lecture notes in physics, Lecture notes of Quantum Physics

Download lecture notes in physics and more Lecture notes Quantum Physics in PDF only on Docsity! Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Los Angeles, CA, USA S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz, Köln, Germany The Editorial Policy for Edited Volumes The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching - quickly, informally but with a high degree of quality. 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It contains the full texts (PDF) of all volumes published since 2000. Abstracts, table of contents and prefaces are accessible free of charge to everyone. Information about the availability of printed volumes can be obtained. −The subscription information. The online archive is free of charge to all subscribers of the printed volumes. −The editorial contacts, with respect to both scientific and technical matters. −The author’s / editor’s instructions. Preface Nanoscale physics, nowadays one of the most topical research subjects, has two major areas of focus. One is the important field of potential applications bearing the promise of a great variety of materials having specific properties that are desirable in daily life. Even more fascinating to the researcher in physics are the fundamental aspects where quantum mechanics is seen at work; most macroscopic phenomena of nanoscale physics can only be understood and described using quantum mechanics. The emphasis of the present volume is on this latter aspect. It fits perfectly within the tradition of the South African Summer Schools in Theoretical Physics and the fifteenth Chris Engelbrecht School was de- voted to this highly topical subject. This volume presents the contents of lectures from four speakers working at the forefront of nanoscale physics. The first contribution addresses some more general theoretical considerations on Fermi liquids in general and quantum dots in particular. The next topic is more experimental in nature and deals with spintronics in quantum dots. The alert reader will notice the close correspondence to the South African Summer School in 2001, published in LNP 587. The following two sections are theoreti- cal treatments of low temperature transport phenomena and electron scatter- ing on normal-superconducting interfaces (Andreev billiards). The enthusiasm and congenial atmosphere created by the speakers will be remembered well by all participants. The beautiful scenery of the Drakensberg surrounding the venue contributed to the pleasant spirit prevailing during the school. A considerable contingent of participants came from African countries out- side South Africa and were supported by a generous grant from the Ford Foundation; the organisers gratefully acknowledge this assistance. The Organising Committee is indebted to the National Research Founda- tion for its financial support, without which such high level courses would be impossible. We also wish to express our thanks to the editors of Lecture Notes in Physics and Springer for their assistance in the preparation of this volume. Stellenbosch WD Heiss February 2005 List of Contributors R. Shankar Sloane Physics Lab, Yale University, New Haven CT 06520 r.shankar@yale.edu J.M. Elzerman Kavli Institute of Nanoscience Delft, PO Box 5046, 2600 GA Delft, The Netherlands ERATO Mesoscopic Correlation Project, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan elzerman@qt.tn.tudelft.nl R. Hanson Kavli Institute of Nanoscience Delft, PO Box 5046, 2600 GA Delft, The Netherlands L.H.W. van Beveren Kavli Institute of Nanoscience Delft, PO Box 5046, 2600 GA Delft, The Netherlands ERATO Mesoscopic Correlation Project, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan S. Tarucha ERATO Mesoscopic Correlation Project, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan NTT Basic Research Laboratories, Atsugi-shi, Kanagawa 243-0129, Japan L.M.K. Vandersypen Kavli Institute of Nanoscience Delft, PO Box 5046, 2600 GA Delft, The Netherlands L.P. Kouwenhoven Kavli Institute of Nanoscience Delft, PO Box 5046, 2600 GA Delft, The Netherlands ERATO Mesoscopic Correlation Project, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan M. Pustilnik School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA L.I. Glazman William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA C.W.J. Beenakker Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands X Contents 3 Thermally-Activated Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4 Activationless Transport through a Blockaded Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Kondo Regime in Transport through a Quantum Dot . . . . . . . . . . . . . 113 6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Andreev Billiards C.W.J. Beenakker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2 Andreev Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3 Minigap in NS Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4 Scattering Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Stroboscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6 Random-Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7 Quasiclassical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8 Quantum-To-Classical Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A Excitation Gap in Effective RMT and Relationship with Delay Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 A Guide for the Reader Quantum dots, often denoted artificial atoms, are the exquisite tools by which quantum behavior can be probed on a scale appreciably larger than the atomic scale, that is, on the nanometer scale. In this way, the physics of the devices is closer to classical physics than that of atomic physics but they are still sufficiently small to clearly exhibit quantum phenomena. The present volume is devoted to some of these fascinating aspects. In the first contribution general theoretical aspects of Fermi liquids are addressed, in particular, the renormalization group approach. The choice of appropriate variables as a result of averaging over “unimportant” variables is presented. This is then aptly applied to large quantum dots. The all impor- tant scales, ballistic dots and chaotic motion are discussed. Nonperturbative methods and critical phenomena feature in this thorough treatise. The tra- ditional phenomenological Landau parameters are given a more satisfactory theoretical underpinning. A completely different approach is encountered in the second contribution in that it is a thorough experimental expose of what can be done or expected in the study of small quantum dots. Here the emphasis lies on the electron spin to be used as a qubit. The experimental steps toward using a single electron spin – trapped in a semiconductor quantum dot – as a spin qubit are described. The introduction contains a resume of quantum computing with quantum dots. The following sections address experimental implementations, the use of different quantum dot architectures, measurements, noise, sensitivity and high-speed performance. The lectures are based on a collaborative effort of research groups in the Netherlands and in Japan. The last two contributions are again theoretical in nature and address particular aspects relating to quantum dots. In the third lecture series, mech- anisms of low-temperature electronic transport through a quantum dot – weakly coupled to two conducting leads – are reviewed. In this case transport is dominated by electron–electron interaction. At moderately low temperatures (comparing with the charging energy) the linear conductance is suppressed by 2 A Guide for the Reader the Coulomb blockade. A further lowering of the temperature leads into the Kondo regime. The fourth series of lectures deals with a very specific and cute aspect of nanophysics: a peculiar property of superconducting mirrors as discovered by Andreev about forty years ago. The Andreev reflection at a superconductor modifies the excitation spectrum of a quantum dot. The difference between a chaotic and integrable billiard (quantum dot) is discussed and relevant clas- sical versus quantum time scales are given. The results are a challenge to experimental physicists as they are not confirmed as yet. RG for Interacting Fermions 5 where the chemical potential µ is introduced to make sure we have a finite density of particles in the ground state: all levels up the Fermi surface, a circle defined by K2F /2m = µ (5) are now occupied and occupying these levels lowers the ground-state energy. Notice that this system has gapless excitations above the ground state. You can take an electron just below the Fermi surface and move it just above, and this costs as little energy as you please. Such a system will carry a dc current in response to a dc voltage. An important question one asks is if this will be true when interactions are turned on. For example the system could develop a gap and become an insulator. What really happens for the d = 2 electron gas? We are going to answer this using the RG. Let us first learn how to do RG for noninteracting fermions. To understand the low energy physics, we take a band of of width Λ on either side of the Fermi surface. This is the first great difference between this problem and the usual ones in relativistic field theory and statistical mechanics. Whereas in the latter examples low energy means small momentum, here it means small deviations from the Fermi surface. Whereas in these older problems we zero in on the origin in momentum space, here we zero in on a surface. The low energy region is shown in Fig. 1. To apply our methods we need to cast the problem in the form of a path integral. Following any number of sources, say [2] we obtain the following expression for the partition function of free fermions: Z0 = ∫ dψdψeS0 (6) where K F ΛK F + Λ K F - Fig. 1. The low energy region for nonrelativistic fermions lies within the annulus concentric with the Fermi circle 6 R. Shankar S0 = ∫ d2K ∫ ∞ −∞ dωψ(ω,K) ( iω − (K 2 − K2F ) 2m ) ψ(ω,K) (7) where ψ and ψ are called Grassmann variables. They are really weird objects one gets to love after some familiarity. Most readers can assume they are ordinary integration variables. The dedicated reader can learn more from [2]. We now adapt this general expression to the annulus to obtain Z0 = ∫ dψdψeS0 (8) where S0 = ∫ 2π 0 dθ ∫ ∞ −∞ dω ∫ Λ −Λ dkψ(iω − v k)ψ . (9) To get here we have had to approximate as follows: K2 − K2F 2m  KF m · k = vF k (10) where k − K − KF and vF is the fermi velocity, hereafter set equal to unity. Thus Λ can be viewed as a momentum or energy cut-off measured from the Fermi circle. We have also replaced KdK by KF dk and absorbed KF in ψ and ψ. It will seen that neglecting k in relation to KF is irrelevant in the technical sense. Let us now perform mode elimination and reduce the cut-off by a factor s. Since this is a gaussian integral, mode elimination just leads to a multiplicative constant we are not interested in. So the result is just the same action as above, but with |k| ≤ Λ/s. Let us now do make the following additional transformations: (ω′, k′) = s(ω, k) (ψ′(ω′, k′), ψ ′ (ω′, k′)) = s−3/2 ( ψ ( ω′ s , k′ s ) , ψ ( ω′ s , k′ s )) . (11) When we do this, the action and the phase space all return to their old values. So what? Recall that our plan is to evaluate the role of quartic in- teractions in low energy physics as we do mode elimination. Now what really matters is not the absolute size of the quartic term, but its size relative to the quadratic term. Keeping the quadratic term identical before and after the RG action makes the comparison easy: if the quartic coupling grows, it is rele- vant; if it decreases, it is irrelevant, and if it stays the same it is marginal. The system is clearly gapless if the quartic coupling is irrelevant. Even a marginal coupling implies no gap since any gap will grow under the various rescalings of the RG. Let us now turn on a generic four-Fermi interaction in path-integral form: S4 = ∫ ψ(4)ψ(3)ψ(2)ψ(1)u(4, 3, 2, 1) (12) RG for Interacting Fermions 7 where ∫ is a shorthand: ∫ ≡ 3∏ i=1 ∫ dθi ∫ Λ −Λ dki ∫ ∞ −∞ dωi (13) At the tree level, we simply keep the modes within the new cut-off, rescale fields, frequencies and momenta, and read off the new coupling. We find u′(k′, ω′, θ) = u(k′/s, ω′/s, θ) (14) This is the evolution of the coupling function. To deal with coupling con- stants with which we are more familiar, we expand the functions in a Taylor series (schematic) u = uo + ku1 + k2u2 . . . (15) where k stands for all the k’s and ω’s. An expansion of this kind is possible since couplings in the Lagrangian are nonsingular in a problem with short range interactions. If we now make such an expansion and compare coefficients in (14), we find that u0 is marginal and the rest are irrelevant, as is any coupling of more than four fields. Now this is exactly what happens in φ44, scalar field theory in four dimensions with a quartic interaction. The difference here is that we still have dependence on the angles on the Fermi surface: u0 = u(θ1, θ2, θ3, θ4) Therefore in this theory we are going to get coupling functions and not a few coupling constants. Let us analyze this function. Momentum conservation should allow us to eliminate one angle. Actually it allows us more because of the fact that these momenta do not come form the entire plane, but a very thin annulus near KF . Look at the left half of Fig. 2. Assuming that the cutoff has been reduced to the thickness of the circle in the figure, it is clear that if two points 1 and 2 are chosen from it to represent the incoming lines in a four point coupling, K K 1 2 K 1 K 2 + K K 1 2 K K 3 4 Fig. 2. Kinematical reasons why momenta are either conserved pairwise or restricted to the BCS channel 10 R. Shankar 3 Large-N Approach to Fermi Liquids Not only did Landau say we could describe Fermi liquids with an F function, he also managed to compute the response functions at small ω and q in terms of the F function even when it was large, say 10, in dimensionless units. Again the RG gives us one way to understand this. To this end we need to recall the the key ideas of “large-N” theories. These theories involve interactions between N species of objects. The large- ness of N renders fluctuations (thermal or quantum) small, and enables one to make approximations which are not perturbative in the coupling constant, but are controlled by the additional small parameter 1/N . As a specific example let us consider the Gross-Neveu model [5] which is one of the simplest fermionic large-N theories. This theory has N identical massless relativistic fermions interacting through a short-range interaction. The Lagrangian density is L = N∑ i=1 ψ̄i
