Quantum computation and the production of entangled photons using coupled quantum dots

Superlattices and Microstructures, Vol. 31, Nos 2–4, 2002 doi:10.1006/spmi.2002.1034 Available online at http://www.idealibrary.com on

Quantum computation and the production of entangled photons using coupled quantum dots O LIVER G YWAT , G UIDO B URKARD† , DANIEL L OSS Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Received 11 March 2002)

We review recent theoretical progress on the use of electron spins as qubits in coupled semiconductor quantum dots for quantum information processing. We discuss the spin exchange mechanism and its microscopic origin in both laterally and vertically tunnelcoupled quantum dots and explain how it can be used to implement the quantum XOR gate which, in combination with single spin rotations, allows to perform arbitrary quantum computations. In addition to their functionality as a quantum gate, coupled quantum dots can act as a source for photon pairs in entangled polarization states which are useful for quantum communication. We describe a mechanism for the production of such entangled photon pairs via a biexciton state in tunnel-coupled quantum dots. c 2002 Elsevier Science Ltd. All rights reserved.

Key words: quantum computation, quantum communication, spin, entanglement, quantum dots.

1. Introduction Quantum computation and quantum communication are new concepts that involve the processing and the transmission of quantum information [1, 2]. The unit of quantum information is the quantum bit (qubit). In contrast to classical bits, qubits have the physical properties of a quantum mechanical two-level system. This implies that a qubit can be in a superposition of its logical basis states |0i and |1i. Moreover, a collection of several quantum bits can be in a superposition of different many-qubit states, which involves entanglement among the qubits. In quantum computation, a set of qubits (the quantum register) is prepared in an initial state |9i i, to which a quantum algorithm, being a unitary operation U , is applied in order to obtain the final state |9 f i = U |9i i, which is measured after the quantum computation has stopped. Powerful quantum algorithms have been proposed, e.g., by Shor [3] and by Grover [4], that would allow a quantum computer to widely surpass the performance of a classical computer in solving certain problems, as for instance the prime factorization of an integer [3] or the search through an unstructured search space [4]. It was shown [5] that all possible quantum algorithms U can be implemented by exclusively concatenating one- and two-qubit gates which is a very useful result in view of practical purposes. However, the experimental implementation of quantum logic operations and finally quantum computation is an extremely demanding task [6]. One can summarize the requirements for quantum computation in five † Author to whom correspondence should be addressed. E-mail: [email protected]

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crucial criteria [7, 8]. First of all, the Hilbert space for a single qubit must be limited (or truncated) to exactly two dimensions and should be extendable for n-qubit processing to a Hilbert space of dimension 2n —most preferably with a simple tensor product structure. In order to outperform a classical computer by algorithms as mentioned earlier, a quantum computer requires on the order of n ∼ 105 qubits. Then, it must be possible to prepare a desired initial n-qubit state. The coupling to the environment must be weak enough to maintain quantum phase coherence over a long time compared to the length of a computational step to allow the carrying out of quantum error correction [9–11]. Essentially, it must be possible to subject the whole computational quantum system to a controlled sequence of precisely defined unitary operations. As mentioned earlier, it is sufficient for quantum computation to experimentally realize single-qubit and twoqubit gates. Finally, the readout of a quantum computation (an ordinary bit string) must be accessible via a sequence of quantum measurements performed on the computational quantum system. The spin of an electron is obviously a natural candidate for the physical realization of a qubit because its Hilbert space is spanned by the two basis states spin up, | ↑i, and spin down, | ↓i, which suggests to assign these states to the two qubit basis states, e.g. by | ↑i = |0i and | ↓i = |1i. In Sections 3–5 we present theoretical work concerned with the scheme of using electron spins in tunnel-coupled semiconductor quantum dots as the qubits of a quantum computer [8]. Several other two-state systems have been proposed to be used as qubits [1]. However, we take the liberty of focusing on our spintronic proposal here and in the following do not discuss other systems. Furthermore, there is also a number of subsequent schemes for using the exchange coupling between spins in solid state structures [12–16]. Besides quantum computation, there has recently been growing interest in quantum communication [2, 17], for which pairs of spatially separated entangled qubits represent an important resource. Two particles are entangled if their wave function cannot be expressed as a tensor product of two single-particle wave functions. Quantum optical methods have made possible the production of pairs of entangled photons and their usage in many experiments, see e.g. [18]. There, the pairs of polarization-entangled photons are generated inside nonlinear crystals by parametric down-conversion. A major disadvantage of these widely used sources is that entangled photons can only be produced in a stochastic manner and at a rather low rate. However, it would be desirable for most applications in quantum communication to have a deterministic and efficient source of Einstein–Podolsky–Rosen (EPR) pairs, i.e. particles in a maximally entangled state. Two tunnel-coupled quantum dots could be used as such a deterministic source of polarization-entangled photons obtained from the recombination of biexciton states, as schematically shown in Fig. 1A. Our calculations predict that the entangled photons are emitted at different dots [19] which could make the spatial separation required for quantum communication more easily achievable. Moreover, the current possibilities of fast switching in solid-state nanodevices promise production efficiencies for the entangled photons that are distinctly larger than for the parametric down-conversion process. These issues will be discussed in Section 5. On the other hand, tunnel-coupled quantum dots also provide powerful deterministic entanglement between qubits of localized and delocalized electrons [8, 17] (see Fig. 1B), i.e. it is possible to create a singlet state (one realization of an EPR pair) out of two uncorrelated electrons and subsequently separate the two electrons spatially by electronic transport while maintaining the entanglement of their spins. The implementation of such devices would allow the study of a new class of quantum phenomena in electronic nanostructures [17] such as the entanglement and nonlocality of electronic EPR pairs, tests of Bell inequalities, quantum teleportation [20], and quantum cryptography [21] which promises secure transmission of information. Semiconductor quantum dots are structures with a spatial extension on the order of the Fermi wavelength of the host material thus providing three-dimensional spatial confinement of charge carriers. The size of semiconductor quantum dots ranges from about 10 nm to 1 µm [22]. Three-dimensional confinement can for instance be achieved by electrical gating of a two-dimensional electron gas (2DEG), possibly combined with etching techniques. Precise control of the number of conduction-band electrons in a quantum dot (starting from zero) has been achieved in GaAs heterostructures [23]. The electronic spectrum of typical quantum

