion of more ions in the trap. Each ion represents a qubit, and gate Fortschr. Phys. 48 (2000) 9±±11 819 operation can in theory be applied to any qubit and to any pair of qubits irrespective of the number of ions in the trap. In [7] eksperimental operation with 4 ions in the trap was presented. 2. The ability to initialize the state of the qubits to a simple fiducial state, such as j000 . . .i. If both qubit states are stable against spontaneous emission, initialization of the ions takes place by optical pumping of all ions by exciting the ions from state j1i to a radiatively unstable state that decays back into j0i or j1i. The duration of the initialization is given by the excited state life time of the unstable state, and it may thus be much faster than the gate operation, and it may be applied to all, or a selected subset, of the ions at the same time. 3. Long relevant coherence times, much longer than the gate operation time. The internal state coherences and populations in the ions may have life times of many seconds, and with a gate duration of, say, 100 times the trap period, this leaves time for thousands of gate operations within the coherence time of the qubits. 4. A `universal' set of quantum gates. Individual access permits simple single-qubit operations, and together with the two-ion transition shown in Fig.1, we have a universal set of gates, sufficient for quantum computing. A C-NOT operation can be obtained by rotating the jz into jy of the control qubit, so that the Jy2 Hamiltonian implemented by Eq. (7), effectively yields a jz jy operation, which, succeeded by the single bit operation jy =2 on the target qubit, can realize a p-rotation of the target spin conditioned on the control bit being in the jz spin-up eigenstate. The ion trap allows to perform two-qubit gates between any pair of ions, not only nearest neighbours in the trap. This is accomplished by the vibrational motion which acts as a `bus' qubit. Our proposal with bichromatic light is actually free of two limitation anticipated in [8]: `Bus'-decoherence does not prevent quantum gate operation: the gate may operate perfectly, even over times much longer than the vibrational coherence time. The use of a `bus' does not limit operation to one gate at a time: using different pairs of frequencies on different pairs of ions, the vibrational coupling can be applied for several gate operations simultaneously. 5. A qubit-specific measurement capability. The electronic states of the ions are read out by the shelving technique, well studied on trapped ions, and with physical separations of tens of microns between the ions, photon fluxes from each ion of zero or houndreds of photons are easily distinguishable in a direct read-out of the entire register. Like the initialization, the read-out involves fluorescence from unstable excited states and it is typically accomplished in less than 1 ms. 6 and 7. The ability to interconvert stationary and flying qubits, and the ability to faithfully transmit flying qubits between specified locations. Proposals exist [16] to transfer the quantum state of an ion to a single photon, which can be transmitted over kilometer distances in an optical fibre (and maybe much further vertically through the atmosphere to a satelite system). Fortunately, we do not have to literally transfer the quantum state from an ion to a photon to a second ion: we can prepare an event-ready entangled pair of ions with one ion sitting in the sending unit and one sitting in the receiving unit, and when the need for communication arises we make the transfer between the two units by teleportation [17]. The event-ready entanglement can be prepared by a less efficient process, which may involve photon transmission over lossy lines and which does not require perfect coupling between single photons and single ions [18, 19, 20]. 5.2. Fidelity of gate operation At a formal level, the ion trap system fulfills the requirements for becoming the central part of a quantum computing system. This does not imply that it is going to be easy. Decoherence mechanisms and different error sources do exist, and the arsenal of stabilizer codes, 820 A. Sùrensen and K. Mùlmer, Ion Trap QC with Bichromatic Light Table 1 ife . . gi Creation of entangled stats od ions p12 eifg jgg . p  ‡ e jee . . . ei† by interaction with a 2 hWJy ‰x cos n d† t ‡ cos n d† t†‡ bichromatic field (5) Hint ˆ 2WJx cos dt p sin n d† t ‡ sin n ‡ d† t†Š obeying nhWd ˆ 2 p1 K , K ˆ 1; 2; 3; . . . and for a duration t ˆ 2pK= n d†. The fidelity of the preparation is reduced by various causes, lited in the table Cause of deviation 1 F Direct off-resonant coupling, Jx term in (5) Deviations from Lamb-Dicke y hnj eih a‡a† jn ‡ 1i p 6ˆ ih n ‡ 1 NW2 2d2 h4 p2 N N 1† 8 Spectator vibrational modes (i) Direct coupling 2 2 Var n† N h nW2 0:8 n ‡ 1† (ii) DebyeWaller 2 h4 p N 8N 1†  0:2 n2 ‡ 0:4 n† Heating towards vibrational number nth with rate Gnth N G 1‡2nth † t 8K error correction and compensating techniques, will be needed to ensure sufficient precision of computation. The results shown in Figs. 5 and 6 were obtained with the idealized Hamiltonian (5) omitting the carrier excitation term 2WJx cos dt† and terms of higher order in the LambDicke parameter h. In [15] we have analyzed the influence of a number of effects left out of this simplified treatment, and in table 1 we summarize the estimated reduction of fidelity due to these different effects. The fidelity is for simplicity defined as the population of the state (9) after application of the bichromiatic fields for the optimum duration. The inclusion of other modes assumes a linear string of ions (all modes involve motion along the axis of the trap) and a thermal distribution of the vibrational excitation, n and Var (n) are the excitation mean and variance of the center of mass mode. The argument K in the right hand column makes the distinction between a slow gate (large K) and a fast gate (K small), and we see that the slow gate is tolerant to heating. 6. Conclusion Quantum computation is a field with several promising perspectives. At the present stage, however, the experiments are lacking behind the theoretical dreams, and there is a need for proposals which may close the gap between experiments and theory. The ion trap is a very promising candidate for a quantum computer, and operation with a few ions is likely to be realized within a few years. We have suggested a method to implement gates in the ion trap quantum computer. Our gate is designed with the specific purpose of reducing the effect of heating of the vibrational motion, which is a major obstacle to quantum computation in this system. At the same time it has the remarkable property that without experimental access to the individual ions it produces multi-particle entangled states. This considerably reduces the experimental requirements for the preparation of such states. It is noteworthy that we are able to perform our slow gate, even though the time required for the gate is much longer than the coherence time of the vibration, which is used for the communication between the ions. This property can be traced back to the fact that we only use virtual excitation of the vibration due to our use of resonance conditions, which ensure that only two-photon excitations are resonant. A similar trick has been proposed to reduce decoherence in a recent solid state implementation [21], and we suggest that the use of virtual excitations might enable the construction of quantum computers within other physi- Fortschr. Phys. 48 (2000) 9±±11 821 cal systems. Also, the fast gate operation, where strong intermediate entanglement with incoherent ancillary degrees of freedom is perfectly allowed, if it is eventually removed, presents new perspectives for the ion trap quantum computer, and we note that it relies on a formal property, Eqs. (4, 7), which may also find application in other implementation schemes. We are grateful to B. King, C. Monroe, D. Wineland, R. Blatt, D. Leibried and F. Schmidt-Kahler for discussions on the their ion trap experiments, and to D. F. V. James for the data needed to compute the reduction of fidelity due to spectator vibrational modes in the ion trap. This research was supported by Thomas B. 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