∂ψi − λ N ( N∑ i=1 ψ̄iψi )2 (21) Note that the kinetic term conserves the internal index, as does the in- teraction term: any index that goes in comes out. You do not have to know much about the GN model to to follow this discussion, which is all about the internal indices. Figure 4 shows the first few diagrams in the expression for the scattering amplitude of particle of isospin index i and j in the Gross-Neveu theory. The “bare” vertex comes with a factor λ/N . The one-loop diagrams all share a factor λ2/N2 from the two vertices. The first one-loop diagram has a free internal summation over the index k that runs over N values, with the con- tribution being identical for each value of k. Thus, this one-loop diagram acquires a compensating factor of N which makes its contribution of order λ2/N , the same order in 1/N as the bare vertex. However, the other one- loop diagrams have no such free internal summation and their contribution = +++ + i i i jii j j j j i i i j i ij j i j j k ik j ... j Fig. 4. Some diagrams from a large-N theory RG for Interacting Fermions 11 is indeed of order 1/N2. Therefore, to leading order in 1/N , one should keep only diagrams which have a free internal summation for every vertex, that is, iterates of the leading one-loop diagram, which are called bubble graphs. For later use remember that in the diagrams that survive (do not survive), the indices i and j of the incoming particles do not (do) enter the loops. Let us assume that the momentum integral up to the cutoff Λ for one bubble gives a factor −Π(Λ, qext), where qext is the external momentum or frequency trans- fer at which the scattering amplitude is evaluated. To leading order in large-N the full expression for the scattering amplitude is Γ (qext) = 1 N λ 1 + λΠ(Λ, qext) (22) Once one has the full expression for the scattering amplitude (to leading order in 1/N) one can ask for the RG flow of the “bare” vertex as the cutoff is reduced by demanding that the physical scattering amplitude Γ remain insensitive to the cutoff. One then finds (with t = ln(Λ0/Λ)) dΓ(qext) dt = 0 ⇒ dλ dt = −λ2 dΠ(Λ, qext) dt (23) which is exactly the flow one would extract at one loop. Thus the one-loop RG flow is the exact answer to leading order in a large-N theory. All higher-order corrections must therefore be subleading in 1/N . 3.1 Large-N Applied to Fermi Liquids Consider now the ψ̄ψ − ψ̄ψ correlation function (with vanishing values of external frequency and momentum transfer). Landau showed that it takes the form χ = χ0 1 + F0 , (24) where F0 is the angular average of F (θ) and χ0 is the answer when F = 0. Note that the answer is not perturbative in F . Landau got this result by working with the ground-state energy as a func- tional of Fermi surface deformations. The RG provides us with not just the ground-state energy, but an effective hamiltonian (operator) for all of low- energy physics. This operator problem can be solved using large N -techniques. The value of N here is of order KF /Λ, and here is how it enters the formalism. Imagine dividing the annulus in (Fig. 1) into N patches of size (Λ) in the radial and angular directions. The momentum of each fermion ki is a sum of a “large” part (O(kF )) centered on a patch labelled by a patch index i = 1, . . . N and a “small” momentum (O(Λ) within the patch [2]. Consider a four-fermion Green’s function, as in (Fig. 4). The incoming momenta are labelled by the patch index (such as i) and the small momentum is not shown but implicit. We have seen that as Λ → 0, the two outgoing 12 R. Shankar momenta are equal to the two incoming momenta up to a permutation. At small but finite Λ this means the patch labels are same before and after. Thus the patch index plays the role of a conserved isospin index as in the Gross-Neveu model. The electron-electron interaction terms, written in this notation, (with k integrals replaced by a sum over patch index and integration over small momenta) also come with a pre-factor of 1/N ( Λ/KF ). It can then be verified that in all Feynman diagrams of this cut-off theory the patch index plays the role of the conserved isospin index exactly as in a theory with N fermionic species. For example in (Fig. 4) in the first dia- gram, the external indices i and j do not enter the diagram (small momentum transfer only) and so the loop momentum is nearly same in both lines and integrated fully over the annulus, i.e., the patch index k runs over all N val- ues. In the second diagram, the external label i enters the loop and there is a large momentum transfer (O(KF )). In order for both momenta in the loop to be within the annulus, and to differ by this large q, the angle of the loop momentum is limited to a range O(Λ/KF ). (This just means that if one mo- mentum is from patch i the other has to be from patch j.) Similarly, in the last loop diagram, the angle of the loop momenta is restricted to one patch. In other words, the requirement that all loop momenta in this cut-off theory lie in the annulus singles out only diagrams that survive in the large N limit. The sum of bubble diagrams, singled out by the usual large-N considera- tions, reproduces Landau’s Fermi liquid theory. For example in the case of χ, one obtains a geometric series that sums to give χ = χ01+F0 . Since in the large N limit, the one-loop β-function for the fermion-fermion coupling is exact, it follows that the marginal nature of the Landau parameters F and the flow of V , (20), are both exact as Λ → 0. A long paper of mine [2] explains all this, as well as how it is to be general- ized to anisotropic Fermi surfaces and Fermi surfaces with additional special features and consequently additional instabilities. Polchinski [6] independently analyzed the isotropic Fermi liquid (though not in the same detail, since it was a just paradigm or toy model for an effective field theory for him). 4 Quantum Dots We will now apply some of these ideas, very successful in the bulk, to two- dimensional quantum dots [7, 9] tiny spatial regions of size L  100−200 nm, to which electrons are restricted using gates. The dot can be connected weakly or strongly to leads. Since many experts on dots are contributing to this volume, I will be sparing in details and references. Let us get acquainted with some energy scales, starting with ∆, the mean single particle level spacing. The Thouless energy is defined as ET =
/τ , where τ is the time it takes to traverse the dot. If the dot is strongly coupled to leads, this is the uncertainty in the energy of an electron as it traverses RG for Interacting Fermions 15 this sum has just one term, at q = 0. Unlike in a clean system, there is no singular behavior associated with q → 0 and this assumption is a good one. Others have asked how one can introduce the Landau interaction that respects momentum conservation in a dot that does not conserve momentum or anything else except energy. To them I say this. Just think of a pair of molecules colliding in a room. As long as the collisions take place in a time scale smaller than the time between collisions with the walls, the interaction will be momentum conserving. That this is true for a collision in the dot for particles moving at vF , subject an interaction of range equal to the Thomas Fermi screening length (the typical range) is readily demonstrated. Like it or not, momentum is a special variable even in a chaotic but ballistic dot since it is tied to translation invariance, and that that is operative for realistic collisions within the dot. • Step 2: Switch to the exact basis states of the chaotic dot, writing the kinetic and interaction terms in this basis. Run the RG by eliminating exact energy eigenstates within ET . While this looks like a reasonable plan, it is not clear how it is going to be executed since knowledge of the exact eigenfunctions is needed to even write down the Landau interaction in the disordered basis: Vαβγδ = ∆4 ∑ kk′ u(θ − θ′) [ φ∗α(k)φ ∗ β(k ′) − φ∗α(k′)φ∗β(k) ] × [φγ(k′)φδ(k) − φγ(k)φδ(k′)] (29) where k and k′ take g possible values. These are chosen as follows. Consider the momentum states of energy within ET of EF . In a dot momentum is defined with an uncertainty ∆k  1/L in either direction. Thus one must form packets in k space obeying this condition. It can be easily shown that g of them will fit into this band. One way to pick such packets is to simply take plane waves of precise k and chop them off at the edges of the dot and normalize the remains. The g values of k can be chosen with an angular spacing 2π/g. It can be readily verified that such states are very nearly orthogonal. The wavefunction φδ(k) is the projection of exact dot eigenstate δ on the state k as defined above. We will see that one can go a long way without detailed knowledge of the wavefunctions φδ(k). First, one can take the view of the Universal Hamiltonian (UH) adherents and consider the ensemble average (enclosed in 〈 〉) of the interactions. RMT tells us that to leading order in 1/g, 〈 φ∗α(k1)φ ∗ β(k2)φγ(k3)φδ(k4) 〉 = δαδδβγδk1k4δk2k3 g2 + δαγδβδδk1k3δk2k4 g2 + δαβδγδδk1,−k2δk3,−k4 g2 (30) It is seen that only matrix elements in (29) for which the indices αβγδ are pairwise equal survive disorder-averaging, and also that the average has no 16 R. Shankar dependence on the energy of αβγδ. In the spinless case, the first two terms on the right hand side make equal contributions and produce the constant charging energy in the Universal Hamiltonian of (26), while in the spinful case they produce the charging and exchange terms. The final term of (30) produces the Cooper interaction of (26). Thus the UH contains the rotationally invariant part of the Landau in- teraction. The others, i.e., those that do not survive ensemble averaging, are dropped because they are of order 1/g. But we have seen before in the BCS instability of the Fermi liquid that a term that is nominally small to begin with can grow under the RG. That this is what happens in this case was shown by the RG calculation of Murthy and Mathur. There was however one catch. The neglected couplings could overturn the UH description for couplings that exceeded a critical value. However the critical value is of order unity and so one could not trust either the location or even the very existence of this crit- ical point based on their perturbative one-loop calculation. Their work also gave no clue as to what lay on the other side of the critical point. Subsequently Murthy and I [24] showed that the methods of the large N theories (with g playing the role of N) were applicable here and could be used to show nonperturbatively in the interaction strength that the phase transition indeed exists. This approach also allowed us to study in detail the phase on the other side of the transition, as well as what is called the quantum critical region, to be described later. Let us now return to Murthy and Mathur and ask how the RG flow is derived. After integrating some of the g states within ET , we end up with g′ = ge−t states. Suppose we compute a scattering amplitude Γαβγδ for the process in which two fermions originally in states αβ are scattered into states γδ. This scattering can proceed directly through the vertex Vαβγδ(t), or via intermediate virtual states higher order in the interactions, which can be clas- sified by a set of Feynman diagrams, as shown in Fig. 5. All the states in the diagrams belong to the g′ states kept. We demand that the entire amplitude be independent of g′, meaning that the physical amplitudes should be the same in the effective theory as in the original theory. This will lead to a set of flow = +++ + α α α βαα ββ β β γ γ γ δ γ γδ δ µ δ δ µ µ νν ν ... Fig. 5. Feynman diagrams for the full four-point amplitude Γαβγδ RG for Interacting Fermions 17 equations for the Vαβγδ. In principle this flow equation will involve all powers of V but we will keep only quadratic terms (the one-loop approximation). Then the diagrams are limited to the ones shown in Fig. 5, leading to the following contributions to the scattering amplitude Γαβγδ Γαβγδ = Vαβγδ + ∑ µ,ν ′NF (ν) − NF (µ) εµ − εν ( VανµδVβµνγ − VανµγVβµνδ ) − ∑ µν ′ 1 − NF (µ) − NF (ν) εµ + εν VαβµνVνµγδ (31) where the prime on the sum reminds us that only the g′ remaining states are to be kept and where NF (α) is the Fermi occupation of the state α. We will confine ourselves to zero temperature where this number can only be zero or one. The matrix element Vαβγδ now explicitly depends on the RG flow parameter t. Now we demand that upon integrating the two states at ±g′∆/2 we recover the same Γαβγδ. Clearly, since g′ = ge−t, d dt = −g′ δ δg′ (32) The effect of this differentiation on the loop diagrams is to fix one of the internal lines of the loop to be at the cutoff ±g′∆/2, while the other one ranges over all smaller values of energy. In the particle-hole diagram, since µ or ν can be at +g′∆/2 or −g′∆/2, and the resulting summations are the same in all four cases, we take a single contribution and multiply by a factor of 4. The same reasoning applies to the Cooper diagram. Let us define the energy cutoff Λ = g′∆/2 to make the notation simpler. Since we are integrating out two states we have δg′ = 2 0 = dVαβγδ dt − g ′ 2 4 ∑ µ=Λ,ν ′ NF (ν) − NF (µ) εµ − εν ( VανµδVβµνγ − VανµγVβµνδ ) + g′ 2 4 ∑ µ=Λ,ν ′ 1 − NF (µ) − NF (ν) εµ + εν VαβµνVνµγδ (33) where µ = Λ means εµ = Λ and so on. The changed sign in front of the 1-loop diagrams reflects the sign of (32) So far we have not made any assumptions about the form of Vαβγδ, and the formulation applies to any finite system. In a generic system such as an atom, the matrix elements depend very strongly on the state being integrated over, and the flow must be followed numerically for each different set αβγδ kept in the low-energy subspace. 20 R. Shankar dVαβγδ dt ∣∣∣∣ sub = −g′∆2 0∑ ν=−Λ 1 Λ + |εν | [∑ kk′ ∑ pp′ u(θk − θp)u(θp′ − θk′) ×φ∗α(p)φ∗β(k′)φγ(k′)φδ(k)φ∗µ(k)φ∗ν(p′)φν(p)φµ(p′) ] (45) Note that the momentum labels of φ∗α and φ ∗ µ have been exchanged com- pared to (36) and there is a minus sign, both coming from the antisymmetriza- tion of (29). Once again we ensemble average the internal µ, ν sum, the wave- function part of which gives 〈 φ∗µ(k)φ ∗ ν(p ′)φν(p)φµ(p′) 〉 = δkp′δpp′ g2 − δkp + δk,−p′δp,−p′ g3 (46) It is clear that there is an extra momentum restriction in each term com- pared to (37), which means that one can no longer sum freely over p to get the factor of g in (39), or the factor of g2 in (40). Therefore this contribution will be down by 1/g compared to that of (36). Turning now to the Cooper diagrams, the internal lines are once again forced to have the same momentum labels as the external lines by the Fermi liquid vertex, therefore they do not make any leading contributions. The general rule is that whenever a momentum label corresponding to an internal line in the diagram (here µ and ν) is forced to become equal to a momentum label corresponding to an external disorder label (here α, β, γ, or δ), the diagram is down by 1/g, exactly as in the 1/N expansion. The fact that 1/g plays the role of 1/N was first realized by Murthy and Shankar [24]. Not only did this mean that the one loop flow of Murthy and Mathur was exact, it meant the disorder-interaction problem of the chaotic dot could be solved exactly in the large g limit. It is the only known case where the problem of disorder and interactions [26, 27, 28] can be handled exactly. From (44) it can be seen that positive initial values of ũm (which are equal to initial values of um inherited from the bulk) are driven to the fixed point at ũm = 0, as are negative initial values as long as um(t = 0) ≥ u∗m = −1/ ln 2. Thus, the Fermi liquid parameters are irrelevant for this range of starting values. Recall that setting all um = 0 for m
= 0 results in the Universal Hamiltonian. Thus, the range um ≥ u∗m is the basin of attraction of the Universal Hamiltonian. On the other hand, for um(t = 0) ≤ u∗m results in a runaway flow towards large negative values of um, signalling a transition to a phase not perturbatively connected to the Universal Hamiltonian. Since we have a large N theory here (with N = g), the one-loop flow and the new fixed point at strong coupling are parts of the final theory.1 What is 1 However the exact location of the critical point cannot be predicted, as pointed out to us by Professor Peet Brower. The reason is that the Landau couplings um are defined at a scale EL much higher than ET (but much smaller than EF ) and their flow till we come down to ET , where our analysis begins, is not within RG for Interacting Fermions 21 the nature of the state for um(t = 0) ≤ u∗m? The formalism and techniques needed to describe that are beyond what was developed in these lectures, which has focused on the RG. Suffice it to say that it is possible to write the partition function in terms of a new collective field σ (which depends on all the particles) and that the action S(σ) has a factor g in front of it, allowing us to evaluate the integral by saddle point(in the limit g → ∞), to confidently predict the strong coupling phase and many of its properties. Our expectations based on the large g analysis have been amply confirmed by detailed numerical studies [25]. For now I will briefly describe the new phenomenon in qualitative terms for readers not accustomed to these ideas and give some references for those who are. In the strong coupling region σ acquires an expectation value in the ground state. The dynamics of the fermions is affected by this variable in many ways: quasi-particle widths become broad very quickly above the Fermi energy, the second difference ∆2 has occasionally very large values and can even be neg- ative2, and the system behaves like one with broken time-reversal symmetry if m is odd. Long ago Pomeranchuk [29] found that if a Landau function of a pure system exceeded a certain value, the fermi surface underwent a shape trans- formation from a circle to an non-rotationally invariant form. Recently this transition has received a lot of attention [30, 31] The transition in question is a disordered version of the same. Details are given in [24, 25]. Details aside, there is another very interesting point: even if the coupling does not take us over to the strong-coupling phase, we can see vestiges of the critical point u∗m and associated critical phenomena. This is a general feature of many quantum critical points [32], i.e., points like u∗m where as a variable in a hamiltonian is varied, the system enters a new phase (in contrast to transitions wherein temperature T is the control parameter). Figure 6 shows what happens in a generic situation. On the x-axis a vari- able (um in our case) along which the quantum phase transition occurs. Along y is measured a new variable, usually temperature T . Let us consider that case first. If we move from right to left at some value of T , we will first encounter physics of the weak-coupling phase determined by the weak-coupling fixed point at the origin. Then we cross into the critical fan (delineated by the V -shaped dotted lines), where the physics controlled by the quantum criti- cal point. In other words we can tell there is a critical point on the x-axis the regime we can control. In other words we can locate u∗ in terms of what couplings we begin with, but these are the Landau parameters renormalized in a nonuniversal way as we come down from EL to ET . 