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B E

A

B

Fig. 1. Coupled quantum dots that are formed inside a 2DEG in the presence of a perpendicular magnetic field B. A, Two entangled photons are emitted at angles θ1 and θ2 with respect to the plane normal n by the decay of a confined biexciton consisting of two electrons e1, e2 (black disks) and two holes h1, h2 (white disks) which can tunnel with amplitudes t between the dots. An in-plane electric field E in the y-direction can be applied to separate the electrons from the holes. B, Formation of a delocalized two-electron state with entangled spins, e.g. a spin singlet, in a tunnel-coupled double dot in an n-doped semiconductor. Given the ability to control the electron tunnelling amplitude t between the dots as a function of time, this structure can be used for the implementation of a quantum gate based on spin qubits.

dots can vary strongly as a consequence of the application of an external magnetic field [22, 23], because the magnetic length pertaining to typical laboratory fields B ≈ 1 T is comparable to typical dot sizes. Experiments on coupled quantum dots have given evidence for Coulomb blockade effects [24], tunnelling between neighboring dots [22, 24], magnetization [25], the Kondo resonance [26], the formation of a delocalized single-particle state [27], and the delocalized character of one single exciton [28, 29].

2. Spin coherence in quantum dots Experimental demonstrations of the creation, coherent manipulation, measurements, and filtering of spins in nanostructures support on the one hand the idea to use the spin as an additional degree of freedom in conventional electronic devices (‘spintronics’) [30, 31] and on the other hand the proposal to use spins in quantum confined structures for quantum information processing. As for any other quantum two-level system, an electron spin in a quantum dot is not completely isolated from its environment. This is the origin of decoherence, i.e. the loss of phase coherence. Thus, only systems with a weak coupling to their environment and therefore with slow decoherence are interesting conditions for the reliable processing of quantum information. Two time scales, T1 and T2 , are relevant for the discussion of spin coherence. The relaxation time T1 gives the time required for the relaxation of a single spin in an external magnetic field from an excited state into the thermal equilibrium. The decoherence time T2 relates to the time over which the relative phase of a superposition of spin-up and spin-down states of a single spin is preserved. Because usually T2 . T1 , the limiting time scale for quantum computing is T2 . Given a small coupling of the spin to the orbital (charge) degree of freedom of an electron, the spin coherence time T2 can be orders of magnitude bigger than the charge coherence time, as recent magnetooptical experiments in bulk GaAs (T2∗ ≥ 100 ns) and a 2DEG [32], and in CdSe quantum dots [33] (T2∗ ≈ 3 ns) have shown. Here, T2∗ ≤ T2 is the ensemble dephasing time. The charge coherence time, i.e. the time over which the relative phase of superpositions of spatial states of the electron (e.g., in the upper and lower arms of an Aharonov–Bohm ring) is preserved, has been measured in experiments like weak localization studies or the Aharonov–Bohm effect. If the materials and the experimental conditions are chosen such that the effects of the spin–orbit coupling are small, the charge coherence times are almost completely irrelevant for the spin coherence times which are important for spin-qubits in quantum dots.

