- Email: [email protected]
Quantum logic gate with microtraps
1 February 2001
Optics Communications 188 (2001) 141±148
www.elsevier.com/locate/optcom
Quantum logic gate with microtraps Erika Andersson *, Stig Stenholm Department of Physics, Royal Institute of Technology, Lindstedtsvagen 24, S-10044 Stockholm, Sweden Received 19 May 2000; received in revised form 1 December 2000; accepted 1 December 2000
Abstract Recent cooling and trapping techniques for atoms allows the fabrication of genuinely microscopic quantum wires and dots for individual particles. We discuss the physical features of a structure which can implement the controlled-not operation in quantum logic. This consists of a network of tube-like microtraps and a laser that acts as a clock pulse. Our work discusses the in¯uence of potential geometry, particle±particle interactions and quantum statistics. The physical features are illustrated by wave packet simulations. Ó 2001 Published by Elsevier Science B.V. PACS: 03.75.Be; 03.67. a Keywords: Quantum information; Quantum computation; Neutral-atom microtraps
1. Introduction The recent advances in technology oer physics research structures where quantum mechanical eects become increasingly signi®cant. In these structures, the spatial dimensions are of the same order as the quantum de Broglie wavelength, which implies that any analysis of the phenomena observed must be based on quantum considerations. These goals have long been achieved in semiconductor heterostructures and metallic nanostructures [1±3]. In the present context, we use the term microstructures to describe devices operating entirely in the quantum regime. The goal can be stated to be structures at the scales of a few nanometers or below. They may be achieved by development of the present manufacturing tech-
*
Corresponding author. Fax: +46-8-200430. E-mail address: [email protected] (E. Andersson).
niques or future chemical design at the molecular level. Such devices can be used both for tests of fundamental physical principles and information storage and processing. Recently many experiments have tested the fundamental features of quantum mechanics using photons; for an example see Refs. [4,5]. These oer many advantages, they are easily produced in controlled ways, they do not interact and they propagate undisturbed over long distances. In addition the detection of photons is ecient and, by now, well understood. It is, however, of interest to carry out fundamental experiments also with massive particles. They have varying velocities, good localization and interactions, which are necessary for information processing. It also becomes possible to perform experiments with atoms obeying Bose or Fermi statistics. In a previous paper [6], we discussed the possibilities to observe eects of quantum statistics on fundamental properties of particles in man-made
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structures. Here we want to discuss the use of similar systems for information processing by considering a simple model of a quantum logical gate. By arranging a network of grooves on a substrate surface, we may launch particles into various inputs, let them propagate through the device and react to its structures and with each other. The advantage is that both the structures and the input states are easy to control in atomic experiments. In experiments with electrons, the structures are easily designed but the control of single electron states has not been eciently achieved. In addition, electrons are likely to lose their quantum coherence much faster than well cooled atomic particles. There has recently been considerable interest in manufacturing the types of structures we envisage in this paper. Such works may well provide an opportunity to design quantum apparatuses, process information and perform computations. An equivalent point of view is expressed by Schmiedmayer in Ref. [7]. Neutral atoms trapped in optical lattices or microtraps might also be used for implementing quantum logic gates [8,9]. The authors of Ref. [10] consider a possible implementation of a quantum gate for neutral atoms in time dependent microtraps based on collisional interactions. Neutral atoms can be stored in magnetic traps, and such traps can be made very small [11,12]. This requires high precision in the fabrication of the solid structures de®ning the dimensions of the trap. Modern lithographic technology suggests that such structures could be made even much smaller, and then we can imagine experiments in traps of genuinely microscopic dimensions, where quantum eects would dominate the particle dynamics. Similar structures can be constructed by combining charged wires with evanescent wave mirrors [13,14] or magnetic mirrors [15,16]. Such combinations can be used to build the structures utilized in nano-electronics. Purely magnetic guides may also be used [17±19]. The use of microfabricated current-carrying structures to guide atomic motion has been investigated by many groups [20±24]. Schmiedmayer has also discussed the use of such structures to construct quantum dots and quantum wires for atoms [7].
Alternative ways to achieve guided motion and possibly controlled interaction between atoms is to utilize hollow optical ®bers with evanescent waves trapping the atoms to narrow channels at the center of the structures [25±30]. These can eventually be fused to provide couplers similar to those used for optical signal transmission in ®bers. Also the pure atomic wave guide achievable by the use of hollow laser modes may be used [31]. In this paper we demonstrate a model of a logical gate by emulating the electronic process of Coulomb blockade [32]. We envisage quantum information carried by atomic particles in the z direction of a man-made structure designed to guide the motion in the x and y directions. Along the path, the particles may thus be transmitted, re¯ected or trapped. Utilizing their interaction we can perform logic operations by directing a target bit particle into dierent outputs depending on the state of a suitably introduced control bit particle. The individual qubits are carried by the presence of particles at speci®c entrance ports of the device. The time evolution takes a sequential stream of particles through the interaction region. As a demonstration, we investigate a simple controllednot (CNOT) gate: Each particle normally passes through the structure unaected. Applying a pulse con®nes one incoming particle to the structure, where its interaction with the next incoming particle prevents this from passing. In analogy with the Coulomb blockade of nanostructures, the interaction shifts the energy necessary to pass out of reach for the incoming one which becomes backscattered. The nanostructure single-electron problem is, however, really a many-body system due to the ever present Fermi seas of the electrodes, whereas we consider a genuine two particle system. In Section 2 we describe the potential structure chosen for our model and describe its action on a single particle, which is found to be transmitted. By considering two particles, and adding an external pulse, we show in Section 3 how the system can operate as a CNOT gate. Here we utilize the particle±particle interaction in a manner reminiscent of the Coulomb blockade. Finally we present a short discussion and some conclusions in Section 4. Details of the numerical computations are given in the Appendix A.
