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VOLUME 75, NUMBER 25 PHYSICAL REVIEW LETTERS 18DECEMBER 1995

Demonstration of a Fundamental Quantum

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland National Institute of Standards and Technology, Boulder, Colorado 80303 (Received 14 July 1995) We demonstrate the operation of a two- “controlled-NOT” gate, which, in conjunction with simple single-bit operations, forms a universal for quantum computation. The two quantum are stored in the internal and external degrees of freedom of a single trapped atom, which is first laser cooled to the zero-point energy. Decoherence effects are identified for the operation, and the possibility of extending the system to more appears promising.

PACS numbers: 89.80.+h, 03.65.–w, 32.80.Pj We report the first demonstration of a fundamental and CN gates involving a dipole-dipole inter- quantum logic gate that operates on prepared quantum action between quantum dots or atomic nuclei [6,7,11,12] states. Following the scheme proposed by Cirac and may suffer from decoherence efforts. The light shifts on Zoller [1], we demonstrate a controlled-NOT gate on a atoms located inside electromagnetic cavities have been pair of quantum bits (qubits). The two qubits comprise shown to be large enough [13,14] that one could construct two internal (hyperfine) states and two external (quantized a quantum gate where a single prepared in the motional harmonic oscillator) states of a single trapped cavity acts as the control [7,15] for the atomic state. atom. Although this minimal system consists of only two However, extension to large quantum registers may be dif- qubits, it illustrates the basic operations necessary for, and ficult. Cirac and Zoller [1] have proposed a very attrac- the problems associated with, constructing a large scale tive quantum architecture based on laser-cooled quantum computer. trapped ions in which the qubits are associated with in- The distinctive feature of a quantum computer is its ternal states of the ions, and information is transferred ability to store and process superpositions of numbers between qubits through a shared motional degree of free- [2]. This potential for parallel has led to dom. The highlights of their proposal are that (i) deco- the discovery that certain problems are more efficiently herence can be small, (ii) extension to large registers is solved on a quantum computer than on a classical relatively straightforward, and (iii) the qubit readout can computer [3]. The most dramatic example is an algorithm have nearly unit efficiency. presented by Shor [4] showing that a quantum computer In our implementation of a quantum CN logic gate, the 2 should be able to factor large numbers very efficiently. target qubit jS͘ is spanned by two S1͞2 hyperfine ground 9 1 This appears to be of considerable interest, since the states of a single Be ion (the jf ෇ 2,mF ෇ 2͘ and security of many data encryption schemes [5] relies on the jF ෇ 1, mF ෇ 1͘ states, abbreviated by the equivalent inability of classical computers to factor large numbers. -1͞2 states j#͘and j"͘) separated in frequency by A quantum computer hosts a register of qubits, each of v0͞2p Ӎ 1.250 GHz. The control qubit jn͘ is spanned which behaves as quantum mechanical two-level systems by the first two quantized harmonic oscillator states and can store arbitrary superposition states of 0 and 1. of the trapped in (j0͘ and j1͘), separated in frequency It has been shown that any computation on a register by the vibrational frequency vx͞2p Ӎ 11 MHz of the of qubits can be broken up into a series of two-bit harmonically trapped ion. Figure 1 displays the relevant operations [6], for example, a series of two-bit 9Be1 energy levels. Manipulation between the four “controlled-NOT” (CN) quantum logic gates, accompa- eigenstates spanning the two-qubit register (jn͘jS͘ ෇ nied by simple rotations on single qubits [7,8]. The CN j0͘j#͘,j0͘j"͘,j1͘j#͘,j1͘j"͘) is achieved by applying a gate transforms the state of two qubits e1 and e2 from pair of off-resonant laser beams to the ion, which drives je1͘je2͘ to je1͘je1 © e2͘, where the © operation is addi- stimulated Raman transitions between basis