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However, none of the experiments have been done on a quantum computing structure as we envision it (see Section 3), but still, the results are encouraging. We have proposed to measure the decoherence time T2 of a single spin in such a device directly via a transport experiment by applying electron spin resonance (ESR) techniques [34]. The stationary current exhibits a resonance with a linewidth determined by T2 . Recent studies on the dominant spin–orbit effects in GaAs quantum dots have led to unusually low phononassisted spin-flip rates, which suggests long spin decoherence times, although more work is required here. Several different mechanisms are responsible for spin-flip processes that accompany phonon-assisted transitions between different discrete energy levels or Zeeman sublevels [35]. It has been shown that in the case of GaAs quantum dots the spin-flip effects related to other mechanisms than the spin–orbit interaction are negligible, among them the modulation of the hyperfine coupling with nuclei by lattice vibrations, spin– spin interaction between the bound electron and the conduction electron in the leads, and the spin–current interaction, when the bound electron spin flip is caused by the fluctuating magnetic field of the conduction electrons. Furthermore, the decoherence of a single electron spin in an isolated quantum dot due to the hyperfine interaction with the nuclei has been studied [36, 37] and was found to be due to the spatial variation of the electron density corresponding to a given discrete electron state in the dot, which leads to a wide distribution of the local values of the hyperfine interaction [36]. The dipole–dipole interaction between nuclei (which is responsible for the nuclear spin relaxation time Tn2 ) was thereby neglected because T2 was found to be shorter than Tn2 . It has also been shown that a perturbative treatment of the electron spin decoherence is not possible in a weak external Zeeman field that is smaller than a typical fluctuating nuclear magnetic field seen by the electron spin through the hyperfine interaction, which is about 100 Gauss in a GaAs dot. The temporal decay of the spin precession amplitude caused by the hyperfine interaction with the nuclei has been found not to be exponential, but either described by a power law (for finite Zeeman field) or an inverse logarithm (for vanishing Zeeman field). There, a characteristic decay time T2 is roughly given by h¯ N /A, where A is the hyperfine interaction constant and N the number of nuclei within the dot (N ∼ 105 ), and is on the order of µs. After the decay, the amplitude of the precession part is of the order one, while the decaying part is 1/N , in agreement with earlier results [37], where we have found in the framework of time-dependent perturbation theory that due to the hyperfine interaction the spin-flip rate can be suppressed by a factor (Bn∗ /B)2 /N , where Bn∗ = AI /gµ B is the maximal magnitude of the effective nuclear field (Overhauser field), N the number of nuclear spins in the vicinity of the electron, and B is either the external magnetic field, or the Overhauser field B = p Bn∗ due to the nuclear spin polarization p, if the external magnetic field is zero. Furthermore, the nuclear spin of both Ga and As is I = 3/2 and A is again the hyperfine coupling constant. A nuclear spin polarization p can be obtained, e.g., by optical pumping [38] or by spin-polarized currents at the edge of a 2DEG [39]. In the latter case, the suppression of the spin-flip rate becomes 1/ p 2 N [37].

3. Coupled quantum dots as quantum gates The application of logical two-qubit operations in a quantum computer requires a controlled interaction among the qubits. We focus here on a mechanism that is able to couple pairs of spin qubits in coupled quantum dots as a consequence of the Coulomb interaction and the Pauli exclusion principle. In the absence of magnetic fields, the ground state of two electrons is given by a spin-singlet state. In the presence of sufficiently strong Coulomb repulsion, the first excited state is given by the spin-triplet state. Higher excited states are separated from these states by an energy gap that is either given by the Coulomb repulsion or the single particle confinement. The low-energy dynamics of this system is described by the effective Heisenberg spin Hamiltonian operator Hs (t) = J (t)S1 · S2 ,

(1)

where J (t) denotes the exchange coupling between the two spins S1 and S2 , i.e. the energy difference

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between the triplet and the singlet state. TheRswap operation, i.e. the interchange of the states in qubit 1 τ and 2, can be applied by a pulse of J (t) with 0 s dt J (t)/h¯ = J0τs /h¯ = π (mod 2π), which leads to a time Rt evolution Usw = U (t) = T exp(i 0 Hs (τ )dτ/h¯ ) [8]. The swap operator Usw is not sufficient for quantum 1/2 1/2 computation. However, any of its square roots Usw , say Usw |φχi = (|φχi + i|χ φi)/(1 + i), turns out to be a universal quantum gate. Thus, it can be used together with single-qubit rotations to assemble any quantum algorithm. This is shown by constructing the known universal quantum gate XOR [40] (also known as CNOT), 1/2 through combination of Usw and single-qubit operations exp(iπ Siz /2), applied in the sequence [8] z

z

1/2

z

1/2

Uxor = ei(π/2)S1 e−i(π/2)S2 Usw eiπ S1 Usw .

(2)

This universal gate allows us to reduce the study of general quantum computation to the study of singlespin rotations (see Section 4) and the exchange mechanism, in particular how J (t) can be controlled experimentally. We have shown that the Heisenberg exchange energy J (t) can be switched by raising or lowering the tunnelling barrier between the dots [37, 41], see Sections 3.1 and 3.2. For adiabatic switching [42], we propose to apply a pulse J (t) = J0 sech(t/1t). For typical numbers J0 = 80 µeV ≈ 1 K and 1t = 4 ps we obtain the switching time τs ≈ 30 ps, while the adiabaticity criterion [43] is h¯ /1t ≈ 150 µeV  δ, where δ is the single-dot level spacing. So far, anisotropies due to the spin–orbit effect have been omitted. In a model that takes the spin–orbit interaction into account, the effective Heisenberg spin Hamiltonian operator (1) acquires anisotropic terms A(t), Hs (t) = J (t)[S1 · S2 + A(t)], A(t) = β(t) · (S1 × S2 ) + γ (t)(β(t) · S1 )(β(t) · S2 ),

