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Coupled quantum dots as quantum exclusive-OR gate
Superlattices and Microstructures, Vol. 22, No. 3, 1997
Coupled quantum dots as quantum exclusive-OR gate ´ A. Brum†, Pawel Hawrylak Jose Institute for Microstructural Sciences, National Research Council, Ottawa (Ont.), Canada, K1A 0R6
(Received 15 July 1996) The concepts relevant to quantum cellular automata and quantum computers are studied using a simple model of a quantum exclusive-OR (QXOR) gate device consisting of four coupled quantum dots. The QXOR device can be charged with up to N = 8 electrons. The quantum bits of the device correspond to states of the device in second quantized form. We use exact diagonalization techniques in the configuration space to calculate physical properties of QXOR as a function of the number of electrons N and external perturbations in the form of electric and magnetic fields. This allows us to investigate the switching of the QXOR gate, and its ability to store and transmit information. c 1997 Academic Press Limited
A significant progress has been achieved in producing low dimensional semiconductor structures (quantum dots) [1, 2], but their integration into quantum circuits appears to be in its infancy. Two quantum logic circuits have been studied theoretically: the quantum cellular automata (QCA) [3] and the quantum computer (QC) [4]. In the QCA individual cells are in contact with reservoirs which guarantee their rapid relaxation into the ground state (GS). The GS can be manipulated by the application of external fields or fields produced by GS charge distribution of neighbouring cells. By contrast, the QC involves both ground and excited states and relies on the coherent superposition of all states. The quantum state of QC is manipulated coherently by unitary transformations due to applied classical external fields. Simple spin models of QC have been studied so far, stimulated by recent developments in, e.g. quantum fourier transform methods [5]. We study here an extension of these ideas to the system of fermions in low dimensional semiconductor structures. To illustrate the principles and problems associated with QCA and QC we study a simple model system of four coupled quantum dots, similar to the one studied by Lent et al. [3]. The quantum dots are placed at the corners of a square, forming a four-dots ring (4-DR). Each dot is independently in contact with external leads. We consider small dots and assume a single orbital per dot. The dots are coupled with each other and with leads through a hopping matrix element ti, j . In the site representation the Hamiltonian of the system can be written as: X X 1 X + + + + e ∗ + H= E i,σ ci,σ ci,σ + [ti, j ci,σ c j,σ + t j,i c j,σ ci,σ ] + hi, j|vee |k, lici,σ c j,σ 0 ck,σ 0 cl,σ (1) 2 0 i,σ i> j,σ i jkl,σ,σ † Leave of absence from: Instituto de Fisica ‘Gleb Wataghin’—DFESCM, Universidade Estadual de Campinas, 13081970 Campinas (SP), Brazil.
0749–6036/97/070431 + 06 $25.00/0
sm960263
c 1997 Academic Press Limited
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Superlattices and Microstructures, Vol. 22, No. 3, 1997
We consider the following Coulomb matrix elements: the on-site Coulomb repulsion, U = hi, i|vee |i, ii, the off-site repulsion, VD = hi, i ± 1|vee |i ± 1, ii, and the exchange interaction, VX = hi, i ± 1|vee |i, i ± 1i. In what follows, VD /U = 0.5. The system can be charged with up to 8 electrons. The critical parameter of the system is the ratio between the Coulomb on-site repulsion U and the hopping parameter between nearest neighbors t, U/t, where t = t1,2 . We apply our 4-DR model to re-examine the QCA [3]. The hopping parameter is ti j = t0 δi, j±1 and the system is charged with two electrons. Our emphasis is on the role of the exchange interaction, VX , which was not considered in the original model [3]. Under the application of an external perturbation V , the ground state 3 −ρ2 −ρ4 , where ρn is the charge density at the ith dot. The external changes polarization P, defined as P = ρρ11 +ρ +ρ3 +ρ2 +ρ4 perturbation is the potential created by a charge distribution of a neighbor cell with a fixed polarization (Pext ). The distance between the