states. When tion modulo 2. Reminiscent of the classical exclusive-OR the difference frequency d of the beams is near d Ӎ (XOR) gate, the CN gate represents a computation at the v0 (the carrier), transitions are coherently driven between most fundamental level: the “target” qubit je2͘ is flipped internal states jS͘ while preserving jn͘. Likewise, for d Ӎ depending on the state of the “control” qubit je1͘. v0 2vx (the red sideband), transitions are coherently Experimental realization of a quantum computer re- driven between j1͘j#͘and j0͘j"͘, and for d Ӎ v0 1vx quires isolated quantum systems that act as the qubits, and (the blue sideband), transitions are coherently driven the presence of controlled unitary interactions between the between j0͘j#͘and j1͘j"͘. Note that when d is tuned to qubits that allow construction of the CN gate. As pointed either sideband, the stimulated Raman transitions entangle out by many authors [6,9,10], if the qubits are not suffi- jS͘ with jn͘, a crucial part of the trapped-ion quantum CN ciently isolated from outside influences, decoherences can gate. destroy the quantum interferences that form the computa- We realize the controlled-NOT gate by sequentially tion. Several proposed experimental schemes for quantum applying three pulses of the Raman beams to the ion 4714 VOLUME 75, NUMBER 25 PHYSICAL REVIEW LETTERS 18DECEMBER 1995

is as follows: Input state ! Output state j0͘j#͘!j0͘j#͘ j0͘j"͘!j0͘j"͘ (2) j1͘j#͘!j1͘j"͘ j1͘j"͘!j1͘j#͘. The experiment apparatus is described elsewhere [16,17]. A single 9Be1 ion is stored in a coaxial- resonator rf- [17], which provides pseudopotential oscillation frequencies of ͑vx, vy, vz͒͞2p Ӎ͑11.2, 18.2, 29.8) MHz along the principal axes of the trap. We cool the ion so that the nx ෇ 0 vibrational is occu- pied Ӎ95% of the time by employing resolved-sideband FIG. 1. 9Be1 energy levels. The levels indicated with stimulated Raman cooling in the x dimension, exactly thick lines form the basis of the quantum register: internal as in Ref. [16]. The two Raman beams each contain 2 levels are jS͘ ෇ j#͘ and j"͑͘S1͞2jF෇2, mF ෇ 2͘ and Ӎ1 mW of power at Ӎ313 nm and are detuned Ӎ50 GHz 2S jF 1, m 1 levels, respectively, separated by 2 1͞2 ෇ F ෇ ͘ red of the P1͞2 . The Raman beams are j 2 j v0͞2p Ӎ 1.250 GHz), and aux͘ ෇ S1͞2 F ෇ 2, mF ෇ 0͘ applied to the ion in directions such that their wave-vector (separated from j# by 2.5 MHz); external vibrational levels ͘ Ӎ difference dk points nearly along the x axis of the are jn͘ ෇ j0͘ and j1͘ (separated by vx͞2p Ӎ 11.2 MHz). 2 trap; thus the Raman transitions are highly insensitive to Stimulated Raman transitions between S1͞2 hyperfine states 2 are driven through the virtual P1͞2 level ͑D Ӎ 50 GHz͒ with in the other two dimensions. The Lamb-Dicke a pair of Ӎ313 nm laser beams. Measurement of S is accom- parameter is hx ෇ dkx0 Ӎ 0.2, where x0 Ӎ 7 nm is 2 plished by driving the cycling j#͘! P3͞2jF෇3, mF ෇ 3͘ 1 the spread of the nx ෇ 0 . The carrier transition with s -polarized light and detecting the resulting j j# !j j" ion fluorescence. ͑ n͘ ͘ n͘ ͒͘ Rabi frequency is V02p Ӎ 140 kHz, the red ͑j1͘j#͘!j0͘j"͒͘ and blue ͑j0͘j#͘!j1͘j"͒͘ sideband Rabi frequencies are h V ͞2p Ӎ 30 kHz, and according to the following format: x 0 the auxiliary transition ͑j1͘j"͘!j0͘j#͒͘ Rabi frequency ͑a͒ A p͞2 pulse is applied on the carrier transition. is hxVaux͞2p Ӎ 12 kHz. The difference frequency of 1 2 The effect is described by the operator V ͞ ͑p͞2͒ the Raman beams is tunable from 1200 to 1300 MHz in the notation of Ref. ͓1͔. with the use of a double pass acousto-optic modulator ͑b͒ A 2p pulse is applied on the blue sideband (AOM), and the Raman pulse durations are controlled transition between j"͘and an auxiliary atomic (1) with additional switching AOMs. Since the Raman beams level jaux͑͘see Fig. 1͒. are generated from a single laser and an AOM, broadening of the Raman transitions due to a finite laser linewidth is ͑c͒ A p͞2 pulse is applied on the carrier transition, negligible [18]. with a p phase shift relative to ͑a͒, leading to the Following Raman cooling to the j0͘j#͘state, but before 1͞2 operator V ͑2p͞2͒ of Ref. ͓1͔ . application of the CN operation, we apply appropriately The p͞2 pulses in steps (a) and (c) cause the spin jS͘ tuned and timed Raman pulses to the ion, which can to undergo 11͞4 and 21͞4 of a complete , prepare an arbitrary state of the two-qubit register. For respectively, while leaving jn͘ unchanged. The auxiliary instance, to prepare a j1͘j#͘ eigenstate, we apply a p transition in step (b) simply reverses the sign of any pulse on the blue sideband followed by a p pulse on component of the register in the j1͘j"͘state by inducing a the carrier ͑j0͘j#͘!j1͘j"͘!j1͘j#͒͘. We perform two complete Rabi cycle from j1͘j"͘!j0͘jaux͘ ! 