(3)

where β is the spin–orbit field. However, it was shown [44] that the spin–orbit effects cancel exactly in the quantum XOR gate (right-hand side of eqn (2)) for the ideal case where the pulse shapes of the exchange and spin–orbit interactions are identical. For the nonideal case, it was shown that the two pulse shapes can be made almost identical and that the gate error is strongly suppressed by two small parameters, the spin–orbit interaction constant and the smallness of the deviation of the two pulse shapes. The scaling requirement for quantum computation is easily achievable with spin-based qubits confined to quantum dots, since the production of arrays of quantum dots [17, 45] is feasible with today’s technology. Certainly, the actual implementation of such arrays including all the needed circuits poses new experimental challenges, but at least we are not aware of physical limitations which would exclude such an upscaling for spin-qubits. An initial n-qubit state with parallel alignment of all spins can be prepared by the application of a strong external magnetic field and allowing all spins to relax into the spin-polarized ground state. For the preparation of a different initial state, we refer here to the methods presented for the rotation of single spins in Section 4. We have proposed several devices for the read out of the final state of each qubit after a computation, like the tunnelling of the electron into a supercooled paramagnetic dot [8, 17], thereby inducing a magnetization nucleation from the metastable phase into a ferromagnetic (FM) domain. The magnetization of the domain points in the same direction as the spin to be read out and can be measured by conventional methods with a reliability of 75%. Another possibility for the read out is given by tunnelling of the electron into another dot through a spin-selective tunnelling barrier, i.e. a spin filter that only permits the transport of one spin direction [47]. The charging of the additional dot can be detected by an electrometer [8, 48]. In the following, we shall review our detailed calculations to describe the mechanism of quantum gate operations in coupled quantum dots. We note that the same principles can also be applied to other spin systems in quantum-confined structures, such as coupled atoms in a crystal, supramolecular structures, overlapping shallow donors in semiconductors [13, 15], etc., using similar methods as explained later. We point out that spins in quantum dots can also be coupled—beyond the mechanisms described in Sections 3.1 and 3.2—on

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a long distance scale by using a cavity-QED scheme [49] or by using superconducting leads to which the quantum dots are attached [50]. 3.1. Laterally coupled quantum dots We have studied [37] a system of two tunnel-coupled quantum dots in a 2DEG, containing one (excess) electron each, see Fig. 1B. The dots are two-dimensional and arranged in a plane, separated by a sufficiently small distance 2a to allow tunnelling of the electrons between the dots (for a lowered barrier) and establishing an exchange interaction J between the two spins. We model this system by the Hamiltonian operator H = P i=1,2 h i + C + H Z = Horb + H Z , where the single-electron dynamics in the 2DEG (x y-plane) is described through  2 1 e pi − A(ri ) + V (ri ), (4) h i (ri ) = 2m c   mω02 1 2 2 2 2 V (x, y) = (x − a ) + y , (5) 2 4a 2 where m is the effective mass of the electron and V (x, y) a confinement potential that models the confinement provided by two laterally tunnel-coupled quantum dots separated by a distance 2a. A magnetic field B = (0, 0, B) is applied along the z-axis, and couples to the electron spin through the Zeeman interaction H Z and to the charge through the vector potential A(r) = B2 (−y, x, 0). Because few-electron quantum dots are almost depleted regions, the screening length λ can be expected to be much larger than the screening length in bulk 2DEG regions (being about 40 nm in GaAs). Thus, for λ  2a ≈ 40 nm, i.e. for small quantum dots, the limit of the bare Coulomb interaction C = e2 /κ|r1 − r2 | can be assumed, where κ is the static dielectric constant. It has been experimentally shown that the low-energy spectrum of single dots is well √ described by a parabolic confinement potential [23]. Considering separated dots with a  a B , where a B = h¯ /mω0 is the effective Bohr radius of the parabolic dot, we can approximate the double dot system by two harmonic wells with frequency ω0 . For the two-electron wave function we make use of the analogy between quantum dots and atoms by applying methods from molecular physics to coupled quantum dots. We start with a variational Heitler– London (HL) ansatz and only consider the two lowest orbital eigenstates of Horb , leaving us with one symmetric (spin-singlet) √and one antisymmetric (spin-triplet) orbital state. The spin√state for the singlet is |Si = (|↑↓i − |↓↑i)/ 2, while the triplet spin states are |T0 i = (|↑↓i + |↓↑i)/ 2, |T+ i = |↑↑i, and |T− i = |↓↓i. For temperatures with kT  h¯ ω0 , higher-lying states are frozen out and Horb can be replaced by the effective Heisenberg spin Hamiltonian (eqn (1)). The exchange energy J = t − s is given as the difference between the triplet and singlet energy. Using only ground-state single-dot orbitals, we obtain [37]   √  −bd 2  3 h¯ ω0 2 2 d 2 (b−1/b) 2 J= (1 + bd ) + c b e I0 (bd ) − e I0 (d [b − 1/b]) , (6) sinh(2d 2 [2b − 1/b]) 4b q where d = a/aB denotes half the dimensionless inter-dot distance, b = ω/ω0 = 1 + ω2L /ω02 is a magnetic compression factor with the Larmor frequency ω L = eB/2mc, and I0 (x) is the zeroth-order modified Bessel function. In eqn (6), the√first term is produced by the confinement potential, while the terms proportional to the parameter c = π/2(e2 /κa B h¯ ω0 ) result from the Coulomb interaction C; the exchange term is recognized by its negative sign. We are mainly interested in the weak coupling limit |J/h¯ ω0 |  1, where the ground-state HL ansatz is self-consistent. The exchange energy J (eqn (6)) is plotted in Fig. 2 as a function of B and d. We underline that J (B = 0) > 0, i.e. the singlet is the ground state, which is generally true for a two-particle system with time-reversal invariance. Over a wide range of the parameters c and d, the