2 How can the cost of adding one particle be negative (after removing the charging energy)? The answer is that adding a new particle sometimes lowers the energy of the collective variable which has a life of its own. However, if we turn a blind eye to it and attribute all the energy to the single particle excitations, ∆2 can be negative. 22 R. Shankar u u∗ critical fan symmetricbroken 1/g Fig. 6. The generic phase diagram for a second-order quantum phase transition. The horizontal axis represents the coupling constant which can be tuned to take one across the transition. The vertical axis is usually the temperature in bulk quantum systems, but is 1/g here, since in our system one of the roles played by g is that of the inverse temperature without actually traversing it. As we move further to the left, we reach the strongly-coupled symmetry-broken phase, with a non-zero order parameter. It can be shown that in our problem, 1/g plays the role of T . One way to see is this that in any large N theory N stands in front of the action when written in terms of the collective variable. That is true in this case well for g. (Here g also enters at a subdominant level inside the action, which makes it hard to predict the exact shape of the critical fan. The bottom line is that we can see the critical point at finite 1/g. In addition one can also raise temperature or bias voltage to see the critical fan. Subsequent work has shown, in more familiar examples that Landau in- teractions, that the general picture depicted here is true in the large g limit: upon adding sufficiently strong interactions the Universal Hamiltonian gives way to other descriptions with broken symmetry [33, 34].3 Acknowledgement It is a pleasure to thank the organizers of this school especially Professors Dieter Heiss, Nithaya Chetty and Hendrik Geyer for their stupendous hos- pitality that made all of decide to revisit South Africa as soon as possible. 3 The only result that is not exact in the large g limit is the critical value u∗m since the input value of um at ET is related in a non-universal way to the Landau parameter. In other words, the Landau coupling um settles down to its fixed point value at an energy scale that is much larger than ET . We do not know how it flows as we reach energies inside ET wherein our RMT assumptions are valid. Semiconductor Few-Electron Quantum Dots as Spin Qubits J.M. Elzerman1,2, R. Hanson1, L.H.W. van Beveren1,2, S. Tarucha2,3, L.M.K. Vandersypen1, and L.P. Kouwenhoven1,2 1 Kavli Institute of Nanoscience Delft, PO Box 5046, 2600 GA Delft, The Netherlands elzerman@qt.tn.tudelft.nl 2 ERATO Mesoscopic Correlation Project, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 3 NTT Basic Research Laboratories, Atsugi-shi, Kanagawa 243-0129, Japan The spin of an electron placed in a magnetic field provides a natural two- level system suitable as a qubit in a quantum computer [1]. In this work, we describe the experimental steps we have taken towards using a single electron spin, trapped in a semiconductor quantum dot, as such a spin qubit [2]. The outline is as follows. Section 1 serves as an introduction into quantum computing and quantum dots. Section 2 describes the development of the “hardware” for the spin qubit: a device consisting of two coupled quantum dots that can be filled with one electron (spin) each, and flanked by two quantum point contacts (QPCs). The system can be probed in two different ways, either by performing conventional measurements of transport through one dot or two dots in series, or by using a QPC to measure changes in the (average) charge on each of the two dots. This versatility has proven to be very useful, and the type of device shown in this section was used for all subsequent experiments. In Sect. 3, it is shown that we can determine all relevant parameters of a quantum dot even when it is coupled very weakly to only one reservoir. In this regime, inaccessible to conventional transport experiments, we use a QPC charge detector to determine the tunnel rate between the dot and the reservoir. By measuring changes in the effective tunnel rate, we can determine the excited states of the dot. In Sect. 4, the QPC as a charge detector is pushed to a faster regime (∼100 kHz), to detect single electron tunnel events in real time. We also de- termine the dominant contributions to the noise, and estimate the ultimate speed and sensitivity that could be achieved with this very simple method of charge detection. In Sect. 5, we develop a technique to perform single-shot measurement of the spin orientation of an individual electron in a quantum dot. This is done by J.M. Elzerman et al.: Semiconductor Few-Electron Quantum Dots as Spin Qubits, Lect. Notes Phys. 667, 25–95 (2005) www.springerlink.com c© Springer-Verlag Berlin Heidelberg 2005 26 J.M. Elzerman et al. combining fast QPC charge detection with “spin-to-charge conversion”. This fully electrical technique to read out a spin qubit is then used to determine the relaxation time of the single spin, giving a value of 0.85 ms at a magnetic field of 8 Tesla. Finally, Sect. 6 puts the results in perspective, arriving at a realistic path towards the experimental demonstration of single- and two-qubit gates and the creation of entanglement of spins in quantum dot systems. 1 Introduction This section gives a brief introduction into quantum computing, continuing with a description of semiconductor quantum dots that covers their fabrication as well as their electronic behavior. We also describe our experimental setup for performing low-temperature transport experiments to probe such quantum dots. 1.1 Quantum Computing More than three quarters of a century after its birth, quantum mechanics re- mains in many ways a peculiar theory [3]. It describes many physical effects and properties with great accuracy, but uses unfamiliar concepts like super- position, entanglement and projection, that seem to have no relation with the everyday world around us. The interpretation of these concepts can still cause controversy. The inherent strangeness of quantum mechanics already emerges in the simplest case: a quantum two-level system. Unlike a classical two-level system, which is always either in state 0 or in state 1, a quantum two-level system can just as well be in a superposition of states |0〉 and |1〉. It is, in some sense, in both states at the same time. Even more exotic states can occur when two such quantum two-level sys- tems interact: the two systems can become entangled. Even if we know the complete state of the system as a whole, for example (|01〉 − |10〉)/ √ 2, which tells us all there is to know about it, we cannot know the state of the two subsystems individually. In fact, the subsystems do not even have a definite state! Due to this strong connection between the two systems, a measurement made on one influences the state of the other, even though it may be arbitrar- ily far away. Such spooky non-local correlations enable effects like “quantum teleportation” [4, 5]. Finally, the concept of measurement in quantum mechanics is rather spe- cial. The evolution of an isolated quantum system is deterministic, as it is governed by a first order differential equation – the Schrödinger equation. However, coupling the quantum system to a measurement apparatus forces it into one of the possible measurement eigenstates in an apparently non- deterministic way: the particular measurement outcome is random, only the Semiconductor Few-Electron Quantum Dots as Spin Qubits 27 probability for each outcome can be determined [3]. The question of what exactly constitutes a measurement is still not fully resolved [6]. These intriguing quantum effects pose fundamental questions about the nature of the world we live in. The goal of science is to explore these questions. At the same time, this also serves a more opportunistic purpose, since it might allow us to actually use the unique features of quantum mechanics to do something that is impossible from the classical point of view. And there are still many things that we cannot do classically. A good ex- ample is prime-factoring of large integers: it is easy to take two prime numbers and compute their product. However, it is difficult to take a large integer and find its prime factors. The time it takes any classical computer to solve this problem grows exponentially with the number of digits. By making the integer large enough, it becomes essentially impossible for any classical computer to find the answer within a reasonable time – such as the lifetime of the universe. This fact is used in most forms of cryptography nowadays [7]. In 1982, Richard Feynman speculated [8] that efficient algorithms to solve such hard computational problems might be found by making use of the unique features of quantum systems, such as entanglement. He envisioned a set of quantum two-level systems that are quantum mechanically coupled to each other, allowing the system as a whole to be brought into a superposition of different states. By controlling the Hamiltonian of the system and therefore its time-evolution, a computation might be performed in fewer steps than is possible classically. Essentially, such a quantum computer could take many computational steps at once; this is known as “quantum parallelism”. A simplified view of the difference between a classical and a quantum computer is shown in Fig. 1. A one-bit classical computer is a machine that f f (0)0 f f (1)1 F10 + 10 +F F f f (00)00 f f (01)01 F0100 + 0100 +F F f f (10)10 f f (11)11 1110 ++ 10+F 11F+ a b1 (qu)bit 2 (qu)bits Fig. 1. Difference between a classical and a quantum computer. (a) To determine the function f for the two possible input states 0 and 1, a one-bit classical computer needs to evaluate the function twice, once for every input state. In contrast, a one- qubit quantum computer can have a superposition of |0〉 and |1〉 as an input, to end up in a superposition of the two output values, F |0〉 and F |1〉. It has taken only half the number of steps as its classical counterpart. (b) Similarly, a two-qubit quantum computer needs only a quarter of the number of steps that are required classically. The computing power of a quantum computer scales exponentially with the number of qubits, for a classical computer the scaling is only linear 30 J.M. Elzerman et al. ee e 2DEG back gatehigh-g layer B Bac e Fig. 2. Schematic picture of the spin qubit as proposed by Loss and DiVincenzo [2]. The array of metal electrodes on top of a semiconductor heterostructure, containing a two-dimensional electron gas (2DEG) below the surface, defines a number of quan- tum dots (dotted circles), each holding a single electron spin (arrow). A magnetic field, B, induces a Zeeman splitting between the spin-up and spin-down states of each electron spin. The spin state is controlled either via an oscillating magnetic field, Bac (on resonance with the Zeeman splitting), or via an oscillating electric field created with the back gates, which can pull the electron wavefunction into a layer with a large g-factor. Coupling between two spins is controlled by changing the voltage on the electrodes between the two dots (Adapted from [2]) very controllable and versatile systems, which can be manipulated and probed electrically. Increasing the number of dots is straightforward, by simply adding more electrodes. Tuning all these gate voltages allows control of the number of electrons trapped on each dot, as well as the tunnel coupling between the dots. With the external magnetic field, B, we can tune the Zeeman splitting, ∆EZ = gµBB, where g ≈ −0.44 is the g-factor of GaAs, and µB = 9.27×10−24 J/T is the Bohr magneton. In this way, we can control the energy levels of the qubit. To perform single-qubit operations, different techniques are available. We can apply a microwave magnetic field on resonance with the Zeeman splitting, i.e. with a frequency f = ∆EZ/h, where h is Planck’s constant. The oscillat- ing magnetic component perpendicular to the static magnetic field B results in a spin nutation. By applying the oscillating field for a fixed duration, a su- perposition of | ↑〉 and | ↓〉 can be created. This magnetic technique is known as electron spin resonance (ESR). A completely electrical alternative might be the emerging technique of g- tensor modulation [20]. In this scheme, an oscillating electric field is created by modulating the voltage applied to a (back) gate. The electric field does not couple to the spin directly, but it can push or pull the electron wavefunction somewhat into another semiconductor layer with a different g-factor. This procedure modulates the effective g-tensor felt by the electron spin. If the modulation frequency is resonant with the Zeeman splitting, the required spin nutation results and superpositions of spin states can again be created. Semiconductor Few-Electron Quantum Dots as Spin Qubits 31 Two-qubit operations can be carried out purely electrically, by varying the gate voltages that control the potential barrier between two dots. It has been shown [2] that the system of two electron spins on neighboring dots, coupled via a tunnel barrier, can be mapped onto the Heisenberg exchange Hamiltonian H = JS1 ·S2. This Hamiltonian describes an indirect interaction between the two spins, S1 and S2, mediated by the exchange interaction, J , which depends on the wavefunction overlap of the electrons. By lowering the tunnel barrier for some time and then raising it again, the effective spin-spin interaction is temporarily turned on. In this way, the two electron spins can be swapped or even entangled. Together with arbitrary single-spin rotations, the exchange interaction can be used to construct a universal set of quantum gates [2]. A last crucial ingredient is a method to read out the state of the spin qubit. This implies measuring the spin orientation of a single electron – a daunting task, since the electron spin magnetic moment is exceedingly small. Therefore, an indirect spin measurement is proposed [2]. First the spin orientation of the electron is correlated with its position, via “spin-to-charge conversion”. Then an electrometer is used to measure the position of the charge, thereby revealing its spin. In this way, the problem of measuring the spin orientation has been replaced by the much easier measurement of charge. The essential advantage of using the electron’s spin degree of freedom to encode a qubit, lies in the fact that the spin is disturbed only weakly by the environment. The main source of spin decoherence and relaxation is predicted to be the phonon bath, which is coupled to the spin via the (weak) spin-orbit interaction [21, 22, 23]. In addition, fluctuations in the nuclear- spin configuration couple to the electron spin via the (even weaker) hyperfine coupling [21, 24]. In contrast, the electron’s charge degree of freedom is much easier to manipulate and read out, but it is coupled via the strong Coulomb interaction to charge fluctuations, which are the source of the ubiquitous 1/f noise in the “dirty” semiconductor environment. This leads to typical charge decoherence times of a few nanoseconds [25, 26]. The spin decoherence and relaxation times are predicted to be about four orders of magnitude longer [22]. Finally, it should be stressed that our efforts to create a spin qubit are not purely application-driven. Aside from the search for a spin quantum com- puter, many interesting questions await exploration. If we have the ability to (coherently) control and read out a single electron spin in a quantum dot, this spin could be used as a local probe of the semiconductor environment. This could shed light for instance on many details of the spin-orbit interaction or the hyperfine coupling. 