2j1͘j"͘. measurements to detect the population of the register 2 The auxiliary level jaux͘ is the S1͞2 jF ෇ 2, mF ෇ 0͘ after an arbitrary sequence of operations. First, we ground state, split from the j#͘ state by virtue of a measure the probability P͕S ෇#͖ that the target qubit Zeeman shift of Ӎ2.5 MHz resulting from a 0.18 mT jS͘ is in the j#͘state by collecting the ion fluorescence applied magnetic field (see Fig. 1). Any component of when s1-polarized laser radiation is applied resonant 2 the quantum register in the jn͘ ෇ j0͘ state is unaffected with the cycling j#͘! P3͞2jF෇3, mF ෇ 3͘ transition by the blue sideband transition of step (b), and the effects (radiative linewidth g͞2p Ӎ 19.4 MHz at l Ӎ 313 nm; of the two Ramsey p͞2 pulses cancel. On the other hand, see Fig. 1). Since this radiation does not appreciably any component of the quantum register in the j1͘j"͘state couple to the j"͘ state (relative excitation probability: acquires a sign change in step (b), and the two Ramsey Ӎ5 3 1025), the fluorescence reading is proportional to pulses add constructively, effectively “flipping” the target P͕S ෇#͖. For S ෇ #, we collect on average Ӎ1 photon qubit by p radians. The truth table of the CN operation per measurement cycle [16]. Once S is measured, we 4715 VOLUME 75, NUMBER 25 PHYSICAL REVIEW LETTERS 18DECEMBER 1995 perform a second independent measurement that provides the probability P͕n ෇ 1͖ that the control bit jn͘ is in the j1͘ state: After the same operation sequence is repeated, an appropriate Raman pulse is added just prior to the detection of S. This “detection preparation” pulse maps jn͘ onto jS͘. For instance, if we first measure S to be #, we repeat the experiment with the addition of a “p pulse” on the red sideband. Subsequent detection of S resulting in the presence (absence) of fluorescence indicates that n ෇ 0 (1). Likewise, if we first measure S to be ",we repeat the experiment with the addition of a “p pulse” on the blue sideband. Subsequent detection of S resulting FIG. 2. Controlled-NOT (CN) truth table measurements for in the presence (absence) of fluorescence indicates that eigenstates. The two horizontal rows give measured final values of n and S with and without operation of the CN gate, n 1 0 . ෇ ͑ ͒ expressed in terms of the probabilities P͕n ෇ 1͖ and P͕S ෇ #͖. In the above measurement scheme, we do not obtain The measurements are grouped according to the initial prepared phase information about the of the register eigenstate of the quantum register (j0͘j#͘, j0͘j"͘, j1͘j#͘,or and therefore do not measure the complete transformation j1͘j"͘). Even without CN operations, the probabilities are not matrix associated with the CN operation. The phase exactly 0 or 1 due to imperfect laser-cooling, state preparation and detection preparation, and decoherence effects. However, information could be obtained by performing additional with high probability, the CN operation preserves the value of operations (similar to those described above) prior to the the control qubit n, and flips the value of the target qubit S measurement of S. Here, we demonstrate the key features only if n ෇ 1. of the CN gate by (i) verifying that the populations of the register follow the truth table given in (2), and (ii) demonstrating the conditional sey pulses. For initial state j0͘j#͘, we obtain the normal associated with the CN operation. Ramsey spectrum since step (b) is inactive. For initial To verify the CN truth table, we separately prepare each state j1͘j#͘, the Ramsey spectrum is inverted indicating of the four eigenstates spanning the register ͑jn͘jS͘ ෇ the conditional control (by