center of the neighbor cells is d = 3a, where a is the distance between the quantum dots in the cell. Figure 1 shows energies as a function of Pext of the 4-DR with U/t0 = 100 for (A) VX /U = 0 and (B) VX /U = 0.01. Figure 1C shows the corresponding GS polarization. For a repulsive exchange VX ≤ 0, Fig. 1A, the GS is a singlet as found in Ref. [3]. The polarization of GS shows a slow response to the external polarization and does not saturate. However, once the attractive exchange interaction is included, Fig. 1B, the triplet becomes the GS. This type of interaction is responsible for singlet–triplet transitions already observed in quantum dots [1, 2]. The polarization of triplet GS shows a strong non-linearity with the external polarization, saturating already at weak external polarization. This is the ideal condition for the storage and transmission of information from cell to cell without loss of information [3]. Clearly, the exchange interaction plays an important role and should be considered in the analysis of these devices. It is important to observe that the triplet excited state shows an opposite polarization (dashed line in Fig. 1C). The energy difference between the triplet states is very small when compared to the value of U/t. This is a major setback for the actual application of the device since a thermal distribution of the electrons in the system might significantly decrease the response to the external polarization. Unfortunately, the energy difference between these two states and the efficiency of the polarization response are intimately connected and cannot be optimized independently. A compromise has to be found in order to have a practical working device. Another way to modify the properties of the GS is through the application of an external magnetic field perpendicular to the plane of the dots, and through a variable number of electrons N . In this case the spin and the total momentum of the GS undergoes transitions induced by the external field. The effect of the magnetic Rj E field is included in our model through Peierl’s substitution, ti, j = |ti, j |ei 2π φi j , where φi j = 2πec¯h i AE · dl 2 Ba 8 (e.g., φ12 = 8π(¯h c/e) = 480 ). Figure 2 shows the energy levels as a function of the magnetic field of the 4-DR charged with N = 2–5 electrons. The numbers indicate the total momentum of the GS for U/t = 2. A number of different transitions in the GS is induced by the magnetic field. They involve the change of the total momentum of the system and also, in some cases, the spin polarization. These transitions will give rise to cusps in the magnetic field dependence of the charging energy and can be used to characterize the system. Associated with these changes of GS are distributions of charge and currents which can be transferred from one dot to the next in a QCA. A quantum computer is a coherent QCA. The coherence between different cells implies that each cell can be in a superposition of its ground and excited states (entanglement) and that the states of the entire system are not a mere product of states of individual cells. The recent observation [5] that a QC can perform some operations practically impossible on a classical computer, like factorization of prime numbers, has led to the explosion of research on QC. The operation of QC depends on quantum gates (QG). The simplest QC is the NOT operation σx which rotates states σx |0i = |1i, σx |1i = |0i. The quantum-XOR (QXOR) acts on pairs of qubits (four-level system) and corresponds to a rotation of a qubit 1 depending on the state of qubit 0: U X O R |00i, U X O R |01i = |01i, U X O R |10i = |11i, U X O R |11i = |10i. It has been shown [6] that any operation can be decomposed into rotations on a single qubit and QXOR between two qubits.
Superlattices and Microstructures, Vol. 22, No. 3, 1997
433
E(2e)/t
36
35
E(2e)/t
34 36
35
P
34 1
0
–1 –1.0
–0.5
0.0 Pext
0.5
1.0
Fig. 1. Energy levels of a 4-DR with two electrons as a function of the polarization of a neighbor cell for (A) VX /U = 0 and (B) VX /U = 0.01. C, The induced polarization of the system for the case V X /U = 0. U/t0 = 100 and VD /U = 0.5. (——) triplet GS; (– – –) singlet states; (− · − · −) triplet first excited state.