1.4 Quantum Dots In this paragraph, the properties of semiconductor quantum dots are described in more detail [27]. In essence, a quantum dot is simply a small box that can be filled with electrons. The box is coupled via tunnel barriers to a source 32 J.M. Elzerman et al. Vg SOURCE DRAIN GATE lateral quantum dot e SOURCE DRAIN vertical quantum dot a b VSD I Fig. 3. Schematic picture of a quantum dot in a lateral (a) and a vertical (b) geometry. The quantum dot (represented by a disk) is connected to source and drain contacts via tunnel barriers, allowing the current through the device, I, to be measured in response to a bias voltage, VSD and a gate voltage, Vg and drain reservoir, with which particles can be exchanged (see Fig. 3). By attaching current and voltage probes to these reservoirs, we can measure the electronic properties of the dot. The box is also coupled capacitively to one or more “gate” electrodes, which can be used to tune the electrostatic potential of the dot with respect to the reservoirs. When the size of the box is comparable to the wavelength of the electrons that occupy it, the system exhibits a discrete energy spectrum, resembling that of an atom. As a result, quantum dots behave in many ways as artificial atoms. Because a quantum dot is such a general kind of system, there exist quan- tum dots of many different sizes and materials: for instance single molecules trapped between electrodes, metallic or superconducting nanoparticles, self- assembled quantum dots, semiconductor lateral or vertical dots, and even semiconducting nanowires or carbon nanotubes between closely spaced elec- trodes. In this work, we focus on lateral (gated) semiconductor quantum dots. These lateral devices allow all relevant parameters to be controlled in the fab- rication process, or tuned in situ. Fabrication of gated quantum dots starts with a semiconductor het- erostructure, a sandwich of different layers of semiconducting material (see Fig. 4a). These layers, in our case GaAs and AlGaAs, are grown on top of each other using molecular beam epitaxy (MBE), resulting in very clean crystals. By doping the n-AlGaAs layer with Si, free electrons are introduced. These ac- cumulate at the interface between GaAs and AlGaAs, typically 100 nm below the surface, forming a two-dimensional electron gas (2DEG) – a thin (10 nm) sheet of electrons that can only move along the interface. The 2DEG can have a high mobility and relatively low electron density (typically 105–106 cm2/Vs and ∼3 × 1015 m−2, respectively). The low electron density results in a large Fermi wavelength (∼40 nm) and a large screening length, which allows us to locally deplete the 2DEG with an electric field. This electric field is created Semiconductor Few-Electron Quantum Dots as Spin Qubits 35 the discrete energy spectrum is independent of the number of electrons on the dot. Under these assumptions the total energy of a N -electron dot with the source-drain voltage, VSD, applied to the source (and the drain grounded), is given by U(N) = [−|e|(N − N0) + CSVSD + CgVg]2 2C + N∑ n=1 En(B) (1) where −|e| is the electron charge and N0 the number of electrons in the dot at zero gate voltage, which compensates the positive background charge originating from the donors in the heterostructure. The terms CSVSD and CgVg can change continuously and represent the charge on the dot that is induced by the bias voltage (through the capacitance CS) and by the gate voltage Vg (through the capacitance Cg), respectively. The last term of (1) is a sum over the occupied single-particle energy levels En(B), which are separated by an energy ∆En = En−En−1. These energy levels depend on the characteristics of the confinement potential. Note that, within the CI model, only these single-particle states depend on magnetic field, B. To describe transport experiments, it is often more convenient to use the electrochemical potential. This is defined as the energy required to add an electron to the quantum dot: µ(N) ≡ U(N)−U(N − 1) = ( N − N0 − 1 2 ) EC − EC |e| (CSVSD +CgVg)+EN where EC = e2/C is the charging energy. The electrochemical potential for different electron numbers N is shown in Fig. 7a. The discrete levels are spaced by the so-called addition energy: Eadd(N) = µ(N + 1) − µ(N) = EC + ∆E . (2) The addition energy consists of a purely electrostatic part, the charging energy EC , plus the energy spacing between two discrete quantum levels, ∆E. Note that ∆E can be zero, when two consecutive electrons are added to the same spin-degenerate level. Of course, for transport to occur, energy conservation needs to be satisfied. This is the case when an electrochemical potential level falls within the “bias window” between the electrochemical potential (Fermi energy) of the source (µS) and the drain (µD), i.e. µS ≥ µ ≥ µD with −|e|VSD = µS − µD. Only then can an electron tunnel from the source onto the dot, and then tunnel off to the drain without losing or gaining energy. The important point to realize is that since the dot is very small, it has a very small capacitance and therefore a large charging energy – for typical dots EC ≈ a few meV. If the electrochemical potential levels are as shown in Fig. 7a, this energy is not available (at low temperatures and small bias voltage). So, the number of 36 J.M. Elzerman et al. µS µD µ( -1)N µ( )N µ( 1)N+ ΓL µ( )N µ( 1)N+ ΓR µ( )N µ( 1)N+ µ( )N a b c d ∆E Eadd eV S D Fig. 7. Schematic diagrams of the electrochemical potential of the quantum dot for different electron numbers. (a) No level falls within the bias window between µS and µD, so the electron number is fixed at N − 1 due to Coulomb blockade. (b) The µ(N) level is aligned, so the number of electrons can alternate between N and N − 1, resulting in a single-electron tunneling current. The magnitude of the current depends on the tunnel rate between the dot and the reservoir on the left, ΓL, and on the right, ΓR. (c) Both the ground-state transition between N − 1 and N electrons (black line), as well as the transition to an N -electron excited state (gray line) fall within the bias window and can thus be used for transport (though not at the same time, due to Coulomb blockade). This results in a current that is different from the situation in (b). (d) The bias window is so large that the number of electrons can alternate between N −1, N and N +1, i.e. two electrons can tunnel onto the dot at the same time electrons on the dot remains fixed and no current flows through the dot. This is known as Coulomb blockade. Fortunately, there are many ways to lift the Coulomb blockade. First, we can change the voltage applied to the gate electrode. This changes the electrostatic potential of the dot with respect to that of the reservoirs, shifting the whole “ladder” of electrochemical potential levels up or down. When a level falls within the bias window, the current through the device is switched on. In Fig. 7b µ(N) is aligned, so the electron number alternates between N −1 and N . This means that the Nth electron can tunnel onto the dot from the source, but only after it tunnels off to the drain can another electron come onto the dot again from the source. This cycle is known as single-electron tunnelling. By sweeping the gate voltage and measuring the current, we obtain a trace as shown in Fig. 8a. At the positions of the peaks, an electrochemical potential level is aligned with the source and drain and a single-electron tunnelling current flows. In the valleys between the peaks, the number of electrons on the dot is fixed due to Coulomb blockade. By tuning the gate voltage from one valley to the next one, the number of electrons on the dot can be precisely controlled. The distance between the peaks corresponds to EC +∆E, and can therefore give information about the energy spectrum of the dot. A second way to lift Coulomb blockade is by changing the source-drain voltage, VSD (see Fig. 7c). (In general, we keep the drain potential fixed, and change only the source potential.) This increases the bias window and also “drags” the electrochemical potential of the dot along, due to the capacitive coupling to the source. Again, a current can flow only when an electrochemical Semiconductor Few-Electron Quantum Dots as Spin Qubits 37 Gate voltage C ur re nt N N+1 N+2N-1 B ia s vo lta ge a b E ad d ∆ E Gate voltage N-1 N N+1 Fig. 8. Transport through a quantum dot. (a) Coulomb peaks in current versus gate voltage in the linear-response regime. (b) Coulomb diamonds in differential conduc- tance, dI/dVSD, versus VSD and Vg, up to large bias. The edges of the diamond- shaped regions (black) correspond to the onset of current. Diagonal lines emanating from the diamonds (gray) indicate the onset of transport through excited states potential level falls within the bias window. By increasing VSD until both the ground state as well as an excited state transition fall within the bias window, an electron can choose to tunnel not only through the ground state, but also through an excited state of the N -electron dot. This is visible as a change in the total current. In this way, we can perform excited-state spectroscopy. Usually, we measure the current or differential conductance while sweeping the bias voltage, for a series of different values of the gate voltage. Such a measurement is shown schematically in Fig. 8b. Inside the diamond-shaped region, the number of electrons is fixed due to Coulomb blockade, and no current flows. Outside the diamonds, Coulomb blockade is lifted and single- electron tunnelling can take place (or for larger bias voltages even double- electron tunnelling is possible, see Fig. 7d). Excited states are revealed as changes in the current, i.e. as peaks or dips in the differential conductance. From such a “Coulomb diamond” the excited-state splitting as well as the charging energy can be read off directly. The simple model described above explains successfully how quantisation of charge and energy leads to effects like Coulomb blockade and Coulomb oscillations. Nevertheless, it is too simplified in many respects. For instance, the model considers only first-order tunnelling processes, in which an electron tunnels first from one reservoir onto the dot, and then from the dot to the other reservoir. But when the tunnel rate between the dot and the leads, Γ , is increased, higher-order tunnelling via virtual intermediate states becomes im- portant. Such processes are known as “cotunnelling”. Furthermore, the simple model does not take into account the spin of the electrons, thereby excluding for instance exchange effects. Also the Kondo effect, an interaction between the spin on the dot and the spins of the electrons in the reservoir, cannot be accounted for. 40 J.M. Elzerman et al. T- S T0 T+ N = 1 N = 2 ∆EZ ∆EZ ∆EZ T+ T0 / / T0 T- S S ∆EZ ∆EZ EST E ne rg y EST a b ↑↔ ↑↔ ↓↔ ↓↔ ↑↔ ↓↔ N = 1 2↔ E le ct ro ch em ic al p ot en tia l Fig. 10. One- and two-electron states and transitions at finite magnetic field. (a) En- ergy diagram for a fixed gate voltage. By changing the gate voltage, the one-electron states (below the dashed line) shift up or down relative to the two-electron states (above the dashed line). The six transitions that are allowed (i.e. not spin-blocked) are indicated by vertical arrows. (b) Electrochemical potentials for the transitions between one- and two-electron states. The six transitions in (a) correspond to only four different electrochemical potentials. By changing the gate voltage, the whole ladder of levels is shifted up or down tunnelling between the dots, with tunnelling matrix element t, ε0 (the “bond- ing state”) and ε1 (the “anti-bonding state”) are split by an energy 2t. By filling the two states with two electrons, we again get a spin singlet ground state and a triplet first excited state (at zero field). However, the singlet ground state is not purely S (Fig. 9a), but also contains a small admixture of the excited singlet S2 (Fig. 9f). The admixture of S2 depends on the compe- tition between inter-dot tunnelling and the Coulomb repulsion, and serves to lower the Coulomb energy by reducing the double occupancy of the dots [33]. If we focus only on the singlet ground state and the triplet first excited states, then we can describe the two spins S1 and S2 by the Heisenberg Hamiltonian, H = JS1 ·S2. Due to this mapping procedure, J is now defined as the energy difference between the triplet state T0 and the singlet ground state, which depends on the details of the double dot orbital states. From a Hund-Mulliken calculation [34], J is approximately given by 4t2/U +V , where U is the on-site charging energy and V includes the effect of the long-range Coulomb interaction. By changing the overlap of the wavefunctions of the two electrons, we can change t and therefore J . Thus, control of the inter- dot tunnel barrier would allow us to perform operations such as swapping or entangling two spins. Semiconductor Few-Electron Quantum Dots as Spin Qubits 41 1.7 Measurement Setup Dilution Refrigerator To resolve small energies such as the Zeeman splitting, the sample has to be cooled down to temperatures well below a Kelvin. We use an Oxford Kelvi- nox 300 dilution refrigerator, which has a base temperature of about 10 mK, and a cooling power in excess of 300 µW (at 100 mK). The sample holder is connected to a cold finger and placed in a copper can (36 mm inner diameter) in the bore of a superconducting magnet that can apply a magnetic field up to 16 T. Measurement Electronics A typical measurement involves applying a source-drain voltage over (a part of) the device, and measuring the resulting current as a function of the volt- ages applied to the gates. The electrical circuits for the voltage-biased current measurement and for applying the gate voltages are shown in Fig. 11 and Fig. 12, respectively. The most important parts of the measurement electron- ics – i.e. the current-to-voltage (IV) convertor, isolation amplifier, voltage source and digital-to-analog convertors (DACs) – were all built by Raymond Schouten at Delft University. The underlying principle of the setup is to isolate the sample electrically from the measurement electronics. This is achieved via optical isolation at both sides of the measurement chain, i.e. in the voltage source, the isolation amplifier, as well as the DACs. In all these units, the electrical signal passes through analog optocouplers, which first convert it to an optical signal using an LED, and then convert the optical signal back using a photodiode. In this way, there is no galvanic connection between the two sides. In addition, all circuitry at the sample side is analog (even the DACs have no clock circuits or microprocessors), battery-powered, and uses a single clean ground (connected to the metal parts of the fridge) which is separated from the ground used by the “dirty” electronics. All these features help to eliminate ground loops and reduce interference on the measurement signal. Measurements are controlled by a computer running LabView. It sends commands via a fiber link to two DAC-boxes, each containing 8 digital-to- analog convertors, and powered by a specially shielded transformer. Most of the DACs are used to generate the voltages applied to the gate electrodes (typically between 0 and −5 V). One of the DACs controls the source-drain voltage for the device. The output voltage of this DAC (typically between +5 and −5 V) is sent to a voltage source, which attenuates the signal by a factor 10, 102, 103 or 104 and provides optical isolation. The attenuated voltage is then applied to one of the ohmic contacts connected to the source reservoir of the device. 