quantum bit jn͘) of the dynam- j0͘j#͘,j0͘j"͘,j1͘j#͘,j1͘j"͘), then apply the CN operation ics of the Ramsey pulses. Appropriate Ramsey curves are given in (1). We measure the resulting register population also obtained for initial states j0͘j"͘and j1͘j"͘. as described above after operation of the CN gate, as The switching speed of the CN gate is approximately shown in Fig. 2. When the control qubit is prepared 20 kHz, limited mainly by the auxiliary 2p pulse in step in the jn͘ ෇ j0͘ state, the measurements show that the (b) given in (1). This rate could be increased with more gate preserves S with high probability, whereas when the Raman beam laser power, although a fundamental limit in initial control qubit is prepared in the jn͘ ෇ j1͘ state, the switching speed appears to be the frequency separation of CN gate flips the value of S with high probability. In the control qubit vibrational energy levels, which can be contrast, the gate preserves the population n of the control as high as 50 MHz in our experiment [17]. qubit jn͘ with high probability, verifying that the register We measure a decoherence rate of a few kHz in the populations follow the CN truth table expressed in (2). experiment, adequate for a single CN gate operating at The fact that the measured probabilities are not exactly a speed of Ӎ20 kHz, but certainly not acceptable for a zero or one is primarily due to imperfect laser-cooling, more extended computation. We identify several sources imperfect state preparation and detection preparation, and responsible for decoherence, including instabilities in the decoherence effects. laser beam power and the relative position of the ion with To illustrate the conditional dynamics of a quantum respect to the beams, fluctuating external magnetic fields logic gate, we desire to perform a unitary transforma- (which can modulate the qubit phases), and instabilities tion on one physical system conditioned upon the quan- in the rf-ion trap drive frequency and voltage amplitude. tum state of another subsystem [19]. To see this in the Substantial reduction of these sources of decoherence can present experiment, it is useful to view steps (a) and (c) be expected. Other sources of decoherence that may of the CN operations given in (1) as Ramsey radiation become important in the future include external heating pulses [20], which drive the jn͘j#͘!jn͘j"͘transition— and dissipation of the ion motion [16,21], and spontaneous with the addition of the perturbation (b) inserted between emission caused by off-resonant transitions. We note that the pulses. If we now vary the frequency of the Ramsey decoherence rates of under 0.001 Hz have been achieved pulses, we obtain the typical sinusoidal Ramsey interfer- for internal-state ion qubits [22]. ence pattern indicative of the coherent evolution between The single-ion quantum register in the experiment com- states jS͘ ෇ j#͘and j"͘. However, the final population prises only two qubits and is therefore not useful for com- S depends on the status of the control qubit jn͘. This putation. However, if the techniques described here are is illustrated in Fig. 3 where we plot the measured prob- applied to a collection of many ions cooled to the n ෇ 0 ability P͕S ෇ #͖ as a function of detuning of the Ram- state of collective motion, it should be possible to imple- 4716 VOLUME 75, NUMBER 25 PHYSICAL REVIEW LETTERS 18DECEMBER 1995

[1] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [2] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982); Opt. News 11, 11 (1985). [3] D. Deutsch, Proc. R. Soc. London A 425, 73 (1989); D. Deutsch and R. Jozsa, Proc. Soc. London A 439, 554 (1992). [4] P. Shor, in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, New York, 1994), p. 124. [5] R. L. Rivest, A. Shamir, and L. Adelman, Comm. ACM, 28, 120 (1978). [6] D. P. DiVincenzo, Phys. Rev. A 51, 1015 (1995). FIG. 3. Ramsey spectra of the controlled-NOT (CN) gate. [7] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. The detuning of the Ramsey p͞2 pulses of the CN gate [steps Rev. Lett. 74, 4083 (1995). (a) and (c)] is swept, and S is measured, expressed in terms of [8] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, the