The QXOR exemplifies conditional quantum dynamics [7] in which the evolution of the target subsystem (cell 1) depends on the state of control subsystem (cell 0). QC based on trapped atoms have recently been proposed [8]. Following these ideas, recent experiments on two-level atoms and Ramsey gates [9] have demonstrated coherent QXOR. These proposals rely heavily on the picture of two level systems: atoms, spins, and photons. Yet existing computers are built from semiconductor components with electrons doing all the work. It is therefore sensible to inquire whether electrons in quantum dots, or an ensemble of them, could form a basis for a QC. Electrons are indistinguishable, interacting particles, which obey Fermi statistics. It is therefore not obvious how to implement conditional quantum dynamics with one subsystem as control and the other as target. The coding of a quantum dot as a quantum computer is most obvious in second quantization. The states of QC are states of a Hilbert space. The dimension of the Hilbert space is controlled by the number of single particle states j = 0, 1, . . . , n (including spin), and the number of electrons N . With c+ j denoting operators creating electrons in state j, the states of noninteracting system can be written in a way analogous to quantum bits: 5 ji ,i=1,...,N c+ ji |vi = |0, 1, 0, 1, 1, 0, . . .i, with N ones distributed over n states. The analogy with two level systems is quite clear but the fermionic character of states and a realistic energy level structure and interaction with external fields must be implemented. We examine the fermionic character of states on a simple model of a QXOR gate [7]. The QXOR proposed by Barenco et al. [7] consists of two quantum dots with two levels each. The first dot is the control dot and the second one is the target dot. Each dot is charged with one electron and charge transfer between the dots is not allowed. A positive background, or a valence hole, is assumed in each dot to preserve charge neutrality. Under the application of an external electric field the
434
Superlattices and Microstructures, Vol. 22, No. 3, 1997
E(2e)/t
0 triplet singlet
–2 3
2
4
E(3e)/t
–4
Sz = ± 3/2 Sz = ± 1/2
–1 4
1
2
–3
E(4e)/t
Sz = ± 2 1
Sz = ± 1
3
Sz = 0
2
E(5e)/t
–1
Sz = ± 3/2 –6
Sz = ± 1/2 1
2
3 –4 0.0
0.2
0.4
0.6
0.8
1.0
Φ/Φ0 Fig. 2. Energy states, E(Ne), as a function of the magnetic field for different number of electrons, N , in the 4-DR. U/t0 = 2 and the other parameters are the same as in Fig. 1.
dots are polarized and a dipole interaction couples the dots. The resonant frequency for transitions between the states of one dot will then depend on the electron occupation of the other dot. This gives the conditional quantum dynamics of the system: a time-dependent perturbation at a certain frequency will induce a transition in the target dot only if the control dot is in a particular state. To simulate this system we consider two different hopping parameters in the 4-DR: t1,2 = t2,1 = t3,4 = t4,3 = t0 and t2,3 = t3,2 = t1,4 = t4,1 = t1 . A static external electric field is applied along the direction dot-2 to dot-1. Figure 3 shows the energy levels as a function of the electric field for (A) t1 = 0 and (B) t1 = 0+ . In the first case no charge transfer is possible between pairs of dots (1, 2) and (3, 4). In the second case, an infinitesimal coupling is assumed and the charge transfer is possible. For t1 = 0 we have a situation similar to the idealized model of Barenco et al. [7]: each pair of dots gives origin to two stastes. The electric field polarizes the pairs of dots. The sign of the polarization depends on the state occupied by the electron. At zero electric field, the system is characterized by four levels. One of the transitions between two of these levels is doubly degenerated, as in Barenco et al.’s case [7]. The degeneracy is broken by the electric field and the energy levels evolve to give origin to a series of transitions. To operate the unitary transformation required for the QXOR the time-dependent perturbation should be resonant with a transition between two levels at a certain electric field. The electric field will then turn on the operation depending on the initial state of the system.