42 J.M. Elzerman et al. O X F O R D K E L V IN O X 30 0 0.4 nF 0.4 nF 0.5 nF 0.5 nF 0.5 nF 0.5 nF 25 0 Ω 25 0 Ω tw is te d p a ir tw is te d p a ir po w de r fil te r po w de r fil te r 0.4 nF 0.4 nF (20 nF) (20 nF) BASE-T ROOM-T sample clean groundS G N G N D G N D S G N 0.22 nF G N D S G N G N D S G N 10 MΩ 100 M 1 G Ω Ω x10 4 DMM R F B SAMPLEx1 fib er G P IB computer computer ELECTRONICS cold ground co ld fin ge r IV co nv er te r V so ur ce 100 V/ V 1 mV/ V 10 mV/ V 100 mV/ V µ DAC 2 IS O am p connector box Fig. 11. Electrical circuit for performing a voltage-biased current measurement. Elements shown in gray are connected to ground. Gray lines indicate the shielding of the measurement electronics and wires Semiconductor Few-Electron Quantum Dots as Spin Qubits 45 the sample heat up due to spurious noise and interference. Several filtering stages are required for different frequency ranges (see Fig. 11 and Fig. 12). In the connector box at room temperature, all wires are connected to ground via 0.22 nF “feedthrough capacitors”. At base temperature, all signal wires run through “copper powder filters” [35]. These are copper tubes filled with cop- per powder, in which 4 signal wires with a length of about 2 meters each are wound. The powder absorbs the high-frequency noise very effectively, leading to an attenuation of more than −60 dB from a few 100 MHz up to more than 50 GHz [36]. To remove the remaining low-frequency noise, we solder a 20 nF capacitor between each signal wire and the cold finger ground. In combination with the ∼100 Ω resistance of the wires, this forms a low-pass RC filter with a cut-off frequency of about 100 kHz (even 10 kHz for the wire connected to the IV convertor, due to its input resistance of about 1.3 kΩ). These filters are used for the wires connecting to ohmic contacts (although they were taken out to perform some of the high-bandwidth measurements described in this work). For the wires connecting to gate electrodes, a 1:3 voltage divider is present (consisting of a 20 MΩ resistance in the signal line and a 10 MΩ resistance to ground). In this way, the gate voltages are filtered by a low-pass RC filter with a cut-off frequency of about 1 Hz. By combining all these filters, the electrons in the sample can be cooled to an effective temperature below 50 mK (if no extra heat loads such as coaxial cables are present). High-Frequency Signals High-frequency signals can be applied to gate electrodes via two coaxial cables. They consist of three parts, connected via standard 2.4 mm Hewlett Packard connectors (specified up to 50 GHz). From room temperature to 1 Kelvin, a 0.085 inch semi-rigid Be-Cu (inner and outer conductor) coaxial cable is used. From 1 Kelvin to the mixing chamber, we use 0.085 inch semi-rigid superconducting Nb. From the mixing chamber to the sample holder, flexible tin plated Cu coaxial cables are present. The coaxes are thermally anchored at 4 K, 1 K, ∼800 mK, ∼100 mK and base temperature, by clamping each cable firmly between two copper parts. To thermalize also the inner conductor of the coax, we use Hewlett Packard 8490D attenuators (typically −20 dB) at 1 K. These attenuators cannot be used at the mixing chamber, as they tend to become superconducting below about 100 mK. We have also tried using Inmet 50EH attenuators at the mixing chamber, but these showed the same problem. To generate the high-frequency signals, we use a microwave source (Hewlett Packard 83650A) that goes up to 50 GHz (or 75 GHz, in combination with a “frequency doubler”); a pulse generator (Hewlett Packard 8133A), which gen- erates simple 10 ns to 1 µs pulses with a rise time of 60 ps; and an arbitrary 46 J.M. Elzerman et al. waveform generator (Sony Tektronix AWS520), which can generate more com- plicated pulses with a rise time of 1.5 ns. With the cables described above, the fastest pulse flank we can transmit to the sample is about 200 ps. Microwave signals are transmitted with about 10 dB loss at 50 GHz. Special care needs to be given to the connection from the coaxial cable to the chip, in order to minimize reflections. The sample holder we use, has an SMA connector that can be connected to the 2.4 mm coaxial cable. At the other end, the pin of the SMA connector sticks through a small hole in the chip carrier. This allows it to be soldered to a metal pad on the chip carrier, from which we can then bond to the chip. This sample holder is used to apply pulses or microwave signals to a gate electrode. 1.8 Sample Stability A severe experimental difficulty that is not related to the measurement setup, but to the sample itself, is the problem of “charge switching”. It shows up in measurements as fluctuations in the position of a Coulomb peak, or as sudden jumps in the QPC-current that are not related to charging or discharging of a nearby quantum dot. Generally, these switches are attributed to (deep) traps in the donor layer that capture or release an electron close to the quantum dot [37]. This well-known but poorly understood phenomenon is a manifesta- tion of 1/f noise in semiconductors, which causes the electrostatic potential landscape in the 2DEG to fluctuate. The strength of the fluctuations can differ enormously. In some samples, switching occurs on a time scale of seconds, making only the most trivial measurements possible, whereas in other samples, no switches are visible on a time scale of hours. It is not clear what exactly determines the stability. It certainly depends on the heterostructure, as some wafers are clearly better than others. A number of growth parameters could be important, such as the Al concentration in the AlGaAs, the doping density and method (modulation doping or delta doping), the thickness of the spacer layer between the n- AlGaAs and GaAs, the depth of the 2DEG below the surface, a possible surface layer, and many more. We have recently started a collaboration with the group of Professor Wegscheider in Regensburg to grow and characterize heterostructures in which some of these parameters are systematically varied. In this way we hope to find out what makes certain heterostructures stable. Even for the same heterostructure, some samples are more quiet than others. The reasons for this are not clear. There are reports that stability is improved if the sample is cooled down slowly, while applying a positive voltage (about +280 mV) on all gates that are going to be used in the experiment. This procedure effectively “freezes in” a negative charge around the gates, such that less negative gate voltages are sufficient to define the quantum dot at low temperatures. Most samples described in this work have been cooled down from room temperature to 4 K slowly (in one to two days) with all gates Semiconductor Few-Electron Quantum Dots as Spin Qubits 47 grounded. We find that in general samples get more quiet during the first week of applying the gate voltages. Finally, sample stability also involves an element of luck: Fig. 13 shows two Coulomb diamonds that were measured im- mediately after each other under identical conditions. Measurement Fig. 13a is reasonably quiet, but in Fig. 13b the effects of an individual two-level fluc- tuator are visible. This particular fluctuator remained active for a week, until the sample was warmed up. 2 1 0 2 1 0 -1 -1 -2 -2 -0.725 -0.725-0.735 -0.735-0.730 -0.730 Gate voltage (V) Gate voltage (V) B ia s vo lta ge (m V ) B ia s vo lt a ge (m V ) a b switching dI dV/ (arb. units)SD -2 -1 1 20 dI dV/ (arb. units)SD -2 -1 1 20 Fig. 13. Charge switching in a large-bias measurement in the few-electron regime, for B = 12 T. (a) Differential conductance, dI/dVSD (in grayscale), as a function of bias voltage and gate voltage. This measurement is considered reasonably stable. (b) Identical measurement, taken immediately after (a). A single two-level fluctuator has become active, causing the effective gate voltage to fluctuate between two values at any position in the figure, and leading to an apparent splitting of all the lines. This is considered a measurement of poor stability Switching has made all experiments we performed more difficult, and has made some experiments that we wanted to perform impossible. Better control over heterostructure stability is therefore essential for the increasingly difficult steps towards creating a quantum dot spin qubit. 2 Few-Electron Quantum Dot Circuit with Integrated Charge Read-Out In this section, we report on the realization of few-electron double quantum dots defined in a two-dimensional electron gas by means of surface gates on top of a GaAs/AlGaAs heterostructure. The double quantum dots are flanked 50 J.M. Elzerman et al. 200 nm200 nmc a b 500 nm Fig. 14. Few-electron quantum dot devices. (a) Scanning electron microscope im- age of the first sample, showing the metal gate electrodes (light) on top of a GaAs/AlGaAs heterostructure (dark) that contains a 2DEG 90nm below the surface (with electron density 2.9× 1011 cm−2). This device was used only as a few-electron single dot. Due to the similarity of the image to characters from the Japanese “Gun- dam” animation, this has become known as the Gundam design. The two gates coming from the top and ending in small circles (the “eyes”) were meant to make the dot confinement potential steeper, by applying a positive voltage to them (up to ∼0.5 V). The gates were not very effective, and were left out in later designs. (The device was fabricated by Wilfred van der Wiel at NTT Basic Research Lab- oratories.) (b) Scanning electron microscope image of the second device, made on a similar heterostructure. It was used only as a few-electron single dot, and was more easily tunable than the first one. (The device was fabricated by Wilfred van der Wiel and Ronald Hanson at NTT Basic Research Laboratories.) (c) Atomic force microscope image of the third device, made on a similar heterostructure. This design, with two extra side gates to form two quantum point contacts, was operated many times as a single dot, and twice as a few-electron double dot. It was used for all subsequent measurements. A zoom-in of the gate structure is shown in Fig. 16a. (The device was fabricated by Ronald Hanson and Laurens Willems van Beveren at NTT Basic Research Laboratories) Semiconductor Few-Electron Quantum Dots as Spin Qubits 51 0 5 10 -5 0 2 -2 Vg (V) V S D (m V ) dI /d V e h S D ( / ) 2 V S D (m V ) VSD(mV) k T eB K/ 0 0.2-0.2-1.0-0.5 N=1 N=0 -0.10.4 a b Vg (V) 0.2 0.3 c N=0 Fig. 15. Kondo effect in a one-electron lateral quantum dot of the type shown in Fig. 14a. (a) Differential conductance (in grayscale) versus source-drain voltage, VSD, and plunger gate voltage, Vg. In the white diamond and the white region to the right (indicated by N = 1 and N = 0, respectively), no current flows due to Coulomb blockade. The N = 0 region opens up to more than 10 mV, indicating that the dot is really empty here. (b) Close-up of the N = 1 diamond for stronger coupling to the reservoirs. A sharp Kondo resonance is visible at zero source-drain voltage. Although charge switching is very severe in this sample, the position of the Kondo resonance is very stable, as it is pinned to the Fermi energy of the reservoirs. (c) Kondo zero-bias peak in differential conductance, taken at the position indicated by the dotted line in (b) PR, are used to change the electrostatic potential of the left and right dot, respectively. The left plunger gate is connected to a coaxial cable, so that we can apply high-frequency signals. In the present experiments we do not apply dc voltages to PL. In order to control the number of electrons on the double dot, we use gate L for the left dot and PR or R for the right dot. All measurements are performed with the sample cooled to a base temperature of about 10 mK inside a dilution refrigerator. We first study sample 1. The individual dots are characterized using stan- dard Coulomb blockade experiments [27], i.e. by measuring IDOT . We find that the energy cost for adding a second electron to a one-electron dot is 3.7 meV. The one-electron excitation energy (i.e. the difference between the ground state and the first orbital excited state) is 1.8 meV at zero magnetic field. For a two-electron dot the energy difference between the spin singlet ground state and the spin triplet excited state is 1.0 meV at zero magnetic field. Increasing the field (perpendicular to the 2DEG) leads to a transition from a singlet to a triplet ground state at about 1.7 Tesla. 52 J.M. Elzerman et al. 2.3 Quantum Point Contact as Charge Detector As an alternative to measuring the current through the quantum dot, we can also measure the charge on the dot using one of the QPCs [44, 45]. To demonstrate this functionality, we first define only the left dot (by grounding gates R and PR), and use the left QPC as a charge detector. The QPC is formed by applying negative voltages to Q − L and L. This creates a narrow constriction in the 2DEG, with a conductance, G, that is quantized when sweeping the gate voltage VQ−L. The last plateau (at G = 2e2/h) and the transition to complete pinch-off (i.e. G = 0) are shown in Fig. 16b. We tune the QPC to the steepest point (G ≈ e2/h), where the QPC-conductance has a maximum sensitivity to changes in the electrostatic environment, including changes in the charge of the nearby quantum dot. To change the number of electrons in the left dot, we make gate volt- age VM more negative (see Fig. 16c). This reduces the QPC current, due to the capacitive coupling from gate M to the QPC constriction. In addition, the changing gate voltage periodically pushes an electron out of the dot. The as- sociated sudden change in charge lifts the electrostatic potential at the QPC constriction, resulting in a step-like feature in IQPC (see the expansion in Fig. 16c, where the linear background is subtracted). This step indicates a change in the electron number. So, even without passing current through the dot, IQPC provides information about the charge on the dot. To enhance the charge sensitivity we apply a small modulation (0.3 mV at 17.7 Hz) to VM and use lock-in detection to measure dIQPC/dVM [45]. The steps in IQPC now appear as dips in dIQPC/dVM . Figure 16d shows the resulting dips, as well as the corresponding Coulomb peaks measured in the current through the dot. The coincidence of the Coulomb peaks and dips demonstrates that the QPC indeed functions as a charge detector. From the height of the step in Fig. 16c (∼50 pA, typically 1–2% of the total current) compared to the noise (∼5 pA for a measurement time of 100 ms), we estimate the sensitivity of the charge detector to be about 0.1e, with e being the single electron charge. The unique advantage of QPC charge detection is that it provides a signal even when the tunnel barriers of the dot are so opaque that IDOT is too small to be measured [44, 45]. This allows us to study quantum dots even when they are virtually isolated from the reservoirs. 2.4 Double Dot Charge Stability Diagram The QPC can also detect changes in the charge configuration of the double dot. To demonstrate this, we use the QPC on the right to measure dIQPC/dVL versus VL and VPR (Fig. 17a), where VL controls (mainly) the number of electrons on the left dot, and VPR (mainly) that on the right. Dark lines in the figure signify a dip in dIQPC/dVL, corresponding to a change in the total number of electrons on the double dot. Together these lines form the so-called “honeycomb diagram” [46, 47]. The almost-horizontal lines correspond to a Semiconductor Few-Electron Quantum Dots as Spin Qubits 55 the modulation to the gate on the left. When an electron is pushed out of the double dot by making VL more negative, the QPC opens up and dIQPC/dVL displays a dip. When VL pushes an electron from the left to the right dot, the QPC is closed slightly, resulting in a peak.) The visibility of all lines in the honeycomb pattern demonstrates that the QPC is sufficiently sensitive to detect even inter-dot transitions. 