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A J [meV]

1.2 J [meV]

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0.6 0

B

1

– 0.6 – 1.2

0 0

2

4

6 B [T]

8

10

0.5

1

1.5

d

Fig. 2. The exchange energy J = t − s (full line) for GaAs quantum dots with confinement energy h¯ ω = 3 meV and c = 2.42. The usual short-range Hubbard result J = 4t 2 /U (dashed–dotted line) and the extended Hubbard result [37] J = 4t 2 /U + V (dashed line) are plotted as well, for comparison. In A, the inter-dot distance is kept at a fixed value d = a/a B = 0.7, and J is plotted as a function of the magnetic field B. In B, J is plotted as a function of the inter-dot distance d = a/a B at B = 0.

sign of J (B) changes from positive to negative at a finite value of B (for the parameters chosen in Fig. 2A at B ≈ 1.3 T). J is suppressed exponentially either by compression of the electron orbitals through large magnetic fields (b  1), or by large distances between the dots (d  1), where in both cases the orbital overlap of the electron wave functions at the two different dots is reduced. This exponential suppression contained in the 1/ sinh prefactor in eqn (6) is partly compensated by the exponentially growing exchange term ∝ exp(2d 2 (b − 1/b)). In total, J decays exponentially as exp(−2d 2 b) for large b or d. Since the sign reversal of J —signalling a singlet–triplet crossing—results from the long-range Coulomb interaction, it is not contained in the standard Hubbard model which takes only short-range interaction into account. In this latter model one finds J = 4t 2 /U > 0 in the limit t/U  1 (see Fig. 2). The HL result (eqn (6)) was refined by introducing sp-hybridized single-dot orbitals and taking double occupancy of the dots into account by a Hund–Mulliken approach, which led to qualitatively similar results [37], in particular concerning the singlet– triplet crossing. These results have been confirmed by numerical calculations which take more single-particle levels into account [51]. We remark that the exponential suppression of J is very desirable since it allows to minimize gate errors. In the absence of tunnelling between the dots we still might have direct Coulomb interaction left between the electrons. However, this has no effect on the spins (qubits) provided the spin–orbit coupling is sufficiently small, which is the case for conduction-band electrons in symmetric GaAs structures, due to the s-wave type of their wave functions. This would not be so for hole-doped systems since the valence-band hole has a much stronger spin–orbit coupling due to its p-wave character. Finally, the vanishing of J can be used for switching by applying a constant homogeneous magnetic field to an array of quantum dots to tune J to zero (or close to some other desirable value). Then, for switching J on and off, only a small gate pulse or a small local magnetic field is needed. 3.2. Vertically coupled quantum dots The case of vertically tunnel-coupled quantum dots has also been investigated [41]. This setup of the dots can be produced in multilayer self-assembled quantum dots [52] as well as in etched mesa heterostructures [53]. We apply the same methods as described in Section 3.1 for laterally coupled dots, but we extend now the Hamiltonian eqn (4) from two to three dimensions. We use a three-dimensional confinement V = 2 Vl + Vv , where the vertical confinement Vv is a quartic potential similar to eqn √ (5), with curvature mωz and minima at z = ±a (see Fig. 3B), implying an effective Bohr radius a B = h¯ /mωz and a dimensionless distance d = a/a B . For the lateral confinement Vl , we have modelled a harmonic potential and have allowed

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A

B

C

Fig. 3. Two vertically coupled quantum dots with different lateral radii a B+ and a B− . A, Setup including the considered magnetic and electric fields, applied either in-plane (Bk , E k ) or perpendicularly (B⊥ , E ⊥ ). B, The quadratic double-well potential used for the vertical confinement Vv . C, Switching of the spin–spin coupling between dots of different size by means of an in-plane electric field E k at B = 0. We have chosen h¯ ωz = 7 meV, d = 1, α0+ = 1/2 and α0− = 1/4. For these parameters, E 0 = h¯ ωz /ea B = 0.56 mV nm−1 2 − α 2 )/2α 2 α 2 = 6. The exchange coupling J decreases exponentially on the scale E /2A = 47 mV µm−1 for the and A = (α0+ 0 0− 0+ 0− electric field. Thus, the exchange coupling is switched ‘on’ for E k = 0 and ‘off’ for E k ≈ 150 mV µm−1 .