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2. Building the model In order to be able to treat the time-dependent problem numerically, we restrict our discussion to two particles in a one-dimensional con®guration. The full two-dimensional Schr odinger equation can then be written as ih
o W z1 ; z2 ; t ot
h2 o2 2M oz21
h2 o2 V z1 2M oz22 V z2 V12 jz1 z2 j W z1 ; z2 ; t: 1
Here the potential V z is given by the structure and the particle±particle eect is described by V12 . The initial state consists of a sequence of wave packets describing particles launched from a source situated at a large negative value of z. Throughout the numerical work we apply the scaling ez z=n;
et t=s;
e p sp=mn
Fig. 1. The gate potential V z. The particles are propagating in the z direction in the matter wave guide, con®ned in the x and y directions. The energies E1 of the bound ground state and E2 of the quasibound tunneling state are indicated with horizontal lines.
2
giving dimensionless variables; in the following treatment we refer to these dimensionless quantities even if this is not indicated in the notation. With the above scaling we ®nd 1 s ez ; e p z; p ie h; 3 n mn which allows us to choose the eective Planck's constant e h 1. The single particle potential V z is chosen as shown in Fig. 1; a double barrier with a dip; the exact shape is given in the Appendix A presenting some details of our numerical work. The potential is designed such that it contains two states: one stable ground state at E1 and one quasibound resonance at E2 . The state at E1 is close to the ground state of the harmonic oscillator part of the e xs bottom of the trap, whose frequency is x e V 0 5. Its energy is thus close to 1=2e h x 2:5; in fact, the numerically obtained value is E1 2:53. The upper state is chosen such that is provides a broad enough resonance to allow a suitably prepared wave packet to tunnel through the structure. In Fig. 2 we plot the transmission of the potential as a function of energy; near
Fig. 2. The transmission coecient of the potential shown in Fig. 1 as a function of the energy of the incoming particle. The tunneling resonance at approximately E2 2:1 is indicated with a dotted line.
E E2 2:1 a reasonably monoenergetic wave packet is totally transmitted. Integrating the one-dimensional version of the Schr odinger equation (1), we ®nd that the wave packet is transmitted to about 90% through the structure with the parameters chosen. In Fig. 3 we show a contour plot of the probability distribution jWj2 during the process. A Gaussian wave packet is incident on the gate and around t 0 it tunnels through. The wave packet dwells in the structure for a period of time undergoing rapid interferences, but transmits nearly without delay. In Fig. 4 we see how the transmission builds up and we also show the probability of re¯ection at about 10%. This number is a numerical compromise between a
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Fig. 3. A contour plot of the probability distribution of a wave packet tunneling through the potential dip of Fig. 1 as a function of position and time. The particle is incident on the gate with the average kinetic energy Ekin E2 . About 90% of the probability tunnels through.
Fig. 4. The probability, as a function of time, to ®nd the wave packet of Fig. 3 to the right (Ð, tunneled) or to the left (- - -, re¯ected or incident) of the potential well.
when a wave packet passes the potential structure, we apply a suitable laser pulse at the resonance frequency E2 E1 . We assume that the induced dipole couples the states. In a simple two-level system such an interaction should bring the population from E2 to E1 by the application of a ppulse. Here we encounter a dynamic situation, the transfer of population from E2 to E1 must compete with the process taking it through the barriers. The catching of the population must occur while the wave packet dwells inside the structure. Hence the competition between a broad transmission band of energy and a sucient dwell time in the structure. The details of the pulse used are given in the Appendix A. In Fig. 5 we show a contour plot of the catching procedure. When the laser pulse hits the incoming wave packet during its sojourn in the structure, it is partly trapped in the state E1 and cannot leave the structure. This ®gure should be compared with Fig. 6. With our choice of parameters we can catch about 63% of the incoming wave packet as shown in Fig. 6. We have not attempted an absolute maximization of this fraction, the present parameters suce to demonstrate our argument. We proceed to exploit the process above to achieve gating operation. To this end we need to invoke the particle±particle interaction. When this is strong enough, the presence of the trapped particle shifts the resonance E2 out of the spectral region of a subsequent particle, which conse-
state of long enough lifetime to work as a gate and a broad enough spectrum to allow the transmission of a wave packet. We could increase the transmission by narrowing the energy distribution, which however would imply a broadening in position space.
3. The gate operation Next we describe how the potential structure may be used to achieve the desired gate operation:
Fig. 5. Catching the passing wave packet by applying a suitable laser pulse. We show a countour plot of the probability distribution of the wave packet attempting to tunnel as a function of position and time.
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Fig. 6. The probability to ®nd the wave packet of Fig. 5 in the ground state of the potential dip as function of time. We see that the particle is trapped with about 63% eciency. The wiggles seen on the ®gure derive from the inclusion of the counterrotating terms in the calculation.
quently is scattered back instead of transmitted. The interaction can be either of the Coulomb form VC
V0 ; r
4
or for neutral particles it may be taken to be of Lennard-Jones type, " 6 # 12 b b VL-J V0 : 5 r r As usual, in one dimension we should use regularized potential variables by setting q r jz21 z22 j e2 : 6 The strength of the potential is regulated by the parameter V0 . The best results are achieved by a repulsive interaction emulating the Coulomb blockade situation. Looking at Fig. 2, this corresponds to moving the transmission peak to the right and attempting tunneling in the low transmission region to its left. On the other hand, an attractive potential would move the transmittance peak to lower energy which cannot be as ecient as is clearly seen in Fig. 2. In order to test the in¯uence of particle statistics, we carry out the two-particle wave function propagation starting from the initial state
Fig. 7. Snapshots of the two-particle (fermion) probability distribution. In the ®rst frame, we see both particles to the left of the well. One of them is moving towards the well at z 0 with the appropriate kinetic energy E 2:1 for resonant tunneling. As it reaches the dip, the laser pulse is applied, and the particle is trapped (frames 2 and 3). The second particle, previously at rest, now gets an appropriate momentum boost in order to start moving towards z 0 (frame 4). If the well were empty, this would give resonant transmission. However, as seen from the last four frames, it is re¯ected back. The interaction is given by Eq. (4) with V0 1:0 and e 0:1.