probability P͕S ෇ #͖. The solid points correspond to initial N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. preparation in the jn͘jS͘ ෇ j0͘j#͘state, and the hollow points Weinfurter, Phys. Rev. A 52, 3457 (1995). correspond to preparation in the jn͘jS͘ ෇ j1͘j#͘ state. The [9] R. Landauer, in Proceedings of the Drexel-4 Symposium resulting patterns are shifted in phase by p rad. This flipping on Quantum Nonintegrability-Quantum Classical Corre- of jS͘ depending on the state of the control qubit indicates the conditional dynamics of the gate. Similar curves are obtained spondence, edited by D. H. Feng and B-L. Hu (Interna- when the jn͘jS͘ ෇ j0͘j"͘and j1͘j"͘states are prepared. The tional Press, Boston, to be published). lines are fits by a sinusoid, and the width of the Ramsey fringes [10] W. G. Unruh, Phys. Rev. A 51, 992 (1995). is consistent with the Ӎ50 msec duration of the CN operation. [11] K. Obermeyer, W. G. Teich, and G. Mahler, Phys. Rev. B 37, 8111 (1988). [12] S. Lloyd, Science 261, 1569 (1993). [13] M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernar- dot, A. Maali, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 72, 3339 (1994); M. Brune, F. Schmidt-Kaler, J. ment computations on larger quantum registers. For ex- Deyer, A. Maali, and J. M. Raimond, “Laser Spectroscopy ample, the CN gate between two ions (m and n) might be XII,” edited by M. Inguscio (World Scientific, Singapore, realized by mapping the internal state of the mth ion onto to be published). the collective vibrational state of all ions, applying the [14] H. J. Kimble, in “Laser Spectroscopy XIII” Ref. [13]. single-ion CN operation demonstrated in this work to the [15] T. Sleator and H. Weinfurter, Phys. Rev. Lett. 74, 4087 nth ion, then returning the vibrational state back to the in- (1995). ternal state of the mth ion. (This mapping may be achieved [16] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev. by simply driving a p pulse on the red of blue sideband of Lett. 75, 4011 (1995). the mth ion.) This approach is equivalent to the scheme [17] S. R. Jefferts, C. Monroe, E. W. Bell, and D. J. Wineland, proposed by Cirac and Zoller [1,23]. An arbitrary compu- Phys. Rev. A 51, 3112 (1995). tation may then be broken into a number of such operations [18] J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. on different pairs of ions, accompanied by single qubit ro- Picard, and C. R. Willis, Phys. Rev. Lett. 48, 867 (1982). tations on each ion (carrier transitions) [6–8]. [19] A. Ekert and R. Jozsa, Rev. Mod. Phys. (to be published). We are currently devoting effort into the multiplexing [20] N. F. Ramsey, Molecular Beams (Oxford University Press, of the register to many ions. Several technical issues London, 1956). remain to be explored in this scaling, including laser- [21] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. cooling efficiency, the coupling of internal vibrational Wineland, Phys. Rev. Lett. 62, 403 (1989). modes due to trap imperfections, and the unique address- [22] J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, IEEE Trans. Instrum. Meas. 40, 126 ing of each ion with laser beams. Although we can trap (1991). and cool a few ions in the current apparatus, other ge- [23] The controlled-NOT operator proposed in Ref. [1] ometries such as the linear rf-ion trap [24] or an array 1͞2 1,0 2,1 1,0 1͞2 is Vn ͑p͞2͒Um Un Um Vn ͑2p͞2͒, adopting their of ion traps each confining a single ion [25] might be notation. This is equivalent to the controlled-NOT considered. operator proposed here between ions m and n, 1,0 1͞2 2,1 1͞2 1,0 This work is supported by the U.S. Office of Naval Um Vn ͑p͞2͒Un Vn ͑2p͞2͒Um , since Vn and Research and the U.S. Army Research Office. We Um commute. acknowledge useful contributions from J. C. Bergquist [24] M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and J. J. Bollinger. We thank Robert Peterson, Dana and D. J. Wineland, Phys. Rev. A 45, 6493 (1992). Berkeland, and Chris Myatt for helpful suggestions on the [25] F. Major, J. Phys. (Paris) Lett. 38, L221 (1977); R. manuscript. Brewer, R. G. DeVoe, and R. Kallenbach, Phys. Rev. A 46, R6781 (1992).

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