√ for different sizes of the two dots, a B± = h¯ /mα0± ωz . Thus, we obtain additional switching mechanisms, as explained later. Since we are considering a three-dimensional setup, the exchange interaction is sensitive to the direction and, of course, the magnitude of the applied fields. We give a brief overview of our results [41] for in-plane (Bk , E k ) and perpendicular (B⊥ , E ⊥ ) fields here, see the setup in Fig. 3A for the particular directions of the mentioned fields. An in-plane magnetic field Bk suppresses J exponentially—similarly as a perpendicular field in laterally coupled dots [37]. A perpendicular magnetic field B⊥ reduces on the one hand the exchange coupling between identically sized dots α0+ = α0− only slightly. On the other hand, for different dot sizes a B+ < a B− , the behaviour of J (B⊥ ) is no longer monotonic: increasing B⊥ from zero amplifies the exchange coupling J until both electronic orbitals are magnetically compressed to approximately the same size, i.e. B ≈ 2mα0+ wz c/e. From this point on, J decreases weakly, as for identically sized dots. Furthermore, a perpendicular electric field E ⊥ detunes the single-dot levels and thus reduces the exchange coupling, as it was also found for laterally coupled dots and an in-plane electric field [37]. Finally, an in-plane electric field E k and different dot sizes provide another switching mechanism for J . The dots are shifted parallel 2 , where E = h ω /ea . Thus, the larger dot is shifted a greater distance to the field by 1x± = E k /E 0 α0± ¯ z 0 B q 1x− > 1x+ and so the mean distance between the electrons grows as d 0 = d 2 + A2 (E k /E 0 )2 > d, taking 2 −α 2 )/2α 2 α 2 . Since the exchange coupling J is exponentially sensitive to the inter-dot distance A = (α0+ 0− 0+ 0− d 0 , it is suppressed exponentially when an in-plane electric field is applied, J ≈ exp[−2A2 (E k /E 0 )2 ], which is illustrated in Fig. 3C. We have thus described an exponential switching mechanism for a quantum gate relying only on a tunable electric field, as an alternative for the magnetically driven switching.

4. Single-qubit rotations In addition to two-qubit gates, quantum computation requires one-qubit operations (as mentioned in Section 1) which correspond to single-spin rotations in spin-based proposals. Relating to spins in quantum dot arrays, it must be feasible to expose any specific spin-qubit to a time-varying Zeeman coupling (gµ B S · B)(t) [37], which is controlled through the magnetic field B and/or the g-factor g. Because only relative phases are relevant, it is sufficient to rotate all spins of the system at the same time by an external magnetic field, but with a different Larmor frequency.

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Quantum dots can thus be individually addressed by a local variation of the magnetic field. Localized magnetic fields can be generated with the magnetic tip of a scanning force microscope, a magnetic disk writing head, by placing the dots above a grid of current-carrying wires, or by placing a small wire coil above the dot. Alternatively one can use ESR techniques [37] to perform single-spin rotations, e.g. if we want to flip a certain qubit (say from |↑i to |↓i) we apply an ac-magnetic field perpendicular to the static field axis that matches the Larmor frequency of that particular electron. Due to paramagnetic resonance the spin can flip. Furthermore, the mean position of the electron in a particular dot can be pushed into a region with a different magnetic field strength or a different effective g-factor by the application of an electric field via an electrical gate. This produces a relative rotation by the angle of φ = (g 0 B 0 − g B)µ B τ/2h¯ if the electron has spent a time τ in the region with different magnetic conditions (g 0 and B 0 ). In bulk semiconductors, the free-electron value of the g-factor (g0 = 2.0023) is modified by the spin–orbit coupling. The g-factor can be drastically enhanced by doping the semiconductor with magnetic impurities [54, 55]. Furthermore, the g-factor is modified in confined structures such as quantum wells, wires, and dots, and becomes sensitive to an external bias voltage [56]. We have numerically analysed a system with a layered structure (AlGaAsGaAs-InAlGaAs-AlGaAs), in which the effective g-factor of electrons is varied by shifting their position from one layer to another by electrical gating [57]. We have found that in this structure the effective gfactor can be changed by about 1geff ≈ 1 [45]. A typical switching time for a rotation by an angle φ = π/2 is τs ≈ 30 ps if 1geff ≈ 1 and B 0 = B ≈ 1 T. Recently, such spatially selective tuning of the spin splitting has experimentally been realized in a specially designed Alx Ga1−x As quantum well in which the Al concentration x was gradually varied across the structure [46]. In the same material, ensemble spin decoherence times T2∗ on the order of hundreds of ps have been measured over a wide range in temperature and for various values of the electron g-factor, promising the feasibility of coherent spin manipulation in quantum dots where even longer spin relaxation times are expected. Another possibility to control a single spin through the charge (i.e. orbital wave function) is given by nearest-neighbour exchange coupling to FM dots [8] or to a FM layer. Regions (layers) with increased magnetic field can be provided by a FM material while an effective magnetic field can be produced e.g. with dynamically polarized nuclear spins (Overhauser effect) [37]. FM dots can be made of magnetic metals (e.g. Co or Dy) or magnetic semiconductors, e.g. Mn-doped GaAs (in this case, exchange effects are equivalent to an effective g-factor description). The magnetic dots should be coupled to the spin-carrying dots through either lateral or vertical [23] exchange coupling. Note that for aligning the magnetic dots, only homogeneous (no localized) magnetic fields are required. Preparing two sets of magnetic dots with different orientations m1 and m2 can be achieved by using two magnets with different Curie temperatures TC1 > TC2 and successively cooling them at two different (weak) external fields, i.e. cooling below TC1 at a field parallel to m1 (fixing m1 for the first set of dots), and then applying the other field parallel to m2 when crossing TC2 , magnetizing the second set of dots without changing the magnetization of the first set. In combination with the two-qubit gates discussed in Section 3, the exchange mechanism would exclusively be used for the physical implementation of a quantum computer. Finally, there is another possibility for exchange-only quantum computation. Using an appropriate encoding for the qubits [58], it turns out that the Heisenberg interaction eqn (1) between the spins representing the qubits alone is sufficient to (exactly) perform any quantum computation if each qubit is encoded using (at least) three spins (instead of only one).