W z1 ; z2 / /1 z1 /2 z2 /1 z2 /2 z1 ;
7
for details of the states chosen see the Appendix A. The results of the calculations indicating two wave packets is shown in Fig. 7, which is drawn for the fermionic case. The frames show consecutive time instants in the con®guration space fz1 , z2 g of the two particles. In the ®rst frame, we see both particles to the left of the well. One of them is moving towards the well at z 0 with the appropriate kinetic energy E 2:1 for resonant tunneling. As it reaches the potential dip, the laser pulse is applied, and the particle is trapped; this is achieved in the third frame. The second particle,
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E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
Fig. 8. A comparison between the cases with and without a waiting particle in the gate well. The full line shows the integrated probability to ®nd the tunneling particle (incident from the left) to the right of an empty gate potential dip, as a function of time. The dotted lines indicate the same probability for the case that another particle is waiting trapped in the central potential well. The bosons are more aected by the interaction than the fermions, since the fermions obey the exclusion principle and stay further apart, thus being less sensitive to the interaction.
previously at rest, is now given an appropriate momentum boost in order to make it start moving towards z 0. If the well were empty, this would result in resonant transmission. However, as seen from the following frames, the particle is re¯ected back. In the last frame we see one particle re¯ected while the other one remains trapped. Fig. 8 shows a comparison between the cases with and without a particle waiting in the gate well. The full line shows the integrated probability to ®nd the tunneling particle to the right of an empty gate potential dip. We see that this probability starts from zero, and, as the particle tunnels, approaches 90% as previously mentioned. The dotted lines indicate the same probability for the case that a previous particle is trapped in the middle potential well. Here we see that the bosons are more aected by the interaction than the fermions, since the fermions obey the exclusion principle and strive to stay further apart, thus being less sensitive to the interaction. The stronger the interaction is, the more complete the re¯ection will be. These calculations were carried out with a repulsive potential given by Eq. (4), with V0 1:0 and e 0:1, which implies
an almost delta-function-like interaction potential. This should be valid whenever the energy is low. Similar results are obtained with a Lennard-Jones potential given by Eq. (5), although the best results are achieved with a purely repulsive potential as previously explained. This implies that CNOT operation might be realized as follows: Let the ®rst particle be the control bit and the second one the target bit. The laser pulse acts as a clock pulse. Let the control bit be equal to one if it is in the wave guide discussed, and equal to zero if it is in another wave guide. Now, if the control bit is equal to one, it will be trapped by the laser, and the well will contain a particle, otherwise not. When the target bit arrives, it will be re¯ected if there is a trapped control bit ± the target bit is ¯ipped ± otherwise it will just tunnel through, which then corresponds to the target bit being the same on exit. Numerically, we are not able to carry out the wave packet calculations for two particles in two dimensions, but in reality, the gate would be two-dimensional so that output and input wave guides would be dierent, as schematically shown in Fig. 9. In our model calculation, the gate bit particle is left in the potential dip. A second laser pulse will obviously take it back to the level E2 , but only a physical realization makes its further fate of any interest. 4. Discussion of real systems and conclusions We have proposed a scheme to realize a quantum CNOT gate using linear matter wave guides, an additional potential dip structure, and a laser which acts as a clock pulse. If the wave guide network is built utilizing evanescent wave or magnetic mirrors with current carrying wires placed on top of these, as described in Ref. [7], the needed gate potential might be realized by placing additional current carrying patterns on the mirror surface. How, then, do the scaled variables relate to experimental values? Our numerical calculations are not done with any speci®c experiment in mind, but the parameters have been chosen to facilitate the numerical computations. For example, if we imagine a length scale n of 100 nm, this would imply a time scale s of about 1.1 ls, ac-
E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
Fig. 9. Although our calculations are performed with only one spatial dimension, where input and output wave guides necessarily are the same, the gate would be realized with a twodimensional pattern of wave guides. In this schematic picture, the thick vertical lines represent the two potential barriers; the control bit may be trapped between them. Input and output channels are indicated by arrows; T refers to the target bit and C to the control bit. If the control bit equals zero it enters and exits in the separate wave guide on the right. In this case the target bit particle passes through the structure unchanged. If the control bit equals one it could enter in either of the target bit input channels, or possibly through a separate channel not shown in this ®gure. In this case, the target bit particle is re¯ected independent of its value. As explained in the text, we have not stated exactly how the control bit should be removed from the dip after gate operation.