2.5 Tunable Tunnel Barriers in the Few-Electron Regime In measurements of transport through lateral double quantum dots, the few- electron regime has never been reached [47]. The problem is that the gates that are used to deplete the dots also strongly influence the tunnel barriers. Reducing the electron number would therefore always lead to the Coulomb peaks becoming unmeasurably small, but not necessarily due to an empty dou- ble dot. The QPC detectors now permit us to compare charge and transport measurements. Figure 18a shows the current through the double dot in the same region as shown in Fig. 17b. In the bottom left region the gates are not very negative, hence the tunnel barriers are quite open. Here the resonant current at the charge transition points is quite high (∼100 pA, dark gray), and lines due to cotunnelling are also visible [47]. Towards the top right corner the gate voltages become more negative, thereby closing off the barriers and reducing the current peaks (lighter gray). The last “triple points” [47] that are visible (<1 pA) are shown in the dashed square. Using the dotted lines, extracted from the measured charge transition lines in Fig. 17b, we label the various regions in the figure according to the charge configuration of the double dot. Apart from a small shift, the dotted lines correspond nicely to the regions where a transport current is visible. This allows us to be confident that the triple points in the dashed square are really the last ones before the double quantum dot is empty. We are thus able to measure transport through a one-electron double quantum dot. Even in the few-electron regime, the double dot remains fully tunable. By changing the voltage applied to gate T , we can make the tunnel barriers more transparent, leading to a larger current through the device. We use this procedure to increase the current at the last set of triple points. For the gate voltages used in Fig. 18b, the resonant current is very small (<1 pA), and the triple points are only faintly visible. By making VT less negative, the resonant current peaks grow to about 5 pA (Fig. 18c). The two triple points are clearly resolved and the cotunnelling current is not visible. By changing VT even more, the current at the last triple points can be increased to ∼70 pA (Fig. 18d). For these settings, the triple points have turned into lines, due to the increased cotunnelling current. This sequence demonstrates that we can tune the few- electron double dot from being nearly isolated from the reservoirs, to being very transparent. 56 J.M. Elzerman et al. 00 10 01 11 22 21 12 a b c d -1.02 -1.00 -0.98 -0.96 -0.15 -0.20 -0.25 -0.30 VPR (V) V L (V ) Fig. 18. Current through the double quantum dot in the few-electron regime. (a) IDOT (in logarithmic grayscale) versus VL and VPR in the same region as shown in Fig. 17b, with VDOT = 100 µV and VSD1 = VSD2 = 0. Dotted lines are extracted from Fig. 17b. Dark gray indicates a current flowing, with the darkest regions (in the bottom left corner) corresponding to ∼100 pA. In the light gray regions current is zero due to Coulomb blockade. Inside the dashed square, the last triple points are faintly visible (∼1 pA). (A smoothly varying background current due to a small leakage current from a gate to the 2DEG has been subtracted from all traces.) (b) Close-up of the region inside the dashed square in (a), showing the last two triple points before the double dot is completely empty. The current at these triple points is very small (<1 pA) since the tunnel barriers are very opaque. (c) Same two triple points for different values of the voltage applied to the gates defining the tunnel barriers. For these settings, the two individual triple points are well resolved, with a height of about 5 pA. The cotunnelling current is not visible. (d) Same two triple points, but now with the gate voltages such that the tunnel barriers are very transparent. The current at the triple points is about 70 pA, and the cotunnelling current is clearly visible We can also control the inter-dot coupling, by changing the voltage applied to gate M . This is demonstrated with a QPC charge measurement (performed on sample 2). We apply a square wave modulation of 3 mV at 235 Hz to the rightmost plunger gate, PR, and measure dIQPC/dVPR using a lock-in am- plifier. Figure 19a shows the familiar honeycomb diagram in the few-electron regime. All lines indicating charge transitions are very straight, implying that for the gate settings used, the tunnnel-coupling between the two dots is negli- gible compared to the capacitive coupling. This is the so-called weak-coupling regime. (We note that the regular shape of the honeycomb pattern demon- strates that the double dot as a whole is still quite well-coupled to the leads, so that the total number of electrons can always find its lowest-energy value, unlike in [48].) By making VM less negative, the tunnel barrier between the Semiconductor Few-Electron Quantum Dots as Spin Qubits 57 00 -0.90 -1.05 VL (V) -0.90 -0.95 -1.00 -0.95 00 00 -0.85 -0.90 -0.95 VL (V) -0.85 -0.90 -1.05 -0.95 -1.10 V R (V ) V R (V ) -1.00 strong coupling intermediate couplingweak coupling a b c -1.00 -1.05 -1.10 VL (V) -1.00 a b c Fig. 19. Controlling the inter-dot coupling (in sample 2) with VM . These charge stability diagrams of the double quantum dot are measured using the QPC on the left. A small modulation (3 mV at 235 Hz) is applied to gate PR, and dIQPC/dVPR is measured with a lock-in amplifier and plotted in grayscale versus VL and VR. A magnetic field of 6 Tesla is applied in the plane of the 2DEG. (a) Weak-coupling regime. VM is such that all dark lines indicating charge transitions are straight. The tunnel-coupling between the two dots is therefore negligible compared to the capacitive coupling. (b) Intermediate-coupling regime. VM is 0.07 V less negative than in (a), such that lines in the bottom left corner are slightly curved. This signifies that here the inter-dot tunnel-coupling is comparable to the capacitive coupling. (c) Strong-coupling regime. VM is 0.1 V less negative than in (b), such that all lines are very curved. This implies that the tunnel-coupling is dominating over the capacitive coupling and the double dot behaves as a single dot 60 J.M. Elzerman et al. voltage [27]. This requires that the quantum dot be connected to two leads with a tunnel coupling large enough to obtain a measurable current [43]. Coupling to the leads unavoidably introduces decoherence of the qubit: even if the number of electrons on the dot is fixed due to Coulomb blockade, an electron can tunnel out of the dot and be replaced by another electron through a second-order tunnelling process, causing the quantum information to be irretrievably lost. Therefore, to optimally store qubits in quantum dots, higher-order tunnelling has to be suppressed, i.e. the coupling to the leads must be made as small as possible. Furthermore, real-time observation of electron tunnelling, important for single-shot read-out of spin qubits via spin- to-charge conversion, also requires a small coupling of the dot to the leads. In this regime, current through the dot would be very hard or even impossible to measure. Therefore an alternative spectroscopic technique is needed, which does not rely on electron transport through the quantum dot. Here we present spectroscopy measurements using charge detection. Our method resembles experiments on superconducting Cooper-pair boxes and semiconductor disks which have only one tunnel junction so that no net cur- rent can flow. Information on the energy spectrum can then be obtained by measuring the energy for adding an electron or Cooper-pair to the box, using a single-electron transistor (SET) operated as a charge detector [51, 52, 53]. We are interested in the excitation spectrum for a given number of electrons on the box, rather than the addition spectra. We use a quantum point con- tact (QPC) as an electrometer [44] and excitation pulses with repetition rates comparable to the tunnel rates to the lead, to measure the discrete energy spectrum of a nearly isolated one- and two-electron quantum dot. 3.2 Tuning the Tunnel Barriers The quantum dot and QPC are defined in the two-dimensional electron gas (2DEG) in a GaAs/Al0.27Ga0.73As heterostructure by dc voltages on gates T,M,R and Q (Fig. 21a). The dot’s plunger gate, P , is connected to a coaxial cable, to which we can apply voltage pulses (rise time 1.5 ns). The QPC charge detector is operated at a conductance of about e2/h with source-drain voltage VSD = 0.2 mV. All data are taken with a magnetic field B// = 10 T applied in the plane of the 2DEG, at an effective electron temperature of about 300 mK. We first describe the procedure for setting the gate voltages such that tunnelling in and out of the dot take place through one barrier only (i.e. the other is completely closed), and the remaining tunnel rate be well controlled. For gate voltages far away from a charge transition in the quantum dot, a pulse applied to gate P (Fig. 21b) modulates the QPC current via the cross- capacitance only (solid trace in Fig. 21c). Near a charge transition, the dot can become occupied with an extra electron during the high stage of the pulse (Fig. 21d). The extra electron on the dot reduces the current through the QPC. The QPC response to the pulse is thus smaller when tunnelling takes place Semiconductor Few-Electron Quantum Dots as Spin Qubits 61 200 nm M R Q T -VP time time a b P τ τ d Γ Γ 0 c EF R E S E R V O IR DRAIN SOURCE QPCI ∆I Q P C B// Fig. 21. QPC response to a pulse train applied to the plunger gate. (a) Scanning electron micrograph of a quantum dot and quantum point contact, showing only the gates used in the present experiment (the complete device is described in [55]) and Sect. 2. (b) Pulse train applied to gate P . (c) Schematic response in QPC current, ∆IQPC , when the charge on the dot is unchanged by the pulse (solid line) or increased by one electron charge during the “high” stage of the pulse (dashed). (d) Schematic electrochemical potential diagrams during the high (left) and low (right) pulse stage, when the ground state is pulsed across the Fermi level in the reservoir, EF (dotted trace in Fig. 21c). We denote the amplitude of the difference between solid and dotted traces as the “electron response”. Now, even when tunnelling is allowed energetically, the electron response is only non-zero when an electron has sufficient time to actually tunnel into the dot during the pulse time, τ . By measuring the electron response as a function of τ , we can extract the tunnel rate, Γ , as demonstrated in Fig. 22a. We apply a pulse train to gate P with equal up and down times, so the repetition rate is f = 1/(2τ) (Fig. 21b). The QPC response is measured using lock-in detection at frequency f [45], and is plotted versus the dc voltage on gate M . For long pulses (lowest curves) the traces show a dip, which is due to the electron response when crossing the zero-to-one electron transition. Here, f  Γ and tunnelling occurs quickly on the scale of the pulse duration. For shorter pulses the dip gradually disappears. We find analytically1 that the dip height is proportional to 1 − π2/(Γ 2τ2 + π2), so the dip height should equal half its maximum value when Γτ = π. From the data (inset to Fig. 22a), we find that this happens for τ ≈ 120 µs, giving Γ ≈ (40 µs)−1. Using this value 1 This expression is obtained by multiplying the probability that the dot is empty, P (t), with a sine-wave of frequency f (as is done in the lock-in amplifier), and averaging the resulting signal over one period. P (t) is given by exp(−Γt)(1 − exp(−Γτ))/(1−exp(−2Γτ)) during the high stage of the pulse, and by 1−P (t−τ) during the low stage. 62 J.M. Elzerman et al. -1.07 -1.40 -0.76 -0.96 f = 4.17 kHz b0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 τ = 15 µs 45 90 180 300 N = 1 N = 0 -1.13-1.12 VM (V) lo ck -in si gn al (a rb .u ni t s ) di p he ig ht (% ) 100 0 τ (µs) 3700 VM (V) a V R (V ) -1.07 -1.40 -0.76 -0.96 f = 41.7 Hz d0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 VM (V) V R (V ) -0.76 -0.96 f = 41.7 kHz c0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 V R (V ) Fig. 22. Lock-in detection of electron tunnelling. (a) Lock-in signal at f = 1/(2τ) versus VM for different pulse times, τ , with VP = 1 mV. The dip due to the electron response disappears for shorter pulses. (Individual traces have been lined up hori- zontally to compensate for a fluctuating offset charge, and have been given a vertical offset for clarity.) (Inset) Height of the dip versus τ , as a percentage of the maximum height (obtained at long τ). Circles: experimental data. Dashed lines indicate the pulse time (τ ≈ 120 µs) for which the dip size is half its maximum value. Solid line: calculated dip height using Γ = (40 µs)−1. (b) Lock-in signal in grayscale versus VM and VR for VP = 1mV and f = 4.17 kHz. Dark lines correspond to dips as in (a), indicating that the electron number changes by one. White labels indicate the absolute number of electrons on the dot. (c) Same plot as in (b), but with larger pulse repetition frequency (f = 41.7 kHz). (d) Same plot as in (b), but with smaller pulse repetition frequency (f = 41.7 Hz) for Γ in the analytical expression given above, we obtain the solid line in the inset to Fig. 22a, which nicely matches the measured data points. We explore several charge transitions in Fig. 22b, which shows the lock-in signal in grayscale for τ = 120 µs, i.e. f = 4.17 kHz. The slanted dark lines correspond to dips as in Fig. 22a. From the absence of further charge tran- sitions past the topmost dark line, we obtain the absolute electron number starting from zero. In the top left region of Fig. 22b, the right tunnel barrier (between gates R and T ) is much more opaque than the left tunnel barrier (between M and T ). Here, charge exchange occurs only with the left reservoir (indicated as “reservoir” in Fig. 21a). Conversely, in the lower right region Semiconductor Few-Electron Quantum Dots as Spin Qubits 65 N = 1N = 2 N = 1N = 2 f = 1.538 kHz -1.160 -1.175VM (V)-1.160 -1.175VM (V) 1 10 ∆EST f = 385 Hz V P (m V ) c Γ d Γeff EF a b S↔ S↔ Fig. 24. Excited state spectroscopy in a two-electron dot. (a) Similar to Fig. 23d, but for the one-to-two electron transition. Again, f = 385Hz. We clearly observe the singlet-triplet splitting ∆EST (individual traces in (a) and (b) have been lined up). (b) Same experiment but with f = 1.538 kHz, which increases the contrast for excited states. An extra slanted line appears (arrow), corresponding to the N = 1ES, spin-down. (c) Schematic electrochemical potential diagram for the case that only the spin-down electron can leave from the two-electron GS (spin singlet). This occurs to the left of the bright line indicated by the arrow in (b). (d) Idem when either the spin-up or the spin-down electron can leave from the spin singlet. This occurs to the right of the arrow in (b), and leads to a larger effective tunnel rate Excitations of the one-electron dot can be made visible at the one-to- two electron transition as well, by changing the pulse frequency to 1.538 kHz (Fig. 24b). This is too fast for electrons to tunnel if only the GS is accessible, so the rightmost line almost vanishes. However, a second slanted line becomes visible (indicated by the arrow in Fig. 24b), corresponding not to an increased tunnel rate into the dot (due to an N = 2 ES), but to an increased tunnel rate out of the dot (due to an N = 1 ES). Specifically, if the pulse amplitude is sufficiently large, either the spin-up or the spin-down electron can tunnel out of the two-electron dot. This is explained schematically in Fig. 24c and d. Similar experiments at the transition between two and three electrons, and for tunnel rates to the reservoir ranging from 12 Hz to 12 kHz, yield similar excitation spectra. The experiments described in this section demonstrate that an electrome- ter such as a QPC can reveal not only the charge state of a quantum dot, but also its tunnel coupling to the outside world and the energy level spectrum of 66 J.M. Elzerman et al. its internal states. We can thus access all the relevant properties of a quantum dot, even when it is almost completely isolated from the leads. 4 Real-Time Detection of Single Electron Tunnelling using a Quantum Point Contact In this section, we observe individual tunnel events of a single electron be- tween a quantum dot and a reservoir, using a nearby quantum point contact (QPC) as a charge meter. The QPC is capacitively coupled to the dot, and the QPC conductance changes by about 1% if the number of electrons on the dot changes by one. The QPC is voltage biased and the current is monitored with an IV-convertor at room temperature. At present, we can resolve tunnel events separated by only 8 µs, limited by noise from the IV-convertor. Shot noise in the QPC sets a 10 ns lower bound on the accessible timescales. 4.1 Charge Detectors Fast and sensitive detection of charge has greatly propelled the study of single-electron phenomena. The most sensitive electrometer known today is the single-electron transistor (SET) [56], incorporated into a radio-frequency resonant circuit [57]. Such RF-SETs can be used for instance to detect charge fluctuations on a quantum dot, capacitively coupled to the SET island [58, 59]. Already, real-time electron tunnelling between a dot and a reservoir has been observed on a sub-µs timescale [58]. A much simpler electrometer is the quantum point contact (QPC). The conductance, GQ, through the QPC channel is quantized, and at the tran- sitions between quantized conductance plateaus, GQ is very sensitive to the electrostatic environment, including the number of electrons, N , on a dot in the vicinity [44]. This property has been exploited to measure fluctuations in N in real-time, on a timescale from seconds [60] down to about 10 ms [61]. Here we demonstrate that a QPC can be used to detect single-electron charge fluctuations in a quantum dot in less than 10 µs, and analyze the fundamental and practical limitations on sensitivity and bandwidth. 