5. Entangled photons from coupled quantum dots We have studied the production of polarization-entangled photons or, alternatively, spin-entangled electrons using biexcitonic states in two laterally tunnel-coupled quantum dots [19]. The use of the electron–hole recombination in a single quantum dot has been suggested for the production of entangled photons [59, 60].

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A quantum-dot-based photon source is expected to yield a considerably higher production rate (about four orders of magnitude) of entangled photons than spontaneous parametric down-conversion sources [60]. Moreover, we have shown that two tunnel-coupled quantum dots can be used as a deterministic source of entangled photons that are spatially separated at production [19]. Since the polarization-entangled photons are emitted from two different quantum dots, their separation, which is required for quantum communication, is more easily achievable than for photon pairs emitted from the same dot. Biexcitons consist of two excitons, which themselves are formed by a bound state of a conduction-band electron and a valence-band hole in a semiconductor. We assume here that the valence band in the semiconductor at k = 0 is split into the heavy hole (spin ±3/2) and light hole (spin ±1/2) band which are sufficiently separated in energy compared to the bandwidth of the exciting laser and the temperature. We thus only consider heavy-hole excitations in the following. For the biexciton wave function, we start from a strong confinement ansatz where a X  ae , ah , with the free exciton radius a X and the effective Bohr radii of the electron and the hole in a single quantum dot, ae and ah . This is equivalent to the statement that the exciton Coulomb energy is small compared to the single-particle confinement energies, and therefore the electrons are assumed to be independent of the holes, and vice versa. Furthermore, it has been measured that the level splittings due to the electron–hole exchange interaction are only on the order of tens of µeV [61] in those GaAs quantum dots that are of interest for our model (see later), and we therefore neglect the electron–hole exchange here. Because the quantum dots we are going to model are rather in an intermediate confinement regime where a X ≈ ae , ah (a X ≈ 10 nm in bulk GaAs), we include the full Coulomb interaction of the biexciton and the tunnelling between the dots by use of a HL ansatz for both a two-electron wave function [37] (as in Section 3.1) and a two-hole wave function. The HL ansatz for one particle type then has the general form |12iα + (−1) I |21iα |I iα = p , 2(1 + (−1) I |α h1|2iα |2 )

(7)

where |siα ≡ |I = 0iα is a spin singlet and |tiα ≡ |I = 1iα a spin triplet. For the two-particle states, we use the notation |D D 0 iα = |Diα ⊗ |D 0 iα , where the quantum dots are labelled by D = 1, 2 and D 0 = 2, 1. The biexciton wave function, then, is formed by a tensor product of the electron and the hole HL wave functions, |I J i = |I ie ⊗ |J ih ,

(8)

where I = 0 (1) for the electron singlet (triplet) and J = 0 (1) for the hole singlet (triplet). Note that this ansatz is completely different from the standard HL method applied to a biexciton in bulk material. In our case, the single-particle orbitals that are linearly combined are exclusively defined by the confinement potentials for electrons and holes and the external fields, and not by the Coulomb interaction. Restricting our model to the low-energy physics of quantum dots that are filled with few carriers, we can assume two-dimensional confinement for electrons and holes. Similarly, as in Section 3.1, we use twodimensional harmonic oscillator states in a perpendicular magnetic field for the single-particle orbitals. Then, an externally applied electric field merely leads to a spatial shift of the energy minima. A biexciton in two coupled dots is modelled by a Hamiltonian H that is an extension of the two-electron Hamiltonian (4) in Section 3.1 to four interacting particles (two electrons and two holes). We choose a double dot potential analogous to eqn (5), but we assume that electrons and holes are both located inside the quantum dots. We also take into account that electrons and holes have different effective masses m e and m h and different confinement energies h¯ ωe and h¯ ωh . The system that is thus modelled can be realized e.g. by quantum dots formed by thickness fluctuations in a quantum well [62] or by self-assembled dots [52, 63]. Using the four HL biexciton states, we have calculated the energy spectrum in the presence of a magnetic field perpendicular to the 2DEG plane and an in-plane electric field either perpendicular or parallel to the inter-dot axis and obtained rather lengthy analytical expressions [19] not reproduced here. Figure 4A shows the biexciton spectrum in the absence of an electric field obtained for two-dimensional GaAs dots for the

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triplet–triplet triplet–singlet

– 1.5

double exciton occupancy

1

2

3 B/T

4

5

ts

ss

0.6

st

0.4 0.2

A

singlet–singlet

0

tt

0.8 f / f0

–1

B

1

singlet–triplet

– 0.5

E

137

6

0

0

50

100 150 E/(mV/µm)