cording to Eq. (2), supposing we consider 7 Li ate 5 would then oms. The scaled trap frequency x correspond to a trap frequency of about 700 kHz. These values are not too far from the parameter ranges listed in Ref. [7]. However, to reach sucient atom±atom interaction strengths, one would have to localize the atoms to a region of a few nanometers. More detailed consideration of the realistic operation ranges of the device have not been contemplated. When experimental realizations emerge, they will no doubt oer their own limitations and opportunities to choose the operation conditions. For the same reason we have not attempted an absolute optimization of the parameters used in the numerical computations. Varying the shape and dimension of the potential, choosing a suitable pulse and possibly tailoring the initial state, we may well achieve results much better than the 90%
147
transmission of Fig. 4, the catching probability of Fig. 6 and the gating eciency of about one order of magnitude as shown in Fig. 8. However, the feasibility of the process proposed appears clearly in Fig. 7, and we feel that more detailed calculations are only justi®ed in modelling a realistic laboratory situation. In the experimental situations, the interaction parameter V0 is determined by nature and cannot easily be varied. However, using the fact that real structures are three-dimensional, suitable engineering may allow one to adjust the particle±particle in¯uence to some degree. The crucial experimental problem is to prepare the particles in the appropriate quantum states, to launch them into to the wave guides and to retain their quantum coherence during the gate operation. With atomic cooling and trapping techniques, we surmise that this may be feasible in the near future. Appendix A. Details of the numerical calculations The potential is given by V z
8 0; > > > > > >
E0 ; 1 xx E0 ; 2 > > > U 12 x2 x2 E0 ; > > > : 0; 1 2 2 xx 2 2 2
x < 1 p= 2C; 1 p= 2C < x < 1 p= 2C < x < 1 1
1 p= 2C; p= 2C;
p= 2C < x < 1 p= 2C;
x > 1 p= 2C;
A:1 where the scaled parameters are x 5, C 5, E0 5 and U sin C x 1 1=2:
A:2
The numerical propagation of the Schr odinger equation (1) is carried out using the split operator method [33,34]. The initial probability distribution for the wave packet in the one-dimensional calculations is given by n o 2 /0 z N exp 12 z z0 =Dz ikz ; A:3 where N is a normalization constant; we take z0 30, Dz 7 and k 2:049 to give resonant transmission. The initial two-particle wave function is given by
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E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
W z1 ; z2 / /1 z1 /2 z2 /1 z2 /2 z1 ;
A:4
where /1 z expf12 z z0 =Dz2 ikzg with z0 20 and k 2:049, and /2 z expf12 z z0 = 2 Dz g with z0 35, and the gives a symmetric or antisymmetric wave function, which is then also normalized. The laser pulse applied in the wave packet calculations is given by F z Az cos Xt cos2 p t 2
exp z=wz ;
t0 = 2wt A:5
when jt t0 j < wt , and F z 0 otherwise. In these calculations we take the amplitude A 20, the laser frequency X 4:6 for resonant coupling of the bound and the quasibound levels, and the widths wt 15 and wz 15. The parameter t0 is adjusted so that the laser pulse is centered around the instant when the particle is tunneling. In the calculations with two particles, because of limitations on the grid size, absorbing boundary conditions are applied in order to deal with the probability that the ®rst particle is not caught in the dip but tunnels through or gets re¯ected. References [1] C. Weisbuch, B. Vinter, Quantum Semiconductor Structures, Academic Press, New York, 1991. [2] J.H. Daves, A.R. Long, Physics of Nanostructures, Proceedings of the Thirty-Eight Scottish Universities Summer School, St. Andrews 1991, Institute of Physics, Bristol, 1991. [3] H. Ehrenreich, D. Turnbull (Eds.), Semiconductor Heterostructures and Nanostructures, Academic Press, New York, 1991. [4] G. Weihs, M. Reck, H. Weinfurter, A. Zeilinger, Phys. Rev. Lett. 54 (1996) 893. [5] K. Mattle, H. Weinfurter, P. Kwiat, A. Zeilinger, Phys. Rev. Lett. 76 (1996) 4656. [6] E. Andersson, M. Fontenelle, S. Stenholm, Phys. Rev. A 59 (1999) 3841. [7] J. Schmiedmayer, Eur. Phys. J. D 4 (1998) 57. [8] G.K. Brennen, C.M. Caves, P.S. Jessen, I. Deutsch, Phys. Rev. Lett. 82 (1999) 1060. [9] H.-J. Briegel, T. Calarco, D. Jaksch, J.I. Cirac, P. Zoller, J. Mod. Opt. 47 (2000) 415.
[10] T. Calarco, E.A. Hinds, D. Jaksch, J. Schmiedmayer, J.I. Cirac, P. Zoller, Phys. Rev. A 61 (2000) 022304. [11] J.D. Weinstein, K.G. Libbrecht, Phys. Rev. A 52 (1995) 4004. [12] V. Vuletic, T.W. Hansch, C. Zimmermann, Europhys. Lett. 36 (1996) 349. [13] T. Esslinger, M. Weidem uller, A. Hemmerich, T. Hansch, Opt. Lett. 18 (1993) 450. [14] W. Seifert, R. Kaiser, A. Aspect, J. Mlynek, Opt. Commun. 111 (1994) 566. [15] T.M. Roach, H. Abele, M.G. Boshier, H.L. Grossman, K.P. Zetie, E.A. Hinds, Phys. Rev. Lett. 75 (1995) 629. [16] I.G. Hughes, P.A. Barton, T.M. Roach, M.G. Boshier, E.A. Hinds, J. Phys. B 30 (1997) 647. [17] J. Denschlag, D. Cassettari, J. Schmiedmayer, Phys. Rev. Lett. 82 (1999) 2014. [18] M. Key, I.G. Hughes, W. Rooijakkers, B.E. Sauer, E.A. Hinds, Phys. Rev. Lett. 84 (2000) 1371. [19] O. Zobay, B.M. Garraway, Opt. Commun. 178 (2000) 93. [20] R. Folman, P. Kr uger, D. Cassettari, B. Hessmo, T. Maier, J. Schmiedmayer, Phys. Rev. Lett. 84 (2000) 4749. [21] D. Cassettari, A. Chenet, R. Folman, A. Haase, B. Hessmo, P. Kr uger, T. Maier, S. Schneider, T. Calarco, J. Scmiedmayer, Appl. Phys. B 70 (2000) 721. [22] D. Cassettari, B. Hessmo, R. Folman, T. Maier, J. Schmiedmayer, quant-ph/0003135. [23] J. Reichel, W. Hansel, T.W. Hansch, Phys. Rev. Lett. 83 (1999) 3398. [24] N.H. Dekker, C.S. Lee, V. Lorent, J.H. Thywissen, S.P. Smith, M. Drndic, R.M. Westerwelt, M. Prentiss, Phys. Rev. Lett. 84 (2000) 1124. [25] M.A. Olshanii, Y.B. Ovchinnikov, V.S. Lethokov, Opt. Commun. 98 (1993) 77. [26] S. Marksteiner, C.M. Savage, P. Zoller, S.L. Rolston, Phys. Rev. A 50 (1994) 2680. [27] M.J. Renn, D. Montgomery, O. Vdovin, D.Z. Anderson, C.E. Wieman, E.A. Cornell, Phys. Rev. Lett. 75 (1995) 3253. [28] M.J. Renn, E.A. Donley, E.A. Cornell, C.E. Wieman, D.Z. Anderson, Phys. Rev. A 53 (1996) R648. [29] M.J. Renn, E.A. Donley, E.A. Cornell, C.E. Wieman, D.Z. Anderson, Phys. Rev. A 55 (1997) 3684. [30] H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K.I. Lee, W. Jhe, Phys. Rev. Lett. 76 (1998) 4500. [31] J. Yin, Y. Zhu, W. Wang, Y. Wang, W. Jhe, J. Opt. Soc. Am. B 15 (1998) 25. [32] K.K. Likharev, IBM J. Res. Dev. 32 (1988) 144. [33] J.A. Fleck, J.A. Morris, M.D. Feit, Appl. Phys. 10 (1976) 129. [34] M.D. Feit, J.A. Fleck, A. Steiger, J. Comput. Phys. 47 (1982) 412.