4.2 Sample and Setup The quantum dot and QPC are defined in the two-dimensional electron gas (2DEG) formed at a GaAs/Al0.27Ga0.73As interface 90 nm below the surface, by applying negative voltages to metal surface gates (Fig. 25a). The device is attached to the mixing chamber of a dilution refrigerator with a base tempera- ture of 20 mK, and the electron temperature is ∼ 300 mK in this measurement. The dot is set near the N = 0 to N = 1 transition, with the gate voltages tuned such that the dot is isolated from the QPC drain, and has a small Semiconductor Few-Electron Quantum Dots as Spin Qubits 67 DRAIN R E S E R V O IR 200 nm T SOURCE a I M P R Q V RFB CL A filter ADC ISO-amp SN i V I I VT V IA R (t)Q ISO-amp room cold sample A Frequency (kHz) 1 100 1 rm s no is e (p A /H z ) 1 10 c b o 1/ 2 1/2 rm s noise (10 /H z ) e -3 IV-convertor shot noise load ref. temp. 01. 01. Fig. 25. Characterization of the experimental setup. (a) Scanning electron micro- graph of a device as used in the experiment (gates which are grounded are hidden). Gates T, M and R define the quantum dot (dotted circle), and gates R and Q form the QPC. Gate P is connected to a pulse source via a coaxial cable. See [55] for a more detailed description. (b) Schematic of the experimental set-up, including the most relevant noise sources. The QPC is represented by a resistor, RQ. (c) Noise spectra measured when the IV-convertor is connected to the sample (top solid trace), and, for reference, to an open-ended 1 m twisted pair of wires (lower solid trace). The latter represents a 300 pF load, if we include the 200 pF measured amplifier input capacitance. The diagram also shows the calculated noise level for the 300 pF reference load (dotted-dashed) and the shot noise limit (dashed). The left and right axes express the noise in terms of current through the QPC and electron charge on the dot respectively tunnel rate, Γ , to the reservoir. Furthermore, the QPC conductance is set at GQ = 1/RQ ≈ (30 kΩ)−1, roughly halfway the transition between GQ = 2e2/h and GQ = 0, where it is most sensitive to the electrostatic environment3. A schematic of the electrical circuit is shown in Fig. 25b. The QPC source and drain are connected to room temperature electronics by signal wires, which run through Cu-powder filters at the mixing chamber to block high fre- quency noise (>100 MHz) coming from room temperature. Each signal wire is twisted with a ground wire from room temperature to the mixing cham- ber. A voltage, Vi, is applied to the source via a home-built opto-coupled isolation stage. The current through the QPC, I, is measured via an IV- convertor connected to the drain, and an opto-coupled isolation amplifier, both 3 Despite a B = 10 T field in the plane of the 2DEG, no spin-split plateau is visible. 70 J.M. Elzerman et al. Time (ms) 10 2 3 4 ∆ I (n A ) 0 1 2 3 Time (ms) 10 2 3 4 0 1 2 ba Fig. 26. Measured changes in the QPC current, ∆I, with the electrochemical po- tential in the dot and in the reservoir nearly equal. ∆I is “high” and “low” for 0 and 1 electrons on the dot respectively (Vi = 1 mV; the steps in ∆I are ≈ 300 pA). Traces are offset for clarity. (a) The dot potential is lowered from top to bottom. (b) The tunnel barrier is lowered from top to bottom Time (ms) ∆I ( nA ) <∆ I> ( nA ) Time (ms) 0.50 1.0 Γ = 60 sµ-1 Γ = 230 sµ-1 pu ls e electron ba 1.5 0.50 1.0 1.5 0.4 0.8 0.0 1.2 -0.4 0.0 0.5 0.0 0.5 1.0 1.0 Fig. 27. QPC pulse response. (a) Measured changes in the QPC current, ∆I, when a pulse is applied to gate P , near the degeneracy point between 0 and 1 electrons on the dot (Vi = 1 mV). (b) Average of 286 traces as in (a). The top and bottom panel are taken with a different setting of gate M . The damped oscillation following the pulse edges is due to the 8th-order 40 kHz filter the time before tunnelling takes place is randomly distributed, and obtain a histogram of this time simply by averaging over many single-shot traces (Fig. 27b). The measured distribution decays exponentially with the tunnel time, characteristic of a Poisson process. The average time before tunnelling corresponds to Γ−1, and can be tuned by adjusting the tunnel barrier. Semiconductor Few-Electron Quantum Dots as Spin Qubits 71 4.5 QPC Versus SET Our measurements clearly demonstrate that a QPC can serve as a fast and sensitive charge detector. Compared to an SET, a QPC offers several practi- cal advantages. First, a QPC requires fabrication and tuning of just a single additional gate when integrated with a quantum dot defined by metal gates, whereas an SET requires two tunnel barriers, and a gate to set the island po- tential. Second, QPCs are more robust and easy to use in the sense that spu- rious, low-frequency fluctuations of the electrostatic potential hardly change the QPC sensitivity to charges on the dot (the transition between quantized conductance plateaus has an almost constant slope over a wide range of elec- trostatic potential), but can easily spoil the SET sensitivity. With an RF-SET, a sensitivity to charges on a quantum dot of ≈2 × 10−4e/ √ Hz has been reached [58], and theoretically even a ten times better sensitivity is possible [57]. Could a QPC reach similar sensitivities? The noise level in the present measurement could be reduced by a factor of two or three using a JFET input-stage which better balances input voltage noise and input current noise. Further improvements can be obtained by low- ering the capacitance of the filters in the line, or the line capacitance itself, by placing the IV-convertor close to the sample, inside the refrigerator. Much more significant reductions in the instrumentation noise could be realized by embedding the QPC in a resonant electrical circuit and measuring the damping of the resonator. We estimate that with an “RF-QPC” and a low-temperature HEMT amplifier, a sensitivity of 2 × 10−4e/ √ Hz could be achieved with the present sample. The noise from the amplifier circuitry is then only 2.5 times larger than the shot noise level. To what extent the signal can be increased is unclear, as we do not yet understand the mechanism through which the dot occupancy is disturbed for Vi > 1 mV4. Certainly, the capacitive coupling of the dot to the QPC channel can easily be five times larger than it is now by optimizing the gate design [60]. Keeping Vi = 1 mV , the sensitivity would then be 4×10−5e/ √ Hz, and a single electron charge on the dot could be measured within a few ns. Finally, we point out that a QPC can reach the quantum limit of detec- tion [63, 64], where the measurement induced decoherence takes the minimum value permitted by quantum mechanics. Qualitatively, this is because (1) in- formation on the charge state of the dot is transferred only to the QPC current and not to degrees of freedom which are not observed, and (2) an external perturbation in the QPC current does not couple back to the charge state of the dot. 4 The statistics of the RTS were altered for Vi > 1 mV, irrespective of (1) whether Vi was applied to the QPC source or drain, (2) the potential difference between the reservoir and the QPC source/drain, and (3) the QPC transmission T . 72 J.M. Elzerman et al. 5 Single-Shot Read-Out of an Individual Electron Spin in a Quantum Dot Spin is a fundamental property of all elementary particles. Classically it can be viewed as a tiny magnetic moment, but a measurement of an electron spin along the direction of an external magnetic field can have only two outcomes: parallel or anti-parallel to the field [65]. This discreteness reflects the quantum mechanical nature of spin. Ensembles of many spins have found diverse appli- cations ranging from magnetic resonance imaging [66] to magneto-electronic devices [67], while individual spins are considered as carriers for quantum in- formation. Read-out of single spin states has been achieved using optical tech- niques [68], and is within reach of magnetic resonance force microscopy [69]. However, electrical read-out of single spins [2, 49, 70, 71, 72, 73, 74, 75] has so far remained elusive. Here, we demonstrate electrical single-shot measurement of the state of an individual electron spin in a semiconductor quantum dot [40]. We use spin-to-charge conversion of a single electron confined in the dot, and detect the single-electron charge using a quantum point contact; the spin measurement visibility is ∼65%. Furthermore, we observe very long single- spin energy relaxation times (up to ∼0.85 ms at a magnetic field of 8 Tesla), which are encouraging for the use of electron spins as carriers of quantum information. 5.1 Measuring Electron Spin in Quantum Dots In quantum dot devices, single electron charges are easily measured. Spin states in quantum dots, however, have only been studied by measuring the average signal from a large ensemble of electron spins [54, 68, 77, 78, 79, 80]. In contrast, the experiment presented here aims at a single-shot measurement of the spin orientation (parallel or antiparallel to the field, denoted as spin-↑ and spin-↓, respectively) of a particular electron; only one copy of the electron is available, so no averaging is possible. The spin measurement relies on spin- to-charge conversion [54, 79] followed by charge measurement in a single-shot mode [58, 59]. Figure 28a schematically shows a single electron spin confined in a quantum dot (circle). A magnetic field is applied to split the spin-↑ and spin-↓ states by the Zeeman energy. The dot potential is then tuned such that if the electron has spin-↓ it will leave, whereas it will stay on the dot if it has spin-↑. The spin state has now been correlated with the charge state, and measurement of the charge on the dot will reveal the original spin state. 5.2 Implementation This concept is implemented using a structure [55] (Fig. 28b) consisting of a quantum dot in close proximity to a quantum point contact (QPC). The quantum dot is used as a box to trap a single electron, and the QPC is Semiconductor Few-Electron Quantum Dots as Spin Qubits 75 E E EF ∆I Q P C V P (m V ) time time in twait Qdot= -e Qdot=0 inject & wait read-out empty out empty a b c Qdot=0 tread out in 10 5 0 Fig. 29. Two-level pulse technique used to inject a single electron and measure its spin orientation. (a) Shape of the voltage pulse applied to gate P . The pulse level is 10 mV during twait and 5 mV during tread (which is 0.5 ms for all measurements). (b) Schematic QPC pulse-response if the injected electron has spin-↑ (solid line) or spin-↓ (dotted line; the difference with the solid line is only seen during the read-out stage). Arrows indicate the moment an electron tunnels into or out of the quantum dot. (c) Schematic energy diagrams for spin-↑ (E↑) and spin-↓ (E↓) during the different stages of the pulse. Black vertical lines indicate the tunnel barriers. The tunnel rate between the dot and the QPC-drain on the right is set to zero. The rate between the dot and the reservoir on the left is tuned to a specific value, Γ . If the spin is ↑ at the start of the read-out stage, no change in the charge on the dot occurs during tread. In contrast, if the spin is ↓, the electron can escape and be replaced by a spin-↑ electron. This charge transition is detected in the QPC-current (dotted line inside gray circle in (b)) 5.4 Tuning the Quantum Dot into the Read-Out Configuration To perform spin read-out, VM has to be fine-tuned so that the position of the energy levels with respect to EF is as shown in Fig. 29c. To find the correct settings, we apply a two-level voltage pulse and measure the QPC-response 76 J.M. Elzerman et al. b 0 1 Time (ms) ∆I Q P C (n A ) 0 1 2 ∆I Q P C (n A ) 0 1 2 ∆I Q P C (n A ) 0 1 2 ∆I Q P C (n A ) 0 1 2 a Time (ms) V M (V ) ∆EZ c EF ∆IQPC (nA)0 2 -1.12 -1.13 0.0 0.5 1.0 twait tread Fig. 30. Tuning the quantum dot into the spin read-out configuration. We apply a two-stage voltage pulse as in Fig. 29a (twait = 0.3 ms, tread = 0.5 ms), and measure the QPC-response for increasingly negative values of VM . (a) QPC-response (in colour-scale) versus VM . Four different regions in VM can be identified (separated by white dotted lines), with qualitatively different QPC-responses. (b) Typical QPC- response in each of the four regions. This behaviour can be understood from the energy levels during all stages of the pulse. (c) Schematic energy diagrams showing E↑ and E↓ with respect to EF before and after the pulse (upper pair), during twait (lower pair) and during tread (middle pair), for four values of VM . For the actual spin read-out experiment, VM is set to the optimum position (indicated by the arrow in a) for increasingly negative values of VM (Fig. 30a). Four different regions in VM can be identified (separated by white dotted lines), with qualitatively different QPC-responses. The shape of the typical QPC-response in each of the four regions (Fig. 30b) allows us to infer the position of E↑ and E↓ with respect to EF during all stages of the pulse (Fig. 30c). In the top region, the QPC-response just mimics the applied two-level pulse, indicating that here the charge on the dot remains constant throughout the pulse. This implies that E↑ remains below EF for all stages of the pulse, thus the dot remains occupied with one electron. In the second region from the top, tunnelling occurs, as seen from the extra steps in ∆IQPC . The dot is empty before the pulse, then an electron is injected during twait, which escapes Semiconductor Few-Electron Quantum Dots as Spin Qubits 77 after the pulse. This corresponds to an energy level diagram similar to before, but with E↑ and E↓ shifted up due to the more negative value of VM in this region. In the third region from the top, an electron again tunnels on the dot during twait, but now it can escape already during tread, irrespective of its spin. Finally, in the bottom region no electron-tunnelling is seen, implying that the dot remains empty throughout the pulse. Since we know the shift in VM corresponding to shifting the energy levels by ∆EZ , we can set VM to the optimum position for the spin read-out experiment (indicated by the arrow). For this setting, the energy levels are as shown in Fig. 29c, i.e. EF is approximately in the middle between E↑ and E↓ during the read-out stage. 5.5 Single-Shot Read-Out of One Electron Spin Figure 31a shows typical experimental traces of the pulse-response recorded after proper tuning of the DC gate voltages (see Fig. 30). We emphasize that each trace involves injecting one particular electron on the dot and subse- quently measuring its spin state. Each trace is therefore a single-shot mea- surement. The traces we obtain fall into two different classes; most traces qualitatively resemble the one in the top panel of Fig. 31a, some resemble the one in the bottom panel. These two typical traces indeed correspond to the signals expected for a spin-↑ and a spin-↓ electron (Fig. 29b), a strong indication that the electron in the top panel of Fig. 31a was spin-↑ and in the bottom panel spin-↓. The distinct signature of the two types of responses in ∆IQPC permits a simple criterion for identifying the spin5: if ∆IQPC goes above the threshold value (red line in Fig. 31a and chosen as explained be- low), we declare the electron “spin-down”; otherwise we declare it “spin-up”. Figure 31b shows the read-out section of twenty more “spin-down” traces, to illustrate the stochastic nature of the tunnel events. The random injection of spin-↑ and spin-↓ electrons prevents us from check- ing the outcome of any individual measurement. Therefore, in order to further establish the correspondence between the actual spin state and the outcome 5 The automated data analysis procedure first corrects for the offset of each trace. This offset, resulting from low-frequency interference signals or charge switches, is found by making a histogram of the QPC current during the read-out stage of a particular trace. The histogram typically displays a peak due to fluctuations around the average value corresponding to an occupied dot. The center of a gaussian fit to the histogram gives the offset. Then each trace is checked to make sure that an electron was injected during the injection stage, by evaluating if the signal goes below the injection threshold (dotted horizontal line in Fig. 33a). If not, the trace is disregarded. Finally, to determine if a trace corresponds to “spin-up” or “spin-down”, we disregard all points that lie below the previous point (since these could correspond to points on the falling pulse flank at the end of the injection stage), and check if any of the remaining points are above the threshold. 