200

250

Fig. 4. A, Biexciton energy spectrum for double dots formed in a GaAs two-dimensional electron system with m e = 0.067 m 0 , m h = 0.112 m 0 , and h¯ ωe = 3 meV in the absence of external electric fields. The energy is given in units of h¯ ωe . The energies of the HL states are indicated with the electron and the hole spin states. The Zeeman splitting of the levels is not shown in this plot. B, Oscillator strength of the transition from a HL state (indicated in the figure) to an exciton that remains on one single dot versus an in-plane electric field that is applied perpendicularly to the inter-dot axis. The same material parameters as in A were chosen.

four HL states and also for the occupancy of the same dot by the two excitons as a function of the magnetic field. In order to keep the plot understandable, the Zeeman splitting is not shown; note that for instance the triplet–triplet level is split into nine Zeeman sublevels if |ge | 6= |gh | for the electron and hole g factors. Let us give an estimate of the Zeeman interaction for the chosen material parameters: if we assume |ge | = |gh | = 1, a level with a triplet state of one particle type is split up by about ±0.02h¯ ωe /T . Note that in contrast to the two-electron case in Section 3, the double exciton occupancy is not affected by the Coulomb blockade due to an overall charge neutrality. It can be seen in Fig. 4A, that at low magnetic fields the biexciton ground state is given by the singlet–singlet state |ssi; even the energy of the state with two excitons on the same dot is higher. At high magnetic fields, the ground state is given by the triplet–triplet state |tti. We have also calculated the oscillator strength of various biexciton–exciton and exciton–vacuum transitions as a function of an external magnetic or electric field. The oscillator strength is a quantity that is proportional to a particular optical transition rate. Figure 4B serves as an example for our results that suggest the fact that the (bi)excitonic transition rates can efficiently be suppressed by the application of an electric field (which gives rise to a spatial shift of electrons and holes in opposite directions). This would in principle allow to ‘switch’ on and off the optical recombination of the biexciton, i.e. the emission of one photon pair, in a deterministic manner. Transformation of the HL biexciton states into a two-exciton basis yields a superposition of bright (Sz = ±1) and dark (Sz = ±2) exciton states. Assuming that the two emitted photons enclose an azimuthal angle 1ϕ = 0 or π, the resulting two-photon state can be written as |χphoton i ∝ | + 1, θ1 i| − 1, θ2 i + (−1) I +J | − 1, θ1 i| + 1, θ2 i,

(9)

where the state of a photon emitted at an angle θ with respect to the normal of the 2DEG plane from the recombination of an exciton with spin −σ in the z-direction is given by |σ, θi = N (θ)(m σ,+1 (θ )|σ+ i + m σ,−1 (θ )|σ− i),

(10)

where the states of right and left circular polarization are given by |σ± i and the (normalized) inter-band momentum matrix element is given by m σ λ (θ ) = (cos θ − σ λ)/2, assuming that the normal of the 2DEG plane coincides with one of the main axes of the cubic crystal. In eqn (9), we have obtained two different states of the polarization-entangled photon pair from the four (HL) biexciton spin states, one for equal symmetry of the electron and the hole spin states, i.e. |ssi and |tti, and one for different ones, i.e. |sti and |tsi. The entanglement of the two-photon state (9) that results from the biexciton recombination cascade can be

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quantified by the von Neumann entropy E = log2 (1 + cos2 θ1 cos2 θ2 ) −

cos2 θ1 cos2 θ2 log2 (cos2 θ1 cos2 θ2 ) . 1 + cos2 θ1 cos2 θ2

(11)

Only the emission of both photons perpendicular to the 2DEG plane (θ1 = θ2 = 0) leads to maximal entanglement (E = 1). Especially, the two photons are not entangled (E = 0) if at least one of them is emitted parallel to the plane (θi = 0). An efficient and deterministic source for entangled photon pairs could be constructed working in a threestep cycle: (1) generation of one biexciton in the double dot, e.g. by optical methods or by subsequent carrier injection [59], (2) application of an electric field to prevent recombination before the biexciton has relaxed into its ground state, (3) release of the photon pair by switching off the electric field. If the biexciton recombines from a HL state, as is predicted by our calculations, the photons are emitted at different dots which would help to separate them for experiments. Single excitons have been stored in coupled self-assembled quantum dot pairs [63] for times on the order of seconds, which is ≈109 times the (unbiased) exciton lifetime. There, an electric field was used to force one of the particles to tunnel to the other dot for the storage and back again for the recombination. Eventually using a similar mechanism, the two holes could be removed from the biexciton in the double dot and be transported to two adjacent dots. A double dot might thus—alternatively—be used as an optically driven electron–spin entangler, if the relaxation to the (biexciton) ground state is faster than the recombination of the involved excitons, which is supported by experiments under low excitation density [64] and could as well be achieved by delaying the recombination of the biexciton by means of an applied electric field. Acknowledgements—We thank D Awschalom, A Imamoglu, and P Petroff for useful discussions. This work is supported by the NCCR Nanoscience, the Swiss NSF, DARPA, and ARO.

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