Optics Communications 188 (2001) 141±148
www.elsevier.com/locate/optcom
Quantum logic gate with microtraps Erika Andersson *, Stig Stenholm Department of Physics, Royal Institute of Technology, Lindstedtsvagen 24, S-10044 Stockholm, Sweden Received 19 May 2000; received in revised form 1 December 2000; accepted 1 December 2000
Abstract Recent cooling and trapping techniques for atoms allows the fabrication of genuinely microscopic quantum wires and dots for individual particles. We discuss the physical features of a structure which can implement the controlled-not operation in quantum logic. This consists of a network of tube-like microtraps and a laser that acts as a clock pulse. Our work discusses the in¯uence of potential geometry, particle±particle interactions and quantum statistics. The physical features are illustrated by wave packet simulations. Ó 2001 Published by Elsevier Science B.V. PACS: 03.75.Be; 03.67. a Keywords: Quantum information; Quantum computation; Neutral-atom microtraps
1. Introduction The recent advances in technology oer physics research structures where quantum mechanical eects become increasingly signi®cant. In these structures, the spatial dimensions are of the same order as the quantum de Broglie wavelength, which implies that any analysis of the phenomena observed must be based on quantum considerations. These goals have long been achieved in semiconductor heterostructures and metallic nanostructures [1±3]. In the present context, we use the term microstructures to describe devices operating entirely in the quantum regime. The goal can be stated to be structures at the scales of a few nanometers or below. They may be achieved by development of the present manufacturing tech-
*
Corresponding author. Fax: +46-8-200430. E-mail address: [email protected] (E. Andersson).
niques or future chemical design at the molecular level. Such devices can be used both for tests of fundamental physical principles and information storage and processing. Recently many experiments have tested the fundamental features of quantum mechanics using photons; for an example see Refs. [4,5]. These oer many advantages, they are easily produced in controlled ways, they do not interact and they propagate undisturbed over long distances. In addition the detection of photons is ecient and, by now, well understood. It is, however, of interest to carry out fundamental experiments also with massive particles. They have varying velocities, good localization and interactions, which are necessary for information processing. It also becomes possible to perform experiments with atoms obeying Bose or Fermi statistics. In a previous paper [6], we discussed the possibilities to observe eects of quantum statistics on fundamental properties of particles in man-made
0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 1 1 6 1 - 5
142
E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
structures. Here we want to discuss the use of similar systems for information processing by considering a simple model of a quantum logical gate. By arranging a network of grooves on a substrate surface, we may launch particles into various inputs, let them propagate through the device and react to its structures and with each other. The advantage is that both the structures and the input states are easy to control in atomic experiments. In experiments with electrons, the structures are easily designed but the control of single electron states has not been eciently achieved. In addition, electrons are likely to lose their quantum coherence much faster than well cooled atomic particles. There has recently been considerable interest in manufacturing the types of structures we envisage in this paper. Such works may well provide an opportunity to design quantum apparatuses, process information and perform computations. An equivalent point of view is expressed by Schmiedmayer in Ref. [7]. Neutral atoms trapped in optical lattices or microtraps might also be used for implementing quantum logic gates [8,9]. The authors of Ref. [10] consider a possible implementation of a quantum gate for neutral atoms in time dependent microtraps based on collisional interactions. Neutral atoms can be stored in magnetic traps, and such traps can be made very small [11,12]. This requires high precision in the fabrication of the solid structures de®ning the dimensions of the trap. Modern lithographic technology suggests that such structures could be made even much smaller, and then we can imagine experiments in traps of genuinely microscopic dimensions, where quantum eects would dominate the particle dynamics. Similar structures can be constructed by combining charged wires with evanescent wave mirrors [13,14] or magnetic mirrors [15,16]. Such combinations can be used to build the structures utilized in nano-electronics. Purely magnetic guides may also be used [17±19]. The use of microfabricated current-carrying structures to guide atomic motion has been investigated by many groups [20±24]. Schmiedmayer has also discussed the use of such structures to construct quantum dots and quantum wires for atoms [7].