Superlattices and Microstructures, Vol. 22, No. 3, 1997
435
15
E(2e)/t0
10
5
A 15
E(2e)/t0
10
5
B 0
1
2
3
∆E/t0 Fig. 3. Energy levels of 4-DR with two alternate first-neighbor hopping parameter, t0 and t1 , for (A) t1 = 0 and (B) t1 = 0+ . U/t0 = 20 and VD /U = 0.5. (——) triplet states; (– – –) singlet states.
Figure 3B shows the case when the electrons are allowed to tunnel between the dots of different pairs. The first striking result is the increase of the number of states of the system, i.e. the ‘inflation of Hilbert space’. These states evolve with the electric field which removes degeneracies. The evolution of the system is described by the time-dependent Schr¨odinger equation. Whether a simple set of unitary operations familiar from the physics of two-level systems is sufficient to describe the conditional quantum dynamics for coupled fermion systems remains to be seen. Acknowledgements—We thank C. R. Leavens for discussions. JAB is partially supported by a RHAE-CNPq fellowship.
References [1] }R. Ashoori, Nature 379, 413 (1996), and references therein. [2] }P. Hawrylak, Phys. Rev. Lett. 71, 3347 (1993). [3] }C. S. Lent, P. D. Tougaw, and W. Porod, Appl. Phys. Lett. 62, 714 (1993), P. D. Tougaw, C. S. Lent, and W. Porod, J. Appl. Phys. 74, 3558 (1993), C. S. Lent and P. D. Tougaw, J. Appl. Phys. 74, 6227 (1993), J. Appl. Phys. 75, 1818 (1994). [4] }For a review, see D. P. DiVincenzo, Science 270, 255 (1995). [5] }P. W. Shor, Proc. of the 35th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, Calif. (1994), p. 124; A. Ekert and R. Jozsa, Rev. Mod. Phys. (in press). For a review, see also C. Bennet, Phys. Today, (October 1995), p. 24. [6] }T. Sleator and H. Weinfurter, Phys. Rev. Lett. 74, 4087 (1995). [7] }A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. Lett. 74, 4083 (1995).
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Superlattices and Microstructures, Vol. 22, No. 3, 1997
[8] }J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [9] }C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 75, 4714 (1995).
Coupled quantum dots as quantum exclusive-OR gate ´ A. Brum†, Pawel Hawrylak Jose Institute for Microstructural Sciences, National Research Council, Ottawa (Ont.), Canada, K1A 0R6
(Received 15 July 1996) The concepts relevant to quantum cellular automata and quantum computers are studied using a simple model of a quantum exclusive-OR (QXOR) gate device consisting of four coupled quantum dots. The QXOR device can be charged with up to N = 8 electrons. The quantum bits of the device correspond to states of the device in second quantized form. We use exact diagonalization techniques in the configuration space to calculate physical properties of QXOR as a function of the number of electrons N and external perturbations in the form of electric and magnetic fields. This allows us to investigate the switching of the QXOR gate, and its ability to store and transmit information. c 1997 Academic Press Limited
A significant progress has been achieved in producing low dimensional semiconductor structures (quantum dots) [1, 2], but their integration into quantum circuits appears to be in its infancy. Two quantum logic circuits have been studied theoretically: the quantum cellular automata (QCA) [3] and the quantum computer (QC) [4]. In the QCA individual cells are in contact with reservoirs which guarantee their rapid relaxation into the ground state (GS). The GS can be manipulated by the application of external fields or fields produced by GS charge distribution of neighbouring cells. By contrast, the QC involves both ground and excited states and relies on the coherent superposition of all states. The quantum state of QC is manipulated coherently by unitary transformations due to applied classical external fields. Simple spin models of QC have been studied so far, stimulated by recent developments in, e.g. quantum fourier transform methods [5]. We study here an extension of these ideas to the system of fermions in low dimensional semiconductor structures. To illustrate the principles and problems associated with QCA and QC we study a simple model system of four coupled quantum dots, similar to the one studied by Lent et al. [3]. The quantum dots are placed at the corners of a square, forming a four-dots ring (4-DR). Each dot is independently in contact with external leads. We consider small dots and assume a single orbital per dot. The dots are coupled with each other and with leads through a hopping matrix element ti, j . In the site representation the Hamiltonian of the system can be written as: X X 1 X + + + + e ∗ + H= E i,σ ci,σ ci,σ + [ti, j ci,σ c j,σ + t j,i c j,σ ci,σ ] + hi, j|vee |k, lici,σ c j,σ 0 ck,σ 0 cl,σ (1) 2 0 i,σ i> j,σ i jkl,σ,σ † Leave of absence from: Instituto de Fisica ‘Gleb Wataghin’—DFESCM, Universidade Estadual de Campinas, 13081970 Campinas (SP), Brazil.