80 J.M. Elzerman et al. Injection threshold (nA) 1.81.61.4 In je ct ed fr ac tio n 1.0 0.5 0.0 100 129 161 195 273 1500 waiting time ( s):µ b ∆ I Q P C (n A ) 0 1 2 0.2 0.3 0.4 Time (ms) a twait Fig. 33. Setting the injection threshold. (a) Example of QPC-signal for the shortest waiting time used (0.1 ms). The dotted horizontal line indicates the injection thresh- old. Injection is declared successful if the QPC-signal is below the injection threshold for a part or all of the last 45 µs before the end of the injection stage (twait). Traces in which injection was not successful, i.e. no electron was injected during twait, are disregarded. (b) Fraction of traces in which injection was successful, out of a total of 625 taken for each waiting time. The threshold chosen for analysing all data is indicated by the vertical line 5.6 Measurement Fidelity For applications in quantum information processing it is important to know the accuracy, or fidelity, of the single-shot spin read-out. The measurement fidelity is characterised by two parameters, α and β (inset to Fig. 34a), which we now determine for the data taken at 10 T. The parameter α corresponds to the probability that the QPC-current ex- ceeds the threshold even though the electron was actually spin-↑, for instance due to thermally activated tunnelling or electrical noise (similar to “dark counts” in a photon detector). The combined probability for such processes is given by the saturation value of the exponential fit in Fig. 31c, α, which depends on the value of the threshold current. We analyse the data in Fig. 31c using different thresholds, and plot α in Fig. 34b. The parameter β corresponds to the probability that the QPC-current stays below the threshold even though the electron was actually spin-↓ at the start of the read-out stage. Unlike α, β cannot be extracted directly from the exponential fit (note that the fit parameter C = p(1 − α − β) contains two unknowns: p = Γ↓/(Γ↑+Γ↓) and β). We therefore estimate β by analysing the two processes that contribute to it. First, a spin-↓ electron can relax to spin- ↑ before spin-to-charge conversion takes place. This occurs with probability β1 = 1/(1 + T1Γ↓). From a histogram (Fig. 34a) of the actual detection time, tdetect (see Fig. 31b), we find Γ−1↓ ≈ 0.11 ms, yielding β1 ≈ 0.17. Second, if the spin-↓ electron does tunnel off the dot but is replaced by a spin-↑ electron within about 8 µs, the resulting QPC-step is too small to be detected. The Semiconductor Few-Electron Quantum Dots as Spin Qubits 81 Threshold (nA) 0.6 0.8 0.0 1.0 1.0 α 1-β2 b 0.8 0.6 0.4 0.2 1-β1-α 1-β α β u d S pi n- do w n co un t s 0 100 200 Detection time (ms) 0.0 0.2 0.4 a P ro ba bi lit y Fig. 34. Measurement fidelity. (a) Histogram showing the distribution of detection times, tdetect, in the read-out stage (see Fig. 31b for definition tdetect). The expo- nential decay is due to spin-↓ electrons tunnelling out of the dot (rate = Γ↓) and due to spin flips during the read-out stage (rate = 1/T1). Solid line: exponential fit with a decay time (Γ↓ + 1/T1) −1 of 0.09 ms. Given that T1 = 0.55 ms, this yields Γ−1↓ ≈ 0.11 ms. Inset: fidelity parameters. A spin-↓ electron is declared “down” (d) or “up” (u) with probability 1 − β or β, respectively. A spin-↑ electron is declared “up” or “down” with probability 1−α or α, respectively. (b) Open squares represent α, obtained from the saturation value of exponential fits as in Fig. 31c for differ- ent values of the read-out threshold. A current of 0.54 nA (0.91 nA) corresponds to the average value of ∆IQPC when the dot is occupied (empty) during tread. Open diamonds: measured fraction of “reverse-pulse” traces in which ∆IQPC crosses the injection threshold (dotted black line in Fig. 31d). This fraction approximates 1−β2, where β2 is the probability of identifying a spin-↓ electron as “spin-up” due to the finite bandwidth of the measurement setup. Filled circles: total fidelity for the spin- ↓ state, 1 − β, calculated using β1 = 0.17. The vertical dotted line indicates the threshold for which the visibility 1 − α − β (separation between filled circles and open squares) is maximal. This threshold value of 0.73 nA is used in the analysis of Fig. 31 probability that a step is missed, β2, depends on the value of the threshold. It can be determined by applying a modified (“reversed”) pulse (Fig. 31d). For such a pulse, we know that in each trace an electron is injected in the dot, so there should always be a step at the start of the pulse. The fraction of traces in which this step is nevertheless missed, i.e. ∆IQPC stays below the threshold (dotted black line in Fig. 31d), gives β2. We plot 1 − β2 in Fig. 34b (open diamonds). The resulting total fidelity for spin-↓ is given by 1− β ≈ (1− β1)(1− β2) + (αβ1). The last term accounts for the case when a spin-↓ electron is flipped to spin-↑, but there is nevertheless a step in ∆IQPC due to the dark-count mechanism6. In Fig. 34b we also plot the extracted value of 1 − β as a function of the threshold. 6 Let us assume there is a spin-↓ electron on the dot at the start of the read-out stage. The probability that the ↓-electron tunnels out (i.e. that it does not relaxto 82 J.M. Elzerman et al. We now choose the optimal value of the threshold as the one for which the visibility 1 − α − β is maximal (dotted vertical line in Fig. 34b). For this setting, α ≈ 0.07, β1 ≈ 0.17, β2 ≈ 0.15, so the measurement fidelity for the spin-↑ and the spin-↓ state is ∼0.93 and ∼0.72 respectively. The measurement visibility in a single-shot measurement is thus at present 65%. Significant improvements in the spin measurement visibility can be made by lowering the electron temperature (smaller α) and especially by making the charge measurement faster (smaller β). Already, the demonstration of single- shot spin read-out and the observation of T1 of order 1 ms are encouraging results for the use of electron spins as quantum bits. 6 Semiconductor Few-Electron Quantum Dots as Spin Qubits In the previous sections we have described experiments aimed at creating a quantum dot spin qubit according to the proposal by Loss and DiVin- cenzo [2] (see also paragraph 1.3). The key ingredients for these experiments – performed over the last two years – are a fully tunable few-electron double quantum dot and a quantum point contact (QPC) charge detector. We have operated the QPC in three different ways: 1. By measuring its DC conductance, changes in the average charge on the double dot are revealed, which can be used to identify the charge configu- ration of the system. 2. By measuring the conductance in real-time (with a bandwidth of ∼100 kHz), we can detect individual electrons tunnelling on or off the dot (in less than 10 µs). 3. By measuring the QPC response to a gate voltage pulse train (with the proper frequency) using a lock-in amplifier, we can determine the tunnel rate between the dot and a reservoir. In addition, by using a large pulse am- plitude and measuring changes in the effective tunnel rate, we can identify excited states of the dot. Using these techniques, we have demonstrated that our GaAs/AlGaAs quan- tum dot circuit is a promising candidate for a spin qubit. However, we do not have a fully functional qubit yet, as coherent manipulation of a single- or a two-spin system has so far remained elusive. In this section, we evaluate the experimental status of the spin qubit project in terms of the DiVincenzo spin-↑) is given by 1− β1. The probability that this tunnel event is detected (i.e. is not too fast) is given by 1−β2. Therefore, the probability that a spin-↓ electron tunnels out and is detected, is (1−β1)(1−β2). In addition, there is the possibility that the ↓-electron relaxes, with probability β1, but a step in the QPC signal is nevertheless detected, with probability α, due to the “dark count” mechanism. Therefore, the total probability that a spin-↓ electron is declared “spin-down” is given by (1 − β1)(1 − β2) + (αβ1) approximately. Semiconductor Few-Electron Quantum Dots as Spin Qubits 85 A particularly convenient way to perform spin-to-charge conversion could be provided by utilizing not a difference in energy between spin-up and spin- down, but a difference in tunnel rate (Fig. 35b). To read out the spin orien- tation of an electron on the dot, we simply raise both dot levels above EF , so that the electron can leave the dot. If the tunnel rate for spin-up electrons, Γ↑, is much larger than that for spin-down electrons, Γ↓, then at a suitably chosen time the dot will have a large probability to be already empty if the spin was up, but a large probability to be still occupied if the spin is down. Measuring the charge on the dot within the spin relaxation time can then reveal the spin state. This scheme is very robust against charge switches, since no precise po- sitioning of the dot levels with respect to the leads is required: both levels simply have to be above EF . Also, switches have a small influence on the tunnel rates themselves, as they tend to shift the whole potential landscape up or down, which does not change the tunnel barrier for electrons in the dot [87]. Of course, the visibility of this spin measurement scheme depends on the difference in tunnel rate we can achieve. A difference in tunnel rate for spin-up and spin-down electrons is provided by the magnetic field. From large-bias transport measurements in a magnetic field parallel to the 2DEG [82], we find that the spin-selectivity (Γ↑/Γ↓) grows roughly linearly from ∼1.5 at 5 Tesla to ∼5 at 14 Tesla. This is in good agreement with the spin-selectivity of about 3 that was found at 10 Tesla using the single-shot spin measurement technique of Sect. 5. We believe that this spin-dependence of the tunnel rates is due to exchange interactions in the reservoirs. If ∆EZ is the same in the dot as in the reservoirs, the tunnel barrier will be the same for | ↑〉 and | ↓〉 electrons, giving Γ↑ = Γ↓ (Fig. 36a). However, close to the dot there is a region with only | ↑〉 electrons, where an electron that is excited from | ↑〉 to | ↓〉 must overcome not only the single-particle Zeeman energy but also the many-body exchange energy between the reservoir electrons [88]. We can describe this situation with an a b ∆EZ ECB ∆E +EZ X ECB EF EF 0↔↑ 0↔↓ ↑ ↓ 0↔↑ 0↔↓ ↑ ↓ Fig. 36. Exchange interaction in the reservoirs leading to spin-selective tunnel rates. (a) Schematic diagram of the conduction band edge ECB near the dot for electrons with spin-up (solid line) and spin-down (dashed line). If ∆EZ in the reservoirs is the same as in the dot, the tunnel rates do not depend on spin. (b) The exchange energy EX in the reservoirs close to the dot induces spin-dependent tunnel rates 86 J.M. Elzerman et al. effective g-factor geff , which can be larger than the bare g-factor (Fig. 36b). In this case, | ↓〉 electrons experience a thicker tunnel barrier than | ↑〉 electrons, resulting in a difference in tunnel rates [43]. In a magnetic field parallel to the 2DEG, the effect only leads to a modest spin-selectivity that does not allow a single-shot measurement. However, a much larger spin-selectivity is possible in a perpendicular magnetic field [88], i.e. in the Quantum Hall regime. Magnetotransport measurements in 2DEGs with odd filling factor have shown that the g-factor can be enhanced by as much as a factor of ten, depending on the field strength. We anticipate that a convenient perpendicular field of ∼4 T could already give enough spin- selectivity to allow high-fidelity spin read-out. Therefore, spin read-out should be feasible not only in a large parallel magnetic field, but also in a somewhat smaller perpendicular field. 6.3 Initialization Initialization of the spin to the pure state | ↑〉 – the desired initial state for most quantum algorithms [1] – has been demonstrated in Sect. 5. There it was shown that by waiting long enough, energy relaxation will cause the the spin on the dot to relax to the | ↑〉 ground state (Fig. 37a). This is a very simple and robust initialization approach, which can be used for any magnetic field orientation (provided that gµBB > 5kBT ). However, as it takes about 5T1 to reach equilibrium, it is also a very slow procedure (≥10 ms), especially at lower magnetic fields, where the spin relaxation time T1 might be very long. A faster initialization method has been used in the “reverse pulse” tech- nique in Sect. 5. By placing the dot in the read-out configuration (Fig. 37b), a spin-up electron will stay on the dot, whereas a spin-down electron will be replaced by a spin-up. After waiting a few times the sum of the typical tun- nel times for spin-up and spin-down (∼1/Γ↑ + 1/Γ↓), the spin will be with large probability in the | ↑〉 state. This initialization procedure can therefore be quite fast (<1 ms), depending on the tunnel rates. EF ∆EZ ↑ ↓ ↑ ↓ ↑ ↓ c ↑ ↓ da b Fig. 37. Schematic energy diagrams depicting initialization procedures in a large parallel or perpendicular magnetic field. (a) Spin relaxation to pure state | ↑〉. (b) The “read-out” configuration can result in | ↑〉 faster. (c) Random spin injection gives a statistical mixture of | ↑〉 and | ↓〉. (d) In a large perpendicular field providing a strong spin-selectivity, injection results mostly in | ↑〉 Semiconductor Few-Electron Quantum Dots as Spin Qubits 87 We also have the possibility to initialize the dot to a mixed state, where the spin is probabilistically in | ↑〉 or | ↓〉. In Sect. 5, mixed-state initialization was demonstrated in a parallel field by first emptying the dot, followed by placing both spin levels below EF during the “injection stage” (Fig. 37c). The dot is then randomly filled with either a spin-up or a spin-down electron. This is very useful, e.g. to test two-spin operations (see paragraph 6.6). In a large perpendicular field providing a strong spin-selectivity, initializa- tion to the | ↑〉 state is possible via spin relaxation (Fig. 37a) or via direct injection (Fig. 37d). Initialization to a mixed state (or in fact to any state other than | ↑〉) is very difficult due to the spin-selectivity. It probably requires the ability to coherently rotate the spin from | ↑〉 to | ↓〉 (see paragraph 6.5). 6.4 Coherence Times The long-term potential of GaAs quantum dots as electron spin qubits clearly depends crucially on the spin coherence times T1 and T2. In Sect. 5, we have shown that the single-spin relaxation time, T1, can be very long – on the order of 1 ms at 8 T. This implies that the spin is only very weakly disturbed by the environment. The dominant relaxation mechanism at large magnetic field is believed to be the coupling of the spin to phonons, mediated by the spin-orbit interaction [22]. The fundamental quantity of interest for spin qubits is the decoherence time of a single electron spin in a quantum dot, T2, which has never been mea- sured. Experiments with electrons in 2DEGs have established an ensemble- averaged decoherence time, T ∗2 , of ∼100 ns [89]. Recently, a similar lower bound on T2 has been claimed for a single trapped electron spin, based on the linewidth of the observed electron spin resonance [90]. Theoretically, it has been suggested that the real value of T2 can be much longer [22], and under certain circumstances could even be given by T2 = 2T1, limited by the same spin-orbit interactions that limit T1. To build a scalable quantum computer, a sufficiently long T2 (correspond- ing to more than 104 times the gate operation time) is essential in order to reach the “accuracy threshold”. However, for experiments in the near future, we only need to perform a few spin rotations within T2, which might already be possible for much shorter T2, on the order of a µs. This should also be long enough to perform two-spin operations, which are likely to be much faster. To find the actual value of T2, the ability to perform coherent spin operations is required. This is discussed in the next paragraphs. 6.5 Coherent Single-Spin Manipulation: ESR We have not yet satisfied the key requirement for an actual spin qubit: coher- ent manipulation of one- and two-spin states. To controllably create super- positions of | ↑〉 and | ↓〉, we can use the well-known electron spin resonance