Alternative ways to achieve guided motion and possibly controlled interaction between atoms is to utilize hollow optical ®bers with evanescent waves trapping the atoms to narrow channels at the center of the structures [25±30]. These can eventually be fused to provide couplers similar to those used for optical signal transmission in ®bers. Also the pure atomic wave guide achievable by the use of hollow laser modes may be used [31]. In this paper we demonstrate a model of a logical gate by emulating the electronic process of Coulomb blockade [32]. We envisage quantum information carried by atomic particles in the z direction of a man-made structure designed to guide the motion in the x and y directions. Along the path, the particles may thus be transmitted, re¯ected or trapped. Utilizing their interaction we can perform logic operations by directing a target bit particle into dierent outputs depending on the state of a suitably introduced control bit particle. The individual qubits are carried by the presence of particles at speci®c entrance ports of the device. The time evolution takes a sequential stream of particles through the interaction region. As a demonstration, we investigate a simple controllednot (CNOT) gate: Each particle normally passes through the structure unaected. Applying a pulse con®nes one incoming particle to the structure, where its interaction with the next incoming particle prevents this from passing. In analogy with the Coulomb blockade of nanostructures, the interaction shifts the energy necessary to pass out of reach for the incoming one which becomes backscattered. The nanostructure single-electron problem is, however, really a many-body system due to the ever present Fermi seas of the electrodes, whereas we consider a genuine two particle system. In Section 2 we describe the potential structure chosen for our model and describe its action on a single particle, which is found to be transmitted. By considering two particles, and adding an external pulse, we show in Section 3 how the system can operate as a CNOT gate. Here we utilize the particle±particle interaction in a manner reminiscent of the Coulomb blockade. Finally we present a short discussion and some conclusions in Section 4. Details of the numerical computations are given in the Appendix A.
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2. Building the model In order to be able to treat the time-dependent problem numerically, we restrict our discussion to two particles in a one-dimensional con®guration. The full two-dimensional Schr odinger equation can then be written as ih
o W z1 ; z2 ; t ot
h2 o2 2M oz21
h2 o2 V z1 2M oz22 V z2 V12 jz1 z2 j W z1 ; z2 ; t: 1
Here the potential V z is given by the structure and the particle±particle eect is described by V12 . The initial state consists of a sequence of wave packets describing particles launched from a source situated at a large negative value of z. Throughout the numerical work we apply the scaling ez z=n;
et t=s;
e p sp=mn
Fig. 1. The gate potential V z. The particles are propagating in the z direction in the matter wave guide, con®ned in the x and y directions. The energies E1 of the bound ground state and E2 of the quasibound tunneling state are indicated with horizontal lines.
2
giving dimensionless variables; in the following treatment we refer to these dimensionless quantities even if this is not indicated in the notation. With the above scaling we ®nd 1 s ez ; e p z; p ie h; 3 n mn which allows us to choose the eective Planck's constant e h 1. The single particle potential V z is chosen as shown in Fig. 1; a double barrier with a dip; the exact shape is given in the Appendix A presenting some details of our numerical work. The potential is designed such that it contains two states: one stable ground state at E1 and one quasibound resonance at E2 . The state at E1 is close to the ground state of the harmonic oscillator part of the e xs bottom of the trap, whose frequency is x e V 0 5. Its energy is thus close to 1=2e h x 2:5; in fact, the numerically obtained value is E1 2:53. The upper state is chosen such that is provides a broad enough resonance to allow a suitably prepared wave packet to tunnel through the structure. In Fig. 2 we plot the transmission of the potential as a function of energy; near
Fig. 2. The transmission coecient of the potential shown in Fig. 1 as a function of the energy of the incoming particle. The tunneling resonance at approximately E2 2:1 is indicated with a dotted line.
E E2 2:1 a reasonably monoenergetic wave packet is totally transmitted. Integrating the one-dimensional version of the Schr odinger equation (1), we ®nd that the wave packet is transmitted to about 90% through the structure with the parameters chosen. In Fig. 3 we show a contour plot of the probability distribution jWj2 during the process. A Gaussian wave packet is incident on the gate and around t 0 it tunnels through. The wave packet dwells in the structure for a period of time undergoing rapid interferences, but transmits nearly without delay. In Fig. 4 we see how the transmission builds up and we also show the probability of re¯ection at about 10%. This number is a numerical compromise between a
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Fig. 3. A contour plot of the probability distribution of a wave packet tunneling through the potential dip of Fig. 1 as a function of position and time. The particle is incident on the gate with the average kinetic energy Ekin E2 . About 90% of the probability tunnels through.
Fig. 4. The probability, as a function of time, to ®nd the wave packet of Fig. 3 to the right (Ð, tunneled) or to the left (- - -, re¯ected or incident) of the potential well.
when a wave packet passes the potential structure, we apply a suitable laser pulse at the resonance frequency E2 E1 . We assume that the induced dipole couples the states. In a simple two-level system such an interaction should bring the population from E2 to E1 by the application of a ppulse. Here we encounter a dynamic situation, the transfer of population from E2 to E1 must compete with the process taking it through the barriers. The catching of the population must occur while the wave packet dwells inside the structure. Hence the competition between a broad transmission band of energy and a sucient dwell time in the structure. The details of the pulse used are given in the Appendix A. In Fig. 5 we show a contour plot of the catching procedure. When the laser pulse hits the incoming wave packet during its sojourn in the structure, it is partly trapped in the state E1 and cannot leave the structure. This ®gure should be compared with Fig. 6. With our choice of parameters we can catch about 63% of the incoming wave packet as shown in Fig. 6. We have not attempted an absolute maximization of this fraction, the present parameters suce to demonstrate our argument. We proceed to exploit the process above to achieve gating operation. To this end we need to invoke the particle±particle interaction. When this is strong enough, the presence of the trapped particle shifts the resonance E2 out of the spectral region of a subsequent particle, which conse-
state of long enough lifetime to work as a gate and a broad enough spectrum to allow the transmission of a wave packet. We could increase the transmission by narrowing the energy distribution, which however would imply a broadening in position space.