0749–6036/97/070431 + 06 $25.00/0
sm960263
c 1997 Academic Press Limited
432
Superlattices and Microstructures, Vol. 22, No. 3, 1997
We consider the following Coulomb matrix elements: the on-site Coulomb repulsion, U = hi, i|vee |i, ii, the off-site repulsion, VD = hi, i ± 1|vee |i ± 1, ii, and the exchange interaction, VX = hi, i ± 1|vee |i, i ± 1i. In what follows, VD /U = 0.5. The system can be charged with up to 8 electrons. The critical parameter of the system is the ratio between the Coulomb on-site repulsion U and the hopping parameter between nearest neighbors t, U/t, where t = t1,2 . We apply our 4-DR model to re-examine the QCA [3]. The hopping parameter is ti j = t0 δi, j±1 and the system is charged with two electrons. Our emphasis is on the role of the exchange interaction, VX , which was not considered in the original model [3]. Under the application of an external perturbation V , the ground state 3 −ρ2 −ρ4 , where ρn is the charge density at the ith dot. The external changes polarization P, defined as P = ρρ11 +ρ +ρ3 +ρ2 +ρ4 perturbation is the potential created by a charge distribution of a neighbor cell with a fixed polarization (Pext ). The distance between the center of the neighbor cells is d = 3a, where a is the distance between the quantum dots in the cell. Figure 1 shows energies as a function of Pext of the 4-DR with U/t0 = 100 for (A) VX /U = 0 and (B) VX /U = 0.01. Figure 1C shows the corresponding GS polarization. For a repulsive exchange VX ≤ 0, Fig. 1A, the GS is a singlet as found in Ref. [3]. The polarization of GS shows a slow response to the external polarization and does not saturate. However, once the attractive exchange interaction is included, Fig. 1B, the triplet becomes the GS. This type of interaction is responsible for singlet–triplet transitions already observed in quantum dots [1, 2]. The polarization of triplet GS shows a strong non-linearity with the external polarization, saturating already at weak external polarization. This is the ideal condition for the storage and transmission of information from cell to cell without loss of information [3]. Clearly, the exchange interaction plays an important role and should be considered in the analysis of these devices. It is important to observe that the triplet excited state shows an opposite polarization (dashed line in Fig. 1C). The energy difference between the triplet states is very small when compared to the value of U/t. This is a major setback for the actual application of the device since a thermal distribution of the electrons in the system might significantly decrease the response to the external polarization. Unfortunately, the energy difference between these two states and the efficiency of the polarization response are intimately connected and cannot be optimized independently. A compromise has to be found in order to have a practical working device. Another way to modify the properties of the GS is through the application of an external magnetic field perpendicular to the plane of the dots, and through a variable number of electrons N . In this case the spin and the total momentum of the GS undergoes transitions induced by the external field. The effect of the magnetic Rj E field is included in our model through Peierl’s substitution, ti, j = |ti, j |ei 2π φi j , where φi j = 2πec¯h i AE · dl 2 Ba 8 (e.g., φ12 = 8π(¯h c/e) = 480 ). Figure 2 shows the energy levels as a function of the magnetic field of the 4-DR charged with N = 2–5 electrons. The numbers indicate the total momentum of the GS for U/t = 2. A number of different transitions in the GS is induced by the magnetic field. They involve the change of the total momentum of the system and also, in some cases, the spin polarization. These transitions will give rise to cusps in the magnetic field dependence of the charging energy and can be used to characterize the system. Associated with these changes of GS are distributions of charge and currents which can be transferred from one dot to the next in a QCA. A quantum computer is a coherent QCA. The coherence between different cells implies that each cell can be in a superposition of its ground and excited states (entanglement) and that the states of the entire system are not a mere product of states of individual cells. The recent observation [5] that a QC can perform some operations practically impossible on a classical computer, like factorization of prime numbers, has led to the explosion of research on QC. The operation of QC depends on quantum gates (QG). The simplest QC is the NOT operation σx which rotates states σx |0i = |1i, σx |1i = |0i. The quantum-XOR (QXOR) acts on pairs of qubits (four-level system) and corresponds to a rotation of a qubit 1 depending on the state of qubit 0: U X O R |00i, U X O R |01i = |01i, U X O R |10i = |11i, U X O R |11i = |10i. It has been shown [6] that any operation can be decomposed into rotations on a single qubit and QXOR between two qubits.