3. The gate operation Next we describe how the potential structure may be used to achieve the desired gate operation:
Fig. 5. Catching the passing wave packet by applying a suitable laser pulse. We show a countour plot of the probability distribution of the wave packet attempting to tunnel as a function of position and time.
E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
145
Fig. 6. The probability to ®nd the wave packet of Fig. 5 in the ground state of the potential dip as function of time. We see that the particle is trapped with about 63% eciency. The wiggles seen on the ®gure derive from the inclusion of the counterrotating terms in the calculation.
quently is scattered back instead of transmitted. The interaction can be either of the Coulomb form VC
V0 ; r
4
or for neutral particles it may be taken to be of Lennard-Jones type, " 6 # 12 b b VL-J V0 : 5 r r As usual, in one dimension we should use regularized potential variables by setting q r jz21 z22 j e2 : 6 The strength of the potential is regulated by the parameter V0 . The best results are achieved by a repulsive interaction emulating the Coulomb blockade situation. Looking at Fig. 2, this corresponds to moving the transmission peak to the right and attempting tunneling in the low transmission region to its left. On the other hand, an attractive potential would move the transmittance peak to lower energy which cannot be as ecient as is clearly seen in Fig. 2. In order to test the in¯uence of particle statistics, we carry out the two-particle wave function propagation starting from the initial state
Fig. 7. Snapshots of the two-particle (fermion) probability distribution. In the ®rst frame, we see both particles to the left of the well. One of them is moving towards the well at z 0 with the appropriate kinetic energy E 2:1 for resonant tunneling. As it reaches the dip, the laser pulse is applied, and the particle is trapped (frames 2 and 3). The second particle, previously at rest, now gets an appropriate momentum boost in order to start moving towards z 0 (frame 4). If the well were empty, this would give resonant transmission. However, as seen from the last four frames, it is re¯ected back. The interaction is given by Eq. (4) with V0 1:0 and e 0:1.
W z1 ; z2 / /1 z1 /2 z2 /1 z2 /2 z1 ;
7
for details of the states chosen see the Appendix A. The results of the calculations indicating two wave packets is shown in Fig. 7, which is drawn for the fermionic case. The frames show consecutive time instants in the con®guration space fz1 , z2 g of the two particles. In the ®rst frame, we see both particles to the left of the well. One of them is moving towards the well at z 0 with the appropriate kinetic energy E 2:1 for resonant tunneling. As it reaches the potential dip, the laser pulse is applied, and the particle is trapped; this is achieved in the third frame. The second particle,
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E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
Fig. 8. A comparison between the cases with and without a waiting particle in the gate well. The full line shows the integrated probability to ®nd the tunneling particle (incident from the left) to the right of an empty gate potential dip, as a function of time. The dotted lines indicate the same probability for the case that another particle is waiting trapped in the central potential well. The bosons are more aected by the interaction than the fermions, since the fermions obey the exclusion principle and stay further apart, thus being less sensitive to the interaction.
previously at rest, is now given an appropriate momentum boost in order to make it start moving towards z 0. If the well were empty, this would result in resonant transmission. However, as seen from the following frames, the particle is re¯ected back. In the last frame we see one particle re¯ected while the other one remains trapped. Fig. 8 shows a comparison between the cases with and without a particle waiting in the gate well. The full line shows the integrated probability to ®nd the tunneling particle to the right of an empty gate potential dip. We see that this probability starts from zero, and, as the particle tunnels, approaches 90% as previously mentioned. The dotted lines indicate the same probability for the case that a previous particle is trapped in the middle potential well. Here we see that the bosons are more aected by the interaction than the fermions, since the fermions obey the exclusion principle and strive to stay further apart, thus being less sensitive to the interaction. The stronger the interaction is, the more complete the re¯ection will be. These calculations were carried out with a repulsive potential given by Eq. (4), with V0 1:0 and e 0:1, which implies
an almost delta-function-like interaction potential. This should be valid whenever the energy is low. Similar results are obtained with a Lennard-Jones potential given by Eq. (5), although the best results are achieved with a purely repulsive potential as previously explained. This implies that CNOT operation might be realized as follows: Let the ®rst particle be the control bit and the second one the target bit. The laser pulse acts as a clock pulse. Let the control bit be equal to one if it is in the wave guide discussed, and equal to zero if it is in another wave guide. Now, if the control bit is equal to one, it will be trapped by the laser, and the well will contain a particle, otherwise not. When the target bit arrives, it will be re¯ected if there is a trapped control bit ± the target bit is ¯ipped ± otherwise it will just tunnel through, which then corresponds to the target bit being the same on exit. Numerically, we are not able to carry out the wave packet calculations for two particles in two dimensions, but in reality, the gate would be two-dimensional so that output and input wave guides would be dierent, as schematically shown in Fig. 9. In our model calculation, the gate bit particle is left in the potential dip. A second laser pulse will obviously take it back to the level E2 , but only a physical realization makes its further fate of any interest. 4. Discussion of real systems and conclusions We have proposed a scheme to realize a quantum CNOT gate using linear matter wave guides, an additional potential dip structure, and a laser which acts as a clock pulse. If the wave guide network is built utilizing evanescent wave or magnetic mirrors with current carrying wires placed on top of these, as described in Ref. [7], the needed gate potential might be realized by placing additional current carrying patterns on the mirror surface. How, then, do the scaled variables relate to experimental values? Our numerical calculations are not done with any speci®c experiment in mind, but the parameters have been chosen to facilitate the numerical computations. For example, if we imagine a length scale n of 100 nm, this would imply a time scale s of about 1.1 ls, ac-
E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
Fig. 9. Although our calculations are performed with only one spatial dimension, where input and output wave guides necessarily are the same, the gate would be realized with a twodimensional pattern of wave guides. In this schematic picture, the thick vertical lines represent the two potential barriers; the control bit may be trapped between them. Input and output channels are indicated by arrows; T refers to the target bit and C to the control bit. If the control bit equals zero it enters and exits in the separate wave guide on the right. In this case the target bit particle passes through the structure unchanged. If the control bit equals one it could enter in either of the target bit input channels, or possibly through a separate channel not shown in this ®gure. In this case, the target bit particle is re¯ected independent of its value. As explained in the text, we have not stated exactly how the control bit should be removed from the dip after gate operation.