Superlattices and Microstructures, Vol. 22, No. 3, 1997
433
E(2e)/t
36
35
E(2e)/t
34 36
35
P
34 1
0
–1 –1.0
–0.5
0.0 Pext
0.5
1.0
Fig. 1. Energy levels of a 4-DR with two electrons as a function of the polarization of a neighbor cell for (A) VX /U = 0 and (B) VX /U = 0.01. C, The induced polarization of the system for the case V X /U = 0. U/t0 = 100 and VD /U = 0.5. (——) triplet GS; (– – –) singlet states; (− · − · −) triplet first excited state.
The QXOR exemplifies conditional quantum dynamics [7] in which the evolution of the target subsystem (cell 1) depends on the state of control subsystem (cell 0). QC based on trapped atoms have recently been proposed [8]. Following these ideas, recent experiments on two-level atoms and Ramsey gates [9] have demonstrated coherent QXOR. These proposals rely heavily on the picture of two level systems: atoms, spins, and photons. Yet existing computers are built from semiconductor components with electrons doing all the work. It is therefore sensible to inquire whether electrons in quantum dots, or an ensemble of them, could form a basis for a QC. Electrons are indistinguishable, interacting particles, which obey Fermi statistics. It is therefore not obvious how to implement conditional quantum dynamics with one subsystem as control and the other as target. The coding of a quantum dot as a quantum computer is most obvious in second quantization. The states of QC are states of a Hilbert space. The dimension of the Hilbert space is controlled by the number of single particle states j = 0, 1, . . . , n (including spin), and the number of electrons N . With c+ j denoting operators creating electrons in state j, the states of noninteracting system can be written in a way analogous to quantum bits: 5 ji ,i=1,...,N c+ ji |vi = |0, 1, 0, 1, 1, 0, . . .i, with N ones distributed over n states. The analogy with two level systems is quite clear but the fermionic character of states and a realistic energy level structure and interaction with external fields must be implemented. We examine the fermionic character of states on a simple model of a QXOR gate [7]. The QXOR proposed by Barenco et al. [7] consists of two quantum dots with two levels each. The first dot is the control dot and the second one is the target dot. Each dot is charged with one electron and charge transfer between the dots is not allowed. A positive background, or a valence hole, is assumed in each dot to preserve charge neutrality. Under the application of an external electric field the
434
Superlattices and Microstructures, Vol. 22, No. 3, 1997
E(2e)/t
0 triplet singlet
–2 3
2
4
E(3e)/t
–4
Sz = ± 3/2 Sz = ± 1/2
–1 4
1
2
–3
E(4e)/t
Sz = ± 2 1
Sz = ± 1
3
Sz = 0
2
E(5e)/t
–1
Sz = ± 3/2 –6
Sz = ± 1/2 1
2
3 –4 0.0
0.2
0.4
0.6
0.8
1.0
Φ/Φ0 Fig. 2. Energy states, E(Ne), as a function of the magnetic field for different number of electrons, N , in the 4-DR. U/t0 = 2 and the other parameters are the same as in Fig. 1.
dots are polarized and a dipole interaction couples the dots. The resonant frequency for transitions between the states of one dot will then depend on the electron occupation of the other dot. This gives the conditional quantum dynamics of the system: a time-dependent perturbation at a certain frequency will induce a transition in the target dot only if the control dot is in a particular state. To simulate this system we consider two different hopping parameters in the 4-DR: t1,2 = t2,1 = t3,4 = t4,3 = t0 and t2,3 = t3,2 = t1,4 = t4,1 = t1 . A static external electric field is applied along the direction dot-2 to dot-1. Figure 3 shows the energy levels as a function of the electric field for (A) t1 = 0 and (B) t1 = 0+ . In the first case no charge transfer is possible between pairs of dots (1, 2) and (3, 4). In the second case, an infinitesimal coupling is assumed and the charge transfer is possible. For t1 = 0 we have a situation similar to the idealized model of Barenco et al. [7]: each pair of dots gives origin to two stastes. The electric field polarizes the pairs of dots. The sign of the polarization depends on the state occupied by the electron. At zero electric field, the system is characterized by four levels. One of the transitions between two of these levels is doubly degenerated, as in Barenco et al.’s case [7]. The degeneracy is broken by the electric field and the energy levels evolve to give origin to a series of transitions. To operate the unitary transformation required for the QXOR the time-dependent perturbation should be resonant with a transition between two levels at a certain electric field. The electric field will then turn on the operation depending on the initial state of the system.
Superlattices and Microstructures, Vol. 22, No. 3, 1997
435
15
E(2e)/t0
10
5
A 15
E(2e)/t0
10
5
B 0
1
2
3
∆E/t0 Fig. 3. Energy levels of 4-DR with two alternate first-neighbor hopping parameter, t0 and t1 , for (A) t1 = 0 and (B) t1 = 0+ . U/t0 = 20 and VD /U = 0.5. (——) triplet states; (– – –) singlet states.
Figure 3B shows the case when the electrons are allowed to tunnel between the dots of different pairs. The first striking result is the increase of the number of states of the system, i.e. the ‘inflation of Hilbert space’. These states evolve with the electric field which removes degeneracies. The evolution of the system is described by the time-dependent Schr¨odinger equation. Whether a simple set of unitary operations familiar from the physics of two-level systems is sufficient to describe the conditional quantum dynamics for coupled fermion systems remains to be seen. Acknowledgements—We thank C. R. Leavens for discussions. JAB is partially supported by a RHAE-CNPq fellowship.
References [1] }R. Ashoori, Nature 379, 413 (1996), and references therein. [2] }P. Hawrylak, Phys. Rev. Lett. 71, 3347 (1993). [3] }C. S. Lent, P. D. Tougaw, and W. Porod, Appl. Phys. Lett. 62, 714 (1993), P. D. Tougaw, C. S. Lent, and W. Porod, J. Appl. Phys. 74, 3558 (1993), C. S. Lent and P. D. Tougaw, J. Appl. Phys. 74, 6227 (1993), J. Appl. Phys. 75, 1818 (1994). [4] }For a review, see D. P. DiVincenzo, Science 270, 255 (1995). [5] }P. W. Shor, Proc. of the 35th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, Calif. (1994), p. 124; A. Ekert and R. Jozsa, Rev. Mod. Phys. (in press). For a review, see also C. Bennet, Phys. Today, (October 1995), p. 24. [6] }T. Sleator and H. Weinfurter, Phys. Rev. Lett. 74, 4087 (1995). [7] }A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. Lett. 74, 4083 (1995).
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Superlattices and Microstructures, Vol. 22, No. 3, 1997
[8] }J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [9] }C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 75, 4714 (1995).