cording to Eq. (2), supposing we consider 7 Li ate 5 would then oms. The scaled trap frequency x correspond to a trap frequency of about 700 kHz. These values are not too far from the parameter ranges listed in Ref. [7]. However, to reach sucient atom±atom interaction strengths, one would have to localize the atoms to a region of a few nanometers. More detailed consideration of the realistic operation ranges of the device have not been contemplated. When experimental realizations emerge, they will no doubt oer their own limitations and opportunities to choose the operation conditions. For the same reason we have not attempted an absolute optimization of the parameters used in the numerical computations. Varying the shape and dimension of the potential, choosing a suitable pulse and possibly tailoring the initial state, we may well achieve results much better than the 90%
147
transmission of Fig. 4, the catching probability of Fig. 6 and the gating eciency of about one order of magnitude as shown in Fig. 8. However, the feasibility of the process proposed appears clearly in Fig. 7, and we feel that more detailed calculations are only justi®ed in modelling a realistic laboratory situation. In the experimental situations, the interaction parameter V0 is determined by nature and cannot easily be varied. However, using the fact that real structures are three-dimensional, suitable engineering may allow one to adjust the particle±particle in¯uence to some degree. The crucial experimental problem is to prepare the particles in the appropriate quantum states, to launch them into to the wave guides and to retain their quantum coherence during the gate operation. With atomic cooling and trapping techniques, we surmise that this may be feasible in the near future. Appendix A. Details of the numerical calculations The potential is given by V z
8 0; > > > > > >
E0 ; 1 xx E0 ; 2 > > > U 12 x2 x2 E0 ; > > > : 0; 1 2 2 xx 2 2 2
x < 1 p= 2C; 1 p= 2C < x < 1 p= 2C < x < 1 1
1 p= 2C; p= 2C;
p= 2C < x < 1 p= 2C;
x > 1 p= 2C;
A:1 where the scaled parameters are x 5, C 5, E0 5 and U sin C x 1 1=2:
A:2
The numerical propagation of the Schr odinger equation (1) is carried out using the split operator method [33,34]. The initial probability distribution for the wave packet in the one-dimensional calculations is given by n o 2 /0 z N exp 12 z z0 =Dz ikz ; A:3 where N is a normalization constant; we take z0 30, Dz 7 and k 2:049 to give resonant transmission. The initial two-particle wave function is given by
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E. Andersson, S. Stenholm / Optics Communications 188 (2001) 141±148
W z1 ; z2 / /1 z1 /2 z2 /1 z2 /2 z1 ;
A:4
where /1 z expf12 z z0 =Dz2 ikzg with z0 20 and k 2:049, and /2 z expf12 z z0 = 2 Dz g with z0 35, and the gives a symmetric or antisymmetric wave function, which is then also normalized. The laser pulse applied in the wave packet calculations is given by F z Az cos Xt cos2 p t 2
exp z=wz ;
t0 = 2wt A:5
when jt t0 j < wt , and F z 0 otherwise. In these calculations we take the amplitude A 20, the laser frequency X 4:6 for resonant coupling of the bound and the quasibound levels, and the widths wt 15 and wz 15. The parameter t0 is adjusted so that the laser pulse is centered around the instant when the particle is tunneling. In the calculations with two particles, because of limitations on the grid size, absorbing boundary conditions are applied in order to deal with the probability that the ®rst particle is not caught in the dip but tunnels through or gets re¯ected. References [1] C. Weisbuch, B. Vinter, Quantum Semiconductor Structures, Academic Press, New York, 1991. [2] J.H. Daves, A.R. Long, Physics of Nanostructures, Proceedings of the Thirty-Eight Scottish Universities Summer School, St. Andrews 1991, Institute of Physics, Bristol, 1991. [3] H. Ehrenreich, D. Turnbull (Eds.), Semiconductor Heterostructures and Nanostructures, Academic Press, New York, 1991. [4] G. Weihs, M. Reck, H. Weinfurter, A. Zeilinger, Phys. Rev. Lett. 54 (1996) 893. [5] K. Mattle, H. Weinfurter, P. Kwiat, A. Zeilinger, Phys. Rev. Lett. 76 (1996) 4656. [6] E. Andersson, M. Fontenelle, S. Stenholm, Phys. Rev. A 59 (1999) 3841. [7] J. Schmiedmayer, Eur. Phys. J. D 4 (1998) 57. [8] G.K. Brennen, C.M. Caves, P.S. Jessen, I. Deutsch, Phys. Rev. Lett. 82 (1999) 1060. [9] H.-J. Briegel, T. Calarco, D. Jaksch, J.I. Cirac, P. Zoller, J. Mod. Opt. 47 (2000) 415.
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